Mathematics of Complexity and Dynamical Systems

Mathematics of Complexity and Dynamical Systems

This book consists of selections from the Encyclopedia of Complexity and Systems Science edited by Robert A. Meyers, published by Springer New York in 2009.

Robert A. Meyers (Ed.)

Mathematics of Complexity and Dynamical Systems With 489 Figures and 25 Tables

123

ROBERT A. MEYERS, Ph. D. Editor-in-Chief RAMTECH LIMITED 122 Escalle Lane Larkspur, CA 94939 USA [email protected]

Library of Congress Control Number: 2011939016

ISBN: 978-1-4614-1806-1 This publication is available also as: Print publication under ISBN: 978-1-4614-1805-4 and Print and electronic bundle under ISBN 978-1-4614-1807-8 © 2012 SpringerScience+Business Media, LLC. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. This book consists of selections from the Encyclopedia of Complexity and Systems Science edited by Robert A. Meyers, published by Springer New York in 2009. springer.com Printed on acid free paper

Preface

Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. They are therefore adaptive as they evolve and may contain self-driving feedback loops. Thus, complex systems are much more than a sum of their parts. Complex systems are often characterized as having extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The conclusion is that a reductionist (bottom-up) approach is often an incomplete description of a phenomenon. This recognition, that the collective behavior of the whole system cannot be simply inferred from the understanding of the behavior of the individual components, has led to many new concepts and sophisticated mathematical and modeling tools for application to many scientific, engineering, and societal issues that can be adequately described only in terms of complexity and complex systems. This compendium, entitled, Mathematics of Complexity and Dynamical Systems, furnishes mathematical solutions to problems, inherent in chaotic complex dynamical systems that are both descriptive and predictive in the same way that calculus is descriptive of continuously-varying processes. There are 113 articles in this encyclopedic treatment which have been organized into 7 sections each headed by a recognized expert in the field and supported by peer reviewers, and also a section of three articles selected by the editor. The sections are:        

Ergodic Theory Fractals and Multifractals Non-Linear Ordinary Differential Equations and Dynamical Systems Non-Linear Partial Differential Equations Perturbation Theory Solitons Systems and Control Theory Three EiC Selections: Catastrophe Theory; Infinite Dimensional Controllability; Philosophy of Science, Mathematical Models In

Descriptions of each of the sections is presented below, together with example articles of exceptional relevance in defining the mathematics of chaotic and dynamical systems.  Ergodic Theory lies at the intersection of many areas of mathematics, including smooth dynamics, statistical mechanics, probability, harmonic analysis, and group actions. Problems, techniques, and results are related to many other areas of mathematics, and ergodic theory has had applications both within mathematics and to numerous other branches of science. Ergodic theory has particularly strong overlap with other branches of dynamical systems (e.g. see articles on  Chaos and Ergodic Theory,  Entropy in Ergodic Theory,  Ergodic Theory: Recurrence and  Kolmogorov–Arnold–Moser (KAM) Theory).  Fractals and Multifractals Fractals generalize Euclidean geometrical objects to non-integer dimensions and allow us, for the first time, to delve into the study of complex systems, disorder, and chaos (e.g. see articles on  Fractals Meet Chaos,  Dynamics on Fractals).  Non-linear Ordinary Differential Equations are n-dimensional ODEs of the form dy/dx D F(x; y) that are not linear. Poincaré developed both quantitative and qualitative methods and his approach has shaped the analysis of non-linear

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Preface









ODEs in the period that followed becoming part of the topic of dynamical systems (see article on  Lyapunov– Schmidt Method for Dynamical Systems). Non-Linear Partial Differential Equations include the Euler and Navier–Stokes equations in fluid dynamics and the Boltzmann equation in gas dynamics. Other fundamental models include reaction-diffusion, porous media, nonlinear Schrödinger, Klein–Gordon, Burger and conservation laws, nonlinear wave Korteweg–de Vries (e.g. see articles on  Hamilton–Jacobi Equations and Weak KAM Theory, and  Scaling Limits of Large Systems of Non-linear Partial Differential Equations). Perturbation Theory involves nonlinear mathematical problems which are not exactly solvable–either due to an inherent impossibility or due to insufficient mathematics. Perturbation theory is often the only way to approach realistic nonlinear systems (e.g. see  Hamiltonian Perturbation Theory (and Transition to Chaos),  Kolmogorov– Arnold–Moser (KAM) Theory, and  Non-linear Dynamics, Symmetry and Perturbation Theory in). Solitons are spatially localized waves in a medium that can interact strongly with other solitons but will afterwards regain original form (e.g. see articles on  Partial Differential Equations that Lead to Solitons,  Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, and  Solitons and Compactons). Systems and Control Theory has a simple and elegant answer in the case of linear systems. However, while much also is known for nonlinear systems, many challenges remain in both the finite dimensional setting (see  Finite Dimensional Controllability) and in the setting of distributed systems modeled by partial differential equations (see  Control of Non-linear Partial Differential Equations).

The complete listing of articles and section editors is presented on pages VII to IX. The articles are written for an audience of advanced university undergraduate and graduate students, professors, and professionals in a wide range of fields who must manage complexity on scales ranging from the atomic and molecular to the societal and global. Each article was selected and peer reviewed by one of our 8 Section Editors with advice and consultation provided by our Board Members: Pierre-Louis Lions, Benoit Mandelbrot, Stephen Wolfram, Jerrod Marsden and Joseph Kung; and the Editor-in-Chief. This level of coordination assures that the reader can have a level of confidence in the relevance and accuracy of the information far exceeding that generally found on the World Wide Web. Accessibility is also a priority and for this reason each article includes a glossary of important terms and a concise definition of the subject. Robert A. Meyers Editor-in-Chief Larkspur, California July 2011

Sections

EiC Selections, Section Editor: Robert A. Meyers Catastrophe Theory Infinite Dimensional Controllability Philosophy of Science, Mathematical Models in Ergodic Theory, Section Editor: Bryna Kra Chaos and Ergodic Theory Entropy in Ergodic Theory Ergodic Theorems Ergodic Theory on Homogeneous Spaces and Metric Number Theory Ergodic Theory, Introduction to Ergodic Theory: Basic Examples and Constructions Ergodic Theory: Fractal Geometry Ergodic Theory: Interactions with Combinatorics and Number Theory Ergodic Theory: Non-singular Transformations Ergodic Theory: Recurrence Ergodic Theory: Rigidity Ergodicity and Mixing Properties Isomorphism Theory in Ergodic Theory Joinings in Ergodic Theory Measure Preserving Systems Pressure and Equilibrium States in Ergodic Theory Smooth Ergodic Theory Spectral Theory of Dynamical Systems Symbolic Dynamics Topological Dynamics Fractals and Multifractals, Section Editor: Daniel ben-Avraham and Shlomo Havlin Anomalous Diffusion on Fractal Networks Dynamics on Fractals Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks Fractal and Multifractal Time Series Fractal and Transfractal Scale-Free Networks Fractal Geometry, A Brief Introduction to Fractal Growth Processes Fractal Structures in Condensed Matter Physics Fractals and Economics Fractals and Multifractals, Introduction to Fractals and Percolation

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Sections

Fractals and Wavelets: What can we Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences? Fractals in Biology Fractals in Geology and Geophysics Fractals in the Quantum Theory of Spacetime Fractals Meet Chaos Phase Transitions on Fractals and Networks Reaction Kinetics in Fractals Non-Linear Ordinary Differential Equations and Dynamical Systems, Section Editor: Ferdinand Verhulst Center Manifolds Dynamics of Parametric Excitation Existence and Uniqueness of Solutions of Initial Value Problems Hyperbolic Dynamical Systems Lyapunov–Schmidt Method for Dynamical Systems Non-linear Ordinary Differential Equations and Dynamical Systems, Introduction to Numerical Bifurcation Analysis Periodic Orbits of Hamiltonian Systems Periodic Solutions of Non-autonomous Ordinary Differential Equations Relaxation Oscillations Stability Theory of Ordinary Differential Equations Non-Linear Partial Differential Equations, Section Editor: Italo Capuzzo Dolcetta Biological Fluid Dynamics, Non-linear Partial Differential Equations Control of Nonlinear Partial Differential Equations Dispersion Phenomena in Partial Differential Equations Hamilton-Jacobi Equations and Weak KAM Theory Hyperbolic Conservation Laws Navier-Stokes Equations: A Mathematical Analysis Non-linear Partial Differential Equations, Introduction to Non-linear Partial Differential Equations, Viscosity Solution Method in Non-linear Stochastic Partial Differential Equations Scaling Limits of Large Systems of Nonlinear Partial Differential Equations Vehicular Traffic: A Review of Continuum Mathematical Models Perturbation Theory, Section Editor: Giuseppe Gaeta Diagrammatic Methods in Classical Perturbation Theory Hamiltonian Perturbation Theory (and Transition to Chaos) Kolmogorov-Arnold-Moser (KAM) Theory n-Body Problem and Choreographies Nekhoroshev Theory Non-linear Dynamics, Symmetry and Perturbation Theory in Normal Forms in Perturbation Theory Perturbation Analysis of Parametric Resonance Perturbation of Equilibria in the Mathematical Theory of Evolution Perturbation of Systems with Nilpotent Real Part Perturbation Theory Perturbation Theory and Molecular Dynamics Perturbation Theory for Non-smooth Systems Perturbation Theory for PDEs

Sections

Perturbation Theory in Celestial Mechanics Perturbation Theory in Quantum Mechanics Perturbation Theory, Introduction to Perturbation Theory, Semiclassical Perturbative Expansions, Convergence of Quantum Bifurcations Solitons, Section Editor: Mohamed A. Helal Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory Inverse Scattering Transform and the Theory of Solitons Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the Korteweg–de Vries Equation (KdV) and Modified Korteweg–de Vries Equations (mKdV), Semi-analytical Methods for Solving the Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the Korteweg–de Vries Equation (KdV) History, Exact N-Soliton Solutions and Further Properties Non-linear Internal Waves Partial Differential Equations that Lead to Solitons Shallow Water Waves and Solitary Waves Soliton Perturbation Solitons and Compactons Solitons Interactions Solitons, Introduction to Solitons, Tsunamis and Oceanographical Applications of Solitons: Historical and Physical Introduction Water Waves and the Korteweg–de Vries Equation Systems and Control Theory, Section Editor: Matthias Kawski Chronological Calculus in Systems and Control Theory Discrete Control Systems Finite Dimensional Controllability Hybrid Control Systems Learning, System Identification, and Complexity Maximum Principle in Optimal Control Mechanical Systems: Symmetries and Reduction Nonsmooth Analysis in Systems and Control Theory Observability (Deterministic Systems) and Realization Theory Robotic Networks, Distributed Algorithms for Stability and Feedback Stabilization Stochastic Noises, Observation, Identification and Realization with System Regulation and Design, Geometric and Algebraic Methods in Systems and Control, Introduction to

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About the Editor-in-Chief

Robert A. Meyers President: RAMTECH Limited Manager, Chemical Process Technology, TRW Inc. Post-doctoral Fellow: California Institute of Technology Ph. D. Chemistry, University of California at Los Angeles B. A., Chemistry, California State University, San Diego

Biography Dr. Meyers has worked with more than 25 Nobel laureates during his career. Research Dr. Meyers was Manager of Chemical Technology at TRW (now Northrop Grumman) in Redondo Beach, CA and is now President of RAMTECH Limited. He is co-inventor of the Gravimelt process for desulfurization and demineralization of coal for air pollution and water pollution control. Dr. Meyers is the inventor of and was project manager for the DOE-sponsored Magnetohydrodynamics Seed Regeneration Project which has resulted in the construction and successful operation of a pilot plant for production of potassium formate, a chemical utilized for plasma electricity generation and air pollution control. Dr. Meyers managed the pilot-scale DoE project for determining the hydrodynamics of synthetic fuels. He is a co-inventor of several thermo-oxidative stable polymers which have achieved commercial success as the GE PEI, Upjohn Polyimides and Rhone-Polenc bismaleimide resins. He has also managed projects for photochemistry, chemical lasers, flue gas scrubbing, oil shale analysis and refining, petroleum analysis and refining, global change measurement from space satellites, analysis and mitigation (carbon dioxide and ozone), hydrometallurgical refining, soil and hazardous waste remediation, novel polymers synthesis, modeling of the economics of space transportation systems, space rigidizable structures and chemiluminescence-based devices. He is a senior member of the American Institute of Chemical Engineers, member of the American Physical Society, member of the American Chemical Society and serves on the UCLA Chemistry Department Advisory Board. He was a member of the joint USA-Russia working group on air pollution control and the EPA-sponsored Waste Reduction Institute for Scientists and Engineers.

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About the Editor-in-Chief

Dr. Meyers has more than 20 patents and 50 technical papers. He has published in primary literature journals including Science and the Journal of the American Chemical Society, and is listed in Who’s Who in America and Who’s Who in the World. Dr. Meyers’ scientific achievements have been reviewed in feature articles in the popular press in publications such as The New York Times Science Supplement and The Wall Street Journal as well as more specialized publications such as Chemical Engineering and Coal Age. A public service film was produced by the Environmental Protection Agency of Dr. Meyers’ chemical desulfurization invention for air pollution control. Scientific Books Dr. Meyers is the author or Editor-in-Chief of 12 technical books one of which won the Association of American Publishers Award as the best book in technology and engineering. Encyclopedias Dr. Meyers conceived and has served as Editor-in-Chief of the Academic Press (now Elsevier) Encyclopedia of Physical Science and Technology. This is an 18-volume publication of 780 twenty-page articles written to an audience of university students and practicing professionals. This encyclopedia, first published in 1987, was very successful, and because of this, was revised and reissued in 1992 as a second edition. The Third Edition was published in 2001 and is now online. Dr. Meyers has completed two editions of the Encyclopedia of Molecular Cell Biology and Molecular Medicine for Wiley VCH publishers (1995 and 2004). These cover molecular and cellular level genetics, biochemistry, pharmacology, diseases and structure determination as well as cell biology. His eight-volume Encyclopedia of Environmental Analysis and Remediation was published in 1998 by John Wiley & Sons and his 15-volume Encyclopedia of Analytical Chemistry was published in 2000, also by John Wiley & Sons, all of which are available on-line.

Editorial Board Members

PIERRE-LOUIS LIONS 1994 Fields Medal Current interests include: nonlinear partial differential equations and applications

STEPHEN W OLFRAM Founder and CEO, Wolfram Research Creator, Mathematica® Author, A New Kind of Science

PROFESSOR BENOIT B. MANDELBROT Sterling Professor Emeritus of Mathematical Sciences at Yale University 1993 Wolf Prize for Physics and the 2003 Japan Prize for Science and Technology Current interests include: seeking a measure of order in physical, mathematical or social phenomena that are characterized by abundant data but wild variability.

JERROLD E. MARSDEN Professor of Control & Dynamical Systems California Institute of Technology

JOSEPH P. S. KUNG Professor Department of Mathematics University of North Texas

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Section Editors

Ergodic Theory

Fractals and Multifractals

BRYNA KRA Professor Department of Mathematics Northwestern University

SHLOMO HAVLIN Professor Department of Physics Bar Ilan University and

DANIEL BEN-AVRAHAM Professor Department of Physics Clarkson University

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Section Editors

Non-linear Ordinary Differential Equations and Dynamical Systems

FERDINAND VERHULST Professor Mathematisch Institut University of Utrecht

Solitons

MOHAMED A. HELAL Professor Department of Mathematics Faculty of Science University of Cairo

Non-linear Partial Differential Equations

Systems and Control Theory

ITALO CAPUZZO DOLCETTA Professor Dipartimento di Matematica “Guido Castelnuovo” Università Roma La Sapienza

MATTHIAS KAWSKI Professor, Department of Mathematics and Statistics Arizona State University

Perturbation Theory

GIUSEPPE GAETA Professor in Mathematical Physics Dipartimento di Matematica Università di Milano, Italy

Table of Contents

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory Abdul-Majid Wazwaz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Anomalous Diffusion on Fractal Networks Igor M. Sokolov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13

Biological Fluid Dynamics, Non-linear Partial Differential Equations Antonio DeSimone, François Alouges, Aline Lefebvre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

26

Catastrophe Theory Werner Sanns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Center Manifolds George Osipenko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

48

Chaos and Ergodic Theory Jérôme Buzzi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

Chronological Calculus in Systems and Control Theory Matthias Kawski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Control of Non-linear Partial Differential Equations Fatiha Alabau-Boussouira, Piermarco Cannarsa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102

Diagrammatic Methods in Classical Perturbation Theory Guido Gentile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

Discrete Control Systems Taeyoung Lee, Melvin Leok, Harris McClamroch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143

Dispersion Phenomena in Partial Differential Equations Piero D’Ancona . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160

Dynamics on Fractals Raymond L. Orbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

175

Dynamics of Parametric Excitation Alan Champneys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183

Entropy in Ergodic Theory Jonathan L. F. King . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205

Ergodicity and Mixing Properties Anthony Quas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225

Ergodic Theorems Andrés del Junco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241

Ergodic Theory: Basic Examples and Constructions Matthew Nicol, Karl Petersen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

264

Ergodic Theory: Fractal Geometry Jörg Schmeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

288

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Table of Contents

Ergodic Theory on Homogeneous Spaces and Metric Number Theory Dmitry Kleinbock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

302

Ergodic Theory: Interactions with Combinatorics and Number Theory Tom Ward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

313

Ergodic Theory, Introduction to Bryna Kra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327

Ergodic Theory: Non-singular Transformations Alexandre I. Danilenko, Cesar E. Silva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

329

Ergodic Theory: Recurrence Nikos Frantzikinakis, Randall McCutcheon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

357

Ergodic Theory: Rigidity Viorel Ni¸tic˘a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

369

Existence and Uniqueness of Solutions of Initial Value Problems Gianne Derks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

383

Finite Dimensional Controllability Lionel Rosier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

395

Fractal Geometry, A Brief Introduction to Armin Bunde, Shlomo Havlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

409

Fractal Growth Processes Leonard M. Sander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

429

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks Sidney Redner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

446

Fractal and Multifractal Time Series Jan W. Kantelhardt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

463

Fractals in Biology Sergey V. Buldyrev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

488

Fractals and Economics Misako Takayasu, Hideki Takayasu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

512

Fractals in Geology and Geophysics Donald L. Turcotte . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

532

Fractals Meet Chaos Tony Crilly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

537

Fractals and Multifractals, Introduction to Daniel ben-Avraham, Shlomo Havlin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

557

Fractals and Percolation Yakov M. Strelniker, Shlomo Havlin, Armin Bunde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

559

Fractals in the Quantum Theory of Spacetime Laurent Nottale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

571

Fractal Structures in Condensed Matter Physics Tsuneyoshi Nakayama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

591

Fractals and Wavelets: What Can We Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences? Alain Arneodo, Benjamin Audit, Edward-Benedict Brodie of Brodie, Samuel Nicolay, Marie Touchon, Yves d’Aubenton-Carafa, Maxime Huvet, Claude Thermes . . . . . . . . . . . . . . . . . . . . . . . . . . .

606

Fractal and Transfractal Scale-Free Networks Hernán D. Rozenfeld, Lazaros K. Gallos, Chaoming Song, Hernán A. Makse . . . . . . . . . . . . . . . . .

637

Table of Contents

Hamiltonian Perturbation Theory (and Transition to Chaos) Henk W. Broer, Heinz Hanßmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

657

Hamilton–Jacobi Equations and Weak KAM Theory Antonio Siconolfi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

683

Hybrid Control Systems Andrew R. Teel, Ricardo G. Sanfelice, Rafal Goebel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

704

Hyperbolic Conservation Laws Alberto Bressan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

729

Hyperbolic Dynamical Systems Vitor Araújo, Marcelo Viana . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

740

Infinite Dimensional Controllability Olivier Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

755

Inverse Scattering Transform and the Theory of Solitons Tuncay Aktosun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

771

Isomorphism Theory in Ergodic Theory Christopher Hoffman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

783

Joinings in Ergodic Theory Thierry de la Rue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

796

Kolmogorov–Arnold–Moser (KAM) Theory Luigi Chierchia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

810

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the Yu-Jie Ren . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

837

Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the Yi Zang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

884

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the Do˘gan Kaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

890

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the Mustafa Inc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

908

Learning, System Identification, and Complexity M. Vidyasagar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

924

Lyapunov–Schmidt Method for Dynamical Systems André Vanderbauwhede . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

937

Maximum Principle in Optimal Control Velimir Jurdjevic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

953

Measure Preserving Systems Karl Petersen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

964

Mechanical Systems: Symmetries and Reduction Jerrold E. Marsden, Tudor S. Ratiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

981

Navier–Stokes Equations: A Mathematical Analysis Giovanni P. Galdi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1009

n-Body Problem and Choreographies Susanna Terracini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1043

Nekhoroshev Theory Laurent Niederman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1070

Non-linear Dynamics, Symmetry and Perturbation Theory in Giuseppe Gaeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1082

XIX

XX

Table of Contents

Non-linear Internal Waves Moustafa S. Abou-Dina, Mohamed A. Helal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1102

Non-linear Ordinary Differential Equations and Dynamical Systems, Introduction to Ferdinand Verhulst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1111

Non-linear Partial Differential Equations, Introduction to Italo Capuzzo Dolcetta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1113

Non-linear Partial Differential Equations, Viscosity Solution Method in Shigeaki Koike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1115

Non-linear Stochastic Partial Differential Equations Giuseppe Da Prato . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1126

Nonsmooth Analysis in Systems and Control Theory Francis Clarke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1137

Normal Forms in Perturbation Theory Henk W. Broer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1152

Numerical Bifurcation Analysis Hil Meijer, Fabio Dercole, Bart Oldeman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1172

Observability (Deterministic Systems) and Realization Theory Jean-Paul André Gauthier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1195

Partial Differential Equations that Lead to Solitons Do˘gan Kaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1205

Periodic Orbits of Hamiltonian Systems Luca Sbano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1212

Periodic Solutions of Non-autonomous Ordinary Differential Equations Jean Mawhin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1236

Perturbation Analysis of Parametric Resonance Ferdinand Verhulst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1251

Perturbation of Equilibria in the Mathematical Theory of Evolution Angel Sánchez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1265

Perturbation of Systems with Nilpotent Real Part Todor Gramchev . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1276

Perturbation Theory Giovanni Gallavotti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1290

Perturbation Theory in Celestial Mechanics Alessandra Celletti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1301

Perturbation Theory, Introduction to Giuseppe Gaeta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1314

Perturbation Theory and Molecular Dynamics Gianluca Panati . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1317

Perturbation Theory for Non-smooth Systems Marco Antônio Teixeira . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1325

Perturbation Theory for PDEs Dario Bambusi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1337

Perturbation Theory in Quantum Mechanics Luigi E. Picasso, Luciano Bracci, Emilio d’Emilio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1351

Perturbation Theory, Semiclassical Andrea Sacchetti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1376

Table of Contents

Perturbative Expansions, Convergence of Sebastian Walcher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1389

Phase Transitions on Fractals and Networks Dietrich Stauffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1400

Philosophy of Science, Mathematical Models in Zoltan Domotor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1407

Pressure and Equilibrium States in Ergodic Theory Jean-René Chazottes, Gerhard Keller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1422

Quantum Bifurcations Boris Zhilinskií . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1438

Reaction Kinetics in Fractals Ezequiel V. Albano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1457

Relaxation Oscillations Johan Grasman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1475

Robotic Networks, Distributed Algorithms for Francesco Bullo, Jorge Cortés, Sonia Martínez . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1489

Scaling Limits of Large Systems of Non-linear Partial Differential Equations D. Benedetto, M. Pulvirenti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1505

Shallow Water Waves and Solitary Waves Willy Hereman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1520

Smooth Ergodic Theory Amie Wilkinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1533

Soliton Perturbation Ji-Huan He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1548

Solitons and Compactons Ji-Huan He, Shun-dong Zhu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1553

Solitons: Historical and Physical Introduction François Marin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1561

Solitons Interactions Tarmo Soomere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1576

Solitons, Introduction to Mohamed A. Helal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1601

Solitons, Tsunamis and Oceanographical Applications of M. Lakshmanan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1603

Spectral Theory of Dynamical Systems Mariusz Lema´nczyk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1618

Stability and Feedback Stabilization Eduardo D. Sontag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1639

Stability Theory of Ordinary Differential Equations Carmen Chicone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1653

Stochastic Noises, Observation, Identification and Realization with Giorgio Picci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1672

Symbolic Dynamics Brian Marcus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1689

System Regulation and Design, Geometric and Algebraic Methods in Alberto Isidori, Lorenzo Marconi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1711

XXI

XXII

Table of Contents

Systems and Control, Introduction to Matthias Kawski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1724

Topological Dynamics Ethan Akin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1726

Vehicular Traffic: A Review of Continuum Mathematical Models Benedetto Piccoli, Andrea Tosin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1748

Water Waves and the Korteweg–de Vries Equation Lokenath Debnath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1771

List of Glossary Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1811

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1819

Contributors

ABOU-DINA , MOUSTAFA S. Cairo University Giza Egypt

AUDIT, BENJAMIN ENS-Lyon CNRS Lyon Cedex France

AKIN, ETHAN The City College New York City USA

BAMBUSI , DARIO Università degli Studi di Milano Milano Italia

AKTOSUN, TUNCAY University of Texas at Arlington Arlington USA

BEN -AVRAHAM, D ANIEL Clarkson University Potsdam USA

ALABAU-BOUSSOUIRA , FATIHA Université de Metz Metz France

BENEDETTO, D. Dipartimento di Matematica, Università di Roma ‘La Sapienza’ Roma Italy

ALBANO, EZEQUIEL V. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA) CCT La Plata La Plata Argentina ALOUGES, FRANÇOIS Université Paris-Sud Orsay cedex France ARAÚJO, VITOR CMUP Porto Portugal IM-UFRJ Rio de Janeiro Brazil ARNEODO, ALAIN ENS-Lyon CNRS Lyon Cedex France

BRACCI , LUCIANO Università di Pisa Pisa Italy Sezione di Pisa Pisa Italy BRESSAN, ALBERTO Penn State University University Park USA BRODIE OF BRODIE, EDWARD-BENEDICT ENS-Lyon CNRS Lyon Cedex France BROER, HENK W. University of Groningen Groningen The Netherlands

XXIV

Contributors

BULDYREV, SERGEY V. Yeshiva University New York USA

CORTÉS, JORGE University of California San Diego USA

BULLO, FRANCESCO University of California Santa Barbara USA

CRILLY, TONY Middlesex University London UK

BUNDE, ARMIN Justus-Liebig-Universität Giessen Germany BUZZI , JÉRÔME C.N.R.S. and Université Paris-Sud Orsay France CANNARSA , PIERMARCO Università di Roma “Tor Vergata” Rome Italy CELLETTI , ALESSANDRA Università di Roma Tor Vergata Roma Italy CHAMPNEYS, ALAN University of Bristol Bristol United Kingdom CHAZOTTES, JEAN-RENÉ CNRS/École Polytechnique Palaiseau France CHICONE, CARMEN University of Missouri-Columbia Columbia USA

D’ANCONA , PIERO Unversità di Roma “La Sapienza” Roma Italy DANILENKO, ALEXANDRE I. Ukrainian National Academy of Sciences Kharkov Ukraine DA PRATO, GIUSEPPE Scuola Normale Superiore Pisa Italy D’AUBENTON -CARAFA , Y VES CNRS Gif-sur-Yvette France

DEBNATH, LOKENATH University of Texas – Pan American Edinburg USA DE LA R UE , T HIERRY CNRS – Université de Rouen Saint Étienne du Rouvray France

CHIERCHIA , LUIGI Università “Roma Tre” Roma Italy

D’E MILIO, E MILIO Università di Pisa Pisa Italy Sezione di Pisa Pisa Italy

CLARKE, FRANCIS Institut universitaire de France et Université de Lyon Lyon France

DERCOLE, FABIO Politecnico di Milano Milano Italy

Contributors

DERKS, GIANNE University of Surrey Guildford UK

GLASS, OLIVIER Université Pierre et Marie Curie Paris France

DESIMONE, ANTONIO SISSA-International School for Advanced Studies Trieste Italy

GOEBEL, RAFAL Loyola University Chicago USA

DOLCETTA , ITALO CAPUZZO Sapienza Universita’ di Roma Rome Italy

GRAMCHEV, TODOR Università di Cagliari Cagliari Italy

DOMOTOR, Z OLTAN University of Pennsylvania Philadelphia USA

GRASMAN, JOHAN Wageningen University and Research Centre Wageningen The Netherlands

FRANTZIKINAKIS, N IKOS University of Memphis Memphis USA

HANSSMANN, HEINZ Universiteit Utrecht Utrecht The Netherlands

GAETA , GIUSEPPE Università di Milano Milan Italy

HAVLIN, SHLOMO Bar–Ilan University Ramat–Gan Israel

GALDI , GIOVANNI P. University of Pittsburgh Pittsburgh USA

HE, JI -HUAN Donghua University Shanghai China

GALLAVOTTI , GIOVANNI Università di Roma I “La Sapienza” Roma Italy

HELAL, MOHAMED A. Cairo University Giza Egypt

GALLOS, LAZAROS K. City College of New York New York USA

HEREMAN, W ILLY Colorado School of Mines Golden USA

GAUTHIER, JEAN-PAUL ANDRÉ University of Toulon Toulon France

HOFFMAN, CHRISTOPHER University of Washington Seattle USA

GENTILE, GUIDO Università di Roma Tre Roma Italy

HUVET, MAXIME CNRS Gif-sur-Yvette France

XXV

XXVI

Contributors

INC, MUSTAFA Fırat University Elazı˘g Turkey

KRA , BRYNA Northwestern University Evanston USA

ISIDORI , ALBERTO University of Rome La Sapienza Italy

LAKSHMANAN, M. Bharathidasan University Tiruchirapalli India

JUNCO, ANDRÉS DEL University of Toronto Toronto Canada

LEE, TAEYOUNG University of Michigan Ann Arbor USA

JURDJEVIC, VELIMIR University of Toronto Toronto Canada

LEFEBVRE, ALINE Université Paris-Sud Orsay cedex France

KANTELHARDT, JAN W. Martin-Luther-University Halle-Wittenberg Halle Germany

´ , MARIUSZ LEMA NCZYK Nicolaus Copernicus University Toru´n Poland

KAWSKI , MATTHIAS Arizona State University Tempe USA

LEOK, MELVIN Purdue University West Lafayette USA

˘ KAYA , DO GAN Firat University Elazig Turkey

MAKSE, HERNÁN A. City College of New York New York USA

KELLER, GERHARD Universität Erlangen-Nürnberg Erlangen Germany

MARCONI , LORENZO University of Bologna Bologna Italy

KING, JONATHAN L. F. University of Florida Gainesville USA

MARCUS, BRIAN University of British Columbia Vancouver Canada

KLEINBOCK, DMITRY Brandeis University Waltham USA

MARIN, FRANÇOIS Laboratoire Ondes et Milieux Complexes, Fre CNRS 3102 Le Havre Cedex France

KOIKE, SHIGEAKI Saitama University Saitama Japan

MARSDEN, JERROLD E. California Institute of Technology Pasadena USA

Contributors

MARTÍNEZ, SONIA University of California San Diego USA

N OTTALE, LAURENT Paris Observatory and Paris Diderot University Paris France

MAWHIN, JEAN Université Catholique de Louvain Maryland USA

OLDEMAN, BART Concordia University Montreal Canada

MCCLAMROCH, HARRIS University of Michigan Ann Arbor USA

ORBACH, RAYMOND L. University of California Riverside USA

MCCUTCHEON, RANDALL University of Memphis Memphis USA MEIJER, HIL University of Twente Enschede The Netherlands N AKAYAMA , TSUNEYOSHI Toyota Physical and Chemical Research Institute Nagakute Japan N ICOLAY, SAMUEL Université de Liège Liège Belgium N ICOL, MATTHEW University of Houston Houston USA N IEDERMAN, LAURENT Université Paris Paris France IMCCE Paris France ˘ , VIOREL N I TIC ¸ A West Chester University West Chester USA Institute of Mathematics Bucharest Romania

OSIPENKO, GEORGE State Polytechnic University St. Petersburg Russia PANATI , GIANLUCA Università di Roma “La Sapienza” Roma Italy PETERSEN, KARL University of North Carolina Chapel Hill USA PICASSO, LUIGI E. Università di Pisa Pisa Italy Sezione di Pisa Pisa Italy PICCI , GIORGIO University of Padua Padua Italy PICCOLI , BENEDETTO Consiglio Nazionale delle Ricerche Rome Italy PULVIRENTI , M. Dipartimento di Matematica, Università di Roma ‘La Sapienza’ Roma Italy

XXVII

XXVIII

Contributors

QUAS, ANTHONY University of Victoria Victoria Canada

SANFELICE, RICARDO G. University of Arizona Tucson USA

RATIU, TUDOR S. École Polytechnique Fédérale de Lausanne Lausanne Switzerland

SANNS, W ERNER University of Applied Sciences Darmstadt Germany

REDNER, SIDNEY Boston University Boston USA

SBANO, LUCA University of Warwick Warwick UK

REN, YU-JIE Shantou University Shantou People’s Republic of China Dalian Polytechnic University Dalian People’s Republic of China Beijing Institute of Applied Physics and Computational Mathematics Beijing People’s Republic of China ROSIER, LIONEL Institut Elie Cartan Vandoeuvre-lès-Nancy France ROZENFELD, HERNÁN D. City College of New York New York USA SACCHETTI , ANDREA Universitá di Modena e Reggio Emilia Modena Italy SÁNCHEZ, ANGEL Universidad Carlos III de Madrid Madrid Spain Universidad de Zaragoza Zaragoza Spain SANDER, LEONARD M. The University of Michigan Ann Arbor USA

SCHMELING, JÖRG Lund University Lund Sweden SICONOLFI , ANTONIO “La Sapienza” Università di Roma Roma Italy SILVA , CESAR E. Williams College Williamstown USA SOKOLOV, IGOR M. Humboldt-Universität zu Berlin Berlin Germany SONG, CHAOMING City College of New York New York USA SONTAG, EDUARDO D. Rutgers University New Brunswick USA SOOMERE, TARMO Tallinn University of Technology Tallinn Estonia STAUFFER, DIETRICH Cologne University Köln Germany

Contributors

STRELNIKER, YAKOV M. Bar–Ilan University Ramat–Gan Israel

VANDERBAUWHEDE, ANDRÉ Ghent University Gent Belgium

TAKAYASU, HIDEKI Sony Computer Science Laboratories Inc Tokyo Japan

VERHULST, FERDINAND University of Utrecht Utrecht The Netherlands

TAKAYASU, MISAKO Tokyo Institute of Technology Tokyo Japan

VIANA , MARCELO IMPA Rio de Janeiro Brazil

TEEL, ANDREW R. University of California Santa Barbara USA TEIXEIRA , MARCO ANTÔNIO Universidade Estadual de Campinas Campinas Brazil TERRACINI , SUSANNA Università di Milano Bicocca Milano Italia THERMES, CLAUDE CNRS Gif-sur-Yvette France TOSIN, ANDREA Consiglio Nazionale delle Ricerche Rome Italy TOUCHON, MARIE CNRS Paris France Université Pierre et Marie Curie Paris France TURCOTTE, DONALD L. University of California Davis USA

VIDYASAGAR, M. Software Units Layout Hyderabad India W ALCHER, SEBASTIAN RWTH Aachen Aachen Germany W ARD, TOM University of East Anglia Norwich UK W AZWAZ, ABDUL-MAJID Saint Xavier University Chicago USA W ILKINSON, AMIE Northwestern University Evanston USA Z ANG, YI Zhejiang Normal University Jinhua China Z HILINSKIÍ , BORIS Université du Littoral Dunkerque France Z HU, SHUN-DONG Zhejiang Lishui University Lishui China

XXIX

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory ABDUL-MAJID W AZWAZ Department of Mathematics, Saint Xavier University, Chicago, USA Article Outline Glossary Definition of the Subject Introduction Adomian Decomposition Method and Adomian Polynomials Modified Decomposition Method and Noise Terms Phenomenon Solitons, Peakons, Kinks, and Compactons Solitons of the KdV Equation Kinks of the Burgers Equation Peakons of the Camassa–Holm Equation Compactons of the K(n,n) Equation Future Directions Bibliography Glossary Solitons Solitons appear as a result of a balance between a weakly nonlinear convection and a linear dispersion. The solitons are localized highly stable waves that retains its identity: shape and speed, upon interaction, and resemble particle like behavior. In the case of a collision, solitons undergo a phase shift. Types of solitons Solitary waves, which are localized traveling waves, are asymptotically zero at large distances and appear in many structures such as solitons, kinks, peakons, cuspons, and compactons, among others. Solitons appear as a bell-shaped sech profile. Kink waves rise or descend from one asymptotic state to another. Peakons are peaked solitary-wave solutions. Cuspons exhibit cusps at their crests. In the peakon structure, the traveling-wave solutions are smooth except for a peak at a corner of its crest. Peakons are the points at which spatial derivative changes sign so that peakons have a finite jump in the first derivative of the solution u(x; t). Unlike peakons, where the derivatives at the peak differ only by a sign, the derivatives at the jump of a cuspon diverge. Compactons are solitons with compact spatial support such that each compacton is a soliton confined to a finite core or a soliton without exponential tails. Com-

pactons are generated due to the delicate interaction between the effect of the genuine nonlinear convection and the genuinely nonlinear dispersion. Adomian method The Adomian decomposition method approaches linear and nonlinear, and homogeneous and inhomogeneous differential and integral equations in a unified way. The method provides the solution in a rapidly convergent series with terms that are elegantly determined in a recursive manner. The method can be used to obtain closed-form solutions, if such solutions exist. A truncated number of the obtained series can be used for numerical purposes. The method was modified to accelerate the computational process. The noise terms phenomenon, which may appear for inhomogeneous cases, can give the exact solution in two iterations only. Definition of the Subject Nonlinear phenomena play a significant role in many branches of applied sciences such as applied mathematics, physics, biology, chemistry, astronomy, plasma, and fluid dynamics. Nonlinear dispersive equations that govern these phenomena have the genuine soliton property. Solitons are pulses that propagate without any change of its identity, i. e., shape and speed, during their travel through a nonlinear dispersive medium [1,5,34]. Solitons resemble properties of a particle, hence the suffix on is used [19,20]. Solitons exist in many scientific branches, such as optical fiber photonics, fiber lasers, plasmas, molecular systems, laser pulses propagating in solids, liquid crystals, nonlinear optics, cosmology, and condensed-matter physics. Based on its importance in many fields, a huge amount of research work has been conducted during the last four decades to make more progress in understanding the soliton phenomenon. A variety of very powerful algorithms has been used to achieve this goal. The Adomian decomposition method introduced in [2,6,21,22,23, 24,25,26], which will be used in this work, is one of the reliable methods that has been used recently. Introduction The aim of this work is to apply the Adomian decomposition method to derive specific types of soliton solutions. Solitons were discovered experimentally by John Scott Russell in 1844. Korteweg and de Vries in 1895 investigated analytically the soliton concept [10], where they derived the pioneer equation of solitons, well known as the KdV equation, that models the height of the surface of shallow water in the presence of solitary waves. Moreover,

1

2

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

Zabusky and Kruskal [35] investigated this phenomenon analytically in 1965. Since 1965, a huge number of research works have been conducted on nonlinear dispersive and dissipative equations. The aim of these works has been to study thoroughly the characteristics of soliton solutions and to study various types of solitons that appear as a result of these equations. Several reliable methods were used in the literature to handle nonlinear dispersive equations. Hirota’s bilinear method [8,9] has been used for single- and multiple-soliton solutions. The inverse scattering method [1] has been widely used. For single-soliton solutions, several methods, such as the tanh method [13,14,15], the pseudospectral method, and the truncated Painlevé expansion, are used. However, in this work, the decomposition method, introduced by Adomian in 1984, will be applied to derive the desired types of soliton solutions. The method approaches all problems in a unified way and in a straightforward manner. The method computes the solution in a fast convergent series with components that are elegantly determined. Unlike other methods, the initial or boundary conditions are necessary for the use of the Adomian method. Adomian Decomposition Method and Adomian Polynomials The Adomian decomposition method, developed by George Adomian in 1984, has been receiving much attention in applied mathematics in general and in the area of initial value and boundary value problems in particular. Moreover, it is also used in the area of series solutions for numerical purposes. The method is powerful and effective and can be used in linear or nonlinear, ordinary or partial differential equations, and integral equations. The decomposition method demonstrates fast convergence of the solution and provides numerical approximation with a high level of accuracy. The method handles applied problems directly and in a straightforward manner without using linearization, perturbation, or any other restrictive assumption that might change the physical behavior of the physical model under investigation. The method is effectively addressed and thoroughly used by many researchers in the literature [1,21,22,23,24, 25,26]. It is important to indicate that well-known methods, such as Backlund transformation, inverse scattering method, Hirota’s bilinear formalism, the tanh method, and many others, can handle problems without using initial or boundary value conditions. For the Adomian decomposition method, the initial or boundary conditions are necessary to conduct the determination of the components recursively. However, some of the standard methods require

huge calculations, whereas the Adomian method minimizes the volume of computational work. The Adomian decomposition method consists in decomposing the unknown function u(x; t) of any equation into a sum of an infinite number of components given by the decomposition series u(x; t) D

1 X

u n (x; t) ;

(1)

nD0

where the components u n (x; t); n  0 are to be determined in a recursive manner. The decomposition method concerns itself with the determination of the components u0 (x; t); u1 (x; t); u2 (x; t); : : : individually. The determination of these components can be obtained through a recursive relation that usually involves evaluation of simple integrals. We now give a clear overview of Adomian decomposition method. Consider the linear differential equation written in an operator form by Lu C Ru D f ;

(2)

where L is the lower-order derivative, which is assumed to be invertible, R is a linear differential operator of order greater than L, and f is a source term. We next apply the inverse operator L1 to both sides of Eq. (2) and use the given initial or boundary condition to get u(x; t) D g  L1 (Ru) ;

(3)

where the function g represents the terms arising from integrating the source term f and from using the given conditions; all are assumed to be prescribed. Substituting the infinite series of components u(x; t) D

1 X

u n (x; t)

(4)

nD0

into both sides of (3) yields 1 X

un D g  L

1

nD0

R

X 1

! un

:

(5)

nD0

The decomposition method suggests that the zeroth component u0 is usually defined by all terms not included under the inverse operator L1 , which arise from the initial data and from integrating the inhomogeneous term. This in turn gives the formal recursive relation u0 (x; t) D g ; u kC1 (x; t) D L1 (R(u k )) ;

k  0;

(6)

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

Adomian assumed that the nonlinear term F(u) can be expressed by an infinite series of the so-called Adomian polynomials An given in the form

or, equivalently, u0 (x; t) D g ; u1 (x; t) D L1 (R(u0 )) ; u2 (x; t) D L1 (R(u1 )) ; u3 (x; t) D L

1

(7)

(R(u2 )) ;

:: :: The differential equation under consideration is now reduced to integrals that can be easily evaluated. Having determined these components u0 (x; t); u1 (x; t); u2 (x; t); : : : , we then substitute the obtained components into (4) to obtain the solution in a series form. The determined series may converge very rapidly to a closed-form solution, if an exact solution exists. For concrete problems, where a closed-form solution is not obtainable, a truncated number of terms is usually used for numerical purposes. Few terms of the truncated series give an approximation with a high degree of accuracy. The convergence concept of the decomposition series is investigated thoroughly in the literature. Several significant studies were conducted to compare the performance of the Adomian method with other methods, such as Picard’s method, Taylor series method, finite differences method, perturbation techniques, and others. The conclusions emphasized the fact that the Adomian method has many advantages and requires less computational work compared to existing techniques. The Adomian method and many others applied this method to many deterministic and stochastic problems. However, the Adomian method, like some other methods, suffers if the zeroth component u0 (x; t) D 0 and makes the integrand of the right side in (7) u1 (x; t) D 0. If the right side integrand in (7) does not vanish, if it is of a form such as eu 0 or ln(˛ C u0 ); ˛ > 0, then the method works effectively. As stated before, the Adomian method decomposes the unknown function u(x; t) into an infinite number of components. However, for nonlinear functions of u(x; t) such as u2 ; u3 ; ln(1 C u); cos u; eu ; uu x , etc., a special representation for nonlinear terms was developed by Adomian and others. Adomian introduced a formal algorithm to establish the proper representation for all forms of nonlinear functions of u(x; t). This representation of nonlinear terms is necessary to apply the Adomian method properly. Several alternative algorithms have been introduced in the literature by researchers to calculate Adomian polynomials. However, the Adomian algorithm remains the most commonly applied because it is simple and practical; therefore, it will be used in this work.

F(u) D

1 X

A n (u0 ; u1 ; u2 ; : : : ; u n ) ;

(8)

nD0

where An can be evaluated for all forms of nonlinearity. Adomian polynomials An for the nonlinear term F(u) can be evaluated using the following expression: " !# n X 1 dn i An D  ui ; n D 0; 1; 2;    : (9) F n! dn iD0

0

The general formula (9) can be simplified as follows. Assuming that the nonlinear function is F(u), using (9), Adomian polynomials are given by A0 D F(u0 ) ; A1 D u1 F 0 (u0 ) ; A2 D u2 F 0 (u0 ) C

1 2 00 u F (u0 ) ; 2! 1

1 A3 D u3 F 0 (u0 ) C u1 u2 F 00 (u0 ) C u13 F 000 (u0 ) ; 3!   1 2 u2 C u1 u3 F 00 (u0 ) A4 D u4 F 0 (u0 ) C (10) 2! 1 1 C u12 u2 F 000 (u0 ) C u14 F (iv) (u0 ) ; 2! 4! A5 D u5 F 0 (u0 ) C (u2 u3 C u1 u4 )F 00 (u0 )   1 1 C u1 u22 C u12 u3 F 000 (u0 ) 2! 2! 1 1 3 C u1 u2 F (iv) (u0 ) C u15 F (v) (u0 ) : 3! 5! Other polynomials can be generated in a similar manner. It is clear that A0 depends only on u0 , A1 depends only on u0 and u1 , A2 depends only on u0 ; u1 , and u2 , and so on. For F(u) D u2 , we find A0 D F(u0 ) D u02 ; A1 D u1 F 0 (u0 ) D 2u0 u1 ; 1 A2 D u2 F 0 (u0 ) C u12 F 00 (u0 ) D 2u0 u2 C u12 ; 2! 1 0 A3 D u3 F (u0 ) C u1 u2 F 00 (u0 ) C u13 F 000 (u0 ) 3! D 2u0 u3 C 2u1 u2 :

(11)

Modified Decomposition Method and Noise Terms Phenomenon A reliable modification of the Adomian decomposition method [22] was developed by Wazwaz in 1999. The modification will further accelerate the convergence of the se-

3

4

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

ries solution. As presented earlier, the standard decomposition method admits the use of the recursive relation u0 D g ; u kC1 D  L1 (Ru k ) ;

k 0:

(12)

The modified decomposition method introduces a slight variation to the recursive relation (12) that will lead to the determination of the components of u in a faster and easier way. For specific cases, the function g can be set as the sum of two partial functions, g 1 and g 2 . In other words, we can set g D g1 C g2 :

(13)

This assumption gives a slight qualitative change in the formation of the recursive relation (12). To reduce the size of calculations, we identify the zeroth component u0 by one part of g, that is, g 1 or g 2 . The other part of g can be added to the component u1 among other terms. In other words, the modified recursive relation can be defined as u0 D g 1 ; u1 D g2  L1 (Ru0 ) ; u kC1 D  L

1

(Ru k ) ;

(14) k 1:

The change occurred in the formation of the first two components u0 and u1 only [22,24,25]. Although this variation in the formation of u0 and u1 is slight, it plays a major role in accelerating the convergence of the solution and in minimizing the size of calculations. It is interesting to point out that by selecting the parts g 1 and g 2 properly, the exact solution u(x; t) may be obtained by using very few iterations, and sometimes by evaluating only two components. Moreover, if g consists of one term only, the standard decomposition method should be employed in this case. Another useful feature of the Adomian method is the noise terms phenomenon. The noise terms may appear for inhomogeneous problems only. This phenomenon was addressed by Adomian in 1994. In 1997, Wazwaz investigated the necessary conditions for the appearance of noise terms in the decomposition series. Noise terms are defined as identical terms with opposite signs [21] that arise in the components u0 and u1 particularly, and in other components as well. By canceling the noise terms between u0 and u1 , even though u1 contains other terms, the remaining noncanceled terms of u0 may give the exact solution of the equation. Therefore, it is necessary to verify that the noncanceled terms of u0 satisfy the equation. The noise terms, if they exist in the components u0 and u1 , will provide the solution in a closed form with only two successive iterations.

It was formally proved by Wazwaz in 1997 that a necessary condition for the appearance of the noise terms is required. The conclusion made is that the zeroth component u0 must contain the exact solution u, among other terms [21]. Moreover, it was shown that the nonhomogeneity condition does not always guarantee the appearance of the noise terms. Solitons, Peakons, Kinks, and Compactons There are many types of solitary waves. Solitons, which are localized traveling waves, are asymptotically zero at large distances [1,5,7,21,22,23,24,25,26,27,28,29,30,31,32, 33]. Solitons appear as a bell-shaped sech profile. Soliton solution u() results as a balance between nonlinearity and dispersion, where u(); u0 (); u00 ();    ! 0 as  ! ˙1, where  D x  ct, and c is the speed of the wave prorogation. The soliton solution either decays exponentially as in the KdV equation, or it converges to a constant at infinity such as the kinks of the sine-Gordon equation. This means that the soliton solutions appear as sech˛ or arctan(e˛(xc t) ). Moreover, one soliton interacts with other solitons preserving its permanent form. Another type of solitary wave is the kink wave, which rises or descends from one asymptotic state to another [18]. The Burgers equation and the sine-Gordon equation are examples of nonlinear wave equations that exhibit kink solutions. It is to be noted that the kink u() converges to ˙˛, where ˛ is a constant. However, u0 (); u00 ();    ! 0 as  ! ˙1. Peakons are peaked solitary-wave solutions and are another type of solitary-wave solution. In this structure, the traveling wave solutions are smooth except for a peak at a corner of its crest. Peakons are the points at which a spatial derivative changes signs so that peakons have a finite jump in the first derivative of the solution u(x; t) [3,4,11,12,32]. This means that the first derivative of u(x; t) has identical values with opposite signs around the peak. A significant type of soliton is the compacton, which is a soliton with compact spatial support such that each compacton is a soliton confined to a finite core or a soliton without exponential tails [16,17]. Compactons were formally derived by Rosenau and Hyman in 1993 where a special kind of the KdV equation was used to derive this new discovery. Unlike a soliton that narrows as the amplitude increases, a compacton’s width is independent of the amplitude [16,17,27,28,29,30]. Classical solitons are analytic solutions, whereas compactons are nonanalytic solutions [16,17]. As will be shown by a graph below, compactons are solitons that are free of exponential wings or

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

tails. Compactons arise as a result of the delicate interaction between genuine nonlinear terms and the genuinely nonlinear dispersion, as is the case with the K(n,n) equation.

Applying L1 t to both sides of (16) and using the initial condition we find 2c 2 e cx u(x; t) D C L1 (20) t (u x x x  6uu x ) : (1 C e cx )2

Solitons of the KdV Equation

Notice that the right-hand side contains a linear term u x x x and a nonlinear term uu x . Accordingly, Adomian polynomials for this nonlinear term are given by

In 1895, Korteweg, together with his Ph.D student de Vriesf, derived analytically a nonlinear partial differential equation, well known now by the KdV equation given in its simplest form by u t C 6uu x C u x x x

2c 2 e cx D 0; u(x; 0) D : (1 C e cx )2

(15)

@ ; @t

D u0 x u3 C u1 x u2 C u2 x u1 C u3 x u0 : Recall that the Adomian decomposition method suggests that the linear function u may be represented by a decomposition series uD

1 X

un ;

(22)

nD0

whereas nonlinear term F(u) can be expressed by an infinite series of the so-called Adomian polynomials An given in the form 1 X A n (u0 ; u1 ; u2 ; : : : ; u n ) : (23) F(u) D nD0

Using the decomposition identification for the linear and nonlinear terms in (20) yields ! 1 1 X X 2c 2 ecx 1 u n (x; t) D  Lt u n (x; t) (1 C ecx )2 nD0 nD0 xxx ! 1 X  6L1 A n : (24) t nD0

(16)

(17)

The components u n ; n  0 can be elegantly calculated by

L1 t

assuming that the integral operator exists and may be regarded as a onefold definite integral defined by Z t (:)dt : (18) L1 t (:) D 0

This means that L1 t L t u(x; t) D u(x; t)  u(x; 0) :

A3 D 12 L x (2u0 u3 C 2u1 u2 )

The Adomian method allows for the use of the recursive relation 2c 2 ecx ; u0 (x; t) D (1 C ecx )2 (25)   1 (A ) ; k  0 : u kC1 (x; t) D  L1 u  6L k k t t xxx

where the differential operator Lt is Lt D

A1 D 12 L x (2u0 u1 ) D u0x u1 C u0 u1x ; A2 D 12 L x (2u0 u2 C u12 ) D u0x u2 C u1x u1 C u2x u0 ; (21)

The KdV equation is the simplest equation embodying both nonlinearity and dispersion [5,34]. This equation has served as a model for the development of solitary-wave theory. The KdV equation is a completely integrable biHamiltonian equation. The KdV equation is used to model the height of the surface of shallow water in the presence of solitary waves. The nonlinearity represented by uu x tends to localize the wave, while the linear dispersion represented by u x x x spreads it out. The balance between the weak nonlinearity and the linear dispersion gives solitons that consist of single humped waves. The equilibrium between the nonlinearity uu x and the dispersion u x x x of the KdV equation is stable. In 1965, Zabusky and Kruskal [35] investigated numerically the nonlinear interaction of a large solitary wave overtaking a smaller one and discovered that solitary waves underwent nonlinear interactions following the KdV equation. Further, the waves emerged from this interaction retaining their original shape and amplitude, and therefore conserved energy and mass. The only effect of the interaction was a phase shift. To apply the Adomian decomposition method, we first write the KdV Eq. (15) in an operator form: L t u D u x x x  6uu x ;

A0 D F(u0 ) D u0 u0x ;

(19)

2c 2 ecx ; (1 C ecx )2   u1 (x; t) D  L1 u0x x x  6L1 t t (A0 )

u0 (x; t) D

2c 5 e cx (ecx  1) t; (1 C ecx )3   u2 (x; t) D  L1 u1x x x  6L1 t t (A1 ) D

D

c 8 ecx (e2cx  4e cx C 1) 2 t ; (1 C e cx )4

5

6

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

  u2x x x  6L1 u3 (x; t) D  L1 t t (A2 ) c 11 ecx (e3cx  11e2cx C 11ecx  1) 3 t ; 3(1 C e cx )5   u3x x x  6L1 u4 (x; t) D  L1 t t (A3 ) D

D

c 14 ecx (e4cx 26e3cx C66e2cx 26ecx C1) 4 t ; 12(1 C e cx )6 (26)

and so on. The series solution is thus given by 2c 2 ecx u(x; t) D (1 C ecx )2  c 3 (e cx  1) c 6 (e2cx  4e cx C 1) 2  1C t t C (1 C ecx ) 2(1 C e cx )2  c 9 (e3cx  11e2cx C 11ecx  1) 3 C t C    ; 6(1 C e cx )3 (27) and in a closed form by u(x; t) D 

2c 2 ec(xc 1C

2 t)

2 ec(xc 2 t)

:

(28)

The last equation emphasizes the fact that the dispersion relation is ! D c 3 . Moreover, the exact solution (28) can be rewritten as hc  i c2 x  c2 t : u(x; t) D (29) sech2 2 2 This in turn gives the bell-shaped soliton solution of the KdV equation. A typical graph of a bell-shaped soliton is given in Fig. 1. The graph shows that solitons are characterized by exponential wings or tails. The graph also confirms that solitons become asymptotically zero at large distances.

Burgers (1895–1981) introduced one of the fundamental model equations in fluid mechanics [18] that demonstrates coupling between nonlinear advection uu x and linear diffusion u x x . The Burgers equation appears in gas dynamics and traffic flow. Burgers introduced this equation to capture some of the features of turbulent fluid in a channel caused by the interaction of the opposite effects of convection and diffusion. The standard form of the Burgers equation is given by

t > 0;

the equation is called an inviscid Burgers equation. The inviscid Burgers equation will not be examined in this work. It is the goal of this work to apply the Adomian decomposition method to the Burgers equation; therefore we write (30) in an operator form L t u D u x x  uu x ;

 ¤0;

(30)

where u(x; t) is the velocity and  is a constant that defines the kinematic viscosity. If the viscosity  D 0, then

(31)

where the differential operator Lt is Lt D

Kinks of the Burgers Equation

u t C uu x D u x x ; 2c ; u(x; 0) D cx (1 C e  )

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory, Figure 1 Figure 1 shows the soliton graph u(x; t) D sech2 (x  ct); c D 1; 5  x; t  5

@ ; @t

(32)

and as a result Z t (:) D (:)dt : L1 t

(33)

0

This means that L1 t L t u(x; t) D u(x; t)  u(x; 0) :

(34)

Applying L1 t to both sides of (31) and using the initial condition we find u(x; t) D

2c cx

(1 C e  )

C L1 t (u x x  uu x ) :

(35)

Notice that the right-hand side contains a linear term and a nonlinear term uu x . The Adomian polynomials for this nonlinear term uu x are the same as in the KdV equation.

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

Using the decomposition identification for the linear and nonlinear terms in (35) yields 1 X

2c

u n (x; t) D

cx

(1 C e  )

nD0

C

L1 t



nD0 1 X

 L1 t

!

1 X

u n (x; t) xx

! :

An

(36)

nD0

The Adomian method allows for the use of the recursive relation 2c

u0 (x; t) D

; cx (1 C e n )   u k x x x  L1 u kC1 (x; t) D L1 t t (A k ) ;

(37) k 0:

The components u n ; n  0 can be elegantly calculated by 2c

; cx (1 C e  )   u0x x x  L1 u1 (x; t) D L1 t t (A0 ) u0 (x; t) D

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory, Figure 2 Figure 2 shows the kink graph u(x; t) D tanh(x  ct); c D 1; 10  x; t  10

cx

2c 3 e 

D

t; cx (1 C e  )2   u2 (x; t) D L1 u1x x x  L1 t t (A1 ) cx cx c 5 e  (e 

 1)

t2 ; e )3   u2x x x  L1 u3 (x; t) D L1 t t (A2 ) D

cx

D

c 7 e  (e

2cx 

3 3 (1 C e )4   u3x x x  L1 u4 (x; t) D L1 t t (A3 ) D

c 9 e (e

3cx 

 11e

12 4 (1

2cx 

Ce

t3 ;

Peakons of the Camassa–Holm Equation

C 11e cx 

cx 

 1)

)5

Camassa and Holm [4] derived in 1993 a completely integrable wave equation

t4 ;

u t C 2ku x  u x x t C 3uu x D 2u x u x x C uu x x x

and so on. The series solution is thus given by cx

u(x; t) D

2ce 

cx

(1 C e  ) cx

c2

cx

 e  C

cx

(1 C e  ) C

c 6 (e

2cx 

tC

c 4 (e   1) cx

2 2 (1 C e  )2

t2

cx

 4e  C 1) cx

6 3 (1 C e  )3

t3 C   

! ;

(39)

and in a closed form by 2c

u(x; t) D 

c

1 C e  (xc t)

;

(41)

Figure 2 shows a kink graph. The graph shows that the kink converges to ˙1 as  ! ˙1.

cx

 4e  C 1) cx 

cx 

(38)

cx 

 2 (1 C

or equivalently i  h c (x  ct) : u(x; t) D c 1  tanh 2

 ¤0;

(40)

(42)

by retaining two terms that are usually neglected in the small-amplitude shallow-water limit [3]. The constant k is related to the critical shallow-water wave speed. This equation models the unidirectional propagation of water waves in shallow water. The CH Eq. (42) differs from the well-known regularized long wave (RLW) equation only through the two terms on the right-hand side of (42). Moreover, this equation has an integrable bi-Hamiltonian structure and arises in the context of differential geometry, where it can be seen as a reexpression for geodesic flow on an infinite-dimensional Lie group. The CH equation admits a second-order isospectral problem and allows for peaked solitary-wave solutions, called peakons.

7

8

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

In this work, we will apply the Adomian method to the CH equation u t u x x t C3uu x D 2u x u x x Cuu x x x ;

u(x; 0) D cejxj ; (43)

The Adomian scheme allows the use of the recursive relation u0 (x; t) D cex ; u kC1 (x; t) D L1 t (u k (x; t)) x x t C L1 t (3 A k C 2B k C C k ) ;

or equivalently

k 0: (51)

u t D u x x t 3uu x C2u x u x x Cuu x x x ;

u(x; 0) D cejxj ; (44)

where we set k D 0. Proceeding as before, the CH Eq. (44) in an operator form is as follows: L t u D u x x t 3uu x C2u x u x x Cuu x x x ;

u(x; 0) D cejxj ; (45)

where the differential operator Lt is as defined above. It is important to point out that the CH equation includes three nonlinear terms; therefore, we derive the following three sets of Adomian polynomials for the terms uu x ; u x u x x , and uu x x x : A 0 D u0 u0 x ; A 1 D u0 x u1 C u0 u1 x ;

(46)

A 2 D u0 x u2 C u1 x u1 C u2 x u0 ; B 0 D u0 x u0 x x ; B 1 D u0 x x u1 x C u0 x u1 x x ;

(47)

B 2 D u0 x x u2 x C u1 x u1 x x C u0 x u2 x x ;

The components u n ; n  0 can be elegantly computed by u0 (x; t) D cex ; 1 u1 (x; t) D L1 t (u0 (x; t)) x x t C L t (3 A0 C 2B0 C C0 )

D c 2 ex t ; 1 u2 (x; t) D L1 t (u1 (x; t)) x x t C L t (3 A1 C 2B1 C C1 ) 1 D c 3 ex t 2 ; 2! 1 u3 (x; t) D L1 t (u2 (x; t)) x x t C L t (3 A2 C 2B2 C C2 ) 1 D c 4 ex t 3 ; 3! 1 u4 (x; t) D L1 t (u3 (x; t)) x x t C L t (3 A3 C 2B3 C C3 ) 1 (52) D c 4 ex t 4 ; 4!

and so on. The series solution for x > 0 is given by u(x; t) D cex   1 1 1  1 C ct C (ct)2 C (ct)3 C (ct)4 C    ; 2! 3! 4! (53) and in a closed form by

and

u(x; t) D ce(xc t) :

C0 D u0 u0 x x x ; (48)

C1 D u0 x x x u1 C u0 u1 x x x ; C2 D u0 x x x u2 C u1 u1 x x x C u0 u2 x x x :

Applying the inverse operator L1 t to both sides of (45) and using the initial condition we find u(x; t) D cejxj CL1 t (u x x t  3uu x C 2u x u x x C uu x x x ) : (49) Case 1. For x > 0, the initial condition will be u(x; 0) D ex . Using the decomposition series (1) and Adomian polynomials in (49) yields ! 1 1 X X x 1 u n (x; t) D ce C L t u n (x; t) nD0

C

nD0

L1 t

3

1 X nD0

An C 2

1 X nD0

Bn C

xxt 1 X nD0

!

Cn

:

(50)

(54)

Case 2. For x < 0, the initial condition will be u(x; 0) D e x . Proceeding as before, we obtain u0 (x; t) D ce x ; 1 u1 (x; t) D L1 t (u0 (x; t)) x x t C L t (3 A0 C 2B0 C C0 )

D  c 2 ex t ; 1 u2 (x; t) D L1 t (u1 (x; t)) x x t C L t (3 A1 C 2B1 C C1 ) 1 D c 3 ex t 2 ; 2! 1 u3 (x; t) D L1 t (u2 (x; t)) x x t C L t (3 A2 C 2B2 C C2 ) 1 D  c4 ex t3 ; 3! 1 u4 (x; t) D L1 t (u3 (x; t)) x x t C L t (3 A3 C 2B3 C C3 ) 1 D c 4 ex t 4 ; 4! (55)

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

KdV equation, named K(m,n), of the form u t C (u m )x C (u n )x x x D 0 ;

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory, Figure 3 Figure 3 shows the peakon graph u(x; t) D ej(xct)j ; c D 1; 2  x; t  2

and so on. The series solution for x < 0 is given by u(x; t) D ce x   1 1 1 2 3 4  1  ct C (ct)  (ct) C (ct) C    ; 2! 3! 4! (56) and in a closed form by u(x; t) D ce(xc t) :

(57)

Combining the results for both cases gives the peakon solution u(x; t) D cejxc tj :

(58)

Figure 3 shows a peakon graph. The graph shows that a peakon with a peak at a corner is generated with equal first derivatives at both sides of the peak, but with opposite signs. Compactons of the K(n,n) Equation The K(n,n) equation [16,17] was introduced by Rosenau and Hyman in 1993. This equation was investigated experimentally and analytically. The K(m,n) equation is a genuinely nonlinear dispersive equation, a special type of the

n>1:

(59)

Compactons, which are solitons with compact support or strict localization of solitary waves, have been investigated thoroughly in the literature. The delicate interaction between the effect of the genuine nonlinear convection (u n )x and the genuinely nonlinear dispersion of (u n )x x x generates solitary waves with exact compact support that are called compactons. It was also discovered that solitary waves may compactify under the influence of nonlinear dispersion, which is capable of causing deep qualitative changes in the nature of genuinely nonlinear phenomena [16,17]. Unlike solitons that narrow as the amplitude increases, a compacton’s width is independent of the amplitude. Compactons such as drops do not possess infinite wings; hence they interact among themselves only across short distances. Compactons are nonanalytic solutions, whereas classical solitons are analytic solutions. The points of nonanalyticity at the edge of a compacton correspond to points of genuine nonlinearity for the differential equation and introduce singularities in the associated dynamical system for the traveling waves [16,17,27,28,29,30]. Compactons were proved to collide elastically and vanish outside a finite core region. This discovery was studied thoroughly by many researchers who were involved with identical nonlinear dispersive equations. It is to be noted that solutions were obtained only for cases where m D n. However, for m ¤ n, solutions have not yet been determined. Without loss of generality, we will examine the special K(3,3) initial value problem p   1 6c u t C (u3 )x C (u3 )x x x D 0 ; u(x; 0) D cos x : 2 3 (60) We first write the K(3,3) Eq. (60) in an operator form: p   1 6c 3 3 cos x ; L t u D (u )x  (u )x x x ; u(x; 0) D 2 3 (61) where the differential operator Lt is as defined above. The differential operator Lt is defined by Lt D

@ : @t

(62)

Applying L1 t to both sides of (61) and using the initial condition we find p    3  6c 1 (u )x C (u3 )x x x : (63) u(x; t) D cos x  L1 t 2 3

9

10

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

Notice that the right-hand side contains two nonlinear terms (u3 )x and (u3 )x x x . Accordingly, the Adomian polynomials for these terms are given by A0 D (u03 )x ; A1 D (3u02 u1 )x ;

(64)

A2 D (3u02 u2 C 3u0 u12 )x ; A3 D (3u02 u3 C 6u0 u1 u2 C u13 )x ;

B0 D (u03 )x x x ; B1 D (3u02 u1 )x x x ;

(65)

B2 D (3u02 u2 C 3u0 u12 )x x x ; B3 D (3u02 u3 C 6u0 u1 u2 C u13 )x x x ;

(69)

u(x; t) p          1 1 1 1 6c D cos x cos ct Csin x sin ct ; 2 3 3 3 3 (70) or equivalently

u(x; t) D

respectively. Proceeding as before, Eq. (63) becomes

nD0

u(x; t) ! p       1 6c 1 ct 4 1 ct 2 C C cos x 1 D 2 3 2! 3 4! 3 ! p       1 6c ct 1 ct 3 1 ct 5 C C C   : sin x  2 3 3 3! 3 5! 5 This is equivalent to

and

1 X

and so on. The series solution is thus given by

8n p  o < 6c cos 1 (x  ct) ; 2 3

j x  ct j

:0

otherwise :

3 2

;

(71)

! p   1 X 1 6c 1 u n (x; t) D An C Bn : cos x L t 2 3 nD0 (66)

This in turn gives the compacton solution of the K(3,3) equation. Figure 4 shows a compacton confined to a finite core without exponential wings. The graph shows a compacton: a soliton free of exponential wings.

This gives the recursive relation p

  1 6c cos x ; 2 3  1  L t (A k C B k ) ; u kC1 (x; t) D  L1 t u0 (x; t) D

(67) k 0:

The components u n ; n  0 can be recursively determined as p   1 6c cos x ; u0 (x; t) D 2 3 p   1 c 6c u1 (x; t) D sin x t; 6 3 p   1 c 2 6c u2 (x; t) D  cos x t2 ; 36 3 (68) p   1 c 3 6c 3 u3 (x; t) D  sin x t ; 324 3 p   1 c 4 6c u4 (x; t) D cos x t4 ; 3888 3 p   1 c 5 6c u5 (x; t) D sin x t5 ; 58320 3

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory, Figure 4 1

Figure 4 shows the compacton graph u(x; t) D cos 2 (x  ct); c D 1; 0  x; t  1

Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory

It is interesting to point out that the generalized solution of the K(n,n) equation, n > 1, is given by the compactons u(x; t) 8n o 1 < 2cn cos2  n1 (x  ct) n1 ; (nC1) 2n D : 0

j x  ct j 

 2

;

otherwise ; (72)

and u3 (x; t) 8n o 1 < 2cn sin2  n1 (x  ct) n1 ; (nC1) 2n D : 0

j x  ct j

 

;

otherwise : (73)

Future Directions The most significant advantage of the Adomian method is that it attacks any problem without any need for a transformation formula or any restrictive assumption that may change the physical behavior of the solution. For singular problems, it was possible to overcome the singularity phenomenon and to attain practically a series solution in a standard way. Moreover, for problems on unbounded domains, combining the obtained series solution by the Adomian method with Padé approximants provides a promising tool to handle boundary value problems. The Padé approximants, which often show superior performance over series approximations, provide a promising tool for use in applied fields. As stated above, unlike other methods such as Hirota and the inverse scattering method, where solutions can be obtained without using prescribed conditions, the Adomian method requires such conditions. Moreover, these conditions must be of a form that does not provide zero for the integrand of the first component u1 (x; t). Such a case should be addressed to enhance the performance of the method. Another aspect that should be addressed is Adomian polynomials. The existing techniques require tedious work to evaluate it. Most importantly, the Adomian method guarantees only one solution for nonlinear problems. This is an issue that should be investigated to improve the performance of the Adomian method compared to other methods. On the other hand, it is important to further examine the N-soliton solutions by simplified forms, such as the form given by Hereman and Nuseir in [7]. The bilinear form of Hirota is not that easy to use and is not always attainable for nonlinear models.

Bibliography Primary Literature 1. Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge 2. Adomian G (1984) A new approach to nonlinear partial differential equations. J Math Anal Appl 102:420–434 3. Boyd PJ (1997) Peakons and cashoidal waves: travelling wave solutions of the Camassa–Holm equation. Appl Math Comput 81(2–3):173–187 4. Camassa R, Holm D (1993) An integrable shallow water equation with peaked solitons. Phys Rev Lett 71(11):1661–1664 5. Drazin PG, Johnson RS (1996) Solitons: an introduction. Cambridge University Press, Cambridge 6. Helal MA, Mehanna MS (2006) A comparison between two different methods for solving KdV-Burgers equation. Chaos Solitons Fractals 28:320–326 7. Hereman W, Nuseir A (1997) Symbolic methods to construct exact solutions of nonlinear partial differential equations. Math Comp Simul 43:13–27 8. Hirota R (1971) Exact solutions of the Korteweg–de Vries equation for multiple collisions of solitons. Phys Rev Lett 27(18):1192–1194 9. Hirota R (1972) Exact solutions of the modified Korteweg– de Vries equation for multiple collisions of solitons. J Phys Soc Jpn 33(5):1456–1458 10. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos Mag 5th Ser 36:422–443 11. Lenells J (2005) Travelling wave solutions of the Camassa– Holm equation. J Differ Equ 217:393–430 12. Liu Z, Wang R, Jing Z (2004) Peaked wave solutions of Camassa–Holm equation. Chaos Solitons Fractals 19:77–92 13. Malfliet W (1992) Solitary wave solutions of nonlinear wave equations. Am J Phys 60(7):650–654 14. Malfliet W, Hereman W (1996) The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys Scr 54:563–568 15. Malfliet W, Hereman W (1996) The tanh method: II. Perturbation technique for conservative systems. Phys Scr 54:569–575 16. Rosenau P (1998) On a class of nonlinear dispersive-dissipative interactions. Phys D 230(5/6):535–546 17. Rosenau P, Hyman J (1993) Compactons: solitons with finite wavelengths. Phys Rev Lett 70(5):564–567 18. Veksler A, Zarmi Y (2005) Wave interactions and the analysis of the perturbed Burgers equation. Phys D 211:57–73 19. Wadati M (1972) The exact solution of the modified Korteweg– de Vries equation. J Phys Soc Jpn 32:1681–1687 20. Wadati M (2001) Introduction to solitons. Pramana J Phys 57(5/6):841–847 21. Wazwaz AM (1997) Necessary conditions for the appearance of noise terms in decomposition solution series. Appl Math Comput 81:265–274 22. Wazwaz AM (1998) A reliable modification of Adomian’s decomposition method. Appl Math Comput 92:1–7 23. Wazwaz AM (1999) A comparison between the Adomian decomposition method and the Taylor series method in the series solutions. Appl Math Comput 102:77–86

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24. Wazwaz AM (2000) A new algorithm for calculating Adomian polynomials for nonlinear operators. Appl Math Comput 111(1):33–51 25. Wazwaz AM (2001) Exact specific solutions with solitary patterns for the nonlinear dispersive K(m,n) equations. Chaos, Solitons and Fractals 13(1):161–170 26. Wazwaz AM (2002) General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive K(n,n) equations in higher dimensional spaces. Appl Math Comput 133(2/3):229–244 27. Wazwaz AM (2003) An analytic study of compactons structures in a class of nonlinear dispersive equations. Math Comput Simul 63(1):35–44 28. Wazwaz AM (2003) Compactons in a class of nonlinear dispersive equations. Math Comput Model l37(3/4):333–341 29. Wazwaz AM (2004) The tanh method for travelling wave solutions of nonlinear equations. Appl Math Comput 154(3):713– 723 30. Wazwaz AM (2005) Compact and noncompact structures for variants of the KdV equation. Int J Appl Math 18(2):213– 221 31. Wazwaz AM (2006) New kinds of solitons and periodic solutions to the generalized KdV equation. Numer Methods Partial Differ Equ 23(2):247–255 32. Wazwaz AM (2006) Peakons, kinks, compactons and solitary patterns solutions for a family of Camassa–Holm equations by using new hyperbolic schemes. Appl Math Comput 182(1):412–424 33. Wazwaz AM (2007) The extended tanh method for new soli-

tons solutions for many forms of the fifth-order KdV equations. Appl Math Comput 184(2):1002–1014 34. Whitham GB (1999) Linear and nonlinear waves. Wiley, New York 35. Zabusky NJ, Kruskal MD (1965) Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243

Books and Reviews Adomian G (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston Burgers JM (1974) The nonlinear diffusion equation. Reidel, Dordrecht Conte R, Magri F, Musette M, Satsuma J, Winternitz P (2003) Lecture Notes in Physics: Direct and Inverse methods in Nonlinear Evolution Equations. Springer, Berlin Hirota R (2004) The Direct Method in Soliton Theory. Cambridge University Press, Cambridge Johnson RS (1997) A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge Kosmann-Schwarzbach Y, Grammaticos B, Tamizhmani KM (2004) Lecture Notes in Physics: Integrability of Nonlinear Systems. Springer, Berlin Wazwaz AM (1997) A First Course in Integral Equations. World Scientific, Singapore Wazwaz AM (2002) Partial Differential Equations: Methods and Applications. Balkema, The Netherlands

Anomalous Diffusion on Fractal Networks

Anomalous Diffusion on Fractal Networks IGOR M. SOKOLOV Institute of Physics, Humboldt-Universität zu Berlin, Berlin, Germany Article Outline Glossary Definition of the Subject Introduction Random Walks and Normal Diffusion Anomalous Diffusion Anomalous Diffusion on Fractal Structures Percolation Clusters Scaling of PDF and Diffusion Equations on Fractal Lattices Further Directions Bibliography Glossary Anomalous diffusion An essentially diffusive process in which the mean squared displacement grows, however, not as hR2 i / t like in normal diffusion, but as hR2 i / t ˛ with ˛ ¤ 0, either asymptotically faster than in normal diffusion (˛ > 1, superdiffusion) or asymptotically slower (˛ < 1, subdiffusion). Comb model A planar network consisting of a backbone (spine) and teeth (dangling ends). A popular simple model showing anomalous diffusion. Fractional diffusion equation A diffusion equation for a non-Markovian diffusion process, typically with a memory kernel corresponding to a fractional derivative. A mathematical instrument which adequately describes many processes of anomalous diffusion, for example the continuous-time random walks (CTRW). Walk dimension The fractal dimension of the trajectory of a random walker on a network. The walk dimension is defined through the mean time T or the mean number of steps n which a walker needs to leave for the first time the ball of radius R around the starting point of the walk: T / R d w . Spectral dimension A property of a fractal structure which substitutes the Euclidean dimension in the expression for the probability to be at the origin at time t, P(0; t) / t d s /2 . Defines the behavior of the network Laplace operator. Alexander–Orbach conjecture A conjecture that the spectral dimension of the incipient infinite cluster in percolation is equal to 4/3 independently on the Eu-

clidean dimension. The invariance of spectral dimension is only approximate and holds within 2% accuracy even in d D 2. The value of 3/4 is achieved in d > 6 and on trees. Compact visitation A property of a random walk to visit practically all sites within the domain of the size of the mean squared displacement. The visitation is compact if the walk dimension dw exceeds the fractal dimension of the substrate df . Definition of the Subject Many situations in physics, chemistry, biology or in computer science can be described within models related to random walks, in which a “particle” (walker) jumps from one node of a network to another following the network’s links. In many cases the network is embedded into Euclidean space which allows for discussing the displacement of the walker from its initial site. The displacement’s behavior is important e. g. for description of charge transport in disordered semiconductors, or for understanding the contaminants’ behavior in underground water. In other cases the discussion can be reduced to the properties which can be defined for a whatever network, like return probabilities to the starting site or the number of different nodes visited, the ones of importance for chemical kinetics or for search processes on the corresponding networks. In the present contribution we only discuss networks embedded in Euclidean space. One of the basic properties of normal diffusion is the fact that the mean squared displacement of a walker grows proportionally to time, R2 / t, and asymptotic deviations from this law are termed anomalous diffusion. Microscopic models leading to normal diffusion rely on the spatial homogeneity of the network, either exact or statistical. If the system is disordered and its local properties vary from one its part to another, the overall behavior at very large scales, larger than a whatever inhomogeneity, can still be diffusive, a property called homogenization. The ordered systems or systems with absent longrange order but highly homogeneous on larger scales do not exhaust the whole variety of relevant physical systems. In many cases the system can not be considered as homogeneous on a whatever scale of interest. Moreover, in some of these cases the system shows some extent of scale invariance (dilatation symmetry) typical for fractals. Examples are polymer molecules and networks, networks of pores in some porous media, networks of cracks in rocks, and so on. Physically all these systems correspond to a case of very strong disorder, do not homogenize even at largest relevant scales, and show fractal properties. The diffusion in such systems is anomalous, exhibiting large deviations

13

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Anomalous Diffusion on Fractal Networks

from the linear dependence of the mean squared displacement on time. Many other properties, like return probabilities, mean number of different visited nodes etc. also behave in a way different from the one they behave in normal diffusion. Understanding these properties is of primary importance for theoretical description of transport in such systems, chemical reactions in them, and in devising efficient search algorithms or in understanding ones used in natural search, as taking place in gene expression or incorporated into the animals’ foraging strategies. Introduction The theory of diffusion processes was initiated by A. Fick who was interested in the nutrients’ transport in a living body and put down the phenomenological diffusion equation (1855). The microscopic, probabilistic discussion of diffusion started with the works of L. Bachelier on the theory of financial speculation (1900), of A. Einstein on the motion of colloidal particles in a quiescent fluid (the Brownian motion) (1905), and with a short letter of K. Pearson to the readers of Nature in the same year motivated by his work on animal motion. These works put forward the description of diffusion processes within the framework of random walk models, the ones describing the motion as a sequence of independent bounded steps in space requiring some bounded time to be completed. These models lead to larger scales and longer times to a typical diffusion behavior as described by Fick’s laws. The most prominent property of this so-called normal diffusion is the fact that the mean squared displacement of the diffusing particle from its initial position at times much longer than the typical time of one step grows as hx 2 (t)i / t. Another prominent property of normal diffusion is the fact that the distribution of the particle’s displacements tends to a Gaussian (of width hx 2 (t)i1/2 ), as a consequence of the independence of different steps. The situations described by normal diffusion are abundant, but looking closer into many other processes (which still can be described within a random walk picture, however, with only more or less independent steps, or with steps showing a broad distribution of their lengths or times) revealed that these processes, still resembling or related to normal diffusion, show a vastly different behavior of the mean squared displacement, e. g. hx 2 (t)i / t ˛ with ˛ ¤ 1. In this case one speaks about anomalous diffusion [11]. The cases with ˛ < 1 correspond to subdiffusion, the ones with ˛ > 1 correspond to superdiffusion. The subdiffusion is typical for the motion of charge carriers in disordered semiconductors, for contaminant transport by underground water or for proteins’ motion in cell

membranes. The models of subdiffusion are often connected either with the continuous-time random walks or with diffusion in fractal networks (the topic of the present article) or with combinations thereof. The superdiffusive behavior, associated with divergent mean square displacement per complete step, is encountered e. g. in transport in some Hamiltonian models and in maps, in animal motion, in transport of particles by flows or in diffusion in porous media in Knudsen regime. The diffusion on fractal networks (typically subdiffusion) was a topic of extensive investigation in 1980s–1990s, so that the most results discussed here can be considered as well-established. Therefore most of the references in this contribution are given to the review articles, and only few of them to original works (mostly to the pioneering publications or to newer works which didn’t find their place in reviews). Random Walks and Normal Diffusion Let us first turn to normal diffusion and concentrate on the mean squared displacement of the random walker. Our model will correspond to a random walk on a regular lattice or in free space, where the steps of the random walker will be considered independent, identically distributed random variables (step lengths s i ) following a distribution with a given symmetric probability density function (PDF) p(s), so that hsi D 0. Let a be the root mean square displacement in one step, hs2i i D a2 . The squared displacement of the walker after n steps is thus * hr2n i

D

n X iD1

!2 + si

D

n n n X X X hs2i i C 2 hs i s j i : iD1

iD1 jDiC1

The first sum is simply na2 , and the second one vanishes if different˝ steps are uncorrelated and have zero mean. ˛ Therefore r2n D a 2 n: The mean squared displacement in a walk is proportional to the number of steps. Provided the mean time  necessary to complete a step exists, this expression can be translated into the temporal dependence of the displacement hr2 (t)i D (a 2 /)t, since the mean number of steps done during the time t is n D t/. This is the time-dependence of the mean squared displacement of a random walker characteristic for normal diffusion. The prefactor a 2 / in the hr2 (t)i / t dependence is connected with the usual diffusion coefficient K, defined through hr2 (t)i D 2dKt, where d is the dimension of the Euclidean space. Here we note that in the theory of random walks one often distinguishes between the situations termed as genuine “walks” (or velocity models) where the particle moves, say, with a constant velocity and

Anomalous Diffusion on Fractal Networks

changes its direction at the end of the step, and “random flight” models, when steps are considered to be instantaneous (“jumps”) and the time cost of a step is attributed to the waiting time at a site before the jump is made. The situations considered below correspond to the flight picture. The are three important postulates guaranteeing that random walks correspond to a normal diffusion process:  The existence of the mean squared displacement per step, hs2 i < 1  The existence of the mean time per step (interpreted as a mean waiting time on a site)  < 1 and  The uncorrelated nature of the walk hs i s j i D hs2 iı i j . These are exactly the three postulates made by Einstein in his work on Brownian motion. The last postulate can be weakened: The persistent (i. e. correlated) random walks P still lead to normal diffusion provided 1 jDi hs i s j i < 1. In the case when the steps are independent, the central limit theorem immediately states that the limiting distribution of the displacements will be Gaussian, of the form   1 dr2 P(r; t) D exp  : 4Kt (4 Kt)d/2 This distribution scales as a whole: rescaling the distance by a factor of , x ! x and of the time by the factor of 2 , t ! 2 t, does not change the form of this distribution. TheRmoments of the displacement scale according to hr k i D P(r; t)r k d d r k D const(k)t k/2 . Another important property of the the PDF is the fact that the return probability, i. e. the probability to be at time t at the origin of the motion, P(0; t), scales as P(0; t) / t d/2 ;

(1)

i. e. shows the behavior depending on the substrate’s dimension. The PDF P(r; t) is the solution of the Fick’s diffusion equation @ P(r; t) D kP(r; t) ; @t

(2)

where  is a Laplace operator. Equation 2 is a parabolic partial differential equation, so that its form is invariant under the scale transformation t ! 2 t, r ! r discussed above. This invariance of the equation has very strong implications. If follows, for example, that one can invert the scaling relation for the mean squared displacement R2 / t and interpret this inverse one T / r2

as e. g. a scaling rule governing the dependence of a characteristic passage time from some initial site 0 to the sites at a distance r from this one. Such scaling holds e. g. for the mean time spent by a walk in a prescribed region of space (i. e. for the mean first passage time to its boundary). For an isotropic system Eq. (2) can be rewritten as an equation in the radial coordinate only:

 @ 1 @ d1 @ P(r; t) D K d1 r P(r; t) : (3) @t @r @r r Looking at the trajectory of a random walk at scales much larger than the step’s length one infers that it is self-similar in statistical sense, i. e. that its whatever portion (of the size considerably larger then the step’s length) looks like the whole trajectory, i. e. it is a statistical, random fractal. The fractal (mass) dimension of the trajectory of a random walk can be easily found. Let us interthe number of steps as the “mass” M of the object and ˝pret ˛1/2 r2n as its “size” L. The mass dimension D is then defined as M / L D , it corresponds to the scaling of the mass of a solid D-dimensional object with its typical size. The scaling of the mean squared displacement of a simple random walk with its number of steps suggests that the corresponding dimension D D dw for a walk is dw D 2. The large-scale properties of the random walks in Euclidean space are universal and do not depend on exact form of the PDF of waiting times and of jumps’ lengths (provided the first possesses at least one and the second one at least two moments). This fact can be used for an interpretation of several known properties of simple random walks. Thus, random walks on a one-dimensional lattice (d D 1) are recurrent, i. e. each site earlier visited by the walk is revisited repeatedly afterwards, while for d  3 they are nonrecurrent. This property can be rationalized through arguing that an intrinsically two-dimensional object cannot be “squeezed” into a one-dimensional space without self-intersections, and that it cannot fill fully a space of three or more dimensions. The same considerations can be used to rationalize the behavior of the mean number of different sites visited by a walker during n steps hS n i. hS n i typically goes as a power law in n. Thus, on a one-dimensional lattice hS n i / n1/2 , in lattices in three and more dimensions hS n i / n, and in a marginal two-dimensional situation hS n i / n/ log n. To understand the behavior in d D 1 and in d D 3, it is enough to note that in a recurrent one-dimensional walk all sites within the span of the walk are visited by a walker at least once, so that the mean number of different sites visited by a one-dimensional walk (d D 1) grows as a span of the walk, which itself is proportional to a mean squared displacement, hS n i / L D hr2n i1/2 , i. e.

15

16

Anomalous Diffusion on Fractal Networks

hS n i / n1/2 D n1/d w . In three dimensions (d D 3), on the contrary, the mean number of different visited sites grows as the number of steps, hS n i / n, since different sites are on the average visited only finite number of times. The property of random walks on one-dimensional and twodimensional lattices to visit practically all the sites within the domain of the size of their mean square displacement is called the compact visitation property. The importance of the number of different sites visited (and its moments like hS n i or hS(t)i) gets clear when one considers chemical reactions or search problems on the corresponding networks. The situation when each site of a network can with the probability p be a trap for a walker, so that the walker is removed whenever it visits such a marked site (reaction scheme ACB ! B, where the symbol A denotes the walker and symbol B the “immortal” and immobile trap) corresponds to a trapping problem of reaction kinetics. The opposite situation, corresponding to the evaluation of the survival probability of an immobile A-particle at one of the network’s nodes in presence of mobile B-walkers (the same A C B ! B reaction scheme, however now with immobile A and mobile B) corresponds to the target problem, or scavenging. In the first case the survival probability ˚(N) for the A-particle after N steps goes for p small as ˚(t) / hexp(pS N )i, while for the second case it behaves as ˚(t) / exp(phS N i). The simple A C B ! B reaction problems can be also interpreted as the ones of the distribution of the times necessary to find one of multiple hidden “traps” by one searching agent, or as the one of finding one special marked site by a swarm of independent searching agents. Anomalous Diffusion The Einstein’s postulates leading to normal diffusion are to no extent the laws of Nature, and do not have to hold for a whatever random walk process. Before discussing general properties of such anomalous diffusion, we start from a few simple examples [10].

consider the random walk to evolve in discrete time; the time t and the number of steps n are simply proportional to each other. Diffusion on a Comb Structure The simplest example of how the subdiffusive motion can be created in a more or less complex (however yet non-fractal) geometrical structure is delivered by the diffusion on a comb. We consider the diffusion of a walker along the backbone of a comb with infinite “teeth” (dangling ends), see Fig. 1. The walker’s displacement in x-direction is only possible when it is on the backbone; entering the side branch switches off the diffusion along the backbone. Therefore the motion on the backbone of the comb is interrupted by waiting times. The distribution of these waiting times is given by the power law (t) / t 3/2 corresponding to the first return probability of a one-dimensional random walk in a side branch to its origin. This situation – simple random walk with broad distribution of waiting times on sites – corresponds to a continuous-time random walk model. In this model, for the case of the power-law waiting time distributions of the form (t) / t 1˛ with ˛ < 1, the mean number of steps hn(t)i (in our case: the mean number of steps over the backbone) as a function of time goes as hn(t)i / t ˛ . Since the mean squared displacement along the backbone is proportional to this number of steps, it is ˝ ˛ also governed by x 2 (t) / t ˛ , i. e. in our ˝ case ˛ we have to do with strongly subdiffusive behavior x 2 (t) / t 1/2 . The PDF of the particle’s displacements in CTRW with the waiting time distribution (t) is governed by a nonMarkovian generalization of the diffusion equation Z 1 @ @ M(t  t 0 )P(x; t 0 ) ; P(x; t) D a 2  @t @t 0 with the memory kernel M(t) which has a physical meaning of the time-dependent density of steps and is given by ˜ the inverse Laplace transform of M(u) D ˜ (u)/[1  ˜ (u)] (with ˜ (u) being the Laplace-transform of (t)). For the

Simple Geometric Models Leading to Subdiffusion Let us first discuss two simple models of geometrical structures showing anomalous diffusion. The discussion of these models allows one to gain some intuition about the emergence and properties of anomalous diffusion in complex geometries, as opposed to Euclidean lattices, and discuss mathematical notions appearing in description of such systems. In all cases the structure on which the walk takes place will be considered as a lattice with the unit spacing between the sites, and the walk corresponds to jumps from a site to one of its neighbors. Moreover, we

Anomalous Diffusion on Fractal Networks, Figure 1 A comb structure: Trapping in the dangling ends leads to the anomalous diffusion along the comb’s backbone. This kind of anomalous diffusion is modeled by CTRW with power-law waiting time distribution and described by a fractional diffusion equation

Anomalous Diffusion on Fractal Networks

case of a power-law waiting-time probability density of the type (t) / t 1˛ the structure of the right-hand side of the equation corresponds to the fractional Riemann– Liouvolle derivative 0 Dˇt , an integro-differential operator defined as Z 1 d t 0 f (t 0 ) ˇ dt ; (4) 0 D t f (t) D  (1  ˇ) dt 0 (t  t 0 )ˇ (here only for 0 < ˇ < 1), so that in the case of the diffusion on a backbone of the comb we have @ P(x; t) ; P(x; t) D K0 D1˛ t @t

(5)

(here with ˛ D 1/2). An exhaustive discussion of such equations is provided in [13,14]. Similar fractional diffusion equation was used in [12] for the description of the diffusion in a percolation system (as measured by NMR). However one has to be cautious when using CTRW and corresponding fractional equations for the description of diffusion on fractal networks, since they may capture some properties of such diffusion and fully disregard other ones. This question will be discussed to some extent in Sect. “Anomalous Diffusion on Fractal Structures”. Equation (5) can be rewritten in a different but equivalent form  ˛ 0 Dt

@ P(x; t) D KP(x; t) ; @t

with the operator 0 D˛t , a Caputo derivative, conjugated to the Riemann–Liouville one, Eq. (4), and different from Eq. (4) by interchanged order of differentiation and integration (temporal derivative inside the integral). The derivation of both forms of the fractional equations and their interpretation is discussed in [21]. For the sake of transparency, the corresponding integrodifferential operators 0 Dˇt and 0 Dˇt are simply denoted as a fractional partial derivative @ˇ /@t ˇ and interpreted as a Riemann– Liouville or as a Caputo derivative depending whether they stand on the left or on the right-hand side of a generalized diffusion equation. Random Walk on a Random Walk and on a Self-Avoiding Walk Let us now turn to another source of anomalous diffusion, namely the nondecaying correlations between the subsequent steps, introduced by the structure of the possible ways on a substrate lattice. The case where the anomalies of diffusion are due to the tortuosity of the path between the two sites of a fractal “network” is illustrated by the examples of random walks on polymer chains, being rather simple, topologically one-dimensional objects. Considering lattice models

for the chains we encounter two typical situations, the simpler one, corresponding to the random walk (RW) chains and a more complex one, corresponding to self-avoiding ones. A conformation of a RW-chain of N monomers corresponds to a random walk on a lattice, self-intersections are allowed. In reality, RW chains are a reasonable model for polymer chains in melts or in -solutions, where the repulsion between the monomers of the same chain is compensated by the repulsion from the monomers of other chains or from the solvent molecules, so that when not real intersections, then at least very close contacts between the monomers are possible. The other case corresponds to chains in good solvents, where the effects of steric repulsion dominate. The end-to-end distance in a random walk chain grows as R / l 1/2 (with l being its contour length). In a self-avoiding-walk (SAW) chain R / l  , with  being a so-called Flory exponent (e. g.   3/5 in 3d). The corresponding chains can be considered as topologically onedimensional fractal objects with fractal dimensions df D 2 and df D 1/  5/3 respectively. We now concentrate on an walker (excitation, enzyme, transcription factor) performing its diffusive motion on a chain, serving as a “rail track” for diffusion. This chain itself is embedded into the Euclidean space. The contour length of the chain corresponds to the chemical coordinate along the path, in each step of the walk at a chain the chemical coordinate changes by ˙1. Let us first consider the situation when a particle can only travel along this chemical path, i. e. cannot change its track at a point of a self-intersection of the walk or at a close contact between the two different parts of the chain. This walk starts with its first step at r D 0. Let K be the diffusion coefficient of the walker along the chain. The typical displacement of a walker along the chain after time t will then be l / t 1/2 D t 1/d w . For a RW-chain this displacement along the chain is translated into the displacement in Euclidean space according to R D hR2 i1/2 / l 1/2 , so that the typical displacement in Euclidean space after time t goes as R / t 1/4 , a strongly subdiffusive behavior known as one of the regimes predicted by a repetition theory of polymer dynamics. The dimension of the corresponding random walk is, accordingly, dw D 4. The same discussion for the SAW chains leads to dw D 1/2. After time t the displacement in the chemical space is given by a Gaussian distribution   l2 1 exp  : pc (l; t) D p 4Kt 4 Kt The change in the contour length l can be interpreted as the number of steps along the random-walk “rail”, so that the Euclidean displacement for the given l is distributed

17

18

Anomalous Diffusion on Fractal Networks

according to  pE (r; l) D



  d dr2 exp  : 2 a2 jlj 2a2 jlj

(6)

We can, thus, obtain the PDF of the Euclidean displacement: Z 1 P(r; l) D pE (r; l)pc (l; t)dl : (7) 0

The same discussion can be pursued for a self-avoiding walk for which

 r   1  r 1/(1)  pE (r; l) '  exp const  ; (8) l l with being the exponent describing the dependence of the number of realizations of the corresponding chains on their lengths. Equation (6) is a particular case of Eq. (8) with D 1 and  D 1/2. Evaluating the integral using the Laplace method, we get " (11/d w )1 #  r P(r; t) ' exp const 1/d t w up to the preexponential. We note that in this case the reason for the anomalous diffusion is the tortuosity of the chemical path. The topologically one-dimensional structure of the lattice allowed us for discussing the problem in considerable detail. The situation changes when we introduce the possibility for a walker to jump not only to the nearest-neighbor in chemical space by also to the sites which are far away along the chemical sequence but close to each other in Euclidean space (say one lattice site of the underlying lattice far). In this case the structure of the chain in the Euclidean space leads to the possibility to jump very long in chemical sequence of the chain, with the jump length distribution going as p(l) / l 3/2 for an RW and p(l) / l 2:2 for a SAW in d D 3 (the probability of a close return of a SAW to its original path). The corresponding distributions lack the second moment (and even the first moment for RW chains), and therefore one might assume that also the diffusion in a chemical space of a chain will be anomalous. It indeed shows considerable peculiarities. If the chain’s structure is frozen (i. e. the conformational dynamics of a chain is slow compared to the diffusion on the chain), the situation in both cases corresponds to “paradoxical diffusion” [22]: Although the PDF of displacements in the chemical space lacks the second (and higher) moments, the width of the PDF (described e. g. as its interquartile distance) grows, according to W / t 1/2 . In the Euclidean

space we have here to do with the diffusion on a static fractal structure, i. e. with a network of fractal dimension df coinciding with the fractal dimension of the chain. Allowing the chain conformation changing in time (i. e. considering the opposite limiting case when the conformation changes are fast enough compared to the diffusion along the chain) leads to another, superdiffusive, behavior, namely to Lévy flights along the chemical sequence. The difference between the static, fixed networks and the transient ones has to be borne in mind when interpreting physical results. Anomalous Diffusion on Fractal Structures Simple Fractal Lattices: the Walk’s Dimension As already mentioned, the diffusion on fractal structures is often anomalous, hx 2 (t)i / t ˛ with ˛ ¤ 1. Parallel to the situation with random walks in Euclidean space, the trajectory of a walk is typically self similar (on the scales exceeding the typical step length), so that the exponent ˛ can be connected with the fractal dimension of the walk’s trajectory dw . Assuming the mean time cost per step to be finite, i. e. t / n, one readily infers that ˛ D 2/dw . The fractal properties of the trajectory can be rather easily checked, and the value of dw can be rather easily obtained for simple regular fractals, the prominent example being a Sierpinski gasket, discussed below. Analytically, the value of dw is often obtained via scaling of the first passage time to a boundary of the region (taken to be a sphere of radius r), i. e. using not the relation R / t 1/d w but the inverse one, T / r d w . The fact that both values of dw coincide witnesses strongly in favor of the self-similar nature of diffusion on fractal structures. Diffusion on a Sierpinski Gasket Serpinski gasket (essentially a Sierpinski lattice), one of the simplest regular fractal network is a structure obtained by iterating a generator shown in Fig. 2. Its simple iterative structure and the ease of its numerical implementation made it a popular toy model exemplifying the properties of fractal networks. The properties of diffusion on this network were investigated in detail, both numerically and analytically. Since the structure does not possess dangling ends, the cause of the diffusion anomalies is connected with tortuosity of the typical paths available for diffusion, however, differently from the previous examples, these paths are numerous and non-equivalent. The numerical method allowing for obtaining exact results on the properties of diffusion is based on exact enumeration techniques, see e. g. [10]. The idea here is to calculate the displacement probability distribution based on the exact number of ways Wi;n the particle, starting at

Anomalous Diffusion on Fractal Networks

Anomalous Diffusion on Fractal Networks, Figure 2 A Sierpinski gasket: a its generator b a structure after 4th iteration. The lattice is rescaled by the factor of 4 to fit into the picture. c A constriction obtained after triangle-star transformation used in the calculation of the walk’s dimension, see Subsect. “The Spectral Dimension”

a given site 0 can arrive to a given site i at step n. For a given network the number Wi;n is simply a sum of the numbers of ways Wj;n1 leading from site 0 to the nearest neighbors j of the site i, Wj;n1 . Starting from W0;0 D 1 and Wi;0 D 0 for i ¤ 0 one simply updates the array P of Wi;n according to the rule Wi;n D jDnn(i) Wj;n1 , where nn(i) denotes the nearest neighbors of the node i in the network. The probability Pi;n is proportional to the number of these equally probably ways, and is obtained from Wi;n by normalization. This is essentially a very old method used to solve numerically a diffusion equation in the times preceding the computer era. For regular lattices this gives us the exact values of the probabilities. For statistical fractals (like percolation clusters) an additional averaging over the realizations of the structure is necessary, so that other methods of calculation of probabilities and exponents might be superior. The value of dw can then be obtained by plotting ln n vs. lnhr2n i1/2 : the corresponding points fall on a straight line with slope dw . The dimension of the walks on a Sierpinski gasket obtained by direct enumeration coincides with its theoretical value dw D ln 5/ ln 2 D 2:322 : : : , see Subsect. “The Spectral Dimension”. Loopless Fractals (Trees) Another situation, the one similar the the case of a comb, is encountered in loopless fractals (fractal trees), as exemplified by a Vicsek’s construction shown in Fig. 3, or by diffusion-limited aggregates (DLA). For the structure depicted in Fig. 3, the walk’s dimension is dw D log 15/ log 3 D 2:465 : : : [3]. Parallel to the case of the comb, the main mechanism leading to subdiffusion is trapping in dangling ends, which, different from the case of the comb, now themselves have a fractal structure. Parallel to the case of the comb, the time of travel

Anomalous Diffusion on Fractal Networks, Figure 3 A Vicsek fractal: the generator (left) and its 4th iteration. This is a loopless structure; the diffusion anomalies here are mainly caused by trapping in dangling ends

from one site of the structure to another one is strongly affected by trapping. However, trapping inside the dangling end does not lead to a halt of the particle, but only to confining its motion to some scale, so that the overall equations governing the PDF might be different in the form from the fractional diffusion ones. Diffusion and Conductivity The random walks of particles on a network can be described by a master equation (which is perfectly valid for the exponential waiting time distribution on a site and asymptotically valid in case of all other waiting time distributions with finite mean waiting time ): Let p(t) be the vector with elements p i (t) being the probabilities to find a particle at node i at time t. The master equation d p D Wp dt

(9)

gives then the temporal changes in this probability. A similar equation with the temporal derivative changed to d/dn

19

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Anomalous Diffusion on Fractal Networks

describes the n-dependence for the probabilities p i;n in a random walk as a function of the number of steps, provided n is large enough to be considered as a continuous variable. The matrix W describes the transition probabilities between the nodes of the lattice or network. The nondiagonal elements of the corresponding matrix are wij , the transition probabilities from site i to site j per unit time or in one step. The diagonal elements are the sums of all non-diagonal elements in the corresponding column P taken with the opposite sign: w i i D  j w ji , which represents the probability conservation law. The situation of unbiased random walks corresponds to a symmetric matrix W: w i j D w ji . Considering homogeneous networks and putting all nonzero wij to unity, one sees that the difference operator represented by each line of the matrix is a symmetric difference approximation to a Laplacian. The diffusion problem is intimately connected with the problem of conductivity of a network, as described by Kirchhoff’s laws. Indeed, let us consider a stationary situation in which the particles enter the network at some particular site A at constant rate, say, one per unit time, and leave it at some site B (or a given set set of sites B). Let us assume that after some time a stationary distribution of the particles over the lattice establishes, and the particles’ concentration on the sites will be described by the vector proportional to the vector of stationary probabilities satisfying Wp D 0 :

(10)

Calculating the probabilities p corresponds then formally to calculating the voltages on the nodes of the resistor network of the same geometry under given overall current using the Kirchhoff’s laws. The conductivities of resistors connecting nodes i and j have to be taken proportional to the corresponding transition probabilities wij . The condition given by Eq. (10) corresponds then to the second Kirchhoff’s law representing the particle conservation (the fact that the sum of all currents to/from the node i is zero), and the fact that the corresponding probability current is proportional to the probability difference is replaced by the Ohm’s law. The first Kirchhoff’s law follows from the uniqueness of the solution. Therefore calculation of the dimension of a walk can be done by discussing the scaling of conductivity with the size of the fractal object. This typically follows a power law. As an example let us consider a Sierpinski lattice and calculate the resistance between the terminals A and B (depicted by thick wires outside of the triangle in Fig. 2c) of a fractal of the next generation, assuming that the conductivity between the corresponding nodes of the lattice

of the previous generation is R D 1. Using the trianglestar transformation known in the theory of electric circuits, i. e. passing to the structure shown in Fig. 2c by thick lines inside the triangle, with the conductivity of each bond r D 1/2 (corresponding to the same value of the conductivity between the terminals), we get the resistance of a renormalized structure R0 D 5/3. Thus, the dependence of R on the spacial scale L of the object is R / L with  D log(5/3)/ log 2. The scaling of the conductivity G is correspondingly G / L . Using the flow-over-population approach, known in calculation of the mean first passage time in a system, we get I / N/hti, where I is the probability current through the system (the number of particles entering A per unit time), N is the overall stationary number of particles within the system and hti is the mean time a particle spends inside the system, i. e. the mean first passage time from A to B. The mean number of particles inside the system is proportional to a typical concentration (say to the probability pA to find a particle at site A for the given current I) and to the number of sites. The first one, for a given current, scales as the system’s resistivity, pA / R / L , and the second one, clearly, as Ld f where L is the system’s size. On the other hand, the mean first passage time scales according to hti / Ld w (this time corresponds to the typical number of steps during which the walk transverses an Euclidean distance L), so that dw D df C  :

(11)

Considering the d-dimensional generalizations of Sierpinski gaskets and using analogous considerations we get  D log[(d C 3)/(d C 1)]/ log 2. Combining this with the fractal dimension df D log(d C 1)/ log 2 of a gasket gives us for the dimension of the walks dw D log(d C 3)/ log 2. The relation between the scaling exponent of the conductivity and the dimension of a random walk on a fractal system can be used in the opposite way, since dw can easily be obtained numerically for a whatever structure. On the other hand, the solutions of the Kirchhoff’s equations on complex structures (i. e. the solution of a large system of algebraic equations), which is typically achieved using relaxation algorithms, is numerically much more involved. We note that the expression for  can be rewritten as  D df (2/ds  1), where ds is the spectral dimension of the network, which will be introduced in Subsect. “The Spectral Dimension”. The relation between the walk dimension and this new quality therefore reads dw D

2df : ds

(12)

Anomalous Diffusion on Fractal Networks

A different discussion of the same problem based on the Einstein’s relation between diffusion coefficient and conductivity, and on crossover arguments can be found in [10,20]. The Spectral Dimension In mathematical literature spectral dimension is defined through the return probability of the random walk, i. e. the probability to be at the origin after n steps or at time t. We consider a node on a network and take it to be an origin of a simple random walk. We moreover consider the probability P(0; t) to be at this node at time t (a return probability). The spectral dimension defines then the asymptotic behavior of this probability for n large: P(0; t) / t d s /2 ;

for t ! 1 :

(13)

The spectral dimension ds is therefore exactly a quantity substituting the Euclidean dimension d is Eq. (1). In the statistical case one should consider an average of p(t) over the origin of the random walk and over the ensemble of the corresponding graphs. For fixed graphs the spectral dimension and the fractal (Hausdorff) dimension are related by 2df  ds  df ; 1 C df

(14)

provided both exist [5]. The same relation is also shown true for some random geometries, see e. g. [6,7] and references therein. The bounds are optimal, i. e. both equalities can be realized in some structures. The connection between the walk dimension dw and the spectral dimension ds of a network can be easily rationalized by the following consideration. Let the random walk have a dimension dw , so that after time t the position of the walker can be considered as more or less homogeneously spread within the spatial region of the linear size R / n1/d w or R / t 1/d w , with the overall number of nodes N / R d f / t d f /d w . The probability to be at one particular node, namely at 0, goes then as 1/N, i. e. as t d f /d w . Comparing this with the definition of the spectral dimension, Eq. (13) we get exactly Eq. (12). The lower bound in the inequality Eq. (14) follows from Eq. (13) and from the the observation that the value of  in Eq. (11) is always smaller or equal to one (the value of  D 1 would correspond to a one-dimensional wire without shunts). The upper bound can be obtained using Eq. (12) and noting that the walks’ dimension never gets less than 2, its value for the regular network: the random walk on a whatever structure is more compact than one on a line.

We note that the relation Eq. (12) relies on the assumption that the spread of the walker within the r-domain is homogeneous, and needs reinterpretation for strongly anisotropic structures like combs, where the spread in teeth and along the spine are very different (see e. g. [2]). For the planar comb with infinite teeth df D 2, and ds as calculated through the return time is ds D 3/2, while the walk’s dimension (mostly determined by the motion in teeth) is dw D 2, like for random walks in Euclidean space. The inequality Eq. (14) is, on the contrary, universal. Let us discuss another meaning of the spectral dimension, the one due to which it got its name. This one has to do with the description of random walks within the master equation scheme. From the spectral (Laplace) representation of the solution of the master equation, Eq. (9), we can easily find the probability P(0; t) that the walker starting at site 0 at t D 0 is found at the same site at time t. This one reads 1 X a i exp( i t) ; P(0; t) D iD1

where i is the ith eigenvalue of the matrix W and ai is the amplitude of its ith eigenvector at site 0. Considering the lattice as infinite, we can pass from discrete eigenvalue decomposition to a continuum Z 1 N ( )a( ) exp( t)d : P(0; t) D 0

For long times, the behavior of P(0; t) is dominated by the behavior of N ( ) for small values of . Here N ( ) is the density of states of a system described by the matrix W. The exact forms of such densities are well known for many Euclidean lattices, since the problem is equivalent to the calculating of spectrum in tight-binding approximation used in the solid state physics. For all Euclidean lattices N ( ) / d/21 for ! 0, which gives us the forms of famous van Hove singularities of the spectrum. Assuming a( ) to be nonsingular at ! 0, we get P(0; t) / t d/2 . For a fractal structure, the value of d changed for the one of the spectral dimension ds . This corresponds to the density of states N ( ) / d s /21 which describes the properties of spectrum of a fractal analog of a Laplace operator. The dimension ds is also often called fracton dimension of the structure, since the corresponding eigenvectors of the matrix (corresponding to eigenstates in tight-binding model) are called fractons, see e. g. [16]. In the examples of random walks on quenched polymer chains the dimension dw was twice the fractal dimension of the underlying structures, so that the spectral dimension of the corresponding structures (without intersections) was exactly 1 just like their topological dimension.

21

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Anomalous Diffusion on Fractal Networks

The spectral dimension of the network governs also the behavior of the mean number of different sites visited by the random walk. A random walk of n steps having a property of compact visitation typically visits all sites within the radius of the order of its typical displacement R n / n1/d w . The number of these sites is S n / R dnf where df is the fractal dimension of the network, so that S n / nd s /2 (provided ds  2, i. e. provided the random walk is recurrent and shows compact visitation). The spectral dimension of a structure plays important role in many other applications [16]. This and many other properties of complex networks related to the spectrum of W can often be easier obtained by considering random walk on the lattice than by obtaining the spectrum through direct diagonalization. Percolation Clusters Percolation clusters close to criticality are one of the most important examples of fractal networks. The properties of these clusters and the corresponding fractal and spectral dimensions are intimately connected to the critical indices in percolation theory. Thus, simple crossover arguments show that the fractal dimension of an incipient infinite percolation cluster in d  6 dimensions is df D d  ˇ/, where ˇ is the critical exponent of the density of the infinite cluster, P1 / (p  pc )ˇ and  is the critical exponent of the correlation length,  / jp  pc j , and stagnates at df D 4 for d > 6, see e. g. [23]. On the other hand, the critical exponent t, describing the behavior of the resistance of a percolation system close to percolation threshold, / (p pc )t , is connected with the exponent  via the following crossover argument. The resistance of the system is the one of the infinite cluster. For p > pc this cluster can be considered as fractal at scales L <  and as homogeneous at larger scales. Our value for the fractal dimension of the cluster follows exactly from this argument by matching the density of the cluster P(L) / L(d f d) at L D  / (p pc ) to its density P1 / (ppc )ˇ at larger scales. The infinite cluster for p > pc is then a dense regular assembly of subunits of size  which on their turn are fractal. If the typical resistivity of each fractal subunit is R then the resistivity of the overall assembly goes as R / R (L/)2d . This is a well-known relation showing that the resistivity of a wire grows proportional to its length, the resistivities of similar flat figures are the same etc. This relation holds in the homogeneous regime. On the other hand, in the fractal regime R L / L , so that overall R /  Cd2 / (p  pc )(Cd2) giving the value of the critical exponent t D ( C d  2). The

value of  can in its turn be expressed through the values of spectral and fractal dimensions of a percolation cluster. The spectral dimension of the percolation cluster is very close to 4/3 in any dimension larger then one. This surprising finding lead to the conjecture by Alexander and Orbach that this value of ds D 4/3 might be exact [1] (it is exact for percolation systems in d  6 and for trees). Much effort was put into proving or disproving the conjecture, see the discussion in [10]. The latest, very accurate simulations of two-dimensional percolation by Grassberger show that the conjecture does not hold in d D 2, and the prediction ds D 4/3 is off by around 2% [9]. In any case it can be considered as a very useful mnemonic rule. Anomalous diffusion on percolation clusters corresponds theoretically to the most involved case, since it combines all mechanisms generating diffusion anomalies. The infinite cluster of the percolation system consists of a backbone, its main current-carrying structure, and the dangling ends (smaller clusters on all scales attached to the backbone through only one point). The anomalous diffusion on the infinite cluster is thus partly caused by trapping in this dangling ends. If one considers a shortest (chemical) way between the two nodes on the cluster, this way is tortuous and has fractal dimension larger than one. The Role of Finite Clusters The exponent t of the conductivity of a percolation system is essentially the characteristic of an infinite cluster only. Depending on the physical problem at hand, one can consider the situations when only the infinite cluster plays the role (like in the experiments of [12], where only the infinite cluster is filled by fluid pressed through the boundary of the system), and situations when the excitations can be found in infinite as well as in the finite clusters, as it is the case for optical excitation in mixed molecular crystals, a situation discussed in [24]. Let us concentrate on the case p D pc . The structure of large but finite clusters at lower scales is indistinguishable from those of the infinite cluster. Thus, the mean squared displacement of a walker on a cluster of size L (one with N / Ld f sites) grows as R2 (t) / t 2/d w until is stagnates at the value of R of the order of the cluster’s size, R2 / N 2/d f . At a given time t we can subdivide all clusters into two classes: those whose size is small compared to t 1/d w (i. e. the ones with N < t d s /2 ), on which the mean square displacement stagnates, and those of the larger size, on which it still grows: ( 2

R (t) /

t 2/d w ;

t 1/d w < N 1/ f f

N 2/d f

otherwise:

;

Anomalous Diffusion on Fractal Networks

The characteristic size Ncross (t) corresponds to the crossover between these two regimes for a given time t. The probability that a particle starts at a cluster of size N is proportional to the number of its sites and goes as P(N) / N p(N) / N 1 with  D (2d ˇ)/(d ˇ), see e. g. [23]. Here p(N) is the probability to find a cluster of N sites among all clusters. The overall mean squared displacement is then given by averaging over the corresponding cluster sizes: hr2 (t)i /

N cross X(t)

N 2/d f N 1 C

ND1

1 X

t 2/d w N 1 :

N cross (t)

Introducing the expression for Ncross (t), we get hr2 (t)i / t 2/d w C(2)d f /d w D t (1/d w )(2d f ˇ /) : A similar result holds for the number of distinct sites visited, where one has to perform the analogous averaging with ( t d s /2 ; t 1/d w < N 1/ ff S(t) / N; otherwise giving S(t) / t (d s /2)(1ˇ/d f ) . This gives us the effective spectral dimension of the system d˜s D (ds /2)(1  ˇ/df ). Scaling of PDF and Diffusion Equations on Fractal Lattices If the probability density of the displacement of a particle on a fractal scales, the overall scaling form of this displacement can be obtained rather easily: Indeed, the typical displacement R during the time t goes as r / t 1/d w , so that the corresponding PDF has to scale as P(r; t) D

r d f d f t d s /2



r t 1/d w

 :

(15)

The prefactor takes into account the normalization of the probability density on a fractal structure (denominator) and also the scaling of the density of the fractal structure in the Euclidean space (enumerator). The simple scaling assumes that all the moments ofR the distance scale P(r; t)r k d d r k D in the same way, so that hr k i D k/d w . The overall scaling behavior of the PDF was const(k)t confirmed numerically for many fractal lattices like Sierpinski gaskets and percolation clusters, and is corroborated in many other cases by the coincidence of the values of dw obtained via calculation of the mean squared displacement and of the mean first passage time (or resistivity of the network). The existence of the scaling form

leads to important consequences e. g. for quantification of anomalous diffusion in biological tissues by characterizing the diffusion-time dependence of the magnetic resonance signal [17], which, for example, allows for differentiation between the healthy and the tumor tissues. However, even in the cases when the scaling form Eq. (15) is believed to hold, the exact form of the scaling function f (x) is hard to get; the question is not yet resolved even for simple regular fractals. For applications it is often interesting to have a kind of a phenomenological equation roughly describing the behavior of the PDF. Such an approach will be analogous to putting down Richardson’s equation for the turbulent diffusion. Essentially, the problem here is to find a correct way of averaging or coarse-graining the microscopic master equation, Eq. (9), to get its valid continuous limit on larger scales. The regular procedure to do this is unknown; therefore, several phenomenological approaches to the problem were formulated. We are looking for an equation for the PDF of the walker’s displacement from its initial position and assume the system to be isotropic on the average. We look for @ an analog of the classical Fick’s equation, @t P(r; t) D rK(r)rP(r; t) which in spherical coordinates and for K D K(r) takes the form of Eq. (3):

 @ 1 @ d1 @ P(r; t) D d1 r K(r) P(r; t) : @t @r @r r On fractal lattices one changes from the Euclidean dimension of space to the fractal dimension of the lattice df , and takes into account that the effective diffusion coefficient K(r) decays with distance as K(r) ' Kr with D dw 2 to capture the slowing down of anomalous diffusion on a fractal compared to the Euclidean lattice situation. The corresponding equation put forward by O’Shaughnessy and Procaccia [18] reads:

 1 @ d f d w C1 @ @ P(r; t) D K d 1 r P(r; t) : (16) @t @r r f @r This equation was widely used in description of anomalous diffusion on fractal lattices and percolation clusters, but can be considered only as a rough phenomenological approximation. Its solution ! r d f d rdw P(r; t) D A d /2 exp B ; t t s (with B D 1/Kdw2 and A being the corresponding normalization constant) corresponds exactly to the type of scaling given by Eq. (15), with the scaling function f (x) D

23

24

Anomalous Diffusion on Fractal Networks

exp(x d w ). This behavior of the scaling function disagrees e. g. with the results for random walks on polymer chains, for which case we had f (x) D exp(x  ) with  D (1  1/dw )1 for large enough x. In literature, several other proposals (based on plausible assumptions but to no extent following uniquely from the models considered) were made, taking into account possible non-Markovian nature of the motion. These are the fractional equations of the type i @1/d w 1 @ h (d s 1)/2 P(r; t) D K P(r; t) ; (17) r 1 @t 1/d w r(d s 1)/2 @r resembling “half” of the diffusion equation and containing a fractional derivative [8], as well as the “full” fractional diffusion equation

 1 @ d s 1 @ @2/d w P(r; t) D K r P(r; t) ; 2 d 1 @r @t 2/d w r s @r

(18)

as proposed in [15]. All these equation are invariant under the scale transformation t ! t ;

r ! 1/d w r ;

and lead to PDFs showing correct overall scaling properties, see [19]. None of them reproduces correctly the PDF of displacements on a fractal (i. e. the scaling function f (x)) in the whole range of distances. Ref. [19] comparing the corresponding solutions with the results of simulation of anomalous diffusion an a Sierpinski gasket shows that the O’Shaughnessy–Procaccia equation, Eq. (16), performs the best for the central part of the distribution (small displacements), where Eq. (18) overestimates the PDF and Eq. (17) shows an unphysical divergence. On the other hand, the results of Eqs. (17) and (18) reproduce equally well the PDF’s tail for large displacements, while Eq. (16) leads to a PDF decaying considerably faster than the numerical one. This fact witnesses favor of strong non-Markovian effects in the fractal diffusion, however, the physical nature of this non-Markovian behavior observed here in a fractal network without dangling ends, is not as clear as it is in a comb model and its more complex analogs. Therefore, the question of the correct equation describing the diffusion in fractal system (or different correct equations for the corresponding different classes of fractal systems) is still open. We note also that in some cases a simple fractional diffusion equation (with the corresponding power of the derivative leading to the correct scaling exponent dw ) gives a reasonable approximation to experimental data, as the ones of the model experiment of [12].

Further Directions The physical understanding of anomalous diffusion due to random walks on fractal substrates may be considered as rather deep and full although it does not always lead to simple pictures. For example, although the spectral dimension for a whatever graph can be rather easily calculated, it is not quite clear what properties of the graph are responsible for its particular value. Moreover, there are large differences with respect to the degree of rigor with which different statements are proved. Mathematicians have recognized the problem, and the field of diffusion on fractal networks has become a fruitful field of research in probability theory. One of the examples of recent mathematical development is the deep understanding of spectral properties of fractal lattices and a proof of inequalities like Eq. (14). A question which is still fully open is the one of the detailed description of the diffusion within a kind of (generalized) diffusion equation. It seems clear that there is more than one type of such equations depending on the concrete fractal network described. However even the classification of possible types of such equations is still missing. All our discussion (except for the discussion of the role of finite clusters in percolation) was pertinent to infinite structures. The recent work [4] has showed that finite fractal networks are interesting on their own and opens a new possible direction of investigations. Bibliography Primary Literature 1. Alexander S, Orbach RD (1982) Density of states on fractals– fractons. J Phys Lett 43:L625–L631 2. Bertacci D (2006) Asymptotic behavior of the simple random walk on the 2-dimensional comb. Electron J Probab 45:1184– 1203 3. Christou A, Stinchcombe RB (1986) Anomalous diffusion on regular and random models for diffusion-limited aggregation. J Phys A Math Gen 19:2625–2636 4. Condamin S, Bénichou O, Tejedor V, Voituriez R, Klafter J (2007) First-passage times in complex scale-invariant media. Nature 450:77–80 5. Coulhon T (2000) Random Walks and Geometry on Infinite Graphs. In: Ambrosio L, Cassano FS (eds) Lecture Notes on Analysis on Metric Spaces. Trento, CIMR, (1999) Scuola Normale Superiore di Pisa 6. Durhuus B, Jonsson T, Wheather JF (2006) Random walks on combs. J Phys A Math Gen 39:1009–1037 7. Durhuus B, Jonsson T, Wheather JF (2007) The spectral dimension of generic trees. J Stat Phys 128:1237–1260 8. Giona M, Roman HE (1992) Fractional diffusion equation on fractals – one-dimensional case and asymptotic-behavior. J Phys A Math Gen 25:2093–2105; Roman HE, Giona M, Fractional diffusion equation on fractals – 3-dimensional case and scattering function, ibid., 2107–2117

Anomalous Diffusion on Fractal Networks

9. Grassberger P (1999) Conductivity exponent and backbone dimension in 2-d percolation. Physica A 262:251–263 10. Havlin S, Ben-Avraham D (2002) Diffusion in disordered media. Adv Phys 51:187–292 11. Klafter J, Sokolov IM (2005) Anomalous diffusion spreads its wings. Phys World 18:29–32 12. Klemm A, Metzler R, Kimmich R (2002) Diffusion on randomsite percolation clusters: Theory and NMR microscopy experiments with model objects. Phys Rev E 65:021112 13. Metzler R, Klafter J (2000) The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep 339:1–77 14. Metzler R, Klafter J (2004) The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J Phys A Math Gen 37:R161–R208 15. Metzler R, Glöckle WG, Nonnenmacher TF (1994) Fractional model equation for anomalous diffusion. Physica A 211:13–24 16. Nakayama T, Yakubo K, Orbach RL (1994) Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Rev Mod Phys 66:381–443 17. Özarslan E, Basser PJ, Shepherd TM, Thelwall PE, Vemuri BC, Blackband SJ (2006) Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal. J Magn Res 183:315–323 18. O’Shaughnessy B, Procaccia I (1985) Analytical solutions for diffusion on fractal objects. Phys Rev Lett 54:455–458 19. Schulzky C, Essex C, Davidson M, Franz A, Hoffmann KH (2000) The similarity group and anomalous diffusion equations. J Phys A Math Gen 33:5501–5511 20. Sokolov IM (1986) Dimensions and other geometrical critical exponents in the percolation theory. Usp Fizicheskikh Nauk 150:221–255 (1986) translated in: Sov. Phys. Usp. 29:924 21. Sokolov IM, Klafter J (2005) From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion. Chaos 15:026103 22. Sokolov IM, Mai J, Blumen A (1997) Paradoxical diffusion in chemical space for nearest-neighbor walks over polymer chains. Phys Rev Lett 79:857–860

23. Stauffer D (1979) Scaling theory of percolation clusters. Phys Rep 54:1–74 24. Webman I (1984) Diffusion and trapping of excitations on fractals. Phys Rev Lett 52:220–223

Additional Reading The present article gave a brief overview of what is known about the diffusion on fractal networks, however this overview is far from covering all the facets of the problem. Thus, we only discussed unbiased diffusion (the effects of bias may be drastic due to e. g. stronger trapping in the dangling ends), and considered only the situations in which the waiting time at all nodes was the same (we did not discuss e. g. the continuous-time random walks on fractal networks), as well as left out of attention many particular systems and applications. Several review articles can be recommended as a further reading, some of them already mentioned in the text. One of the best-known sources is [10] being a reprint of the text from the “sturm und drang” period of investigation of fractal geometries. A lot of useful information on random walks models in general an on walks on fractals is contained in the review by Haus and Kehr from approximately the same time. General discussion of the anomalous diffusion is contained in the work by Bouchaud and Georges. The classical review of the percolation theory is given in the book of Stauffer and Aharony. Some additional information on anomalous diffusion in percolation systems can be found in the review by Isichenko. A classical source on random walks in disordered systems is the book by Hughes. Haus JW, Kehr K (1987) Diffusion in regular and disordered lattices. Phys Rep 150:263–406 Bouchaud JP, Georges A (1990) Anomalous diffusion in disordered media – statistical mechanisms, models and physical applications. Phys Rep 195:127–293 Stauffer D, Aharony A (2003) Introduction to Percolation Theory. Taylor & Fransis, London Isichenko MB (1992) Percolation, statistical topography, and transport in random-media. Rev Mod Phys 64:961–1043 Hughes BD (1995) Random Walks and random Environments. Oxford University Press, New York

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Biological Fluid Dynamics, Non-linear Partial Differential Equations

Biological Fluid Dynamics, Non-linear Partial Differential Equations ANTONIO DESIMONE1 , FRANÇOIS ALOUGES2 , ALINE LEFEBVRE2 1 SISSA-International School for Advanced Studies, Trieste, Italy 2 Laboratoire de Mathématiques, Université Paris-Sud, Orsay cedex, France Article Outline Glossary Definition of the Subject Introduction The Mathematics of Swimming The Scallop Theorem Proved Optimal Swimming The Three-Sphere Swimmer Future Directions Bibliography Glossary Swimming The ability to advance in a fluid in the absence of external propulsive forces by performing cyclic shape changes. Navier–Stokes equations A system of partial differential equations describing the motion of a simple viscous incompressible fluid (a Newtonian fluid)   @v C (v  r)v D r p C v

@t div v D 0 where v and p are the velocity and the pressure in the fluid, is the fluid density, and  its viscosity. For simplicity external forces, such as gravity, have been dropped from the right hand side of the first equation, which expresses the balance between forces and rate of change of linear momentum. The second equation constrains the flow to be volume preserving, in view of incompressibility. Reynolds number A dimensionless number arising naturally when writing Navier–Stokes equations in nondimensional form. This is done by rescaling position and velocity with x  D x/L and v  D v/V , where L and V are characteristic length scale and velocity associated with the flow. Reynolds number (Re) is defined by Re D

VL V L D  

where  D / is the kinematic viscosity of the fluid, and it quantifies the relative importance of inertial versus viscous effects in the flow. Steady Stokes equations A system of partial differential equations arising as a formal limit of Navier–Stokes equations when Re ! 0 and the rate of change of the data driving the flow (in the case of interest here, the velocity of the points on the outer surface of a swimmer) is slow v C r p D 0 div v D 0 : Flows governed by Stokes equations are also called creeping flows. Microscopic swimmers Swimmers of size L D 1 μm moving in water ( 1 mm2 /s at room temperature) at one body length per second give rise to Re 106 . By contrast, a 1 m swimmer moving in water at V D 1 m/s gives rise to a Re of the order 106 . Biological swimmers Bacteria or unicellular organisms are microscopic swimmers; hence their swimming strategies cannot rely on inertia. The devices used for swimming include rotating helical flagella, flexible tails traversed by flexural waves, and flexible cilia covering the outer surface of large cells, executing oar-like rowing motion, and beating in coordination. Self propulsion is achieved by cyclic shape changes described by time periodic functions (swimming strokes). A notable exception is given by the rotating flagella of bacteria, which rely on a submicron-size rotary motor capable of turning the axis of an helix without alternating between clockwise and anticlockwise directions. Swimming microrobots Prototypes of artificial microswimmers have already been realized, and it is hoped that they can evolve into working tools in biomedicine. They should consist of minimally invasive, small-scale self-propelled devices engineered for drug delivery, diagnostic, or therapeutic purposes. Definition of the Subject Swimming, i. e., being able to advance in a fluid in the absence of external propulsive forces by performing cyclic shape changes, is particularly demanding at low Reynolds numbers (Re). This is the regime of interest for micro-organisms and micro-robots or nano-robots, where hydrodynamics is governed by Stokes equations. Thus, besides the rich mathematics it generates, low Re propulsion is of great interest in biology (How do microorganism swim? Are their strokes optimal and, if so, in which sense? Have

Biological Fluid Dynamics, Non-linear Partial Differential Equations

these optimal swimming strategies been selected by evolutionary pressure?) and biomedicine (can small-scale selfpropelled devices be engineered for drug delivery, diagnostic, or therapeutic purposes?). For a microscopic swimmer, moving and changing shape at realistically low speeds, the effects of inertia are negligible. This is true for both the inertia of the fluid and the inertia of the swimmer. As pointed out by Taylor [10], this implies that the swimming strategies employed by bacteria and unicellular organism must be radically different from those adopted by macroscopic swimmers such as fish or humans. As a consequence, the design of artificial microswimmers can draw little inspiration from intuition based on our own daily experience. Taylor’s observation has deep implications. Based on a profound understanding of low Re hydrodynamics, and on a plausibility argument on which actuation mechanisms are physically realizable at small length scales, Berg postulated the existence of a sub-micron scale rotary motor propelling bacteria [5]. This was later confirmed by experiment.

cal simplification is obtained by looking at axisymmetric swimmers which, when advancing, will do so by moving along the axis of symmetry. Two such examples are the three-sphere-swimmer in [7], and the push-me–pull-you in [3]. In fact, in the axisymmetric case, a simple and complete mathematical picture of low Re swimming is now available, see [1,2]. The Mathematics of Swimming This article focuses, for simplicity, on swimmers having an axisymmetric shape ˝ and swimming along the axis of symmetry, with unit vector E{ . The configuration, or state s of the system is described by N C 1 scalar parameters: s D fx (1) ; : : : ; x (NC1) g. Alternatively, s can be specified by a position c (the coordinate of the center of mass along the symmetry axis) and by N shape parameters  D f (1) ; : : : ;  (N) g. Since this change of coordinates is invertible, the generalized velocities u(i) : D x˙ (i) can be represented as linear functions of the time derivatives of position and shape: (u(1) ; : : : ; u(NC1) ) t D A( (1) ; : : : ;  (N) )(˙(1) ; : : : ; ˙(N) ; c˙) t

Introduction In his seminal paper Life at low Reynolds numbers [8], Purcell uses a very effective example to illustrate the subtleties involved in microswimming, as compared to the swimming strategies observable in our mundane experience. He argues that at low Re, any organism trying to swim adopting the reciprocal stroke of a scallop, which moves by opening and closing its valves, is condemned to the frustrating experience of not having advanced at all at the end of one cycle. This observation, which became known as the scallop theorem, started a stream of research aiming at finding the simplest mechanism by which cyclic shape changes may lead to effective self propulsion at small length scales. Purcell’s proposal was made of a chain of three rigid links moving in a plane; two adjacent links swivel around joints and are free to change the angle between them. Thus, shape is described by two scalar parameters (the angles between adjacent links), and one can show that, by changing them independently, it is possible to swim. It turns out that the mechanics of swimming of Purcell’s three-link creature are quite subtle, and a detailed understanding has started to emerge only recently [4,9]. In particular, the direction of the average motion of the center of mass depends on the geometry of both the swimmer and of the stroke, and it is hard to predict by simple inspection of the shape of the swimmer and of the sequence of movements composing the swimming stroke. A radi-

(1) where the entries of the N C 1  N C 1 matrix A are independent of c by translational invariance. Swimming describes the ability to change position in the absence of external propulsive forces by executing a cyclic shape change. Since inertia is being neglected, the total drag force exerted by the fluid on the swimmer must also vanish. Thus, since all the components of the total force in directions perpendicular to E{ vanish by symmetry, self-propulsion is expressed by Z 0D  n  E{ (2) @˝

where  is the stress in the fluid surrounding ˝, and n is the outward  unit normal to @˝. The stress  D  rv C (rv) t  pId is obtained by solving Stokes equation outside ˝ with prescribed boundary data v D v¯ on @˝. In turn, v¯ is the velocity of the points on the boundary @˝ of the swimmer, which moves according to (1). By linearity of Stokes equations, (2) can be written as 0D

NC1 X iD1 t

' (i) ( (1) ; : : : ;  (N) )u(i)

D A ˚  (˙(1) ; : : : ; ˙(N) ; c˙) t

(3)

where ˚ D (' (1) ; : : : ; ' (N) ) t , and we have used (1). Notice that the coefficients ' (i) relating drag force to velocities are

27

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Biological Fluid Dynamics, Non-linear Partial Differential Equations

Biological Fluid Dynamics, Non-linear Partial Differential Equations, Figure 1 A mirror-symmetric scallop or an axisymmetric octopus

independent of c because of translational invariance. The coefficient of c˙ in (3) represents the drag force corresponding to a rigid translation along the symmetry axis at unit speed, and it never vanishes. Thus (3) can be solved for c˙, and we obtain c˙ D

N X

Vi ( (1) ; : : : ;  (N) )˙(i) D V ()  ˙ :

(4)

iD1

Equation (4) links positional changes to shape changes through shape-dependent coefficients. These coefficients encode all hydrodynamic interactions between ˝ and the surrounding fluid due to shape changes with rates ˙(1) ; : : : ; ˙(N) . A stroke is a closed path in the space S of admissible shapes given by [0; T] 3 t 7! ( (1) ; : : :  (N1) ). Swimming requires that Z

N TX

0 ¤ c D 0

Vi ˙(i) dt

(5)

iD1

i. e., that the differential form

PN

iD1 Vi

d (i) is not exact.

The Scallop Theorem Proved Consider a swimmer whose motion is described by a parametrized curve in two dimensions (N D 1), so that (4) becomes ˙ ; c˙(t) D V((t))(t)

t 2R;

(6)

and assume that V 2 L1 (S) is an integrable function in the space of admissible shapes and  2 W 1;1 (R; S) is a Lipschitz-continuous and T-periodic function for some T > 0, with values in S. Figure 1 is a sketch representing concrete examples compatible with these hypotheses. The axisymmetric case consists of a three-dimensional cone with axis along E{ and opening angle  2 [0; 2] (an axisymmetric octopus). A non-axisymmetric example is also allowed in

this discussion, consisting of two rigid parts (valves), always maintaining mirror symmetry with respect to a plane (containing E{ and perpendicular to it) while swiveling around a joint contained in the symmetry plane and perpendicular to E{ (a mirror-symmetric scallop), and swimming parallel to E{ . Among the systems that are not compatible with the assumptions above are those containing helical elements with axis of rotation E{ , and capable of rotating around E{ always in the same direction (call the rotation angle). Indeed, a monotone function t 7! (t) is not periodic. The celebrated “scallop theorem” [8] states that, for a system like the one depicted in Fig. 1, the net displacement of the center of mass at the end of a periodic stroke will always vanish. This is due to the linearity of Stokes equation (which leads to symmetry under time reversals), and to the low dimensionality of the system (a one-dimensional periodic stroke is necessarily reciprocal). Thus, whatever forward motion is achieved by the scallop by closing its valves, it will be exactly compensated by a backward motion upon reopening them. Since the low Re world is unaware of inertia, it will not help to close the valves quickly and reopen them slowly. A precise statement and a rigorous short proof of the scallop theorem are given below. Theorem 1 Consider a swimmer whose motion is described by ˙ ; c˙(t) D V((t))(t)

t 2 R;

(7)

with V 2 L1 (S). Then for every T-periodic stroke  2 W 1;1 (R; S), one has Z

T

c˙(t)dt D 0 :

c D

(8)

0

Proof Define the primitive of V by Z s  (s) D V( ) d 0

(9)

Biological Fluid Dynamics, Non-linear Partial Differential Equations

so that  0 () D V(). Then, using (7), Z

T

c D

Substituting (13) in (11), the expended power becomes a quadratic form in ˙

˙ V ((t))(t)dt

Z

0

Z

T

@˝

d  ((t))dt dt

D 0

Z

by the T-periodicity of t 7! (t).

G i j () D

Optimal Swimming A classical notion of swimming efficiency is due to Lighthill [6]. It is defined as the inverse of the ratio between the average power expended by the swimmer during a stroke starting and ending at the shape 0 D (0(1) ; : : : ; 0(N) ) and the power that an external force would spend to translate the system rigidly at the same average speed c¯ D c/T : Eff

D

1 T

RTR

R1R n  v n  v D 0 @˝ 6L c¯2 6L(c)2

0

@˝

(10)

@˝

At a point p 2 @˝, the velocity v(p) accompanying a change of state of the swimmer can be written as a linear combination of the u(i) v(p) D

NC1 X

V i (p; )u(i)

(12)

W i (p; )˙(i) :

(13)

iD1

D

N X iD1

Indeed, the functions V i are independent of c by translational invariance, and (4) has been used to get (13) from the line above.

@˝

DN(W i (p; ))  W j (p; )dp :

(15)

Strokes of maximal efficiency may be defined as those producing a given displacement c of the center of mass with minimal expended power. Thus, from (10), maximal efficiency is obtained by minimizing Z

Z

1Z 0

@˝

1

n  v D

˙ ) ˙ (G();

(16)

0

subject to the constraint Z

where  is the viscosity of the fluid, L D L(0 ) is the effective radius of the swimmer, and time has been rescaled to a unit interval to obtain the second identity. The expression in the denominator in (10) comes from a generalized version of Stokes formula giving the drag on a sphere of radius L moving at velocity c¯ as 6L c¯. Let DN: H 1/2 (@˝) ! H 1/2 (@˝) be the Dirichlet to Neumann map of the outer Stokes problem, i. e., the map such that  n D DNv, where  is the stress in the fluid, evaluated on @˝, arising in response to the prescribed velocity v on @˝, and obtained by solving the Stokes problem outside ˝. The expended power in (10) can be written as Z Z n  v D DN(v)  v : (11) @˝

(14)

where the symmetric and positive definite matrix G() is given by

D  ((T))   ((0)) D 0

1

˙ ) ˙  n  v D (G();

c D

1

V()  ˙

(17)

0

among all closed curves  : [0; 1] ! S in the set S of admissible shapes such that (0) D (1) D 0 . The Euler–Lagrange equations for this optimization problem are 0

@G ˙ ˙ ;  @ (1) :: :

1

B C C   d ˙ 1B B C  (G )C B C C r V  rt V ˙ D 0 dt 2 B C @ @G A ˙ ˙ ; (N) @ (18) where r V is the matrix (r V ) i j D @Vi /@ j , rt V is its transpose, and  is the Lagrange multiplier associated with the constraint (17). Given an initial shape  0 and an initial position c0 , the solutions of (18) are in fact sub-Riemannian geodesics joining the states parametrized by (0 ; c0 ) and (0 ; c0 Cc) in the space of admissible states X , see [1]. It is well known, and easy to prove using (18), that along such geodesics ˙ is constant. This has interesting consequences, (G( ) ˙ ; ) because swimming strokes are often divided into a power phase, where jG( )j is large, and a recovery phase, where jG( )j is smaller. Thus, along optimal strokes, the recovery phase is executed quickly while the power phase is executed slowly.

29

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Biological Fluid Dynamics, Non-linear Partial Differential Equations

Biological Fluid Dynamics, Non-linear Partial Differential Equations, Figure 2 Swimmer’s geometry and notation

The Three-Sphere Swimmer For the three-sphere-swimmer of Najafi and Golestanian [7], see Fig. 2, ˝ is the union of three rigid disjoint balls B(i) of radius a, shape is described by the distances x and y, the space of admissible shapes is S D (2a; C1)2 , and the kinematic relation (1) takes the form 1 u(1) D c˙  (2x˙ C y˙) 3 1 u(2) D c˙ C (x˙  y˙) 3 1 (3) u D c˙ C (2 y˙ C x˙ ) : 3

(19)

Consider, for definiteness, a system with a D 0:05 mm, swimming in water. Calling f (i) the total propulsive force on ball B(i) , we find that the following relation among forces and ball velocities holds 0 (1) 1 0 (1) 1 f u @ f (2) A D R(x; y) @ u(2) A (20) f (3) u(3) where the symmetric and positive definite matrix R is known as the resistance matrix. From this last equation, using also (19), the condition for self-propulsion f (1) C f (2) C f (3) D 0 is equivalent to c˙ D Vx (x; y)x˙ C Vy (x; y) y˙ ;

(21)

where

Biological Fluid Dynamics, Non-linear Partial Differential Equations, Table 1 Energy consumption (10–12 J) for the three strokes of Fig. 3 inducing the same displacement c D 0:01 mm in T D 1 s Optimal stroke Small square stroke Large square stroke 0.229 0.278 0.914

which is guaranteed, in particular, if curl V is bounded away from zero. Strokes of maximal efficiency for a given initial shape (x0 ; y0 ) and given displacement c are obtained by solving Eq. (18). For N D 2, this becomes   1 (@x G ˙ ; ˙ ) d C curlV( ) ˙ ? D 0 (25)  (G ˙ ) C ˙ dt 2 (@ y G ˙ ; ) where @x G and @ y G stand for the x and y derivatives of the 2  2 matrix G(x; y). It is important to observe that, for the three-sphere swimmer, all hydrodynamic interactions are encoded in the shape dependent functions V(x; y) and G(x; y). These can be found by solving a two-parameter family of outer Stokes problems, where the parameters are the distances x and y between the three spheres. In [1], this has been done numerically via the finite element method: a representative example of an optimal stroke, compared to two more naive proposals, is shown in Fig. 3. Future Directions

Re c  (e c  e y ) Vx (x; y) D Re c  (e x  e y ) Re c  (e c  e x ) Vy (x; y) D  : Re c  (e x  e y )

(22) (23)

Moreover, e x D (1; 1; 0) t , e y D (0; 1; 1) t , e c D (1/3; 1/3; 1/3) t . Given a stroke D @! in the space of admissible shapes, condition (5) for swimming reads Z T   Vx x˙ C Vy y˙ dt 0 ¤ c D Z0 curlV(x; y)dxdy (24) D !

The techniques discussed in this article provide a head start for the mathematical modeling of microscopic swimmers, and for the quantitative optimization of their strokes. A complete theory for axisymmetric swimmers is already available, see [2], and further generalizations to arbitrary shapes are relatively straightforward. The combination of numerical simulations with the use of tools from sub-Riemannian geometry proposed here may prove extremely valuable for both the question of adjusting the stroke to global optimality criteria, and of optimizing the stroke of complex swimmers. Useful inspiration can come from the sizable literature on the related field dealing with control of swimmers in a perfect fluid.

Biological Fluid Dynamics, Non-linear Partial Differential Equations

3. Avron JE, Kenneth O, Oakmin DH (2005) Pushmepullyou: an efficient micro-swimmer. New J Phys 7:234-1–8 4. Becker LE, Koehler SA, Stone HA (2003) On self-propulsion of micro-machines at low Reynolds numbers: Purcell’s three-link swimmer. J Fluid Mechanics 490:15–35 5. Berg HC, Anderson R (1973) Bacteria swim by rotating their flagellar filaments. Nature 245:380–382 6. Lighthill MJ (1952) On the Squirming Motion of Nearly Spherical Deformable Bodies through Liquids at Very Small Reynolds Numbers. Comm Pure Appl Math 5:109–118 7. Najafi A, Golestanian R (2004) Simple swimmer at low Reynolds numbers: Three linked spheres. Phys Rev E 69:062901-1–4 8. Purcell EM (1977) Life at low Reynolds numbers. Am J Phys 45:3–11 9. Tan D, Hosoi AE (2007) Optimal stroke patterns for Purcell’s three-link swimmer. Phys Rev Lett 98:068105-1–4 10. Taylor GI (1951) Analysis of the swimming of microscopic organisms. Proc Roy Soc Lond A 209:447–461 Biological Fluid Dynamics, Non-linear Partial Differential Equations, Figure 3 Optimal stroke and square strokes which induce the same displacement c D 0:01 mm in T D 1 s, and equally spaced level curves of curl V. The small circle locates the initial shape 0 D (0:3 mm; 0:3 mm)

Bibliography Primary Literature 1. Alouges F, DeSimone A, Lefebvre A (2008) Optimal strokes for low Reynolds number swimmers: an example. J Nonlinear Sci 18:277–302 2. Alouges F, DeSimone A, Lefebvre A (2008) Optimal strokes for low Reynolds number axisymmetric swimmers. Preprint SISSA 61/2008/M

Books and Reviews Agrachev A, Sachkov Y (2004) Control Theory from the Geometric Viewpoint. In: Encyclopaedia of Mathematical Sciences, vol 87, Control Theory and Optimization. Springer, Berlin Childress S (1981) Mechanics of swimming and flying. Cambridge University Press, Cambridge Happel J, Brenner H (1983) Low Reynolds number hydrodynamics. Nijhoff, The Hague Kanso E, Marsden JE, Rowley CW, Melli-Huber JB (2005) Locomotion of Articulated Bodies in a Perfect Fluid. J Nonlinear Sci 15: 255–289 Koiller J, Ehlers K, Montgomery R (1996) Problems and Progress in Microswimming. J Nonlinear Sci 6:507–541 Montgomery R (2002) A Tour of Subriemannian Geometries, Their Geodesics and Applications. AMS Mathematical Surveys and Monographs, vol 91. American Mathematical Society, Providence

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Catastrophe Theory

Catastrophe Theory W ERNER SANNS University of Applied Sciences, Darmstadt, Germany Article Outline Glossary Definition of the Subject Introduction Example 1: The Eccentric Cylinder on the Inclined Plane Example 2: The Formation of Traffic Jam Unfoldings The Seven Elementary Catastrophes The Geometry of the Fold and the Cusp Further Applications Future Directions Bibliography Glossary Singularity Let f : Rn ! Rm be a differentiable map defined in some open neighborhood of the point p 2 Rn and J p f its Jacobian matrix at p, consisting of the partial derivatives of all components of f with respect to all variables. f is called singular in p if rank J p f < min fn; mg. If rank J p f D min fm; ng, then f is called regular in p. For m D 1 (we call the map f : Rn ! R a differentiable function) the definition implies: A differentiable function f : Rn ! R is singular in p 2 Rn , if grad f (p) D 0. The point p, where the function is singular, is called a singularity of the function. Often the name “singularity” is used for the function itself if it is singular at a point p. A point where a function is singular is also called critical point. A critical point p of a function f is called a degenerated critical point if the Hessian (a quadratic matrix containing the second order partial derivatives) is singular at p; that means its determinant is zero at p. Diffeomorphism A diffeomorphism is a bijective differentiable map between open sets of Rn whose inverse is differentiable, too. Map germ Two continuous maps f : U ! R k and g : V ! R k , defined on neighborhoods U and V of p 2 Rn , are called equivalent as germs at p if there exists a neighborhood W U \ V on which both coincide. Maps or functions, respectively, that are equivalent as germs can be considered to be equal regarding local features. The equivalence classes of this equivalence relation are called germs. The set of all germs of

differentiable maps Rn ! R k at a point p is named " p (n; k). If p is the origin of Rn , one simply writes "(n; k) instead of "0 (n; k). Further, if k D 1, we write "(n) instead of "(n; 1) and speak of function germs (also simplified as “germs”) at the origin of Rn . "(n; k) is a vector space and "(n) is an algebra, that is, a vector space with a structure of a ring. The ring "(n) contains a unique maximal ideal (n) D f f 2 "(n)j f (0) D 0g. The ideal (n) is generated by the germs of the coordinate functions x1 ; : : : ; x n . We use the form (n) D hx1 ; : : : ; x n i"(n) to emphasize, that this ideal is generated over the ring "(n). So a function germ in (n) is of P the form niD1 a i (x)  x i , with certain function germs a i (x) 2 "(n). r-Equivalence Two function germs f ; g 2 "(n) are called r-equivalent if there exists a germ of a local diffeomorphism h : Rn ! Rn at the origin, such that g D f ı h. Unfolding An unfolding of a differentiable function germ f 2 (n) is a germ F 2 (nCr) with FjRn D f (here j means the restriction. The number r is called an unfolding dimension of f . An unfolding F of a germ f is called universal if every other unfolding of f can be received by suitable coordinate transformations, “morphisms of unfoldings” from F, and the number of unfolding parameters of F is minimal (see “codimension”). Unfolding morphism Suppose f 2 "(n) and F : Rn  R k ! R and G : Rn  Rr ! R be unfoldings of f . A right-morphism from F to G, also called unfolding morphism, is a pair (˚; ˛), with ˚ 2 "(n C k; n C r) and ˛ 2 (k), such that: 1. ˚jRn D id(Rn ), that is ˚(x; 0) D (x; 0), 2. If ˚ D (; ), with  2 "(n C k; n), k; r), then 2 "(k; r),

2 "(n C

3. For all (x; u) 2 Rn  R k we get F(x; u) D G(˚(x; u)) C ˛(u). Catastrophe A catastrophe is a universal unfolding of a singular function germ. The singular function germs are called organization centers of the catastrophes. Codimension The codimension of a singularity f is given by codim( f ) D dimR (n)/h@x f i (quotient space). Here h@x f i is the Jacobian ideal generated by the partial derivatives of f and (n) D f f 2 "(n)j f (0) D 0g. The codimension of a singularity gives of the minimal number of unfolding parameters needed for the universal unfolding of the singularity. Potential function Let f : Rn ! Rn be a differentiable map (that is, a differentiable vector field). If there exists a function ' : Rn ! R with the property that

Catastrophe Theory

grad ' D f , then f is called a gradient vector field and ' is called a potential function of f . Definition of the Subject Catastrophe theory is concerned with the mathematical modeling of sudden changes – so called “catastrophes” – in the behavior of natural systems, which can appear as a consequence of continuous changes of the system parameters. While in common speech the word catastrophe has a negative connotation, in mathematics it is neutral. You can approach catastrophe theory from the point of view of differentiable maps or from the point of view of dynamical systems, that is, differential equations. We use the first case, where the theory is developed in the mathematical language of maps and functions (maps with range R). We are interested in those points of the domain of differentiable functions where their gradient vanishes. Such points are called the “singularities” of the differentiable functions. Assume that a system’s behavior can be described by a potential function (see Sect. “Glossary”). Then the singularities of this function characterize the equilibrium points of the system under consideration. Catastrophe theory tries to describe the behavior of systems by local properties of corresponding potentials. We are interested in local phenomena and want to find out the qualitative behavior of the system independent of its size. An important step is the classification of catastrophe potentials that occur in different situations. Can we find any common properties and unifying categories for these catastrophe potentials? It seems that it might be impossible to establish any reasonable criteria out of the many different natural processes and their possible catastrophes. One of the merits of catastrophe theory is the mathematical classification of simple catastrophes where the model does not depend on too many parameters. Classification is only one of the mathematical aspects of catastrophe theory. Another is stability. The stable states of natural systems are the ones that we can observe over a longer period of time. But the stable states of a system, which can be described by potential functions and their singularities, can become unstable if the potentials are changed by perturbations. So stability problems in nature lead to mathematical questions concerning the stability of the potential functions. Many mathematical questions arise in catastrophe theory, but there are also other kinds of interesting problems, for example, historical themes, didactical questions and even social or political ones. How did catastrophe theory come up? How can we teach catastrophe theory to the stu-

dents at our universities? How can we make it understandable to non-mathematicians? What can people learn from this kind of mathematics? What are its consequences or insights for our lives? Let us first have a short look at mathematical history. When students begin to study a new mathematical field, it is always helpful to learn about its origin in order to get a good historical background and to get an overview of the dependencies of inner-mathematical themes. Catastrophe theory can be thought of as a link between classical analysis, dynamical systems, differential topology (including singularity theory), modern bifurcation theory and the theory of complex systems. It was founded by the French mathematician René Thom (1923–2002) in the sixties of the last century. The name ‘catastrophe theory’ is used for a combination of singularity theory and its applications. In the year 1972 Thom’s famous book “Stabilité structurelle et morphogénèse” appeared. Thom’s predecessors include Marston Morse, who developed his theory of the singularities of differentiable functions (Morse theory) in the thirties, Hassler Whitney, who extended this theory in the fifties to singularities of differentiable maps of the plane into the plane, and John Mather (1960), who introduced algebra, especially the theory of ideals of differentiable functions, into singularity theory. Successors to Thom include Christopher Zeeman (applications of catastrophe theory to physical systems), Martin Golubitsky (bifurcation theory), John Guckenheimer (caustics and bifurcation theory), David Schaeffer (shock waves) and Gordon Wassermann (stability of unfoldings). From the point of view of dynamical systems, the forerunners of Thom are Jules Henry Poincaré (1854–1912) who looked at stability problems of dynamical systems, especially the problems of celestial mechanics, Andrei Nikolaevic Kolmogorov, Vladimir I. Arnold and Jürgen Moser (KAM), who influenced catastrophe theory with their works on stability problems (KAM-theorem). From the didactical point of view, there are two main positions for courses in catastrophe theory at university level: Trying to teach the theory as a perfect axiomatic system consisting of exact definitions, theorems and proofs or trying to teach mathematics as it can be developed from historical or from natural problems (see [9]). In my opinion the latter approach has a more lasting effect, so there is a need to think about simple examples that lead to the fundamental ideas of catastrophe theory. These examples may serve to develop the theory in a way that starts with intuitive ideas and goes forward with increasing mathematical precision. When students are becoming acquainted with catastrophe theory, a useful learning tool is the insight that con-

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tinuous changes in influencing system parameters can lead to catastrophes in the system behavior. This phenomenon occurs in many systems. Think of the climate on planet earth, think of conflicts between people or states. Thus, they learn that catastrophe theory is not only a job for mathematical specialists, but also a matter of importance for leading politicians and persons responsible for guiding the economy. Introduction Since catastrophe theory is concerned with differentiable functions and their singularities, it is a good idea to start with the following simple experiment: Take a piece of wire, about half a meter in length, and form it as a ‘curve’ as shown in Fig. 1. Take a ring or screw-nut and thread the wire through it so that the object can move along the wire easily. The wire represents the model of the graph of a potential function, which determines the reactions of a system by its gradient. This means the ring’s horizontal position represents the actual value of a system parameter x that characterizes the state of the system, for example, x might be its position, rotation angle, temperature, and so on. This parameter x is also called the state parameter. The form of the function graph determines the behavior of the system, which tries to find a state corresponding to a minimum of the function (a zero of the gradient of f ) or to stay in

such a ‘stable’ position. (Move the ring to any position by hand and then let it move freely.) Maxima of the potential function are equilibrium points too, but they are not stable. A small disturbance makes the system leave such an equilibrium. But if the ring is in a local minimum, you may disturb its position by slightly removing it from that position. If this disturbance is not too big, the system will return to its original position at the minimum of the function when allowed to move freely. The minimum is a stable point of the function, in the sense that small disturbances of the system are corrected by the system’s behavior, which tries to return to the stable point. Observe that you may disturb the system by changing its position by hand (the position of the ring) or by changing the form of the potential function (the form of the wire). If the form of the potential function is changed, this corresponds to the change of one or more parameters in the defining equation for the potential. These parameters are called external parameters. They are usually influenced by the experimenter. If the form of the potential (wire) is changed appropriately (for example, if you pull upward at one side of the wire), the ring can leave the first disappearing minimum and fall into the second stable position. This is the way catastrophes happen: changes of parameters influence the potentials and thus may cause the system to suddenly leave a stable position. Some typical questions of catastrophe theory are:  How can one find a potential function that describes the system under consideration and its catastrophic jumps?  What does this potential look like locally? How can we describe the changes in the potential’s parameters?  Can we classify the potentials into simple categories?  What insights can such a classification give us? Now we want to consider two simple examples of systems where such sudden changes become observable. They are easy to describe and lead to the simplest forms of catastrophes which Thom listed in the late nineteen-sixties. The first is a simple physical model of an eccentric cylinder on an inclined plane. The second example is a model for the formation of a traffic jam. Example 1: The Eccentric Cylinder on the Inclined Plane

Catastrophe Theory, Figure 1 Ring on the wire

We perform some experiments with a cylinder or a wide wheel on an inclined plane whose center of gravity is eccentric with respect to its axes (Fig. 2). You can build such a cylinder from wood or metal by taking two disks of about 10 cm in radius. Connect them

Catastrophe Theory

Catastrophe Theory, Figure 2 The eccentric cylinder on the inclined plane. The inclination of the plane can be adjusted by hand. The position of the disc center and the center of gravity are marked by colored buttons

Catastrophe Theory, Figure 3 Two equilibrium positions (S1 and S2 ) are possible with this configuration. If the point A lies “too close” to M, there is no equilibrium position possible at all when releasing the wheel, since the center of gravity can never lie vertically over P

with some sticks of about 5 cm in length. The eccentric center of gravity is generated by fixing a heavy piece of metal between the two discs at some distance from the axes of rotation. Draw some lines on the disks, where you can read or estimate the angle of rotation. (Hint: It could be interesting for students to experimentally determine the center of gravity of the construction.) If mass and radius are fixed, there are two parameters which determine the system: The position of the cylinder can be specified by the angle x against a zero level (that is, against the horizontal plane; Fig. 4). This angle is the “inner” variable of the system, which means that it is the “answer” of the cylinder to the experimenter’s changes of

the inclination u of the plane, which is the “outer” parameter. In this example you should determine experimentally the equilibrium positions of the cylinder, that is, the angle of rotation where the cylinder stays at rest on the inclined plane (after releasing it cautiously and after letting the system level off a short time). We are interested in the stable positions, that is, the positions where the cylinder returns after slight disturbances (slightly nudging it or slightly changing the inclination of the plane) without rolling down the whole plane. The search for the equilibrium position x varies with the inclination angle u. You can make a table and a graphic showing the dependence of x and u.

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Catastrophe Theory, Figure 4 The movement of S is a composite of the movement of the cylinder represented by the movement of point M and the rotation movement with angle x

The next step is to find an analytical approach. Refer to Fig. 3: Point P is the supporting point of the cylinder on the plane. Point A marks the starting position of the center of gravity. When the wheel moves and the distance between M and A is big enough, this center can come into two equilibrium positions S1 and S2 , which lie vertically over P. Now, what does the potential function look like? The variables are: R: radius of the discs, u: angle of the inclined plane against the horizontal plane measured in radians, x: angle of rotation of the cylinder measured against the horizontal plane in radians, r: distance between the center of gravity S of the cylinder from the axes marked by point M, x0 (resp. u0 ): angles where the cylinder suddenly starts to roll down the plane. Refer to Fig. 4. If the wheel is rolled upward by hand from the lower position on the right to the upper position, the center of gravity moves upward with the whole wheel, or as the midpoint M or the point P. But point S also moves downward by rotation of the wheel with an amount of r  sin(x). The height gained is h D R  x  sin(u)  r  sin(x) :

(1)

From physics one knows the formula for the potential energy Epot of a point mass. It is the product m  g  h, where m D mass, g D gravitational constant, h D height, where the height is to be expressed by the parameters x and u and the given data of the cylinder.

To a fixed angle u1 of the plane, by formula (1), there corresponds the following potential function: f (x) D Epot D m  g  h D m  g (R x sin(u1 )r sin(x)) : If the inclination of the plane is variable, the angle u of inclination serves as an outer parameter and we obtain a whole family of potential functions as a function F in two variables x and u: F(x; u) D Fu (x) D m  g  (R  x  sin(u)  r  sin(x)) : Thus, the behavior of the eccentric cylinder is described by a family of potential functions. The extrema of the single potentials characterize the equilibria of the system. The saddle point characterizes the place where the cylinder suddenly begins to roll down due to the slightest disturbance, that is, where the catastrophe begins (see Fig. 5). Example. The data are m D 1 kg, g D 10 m/sec2 , R D 0:1 m, r D 0:025 m. For constant angle of inclination u, each of the functions f (x) D F(x; u D const) is a section of the function F. To determine the minima of each of the section graphs, we calculate the zeros of the partial derivative of F with respect to the inner variable x. In our particular example above, the result is @x F D 0:25 cos(x) C sin(u). The solution for @x F D 0 is u D arcsin(0:25 cos(x)) (see Fig. 6). Calculating the point in x–u-space at which both partial derivatives @x F and @x x F vanish (saddle point of the section) and using only positive values of angle u gives the point (x0 ; u0 ) D (0; 0:25268). In general: x0 D 0 and u0 D arcsin(r/R). In our example this means the catastrophe begins at an inclination angle of u0 D 0:25268 (about

Catastrophe Theory

Catastrophe Theory, Figure 6 Curve in x–u-space, for which F(x; u) has local extrema

Catastrophe Theory, Figure 5 Graph of F(x; u) for a concrete cylinder

14.5 degrees). At that angle, sections of F with u > u0 do not possess any minimum where the system could stably rest. Making u bigger than u0 means that the cylinder must leave its stable position and suddenly roll down the plane. The point u0 in the u-parameter space is called catastrophe point. What is the shape of the section of F at u0 ? Since in our example f (x) D F(x; u0 ) D 0:25x  0:25 sin(x) ; the graph appears as shown in Fig. 7. The point x D 0 is a degenerated critical point of f (x), since f 0 (x) D f 00 (x) D 0. If we expand f into its Taylor polynomial t(x) around that point we get t(x) D 0:0416667x 3  0:00208333x 5 C 0:0000496032x 7 6:88933  107 x 9 C O[x]11 :

Catastrophe Theory, Figure 7 Graph of the section F(x; u0 )

Catastrophe theory tries to simplify the Taylor polynomials of the potentials locally about their degenerated critical points. This simplification is done by coordinate transformations, which do not change the structure of the singularities. These transformations are called diffeomorphisms (see Sect. “Glossary”). A map h : U ! V, with U and V open in Rn , is a local diffeomorphism if the Jacobian determinant det(J p h) ¤ 0 at all points p in U. Two functions f and g, both defined in some neighborhood of the origin of Rn , are called r-equivalent if there exists a local diffeomorphism h : U ! V at the origin, such that g D f ı h.

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Catastrophe Theory, Figure 8 Characteristic surface S, initial density, fold curve and cusp curve

The Taylor polynomial t(x) in our example starts with a term of 3rd order in x. Thus, there exists a diffeomorphism (a non-singular coordinate transformation), such that in the new coordinates t is of the form x3 . To see this, we write t(x) D x 3  a(x), with a(0) ¤ 0. We define h(x) D x a(x)1/3 . Note that the map h is a local diffeomorphism near the origin since h0 (0) D a(0)1/3 ¤ 0. (Hint: Modern computer algebra systems, such as Mathematica, are able to expand expressions like a(x)1/3 and calculate the derivative of h easily for the concrete example.) Thus for the eccentric cylinder the intersection of the graph of the potential F(x; u) with the plane u D u0 is the graph of a function, whose Taylor series at its critical point x D 0 is t D x 3 ı h, in other words: t is r-equivalent to x3 . Before we continue with this example and its relation to catastrophe theory, let us introduce another example which leads to a potential of the form x4 . Then we will have two examples which lead to the two simplest catastrophes in René Thom’s famous list of the seven elementary catastrophes. Example 2: The Formation of Traffic Jam In the first example we constructed a family F of potential functions directly from the geometric properties of the model of the eccentric cylinder. We then found its critical points by calculating the derivatives of F. In this second example the method is slightly different. We start with a partial differential equation which occurs in traffic flow modeling. The solution surface of the initial value problem

will be regarded as the surface of zeros of the derivative of a potential family. From this, we then calculate the family of potential functions. Finally, we will see that the Taylor series of a special member of this family, the one which belongs to its degenerated critical point, is equivalent to x4 . Note: When modeling a traffic jam we use the names of variables that are common in literature about this subject. Later we will rename the variables into the ones commonly used in catastrophe theory. Let u(x; t) be the (continuous) density of the traffic at a street point x at time t and let a(u) be the derivative of the traffic flow f (u). The mathematical modeling of simple traffic problems leads to the well known traffic flow equation: @u(x; t) @u(x; t) C a(u) D0: @t @x We must solve the Cauchy problem for this quasilinear partial differential equation of first order with given initial density u0 D u(x; 0) by the method of characteristics. The solution is constant along the (base) characteristics, which are straight lines in xt-plane with slope a(u0 (y)) against the t-axes, if y is a point on the x-axes, where the characteristic starts at t D 0. For simplicity we write “a” instead of a(u0 (y)). Thus the solution of the initial value problem is u(x; t) D u0 (x  a  t) and the characteristic surface S D f(x; t; u)ju  u0 (x  a  t) D 0g is a surface in xtuspace. Under certain circumstances S may not lie uniquely above the xt-plane but may be folded as is shown in Fig. 8.

Catastrophe Theory

There is another curve to be seen on the surface: The border curve of the fold. Its projection onto the xt-plane is a curve with a cusp, specifying the position and the beginning time of the traffic jam. The equations used in the example, in addition to the traffic flow equation, can be deduced in traffic jam modeling from a simple parabolic model for a flow-density relation (see [5]). The constants in the following equations result from considerations of maximal possible density and an assumed maximal allowed velocity on the street. In our example a model of a road of 2 km in length and maximal traffic velocity of 100 km/h is chosen. Maximal traffic density is umax D 0:2. The equations are f (u) D 27:78u  138:89u2 and u(x; 0) D u0 (x) D 0:1 C 0:1  Exp(((x  2000)/700)2 ). The latter was constructed to simulate a slight increase of the initial density u0 along the street at time t D 0. The graph of u0 can be seen as the front border of the surface. To determine the cusp curve shown in Fig. 8, consider the following parameterization ˚ of the surface S and the projection map  : S ! R2 : ˚ : R  R0 ! S R

3

(x; t) 7! (x C a(u0 (x))  t; t; u0 (x)) : The Jacobian matrix of  ı ˚ is 0 1 @( ı ˚ )1 @( ı ˚)1 B C @x @t J( ı ˚)(x; t) D @ A @( ı ˚ )2 @( ı ˚)2 @x @t   0 1 C a(u0 (x))  t a(u0 (x)) : D 0 1 It is singular (that is, its determinate vanishes) if 1 C (a(u0 (x)))0  t D 0. From this we find the curve c : R ! R2   1 c(x) D x;  0 a (u0 (x))  u00 (x) which is the pre-image of the cusp curve and the cusp curve in the Fig. 8 is the image of c under  ı ˚. Whether the surface folds or not depends on both the initial density u0 and the properties of the flow function f (u). If such folding happens, it would mean that the density would have three values at a given point in xt-space, which is a physical impossibility. Thus, in this case, no continuous solution of the flow equation can exist in the whole plane. “Within” the cusp curve, which is the boundary for the region where the folding happens, there exists a curve with its origin in the cusp point. Along this curve the surface must be “cut” so that a discontinuous solution occurs. The curve within the cusp region is called

Catastrophe Theory, Figure 9 The solution surface is “cut” along the shock curve, running within the cusp region in the xt-plane

a shock curve and it is characterized by physical conditions (jump condition, entropy condition). The density makes a jump along the shock. The cusp, together with the form of the folding of the surface, may give an association to catastrophe theory. One has to find a family of potential functions for this model, that is, functions F(x;t) (u), whose negative gradient, that is, its derivative by u (the inner parameter), describes the characteristic surface S by its zeroes. The family of potentials is given by   xa(u)t Z Z u0 (x)dx F(x; t; u) D t  u  a(u)  a(u)du  0

(see [6]). Since gradFx;t (u) D 

@F D 0 , u  u0 (x  a(u)  t) D 0 @u

the connection of F and S is evident. One can show that a member f of the family F, the function f (u) D F(; ; u), is locally in a traffic jam formation point (; ) of the form f (u) D u4 (after Taylor expansion and after suitable coordinate transformations, similar to the first example). The complicated function terms of f and F thus can be replaced in qualitative investigations by simple polynomials. From the theorems of catastrophe theory, more properties of the models under investigation can be discovered. Now we want to use the customary notation of catastrophe theory. We shall rename the variables: u is the inner variables, which in catastrophe theory usually is x, and x, t (the outer parameters) get the names u and v respectively.

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Thus F D F(x; u; v) is a potential family with one inner variable (the state variable x) and two outer variables (the control variables u and v). Unfoldings In the first examples above we found a family F(x; u) of potential functions for the eccentric cylinder. For u fixed we found single members of that family. The member which belonged to the degenerated critical point of F, that is, to the point (x; u) where @x F D @x x F D 0, turned out to be equivalent to x3 . In the second example the corresponding member is equivalent to x4 (after renaming the variables). These two singularities are members of potential families which also can be transformed into simple forms. In order to learn how such families, in general, arise from a single function germ, let us look at the singularity f (x) D x 3 . If we add to it a linear “disturbance” ux, where the factor (parameter) u may assume different values, we qualitatively have one of the function graphs in Fig. 10. The solid curve in Fig. 10 represents the graph of f (x) D x 3 . The dashed curve is the graph of g(x) D x 3  u  x; u > 0. The dotted line is h(x) D x 3 C u  x; u > 0. Please note: While for positive parameter values, the disturbed function has no singularity, in the case of negative u-values a relative maximum and a relative minimum exist. The origin is a singular point only for u D 0, that is, for the undisturbed function f . We can think of f (x) D x 3 as a member of a function family, which contains disturbances with linear terms. The function f (x) D x 3 changes the type of singularity with the addition of an appropriate small disturbing function, as we have seen. Therefore f is called “structurally unstable”. But we have also learned that f (x) D x 3 can be seen as a member of a whole family of functions F(x; u) D x 3 C u  x, because F(x; 0) D f (x). This fam-

Catastrophe Theory, Figure 10 Perturbations of x 3

ily is stable in the sense that all kinds of singularities of the perturbed function are included by its members. This family is only one of the many possibilities to “unfold” the function germ f (x) D x 3 by adding disturbing terms. For example F(x; u; v) D x 3 C u  x 2 C v  x would be another such unfolding (see Sect. “Glossary”) of f . But this unfolding has more parameters than the minimum needed. The example of a traffic jam leads to an unfolding of f (x) D x 4 which is F(x; u; v) D x 4 C u  x 2 C v  x. How can we find unfoldings, which contain all possible disturbances (all types of singularities), that is, which are stable and moreover have a minimum number of parameters? Examine the singularity x4 (Fig. 12). It has a degenerated minimum at the origin. We disturb this function by a neighboring polynomial of higher degree, for example, by u  x 5 , where the parameter u with small values (here and in what follows “small” means small in amount), ensures that the perturbed function is “near” enough to x4 .

Catastrophe Theory, Figure 11 The function f(x) D x 3 embedded in a function family

Catastrophe Theory

without a term of degree n  1. Indeed, the expression F(x; u; v) D x 4 C u  x 2 C v  x is, as we shall see later, the “universal” unfolding of x4 . Here an unfolding F of a germ f is called universal, if every other unfolding of f can be received by suitable coordinate transformations “morphism of unfoldings” from F, and the number of unfolding parameters of F is minimal. The minimal number of unfolding parameters needed for the universal unfolding F of the singularity f is called the codimension of f . It can be computed as follows: Catastrophe Theory, Figure 12 The function f(x) D x4 C u  x5, u D 0 (solid), u D 0:2 (dotted) and u D 0:3 (dashed)

For f (x) D x 4 C u  x 5 , we find that the equation D 0 has the solutions x1 D x2 D x3 D 0, x4 D 4/(5u). To the threefold zero of the derivative, that is, the threefold degenerated minimum, another extremum is added which is arbitrarily far away from the origin for values of u that are small enough in amount due to the term 4/(5u). This is so, because the term 4/(5u) increases as u decreases. (This is shown in Fig. 12 with u D 0, u D 0:2 and u D 0:3). The type of the singularity at the origin thus is not influenced by neighboring polynomials of the form u  x5. If we perturb the polynomial x4 by a neighboring polynomial of lower degree, for example, u  x 3 , then for small amounts of u a new minimum is generated arbitrarily close to the existing singularity at the origin. If disturbed by a linear term, the singularity at the origin can even be eliminated. The only real solution of h0 (x) D 0, where h(x) D x 4 C ux, is x D (u/4)(1/3) . Note: Only for u D 0, that is, for a vanishing disturbance, is h singular at the origin. For each u ¤ 0 the linear disturbance ensures that no singularity at the origin occurs. The function f (x) D x 4 is structurally unstable. To find a stable unfolding for the singularity f (x) D x 4 , we must add terms of lower order. But we need not take into account all terms x 4 C u  x 3 C v  x 2 C w  x C k, since the absolute term plays no role in the computation of singular points, and each polynomial of degree 4 can be written by a suitable change of coordinates without a cubic term. Similarly, a polynomial of third degree after a coordinate transformation can always be written without a quadratic term. The “Tschirnhaus-transformation”: x 7! x  a1 /n, if a1 is the coefficient of x n1 in a polynomial p(x) D x n C a1 x n1 C : : : C a n1 x C a n of degree n, leads to a polynomial q(x) D x n C b1 x n2 C : : : C b n2 x C b n1 f 0 (x)

codim( f ) D

dimR (n) : h@x f i

Here h@x f i is the Jacobian ideal generated by the partial derivatives of f and (n) D f f 2 "(n)j f (0) D 0g. "(n) is the vector space of function germs at the origin of Rn . The quotient (n)/h@x f i is the factor space. What is the idea behind that factor space? Remember, that the mathematical description of a plane (a two dimensional object) in space (3 dimensional) needs only one equation, while the description of a line (1 dimensional) in space needs two equations. The number of equations that are needed for the description of these geometrical objects is determined by the difference between the dimensions of the surrounding space and the object in it. This number is called the codimension of the object. Thus the codimension of the plane in space is 1 and the codimension of the line in space is 2. From linear algebra it is known that the codimension of a subspace W of a finite dimensional vector space V can be computed as codim W D dim V  dim W. It gives the number of linear independent vectors that are needed to complement a basis of W to get a basis of the whole space V. Another well known possibility for the definition is codimW D dimV/W. This works even for infinite dimensional vector spaces and agrees in finite dimensional cases with the previous definition. In our example V D (n) and W should be the set O of all functions germs, which are r-equivalent to f (O is the orbit of f under the action of the group of local diffeomorphisms preserving the origin of Rn ). But this is an infinite dimensional manifold, not a vector space. Instead, we can use its tangent space T f O in the point f , since spaces tangent to manifolds are vector spaces with the same dimension as the manifold itself and this tangent space is contained in the space tangent to (n) in an obvious way. It turns out, that the tangent space T f O is h@x f i. The space tangent to (n) agrees with (n), since it is a vector space. The details for the computations together with some examples can be found in the excellent article by M. Golubitsky (1978).

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Catastrophe Theory

The Seven Elementary Catastrophes Elementary catastrophe theory works with families (unfoldings) of potential functions F : Rn  R k ! R (with k  4) (x; u) 7! F(x; u) : is called the state space, R k is called the parameter space or control space. Accordingly x D (x1 ; : : : ; x n ) is called a state variable or endogenous variable and u D (u1 ; : : : ; u k ) is called a control variable (exogenous variable, parameter). We are interested in the singularities of F with respect to x and the dependence of F with respect to the control variables u. We regard F as the unfolding of a function germ f and want to classify F. In analogy to the r-equivalence of function germs, we call two unfoldings F and G of the same function germ r-equivalent as unfoldings if there exists an unfoldingmorphism (see Sect. “Glossary”) from F to G. An unfolding F of f is called versal (or stable), if each other unfolding of f is right-equivalent as unfolding to F, that is, if there exists a right-morphism between the unfoldings. A versal unfolding with minimal unfolding dimensions is called universal unfolding of f . Thus we have the following theorem (all proofs of the theorems cited in this article can be found in [1,2] or [12].) Rn

Theorem 1 (Theorem on the existence of a universal unfolding) A singularity f has a universal unfolding iff codim( f ) D k < 1. Examples (see also the following theorem with its examples): If f (x) D x 3

it follows

F(x; u; v) D x 3 C ux 2 C vx 3

F(x; u) D x C ux 3

F(x; u) D x C ux

is versal;

is universal; 2

is not versal :

The following theorem states how one can find a universal unfolding of a function germ: Theorem 2 (Theorem on the normal form of universal unfoldings) Let f 2 (n) be a singularity with codim( f ) D k < 1. Let u D (u1 ; : : : ; u k ) be the parameters of the unfolding. Let b i (x), i D 1; : : : ; k, be the elements of (n), whose cosets modulo h@x f i generate the vector space (n)/h@x f i. Then F(x; u) D f (x) C

k X

u i b i (x)

iD1

is a universal unfolding of f .

Examples: 1. Consider f (x) D x 4 . Here n D 1 and (n) D (1) D hxi. The derivative of f is 4x 3 , so h@x f i D hx 3 i and we get hxi/hx 3 i D hx; x 2 i. Thus F(x; u1 ; u2 ) D x 4 C u1 x 2 C u2 x is the universal unfolding of f . 2. Consider f (x; y) D x 3 C y 3 . Here n D 2 and (n) D (2) D hx; yi. The partial derivatives of f with respect to x and y are 3x 2 and 3y 2 , thus h@x f i D hx 2 ; y 2 i and we get hx; yi/hx 2 ; y 2 i D hx; x y; yi. Therefore F(x; y; u1 ; u2 ; u3 ) D x 3 C y 3 C u1  x  y C u2  x C u3  y is the universal unfolding of f . Hint: There exists a useful method for the calculation of the quotient space which you will find in the literature as “Siersma’s trick” (see for example [7]). We now present René Thom’s famous list of the seven elementary catastrophes. Classification Theorem (Thom’ s List) Up to addition of a non degenerated quadratic form in other variables and up to multiplication by ˙1 a singularity f of codimension k (1  k  4) is right-equivalent to one of the following seven: f x3 x4 x5 x3 C y3

codim f 1 2 3 3

x3  xy2 3 x6 4 x2 y C y4 4

universal unfolding x3 C ux x4 C ux2 C vx x5 C ux3 C vx2 C wx x3 C y3 C uxy C vx C wy

name fold cusp swallowtail hyperbolic umbilic x3  xy2 C u(x2 C y2 ) elliptic C vx C wy umbilic x6 C ux4 C vx3 C wx2 C tx butterfly x2 y C y4 C ux2 C vy2 parabolic C wx C ty umbilic

In a few words: The seven elementary catastrophes are the seven universal unfoldings of singular function germs of codimension k (1  k  4). The singularities itself are called organization centers of the catastrophes. We will take a closer look at the geometry of the simplest of the seven elementary catastrophes: the fold and the cusp. The other examples are discussed in [9]. The Geometry of the Fold and the Cusp The fold catastrophe is the first in Thom’s list of the seven elementary catastrophes. It is the universal unfolding of the singularity f (x) D x 3 which is described by the equation F(x; u) D x 3 C u  x. We are interested in the set of singular points of F relative to x, that is, for those points where the first partial derivative with respect to x (the sys-

Catastrophe Theory

tem variable) vanishes. This gives the parabola u D 3x 2 . Inserting this into F gives a space curve. Figure 13 shows the graph of F(x; u). The curve on the surface joins the extrema of F and the projection curve represents their x; u values. There is only one degenerated critical point of F (at the vertex of the Parabola), that is, a critical point where the second derivative by x also vanishes. The two branches of the parabola in the xu-plane give the positions of the maxima and the minima of F, respectively. At the points belonging to a minimum, the system is stable, while in a maximum it is unstable. Projecting x–u space, and with it the parabola, onto parameter space (u-axis), one gets a straight line shown within the interior of the parabola, which represents negative parameter values of u. There, the system has a stable minimum (and an unstable maximum, not observable in nature). For parameter values u > 0 the system has no stability points at all. Interpreting the parameter u as time, and “walking along” the u-axis, the point u D 0 is the beginning or the end of a stable system behavior. Here the system shows catastrophic behavior. According to the interpretation of the external parameter as time or space, the morphology of the fold is a “beginning,” an “end” or a “border”, where something new occurs. What can we say about the fold catastrophe and the modeling of our first example, the eccentric cylinder? We have found that the potential related to the catastrophe point of the eccentric cylinder is equivalent to x3 , and F(x; u) D x 3 C ux is the universal unfolding. This is independent of the size of our ‘machine’. The behavior of the machine should be qualitatively the same as that of other machines which are described by the fold catastrophe as a mechanism. We can expect that the begin (or end) of a catastrophe depends on the value of a single outer parameter. Stable states are possible (for u < 0 in the standard fold. In the untransformed potential function of our model of the eccentric cylinder, this value is u D 0:25268) corresponding to the local minima of F(:; u). This means that it is possible for the cylinder to stay at rest on the inclined plane. There are unstable equilibria (local maxima of F(:; u) and the saddle point, too). The cylinder can be turned such that the center of gravity lies in the upper position over the supporting point (see Fig. 3). But a tiny disturbance will make the system leave this equilibrium. Let us now turn to the cusp catastrophe. The cusp catastrophe is the universal unfolding of the function germ f (x) D x 4 . Its equation is F(x; u; v) D x 4 Cu  x 2 Cv  x. In our second example (traffic jam), the function f is equivalent to the organization center of the cusp catastrophe, the second in René Thom’s famous list of the seven elementary catastrophes. So the potential family is equivalent to

Catastrophe Theory, Figure 13 Graph of F(x; u) D x 3 C u  x

the universal unfolding F(x; u; v) D x 4 Cux 2 Cvx of the function x4 . We cannot draw such a function in three variables in three-space, but we can try to draw sections giving u or v certain constant values. We want to look at its catastrophe surface S, that is, the set of those points in (x; u; v)space, where the partial derivative of F with respect to the system variable x is zero. It is a surface in three-space called catastrophe manifold, stability surface or equilibrium surface, since the surface S describes the equilibrium points of the system. If S is not folded over a point in u; v-space there is exactly one minimum of the potential F. If it is folded in three sheets, there are two minima (corresponding to the upper and lower sheet of the fold) and a maximum (corresponding to the middle sheet). Stable states of the system belong to the stable minima, that is, to the upper and lower sheets of the folded surface. Other points in three-space that do not lie on the surface S correspond to the states of the system that are not equilibrium. The system does not rest there. Catastrophes do occur when the

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Catastrophe Theory

Catastrophe Theory, Figure 14 The catastrophe manifold of the cusp catastrophe

system jumps suddenly from one stable state to another stable state, that is, from one sheet to the other. There are two principal possibilities, called “conventions”, where this jump can happen. The “perfect delay convention” says that jumps happen at the border of the folded surface. The “Maxwell convention” says that jumps can happen along a curve in uv-space (a “shock-curve”) which lays inside the cusp area. This curve consists of those points (u; v) in parameter space, where F(:; u; v) has two critical points with the same critical value. In our traffic jam model a shock curve is a curve in xt-space that we can interpret dynamically as the movement of the jam formation front along the street (see Fig. 9). The same figure shows what we call the morphology of the cusp: according to the convention, the catastrophe manifold with its fold or its cut can be interpreted as a fault or slip (as in geology) or a separation (if one of the parameter represents time). In our traffic jam model, there is a “line” (actually a region) of separation (shock wave) between the regions of high and low traffic density. Besides the catastrophe manifold (catastrophe surface) there are two other essential terms: catastrophe set and bifurcation set (see Fig. 15). The catastrophe set can be viewed as the curve which goes along the border of the fold on the catastrophe manifold. It is the set of degenerated critical points and is described mathematically as the set f(x; u; v)j@x F D @2x F D 0g. Its projection into the uv-parameter space is the cusp curve, which is called the bifurcation set of the cusp catastrophe.

Catastrophe Theory, Figure 15 The catastrophe manifold (CM), catastrophe set (CS) and bifurcation set (BS) of the cusp catastrophe

The cusp catastrophe has some special properties which we discuss now with the aid of some graphics. If the system under investigation shows one or more of these properties, the experimenter should try to find a possible cusp potential that accompanies the process. The first property is called divergence. This property can be shown by the two curves (black and white) on the catastrophe manifold which describes the system’s equilibrium points. Both curves initially start at nearby positions on the surface, that is, they start at nearby stable system states. But the development of the system can proceed quite differently. One of the curves runs on the upper sheet of the surface while the other curve runs on the lower sheet. The system’s stability positions are different and thus its behavior is different. The next property is called hysteresis. If the system development is running along the path shown in Fig. 17 from P1 to P2 or vice versa, the jumps in the system behavior, that is, the sudden changes of internal system variables, occur at different parameter constellations, depending on the direction the path takes, from the upper to the lower sheet or vice versa. Jumps upward or downward happen

Catastrophe Theory

Catastrophe Theory, Figure 16 The divergence property of the cusp catastrophe

Catastrophe Theory, Figure 18 The bifurcation property of the cusp catastrophe

Further Applications

Catastrophe Theory, Figure 17 The hysteresis property of the cusp catastrophe

at the border of the fold along the dotted lines. The name of this property of the cusp comes from the characteristic hysteresis curve in physics, occurring for example at the investigation of the magnetic properties of iron. Figure 18 shows the bifurcation property of the cusp catastrophe: the number of equilibrium states of the system splits from one to three along the path beginning at the starting point (SP) and running from positive to negative u values as shown. If the upper and the lower sheet of the fold correspond to the local minima of F and the middle sheet corresponds to the local maxima, then along the path shown, one stable state splits into two stable states and one unstable state.

Many applications of catastrophe theory are attributable to Christopher Zeeman. For example, his catastrophe machine, the “Zeeman wheel,” is often found in literature. This simple model consists of a wheel mounted flat against a board and able to turn freely. Two elastics are attached at one point (Fig. 19, point B) close to the periphery of the wheel. One elastic is fixed with its second end on the board (point A). The other elastic can be moved with its free end in the plane (point C). Moving the free end smoothly, the wheel changes its angle of rotation smoothly almost everywhere. But at certain positions of C, which can be marked with a pencil, the wheel suddenly changes this angle dramatically. Joining the marked points where these jumps occur, you will find a cusp curve in the plane. In order to describe this behavior, the two coordinates of the position of point C serve as control parameters in the catastrophe model. The potential function for this model results from Hook’s law and simple geometric considerations. Expanding the potential near its degenerated critical point into its Taylor series, and transforming the series with diffeomorphisms similar to the examples we have already calculated, this model leads to the cusp catastrophe. Details can be found in the book of Poston and Stewart [7] or Saunders (1986). Another example of an application of catastrophe theory is similar to the traffic jam model we used here. Catastrophe theory can describe the formation of shockwaves in hydrodynamics. Again the cusp catastrophe describes

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Catastrophe Theory

Catastrophe Theory, Figure 19 The Zeeman wheel

this phenomenon. The mathematical frame is given in the works of Lax [6], Golubitsky and Schaeffer[3] and Guckenheimer [4]. In geometrical optics light rays are investigated which are reflected on smooth surfaces. Caustics are the envelopes of the light rays. They are the bright patterns of intensive light, which can be seen, for example, on a cup of coffee when bright sunlight is reflected on the border of the cup. Catastrophe theory can be applied to light caustics, because the light rays obey a variational principle. According to Fermat’s principle, light rays travel along geodesics and the role of the potential functions in catastrophe theory is played by geodesic distance functions. Caustics are their bifurcation sets. The calculations are given for example in the work of Sinha [11]. A purely speculative example, given by Zeeman, is an aggression model for dogs. Suppose that fear and rage of a dog can be measured in some way from aspects of its body language (its attitude, ears, tail, mouth, etc.). Fear and rage are two conflicting parameters, so that a catastrophe in the dog’s behavior can happen. Look at Fig. 20. The vertical axis shows the dog’s aggression potential, the other axes show its fear and rage. Out of the many different ways the dog can behave, we choose one example. Suppose we start at point A on the catastrophe manifold (behavioral surface). Maybe the dog is dozing in the sun without any thought of aggression. The dog’s aggression behavior is neutral (point A). Another dog is coming closer and our dog, now awakened, is becoming more and more angry by this breach of his territory. Moving along the path on the behavior surface, the dog is on a trajectory to attack his opponent. But it suddenly notices that the enemy is very big (point B). So its fear is growing, its

Catastrophe Theory, Figure 20 A dog’s aggression behavior depending on its rage and fear

rage diminishes. At point C the catastrophe happens: Attack suddenly changes to flight (point D). This flight continues until the dog notices that the enemy does not follow (point E). Fear and rage both get smaller until the dog becomes calm again. The example can be modified a little bit to apply to conflicts between two states. The conflicting factors can be the costs and the expected gain of wars, and so on. Among the variety of possible examples, those worthy of mention include investigations of the stability of ships, the gravitational collapse of stars, the buckling beam (Euler beam), the breaking of ocean waves, and the development of cells in biology. The latter two problems are examples for the occurrence of higher catastrophes, for example

Catastrophe Theory

the hyperbolic umbilic (see Thom’s List of the seven elementary catastrophes). Future Directions When catastrophe theory came up in the nineteen-sixties, much enthusiasm spread among mathematicians and other scientists about the new tool that was expected to explain many of the catastrophes in natural systems. It seemed to be the key for the explanation of discontinuous phenomena in all sciences. Since many applications were purely qualitative and speculative, a decade later this enthusiasm ebbed away and some scientists took this theory for dead. But it is far from that. Many publications in our day show that it is still alive. The theory has gradually become a ‘number producing’ theory, so that it is no longer perceived as purely qualitative. It seems that it will be applied the sciences increasingly, producing numerical results. Thom’s original ideas concerning mathematical biology seem to have given the basis for the trends in modern biology. This is probably the most promising field of application for catastrophe theory. New attempts are also made, for example, in the realm of decision theory or in statistics, where catastrophe surfaces may help to clarify statistical data. From the point of view of teaching and learning mathematics, it seems that catastrophe theory is becoming and increasingly popular part of analysis courses at our universities. Thus, students of mathematics meet the basic ideas of catastrophe theory within their first two years of undergraduate studies. Perhaps in the near future its basic principles will be taught not only to the mathematical specialists but to students of other natural sciences, as well. Bibliography Primary Literature 1. Bröcker T (1975) Differentiable germs and catastrophes. London Mathem. Soc. Lecture Notes Series. Cambridge Univ Press, Cambridge 2. Golubitsky M, Guillemin V (1973) Stable mappings and their singularities. Springer, New York, GTM 14 3. Golubitsky M, Schaeffer DG (1975) Stability of shock waves for a single conservation law. Adv Math 16(1):65–71 4. Guckenheimer J (1975) Solving a single conservation law. Lecture Notes, vol 468. Springer, New York, pp 108–134 5. Haberman R (1977) Mathematical models – mechanical vibrations, population dynamics, and traffic flow. Prentice Hall, New Jersey 6. Lax P (1972) The formation and decay of shockwaves. Am Math Monthly 79:227–241 7. Poston T, Stewart J (1978) Catastrophe theory and its applications. Pitman, London

8. Sanns W (2000) Catastrophe theory with mathematica – a geometric approach. DAV, Germany 9. Sanns W (2005) Genetisches Lehren von Mathematik an Fachhochschulen am Beispiel von Lehrveranstaltungen zur Katastrophentheorie. DAV, Germany 10. Saunders PT (1980) An introduction to catastrophe theory. Cambridge University Press, Cambridge. Also available in German: Katastrophentheorie. Vieweg (1986) 11. Sinha DK (1981) Catastrophe theory and applications. Wiley, New York 12. Wassermann G (1974) Stability of unfoldings. Lect Notes Math, vol 393. Springer, New York

Books and Reviews Arnold VI (1984) Catastrophe theory. Springer, New York Arnold VI, Afrajmovich VS, Il’yashenko YS, Shil’nikov LP (1999) Bifurcation theory and catastrophe theory. Springer Bruce JW, Gibblin PJ (1992) Curves and singularities. Cambridge Univ Press, Cambridge Castrigiano D, Hayes SA (1993) Catastrophe theory. Addison Wesley, Reading Chillingworth DRJ (1976) Differential topology with a view to applications. Pitman, London Demazure M (2000) Bifurcations and catastrophes. Springer, Berlin Fischer EO (1985) Katastrophentheorie und ihre Anwendung in der Wirtschaftswissenschaft. Jahrb f Nationalök u Stat 200(1):3–26 Förster W (1974) Katastrophentheorie. Acta Phys Austr 39(3):201– 211 Gilmore R (1981) Catastrophe theory for scientists and engineers. Wiley, New York Golubistky M (1978) An introduction to catastrophe theory. SIAM 20(2):352–387 Golubitsky M, Schaeffer DG (1985) Singularities and groups in bifurcation theory. Springer, New York Guckenheimer J (1973) Catastrophes and partial differential equations. Ann Inst Fourier Grenoble 23:31–59 Gundermann E (1985) Untersuchungen über die Anwendbarkeit der elementaren Katastrophentheorie auf das Phänomen “Waldsterben”. Forstarchiv 56:211–215 Jänich K (1974) Caustics and catastrophes. Math Ann 209:161–180 Lu JC (1976) Singularity theory with an introduction to catastrophe theory. Springer, Berlin Majthay A (1985) Foundations of catastrophe theory. Pitman, Boston Mather JN (1969) Stability of C1 -mappings. I: Annals Math 87:89– 104; II: Annals Math 89(2):254–291 Poston T, Stewart J (1976) Taylor expansions and catastrophes. Pitman, London Stewart I (1977) Catastrophe theory. Math Chronicle 5:140–165 Thom R (1975) Structural stability and morphogenesis. Benjamin Inc., Reading Thompson JMT (1982) Instabilities and catastrophes in science and engineering. Wiley, Chichester Triebel H (1989) Analysis und mathematische Physik. Birkhäuser, Basel Ursprung HW (1982) Die elementare Katastrophentheorie: Eine Darstellung aus der Sicht der Ökonomie. Lecture Notes in Economics and Mathematical Systems, vol 195. Springer, Berlin Woodcock A, Davis M (1978) Catastrophe theory. Dutton, New York Zeeman C (1976) Catastrophe theory. Sci Am 234(4):65–83

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Center Manifolds

Center Manifolds GEORGE OSIPENKO State Polytechnic University, St. Petersburg, Russia Article Outline Glossary Definition of the Subject Introduction Center Manifold in Ordinary Differential Equations Center Manifold in Discrete Dynamical Systems Normally Hyperbolic Invariant Manifolds Applications Center Manifold in Infinite-Dimensional Space Future Directions Bibliography Glossary Bifurcation Bifurcation is a qualitative change of the phase portrait. The term “bifurcation” was introduced by H. Poincaré. Continuous and discrete dynamical systems A dynamical system is a mapping X(t; x), t 2 R or t 2 Z, x 2 E which satisfies the group property X(t C s; x) D X(t; X(s; x)). The dynamical system is continuous or discrete when t takes real or integer values, respectively. The continuous system is generated by an autonomous system of ordinary differential equations x˙ 

dx D F(x) dt

(1)

as the solution X(t, x) with the initial condition X(0; x) D x. The discrete system generated by a system of difference equations x mC1 D G(x m )

(2)

as X(n; x) D G n (x). The phase space E is the Euclidean or Banach. Critical part of the spectrum Critical part of the spectrum for a differential equation x˙ D Ax is  c = {eigenvalues of A with zero real part}. Critical part of the spectrum for a diffeomorphism x ! Ax is  c = {eigenvalues of A with modulus equal to 1}. Eigenvalue and spectrum If for a matrix (linear mapping) A the equality Av D v, v ¤ 0 holds then v and  are called eigenvector and eigenvalue of A. The set of the eigenvalues is the spectrum of A. If there exists k such that (A  I) k v D 0, v is said to be generating vector.

Equivalence of dynamical systems Two dynamical systems f and g are topologically equivalent if there is a continuous one-to-one correspondence (homeomorphism) that maps trajectories (orbits) of f on trajectories of g: It should be emphasized that the homeomorphism need not be differentiable. Invariant manifold In applications an invariant manifold arises as a surface such that the trajectories starting on the surface remain on it under the system evolution. Local properties If F(p) D 0, the point p is an equilibrium of (1). If G(q) D q; q is a fixed point of (2). We study dynamics near an equilibrium or a fixed point of the system. Thus we consider the system in a neighborhood of the origin which is supposed to be an equilibrium or a fixed point. In this connection we use terminology local invariant manifold or local topological equivalence. Reduction principle In accordance with this principle a locally invariant (center) manifold corresponds to the critical part of the spectrum of the linearized system. The behavior of orbits on a center manifold determines the dynamics of the system in a neighborhood of its equilibrium or fixed point. The term “reduction principle” was introduced by V. Pliss [59]. Definition of the Subject Let M be a subset of the Euclidean space Rn . Definition 1 A set M is said to be smooth manifold of dimension d  n if for each point p 2 M there exists a neighborhood U M and a smooth mapping g : U ! U0 R d such that there is the inverse mapping g –1 and its differential Dg –1 (matrix of partial derivatives) is an injection i. e. its maximal rang is d. For sphere in R3 which is a two-dimensional smooth manifold the introduction of the described neighborhoods is demonstrated in [77]. A manifold M is Ck -smooth, k  1 if the mappings g and g –1 have k continuous derivatives. Moreover, chosing the mappings g and g –1 as C 1 smooth or analytical, we obtain the manifold with the same smoothness. Consider a continuous dynamical system x˙ D F(x) ; x 2 R n ; Rn

Rn

is a Ck -smooth vector field,

(3)

! k  1; i. e. where F : the mapping F has k continuous derivatives. Let us denote the solution of (3) passing through the point p 2 R n at t D 0 by X(t, p). Suppose that the solution is determined for all t 2 R. By the fundamental theorems of the differential equations theory, the hypotheses imposed on F guarantee the existence, uniqueness, and smoothness of X(t, p)

Center Manifolds

for all p 2 R n . In this case the mapping X t (p) D X(t; p) under fixed t is a diffeomorphism of Rn . Thus, F generates the smooth flow X t : R n ! R n , X t (p) D X(t; p). The differential DX t (0) is a fundamental matrix of the linearized system v˙ D DF(0)v. A trajectory (an orbit) of the system (3) through x0 is the set T(x0 ) D fx D X(t; x0 ); t 2 Rg. Definition 2 A manifold M is said to be invariant under (3) if for any p 2 M the trajectory through p lies in M. A manifold M is said to be locally invariant under (3) if for every p 2 M there exists, depending on p, an interval T1 < 0 < T2 , such that X t (p) 2 M as t 2 (T1 ; T2 ). It means that the invariant manifold is formed by trajectories of the system and the locally invariant manifold consists of arcs of trajectories. An equilibrium is 0-dimensional invariant manifold and a periodic orbit is 1-dimensional one. The concept of invariant manifold is a useful tool for simplification of dynamical systems. Definition 3 A manifold M is said to be invariant in T a neighborhood U if for any p 2 M U the moving point X t (p) remains on M as long as X t (p) 2 U. In this case the manifold M is locally invariant. Near an equilibrium point O the system (3) can be rewritten in the form x˙ D Ax C f (x) ;

(4)

It is evident that the dynamics of (4) and (6) is the same on U D fjxj < /2g. If the constructed system (6) has T an invariant manifold M then M U is locally invariant manifold for the initial system (4) or, more precisely, the manifold M is invariant in U. It should be noted that for the case of infinite-dimensional phase space the described construction is more delicate (see details below). Let us consider a Ck -smooth mapping (k  1) G : R n ! R n which has the inverse Ck -smooth mapping G–1 . The mapping G generates the discrete dynamical system of the form x mC1 D G(x m ) ;

m D : : : ; 2; 1; 0; 1; 2; : : : : (7)

Each continuous system X(t, p) gives rise to the discrete system G(x) D X(1; x) which is the shift operator on unit time. Such a system preserves the orientation of the phase space,where as an arbitrary discrete system may change it. Hence, the space of discrete systems is more rich than the space of continuous ones. Moreover, usually investigation of a discrete system is, as a rule, simpler than study of a differential equations one. In many instances the investigation of a differential equation may be reduced to study of a discrete system by Poincaré (first return) mapping. An orbit of (7) is the set T D fx D x m ; m 2 Zg; where xm satisfies (7). Near a fixed point O the mapping x ! G(x) can be rewritten in the form x ! Ax C f (x) ;

where O D fx D 0g, A D DF(O), f is second-order at the origin, i. e. f (0) D 0 and D f (0) D 0. Let us show that the system (4) near O can be considered as a perturbation of the linearized system x˙ D Ax :

(5)

For this we construct a Ck -smooth mapping g which coincides with f in a sufficiently small neighborhood of the origin and is C1 -close to zero. To construct the mapping g one uses the C 1 -smooth cut-off function ˛ : RC ! RC 8 r < 1/2 ; ˆ <1; ˛(r) D > 0; 1/2  r  1 ; ˆ : 0; r >1: Then set g(x) D ˛(jxj/ ) f (x) where " is a parameter. If jxj > , g(x) D 0 and if jxj < /2, g(x) D f (x) and g is C1 -close to zero. Consider the system x˙ D Ax C g(x) :

(6)

(8)

where O D fx D 0g, A D DG(O), f is second-order at the origin. It is shown above that near O (8) may be considered as a perturbation of the linearized system x ! Ax :

(9)

Motivational Example Consider the discrete dynamical system     x X(x; y) D x C x y ! : (10) y Y(x; y) D 12 y C 12 x 2 C 2x 2 y C y 3 The origin (0; 0) is fixed point. Our task is to examine the stability of the fixed point. The linearized system at 0 is defined by the matrix   1 0 0 1/2 which has two eigenvalues 1 and 1/2. Hence, the first approximation system contracts along y-axis, whereas its action along x-axis is neutral. So the stability of the fixed

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Center Manifolds, Figure 1 Dynamics on the invariant curve y D x 2

point depends on nonlinear terms. Let us show that the curve y D x 2 is invariant for the mapping (10). It is enough to check that if a point (x, y) is on the curve then the image (X, Y) is on the curve, i. e. Y D X 2 as y D x 2 . We have Yj yDx 2 D 12 y C 12 x 2 C 2x 2 y C y 3 j yDx 2 D x 2 C 2x 4 C x 6 ; X 2 j yDx 2 D (x C x y)2 j yDx 2

(11)

D x 2 C 2x 4 C x 6 : Thus, the curve y D x 2 is an invariant one-dimensional manifold which is a center manifold W c for the system (10), see Fig. 1. The fixed point O is on W c . The restriction of the system on the manifold is x ! x C x yj yDx 2 D x(1 C x 2 ) :

(12)

It follows that the fixed point 0 is unstable and x m ! 1 as m ! 1, see Fig. 1. Hence, the origin is unstable fixed point for the discrete system (10). It turns out the system (10) near O is topologically equivalent to the system x ! x(1 C x 2 ) ; y ! 12 y

(13)

which is simpler than (10). The center manifold y D x 2 is tangent to the x-axis at the origin and W c near 0 can be considered as a perturbed manifold of fy D 0g. Introduction In his famous dissertation “General Problem on Stability of Motion” [41] published at 1892 in Khar’kov (Ukraine) A.M. Lyapunov proved that the equilibrium O of the system (4) is stable if all eigenvalues of matrix A have negative real parts and O is unstable if there exists an eigenvalue with positive real part. He studied the case when

some eigenvalues of A have negative real part and the rest of them have zero real one. Lyapunov proved that if the matrix A has a pair of pure imaginary eigenvalues and the other eigenvalues have negative real parts then there exists a two-dimensional invariant surface M through O, and the equilibrium O is stable if O is stable for the system restricted on M. Speaking in modern terms, Lyapunov proved the existence of “center manifold” and formulated “reduction principle” under the described conditions. Moreover, he found the stability condition by using power series expansions, which is an extension of his first method to evaluate stability of systems whose eigenvalues have zero real part. Now we use the same method to check stability [10,25,31,36,43,76]. The general reduction principle in stability theory was established by V. Pliss [59] in 1964, the term “center manifold” was introduced by A. Kelley at 1967 in the paper [38] where the existence of a family of invariant manifolds through equilibrium was proved in general case. As we see the center manifold can be considered as a perturbation of the center subspace of linearized system. H. Poincaré [61] was probably the first who perceived the importance of the perturbation problem and began to study conditions ensuring the preservation of the equilibriums and periodic orbits under a perturbation of differential equations. Hadamard [26] and Perron [57] proved the existence of stable and unstable invariant manifolds for a hyperbolic equilibrium point and, in fact, showed their preservation. The conditions necessary and sufficient for the preservation of locally invariant manifolds passing through an equilibrium have been obtained in [51]. Many results on the center manifold follow from theory of normal hyperbolicity [77] which studies dynamics of a system near a compact invariant manifold. They will be considered below. The books by Carr [10], Guckenheimer and Holmes [25], Marsden and McCracken [43], Iooss [36], Hassard, Kazarinoff and Wan [31], and Wiggins [76] are popular sources of the information about center manifolds. Center Manifold in Ordinary Differential Equations Consider a linear system of differential equations v˙ D Lv ;

(14)

where v 2 R n and L is a matrix. Divide the eigenvalues of the matrix L into three parts: stable  s = {eigenvalues with negative real part}, unstable  u = {eigenvalues with positive real part} and central (neutral, critical)  c = {eigenvalues with zero real part}. If the matrix does not have the eigenvalues with zero real part, the system is called

Center Manifolds

hyperbolic. The matrix L has three eigenspaces: the stable subspace Es , the unstable subspace Eu , and the central subspace Ec correspond to these parts of spectrum. The subspaces Es , Eu , and Ec are invariant under the system (14), that is, the solution starting on the subspace stays on it [33]. The solutions on Es have exponential tending to 0, the solutions on Eu have exponential growth. The subspaces E s ; E c ; E u meet in pairs only at the origin and E s C E c C E u D R n , i. e. we have the invariant decomposition R n D E s ˚ E c ˚ E u . There exists a linear change of coordinates (see [33]) transforming (14) to the form x˙ D Ax ; y˙ D By ;

(15)

Center Manifolds, Figure 2 The classical five invariant manifolds

z˙ D Cz ; where the matrix A has eigenvalues with zero real part, the matrix B has eigenvalues with negative real part, and the matrix C has eigenvalues with positive real part. So, (14) decomposes into three independent systems of differential equations. It is known [1,23,24] that the system (15) is topologically equivalent to the system x˙ D Ax ; y˙ D y ;

(16)

z˙ D z : Thus, the dynamics of the system (14) is determined by the system x˙ D Ax which is the restriction of (14) on the center subspace Ec . Our goal is to justify a similar “reduction principle” for nonlinear systems of differential equations. Summarizing the results of V. Pliss [59,60], A. Kelley [38], N. Fenichel [19], and M. Hirsch, C. Pugh, M. Shub [34] we obtain the following theorem. Theorem 1 (Existence Theorem) Consider a Ck -smooth (1  k < 1) system of differential equations v˙ D F(v), F(O) D 0 and L D DF(O). Let E s ; E u and E c be the stable, unstable, and central eigenspaces of L. Then near the equilibrium O there exist the following five Ck -smooth invariant manifolds (see Fig. 2): The center manifold W c , tangent to E c at O; The stable manifold W s , tangent to E s at O; The unstable manifold W u , tangent to E u at O; The center-stable manifold W cs , tangent to E c C E s at O;  The center-unstable manifold W cu , tangent to E c C E u at O;

   

The stable and unstable manifolds are unique, but center, center-stable, and center-unstable manifolds are not necessarily unique.

Solutions on W s have exponential tending to O as t ! 1. Solutions on W u have exponential tending to O as t ! 1.

Remark 1 The expression “near the equilibrium O” means that there exists a neighborhood U(O), depending on F and k, where the statements of the Existence Theorem hold. Remark 2 The linearized system v˙ D DF(O)v has the invariant subspace E s C E u whereas the complete system v˙ D F(v) may not have any smooth invariant manifolds with the tangent space E s C E u at O, see [5,30]. Representation of the Center Manifold At first we suppose that our system does not have the unstable eigenspace and is transformed by the linear change of coordinates mentioned above to the form x˙ D Ax C f (x; y) ; y˙ D By C g(x; y) ;

(17)

where E c D f(x; 0)g, E s D f(0; y)g. Since the center manifold W c is tangent to E c D f(x; 0)g at the origin O D (0; 0), it can be represented near O in the form W c D f(x; y) j jxj < "; y D h(x)g : In other words the center manifold is represented as the graph of a smooth mapping h : V Rc ! Rs , where c and s are dimensions of the center and stable subspaces, see Fig. 3. Since W c goes through the origin and is tangent to Es , we have the equalities h(0) D 0 ;

Dh(0) D 0 :

(18)

51

52

Center Manifolds

Center Manifolds, Figure 3 Representation of a center manifold

The invariance of W c means that if an initial point (x, y) is in W c , i. e. y D h(x) then the solution (X(t; x; y); Y(t; x; y)) of (17) is in W c , i. e. Y(t; x; y)) D h(X(t; x; y)). Thus, we get the invariance condition Y(t; x; h(x))) D h(X(t; x; h(x))) :

(19)

Differentiating (19) with respect to t and putting t D 0 we get Bh(x) C g(x; h(x)) D Dh(x)(Ax C f (x; h(x))) : (20) Thus, the mapping h have to be a solution of the partial differential equation (20) and satisfy (18). Suppose that the system has the form x˙ D Ax C f (x; y; z) ; y˙ D By C g 1 (x; y; z) ;

(21)

z˙ D Cz C g2 (x; y; z) ; where A has the eigenvalues with zero real part, B has the eigenvalues with negative real part, C has the eigenvalues with positive real part. Analogously to the previous case we can show that the center manifold is represented in the form W c D f(x; y; z) j jxj < ; y D h1 (x); z D h2 (x)g

Center Manifolds, Figure 4 Nonuniqueness of the center manifold

Uniqueness and Nonuniqueness of the Center Manifold The center manifold is nonunique in general. Let us consider the illustrating example from [38]. The system has the form x˙ D x 2 ;

(24)

y˙ D y ;

where (x; y) 2 R2 : It is obviously that the origin (0; 0) is the equilibrium with single stable manifold W s D f(0; y)g: There is a center manifold of the form W c D f(x; 0)g. Moreover, there are the center manifolds that can be obtained when solving the equation dy y D 2 : dx x

(25)

The solution of (24) has the form   1 y(x) D a exp x for x ¤ 0 and any constant a 2 R. It follows that

where the mappings h1 and h2 satisfy the equations W c (a) D f(x; y)jy D a exp

Bh1 (x) C g1 (x; h1 (x); h2 (x)) D Dh1 (x)(Ax C f (x; h1 (x); h2 (x))) Ch2 (x) C g2 (x; h1 (x); h2 (x))

(22)

D Dh2 (x)(Ax C f (x; h1 (x); h2 (x))) and the conditions h1 (0) D 0 ;

Dh1 (0) D 0

h2 (0) D 0 ;

Dh2 (0) D 0 :

(26)

(23)

  1 x

as x < 0 and y D 0 as x  0g is center manifold for each a, being C 1 -smooth and pasting together the curve fy D a exp(1/x)g as x < 0 and the straight line fy D 0g as x  0:, see Fig. 4. However, the center manifolds possess the following weak uniqueness property. Let U be a neighborhood of the W c : It turns out that the maximal invariant set I in U

Center Manifolds

that the center manifold has to be given by the series y P n D 1 nD2 (n  1)!x which diverges when x ¤ 0. If the system is C 1 -smooth then the Existence Theorem guaranties the existence of a Ck -smooth center manifold for any k < 1. However, the center manifold may not be C 1 -smooth. Van Strien S.J. [71] showed that the system x˙ D x 2 C 2 ; y˙ D y  (x 2  2 ) ;

(28)

˙ D0: 

Center Manifolds, Figure 5 Weak uniqueness of the center manifold

must be in W c , that follows from the Reduction Theorem, see below. Consequently, all center manifolds contact on I. In this case the center manifold is called locally maximal. Thus, the center manifolds may differ on the trajectories leaving the neighborhood as t ! 1 or 1. For example, suppose that a center manifold is twodimensional and a limit cycle is generated from the equilibrium through the Hopf bifurcation [31,35]. In this case the invariant set I is a disk bounded by the limit cycle and all center manifolds will contain this disk, see Fig. 5. Smoothness M. Hirsch, C. Pugh, M. Shub [34] proved that the center manifold is Ck -smooth if the system is Ck -smooth and k < 1. Moreover if the kth derivative of the vector field is ˛Hölder or Lipschitz mapping, the center manifold is the same. However, if the system is C 1 or analytic then the center manifold is not necessary the same. First consider the analytic case. It is clear that if the analytic center manifold exists then it is uniquely determined by applying the Taylor power series expansion. Consider the illustrating example [31] x˙ D x 2 ; y˙ D y C x 2

(27)

which does not have an analytic center manifold. In fact, applying the Taylor power series expansions we obtain

does not have a C 1 -smooth center manifold. In fact if the system is C 1 -smooth then for each k there is a neighborhood U k (O) of a Ck -smooth central manifold W c : There are the systems for which the sequence fU k g can shrink to O as k ! 1 [25,71]. The results of the papers [19,34] show that the smoothness of the invariant manifold depends on the relation between Lyapunov exponents on the center manifold and on the normal subspace (stable plus unstable subspaces). This relation  is included in the concept of the normal hyperbolicity. At an equilibrium  D min(Ren /Rec ), where Ren /Rec > 0; c is the eigenvalue on the center subspace and n is the eigenvalue on the normal subspace. The condition  > 1 is necessary for a persistence of smooth invariant manifold. One can guarantee degree of smoothness k of the manifold provided k <  . Moreover, there exists examples showing that the condition k <  is essential. It means that the system has to contract (expand) along E s (E u )k-times stronger than along the center manifold. If a center manifold to the equilibrium O exists then Re c D 0 and  D 1 at O, but near O may be other equilibrium (or other orbits) on the center manifold where  < 1, and as a consequence, the center manifold may not be C 1 -smooth. Let us consider the illustrating example [25] x˙ D x  x 3 ; y˙ D y C x 4 ;

(29)

˙ D0:  The point O D (0; 0; 0) is equilibrium, the system linearized at O has the form x˙ D 0 ; y˙ D y ;

(30)

˙ D0:  The system (30) has the following invariant subspace: E s D f0; 0; 0g, E c D fx; 0; g, E u D f0; y; 0g. The -axis

53

54

Center Manifolds

consists of the equilibriums (0; 0; 0 ) for the system (29). The system linearized at (0; 0 ; 0) is the following x˙ D 0 x ; y˙ D y ;

(31)

˙ D0:  The eigenvalues on the center subspace are 0 and 0 , the eigenvalue on the normal (unstable) subspace is 1. Therefore the degree of smoothness is bounded by  D 1/0 : Detailed information and examples are given in [10,25,31,36,43,76].

tigation of the dynamics near a nonhyperbolic fixed point to the study of the system on the center manifold. Construction of the Center Manifold The center manifold may be calculated by using simple iterations [10]. However, this method does not have wide application and we consider a method based on the Taylor power series expansion. First it was proposed by A. Lyapunov in [41]. Suppose that the unstable subspace is 0, that is the last equation of the system (32) is absent. Such systems have a lot of applications in stability theory. So we consider a system

Reduction Principle

x˙ D Ax C f (x; y) ;

As we saw above, a smooth system of differential equations near an equilibrium point O can be written in the form

y˙ D By C g(x; y) ;

The center manifold is the graph of the mapping y D h(x) which satisfies the equation

x˙ D Ax C f (x; y; z) ; y˙ D By C g(x; y; z) ;

(32) Dh(x)(Ax C f (x; h(x)))  Bh(x)  g(x; h(x)) D 0 (36)

z˙ D Cz C q(x; y; z) ; where O D (0; 0; 0), A has eigenvalues with zero real part, B has eigenvalues with negative real part and C has eigenvalues with positive real part; f (0; 0; 0) D g(0; 0; 0) D q(0; 0; 0) D 0 and D f (0; 0; 0) D Dg(0; 0; 0) D Dq(0; 0; 0) D 0. In this case the invariant subspaces at O are E c D f(x; 0; 0)g, E s D f(0; y; 0)g, and E u D f(0; 0; z)g. The center manifold has the form ˚ W c D (x; y; z) j x 2 V Rc ;

y D h1 (x); z D h2 (x) ;

(33)

where c is the dimension of the center subspace Ec , the mappings h1 (x) and h2 (x) are Ck -smooth, k  1, h1 (0) D h2 (0) D 0 and Dh1 (0) D Dh2 (0) D 0. The last equalities mean that the manifold W c goes through O and is tangent to Ec at O. Summarizing the results of V. Pliss [59,60], A. Shoshitaishvili [68,69], A. Reinfelds [63], K. Palmer [55], Pugh C.C., Shub M. [62], Carr J. [10], Grobman D.M. [24], and F. Hartman [29] we obtain the following theorem. Theorem 2 (Reduction Theorem) The system of differential Equations (32) near the origin is topologically equivalent to the system x˙ D Ax C f (x; h1 (x); h2 (x)) ; y˙ D y ;

(35)

(34)

and the conditions h(0) D 0 and Dh(0) D 0: Let us try to solve the equation by applying the Taylor power series expansion. Denote the left part of (36) by N(h(x)). J. Carr [10] and D. Henry [32] proved the following theorem. Theorem 3 Let  : V(0) Rc ! Rs be a smooth mapping, (0) D 0 and D(0) D 0 such that N((x)) D o(jxjm ) for some m > 0 as jxj ! 0 then h(x)  (x) D o(jxj m ) as jxj ! 0 where r(x) D o(jxj m ) means that r(x)/jxj m ! 0 as jxj ! 0. Thus, if we solve the Eq. (36) with a desired accuracy we construct h with the same accuracy. Theorem 3 substantiates the application of the described method. Let us consider a simple example from [76] x˙ D x 2 y  x 5 ;

(x; y) 2 R2 , the equilibrium is at the origin (0; 0). The eigenvalues of the linearized system are 0 and 1. According to the Existence Theorem the center manifold is locally represented in the form ˚ W c D (x; y) j y D h(x); jxj < ı;

h(0) D 0; Dh(0) D 0) ;

z˙ D z : The first equation is the restriction of the system on the center manifold. The theorem allows to reduce the inves-

(37)

y˙ D y C x 2 ;

where ı is sufficiently small. It follows that h has the form h D ax 2 C bx 3 C    :

(38)

Center Manifolds

The equation for the center manifold is given by Dh(x)(Ax C f (x; h(x)))  Bh(x)  g(x; h(x)) D 0 ; (39)

their Taylor expansions coincide up to all existing orders. For example, the system (24) has the center manifolds of the form ˚   W c (a) D (x; y)jy D a exp x1

where A D 0, B D 1, f (x; y) D x 2 y  x 5 , g(x; y) D x 2 . Substituting (38) in (39) we obtain the equality (2ax C 3bx 2 C    )(ax 4 C bx 5  x 5 C    ) C ax 2 C bx 3  x 2 C    D 0 :

: :

a  1 D 0; ) a D 1 b D 0; :: ::

for any a. However, each manifold has null-expansion at 0.

y˙ D y ;

(42)

(43)

(44)

Hence the equilibrium is unstable. It should be noted that the calculation of bx3 is unnecessary, since substituting h(x) D x 2 C bx 3 C    in the first equation of the system (43), we also obtain (44). Thus, it is enough to compute the first term ax2 of the mapping h. This example brings up the following question. The system (37) is topologically equivalent to the system x˙ D x 4 C    ; y˙ D y ;

(46)

where v 2 R n , L is a matrix and G is second-order at the origin. Without loss of generality we consider the system (46) as a perturbation of the linear system

where h(x) D x 2 C 0x 3 C    : Substituting h we obtain the equation x˙ D x 4 C : : : :

v ! Lv C G(v) ;

(41)

In this connection we have to decide how many terms (powers of x) have be computed? The solution depends on our goal. For example, suppose that we study the Lyapunov stability of the equilibrium (0; 0) for (37). According to the Reduction Theorem the system (37) is topologically equivalent to the system x˙ D x 2 h(x)  x 5 ;

Center Manifold in Discrete Dynamical Systems Consider a dynamical system generated by the mapping

Therefore we have h(x) D x 2 C 0x 3 C    :



(40)

Equating coefficients on each power of x to zero we obtain x2 x3 :: :

as x < 0 and y D 0 as x  0

(45)

which has many center manifolds. From this it follows that the initial system has a lot of center manifolds. Which of center manifolds is actually being found when approximating the center manifold via power series expansion? It turns out [10,70,74] that any two center manifolds differ by transcendentally small terms, i. e. the terms of

x ! Lx :

(47)

Divide the eigenvalues of L into three parts: stable s D feigenvalues with modulus less than 1g, unstable u D feigenvalues with modulus greater than 1g, critical c D feigenvalues with modulus equal to 1g. The matrix L has three eigenspaces: the stable subspace Es , the unstable subspace Eu , and the central subspace Ec , which correspond to the mentioned spectrum parts respectively, being E s ˚ E u ˚ E c D R n . The next theorem follows from the theorem on perturbation of -hyperbolic endomorphism on a Banach space [34]. Theorem 4 (Existence Theorem) Consider a Ck -smooth (1  k < 1) discrete system v ! Lv C G(v) ; G(0) D 0 and G is C1 -close to zero. Let E s ; E u and Ec be the stable, unstable, and central eigenspaces of the linear mapping L. Then near the fixed point O there exist the following Ck -smooth invariant manifolds:     

The center manifold W c tangent to E c at O, The stable manifold W s tangent to E s at O, The unstable manifold W u tangent to E u at O, The center-stable manifold W cs tangent to E c C E s at O, The center-unstable manifold W cu tangent to E c C E u at O.

The stable and unstable manifolds are unique, but center, center-stable, and center-unstable manifolds may not be unique.

55

56

Center Manifolds

The orbits on W s have exponential tending to O as m ! 1. The orbits on W u have exponential tending to O as m ! 1. There exists a linear change of coordinates transforming (46) to the form 1 1 0 x Ax C f (x; y; z) @ y A ! @ By C g1 (x; y; z) A ; Cz C g2 (x; y; z) z 0

(48)

where A has eigenvalues with modulus equal 1, B has eigenvalues with modulus less than 1, C has eigenvalues with modulus greater than 1. The system (48) at the origin has the following invariant eigenspaces: E c D f(x; 0; 0)g, E s D f(0; y; 0)g, and E u D f(0; 0; z)g. Since the center manifold W c is tangent to Ec at the origin O D (0; 0), it can be represented near O in the form W c D f(x; y; z) j jxj < ; y D h1 (x); z D h2 (x)g ; where h1 and h2 are second-order at the origin, i. e. h1;2 (0) D 0; Dh1;2 (0) D 0: The invariance of W c means that if a point (x; y; z) 2 W c ; i. e. y D h1 (x); z D h2 (x), then (Ax C f (x; y; z); By C g1 (x; y; z); Cz C g2 (x; y; z)) 2 W c ; i. e. By C g1 (x; y; z) D h1 (Ax C f (x; y; z)); Cz C g2 (x; y; z) D h2 (Ax C f (x; y; z)). Thus, we get the invariance property Bh1 (x) C g1 (x; h1 (x); h2 (x)) D h1 (Ax C f (x; h1 (x); h2 (x)) ; Ch2 (x) C g2 (x; h1 (x); h2 (x))

(49)

D h2 (Ax C f (x; h1 (x); h2 (x)) : Results on smoothness and uniqueness of center manifold for discrete systems are the same as for continuous systems. Theorem 5 (Reduction Theorem [55,62,63,64]) The discrete system (48) near the origin is topologically equivalent to the system x ! Ax C f (x; h1 (x); h2 (x)) ; y ! By ;

(50)

z ! Cz : The first mapping is the restriction of the system on the center manifold. The theorem reduces the investigation of dynamics near a nonhyperbolic fixed point to the study of the system on the center manifold.

Normally Hyperbolic Invariant Manifolds As it is indicated above, a center manifold can be considered as a perturbed invariant manifold for the center subspace of the linearized system. The normal hyperbolicity conception arises as a natural condition for the persistence of invariant manifold under a system perturbation. Informally speaking, an f -invariant manifold M, where f -diffeomorphism, is normally hyperbolic if the action of Df along the normal space of M is hyperbolic and dominates over its action along the tangent one. (the rigorous definition is below). Many of the results concerning center manifold follows from the theory of normal hyperbolic invariant manifolds. In particular, the results related to center eigenvalues of Df with jj ¤ 1 may be obtained using this theory. The problem of existence and preservation for invariant manifolds has a long history. Initial results on invariant (integral) manifolds of differential equations were obtained by Hadamard [26] and Perron [57], Bogoliubov, Mitropolskii [8] and Lykova [48], Pliss [60], Hale [28], Diliberto [16] and other authors (for references see [39,77]). In the late 1960s and throughout 1970s the theory of perturbations of invariant manifolds began to assume a very general and well-developed form. Sacker [66] and Neimark [50] proved (independently and by different methods) that a normally hyperbolic compact invariant manifold is preserved under C1 perturbations. Final results on preservation of invariant manifolds were obtained by Fenichel [19,20] and Hirsch, Pugh, Shub [34]. The linearization theorem for a normally hyperbolic manifold was proved in [62]. Let f : R n ! R n be a Cr -diffeomorphism, 1  r < 1, and a compact manifold M is f -invariant. Definition 4 An invariant manifold M is called r-normally hyperbolic if there exists a continuous invariant decomposition TR n j M D TM ˚ E s ˚ E u

(51)

tangent bundle TRn

into a direct sum of subbundles of the TM; E s ; E u and constants a;  > 0 such that for 0  k  r and all p 2 M, jD s f n (p)j j(D0 f n (p))1 j k  a exp(n) u n

0 n

1 k

jD f (p)j j(D f (p)) j  a exp(n)

for n  0 ; for n  0 : (52)

Here D0 f ; Ds f and Du f are restrictions of Df on TM; E s and Eu , respectively. The invariance of the bundle E  means that for every p 2 M0 , the image of the fiber E  (p) at p under Df (p)

Center Manifolds

is the fiber E  (q) at q D f (p): The bundles Es and Eu are called stable and unstable, respectively. In other words, M is normally hyperbolic if the differential Df contracts (expands) along the normal (to M) direction and this contraction (expansion) is stronger in r times than a conceivable contraction (expansion) along M: Summarizing the results of [19,20,34,50,62,66] we obtain the following theorem. Theorem 6 Let a Cr diffeomorphism f be r-normally hyperbolic on a compact invariant manifold M with the decomposition TR n j M D TM ˚ E s ˚ E u . Then  there exist invariant manifolds W s and W u near M, which are tangent at M to TM ˚ E s and TM ˚ E u ;  the manifolds M, W s and W u are Cr -smooth;  if g is another Cr -diffeomorphism Cr -close to f ; then there exists the unique manifold M g which is invariant and r-normally hyperbolic for g;

The reduced system is of the form x˙ D ax 3 C : : : : Thus, if a < 0, the origin is stable. The book [25] contains other examples of investigation of stability. Bifurcations Consider a parametrized system of differential equations x˙ D A()x C f (x; ) ;

(55)

where  2 R k , f is C1 -small when  is small. To study the bifurcation problem near  D 0 it is useful to deal with the extended system of the form x˙ D A()x C f (x; ) ; ˙ D0: 

(56)

Similarly result takes place for flows. Mañé [44] showed that a locally unique, preserved, compact invariant manifold is necessarily normally hyperbolic. Local uniqueness means that near M, an invariant set I of the perturbation g is in M g . Conditions for the preservation of locally nonunique compact invariant manifolds and linearization were found by G. Osipenko [52,53].

Suppose that (55) has a n-dimensional center manifold for  D 0. Then the extended system (56) has a n C k-dimensional center manifold. The reduction principle guarantees that all bifurcations lie on the center manifold. Moreover, the center manifold has the form y D h(x; 0 ) for every ˙ D 0 and the map0 which is a solution of the equation  ping h may be presented as a power series in . Further interesting information regarding to recent advancements in the approximation and computation of center manifolds is in [37]. The book [31] deals with Hopf bifurcation and contains a lot of examples and applications. Invariant manifolds and foliations for nonautonomous systems are considered in [3,13]. Partial linearization for noninvertible mappings is studied in [4].

Applications

Center Manifold in Infinite-Dimensional Space

Stability

Center manifold theory is a standard tool to study the dynamics of discrete and continuous dynamical systems in infinite dimensional phase space. We start with the discrete dynamical system generated by a mapping of a Banach space. The existence of the center manifold of the mappings in Banach space was proved by Hirsch, Pugh and Shub [34]. Let T : E ! E be a linear endomorphism of a Banach space E.

 near M f is topologically equivalent to D f j E s˚E u , which in local coordinates has the form (x; y; z) ! ( f (x); D f (x)y; D f (x)z) ; x2M;

y 2 E s (x) ;

z 2 E u (x) :

(53)

As it was mentioned above, it was Lyapunov who pioneered in applying the concept of center manifold to establish the stability of equilibrium. V. Pliss [59] proved the general reduction principle in stability theory. According to this principle the equilibrium is stable if and only if it is stable on the center manifold. We have considered the motivating example where the center manifold concept was used. The system x˙ D x y ; y˙ D y C ax 2 ;

(54)

has the center manifold y D h(x): Applying Theorem 3, we obtain h(x) D ax 2 C    :

Definition 5 A linear endomorphism T is said to be -hyperbolic, > 0, if no eigenvalue of T has modulus , i. e., its spectrum Spect(T) has no points on the circle of radius

in the complex plane C. A linear 1-hyperbolic isomorphism T is called hyperbolic. For a -hyperbolic linear endomorphism there exists a T-invariant splitting of E D E1 ˚ E2 such that the spectrum of the restriction T1 D Tj E 1 lies outside of the disk of

57

58

Center Manifolds

the radius , where as the spectrum of T2 D Tj E 2 lies inside it. So T 1 is automorphism and T11 exists. It is known (for details see [34]) that one can define norms in E1 and E2 such that associated norms of T11 and T 2 may be estimated in the following manner jT11 j <

1

;

jT2 j < :

(57)

Conversely, if T admits an invariant splitting T D T1 ˚ T2 with jT11 j  a; jT2 j  b; ab < 1 then Spect(T1 ) lies outside the disk f 2 C : jj < 1/ag, and Spect(T2 ) lies in the disk f 2 C : jj  bg: Thus, T is -hyperbolic with b < < 1/a: Theorem 7 Let T be a -hyperbolic linear endomorphism,  1. Assume that f : E ! E is C r ; r  1; f D T C g; f (0) D 0 and there is ı > 0 such that if g is a Lipschitz mapping and Lip(g) D Lip( f  T)  ı ;

(58)

then there exists a manifold Wf ; which is a graph of a C1 map ' : E1 ! E2 , i. e. Wf D fx C y j y D '(x); x 2 E1 ; y 2 E2 g ; with the following properties:  W f is f -invariant;  if kT11 k j kT2 k < 1; j D 1; : : : ; r then W f is Cr and depends continuously on f in the Cr topology;  WT D E1 ;  for x 2 Wf j f 1 (x)j  (a C 2") jxj ;

(59)

where a < 1/ , and " is small provided ı is small;  if D f (0) D T then W f is tangent to E1 at 0. Suppose that the spectrum of T is contained in A1 where A1 D fz 2 C; jzj  1g ;

S

A2

A2 D fz 2 C; jzj  a < 1g :

Corollary 1 If f : E ! E is C r ; 1  r < 1; f (0) D 0 and Lip( f  T)  is small, then there exists a centerunstable manifold W cu D Wf which is the graph of a Cr function E1 ! E2 : The center-unstable manifold is attractor, i. e. for any x 2 E the distance between f n (x) and W cu tends to zero as n ! 1. The manifold Wf D W c is center manifold when A1 D fz 2 C; jzj D 1g. Apply this theorem to a mapping of the general form u ! Au C f (u) ;

(60)

where E is a Banach space, u 2 E, A: E ! E is a linear operator, f is second-order at the origin. The question arises of whether there exists a smooth mapping g such that g coincides with f in a sufficiently small neighborhood of the origin and g is C1 -close to zero? The answer is positive if E admits C 1 -norm, i. e. the mapping u ! kuk is C 1 -smooth for u ¤ 0. It is performed for Hilbert space. The desired mapping g may be constructed with the use of a cut-off function. To apply Theorem 7 to (60) we have to suppose Hypothesis on C 1 -norm: the Banach space E admits C 1 -norm. Thus, Theorem 7 together with Hypothesis on C 1 norm guarantee an existence of center manifold for the system (60) in a Banach space. A. Reinfelds [63,64] prove a reduction theorem for homeomorphism in a metric space from which an analog of Reduction Theorem 5 for Banach space follows. Flows in Banach Space Fundamentals of center manifold theory for flows and differential equations in Banach spaces can be found in [10,21,43,73]. In Euclidean space a flow is generated by a solution of a differential equation and the theory of ordinary differential equations guarantees the clear connection between flows and differential equations. In infinite dimensional (Banach) space this connection is more delicate and at first, we consider infinite dimensional flows and then (partial) differential equations. Definition 6 A flow (semiflow) on a domain D is a continuous family of mappings F t : D ! D where t 2 R (t  0) such that F0 D I and F tCs D F t Fs for t; s 2 R (t; s  0): Theorem 8 (Center Manifold for Infinite Dimensional Flows [43]) Let the Hypothesis on C 1 -norm be fulfilled and F t : E ! E be a semiflow defined near the origin for 0  t  , Ft (0) D 0. Suppose that the mapping F t (x) is C0 in t and C kC1 in x and the spectrum of the linear semigroup DF t (0) is of the form e t(1 [2 ) ; where e t(1 ) is on the unit circle (i. e. 1 is on the imaginary axis) and e t(2 ) is inside the unit circle with a positive distance from one as t > 0 (i. e. 2 is in the left half-plane). Let E D E1 ˚ E2 be the DF t (0)-invariant decomposition corresponding to the decomposition of the spectrum e t(1 [2 ) and dimE1 D d < 1. Then there exists a neighborhood U of the origin and Ck -submanifold W c U of the dimension d such that  W c is invariant for F t in U, 0 2 W c and W c is tangent to E1 at 0;  W c is attractor in U, i. e. if F tn (x) 2 U; n D 0; 1; 2; : : : then F tn (x) ! W c as n ! 1.

Center Manifolds

In infinite dimensional dynamics it is also popular notion of “inertial manifold” which is an invariant finite dimensional attracting manifold corresponding to W c of the Theorem 8. Partial Differential Equations At the present the center manifold method takes root in theory of partial differential equations. Applying center manifold to partial differential equations leads to a number of new problems. Consider an evolution equation u˙ D Au C f (u) ;

(61)

where u 2 E, E is a Banach space, A is a linear operator, f is second-order at the origin. Usually A is a differential operator defined on a domain D E; D ¤ E. To apply Theorem 8 we have to construct a semiflow F t defined near the origin of E. As a rule, an infinite dimensional phase space E is a functional space where the topology may be chosen by the investigator. A proper choice of functional space and its topology may essentially facilitate the study. As an example we consider a nonlinear heat equation u0t D u C f (u) ;

(62)

where u is defined on a bounded domain ˝ R n with Dirichlet condition on the boundary uj@˝ D 0, u D (@2 /@x12 C    C @2 /@x n2 )u is the Laplace operator defined on the set C02 D fu 2 C 2 (˝); u D 0 on @˝g : According to [49] we consider a HilbertR space E D L2 (˝) with the inner product < u; v >D ˝ uvdx. The operator  may be extended to a self-conjugate operator A: D A ! L2 (˝) with the domain DA that is the closure of C02 in L2 (˝). Consider the linear heat equation u0t D u and its extension u0t D Au;

u 2 DA :

(63)

What do we mean by a flow generated by (63)? Definition 7 A linear operator A is the infinitisemal generator of a continuous semigroup U(t); t  0 if: U(t) is a linear mapping for each t; U(t) is a semigroup and kU(t)u  uk ! 0 as t ! C0; U(t)u  u t!C0 t

Au D lim

(64)

whenever the limit exists. For Laplace operator we have < u; u > 0 for u 2 C02 ; hence the operator A is dissipative. It guarantees [6] the existence of the semigroup U(t) with the infinitesimal gen-

erator A. The mapping U(t) is the semigroup DF t (0) in Theorem 8. The next problem is the relation between the linearized system u0 D Au and full nonlinear Equation (61). To consider the mapping f in (62) as C1 -small perturbation, the Hypothesis on C 1 -norm has to be fulfilled. To prove the existence of the solution of the nonlinear equation with u(0) D u0 one uses the Duhamel formula Zt u(t) D U t u0 C

U ts f (u(s))ds 0

and the iterative Picard method. By this way we construct the semiflow required to apply Theorem 8. The relation between the spectrum of a semigroup U(t) and the spectrum of its infinitisemal generator gives rise to an essential problem. All known existence theorems are formulated in terms of the semigroup as in Theorem 8. The existence of the spectrum decompositions of a semigroup U(t) and the corresponding estimations on the appropriate projections are supposed. This framework is inconvenient for many applications, especially in partial differential equations. In finite dimensions, a linear system x˙ D Ax has a solution of the form U(t)x D e At x and  is an eigenvalue of A if and only if et is an eigenvalue of U(t). We have the spectral equality Spect(U(t)) D eSpect(A)t : In infinite dimensions, relating the spectrum of the infinitesimal generator A to that of the semigroup U(t) is a spectral mapping problem which is often nontrivial. The spectral inclusion Spect(U(t)) eSpect(A)t always holds and the inverse inclusion is a problem that is solved by the spectral mapping theorems [12,18,22,65]. Often an application of center manifold theory needs to prove an appropriate version of the reduction theorem [7,11,12,14,15,18]. Pages 1–5 of the book [40] give an extensive list of the applications of center manifold theory to infinite dimensional problems. Here we consider a few of typical applications. Reaction-diffusion equations are typical models in chemical reaction (Belousov– Zhabotinsky reaction), biological systems, population dynamics and nuclear reaction physics. They have the form u0t D (K() C D)u C f (u; ) ;

(65)

where K is a matrix depending on a parameter, D is a symmetric, positive semi-definite, often diagonal matrix,  is the Laplace operator,  is a control parameter. The pa-

59

60

Center Manifolds

pers [2,27,31,67,78] applied center manifold theory to the study of the reaction-diffusion equation. Invariant manifolds for nonlinear Schrodinger equations are studied in [22,40]. In the elliptic quasilinear equations the spectrum of the linear operator is unbounded in unstable direction and in this case the solution does not generate even a semiflow. Nevertheless, center manifold technique is successfully applied in this case as well [45]. Henry [32] proved the persistence of normally hyperbolic closed linear subspace for semilinear parabolic equations. It follows the existence of the center manifold which is a perturbed manifold for the center subspace. I. Chueshov [14,15] considered a reduction principle for coupled nonlinear parabolic-hyperbolic partial differential equations that has applications in thermoelastisity. Mielke [47] has developed the center manifold theory for elliptic partial differential equations and has applied it to problems of elasticity and hydrodynamics. The results for non-autonomous system in Banach space can be found in [46]. The reduction principle for stochastic partial differential equations is considered in [9,17,75]. Future Directions Now we should realize that principal problems in the center manifolds theory of finite dimensional dynamics have been solved and we may expect new results in applications. However the analogous theory for infinite dimensional dynamics is far from its completion. Here we have few principal problems such as the reduction principle (an analogue of Theorem 2 or 5) and the construction of the center manifold. The application of the center manifold theory to partial differential equations is one of promising and developing direction. In particular, it is very important to investigate the behavior of center manifolds when Galerkin method is applied due to a transition from finite dimension to infinite one. As it was mentioned above each application of center manifold methods to nonlinear infinite dimensional dynamics is nontrivial and gives rise to new directions of researches. Bibliography Primary Literature 1. Arnold VI (1973) Ordinary Differential Equations. MIT, Cambridge 2. Auchmuty J, Nicolis G (1976) Bifurcation analysis of reactiondiffusion equation (III). Chemical Oscilations. Bull Math Biol 38:325–350 3. Aulbach B (1982) A reduction principle for nonautonomous differential equations. Arch Math 39:217–232 4. Aulbach B, Garay B (1994) Partial linearization for noninvertible mappings. J Appl Math Phys (ZAMP) 45:505–542

5. Aulbach B, Colonius F (eds) (1996) Six Lectures on Dynamical Systems. World Scientific, New York 6. Balakrishnan AV (1976) Applied Functional Analysis. Springer, New York, Heidelberg 7. Bates P, Jones C (1989) Invariant manifolds for semilinear partial differential equations. In: Dynamics Reported 2. Wiley, Chichester, pp 1–38 8. Bogoliubov NN, Mitropolsky YUA (1963) The method of integral manifolds in non-linear mechanics. In: Contributions Differential Equations 2. Wiley, New York, pp 123–196 9. Caraballo T, Chueshov I, Landa J (2005) Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations. Nonlinearity 18:747–767 10. Carr J (1981) Applications of Center Manifold Theory. In: Applied Mathematical Sciences, vol 35. Springer, New York 11. Carr J, Muncaster RG (1983) Applications of center manifold theory to amplitude expansions. J Diff Equ 59:260–288 12. Chicone C, Latushkin Yu (1999) Evolution Semigroups in Dynamical Systems and Differential Equations. Math Surv Monogr 70. Amer Math Soc, Providene 13. Chow SN, Lu K (1995) Invariant manifolds and foliations for quasiperiodic systems. J Diff Equ 117:1–27 14. Chueshov I (2007) Invariant manifolds and nonlinear masterslave synchronization in coupled systems. In: Applicable Analysis, vol 86, 3rd edn. Taylor and Francis, London, pp 269–286 15. Chueshov I (2004) A reduction principle for coupled nonlinesr parabolic-hyperbolic PDE. J Evol Equ 4:591–612 16. Diliberto S (1960) Perturbation theorem for periodic surfaces I, II. Rend Cir Mat Palermo 9:256–299; 10:111–161 17. Du A, Duan J (2006) Invariant manifold reduction for stochastic dynamical systems. http://arXiv:math.DS/0607366 18. Engel K, Nagel R (2000) One-parameter Semigroups for Linear Evolution Equations. Springer, New York 19. Fenichel N (1971) Persistence and smoothness of invariant manifolds for flows. Ind Univ Math 21:193–226 20. Fenichel N (1974) Asymptotic stability with rate conditions. Ind Univ Math 23:1109–1137 21. Gallay T (1993) A center-stable manifold theorem for differential equations in Banach space. Commun Math Phys 152:249–268 22. Gesztesy F, Jones C, Latushkin YU, Stanislavova M (2000) A spectral mapping theorem and invariant manifolds for nonlinear Schrodinger equations. Ind Univ Math 49(1):221–243 23. Grobman D (1959) Homeomorphism of system of differential equations. Dokl Akad Nauk SSSR 128:880 (in Russian) 24. Grobman D (1962) The topological classification of the vicinity of a singular point in n-dimensional space. Math USSR Sbornik 56:77–94; in Russian 25. Guckenheimer J, Holmes P (1993) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vectors Fields. Springer, New York 26. Hadamard J (1901) Sur l’etaration et les solution asymptotiques des equations differentielles. Bull Soc Math France 29:224–228 27. Haken H (2004) Synergetics: Introduction and Advanced topics. Springer, Berlin 28. Hale J (1961) Integral manifolds of perturbed differential systems. Ann Math 73(2):496–531 29. Hartman P (1960) On local homeomorphisms of Euclidean spaces. Bol Soc Mat Mex 5:220

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30. Hartman P (1964) Ordinary Differential Equations. Wiley, New York 31. Hassard BD, Kazarinoff ND, Wan YH (1981) Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge 32. Henry D (1981) Geometric theory of semilinear parabolic equations. Lect Notes Math 840:348 33. Hirsch M, Smale S (1974) Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, Orlando 34. Hirsch M, Pugh C, Shub M (1977) Invariant manifolds. Lect Notes Math 583:149 35. Hopf E (1942) Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber Verh Sachs Akad Wiss Leipzig Math-Nat 94:3–22 36. Iooss G (1979) Bifurcation of Maps and Application. N-Holl Math Stud 36:105 37. Jolly MS, Rosa R (2005) Computation of non-smooth local centre manifolds. IMA J Numer Anal 25(4):698–725 38. Kelley A (1967) The stable, center stable, center, center unstable and unstable manifolds. J Diff Equ 3:546–570 39. Kirchgraber U, Palmer KJ (1990) Geometry in the Neighborhood of Invariant Manifolds of the Maps and Flows and Linearization. In: Pitman Research Notes in Math, vol 233. Wiley, New York 40. Li C, Wiggins S (1997) Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrodinger Equations. Springer, New York 41. Lyapunov AM (1892) Problémé Générale de la Stabilité du Mouvement, original was published in Russia 1892, transtalted by Princeton Univ. Press, Princeton, 1947 42. Ma T, Wang S (2005) Dynamic bifurcation of nonlinear evolution equations and applications. Chin Ann Math 26(2): 185–206 43. Marsden J, McCracken M (1976) Hopf bifurcation and Its Applications. Appl Math Sci 19:410 44. Mañé R (1978) Persistent manifolds are normally hyperbolic. Trans Amer Math Soc 246:261–284 45. Mielke A (1988) Reduction of quasilinear elliptic equations in cylindrical domains with application. Math Meth App Sci 10:51–66 46. Mielke A (1991) Locally invariant manifolds for quasilinear parabolic equations. Rocky Mt Math 21:707–714 47. Mielke A (1996) Dynamics of nonlinear waves in dissipative systems: reduction, bifurcation and stability. In: Pitman Research Notes in Mathematics Series, vol 352. Longman, Harlow, pp 277 48. Mitropolskii YU, Lykova O (1973) Integral Manifolds in Nonlinear Mechanics. Nauka, Moscow 49. Mizohata S (1973) The Theory of Partial Differential Equations. Cambridge University Press, Cambridge 50. Neimark Y (1967) Integral manifolds of differential equations. Izv Vuzov, Radiophys 10:321–334 (in Russian) 51. Osipenko G, Ershov E (1993) The necessary conditions of the preservation of an invariant manifold of an autonomous system near an equilibrium point. J Appl Math Phys (ZAMP) 44:451–468 52. Osipenko G (1996) Indestructibility of invariant non-unique manifolds. Discret Contin Dyn Syst 2(2):203–219 53. Osipenko G (1997) Linearization near a locally non-unique invariant manifold. Discret Contin Dyn Syst 3(2):189–205

54. Palis J, Takens F (1977) Topological equivalence of normally hyperbolic dynamical systems. Topology 16(4):336–346 55. Palmer K (1975) Linearization near an integral manifold. Math Anal Appl 51:243–255 56. Palmer K (1987) On the stability of center manifold. J Appl Math Phys (ZAMP) 38:273–278 57. Perron O (1928) Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystem. Math Z 29:129–160 58. Pillet CA, Wayne CE (1997) Invariant manifolds for a class of dispersive. Hamiltonian, partial differential equations. J Diff Equ 141:310–326 59. Pliss VA (1964) The reduction principle in the theory of stability of motion. Izv Acad Nauk SSSR Ser Mat 28:1297–1324; translated (1964) In: Soviet Math 5:247–250 60. Pliss VA (1966) On the theory of invariant surfaces. In: Differential Equations, vol 2. Nauka, Moscow pp 1139–1150 61. Poincaré H (1885) Sur les courbes definies par une equation differentielle. J Math Pure Appl 4(1):167–244 62. Pugh C, Shub M (1970) Linearization of normally hyperbolic diffeomorphisms and flows. Invent Math 10:187–198 63. Reinfelds A (1974) A reduction theorem. J Diff Equ 10:645–649 64. Reinfelds A (1994) The reduction principle for discrete dynamical and semidynamical systems in metric spaces. J Appl Math Phys (ZAMP) 45:933–955 65. Renardy M (1994) On the linear stability of hyperbolic PDEs and viscoelastic flows. J Appl Math Phys (ZAMP) 45:854–865 66. Sacker RJ (1967) Hale J, LaSalle J (eds) A perturbation theorem for invariant Riemannian manifolds. Proc Symp Diff Equ Dyn Syst Univ Puerto Rico. Academic Press, New York, pp 43–54 67. Sandstede B, Scheel A, Wulff C (1999) Bifurcations and dynamics of spiral waves. J Nonlinear Sci 9(4):439–478 68. Shoshitaishvili AN (1972) Bifurcations of topological type at singular points of parameterized vector fields. Func Anal Appl 6:169–170 69. Shoshitaishvili AN (1975) Bifurcations of topological type of a vector field near a singular point. Trudy Petrovsky seminar, vol 1. Moscow University Press, Moscow, pp 279–309 70. Sijbrand J (1985) Properties of center manifolds. Trans Amer Math Soc 289:431–469 71. Van Strien SJ (1979) Center manifolds are not C 1 . Math Z 166:143–145 72. Vanderbauwhede A (1989) Center Manifolds, Normal Forms and Elementary Bifurcations. In: Dynamics Reported, vol 2. Springer, Berlin, pp 89–169 73. Vanderbauwhede A, Iooss G (1992) Center manifold theory in infinite dimensions. In: Dynamics Reported, vol 1. Springer, Berlin, pp 125–163 74. Wan YH (1977) On the uniqueness of invariant manifolds. J Diff Equ 24:268–273 75. Wang W, Duan J (2006) Invariant manifold reduction and bifurcation for stochastic partial differential equations. http://arXiv: math.DS/0607050 76. Wiggins S (1992) Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York 77. Wiggins S (1994) Normally Hyperbolic Invariant Manifolds of Dynamical Systems. Springer, New York 78. Wulff C (2000) Translation from relative equilibria to relative periodic orbits. Doc Mat 5:227–274

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Books and Reviews Bates PW, Lu K, Zeng C (1998) Existence and persistence of invariant manifolds for semiflows in Banach spaces. Mem Amer Math Soc 135:129 Bates PW, Lu K, Zeng C (1999) Persistence of overflowing manifolds for semiflow. Comm Pure Appl Math 52(8):983–1046 Bates PW, Lu K, Zeng C (2000) Invariant filiations near normally hyperbolic invariant manifolds for semiflows. Trans Amer Math Soc 352:4641–4676 Babin AV, Vishik MI (1989) Attractors for Evolution Equations. Nauka. Moscow; English translation (1992). Elsevier Science, Amsterdam Bylov VF, Vinograd RE, Grobman DM, Nemyskiy VV (1966) The Theory of Lyapunov Exponents. Nauka, Moscow (in Russian) Chen X-Y, Hale J, Tan B (1997) Invariant foliations for C 1 semigroups in Banach spaces. J Diff Equ 139:283–318 Chepyzhov VV, Goritsky AYU, Vishik MI (2005) Integral manifolds and attractors with exponential rate for nonautonomous hyperbolic equations with dissipation. Russ J Math Phys 12(1): 17–39 Chicone C, Latushkin YU (1997) Center manifolds for infinite dimensional nonautonomous differential equations. J Diff Equ 141:356–399 Chow SN, Lu K (1988) Invariant manifolds for flows in Banach spaces. J Diff Equ 74:285–317 Chow SN, Lin XB, Lu K (1991) Smooth invariant foliations in infinite dimensional spaces. J Diff Equ 94:266–291 Chueshov I (1993) Global attractors for non-linear problems of mathematical physics. Uspekhi Mat Nauk 48(3):135–162; English translation in: Russ Math Surv 48:3 Chueshov I (1999) Introduction to the Theory of Infinite-Dimensional Dissipative Systems. Acta, Kharkov (in Russian); English translation (2002) http://www.emis.de/monographs/ Chueshov/. Acta, Kharkov Constantin P, Foias C, Nicolaenko B, Temam R (1989) Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations. Appl Math Sci, vol 70. Springer, New York Gonçalves JB (1993) Invariant manifolds of a differentiable vector field. Port Math 50(4):497–505 Goritskii AYU, Chepyzhov VV (2005) Dichotomy property of solu-

tions of quasilinear equations in problems on inertial manifolds. SB Math 196(4):485–511 Hassard B, Wan Y (1978) Bifurcation formulae derived from center manifold theory. J Math Anal Appl 63:297–312 Hsia C, Ma T, Wang S (2006) Attractor bifurcation of three-dimensional double-diffusive convection. http://arXiv:nlin.PS/ 0611024 Knobloch HW (1990) Construction of center manifolds. J Appl Math Phys (ZAMP) 70(7):215–233 Latushkin Y, Li Y, Stanislavova M (2004) The spectrum of a linearized 2D Euler operator. Stud Appl Math 112:259 Leen TK (1993) A coordinate independent center manifold reduction. Phys Lett A 174:89–93 Li Y (2005) Invariant manifolds and their zero-viscosity limits for Navier–Stokes equations. http://arXiv:math.AP/0505390 Osipenko G (1989) Examples of perturbations of invariant manifolds. Diff Equ 25:675–681 Osipenko G (1985, 1987, 1988) Perturbation of invariant manifolds I, II, III, IV. Diff Equ 21:406–412, 21:908–914, 23:556–561, 24:647–652 Podvigina OM (2006) The center manifold theorem for center eigenvalues with non-zero real parts. http://arXiv:physics/ 0601074 Sacker RJ, Sell GR (1974, 1976, 1978) Existence of dichotomies and invariant splitting for linear differential systems. J Diff Equ 15:429-458, 22:478–522, 27:106–137 Scarpellini B (1991) Center manifolds of infinite dimensional. Main results and applications. J Appl Math Phys (ZAMP) 43: 1–32 Sell GR (1983) Vector fields on the vicinity of a compact invariant manifold. Lect Notes Math 1017:568–574 Swanson R (1983) The spectral characterization of normal hyperbolicity. Proc Am Math Soc 89(3):503–508 Temam R (1988) Infinite Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin Zhenquan Li, Roberts AJ (2000) A flexible error estimate for the application of center manifold theory. http://arXiv.org/abs/math. DS/0002138 Zhu H, Campbell SA, Wolkowicz (2002) Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM J Appl Math 63:636–682

Chaos and Ergodic Theory

Chaos and Ergodic Theory JÉRÔME BUZZI C.N.R.S. and Université Paris-Sud, Orsay, France

Article Outline Glossary Definition of the Subject Introduction Picking an Invariant Probability Measure Tractable Chaotic Dynamics Statistical Properties Orbit Complexity Stability Untreated Topics Future Directions Acknowledgments Bibliography

Glossary For simplicity, definitions are given for a continuous or smooth self-map or diffeomorphism T of a compact manifold M. Entropy, measure-theoretic (or: metric entropy) For an ergodic invariant probability measure , it is the smallest exponential growth rates of the number of orbit segments of given length, with respect to that length, after restriction to a set of positive measure. We denote it by h(T; ). See  Entropy in Ergodic Theory and Subsect. “Local Complexity” below. Entropy, topological It is the exponential growth rates of the number of orbit segments of given length, with respect to that length. We denote it by htop ( f ). See  Entropy in Ergodic Theory and Subsect. “Local Complexity” below. Ergodicity A measure is ergodic with respect to a map T if given any measurable subset S which is invariant, i. e., such that T 1 S D S, either S or its complement has zero measure. Hyperbolicity A measure is hyperbolic in the sense of Pesin if at almost every point no Lyapunov exponent is zero. See  Smooth Ergodic Theory. Kolmogorov typicality A property is typical in the sense of Kolmogorov for a topological space F of parametrized families f D ( f t ) t2U , U being an open subset of Rd for some d  1, if it holds for f t for Lebesgue almost every t and topologically generic f 2 F .

Lyapunov exponents The Lyapunov exponents ( Smooth Ergodic Theory) are the limits, when they exist, lim n!1 n1 log k(T n )0 (x):vk where x 2 M and v is a non zero tangent vector to M at x. The Lyapunov exponents of an ergodic measure is the set of Lyapunov exponents obtained at almost every point with respect to that measure for all non-zero tangent vectors. Markov shift (topological, countable state) It is the set of all infinite or bi-infinite paths on some countable directed graph endowed with the left-shift, which just translates these sequences. Maximum entropy measure It is a measure  which maximizes the measured entropy and, by the variational principle, realized the topological entropy. Physical measure It is a measure  whose basin, P fx 2 M : 8 : M ! R continuous limn!1 n1 n1 kD0 R ( f k x) D  dg has nonzero volume. Prevalence A property is prevalent in some complete metric, separable vector space X if it holds outside of a set N such that, for some Borel probability measure  on X: (A C v) D 0 for all v 2 X. See [76,141,239]. Sensitivity on initial conditions T has sensitivity to initial conditions on X 0 X if there exists a constant

> 0 such that for every x 2 X 0 , there exists y 2 X, arbitrarily close to x, and n  0 such that d(T n y; T n x) > . Sinai–Ruelle–Bowen measures It is an invariant probability measure which is absolutely continuous along the unstable foliation (defined using the unstable manifolds of almost every x 2 M, which are the sets W u (x), of points y such that limn!1 n1 log d(T n y; T n x) < 0). Statistical stability T is statistically stable if the physical measures of nearby deterministic systems are arbitrarily close to the convex hull of the physical measures of T. Stochastic stability T is stochastically stable if the invariant measures of the Markov chains obtained from T by adding a suitable, smooth noise with size ! 0 are arbitrarily close to the convex hull of the physical measures of T. Structural stability T is structurally stable if any S close enough to T is topologically the same as T: there exists a homeomorphism h : M ! M such that hı T D S ı h (orbits are sent to orbits). Subshift of finite type It is a closed subset ˙F of ˙ D AZ or ˙ D AN where A is a finite set satisfying: ˙F D fx 2 ˙ : 8k < ` : x k x kC1 : : : x` … Fg for some finite set F.

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Topological genericity Let X be a Baire space, e. g., a complete metric space. A property is (topologically) generic in a space X (or holds for the (topologically) generic element of X) if it holds on a nonmeager set (or set of second Baire category), i. e., on a dense Gı subset.

T : X ! X on a compact metric space whose distance is denoted by d: T has sensitivity to initial conditions on X 0 X if there exists a constant > 0 such that for every x 2 X 0 , there exists y 2 X, arbitrarily close to x with a finite separating time: 9n  0 such that d(T n y; T n x) > :

Definition of the Subject Chaotic dynamical systems are those which present unpredictable and/or complex behaviors. The existence and importance of such systems has been known at least since Hadamard [126] and Poincaré [208], however it became well-known only in the sixties. We refer to [36,128,226,236] and [80,107,120,125,192,213] for the relevance of such dynamics in other fields, mathematical or not (see also  Ergodic Theory: Interactions with Combinatorics and Number Theory,  Ergodic Theory: Fractal Geometry). The numerical simulations of chaotic dynamics can be difficult to interpret and to plan, even misleading, and the tools and ideas of mathematical dynamical system theory are indispensable. Arguably the most powerful set of such tools is ergodic theory which provides a statistical description of the dynamics by attaching relevant probability measures. In opposition to single orbits, the statistical properties of chaotic systems often have good stability properties. In many cases, this allows an understanding of the complexity of the dynamical system and even precise and quantitative statistical predictions of its behavior. In fact, chaotic behavior of single orbits often yields global stability properties. Introduction The word chaos, from the ancient Greek ˛o, “shapeless void” [131] and “raw confused mass” [199], has been used [˛o also inspired Van Helmont to create the word “gas” in the seventeenth century and this other thread leads to the molecular chaos of Boltzmann in the nineteenth century and therefore to ergodic theory itself .] since a celebrated paper of Li and Yorke [169] to describe evolutions which however deterministic and defined by rather simple rules, exhibit unpredictable or complex behavior. Attempts at Definition We note that, like many ideas [237], this is not captured by a single mathematical definition, despite several attempts (see, e. g., [39,112,158,225] for some discussions as well as the monographs on chaotic dynamics [15,16,58,60,78,104, 121,127,215,250,261]). Let us give some of the most wellknown definitions, which have been given mostly from the topological point of view, i. e., in the setting of a self-map

In other words, any uncertainty on the exact value of the initial condition x makes T n (x) completely unknown for n large enough. If X is a manifold, then sensitivity to initial conditions in the sense of Guckenheimer [120] means that the previous phenomenon occurs for a set X 0 with nonzero volume. T is chaotic in the sense of Devaney [94] if it admits a dense orbit and if the periodic points are dense in X. It implies sensitivity to initial conditions on X. T is chaotic in the sense of Li and Yorke [169] if there exists an uncountable subset X 0 X of points, such that, for all x ¤ y 2 X 0 , lim inf d(T n x; T n y) D 0 and n!1

lim sup d(T n x; T n y) > 0 : n!1

T has generic chaos in the sense of Lasota [206] if the set f(x; y) 2 X  X : lim inf d(T n x; T n y) D 0 n!1

< lim sup d(T n x; T n y)g n!1

is topologically generic (see glossary.) in X  X. Topological chaos is also sometimes characterized by nonzero topological entropy ( Entropy in Ergodic Theory): there exist exponentially many orbit segments of a given length. This implies chaos in the sense of Li and Yorke by [39]. As we shall see ergodic theory describes a number of chaotic properties, many of them implying some or all of the above topological ones. The main such property for a smooth dynamical system, say a C 1C˛ -diffeomorphism of a compact manifold, is the existence of an invariant probability measure which is: 1. Ergodic (cannot be split) and aperiodic (not carried by a periodic orbit); 2. Hyperbolic (nearby orbits converge or diverge at a definite exponential rate); 3. Sinai–Ruelle–Bowen (as smooth as it is possible). (For precise definitions we refer to  Smooth Ergodic Theory or to the discussions below.) In particular such

Chaos and Ergodic Theory

a situation implies nonzero entropy and sensitivity to initial condition of a set of nonzero Lebesgue measure (i. e., positive volume). Before starting our survey in earnest, we shall describe an elementary and classical example, the full tent map, on which the basic phenomena can be analyzed in a very elementary way. Then, in Sect. “Picking an Invariant Probability Measure”, we shall give some motivations for introducing probability theory in the description of chaotic but deterministic systems, in particular the unpredictability of their individual orbits. We define two of the most relevant classes of invariant measures: the physical measures and those maximizing entropy. It is unknown in which generality these measures exist and can be analyzed but we describe in Sect. “Tractable Chaotic Dynamics” the major classes of dynamics for which this has been done. In Sect. “Statistical Properties” we describe some of the finer statistical properties that have been obtained for such good chaotic systems: sums of observables along orbits are statistically undistinguishable from sums of independent and identically distributed random variables. Sect. “Orbit Complexity” is devoted to the other side of chaos: the complexity of these dynamics and how, again, this complexity can be analyzed, and sometimes classified, using ergodic theory. Sect “Stability” describes perhaps the most striking aspect of chaotic dynamics: the unstability of individual orbit is linked to various forms of stability of the global dynamics. Finally we conclude by mentioning some of the most important topics that we could not address and we list some possible future directions. Caveat. The subject-matter of this article is somewhat fuzzy and we have taken advantage of this to steer our path towards some of our favorite theorems and to avoid the parts we know less (some of which are listed below). We make no pretense at exhaustivity neither in the topics nor in the selected results and we hope that our colleagues will excuse our shortcomings. Remark 1 In this article we only consider compact, smooth and finite-dimensional dynamical systems in discrete time, i. e., defined by self-maps. In particular, we have omitted the natural and important variants applying to flows, e. g., evolutions defined by ordinary differential equations but we refer to the textbooks (see, e. g.,[15,128,148]) for these. Elementary Chaos: A Simple Example We start with a toy model: the full tent map T of Fig. 1. Observe that for any point x 2 [0; 1], T n (x) D f((k; n)x C k)2n : k D 0; 1; : : : ; 2n 1g, where (k; n) D ˙1.

Chaos and Ergodic Theory, Figure 1 The graph of the full tent map f(x) D 1  j1  2xj over [0; 1]

S Hence n0 T n (x) is dense in [0; 1]. It easily follows that T exhibits sensitive dependence to initial conditions. Even worse in this example, the qualitative asymptotic behavior can be completely changed by this arbitrarily small perturbation: x may have a dense orbit whereas y is eventually mapped to a fixed point! This is Devaney chaos [94]. This kind of unstability was first discovered by J. Hadamard [126] in his study of the geodesic flow (i. e., the frictionless movement of a point mass constrained to remain on a surface). At that time, such an unpredictability was considered a purely mathematical pathology, necessarily devoid of any physical meaning [Duhem qualified Hadamard’s result as “an example of a mathematical deduction which can never be used by physics” (see pp. 206–211 in [103])!]. Returning to out tent map, we can be more quantitative. At any point x 2 [0; 1] whose orbit never visits 1/2, the Lyapunov exponent limn!1 n1 log j(T n )0 (x)j is log 2. (See the glossary.) Such a positive Lyapunov exponent corresponds to infinitesimally close orbits getting separated exponentially fast. This can be observed in Fig. 2. Note how this exponential speed creates a rather sharp transition. It follows in particular that experimental or numerical errors can grow very quickly to size 1, [For simple precision arithmetic the uncertainty is 1016 which grows to size 1 in 38 iterations of T.] i. e., the approximate orbit may contain after a while no information about the true orbit. This casts a doubt on the reliability of simulations. Indeed, a simulation of T on most computers will suggest that all orbits quickly converge to 0, which is completely false [Such a collapse to 0 does really occurs but only for a countable

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Chaos and Ergodic Theory, Figure 2 jT n (x)  T n (y)j for T(x) D 1  j1  2xj, jx  yj D 1012 and 0  n  100. The vertical red line is at n D 28 and shows when jT n x  T n yj  0:5  104 for the first time

subset of initial conditions in [0; 1] whereas the points with dense orbit make a subset of [0; 1] with full Lebesgue measure (see below). This artefact comes from the way numbers are represented – and approximated – on the computer: multiplication by even integers tends to “simplify” binary representations. Thus the computations involved in drawing Fig. 2 cannot be performed too naively.]. Though somewhat atypical in its dramatic character, this failure illustrates the unpredictability and unstability of individual orbits in chaotic systems. Does this mean that all quantitative predictions about orbits of T are to be forfeited? Not at all, if we are ready to change our point of view and look beyond a single orbit. This can be seen easily in this case. Let us start with such a global analysis from the topological point of view. Associate to x 2 [0; 1], a sequence i(x) D i D i0 i1 i2 : : : of 0s and 1s according to i k D 0 if T k x  1/2, i k D 1 otherwise. One can check that [Up to a countable set of exceptions.] fi(x) : x 2 [0; 1]g is the set ˙2 :D f0; 1gN of all infinite sequences of 0s and 1s and that at most one x 2 [0; 1] can realize a given sequence as i(x). Notice how the transformation f becomes trivial in this representation:

This can be a very powerful tool. Observe for instance how here it makes obvious that we have complete combinatorial freedom over the orbits of T: one can easily build orbits with various asymptotic behaviors: if a sequence of ˙ 2 contains all the finite sequences of 0s and 1s, then the corresponding point has a dense orbit; if the sequence is periodic, then the corresponding point is itself periodic, to give two examples of the richness of the dynamics. More quantitatively, the number of distinct subsequences of length n appearing in sequences i(x), x 2 [0; 1], is 2n . It follows that the topological entropy ( Entropy in Ergodic Theory) of T is htop (T) D log 2. [For the coincidence of the entropy and the Lyapunov exponent see below.] The positivity of the topological entropy can be considered as the signature of the complexity of the dynamics and considered as the definition, or at least the stamp, of a topologically chaotic dynamics. Let us move on to a probabilistic point of view. Pick x 2 [0; 1] randomly according to, say, the uniform law in [0; 1]. It is then routine to check that i(x) follows the (1/2; 1/2)-Bernoulli law: the probability that, for any given k, i k (x) D 0 is 1/2 and the ik s are independent. Thus i(x), seen as a sequence of random 0 and 1 when x is subject to the uniform law on [0; 1], is statistically undistinguishable from coin tossing! This important remark leads to quantitative predictions. For instance, the strong law of large numbers implies that, for Lebesgue-almost every x 2 [0; 1] (i. e., for all x 2 [0; 1] except in a set of Lebesgue measure zero), the fraction of the time spent in any dyadic interval I D [k  2N ; `  2N ] [0; 1], k; `; N 2 N, by the orbit of x, lim

n!1

1 #f0  k < n : T k x 2 Ig n

(1)

exists and is equal to the length, 2N , of that interval. [Eq. (1) in fact holds for any interval I. This implies that the orbit of almost every x 2 [0; 1] visits all subintervals of [0; 1], i. e., the orbit is dense: in complete contradiction with the above mentioned numerical simulation!] More generally, we shall see that, if  : [0; 1] ! R is any continuous function, then, for Lebesgue almost every-x, Z n1 1X k (T x) exists and is equal to (x) dx : lim n!1 n kD0

i( f (x)) D i1 i2 i3 : : :

if i(x) D i0 i1 i2 i3 : : :

Thus f is represented by the simple and universal “leftshift” on sequences, which is denoted by . This representation of a rather general dynamical system by the leftshift on a space of sequences is called symbolic dynamics ( Symbolic Dynamics), [171].

(2) Using strong mixing properties ( Ergodicity and Mixing Properties) of the Lebesgue measure under T, one can prove further properties, e. g., sensitivity on initial conditions in the sense of Guckenheimer [The Lebesgue measure is weak-mixing: Lebesgue-almost all couples of points (x; y) 2 [0; 1]2 get separated. Note that it is not true

Chaos and Ergodic Theory

of every couple off the diagonal: Counter-examples can be found among couples (x; y) with T n x D T n y arbitrarily close to 1.] and study the fluctuations of the averages 1 Pn1 kD0 (Tx) by the way of limit theorems. n The above analysis relied on the very special structure of T but, as we shall explain, the ergodic theory of differentiable dynamical systems shows that all of the above (and much more) holds in some form for rather general classes of chaotic systems. The different chaotic properties are independent in general (e. g., one may have topological chaos whereas the asymptotic behavior of almost all orbits is periodic) and the proofs can become much more difficult. We shall nonetheless be rewarded for our efforts by the discovery of unexpected links between chaos and stability, complexity and simplicity as we shall see. Picking an Invariant Probability Measure One could think that dynamical systems, such as those defined by self-maps of manifolds, being completely deterministic, have nothing to do with probability theory. There are in fact several motivations for introducing various invariant probability measures. Statistical Descriptions An abstract goal might be to enrich the structure: a smooth self-map is a particular case of a Borel self-map, hence one can canonically attach to this map its set of all invariant Borel probability measures [From now on all measures will be Borel probability measures except if it is explicitly stated otherwise.], or just the set of ergodic [A measure is ergodic if all measurable invariant subsets have measure 0 or 1. Note that arbitrary invariant measures are averages of ergodic ones, so many questions about invariant measures can be reduced to ergodic ones.] ones. By the Krylov–Bogoliubov theorem (see, e. g., [148]), this set is non-empty for any continuous self-map of a compact space. By the following fundamental theorem  Ergodic Theorems, each such measure is the statistical description of some orbit: Birkhoff Pointwise Ergodic Theorem Let (X; F ; ) be a space with a -field and a probability measure. Let f : X ! X be a measure-preserving map, i. e., f 1 (F )

F and  ı f 1 D . Assume ergodicity of  (See the glossary.) and (absolute) integrability of  : X ! R with respect to . Then for -almost every x 2 X, n1 1X lim ( f k x) exists and is n!1 n kD0

Z  d :

This theorem can be interpreted as saying that “time averages” coincide almost surely with “ensemble averages” (or “phase space average”), i. e., that Boltzmann’s Ergodic Hypothesis of statistical mechanics [110] holds for dynamical systems that cannot be split in a measurable and non trivial way. [This indecomposability is however often difficult to establish. For instance, for the hard ball model of a gas it is known only under some generic assumption (see [238] and the references therein).] We refer to [161] for background. Remark 2 One should observe that the existence of the above limit is not at all obvious. In fact it often fails from other points of view. One can show that for the full tent map T(x) D 1  j1  2xj analyzed above and many functions , the set of points for which it fails is large both from the topological point of view (it contains a dense Gı set) and from the dimension point of view (it has Hausdorff dimension 1 [28]). This is an important point: the introduction of invariant measures allows one to avoid some of the wilder pathologies. To illustrate this let us consider the full tent map T(x) D 1  j1  2xj again and the two ergodic invariant measures: ı 0 (the Dirac measure concentrated at the fixed point 0) and the Lebesgue measure dx. In the first case, we obtain a complex proof of the obvious fact that the time average at x D 0 (some set of full measure!) and the ensemble average with respect to ı 0 are both equal to f (0). In the second case, we obtain a very general proof of the above Eq. (2). Another type of example is provided by the contracting map S : [0; 1] ! [0; 1], S(x) D x/2. S has a unique invariant probability measure, ı 0 . For Birkhoff theorem the situation is the same as that of T and ı 0 : it asserts only that the orbit of 0 is described by ı 0 . One can understand Birkhoff theorem as a (first and rather weak) stability result: the time averages are independent of the initial condition, almost surely with respect to . Physical Measures In the above silly example S, much more is true than the conclusion of Birkhoff Theorem: all points of [0; 1] are described by ı 0 . This leads to the definition of the basin of a probability measure  for a self-map f of a space M: ( B() :D

x 2 M : 8 :

) Z n1 1X k ( f x) D  d : M ! R continuous lim n!1 n kD0

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If M is a manifold, then there is a notion of volume and one can make the following definition. A physical measure is a probability measure whose basin has nonzero volume in M. Say that a dynamical system f : M ! M on a manifold has a finite statistical description if there exists finitely many invariant probability measures 1 ; : : : ; n the union of whose basins is the whole of M, up to a set of zero Lebesgue measure. Physical measures are among the main subject of interest as they are expected to be exactly those that are “experimentally visible”. Indeed, if x0 2 B() and 0 > 0 is small enough, then, by Lebesgue density theorem, a point x picked according to, say, the uniform law in the ball B(x0 ; 0 ) of center x0 and radius 0 , will be in B() with probability almost 1 and therefore its ergodic averages will be described by . Hence “experiments” can be expected to follow the physical measures and this is what is numerically observed in most of the situations (see however the caveat in the discussion of the full tent map). The existence of a finite statistical description (or even of a physical measure) is, as we shall see, not automatic nor routine to prove. Attracting periodic points as in the above silly example provide a first type of physical measures. Birkhoff ergodic theorem asserts that absolutely continuous ergodic invariant measures, usually obtained from some expansion property, give another class of physical measures. These contracting and expanding types can be combined in the class of Sinai–Ruelle–Bowen measures [166] which are the invariant measures absolutely continuous “along expanding directions” (see for the precise but technical definition  Smooth Ergodic Theory). Any ergodic Sinai–Ruelle–Bowen measure which is ergodic and without zero Lyapunov exponent [That is, the set of points x 2 M such that lim n!1 n1 log k( f n )0 (x):vk D 0 for some v 2 Tx M has zero measure.] is a physical measure. Conversely, “most” physical measures [For counter-examples see [136].] are of this type [243,247]. Measures of Maximum Entropy For all parameters t 2 [3:96; 4], the quadratic maps Q t (x) D tx(1  x), Q t : [0; 1] ! [0; 1], have nonzero topological entropy [91] and exponentially many periodic points [134]: lim

n!1

#fx 2 [0; 1] : Q tn (x) D xg e n htop (Q t )

D1:

On the other hand, by a deep theorem [117,178] there is an open and dense subset of t 2 [0; 4], such that Qt has a unique physical measure concentrated on a periodic or-

bit! Thus the physical measures can completely miss the topological complexity (and in particular the distribution of the periodic points). Hence one must look at other measures to get a statistical description of the complexity of such Qt . Such a description is often provided by measures of maximum entropy  M whose measured entropy [The usual phrases are “measure-theoretic entropy”, “metric entropy”.] ( Entropy in Ergodic Theory) satisfies: h( f ;  M ) D sup h( f ; ) D1 htop ( f ) : 2M( f )

M( f ) is the set of all invariant measures. [One can restrict this to the ergodic invariant measures without changing the value of the supremum ( Entropy in Ergodic Theory).] Equality 1 above is the variational principle: it holds for all continuous self-maps of compact metric spaces. One can say that the ergodic complexity (the complexity of f as seen by its invariant measures) captures the full topological complexity (defined by counting all orbits). Remark 3 The variational principle implies the existence of “complicated invariant measures” as soon as the topological entropy is nonzero (see [47] for a setting in which this is of interest). Maximum entropy measures do not always exist. However, if f is C 1 smooth, then maximum entropy measures exist by a theorem of Newhouse [195] and they indeed describe the topological complexity in the following sense. Consider the probability measures: n; :D

n1 1X n

X

ıf kx

kD0 x2E(n; )

where E(n; ) is an arbitrary ( ; n)-separated subset [See  Entropy in Ergodic Theory: 8x; y 2 E(n; ) x ¤ y H) 90  k < n d(T k x; T k y)  .] of M with maximum cardinality. Then accumulation points for the weak star topology on the space of probability measures on M of n; when n ! 1 and then ! 0 are maximum entropy measures [185]. Let us quote two important additional properties, discovered by Margulis [179], that often hold for the maximum entropy measures:  The equidistribution of periodic points with respect to some maximum entropy measure  M : X 1 ıx :  M D lim n!1 #fx 2 X : x D f n xg n xD f x

 The holonomy invariance which can be loosely interpreted by saying that the past and the future are independent conditionally on the present.

Chaos and Ergodic Theory

Other Points of View Many other invariant measures are of interest in various contexts and we have made no attempt at completeness: for instance, invariant measures maximizing dimension [111,204], or pressure in the sense of the thermodynamical formalism [148,222], or some energy [9,81,146], or quasi-physical measures describing the dynamics around saddle-type invariant sets [104] or in systems with holes [75]. Tractable Chaotic Dynamics The Palis Conjecture There is, at this point, no general theory allowing the analysis of all dynamical systems or even of most of them despite many recent and exciting developments in the theory of generic C1 -diffeomorphisms [51,84]. In particular, the question of the generality in which physical measures exist remains open. One would like generic systems to have a finite statistical description (see Subsect. “Physical Measures”). This fails in some examples but these look exceptional and the following question is asked by Palis [200]: Is it true that any dynamical system defined by a Cr diffeomorphism on a compact manifold can be transformed by an arbitrarily small Cr -perturbation to another dynamical system having a finite statistical description? This is completely open though widely believed [Observe, however, that such a statement is false for conservative diffeomorphisms with high order smoothness as KAM theory implies stable existence of invariant tori foliating a subset of positive volume.]. Note that such a good description is not possible for all systems (see, e. g., [136,194]). Note that one would really like to ask about unperturbed “typical” [The choice of the notion of typicality is a delicate issue. The Newhouse phenomenon shows that among C2 -diffeomorphisms of multidimensional compact manifolds, one cannot use topological genericity and get a positive answer. Popular notions are prevalence and Kolmogorov genericity – see the glossary.] dynamical systems in a suitable sense, but of course this is even harder. One is therefore led to make simplifying assumptions: typically of small dimension, uniform expansion/ contraction or geometry. Uniformly Expanding/Hyperbolic Systems The most easily analyzed systems are those with uniform expansion and/or contraction, namely the uniformly

expanding maps and uniformly hyperbolic diffeomorphisms, see  Smooth Ergodic Theory. [We require uniform hyperbolicity on the so-called chain recurrent set. This is equivalent to the usual Axiom-A and no-cycle condition.] An important class of example is obtained as follows. Consider A: Rd ! Rd , a linear map preserving Zd (i. e., A is a matrix with integer coefficients in the canonical basis) so ¯ T d ! T d on the torus. If there is that it defines a map A: a constant  > 1 such that for all v 2 Rd , kA:vk  kvk then A¯ is a uniformly expanding map. If A has determinant ˙1 and no eigenvalue on the unit circle, then A¯ is a uniformly hyperbolic diffeomorphism ( Smooth Ergodic Theory) (see also [60,148,215,233]). Moreover all C1 -perturbations of the previous examples are again uniformly expanding or uniformly hyperbolic. [One can define uniform hyperbolicity for flows and an important class of examples is provided by the geodesic flow on compact manifolds with negative sectional curvature [148].] These uniform systems are sometimes called “strongly chaotic”. Remark 4 Mañé Stability Theorem (see below) shows that uniform hyperbolicity is a very natural notion. One can also understand on a more technical level uniform hyperbolicity as what is needed to apply an implicit function theorem in some functional space (see, e. g., [233]). The existence of a finite statistical description for such systems has been proved since the 1970s by Bowen, Ruelle and Sinai [54,218,235] (the expanding case is much simpler [162]). Theorem 1 Let f : M ! M be a C 1C˛ map of a compact manifold. Assume f to be (i) a uniformly expanding map on M or (ii) a uniformly hyperbolic diffeomorphism.  f admits a finite statistical description by ergodic and hyperbolic Sinai–Ruelle–Bowen measures (absolutely continuous in case (i)).  f has finitely many ergodic maximum entropy measures, each of which makes f isomorphic to a finite state Markov chain. The periodic points are uniformly distributed according to some canonical average of these ergodic maximum entropy measures.  f is topologically conjugate [Up to some negligible subset.] to a subshift of finite type (See the glossary.) The construction of absolutely continuous invariant measures for a uniformly expanding map f can be done in a rather direct way by considering the pushed forP k ward measures n1 n1 kD0 f Leb and taking weak star limits while preventing the appearance of singularities, by, e. g., bounding some Hölder norm of the density using expansion and distortion of f .

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The classical approach to the uniformly hyperbolic dynamics [52,222,233] is through symbolic dynamics and coding. Under the above hypothesis one can build a finite partition of M which is tailored to the dynamics (a Markov partition) so that the corresponding symbolic dynamics has a very simple structure: it is a full shift f1; : : : ; dgZ , like in the example of the full tent map, or a subshifts of finite type. The above problems can then be solved using the thermodynamical formalism inspired from the statistical mechanics of one-dimensional ferromagnets [217]: ergodic properties are obtained through the spectral properties of a suitable transfer operator acting on some space of regular functions, e. g., the Hölder-continuous functions defined over the symbolic dynamics with respect to the P n distance d(x; y) :D n2Z 2 1 x n ¤y n where 1s¤t is 1 if s ¤ t, 0 otherwise. A recent development [24,43,116] has been to find suitable Banach spaces to apply the transfer operator technique directly in the smooth setting, which not only avoids the complication of coding (or rather replace them with functional analytic preliminaries) but allows the use of the smoothness beyond Hölder-continuity which is important for finer ergodic properties. Uniform expansion or hyperbolicity can easily be obstructed in a given system: a “bad” point (a critical point or a periodic point with multiplier with an eigenvalue on the unit circle) is enough. This leads to the study of other systems and has motivated many works devoted to relaxing the uniform hyperbolicity assumptions [51]. Pesin Theory The most general such approach is Pesin theory. Let f be a C 1C˛ -diffeomorphism [It is an important open problem to determine to which extent Pesin theory could be generalized to the C1 setting.] f with an ergodic invariant measure . By Oseledets Theorem  Smooth Ergodic Theory, for almost every x with respect to any invariant measure, the behavior of the differential Tx f n for n large is described by the Lyapunov exponents 1 ; : : : ; d , at x. Pesin is able to build charts around almost every orbit in which this asymptotic linear behavior describes that of f at the first iteration. That is, there are diffeomorphisms ˚x : U x M ! Vx Rd with a “reasonable dependence on x” such that the differential of ˚ f n x ı f n ı ˚x1 at any point where it is defined is close to a diagonal matrix with entries (e(1 ˙ )n ; e(2 ˙ )n ; : : : ; e(d ˙ )n ). In this full generality, one already obtains significant results:  The entropy is bounded by the expansion: h( f ; )  Pd C iD1  i () [219]

 At almost every point x, there are strong stable resp. unstable manifolds W ss (x), resp. W uu (x), coinciding with the sets of points y such that d(T n x; T n y) ! 0 exponentially fast when n ! 1, resp. n ! 1. The corresponding holonomies are absolutely continuous (see, e. g., [59]) like in the uniform case. This allows Ledrappier’s definition of Sinai–Ruelle–Bowen measures [166] in that setting.  Equality in the above formula holds if and only if  is a Sinai–Ruelle–Bowen measure [167]. More generP ally the entropy can be computed as diD1 i ()C i () where the i are some fractal dimensions related to the exponents. Under the only assumption of hyperbolicity (i. e., no zero Lyapunov exponent almost everywhere), one gets further properties:  Existence of an hyperbolic measure which is not periodic forces htop ( f ) > 0 [However h( f ; ) can be zero.] by [147].   is exact dimensional [29,256]: the limit limr!0 log (B(x; r))/ log r exists -almost everywhere and is equal to the Hausdorff dimension of  (the infimum of the Hausdorff dimension of the sets with full -measure). This is deduced from a more technical “asymptotic product structure” property of any such measure. For hyperbolic Sinai–Ruelle–Bowen measures , one can then prove, e. g.,:  local ergodicity [202]:  has at most countably many ergodic components and -almost every point has a neighborhood whose Lebesgue-almost every point are contained in the basin of an ergodic component of .  Bernoulli [198]: Each ergodic component of  is conjugate in a measure-preserving way, up to a period, to a Bernoulli shift, that is, a full shift f1; : : : ; NgZ equipped with a product measure. This in particular implies mixing and sensitivity on initial conditions for a set of positive Lebesgue measure. However, establishing even such a weak form of hyperbolicity is rather difficult. The fragility of this condition can be illustrated by the result [44,45] that the topologically generic area-preserving surface C1 -diffeomorphism is either uniformly hyperbolic or has Lebesgue almost everywhere vanishing Lyapunov exponents, hence is never non-uniformly hyperbolic! (but this is believed to be very specific to the very weak C1 topology). Moreover, such weak hyperbolicity is not enough, with the current techniques, to build Sinai–Ruelle–Bowen measures or ana-

Chaos and Ergodic Theory

lyze maximum entropy measures only assuming non-zero Lyapunov exponents. Let us though quote two conjectures. The first one is from [251] [We slightly strengthened Viana’s statement for expository reasons.]: Conjecture 1 Let f be a C 1C -diffeomorphism of a compact manifold. If Lebesgue-almost every point x has welldefined Lyapunov exponents in every direction and none of these exponents is zero, then there exists an absolutely continuous invariant -finite positive measure. The analogue of this conjecture has been proved for C3 interval maps with unique critical point and negative Schwarzian derivative by Keller [150], but only partial results are available for diffeomorphisms [168]. We turn to measures of maximum entropy. As we said, C 1 smoothness is enough to ensure their existence but this is through a functional-analytic argument (allowed by Yomdin theory [254]) which says nothing about their structure. Indeed, the following problem is open: Conjecture 2 Let f be a C 1C -diffeomorphism of a compact surface. If the topological entropy of f is nonzero then f has at most countably many ergodic invariant measures maximizing entropy. The analogue of this conjecture has been proved for C 1C interval maps [64,66,70]. In the above setting a classical result of Katok shows the existence of uniformly hyperbolic compact invariant subsets with topological entropy arbitrarily close to that of f implying the existence of many periodic points: lim sup n!1

1 log #fx 2 M : f n (x) D xg  htop ( f ) : n

The previous conjecture would follow from the following one: Conjecture 3 Let f be a C 1C -diffeomorphism of a compact manifold. There exists an invariant subset X M, carrying all ergodic measures with maximum entropy, such that the restriction f jX is conjugate to a countable state topological Markov shift (See the glossary.). Systems with Discontinuities We now consider stronger assumptions to be able to build the relevant measures. The simplest step beyond uniformity is to allow discontinuities, considering piecewise expanding maps. The discontinuities break the rigidity of the uniformly expanding situation. For instance, their symbolic dynamics are usually no longer subshifts of finite type though they still retain some “simplicity” in good cases (see [68]).

To understand the problem in constructing the absolutely continuous invariant measures, it is instructive to consider the pushed forwards of a smooth measure. Expansion tends to keep the measure smooth whereas discontinuities may pile it up, creating non-absolute continuity in the limit. One thus has to check that expansion wins. In dimension 1, a simple fact resolves the argument: under a high enough iterate, one can make the expansion arbitrarily large everywhere, whereas a small interval can be chopped into at most two pieces. Lasota and Yorke [165] found a suitable framework. They considered C2 interval maps with j f 0 (x)j  const > 1 except at finitely many points. They used the Ruelle transfer operator directly on the interval. Namely they studied X (y) (L)(x) D jT 0 (y)j 1 y2T

x

acting on functions  : [0; 1] ! R with bounded variation and obtained the invariant density as the eigenfunction associated to the eigenvalue 1. One can then prove a Lasota– Yorke inequality (which might more accurately be called Doeblin–Fortet since it was introduced in the theory of Markov chains much earlier): kLkBV  ˛kkBV C ˇkk1

(3)

where k  kBV , k  k1 are a strong and a weak norm, respectively and ˛ < 1 and ˇ < 1. One can then apply general theorems [143] or [196] (see [21] for a detailed presentation of this approach and its variants). Here ˛ can essentially be taken as 2 (reflecting the locally simple discontinuities) divided by the minimum expansion: so ˛ < 1, perhaps after replacing T with an iterate. In particular, the existence of a finite statistical description then follows (see [61] for various generalizations and strengthenings of this result on the interval). The situation in higher dimension is more complex for the reason explained above. One can obtain inequalities such as (3) on suitable if less simple functional spaces (see, e. g., [231]) but proving ˛ < 1 is another matter: discontinuities can get arbitrarily complex under iteration. [67,241] show that indeed, in dimension 2 and higher, piecewise uniform expansion (with a finite number of pieces) is not enough to ensure a finite statistical description if the pieces of the map have only finite smoothness. In dimension 2, resp. 3 or more, piecewise real-analytic, resp. piecewise affine, is enough to exclude such examples [65,240], resp. [242]. [82] has shown that, for any r > 1, an open and dense subset of piecewise Cr and expanding maps have a finite statistical description.

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Piecewise hyperbolic diffeomorphisms are more difficult to analyze though several results (conditioned on technical assumptions that can be checked in many cases) are available [22,74,230,257]. Interval Maps with Critical Points A more natural but also more difficult situation is a map for which the uniformity of the expansion fails because of the existence of critical points. [Note that, by a theorem of Mañé a circle map without critical points or indifferent periodic point is either conjugate to a rotation or uniformly expanding [181].] A class which has been completely analyzed at the level of the above conjecture is that of real-analytic families of maps of the interval f t : [0; 1] ! [0; 1], t 2 I, with a unique critical point, the main example being the quadratic family Q t (x) D tx(1  x) for 0  t  4. It is not very difficult to find quadratic maps with the following two types of behavior: (stable) the orbit of Lebesgue-almost every x 2 [0; 1] tends to an attracting periodic orbit; (chaotic) there is an absolutely continuous invariant probability measure  whose basin contains Lebesguealmost every x 2 [0; 1]. To realize the first it is enough to arrange the critical point to be periodic. One can easily prove that this stable behavior occurs on an open set of parameters – thus it is stable with respect to the parameter or the dynamical system. p The second occurs for Q4 with  D dx/  x(1  x). It is much more difficult to show that this chaotic behavior occurs for a set of parameters of positive Lebesgue measure. This is a theorem of Jakobson [145] for the quadratic family (see for a recent variant [265]). Let us sketch two main ingredients of the various proofs of this theorem. The first is inducing: around Lebesgue-almost every point x 2 [0; 1] one tries to find a time (x) and an interval J(x) such that f (x) : J(x) ! f (x) (J(x)) is a map with good expansion and distortion properties. This powerful idea appears in many disguises in the non-uniform hyperbolic theory (see for instance [133,262]). The second ingredient is parameter exclusion: one removes the parameters at which a good inducing scheme cannot be built. More precisely one proceeds inductively, performing simultaneously the inducing and the exclusion, the good properties of the early stage of the inducing allowing one to control the measure of the parameters that need to be excluded to continue [30,145]. Indeed, the expansion established at a given stage allows to transfer estimates in the dynamical space to the parameter space. Using methods from complex analysis and renormalization theory one can go much further and prove the fol-

lowing difficult theorems (actually the product of the work of many people, including Avila, Graczyk, Kozlovski, Lyubich, de Melo, Moreira, Shen, van Strien, Swiatek), which in particular solves Palis conjecture in this setting: Theorem 2 [117,160,178] Stable maps (that is, such that Lebesgue almost every orbit converges to one of finitely many periodic orbits) form an open and dense set among Cr interval maps, for any r  2. [In fact this is even true for polynomials.]. The picture has been completed in the unimodal case (that is, with a unique critical point): Theorem 3 [19,20,117,159,178] Let f t : [0; 1] ! [0; 1], t 2 [t0 ; t1 ], be a real-analytic family of unimodal maps. Assume that it is not degenerate [ f t0 and f t1 are not conjugate]. Then:  The set of t such that f t is chaotic in the above sense has positive Lebesgue measure;  The set of t such that f t is stable is open and dense;  The remaining set of parameters has zero Lebesgue measure. [This set of “strange parameters” of zero Lebesgue measure has however positive Hausdorff dimension according to work of Avila and Moreira. In particular each of the following situations is realized on a set of parameters t of positive Hausdorff dimension: non-existence of the Birkhoff limit at Lebesgue-almost every point, the physical measure is ı p for p a repelling fixed point, the physical measure is non-ergodic.] We note that the theory underlying the above theorem yields much more results, including a very paradoxical rigidity of typical analytic families as above. See [19]. Non-uniform Expansion/Contraction Beyond the dimension 1, only partial results are available. The most general of those assume uniform contraction or expansion along some direction, restricting the non-uniform behavior to an invariant sub-bundle often one-dimensional, or “one-dimensional-like”. A first, simpler situation is when there is a dominated decomposition with a uniformly expanding term: there is a continuous and invariant splitting of the tangent bundle, T M D E uu ˚ E cs , over some  an attracting set: for all unit vectors v u 2 E uu , v c 2 E cs , k( f n )0 (x):v u k  Cn n 0

and

k( f ) (x):v k  C k( f n )0 (x):v u k : c

n

Standard techniques (pushing the Riemannian volume of a piece of unstable leaf and taking limits) allow the construction of Gibbs u-states as introduced by [205].

Chaos and Ergodic Theory

Theorem 4 (Alves–Bonatti–Viana [8]) [A slightly different result is obtained in [49].] Let f : M ! M be a C2 diffeomorphism with an invariant compact subset . Assume that there is a dominated splitting T M D E u ˚ E cs such that, for some c > 0, n1

lim sup n!1

Y 1 k f 0 ( f x )jE cs k  c < 0 log n kD0

on a subset of  of positive Lebesgue measure. Then this subset is contained, up to a set of zero Lebesgue measure, in the union of the basins of finitely many ergodic and hyperbolic Sinai – Ruelle – Bowen measures. The non-invertible, purely expansive version of the above theorem can be applied in particular to the following maps of the cylinder (d  16, a is properly chosen close to 2 and ˛ is small): f ( ; x) D (dx mod 2; a  x 2 C sin( )) which are natural examples of maps with multidimensional expansion and critical lines considered by Viana [249]. A series of works have shown that the above maps fit in the above non-uniformly expanding setting with a proper control of the critical set and hence can be thoroughly analyzed through variants of the above theorem [4,249] and the references in [5]. For a; b properly chosen close to 2 and small , the following maps should be even more natural examples: f (x; y) D (a  x 2 C y; b  y 2 C x) :

(4)

However the inexistence of a dominated splitting has prevented the analysis of their physical measures. See [66,70] for their maximum entropy measures. Cowieson and Young have used completely different techniques (thermodynamical formalism, Ledrappier– Young formula and Yomdin theory on entropy and smoothness) to prove the following result (see [13] for related work): Theorem 5 (Cowieson–Young [83]) Let f : M ! M be a C 1 diffeomorphism of a compact manifold. Assume that f admits an attractor  M on which the tangent bundle has an invariant continuous decomposition T M D E C ˚ E  such that all vectors of E C n f0g, resp. E  n f0g, have positive, resp. negative, Lyapunov exponents. Then any zero-noise limit measure  of f is a Sinai–Ruelle– Bowen measure and therefore, if it is ergodic and hyperbolic, a physical measure. One can hope, that typically, the latter ergodicity and hyperbolicity assumptions are satisfied (see, e. g., [27]).

By pushing classical techniques and introducing new ideas for generic maps with one expanding and one weakly contracting direction, Tsujii has been able to prove the following generic result (which can be viewed as a 2-dimensional extension of some of the one-dimensional results above: one adds a uniformly expanding direction): Theorem 6 (Tsujii [243]) Let M be a compact surface. Consider the space of C20 self-maps f : M ! M which admits directions that are uniformly expanded [More precisely, there exists a continuous, forward invariant cone field which is uniformly expanded under the differential of f .] Then existence of a finite statistical description is both topologically generic and prevalent in this space of maps. Hénon-Like Maps and Rank One Attractors In 1976, Hénon [130] observed that the diffeomorphism of the plane H a;b (x; y) D (1  ax 2 C y; bx) seemed to present a “strange attractor” for a D 1:4 and b D 0:3, that is, the points of a numerically simulated orbit seemed to draw a set looking locally like the product of a segment with a Cantor set. This attractor seems to be supported by the unstable manifold W u (P) of the hyperbolic fixed point P with positive absciss. On the other hand, the Plykin classification [207] excluded the existence of a uniformly hyperbolic attractor for a dissipative surface diffeomorphism. For almost twenty years the question of the existence of such an attractor (by opposition to an attracting periodic orbit with a very long period) remained open. Indeed, one knew since Newhouse that, for many such maps, there exist infinitely many such periodic orbits which are very difficult to distinguish numerically. But in 1991 Benedicks and Carleson succeeded in proposing an argument refining (with considerable difficulties) their earlier proof of Jakobson one-dimensional theorem and established the first part of the following theorem: Theorem 7 (Benedicks–Carleson [31]) For any > 0, for jbj small enough, there is a set A with Leb(A) > 0 satisfying: for all a 2 A, there exists z 2 W u (P) such that  The orbit of z is dense in W u (P);  lim infn!1 n1 log k( f n )0 (z)k > 0. Further properties were then established, especially by Benedicks, Viana, Wang, Young [32,34,35,264]. Let us quote the following theorem of Wang and Young which includes the previous results:

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Theorem 8 [264] Let Tab : S 1  [1; 1] ! S 1  [1; 1] be such that  Ta0 (S 1  [1; 1]) S 1  f0g;  For b > 0, Tab is a diffeomorphism on its image with c 1 b  j det Tab (x; y)j  c  b for some c > 1 and all (x; y) 2 S 1  [1; 1] and all (a; b). Let f a : S 1 ! S 1 be the restriction of Ta0 . Assume that f D f0 satisfies:  Non-degenerate critical points: f 0 (c) D 0 ) f 00(c) ¤ 0;  Negative Schwarzian derivative: for all x 2 S 1 non-critical, f 000 (x)/ f 0 (x)  3/2( f 00 (x)/ f 0 (x))2 < 0;  No indifferent or attracting periodic point, i. e., x such that f n (x) D x and j( f )0 (x)j  1;  Misiurewicz condition: d( f n c; d) > c > 0 for all n  1 and all critical points c; d. Assume the following transversality condition on f at a D 0: for every critical point c, ( d/da)( f a (c a )  p a ) ¤ 0 if ca is the critical point of f a near c and pa is the point having the same itinerary under f a as f (c) under c. Assume the following non-degeneracy of T : f00 (c) D 0 H) @T00 (c; 0)/@y ¤ 0.  Tab restricted to a neighborhood of S 1  f0g has a finite statistical description by a number of hyperbolic Sinai– Ruelle–Bowen measures bounded by the number of critical points of f ;  There is exponential decay of correlations and a Central Limit Theorem (see below) – except, in an obvious way, if there is a periodic interval with period > 1;  There is a natural coding of the orbits that remains for ever close to S 1 f0g by a closed invariant subset of a full shift. Very importantly, the above dynamical situation has been shown to occur near typical homoclinic tangencies: [190] proved that there is an open and dense subset of the set of all C3 families of diffeomorphisms unfolding a first homoclinic tangency such that the above holds. However [201] shows that the set of parameters with a Henon-like attractor has zero Lebesgue density at the bifurcation itself, at least under an assumption on the so-called stable and unstable Hausdorff dimensions. [95] establishes positive density for another type of bifurcation. Furthermore [191] has related the Hausdorff dimensions to the abundance of uniformly hyperbolic dynamics near the tangency. [248] is able to treat situations with more than one contracting direction. More recently [266] has proposed

a rather general framework, with easily checkable assumptions in order to establish the existence of such dynamics in various applications. See also [122,252] for applications. Statistical Properties The ergodic theorem asserts that time averages of integrable functions converge to phase space averages for any ergodic system. The speed of convergence is quite arbitrary in that generality [161] (only upcrossing inequalities seem to be available [38,132]), however many results are available under very natural hypothesis as we are going to explain in this section. The underlying idea is that for sufficiently chaotic dynamics T and reasonably smooth observables , the time averages A n (x) :D

n1 1X  ı T k (x) n kD0

should behave as averages of independent and identically distributed random variables and therefore satisfy the classical limit theorems of probability theory. The dynamical systems which are amenable to current technology are in a large part [But other approaches are possible. Let us quote the work [98] on partially hyperbolic systems, for instance.] those that can be reduced to the following type: Definition 1 Let T : X ! X be a nonsingular map on a probability, metric space (X; B; ; d) with bounded diameter, preserving the probability measure . This map is said to be Gibbs–Markov if there exists a countable (measurable) partition ˛ of X such that: 1. For all a 2 ˛, T is injective on a and T(a) is a union of elements of ˛. 2. There exists  > 1 such that, for all a 2 ˛, for all points x; y 2 a, d(Tx; Ty)  d(x; y). 3. Let Jac be the inverse of the Jacobian of T. There exists C > 0 such that, for all a 2 ˛, for all points x; y 2 a, j1  Jac(x)/Jac(y)j  Cd(Tx; Ty). 4. The map T has the “big image property”: inf a2˛ (Ta) > 0. Some piecewise expanding and C2 maps are obviously Gibbs–Markov but the real point is that many dynamics can be reduced to that class by the use of inducing and tower constructions as in [262], in particular. This includes possibly piecewise uniformly hyperbolic diffeomorphisms, Collet–Eckmann maps of the interval [21] (typical chaotic maps in the quadratic family), billiards with convex scatterers [262], the stadium billiard [71], Hénon-like mappings [266].

Chaos and Ergodic Theory

We note that in many cases one is led to first analyze mixing properties through decay of correlations, i. e., to prove inequalities of the type [21]: ˇZ ˇ ˇ  ˇ

ı T n d 

X

Z

ˇ ˇ dˇˇ  kkk kw a n

Z  d X

X

(5) where (a n )n1 is some sequence converging to zero, e. g., a n D en , 1/n˛ , . . . and k  k, k  kw a strong and a weak norm (e. g., the variation norm and the L1 norm). These rates of decay are often linked with return times statistics [263]. Rather general schemes have been developed to deduce various limit theorems such as those presented below from sufficiently quick decay of correlations (see notably [175] based on a dynamical variant of [113]). Probabilistic Limit Theorems The foremost limit property is the following: Definition 2 A class C of functions  : X ! R is said to satisfy the Central Limit Theorem if the following holds: for all  2 C , there is a number  D () > 0 such that: R

 lim 

n!1

x 2 X:



A n (x)   d t  n1/2 Z t dx 2 2 D ex /2 p 2 1

except for the degenerate case when (x) D (x) C const.

(6)

(Tx) 

The Central Limit Theorem can be seen in many cases as essentially a by-product of fast decay of correlations [175], P i. e., if n0 a n < 1 in the notations of Eq. (5). It has been established for Hölder-continuous observables for many systems together with their natural invariant measures including: uniformly hyperbolic attractors, piecewise expanding maps of the interval [174], Collet–Eckmann unimodal maps on the interval [152,260], piecewise hyperbolic maps [74], billiards with convex scatterers [238], Hénon-like maps [35]. Remark 5 The classical Central Limit Theorem holds for square-integrable random variables [193]. For maps exhibiting intermittency (e.g, interval maps like f (x) D x C x 1C˛ mod 1 with an indifferent fixed point at 0) the invariant density has singularities and the integrability condition is non longer automatic for smooth functions. One can then observe convergence to stable laws, instead of the normal law [114].

A natural question is the speed of the convergence in (6). The Berry–Esseen inequality: R ˇ  ˇ  Z t ˇ ˇ A n (x)   d x 2 /2 2 dx ˇ ˇ  t  e p ˇ 1/2 n 2 ˇ 1 C  ı/2 n for some ı > 0. It holds with ı D 1 in the classical, probabilistic setting. The Local Limit Theorem looks at a finer scale, asserting that for any finite interval [a; b], any t 2 R, n p p lim n x 2 X : nA n (x) 2 [a; b] C nt n!1

Z Cn

2 2 o et /2 :  d D jb  aj p 2

Both the Berry–Esseen inequality and the local limit theorem have been shown to hold for non-uniformly expanding maps [115] (also [62,216]). Almost Sure Results It is very natural to try and describe the statistical properties of the averages A n (x) for almost every x, instead of the weaker above statements in probability over x. An important such property is the almost sure invariance principle. It asks for the discrete random walk defined by the increments of  ı T n (x) to converge, after a suitable renormalization, to a Brownian motion. This has been proved for systems with various degrees of hyperbolicity [92,98,135,183]. Another one is the almost sure Central Limit Theorem. In the independent case (e. g., if X1 ; X2 ; : : : are independent and identically distributed random variables in L2 with zero average and unit variance), the almost sure Central Limit Theorem states that, almost surely: n 1 X1 ıPk1 p log n k jD0 X j / k kD1

converges in law to the normal distribution. This implies, that, almost surely, for any t 2 R: n 1 X1 n ı Pk1 p o n!1 log n k jD0 X j / kt kD1 Z t dx 2 2 D ex /2 p 2 1

lim

compare to (6).

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A general approach is developed in [73], covering Gibbs–Markov maps and those that can be reduced to it. They show in particular that the dynamical properties needed for the classical Central Limit Theorem in fact suffice to prove the above almost invariance principle and even the almost sure version of the Central Limit Theorem (using general probabilistic results, see [37,255]). Other Statistical Properties Essentially all the statistical properties of sums of independent identically distributed random variables can be established for tractable systems. Thus one can also prove large deviations [156,172,259], iterated law of the logarithm, etc. We note that the monograph [78] contains a nice introduction to the current work in this area. Orbit Complexity The orbit complexity of a dynamical system f : M ! M is measured by its topological and measured entropies. We refer to  Entropy in Ergodic Theory for detailed definitions. The Variational Principle Bowen–Dinaburg and Katok formulae can be interpreted as meaning that the topological entropy counts the number of arbitrary orbits whereas the measured entropy counts the number of orbits relevant for the given measure. In most situations, and in particular for continuous self-map of compact metric spaces, the following variational principle holds: htop ( f ) D sup h( f ; ) 2M( f )

where M( f ) is the set of all invariant probability measures. This is all the more striking in light of the fact that for many systems, the set of points which are typical from the point of view of ergodic theory [for instance, those x P k such that limn!1 n1 n1 kD0 (T x) exists for all continuous functions .] is topologically negligible [A subset of a countable union of closed sets with empty interior, that is, meager or first Baire category.]. Strict Inequality For a fixed invariant measure, one can only assert that h( f ; )  htop ( f ). One should be aware that this inequality may be strict even for a measure with full support. For instance, it is not difficult to check that the full tent map with htop ( f ) D log 2, admits ergodic invariant measures with full support and zero entropy.

There are also examples of dynamical systems preserving Lebesgue measure which have simultaneously positive topological entropy and zero entropy with respect to Lebesgue measure. That this occurs for C1 surface diffeomorphisms preserving area is a simple consequence of a theorem of Bochi [44] according to which, generically in the C1 topology, such a diffeomorphism is either uniformly hyperbolic or with Lyapunov exponents Lebesguealmost everywhere zero. [Indeed, it is easy to build such a diffeomorphism having both a uniformly hyperbolic compact invariant subset which will have robustly positive topological entropy and a non-degenerate elliptic fixed point which will prevent uniform hyperbolicity and therefore force all Lyapunov exponents to be zero. But Ruelle–Margulis inequality then implies that the entropy with respect to Lebesgue measure is zero.] Smooth examples also exist [46]. Remark 6 Algorithmic complexity [170] suggests another way to look at orbit complexity. One obtains in fact in this way another formula for the entropy. However this point of view becomes interesting in some settings, like extended systems defined by partial differential equations in unbounded space. Recently, [47] has used this approach to build interesting invariant measures. Orbit Complexity on the Set of Measures We have considered the entropies of each invariant measure separately, sharing only the common roof of topological entropy. One may ask how these different complexity sit together. A first answer is given by the following theorem in the symbolic and continuous setting. Let . Theorem 9 (Downarowicz–Serafin [102]) Let K be a Choquet simplex and H : K ! R be a convex function. Say that H is realized by a self-map f : X ! X and its set M( f ) of f -invariant probability measures equipped with the weak star topology if the following holds. There exists an affine homeomorphism  : M( f ) ! K such that, if h : M( f ) ! [0; 1] is the entropy function, H D h ı  . Then  H is realized by some continuous self-map of a compact space if and only if it is an increasing limit of upper semicontinuous and affine functions.  H is realized by some subshift on a finite alphabet, i. e., by the left shift on a closed invariant subset ˙ of f1; 2; : : : ; NgZ for some N < 1, if and only if it is upper semi-continuous Thus, in both the symbolic and continuous settings it is possible to have a unique invariant measure with any prescribed entropy. This stands in contrast to surface C 1C -

Chaos and Ergodic Theory

diffeomorphisms for which the set of the entropies of ergodic invariant measures is always the interval [0; htop ( f )] as a consequence of [147]. Local Complexity Recall that the topological entropy can be computed as: htop ( f ) D lim !0 htop ( f ; ) where: htop ( f ; ) :D lim

n!1

hloc ( f ) D lim sup h( f ; )  h( f ; ; ı) ı!0

1 log s(ı; n; X) n

x ¤ y H) 90  k < n d( f k x; f k y)  (see Bowen’s formula of the topological entropy  Entropy in Ergodic Theory). Likewise, the measure-theoretic entropy h(T; ) of an ergodic invariant probability measure  is lim !0 h( f ; ; ) where: 1 h(T; ) :D lim log r(ı; n; ) n!1 n where r(ı; n; ) is the minimum cardinality of C X such that ˚  x 2 X : 9y 2 C such that  80  k < n d( f k x; f k y) < > 1/2 : One can ask at which scales does entropy arise for a given dynamical system?, i. e., how the above quantities h(T; ), h(T; ; ) converge when ! 0. An answer is provided by the local entropy. [This quantity was introduced by Misiurewicz [186] under the name conditional topological entropy and is called tail entropy by Downarowicz [100].] For a continuous map f of a compact metric space X, it is defined as: hloc ( f ) :D lim hloc ( f ; ) with !0

hloc ( f ; ) :D sup hloc ( f ; ; x) and hloc ( f ; ; x) :D lim lim sup ı!0 n!1

 ˚ 1 log s ı; n; y 2 X : n

8k  0 d( f k y; f k x) <



D sup lim sup h( f ; )  h( f ; )

where s(ı; n; E) is the maximum cardinality of a subset S of E such that:

x2X

h( f ; ) D limı!0 h( f ; ; ı). An exercise in topology shows that the local entropy therefore also bounds the defect in upper semicontinuity of  7! h( f ; ). In fact, by a result of Downarowicz [100] (extended by David Burguet to the non-invertible case), there is a local variational principle: !



Clearly from the above formulas: htop ( f )  htop ( f ; ı) C hloc ( f ; ı) and h( f ; )  h( f ; ; ı) C hloc ( f ; ı) : Thus the local entropy bounds the defect in uniformity with respect to the measure of the pointwise limit



!

for any continuous self-map f of a compact metric space. The local entropy is easily bounded for smooth maps using Yomdin’s theory: Theorem 10 ([64]) For any Cr map f of a compact manifold, hloc ( f )  dr log supx k f 0 (x)k. In particular, hloc ( f ) D 0 if r D 1. Thus C 1 smoothness implies the existence of a maximum entropy measure (this was proved first by Newhouse) and the existence of symbolic extension: a subshift over a finite alphabet  : ˙ ! ˙ and a continuous and onto map  : ˙ ! M such that  ı  D f ı  . More precisely, Theorem 11 (Boyle, Fiebig, Fiebig [56]) Given a homeomorphism f of a compact metric space X, there exists a principal symbolic extension  : ˙ ! ˙ , i. e., a symbolic extension such that, for every  -invariant probability measure , h(; ) D h( f ;  ı  1 ), if and only if hloc ( f ) D 0. We refer to [55,101] for further results, including a realization theorem showing that the continuity properties of the measured entropy are responsible for the properties of symbolic extensions and also results in finite smoothness. Global Simplicity One can marvel at the power of mathematical analysis to analyze such complex evolutions. Of course another way to look at this is to remark that this analysis is possible once this evolution has been fitted in a simple setting: one had to move focus away from an individual, unpredictable orbit, of, say, the full tent map to the set of all the orbits of that map, which is essentially the set of all infinite sequences over two symbols: a very simple set indeed corresponding to full combinatorial freedom [[253] describes a weakening of this which holds for all positive entropy symbolic dynamics.]. The complete description of a given typical orbit requires an infinite amount of information, whereas the set of all orbits has a finite and very tractable definition. The complexity of the individual orbits is seen now as coming from purely random choices inside a simple structure.

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The classical systems, namely uniformly expanding maps or hyperbolic diffeomorphisms of compact spaces, have a simple symbolic dynamics. It is not necessarily a full shift like for the tent map, but it is a subshift of finite type, i. e., a subshift obtained from a full shift by forbidding finitely many finite subwords. What happens outside of the uniform setting? A fundamental example is provided by piecewise monotone maps, i. e., interval maps with finitely many critical points or discontinuities. The partition cut by these points defines a symbolic dynamics. This subshift is usually not of finite type. Indeed, the topological entropy taking arbitrary finite nonnegative values [For instance, the topological entropy of the ˇ-transformation, x 7! ˇx mod 1, is log ˇ for ˇ  1.], a representation that respects it has to use an uncountable class of models. In particular models defined by finite data, like the subshifts of finite type, cannot be generally adequate. However there are tractable “almost finite representations” in the following senses: Most symbolic dynamics ˙ (T) of piecewise monotone maps T can be defined by finitely many infinite sequences, the kneading invariants of Milnor and Thurston:  0C ; 1 ; 1C ; : : : ; dC1 2 f0; : : : ; dgN if d is the number of critical/discontinuity points. [The kneading invariants are the suitably defined (left and right) itineraries of the critical/discontinuity points and endpoints.] Namely, ˚ ˙ (T) D ˛ 2 f0; : : : ; dgN : 8n  0 ˛Cn  n ˛ ˛n C1



where is a total order on f0; : : : ; dgN making the coding x 7! ˛ non-decreasing. Observe how the kneading invariants determine ˙ (T) in an effective way: knowing their first n symbols is enough to know the sequences of length n which begin sequences of ˙ (T). We refer to [91] for the wealth of information that can be extracted from these kneading invariants following Milnor and Thurston [184]. This form of global simplicity can be extended to other classes of non-uniformly expanding maps, including those like Eq. (4) using the notions of subshifts and puzzles of quasi-finite type [68,69]. This leads to the notion and analysis of entropy-expanding maps, a new open class of nonuniformly expanding maps admitting critical hypersurfaces, defined purely in terms of entropies including the otherwise untractable examples of Eq. (4). A generalization of the representation of uniform systems by subshifts of finite type is provided by strongly positive recurrent countable state Markov shifts, a subclass of Markov shifts (see glossary.) which shares many properties with the subshifts of finite type [57,123,124,224,229].

These “simple” systems admit a classification result which in particular identifies their measures with entropy close to the maximum [57]. Such a classification generalizes [164]. The “ideology” here is that complexity of individual orbits in a simple setting must come from randomness, but purely random systems are classified by their entropy according to Ornstein [197]. Stability By definition, chaotic dynamical systems have orbits which are unstable and numerically unpredictable. It is all the more surprising that, once one accepts to consider their dynamics globally, they exhibit very good stability properties. Structural Stability A simple form of stability is structural stability: a system f : M ! M is structurally Cr -stable if any system g sufficiently Cr -close to f is topologically the same as f , formally: g is topologically conjugate, i. e., there is some homeomorphism [If h were C1 , the conjugacy would imply, among other things, that for every p-periodic point: det(( f p )0 (x)) D det((g p )0 (h(p))), a much too strong requirement.] h : M ! M mapping the orbits of f to those of g, i. e., g ı h D h ı f . Andronov and Pontryaguin argued in the 1930’s that only such structurally stable systems are physically relevant. Their idea was that the model of a physical system is always known only to some degree of approximation, hence mathematical model whose structure depends on arbitrarily small changes should be irrelevant. A first question is: What are these structurally stable systems? The answer is quite striking: Theorem 12 (Mañé [182]) Let f : M ! M be a C1 diffeomorphism of a compact manifold. f is structurally stable among C1 -diffeomorphisms of M if and only if f is uniformly hyperbolic on its chain recurrent set. [A point x is chain recurrent if, for all > 0, there exists a finite sequence x0 ; x1 ; : : : ; x n such that x0 D x n D x and d( f (x k ); x kC1 ) < . The chain recurrent set is the set of all chain recurrent points.] A basic idea in the proof of the theorem is that failure of uniform hyperbolicity gives the opportunity to make an arbitrarily small perturbation contradicting the structural stability. In higher smoothness the required perturbation lemmas (e. g., the closing lemma [14,148,180]) are not available. We note that uniform hyperbolicity without invertibility does not imply C1 -stability [210].

Chaos and Ergodic Theory

A second question is: are these stable systems dense? (So that one could offer structurally stable models for all physical situations). A deep discovery around 1970 is that this is not the case: Theorem 13 (Abraham–Smale, Simon [3,234]) For any r  1 and any compact manifold M of dimension  3, the set of uniformly hyperbolic diffeomorphisms is not dense in the space of Cr diffeomorphisms of M. [They use the phenomenon called “heterodimensional homoclinic intersections”.] Theorem 14 (Newhouse [194]) For any r  2, for any compact manifold M of dimension  2, the set of uniformly hyperbolic diffeomorphisms is not dense in the space of Cr diffeomorphisms of M. More precisely, there exists a nonempty open subset in this space which contains a dense Gı subset of diffeomorphisms with infinitely many periodic sinks. [So these diffeomorphisms have no finite statistical description.] Observe that it is possible that uniform hyperbolicity could be dense among surface C1 -diffeomorphisms (this is the case for C1 circle maps by a theorem of Jakobson [145]). In light of Mañé C1 -stability theorem this implies that structurally stable systems are not dense, thus one can robustly see behaviors that are topologically modified by arbitrarily small perturbations (at least in the C1 -topology)! So one needs to look beyond these and face that topological properties of relevant dynamical system are not determined from “finite data”. It is natural to ask whether the dynamics is almost determined by “sufficient data”. Continuity Properties of the Topological Dynamics Structural stability asks the topological dynamics to remain unchanged by a small perturbation. It is probably at least as interesting to ask it to change continuously. This raises the delicate question of which topology should be put on the rather wild set of topological conjugacy classes. It is perhaps more natural to associate to the system a topological invariant taking value in a more manageable set and ask whether the resulting map is continuous. A first possibility is Zeeman’s Tolerance Stability Conjecture. He associated to each diffeomorphism the set of all the closures of all of its orbits and he asked whether the resulting map is continuous on a dense Gı subset of the class of Cr -diffeomorphisms for any r  0. This conjecture remains open, we refer to [85] for a discussion and related progress. A simpler possibility is to consider our favorite topological invariant, the topological entropy, and thus ask whether the dynamical complexity as measured by the

entropy is a stable phenomenon. f 7! htop ( f ) is lower semicontinuous for f among C0 maps of the interval [On the set of interval maps with a bounded number of critical points, the entropy is continuous [188]. Also t 7! htop (Q t ) is non-decreasing by complex arguments [91], though it is a non-smooth function.] [187] and for f among C 1C diffeomorphisms of a compact surface [147]. [It is an important open question whether this actually holds for C1 diffeomorphisms. It fails for homeomorphisms [214].] In both cases, one shows the existence of structurally stable invariant uniformly expanding or hyperbolic subsets with topological entropy close to that of the whole dynamics. On the other hand f 7! htop ( f ) is upper semi-continuous for C 1 maps [195,254]. Statistical Stability Statistical stability is the property that deterministic perturbations of the dynamical system cause only small changes in the physical measures, usually with respect to the weak star topology on the space of measures. When the physical measure  g is uniquely defined for all systems g near f , statistic stability is the continuity of the map g 7!  g thus defined. Statistical stability is known in the uniform setting and also in the piecewise uniform case, provided the expansion is strong enough [25,42,151] (otherwise there are counterexamples, even in the one-dimensional case). It also holds in some non-uniform settings without critical behavior, in particular for maps with dominated splitting satisfying robustly a separation condition between the positive and negative Lyapunov exponents [247] (see also [7]). However statistical unstability occurs in dynamics with critical behaviors [18]: Theorem 15 Consider the quadratic family Q t (x) D tx(1  x), t 2 [0; 4]. Let I [0; 4] be the full measure subset of [0; 4] of good parameters t such that in particular Qt admits a physical measure (necessarily unique). For t 2 I, let  t be this physical measure. Lebesgue almost every t 2 I such that  t is not carried by a periodic orbit [At such parameters, statistical stability is easily proved.], is a discontinuity point of the map t 7!  t [M([0; 1]) is equipped with the vague topology.]. However, for Lebesgue almost every t, there is a subset I t I for which t is a Lebesgue density point [t is a Lebesgue density point if for all r > 0, the Lebesgue measure m(I t \ [t  r; t C r]) > 0 and lim !0 m(I t \ [t  ; t C ])/2 D 1.] and such that  : I t ! M([0; 1]) is continuous at t.

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Stochastic Stability A physically motivated and a technically easier approach is to study stability of the physical measure under stochastic perturbations. For simplicity let us consider a diffeomorphism of a compact subset of Rd allowing for a direct definition of additive noise. Let (x)dx be an absolutely continuous probability law with compact support. [Sometimes additional properties are required of the density, e. g., (x) D (x)1B where 1B is the characteristic function of the unit ball and C 1  (x)  C for some C < 1.] For > 0, consider the Markov chain f with state space M and transition probabilities:   Z y  f (x) dy P (x; A) D : d A The evolution of measures is given by: ( f )(A) D R P (x; A) d. Under rather weak irreducibility assump M tions on f , f has a unique invariant measure  (contrarily to f ) and  is absolutely continuous. When f has a unique physical measure , it is said to be stochastically stable if lim !0  D  in the appropriate topology (the weak star topology unless otherwise specified). It turns out that stochastic stability is a rather common property of Sinai–Ruelle–Bowen measures. It holds not only for uniformly expanding maps or hyperbolic diffeomorphisms [153,258], but also for most interval maps [25], for partially hyperbolic systems of the type E u ˚ E cs or Hénon-like diffeomorphisms [6,13,33,40]. We refer to the monographs [21,41] for more background and results. Untreated Topics For reasons of space and time, many important topics have been left out of this article. Let us list some of them. Other phenomena related to chaotic dynamics have been studied: entrance times [77,79], spectral properties and dynamical zeta functions [21], escape rates [104], dimension [204], differentiability of physical measures with respect to parameters [99,223], entropies and volume or homological growth rates [118,139,254]. As far as the structure of the setting is concerned, one can go beyond maps or diffeomorphisms or flows and study: more general group actions [128]; holomorphic and meromorphic structures [72] and the references therein; symplectic or volume-preserving [90,137] and in particular the Pugh–Shub program around stable ergodicity of partially hyperbolic systems [211]; random iterations [17, 154,155]. A number of important problems have motivated the study of special forms of chaotic dynamics: equidistribution in number theory [105,109,138] and geometry [236];

quantum chaos [10,125]; chaotic control [227]; analysis of algorithms [245]. We have also omitted the important problem of applying the above results. Perhaps because of the lack of a general theory, this can often be a challenge (see for instance [244] for the already complex problem of verifying uniform hyperbolicity for a singular flow). Liverani has shown how theoretical results can lead to precise and efficient estimates for the toy model of piecewise expanding interval maps [176]. Ergodic theory implies that, in some settings at least, adding noise may make some estimates more precise (see [157]). We refer to [106] and the reference therein. Future Directions We conclude this article by a (very partial) selection of open problems. General Theory In dimension 1, we have seen that the analogue of the Palis conjecture (see above) is established (Theorem 2). However the description of the typical dynamics in Kolmogorov sense is only known in the unimodal, non-degenerate case by Theorem 3. Indeed, results like [63] suggest that the multimodal picture could be more complex. In higher dimensions, our understanding is much more limited. As far as a general theory is concerned, a deep problem is the paucity of results on the generic dynamics in Cr smoothness with r > 1. The remarkable current progress in generic dynamics (culminating in the proof of the weak Palis conjecture, see [84], the references therein and [51] for background) seems restricted to the C1 topology because of the lack of fundamental tools (e. g., closing lemmas) in higher smoothness. But Pesin theory requires higher smoothness at least technically. This is not only a hard technical issue but generic properties of physical measures, when they have been analyzed are often completely different between the C1 case and higher smoothness [45]. Physical Measures In higher dimensions, Benedicks and Carleson analysis of the Hénon map has given rise to a rather general theory of Hénon-like maps and more generally of the dynamical phenomena associated to homoclinic tangencies. However, the proofs are extremely technical. Could they be simplified? Current attempts like [266] center on the introduction of a simpler notion of critical points, possibly a non-inductive one [212].

Chaos and Ergodic Theory

Can this Hénon theory be extended to the weakly dissipative situation? to the conservative situation (for which the standard map is a well-known example defying analysis)? In the strongly dissipative setting, what are the typical phenomena on the complement of the Benedicks– Carleson set of parameters? From a global perspective one of the main questions is the following: Can infinitely many sinks coexist for a large set of parameters in a typical family or is Newhouse phenomenon atypical in the Kolmogorov or prevalent sense? This seems rather unlikely (see however [11]). Away from such “critical dynamics”, there are many results about systems with dominated splitting satisfying additional conditions. Can these conditions be weakened so they would be satisfied by typical systems satisfying some natural conditions (like robust transitivity)? For instance: Could one analyze the physical measures of volumehyperbolic systems? A more specific question is whether Tsujii’s striking analysis of surface maps with one uniformly expanding direction can be extended to higher dimensions? can one weaken the uniformity of the expansion? The same questions for the corresponding invertible situation is considered in [83]. Maximum Entropy Measures and Topological Complexity As we explained, C 1 smoothness, by a Newhouse theorem, ensures the existence of maximum entropy measures, making the situation a little simpler than with respect to physical measures. This existence results allow in particular an easy formulation of the problem of the typicality of hyperbolicity: Are maximum entropy ergodic measures of systems with positive entropy hyperbolic for most systems? A more difficult problem is that of the finite multiplicity of the maximum entropy measures. For instance: Do typical systems possess finitely many maximum entropy ergodic measures? More specifically, can one prove intrinsic ergodicity (ie, uniqueness of the measure of maximum entropy) for an

isolated homoclinic class of some diffeomorphisms (perhaps C1 -generic)? Can a generic C1 -diffeomorphism carry an infinite number of homoclinic classes, each with topological entropy bounded away from zero? A perhaps more tractable question, given the recent progress in this area: Is a C1 -generic partially hyperbolic diffeomorphisms, perhaps with central dimension 1 or 2, intrinsically ergodic? We have seen how uniform systems have simple symbolic dynamics, i. e., subshifts of finite type, and how interval maps and more generally entropy-expanding maps keep some of this simplicity, defining subshifts or puzzles of quasi-finite type [68,69]. [264] have defined symbolic dynamics for topological Hénon-like map which seems close to that of that of a one-dimensional system. Can one describe Wang and Young symbolic dynamics of Hénon-like attractors and fit it in a class in which uniqueness of the maximum entropy measure could be proved? More generally, can one define nice combinatorial descriptions, for surface diffeomorphisms? Can one formulate variants of the entropy-expansion condition [For instance building on our “entropy-hyperbolicity”.] of [66,70], that would be satisfied by a large subset of the diffeomorphisms? Another possible approach is illustrated by the pruning front conjecture of [87] (see also [88,144]). It is an attempt to build a combinatorial description by trying to generalize the way that, for interval maps, kneading invariants determine the symbolic dynamics by considering the bifurcations from a trivial dynamics to an arbitrary one. We hope that our reader has shared in our fascination with this subject, the many surprising and even paradoxical discoveries that have been made and the exciting current progress, despite the very real difficulties both in the analysis of such non-uniform systems as the Henon map and in the attemps to establish a general (and practical) ergodic theory of chaotic dynamical systems. Acknowledgments I am grateful for the advice and/or comments of the following colleagues: P. Collet, J.-R. Chazottes, and especially S. Ruette. I am also indebted to the anonymous referee. Bibliography Primary Literature 1. Aaronson J, Denker M (2001) Local limit theorems for partial sums of stationary sequences generated by Gibbs–Markov maps. Stoch Dyn 1:193–237

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Chronological Calculus in Systems and Control Theory MATTHIAS KAWSKI Department of Mathematics, Arizona State University, Tempe, USA Article Outline Glossary Definition of the Subject Introduction, History, and Background Fundamental Notions of the Chronological Calculus Systems That Are Affine in the Control Future Directions Acknowledgments Bibliography Glossary Controllability A control system is controllable if for every pair of points (states) p and q there exists an admissible control such that the corresponding solution curve that starts at p ends at q. Local controllability about a point means that all states in some open neighborhood can be reached. Pontryagin Maximum Principle of optimal control Optimality of a control-trajectory pair geometrically is a property dual to local controllability in the sense that an optimal trajectory (endpoint) lies on the boundary of the reachable sets (after possibly augmenting the state of the system by the running cost). The maximum principle is a necessary condition for optimality. Geometrically it is based on analyzing the effect of families of control variations on the endpoint map. The chronological calculus much facilitates this analysis. E (M) D C 1 (M) The algebra of smooth functions on a finite dimensional manifold M, endowed with the topology of uniform convergence of derivatives of all orders on compact sets.  1 (M) The space of smooth vector fields on the manifold M. Chronological calculus An approach to systems theory based on a functional analytic operator calculus, that replaces nonlinear objects such as smooth manifolds by infinite dimensional linear ones, by commutative algebras of smooth functions. Chronological algebra A linear space with a bilinear product F that satisfies the identity a ? (b ? c)  b ? (a ? c) D (a ? b) ? c  (b ? a) ? c. This structure arises

Rt naturally via the product ( f ? g) t D 0 [ f s ; g s0 ]ds of time-varying vector fields f and g in the chronological calculus. Here [ ; ] denotes the Lie bracket. Zinbiel algebra A linear space with a bilinear product that satisfies the identity a (b c) D (a b) c C (b a) c. This structure arises naturally in the special case ofR affine control systems for the product t (U V)(t) D 0 U(s)V 0 (s)ds of absolutely continuous scalar valued functions U and V. The name Zinbiel is Leibniz read backwards, reflecting the duality with Leibniz algebras, a form of noncommutative Lie algebras. There has been some confusion in the literature with Zinbiel algebras incorrectly been called chronological algebras. IIF (U Z ) For a suitable space U of time-varying scalars, e. g. the space of locally absolutely continuous real-valued functions defined on a fixed time interval, and an indexing set Z, IIF (U Z ) denotes the space of iterated integral functionals from the space of Z-tuples with values in U to the space U. Definition of the Subject The chronological calculus is a functional analytic operator calculus tool for nonlinear systems theory. The central idea is to replace nonlinear objects by linear ones, in particular, smooth manifolds by commutative algebras of smooth functions. Aside from its elegance, its main virtue is to provide tools for problems that otherwise would effectively be untractable, and to provide new avenues to investigate the underlying geometry. Originally conceived to investigate problems in optimization and control, specifically for extending Pontryagin’s Maximum Principle, the chronological calculus continues to establish itself as the preferred language of geometric control theory, and it is spawning new sets of problems, including its own set of algebraic structures that are now studied in their own right. Introduction, History, and Background This section starts with a brief historical survey of some landmarks that locate the chronological calculus at the interface of systems and control theory with functional analysis. It is understood that such brief survey cannot possibly do justice to the many contributors. Selected references given are meant to merely serve as starting points for the interested reader. Many problems in modern systems and control theory are inherently nonlinear and e. g., due to conserved quantities or symmetries, naturally live on manifolds rather than on Euclidean spaces. A simple example is the problem of stabilizing the attitude of a satellite via feedback con-

Chronological Calculus in Systems and Control Theory

trols. In this case the natural state space is the tangent bundle T SO(3) of a rotation group. The controlled dynamics are described by generally nonautonomous nonlinear differential equations. A key characteristic of their flows is their general lack of commutativity. Solutions of nonlinear differential equations generally do not admit closed form expressions in terms of the traditional sets of elementary functions and symbols. The chronological calculus circumvents such difficulties by reformulating systems and control problems in a different setting which is infinite dimensional, but linear. Building on well established tools and theories from functional analysis, it develops a new formalism and a precise language designed to facilitate studies of nonlinear systems and control theory. The basic plan of replacing nonlinear objects by linear ones, in particular smooth manifolds by commutative algebras of smooth functions, has a long history. While its roots go back even further, arguably this approach gained much of its momentum with the path-breaking innovations by John von Neumann’s work on the “Mathematical Foundations of Quantum Mechanics” [104] and Marshall Stone’s seminal work on “Linear Transformations in Hilbert Space” [89], quickly followed by Israel Gelfand’s dissertation on “Abstract Functions and Linear Operators” [34] (published in 1938). The fundamental concept of maximal ideal justifies this approach of identifying manifolds with commutative normed rings (algebras), and vice versa. Gelfand’s work is described as uniting previously uncoordinated facts and revealing the close connections between classical analysis and abstract functional analysis [102]. In the seventy years since, the study of Banach algebras, C -algebras and their brethren has continued to develop into a flourishing research area. In the different arena of systems and control theory, major innovations at formalizing the subject were made in the 1950s. The Pontryagin Maximum Principle [8] of optimal control theory went far beyond the classical calculus of variations. At roughly the same time Kalman and his peers introduced the Kalman filter [51] for extracting signals from noisy observations, and pioneered statespace approaches in linear systems theory, developing the fundamental concepts of controllability and observability. Linear systems theory has bloomed and grown into a vast array of subdisciplines with ubiquitous applications. Via well-established transform techniques, linear systems lend themselves to be studied in the frequency domain. In such settings, systems are represented by linear operators on spaces of functions of a complex variable. Starting in the 1970s new efforts concentrated on rigorously extending linear systems and control theory to nonlinear settings. Two complementary major threads emerged

that rely on differential geometric methods, and on operators represented by formal power series, respectively. The first is exemplified in the pioneering work of Brockett [10,11], Haynes and Hermes [43], Hermann [44], Hermann and Krener [45], Jurdjevic and Sussmann [50], Lobry [65], and many others, which focuses on state-space representations of nonlinear systems. These are defined by collections of vector fields on manifolds, and are analyzed using, in particular, Lie algebraic techniques. On the other side, the input-output approach is extended to nonlinear settings primarily through a formal power series approach as initiated by Fliess [31]. The interplay between these approaches has been the subject of many successful studies, in which a prominent role is played by Volterra series and the problem of realizing such input-output descriptions as a state-space system, see e. g. Brockett [11], Crouch [22], Gray and Wang [37], Jakubczyk [49], Krener and Lesiak [61], and Sontag and Wang [87]. In the late 1970s Agrachëv and Gamkrelidze introduced into nonlinear control theory the aforementioned abstract functional analytic approach, that is rooted in the work of Gelfand. Following traditions from the physics community, they adopted the name chronological calculus. Again, this abstract approach may be seen as much unifying what formerly were disparate and isolated pieces of knowledge and tools in nonlinear systems theory. While originally conceived as a tool for extending Pontryagin’s Maximum Principle in optimal control theory [1], the chronological calculus continues to yield a stream of new results in optimal control and geometry, see e. g. Serres [85], Sigalotti [86], Zelenko [105] for very recent results utilizing the chronological calculus for studying the geometry. The chronological calculus has led to a very different way of thinking about control systems, epitomized in e. g. the monograph on geometric control theory [5] based on the chronological calculus, or in such forceful advocacies for this approach for e. g. nonholonomic path finding by Sussmann [95]. Closely related are studies of nonlinear controllability, e. g. Agrachëv and Gamkrelidze [3,4], Tretyak [96,97,98], and Vakhrameev [99], including applications to controllability of the Navier–Stokes equation by Agrachëv and Sarychev [6]. The chronological calculus also lends itself most naturally to obtaining new results in averaging theory as in Sarychev [83], while Cortes and Martinez [20] used it in motion control of mechanical systems with symmetry and Bullo [13,68] for vibrational control of mechanical systems. Noteworthy applications include locomotion of robots by Burdick and Vela [100], and even robotic fish [71]. Instrumental is its interplay with series expansion as in Bullo [12,21] that uti-

89

90

Chronological Calculus in Systems and Control Theory

lize affine connections of mechanical systems. There are further applications to stability and stabilization Caiado and Sarychev [15,82], while Monaco et.al. [70] extended this approach to discrete-time dynamics and Komleva and Plotnikov used it in differential game theory [60]. Complementing such applications are new subjects of study such as the abstract algebraic structures that underlie the chronological calculus. The chronological algebra itself has been the subject of study as early as Agrachëv and Gamkrelidze [2]. The closely related structure of Zinbiel structures has recently found more attention, see work by Dzhumadil’daev [25,26], Kawski and Sussmann [57] and Kawski [54,55]. Zinbiel algebras arise in the special case when the dynamics (“time-varying vector fields”) splits into a sum of products of time-varying coefficients and autonomous vector fields. There is an unfortunate confusion of terms in the literature as originally Zinbiel algebras had also been called chronological algebras. Recent usage disentangles these closely related, but distinct, structures and reflects the primacy of the latter term coined by Loday [66,67] who studies Leibniz algebras (which appear in cyclic homology). Zinbiel is simply Leibniz spelled backwards, a choice which reflect that Leibniz and Zinbiel are dual operands in the sense of Koszul duality as investigated by Ginzburg and Kapranov [36]. Fundamental Notions of the Chronological Calculus From a Manifold to a Commutative Algebra This and the next sections very closely follow the introductory exposition of Chapter 2 of Agrachëv and Sachkov [5], which is also recommended for the full technical and analytical details regarding the topology and convergence. The objective is to develop the basic tools and formalism that facilitate the analysis of generally nonlinear systems that are defined on smooth manifolds. Rather than primarily considering points on a smooth manifold M, the key idea is to instead focus on the commutative algebra of E (M) D C 1 (M; R) of real-valued smooth functions on M. Note that E (M) not only has the structure of a vector space over the field R, but it also inherits the structure of a commutative ring under pointwise addition and multiplication from the codomain R. Every point p 2 M gives rise to a functional pˆ : E (M) 7! R defined by pˆ(') D '(p). This functional is linear and multiplicative, and is a homomorphism of the algebras E (M) and R: For every p 2 M, for '; 2 E (M) and t 2 R the following hold pˆ(' C and

) D pˆ' C pˆ'; pˆ('  pˆ (t') D t  ( pˆ ') :

) D ( pˆ ')  ( pˆ ) ;

Much of the power of this approach derives from the fact that this correspondence is invertible: For a nontrivial multiplicative linear functional : E (M) 7! R consider its kernel ker D f' 2 E (M) : ' D 0g. A critical observation is that this is a maximal ideal, and that it must be of the form f' 2 E (M) : '(p) D 0g for some, uniquely defined, p 2 M. For the details of a proof see appendix A.1. of [5]. Proposition 1 For every nontrivial multiplicative linear functional : E (M) 7! R there exists p 2 M such that D pˆ. Note on the side, that there may be maximal ideals in the space of all multiplicative linear functionals on E (M) that do not correspond to any point on M – e. g. the ideal of all linear functionals that vanish on every function with compact support. But this does not contradict the stated proposition. Not only can one recover the manifold M as a set from the commutative ring E (M), but using the weak topology on the space of linear functionals on E (M) one also recovers the topology on M p n ! p if and only if 8 f 2 E (M); pˆn f ! pˆ f : (1) The smooth structure on M is recovered from E (M) in a trivial way: A function g on the space of multiplicative linear functionals pˆ : p 2 M is smooth if and only if there exists f 2 E (M) such that for every pˆ; g( pˆ ) D pˆ f . In modern differential geometry it is routine to identify tangent vectors to a smooth manifold with either equivalence classes of smooth curves, or with first order partial differential operators. In this context, tangent vectors at a point q 2 M are derivations of E (M), that is, linear functionals fˆq on E (M) that satisfy the Leibniz rule. ˆ this means for every '; 2 E (M), Using q, ˆ Xˆ q ) : Xˆ q (' ) D ( Xˆ q ')(qˆ ) C (q')(

(2)

Smooth vector fields on M correspond to linear functionals fˆ : E (M) 7! E (M) that satisfy for all '; 2 E (M) fˆ(' ) D ( fˆ') 

C '  ( fˆ ) :

(3)

Again, the correspondence between tangent vectors and vector fields and the linear functionals as above is invertible. Write  1 (M) for the space of smooth vector fields on M. Finally, there is a one-to-one correspondence between smooth diffeomorphisms ˚ : M 7! M and automorphisms of E (M). The map ˚ˆ : E (M) 7! E (M) defined for ˆ D '(˚(p)) clearly has p 2 M and ' 2 E (M) by ˚(')(p)

Chronological Calculus in Systems and Control Theory

the desired properties. For the reverse direction, suppose  : E (M) 7! E (M) is an automorphism. Then for every p 2 M the map pˆ ı  : E (M) 7! R is a nontrivial linear multiplicative functional, and hence equals qˆ for some q 2 M. It is easy to see that the map ˚ : M 7! M is indeed a diffeomorphism, and  D ˚ˆ . In the sequel we shall omit the hats, and simply write, say, p for the linear functional pˆ. The role of each object is usually clear from the order. For example, for a point p, a smooth function ', smooth vector  fields f ; g; h, with flow e th and its tangent map e th  , what in traditional   format might be expressed as ( e th  g)'(e f (p)) is simply written as pe f ge th ', not requiring any parentheses. Frechet Space and Convergence The usual topology on the space E (M) is the one of uniform convergence of all derivatives on compact sets, i. e., a sequence of functions f' k g1 kD1  E (M) converges to  2 E (M) if for every finite sequence f1 ; f2 ; : : : ; f s of smooth vector fields on M and every compact set K  M the sequence f f s : : : f2 f1 ' k g1 kD1 converges uniformly on K to f s : : : f2 f1 '. This topology is also obtained by a countable family of semi-norms k  ks;K defined by k'ks;K D supf jp f s : : : f2 f1 'j : p 2 K; f i 2  1 (M); s 2 ZC g

(4)

where K ranges over a countable collection of compact subsets whose union is all of M. In an analogous way define semi-norms of smooth vector fields f 2  1 (M) k f ks;K D supf k f 'ks;K : k f 'ksC1;K D 1 g :

(5)

Finally, for every smooth diffeomorphism ˚ of M, s 2 ZC and K  M compact there exist Cs;K;˚ 2 R such that for all ' 2 E (M) k˚ 'ks;K  Cs;K;˚ k'ks;'(K) :

(6)

Regularity properties of one-parameter families of vector fields and diffeomorphisms (“time-varying vector fields and diffeomorphisms”) are understood in the weak sense. In particular, for a family of smooth vector fields f t 2  1 (M), its integral and derivative (if they exist) are defined as the operators that satisfy for every ' 2 E (M) 

The Chronological Exponential This section continues to closely follow [5] which contains full details and complete proofs. On a manifold M consider generally time-varying differential equations of the d form dt q t D f t (q t ). To assure existence and uniqueness of solutions to initial value problems, make the typical regularity assumptions, namely that in every coordinate chart U  M ; x : U 7! Rn the vector field (x f t ) is (i) measurable and locally bounded with respect to t for every fixed x and (ii) smooth with locally bounded partial derivatives with respect to x for every fixed t. For the purposes of this article and for clarity of exposition, also assume that vector fields are complete. This means that solutions to initial value problems are defined for all times t 2 R. This is guaranteed if, for example, all vector fields considered vanish identically outside a common compact subset of M. As in the previous sections, for each fixed t interpret q t : E (M) 7! R as a linear functional on E (M), and note that this family satisfies the time-varying, but linear differential equation, suggestively written as q˙ t D q t ı f t :

d dt

   Z t Z ! ! exp f d D exp 0

0

d dt

 Z  exp

a

 f t ' dt :



t

f d

0

(9)

 Z  D f t ı exp



t 0

f d

: (10)

t

q() ı f d

Z t (7)

ı ft :

f d

Formally, one obtains a series expansion for the chronological exponentials by rewriting the differential equation as an integral equation and solving it by iteration

0





t

Analogously the left chronological exponential satisfies

q t D q0 C b

(8)

on the space of linear functionals on E (M). It may be shown that under the above assumptions it has a unique solution, called the right chronological exponential of the vector field f t as the corresponding flow. Formally, it satisfies for almost all t 2 R

Z

 d d   f t ' and ft ' D dt dt ! Z b Z f t dt ' D a

The convergence of series expansions of vector fields and of diffeomorphisms encountered in the sequel are to be interpreted in the analogous weak sense.

D q0 C

Z

0





q0 C 0

(11)

q( ) ı f d

ı f d

(12)

91

92

Chronological Calculus in Systems and Control Theory

to eventually, formally, obtain the expansion !

Z

t

f d Id C

exp 0

1 Z tZ X kD1 0

tk

Z 

0

Variation of Parameters and the Chronological Logarithm

t2

0

 fk ı    ı f2 ı f1 d k : : : d2 d1

(13)

and analogously for the left chronological exponential 

Z

t

f d Id C

exp 0

1 Z tZ X kD1 0

tk

Z 

0

t2

0

 f1 ı f2 ı    ı fk d k : : : d2 d1 :

(14)

While this series never converges, not even in a weak sense, it nonetheless has an interpretation as an asymptotic expansion. In particular, for any fixed function  2 E (M) and any semi-norm as in the previous section, on any compact set one obtains an error estimate for the remainder after truncating the series at any order N which establishes that the truncation error is of order O(t N ) as t ! 0. When one considers the restrictions to any f t -invariant normed linear subspace L  E (M), i. e. for all t, f t (L)  L, on which f t is bounded, then the asymptotic series converges to the chronological exponential and it satisfies for every '2L    Z Rt   ! t  exp 0 k f  k d k' k : f d '    e 

(15)

0

In the case of analytic vector fields f and analytic functions ' one obtains convergence of the series for sufficiently small t. While in general there is no reason why the vector fields f t should commute at different times, the chronological exponentials nonetheless still share some of the usual properties of exponential and flows, for example the composition of chronological exponentials satisfies for all t i 2 R  Z ! exp

t2 t1

  Z ! f d ı exp



t3

f d

t2

 Z ! D exp



t3 t1

f d

:

Moreover, the left and right chronological exponentials are inverses of each other in the sense that for all t0 ; t1 2 R 1

t1 t0

f d

!

Z

t0



Z

t1 t1

D exp D exp t0

f d (17) ( f ) d :

q˙ D f t (q) C g t (q) :

(18)

The objective is to obtain a formula for the flow of the field ( f t C g t ) as a perturbation of the flow of f t . Writing !

Z

t

˚ t D exp 0

!

f d and  t D exp

Z

t 0

( f Cg ) d; (19)

this means one is looking for a family of operators  t such that for all t t D t ı ˚t :

(20)

Differentiating and using the invertibility of the flows one obtains a differential equation in the standard form ˙ (t) D  t ı ˚ t ı g t ı ˚ t1 D  t ı (Ad˚ t ) g t   Z t ! D  t ı exp ad f d g t ; (21) 0

which has the unique solution !

 t D exp

Z t Z ! exp 0



 0

ad f d

g d ;

(22)

and consequently one has the variations formula !

Z

t

exp 0

(16)

 Z ! exp

In control one rarely considers only a single vector field, and, instead, for example, is interested in the interaction of a perturbation or control vector field with a reference or drift vector field. In particular, in the nonlinear setting, this is where the chronological calculus very much facilitates the analysis. For simplicity consider a differential equation defined by two, generally time varying, vector fields (with the usual regularity assumptions)

 Z t Z ! ! ( f C g ) d D exp exp 0

 Z  !  g d ı exp



 0

ad f d

t 0

f d

 :

(23)

This formula is of fundamental importance and used ubiquitously for analyzing controllability and optimality: One typically considers f t as defining a reference dynamical system and then considers the effect of adding a control term g t . A special application is in the theory of averaging where g t is considered a perturbation of the field f t , which in the most simple form may be assumed to be time-periodic. The monograph [81] by Sanders and Verhulst is the

Chronological Calculus in Systems and Control Theory

classical reference for the theory and applications of averaging using traditional language. The chronological calculus, in particular the above variations formula, have been used successfully by Sarychev [83] to investigate in particular higher order averaging, and by Bullo [13] for averaging in the context of mechanical systems and its interplay with mechanical connections. Instrumental to the work [83] is the notion of the chronological logarithm, which is motivated by rewriting the defining Eqs. (9) and (10) [2,80] as  1   Z t Z d ! t ! f d ı exp f d f t D exp dt 0 0   1  Z t Z t d   D exp f d ı exp f d : (24) dt 0 0 While in general the existence of the logarithm is a delicate question [62,83], under suitable hypotheses it is instrumental for the derivation of the nonlinear Floquet theorem in [83]. Theorem 1 (Sarychev [83]) Suppose f t D f tC1 is a period-one smooth vector field generating the flow ˚ t and  D log ˚1 , then there exists a period-one family of diffeomorphisms  t D  tC1 such that for all t ˚ t D e t ı  t . The logarithm of the chronological exponential of a timevarying vector field " f t may be expanded as a formal chronological series Z 1  1 X ! " f d D " j j : (25) log exp 0

jD1

The first three terms of which are given explicitly [83] as  Z 1 Z Z  1 1 1 2 f d ;  D f d ; f d;  D 2 0 0 0 1 and 3 D  [1 ; 2 ] 2  Z  Z 1 1 2 ad f d f d : (26) C 3 0 0 Aside from such applications as averaging, the main use of the chronological calculus has been for studying the dual problems of optimality and controllability, in particular controllability of families of diffeomorphisms [4]. In particular, this approach circumvents the limitations encountered by more traditional approaches that use parametrized families of control variations that are based on a finite number of switching times. However, since [52] it is known that a general theory must allow also, e. g., for increasing numbers of switchings and other more general families of control variations. The chronological calculus is instrumental to developing such general theory [4].

Chronological Algebra When using the chronological calculus for studying controllability, especially when differentiating with respect to a parameter, and also in the aforementioned averaging theory, the following chronological product appears almost everywhere: For two time-varying vector fields f t and g t that are absolutely continuous with respect to t, define their chronological product as Z t [ f s ; g s0 ]ds (27) ( f ? g) t D 0

where [; ] denotes the Lie bracket. It is easily verified that this product is generally neither associative nor satisfies the Jacobi identity, but instead it satisfies the chronological identity x ? (y ? z)  y ? (x ? z) D (x ? y) ? z  (y ? x) ? z : (28) One may define an abstract chronological algebra (over a field k) as a linear space that is equipped with a bilinear product that satisfies the chronological identity (28). More compactly, this property may be written in terms of the left translation  that associates with any element x 2 A of a not-necessarily associative algebra A the map x : y 7! x y. Using this , the chronological identity (28) is simply [x;y] D [x ;  y ] ;

(29)

i. e., the requirement that the map x 7! x is a homomorphism of the algebra A with the commutator product (x; y) 7! x y  yx into the Lie algebra of linear maps from A into A under its commutator product. Basic general properties of chronological algebras have been studied in [2], including a discussion of free chronological algebras (over a fixed generating set S). Of particular interest are bases of such free chronological algebras as they allow one to minimize the number of terms in series expansions by avoiding redundant terms. Using the graded structure, it is shown in [2] that an ordered basis of a free chronological algebra over a set S may be obtained recursively from products of the form b 1 b 2    b k s where s 2 S is a generator and b1 4 b2 4    4 b k are previously constructed basis elements. According to [2], the first elements of such a basis over the single-element generating set S D fsg may be chosen as (using juxtaposition for multiplication) b1 D s ; b 2 D b 1 s D s 2 ; b3 D b 2 s D s 2 s ; b4 D b 1 b 1 s D ss 2 ; b5 D b 3 s D (s 2 s)s ; b6 D b 4 s D (ss 2 )s ; b7 D b 1 b 2 s D s(s 2 s) ; b8 D b 2 b 1 s D s(ss 2 ) ; : : :

(30)

93

94

Chronological Calculus in Systems and Control Theory

Using only this algebraic and combinatorial structure, one may define exponentials (and logarithms) and analyze the group of formal flows [2]. Of particular interest systems and control is an explicit formula that allows one to express products (compositions) of two or more exponentials (flows) as a single exponential. This is tantamount to a formula for the logarithm of a product of exponentials of two noncommuting indeterminates X and Y. The classical version is known as the Campbell–Baker–Hausdorff formula [16], the first few terms being 1

1

1

e X  eY D e XCYC 2 [X;Y]C 12 [XY;[X;Y]] 24 [X;[Y;[X;Y]]]::: : (31) A well-known short-coming of the classical formula is that the higher-order iterated Lie brackets appearing in this formula are always linearly dependent and hence the coefficients are never well-defined. Hence one naturally looks for more compact, minimal formulas which avoid such redundancies. The chronological algebra provides one such alternative and has been developed and utilized in [2]. The interested reader is referred to the original reference for technical details. Systems That Are Affine in the Control Separating Time-Varying Control and Geometry The chronological calculus was originally designed for work with nonstationary (time-varying) families of vector fields, and it arguably reaps the biggest benefits in that setting where there are few other powerful tools available. Nonetheless, the chronological calculus also much streamlines the analysis of autonomous vector fields. The best studied case is that of affine control systems in which the time-varying control can be separated from the geometry determined by autonomous vector fields. In classical notation such control systems are written in the form x˙ D u1 (t) f1 (x) C    C u m (t) f m (x) :

(32)

where f i are vector fields on a manifold M and u is a measurable function of time taking values typically in a compact subset U  Rm . This description allows one to also accommodate physical systems that have an uncontrolled drift term by simply fixing u1  1. For the sake of clarity, and for best illustrating the kind of results available, assume that the fields f i are smooth and complete. Later we specialize further to real analytic vector fields. This set-up does not necessarily require an interpretation of the ui as controls: they may equally well be disturbances, or it may simply be a dynamical system which splits in this convenient way.

As a starting point consider families of piecewise constant control functions u : [0; T] 7! U  Rm . On each interval [t j ; t jC1 ] on which the control is constant u[t j ;t jC1] D u( j) , the right hand side of (32) is a fixed vector ( j)

( j)

field g j D u1 f1 C : : : u m f m . The endpoint of the solution curve starting from x(0) D p is computed as a directed product of ordinary exponentials (flows) of vector fields pe(t1 t0 )g 1 ı e(t2 t1 )g 2 ı    e(Tt s1 )g s :

(33)

But even in this case of autonomous vector fields the chronological calculus brings substantial simplifications. Basically this means that rather than considering the expression (33) as a point on M, consider this product as a functional on the space E (M) of smooth functions on M. To reiterate the benefit of this approach [5,57] consider the simple example of a tangent vector to a smooth curve

: ("; ") 7! M which one might want to define as 1

˙ (0) D lim ( (t)  (0)) : t!0 t

(34)

Due to the lack of a linear structure on a general manifold, with a classical interpretation this does not make any sense at all. Nonetheless, when interpreting (t) ; (0) ; 0 (0), and 1t ( (t) (0)) as linear functionals on the space E (M) of smooth functions on M this is perfectly meaningful. The meaning of the limit can be rigorously justified as indicated in the preceding sections, compare also [1]. A typical example to illustrate how this formalism almost trivializes important calculations involving Lie brackets is the following, compare [57]. Suppose f and g are smooth vector fields on the manifold M, p 2 M, and t 2 R is sufficiently small in magnitude. Using that d tf tf dt pe D pe f and so on, one immediately calculates d  t f t g t f t g  pe e e e dt D pe t f f e t g et f et g C pe t f e t g get f et g  pe t f e t g et f f et g  pe t f e t g et f et g g :

(35)

In particular, at t D 0 this expression simplifies to p f C pg  p f  pg D 0. Analogously the second derivative is calculated as d2  t f t g t f t g  pe e e e dt 2 D pe t f f 2 e t g et f et g C pe t f f  e t g get f et g  pe t f f et g et f f et g  pe t f f e t g et f  et g g C pe t f f e t g get f et g C pe t f e t g g 2 et f et g  pe t f e t g get f f et g  pe t f e t g get f et g g  pe t f f

Chronological Calculus in Systems and Control Theory

 e t g et f f et g  pe t f e t g get f f et g C pe t f e t g et f 2 t g

f e

t f t g t f

C pe e e

t f tg

 pe e ge

t f t g

e

t g

fe

t g t f t g

tf

g  pe f e e

t f t g t f

g C pe e e

t g

fe

g C pe

t g t f t g 2

e e

e

e

g

g

tf

(36)

which at t D 0 evaluates to p f 2 C p f g  p f 2  p f g C p f g C pg 2  pg f  pg 2  p f 2  pg f C p f 2 C p f g  p f g  pg 2 C p f g C pg 2

where the sum ranges over all multi-indices I D (i1 ; : : : ; i s ); s  0 with i j 2 f1; 2 : : : mg. This series gives rise to an asymptotic series for solution curves of the system (32). For example, in the analytic setting, following [91], one has: Theorem 2 Suppose f i are analytic vector fields on Rn  : Rn 7! R is analytic and U Rm is compact. Then for every compact set K  Rn , there exists T > 0 such that the series

D 2p f g  2pg f D 2p[ f ; g] : While this simple calculation may not make sense on a manifold with a classical interpretation in terms of points and vectors on a manifold, it does so in the context of the chronological calculus, and establishes the familiar formula for the Lie bracket as the limit of a composition of flows pe t f e t g et f et g D pCt 2 p[ f ; g]CO(t 2 ) as t ! 0: (37) Similar examples may be found in [5] (e. g. p. 36), as well as analogous simplifications for the variations formula for autonomous vector fields ([5] p. 43). Asymptotic Expansions for Affine Systems Piecewise constant controls are a starting point. But a general theory requires being able to take appropriate limits and demands completeness. Typically, one considers measurable controls taking values in a compact subspace U  Rm , and uses L1 ([0; t]; U) as the space of admissible controls. Correspondingly the finite composition of exponentials as in (33) is replaced by continuous formulae. These shall still separate the effects of the time-varying controls from the underlying geometry determined by the autonomous vector fields. This objective is achieved by the Chen–Fliess series which may be interpreted in a number of different ways. Its origins go back to the 1950s when K. T. Chen [18], studying geometric invariants of curves in Rn , associated to each smooth curve a formal noncommutative power series. In the early 1970s, Fliess [30,31] recognized the utility of this series for studying control systems. Using a set X 1 ; X2 ; : : : ; X m of indeterminates, the Chen– Fliess series of a measurable control u 2 L1 ([0; t]; U) as above is the formal power series SCF (T; u) D

XZ 0

I

Z  0

Z u i s (ts )

T

Z t3    u i 2 (t2 )

t s1 0

0

t2

u i 1 (t1 ) dt1 : : : dts  X i 1 : : : X i s

(38)

SCF; f (T; u; p)(') D Z  0

Z u i s (ts )

XZ

T

0

I

0

Z t3    u i 2 (t2 )

t s1

0

t2

u i 1 (t1 ) dt1 : : : dts  p f i 1 : : : f i s '

(39)

converges uniformly to the solution of (32) for initial conditions x(0) D p 2 K and u : [0; T] 7! U measurable. Here the series SCF; f (T; u; p) is, for every fixed triple ( f ; u; p), interpreted as an element of the space of linear functionals on E (M). But one may abstract further. In particular, for each fixed multi-index I as above, the iterated integral coefficient is itself a functional that takes as input a control function u 2 Um and maps it to the corresponding iterated integral. R t It is convenient to work with the primitives U j : t 7! 0 u j (s) ds of the control functions uj rather than the controls themselves. More specifically, if e. g. U D AC([0; T]; [1; 1]) then for every multi-index I D (i1 ; : : : ; i s ) 2 f1; : : : ; mgs as above one obtains the iterated integral functional I : Um 7! U defined by Z

T

I : U 7! 0

U i0s (ts )

Z

t s1

Z

 0

0

Z

t3

U i02 (t2 )

t2

 0

U i01 (t1 ) dt1 dt2 : : : dts :

(40)

Denoting by IIF (U Z ) the linear space of iterated integral functionals spanned by the functionals of the above form, the map  maps multi-indices to IIF (U Z ). The algebraic and combinatorial nature of these spaces and this map will be explored in the next section. On the other side, there is the map F that maps a multi-index (i1 ; i2 ; : : : ; i s ) to the partial differential operator f i 1 : : : f i s : E (M) 7! E (M), considered as a linear transformation of E (M). In the analytic case one may go further, since as a basic principle, the relations between the iterated Lie brackets of the vector fields f j completely determine the geometry and dynamical behavior [90,91]. Instead of considering actions on E (M), it suffices to consider the action of these partial differential operators, and

95

96

Chronological Calculus in Systems and Control Theory

Zinbiel Algebra and Combinatorics

w a D wa and w (za) D (w z C z w)a (41) and extending bilinearly to AC (Z)  AC (Z). One easily verifies that this product satisfies the Zinbiel identity for all r; s; t 2 AC (Z) r (s t) D (r s) t C (s r) t :

(42)

The symmetrization of this product is the better known associative shuffle product w z D w z C z w which may be defined on all of A(Z)  A(Z). Algebraically, the shuffle product is characterized as the transpose of the coproduct  : A(Z) 7! A(Z) ˝ A(Z) which is the concatenation product homomorphism that on letters a 2 Z is defined as  : a 7! a ˝ 1 C 1 ˝ a :

(43)

This relation between this coproduct and the shuffle is that for all u; v; w 2 A(Z) 9

h(w); u ˝ vi D hw; u

vi :

(44)

After this brief side-trip into algebraic and combinatorial objects we return to the control systems. The shuffle product has been utilized for a long time in this and related contexts. But the Zinbiel product structure was only recently recognized as encoding important information. Its most immediate role is seen in the fact that the aforementioned map  from multi-indices to iterated integral functionals, now easily extends to a map (using the same name)  : A(Z) 7! IIF (U Z ) is a Zinbiel algebra homomorphism when the latter is equipped with a pointwise product induced by the product on U D AC([0; T]; [1; 1]) defined by Z t U(s)V 0 (s)ds : (45) (U V)(t) D 9

Just as the chronological calculus of time-varying vector fields is intimately connected to chronological algebras, the calculus of affine systems gives rise to its own algebraic and combinatorial structure. This, and the subsequent section, give a brief look into this algebra structure and demonstrate how, together with the chronological calculus, it leads to effective solution formulas for important control problems and beyond. It is traditional and convenient to slightly change the notation and nomenclature. For further details see [78] or [57]. Rather than using small integers 1; 2; : : : ; m to index the components of the control and the vector fields in (32), use an arbitrary index set Z whose elements will be called letters and denoted usually by a; b; c; : : : For the purposes of this note the alphabet Z will be assumed to be finite, but much of the theory can be developed for infinite alphabets as well. Multi-indices, i. e. finite sequences with values in Z are called words, and they have a natural associative product structure denoted by juxtaposition. For example, if w D a1 a2 : : : ar and z D b1 b2 : : : bs then wz D a1 a2 : : : ar b1 b2 : : : bs . The empty word is denoted e n or 1. The set of all words of length n is Zn , Z  D [1 nD0 Z C 1 n and Z D [nD1 Z are the sets of all words and all nonempty words. Endow the associative algebra A(Z), generated by the alphabet Z with coefficients in the field k D R, with an inner product h; i so that Z  is an orthonormal ˆ basis. Write A(Z) for the completion of A(Z) with respect to the uniform structure in which a fundamental system of basic neighborhoods of a polynomial p 2 A(Z) is the family of subsets fq 2 A(Z) : 8w 2 W; hw; p  qi D 0g indexed by all finite subsets W Z  .

The smallest linear subspace of the associative algebra A(Z) that contains the set Z and is closed under the commutator product (w; z) 7! [w; z] D wzzw is isomorphic to the free Lie algebra L(Z) over the set Z. On the other side, one may equip AC (Z) with a Zinbiel algebra structure by defining the product : AC (Z)  AC (Z 7! AC (Z) for a 2 Z and w; z 2 Z C by

9

of the dynamical system (32) on a set of polynomial functions. In a certain technical sense, an associative algebra (of polynomial functions) is dual to the Lie algebra (generated by the analytic vector fields f j ). However, much further simplifications and deeper insights are gained by appropriately accounting for the intrinsic noncommutative structure of the flows of the system. Using a different product structure, the next section will lift the system (32) to a universal system that has no nontrivial relations between the iterated Lie brackets of the formal vector fields and which is modeled on an algebra of noncommutative polynomials (and noncommutative power series) rather than the space E (M) of all smooth functions on M. An important feature of this approach is that it does not rely on polynomial functions with respect to an a-priori chosen set of local coordinates, but rather uses polynomials which map to intrinsically geometric objects.

0

It is straightforward to verify that this product indeed equips U with a Zinbiel structure, and hence also IIF (U Z ). On the side, note that the map  maps the shuffle product of words (noncommuting polynomials) to the associative pointwise product of scalar functions. The connection of the Zinbiel product with differential equations and the Chen–Fliess series is immediate. First lift

Chronological Calculus in Systems and Control Theory

the system (32) from the manifold M to the algebra A(Z), roughly corresponding to the linear space of polynomial functions in an algebra of iterated integral functionals. Formally consider curves S : [0; T] 7! A(Z) that satisfy S˙ D S

X

au a (t) :

(46)

a2Z

the Zinbiel product (45), and writing U a (t) D RUsing t u () d for the primitives of the controls, the inte0 a grated form of the universal control system (46) Z S(t) D 1 C

t

S()F 0 () d with F D

0

X

U a ; (47)

a2Z

is most compactly written as S D1CS F :

(48)

Iteration yields the explicit series expansion S D 1 C (1 C S F) F D 1 C F C ((1 C S F) F) F D 1 C F C (F F) C (((1 C S F) F) F) F D 1 C F C (F F) C ((F F) F) C ((((1 C S F) F) F) F) F :: : D 1 C F C (F F) C ((F F) F) C (((F F) F) F) : : : Using intuitive notation for Zinbiel powers this solution formula in the form of an infinite series is compactly written as SD

1 X

necessary conditions and sufficient conditions for local controllability and optimality, compare e. g. [53,88,94], this series expansion has substantial shortcomings. For example, it is clear from Ree’s theorem [77] that the series is an exponential Lie series. But this is not at all obvious from the series as presented (38). Most detrimental for practical purposes is that finite truncations of the series never correspond to any approximating systems. Much more convenient, in particular for path planning algorithms [47,48,63,64,75,95], but also in applications for numerical integration [17,46] are expansions as directed infinite products of exponentials or as an exponential of a Lie series. In terms of the map F that substitutes for each letter a 2 Z the corresponding vector field f a , one finds that the Chen–Fliess series (39) is simply the image of a natural object under the map  ˝ F . Indeed, under the usual identification of the space Hom(V ; W) of linear maps between vector spaces V and W with the product V  ˝ W, and noting that Z is an orthonormal basis for A(Z), the identity map from A(Z) to A(Z) is identified with the series X ˆ w ˝ w 2 A(Z) ˝ A(Z) : (50) IdA(Z)  w2Z 

Thus, any rewriting of the combinatorial object on the right hand side will, via the map  ˝ F , give an alternative presentation of the Chen–Fliess series. In particular, from elementary consideration, using also the Hopf-algebraic structures of A(Z), it is a-priori clear that there exist expansions in the forms ! X X w ˝ w D exp  h ˝ [h] w2Z 

h2H

D F

n

2

D 1C F C F C F

3

4

CF CF

5

6

C F C  

nD0

(49) The reader is encouraged to expand all of the terms and match each step to the usual computations involved in obtaining Volterra series expansions. Indeed this formula (49) is nothing more than the Chen–Fliess series (38) in very compact notation. For complete technical details and further discussion we refer the interested reader to the original articles [56,57] and the many references therein. Application: Product Expansions and Normal Forms While the Chen–Fliess series as above has been extremely useful to obtain precise estimates that led to most known

Y

exp ( h ˝ [h])

(51)

h2H

where H indexes an ordered basis f[h] : H 2 H g of the free Lie algebra L(Z) and for each h 2 H ;  h and  h are polynomials in A(Z) that are mapped by  to corresponding iterated integral functionals. The usefulness of such expression depends on the availability of simple formulas for H ,  h and  h . Bases for free Lie algebras are well-known since Hall [42], and have been unified by Viennot [101]. Specifically, a Hall set over a set Z is any strictly ordered subset ˜  T (Z) from the set M(Z) labeled binary trees (with H leaves labeled by Z) that satisfies ˜ (i) Z  H ˜ iff t 0 2 H ˜ ; t0 < a (ii) Suppose a 2 Z. Then (t; a) 2 H 0 and a < (t ; a).

97

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Chronological Calculus in Systems and Control Theory

˜ . Then (t 0 ; (t 000; t 0000 )) 2 H ˜ (iii) Suppose u; v; w; (u; v) 2 H iff t 000  t 0  (t 000 ; t 0000 ) and t 0 < (t 0 ; (t 000 ; t 0000 )). There are natural mappings : T (Z) 7! L(Z)  A(Z) and # : T (Z) 7! A(Z) (the foliage map) that map labeled binary trees to Lie polynomials and to words, and which are defined for a 2 Z, (t 0 ; t 00 ) 2 T (Z) by (a) D #(a) D a and recursively ((t 0 ; t 00 )) D [ (t 0 ); (t 00 )] and #((t 0 ; t 00 )) D #(t 0 )#(t 00 ). The image of a Hall set under the map is a basis for the free Lie algebra L(Z). Moreover, the restriction of the map # to a Hall-set is one-to-one which leads to a fundamental unique factorization property of words into products of Hall-words, and a unique way of recovering Hall trees from the Hall words that are their images under # [101]. This latter property is very convenient as it allows one to use Hall words rather than Hall trees for indexing various objects. The construction of Hall sets, as well as the proof that they give rise to bases of free Lie algebras, rely in an essential way on the process of Lazard elimination [9] which is intimately related to the solution of differential Eqs. (32) or (46) by a successive variation of parameters [93]. As a result, using Hall sets it is possible to obtain an extremely simple and elegant recursive formula for the coefficients  h in (51), whereas formulas for the  h are less immediate, although they may be computed by straightforward formulas in terms of natural maps of the Hopf algebra structure on A(Z) [33]. Up to a normalization factor  in terms of multi-factorials, the formula for the  h is extremely simple in terms of the Zinbiel product on A(Z). For letters a 2 Z and Hall words h; k; hk 2 H one has a D a

and

 hk D  hk   h  k :

(52)

Using completely different notation, this formula appeared originally in the work of Schützenberger [84], and was later rediscovered in various settings by Grayson and Grossman [38], Melancon and Reutenauer [69] and Sussmann [93]. In the context of control, it appeared first in [93]. To illustrate the simplicity of the recursive formula, which is based on the factorization of Hall words, consider the following normal form for a free nilpotent system (of rank r D 5) using a typical Hall set over the alphabet Z D f0; 1g. This is the closest nonlinear analogue to the Kalman controller normal form of a linear system, which is determined by Kronecker indices that classify the lengths of chains of integrators. For practical computations nilpotent systems are commonly used as approximations of general nonlinear systems – and every nilpotent system (that is, a system of form (32) whose vector fields f a generate a nilpotent Lie algebra) is the image of a free

nilpotent system as below under some projection map. For convenience, the example again uses the notation xh instead of  h . x˙0 D u0 x˙1 D u1 x˙01 D x0  x˙1 D x0 u1 x˙001 D x0  x˙01 D x02 u1 using # 1 (001) D (0(01)) x˙101 D x1  x˙01 D x1 x0 u1 using # 1 (101) D (1(01)) x˙0001 D x0  x˙001 D x03 u1 using # 1 (0001) D (0(0(01))) x˙1001 D x1  x˙001 D x1 x02 u1 using # 1 (1001) D (1(0(01))) x˙1101 D x1  x˙101 D x12 x0 u1 using # 1 (1101) D (1(1(01))) x˙00001 D x0  x˙001 D x04 u1 using # 1 (00001) D (0(0(0(01)))) x˙10001 D x1  x˙0001 D x1 x03 u1 using # 1 (10001) D (1(0(0(01)))) x˙11001 D x1  x˙1001 D x12 x02 u1 using # 1 (11001) D (1(1(0(01)))) x˙01001 D x01  x˙001 D x01 x03 u1 using # 1 (01001) D ((01)(0(01))) x˙01101 D x01  x˙101 D x01 x12 x0 u1 using # 1 (01101) D ((01)(1(01))) : This example may be interpreted as a system of form (32) with the xh being a set of coordinate functions on the manifold M, or as the truncation of the lifted system (46) with the xh being a basis for a finite dimensional subspace of the linear space E (M). For more details and the technical background see [56,57]. Future Directions The chronological calculus is still a young methodology. Its intrinsically infinite dimensional character demands a certain sophistication from its users, and thus it may take some time until it reaches its full potential. From a different point of view this also means that there are many opportunities to explore its advantages in ever new areas of applications. A relatively straightforward approach simply checks classical topics of systems and control for whether

Chronological Calculus in Systems and Control Theory

they are amenable to this approach, and whether this will be superior to classical techniques. The example of [70] which connects the chronological calculus with discrete time systems is a model example for such efforts. There are many further opportunities, some more speculative than others. Control The chronological calculus appears ideally suited for the investigation of fully nonlinear systems – but in this area there is still much less known than in the case of nonlinear control systems that are affine in the control, or the case of linear systems. Specific conditions for the existence of solutions, controllability, stabilizability, and normal forms are just a few subjects that, while studies have been initiated [1,2,4], deserve further attention. Computation In recent years much effort has been devoted to understanding the foundations of numerical integration algorithms in nonlinear settings, and to eventually design more efficient algorithms and eventually prove their superiority. The underlying mathematical structures, such as the compositions of noncommuting flows are very similar to those studies in control, as observed in e. g. [23], one of the earlier publications in this area. Some more recent work in this direction is [17,46,72,73,74]. This is also very closely related to the studies of the underlying combinatorial and algebraic structures – arguably [27,28,29,76] are some of the ones most closely related to the subject of this article. There appears to be much potential for a two-way exchange of ideas and results. Geometry Arguably one of the main thrusts will continue to be the use of the chronological calculus for studying the differential geometric underpinnings of systems and control theory. But it is conceivable that such studies may eventually abstract further – just like after the 1930s – to studies of generalizations of algebras that stem from systems on classical manifolds, much in the spirit of modern noncommutative geometry with [19] being the best-known proponent. Algebra and combinatorics In general very little is known about possible ideal and subalgebra structures of chronological and Zinbiel algebras. This work was initiated in [2], but relatively little progress has been made since. (Non)existence of finite dimensional (nilpotent) Zinbiel algebras has been established, but only over a complex field [25,26]. Bases of free chronological algebras are discussed in [2], but many other aspects of this algebraic structure remain unexplored. This also intersects with the aforementioned efforts in foundations of computation – on one side there are

Hopf-algebraic approaches to free Lie algebras [78] and nonlinear control [33,40,41], while the most recent breakthroughs involve dendriform algebras and related subjects [27,28,29]. This is an open field for uncovering the connections between combinatorial and algebraic objects on one side and geometric and systems objects on the other side. The chronological calculus may well serve as a vehicle to elucidate the correspondences. Acknowledgments This work was partially supported by the National Science Foundation through the grant DMS 05-09030 Bibliography 1. Agrachëv A, Gamkrelidze R (1978) Exponential representation of flows and chronological calculus. Math Sbornik USSR (Russian) 107(N4):487–532. Math USSR Sbornik (English translation) 35:727–786 2. Agrachëv A, Gamkrelidze R (1979) Chronological algebras and nonstationary vector fields. J Soviet Math 17:1650–1675 3. Agrachëv A, Gamkrelidze R, Sarychev A (1989) Local invariants of smooth control systems. Acta Appl Math 14:191–237 4. Agrachëv A, Sachkov YU (1993) Local controllability and semigroups of diffeomorphisms. Acta Appl Math 32:1–57 5. Agrachëv A, Sachkov YU (2004) Control Theory from a Geometric Viewpoint. Springer, Berlin 6. Agrachëv A, Sarychev A (2005) Navier Stokes equations: controllability by means of low modes forcing. J Math Fluid Mech 7:108–152 7. Agrachëv A, Vakhrameev S (1983) Chronological series and the Cauchy–Kowalevski theorem. J Math Sci 21:231–250 8. Boltyanski V, Gamkrelidze R, Pontryagin L (1956) On the theory of optimal processes (in Russian). Doklady Akad Nauk SSSR, vol.10, pp 7–10 9. Bourbaki N (1989) Lie Groups and Lie algebras. Springer, Berlin 10. Brockett R (1971) Differential geometric methods in system theory. In: Proc. 11th IEEE Conf. Dec. Cntrl., Berlin, pp 176–180 11. Brockett R (1976) Volterra series and geometric control theory. Autom 12:167–176 12. Bullo F (2001) Series expansions for the evolution of mechanical control systems. SIAM J Control Optim 40:166–190 13. Bullo F (2002) Averaging and vibrational control of mechanical systems. SIAM J Control Optim 41:542–562 14. Bullo F, Lewis A (2005) Geometric control of mechanical systems: Modeling, analysis, and design for simple mechanical control systems. Texts Appl Math 49 IEEE 15. Caiado MI, Sarychev AV () On stability and stabilization of elastic systems by time-variant feedback. ArXiv:math.AP/0507123 16. Campbell J (1897) Proc London Math Soc 28:381–390 17. Casas F, Iserles A (2006) Explicit Magnus expansions for nonlinear equations. J Phys A: Math General 39:5445–5461 18. Chen KT (1957) Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula. Ann Math 65:163– 178

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control problems with general boundary conditions. J Dyn Control Syst 11:91–123 Sontag E, Wang Y (1992) Generating series and nonlinear systems: analytic aspects, local realizability and i/o representations. Forum Math 4:299–322 Stefani G (1985) Polynomial approximations to control systems and local controllability. In: Proc. 25th IEEE Conf. Dec. Cntrl., pp 33–38, New York Stone M (1932) Linear Transformations in Hilbert Space. Amer Math Soc New York Sussmann H (1974) An extension of a theorem of Nagano on transitive Lie algebras. Proc Amer Math Soc 45:349–356 Sussmann H (1983) Lie brackets and local controllability: a sufficient condition for scalar-input systems. SIAM J Cntrl Opt 21:686–713 Sussmann H (1983) Lie brackets, real analyticity, and geometric control. In: Brockett RW, Millman RS, Sussmann HJ (eds) Differential Geometric Control. pp 1–116, Birkhauser Sussmann H (1986) A product expansion of the Chen series. In: Byrnes C, Lindquist A (eds) Theory and Applications of Nonlinear Control Systems. Elsevier, North-Holland, pp 323– 335 Sussmann H (1987) A general theorem on local controllability. SIAM J Control Opt 25:158–194 Sussmann H (1992) New differential geometric methods in nonholonomic path finding. In: Isidori A, Tarn T (eds) Progr Systems Control Theory 12. Birkhäuser, Boston, pp 365– 384 Tretyak A (1997) Sufficient conditions for local controllability and high-order necessary conditions for optimality. A differential-geometric approach. J Math Sci 85:1899–2001 Tretyak A (1998) Chronological calculus, high-order necessary conditions for optimality, and perturbation methods. J Dyn Control Syst 4:77–126 Tretyak A (1998) Higher-order local approximations of smooth control systems and pointwise higher-order optimality conditions. J Math Sci 90:2150–2191 Vakhrameev A (1997) A bang-bang theorem with a finite number of switchings for nonlinear smooth control systems. Dynamic systems 4. J Math Sci 85:2002–2016 Vela P, Burdick J (2003) Control of biomimetic locomotion via averaging theory. In: Proc. IEEE Conf. Robotics and Automation. pp 1482–1489, IEEE Publications, New York Viennot G (1978) Algèbres de Lie Libres et Monoïdes Libres. Lecture Notes in Mathematics, vol 692. Springer, Berlin Visik M, Kolmogorov A, Fomin S, Shilov G (1964) Israil Moiseevich Gelfand, On his fiftieth birthday. Russ Math Surv 19:163– 180 Volterra V (1887) Sopra le funzioni che dipendono de altre funzioni. In: Rend. R Academia dei Lincei. pp 97–105, 141– 146, 153–158 von Neumann J (1932) Mathematische Grundlagen der Quantenmechanik. Grundlehren Math. Wissenschaften 38. Springer, Berlin Zelenko I (2006) On variational approach to differential invariants of rank two distributions. Diff Geom Appl 24:235–259

101

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Control of Non-linear Partial Differential Equations FATIHA ALABAU-BOUSSOUIRA 1 , PIERMARCO CANNARSA 2 1 L.M.A.M., Université de Metz, Metz, France 2 Dipartimento di Matematica, Università di Roma “Tor Vergata”, Rome, Italy Article Outline Glossary Definition of the Subject Introduction Controllability Stabilization Optimal Control Future Directions Bibliography Glossary R denotes the real line, Rn the n-dimensional Euclidean space, x  y stands for the Euclidean scalar product of x; y 2 Rn , and jxj for the norm of x. State variables quantities describing the state of a system; in this note they will be denoted by u; in the present setting, u will be either a function defined on a subset of R  Rn , or a function of time taking its values in an Hilbert space H. Space domain the subset of Rn on which state variables are defined. Partial differential equation a differential equation containing the unknown function as well as its partial derivatives. State equation a differential equation describing the evolution of the system of interest. Control function an external action on the state equation aimed at achieving a specific purpose; in this note, control functions they will be denoted by f ; f will be used to denote either a function defined on a subset of R  Rn , or a function of time taking its values in an Hilbert space F. If the state equation is a partial differential equation of evolution, then a control function can be: 1. distributed if it acts on the whole space domain; 2. locally distributed if it acts on a subset of the space domain; 3. boundary if it acts on the boundary of the space domain; 4. optimal if it minimizes (together with the corresponding trajectory) a given cost;

5. feedback if it depends, in turn, on the state of the system. Trajectory the solution of the state equation uf that corresponds to a given control function f . Distributed parameter system a system modeled by an evolution equation on an infinite dimensional space, such as a partial differential equation or a partial integro-differential equation, or a delay equation; unlike systems described by finitely many state variables, such as the ones modeled by ordinary differential equations, the information concerning these systems is “distributed” among infinitely many parameters. 1A denotes the characteristic function of a set A Rn , that is, ( 1 x2A 1A (x) D 0 x 2 Rn n A @ t ; @x i denote partial derivatives with respect to t and xi , respectively. L2 (˝) denotes the Lebesgue space of all real-valued square integrable functions, where functions that differ on sets of zero Lebesgue measure are identified. H01 (˝) denotes the Sobolev space of all real-valued functions which are square integrable together with their first order partial derivatives in the sense of distributions in ˝, and vanish on the boundary of ˝; similarly H 2 (˝) denotes the space of all functions which are square integrable together with their second order partial derivatives. H 1 (˝) denotes the dual of H01 (˝). H n1 denotes the (n  1)-dimensional Hausdorff measure. H denotes a normed spaces over R with norm k  k, as well as an Hilbert space with the scalar product h; i and norm k  k. L2 (0; T; H) is the space of all square integrable functions f : [0; T] ! H; C([0; T]; H) (continuous functions) and H 1 (0; T; H) (Sobolev functions) are similarly defined . Given Hilbert spaces F and H, L(F; H) denotes the (Banach) space of all bounded linear operators  : F ! H with norm kk D supkxkD1 kxk (when F D H, we use the abbreviated notation L(H));  : H ! F denotes the adjoint of  given by h u; i D hu; i for all u 2 H,  2 F. Definition of the Subject Control theory (abbreviated, CT) is concerned with several ways of influencing the evolution of a given system by an external action. As such, it originated in the nine-

Control of Non-linear Partial Differential Equations

teenth century, when people started to use mathematics to analyze the perfomance of mechanical systems, even though its roots can be traced back to the calculus of variation, a discipline that is certainly much older. Since the second half of the twentieth century its study was pursued intensively to address problems in aerospace engineering, and then economics and life sciences. At the beginning, CT was applied to systems modeled by ordinary differential equations (abbreviated, ODE). It was a couple of decades after the birth of CT—in the late sixties, early seventies—that the first attempts to control models described by a partial differential equation (abbreviated, PDE) were made. The need for such a passage was unquestionable: too many interesting applications, from diffusion phenomena to elasticity models, from fluid dynamics to traffic flows on networks and systems biology, can be modeled by a PDE. Because of its peculiar nature, control of PDE’s is a rather deep and technical subject: it requires a good knowledge of PDE theory, a field of enormous interest in its own right, as well as familiarity with the basic aspects of CT for ODE’s. On the other hand, the effort put into this research direction has been really intensive. Mathematicians and engineers have worked together in the construction of this theory: the results—from the stabilization of flexible structures to the control of turbulent flows—have been absolutely spectacular. Among those who developed this subject are A. V. Balakrishnan, H. Fattorini, J. L. Lions, and D. L. Russell, but many more have given fundamental contributions. Introduction The basic examples of controlled partial differential equations are essentially two: the heat equation and the and the wave equation. In a bounded open domain ˝ Rn with sufficiently smooth boundary  the heat equation @ t u D u C f

: in Q T D (0; T)  ˝

(1)

describes the evolution in time of the temperature u(t; x) at any point x of the body ˝. The term u D @2x 1 u C    C @2x n u, called the Laplacian of u, accounts for heat diffusion in ˝, whereas the additive term f represents a heat source. In order to solve the above equation uniquely one needs to add further data, such as the initial distribution u0 and the temperature of the boundary surface  of ˝. The fact that, for any given data u0 2 L2 (˝) and f 2 L2 (Q T ) Eq. (1) admits a unique weak solution uf satisfying the boundary condition : u D 0 on ˙T D (0; T)  

(2)

and the initial condition u(0; x) D u0 (x) 8x D (x1 ; : : : ; x n ) 2 ˝

(3)

is well-known. So is the maximal regularity result ensuring that     u f 2 H 1 0; T; L2 (˝) \ C [0; T]; H01 (˝)   \ L2 0; T; H 2 (˝)

(4)

whenever u0 2 H01 (˝). If problem (1)–(3) possesses a unique solution which depends continuously on data, then we say that the problem is well-posed. Similarly, the wave equation @2t u D u C f

in

QT

(5)

describes the vibration of an elastic membrane (when n D 2) subject to a force f . Here, u(t; x) denotes the displacement of the membrane at time t in x. The initial condition now concerns both initial displacement and velocity: ( u(0; x) D u0 (x) 8x 2 ˝ (6) @ t u(0; x) D u1 (x) : It is useful to treat the above problems as a first order evolution equation in a Hilbert space H u0 (t) D Au(t) C B f (t)

t 2 (0; T) ;

(7)

where f (t) takes its valued in another Hilbert space F, and B 2 L(F; H). In this abstract set-up, the fact that (7) is related to a PDE translates into that the closed linear operator A is not defined on the whole space but only on a (dense) subspace D(A) H, called the domain of A; such a property is often referred to as the unboundedness of A. For instance, in the case of the heat equation (1), H D L2 (˝) D F, and A is defined as ( D(A) D H 2 (˝) \ H01 (˝) (8) Au D u ; 8u 2 D(A) ; whereas B D I. As for the wave equation, since it is a second order differential equation with respect to t, the Hilbert space H should be given by the product H01 (˝)  L2 (˝). Then, problem (5) is turned into the first order equation U 0 (t) D AU(t) C B f (t)

t 2 (0; T) ;

where  UD

u v



 ;

BD

0 I

 ;

F D L2 (˝) :

103

104

Control of Non-linear Partial Differential Equations

Accordingly, A : D(A) H ! H is given by 8   D(A) D H 2 (˝) \ H01 (˝)  H01 (˝) ˆ ˆ < ! ! 0 I v ˆ ˆ UD 8U 2 D(A) ; :AU D A 0 Au where A is taken as in (8). Another advantage of the abstract formulation (7) is the possibility of considering locally distributed or boundary source terms. For instance, one can reduce to the same set-up the equation @ t u D u C 1! f

in Q T ;

(9)

where 1! denotes the characteristic function of an open set ! ˝, or the nonhomogeneus boundary condition of Dirichlet type uD f

on ˙T ;

(10)

or Neumann type @u D f @

on ˙T ;

(11)

where  is the outward unit normal to  . For Eq. (9), B reduces to multiplication by 1! —a bounded operator on L2 (˝); conditions (10) and (11) can also be associated to suitable linear operators B—which, in this case, turn out to be unbounded. Similar considerations can be adapted to the wave equation (5) and to more general problems. Having an efficient way to represent a source term is essential in control theory, where such a term is regarded as an external action, the control function, exercised on the state variable u for a purpose, of which there are two main kinds:  positional: u(t) is to approach a given target in X, or attain it exactly at a given time t > 0;  optimal: the pair (u; f ) is to minimize a given functional. The first criterion leads to approximate or exact controllability problems in time t, as well as to stabilization problems as t ! 1. Here, the main tools will be provided by certain estimates for partial differential operators that allow to study the states that can be attained by the solution of a given controlled equation. These issues will be addressed in Sects. “Controllability” and “Stabilization” for linear evolution equations. Applications to the heat and wave equations will be discussed in the same sections. On the other hand, optimal control problems require analyzing the typical issues of optimizations: existence results, necessary conditions for optimality, sufficient conditions, robustness. Here, the typical problem that has been

successfully studied is the Linear Quadratic Regulator that will be discussed in Sect. “Linear Quadratic Optimal Control”. Control problems for nonlinear partial differential equations are extremely interesting but harder to deal with, so the literature is less rich in results and techniques. Nevertheless, among the problems that received great attention are those of fluid dynamics, specifically the Euler equations @ t u C (u  r)u C r p D 0 and the Navier–Stokes equations @ t u  u C (u  r)u C r p D 0 subject to a boundary control and to the incompressibility condition div u D 0. Controllability We now proceed to introduce the main notions of controllability for the evolution equation (7). Later on in this section we will give interpretations for the heat and wave equations. In a given Hilbert space H, with scalar product h; i and norm k  k, let A: D(A) H ! H be the infinitesimal generator of a strongly continuous semigroup etA , t  0, of bounded linear operators on X. Intu: itively, this amounts to saying that u(t) D e tA u0 is the unique solution of the Cauchy problem ( u0 (t) D Au(t) t  0 u(0) D u0 ; in the classical sense for u0 2 D(A), and in a suitable generalized sense for all u0 2 H. Necessary and sufficient conditions in order for an unbounded operator A to be the infinitesimal generator of a strongly continuous semigroup are given by the celebrated Hille–Yosida Theorem, see, e. g. [99] and [55]. Abstract Evolution Equations Let F be another Hilbert space (with scalar product and norm denoted by the same symbols as for H), the so-called control space, and let B : F ! H be a linear operator, that we will assume to be bounded for the time being. Then, given T > 0 and u0 2 H, for all f 2 L2 (0; T; F) the Cauchy problem ( u0 (t) D Au(t) C B f (t) t  0 (12) u(0) D u0

Control of Non-linear Partial Differential Equations

has a unique mild solution u f 2 C([0; T]; H) given by Z t e(ts)A B f (s) 8t  0 (13) u f (t) D e tA u0 C 0

Note 1 Boundary control problems can be reduced to the same abstract form as above. In this case, however, B in (12) turns out to be an unbounded operator related to suitable fractional powers of A, see, e. g., [22]. For any t  0 let us denote by  t : L2 (0; t; F) ! H the bounded linear operator

Such a characterization is the object of the following theorem. Notice that the above identity and (14) yield 

T u(s) D B e(Ts)A u Theorem 1 System (7) is:

 exactly controllable in time T if and only if there is a constant C > 0 such that Z T   tA 2 B e u dt  Ckuk2 8u 2 H ; (15) 0

Z

t

t f D

e(ts)A B f (s) ds

8 f 2 L2 (0; t; F) :

(14)

0

The attainable (or reachable) set from u0 at time t, A(u0 ; t) is the set of all points in H of the form u f (t) for some control function f , that is : A(u0 ; t) D e tA u0 C  t L2 (0; t; F) :

 null controllable in time T if and only if there is a constant C > 0 such that Z T     tA 2 B e u dt  C eTA u2 8u 2 H ; (16) 0

 approximately controllable in time T if and only if, for every u 2 H, 

We introduce below the main notions of controllability for (7). Let T > 0. Definition 1 System (7) is said to be:  exactly controllable in time T if A(u0 ; T) D H for all u0 2 H, that is, if for all u0 ; u1 2 H there is a control function f 2 L2 (0; T; F) such that u f (T) D u1 ;  null controllable in time T if 0 2 A(u0 ; T) for all u0 2 H, that is, if for all u0 2 H there is a control function f 2 L2 (0; T; F) such that u f (T) D 0;  approximately controllable in time T if A(u0 ; T) is dense in H for all u0 2 H, that is, if for all u0 ; u1 2 H and for any " > 0 there is a control function f 2 L2 (0; T; F) such that ku f (T)  u1 k < ". Clearly, if a system is exactly controllable in time T, then it is also null and approximately controllable in time T. Although these last two notions of controllability are strictly weaker than strong controllability, for specific problems—like when A generates a strongly continuous group—some of them may coincide. Since controllability properties concern, ultimately, the range of the linear operator T defined in (14), it is not surprising that they can be characterized in terms of the adjoint operator T : H ! L2 (0; T; F), which is defined by Z 0

T

˝

8s 2 [0; T] :

T u(s); f (s)ids D hu0 ; T f i 8u 2 H ; 8 f 2 L2 (0; T; F) :

B e tA u D 0

t 2 [0; T] a.e. H) u D 0 :

(17)

To benefit the reader who is more familiar with optimization theory than abstract functional analysis, let us explain, by a variational argument, why estimate (16) implies null controllability. Consider, for every " > 0, the penalized problem ˚ min J" ( f ) : f 2 L2 (0; T; H) ; where 1 J" ( f ) D 2

Z

T

k f (t)k2 dt C

0

1 ku f (T)k2 2" 8 f 2 L2 (0; T; H) :

Since J" is strictly convex, it admits a unique minimum point f" . Set u" D u f" . Recalling (13) we have, By Fermat’s rule, 0 D J"0 ( f" )g D

Z

T 0

h f" (t); g(t)i dt

1 C hu" (T); T gi "

8g 2 L2 (0; T; H) :

(18)

Therefore, passing to the adjoint of T , Z

T 0

D

f" (t) C

E  1  T u" (T) (t); g(t) dt D 0 " 8g 2 L2 (0; T; H) ;

105

106

Control of Non-linear Partial Differential Equations

Taking

whence, owing to (14), f" (t) D 

 1   u" (T) (t) D B v" (t) " T 8t 2 [0; T] ;

H D L2 (˝) D F ; (19)

 : where v" (t) D "1 e(Tt)A u" (T) is the solution of the dual problem ( t 2 [0; T] v 0 C A v D 0

v(T) D 1" u" (T) : Z

T

0

1 k f" (t)k2 dt C ku" (T)k2  Cku0 k2 " 8" > 0

0

hence Z 0

T

  2 B v"  dt D hu0 ; v" (0)i :

Now, apply estimate (16) with u D  v" (T  t) D e tA u ""(T) to obtain Z 0

T

(22)

on ˙T x2˝

8i D 1; : : : ; n :

(23)

Notice that the above property already suffices to explain why the heat equation cannot be exactly controllable: it is impossible to attain a state u1 2 L2 (˝) which is not compatible with (23). On the other hand, null controllability holds true in any positive time.

(21)

“

2

Z

j f j dxdt  C T u " (T) "

in Q T

Theorem 2 Let T > 0 and let ! be an open subset of ˝ such that ! ˝. Then the heat equation (9) with homogeneous Dirichlet boundary conditions is null controllable in time T, i. e., for every initial condition u0 2 L2 (˝) there is a control function f 2 L2 (Q T ) such that the solution uf of (22) satisfies u f (T; )  0. Moreover,



 2 d hu" ; v" i C B v"  dt D 0 ; dt

1 ku" (T)k2 C "

8 ˆ <@ t u D u C 1! f uD0 ˆ : u(0; x) D u0 (x)

@x i u 2 L2 (Q T )

So, T

and A as in (8), one obtains that, for any u0 2 L2 (˝) and f 2 L2 (Q T ), the initial-boundary value problem

(20)

for some positive constant C. Indeed, observe that, in view of (19), (˝ ˛ u"0  Au" C BB v" ; v" D 0 ; u" (0) D u0 ˝ 0 ˛ v" C A v" ; u" D 0 ; v" (T) D 1" u" (T) : Z

8 f 2 L2 (˝)

has a unique mild solution u f 2 C([0; T]; L2 (˝)). Moreover, multiplying both sides of equation (9) by u and integrating by parts, it is easy to see that

It turns out that 1 2

B f D 1! f

QT

and note that

   B v" (t)2 dt  C kv" (0)k2

for some positive constant C. Hence, (20) follows from (21) and (19). Finally, from (20) one deduces the existence of a weakly convergent subsequence f" j in L2 (0; T; F). Then, called f 0 the weak limit of f" j , u" j (t) ! u f0 (t) for all t 2 [0; T]. So, owing to (20), u f0 (T) D 0. Heat Equation It is not hard to see that the heat equation (9) with Dirichlet boundary conditions (2) fails to be exactly controllable. On the other hand, one can show that it is null controllable in any time T > 0, hence approximately controllable. Let ! be an open subset of ˝ such that ! ˝.

˝

ju0 j2 dx

for some positive constant CT . The above property is a consequence of the abstract result in Theorem 1 and of concrete estimates for solutions of parabolic equations. Indeed, in order to apply Theorem 1 one has to translate (16) into an estimate for the heat operator. Now, observing that both A and B are self-adjoint, one promptly realizes that (16) reduces to Z

T 0

Z !

jv(t; x)j2 dxdt  C

Z ˝

jv(T; x)j2 dx

for every solution v of the problem ( @ t v D v in Q T vD0

on ˙T :

(24)

(25)

Estimate (24) is called an observability inequality for the heat operator for obvious reasons: problem (25) is not

Control of Non-linear Partial Differential Equations

well-posed since the initial condition is missing. Nevertheless, if, “observing” a solution v of such a problem on the “small” cylinder (0; T)  !, you find that it vanishes, then you can conclude that v(T; )  0 in the whole domain ˝. Thus, v(0; )  0 by backward uniqueness. In conclusion, as elegant as the abstract approach to null controllability may be, one is confronted by the difficult task of proving observability estimates. In fact, for the heat operator there are several ways to prove inequality (24). One of the most powerful, basically due to Fursikov and Imanuvilov [65], relies on global Carleman estimates. Compared to other methods that can be used to derive observability, such a technique has the advantage of applying to second order parabolic operators with variable coefficients, as well as to more general operators. Global Carleman estimates are a priori estimates in weighted norms for solutions of the problem ( @ t v D v C f in Q T (26) vD0 on ˙T : regardless of initial conditions. The weight function is usually of the form   2rkk : 1;˝  er(x) (t; x) 2 Q T ; (27) r (t; x) D (t) e where r is a positive constant,  is a given function in C 2 (˝) such that r(x) ¤ 0 8x 2 ˝ ;

(28)

and : (t) D

1 t(T  t)

0< t< T:

Note that > 0; r

> 0;

(t) ! 1 r (t; x)

are positive constants r; s0 and C such that, for any s > s0 , “ s3 3 (t)jv(t; x)j2 e2s r dxdt QT

“

2 2s

j f (t; x)j e

C QT

Z 



Z r

T

dxdt C Cs

ˇ ˇ2 ˇ @v ˇ @ ˇ ˇ (x) ˇ (t; x)ˇ e2s ˇ @ ˇ @

(t)dt 0

r

dH n1 (x)

(29)

It is worth underlying that, thanks to the singular behavior of near 0 and T, the above result is independent of the initial value of v. Therefore, it can be applied, indifferently, to any solution of (26) as well as to any solution of the backward problem ( @ t v C v D f in Q T vD0

on ˙T :

Moreover, inequality (29) can be completed adding first and second order terms to its left-hand side, each with its own adapted power of s and . Instead of trying to sketch the proof of Theorem 3, which would go beyond the scopes of this note, it is interesting to explain how it can be used to recover the observability inequality (24), which is what is needed to show that the heat equation is null controllable. The reasoning—not completely straightforward—is based on the following topological lemma, proved in [65]. Lemma 1 Let ˝ Rn be a bounded domain with boundary  of class Ck , for some k  2, and let ! ˝ be an open set such that ! ˝. Then there is function  2 C k (˝) such that ( (x) < 0 8x 2  (i) (x) D 0 and @ @ (30) (ii) fx 2 ˝jr(x) D 0g ! :

t ! 0;T

!1

t # 0;t " T :

Using the above notations, a typical global Carleman estimate for the heat operator is the following result obtained in [65]. Let us denote by (x) the outword unit normal to  at a point x 2  , and by @ (x) D r(x)  (x) @ the normal derivative of  at x. Theorem 3 Let ˝ be a bounded domain of Rn with boundary of class C2 , let f 2 L2 (Q T ), and let  be a function satisfying (28). Let v be a solution of (26). Then there

Now, given a solution v of (25) and an open set ! such that ! ˝, let ! 0

! 00

! be subdomains with smooth boundary. Then the above lemma ensures the existence of a function  such that fx 2 ˝jr(x) D 0g ! 0 : : “Localizing” problem (25) onto ˝ 0 D ˝ n ! 0 by a cutoff function  2 C 1 (Rn ) such that 0    1;

1

on Rn n ! 00 ;

0

that is, taking w D v, gives ( : @ t w D w C h in Q T0 D (0; T)  ˝ 0 w(t; ) D 0

on @˝ 0 D @˝ [ @! 0 ;

on ! 0 ;

(31)

107

108

Control of Non-linear Partial Differential Equations

with h :D v C 2r  ru. Since r ¤ 0 on ˝ 0 , Theorem 3 can be applied to w on Q T0 to obtain s3

“

3 jwj2 e2s

Q 0T

C Z

2 2s

Q 0T

jhj e

Z

T

C Cs

dt 0

Z



Z

T

C Cs

dt

r

dxdt

r

dxdt

ˇ ˇ2 ˇ @w ˇ 2s r ˇ ˇ e d H n1 ˇ @ ˇ ˇ ˇ @ ˇˇ @w ˇˇ2 2s r e d H n1 @ ˇ @ ˇ “ C jhj2 e2s r dxdt

for s sufficiently large. On the other hand, for any 0 < T0 < T1 < T, “

3 jwj2 e2s

Q 0T

 s3

Z

Z

T1

dt

˝n!

T0

r

dxdt

3 jwj2 e2s r dxdt Z

Z

T1



dt

˝n!

T0

jvj2 dx

Therefore, recalling the definition of h, Z

T1

dt Z

˝n!

Z

T

C

dt Z

0

dt 0

! 00 n! 0

Z

T

C

jvj2 dx  C

!

“

jhj2 e2s

Q 0T

Z

r

dxdt

dt

Z dt

0

! 00 n! 0

Z

jrvj2 e2s

2 2s

jrvj e

! 00 n! 0

r

r

Z

Z

T

dt 0

to conclude that Z 2T/3 Z Z dt jvj2 dx  C ˝n!

dt

!

jvj2 dx

0

!

dt

Z

Z

2T/3

dt T/3

Z

˝ T

v 2 (t; x) dx Z dt

0

!

v 2 (t; x) dx ;

Wave Equation Compared to the heat equation, the wave equation (5) exhibits a quite different behavior from the point of view of exact controllability. Indeed, on the one hand, there is no obstruction to exact controllability since no regularizing effect is connected with wave propagation. On the other hand, due to the finite speed of propagation, exact controllability cannot be expected to hold true in arbitrary time, as null controllability does for the heat equation. In fact, a typical result that holds true for the wave equation is the following, where a boundary control of Dirichlet type acts on a part 1  , while homogeneous boundary conditions are imposed on 0 D  n 1 : 8 2 in Q T ˆ <@ t u D u u D f 11 on ˙T ˆ : u(0; x) D u0 (x) ; @ t u(0; x) D u1 (x) x 2 ˝

jvj2 e2s

r

dx ;

u0 2L2 (˝) ;

(32)

u1 2 H 1 (˝)

f 2L2 (0; T; L2 ( )) u 2C([0; T]; L2 (˝)) \ C 1 ([0; T]; H 1 (˝)) : Theorem 4 Let ˝ be a bounded domain of Rn with boundary of class C2 and suppose that, for some point x0 2 Rn , ( (x  x0 )  (x) > 0 8x 2 1 (x  x0 )  (x)  0 8x 2 0 : Let

Z

T

3 v (T; x) dx  T ˝ 2

which is exactly (24).

dx :

dx

C

T/3

Z

T

Observe that problem (32) is well-posed taking

0

Now, fix T0 D T/3 ; T1 D 2T/3 and use Caccioppoli’s inequality (a well-known estimate for solution of elliptic and parabolic PDE’s) T

Z 0

3  (1 C C) T

T



Z

Z

 2 2 2 jr j v C jrj2jrvj2 e2s r dx

jvj2 dx C C

˝

jvj2 dx  (1 C C)

for some constant C. Then, the R dissipativity of the heat operator (that is, the fact that ˝ jv(t; x)j2 dx is decreasing with respect to t) implies that

Q 0T

T0

Z

2T/3 T/3

@ @

@! 0

0

Z

Z

dt “

s3

or

!

jvj2 dx

R D sup jx  x0 j : x2˝

Control of Non-linear Partial Differential Equations

If T > 2R, then, for all (u0 ; u1 ); (v0 ; v1 ) 2 L2 (˝)  H 1 (˝) there is a control function f 2 L2 (0; T; L2 ( )) such that the solution uf of (32) satisfies u f (T; x) D v0 (x) ;

@ t u f (T; x) D v1 (x) :

As we saw for abstract evolution equations, the above exact controllability property is proved to be equivalent to an observability estimate for the dual homogeneous problem using, for instance, the Hilbert Uniqueness Method (HUM) by J.-L. Lions [86]. Bibliographical Comments The literature on controllability of parabolic equations and related topics is so huge, that no attempt to provide a comprehensive account of it would fit within the scopes of this note. So, the following comments have to be taken as a first hint for the interested reader to pursue further bibliographical research. The theory of exact controllability for parabolic equations was initiated by the seminal paper [58] by Fattorini and Russell. Since then, it has experienced an enormous development. Similarly, the multiplier method to obtain observability inequalities for the wave equation was developed in [17,73,74,77,86]. Some fundamental early contributions were surveyed by Russell [108]. The next essential progress was made in the work by Lebeau and Robbiano [83] and then by Fursikov and Imanuvilov in a series of papers. In [65] one can find an introduction to global Carleman estimates, as well as applications to the controllability of several ODE’s. In particular, the presentation of this paper as for observability inequalities and Carleman estimates for the heat operator is inspired by the last monograph. General perspectives for the understanding of global Carleman estimates and their applications to unique continuation and control problems for PDE’s can be found in the works by Tataru [113,114,115,116]. Usually, the above approach requires coefficients to be sufficiently smooth. Recently, however, interesting adaptations of Carleman estimates to parabolic operators with discontinuous coefficients have been obtained in [21,82]. More recently, interest has focussed on control problems for nonlinear parabolic equations. Different approaches to controllability problems have been proposed in [57] and [44]. Then, null and approximate controllability results have been improved by Fernandez–Cara and Zuazua [61,62]. Techniques to produce insensitizing controls have been developed in [117]. These techniques have been successfully applied to the study of Navier–Stokes equations by several authors, see e. g. [63].

Fortunately, several excellent monographs are now available to help introduce the reader to this subject. For instance, the monograph by Zabczyk [121] could serve as a clean introduction to control and stabilization for finiteand infinite-dimensional systems. Moreover, [22,50,51], as well as [80,81] develop all the basic concepts of control and system theory for distributed parameter systems with special emphasis on abstract formulation. Specific references for the controllability of the wave equation by HUM can be found in [86] and [74]. More recent results related to series expansion and Ingham type methods can be found in [75]. For the control of Navier–Stokes equations the reader is referred to [64], as well as to the book by Coron [43], which contains an extremely rich collection of classical results and modern developments.

Stabilization Stabilization of flexible structures such as beams, plates, up to antennas of satellites, or of fluids as, for instance, in aeronautics, is an important part of CT. In this approach, one wants either to derive feedback laws that will allow the system to autoregulate once they are implemented, or study the asymptotic behavior of the stabilized system i. e. determine whether convergence toward equilibrium states as times goes to infinity holds, determine its speed of convergence if necessary or study how many feedback controls are required in case of coupled systems. Different mathematical tools have been introduced to handle such questions in the context of ODE’s and then of PDE’s. Stabilization of ODE’s goes back to the work of Lyapunov and Lasalle. The important property is that trajectories decay along Lyapunov functions. If trajectories are relatively compact in appropriate spaces and the system is autonomous, then one can prove that trajectories converge to equilibria asymptotically. However, the construction of Lyapunov functions is not easy, in general. This section will be concerned with some aspects of the stabilization of second order hyperbolic equations, our model problem being represented by the wave equation with distributed damping 8 ˆ <@ t t u  u C a(x)u t D 0 in ˝  R ; uD0 on ˙ D (0; 1)   (33) ˆ : 0 1 on ˝ ; (u; @ t u)(0) D (u ; u ) in a bounded domain ˝ Rn with a smooth boundary  . For n D 2, u(t; x) represents the displacement of point x of the membrane at time t. Therefore, equation (33) describes an elastic system. The energy of such

109

110

Control of Non-linear Partial Differential Equations

a system is given by E(t) D

1 2

Z ˝

Geometrical Aspects

 ju t (t; x)j2 C jru(t; x)j2 dx :

When a  0, the feedback term a(x)u t models friction: it produces a loss of energy through a dissipation phenomenon. More precisely, multiplying the equation in (33) by ut and integrating by parts on ˝, it follows that E 0 (t) D 

Z ˝

a(x)ju t j2 dx  0 ;

8t  0 :

(34)

On the other hand, if a  0, then the system is conservative, i. e., E(t) D E(0) for all t  0. Another well-investigated stabilization problem for the wave equation is when the feedback is localized on a part  0 of the boundary  , that is, 8 ˆ @ t t u  u D 0 ˆ ˆ ˆ < @u C u D 0

in ˝  R

on ˙0 D (0; 1)  0 t @ ˆ u D 0 on ˙1 D (0; 1)  ( n 0 ) ˆ ˆ ˆ :(u; @ u)(0) D (u0 ; u1 ) t

(35) In this case, the dissipation relation (34) takes the form 0

E (t) D 

Z 0

ju t j2 dH n1  0 ;

8t  0 :

In many a situation—such as to improve the quality of an acoustic hall—one seeks to reduce vibrations to a minimum: this is why stabilization is an important issue in CT. We note that the above system has a unique stationary solution—or, equilibrium—given by u  0. Stabilization theory studies all questions related to the convergence of solutions to such an equilibrium: existence of the limit, rate of convergence, different effects of nonlinearities in both displacement and velocity, effects of geometry, coupled systems, damping effects due to memory in viscoelastic materials, and so on. System (33) is said to be:  strongly stable if E(t) ! 0 as t ! 1;  (uniformly) exponentially stable if E(t)  Ce˛t E(0) for all t  0 and some constants ˛ > 0 and C  0, independent of u0 ; u1 . This note will focus on some of the above issues, such as geometrical aspects, nonlinear damping, indirect damping for coupled systems and memory damping.

A well-known property of the wave equation is the socalled finite speed of propagation, which means that, if the initial conditions u0 ; u1 have compact support, then the support of u(t; ) evolves in time at a finite speed. This explains why, for the wave equation, the geometry of ˝ plays an essential role in all the issues related to control and stabilization. The size and localization of the region in which the feedback is active is of great importance. In this paper such a region, denoted by !, is taken as a subset of ˝ of positive Lebesgue measure. More precisely, a is assumed to be continuous on ˝ and such that a0

on

˝

and

a  a0

on

!;

(36)

for some constant a0 > 0. In this case, the feedback is said to be distributed. Moreover, it is said to be globally distributed if ! D ˝ and locally distributed if ˝ n ! has positive Lebesgue measure. Two main methods have been used or developed to study stabilization, namely the multiplier method and microlocal analysis. The one that gives the sharpest results is based on microlocal analysis. It goes back to the work of Bardos, Lebeau and Rauch [17], giving geodesics sufficient conditions on the region of active control for exact controllability to hold. These conditions say that each ray of geometric optics should meet the control region. Burq and Gérard [25] showed that these results hold under weaker regularity assumptions on the domain and coefficients of the operators (see also [26,27]). These geodesics conditions are not explicit, in general, but they allow to get decay estimates of the energy under very general hypotheses. The multiplier method is an explicit method, based on energy estimates, to derive decay rates (as well as observability and exact controllability results). For boundary control and stabilization problems it was developed in the works of several authors, such as Ho [38,73], J.-L. Lions [86], Lasiecka–Triggiani, Komornik–Zuazua [76], and many others. Zuazua [123] gave an explicit geometric condition on ! for a semilinear wave equation subject to a locally distributed damping. Such a condition was then relaxed K. Liu [87] (see also [93]) who introduced the so-called piecewise multiplier method. Lasiecka and Triggiani [80,81] introduced a sharp trace regularity method which allows to estimate boundary terms in energy estimates. There also exist intermediate results between the geodesics conditions of Bardos–Lebeau–Rauch and the multiplier method, obtained by Miller [95] using differentiable escape functions.

Control of Non-linear Partial Differential Equations

Zuazua’s multiplier geometric condition can be described as follows. If a subset O of ˝ is given, one can define an "-neighborhood of O in ˝ as the subset of points of ˝ which are at distance at most " of O. Zuazua proved that if the set ! is such that there exists a point x0 2 Rn —an observation point—for which ! contains an "neighborhood of  (x 0 ) D fx 2 @˝ ; (x  x 0 )  (x)  0g, then the energy decays exponentially. In this note, we refer to this condition as (MGC). If a vanishes for instance in a neighborhood of the two poles of a ball ˝ in Rn , one cannot find an observation point x0 such that (MGC) holds. K. Liu [87] (see also [93]) introduced a piecewise multiplier method which allows to choose several observation points, and therefore to handle the above case. Introduce disjoint lipschitzian domains ˝ j of ˝, j D 1; : : : ; J, and observation points x j 2 R N , j D 1; : : : ; J and define

j (x j ) D fx 2 @˝ j ; (x  x j )   j (x)  0g Here  j stands for the unit outward normal vector to the boundary of ˝ j . Then the piecewise multiplier geometrical condition for ! is:    (PWMGC) !  N" [ JjD1 j (x j ) [ ˝n [ JjD1 ˝ j It will be denoted by (PWMGC) condition in the sequel. Assume now that a vanishes in a neighborhood of the two poles of a ball in Rn . Then, one can choose two subsets ˝ 1 and ˝ 2 containing, respectively, the two regions where a vanishes and apply the piecewise multiplier method with J D 2 and with the appropriate choices of two observation points and ". The multiplier method consists of integrating by parts expressions of the form Z TZ  t

˝

@2t u

otherwise. One can then prove that the energy satisfies an estimate of the form Z

T

E(s) ds t

Z

Z

T

 cE(t) C

2

˝

t

 u C a(x)u t Mu dx dt D 0 80  t  T ;

where u stands for a (strong) solution of (33), with an appropriate choice of Mu. Multipliers have generally the form Mu D (m(x)  ru C c u) (x) ; where m depends on the observation points and is a cut-off function. Other multipliers of the form Mu D 1 (ˇu), where ˇ is a cut-off function and 1 is the inverse of the Laplacian operator with homogeneous Dirichlet boundary conditions, have also used. The geometric conditions (MGC) or (PWMGC) serve to bound above by zero terms which cannot be controlled

!

ju t j

2

ds

8t  0 :

(37)

Once this estimate is proved, one can use the dissipation relation to prove that the energy satisfies integral inequalities of Gronwall type. This is the subject of the next section. Decay Rates, Integral Inequalities and Lyapunov Techniques The Linear Feedback Case Using the dissipation relation (34), one has Z TZ ˝

t

aju t j2 dx ds 

Z

T

E 0 (s) ds  E(t)

t

8 0  t  T: On the other hand, thanks to assumption (36) on a Z TZ !

t

u2t dx ds 

1 a0

Z TZ ˝

t



aju t j2 dx ds 1 E(t) a0

80  t  T :

By the above inequalities and (37), E satisfies Z

T

E(s) ds  cE(t) ;



a(x)ju t j C

!

Z

80  t  T :

(38)

t

Since E is a nonincreasing function and thanks to this integral inequality, Haraux [71] (see also Komornik [74]) proved that E decays exponentially at infinity, that is  E(t)  E(0) exp 1  t/c) ;

8t  c :

This proof is as follows. Define Z

1

(t) D exp(t/c)

E(s) ds

8t  0 :

t

Thanks to (38)  is nonincreasing on [0; 1), so that Z

1

(t)  (0) D

E(s) ds : 0

(39)

111

112

Control of Non-linear Partial Differential Equations

Using once again (38) with t D 0 in this last inequality and the definition of , one has Z 1 E(s) ds  cE(0) exp(t/c) 8t  0 :

one can prove that the energy E of solutions satisfies the following inequality for all 0  t  T Z

t

E (pC1)/2 (s) dt

t

Since E is a nonnegative and nonincreasing function Z t Z 1 cE(t)  E(s) ds  E(s) ds tc

T

 cE (pC1)/2 (t) C c

Z

T t

tc

 cE(0) exp((t  c)/c) ; so that (39) is proved. An alternative method is to introduce a modified (or perturbed) energy E" which is equivalent to the natural one for small values of the parameter " as in Komornik and Zuazua [76]. Then one shows that this modified energy satisfies a differential Gronwall inequality so that it decays exponentially at infinity. The exponential decay of the natural energy follows then at once. In this case, the modified energy is indeed a Lyapunov function for the PDE. The natural energy cannot be in general such a Lyapunov function due to the finite speed of propagation (consider initial data which have compact support compactly embedded in ˝n!). There are also very interesting approaches using the frequency domain approach, or spectral analysis such as developed by K. Liu [87] Z. Liu and S. Zheng [88]. In the sequel, we concentrate on the integral inequality method. This method has been generalized in several directions and we present in this note some results concerning extensions to  nonlinear feedback  indirect or single feedback for coupled system  memory type feedbacks Generalizations to Nonlinear Feedbacks Assume now that the feedback term a(x)u t in (33) is replaced by a nonlinear feedback a(x) (u t ) where is a smooth, increasing function satisfying v (v)  0 for v 2 R, linear at 1 and with polynomial growth close to zero, that is: (v) D jvj p for jvj  1 where p 2 (1; 1). Assume moreover that ! satisfies Zuazua’s multiplier geometric condition (MGC) or Liu’s piecewise multiplier method (PWMGC). Then using multipliers of the space and time variables defined as E(s)(p1)/2 Mu(x) where Mu(x) are multipliers of the form described in section 5.1 and integrating by parts expressions of the form Z T E(s)(p1)/2 t Z   2  @ t u  u C a(x) (u t ) Mu(x) dx ds D 0 ; ˝

E (p1)/2 (s) Z  Z 2 2

(u t ) C ju t j :  ˝

!

One can remark than an additional multiplicative weight in time depending on the energy has to be taken. This weight is E (p1)/2 . Then as in the linear case, but in a more involved way, thanks to the dissipation relation E 0 (t) D 

Z ˝

a(x)u t (u t ) ;

(40)

one can prove that E satisfies the following nonlinear integral inequality Z

T

E (pC1)/2 (s) ds  cE(t) ;

80  t  T :

t

Thanks to the fact that E is nonincreasing, a wellknown result by Komornik [74] shows that E is polynomially decaying, as t 2/(p1) at infinity. The above type results have been obtained by many authors under weaker form (see also [40,41,71,98,122]). Extensions to nonlinear feedbacks without growth conditions close to zero have been studied by Lasiecka and Tataru [78], Martinez [93], W. Liu and Zuazua [89], Eller Lagnese and Nicaise [56] and Alabau–Boussouira [5]. We present the results obtained in this last reference since they provide optimal decay rates. The method is as follows. Define respectively the linear and nonlinear kinetic energies (R

R!

˝

ju t j2 dx ; ja(x) (u t )j2 dx ;

and use a weight function in time f (E(s)) which is to be determined later on in an optimal way. Integrating by parts expressions of the form Z

Z

T

f (E(s)) t

 ˝

 @2t uuCa(x) (u t ) Mu(x) dx ds D 0 ;

one can prove that the energy E of solutions satisfies the

Control of Non-linear Partial Differential Equations

following inequality for all 0  t  T Z

T t

Z T E(s) f (E(s)) ds  c f (E(t)) C c f (E(s)) t Z  Z 2 2  ja(x) (u t )j C ju t j : ˝

!

types of the feedback, one can assume convexity of H only in a neighborhood of 0. One can prove from (41) that there exists an (explicit) T0 > 0 such that for all initial data, E satisfies the following nonlinear integral inequality (41)

The difficulty is to determine the optimal weight under general growth conditions on the feedback close to 0, in particular for cases for which the feedback decays to 0 faster than polynomials. Assume now that the feedback satisfies g(jvj)  j (v)j  Cg 1 (jvj) ;

8jvj  1 ;

(42)

Z

E(s) f (E(s)) ds  T0 E(t)

then, define a function F as follows: 8  ˆ < Hˆ (y) if y 2 (0; C1) ; F(y) D y ˆ :0 if y D 0 ; ˆ that is where Hˆ  stands for the convex conjugate of H, ˆ ˆ  (y) D sup fx y  H(x)g : H

Z ˝t

where ˇ is of the form max(1 ; 2 E(0)), 1 and 2 being explicit positive constants. One can prove that the above formulas make sense, and in particular that F is invertible and smooth. More precisely, F is twice continuously differentiable strictly increasing, one-to-one function from [0; C1) onto [0; r02 ). Note that since the feedback is supposed to be linear at infinity, if one wants to obtain results for general growth

ja(x) (u t )j2 dx  1 (t)Hˆ 1   Z 1  a(x)u t (u t ) dx

1 (t) ˝

In a similar way, one proves that Z

ju t j2 dx  2 (t)Hˆ 1



1

2 (t)



Z ˝

a(x)u t (u t ) dx

where ˝ t and ! t are time-dependent sets of respective Lebesgue measures 1 (t) and 2 (t) on which the velocity u t (t; x) is sufficiently small. Using the above two estimates, together with the linear growth of at infinity, one proves Z

Z

T

f (E(s)) t

˝

Z

T

 t

ja(x) (u t )j2 C

Z !

ju t j2



  Z 1 f (E(s))Hˆ 1 a(x)u t (u t ) dx c ˝

Using then Young’s inequality, together with the dissipation relation (40) in the above inequality, one obtains Z

Z

T

f (E(s)) t

x2R

Then the optimal weight function f is determined in the following way   f (s) D F 1 (s/2ˇ) s 2 0; 2ˇr02 ;

(43)

This inequality is proved thanks to convexity arguˆ one can ments as follows. Thanks to the convexity of H, use Jensen’s inequality and (42), so that

!t

Moreover, is assumed to have a linear growth at infinity. We define the optimal weight function f as follows. We first extend H to a function Hˆ define on all R (  H(x) if x 2 0; r02 ; ˆ H(x) D C1 otherwise ;

80  t  T :

t

where g is continuously differentiable on R strictly increasing with g(0) D 0 and 8 2 ˆ < g 2 C ([0; r0 ]) ; r0 sufficiently small ;  p p H(:) D :g( :) is strictly convex on 0; r02 ; ˆ : g is odd :

T

˝ T

Z

ja(x) (u t )j2 C

Hˆ

 C1

 ?

Z !

ju t j2



 f (E(s) ds C C2 E(t) ;

(44)

t

where C i > 0 i D 1; 2 is a constant independent of the initial data. Using the dissipation relation (40) in the above inequality, this gives for all 0  t  T Combining this last inequality with (41) gives Z

Z

T

T

E(s) f (E(s)) ds  ˇ t

  ˆ ? f (E(s) ds C C2 E(t) (H)

t

where ˇ is chosen of the form max(1 ; 2 E(0)), 1 and 2 being explicit positive constants to guarantee that the argument E of f stays in the domain of definition of f .

113

114

Control of Non-linear Partial Differential Equations

Thus (43) is proved, thanks to the fact that the weight function has been chosen so that ˆ ? ( f (E(s)) D ˇH

1 E(s) f (E(s)) 2

80  s:

Therefore E satisfies a nonlinear integral inequality with a weight function f (E) which is defined in a semi-explicit way in general cases of feedback growth. The last step is to prove that a nonincreasing and nonnegative absolutely continuous function E satisfying a nonlinear integral inequality of the form (43) is decaying at infinity, and to establish at which rate this holds. For this, one proceeds as in [5]. Let  > 0 and T0 > 0 be fixed given real numbers and F be a strictly increasing function from [0; C1) on [0; ), with F(0) D 0 and lim y!C1 F(y) D . For any r 2 (0; ), we define a function K r from (0; r] on [0; C1) by Z r dy ; (45) K r () D 1  yF (y) and a function r which is a strictly increasing function from [ F 11 (r) ; C1) onto [ F 11 (r) ; C1) by r (z)

1 D z C K r (F( ))  z; z

8z 

1 ; F 1 (r)

(46)

One can prove that if E is a nonincreasing, absolutely continuous function from [0; C1) on [0; C1), satisfying 0 < E(0) <  and the inequality Z T E(s)F 1 (E(s)) ds  T0 E(S) ; 8 0  t  T ; (47) t

then E satisfies the following estimate: ! T0 1 ; 8t  1 ; E(t)  F 1 ( t ) F (r) r T

(48)

0

where r is any real such that Z C1 1 E()F 1 (E()) d  r   : T0 0 Thus, one can apply the above result to E with  D r02 and show that lim t ! C1E(t) D 0, the decay rate being given by estimate (48). If g is polynomial close to zero, one gets back that the 2 energy E(t) decays as t p1 at infinity. If g(v) behaves as exp(1/jvj) close to zero, then E(t) decays as 1/(ln(t))2 at infinity. The usefulness of convexity arguments has been first pointed out by Lasiecka and Tataru [78] using Jensen’s

inequality and then in different ways by Martinez [93] (the weight function does not depend on the energy) and W. Liu and Zuazua [89] and Eller Lagnese and Nicaise [56]. Optimal decay rates have been obtained by Alabau–Boussouira [5,6] using a weight function determined through the theory of convex conjugate functions and Young’s (named also as Fenchel–Moreau’s) inequality. This argument was also used by W. Liu and Zuazua [89] in a slightly different way and combined to a Lyapunov technique. Optimality of estimates in [5] is proved in one-dimensional situation and for boundary dampings applying optimality results of Vancostenoble [119] (see also Martinez and Vancostenoble [118]). Indirect Damping for Coupled Systems Many complex phenomena are modeled through coupled systems. In stabilizing (or controlling) energies of the vector state, one has very often access only to some components of this vector either due to physical constraints or to cost considerations. In this case, the situation is to stabilize a full system of coupled equation through a reduced number of feedbacks. This is called indirect damping. This notion has been introduced by Russell [109] in 1993. As an example, we consider the following system: (

@2t u  u C @ t u C ˛v D 0 @2t v  v C ˛u D 0 in ˝  R ;

uD0Dv

on @˝  R :

(49)

Here, the first equation is damped through a linear distributed feedback, while no feedback is applied to the second equation. The question is to determine if this coupled system inherits any kind of stability for nonzero values of the coupling parameter ˛ from the stabilization of the first equation only. In the finite dimensional case, stabilization (or control) of coupled ODE’s can be analyzed thanks to a powerful rank type condition named Kalman’s condition. The situation is much more involved in the case of coupled PDE’s. One can show first show that the above system fails to be exponentially stable (see also [66] for related results). More generally, one can study the stability of the system ( u00 C A1 u C Bu0 C ˛v D 0 (50) v 00 C A2 v C ˛u D 0 in a separable Hilbert space H with norm jj, where A1 ; A2 and B are self-adjoint positive linear operators in H. Moreover, B is assumed to be a bounded operator. So, our analysis applies to systems with internal damping supported

Control of Non-linear Partial Differential Equations

in the whole domain ˝ such as (49); the reader is referred to [1,2] for related results concerning boundary stabilization problems (see also Beyrath [23,24] for localized indirect dampings). In light of the above observations, system (50) fails to be exponentially stable, at least when H is infinite dimensional and A1 has a compact resolvent as in (49). Indeed it is shown in Alabau, Cannarsa and Komornik [8] that the total energy of sufficiently smooth solutions of (50) decays polynomially at infinity whenever j˛j is small enough but nonzero. From this result we can also deduce that any solution of (50) is strongly stable regardless of its smoothness: this fact follows by a standard density argument since the semigroup associated with (50) is a contraction semigroup. A brief description of the key ideas of the approach developed in [2,8] is as follows. Essentially, one uses a finite iteration scheme and suitable multipliers to obtain an estimate of the form Z

j

T

E(u(t); v(t))dt  c 0

X kD0

(51)

where j is a positive integer and E denotes the total energy of the system  1  1/2 2 jA1 uj C ju0 j2 2  1  1/2 2 jA2 vj C jv 0 j2 C ˛hu; vi : C 2 Once (51) is proved, an abstract lemma due to Alabau [1,2] shows that E(u(t); v(t)) decays polynomially at 1. This abstract lemma can be stated as follows. Let A be the infinitesimal generator of a continuous semi-group exp(tA) on an Hilbert space H , and D(A) its domain. For U 0 in H we set in all the sequel U(t) D exp(tA)U 0 and assume that there exists a functional E defined on C([0; C1); H ) such that for every U 0 in H , E(exp(:A)) is a non-increasing, locally absolutely continuous function from [0; C1) on [0; C1). Assume moreover that there exist an integer k 2 N ? and nonnegative constants cp for p D 0; : : : k such that E(u; v) D

Z

T

E(U(t))dt  S

k X

c p E(U (p) (S))

pD0

80  S  T ; 8U 0 2 D(Ak ) :

T

E(U()) S

(52) U0

in Then the following inequalities hold for every D(Akn ) and all 0  S  T where n is any positive integer:

kn X   (  S)n1 E U (p) (S) ; (53) d  c (n  1)! pD0

and

E(U(t))  c

kn X   E U (p) (0) t n pD0

8t > 0 ;

  8U 0 2 D Akn ;

where c is a constant which depends on n. First (53) is proved by induction on n. For n D 1, it reduces to the hypothesis (52). Assume now that (53) holds for n and let U 0 be given in D(Ak(nC1) ). Then we have Z TZ

T

E(U()) S

c

t kn Z X pD0 S

E(u(k) (0); v (k) (0)) 8T  0;

Z

T

(  t)n1 ddt (n  1)!

  E U (p) (t) dt   8 0  S  T ; 8 U 0 2 D Akn :

Since U 0 is in D(Ak(nC1) ) we deduce that U (p) (0) D A p U 0 is in D(Ak ) for p 2 f0; : : : kng. Hence we can apply the assumption (52) to the initial data U (p) (0). This together with Fubini’s Theorem applied on the left hand side of the above inequality give (53) for n C 1. Using the property that E(U(t)) is non increasing in (53) we easily obtain the last desired inequality. Applications to wave-wave, wave-Petrowsky equations and various concrete examples hold. The above results have been studied later on by Batkai, Engel, Prüss and Schnaubelt [18] using very interesting resolvent and spectral criteria for polynomial stability of abstract semigroups. The above abstract lemma in [2] has also been generalized using interpolation theory. One should note that this integral inequality involving higher order energies of solutions is not of differential nature, unlike Haraux’s and Komornik’s integral inequalities. Another approach based on decoupling techniques and for slightly different abstract systems have been introduced by Ammar Khodja Bader and Ben Abdallah [12]. Spectral conditions have also been studied by Z. Liu [88] and later on by Z. Liu and Rao [90], Loreti and Rao [92] for peculiar abstract systems and in general for coupled equations only of the same nature (wave-wave for instance), so that a dispersion relation for the eigenvalues of the coupled system can be derived. Also these last results are given for internal stabilization only. Because of the above limitations, Z. Liu–Rao and Loreti–Rao’s results

115

116

Control of Non-linear Partial Differential Equations

are less powerful in generality than the ones given by Alabau, Cannarsa and Komornik [8] and Alabau [2]. Moreover results through energy type estimates and integral inequalities can be generalized to include nonlinear indirect dampings as shown in [7]. On the other side spectral methods are very useful to obtain optimal decay rates provided that one can determine at which speed the eigenvalues approach the imaginary axis for high frequencies.

in a Hilbert space X, where A: D(A) X ! X is an accretive self-adjoint linear operator with dense domain, and rF denotes the gradient of a Gâteaux differentiable functional F : D(A1/2 ) ! R. In particular, equation (54) fits into this framework as well as several other classical equations of mathematical physics such as the linear elasticity system. We consider the following assumptions.

Memory Dampings

Assumptions (H1)

We consider the following model problem

1. A is a self-adjoint linear operator on X with dense domain D(A), satisfying

8 Rt ˆ ˆ ˆu t t (t; x)  u(t; x) C 0 ˇ(t  s)u(s; x) ds D ˆ < ju(t; x)j u(t; x) ˆ u(t; )j@˝ D 0 ˆ ˆ ˆ :(u(0; ); u (0; )) D (u ; u ) t 0 1 (54) 2 where 0 <  N2 holds. The second member is a source term. The damping

Z

hAx; xi  Mkxk2

8x 2 D(A)

(56)

for some M > 0. 2. ˇ : [0; 1) ! [0; 1) is a locally absolutely continuous function such that Z

1

ˇ(t)dt < 1 ˇ(0) > 0

ˇ 0 (t)  0

0

for a.e. t  0 :

t

ˇ(t  s)u(s; x) ds

3. F : D(A1/2 ) ! R is a functional such that 1. F is Gâteaux differentiable at any point x 2 D(A1/2 ); 2. for any x 2 D(A1/2 ) there exists a constant c(x) > 0 such that

0

is of memory type. The energy is defined by Eu (t) D

1 ku t (t)k2L 2 (˝) dx 2   Z t 1 ˇ(s) ds kru(t)k2L 2 (˝) 1 C 2 0 1  C2  ku(t)kL  C2 (˝)

C2 Z 1 t C ˇ(t  s)kru(t)  ru(s)k2L 2 (˝) ds 2 0

The damping term produces dissipation of the energy, that is (for strong solutions) 1 Eu0 (t) D  ˇ(t)kru(t)k2 2 Z 1 t 0 C ˇ (t)kru(s)  ru(t)k2 ds  0 2 0 One can consider more general abstract equations of the form u00 (t) C Au(t) 

Z

t

jDF(x)(y)j  c(x)kyk;

where DF(x) denotes the Gâteaux derivative of F in x; consequently, DF(x) can be extended to the whole space X (and we will denote by rF(x) the unique vector representing DF(x) in the Riesz isomorphism, that is, hrF(x); yi D DF(x)(y), for any y 2 X); 3. for any R > 0 there exists a constant C R > 0 such that krF(x)  rF(y)k  C R kA1/2 x  A1/2 yk for all x; y 2 D(A1/2 ) satisfying kA1/2 xk ; kA1/2 yk  R. Assumptions (H2) 1. There exist p 2 (2; 1] and k > 0 such that ˇ 0 (t)  kˇ

ˇ(t  s)Au(s) ds D rF(u(t))

for any y 2 D(A1/2 );

1C 1p

(t) for a.e. t  0

0

t 2 (0; 1)

(55)

(here we have set

1 p

D 0 for p D 1).

Control of Non-linear Partial Differential Equations

2. F(0) D 0, rF(0) D 0, and there is a strictly increasing continuous function : [0; 1) ! [0; 1) such that (0) D 0 and jhrF(x); xij 

1/2

(kA

1/2

xk)kA

2

xk

T

m p

Z

t

Eu (t) S

8x 2 D(A ) :

ˇ(t  s)kA1/2 u(s)  A1/2 u(t)k2 dsdt

0 p pCm

 CEu

Z

T

(S)

1C mp Eu (t)'m (t)dt

!

m pCm

(58)

S

for some constant C > 0. Suppose that, for some m  1, the function ' m defined in (57) is bounded. Then, for any S0 > 0 there is a positive constant C such that Z

1

1C mp

Eu

 m m  p p (t)dt  C Eu (0) C k'm k1 Eu (S)

S

8 S  S0 :

(59)

One uses this last result first with m D 2 noticing that ' 2 is bounded and  D E 2/p . This gives a first energy decay rate as (t C1)p/2 . This estimate shows that ' 1 is bounded. Then one applies once again the last result with m D 1 and  D E 1/p . One deduces then that E decays as (t C 1)p which is the optimal decay rate expected.

0

where the multipliers Mu are of the form (s)(c1 (ˇ u)(s) C c2 (s)u) with  which is a differentiable, nonincreasing and nonnegative function, and c1 being a suitable constant, whereas c2 may be chosen dependent on ˇ. Integrating by parts the resulting relations and performing some involved estimates, one can prove that for all t0 > 0 and all T  t  t0 Z T Z T (s)E(s) ds  C(0)E(t) C (s) t t Z s  2  ˇ(s  ) A1/2 u(s)  A1/2 u() d ds ; 0

If p D 1, that is if the kernel ˇ decays exponentially, one can easily bound the last term of the above estimate by cE(t) thanks to the dissipation relation. If p 2 (2; 1), one has to proceed differently since the term Z T Z s  2 (s) ˇ(s  )  A1/2 u(s)  A1/2 u() d ds t

Z

1/2

Under these assumptions, global existence for sufficiently small (resp. all) initial data in the energy space can be proved for nonvanishing (resp. vanishing) source terms. It turns out that the above energy methods based on multiplier techniques combined with linear and nonlinear integral inequalities can be extended to handle memory dampings and applied to various concrete examples such as wave, linear elastodynamic and Petrowsky equations for instance. This allows to show in [10] that exponential as well as polynomial decay of the energy holds if the kernel decays respectively exponentially or polynomially at infinity. The method is as follows. One evaluates expressions of the form Z T Z t hu00 (s) C Au(s)  ˇ Au(s)  rF(u(s); Mui ds t

Then, we have for any T  S  0

0

cannot be directly estimated thanks to the dissipation relation. To bound this last term, one can generalize an argument of Cavalcanti and Oquendo [37] as follows. Define, for any m  1, Z t 1 'm (t) :D ˇ 1 m (t  s)kA1/2 u(s)  A1/2 u(t)k2 ds ; 0

t  0:

(57)

Bibliographical Comments For an introduction to the multiplier method, we refer the interested reader to the books of J.-L. Lions [86], Komornik [74] and the references therein. The celebrated result of Bardos Lebeau and Rauch is presented in [86]. A general abstract presentation of control problems for hyperbolic and parabolic equations can be found in the book of Lasiecka and Triggiani [80,81]. Results on spectral methods and the frequency domain approach can be found in the book of Z. Liu [88]. There also exists an interesting approach developed for bounded feedback operators by Haraux and extended to the case of unbounded feedbacks by Ammari and Tucsnak [11]. In this approach, the polynomial (or exponential) stability of the damped system is proved thanks to the corresponding observability for the undamped (conservative) system. Such observability results for weakly coupled undamped systems have been obtained for instance in [3]. Many other very interesting issues have been studied in connection to semilinear wave equations, see [34,123] and the references therein, and damped wave equations with nonlinear source terms [39]. Well-posedness and asymptotic properties for PDE’s with memory terms have first been studied by Dafermos [53,54] for convolution kernels with past history (convolution up to t D 1), by Prüss [103] and Prüss and Propst [102] in which the efficiency of different models of dampings are compared to experiments (see also

117

118

Control of Non-linear Partial Differential Equations

Londen Petzeltova and Prüss [91]). Decay estimates for the energy of solutions using multiplier methods combined with Lyapunov type estimates for an equivalent energy are proved in Munoz Rivera [97], Munoz Rivera and Salvatierra [96], Cavalcanti and Oquendo [37] and Giorgi Naso and Pata [67] and many other papers. Optimal Control As for positional control, also for optimal control problems it is convenient to adopt the abstract formulation introduced in Sect. “Abstract Evolution Equations”. Let the state space be represented by the Hilbert space H, and the state equation be given in the form (12), that is (

u0 (t) D Au(t) C B f (t)

t 2 [0; T]

u(0) D u0 :

(60)

Recall that A is the infinitesimal of a strongly continuous semigroup, etA , in H, B is a (bounded) linear operator from F (the control space) to H, and uf stands for the unique (mild) solution of (60) for a given control function f 2 L2 (0; T; H). A typical optimal control problem of interest for PDE’s is the Bolza problem which consists in 8 the cost functional ˆ < minimizing   : RT J( f ) D 0 L(t; u f (t); f (t))dt C ` u f (T) ˆ : over all controls f 2 L2 (0; T; F) :

(61)

Here, T is a positive number, called the horizon, whereas L and ` are given functions, called the running cost and final cost, respectively. Such functions are usually assumed to be bounded below. A control function f 2 L2 (0; T; F) at which the above minimum is attained is called an optimal control for problem (61) and the corresponding solution u f of (60) is said to be an optimal trajectory. Alltogether, fu f ; f g is called an optimal (trajectory/control) pair. For problem (61) the following issues will be addressed in the sections below:  the existence of controls minimizing functional J;  necessary conditions that a candidate solution must satisfy;  sufficient conditions for optimality provided by the dynamic programming method. Other problems of particular interest to CT for PDE’s are problems with an infinite horizon (T D 1), problems with a free horizon T and a final target, and problems with constraints on both control variables and state variables.

Moreover, the study of nonlinear variants of (60), including semilinear problems of the form ( u0 (t) D Au(t) C h(t; u(t); f (t)) t 2 [0; T] (62) u(0) D u0 ; is strongly motivated by applications. The discussion of all these variants, however, will not be here pursued in detail. Traditionally, in optimal control theory, state variables are denoted by the letters x; y; : : : , whereas u; v; : : : are reserved for control variables. For notational consistency, in this section u() will still denote the state of a given system and f () a control function, while  will stand for a fixed element of control space F. Existence of Optimal Controls From the study of finite dimensional optimization it is a familiar fact that the two essential ingredients to guarantee the existence of minima are compactness and lower semicontinuity. Therefore, it is clear that, in order to obtain a solution of the optimal control problem (60)–(61), one has to make assumptions that allow to recover such properties. The typical hypotheses that are made for this purpose are the following:  coercivity: there exist constants c0 > 0 and c1 2 R such that `()  c1

and L(t; u; )  c0 kk2 C c1 8(t; u; ) 2 [0; T]  H  F

(63)

 convexity: for every (t; u) 2 [0; T]  H  7! L(t; u; )

is convex on

F:

(64)

Under the above hypotheses, assuming lower semicontinuity of ` and of the map L(t; ; ), it is not hard to show that problem (60)–(61) has at least one solution. Indeed, assumption (63) allows to show that any minimizing sequence of controls f f k g is bounded in L2 (0; T; H). So, it admits a subsequence, still denoted by f f k g which converges weakly in L2 (0; T; H) to some function f . Then, by linearity, u f k (t) converges to u f (t) for every t 2 [0; T]. So, using assumption (64), it follows that f is a solution of (60)–(61). The problem becomes more delicate when the Tonelli type coercivity condition (63) is relaxed, or the state equation is nonlinear as in (62). Indeed, the convergence of u f k (t) is no longer ensured, in general. So, in order to recover compactness, one has to make further assumptions, such as the compactness of etA , or structural properties of L

Control of Non-linear Partial Differential Equations

and h. For further reading, one may consult the monographs [22,85], and [79], for problems where the running and final costs are given by quadratic forms (the so-called Linear Quadratic problem), or [84] and [59] for more general optimal control problems. Necessary Conditions Once the existence of a solution to problem (60)–(61) has been established, the next important step is to provide conditions to detect a candidate solution, possibly showing that it is, in fact, optimal. By and large the optimality conditions of most common use are the ones known as Pontryagin’s Maximum Principle named after the Russian mathematician L.S. Pontryagin who greatly contributed to the development of control theory, see [100,101]. So, suppose fu ; f g, where u D u f is a candidate optimal pair and consider the so-called adjoint system 8 0  ˆ <p (t) D A p(t) C @u L(t; u (t); f (t)) D 0 ˆ :

t 2 [0; T] a.e.

p(T) D @`(u (T)) ;

where @u L(t; u; ) and @`(u) denote the Fréchet gradients of the maps L(t; ; ) and ` at u, respectively. Observe that the above is a backward linear Cauchy problem with terminal condition, which can obviously be reduced to a forward one by the change of variable t ! T  t. So, it admits a unique mild solution, labeled p , which is called the adjoint state associated with fu ; f g. Pontryagin’s Maximum Principle states that, if fu ; f g is optimal, then hp (t); B f (t)i C L(t; u (t); f (t)) D  min hp (t); Bi C L(t; u (t); ) 2F

t 2 [0; T] a.e.

(65)

The name Maximum Principle rather than Minimum Principle, as it would be more appropriate, is due to the fact that, traditionally, attention was focussed on the maximization—instead of minimization—of the functional in (61). Even today, in most models from economics, one is interested in maximizing payoffs, such as revenues, utility, capital and so on. In that case, (65) would still be true, with a “max” instead of a “min”. At first glance, it might be hard to understand the revelance of (65) to problem (61). To explain this, introduce the function, called the Hamiltonian,  H (t; u; p) D min hp; Bi C L(t; u; ) 2F

(t; u; p) 2 [0; T]  H  H :

(66)

Then, Fermat’s rule yields B p C @ L(t; u; ) D 0 at every  2 F at which the minimum in (66) is attained. Therefore, from (65) it follows that B p (t)C@ L(t; u (t); f (t)) D 0

t 2 [0; T] a.e. (67)

which provides a much-easier-to-use optimality condition. There is a vast literature on necessary conditions for optimality for distributed parameter systems. The set-up that was considered above can be generalized in several ways: one can consider nonlinear state equations as in (62), nonsmooth running and finals costs, constraints on both state and control, problems with infinite horizon or exit times. Further reading and useful references on most of these extensions can be found in the aforementioned monographs [22,79,84,85], and in [59] which is mainly concerned with time optimal control problems. Dynamic Programming Though useful as it may be, Pontryagin’s Maximum Principle remains a necessary condition. So, without further information, it does not suffice to prove the optimality of a give trajectory/control pair. Moreover, even when the map  7! @ L(t; u; ) turns out to be invertible, the best result identity (67) can provide, is a representation of f (t) in terms of u (t) and p (t): not enough to determine f (t), in general. This is why other methods to construct optimal controls have been proposed over the years. One of the most interesting ones is the so-called dynamic programming method (abbreviated, DP), initiated by the work of R. Bellman [20]. Such a method will be briefly described below in the set-up of distributed parameter systems. Fix T > 0, s such that 0  s  T, and consider the optimal control problem to minimize Z J s;v ( f ) D

T s

    s;v L t; u s;v (t); f (t) dt C ` u (T) f f (68)

over all control functions f 2 L2 (s; T; F), where u s;v f (t) is the solution of the controlled system ( u0 (t) D Au(t) C B f (t) t 2 [s; T] (69) u(s) D v : The value function U associated to (68)-(69) is the realvalued function defined by U(s; v) D

inf

f 2L 2 (s;T;F)

J s;v ( f ) 8(s; v) 2 [0; T]  H : (70)

119

120

Control of Non-linear Partial Differential Equations

A fundamental step of DP is the following result, known as Bellman’s optimality principle. Theorem 5 For any (s; v) 2 [0; T]  H and any f 2 L2 (s; T; F) Z

r

U(s; v)  s



   s;v L t; u s;v (t); f (t) dt C U r; u (r) f f 8r 2 [s; T] :

Moreover, f  () is optimal if and only if

Now, suppose there is a control function f 2 L2 (s; T; F) such that, for all t 2 [s; T], h@v W(t; u (t)); B f (t)i C L(t; u (t); f (t)) D H (t; u (t); @v W(t; u (t))) ;

(73)

where u () D u s;v f  (). Then, from (71) and (73) it follows that d W(t; u (t)) D L(t; u(t); f (t)) ; dt whence

Z

r

U(s; v) D s



   s;v L t; u s;v f (t); f (t) dt C U r; u f (r) 8r 2 [s; T] :

The connection between DP and optimal control is based on the properties of the value function. Indeed, applying Bellman’s optimality principle, one can show that, if U is Fréchet differentiable, then 8 ˆ <@s U(s; v) C hAv; @v U(s; v)i C H (s; v; @v U(s; v)) D 0 (s; v) 2 (0; T)  D(A) ˆ : U(T; v) D `(v) v 2 H where H is the Hamiltonian defined in (66). The above equation is the celebrated Hamilton–Jacobi equation of DP. To illustrate its connections with the original optimal control problem, a useful formal argument—that can, however, be made rigorous—is the following. Consider a sufficiently smooth solution W of the above problem and let (s; v) 2 (0; T)  D(A). Then, for any trajectory/control pair fu; f g, d W(t; u(t)) D @s W(t; u(t)) C h@v W(t; u(t)); Au(t) dt C B f (t)i D h@v W(t; u(t)); B f (t)i  H (t; u(t); @v W(t; u(t)))  L(t; u(t); f (t))

(71)

W(s; v) D J s;v ( f )  U(s; v) : From the above inequality and (72) it follows that W(s; v) D U(s; v) for all (s; v) 2 (0; T)  D(A), hence for all (s; v) 2 (0; T)  H since D(A) is dense in H. So, f is an optimal control. Note 2 The above considerations lead to the following procedure to obtain optimal an optimal trajectory:  find a smooth solution of the Hamilton–Jacobi equation;  for every (t; v) 2 (0; T)  D(A) provide a feedback f (t; v) such that h@v W(t; v); B f (t; v)i C L(t; v; f (t; v)) D H (t; v; @v W(t; v))  solve the so-called closed loop equation ( u0 (t) D Au(t) C B f (t; u(t)) t 2 [s; T] u(s) D v Notice that not only is trajectory u optimal, but the corresponding control f is given in feedback form as well. Linear Quadratic Optimal Control One of the most successful applications of DP is the socalled Linear Quadratic optimal control problem. Consider problem (68)–(69) with costs L and ` given by L(t; u; ) D hM(t)u; ui C hN(t); i

by the definition of H . Therefore, integrating from s to T, Z

T

`(u(T))  W(s; v)  

and

L(t; u(t); f (t))dt; s

`(u) D hDu; ui

whence J s;v ( f )  W(s; v). Thus, taking the infimum over all f 2 L2 (s; T; F), W(s; v)  U(s; v)

8(s; v) 2 (0; T)  D(A) :

8(T; u; ) 2 [0; T]  H  F

(72)

8u 2 H ;

where  M : [0; T] ! L(H) is continuous, M(t) is symmetric and hM(t)u; ui  0 for every (t; u) 2 [0; T]  H;

Control of Non-linear Partial Differential Equations

 N : [0; T] ! F is continuous, N(t) is symmetric and hN(t); i  c0 jj2 for every (t; ) 2 [0; T]  F and some constant c0 > 0;  D 2 L(H) is symmetric and hDu; ui  0 for every u 2 H. Then, assumptions (63) and (64) are satisfied. So, a solution to (68)–(69) does exist. Moreover, it is unique because of the strict convexity of functional J s;v . In order to apply DP, one computes the Hamiltonian h i H (t; u; p) D min hp; Bi C hM(t)u; ui C hN(t); i

So, solving the closed loop equation ( u0 (t) D [A  BN 1 (t)B P(t)]u(t)

t 2 (s; T)

u(s) D v one obtains the unique optimal trajectory of problem (68)–(69). In sum, by DP one reduces the original Linear Quadratic optimal control problem to the problem of finding the solution of the Riccati equation, which is easier to solve than the Hamilton–Jacobi equation.

2F

1 D hM(t)u; ui  hBN 1 (t)B p; pi ; 4

Bibliographical Comments

where the above minimum is attained at 1  (t; p) D  N 1 (t)B p : 2

(74)

Therefore, the Hamilton–Jacobi equation associated to the problem is 8 ˆ ˆ ˆ@s W(s; v) C hAv; @v W(s; v)i C hM(s)v; vi ˆ <  1 hBN 1 (s)B @ W(s; v); @ W(s; v)i D 0 v v 4 ˆ 8(s; v) 2 (0; T)  D(A) ˆ ˆ ˆ :w(T; v) D hDv; vi 8v 2 H It is quite natural to search a solution of the above problem in the form W(s; v) D hP(s)v; vi

8(s; v) 2 [0; T]  H ;

with P : [0; T] ! L(H) continuous, symmetric and such that hP(t)u; ui  0. Substituting into the Hamilton–Jacobi equation yields 8 0  ˆ ˆhP (s)v; vi C h[A P(s) C P(s)A]v; vi C hM(s)v; vi ˆ ˆ < hBN 1 (s)B P(s)v; P(s)vi D 0 ˆ 8(s; v) 2 (0; T)  D(A) ˆ ˆ ˆ :hP(T)v; vi D hDv; vi 8v 2 H Therefore, P must be a solution of the so-called Riccati equation 8 0  ˆ

P(s)BN 1 (s)B P(s) D 0

8s 2 (0; T)

P(T) D D

Once a solution P() of the Riccati equation is known, the procedure described in Note 2 can be applied. Indeed, recalling (74) and the fact that @v W(t; v) D 2P(t)v, one concludes that f (t; v) D N 1 (t)B P(t)v is a feedback law.

Different variants of the Riccati equation have been successfully studied by several authors in connection with different state equations and cost functionals, including boundary control problems and problems for other functional equations, see [22,79] and the references therein. Sometimes, the solution of the Riccati equation related to a linearized model provides feedback stabilization for nonlinear problems as in [104]. Unfortunately, the DP method is hard to implement for general optimal control problems, because of several obstructions: nonsmoothness of solutions to Hamilton– Jacobi equations, selection problems that introduce discontinuities, unboundedness of the coefficients, numerical complexity. Besides the Linear Quadratic case, the so-called Linear Convex case is the other example that can be studied by DP under fairly general conditions, see [14]. For nonlinear optimal control problems some of the above difficulties have been overcome extending the notion of viscosity solutions to infinite dimensional spaces, see [45,46,47,48,49], see also [28,29,30,31,32,33] and [112]. Nevertheless, finding additional ideas to make a generalized use of DP for distributed parameter systems possible, remains a challenging problem for the next generations. Future Directions In addition to all considerations spread all over this article on promising developments of recent—as well as established—research lines, a few additional topics deserve to be mentioned. The one subject that has received the highest attention, recently, is that of numerical approximation of control problems, from the point of view of both controllability and optimal control. Here the problem is that, due to high frequency spurious numerical solutions, stable algorithms for solving initial-boundary value problems do not necessarily yield convergent algorithms for computing

121

122

Control of Non-linear Partial Differential Equations

controls. This difficulty is closely related to the existence of concentrated numerical solutions that escape the observation mechanisms. Nevertheless, some interesting results have been obtained so far, see, e. g., [124,125]. Several interesting results for nonlinear control problems have been obtained by the return method, developed initially by Coron [42] for a stabilization problem. This and other techniques have then been applied to fluid models ([68,69]), the Korteweg–de Vries equation ([105,106,107]), and Schrödinger type equations ([19]), see also [43] and the references therein. It seems likely that these ideas, possibly combined with other techniques like Carleman estimates as in [70], will lead to new exiting results in the years to come. A final comment on null controllability for degenerate parabolic equations is in order. Indeed, many problems that are relevant for applications are described by parabolic equation equations in divergence form @ t u D r (A(x)ru)Cb(x)ruCc(t; x)uC f

in

QT ;

or in the general form @ t u D Tr [A(x)r 2 u]Cb(x)ruCc(t; x)uC f

4.

5.

6.

7.

8. 9.

10.

11.

12.

in

QT ; 13.

where A(x) is a symmetric matrix, positive definite in ˝ but possibly singular on  . For instance, degenerate parabolic equations arise in fluid dynamics as suitable transformations of the Prandtl equations, see, e. g., [94]. They can also be obtained as Kolmogorov equations of diffusions processes on domains that are invariant for stochastic flows, see, e. g., [52]. The latter interpretation explains why they have been applied to biological problems, such as gene frequency models for population genetics (see, e. g., the Wright–Fischer model studied in [111]). So far, null controllability properties of degenerate parabolic equations have been fully understood only in dimension one: for some kind of degeneracy, null controllability holds true (see [36] and [9]), but, in general, one can only expect regional null controllability (see [35]). Since very little is known on null controllability for degenerate parabolic equations in higher space dimensions, it is conceivable that such a topic will provide interesting problems for future developments. Bibliography 1. Alabau F (1999) Stabilisation frontière indirecte de systèmes faiblement couplés. C R Acad Sci Paris Sér I 328:1015–1020 2. Alabau F (2002) Indirect boundary stabilization of weakly coupled systems. SIAM J Control Optim 41(2):511–541 3. Alabau-Boussouira F (2003) A two-level energy method for indirect boundary observability and controllability of weakly

14. 15.

16.

17.

18.

19. 20. 21.

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Diagrammatic Methods in Classical Perturbation Theory GUIDO GENTILE Dipartimento di Matematica, Università di Roma Tre, Roma, Italy Article Outline Glossary Definition of the Subject Introduction Examples Trees and Graphical Representation Small Divisors Multiscale Analysis Resummation Generalizations Conclusions and Future Directions Bibliography Glossary Dynamical system Let W  R N be an open set and f : W  R ! R N be a smooth function. The ordinary differential equation x˙ D f (x; t) on W defines a continuous dynamical system. A discrete dynamical system on W is defined by a map x ! x0 D F(x), with F depending smoothly on x. Hamiltonian system Let A  Rd be an open set and H : A  Rd  R ! R be a smooth function (A  Rd is called the phase space). Consider the system of ordinary differential equations q˙k D @H (q; p; t)/@p k , p˙ k D @H (q; p; t)/@q k , for k D 1; : : : ; d. The equations are called Hamilton equations, and H is called a Hamiltonian function. A dynamical system described by Hamilton equations is called a Hamiltonian system. Integrable system A Hamiltonian system is called integrable if there exists a system of coordinates (˛; A) 2 T d  Rd , called angle-action variables, such that in these coordinates the motion is (˛; A) ! (˛ C !(A)t; A), for some smooth function !(A). Hence in these coordinates the Hamiltonian function H depends only on the action variables, H D H0 (A). Invariant torus Given a continuous dynamical system we say that the motion occurs on an invariant d-torus if it takes place on a d-dimensional manifold and its position on the manifold is identified through a coordinate in T d . In an integrable Hamiltonian system all phase space is filled by invariant tori. In a quasi-integrable

system the KAM theorem states that most of the invariant tori persist under perturbation, in the sense that the relative Lebesgue measure of the fraction of phase space filled by invariant tori tends to 1 as the perturbation tends to disappear. The persisting invariant tori are slight deformations of the unperturbed invariant tori. Quasi-integrable system A quasi-integrable system is a Hamiltonian system described by a Hamiltonian function of the form H D H0 (A) C " f (˛; A), with (˛; A) angle-action variables, " a small real parameter and f periodic in its arguments ˛. Quasi-periodic motion Consider the motion ˛ ! ˛ C ! t on T 2 , with ! D (!1 ; !2 ). If !1 /!2 is rational, the motion is periodic, that is there exists T > 0 such that !1 T D !2 T D 0 mod 2. If !1 /!2 is irrational, the motion never returns to its initial value. On the other hand it densely fills T 2 , in the sense that it comes arbitrarily close to any point of T 2 . We say in that case that the motion is quasi-periodic. The definition extends to T d , d > 2: a linear motion ˛ ! ˛ C ! t on T d is quasi-periodic if the components of ! are rationally independent, that is if ! D !1 1 C  C!d d D 0 for  2 Zd if and only if  D 0 (a  b is the standard scalar product between the two vectors a, b). More generally we say that a motion on a manifold is quasi-periodic if, in suitable coordinates, it can be described as a linear quasi-periodic motion. The vector ! is usually called the frequency or rotation vector. Renormalization group By renormalization group one denotes the set of techniques and concepts used to study problems where there are some scale invariance properties. The basic mechanism consists in considering equations depending on some parameters and defining some transformations on the equations, including a suitable rescaling, such that after the transformation the equations can be expressed, up to irrelevant corrections, in the same form as before but with new values for the parameters. Torus The 1-torus T is defined as T D R/2Z, that is the set of real numbers defined modulo 2 (this means that x is identified with y if x  y is a multiple of 2). So it is the natural domain of an angle. One defines the d-torus T d as a product of d 1-tori, that is T d D T      T . For instance one can imagine T 2 as a square with the opposite sides glued together. Tree A graph is a collection of points, called nodes, and of lines which connect the nodes. A walk on the graph is a sequence of lines such that any two successive lines in the sequence share a node; a walk is nontrivial if it

Diagrammatic Methods in Classical Perturbation Theory

contains at least one line. A tree is a planar graph with no closed loops, that is, such that there is no nontrivial walk connecting any node to itself. An oriented tree is a tree with a special node such that all lines of the tree are oriented toward that node. If we add a further oriented line connecting the special node to another point, called the root, we obtain a rooted tree (see Fig. 1 in Sect. “Trees and Graphical Representation”). Definition of the Subject Recursive equations naturally arise whenever a dynamical system is considered in the regime of perturbation theory; for an introductory article on perturbation theory see  Perturbation Theory. A classical example is provided by Celestial Mechanics, where perturbation series, known as Lindstedt series, are widely used; see Gallavotti [21] and  Perturbation Theory in Celestial Mechanics. A typical problem in Celestial Mechanics is to study formal solutions of given ordinary differential equations in the form of expansions in a suitable small parameter, the perturbation parameter. In the case of quasi-periodic solutions, the study of the series, in particular of its convergence, is made difficult by the presence of the small divisors – which will be defined later on. Under some non-resonance condition on the frequency vector, one can show that the series are well-defined to any order. The first proof of such a property was given by Poincaré [53], even if the convergence of the series remained an open problem up to the advent of KAM theory – an account can be found in Gallavotti [17] and in Arnold et al. [2]; see also  Kolmogorov–Arnold–Moser (KAM) Theory. KAM is an acronym standing for Kolmogorov [47], Arnold [1] and Moser [51], who proved in the middle of last century the persistence of most of invariant tori for quasi-integrable systems. Kolmogorov and Arnold proofs apply to analytic Hamiltonian systems, while Moser’s approach deals also with the differentiable case; the smoothness condition on the Hamiltonian function was thereafter improved by Pöschel [54]. In the analytic case, the persisting tori turn out to be analytic in the perturbation parameter, as explicitly showed by Moser [52]. In particular, this means that the perturbation series not only are well-defined, but also converge. However, a systematic analysis with diagrammatic techniques started only recently after the pioneering, fundamental works by Eliasson [16] and Gallavotti [18], and were subsequently extended to many other problems with small divisors, including dynamical systems with infinitely many degrees of freedom, such as nonlinear partial differential equations, and non-Hamiltonian systems.

Some of these extensions will be discussed in Sect. “Generalizations”. From a technical point of view, the diagrammatic techniques used in classical perturbation theory are strongly reminiscent of the Feynman diagrams used in quantum field theory: this was first pointed out by Gallavotti [18]. Also the multiscale analysis used to control the small divisors is typical of renormalization group techniques, which have been successfully used in problems of quantum field theory, statistical mechanics and classical mechanics; see Gallavotti [20] and Gentile & Mastropietro [38] for some reviews. Note that there exist other renormalization group approaches to the study of dynamical systems, and of KAMlike problems in particular, different from that outlined in this article. By confining ourselves to the framework of problems of KAM-type, we can mention the paper by Bricmont et al. [11], which also stressed the similarity of the technique with quantum field theory, and the so called dynamical renormalization group method – see MacKay [50] – which recently produced rigorous proofs of persistence of quasi-periodic solutions; see for instance Koch [46] and Khanin et al. [45]. Introduction Consider the ordinary differential equation on Rd Du D G(u) C "F(u) ;

(1)

where D is a pseudo-differential operator and G, F are real analytic functions. Assume that (1) admits a solution u(0) (t) for " D 0, that is Du(0) D G(u(0) ). The problem we are interested in is to investigate whether there exists a solution of (1) which reduces to u(0) as " ! 0. For simplicity assume G D 0 in the following. The first attempt one can try is to look for solutions in the form of power series in ", u(t) D

1 X

" k u(k) (t) ;

(2)

kD0

which, inserted into (1), when equating the left and right hand sides order by order, gives the list of recursive equations Du(0) D 0, Du(1) D F(u(0) ), Du(2) D @u F(u(0) )u(1) , and so on. In general to order k  1 one has Du(k) D

k1 X 1 s @u F(u(0) ) s! sD0

X

u(k 1 ) : : : u(k s ) ;

(3)

k 1 CCk s Dk1 k i 1

where @su F, the sth derivative of F, is a tensor with s C 1 indices (s must be contracted with the vectors

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u(k 1 ) ; : : : ; u(k s ) ), and the term with s D 0 in the sum has to be interpreted as F(u(0) ) and appears only for k D 1. For instance for F(u) D u3 the first orders give Du(1) D u(0)3 ; Du(2) D 3u(0)2 u(1) ; Du(3) D 3u(0)2 u(2) C 3u(0) u(1)2 ;

(4)

Du(4) D 3u(0)2 u(3) C 6u(0) u(1) u(2) C u(1)3 ; as is easy to check. If the operator D can be inverted then the recursions (3) provide an algorithm to compute the functions u(k) (t). In that case we say that (2) defines a formal power series: by this we mean that the functions u(k) (t) are well-defined for all k  0. Of course, even if this can be obtained, there is still the issue of the convergence of the series that must be dealt with. Examples In this section we consider a few paradigmatic examples of dynamical systems which can be described by equations of the form (1). A Class of Quasi-integrable Hamiltonian Systems Consider the Hamiltonian system described by the Hamiltonian function H (˛; A) D 12 A2 C " f (˛) ;

Td

(5)

Rd

where (˛; A) 2  are angle-action variables, with T D R/2Z, f is a real analytic function, 2-periodic in each of its arguments, and A2 D A  A, if (here and henceforth)  denotes the standard scalar product in Rd , that is a  b D a1 b1 C    C ad bd . Assume also for simplicity that f is a trigonometric polynomial of degree N. The corresponding Hamilton equations are (we shorten @x D @/@x )

 (k) ˛¨ (k) D  " f (˛) :D 

k1 X 1 sC1 @˛ f (! t) s! sD0

X

˛ (k 1 ) : : : ˛ (k s ) : (7)

k 1 CCk s Dk1 k i 1

We look for a quasi-periodic solution of (6), that is a solution of the form ˛(t) D ! t C h(! t), with h(! t) D O("). We call h the conjugation function, as it “conjugates” (that is, maps) the perturbed solution ˛(t) to the unperturbed solution ! t. In terms of the function h (6) becomes h¨ D "@˛ f (! t C h) ;

(8)

where @˛ denotes derivative with respect to the argument. Then (8) can be more conveniently written in Fourier space, where the operator D acts as a multiplication operator. If we write h(! t) D

X

e i!t h ;

2Z d

h D

1 X

" k h(k) ;

which can be written as an equation involving only the angle variables: (6)

which is of the form (1) with u D ˛, G D 0, F D @˛ f , and D D d2 /dt 2 . For " D 0, ˛ (0) (t) D ˛0 C! t is a solution of (6) for any choice of ˛0 2 T d and ! 2 Rd . Take for simplicity ˛0 D

(9)

kD1

and insert (9) into (8) we obtain  (k) (!  )2 h(k) D "@˛ f (˛)  :D

k1 X

X

X

sD0 k 1 CCk s Dk1 0 C1 CC s D  i 2Z n k i 1

 (i0 )sC1 f0 h(k11 ) : : : h(ks s ) :

˛˙ D @A H (˛; A) D A ; A˙ D @˛ H (˛; A) D "@˛ f (˛) ;

˛¨ D "@˛ f (˛) ;

0: we shall see that this choice makes sense. We say that for " D 0 the Hamiltonian function (5) describes a system of d rotators. We call ! the frequency vector, and we say that ! is irrational if its components are rationally independent, that is if !   D 0 for  2 Zd if and only if  D 0. For irrational ! the solution ˛ (0) (t) describes a quasi-periodic motion with frequency vector !, and it densely fills T d . Then (3) becomes

1 s! (10)

These equations are well-defined to all orders provided ["@˛ f (˛)](k)  D 0 for all  such that !   D 0. If ! is an irrational vector we need ["@˛ f (˛)](k) 0 D 0 for the equations to be well-defined. In that case the coefficients h0(k) are left undetermined, and we can fix them arbitrarily to vanish (which is a convenient choice). We shall see that under some condition on ! a quasiperiodic solution ˛(t) exists, and densely fills a d-dimensional manifold. The analysis carried out above for ˛0 D 0 can be repeated unchanged for all values of ˛0 2 T d : ˛0

Diagrammatic Methods in Classical Perturbation Theory

represents the initial phase of the solution, and by varying ˛0 we cover all the manifold. Such a manifold can be parametrized in terms of ˛0 , so it represents an invariant torus for the perturbed system. A Simplified Model with No Small Divisors Consider the same equation as (8) with D D d2 /dt 2 replaced by 1, that is h D "@˛ f (! t C h) :

(11)

Of course in this case we no longer have a differential equation; still, we can look again for quasi-periodic solutions h(! t) D O(") with frequency vector !. In such a case in Fourier space we have  (k) h(k) D "@˛ f (˛)  :D

k1 X

X

X

sD0 k 1 CCk s Dk1 0 C1 CC s D  i 2Z n k i 1

1 (i0 )sC1 s!

 f0 h(k11 ) : : : h(ks s ) :

(12)

For instance if d D 1 and f (˛) D cos ˛ the equation, which is known as the Kepler equation, can be explicitly solved by the Lagrange inversion theorem [55], and gives

h(k)

D

8 ˆ < ˆ :

jj  k;

i(1) kC(k)/2 ; 2 ((k  )/2)!((k C )/2)!

 C k even ;

0;

otherwise :

k

(13) We shall show in Sect. “Small Divisors” that a different derivation can be provided by using the forthcoming diagrammatic techniques. The Standard Map Consider the finite difference equation D˛ D " sin ˛ ;

(14)

on T , where now D is defined by D˛( ) :D 2˛( )  ˛( By writing ˛ D Dh D " sin(

C !)  ˛(

k1

h(k) D

X 1 2 4 sin (!/2) sD0

X

X

k 1 CCk s Dk1 0 C1 CC s D  i 2Z k i 1

 (i0 )sC1 f0 h(k11 ) : : : h(ks s ) ;

1 s!

(18)

where 0 D ˙1 and f˙1 D 1/2. Note that (17) is a discrete dynamical system. However, when passing to Fourier space, (18) acquires the same form as for the continuous dynamical systems previously considered, simply with a different kernel for D. In particular if we replace D with 1 we recover the Kepler equation. The number ! is called the rotation number. We say that ! is irrational if the vector (2; !) is irrational according to the previous definition. Trees and Graphical Representation Take ! to be irrational. We study the recursive equations (  (k) h(k) D g(!  ) "@˛ f (˛)  ;  ¤ 0 ; (19) (k)   D0; "@˛ f (˛) 0 D 0 ; where the form of g depends on the particular model we are investigating. Hence one has either g(!  ) D (!  )2 or g(!  ) D 1 or g(!  ) D (2 sin(!/2))2 according to models described in Sect. “Examples”. For  ¤ 0 we have equations which express the co0 efficients h(k) ,  2 Zd , in terms of the coefficients h(k ) ,  2 Zd , with k 0 < k, provided the equations for  D 0 are satisfied for all k  1. Recursive equations, such as (19), naturally lead to a graphical representation in terms of trees. Trees

 !) :

(15)

C h( ), (14) becomes C h) ;

In other words, by writing x D C h( ) and y D ! C h( ) h( !), with ( ; !) solving (17) for " D 0, that is ( 0 ; ! 0 ) D ( C !; !), we obtain a closed-form equation for h, which is exactly (16). In Fourier space the operator D acts as D : e i ! 4 sin2 (!/2)e i , so that, by expanding h according to (9), we can write (16) as

(16)

which is the functional equation that must be solved by the conjugation function of the standard map ( x 0 D x C y C " sin x ; (17) y 0 D y C " sin x :

A connected graph G is a collection of points (nodes) and lines connecting all of them. Denote with N(G ) and L(G ) the set of nodes and the set of lines, respectively. A path between two nodes is the minimal subset of L(G ) connecting the two nodes. A graph is planar if it can be drawn in a plane without graph lines crossing. A tree is a planar graph G containing no closed loops. Consider a tree G with a single special node v0 : this introduces a natural partial ordering on the set of lines and nodes, and one can imagine that each line carries an arrow

129

130

Diagrammatic Methods in Classical Perturbation Theory

pointing toward the node v0 . We add an extra oriented line `0 exiting the special node v0 ; the added line will be called the root line and the point it enters (which is not a node) will be called the root of the tree. In this way we obtain a tree defined by N( ) D N(G ) and L( ) D L(G ) [ `0 . A labeled tree is a rooted tree together with a label function defined on the sets L( ) and N( ). We call equivalent two rooted trees which can be transformed into each other by continuously deforming the lines in the plane in such a way that the lines do not cross each other. We can extend the notion of equivalence also to labeled trees, by considering equivalent two labeled trees if they can be transformed into each other in such a way that the labels also match. In the following we shall deal mostly with nonequivalent labeled trees: for simplicity, where no confusion can arise, we call them just trees. Given two nodes v; w 2 N( ), we say that w  v if v is on the path connecting w to the root line. We can identify a line ` through the node v it exits by writing ` D `v . We call internal nodes the nodes such that there is at least one line entering them, and end-points the nodes which have no entering line. We denote with L( ), V ( ) and E( ) the set of lines, internal nodes and end-points, respectively. Of course N( ) D V( ) [ E( ). The number of unlabeled trees with k nodes (and hence with k lines) is bounded by 22k , which is a bound on the number of random walks with 2k steps [38]. For each node v denote by S(v) the set of the lines entering v and set sv D jS(v)j. Hence sv D 0 if v is an endnode, and sv  1 if v is an internal node. One has X X sv D sv D k  1 ; (20) v2N( )

v2V ( )

this can be easily checked by induction on the order of the tree. An example of unlabeled tree is represented in Fig. 1. For further details on graphs and trees we refer to the literature; cf. for instance Harary [43]. Labels and Diagrammatic Rules We associate with each node v 2 N( ) a mode label v 2 Zd , and with each line ` 2 L( ) a momentum label ` 2 Zd , with the constraint X X w D v C ` ; (21) `v D w2N( ) wv

`2S(v)

which represents a conservation rule for each node. Call T k; the set of all trees with k nodes and momentum  associated with the root line. We call k and  the order and the momentum of , respectively.

Diagrammatic Methods in Classical Perturbation Theory, Figure 1 An unlabeled tree with 17 nodes

Diagrammatic Methods in Classical Perturbation Theory, Figure 2 Graph element

Diagrammatic Methods in Classical Perturbation Theory, Figure 3 Graphical representation of the recursive equations

We want to show that trees naturally arise when studying (19). Let h(k) be represented with the graph element in Fig. 2 as a line with label  exiting from a ball with label (k). Then we can represent (19) graphically as depicted in Fig. 3. Simply represent each factor h(ki i ) on the right hand side as a graph element according to Fig. 2. The lines of all such graph elements enter the same node v0 . This is a graphical expedient to recall the conservation rule: the momentum  of the root line is the sum of the mode label 0 of the node v0 plus the sum of the momenta of the lines entering v0 . The first few orders k  4 are as depicted in Fig. 4. For each node the conservation rule (21) holds: for instance for k D 2 one has  D 1 C 2 , for k D 3 one has  D 1 C `1 and `1 D 2 C 3 in the first tree and  D 1 C 2 C 3 in

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 4 Trees of lower orders

the second tree, and so on. Moreover one has to sum over all possible choices of the labels v , v 2 N( ), which sum up to . Given any tree 2 T k; we associate with each node v 2 N( ) a node factor F v and with each line ` 2 L( ) a propagator g` , by setting Fv :D

1 (iv )s v C1 fv ; sv !

g` :D g(!  ` ) ;

and define the value of the tree as  Y   Y Fv g` : Val( ) :D v2N( )

(22)

(23)

`2L( )

The propagators g` are scalars, whereas each F v is a tensor with sv C 1 indices, which can be associated with the sv C 1 lines entering or exiting v. In (23) the indices of the tensors F v must be contracted: this means that if a node v is connected to a node v0 by a line ` then the indices of F v and Fv 0 associated with ` are equal to each other, and eventually one has to sum over all the indices except that associated with the root line. For instance the value of the tree in Fig. 4 contributing to h(2) is given by Val( ) D (i1 )2 f1 (i2 ) f2 g(!  )g(!  2 ) ;

with 1 C 2 D , while the value of the last tree in Fig. 4 contributing to h(4) is given by Val( ) D

(i1 )4 f1 (i2 ) f2 (i3 ) f3 (i4 ) f4 3!  g(!  )g(!  2 )g(!  3 )g(!  4 ) ;

with 1 C 2 C 3 C 4 D . It is straightforward to prove that one can write X h(k) D Val( ) ;  ¤0; k  1:

(24)

2T k;

This follows from the fact that the recursive equations (19) can be graphically represented through Fig. 3: one iterates the graphical representation of Fig. 3 until only graph elements of order k D 1 appear, and if is of order 1 (cf. Fig. 4) then Val( ) D (i) f g(!  ). Each line ` 2 L( ) can be seen as the root line of the tree consisting of all nodes and lines preceding `. The choice h0(k) D 0 for all k  1 implies that no line can have zero momentum: in other words we have ` ¤ 0 for all ` 2 L( ). Therefore in order to prove that (9) with h(k) given by (24) solves formally, that is order by order, the Eqs. (19),

131

132

Diagrammatic Methods in Classical Perturbation Theory

we have only to check that ["@˛ f (! t C h(! t))](k) 0 D 0 for all k  1. If we define g` D 1 for ` D 0, then also the second relation in (19) can be graphically represented as in Fig. 3 by setting  D 0 and requiring h0(k) D 0, which yields that the sum of the values of all trees on the right hand side must vanish. Note that this is not an equation to solve, but just an identity that has to be checked to hold at all orders. For instance for k D 2 (the case k D 1 is trivial) the identity ["@˛ f (! t C h(! t))](2) 0 D 0 reads (cf. the second line in Fig. 4) X (i1 )2 f1 (i2 ) f2 g(!  2 ) D 0 ; 1 C2 D0

which is found to be satisfied because the propagators are even in their arguments. Such a cancellation can be graphically interpreted as follows. Consider the tree with mode labels 1 and 2 , with 1 C 2 D 0: its value is (i1 )2 f1 (i2 ) f2 g(!  2 ). One can detach the root line from the node with mode label 1 and attach it to the node with mode label 2 , and reverse the arrow of the other line so that it points toward the new root line. In this way we obtain a new tree (cf. Fig. 5): the value of the new tree is (i1 ) f1 (i2 )2 f2 g(!  1 ), where g(!  1 ) D g(!  2 ) D g(!  2 ), so that the values of the two trees contain a common factor (i1 ) f1 (i2 ) f2 g(!  2 ) times an extra factor which is (i1 ) for the first tree and (i2 ) for the second tree. Hence the sum of the two values gives zero. The cancellation mechanism described above can be generalized to all orders. Given a tree one considers all trees which can be obtained by detaching the root line and attaching to the other nodes of the tree, and by reversing the arrows of the lines (when needed) to make them point toward the root line. Then one sums together the values of all the trees so obtained: such values contain a common factor times a factor iv , if v is the node which the root line exits (the only nontrivial part of the proof is to check that the combinatorial factors match each other: we refer to Gentile & Mastropietro [37] for details). Hence the sum gives zero, as the sum of all the mode labels vanishes. For instance for k D 3 the cancellation operates by considering the three trees in Fig. 5: such trees can be considered to be obtained from each other by shifting the root line and consistently reversing the arrows of the lines.

In such a case the combinatorial factors of the node factors are different, because in the second tree the node factor associated with the node with mode label 2 contains a factor 1/2: on the other hand if 1 ¤ 3 there are two nonequivalent trees with that shape (with the labels 1 and 3 exchanged between themselves), whereas if 1 D 3 there is only one such tree, but then the first and third trees are equivalent, so that only one of them must be counted. Thus, by using that 1 C 2 C 3 D 0 – which implies g(!  (2 C 3 )) D g(!  1 ) and g(!  (1 C 2 )) D g(!  3 ) – in all cases we find that the sum of the values of the trees gives a common factor (i1 ) f1 (i2 )2 f2 (i3 ) f3 g(!  3 )g(!  1 ) times a factor 1 or 1/2 times i(1 C2 C3 ), and hence vanishes: once more the property that g is even is crucial. Small Divisors We want to study the convergence properties of the series h(! t) D

X

e i!t h ;

h D

2Z d

" k h(k) ;

(25)

kD1

which has been shown to be well–defined as a formal power series for the models considered in Sect. “Examples”. Recall that the number of unlabeled trees of order k is bounded by 22k . To sum over the labels we can confine ourselves to the mode labels, as the momenta are uniquely determined by the mode labels. If f is a trigonometric polynomial of degree N, that is f D 0 for all  such that jj :D j1 j C    C jd j > N, we have that h(k) D 0 for all jj > kN (which can be easily proved by induction), and moreover we can bound the sum over the mode labels of any tree of order k by (2N C 1)d k . Finally we can bound Y Y jv js vC1  N s vC1  N 2k ; (26) v2N( )

v2N( )

because of (20). For the model (11), where g` D 1 in (22), we can bound ˇ ˇ ˇ X ˇˇ ˇ ˇ (k) ˇ ˇh(k) ˇ  22k (2N C 1)d k N 2k ˚ k ; ˇh ˇ  2Z d

(27) ˚ D max j f j ; jjN

which shows that the series (25) converges for " small enough, more precisely for j"j < "0 , with "0 :D C0 (4N 2 ˚(2N C 1)d )1 ;

Diagrammatic Methods in Classical Perturbation Theory, Figure 5 Trees to be considered together to prove that ["@f(˛)]20 D 0

1 X

(28)

where C0 D 1. Hence the function h(! t) in that case is analytic in ". For d D 1 and f (˛) D cos ˛, we can eas-

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 6 Trees to be considered together to prove that ["@f(˛)]30 D 0

ily provide an exact expression for the coefficients h(k) : all the computational difficulties reduce to a combinatorial check, which can be found in Gentile & van Erp [42], and the formula (13) is recovered. However for the models where g` ¤ 1, the situation is much more involved: the propagators can be arbitrarily close to zero for  large enough. This is the so-called small divisor problem. The series (25) is formally well–defined, assuming only an irrationality condition on !. But to prove the convergence of the series, we need a stronger condition. For instance one can require the standard Diophantine condition j!  j >

jj

8 ¤ 0 ;

(29)

for suitable positive constants and . For fixed  > d 1, the sets of vectors which satisfy (29) for some constant

> 0 has full Lebesgue measure in Rd [17]. We can also impose a weaker condition, known as the Bryuno condition, which can be expressed by requiring B(!) :D

1 X 1 1 log <1: k min0<jj2k j!  j 2

(30)

kD0

We call Diophantine vectors and Bryuno vectors the vectors satisfying (29) and (30), respectively. Note that any Diophantine vector satisfies (30). If we assume only analyticity on f – that is, we remove the assumption that f be a trigonometric polynomial, – then we need a Diophantine condition, such as (29) or (30), also to show that the series (25) are well-defined as formal power series: the condition is needed, as one can easily check, in order to sum over the mode labels. In the case of the standard map j!  j must be replaced with min p2Z j!  pj, and the function B(!) can be expressed in terms of the best approximants of the number ! [15]. For the models with small divisors considered in Sect. “Examples” convergence of the series (25) can be proved for ! satisfying (29) or (30). But this requires more work, and, in particular, a detailed discussion of the product of propagators in (24). In the following we shall consider the case of Diophantine vectors, by referring to the

bibliography for the Bryuno vectors (cf. Sect. “Generalizations”).

Multiscale Analysis Consider explicitly the case g(!  ) D (!  )2 in (19) and ! satisfying (29). We can introduce a label characterizing the size of the propagator: we say that  2 Zd is on scale (

n 1;

if 2n  j!  j < 2(n1) ;

n D0;

if  j!  j ;

(31)

where is the constant appearing in (29), and we say that a line ` has a scale label n` D n if ` is on scale n. If Nn ( ) denotes the number of lines ` 2 L( ) with scale n` D n, then we can bound in (23) 1 ˇ Y ˇ Y ˇ ˇ g` ˇ  2k 22nNn ( ) ; ˇ `2L( )

(32)

nD0

so that the problem is reduced to bounding N n ( ). The product of propagators gives problems when the small divisors “accumulate”. To make more precise the idea of accumulation we introduce the notion of cluster. Once all lines of a tree have been given their scale labels, for any n  0 we can identify the maximal connected sets of lines with scale not larger than n. If at least one among such lines has scale equal to n we say that the set is a cluster on scale n. For instance consider the tree in Fig. 1, and assign the mode labels to the nodes: this uniquely fixes the momenta, and hence the scale labels, of the lines. Suppose that we have found the scales as in Fig. 7a. Then a cluster decomposition as in Fig. 7b follows. Given a cluster T call L(T) the set of lines of contained in T, and denote by N(T) the set of nodes connected by such lines. We define k T D jN(T)j the order of the cluster T. Any cluster has either one or no exiting line, and can have an arbitrary number of entering lines. We call selfenergy clusters the clusters which have one exiting line and only one entering line and are such that both lines have the same momentum. This means that if T is a self-energy

133

134

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 7 Example of clusters with their scales

cluster and `1 and `2 are the lines entering and exiting T, respectively, then `1 D `2 , so that X

v D 0 :

(33)

v2N(T)

By definition of scale, both lines `1 and `2 have the same scale, say n, and that, by definition of cluster, one has n` < n for all ` 2 L(T). We define the value of the self-energy cluster T whose entering line has momentum  as the matrix VT (!  ) :D

 Y v2N(T)

Fv

 Y

 g` ;

(34)

`2L(T)

where all the indices of the node factors must be contracted except those associated with the line `1 entering T and with the line `2 exiting T. We can extend the notion of self-energy clusters also to a single node, by saying that v is a self-energy cluster if sv D 1 and the line entering v has the same momentum as the exiting line. In that case (34) has to be interpreted as VT (!  ) D Fv : in particular it is independent of !  . The simplest self-energy cluster one can think of consists of only one node v, but then (33) implies v D 0,

Diagrammatic Methods in Classical Perturbation Theory, Figure 8 Self-energy clusters of order k D 2

so that Fv D 0, see (22), hence the corresponding value is zero. Thus the simplest non-trivial self-energy clusters contain at least two nodes, and are represented by the clusters T 1 , T 2 and T 3 of Fig. 8. In all cases one has 1 C2 D 0. By using the definition (34) one has VT1 (x) D (i1 )2 f1 (i2 )2 f2 g(!  2 C x) and VT2 (x) D VT3 (x) D (i1 )3 f1 (i2 ) f2 g(!  2 )/2, with x D !  . Hence for x D 0 the sum of the three values gives zero. It is not difficult to see that also the first derivatives of the values of all self-energy clusters of order 2 sum up to zero, that is the sum of the values of all possible self-energy clusters of order k D 2 gives zero up to order x2 . (The only caveat is that, if we want to derive VT (x), the sharp multiscale decomposition in (31) can be a little annoying, hence it can be more convenient to replace it with a smooth decomposition through C 1 compact support functions; we refer to the literature for details). This property generalizes to all orders k, and the underlying cancellation mechanism is essentially the same that ensures the validity of the second relation in (19). The reason why it is important to introduce the selfenergy clusters is that if we could neglect them then the product of small divisors would be controlled. Indeed, let us denote by Rn ( ) the number of lines on scale n which do exit a self-energy cluster, and set Nn ( ) D N n ( ) 

Diagrammatic Methods in Classical Perturbation Theory

Rn ( ). Then an important result, known as the Siegel– Bryuno lemma, is that Nn ( )  c2n/ k ;

(35)

for some constant c, where k is the order of and  is the Diophantine exponent in (29). The bound (35) follows from the fact that if Nn ( ) ¤ 0 then Nn ( )  E(n; k) :D 2N k2(n2)/  1, which can be proved by induction on k as follows. Given a tree let `0 be its root line, let `1 ; : : : ; `m , m  0, be the lines on scales  n which are the closest to `0 , and let 1 ; : : : ; m the trees with root lines `1 ; : : : ; `m , respectively (cf. Fig. 9 – note that by construction all lines ` in the subgraph T have scales n` < n, so that if n`0  n then T is necessarily a cluster). If either `0 is not on scale n or it is on scale n but exits a self-energy cluster then Nn ( ) D Nn ( 1 ) C    C Nn ( m ) and the bound Nn ( )  E(n; k) follows by the inductive hypothesis. If `0 does not exit a self-energy cluster and n`0 D n then Nn ( ) D 1 C Nn ( 1 ) C    C Nn ( m ), and the lines `1 ; : : : ; `m enter a cluster T with k T D k  (k1 C    C k m ), where k1 ; : : : ; k m are the orders of 1 ; : : : ; m , respectively. If m  2 the bound Nn ( )  E(n; k) follows once more by the inductive hypothesis. If m D 0 then Nn ( ) D 1; on the other hand for `0 to be on scale n`0 D n one must have j!  `0 j < 2nC1 (see (31)), which, by the Diophantine condition (29), implies N k  j`0 j > 2(n1)/ , hence E(n; k) > 1. If m D 1 call 1 and 2 the momenta of the lines `0 and `1 , respectively. By construction T cannot be a self-energy cluster, hence 1 ¤ 2 , so that, by the Diophantine condition (29), 2nC2  j!  1 j C j!  2 j  j!  (1  2 )j

; > j1  2 j because n`0 D n and n`1  n. Thus, one has X N kT  jv j  j1  2 j > 2(n2)/ ;

hence T must contain “many nodes”. In particular, one finds also in this case Nn ( ) D 1 C Nn ( 1 )  1 C E(n; k1 )  1 C E(n; k)  E(n; k T )  E(n; k), where we have used that E(n; k T )  1 by (37). The argument above shows that small divisors can accumulate only by allowing self-energy clusters. That accumulation really occurs is shown by the example in Fig. 10, where a tree of order k containing a chain of p self-energy clusters is depicted. Assume for simplicity that k/3 is an integer: then if p D k/3 the subtree 1 with root line ` is of order k/3. If the line ` entering the rightmost self-energy cluster T p has momentum , also the lines exiting the p self-energy clusters have the same momentum . Suppose that jj  N k/3 and j!  j  /jj (this is certainly possible for some ). Then the value of the tree grows like a1k (k!) a 2 , for some constants a1 and a2 : a bound of this kind prevents the convergence of the perturbation series (25). If no self-energy clusters could occur (so that Rn ( ) D 0) the Siegel–Bryuno lemma would allow us to bound in (32) 1 Y

22nNn ( ) D

nD0

1 Y



22nNn ( )

nD0 1   X  exp C1 k n2n/  C2k ;

(38)

nD0

for suitable constants C1 and C2 . In that case convergence of the series for j"j < "0 would follow, with "0 defined as in (26) with C0 D 2 /C2 . However, there are selfenergy clusters and they produce factorials, as the example in Fig. 10 shows, so that we have to deal with them.

(36) Resummation (37)

v2N(T)

Let us come back to the Eq. (10). If we expand g(!  )["@˛ f (˛)](k)  in trees according to the diagrammatic rules described in Sect. “Trees and Graphical Representation”, we can distinguish between contributions in which the root line exits a self-energy cluster T, that we can write as X g(!  ) VT (!  )h(kk T ) ; (39) T : k T
Diagrammatic Methods in Classical Perturbation Theory, Figure 9 Construction for the proof of the Siegel–Bryuno lemma

and all the other contributions, that we denote by ["@˛ f (˛)](k)  . We can shift the contributions (39) to the left hand side of (10) and divide by g(!  ), so to obtain X VT (!  )h(kk T ) D(!  )h(k)  T : k T
D ["@˛ f (! t C h)](k)  :

(40)

135

136

Diagrammatic Methods in Classical Perturbation Theory

Diagrammatic Methods in Classical Perturbation Theory, Figure 10 Example of accumulation of small divisors because of the self-energy clusters

where D(!  ) D 1/g(!  ) D (!  )2 . By summing over k and setting M(!  ; ") D

1 X kD1

"k

X

VT (!  ) ;

(41)

T : k T Dk

 1 g` :D g [n] (!; ") D D(!  )  M[n] (!  ; ") ; (44)

then (40) gives D(!  )h D ["@˛ f (! t C h)] ;

with

D(!  ) :D D(!  )  M(!  ; ") :

(42)

The motivation for proceeding in this way is that, at the price of changing the operator D into D, hence of changing the propagators, lines exiting self-energy clusters no longer appear. Therefore, in the tree expansion of the right hand side of the equation, we have eliminated the selfenergy clusters, that is the source of the problem of accumulation of small divisors. Unfortunately the procedure described above has a problem: M(!  ; ") itself is a sum of self-energy clusters, which can still contain some other self-energy clusters on lower scales. So finding a good bound for M(!  ; ") could have the same problems as for the values of the trees. To deal with such a difficulty we modify the prescription by proceeding recursively, in the following sense. Let us start from the momenta  which are on scale n D 0. Since there are no self-energy clusters with exiting line on scale n D 0, for such  one has M(!  ; ") D 0. Next, we consider the momenta  which are on scale n D 1, and we can write (42), where now all self-energy clusters T whose values contribute to M(!  ; ") cannot contain any self-energy clusters, because the lines ` 2 T are on scale n` D 0. Then, we consider the momenta  which are on scale n D 2: again all the self-energy clusters contributing to M(!  ; ") do not contain any self-energy clusters, because the lines on scale n D 0; 1 cannot exit self-energy clusters by the construction of the previous step, and so on. The conclusion is that we have obtained a different expansion for h(! t), that we call a resummed series, h(! t) D

X 2Z d

e i!t h ;

where the self-energy clusters do not appear any more in the tree expansion and the propagators must be defined recursively: the propagator g` of a line ` on scale n` D n and momentum ` D  is the matrix

h D

1 X kD1

" k h[k] (") ; (43)

X

M[n] (!  ; ") :D

" k T VT (!  ) ;

(45)

T : n T
where the value VT (!  ) is written in accord with (34), with all the lines `0 2 L(T) on scales n`0 < n and the corresponding propagators g`0 expressed in terms of matrices M[n `0 ] (! `0 ; ") as in (44). By construction, the new propagators depend on ", so that the coefficients h[k] (") depend explicitly on ": hence (43) is not a power series expansion. The coefficients h[k] (") still admit a tree expansion h[k] (") D

X

Val( );

R 2T k;

Val( ) :D

 Y v2N( )

Fv

 ¤ 0; k  1 ;

 Y

 (46) g [n ` ] (!  ` ; ") ;

`2L( )

R which replaces (24). In particular T k; is defined as the set of renormalized trees of order k and momentum , where “renormalized” means that the trees do not contain any self-energy clusters. Now the propagators are matrices: the contractions of the indices in (46) yields that if ` connects v to v0 the indices of F v and Fv 0 associated with ` must be equal to the column and row indices of g` , respectively. R Since for any tree 2 T k; one has Nn ( ) D  Nn ( ), we can bound the product of propagators according to (36) provided the propagators on scale n can still be bounded proportionally to 22n . This is certainly not obvious, because of the extra term M[n] (!  ; ") appearing in (44). It is a remarkable cancellation that M[n] (x; ") vanishes up to second order, that is M[n] (x; ") D O(x 2 ), so that, by

Diagrammatic Methods in Classical Perturbation Theory

taking into account also that VT (x) ¤ 0 requires k T  2, we can write, for some constant C, [n] M[n] (x; ") D "2 x 2 M (x; ") ;

   [  M n](x; ")  C ; (47)

where k  k denotes – say – the uniform norm. The cancellation leading to (47) can be proved as discussed in Sect. “Multiscale Analysis” – where it has been explicitly proved for k D 2 in the absence of resummation. Now the propagators are matrices, but one can prove (by induction on n) that M[n] (x; ") D (M[n] (x; "))T D (M[n] (x; ")) , with T and † denoting transposition and adjointness, respectively, and this is enough to see that the same cancellation mechanism applies [23]. Thus (47) implies that  1     [n]   g (x; ") D  x 2  "2 x 2 M[n] (x; ")   2 ; (48)  x2  and the same argument as used in Sect. “Small Divisors” implies that the series in (43) for h converges for j"j < "0 , where "0 is defined as in (26) with C0 D 2 2 /C2 . Moreover (47) also implies that h(! t) is analytic in " (notwithstanding that the expansion is not a power expansion), so that we can say a posteriori that the original power series (25) also converges. Generalizations The case where the function G(u) in (1) does not vanish is notationally more involved, and it is explicitly discussed in Gentile [28]. Extensions of the results described above to Hamiltonian functions more general than (5) require only some notational complications, and can be found in Gentile & Mastropietro [37] for anisochronous systems and in Bartuccelli & Gentile [3] for isochronous systems. The case (5) with f a real analytic function was first discussed by using trees in Chierchia & Falcolini [12]. Lower-Dimensional Tori A tree formalism for hyperbolic lower-dimensional tori was introduced and studied in Gallavotti [19], Gentile [26] and Gallavotti, Gentile & Mastropietro [25], for systems consisting of a set of rotators interacting with a pendulum. In that case also the stable and unstable manifolds (whiskers) of the tori were studied with diagrammatic techniques. For Hamiltonian functions of the form (5) solutions of the form (43) describe quasi-periodic motions with

frequency vector ! on a d-dimensional manifold. If we fix t, say t D 0, and keep ˛0 as a parameter, we obtain a parametrization of the manifold in terms of ˛0 2 T d , hence the manifold is an invariant torus. We say that the torus is a maximal torus. We can study the problem of persistence of lowerdimensional tori by considering unperturbed solutions ˛(t) D ˛0 C ! t, where the components of ! are rationally dependent. For instance we can imagine that there exist s linearly independent vectors b 1 ; : : : ;b s 2 Zd such that !  b  k D 0 for k D 1; : : : ; s. In that case, we say that the unperturbed torus is a resonant torus of order s. We can imagine performing a linear change of variables which transforms the frequency vector into a new vector, that we ¯ 0), where !¯ 2 Rr still denote with !, such that ! D (!; s and 0 2 R , with r C s D d, and !¯ is irrational. This nat¯ ˇ), with ˛¯ 2 T r and urally suggests that we write ˛ D (˛; s ˇ 2 T . More generally, here and henceforth in this subsection for any vector v 2 Rd we denote by v¯ the vector in Rr whose components are the first r components of v. For instance for the initial phase ˛0 we write ˛0 D (˛¯ 0 ; ˇ0 ). In general the resonant torus is destroyed by the perturbation, and only some lower-dimensional tori persist under perturbation. We assume on !¯ one of the Diophantine conditions of Sect. “Small Divisors” in Rr , for instance j!¯  ¯ j > /j¯j for all ¯ 2 Zr , ¯ ¤ 0. To prove the existence of a maximal torus for the system with Hamiltonian function (5) we needed no condition on the perturbation f . On the contrary to prove existence of lower-dimensional tori, we need some nondegeneracy condition: by defining Z f0 (ˇ) D

Tr

d˛¯ ¯ ˇ) ; f (˛; (2)r

(49)

if @ˇ f0 (ˇ ) D 0 for some ˇ then we assume that the matrix @2ˇ f0 (ˇ ) is positive definite (more generally one could assume it to be non-singular, that is det @2ˇ f0 (ˇ ) ¤ 0). The formal analysis can be carried out as in Sect. “Trees and Graphical Representation”, with the only difference being that now the compatibility conditions ["@˛ f (˛)](k)  D 0 have to be imposed for all  such that ¯ D 0, because !   D !¯  ¯ . It turns out to be an identity only for the first r components (this is trivial for k D 1, whereas it requires some work for k > 1). For the last s components, for k D 1 it reads @ˇ f0 (ˇ0 ) D 0, hence it fixes ˇ0 to be a stationary point for f0 (ˇ), while for higher values of k it fixes the corrections of higher order of these values (to do this we need the non-degeneracy condition). Thus, we are free to choose only ˛¯ 0 as a free parameter, since the last s components of ˛0 have to be fixed.

137

138

Diagrammatic Methods in Classical Perturbation Theory

Clusters and self-energy clusters are defined as in Sect. “Multiscale Analysis”. Note that only the first r components ¯ of the momenta  intervene in the definition of the scales – again because !   D !¯  ¯ . In particular, in the definition of self-energy clusters, in (33) we must replace v with ¯ v . Thus, already to first order the value of a self-energy cluster can be non-zero: for k T D 1, that is for T containing only a node v with mode label (¯v ; ˜v ) D (0; ˜ v ), the matrix VT (x; ") is of the form  0 VT (x) D 0 (b˜ v ) i; j D e



0 ; b˜ v

i ˜ v ˇ0

(50)

(i˜v;i )(i˜v; j ) f(0;˜v ) ;

with i; j D r C 1; : : : ; d. If we sum over ˜ v 2 Zs and multiply times ", we obtain 

 0 0 ; 0 "B (51) X ˜ D e i ˇ0 (i˜ i )(i˜ j ) f(0;˜) D @ˇ i @ˇ j f0 (ˇ0 ) :

M0 :D B i; j

˜ 2Z s

The s  s block B is non-zero: in fact, the non-degeneracy condition yields that it is invertible. To higher orders one finds that the matrix M[n] (x; "), with x D !   D !¯  ¯ , is a self-adjoint matrix and M[n] (x; ") D (M[n] (x; "))T , as in the case of maximal tori. Moreover the corresponding eigenvalues [n] i (x; ") 2 x 2 ) for i D 1; : : : ; r and [n] (x; (x; ") D O(" satisfy [n] i i ") D O(") for i D r C 1; : : : ; d; this property is not trivial because of the off-diagonal blocks (which in general do not vanish at orders k  2), and to prove it one has to use the self-adjointness of the matrix M[n] (x; "). More 2 precisely one has [n] i (x; ") D "a i C O(" ) for i > r, where a rC1 ; : : : ; ad are the s eigenvalues of the matrix B in (51). From this point on the discussion proceeds in a very different way according to the sign of " (recall that we are assuming that a i > 0 for all i > r). 2 For " < 0 one has [n] i (x; ") D a i " C O(" ) < 0 for i > r, so that we can bound the last s eigenvalues of x 2  M[n] (x; ") with x2 , and the first r with x 2 /2 by the same argument as in Sect. “Resummation”. Hence we obtain easily the convergence of the series (43); of course, analyticity at the origin is prevented because of the condition " < 0. We say in that case that the lower-dimensional tori are hyperbolic. We refer to Gallavotti & Gentile [22] and Gallavotti et al. [23] for details. The case of elliptic lower-dimensional tori – that is " > 0 when a i > 0 for all i > r – is more difficult. Essentially the idea is as follows (we only sketch the strat-

egy: the details can be found in Gentile & Gallavotti [36]). One has to define the scales recursively, by using a variant, first introduced in Gentile [27], of the resummation technique described in Sect. “Resummation”. We say that  is on scale 0 if j!¯  ¯ j  and on scale [ 1] otherwise: for  on scale 0 we write (42) with M D M0 , as given in (51). This defines the propagators of the lines ` on scale n D 0 as 1  g` D g [0] (!  ` ) D (!¯  ¯ ` )2  M0 :

(52)

Denote by  i the eigenvalues of M 0 : given  on scale [ 1] wepsay that  is on scale 1 if 21  min iD1;:::;d pj(!¯  ¯ )2   i j, and on scale [ 2] if min iD1;:::;d j(!¯  ¯ )2   i j < 21 . For  on scale 1 we write (42) with M replaced by M[0] (!  ¯ ; "), which is given by M 0 plus the sum of the values of all self-energy clusters T on scale n T D 0. Then the propagators of the lines ` on scale n` D 1 is defined as  1 g` D g [1] (! ` ) D (!¯  ¯ ` )2  M[0] (!¯  ¯ ` ; ") : (53) [n] Call [n] i (x; ") the eigenvalues of M (x; "): given  on scale [ 2]qwe say that  is on scale 2 if 22 

¯  ¯ ; ")j, and on scale [ 3] j(!¯  ¯ )2  [0] i (! q ¯  ¯ ; ")j < 22 . For  if min iD1;:::;d j(!¯  ¯ )2  [0] i (! on scale 2 we write (42) with M replaced by M[1] (!¯  ¯ ; "), which is given by M[0] (!¯  ¯ ; ") plus the sum of the values of all self-energy clusters T on scale n T D 1. Thus, the propagators of the lines ` on scale n` D 2 will be defined as min iD1;:::;d

 1 ; (54) g` D g [2] (! ` ) D (!¯  ¯ ` )2  M[1] (!¯  ¯ ` ; ") and so on. The propagators are self-adjoint matrices, hence their norms can be bounded through the corresponding eigenvalues. In order to proceed as in Sects. “Multiscale Analysis” and “Resummation” we need some Diophantine conditions on these eigenvalues. We can assume for some  0 >  ˇ ˇ q ˇ ˇ ˇj!¯  ¯ j  j[n] (!¯  ¯ ; ")jˇ >

i ˇ ˇ j¯j0

8¯ ¤ 0 ; (55)

for all i D 1; : : : ; d and n  0. These are known as the first Melnikov conditions. Unfortunately, things do not proceed so plainly. In order to prove a bound like (35), possibly with a different  0 replacing , we need to compare the propagators of the

Diagrammatic Methods in Classical Perturbation Theory

lines entering and exiting clusters T which are not selfenergy clusters. This requires replacing (36) with 2nC2  ˇ ˇ q q ˇ ˇ ˇ!¯  (¯1 ¯2 ) ˙ j[n] (!¯  ¯ 1 ; ")j ˙ j[n] (!¯  ¯ 2 ; ")jˇ i j ˇ ˇ

; > j¯1  ¯ 2 j0 (56) for all i; j D 1; : : : ; d and choices of the signs ˙, and hence introduces further Diophantine conditions, known as the second Melnikov conditions. The conditions in (56) turn out to be too many, because for all n  0 and all ¯ 2 Zr such that ¯ D ¯ 1  ¯ 2 there are infinitely many conditions to be considered, one per pair (¯1 ; ¯2 ). However we can impose both the condi¯  ¯ ; "), tions (55) and (56) not for the eigenvalues [n] i (! ¯ (") independent of  and then but for some quantities [n] i use the smoothness of the eigenvalues in x to control ¯  ¯ ; ") in terms of (!¯  ¯ )2  [n] (!¯  ¯ )2  [n] i (! i ("). Even¯ we have to tually, beside the Diophantine condition on !, impose the Melnikov conditions ˇ ˇ q ˇ ˇ ˇj!¯  ¯ j  j[n] (")jˇ > 0 ; i ˇ ˇ j¯j ˇ ˇ (57) q q ˇ ˇ ˇj!¯  ¯ j ˙ j[n] (")j ˙ j[n] (")jˇ > 0 ; i j ˇ ˇ j¯j for all ¯ ¤ 0 and all n  0. Each condition in (57) leads us to eliminate a small interval of values of ". For the values of " which are left define h(! t) according to (43) and (46), with the new definition of the propagators. If " is small enough, say j"j < "0 , then the series (43) converges. Denote by E [0; "0 ] the set of values of " for which the conditions (57) are satisfied. One can prove that E is a Cantor set, that is a perfect, nowhere dense set. Moreover E has large relative Lebesgue measure, in the sense that lim

"!0

meas(E \ [0; "]) D1; "

(58)

provided  0 in (57) is large enough with respect to . The property (58) yields that, notwithstanding that we are eliminating infinitely many intervals, the measure of the union of all these intervals is small. If a i < 0 for all i > r we reason in the same way, simply exchanging the role of positive and negative ". On the contrary if a i D 0 for some i > r, the problem becomes much more difficult. For instance if s D 1 and arC1 D 0, then in general perturbation theory in " is not possible, not even at a formal level. However, under some conditions,

one can still construct fractional series in ", and prove that the series can be resummed [24]. Other Ordinary Differential Equations The formalism described above extends to other models, such as skew-product systems [29] and systems with strong damping in the presence of a quasi-periodic forcing term [32]. As an example of skew-product system  one can con sider the linear differential equation x˙ D A C " f (! t) x on SL(2; R), where  2 R, " is a small real parameter, ! 2 Rn is an irrational vector, and A; f 2 sl(2; R), with A is a constant matrix and f an analytic function periodic in its arguments. Trees for skew-products were considered by Iserles and Nørsett [44], but they used expansions in time, hence not suited for the study of global properties, such as quasi-periodicity. Quasi-periodically forced one-dimensional systems with strong damping are described by the ordinary differential equations x¨ C x˙ C g(x) D f (! t), where x 2 R, " D 1/ is a small real parameter, ! 2 Rn is irrational, and f ,g are analytic functions (g is the “force”), with f periodic in its arguments. We refer to the bibliography for details and results on the existence of quasi-periodic solutions. Bryuno Vectors The diagrammatic methods can be used to prove the any unperturbed maximal torus with frequency vector which is a Bryuno vector persists under perturbation for " small enough [31]. One could speculate whether the Bryuno condition (30) is optimal. In general the problem is open. However, in the case of the standard map – see Sect. “The Standard Map”, – one can prove [6,15] that, by considering the radius of convergence "0 of the perturbation series as a function of !, say "0 D 0 (!), then there exists a universal constant C such that jlog 0 (!) C 2B(!)j  C :

(59)

In particular this yields that the invariant curve with rotation number ! persists under perturbation if and only if ! is a Bryuno number. The proof of (59) requires the study of a more refined cancellation than that discussed in Sect. “Resummation”. We refer to Berretti & Gentile [6] and Gentile [28] for details. Extensions to Bryuno vectors for lower-dimensional tori can also be found in Gentile [31]. For the models considered in Sect. “Other Ordinary Differential Equations” we refer to Gentile [29] and Gentile et al. [33].

139

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Diagrammatic Methods in Classical Perturbation Theory

Partial Differential Equations Existence of quasi-periodic solutions in systems described by one-dimensional nonlinear partial differential equations (finite-dimensional tori in infinite-dimensional systems) was first studied by Kuksin [48], Craig and Wayne [14] and Bourgain [9]. In these systems, even the case of periodic solutions yields small divisors, and hence requires a multiscale analysis. The study of persistence of periodic solutions for nonlinear Schrödinger equations and nonlinear wave equations, with the techniques discussed here, can be found in Gentile & Mastropietro [39], Gentile et al. [40], and Gentile & Procesi [41]. The models are still described by (1), with G(u) D 0, but now D is given by D D @2t   C  in the case of the wave equation and by D D i@ t   C  in the case of the Schrödinger equation, where  is the Laplacian and  2 R. In dimension 1, one has  D @2x . If we look for periodic solutions with frequency ! it can be convenient to pass to Fourier space, where the operator D acts as   D : e i!ntCi mx ! ! 2 n2 C m2 C  e i!ntCi mx ; (60) for the wave equation; a similar expression holds for the Schrödinger equation. Therefore the kernel of D can be arbitrarily close to zero for n and m large enough. Then one can consider, say, (1) for x 2 [0; ] and F(u) D u3 , with Dirichlet boundary conditions u(0) D u() D 0, and study the existence of periodic solutions with frequency ! close to some of the unperturbed frequencies. We refer to the cited bibliography for results and proofs.

Conclusions and Future Directions The diagrammatic techniques described above have been applied also in cases where no small divisors appear; cf. Berretti & Gentile [4] and Gentile et al. [34]. Of course, such problems are much easier from a technical point of view, and can be considered as propaedeutic examples to become familiar with the tree formalism. Also the study of lower-dimensional tori becomes easy for r D 1 (periodic ¯ j  j!j ¯ for all ¯ ¤ 0, so solutions): in that case one has j!¯ ¯ 2k , that the product of the propagators is bounded by j!j and one can proceed as in Sect. “Small Divisors” to obtain analyticity of the solutions. In the case of hyperbolic lower-dimensional tori, if ! is a two-dimensional Diophantine vector of constant type (that is, with  D 1) the conjugation function h can be proved to be Borel summable [13]. Analogous considerations hold for the one-dimensional systems in the presence

of friction and of a quasiperiodic forcing term described in Sect. “Other Ordinary Differential Equations”; in that case one has Borel summability also for one-dimensional !, that is for periodic forcing [33]. It would be interesting to investigate whether Borel summability could be obtained for higher values of . Recently existence of finite-dimensional tori in the nonlinear Schrödinger equation in higher dimensions was proved by Bourgain [10]. It would be interesting to investigate how far the diagrammatic techniques extend to deal with such higher dimensional generalizations. The main problem is that (the analogues of) the second Melnikov conditions in (57) cannot be imposed. In certain cases the tree formalism was extended to non-analytic systems, such as some quasi-integrable systems of the form (5) with f in a class of Cp functions for some finite p [7,8]. However, up to exceptional cases, the method described here seems to be intrinsically suited in cases in which the vector fields are analytic. The reason is that in order to exploit the expansion (3), we need that F be infinitely many times differentiable and we need a bound on the derivatives. It is a remarkable property that the perturbation series can be given a meaning also in cases where the solutions are not analytic in ". An advantage of the diagrammatic method is that it allows rather detailed information about the solutions, hence it could be more convenient than other techniques to study problems where the underlying structure is not known or too poor to exploit general abstract arguments. Another advantage is the following. If one is interested not only in proving the existence of the solutions, but also in explicitly constructing them with any prefixed precision, this requires performing analytical or numerical computations with arbitrarily high accuracy. Then high perturbation orders have to be reached, and the easiest and most direct way to proceed is just through perturbation theory: so the approach illustrated here allows a unified treatment for both theoretical investigations and computational ones. The resummation technique described in Sect. “Resummation” can also be used for computational purposes. With respect to the naive power series expansion it can reduce the computation time required to approximate the solution within a prefixed precision. It can also provide accurate information on the analyticity properties of the solution. For instance, for the Kepler equation, Levi-Civita at the beginning of the last century described a resummation rule (see [49]), which gives immediately the radius of convergence of the perturbation series. Of course, in the case of small divisor problems, everything becomes much more complicated.

Diagrammatic Methods in Classical Perturbation Theory

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21. Gallavotti G (2006) Classical mechanics. In: Françoise JP, Naber GL, Tsun TS (eds) Encyclopedia of Mathematical Physics, vol 1. Elsevier, Oxford 22. Gallavotti G, Gentile G (2002) Hyperbolic low-dimensional invariant tori and summations of divergent series. Comm Math Phys 227(3):421–460 23. Gallavotti G, Bonetto F, Gentile G (2004) Aspects of ergodic, qualitative and statistical theory of motion. Texts and Monographs in Physics. Springer, Berlin 24. Gallavotti G, Gentile G, Giuliani A (2006) Fractional Lindstedt series. J Math Phys 47(1):1–33 25. Gallavotti G, Gentile G, Mastropietro V (1999) A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems. J Math Phys 40(12):6430–6472 26. Gentile G (1995) Whiskered tori with prefixed frequencies and Lyapunov spectrum. Dyn Stab Syst 10(3):269–308 27. Gentile G (2003) Quasi-periodic solutions for two-level systems. Comm Math Phys 242(1-2):221–250 28. Gentile G (2006) Brjuno numbers and dynamical systems. Frontiers in number theory, physics, and geometry. Springer, Berlin 29. Gentile G (2006) Diagrammatic techniques in perturbation theory. In: Françoise JP, Naber GL, Tsun TS (eds) Encyclopedia of Mathematical Physics, vol 2. Elsevier, Oxford 30. Gentile G (2006) Resummation of perturbation series and reducibility for Bryuno skew-product flows. J Stat Phys 125(2): 321–361 31. Gentile G (2007) Degenerate lower-dimensional tori under the Bryuno condition. Ergod Theory Dyn Syst 27(2):427–457 32. Gentile G, Bartuccelli MV, Deane JHB (2005) Summation of divergent series and Borel summability for strongly dissipative differential equations with periodic or quasiperiodic forcing terms. J Math Phys 46(6):1–21 33. Gentile G, Bartuccelli MV, Deane JHB (2006) Quasiperiodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems. J Math Phys 47(7):1–10 34. Gentile G, Bartuccelli MV, Deane JHB (2007) Bifurcation curves of subharmonic solutions and Melnikov theory under degeneracies. Rev Math Phys 19(3):307–348 35. Gentile G, Cortez DA, Barata JCA (2005) Stability for quasiperiodically perturbed Hill’s equations. Comm Math Phys 260(2):403–443 36. Gentile G, Gallavotti G (2005) Degenerate elliptic tori. Comm Math Phys 257(2):319–362 37. Gentile G, Mastropietro V (1996) Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev Math Phys 8(3):393–444 38. Gentile G, Mastropietro V (2001) Renormalization group for one-dimensional fermions. A review on mathematical results. Renormalization group theory in the new millennium III. Phys Rep 352(4–6):273–437 39. Gentile G, Mastropietro V (2004) Construction of periodic solutions of nonlinear wave equations with Dirichlet boundary conditions by the Lindstedt series method. J Math Pures Appl 83(8):1019–1065 40. Gentile G, Mastropietro V, Procesi M (2005) Periodic solutions for completely resonant nonlinear wave equations with Dirichlet boundary conditions. Comm Math Phys 256(2):437– 490 41. Gentile G, Procesi M (2006) Conservation of resonant peri-

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Books and Reviews Berretti A, Gentile G (2001) Renormalization group and field theoretic techniques for the analysis of the Lindstedt series. Regul Chaotic Dyn 6(4):389–420 Gentile G (1999) Diagrammatic techniques in perturbations theory, and applications. Symmetry and perturbation theory. World Science, River Edge

Discrete Control Systems

Discrete Control Systems TAEYOUNG LEE1 , MELVIN LEOK2 , HARRIS MCCLAMROCH1 1 Department of Aerospace Engineering, University of Michigan, Ann Arbor, USA 2 Department of Mathematics, Purdue University, West Lafayette, USA Article Outline Glossary Definition of the Subject Introduction Discrete Lagrangian and Hamiltonian Mechanics Optimal Control of Discrete Lagrangian and Hamiltonian Systems Controlled Lagrangian Method for Discrete Lagrangian Systems Future Directions Acknowledgments Bibliography Glossary Discrete variational mechanics A formulation of mechanics in discrete-time that is based on a discrete analogue of Hamilton’s principle, which states that the system takes a trajectory for which the action integral is stationary. Geometric integrator A numerical method for obtaining numerical solutions of differential equations that preserves geometric properties of the continuous flow, such as symplecticity, momentum preservation, and the structure of the configuration space. Lie group A differentiable manifold with a group structure where the composition is differentiable. The corresponding Lie algebra is the tangent space to the Lie group based at the identity element. Symplectic A map is said to be symplectic if given any initial volume in phase space, the sum of the signed projected volumes onto each position-momentum subspace is invariant under the map. One consequence of symplecticity is that the map is volume-preserving as well. Definition of the Subject Discrete control systems, as considered here, refer to the control theory of discrete-time Lagrangian or Hamiltonian systems. These discrete-time models are based on a discrete variational principle, and are part of the broader

field of geometric integration. Geometric integrators are numerical integration methods that preserve geometric properties of continuous systems, such as conservation of the symplectic form, momentum, and energy. They also guarantee that the discrete flow remains on the manifold on which the continuous system evolves, an important property in the case of rigid-body dynamics. In nonlinear control, one typically relies on differential geometric and dynamical systems techniques to prove properties such as stability, controllability, and optimality. More generally, the geometric structure of such systems plays a critical role in the nonlinear analysis of the corresponding control problems. Despite the critical role of geometry and mechanics in the analysis of nonlinear control systems, nonlinear control algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. The field of discrete control systems aims to address this deficiency by restricting the approximation to the choice of a discrete-time model, and developing an associated control theory that does not introduce any additional approximation. In particular, this involves the construction of a control theory for discrete-time models based on geometric integrators that yields numerical implementations of nonlinear and geometric control algorithms that preserve the crucial underlying geometric structure. Introduction The dynamics of Lagrangian and Hamiltonian systems have unique geometric properties; the Hamiltonian flow is symplectic, the total energy is conserved in the absence of non-conservative forces, and the momentum maps associated with symmetries of the system are preserved. Many interesting dynamics evolve on a non-Euclidean space. For example, the configuration space of a spherical pendulum is the two-sphere, and the configuration space of rigid body attitude dynamics has a Lie group structure, namely the special orthogonal group. These geometric features determine the qualitative behavior of the system, and serve as a basis for theoretical study. Geometric numerical integrators are numerical integration algorithms that preserve structures of the continuous dynamics such as invariants, symplecticity, and the configuration manifold (see [14]). The exact geometric properties of the discrete flow not only generate improved qualitative behavior, but also provide accurate and efficient numerical techniques. In this article, we view a geometric integrator as an intrinsically discrete dynamical system, instead of concentrating on the numerical approximation of a continuous trajectory.

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Numerical integration methods that preserve the simplecticity of a Hamiltonian system have been studied (see [28,36]). Coefficients of a Runge–Kutta method are carefully chosen to satisfy a simplecticity criterion and order conditions to obtain a symplectic Runge–Kutta method. However, it can be difficult to construct such integrators, and it is not guaranteed that other invariants of the system, such as a momentum map, are preserved. Alternatively, variational integrators are constructed by discretizing Hamilton’s principle, rather than discretizing the continuous Euler–Lagrange equation (see [31,34]). The resulting integrators have the desirable property that they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long times (see [2]). Lie group methods are numerical integrators that preserve the Lie group structure of the configuration space (see [18]). Recently, these two approaches have been unified to obtain Lie group variational integrators that preserve the geometric properties of the dynamics as well as the Lie group structure of the configuration space without the use of local charts, reprojection, or constraints (see [26,29,32]). Optimal control problems involve finding a control input such that a certain optimality objective is achieved under prescribed constraints. An optimal control problem that minimizes a performance index is described by a set of differential equations, which can be derived using Pontryagin’s maximum principle. Discrete optimal control problems involve finding a control input for a discrete dynamic system such that an optimality objective is achieved with prescribed constraints. Optimality conditions are derived from the discrete equations of motion, described by a set of discrete equations. This approach is in contrast to traditional techniques where a discretization appears at the last stage to solve the optimality condition numerically. Discrete mechanics and optimal control approaches determine optimal control inputs and trajectories more accurately with less computational load (see [19]). Combined with an indirect optimization technique, they are substantially more efficient (see [17,23,25]). The geometric approach to mechanics can provide a theoretical basis for innovative control methodologies in geometric control theory. For example, these techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion. While the geometric structure of mechanical systems plays a critical role in the construction of geometric control algorithms, these algorithms have typically been implemented using numerical schemes that ignore the underlying geometry. By applying geometric control algorithms to discrete mechanics that preserve geometric properties, we

obtain an exact numerical implementation of the geometric control theory. In particular, the method of controlled Lagrangian systems is based on the idea of adopting a feedback control to realize a modification of either the potential energy or the kinetic energy, referred to as potential shaping or kinetic shaping, respectively. These ideas are applied to construct a real-time digital feedback controller that stabilizes the inverted equilibrium of the cart-pendulum (see [10,11]). In this article, we will survey discrete Lagrangian and Hamiltonian mechanics, and their applications to optimal control and feedback control theory. Discrete Lagrangian and Hamiltonian Mechanics Mechanics studies the dynamics of physical bodies acting under forces and potential fields. In Lagrangian mechanics, the trajectory of the object is derived by finding the path that extremizes the integral of a Lagrangian over time, called the action integral. In many classical problems, the Lagrangian is chosen as the difference between kinetic energy and potential energy. The Legendre transformation provides an alternative description of mechanical systems, referred to as Hamiltonian mechanics. Discrete Lagrangian and Hamiltonian mechanics has been developed by reformulating the theorems and the procedures of Lagrangian and Hamiltonian mechanics in a discrete time setting (see, for example, [31]). Therefore, discrete mechanics has a parallel structure with the mechanics described in continuous time, as summarized in Fig. 1 for Lagrangian mechanics. In this section, we describe discrete Lagrangian mechanics in more detail, and we derive discrete Euler–Lagrange equations for several mechanical systems. Consider a mechanical system on a configuration space Q, which is the space of possible positions. The Lagrangian depends on the position and velocity, which are elements of the tangent bundle to Q, denoted by TQ. Let L : TQ ! R be the Lagrangian of the system. The discrete Lagrangian, Ld : Q  Q ! R is an approximation to the exact discrete Lagrangian, Z h (q ; q ) D L(q01 (t); q˙01 (t)) dt ; (1) Lexact 0 1 d 0

where q01 (0) D q0 , q01 (h) D q1 ; and q01 (t) satisfies the Euler–Lagrange equation in the time interval (0; h). A discrete action sum Gd : Q NC1 ! R, analogous to the action integral, is given by Gd (q0 ; q1 ; : : : ; q N ) D

N1 X kD0

Ld (q k ; q kC1 ) :

(2)

Discrete Control Systems

Discrete Control Systems, Figure 1 Procedures to derive continuous and discrete equations of motion

The discrete Hamilton’s principle states that ıGd D 0 for any ıq k , which yields the discrete Euler–Lagrange (DEL) equation, D2 Ld (q k1 ; q k ) C D1 Ld (q k ; q kC1 ) D 0 :

(3)

This yields a discrete Lagrangian flow map (q k1 ; q k ) 7! (q k ; q kC1 ). The discrete Legendre transformation, which from a pair of positions (q0 ; q1 ) gives a position-momentum pair (q0 ; p0 ) D (q0 ; D1 Ld (q0 ; q1 )) provides a discrete Hamiltonian flow map in terms of momenta. The discrete equations of motion, referred to as variational integrators, inherit the geometric properties of the continuous system. Many interesting Lagrangian and Hamiltonian systems, such as rigid bodies evolve on a Lie group. Lie group variational integrators preserve the nonlinear structure of the Lie group configurations as well as geometric properties of the continuous dynamics (see [29,32]). The basic idea for all Lie group methods is to express the update map for the group elements in terms of the group operation, g1 D g0 f0 ;

(4)

where g0 ; g1 2 G are configuration variables in a Lie group G, and f0 2 G is the discrete update represented

by a right group operation on g 0 . Since the group element is updated by a group operation, the group structure is preserved automatically without need of parametrizations, constraints, or re-projection. In the Lie group variational integrator, the expression for the flow map is obtained from the discrete variational principle on a Lie group, the same procedure presented in Fig. 1. But, the infinitesimal variation of a Lie group element must be carefully expressed to respect the structure of the Lie group. For example, it can be expressed in terms of the exponential map as ˇ d ˇˇ ıg D g exp  D g ; d ˇ D0 for a Lie algebra element  2 g. This approach has been applied to the rotation group SO(3) and to the special Euclidean group SE(3) for dynamics of rigid bodies (see [24,26,27]). Generalizations to arbitrary Lie groups gives the generalized discrete Euler–Poincaré (DEP) equation,   Te L f0  D2 Ld (g0 ; f0 )  Adf0  Te L f1  D2 Ld (g1 ; f1 ) (5) CTe L g 1  D1 Ld (g1 ; f1 ) D 0 ; for a discrete Lagrangian on a Lie group, Ld : G G ! R. Here L f : G ! G denotes the left translation map given by L f g D f g for f ; g 2 G, Tg L f : Tg G ! T f g G is the

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tangential map for the left translation, and Ad g : g ! g is the adjoint map. A dual map is denoted by a superscript  (see [30] for detailed definitions). We illustrate the properties of discrete mechanics using several mechanical systems, namely a mass-spring system, a planar pendulum, a spherical pendulum, and a rigid body. Example 1 (Mass-spring System) Consider a mass-spring system, defined by a rigid body that moves along a straight frictionless slot, and is attached to a linear spring. Continuous equation of motion: The configuration space is Q D R, and the Lagrangian L : R  R ! R is given by ˙ D L(q; q)

1 2 1 2 m q˙  q ; 2 2

(6)

where q 2 R is the displacement of the body measured from the point where the spring exerts no force. The mass of the body and the spring constant are denoted by m;  2 R, respectively. The Euler–Lagrange equation yields the continuous equation of motion. m q¨ C q D 0 :

(7)

Discrete equation of motion: Let h > 0 be a discrete time step, and a subscript k denotes the kth discrete variable at t D kh. The discrete Lagrangian Ld : RR ! R is an approximation of the integral of the continuous Lagrangian (6) along the solution of (7) over a time step. Here, we choose the following discrete Lagrangian. 

 q k C q kC1 q kC1  q k Ld (q k ; q kC1 ) D hL ; 2 h 1 h D m(q kC1  q k )2  (q k C q kC1 )2 : 2h 8 (8) Direct application of the discrete Euler–Lagrange equation to this discrete Lagrangian yields the discrete equations of motion. We develop the discrete equation of motion using the discrete Hamilton’s principle to illustrate the principles more explicitly. Let Gd : R NC1 ! R be the discrete P N1 action sum defined as Gd D kD0 Ld (q k ; q kC1 ), which approximates the action integral. The infinitesimal variation of the action sum can be written as ıGd D

 m h (q kC1  q k )  (q k C q kC1 ) h 4 kD0

 m h C ıq k  (q kC1  q k )  (q k C q kC1 ) : h 4 N1 X

ıq kC1

Since ıq0 D ıq N D 0, the summation index can be rewritten as ıGd D

N1 X kD1

n m ıq k  (q kC1  2q k C q k1 ) h 

o h (q kC1 C 2q k C q k1 ) : 4

From discrete Hamilton’s principle, ıGd D 0 for any ıq k . Thus, the discrete equation of motion is given by m (q kC1  2q k C q k1 ) h h (q kC1 C 2q k C q k1 ) D 0 : C 4

(9)

For a given (q k1 ; q k ), we solve the above equation to obtain q kC1 . This yields a discrete flow map (q k1 ; q k ) 7! (q k ; q kC1 ), and this process is repeated. The discrete Legendre transformation provides the discrete equation of motion in terms of the velocity as     h2  h2  ˙ q kC1 D h q k C 1  qk ; 1C 4m 4m h h q˙ kC1 D q˙ k  qk  q kC1 : 2m 2m

(10) (11)

For a given (q k ; q˙ k ), we compute q kC1 and q˙ kC1 by (10) and (11), respectively. This yields a discrete flow map (q k ; q˙k ) 7! (q kC1 ; q˙kC1 ). It can be shown that this variational integrator has second-order accuracy, which follows from the fact that the discrete action sum is a second-order approximation of the action integral. Numerical example: We compare computational properties of the discrete equations of motion given by (10) and (11) with a 4(5)th order variable step size Runge– Kutta method. We choose m D 1 kg,  D 1 kg/s2 so that the natural p frequency is 1 rad/s. The initial conditions are q0 D 2 m, q˙0 D 0, and the total energy is E D 1 Nm. The simulation time is 200 sec, and the stepsize h D 0:035 of the discrete equations of motion is chosen such that the CPU times are the same for both methods. Figure 2a shows the computed total energy. The variational integrator preserves the total energy well. The mean variation is 2:73271013 Nm. But, there is a notable dissipation of the computed total energy for the Runge–Kutta method Example 2 (Planar Pendulum) A planar pendulum is a mass particle connected to a frictionless, one degree-offreedom pivot by a rigid massless link under a uniform gravitational potential. The configuration space is the one-

Discrete Control Systems

Discrete Control Systems, Figure 2 Computed total energy (RK45: blue, dotted, VI: red, solid)

sphere S1 D fq 2 R2 j kqk D 1g. While it is common to parametrize the one-sphere by an angle, we develop parameter-free equations of motion in the special orthogonal group SO(2), which is a group of 22 orthogonal matrices with determinant 1, i. e. SO(2) D fR 2 R22 j RT R D I22 ; det[R] D 1g. SO(2) is diffeomorphic to the one-sphere. It is also possible to develop global equations of motion on the one-sphere directly, as shown in the next example, but here we focus on the special orthogonal group in order to illustrate the key steps to develop a Lie group variational integrator. We first exploit the basic structures of the Lie group SO(2). Define a hat map ˆ, which maps a scalar ˝ to a 22 skew-symmetric matrix ˝ˆ as ˝ˆ D



0 ˝

 ˝ : 0

The set of 22 skew-symmetric matrices forms the Lie algebra so(2). Using the hat map, we identify so(2) with R. An inner product on so(2) can be induced from the inner product on R as h˝ˆ 1 ; ˝ˆ 2 i D 1/2tr[˝ˆ 1T ˝ˆ 2 ] D ˝1 ˝2 for any ˝1 ; ˝2 2 R. The matrix exponential is a local diffeomorphism from so(2) to SO(2) given by

cos ˝ exp ˝ˆ D sin ˝

 sin ˝ cos ˝



ˆ D 1 ml 2 ˝ 2 C mg l e T Re2 L(R; ˝) 2 2 (13) 1 2 ˆ ˆ D ml h˝; ˝i C mg l e2T Re2 ; 2 where the constant g 2 R is the gravitational acceleration. The mass and the length of the pendulum are denoted by m; l 2 R, respectively. The second expression is used to define a discrete Lagrangian later. We choose the bases of the inertial frame and the body-fixed frame such that the unit vector along the gravity direction in the inertial frame, and the unit vector along the pendulum axis in the body-fixed frame are represented by the same vector e2 D [0; 1] 2 R2 . Thus, for example, the hanging attitude is represented by R D I22 . Here, the rotation matrix R 2 SO(2) represents the linear transformation from a representation of a vector in the body-fixed frame to the inertial frame. Since the special orthogonal group is not a linear vector space, the expression for the variation should be carefully chosen. The infinitesimal variation of a rotation matrix R 2 SO(2) can be written in terms of its Lie algebra element as ˇ ˇ ˇ d ˇˇ R exp ˆ D Rˆ exp ˆ ˇˇ D Rˆ ; (14) ıR D d ˇ D0

:

The kinematics equation for R 2 SO(2) can be written in terms of a Lie algebra element as R˙ D R˝ˆ :

as

(12)

Continuous equations of motion: The Lagrangian for a planar pendulum L : SO(2)so(2) ! R can be written

D0

where  2 R so that ˆ 2 so(2). The infinitesimal variation of the angular velocity is induced from this expression and (12) as ı ˝ˆ D ıRT R˙ C RT ı R˙ ˆ˙ ˆ T R˙ C RT (R˙ ˆ C R) D (R)

(15)

D ˆ ˝ˆ C ˝ˆ ˆ C ˆ˙ D ˆ˙ ; where we used the equality of mixed partials to compute ı R˙ as d/dt(ıR).

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RT ˆ Define the action integral to be G D 0 L(R; ˝)dt. The infinitesimal variation of the action integral is obtained by using (14) and (15). Hamilton’s principle yields the following continuous equations of motion. g ˝˙ C e2T Re1 D 0 ; (16) l R˙ D R˝ˆ :

(17)

where we use the fact that F ˆ F T D ˆ for any F 2 SO(2) and ˆ 2 so(2). Define an action sum Gd : SO(2) NC1 ! R as Gd D P N1 kD0 Ld (R k ; F k ). Using (21) and (22), the variation of the action sum is written as ıGd D

N1 X kD1

If we parametrize the rotation matrix as

 cos  sin RD ; sin cos

  1 T ml 2 F k1  F k1 2h 

these equations are equivalent to g (18) ¨ C sin D 0 : l Discrete equations of motion: We develop a Lie group variational integrator on SO(2). Similar to (4), define F k 2 SO(2) such that R kC1 D R k F k :

(19)

Thus, F k D RTk R kC1 represents the relative update between two integration steps. If we find F k 2 SO(2), the orthogonal structure is preserved through (19) since multiplication of orthogonal matrices is also orthogonal. This is a key idea of Lie group variational integrators. Define the discrete Lagrangian Ld : SO(2)SO(2) ! R to be 1 ml 2 hF k  I22 ; F k  I22 i Ld (R k ; F k ) D 2h h h C mg l e2T R k e2 C mg l e2T R kC1 e2 2 2 (20) 1 h 2 D ml tr[I22  F k ] C mg l e2T R k e2 2h 2 h C mg l e2T R kC1 e2 ; 2 which is obtained by an approximation h˝ˆ k ' RT (R kC1

2

  1  F k  F kT  hmg l e2T R k e1 ; ˆ k : 2h

From the discrete Hamilton’s principle, ıGd D 0 for any ˆ k . Thus, we obtain the Lie group variational integrator on SO(2) as 

3

   2h2 g T T  F k  F kT  F kC1  F kC1 e R kC1 e1 D 0; (23) l 2 R kC1 D R k F k :

(24)

For a given (R k ; F k ) and R kC1 D R k F k , (23) is solved to find F kC1 . This yields a discrete map (R k ; F k ) 7! (R kC1 ; F kC1 ). If we parametrize the rotation matrices R and F with and  and if we assume that   1, these equations are equivalent to 1 hg ( kC1  2 k C k ) C sin k D 0 : h l The discrete version of the Legendre transformation provides the discrete Hamiltonian map as follows. F k  F kT D 2h˝ˆ 

2

h2 g T e R k e1 ; l 2

R kC1 D R k F k ; ˝ kC1 D ˝ k 

(25) (26)

hg T hg T e2 R k e1  e R kC1 e1 : 2l 2l 2

(27)

k

 R k ) D F k  I22 , applied to the continuous Lagrangian given by (13). As for the continuous time case, expressions for the infinitesimal variations should be carefully chosen. The infinitesimal variation of a rotation matrix is the same as (14), namely ıR k D R k ˆ k ;

(21)

for  k 2 R, and the constrained variation of F k is obtained from (19) as ıF k D ıRTk R kC1 C RTk ıR kC1 D ˆ k F k C F k ˆ kC1 D F k (ˆ  kC1  F kT ˆ k F k ) (22)  kC1  ˆ k ) ; D F k (ˆ

For a given (R k ; ˝ k ), we solve (25) to obtain F k . Using this, (R kC1 ; ˝ kC1 ) is obtained from (26) and (27). This yields a discrete map (R k ; ˝ k ) 7! (R kC1 ; ˝ kC1 ). Numerical example: We compare the computational properties of the discrete equations of motion given by (25)–(27) with a 4(5)th order variable step size Runge– Kutta method. We choose m D 1 kg, l D 9:81 m. The initial conditions are 0 D /2 rad, ˝ D 0, and the total energy is E D 0 Nm. The simulation time is 1000 sec, and the step-size h D 0:03 of the discrete equations of motion is chosen such that the CPU times are identical. Figure 2b shows the computed total energy for both methods. The variational integrator preserves the total energy well. There is no drift in the computed total energy, and the

Discrete Control Systems

mean variation is 1:0835102 Nm. But, there is a notable dissipation of the computed total energy for the Runge– Kutta method. Note that the computed total energy would further decrease as the simulation time increases.

Since q˙ D !  q for some angular velocity ! 2 R3 satisfying !  q D 0, this can also be equivalently written as

Example 3 (Spherical Pendulum) A spherical pendulum is a mass particle connected to a frictionless, two degree-offreedom pivot by a rigid massless link. The mass particle acts under a uniform gravitational potential. The configuration space is the two-sphere S2 D fq 2 R3 j kqk D 1g. It is common to parametrize the two-sphere by two angles, but this description of the spherical pendulum has a singularity. Any trajectory near the singularity encounters numerical ill-conditioning. Furthermore, this leads to complicated expressions involving trigonometric functions. Here we develop equations of motion for a spherical pendulum using the global structure of the two-sphere without parametrization. In the previous example, we develop equations of motion for a planar pendulum using the fact that the one-sphere S1 is diffeomorphic to the special orthogonal group SO(2). But, the two-sphere is not diffeomorphic to a Lie group. Instead, there exists a natural Lie group action on the two-sphere. That is the 3-dimensional special orthogonal group SO(3), a group of 33 orthogonal matrices with determinant 1, i. e. SO(3) D fR 2 R33 j RT R D I33 ; det[R] D 1g. The special orthogonal group SO(3) acts on the two-sphere in a transitive way; for any q1 ; q2 2 S2 , there exists a R 2 SO(3) such that q2 D Rq1 . Thus, the discrete update for the two-sphere can be expressed in terms of a rotation matrix as (19). This is a key idea to develop a discrete equations of motion for a spherical pendulum. Continuous equations of motion: Let q 2 S2 be a unit vector from the pivot point to the point mass. The Lagrangian for a spherical pendulum can be written as

(32)

1 ˙ D ml 2 q˙  q˙ C mg l e3  q ; L(q; q) 2

(28)

where the gravity direction is assumed to be e3 D [0; 0; 1] 2 R3 . The mass and the length of the pendulum are denoted by m; l 2 R, respectively. The infinitesimal variation of the unit vector q can be written in terms of the vector cross product as ıq D   q ;

(29)

where  2 R3 is constrained to be orthogonal to the unit vector, i. e.  q D 0. Using this expression for the infinitesimal variation, Hamilton’s principle yields the following continuous equations of motion. g ˙ C (q  (q  e3 )) D 0 : q¨ C (q˙  q)q l

(30)

g q  e3 D 0 ; l q˙ D !  q :

!˙ 

(31)

These are global equations of motion for a spherical pendulum; these are much simpler than the equations expressed in term of angles, and they have no singularity. Discrete equations of motion: We develop a variational integrator for the spherical pendulum defined on S2 . Since the special orthogonal group acts on the two-sphere transitively, we can define the discrete update map for the unit vector as q kC1 D F k q k

(33)

for F k 2 SO(3). The rotation matrix F k is not uniquely defined by this condition, since exp(qˆ k ), which corresponds to a rotation about the qk direction fixes the vector qk . Consequently, if F k satisfies (33), then F k exp(qˆ k ) does as well. We avoid this ambiguity by requiring that F k does not rotate about qk , which can be achieved by letting F k D exp( fˆk ), where f k  q k D 0. Define a discrete Lagrangian Ld : S2  S2 ! R to be Ld (q k ; q kC1 ) D

1 ml 2 (q kC1  q k )  (q kC1  q k ) 2h h h C mg l e3  q k C mg l e3  q kC1 : 2 2

The variation of qk is the same as (29), namely ıq k D  k  q k

(34)

for  k 2 R3 with a constraint  k  q k D 0. Using this discrete Lagrangian and the expression for the variation, discrete Hamilton’s principle yields the following discrete equations of motion for a spherical pendulum.   h2 g q k  e3  q k q kC1 D h! k C 2l 2 !1/2    h2 g  C 1   h! k C qk ; q k  e3   2l hg hg ! kC1 D ! k C q k  e3 C q kC1  e3 : 2l 2l

(35) (36)

Since an explicit solution for F k 2 SO(3) can be obtained in this case, the rotation matrix F k does not appear in the equations of motion. This variational integrator on S2 exactly preserves the unit length of qk , the constraint

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Discrete Control Systems, Figure 3 Numerical simulation of a spherical pendulum (RK45: blue, dotted, VI: red, solid)

q k  ! k D 0, and the third component of the angular velocity ! k  e3 which is conserved since gravity exerts no moment along the gravity direction e3 . Numerical example: We compare the computational properties of the discrete equations of motion given by (35) and (36) with a 4(5)th order variable step size Runge–Kutta method for (31) and (32). We choose m D 1p kg, l D 9:81 m. The p initial conditions are q0 D [ 3/2; 0; 1/2], !0 D 0:1[ 3; 0; 3] rad/sec, and the total energy is E D 0:44 Nm. The simulation time is 200 sec, and the step-size h D 0:05 of the discrete equations of motion is chosen such that the CPU times are identical. Figure 3 shows the computed total energy and the unit length errors. The variational integrator preserves the total energy and the structure of the two-sphere well. The mean total energy deviation is 1:5460104 Nm, and the mean unit length error is 3:2476  1015 . But, there is a notable dissipation of the computed total energy for the Runge–Kutta method. The Runge–Kutta method also fails to preserve the structure of the two-sphere. The mean unit length error is 1:0164102 . Example 4 (Rigid Body in a Potential Field) Consider a rigid body under a potential field that is dependent on the position and the attitude of the body. The configuration space is the special Euclidean group, which is a semidirect product of the special orthogonal group and Eus R3 . clidean space, i. e. SE(3) D SO(3)  Continuous equations of motion: The equations of motion for a rigid body can be developed either from Hamilton’s principle (see [26]) in a similar way as Example 2, or directly from the generalized discrete Euler–Poincaré equation given at (5). Here, we summarize the results. Let m 2 R and J 2 R33 be the mass and the moment of inertia matrix of a rigid body. For (R; x) 2 SE(3), the linear

transformation from the body-fixed frame to the inertial frame is denoted by the rotation matrix R 2 SO(3), and the position of the mass center in the inertial frame is denoted by a vector x 2 R3 . The vectors ˝; v 2 R3 are the angular velocity in the body-fixed frame, and the translational velocity in the inertial frame, respectively. Suppose that the rigid body acts under a configuration-dependent potential U : SE(3) ! R. The continuous equations of motion for the rigid body can be written as R˙ D R˝ˆ ;

(37)

x˙ D v ;

(38)

J ˝˙ C ˝  J˝ D M ;

(39)

@U ; @x

(40)

m˙v D 

where the hat map ˆ is an isomorphism from R3 to 3  3 skew-symmetric matrices so(3), defined such that xˆ y D x  y for any x; y 2 R3 . The moment due to the potential M 2 R3 is obtained by the following relationship: T ˆ D @U R  RT @U : M @R @R

(41)

The matrix @U/@R 2 R33 is defined such that [@U/@R] i j D@U/@[R] i j for i; j 2 f1; 2; 3g, where the i, jth element of a matrix is denoted by [] i j . Discrete equations of motion: The corresponding discrete equations of motion are given by

b

h J ˝k C

h2 ˆ M k D F k Jd  Jd F kT ; 2

R kC1 D R k F k ;

(42) (43)

Discrete Control Systems

Discrete Control Systems, Figure 4 Numerical simulation of a dumbbell rigid body (LGVI: red, solid, RK45 with rotation matrices: blue, dash-dotted, RK45 with quaternions: black, dotted)

h2 @U k ; 2m @x k h h D F kT J˝ k C F kT M k C M kC1 ; 2 2 h Uk h @U kC1 D mv k   ; 2 @x k 2 @x kC1

x kC1 D x k C hv k 

(44)

J˝ kC1

(45)

mv kC1

(46)

where Jd 2 R33 is a non-standard moment of inertia matrix defined as Jd D1/2tr[J]I33  J. For a given (R k ; x k ; ˝ k ; v k ), we solve the implicit Eq. (42) to find F k 2 SO(3). Then, the configuration at the next step R kC1 ;x kC1 is obtained by (43) and (44), and the moment and force M kC1;(@U kC1 )/(@x kC1 ) can be computed. Velocities ˝ kC1 ;v kC1 are obtained from (45) and (46). This defines a discrete flow map, (R k ; x k ; ˝ k ; v k ) 7! (R kC1 ; x kC1 ; ˝ kC1 ; v kC1 ), and this process can be repeated. This Lie group variational integrator on SE(3) can be generalized to multiple rigid bodies acting under their mutual gravitational potential (see [26]). Numerical example: We compare the computational properties of the discrete equations of motion given by (42)–(46) with a 4(5)th order variable step size Runge– Kutta method for (37)–(40). In addition, we compute the attitude dynamics using quaternions on the unit threesphere S3 . The attitude kinematics Eq. (37) is rewritten in terms of quaternions, and the corresponding equations are integrated by the same Runge–Kutta method. We choose a dumbbell spacecraft, that is two spheres connected by a rigid massless rod, acting under a central gravitational potential. The resulting system is a restricted full two body problem. The dumbbell spacecraft model has an analytic expression for the gravitational potential, resulting in a nontrivial coupling between the attitude dynamics and the orbital dynamics. As shown in Fig. 4a, the initial conditions are chosen such that the resulting motion is a near-circular orbit

combined with a rotational motion. Figure 4b and c show the computed total energy and the orthogonality errors of the rotation matrix. The Lie group variational integrator preserves the total energy and the Lie group structure of SO(3). The mean total energy deviation is 2:5983104 , and the mean orthogonality error is 1:8553  1013 . But, there is a notable dissipation of the computed total energy and the orthogonality error for the Runge–Kutta method. The mean orthogonality errors for the Runge– Kutta method are 0.0287 and 0.0753, respectively, using kinematics equation with rotation matrices, and using the kinematics equation with quaternions. Thus, the attitude of the rigid body is not accurately computed for Runge– Kutta methods. It is interesting to see that the Runge– Kutta method with quaternions, which is generally assumed to have better computational properties than the kinematics equation with rotation matrices, has larger total energy error and orthogonality error. Since the unit length of the quaternion vector is not preserved in the numerical computations, orthogonality errors arise when converted to a rotation matrix. This suggests that it is critical to preserve the structure of SO(3) in order to study the global characteristics of the rigid body dynamics. The importance of simultaneously preserving the symplectic structure and the Lie group structure of the configuration space in rigid body dynamics can be observed numerically. Lie group variational integrators, which preserve both of these properties, are compared to methods that only preserve one, or neither, of these properties (see [27]). It is shown that the Lie group variational integrator exhibits greater numerical accuracy and efficiency. Due to these computational advantages, the Lie group variational integrator has been used to study the dynamics of the binary near-Earth asteroid 66391 (1999 KW 4 ) in joint work between the University of Michigan and the Jet Propulsion Laboratory, NASA (see [37]).

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Optimal Control of Discrete Lagrangian and Hamiltonian Systems Optimal control problems involve finding a control input such that a certain optimality objective is achieved under prescribed constraints. An optimal control problem that minimizes a performance index is described by a set of differential equations, which can be derived using Pontryagin’s maximum principle. The equations of motion for a system are constrained by Lagrange multipliers, and necessary conditions for optimality is obtained by the calculus of variations. The solution for the corresponding two point boundary value problem provides the optimal control input. Alternatively, a sub-optimal control law is obtained by approximating the control input history with finite data points. Discrete optimal control problems involve finding a control input for a given system described by discrete Lagrangian and Hamiltonian mechanics. The control inputs are parametrized by their values at each discrete time step, and the discrete equations of motion are derived from the discrete Lagrange–d’Alembert principle [21], ı

N1 X kD0

Ld (q k ; q kC1 ) D

N1 X

[Fd (q k ; q kC1 )  ıq k

kD0

C FdC (q k ; q kC1 )  ıq kC1 ] ; which modifies the discrete Hamilton’s principle by taking into account the virtual work of the external forces. Discrete optimal control is in contrast to traditional techniques such as collocation, wherein the continuous equations of motion are imposed as constraints at a set of collocation points, since this approach induces constraints on the configuration at each discrete timestep. Any optimal control algorithm can be applied to the discrete Lagrangian or Hamiltonian system. For an indirect method, our approach to a discrete optimal control problem can be considered as a multiple stage variational problem. The discrete equations of motion are derived by the discrete variational principle. The corresponding variational integrators are imposed as dynamic constraints to be satisfied by using Lagrange multipliers, and necessary conditions for optimality, expressed as discrete equations on multipliers, are obtained from a variational principle. For a direct method, control inputs can be optimized by using parameter optimization tools such as a sequential quadratic programming. The discrete optimal control can be characterized by discretizing the optimal control problem from the problem formulation stage. This method has substantial computational advantages when used to find an optimal control law. As discussed in the previous section, the discrete dynamics are

more faithful to the continuous equations of motion, and consequently more accurate solutions to the optimal control problem are obtained. The external control inputs break the Lagrangian and Hamiltonian system structure. For example, the total energy is not conserved for a controlled mechanical system. But, the computational superiority of the discrete mechanics still holds for controlled systems. It has been shown that the discrete dynamics is more reliable even for controlled system as it computes the energy dissipation rate of controlled systems more accurately (see [31]). For example, this feature is extremely important in computing accurate optimal trajectories for long term spacecraft attitude maneuvers using low energy control inputs. The discrete dynamics does not only provide an accurate optimal control input, but also enables us to find it efficiently. For the indirect optimal control approach, optimal solutions are usually sensitive to a small variation of multipliers. This causes difficulties, such as numerical ill-conditioning, when solving the necessary conditions for optimality expressed as a two point boundary value problem. Sensitivity derivatives along the discrete necessary conditions do not have numerical dissipation introduced by conventional numerical integration schemes. Thus, they are numerically more robust, and the necessary conditions can be solved computationally efficiently. For the direct optimal control approach, optimal control inputs can be obtained by using a larger discrete step size, which requires less computational load. We illustrate the basic properties of the discrete optimal control using optimal control problems for the spherical pendulum and the rigid body model presented in the previous section. Example 5 (Optimal Control of a Spherical Pendulum) We study an optimal control problem for the spherical pendulum described in Example 3. We assume that an external control moment u 2 R3 acts on the pendulum. Control inputs are parametrized by their values at each time step, and the discrete equations of motion are modified to include the effects of the external control inputs by using the discrete Lagrange–d’Alembert principle. Since the component of the control moment that is parallel to the direction along the pendulum has no effect, we parametrize the control input as u k D q k w k for w k 2 R3 . The objective is to transfer the pendulum from a given initial configuration (q0 ; !0 ) to a desired configuration (qd ; ! d ) during a fixed maneuver time N, while minimizing the square of the weighted l2 norm of the control moments. N N X X h T h min J D uk uk D (q k  w k )T (q k  w k ) : wk 2 2 kD0

kD0

Discrete Control Systems

Discrete Control Systems, Figure 5 Optimal control of a spherical pendulum

We solve this optimal control problem by using a direct numerical optimization method. The terminal boundary condition is imposed as an equality constraint, and the 3(N C 1) control input parameters fw k g N kD0 are numerically optimized using sequential quadratic programming. This method is referred to as a DMOC (Discrete Mechanics and Optimal Control) approach (see [19]). Figure 5 shows a optimal solution transferring the spherical pendulum from a hanging configuration given by (q0 ; !0 ) D (e3 ; 031 ) to an inverted configuration (qd ; ! d ) D (e3 ; 031 ) during 1 second. The time step size is h D 0:033. Experiment have shown that the DMOC approach can compute optimal solutions using larger step sizes than typical Runge–Kutta methods, and consequently, it requires less computational load. In this case, using a second-order accurate Runge–Kutta method, the same optimization code fails while giving error messages of inaccurate and singular gradient computations. It is presumed that the unit length errors of the Runge– Kutta method, shown in Example 3, cause numerical instabilities for the finite-difference gradient computations required for the sequential quadratic programming algorithm. Example 6 (Optimal Control of a Rigid Body in a Potential Field) We study an optimal control problem of a rigid body using a dumbbell spacecraft model described in Example 4 (see [25] for detail). We assume that external control forces u f 2 R3 , and control moment u m 2 R3 act on the dumbbell spacecraft. Control inputs are parametrized by their values at each time step, and the Lie group variational integrators are modified to include the effects of the external control inputs by using discrete Lagrange– d’Alembert principle. The objective is to transfer the dumbbell from a given initial configuration (R0 ; x0 ; ˝0 ; v0 ) to a desired configuration (R d ; x d ; ˝ d ; v d ) during a fixed maneuver

time N, while minimizing the square of the l2 norm of the control inputs. N1 X

min J D

u kC1

kD0

T h  f T h f Wm u m u kC1 Wf u kC1C u m kC1 kC1 ; 2 2

where Wf ; Wm 2 R33 are symmetric positive definite matrices. Here we use a modified version of the discrete equations of motion with first order accuracy, as it yields a compact form for the necessary conditions. Necessary conditions for optimality: We solve this optimal control problem by using an indirect optimization method, where necessary conditions for optimality are derived using variational arguments, and a solution of the corresponding two-point boundary value problem provides the optimal control. This approach is common in the optimal control literature; here the optimal control problem is discretized at the problem formulation level using the Lie group variational integrator presented in Sect. “Discrete Lagrangian and Hamiltonian Mechanics”. Ja D

N1 X kD0

h  f T f f h  m T m m u W u kC1 C W u kC1 u 2 kC1 2 kC1

C 1;T k fx kC1 C x k C hv k g

 @U kC1 f 2;T mv kC1 C mv k  h C hu kC1 C k @x kC1  _ logm(F k  RTk R kC1 ) C 3;T k  ˚  C 4;T J˝ kC1 C F kT J˝ k C h M kC1 C u m kC1 ; k where 1k ; 2k ; 3k ; 4k 2 R3 are Lagrange multipliers. The matrix logarithm is denoted by logm : SO(3) ! so(3) and the vee map _ : so(3) ! R3 is the inverse of the hat map introduced in Example 4. The logarithm form of (43) is used, and the constraint (42) is considered implicitly using constrained variations. Using similar expressions for the

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Discrete Control Systems, Figure 6 Optimal control of a rigid body

variation of the rotation matrix and the angular velocity given in (14) and (15), the infinitesimal variation can be written as N1 n o X f ;T f hıu k W f u k C 2k1 ıJa D kD1

˚ 4 C hıu m;T Wm u m k C  k1 k ˚ C zTk  k1 C ATk  k ; where  k D [1k ; 2k ; 3k ; 4k ] 2 R12 , and z k 2 R12 represents the infinitesimal variation of (R k ; x k ; ˝ k ; v k ), given by z k D [logm(RTk ıR k )_ ; ıx k ; ı˝ k ; ıv k ]. The matrix A k 2 R1212 is defined in terms of (R k ; x k ; ˝ k ; v k ). Thus, necessary conditions for optimality are given by f

u kC1 D Wf1 2k ;

(47)

1 4 um kC1 D Wm  k ;

(48)

 k D ATkC1  kC1

(49)

together with the discrete equations of motion and the boundary conditions. Computational approach: Necessary conditions for optimality are expressed in terms of a two point boundary problem. The problem is to find the optimal discrete flow, multipliers, and control inputs to satisfy the equations of motion, optimality conditions, multiplier equations, and boundary conditions simultaneously. We use a neighboring extremal method (see [12]). A nominal solution satisfying all of the necessary conditions except the boundary conditions is chosen. The unspecified initial multiplier is updated by successive linearization so as to satisfy the specified terminal boundary conditions in the limit. This is also referred to as the shooting method. The main advantage of the neighboring extremal method is that the number of iteration variables is small.

The difficulty is that the extremal solutions are sensitive to small changes in the unspecified initial multiplier values. The nonlinearities also make it hard to construct an accurate estimate of sensitivity, thereby resulting in numerical ill-conditioning. Therefore, it is important to compute the sensitivities accurately to apply the neighboring extremal method. Here the optimality conditions (47) and (48) are substituted into the equations of motion and the multiplier equations, which are linearized to obtain sensitivity derivatives of an optimal solution with respect to boundary conditions. Using this sensitivity, an initial guess of the unspecified initial conditions is iterated to satisfy the specified terminal conditions in the limit. Any type of Newton iteration can be applied. We use a line search with backtracking algorithm, referred to as Newton–Armijo iteration (see [22]). Figure 6 shows optimized maneuvers, where a dumbbell spacecraft on a reference circular orbit is transferred to another circular orbit with a different orbital radius and inclination angle. Figure 6a shows the violation of the terminal boundary condition according to the number of iterations on a logarithmic scale. Red circles denote outer iterations in the Newton–Armijo iteration to compute the sensitivity derivatives. The error in satisfaction of the terminal boundary condition converges quickly to machine precision after the solution is close to the local minimum at around the 20th iteration. These convergence results are consistent with the quadratic convergence rates expected of Newton methods with accurately computed gradients. The neighboring extremal method, also referred to as the shooting method, is numerically efficient in the sense that the number of optimization parameters is minimized. But, in general, this approach may be prone to numerical ill-conditioning (see [3]). A small change in the initial multiplier can cause highly nonlinear behavior of the terminal attitude and angular momentum. It is difficult

Discrete Control Systems

to compute the gradient for Newton iterations accurately, and the numerical error may not converge. However, the numerical examples presented in this article show excellent numerical convergence properties. The dynamics of a rigid body arises from Hamiltonian mechanics, which have neutral stability, and its adjoint system is also neutrally stable. The proposed Lie group variational integrator and the discrete multiplier equations, obtained from variations expressed in the Lie algebra, preserve the neutral stability property numerically. Therefore the sensitivity derivatives are computed accurately. Controlled Lagrangian Method for Discrete Lagrangian Systems The method of controlled Lagrangians is a procedure for constructing feedback controllers for the stabilization of relative equilibria. It relies on choosing a parametrized family of controlled Lagrangians whose corresponding Euler–Lagrange flows are equivalent to the closed loop behavior of a Lagrangian system with external control forces. The condition that these two systems are equivalent results in matching conditions. Since the controlled system can now be viewed as a Lagrangian system with a modified Lagrangian, the global stability of the controlled system can be determined directly using Lyapunov stability analysis. This approach originated in Bloch et al. [5] and was then developed in Auckly et al. [1]; Bloch et al. [6,7,8,9]; Hamberg [15,16]. A similar approach for Hamiltonian controlled systems was introduced and further studied in the work of Blankenstein, Ortega, van der Schaft, Maschke, Spong, and their collaborators (see, for example, [33,35] and related references). The two methods were shown to be equivalent in Chang et al. [13] and a nonholonomic version was developed in Zenkov et al. [40,41], and Bloch [4]. Since the method of controlled Lagrangians relies on relating the closed-loop dynamics of a controlled system with the Euler–Lagrange flow associated with a modified Lagrangian, it is natural to discretize this approach through the use of variational integrators. In Bloch et al. [10,11], a discrete theory of controlled Lagrangians was developed for variational integrators, and applied to the feedback stabilization of the unstable inverted equilibrium of the pendulum on a cart. The pendulum on a cart is an example of an underactuated control problem, which has two degrees of freedom, given by the pendulum angle and the cart position. Only the cart position has control forces acting on it, and the stabilization of the pendulum has to be achieved indirectly through the coupling between the pendulum and the cart. The controlled Lagrangian is obtained by modifying the

kinetic energy term, a process referred to as kinetic shaping. Similarly, it is possible to modify the potential energy term using potential shaping. Since the pendulum on a cart model involves both a planar pendulum, and a cart that translates in one-dimension, the configuration space is a cylinder, S1 R. Continuous kinetic shaping: The Lagrangian has the form kinetic minus potential energy ˙ D L(q; q)

1  ˙2 ˛ C 2ˇ( ) ˙ s˙ C s˙2  V(q); 2

(50)

and the corresponding controlled Euler–Lagrange dynamics is d @L @L D0;  dt @ ˙ @ d @L Du; dt @˙s

(51) (52)

where u is the control input. Since the potential energy is translation invariant, i. e., V(q) D V( ), the relative equilibria D e , s˙ D const are unstable and given by non-degenerate critical points of V( ). To stabilize the relative equilibria D e , s˙ D const with respect to , kinetic shaping is used. The controlled Lagrangian in this case is defined by 1 ˙ D L( ; ˙ ; s˙ C ( ) ˙ ) C  (( ) ˙ )2 ; (53) L; (q; q) 2 where ( ) D ˇ( ). This velocity shift corresponds to a new choice of the horizontal space (see [8] for details). The dynamics is just the Euler–Lagrange dynamics for controlled Lagrangian (53), d @L; @L;  D0; dt @ ˙ @ d @L; D0: dt @˙s

(54) (55)

The Lagrangian (53) satisfies the simplified matching conditions of Bloch et al. [9] when the kinetic energy metric coefficient in (50) is constant.  Setting u D d ( ) ˙ /dt defines the control input, makes Eqs. (52) and (55) identical, and results in controlled momentum conservation by dynamics (51) and (52). Setting  D 1/  makes Eqs. (51) and (54) identical when restricted to a level set of the controlled momentum. Discrete kinetic shaping: Here, we adopt the following notation: q kC1/2 D

q k C q kC1 ; 2 q k D q kC1  q k ;

q k D ( k ; s k ) :

155

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Discrete Control Systems

Then, a second-order accurate discrete Lagrangian is given by, Ld (q k ; q kC1 ) D hL(q kC1/2 ; q k /h) : The discrete dynamics is governed by the equations @Ld (q k ; q kC1 ) @Ld (q k1 ; q k ) C D0; @ k @ k @Ld (q k ; q kC1 ) @Ld (q k1 ; q k ) C D u k ; @s k @s k

(56) (57)

where u k is the control input. Similarly, the discrete controlled Lagrangian is, ; (q kC1/2 ; q k /h) ; L; d (q k ; q kC1 ) D hL

with discrete dynamics given by, @L; @L; (q k1 ; q k ) d (q k ; q kC1 ) C d D0; @ k @ k

@L; @L; (q k1 ; q k ) d (q k ; q kC1 ) C d D0: @s k @s k

Numerical example: Simulating the behavior of the discrete controlled Lagrangian system involves viewing Eqs. (58)–(59) as an implict update map (q k2 ; q k1 ) 7! (q k1 ; q k ). This presupposes that the initial conditions are given in the form (q0 ; q1 ); however it is generally preferable to specify the initial conditions as (q0 ; q˙0 ). This is achieved by solving the boundary condition,

(58) (59)

@L (q0 ; q˙0 ) C D1 Ld (q0 ; q1 ) C Fd (q0 ; q1 ) D 0 ; @q˙ for q1 . Once the initial conditions are expressed in the form (q0 ; q1 ), the discrete evolution can be obtained using the implicit update map. We first consider the case of kinetic shaping on a level surface, when  is twice the critical value, and without dissipation. Here, h D 0:05 sec, m D 0:14 kg, M D 0:44 kg, and l D 0:215 m. As shown in Fig. 7, the dynamics is stabilized, but since there is no dissipation, the oscillations are sustained. The s dynamics exhibits both a drift and oscillations, as potential shaping is necessary to stabilize the translational dynamics.

Equation (59) is equivalent to the discrete controlled momentum conservation: Future Directions pk D  ; where @L; d (q k ; q kC1 ) @s k (1 C )ˇ( kC1/2 ) k C s k D : h

pk D 

Setting uk D 

 k ( kC1/2 )   k1 ( k1/2 ) h

makes Eqs. (57) and (59) identical and allows one to represent the discrete momentum equation (57) as the discrete momentum conservation law p k D p. The condition that (56)–(57) are equivalent to (58)– (59) yield the discrete matching conditions. The dynamics determined by Eqs. (56)–(57) restricted to the momentum level p k D p is equivalent to the dynamics of Eqs. (58)– (59) restricted to the momentum level p k D  if and only if the matching conditions  D hold.

1 ;



D

p ; 1 C 

Discrete Receding Horizon Optimal Control: The existing work on discrete optimal control has been primarily focused on constructing the optimal trajectory in an open loop sense. In practice, model uncertainty and actuation errors necessitate the use of feedback control, and it would be interesting to extend the existing work on optimal control of discrete systems to the feedback setting by adopting a receding horizon approach. Discrete State Estimation: In feedback control, one typically assumes complete knowledge regarding the state of the system, an assumption that is often unrealistic in practice. The general problem of state estimation in the context of discrete mechanics would rely on good numerical methods for quantifying the propagation of uncertainty by solving the Liouville equation, which describes the evolution of a phase space distribution function advected by a prescribed vector field. In the setting of Hamiltonian systems, the solution of the Liouville equation can be solved by the method of characteristics (Scheeres et al. [38]). This implies that a collocational approach (Xiu [39]) combined with Lie group variational integrators, and interpolation based on noncommutative harmonic analysis on Lie groups could yield an efficient means of propagating uncertainty, and serve as the basis of a discrete state estimation algorithm.

Discrete Control Systems

Discrete Control Systems, Figure 7 Discrete controlled dynamics with kinetic shaping and without dissipation. The discrete controlled system stabilizes the  motion about the equilibrium, but the s dynamics is not stabilized; since there is no dissipation, the oscillations are sustained

Forced Symplectic–Energy-Momentum Variational Integrators: One of the motivations for studying the control of Lagrangian systems using the method of controlled Lagrangians is that the method provides a natural candidate Lyapunov function to study the global stability properties of the controlled system. In the discrete theory, this approach is complicated by the fact that the energy of a discrete Lagrangian system is not exactly conserved, but rather oscillates in a bounded fashion. This can be addressed by considering the symplectic-energy-momentum [20] analogue to the discrete Lagrange-d’Alembert principle, ı

N1 X kD0

Ld (q k ; q kC1 ; h k ) D

N1 X

[Fd (q k ; q kC1 ; h k )  ıq k

kD0

C FdC (q k ; q kC1 ; h k )  ıq kC1 ] ; where the timestep hk is allowed to vary, and is chosen to satisfy the variational principle. The variations in hk yield an Euler–Lagrange equation that reduces to the conservation of discrete energy in the absence of external forces. By developing a theory of controlled Lagrangians around a geometric integrator based on the symplecticenergy-momentum version of the Lagrange–d’Alembert principle, one would potentially be able to use Lyapunov techniques to study the global stability of the resulting numerical control algorithms.

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cations to the 3D pendulum. In: Proceedings of the IEEE Conference on Control Applications. Toronto, Canada, 28–31 August 2005, pp 962–967 Lee T, Leok M, McClamroch NH (2006) Optimal control of a rigid body using geometrically exact computations on SE(3). In: Proceedings of the IEEE Conference on Decision and Control. Manchester Grand Hyatt, San Diego, CA, 13–15 December 2006, pp 2710–2715 Lee T, Leok M, McClamroch NH (2007) Lie group variational integrators for the full body problem. Comput Method Appl Mech Eng 196:2907–2924 Lee T, Leok M, McClamroch NH (2007) Lie group variational integrators for the full body problem in orbital mechanics. Celest Mech Dyn Astron 98(2):121–144 Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol 14. Cambridge University Press, Cambridge Leok M (2004) Foundations of Computational Geometric Mechanics. Ph D thesis, California Instittute of Technology Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol 17. Springer, New York Marsden JE, West M (2001) Discrete mechanics and variational integrators. In: Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 317–514 Marsden JE, Pekarsky S, Shkoller S (1999) Discrete Euler–Poincarée and Lie–Poisson equations. Nonlinearity 12(6):1647–1662 Maschke B, Ortega R, van der Schaft A (2001) Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans Autom Contr 45:1498–1502 Moser J, Veselov AP (1991) Discrete versions of some classical integrable systems and factorization of matrix polynomials. Commun Math Phys 139:217–243 Ortega R, Spong MW, Gómez-Estern F, Blankenstein G (2002) Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans Autom Contr 47:1218–1233 Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. In: Acta Numerica, vol 1. Cambridge University Press, Cambridge, pp 243–286 Scheeres DJ, Fahnestock EG, Ostro SJ, Margot JL, Benner LAM, Broschart SB, Bellerose J, Giorgini JD, Nolan MC, Magri C, Pravec P, Scheirich P, Rose R, Jurgens RF, De Jong EM, Suzuki S (2006) Dynamical configuration of binary near-Earth asteroid (66391) 1999 KW4. Science 314:1280–1283 Scheeres DJ, Hsiao F-Y, Park RS, Villac BF, Maruskin JM (2006) Fundamental limits on spacecraft orbit uncertainty and distribution propagation. J Astronaut Sci 54:505–523 Xiu D (2007) Efficient collocational approach for parametric uncertainty analysis. Comm Comput Phys 2:293–309 Zenkov DV, Bloch AM, Leonard NE, Marsden JE (2000) Matching and stabilization of low-dimensional nonholonomic systems. In: Proceedings of the IEEE Conference on Decision and Control. Sydney Convention and Exhibition Centre, Sydney, NSW Australia; 12–15 December 2000, pp 1289–1295 Zenkov DV, Bloch AM, Leonard NE, Marsden JE (2002) Flat nonholonomic matching. In: Proceedings of the American Control Conference. Hilton Anchorage, Anchorage, AK, 8–10 May 2002, pp 2812–2817

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Books and Reviews Bloch AM (2003) Nonholonomic Mechanics and Control. Interdisciplinary Appl Math, vol 24. Springer Bullo F, Lewis AD (2005) Geometric control of mechanical systems. Texts in Applied Mathematics, vol 49. Springer, New York Hairer E, Lubich C, Wanner G (2006) Geometric Numerical Integration, 2nd edn. Springer Series in Computational Mathematics, vol 31. Springer, Berlin Iserles A, Munthe-Kaas H, Nørsett SP, Zanna A (2000) Lie-group methods. In: Acta Numerica, vol 9. Cambridge University Press, Cambridge, pp 215–365

Leimkuhler B, Reich S (2004) Simulating Hamiltonian Dynamics. Cambridge Monographs on Applied and Computational Mathematics, vol 14. Cambridge University Press, Cambridge Marsden JE, Ratiu TS (1999) Introduction to Mechanics and Symmetry, 2nd edn. Texts in Applied Mathematics, vol 17. Springer Marsden JE, West M (2001) Discrete mechanics and variational integrators. In Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 317–514 Sanz-Serna JM (1992) Symplectic integrators for Hamiltonian problems: an overview. In: Acta Numerica, vol 1. Cambridge University Press, Cambridge, pp 243–286

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Dispersion Phenomena in Partial Differential Equations PIERO D’ANCONA Dipartimento di Matematica, Unversità di Roma “La Sapienza”, Roma, Italy

  kukH s D (1  )s/2 uL q : Recall that this definition does not reduce to the preceding one when q D 1. We shall also use the homogeneous space H˙ qs , with norm   kukH˙ s D ()s/2 u L q :

Article Outline Glossary Definition of the Subject Introduction The Mechanism of Dispersion Strichartz Estimates The Nonlinear Wave Equation The Nonlinear Schrödinger Equation Future Directions Bibliography Glossary Notations Partial derivatives are written as ut or @ t u, @˛ D @˛x 11  @˛x nn , the Fourier transform of a function is defined as Z F f () D b f () D ei x f (x)dx and we frequently use the mute constant notation A . B to mean A  CB for some constant C (but only when the precise dependence of C from the other quantities involved is clear from the context). Evolution equations Partial differential equations describing physical systems which evolve in time. Thus, the variable representing time is distinguished from the others and is usually denoted by t. Cauchy problem A system of evolution equations, combined with a set of initial conditions at an initial time t D t0 . The problem is well posed if a solution exists, is unique and depends continuously on the data in suitable norms adapted to the problem. Blow up In general, the solutions to a nonlinear evolution equation are not defined for all times but they break down after some time has elapsed; usually the L1 norm of the solution or some of its derivatives becomes unbounded. This phenomenon is called blow up of the solution. Sobolev spaces we shall use two instances of Sobolev space: the space W k;1 with norm kukW k;1 D

and the space H qs with norm

X j˛jk

k@˛ ukL 1

Dispersive estimate a pointwise decay estimate of the form ju(t; x)j  Ct ˛ , usually for the solution of a partial differential equation. Definition of the Subject In a very broad sense, dispersion can be defined as the spreading of a fixed amount of matter, or energy, over a volume which increases with time. This intuitive picture suggests immediately the most prominent feature of dispersive phenomena: as matter spreads, its size, defined in a suitable sense, decreases at a certain rate. This effect should be contrasted with dissipation, which might be defined as an actual loss of energy, transferred to an external system (heat dissipation being the typical example). This rough idea has been made very precise during the last 30 years for most evolution equations of mathematical physics. For the classical, linear, constant coefficient equations like the wave, Schrödinger, Klein–Gordon and Dirac equations, the decay of solutions can be measured in the Lp norms, and sharp estimates are available. In addition, detailed information on the profile of the solutions can be obtained, producing an accurate description of the evolution. For nonlinear equations, the theory now is able to explain and quantify several complex phenomena such as the splitting of solutions into the sum of a train of solitons, plus a radiation part which disperses and decays as time increases. Already in the 1960s it was realized that precise information on the decay of free waves could be a powerful tool to investigate the effects of linear perturbations (describing interactions with electromagnetic fields) and even nonlinear perturbations of the equations. One of the first important applications was the discovery that a nonlinear PDE may admit global small solutions, provided the rate of decay is sufficiently strong compared with the concentration effect produced by the nonlinear terms. This situation is very different from the ODE setting, where dispersion is absent. Today, dispersive and Strichartz estimates represent the backbone of the theory of nonlinear evolution equations. Their applications include local and global existence

Dispersion Phenomena in Partial Differential Equations

results for nonlinear equations, existence of low regularity solutions, scattering, qualitative study of evolution equations on manifolds, and many others. Introduction To a large extent, mathematical physics is the study of qualitative and quantitative properties of the solutions to systems of differential equations. Phenomena which evolve in time are usually described by nonlinear evolution equations, and their natural setting is the Cauchy problem: the state of the system is assigned at an initial time, and the following evolution is, or should be, completely determined by the equations. The most basic question concerns the local existence and uniqueness of the solution; we should require that, on one hand, the set of initial conditions is not too restrictive, so that a solution exists, and on the other hand it is not too lax, so that a unique solution is specified. For physically meaningful equations it is natural to require in addition that small errors in the initial state propagate slowly in time, so that two solutions corresponding to slightly different initial data remain close, at least for some time (continuous dependence on the data). This set of requirements is called local well posedness of the system. Global well posedness implies the possibility of extending these properties to arbitrary large times. However, local well posedness is just the first step in understanding a nonlinear equation; basically, it amounts to a check that the equation and its Cauchy problem are meaningful from the mathematical and physical point of view. But the truly interesting questions concern global existence, asymptotic behavior and more generally the structure and shape of solutions. Indeed, if an equation describes a microscopic system, all quantities measured in an experiment correspond to asymptotic features of the solution, and local properties are basically useless in this context. The classical tool to prove local well posedness is the energy method; it can be applied to many equations and leads to very efficient proofs. But the energy method is intrinsically unable to give answers concerning the global existence and behavior of solutions. Let us illustrate this point more precisely. Most nonlinear evolution equations are of the form Lu D N(u) with a linear part L (e. g., the d’Alembert or Schrödinger operator) and a nonlinear term N(u). The two terms are in competition: the linear flow is well behaved, usually with

several conserved quantities bounded by norms of the initial data, while the nonlinear term tends to concentrate the peaks of u and make them higher, increasing the singularity of the solution. At the same time, a small function u is made even smaller by a power-like nonlinearity N(u). The energy method tries to use only the conserved quantities, which remain constant during the evolution of the linear flow. Essentially, this means to regard the equation as an ODE of the form y 0 (t) D N(y(t)). This approach is very useful if the full (nonlinear) equation has a positive conserved quantity. But in general the best one can prove using the energy method is local well posedness, while a blow up in finite time can not be excluded, even if the initial data are assumed to be small, as the trivial example y 0 D y 2 clearly shows. To improve on this situation, the strength of the linear flow must be used to a full extent. This is where dispersive phenomena enter the picture. A finer study of the operator L shows that there are quantities, different from the standard L2 type norms used in the energy estimates, which actually decay during the evolution. Hence the term L has an additional advantage in the competition with N which can lead to global existence of small solutions. See Sect. “The Nonlinear Wave Equation” for more details. This circle of ideas was initiated at the beginning of the 1970s and rapidly developed during the 1980s in several papers, (see [26,27,31,32,33,34,38,40,41,42,50] to mention just a few of the fundamental contributions on the subject). Global existence of small solutions was proved for many nonlinear evolution equations, including the nonlinear wave, Klein–Gordon and Schrödinger equations. The proof of the stability of vacuum for the Einstein equations [15,35] can be regarded as an extreme development of this direction of research. Another early indication of the importance of Lp estimates came from the separate direction of harmonic analysis. In 1970, at a time when general agreement in the field was that the L2 framework was the only correct one for the study of the wave equation, R. Strichartz ([44,45], see also [10,11,46]) proved the estimate u t t  u D F ;

u(0; x) D u t (0; x) D 0

H)

kukL p (RnC1 ) . kFkL p0 (RnC1 )

for p D 2(n C 1)/(n  1). This was a seminal result, both for the emphasis on the Lp approach, and for the idea of regarding all variables including time on the same level, an idea which would play a central role in the later developments. At the same time, it hinted at a deep connection

161

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between dispersive properties and the techniques of harmonic analysis. The new ideas were rapidly incorporated in the mainstream theory of partial differential equations and became an important tool ([22,23,52]); the investigation of Strichartz estimates is still in progress. The freedom to work in an Lp setting was revealed as useful in a wide variety of contexts, including linear equations with variable coefficients, scattering, existence of low regularity solutions, and several others. We might say that this point of view created a new important class of dispersive equations, complementing and overlapping with the standard classification into elliptic, hyperbolic, parabolic equations. Actually this class is very wide and contains most evolution equations of mathematical physics: the wave, Schrödinger and Klein–Gordon equations, Dirac and Maxwell systems, their nonlinear counterparts including wave maps, Yang–Mills and Einstein equations, Korteweg de Vries, Benjamin–Ono, and many others. In addition, many basic physical problems are described by systems obtained by coupling these equations; among the most important ones we mention the Dirac–Klein–Gordon, Maxwell–Klein–Gordon, Maxwell– Dirac, Maxwell–Schrödinger and Zakharov systems. Finally, a recent line of research pursues the extension of the dispersive techniques to equations with variable coefficients and equations on manifolds. The main goal of this article is to give a quick overview of the basic dispersive techniques, and to show some important examples of their applications. It should be kept in mind however that the field is evolving at an impressive speed, mainly due to the contribution of ideas from Fourier and harmonic analysis, and it would be difficult to give a self-contained account of the recent developments, for which we point at the relevant bibliography. In Sect. “The Mechanism of Dispersion” we analyze the basic mechanism of dispersion, with special attention given to two basic examples (Schrödinger and wave equation). Section “Strichartz Estimates” is a coincise exposition of Strichartz estimates, while Sects. “The Nonlinear Wave Equation” and “The Nonlinear Schrödinger Equation” illustrate some typical applications of dispersive techniques to problems of global existence for nonlinear equations. In the last Sect. “Future Directions” we review some of the more promising lines of research in this field.

The same mechanism lies behind dispersive phenomena, and indeed two dual explanations are possible: a) in physical space, dispersion is characterized by a finite speed of propagation of the signals; data concentrated on some region tend to spread over a larger volume following the evolution flow. Correspondingly, the L1 and other norms of the solution tend to decrease; b) in Fourier space, dispersion is characterized by oscillations in the Fourier variable , which increase for large jj and large t. This produces cancelation and decay in suitable norms. We shall explore these dual points of view in two fundamental cases: the Schrödinger and the wave equation. These are the most important and hence most studied dispersive equations, and a fairly complete theory is available for both. The Schrödinger Equation in Physical Space The solution u(t; x) of the Schrödinger equation iu t C u D 0 ;

u(0; x) D f (x)

admits several representations which are not completely equivalent. From spectral theory we have an abstract representation as a group of L2 isometries u(t; x) D e i t f ;

(1)

then we have an explicit representation using the fundamental solution 2 1 i jxj 4t f e (4 it)n/2 Z jx yj2 1 i 4t e f (y)dy ; D (4 it)n/2 Rn

u(t; x) D

finally, we can represent the solution via Fourier transform as Z 2 n f ()d : (3) e i(tjj Cx)b u(t; x) D (2) Notice for instance that the spectral and Fourier representations are well defined for f 2 L2 , while it is not immediately apparent how to give a meaning to (2) if f 62 L1 . From (1) we deduce immediately the dispersive estimate for the Schrödinger equation, which has the form 1

The Mechanism of Dispersion

je i t f j  (4)n/2

The Fourier transform puts in duality oscillations and translations according to the elementary rule

Since the L2 norm of u is constant in time

F ( f (x C h)) D ei xh F f :

(2)

t n/2

ke i t f kL 2 D k f kL 2 ;

k f kL 1 :

(4)

(5)

Dispersion Phenomena in Partial Differential Equations

this might suggest that the constant L2 mass spreads on a region of volume t n i. e. of diameter t. Apparently, this is in contrast with the notion that the Schrödinger equation has an infinite speed of propagation. Indeed, this is a correct but incomplete statement. To explain this point with an explicit example, consider the evolution of a wave packet 1

from zero, and this already suggests a decay of the order t n/2 . This argument can be made precise and gives an alternative proof of (4). We also notice that the Fourier transform in both space and time of u(t; x) D e i t f is a temperate distribution in S0 (RnC1 ) which can be written, apart from a constant, ˜ ) D ı(  jj2) u(;

2

x 0 ;0 D e 2 jxx 0 j e i(xx 0 )0 : A wave packet is a mathematical representation of a particle localized near position x0 , with frequency localized near  0 ; notice indeed that its Fourier transform is 1

2

b  x 0 ;0 D ce 2 j0 j e ix 0 for some constant c. Exact localization both in space and frequency is of course impossible by Heisenberg’s principle; however the approximate localization of the wave packet is a very good substitute, since the gaussian tails decay to 0 extremely fast. The evolution of a wave packet is easy to compute explicitly:   1 (x  x0  2t0 )2 i i t  n2 e x 0 ;0 D (1C2it) exp  e 2 (1 C 4t 2 )

and is supported on the paraboloid  D jj2. This representation of the solution gives additional insight to its behavior in certain frequency regimes, and has proved to be of fundamental importance when studying the existence of low regularity solutions to the nonlinear Schrödinger equation [6,9]. The Wave Equation in Physical Space For the wave equation, the situation is reversed: the most efficient proof of the dispersive estimate is obtained via the Fourier representation of the solution. This is mainly due to the complexity of the fundamental solution; we recall the relevant formulas briefly. The solution to the homogeneous wave equation

where the real valued function  is given by  D

t(x  x0 )2 C (x  x0  t0 )  0 : (1 C 4t 2 )

We notice that the particle “moves” with velocity 20 and the height of the gaussian spreads precisely at a rate t n/2 . The mass is essentially concentrated in a ball of radius t, as suspected. This simple remark is at the heart of one of the most powerful techniques to have appeared in harmonic analysis in recent years, the wave packet decomposition, which has led to breakthroughs in several classical problems (see e. g. [7,8,48,51]).

w t t  w D 0 ;

2

(t; x; ) D tjj C x  

H)

r  D 2t C x

so that for each (t; x) we have a unique stationary point  D x/2t; the n curvatures at that point are different

w t (0; x) D g(x)

can be written as u(t; x) D cos(tjDj) f C

sin(tjDj) g jDj

where we used the symbolic notations cos(tjDj) f D F 1 cos(tjj)F f ; sin(tjDj) sin(tjj) Fg g D F 1 jDj jj (recall that F denotes the Fourier transform in space variables). Notice that

The Schrödinger Equation in Fourier Space In view of the ease of proof of (4), it may seem unnecessary to examine the solution in Fourier space, however a quick look at this other representation is instructive and prepares the discussion for the wave equation. If we examine (3), we see that the integral is a typical oscillatory integral which can be estimated using the stationary phase method. The phase here is

w(0; x) D f (x) ;

cos(tjDj) f D

@ sin(tjDj) f; @t jDj

hence the properties of the first operator can be deduced from the second. Moreover, by Duhamel’s formula, we can express the solution to the nonhomogeneous wave equation w t t  w D F(t; x) ;

w(0; x) D 0 ;

w t (0; x) D 0

as Z

t

u(t; x) D 0

sin((t  s)jDj) F(u(s; ))ds : jDj

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Dispersion Phenomena in Partial Differential Equations

of the solution remains constant, in view of the energy estimate

In other words, a study of the operator S(t)g D

sin(tjDj) g jDj

ku t (t; )k2L 2 Ckru(t; )k2L 2 D ku t (0; )k2L 2 Ckru(0; )k2L 2 ;

is sufficient to deduce the behavior of the complete solution to the nonhomogeneous wave equation with general data. The operator S(t) can be expressed explicitly using the fundamental solution of the wave equation. We skip the one-dimensional case since then from the expression of the general solution u(t; x) D (t C x) C

(t  x)

it is immediately apparent that no dispersion can occur. In odd space dimension n  3 we have !   n3 Z 2 1 sin(tjDj) 1 g(x  y)d y ; g D cn @t jDj t t jyjDt

on the other hand, the velocity of propagation for the wave equation is exactly 1, so that the constant L2 energy is spread on a volume which increases at a rate t n ; as a consequence, we would expect the same rate of decay t n/2 as for the Schrödinger equation. However, a more detailed study of the shape of the solution reveals that it tends to concentrate in a shell of constant thickness around the light cone. For instance, in odd dimension larger than 3 initial data with support in a ball B(0; R) produce a solution which at time t is supported in the shell B(0; t C R) n B(0; t  R). Also in even dimension the solution tends to concentrate along the light cone jtj D jxj, with a faster decreasing tail far from the cone. Thus, the energy spreads over a volume of size t n1 , in accordance with the rate of decay in the dispersive estimate.

and in even space dimension n  2 sin(tjDj) g D cn jDj



1 @t t

 n2 2

1 t

Z jyj
g(x  y) p dy t 2  jyj2

!

for some constant cn . These formulas can be expanded and the resulting integrals estimated in order to compute the explicit decay rate of the solution; this approach was followed in [50] (see also [43] for a more concise proof based on the fundamental solution). We shall prove the dispersive estimate using Fourier methods in the next section, but here it is instructive to consider the particularly simple case n D 3, when the above representation reduces to Kirchhoff’s formula Z sin(tjDj) c g(x  y)d y : gD jDj t jyjDt Using the divergence theorem it is easy to deduce that ˇ ˇ ˇ sin(tjDj) ˇ 1 ˇ ˇ . kr f kL 1 : g ˇ jDj ˇ t Recall that the rate of decay for the Schrödinger equation in dimension 3 is t 3/2 ; for the wave equation the rate of decay is slower, in addition the estimate shows a loss of one derivative with respect to the initial data. Actually this estimate is optimal, as it is evident from the explicit representation of the solution. In dimension n the correct n1 rate of decay is t  2 and the loss of derivative is of order (n  1)/2, as we shall see in the next section. This result is in apparent contrast with our physical space picture. Indeed the L2 norm of the first derivatives

The Wave Equation in Fourier Space Since 2i

e i tjDj ei tjDj sin(tjDj) gD g g; jDj jDj jDj

2 cos(tjDj) f D e i tjDj f C ei tjDj f ; we can unify the study of the various components of the solution by considering the operator Z f d : e i tjDj f D F 1 e i tjj F f D (2)n e i(xCtjj)b The phase of the last integral is (t; x; ) D tjj C x  

H)

r  D t

 Cx jj

and we notice immediately that the structure of stationary points is more complex than for Schrödinger: if jtj D jxj, i. e. if the point (t; x) is on the light cone, the phase has a half line of critical points jtj D jxj

H)

 x D jj t

which of course are degenerate since only n  1 curvatures are different from zero; on the other hand, when (t; x) is not on the light cone, the phase is nondegenerate at all points. In the nondegenerate case jtj ¤ jxj, we notice that r  D t

 Cx jj

H)

jr j  jjtj  jxjj

Dispersion Phenomena in Partial Differential Equations

hence by direct integration by parts we obtain ˇ ˇZ ˇ ˇ ˇ  C( f )jjtj  jxjj(n1) : ˇ e i(xCtjj)b f d ˇ ˇ

(6)

Notice that the singularity jj of the phase makes it impossible to perform the integration by parts more than n  1 times, but this is largely sufficient for our purposes here. In the degenerate case jtj D jxj, we can eliminate the degeneracy by passing to polar coordinates: Z 1 Z i(xCtjj)b f d D e i t n1 e 0 Z f ( !)d!d :  e i x !b j!jD1

Now the phase in the inner integral has exactly two isolated critical points for each value of x; by a standard application of the stationary phase method we can obtain an estimate of the form ˇZ ˇ ˇ ˇ ˇ e i(xCtjj)b ˇ  C( f )jxj n1 2 : f d (7) ˇ ˇ The power (n  1)/2 is directly related to the dimension n  1 of the unit sphere. From the above computations it is possible to obtain a very detailed description of the shape of the solution (see e. g. [20]). However, here we are only interested in the rate of decay of the L1 norm of the solution. Thus, using estimate (6) when jxj < jtj/2 and (7) when jxj > jtj/2 we obtain je i tjDj f j  C( f )jtj

n1 2

:

It is also possible to make precise the form of the constant C( f ) appearing in the estimate. This can be done in several ways; one of the most efficient exploits the scaling properties of the wave equation (see [10,23]). In order to state the optimal estimate we briefly recall the definition of some function spaces. The basic ingredient is the homogeneous Paley–Littlewood partition of unity which has the form X  j () D 1 ;  j () D 0 (2 j ) ; j2Z

sup  j D f : 2 j1  jj  2 jC1 g for a fixed 0 2 C01 (Rn ) (the following definitions are independent of the precise choice of 0 ). To each symbol  j () we associate by the standard calculus the operator  j (D) D F 1  j ()F . Now the homogeneous Besov norm s can be defined as follows: B˙ 1;1 X s D 2 js k j (D) f kL 1 (8) k f kB˙ 1;1 j2Z

and the corresponding Besov space can be obtained by completing C01 (see [4] for a thorough description of these definitions). Then the optimal dispersive estimate for the wave equation is je i tjDj f j  Ck f k

n1

B˙ 1;12

jtj

n1 2

:

(9)

We might add that in odd space dimension a slightly better estimate can be proved involving only the standard ˙ n1 2 instead of the Besov norm. Sobolev norm W The singularity in t at the right hand side of (9) can be annoying in applications, thus it would be natural to try to replace jtj by (1 C jtj) by some different proof. However, this is clearly impossible since both sides have the same scaling with respect to the transformation u(t; x) ! u(t; x) which takes a solution of the homogeneous wave equation into another solution. Moreover, if such an estimate were true, taking the limit as t ! 0 we would obtain a false Sobolev embedding. Thus, the only way to get a nonsingular estimate is to replace the Besov norm of the data by some stronger norm. Indeed, for jtj < 1 we can use the L2 conservation which implies, for any > 0, je i tjDj f j  ke i tjDj f kH n/2C D k f kH n/2C : Combining the two estimates, we arrive at   n1 je i tjDj f j . k f k n1 C k f k (1Cjtj) 2 : (10) n/2C H 2 B˙ 1;1

Other Dispersive Equations Dispersive estimates can be proved for many other equations; see for instance [3] for general results concerning the equations of the form iu t D P(D)u ;

P polynomial

which correspond to integrals of the form Z e i(xx iCP())b f ()d : Here we would like to add some more details on two equations of particular importance for physics. The Klein– Gordon equation u t t  u C u D 0 ;

u(0; x) D f (x) ;

u t (0; x) D g(x)

shows the same decay rate as the Schrödinger equation, although for most other properties it is very close to the wave

165

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Dispersion Phenomena in Partial Differential Equations

equation. This phenomenon can be easily explained. In physical space, we notice that the solution has finite speed of propagation less or equal to 1, so that for compactly supported initial data the solution spreads uniformly on a volume of order tn , and there is no concentration along the light cone as for the wave equation. On the other hand, the Fourier representation of the solution is sin(thDi) u(t; x) D cos(thDi) f C g hDi

cos(thDi) f D F 1 cos(tjhij)F f ;

with hi D (1 C i. e. hDi D (1  Thus, we see that, similarly to the wave equation, the study of the solutions can be reduced to the basic operator Z f ()d : e i thDi f D F 1 e i thi F f D (2)n e i(thiCx)b It is easy to check that the phase has an isolated stationary point if and only if jtj < jxj, and no stationary point if jtj  jj:  Cx; hi

in accord with the claimed dispersion rate of order t n/2 . However, the proof is not completely elementary (for a very detailed study, see Section 7.2 of [25]). The optimal estimate is now je

i thDi

f j  Ck f k

n

2 B 1;1

jtj

 n2

sup  j D f : 2

and the 4  4 Dirac matrices can be written     I 0 k 0 ˛k D ; ˇD 2 ; k 0 0 I2 in terms of the Pauli matrices    1 0 0 I2 D ; 1 D 0 1 1    0 i 1 2 D ; 3 D i 0 0

j  0;

 jj  2 jC1 g ;

sup 1 D fjxj  1g for some 1 ; 0 2 C01 (Rn ). A second important example is the Dirac system. From a mathematical point of view (and setting the values of the

1 0

k D 1; 2; 3

 ;

 0 : 1

Then the solution u(t; x) D e i t D f of the massless Dirac system with initial value u(0; x) D f (x) satisfies the dispersive estimate ju(t; x)j 

C k f kB˙ 2 1;1 t

while in the massive case we have ju(t; x)j 

j2N

j1

3 1X ˛k @k i

(11)

(see the Appendix to [18]). Here we used the nonhomoges which is defined exactly as the honeous Besov norm B1;1 mogeneous one (8) by using the inhomogeneous version of the Palye–Littlewood partition of unity: X 1 () C  j () D 1 ;  j () D 0 (2 j ) for

iu t C Du C ˇu D 0

kD1

)1/2 .

r  D t

in the massless case, and

DD

sin(thDi) sin(tjhij) Fg g D F 1 hDi jhij

(t; x; ) D thi C x   ;

iu t C Du D 0

in the massive case. Here u : R t  R3x ! C 4 , the operator D is defined as

where we used the symbolic notations

jj2)1/2

speed of light and Planck’s constant equal to 1 for simplicity of notation) this is a 4  4 constant coefficient system of the form

C 3

t2

kfk

5

2 B 1;1

(see Sect. 7 of [16]). Strichartz Estimates Since the Schrödinger flow e i t f is an L2 isometry, it may be tempting to regard it as some sort of “rotation” of the Hilbert space L2 . This picture is far from true: there exist small subspaces of L2 which contain the flow for almost all times. This surprising phenomenon, which holds in greater generality for any unbounded selfadjoint operator on an abstract Hilbert space, is known as Kato smoothing and was discovered by T. Kato in 1966 [29], see also [39]. This fact is just a corollary of a quantitative estimate, the Kato smoothing estimate, which in the case of the Laplace

Dispersion Phenomena in Partial Differential Equations

operator takes the following form: there exist closed unbounded operators A on L2 such that kAe

i t

f kL 2 L 2 . k f kL 2 :

As a consequence, the flow belongs for a. e. t to the domain of the operator A, which can be a very small subspace of L2 . Two simple examples of smoothing estimates are the following: V(x) 2 L2 \ L n \ L nC H)

kV(x)e i t f kL 2 L 2 . k f kL 2

(12)

and kjxj1 e i t f kL 2 L 2 . k f kL 2 :

(13)

On the other hand, Strichartz proved in 1977 [46] that pD

2(n C 2) n H)

2n

L 2 L n2

ke

i t

f kL p (RnC1 ) . k f kL 2

(14)

. k f kL 2

(15)

while (13) would follow from the slightly stronger ke i t f k

2n

L 2 L n2 ;2

The homogeneous Strichartz estimates have the general form ke i t f kL p L q . k f kL 2 for suitable values of the couple (p; q). Notice that we can restrict the L p L q norm at the right hand side to an arbitrary time interval I (i. e. to L p (I; L q (Rn ))). Notice also that by the invariance of the flow under the scaling u(t:x) 7! u(2 t; x) the possible couples (p; q) are bound to satisfy n n 2 C D : p q 2

. k f kL 2

(16)

where L p;2 denotes a Lorentz space. Both (15) and (16) are true, although their proof is far from elementary. Such estimates are now generally called Strichartz estimates. They can be deduced from the dispersive estimates, hence they are in a sense weaker and contain less information. However, for this same reason they can be obtained for a large number of equations, even with variable coefficients, and their flexibility and wide range of applications makes them extremely useful. The full set of Strichartz estimates, with the exception of the endpoint (see below) was proved in [22], where it was also applied to the problem of global well posedness for the nonlinear Schrödinger equation. The corresponding estimates for the wave equation were obtained in [23], however they turned out to be less useful because of the loss of derivatives. A fundamental progress was made in [30], where not only the most difficult endpoint case was solved, but it also outlined a general procedure to deduce the Strichartz estimates from the dispersive estimates, with applications to any evolution equation. In the following we shall focus on the two main examples, the Schrödinger and the wave equation, and we refer to the original papers for more details and the proofs.

(17)

The range of possible values of q is expressed by the inequalities 2q

thus showing that the smoothing effect could be measured also in the Lp norms. If we try to reformulate (12) in terms of an Lp norm, we see that it would follow from ke i t f k

The Schrödinger Equation

2n n2

2  q  1 if

if

n  2;

n D1;

q ¤1;

(18) (19)

the index p varies accordingly between 1 and 2 (between 4 and 2 in one space dimension). Such couples (p; q) are called (Schrödinger) admissible. It is easy to visualize the admissible couples by plotting the usual diagram in the (1/p; 1/q) space (see Fig 1–3). Note that the choice (p; q) D (1; 2), the point E in the diagrams, corresponds to the conservation of the L2 norm, while at the other extreme we have for n  2 point P corresponding to the couple   2n (p; q) D 2; : (20) n2 This is usually called the endpoint and is excluded in dimension 2, allowed in dimension  3. The original Strichartz estimate corresponds to the case p D q and is denoted by S. After the proof in [22] of the general Strichartz estimates, it was not clear if the endpoint case P could be reached or not; the final answer was given in [30] where the endpoint estimates were proved to be true for all dimension larger than three. For the two-dimensional case and additional estimates, see [47]. Notice that all admissible couples can be obtained by interpolation between the endpoint and the L2 conservation, thus in a sense the endpoint contains the maximum of information concerning the Lp smoothing properties of the flow. There exist an inhomogeneous form of the estimates, which is actually equivalent to the homogeneous one, and can be stated as follows:  Z t   i(ts)   e F(s)ds . kFkL p˜ 0 L q˜0 (21)   0

L p Lq

167

168

Dispersion Phenomena in Partial Differential Equations

Dispersion Phenomena in Partial Differential Equations, Figure 3 Schrödinger admissible couples, n  3 Dispersion Phenomena in Partial Differential Equations, Figure 1 Schrödinger admissible couples, n D 1

neous global estimates are optimal since other combination of indices are excluded by scale invariance; however, the local Strichartz estimates can be slightly improved allowing for indices outside the admissible region (see [21]). We also mention that a much larger set of estimates can be deduced from the above ones by combining them with Sobolev embeddings; in the above diagrams, all couples in the region bounded by the line of admissibility, the axes and the line 1/p D 2 can be reached, while the external region is excluded by suitable counterexamples (see [30]). Strichartz Estimates for the Wave Equation

Dispersion Phenomena in Partial Differential Equations, Figure 2 Schrödinger admissible couples, n D 2

Most of the preceding section has a precise analogue for the wave equation. Since the decay rate for the dispersive estimates is t (n1)/2 for the wave equation instead of t n/2 as for the Schrödinger equation, this causes a shift n ! n 1 in the numerology of indices. A more serious difference is the loss of derivatives: this was already apparent in the original estimate by Strichartz [44], which can be written ke i tjDj f k

˜ which can be for all admissible couples (p; q) and ( p˜ ; q), chosen independently. Here p0 denotes the conjugate exponent to p. These estimates continue to be true in their local version, with the norms at both sides restricted to a fixed time interval I R. Notice that the inhomoge-

L

2(nC1) n1

. k f kH˙ 1/2 ;

n  2:

The general homogeneous Strichartz estimates for the wave equation take the form ke i tjDj f kL p L q . k f k

1 nC1 n1

˙p H

;

Dispersion Phenomena in Partial Differential Equations

now the (wave) admissible couples of indices must satisfy the scaling condition 2 n1 n1 C D p q 2

(22)

and the constraints 2  p  1, 2q

2(n  1) n3

2q1

if

n  3;

if

q¤1

(23)

n D2;

(24)

The Nonlinear Wave Equation In this section we shall illustrate, using the semilinear wave equation as a basic example, what kind of improvements can be obtained over the classical energy method via the additional information given by dispersive techniques. From the linear theory it is well known that the correct setting of the Cauchy problem for the wave equation u t t  u D F(t; x) ;

requires two initial data at t D 0: u(0; x) D u0 (x) ;

while p varies accordingly between 2 and 1 (between 4 and 1 in dimension 2). Recall that in dimension 1 the solutions of the wave equation do not decay since they can be decomposed into traveling waves. We omit the graphic representation of these sets of conditions since it is completely analogous to the Schrödinger case (apart from the shift n ! n  1). The inhomogeneous estimates may be written  Z t   i(ts)   e F(s)ds   0

nC1  1p n1

˙q LpH

. kFk

nC1  1 n1

0 ˙ p˜ L p˜ H q˜0

(25)

u(t; x) : R  Rn ! C

u t (0; x) D u1 (x) :

For this equation the energy method is particularly simple. If we multiply the equation by u t we can recast it in the form   @ t ju t j2 C jruj2 D 2
. ku1 kL 2 C kru0 kL 2 C kFkL 1 L 2 T

in terms of the homogeneous Sobolev norms kgkH˙ sq D kjDjs gkL q ;

jDjs D (1  )s/2 :

Other Equations The general procedure of Keel and Tao can be applied to a large variety of equations. For the convenience of the reader we list here the precise form of the homogeneous Strichartz estimates in some important cases. For the Klein–Gordon equation we have ke i thDi f kL p L q . k f k

1 nC2 n

where we are using the shorthand notation p

L T L q D L p (0; T; L q (Rn )) : Estimate (26) is the basic energy estimate, which is enough to prove the existence of a global solution u 2 C(R; H 1 (Rn )) \ C 1 (R; L2 (Rn )) of the linear wave equation for data in (u0 ; u1 ) 2 H 1  L2 and a forcing term in F 2 L1T L2 for all T. Uniqueness and continuous dependence on the data are simple consequences of the energy estimate. Moreover, if we differentiate the equation k times, the same computations lead to the higher order energy estimates

˙p H

with (p; q) Schrödinger admissible (recall that the rate of dispersion is t n/2 for Klein–Gordon). For the massless Dirac system we have ke i t D f kL p L q . k f k

(26)

2

˙p H

;

n D3;

T

5

H 3p

;

nD3

with (p; q) Schrödinger admissible in dimension 3.

T

. ku1 kH k Cku0 kH kC1 CkFkL 1 H k ; T

k D 0; 1; 2; : : : (27)

Consider now the nonlinear wave equation u t t  u D F(u)

with (p; q) wave admissible in dimension 3, while for the massive Dirac system the estimate takes the form ke i t(DCˇ ) f kL p L q . k f k

ku t kL 1 H k C kukL 1 H kC1

(28)

for some function F(s) sufficiently smooth. The classical approach to solve it is to look for a fixed point of the mapping u D ˚ (v), where u(t; x) solves the linearized equation u t t  u D F(v) :

(29)

169

170

Dispersion Phenomena in Partial Differential Equations

If we apply the energy estimate to the difference of two solutions u1 D ˚(v1 ) and u2 D ˚(v2 ) with the same initial data, we obtain k@ t (u1  u2 )kL 1 H k C ku1  u2 kL 1 H kC1 T

T

. kF(v1 )  F(v2 )kL 1 H k : T

Using a standard Moser-type estimate (see e. g. [25,49]) k f (v1 )  f (v2 )kH k . C(kv1 kL 1 C kv2 kL 1 )kv1  v2 kH k we arrive at the inequality k@ t (u1  u2 )kL 1 H k C ku1  u2 kL 1 H kC1 T

T

. C(kv1 kL 1 1 C kv2 k L 1 L 1 )  kv1  v2 k L 1 H k : T L T T

k In particular, writing X T D L1 T H , and noticing that

kGkL 1 H k  kGk X T  T ; T

Actually, the two difficulties are related to each other. Indeed, from the above argument one can check that the lifespan of the local solution is a function of the H k norm of the initial data. Most equations with a physical origin possess conserved quantities, which are essentially H k norms of the solution with k low, e. g. k D 0 or 1. Assume now that we can prove the local existence for data say in H 1 , on some interval [0; T0 ] with T 0 depending on the H 1 norm of the data. Since this norm is conserved, we can take T 0 as a new initial time and construct a solution on [T0 ; 2T0 ], and so on. This is the essence of the continuation method. As a consequence, the local existence of low regularity solution with the addition of suitable conservation laws can lead to global existence. We shall now apply the dispersive information to improve on the preceding result and obtain the global existence of small solution, under suitable assumptions on the nonlinear term. Consider a free wave w(t; x), solution of w t t  w D 0 ;

w(0; x) D f (x) ;

w t (0; x) D g(x) :

we obtain Recalling the dispersive estimate (10), here the following simplified version

ku1  u2 k X T . C(kv1 kL 1 1 C kv2 k L 1 L 1 )  kv1  v2 k X T  T : T L T

jw(t; x)j  C(1 C t)

n1 2

L1

norm of vj , which can be We still have to handle the controlled by the H k norm provided k > n/2. In conclusion k>

n 2

H)

ku1  u2 k X T

. C(kv1 k X T C kv2 k X T )  kv1  v2 k X T  T : At this point the philosophy of the method should be clear: if we agree to differentiate n/2 times the equation, we can prove a nonlinear estimate, with a coefficient which can be made small for T small. Thus, for T  1 the mapping v 7! u is a contraction, and we obtain a local (in time) solution of the nonlinear equation. The energy method is quite elementary, yet it has the remarkable property that it can be adapted to a huge number of equations: it can be applied to equations with variable coefficients, and even to fully nonlinear equations. An outstanding example of its flexibility is the proof of the local solvability of Einstein’s equations ([14]). The main drawbacks of the energy method are: 1) A high regularity of the data and of the solution is required; 2) The nonlinear nature of the estimate leads to a local existence result, leaving completely open the question of global existence.

k f kW N;1 ;

N>n

will be sufficient. This suggests modification of the energy norm by adding a term that incorporates the decay information. To this end, we introduce the time-dependent quantity M(t) D sup

n

0t

kD

hni 2

ku(; )kH kC1 C (1 C )

n1 2

o ku(; )kL 1 ;

:

Consider now the nonlinear Eq. (28) with initial data u0 ; u1 . We already know that a local solution exists, provided the initial data are smooth enough; we shall now try to extend this solution to a global one. Using Duhamel’s principle we rewrite the equation as Z u(t; x) D w(t; x) C 0

t

sin((t  s)jDj) F(u(s; ))ds jDj

where we are using the symbolic notations of Sect. “Other Dispersive Equations”. F being the space Fourier transform. Note that the above dispersive estimate can be applied in particular to the last term; we obtain ˇ ˇ ˇ sin((t  s)jDj) ˇ ˇ ˇ  C(1 C t  s) n1 2 kFk N;1 : F(s) W ˇ ˇ jDj

Dispersion Phenomena in Partial Differential Equations

By this inequality and the energy estimate already proved, in a few steps we arrive at the estimate n1 2

M(T) . C0 C kFkL 1 H k C (1 C T) T Z T n1 (1 C T  s) 2 kF(u(s))kW N;1 ds 0

where C0 D ku0 kW N;1 C ku1 kW N;1 and the mute constant is independent of T. By suitable nonlinear estimates of Moser type, we can bound the derivatives of the nonlinear term using only the L1 and the H k norm of u, i. e., the quantity M(t) itself. If we assume that F(u) juj for small u, we obtain

Consider the Cauchy problem iu t  u D juj ;

u(0; x) D f (x)

(31)

for some  1. We want to study the local and global solvability in L2 of this Cauchy problem. Recall that this is a difficult question, still not completely solved; however, at least in the subcritical range, a combination of dispersive techniques and the contraction mapping principle gives the desired result easily. Instead of Eq. (31), we are actually going to solve the corresponding integral equation Z t e i(ts) ju(s)j ds : u(t; x) D e i t f C i 0

n1

M(T) . C0 C M(T) (1 C T) 2 Z T n1 n1 (1 C T  s) 2 (1 C s) 2 ( 2) ds : 0

Our standard approach is to look for a fixed point of the mapping u D ˚(v) defined by Z t ˚(v) D e i t f C i e i(ts) jv(s)j ds ; 0

Now we have Z

t

(1 C t  )

n1 2

 (1 C )

n1 2 ( 2)

d

0

 C(1 C t)

n1 2

k˚(v)kL p L q . k f kL 2 C kj˚(v)j kL p˜ 0 L q˜0

provided

> 2C

it is easy to check, using Strichartz estimates, that the mapping ˚ is well defined, provided the function v is in the suitable L p L q . Indeed we can write, for all admissible cou˜ ples (p; q) and ( p˜; q)

2 n1

so that, for such values of , we obtain M(T) . C0 CM(T) : For small initial data, which means small values of the constant C0 , (30) implies that M(T) is uniformly bounded as long as the smooth solution exists. By a simple continuation argument, this implies that the solution is in fact global, provided the initial data are small enough.

and by Hölder’s inequality this implies  0 0 L  p˜ L  q˜

k˚(v)kL p L q  C1 k f kL 2 C C1 kvk

:

(32)

Thus, we see that the mapping ˚ operates on the space L p L q , provided we can satisfy the following set of conditions: (30) p˜0 D p ; q˜0 D q; 2 n n n n 2˜ C D ; C D ; ˜ p q˜ p q 2 2

 2n p; p˜ 2 [2; 1] ; q˜ 2 2; : n2 Notice that from the first two identities we have

The Nonlinear Schrödinger Equation Strichartz estimates for the Schrödinger equation are expressed entirely in terms of Lp norms, without loss of derivatives, in contrast with the case of the wave equations. For this reason they represent a perfect tool to handle semilinear perturbations. Using Strichartz estimates, several important results can be proved with minimal effort. As an example, we chose to illustrate in this Section the critical and subcritical well posedness in L2 in the case of power nonlinearities (see [13] for a systematic treatment).

n 2

n

2 C D˛ Cn p˜ q˜ p q and by the admissibility conditions this implies n n

D2Cn ; 2 2 which forces the value of to be

D1C

4 : n

This value of is called L2 critical.

(33)

171

172

Dispersion Phenomena in Partial Differential Equations

Let us assume for the moment that is exactly the critical exponent (33). Then it is easy to check that there exist ˜ which satisfy all the above conchoices of (p; q) and ( p˜ ; q) straints. To prove that ˚ is a contraction on the space

p˜0 < p ;

we take any v1 ; v2 2 X and compute   k˚(v1 )  ˚(v2 )k X . jv1 j  jv2 j  L p˜ 0 L q˜0   . jv1  v2 j(jv1 j 1 C jv2 j 1 )

0

L p˜ L q˜

 1

k˚ (v1 )  ˚(v2 )k X  C2 kv1  v2 k X kjv1 j C jv2 jk X

0

;

(34)

for some constant C2 . Now assume that

 C2 T  kv1  v2 k X T kjv1 j C jv2 jkX1 : T

provided "; ı are so small that

k f kL 2 <

On the other hand, by (34) we get 1 kv1 v2 k X 2

provided " is so small that 1 : 2

In conclusion, if the initial data are small enough, the mapping ˚ is a contraction on a ball B(0; ") X and the unique fixed point is a global small solution of the critical L2 equation 4

iu t  u D juj1C n : Notice that in estimate (32) the first couple can also be taken equal to (1; 2), thus the solution we have constructed belongs to L1 L2 . A limit argument (dominated convergence) in the identity t

(35)

(36)

Now let M be so large and T so small that

" C1 " < : 2 

Z

and proceeding as above and using Hölder inequality in time, we obtain the inequality. Indeed, choosing the indices as above and applying Hölder’s inequality in time we have

k˚ (v1 )  ˚(v2 )k X T

k˚(v)k X  C1 (ı C " ) < "

u D e i t f C i

q˜0 D q

for some strictly positive  > 0. Analogously, estimate (34) will be replaced by

using (32) we can write

C2 (2") 1 <

p˜0 < p ;

k˚(v)k X T  C1 k f kL 2 C C1 T  kvk X T

k f kL 2 < ı ;

k˚(v1 )˚(v2 )k X  C2 kv1 v2 k X (2") 1 <

q˜0 < q :

˜ so that It is easy to check that we can shift the point ( p˜ ; q)

the same computation as above gives

" C1 ı < ; 2

p

X T D L T L q D L p (0; T; L q (Rn )) : If we follow the steps of the preceding proof and choose the same indices, we have now

X D L p Lq

kv i k X < " ;

We briefly sketch a proof of the local existence in the subcritical case 1 < < 1 C n4 . Consider the local space

e i(ts) ju(s)j ds ;

0

proves that we have actually u 2 C(R; L2 (Rn )).

M ; 2C1

C1 T  <

M 2

we obtain from (35) that ˚ takes the ball B(0; M) X T into itself; if in addition we have also, by possibly taking T smaller, C2 T  (2M) 1 <

1 2

we see by (36) that ˚ is a contraction, and in conclusion we obtain the existence of a unique local solution to the subcritical equation. By the same argument as above this solution is in C([0; T]; L2 (Rn )). We notice that in the above proof the quantity M is only determined by the L2 norm of f , and hence also the lifespan of the local solution is a function of k f kL 2 . If for some reason we know a priori that the L2 norm of the solution is conserved by the evolution, this remark implies immediately that the solution is in fact global, by an elementary continuation argument. This is the case for instance of the gauge invariant equation iu t  u D ˙juj 1 u for which any solution continuous with values in L2 satisfies ku(t)kL 2 D ku(0)kL 2 . The above computations apply without modification and we obtain the global well posedness for all subcritical exponents .

Dispersion Phenomena in Partial Differential Equations

Future Directions The dispersive properties of constant coefficient equations are now fairly well understood. In view of the wide range of applications, it would be very useful to extend the techniques to more general classes of equations, in particular for equations with variable coefficients and on manifolds. These problems have been the subject of intense interest in recent years and are actively pursued. Strichartz estimates have been extended to very general situations, including the fully variable coefficients case, under suitable smoothness, decay and spectral assumptions on the coefficients (see e. g. [36,37] and, for singular coefficients, [18]). The results concerning dispersive estimates are much less complete. They have been proved for perturbations of order zero iu t  u C V(x)u D 0 ;

u t t  u C V (x)u D 0

with a potential V(x) sufficiently smooth and decaying at infinity (see e. g. [2,19,28,53,54,55]) while for perturbation of order one iu t  u C a(x)  ru x j C V(x)u D 0 ; u t t  u C a(x)  ru x j C V(x)u D 0 very few results are available, (see [16] for the threedimensional wave and Dirac equations, and [1,17] for the one-dimensional case where quite general results are available). A closely connected problem is the study of the dispersive properties of equations on manifolds. This part of the theory is advancing rapidly and will probably be a very interesting direction of research in the next years (see e. g. [12] and related papers for the case of compact manifolds, and [5,24] for noncompact manifolds). Bibliography 1. Artbazar G, Yajima K (2000) The Lp -continuity of wave operators for one dimensional Schrödinger operators. J Math Sci Univ Tokyo 7(2):221–240 2. Beals M (1994) Optimal L1 decay for solutions to the wave equation with a potential. Comm Partial Diff Eq 19(7/8):1319– 1369 3. Ben-Artzi M, Trèves F (1994) Uniform estimates for a class of evolution equations. J Funct Anal 120(2):264–299 4. Bergh J, Löfström J (1976) Interpolation spaces. An introduction. Grundlehren der mathematischen Wissenschaften, No 223. Springer, Berlin 5. Bouclet J-M, Tzvetkov N (2006) On global strichartz estimates for non trapping metrics. 6. Bourgain J (1993) Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom Funct Anal 3(2):107–156

7. Bourgain J (1995) Some new estimates on oscillatory integrals. In: (1991) Essays on Fourier analysis in honor of Elias M. Stein. Princeton Math Ser vol. 42. Princeton Univ Press, Princeton, pp 83–112 8. Bourgain J (1995) Estimates for cone multipliers. In: Geometric aspects of functional analysis. Oper Theory Adv Appl, vol 77. Birkhäuser, Basel, pp 41–60 9. Bourgain J (1998) Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity. Int Math Res Not 5(5):253–283 10. Brenner P (1975) On Lp  Lp0 estimates for the wave-equation. Math Z 145(3):251–254 11. Brenner P (1977) Lp  Lp0 -estimates for Fourier integral operators related to hyperbolic equations. Math Z 152(3):273–286 12. Burq N, Gérard P, Tzvetkov N (2003) The Cauchy problem for the nonlinear Schrödinger equation on a compact manifold. J Nonlinear Math Phys 10(1):12–27 13. Cazenave T (2003) Semilinear Schrödinger equations. Courant Lecture Notes in Mathematics, vol 10. New York University Courant Institute of Mathematical Sciences, New York 14. Choquet-Bruhat Y (1950) Théorème d’existence pour les équations de la gravitation einsteinienne dans le cas non analytique. CR Acad Sci Paris 230:618–620 15. Christodoulou D, Klainerman S (1989) The nonlinear stability of the Minkowski metric in general relativity. In: Bordeaux (1988) Nonlinear hyperbolic problems. Lecture Notes in Math, vol 1402. Springer, Berlin, pp 128–145 16. D’Ancona P, Fanelli L (2006) Decay estimates for the wave and dirac equations with a magnetic potential. Comm Pure Appl Anal 29:309–323 17. D’Ancona P, Fanelli L (2006) Lp – boundedness of the wave operator for the one dimensional Schrödinger operator. Comm Math Phys 268:415–438 18. D’Ancona P, Fanelli L (2008) Strichartz and smoothing estimates for dispersive equations with magnetic potentials. Comm Partial Diff Eq 33(6):1082–1112 19. D’Ancona P, Pierfelice V (2005) On the wave equation with a large rough potential. J Funct Anal 227(1):30–77 20. D’Ancona P, Georgiev V, Kubo H (2001) Weighted decay estimates for the wave equation. J Diff Eq 177(1):146–208 21. Foschi D (2005) Inhomogeneous Strichartz estimates. J Hyperbolic Diff Eq 2(1):1–24 22. Ginibre J, Velo G (1985) The global Cauchy problem for the nonlinear Schrödinger equation revisited. Ann Inst Poincaré H Anal Non Linéaire 2(4):309–327 23. Ginibre J, Velo G (1995) Generalized Strichartz inequalities for the wave equation. J Funct Anal 133(1):50–68 24. Hassell A, Tao T, Wunsch J (2006) Sharp Strichartz estimates on nontrapping asymptotically conic manifolds. Am J Math 128(4):963–1024 25. Hörmander L (1997) Lectures on nonlinear hyperbolic differential equations. Mathématiques & Applications (Mathematics & Applications), vol 26. Springer, Berlin 26. John F (1979) Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscr Math 28(1–3):235– 268 27. John F, Klainerman S (1984) Almost global existence to nonlinear wave equations in three space dimensions. Comm Pure Appl Math 37(4):443–455 28. Journé J-L, Soffer A, Sogge CD (1991) Decay estimates for Schrödinger operators. Comm Pure Appl Math 44(5):573–604

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43. Shatah J, Struwe M (1998) Geometric wave equations. In: Courant Lecture Notes in Mathematics, vol 2. New York University Courant Institute of Mathematical Sciences, New York 44. Strichartz RS (1970) Convolutions with kernels having singularities on a sphere. Trans Am Math Soc 148:461–471 45. Strichartz RS (1970) A priori estimates for the wave equation and some applications. J Funct Anal 5:218–235 46. Strichartz RS (1977) Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations. J Duke Math 44(3):705–714 47. Tao T (2000) Spherically averaged endpoint Strichartz estimates for the two-dimensional Schrödinger equation. Comm Part Diff Eq 25(7–8):1471–1485 48. Tao T (2003) Local well-posedness of the Yang–Mills equation in the temporal gauge below the energy norm. J Diff Eq 189(2):366–382 49. Taylor ME (1991) Pseudodifferential operators and nonlinear PDE. Progr Math 100. Birkhäuser, Boston 50. von Wahl W (1970) Über die klassische Lösbarkeit des CauchyProblems für nichtlineare Wellengleichungen bei kleinen Anfangswerten und das asymptotische Verhalten der Lösungen. Math Z 114:281–299 (in German) 51. Wolff T (2001) A sharp bilinear cone restriction estimate. Ann Math (2) 153(3):661–698 52. Yajima K (1987) Existence of solutions for Schrödinger evolution equations. Comm Math Phys 110(3):415–426 53. Yajima K (1995) The W k;p -continuity of wave operators for Schrödinger operators. J Math Soc Japan 47(3):551–581 54. Yajima K (1995) The W k;p -continuity of wave operators for schrödinger operators. III. even-dimensional cases m  4. J Math Sci Univ Tokyo 2(2):311–346 55. Yajima K (1999) Lp -boundedness of wave operators for two-dimensional schrödinger operators. Comm Math Phys 208(1):125–152

Dynamics on Fractals

˜

Dynamics on Fractals RAYMOND L. ORBACH Department of Physics, University of California, Riverside, USA Article Outline Glossary Definition of the Subject Introduction Fractal and Spectral Dimensions Nature of Dynamics on Fractals – Localization Mapping of Physical Systems onto Fractal Structures Relaxation Dynamics on Fractal Structures Transport on Fractal Structures Future Directions Bibliography Glossary Fractal Fractal structures are of two types: deterministic fractals and random fractals. The former involves repeated applications of replacing a given structural element by the structure itself. The process proceeds indefinitely, leading to dilation symmetry: if we magnify part of the structure, the enlarged portion looks the same as the original. Examples are the Mandelbrot– Given fractal and the Sierpinski gasket. A random fractal obeys the same properties (e. g. dilation symmetry), but only in terms of an ensemble average. The stereotypical example is the percolating network. Fractals can be constructed in any dimension, d, with for example, d D 6 being the mean-field dimension for percolating networks. Fractal dimension The fractal dimension represents the “mass” dependence upon length scale (measuring length). It is symbolized by Df , with the number of sites on a fractal as a function of the measurement length L being proportional to L D f , in analogy to a homogeneous structure embedded in a dimension d having mass proportional to the volume spanned by a length L proportional to Ld . Spectral (or fracton) dimension The spectral (or fracton) dimension refers to the dynamical properties of fractal networks. It is symbolized by d˜s and can be most easily thought of in terms of the density of states of a dynamical fractal structure (e. g. vibrations of a fractal network). Thus, if the excitation spectrum is measured as a function of energy !, the density of states for excitations of a fractal network would be propor-

tional to ! (d s 1) , in analogy to a homogeneous structure embedded in a dimension d having a density of states proportional to ! (d1) . Localization exponent Excitations on a fractal network are in general strongly localized in the sense that wave functions fall off more rapidly than a simple exponential,  (r) exp[f[r/(!)]d  g] where (!) is an energy dependent localization length, and the exponent d is in general greater than unity. Definition of the Subject The dynamical properties of fractal networks are very different from homogeneous structures, dependent upon strongly localized excitations. The thermal properties of fractals depend upon a “spectral dimension” d˜s less than the “Euclidean” or embedding dimension d. Fractal randomness introduces statistical properties into relaxation dynamics. Transport takes place via hopping processes, reminiscent of Mott’s variable range-rate hopping transport for electronic impurity states in semi-conductors. Fractal dynamics serve as a guide for the behavior of random structures where the short length scale excitations are localized. Introduction The study of the dynamics on and of fractal networks [29, 30,31] is not an arcane investigation, with little application to real physical systems. Rather, there are many examples of real materials that exhibit fractal dynamics. Examples are the vibrational properties of silica aerogels [21,45], and magnetic excitations in diluted antiferromagnets [18,19,43]. Beyond these explicit physical realizations, one can learn from the very nature of the excitations on fractal structures how localized excitations in homogeneous materials behave. That is, the dynamics of fractal networks serve as a calculable model for transport of localized excitations in random structures that are certainly not mass fractals. Examples are thermal transport in glasses above the so-called “plateau” temperature [2,20,26], and the lifetime of high-energy lattice vibrations in a-Si [32,33,36,37]. As we shall show in subsequent sections, the former is an example of vibrational hopping-type transport (similar in nature to Mott’s variable-range hopping [22,23] for localized electronic states), while the latter exhibits counter-intuitive energy dependences for vibrational lifetimes. The following Sect. “Fractal and Spectral Dimensions” describes in brief terms the concept of the fractal (or mass) dimension, Df [7,8,9,10,15,40,41,46], and the spectral (or fracton) dimension d˜s [5,14,17,34,35]. The latter is re-

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lated to the anomalous diffusion characteristics of fractals (the so-called “ant in a labyrinth” introduced by de Gennes [8]), and is at the heart of the thermal properties of fractal structures. A conjecture introduced by Alexander and Orbach [5] suggested that the mean-field value (exact at d D 6) for d˜s for percolating networks, d˜s D 4/3, might be universal, independent of dimension. It is now generally regarded that this conjecture is only approximate, though a very good approximation for percolating networks in 2 D d < 6. Were it to be exact, it would mean that the dynamical properties of percolating networks could be expressed in terms of their geometrical properties. Section “Nature of Dynamics on Fractals–Localization” introduces the nature of the wave function for an excitation embedded in a fractal structure. In general, random self-similar structures generate “super-localized” wave functions, falling off faster than exponential. We will characterize the envelope of the wave function by the functional form [1]  (r) exp[f[r/(!)]d  g] ; where (!) is an energy-dependent localization length, and the exponent d is in general greater than unity. For excitations on fractal structures, the localization length (!) can be calculated analytically, allowing for explicit computation of physical phenomena. This allows analytic expressions for scattering lifetimes and excitation transport. Mapping of the diffusion problem onto the secular equation for scalar lattice vibrations allows the extraction of the density of vibrational states [5]. It is given ˜ by the simple relationship D(!) ! (d s 1) displaying the utility of the spectral dimension d˜s . A similar though slightly more complicated mapping can be applied to diluted Heisenberg antiferromagnets. The silica aerogels [13,16,42,44,45], and Mnx Zn1x F2 [43] and RbMnx Mg1x F3 [18,19] are realizations of such systems, respectively, and have been extensively investigated experimentally. The random nature of fractal structures introduces statistical complexity into relaxation processes. The direct (single vibration) [38] and Raman (scattering of two vibrations) [24,39] spin-lattice relaxation processes are discussed for random fractal networks in Sect. “Relaxation Dynamics on Fractal Structures”. The theoretical results are complex, related to the “Devil’s staircase” for Raman relaxation processes. Experiments probing these rather bizarre behaviors would be welcome. Given the “super-localization” of excitations on fractal networks [1,42], transport properties need to be ex-

pressed according to the same concepts as introduced by Mott [22,23] for the insulating phase of doped semiconductors—the so-called “variable range-rate hopping”. This is developed in Sect. “Transport on Fractal Structures”, and applied to the thermal transport of fractal structures [2,20,26]. The “lessons learned” from these applications suggest an interpretation of the thermal transport of glassy materials above the so-called “plateau” temperature range. This instance of taking the insights from fractal structures, and applying it to materials that are certainly not fractal yet possess localized vibrational states, is an example of the utility of the analysis of fractal dynamics. Finally, some suggestions, hinted at above, for future experimental investigations are briefly discussed in Sect. “Future Directions”. Fractal and Spectral Dimensions Fractals can be defined by the nature of the symmetry they exhibit: self-similar geometry or equivalently dilation symmetry [7,8,9,10,15,40,41,46]. This symmetry is most assuredly not translational, leading to a very different set of behaviors for physical realizations that shall be explored in subsequent Sections. Stated most simply, self-similar structures “look the same” on any length scale. Conversely, one cannot extract a length scale from observation of a self-similar structure. There are clearly limits to self-similarity for physical realizations. For example, at the atomic level, dilation symmetry must come to an end. And for finite structures, there will be a largest length scale. There are other length scales that bracket the self-similar regime. Thus, for percolating networks, an example of a random fractal [38,39], the percolation correlation length, (p), where p is the concentration of occupied sites or bonds, defines an upper limit to the regime of fractal geometry. In general, for percolating networks, the range of length scales ` over which the structure is fractal is sandwiched between the atomic length a, and the percolation correlation length (p) : a < ` < (p). Fractal structures can be deterministic or random. The former is a result of applying a structural rule indefinitely, replacing an element of a design by the design itself. Examples are the Mandelbrot–Given fractal and the Sierpinski gasket. The latter arises from a set of probabilistic rules. A simple example is the percolating network where a site or bond (e. g. in d D 2, a square grid) is occupied randomly with probability p. The resulting clusters contain a random number of sites or bonds, where the distribution function for the number of finite clusters of size s is denoted by ns (p). There exists a critical value p D pc where a connected cluster extends over all space (the “in-

Dynamics on Fractals

finite” cluster). For p < pc only finite clusters exist, with the largest cluster spanned by the length scale (p). (p) diverges at p D pc as jp  pc j . For p > pc finite clusters and the infinite cluster coexist, the number of finite clusters vanishing at p approaches unity. The probability that a site belongs to the infinite cluster varies with occupation probability p as (p  pc )ˇ . The characteristic length (p) for p > pc is a measure of the largest finite cluster size. Hence, for length scales ` > (p) the systems looks homogeneous. Thus, the fractal regime is sandwiched between the lattice constant a and the percolation correlation length (p). The remarkable utility of the percolating network becomes clear from these properties: there is a crossover length scale between fractal and homogeneous structural behavior that can be chosen at will by choosing a value for the site or bond occupation probability p > pc . Excitations of structures that map onto percolating networks can change their nature from fractal to continuous as a function of their length scale `, the transition occurring when `c (p). This property makes the percolating network the “fruit fly” of random structures. The flexibility of adjusting the crossover length `c enables a fit between those properties calculated for a percolating network with the properties of those physical systems that map onto percolating networks. The fractal dimension, Df , is defined from the dependence of the “mass” on length scale. For fractals, the number of points or bonds on the infinite cluster within a length scale ` (hence, the “mass”), for `  (p), depends upon ` as M(`) ` D f . From above it is straight forward to show Df D d  (ˇ/). For a percolation network [29] in d D 2; Df D 91/48. For d D 3; Df D 2:48 ˙ 0:09. In mean field, d D 6; Df D 4. The spectral dimension, d˜s , follows from the analysis of diffusion on a fractal network, as first postulated by de Gennes [14]: “An ant parachutes down onto an occupied site of the infinite cluster of a percolating network. At every time unit, the ant makes one attempt to jump to one of its adjacent sites. If that site is occupied, it moves there. If it is empty, the ant stays at its original site. What is the ensemble-averaged square distance that the ant travels in time t?” Gefen et al. [17] found that hr2 (t)i t 2C where depends upon the scaling exponent  (conductivity exponent), the probability that a site belongs to the infinite cluster scaling exponent ˇ, and the scaling exponent  for (p) : D (  ˇ)/. Alexander and Orbach [5] defined the spectral (or fracton) dimension as d˜s D 2Df /(2 C ). The importance of d˜s can be seen from the calculation of the probability of finding a diffusing particle at the starting point at time t, P0 (t), which for compact diffusion is the inverse of the number of visited sites in

˜

time t, V(t) t (d s /2) . For scalar elasticity, the vibrational density of states ˜

D(!) D (2!/)Im P˜0 (! 2 C i0C ) ! (d s 1) ; where P˜0 (!) is the Laplace transform of P0 (t). Noting that for homogeneous systems, D(!) ! (d1) , the name spectral dimension for d˜s becomes evident. Alexander and Orbach [5] named the vibrational excitations of fractal structures, fractons, in analogy with the vibrational excitations of homogeneous structures, phonons. Hence, they termed d˜s the fracton dimension. Alexander and Orbach noted that for percolation structures in the mean-field limit, d D 6, d˜s D 4/3 precisely. At that time (1982), values for d˜s for d < 6 appeared close to that value, and they conjectured that d˜s D 4/3 for 2  d  6. We now know that this is only approximate (but remarkably close [24]): d˜s D 1:325 ˙ 0:002 in d D 2, and 1:317 ˙ 0:003 in d D 3. Were the conjecture exact, a numerical relationship between structure and transport would be exact [ D (3Df /2)  2], dictated solely by fractal geometry. For percolating networks, the structure appears fractal for length scales `  (p) and homogeneous for `  (p). The vibrational excitations in the fractal regime are termed fractons, and in the homogeneous regime phonons. We define the crossover frequency !c as that for which the excitation length scale equals (p). Then !c (p  ˜ pc )( D f /d s ) . Continuity leads to a phonon velocity v(p) (p  pc ) [(Df /d˜s )  1]. This then leads to the dispersion relations: [`  (p); !  !c ; phonon regime]! v(p)k ; and ˜

[`  (p); !  !c ; fracton regime]! k (D f /d s ) : In the latter case, as we shall show in the next Section, vibrational excitations in the fractal regime are localized, so that k should not be thought of as a wave vector but rather the inverse of the localization length (!). Nature of Dynamics on Fractals – Localization The conductance of a d dimensional percolating network of size L; G(L), can be shown to be proportional to Lˇ where the exponent ˇ D (Df /d˜s )(d˜s  2). In the Anderson sense [6], localization occurs when ˇ  0 (marginal at ˇ D 0). The Alexander–Orbach conjecture [5], d˜s  4/3, leads to ˇ well less than zero for all embedding dimensions d. One can think of this form of localization as geometrical, as opposed to the scattering localization that traditionally is associated with Anderson localization. The lo-

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calization is strong, the wave function falling off faster than simple exponential. The wave function has the form [1]: h (r)i exp[ f[r/(!)]d  g]. The exponent d is a geometrical exponent describing the fact that an exponential decay along the fractal will be distorted when viewed in real space. In general, 1  d  dmin , where dmin is defined by ` R d min , with ` being the shortest path along the network between two points separated by a Pythagorean distance R. Bunde and Roman [11] have found that for percolating networks d D 1. The decay length (!) is the localization length ˜ from Sect. “Introduction”, (!) ! (d s /D f ) . For a random system such as a percolating network, a given realization of  (r) is badly behaved, depending upon the particular choice of origin taken for the position coordinate r. The ensemble average takes the average over all realizations of  (r), and is denoted by h (r)i. This has its dangers, as the calculation of physical properties should be performed using a specific realization for  (r), and then ensemble averaged. Calculation of physical properties utilizing an ensemble averaged wave function h (r)i will not yield the same result, and could be very misleading. Specific realizations for  (r) for two-dimensional bond percolation vibrational networks have been calculated by Nakayama and Yakubo [25,28,47]. The fracton “core” (or largest amplitude) possesses very clear boundaries for the edges of the excitation, with an almost steplike character and a long tail. The tail extends over large distances and the amplitude oscillates in sign. This is required for orthogonality to the uniform translational mode with ! D 0. Clearly these modes do not look like simple exponentials, hence the warning about the use of h (r)i as compared to  (r) for the calculation of matrix elements. Mapping of Physical Systems onto Fractal Structures The foregoing description of dynamics of fractal networks would be interesting in and of itself, but its applicability to physical systems makes it of practical importance. Further, the nature of localization, and computation of “hopping-type” transport on fractal structures, has application to materials that are not fractal. Thus, it is worthwhile to consider what systems are fractal, the nature of dynamics on such systems, and the lessons we can learn from them for structures that possess similar characteristics but are not fractal. Specifically, Mott’s [22,23] variable range-rate hopping for localized electronic impurity states has direct application to transport of vibrational energy via fracton hopping on fractal networks [2], and we shall argue to

thermal transport in glassy materials above the “plateau” temperature [20,26]. The fundamental equations for fractal dynamics spring from the equation for diffusion of a random walker P on a network: dPi /dt D j¤I w i j (P j  Pi ) where Pi is the probability that the ith site is occupied, wij is the probability per unit time for hopping from site i to site j, and w i j D w ji . Introducing new quantities Wi j D w i j for P i ¤ j, and Wi i D  j¤I w i j , one can rewrite the difP fusion equation as dPi /dt D j Wi j P j with the defined P relation j Wi j D 0. For scalar elasticity, the equation of motion for atomic P vibrations is d2 u i /dt 2 D j K i j u j , with ui the atomic displacement at position i, and K i j (i ¤ j) is given by K i j D k i j /m i with kij the force constant connecting two sites i and j, and mi the mass of the ith site. The diagonal element W ii is obtained from the balancing of forces at P the ith site: j K i j D 0. The only difference between these two equations is the order of the time derivatives, leading to the equivalence of the eigenvalue problems. We have used this equivalence to extract the density of vibrational states for scalar elasticity, and the definition of the spectral (or fracton) dimension in Sect. “Introduction”. The most direct physical example of a vibrating fractal network is the silica aerogel [13,16,42,44]. Different preparation conditions can change the density of these materials to very small values, indicative of the open porous structure of these materials. Two length scales are present: the fundamental building block of the structure (the silica), and the correlation length of the gel. In between, the clusters possess a fractal structure, while at length scales beyond the correlation length the gel behaves as a homogeneous porous glass. This crossover from fractal to homogeneous is the physical example of our previous discussion for percolating networks when the length scale of excitations passes through the percolation correlation length (p). The thermal, transport, and scattering properties of aerogels have been well studied, and are the classic example of fractal behavior. There are other examples of structures that map onto the eigenvalue spectrum of the diffusion equation. Ferromagnetic and antiferromagnetic materials can be diluted by non-magnetic impurities. This leads to random site dilution, mapping directly onto the site-diluted percolating network for magnetic interactions. This can be seen from the Heisenberg Hamiltonian for spin systems: P H D 1/2 i; j J i j S i S j , where S i is the vector spin at the ith site, and J ij the isotropic exchange interaction between sites i and j. The equation of motion for the spin operx x ator S C i D S i C iS i becomes for diluted ferromagnets

Dynamics on Fractals

P z C z C i„@S C j¤i J i j (S i S j  S j S i ). For low levels of i /@t D spin wave excitations, S zi  S zj  S, so that one obP tains the linear equation of motion i„@S C j¤I J i j i /@t D C C (S j  S i ). This is the same form as that for diffusion of a random walker on a network, so that, in analogy with the scalar vibrational network, the eigenvalue spectrum is the same. This mapping allows the spin wave spectrum to exhibit magnons and fracton waves obeying the same properties that phonons and fractons exhibit for scalar vibrations. For antiferromagnets, the change in sign of S zi for the two sublattices complicates the linearized equations of P C C motion. They become, i„@S C j¤I J i j (S j CS i ), i /@t D  i where  i D 1 for down spins and  i D 1 for up spins. This alters the dynamical equations for diluted antiferromagnets, and requires a separate calculation of the eigenvalue spectrum. Neutron diffraction experiments exhibit both conventional magnon and fracton excitations. In the case of the site-diluted antiferromagnet Mnx Zn1x F2 both excitations are observed simultaneously for length scales in the vicinity of (p) [43]. Relaxation Dynamics on Fractal Structures Electron spin resonance and non-radiative decay experiments can be performed in fractal materials. It is of interest to compare these dynamical properties with those found in translationally ordered structures. At first sight, for vibrational relaxation, it might be thought that the only difference would originate with the differences in the densities of vibrational states D(!). However, the localized nature of the vibrational states (fractons) and their random positions in the fractal network, introduces properties that are specific to fractal lattices. There are two cases of interest: one-fracton relaxation, and two-fracton inelastic scattering. The former is referred to as the “direct” relaxation process; the latter the “Raman” relaxation process. In both cases, the main effect of vibrational localization is on the relaxation time profile. Different spatial sites can have very different relaxation rates because of the randomness of their distance from the appropriate fracton sites. The result is a strongly non-exponential time decay. For one-fracton relaxation [1], a straightforward generalization of the direct relaxation process rate, W(!0 ; L), for relaxation energy ! 0 arising from interaction with a fracton of the same energy centered a distance L away, is proportional to 2q1

W(!0 ; L) !0

f[(!0 )](D f ) g

 [coth(ˇ!0 /2)](1/ıL ) expf[L/(!0 )]d  g :

Here, ˇ D 1/kB T and the factor ı L represents the energy width of the fracton state. In the homogeneous limit, where phonons represent the extended vibrational states, an energy conserving delta function would replace ı L . There are two limits for ı L , case (a) when the fracton relaxation rate ı caused by anharmonicity dominates; and case (b) when the electronic relaxation rate caused by the electron-fracton interaction is greater than ı. In the former case, ıL D ı; while in the latter, ıL D W(!0 ; L) itself, leading to a self-consistent determination of the relaxation rate. The latter case requires explicit consideration of the L dependence of ı L , further complicating the calculation of W(!0 ; L). The calculation of the time profile of the electronic state population and the average relaxation rate are complex, and the reader is referred to [32] for details. For case (a), the population of the initial electronic state is found to vary as h i [(D f /d  )1] g : (ln t)[(D f d  )/2d  ] t fc 1 (ln t) Here, c1 is a constant independent of time, but dependent upon ! 0 and ı. The population of the initial electronic state decays is thus found to be faster than a power law, but slower than exponential or stretched exponential. For case (b), the population of the initial electronic state is found to vary as n o (1/t) (ln t)[(D f /d  )1] : The decay is slower than in case (a), and is closer to a power law. The average relaxation rates for cases (a) and (b) have the same functional form: hWi D(!0 )[(!0 )(2q1) ] coth(ˇ!0 /2) where q D d˜s (d /Df ). The temperature dependence is the usual one found for the direct process, but the energy dependence differs, depending on the values of the dimensions and parameters for fractal networks. Two fracton relaxation is considerably more complex [3,4]. In summary, the time profile of the initial electronic state in the long time regime begins as a stretched exponential, then crosses over into a form similar to case (a) for the one fracton decay process. In the presence of rapid electronic cross relaxation, the time profile is exponential, with a low-temperature relaxation time h1/T1 i ˜ proportional to T f2d s [1C2(d  /D f )]1g for Kramers transitions (between half-integer time-reversed spin states), and ˜ T f2d s [1C2(d  /D f )]3g for non-Kramers transitions (between integer time-reversed spin states). The complexity of random systems adds an interesting consideration: the nature of the relaxation rate at a specific site as compared to the average relaxation rate calculated above. The decay profile of single-site spin-lattice

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relaxation is always exponential. However, the temperature dependence does not simply follow that of the average relaxation rate. Instead, it exhibits irregular statistical fluctuations reflecting the environment of the chosen site. As the temperature is raised, new relaxation channels, involving higher-frequency localized vibrations, will become activated, adding to the relaxation processes responsible for the relaxation at low temperatures. The temperature dependence of the relaxation rate W(T) at different sites will fluctuate with strong correlations in the temperature dependence between them, underlying large accumulative fluctuations. Using a step function for the vibrational Bose functions, the relaxation rate has a T 2 temperature dependence between randomly spaced steps that occur when new relaxation channels are opened: a “devil’s staircase” structure. The full Bose function smoothes out these steps, but the qualitative features remain. All of these features point to the complexity and richness of electronic relaxation associated with localized vibrational spectra. Their remarkable complexity is worthy of investigation. Transport on Fractal Structures The localization of excitations discussed in Sect. “Fractal and Spectral Dimensions” has profound influence on the transport properties of and on fractal structures. For example, in the case of the silica aerogels, the “conventional” form for glasses or amorphous materials for the thermal conductivity (T) is found [12]: a rapid rise at low temperatures leveling off to a “plateau” as a function of temperature T, and then a rise in (T) roughly as T 2 . At first sight, this is to be “expected” as the aerogels are random, and one could make an analogy with glassy systems. However, there is a fundamental inconsistency in such an analysis. The fracton excitations are localized, and therefore cannot contribute to transport. Thus, one would expect instead a rapid rise in (T) at low temperatures where the long length scale excitations are phonon-like (and hence delocalized), flattening off to a constant value when all the extended state density of states have reached their duLong– Petit value. The specific heat for these states is then a constant, and (T) would be a constant independent of temperature T. So far, this behavior is consistent with experiment. But what happens for temperatures T greater than the plateau temperature region? How can (T) increase when the only excitations are fractons and therefore localized? The solution to this conundrum lies in the anharmonic forces connecting the silica clusters in the aerogel network. The

presence of the anharmonicity allows for a process of fracton hopping [2,20,26], in precisely the same fashion as in Mott’s [22,23] variable range-rate hopping for localized electronic impurity states in semiconductors. The fracton can absorb/emit a lower frequency extended state phonon, and hop to another fracton location, thereby transmitting excitation energy spatially. This possibility is amusing, for it means that for fractal networks the anharmonicity contributes to thermal transport, whereas in translationally invariant systems the anharmonicity inhibits thermal transport. In fact, were there no anharmonicity in a fractal network, the thermal conductivity (T) would stay at the plateau value for all T greater than the temperature T p , the onset of the plateau, for all T > Tp . In this sense, anharmonicity stands on its head. It is essential for transport in the fracton regime, defined by T  Tp  „!c /kB . The localized vibrational excitation hopping contribution to the thermal conductivity is hop (T) D (kB /2V)˙0 R2 (!0 )/sl (!0 ; T). Here sl means strongly localized modes, R(!0 ) is the hopping distance associated with the sl mode, 0 ; sl (!0 ; T) the hopping lifetime of the sl mode 0 caused by anharmonic interactions at temperature T, and V the volume of the system. We have already derived the ensemble averaged wave function for fractons, h (r)i, in Sect. “Fractal and Spectral Dimensions”. We adopt the sl notation [26] because the hopping transport arguments apply to strongly localized states, independent of whether or not they are fractons. The dynamical process for hot (T) is an sl mode at one site coupling with a low energy extended state to hop to an sl mode at another site, thereby transferring excitation energy spatially. The evaluation of sl (!0 ; T) will depend upon the hopping distance for transport of the sl modes, R(!0 ). The argument of Mott [22,23], derived by him for localized electronic states, can be taken over for sl vibrational modes [2]. The volume that contains at least one sl mode is given by (4/3)D(!0 )!sl [R(!0 )]3 D 1, where D(!0 ) is the density of sl states per unit energy. This condition assures that, for an sl mode at the origin, a second sl mode can be found within a hopping distance R(!0 ). We find R(!0 )  (!0 /!sl )(!0 ). Thus, the most probable hopping distance is at least the sl localization length. The hopping rate 1/sl (!0 ; T) caused by anharmonicity, with anharmonic coupling constant Ceff can be found from the Golden Rule, leading to the hopping contribution to the thermal conductivity, with ( M )3 the volume for finding a single sl mode, hop (T) D [43 3 (Ceff )2 (kB )2 T]/[ 3 (vs )5 3 ( M )3 h(! )i2 ]. Here h(! )i is the average sl length scale, vs the velocity of sound, and the density. There are very few undetermined parameters in this expression for hop (T). The anharmonic coupling

Dynamics on Fractals

constant can be found from the shift of the “boson peak” in the Raman spectra with pressure [48,49], typically a factor of 25 times larger than the long length scale third-order elastic coupling constant for extended modes. Making use of experiments on amorphous a-GeS2 one finds [26,27] hop (T)  0:0065T (W/mK) using the appropriate values for and vs , and  M  h(! )i D 15 Å. This contribution is comparable to that observed for other networkforming glasses above the plateau value. There is a quantum phenomenon associated with sl mode hopping. The hopping vertex is associated with a low energy extended mode coupling with an sl mode to hop to another (spatially separated) sl mode. As the temperature increases, the lifetime of the sl mode will become shorter than 1/! of the low energy extended mode. This quantum effect leads to a breakdown of the Golden Rule expression, causing a leveling off of the linear-in-T thermal conductivity above the plateau value. The sl mode hopping contribution to (T) above the plateau temperature for glasses and amorphous materials appears to provide a quantitative explanation of the observed phenomena. It is a specific example of how dynamics on a fractal network can be used to determine the properties of materials that, while certainly not fractal, have properties that map onto the behavior of fractal structures. Future Directions As stated in the Introduction, studying the dynamics on fractal networks is a convenient structure for analyzing the properties of localized states, with all of the frequency and temperature-dependent physical properties determined without arbitrary parameters. For example, the frequency dependence of the localization length scale, (!), is known precisely on fractal networks, and is used for precise calculation of the hopping transport in Sect. “Relaxation Dynamics on Fractal Structures”, without adjustable parameters. If one extrapolates the properties of fractal networks to those random systems that are certainly not fractal, but which exhibit localization, the transport and relaxation properties can be understood. Examples are the thermal transport of glasses and amorphous materials above the plateau temperature, and the lifetime of high energy vibrational states in a-Si. Measurements of these properties may offer opportunities for practical devices in frequency regimes beyond extended state frequencies. For example, the lifetime of high energy vibrational states in a-Si has been shown to increase with increasing vibrational energies [36,37]. Such behavior is opposite to that expected from extended vibrational states interacting anharmoni-

cally. But it is entirely consistent if these states are localized and behave as shown for fractal networks [32,33]. The thesis presented here points to a much broader application of the consequences of fractal geometry than simply of those physical systems which exhibit such structures. The claim is made that using what has been learned from fractal dynamics can point to explication of physical phenomena in disordered systems that are certainly not fractal, but display properties analogous to fractal structures. In that sense, the future lies in application of the ideas contained in the study of fractal dynamics to that of random systems in general. This very broad class of materials may well have properties of great practical importance, predicted from the dynamics on fractals.

Bibliography Primary Literature 1. Alexander S, Entin-Wohlman O, Orbach R (1985) Relaxation and nonradiative decay in disordered systems, I. One-fracton emission. Phys Rev B 32:6447–6455 2. Alexander S, Entin-Wohlman O, Orbach R (1986) Phonon-Fracton anharmonic interaction: the thermal conductivity of amorphous materials. Phys Rev B 34:2726–2734 3. Alexander S, Entin-Wohlman O, Orbach R (1986) Relaxation and non-radiative decay in disordered systems, II. Two-fracton inelastic scattering. Phys Rev B 33:3935–3946 4. Alexander S, Entin-Wohlman O, Orbach R (1987) Relaxation and non-radiative decay in disordered systems, III. Statistical character of Raman (two-quanta) spin-lattice relaxation. Phys Rev B 35:1166–1173 5. Alexander S, Orbach R (1982) Density of states on fractals, “fractons”. J Phys Lett (Paris) 43:L625–L631 6. Anderson PW (1958) Absence of diffusion in certain random lattices. Phys Rev 109:1492–1505 7. Avnir D (ed) (1989) The Fractal Approach to Heterogeneous Chemistry. Wiley, Chichester 8. Barabaási A-L, Stanley HE (1995) Fractal Concepts in Crystal Growth. Cambridge University Press, Cambridge 9. Bunde A, Havlin S (eds) (1991) Fractals and Disordered Systems. Springer, Berlin 10. Bunde A, Havlin S (eds) (1994) Fractals in Science. Springer-Verlag, Berlin 11. Bunde A, Roman HE (1992) Vibrations and random walks on random fractals: anomalous behavior and multifractality. Philos Magazine B 65:191–211 12. Calemczuk R, de Goer AM, Salce B, Maynard R, Zarembowitch A (1987) Low-temperature properties of silica aerogels. Europhys Lett 3:1205–1211 13. Courtens E, Pelous J, Phalippou J, Vacher R, Woignier T (1987) Brillouin-scattering measurements of phonon-fracton crossover in silica aerogels. Phys Rev Lett 58:128–131 14. de Gennes PG (1976) La percolation: un concept unificateur. Recherche 7:919–927 15. Feder J (1988) Fractals. Plenum Press, New York

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16. Freltoft T, Kjems J, Richter D (1987) Density of states in fractal silica smoke-particle aggregates. Phys Rev Lett 59:1212–1215 17. Gefen Y, Aharony A, Alexander S (1983) Anomalous diffusion on percolating clusters. Phys Rev Lett 50:77–80 18. Ikeda H, Fernandez-Baca JA, Nicklow RM, Takahashi M, Iwasa I (1994) Fracton excitations in a diluted Heisenberg antiferromagnet near the percolation threshold: RbMn0.39 Mg0.61 F3 . J Phys: Condens Matter 6:10543–10549 19. Ikeda H, Itoh S, Adams MA, Fernandez-Baca JA (1998) Crossover from Homogeneous to Fractal Excitations in the Near-Percolating Heisenberg Antiferromagnet RbMn0.39 Mg0.61 F3 . J Phys Soc Japan 67:3376–3379 20. Jagannathan A, Orbach R, Entin-Wohlman O (1989) Thermal conductivity of amorphous materials above the plateau. Phys Rev B 39:13465–13477 21. Kistler SS (1932) Coherent Expanded–Aerogels. J Phys Chem 36:52–64 22. Mott NF (1967) Electrons in disordered structures. Adv Phys 16:49–144 23. Mott NF (1969) Conduction in non-crystalline materials III. Localized states in a pseudogap and near extremities of conduction and valence bands. Philos Mag 19:835–852 24. Nakayama T (1992) Dynamics of random fractals: large-scale simulations. Physica A 191:386–393 25. Nakayama T (1995) Elastic vibrations of fractal networks. Japan J Appl Phys 34:2519–2524 26. Nakayama T, Orbach R (1999) Anharmonicity and thermal transport in network glasses. Europhys Lett 47:468–473 27. Nakayama T, Orbach RL (1999) On the increase of thermal conductivity in glasses above the plateau region. Physica B 263– 264:261–263 28. Nakayama T, Yakubo K (2001) The forced oscillator method: eigenvalue analysis and computing linear response functions. Phys Rep 349:239–299 29. Nakayama T, Yakubo K (2003) Fractal Concepts in Condensed Matter Physics. Solid-State Sciences. Springer, Berlin 30. Nakayama T, Yakubo K, Orbach R (1994) Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Rev Mod Phys 66:381–443 31. Orbach R (1986) Dynamics of Fractal Networks. Science 231:814–819 32. Orbach R (1996) Transport and vibrational lifetimes in amorphous structures. Physica B 219–220:231–234 33. Orbach R, Jagannathan A (1994) High energy vibrational lifetimes in a-Si:H. J Phys Chem 98:7411–7413 34. Rammal R (1984) Spectrum of harmonic excitations on fractals. J Phys (Paris) 45:191–206 35. Rammal R, Toulouse G (1983) Random walks on fractal structures and percolation clusters. J Phys Lett (Paris) 44:L13–L22

36. Scholten AJ, Dijkhuis JI (1996) Decay of high-frequency phonons in amorphous silicon. Phys Rev B 53:3837–3840 37. Scholten AJ, Verleg PAWE, Dijkjuis JI, Akimov AV, Meltzer RS, Orbach R (1995) The lifetimes of high-frequency phonons in amorphous silicon: evidence for phonon localization. Solid State Phenom 44–46:289–296 38. Stanley HE (1977) Cluster shapes at the percolation threshold: and effective cluster dimensionality and its connection with critical-point exponents. J Phys A 10:L211–L220 39. Stauffer D, Aharony A (1992) Introduction to Percolation Theory, 2nd edn. Taylor and Francis, London/Philadelphia 40. Stauffer D, Stanley HE (1995) From Newton to Mandelbrot: A primer in Theoretical Physics, 2nd edn. Springer, Berlin 41. Takayasu H (1990) Fractals in the Physical Sciences. Manchester University Press, Manchester 42. Tsujimi Y, Courtens E, Pelous J, Vacher R (1988) Raman-scattering measurements of acoustic superlocalization in silica aerogels. Phys Rev Lett 60:2757–2760 43. Uemura YJ, Birgeneau RJ (1987) Magnons and fractons in the diluted antiferromagnet Mnx Zn1x F2 . Phys Rev B 36:7024– 7035 44. Vacher R, Courtens E, Coddens G, Pelous J, Woignier T (1989) Neutron-spectroscopy measurement of a fracton density of states. Phys Rev B 39:7384–7387 45. Vacher R, Woignier T, Pelous J, Courtens E (1988) Structure and self-similarity of silica aerogels. Phys Rev B 37:6500–6503 46. Vicsek T (1993) Fractal Growth Phenomena. World Scientific, Singapore 47. Yakubo K, Nakayama T (1989) Direct observation of localized fractons excited on percolating nets. J Phys Soc Japan 58:1504–1507 48. Yamaguchi M, Nakayama T, Yagi T (1999) Effects of high pressure on the Bose peak in a-GeS2 studied by light scattering. Physica B 263–264:258–260 49. Yamaguchi M, Yagi T (1999) Anharmonicity of low-frequency vibrations in a-GeS2 studied by light scattering. Europhys Lett 47:462–467

Recommended Reading While many of the original articles are somewhat difficult to access, the reader is nevertheless recommended to read them as there are subtle issues that tend to disappear as references are quoted and re-quoted over time. For a very helpful introduction to percolation theory, Ref. 39 is highly recommended. A thorough review of dynamics of/on fractal structures can be found in Ref. 30. Finally, a comprehensive treatment of localization and multifractals, and their relevance to dynamics of/on fractal networks, can be found in Ref. 29.

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation ALAN CHAMPNEYS Department of Engineering Mathematics, University of Bristol, Bristol, United Kingdom Article Outline Glossary Definition of the Subject Introduction Linear Resonance or Nonlinear Instability? Multibody Systems Continuous Systems Future Directions Bibliography Glossary Parametric excitation Explicit time-dependent variation of a parameter of a dynamical system. Parametric resonance An instability that is caused by a rational relationship between the frequency of parametric excitation and the natural frequency of free oscillation in the absence of the excitation. If ! is the excitation frequency, and ! 0 the natural frequency, then parametric resonance occurs when ! D (n/2)!0 for any positive integer n. The case n D 1 is usually the most prominent form of parametric resonance, and is sometimes called the principle subharmonic resonance. Autoparametric resonance A virtual parametric resonance that occurs due to the coupling between two independent degrees of freedom within a system. The output of one degree of freedom acts like the parametric excitation of the other. Ince–Strutt diagram A two-parameter bifurcation diagram indicating stable and unstable regions, specifically plotting the instability boundaries as the required amplitude of parametric excitation against the square of the ratio of natural to excitation frequency. Floquet theory The determination of the eigenvalue spectrum that governs the stability of periodic solutions to systems of ordinary differential equations. Bifurcation A qualitative change in a system’s dynamics as a parameter is varied. One-parameter bifurcation diagrams often depict invariant sets of the dynamics against a single parameter, indicating stability and any bifurcation points. Two-parameter bifurcation diagrams depict curves in a parameter plane on which one-parameter bifurcations occur.

Modal analysis The study of continuum models by decomposing their spatial parts into eigenmodes of the dynamic operator. The projection of the full system onto each mode gives an infinite system of differential equations in time, one for each mode. Monodromy matrix The matrix used in Floquet theory, whose eigenvalues (also known as Floquet multipliers) determine stability of a periodic solution. Definition of the Subject Parametric excitation of a system differs from direct forcing in that fluctuations appear as temporal modulation of a parameter rather than as a direct additive term. A common paradigm is that of a pendulum hanging under gravity whose support is subjected to a vertical sinusoidal displacement. In the absence of any dissipative effects, instabilities occur to the trivial equilibrium whenever the natural frequency is a multiple of half the excitation frequency. At amplitude levels beyond the instability, further bifurcations (dynamical transitions) can lead to more complex quasi-periodic or chaotic dynamics. In multibody mechanical systems, one mode of vibration can effectively act as the parametric excitation of another mode through the presence of multiplicative nonlinearity. Thus autoparametric resonance occurs when one mode’s frequency is a multiple of half of the other. Other effects include combination resonance, where the excitation is at a sum or difference of two modal frequencies. Parametric excitation is important in continuous systems too and can lead to counterintuitive effects such as stabilization “upside-down” of slender structures, complex wave patterns on a fluid free surface, stable pulses of light that overcome optical loss, and fluid-induced motion from parallel rather than transverse flow. Introduction Have you observed a child on a playground swing? In either a standing or a sitting position, generations of children have learned how to “pump” with their legs to set the swing in motion without making contact with the ground, and without external forcing. Note that pumping occurs at twice the natural frequency of swing of the pendulum, since the child extends her legs maximally at the two extremities of the swing’s cycle, that is, two times per period. What is happening here? The simplest case to analyze is where the child is standing. Then, it has been argued by Curry [17] that the child plus swing is effectively a simple pendulum system, where the child’s pumping has the effect of periodic variation of the position of the center of the mass along the arm of the pendulum (see Fig. 1a,b).

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Dynamics of Parametric Excitation, Figure 1 a A child standing on a swing modeled as b a parametrically excited pendulum c a directly forced pendulum

We shall return to the self-propelled swing problem in Sect. “Future Directions” below, where we shall find out that all is not what it seems with this motivating example, but for the time being lets make the simple hypothesis that the child’s pumping causes the center of mass to move up and down. Using the standard model of a pendulum that consists of a lumped mass suspended by a rigid massless rod of length l from a pivot, the equations of motion for the angular displacement (t) of a pendulum hanging under gravity can be written in the form g ¨ (t) C sin[ (t)] D 0 : l Here g is the acceleration due to gravity, a dot denotes differentiation with respect to time t and, for the time being, we have ignored any damping that must inevitably be present. Now, the moving center of mass means that the effective length of the rigid rod oscillates, and let us suppose for simplicity that this oscillation is sinusoidal about some mean length l0 : l(t) D l0 [1  " cos(! t)] : Here " > 0 assumed to be small and as yet we haven’t specified the frequency !. Then, upon Taylor expansion in ", we find to leading order ¨ C !02 [1 C " cos(! t)] sin D 0 ; p where !0 D (g/l0 ) is the natural frequency of small amplitude swings in the absence of the pump. Finally, upon rescaling time ˜t D ! t and dropping the tildes we find ¨ C [˛ C ˇ cos t] sin D 0 ;

(1)

where ˛D

!02 !2

and ˇ D "

!02 !2

(2)

are dimensionless parameters that respectively describe the square of the ratio of natural frequency to excitation frequency, and the amplitude of the excitation. Equation (1) is a canonical example of a parametrically excited system. Note that (1) also describes, at least to leading order, the case of a pendulum hanging under gravity that is excited by a vertical force of size "!02 cos t which we can think of as a periodic modulation of the gravity. This would be in contrast to a directly forced simple pendulum where the forcing would act in the direction of swing for small amplitude motion; see Fig. 1c. For the forced pendulum, the equivalent equations of motion would look more like ¨ C ˛ sin D ˇ cos t ;

(3)

where the periodic forcing occurs as an additive input into the differential equation. In contrast, for (1) the periodic excitation occurs as a modulation of a parameter (which you can think of as gravity), hence the name ‘parametric’. Another feature of (1) that makes it a good paradigm for this article is that it is nonlinear and, as we shall see, a full appreciation of the dynamics resulting from parametric excitation requires treatment of nonlinear terms. An interesting contrast between parametrically excited and directly forced systems is in the nature of the trivial response to a non-zero input. In particular, for any ˛ and ˇ, Eq. (1) admits the trivial solution (t)  0, whereas there is no such solution to (3) for non-zero ˇ. The simplest “steady” solution to the directly forced system for small ˇ is a small-amplitude periodic oscillation of period 2 (in our dimensionless time units). So any analysis of a parametrically excited system should start with an analysis of the steady state D 0, which corresponds to the pendulum just hanging under gravity. To analyze stability of this equilibrium state let D 0C x where x is small, then x sat-

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 2 Possible behavior of Floquet multipliers  (eigenvalues of the monodromy matrix M) for a a conservative Hill equation and b in the presence of small damping

isfies linearization (1) x¨ C (˛ C ˇ cos t)x D 0 :

(4)

The equilibrium position is stable if all solutions x(t) of this equation, remain bounded as t ! 1 and it is unstable if there are solutions x(t) ! 1 as t ! 1.

Equation (4) is known as the Mathieu equation [42], and is a specific example of Hill’s equation [29,40] x¨ C [˛ C V(t)]x D 0 ;

(5)

where V(t) is any periodic function with period 2. Hill’s equation can be solved using classical methods for linear

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time-dependent ordinary differential equations (ODEs), see e. g. [31,33]. In particular, by writing y1 D x and y2 D x˙ , Hill’s equation (5) is just a two-dimensional example of a linear first-order time-periodic equation y˙ D P(t)y ;

y 2 Rn :

(6)

The general solutions to (6) for any initial condition can be expressed in terms of a fundamental matrix ˚ (t), with ˚(0) D I n (the identity matrix of dimension n): y(t) D ˚(t)y(0) : Note that just because P(t) has period 2 it does not follow that ˚(t) does. Also ˚(t) cannot be computed exactly, except in highly specialized cases, and one has to rely on approximate methods as described in the next Sect. “Floquet Analysis” and in the companion article to this one  Perturbation Analysis of Parametric Resonance. However, in order to study stability we don’t actually need to construct the full solution ˚ (t), it is sufficient to consider M D ˚(2), which is called the monodromy matrix associated with Hill’s equation. Stability is determined by

studying the eigenvalues of M: solutions to (5) with small initial conditions remain small for all time if all eigenvalues of M lie on or within the unit circle in the complex plane; whereas a general initial condition will lead to unbounded motion if there are eigenvalues outside this circle. See Fig. 2a. Eigenvalues of the Monodromy matrix are also known as Floquet multipliers of the system, and the general study of the stability of periodic systems is called Floquet theory. In the case of the Mathieu equation, where V (t) D ˇ cos(t), we have

 0 1 P(t) D : (7) 0 ˛ C ˇ cos(t) Figure 3 depicts graphically the behavior of Floquet multipliers for this case. Here, in what is known as an Ince– Strutt diagram [31,63], the shaded regions represent the values of parameters at which the origin x D 0 is stable (with Floquet multipliers on the unit circle) and the white regions are where the origin is unstable (at least one multiplier outside the unit circle). We describe how to calculate such a diagram in the next section. For the time be-

Dynamics of Parametric Excitation, Figure 3 Ince–Strutt diagram showing resonance tongues of the Mathieu equation (4). Regions of instability of the trivial solution x D 0 are shaded and a distinction is drawn between instability curves corresponding to a period-T (T D 2) and to a period-2T orbit. After (2nd edn. in [33]), reproduced with permission from Oxford University Press

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 4 Similar to Fig. 3 but for the damped Mathieu equation (8) with ı D 0:1. After http://monet.unibas.ch/~elmer/pendulum/parres.htm, reproduced with permission

ing we note that there are resonance tongues that emanate from ˇ D 0 whenever ˛ is the square of a half-integer: ˛ D 1/4, 1, 9/4, 4, 25/4, etc. The width (in ˛) of the tongue for small ˇ turns out to be proportional to ˇ to the power of twice the half-integer that is being squared. That is, the ˛ D 1/4 tongue has width O(ˇ), the ˛ D 1 tongue has width O(ˇ)2 , the ˛ D 9/4 tongue has width O(ˇ 3 ) etc. On the tongue boundaries (between the shaded and the unshaded regions of Fig. 3), the Mathieu equation admits non-trivial periodic solutions. For the tongues that are rooted at square integers, ˛ D 1, 9, 16, etc., these solutions have period 2 (the same period as that of the excitation) and the instability that occurs upon crossing such a tongue boundary is referred to as a harmonic instability. On the other boundaries, those of tongues rooted at ˛ D 1/4, 9/4, 25/4 etc., the periodic solution has period 4 (twice that of the excitation). Hence such instabilities are characterized by frequencies that are half that of the excitation and are hence referred to as subharmonic instabilities. So far, we have ignored any dissipative effects, so that Eq. (5) can be regarded as a conservative or Hamiltonian system. Damping can be introduced via a dissipative force that is proportional to velocity, hence (4) becomes x¨ C ı x˙ C (˛ C ˇ cos t)x D 0 ;

(8)

where ı is a dimensionless damping coefficient. One can also apply Floquet theory to this damped Mathieu equation; see Fig. 2b. The results for Eq. (8) are plotted in Fig. 4, where we can see the intuitive effect that damping

increases the areas of stability. In particular, in comparison with Fig. 3, note that the resonance tongues have been lifted from the ˛-axis, with the amount of the raise being inversely proportional to the width of the undamped tongue. Thus, the resonance tongue corresponding to ˛ D 1/4 is easily the most prominent. Not only for fixed forcing amplitude ˇ does this tongue occupy for the greatest interval of square frequency ratio ˛, but it is accessible with the least forcing amplitude ˇ. Thus, practicing engineers often think of this particular subharmonic instability close to ˛ D 1/4 (i. e. where the forcing frequency is approximately half the natural frequency) as being the hallmark of parametric resonance. Such an instability is sometimes called the principal parametric resonance. Nevertheless, we shall see that other instability tongues, in particular the fundamental or first harmonic parametric resonance near to ˛ D 1 can also sometimes be of significance. Returning briefly to the problem of how children swing, it would seem that the choice ˛  1/4 is in some sense preferable. This is the largest instability tongue for small ˇ. That is, the smallest effort is required in order to cause an exponential instability of the downwards hanging solution, and set the swing in motion. And, according to (2), ˛ D 1/4 corresponds to ! D 2!0 . Thus generations of children seem to have hit upon an optimal strategy [48], by pumping their legs at precisely twice the frequency of the swing. Finally, looking at Fig. 4, note the curious feature that we have continued the diagrams into negative ˛, despite the fact that ˛ was defined as a square term. In the case of

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a pendulum though, ˛ D l/(g! 2 ), and so we can think of negative ˛ as representing negative g. That is, a pendulum with negative ˛ is defying gravity by ‘hanging upwards’. Clearly without any excitation, such a situation would be violently unstable. The big surprise then is that there is a thin wedge of stability (which is present either with or without damping) for ˛ < 0. Note though that the wedge is thickest for small ˇ and small j˛j, which limit corresponds to small amplitude, high-frequency oscillation. This remarkable stabilization of an upside-down pendulum by tiny, rapid vertical vibration was first described by Stephenson [53] in 1908, re-discovered by Kapitza [34] in 1951 (and hence often called the Kapitza pendulum problem) and has been experimentally verified many times; see e. g. [3]. We shall return to this upside down stability in Sect. “Upside-Down Stability”. The rest of this article is outlined as follows. In Sect. “Linear Resonance or Nonlinear Instability?” we shall examine how the Ince–Strutt diagram is constructed and see that this is just an example of the linearized analysis around any periodic state of a nonlinear system. Further quasi-periodic or chaotic motion can ensue as we push beyond this initial instability. Section “Multibody Systems” concerns mechanical systems with multiple degrees of freedom. We consider autoparametric resonance where the motion of one degree of freedom acts as the parametric excitation of another. We also look at conditions in multibody systems where a combination of two natural frequencies can be excited parametrically, focusing in particular on the somewhat illusive phenomenon of difference combination resonances. Section “Continuous Systems” then looks at parametric excitation in continuous systems, looking at phenomena as diverse as pattern formation on the surface of a liquid, flow-induced oscillation of pipes, a clever way of overcoming loss in optical fibers and the oscillation-induced stiffening of structures. Finally, Sect. “Future Directions” draws conclusions by looking at future and emerging ideas. Linear Resonance or Nonlinear Instability? Let us now consider a more detailed analysis of parametric resonance of the Mathieu equation and of the implications for a nonlinear system, using the parametrically excited pendulum as a canonical example.

behave as parameters are varied. Essentially, we explain the behavior of Fig. 2a. Our treatment follows the elementary account in [33], to which the interested reader is referred for more details. For a general linear system of the form (6), a lot of insight can be gained by considering the so-called Wronskian W(t) D det(˚(t)). From the elementary theory of differential equations (e. g. [33]) we have Z t  trace[P(s)] ds ; (9) W(t) D W(t0 ) exp t0

where the trace of a matrix is the sum of its diagonal elements. Note that the specific form of P(t) for the Mathieu equation (7) has trace[P(t)] D 0, hence, from (9) that that det(M) D det[W(2)] D 1. But from elementary algebra we know that the trace of the matrix is the product of its eigenvalues. So we know that the two Floquet multipliers 1;2 satisfy 1 2 D 1 and must therefore solve a characteristic equation of the form 2  2(˛; ˇ) C 1 D 0 ; for some unknown real function . There are two generic cases jj < 1 and jj > 1. If  > 1, then the Floquet multipliers are complex and lie on the unit circle. This corresponds to stability of the zero-solution of the Mathieu equation. Conversely if jj < 1, then the Floquet multipliers are real and satisfy j1 j < 1 and j2 j D 1/j1 j > 1. The larger multiplier represents an exponentially growing solution and hence corresponds to an instability. The two boundary cases are:  D 1, in which case we have a double Floquet multiplier at  D 1; or  D 1, which corresponds to a double multiplier at  D 1. These represent respectively the harmonic and the subharmonic stability boundaries in the Ince–Strutt diagram. But, how do we compute ? The answer is that we don’t. We simply recognize that  D 1 corresponds to the existence of a pure 2-periodic solution of the fundamental matrix M D ˚ (2), and that  D 1 represents a periodic solution of ˚(2)2 D ˚(4) and hence a 4periodic solution of the fundamental matrix. To look for a 2-periodic solution of the Mathieu equation, it is convenient to use Fourier series. That is, we seek a solution (4) in the form x(t) D

nDC1 X

c n e i nt :

nD1

Floquet Analysis We begin by giving a reasoning as to why the Ince–Strutt diagram Fig. 3 for the Mathieu equation (ignoring damping for the time being) looks the way it does, by making a more careful consideration of how Floquet multipliers

Substitution into (4), using cos(t) D (1/2)(e i t Cei t ) leads to infinitely many equations of the form (1/2)ˇc nC1 C (˛  n2 )c n C (1/2)ˇc n1 D 0 ; n D 1; : : : ; 1 ;

with cn D c n ;

(10)

Dynamics of Parametric Excitation

where an asterisk represents complex conjugation. Note that the final condition implies that x is real. Clearly, for ˇ D 0, a nontrivial solution can be found for any integer p which has c n D 0 for all n ¤ p, provided ˛ 2 D p2 . That is, solution curves bifurcate from ˇ D 0 for each ˛-value that is the square of a whole integer. Furthermore, nonlinear analysis can be used to show that for small positive ˇ, there are two solution branches that emerge from each such point (provided p ¤ 0); one corresponding to cp being real, which corresponds to motion x(t) cos(pt) in phase with the excitation; and another corresponding to cp imaginary, which corresponds to motion x(t) sin(pt) out of phase with the excitation. Similarly, if we look for solutions which are 4-periodic by making the substitution x(t) D

mDC1 X

d m e i mt/2 ;

(11)

mD1

we arrive at an infinite system of equations (1/2)ˇd mC2 C (˛  (1/4)m2 )d m C (1/2)ˇd m1 D 0 ; m D 1; : : : ; 1 ;

 with dm D d m :

(12)

Note that the equations for m even and m odd in (12) are decoupled. The equations for even m become, on setting m D 2n, precisely the system (10) with d2m D c n . For m D 2n C 1, however, we arrive at new points of bifurcation, by exactly the same logic as we used to analyze (10). Namely whenever ˛ 2 D (p/2)2 for any odd integer p we have the bifurcation from ˇ D 0 of two 4-periodic solutions (one like cos(pt/2) and one like sin(pt/2)). Perturbation analysis can be used to obtain asymptotic expressions for the transition curves that bifurcate from ˇ D 0 at these special ˛-values; see the companion article  Perturbation Analysis of Parametric Resonance for the details. In particular it can be shown that the width of the stability tongue (separation in ˛-values from the sine and the cosine instability curves) originating at ˛ D (p/2)2 (for p being odd or even) scales like tongue width p ˇ p C h.o.t : Hence the principal subharmonic instability arising from ˛ D 1/4 has width O(ˇ) (actually the transition curves are given by ˛ D 1/4 ˙ (1/2)ˇ C O(ˇ 2 )), and the fundamental harmonic instability arising from ˛ D 1 has width O(ˇ 2 ), etc. Hence the principal subharmonic instability is by far the most prevalent as it occupies the greatest frequency width for small ˇ, and is often thought to be the hallmark of parametric resonance in applications.

Note that we have not as yet mentioned the transition curve emerging from ˛ D ˇ D 0, which corresponds to n D 0 in the above analysis. This case can also be analyzed asymptotically, and the single transition curve can be shown to scale like ˛ D (1/2)ˇ 2 C O(ˇ 3 ) for small ˇ. It is this bending back into ˛ < 0 that leads to the Kapitza pendulum effect, which we shall return to in Sect. “Upside-Down Stability” below. Finally, the addition of small damping can be shown to narrow the tongues and lift them off from the ˇ D 0 axis as shown in Fig. 4. The amount of lift is in proportion to the width of the tongue. This property further enhances the pre-eminence of the principal subharmonic instability for parametrically excited systems in practice, as they thus require the least amplitude of vibration in order to overcome the damping forces. Bifurcation Theory We have just analyzed the Mathieu equation, which is purely linear. But, for example as shown in the Introduction, such an equation is often derived in the course of analyzing the stability of a trivial solution to a nonlinear parametrically excited system. It is worthwhile therefore to consider the nonlinear implications of the harmonic and subharmonic instabilities. The time-periodic Mathieu equation can be written as an example of a three-dimensional autonomous system of ODEs x˙ D y ;

y˙ D f (x; y; s) ;

s˙ D 1 (mod 2) ;

for (x; y; s) 2 R2  S 1 , where f (x; y; s) D ı y  (˛ C ˇ cos(s))x. As such, the trivial solution x D 0 should actually be seen as the periodic solution x D y D 0, s D t (mod 2). Hence we can understand instability of the trivial solution as points of bifurcation of periodic solutions, for which there is a mature theory, see for example [27,37,65,67]. In particular, the subharmonic instability with Floquet multiplier  1 is a period doubling bifurcation, and the harmonic bifurcation corresponding to multiplier + 1 is a pitchfork bifurcation. Note that both these bifurcations come in super- and sub-critical forms. Hence, depending on the nature of the nonlinearity we may find either stable bounded motion close to the instability inside the resonant tongue close to the boundary edge, or motion that is not of small amplitude. See Fig. 5. The case of the Mathieu or Hill equations without damping have received special attention in the literature. This is because of the extra mathematical structure they posses, namely that they can be expressed as reversible and Hamiltonian systems. Specifically, Hill’s equation of the form (5) can be shown to conserve the energy function

189

190

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 5 Contrasting a super-critical and b sub-critical bifurcations close to the resonance tongue boundary of a subharmonic (period-doubling) instability, here represented by the abbreviation P.D. Solid lines represent stable solutions, dashed lines represent unstable solutions. c A one-parameter bifurcation diagram close to the super/sub-critical boundary. d A two parameter unfolding of the codimension-two point P where the super/sub-critical transition occurs. Shading is used to represent where a stable period-two cycle exists. Note in this case that for higher ˇ-values the right-hand boundary of this existence region is therefore the fold curve and not the period-doubling point. Between the period-doubling and the fold there is hysteresis (coexistence) between the stable trivial solution and the stable period-two cycle

H(x; y; t) D (1/2)y 2 C (1/2)(˛ C V(t))x 2 ; y D x˙ , and to be expressible in Hamiltonian form x˙ D

@H ; @y

y˙ D 

@H : @x

Reversibility of Hill’s equation is expressed in the fact that the model is invariant under the transformation t ! t, y ! y. Broer and Levi [7] used Hamiltonian and reversible theory to look at Hill’s equations for general periodic functions V(t). They showed that under certain conditions, the resonance tongues may become of zero width for nonzero ˇ. Effectively, the left and right-hand boundaries of the tongue pass through each other. This is a generic phenomenon, in that a small perturbation within the class of Hill’s equations will lead not destroy such an intersection. However, the Mathieu equation for which V(t) D cos(t) does not posses this property. Many nonlinear models that undergo parametric resonance also preserve the properties of being Hamiltonian and reversible. Special tools from reversible or Hamiltonian systems theory can sometimes be used to analyze the bifurcations that occur under parametric resonance in nonlinear systems, see for example [8,9]. Points outside

the resonance tongues correspond to where the trivial solution is an elliptic (weakly stable) fixed point of the time2 map, and those inside the resonance tongue to where it is a hyperbolic (unstable) fixed point. One therefore does not generally see asymptotically stable non-trivial solutions being born at a resonance tongue boundary, rather one sees elliptic nontrivial dynamics. In fact, for a general periodic solution in a Hamiltonian system, pitchfork and period-doubling bifurcations are not the only kind of bifurcations that produce non-trivial elliptic periodic orbits. If the Floquet multipliers pass through any nth root of unity, there is a period multiplying (subharmonic) bifurcation at which a periodic orbit of period 2n is born (see, e. g. [62]). A period-doubling is just the case n D 2. Such bifurcations can also occur for the non-trivial elliptic periodic orbits within a resonance tongue, but they are all destroyed by the presence of even a small amount of damping. Beyond the Initial Instability For a nonlinear system, the pitchfork or period-doubling bifurcation that occurs upon crossing a resonance tongue might just be the first in a sequence of bifurcations that is

Dynamics of Parametric Excitation

encountered as a parameter is varied. As a general rule, the more ˇ is increased for fixed ˛, the more irregular the stable dynamics becomes. For example, the period-doubling bifurcation that is encountered upon increasing ˇ for ˛ close to 1/4 is typically just the first in a sequence of perioddoublings, that accumulate in the creation of a chaotic attractor, obeying the famous Feigenbaum scaling of this route to chaos (see e. g. [18]). Van Noort [64] produced a comprehensive numerical and analytical study into the nonlinear dynamics of the parametrically excited pendulum (1) without damping. Of interest in that study was to complete the nonlinear dynamics of one of the simplest non-integrable Hamiltonian and reversible dynamical systems. The parameter ˇ acts like a perturbation to integrability. Certain structures that can be argued to exist in the integrable system, like closed curves of quasiperiodic motion in a Poincaré section around an elliptic fixed point, serve as approximate organizers of the dynamics for non-zero ˇ. However, KAM theory (see e. g. [55]) shows that these closed curves become invariant tori that typically breakup as ˇ is increased. At the heart of the regions of chaotic dynamics lie the resonance tongues associated with a period-multiplying bifurcation. Inside the resonance tongues are nontrivial fixed points, whose surrounding invariant tori break up into heteroclinic chains, which necessarily imply chaos. Acheson [2] studied the dynamics of the parametrically forced pendulum in the presence of damping (i. e. Eq. (1) with an additional term x˙ on the right-hand side) via computer simulation. He found parameter regions inside the principal resonance tongue of the pendulum where it undergoes what he referred to as multiple nodding oscillations. The subharmonic motion inside the tongue should be such that during one cycle of the excitation, the pendulum swings to the left and back, and during the next cycle it swings to the right and back. That is, one complete oscillation per two cycles. Multiple nodding motion by contrast is asymmetric and involves two swings to the left, say, (one swing per cycle of the excitation), followed by one swing to the right, with the motion repeating not every two but every three cycles. He also found similar four-cycle and five-cycle asymmetric motion. This behavior can be explained as pockets of stable behavior that survives from the period-multiplying bifurcations that are only present in the case of zero damping [21]. Clearly, beyond the initial instability the universality of the Ince–Strutt diagram for the Mathieu equation, or for any other Hill equation for that matter, disappears and the precise features of the nonlinear dynamics depends crucially on the details of the particular nonlinear system being investigated.

Multibody Systems Of course, there is little in the real world that can be accurately described by deterministic, parametrically excited single degree-of-freedom ODEs. In this section we shall consider slightly more realistic situations that are governed by higher-dimensional systems, albeit still deterministic, and still described by finitely many dynamic variables. For simplicity, we shall restrict attention to rigidbody mechanisms that can be described by a system of (finitely many) coupled modes of vibration. Furthermore, we shall adopt a Lagrangian framework where the dynamics of mode i can be represented by a generalized coordinate qi and its associated generalized velocity q˙i , such that we arrive at a system of (generally nonlinear) coupled second-order ODEs; one equation for each acceleration q¨i . Autoparametric Resonance Autoparametric resonance occurs when there is asymmetric nonlinear multiplicative coupling between modes of a multibody system. Specifically, suppose that a mode (q1 ; q˙1 ) is coupled to a mode (q2 ; q˙2 ) through a term proportional to q1 q2 in the equation of motion for the 2nd mode (i. e. the q¨2 equation), yet there is no corresponding q2 q1 term in the q¨1 equation. For example, consider q¨1 C ı1 q˙1 C ˛1 q1 D f1 (q1 ; t)

(13)

q¨2 C ı2 q˙2 C ˛2 q2 C ˇq1 q2 D f2 (q2 ) ;

(14)

where f 1 and f 2 are nonlinear functions of their arguments and the dependence of f 1 on t is periodic. Furthermore suppose that mode 1 undergoes a periodic motion (either as a result of direct forcing, its own parametric resonance, or self-excitation). For example, if f2 D 0 and f 1 were simply cos(! t) for some ! 2 sufficiently detuned from ˛ 1 , then the solution q1 (t) to (13) would, after allowing sufficient time for transients to decay, be simply proportional to cos(! t C ) for some phase . Then, the q1 q2 term in (14) would be effectively act like ˇ cos(! t C )q2 and so (14) would be precisely the Mathieu equation. Values of ! 2 D 4˛2 /n for any positive integer n correspond to the root points of the resonance tongues of the Ince–Strutt diagram. Such points are referred to as internal or autoparametric resonances. A canonical example of an autoparametrically resonant system is the simple spring-mass-pendulum device depicted in Fig. 6, which we shall use to discuss the some key generic features of autoparametric resonance; see the book by Tondl et al. [58] for more details. In the figure, a mass M is mounted on a linear spring with stiffness k,

191

192

Dynamics of Parametric Excitation

x(t) D R0 cos(t C where R0 D q and tan

0

D

0)

; A

ı12 C (˛1  1)2

ı1 : ˛1  1

(17)

Note that the solution (17) exists for all values of the physical parameters, in particular for all positive ˛ 1 provided ı1 > 0. Substitution of the semi-trivial solution (17) into the pendulum Eq. (16), leads to ¨ C ı2 ˙ C [˛2 C R0 cos(t C

0 )] sin 

D0;

which, after a time shift t ! t  0 is precisely the Eq. (1) for a parametrically excited pendulum in the presence of damping. Thus, the Ince–Struttqdiagram of the damped

Dynamics of Parametric Excitation, Figure 6 A mass-spring-pendulum system

is forced by an applied displacement P cos(! ) and is attached to the pivot of a pendulum of length l and lumped mass m that hangs under gravity. The motion of the mass and the pendulum are assumed to be subject to proportional viscous damping with coefficients b and c respectively. Using standard techniques, the equations of motion can of such a device can be written in the dimensionless form x¨ C ı1 x˙ C ˛1 C (¨ sin() C ˙ 2 cos ) D A cos t (15) ¨ C ı2 ˙ C ˛2 sin  C x¨ sin  D 0 ;

(16)

where x D y/l, time t has been scaled so that t D !, and the dimensionless parameters are b k m ; ˛1 D 2 ; D ; !(M C m) ! (M C m) MCm c g P ; ˛2 D : ; ı2 D AD 2 l! (M C m) !ml 2 l! 2

ı1 D

The simplest form of output of such a model is the socalled semi-trivial solution in which the pendulum is at rest, ˙ D   0, and the mass-spring system alone satisfies the simple equation x¨ C ı1 x˙ C 1 x D A cos t : After transients have died out, this single equation has the attracting solution

Mathieu equation, with ˇ D A/( ı12 C (˛1  1)2 ), ı D ı2 and ˛ D ˛2 tell us precisely the parameter values for which the semi-trivial solution is stable. Inside the resonance tongues we have an instability of the semi-trivial solution, which leads to stable non-trivial motion with the pendulum swinging. To determine precisely what happens away from the boundaries of the stability tongue, one again has to perform fully nonlinear analysis, as outlined in Sect. “Linear Resonance or Nonlinear Instability?”. Note now though that for nonzero pendulum motion,  6 0, the Eqs. (15) and (16) become a fully coupled two-degreeof-freedom system, whose dynamics can be even more complex than that of the one-degree-of-freedom parametrically excited pendulum. Indeed in [58], a further destabilizing secondary-Hopf (Neimark–Sacker) bifurcation is identified within the principal (˛2 D 1/4) subharmonic resonance tongue, and numerical evidence is found for several different kinds of quasiperiodic motion. There is one difference though between autoparametric resonance and pure parametric resonance. The paradigm for parametric resonance is that the external parametric excitation is held constant (its amplitude and frequency are parameters of the system). Here, however, for nonzero , the system is fully coupled, so that the parametric term (17) can no longer be thought of as being independent of , effectively the excitation becomes a dynamic variable itself. Thus, motion of the pendulum can affect the motion of the mass-spring component of the system. In particular if there is an autoparametric instability where the pendulum is set in motion, then energy must be transferred from the mass-spring component to the pendulum, a phenomenon known as quenching. But the pendulum is only set in motion close to certain resonant frequency ratios ˛ 2 . Thus, we have a tuned damper that is designed to

Dynamics of Parametric Excitation

take energy out of the motion of the mass at certain input frequencies ! frequencies. To see how such a damper might work in practice, suppose we build a civil engineering structure such as a bridge or a building, that has a natural frequency ! 1 in some mode, whose amplitude we call x(t). Suppose the system is subjected to an external dynamic loading (perhaps from traffic, wind or earthquakes) that is rich in a frequency ! which is close to ! 1 . Then, if the damping is small (as is typical in large structures) using (15) with  D 0, we find that the response given by (17) is large since ˛ 1 is close to 1 and ı 1 is small. This is the fundamental resonant response of the structure close to its natural frequency. Now suppose we add the pendulum as a tuned damper. In particular we design the pendulum so that ˛2  ˛1 . The pendulum now sees a large amplitude parametric excitation within its main subharmonic resonance tongue. It becomes violently unstable, thus sucking energy from the structure. To see just how effective this quenching effect can be, consider the following asymptotic calculation [58]. Supˆ for some pose the external forcing is very small, A D "2 A, small parameter ", and that the damping and frequency detunings scale with ": ı1 D "ıˆ1 , ı2 D "ıˆ2 , ˛1 D 1 C "ˆ 1 , ˛2 D (1/4) C "ˆ 2 , where the hatted quantities are all assumed to be O(1) as " ! 0. Upon making these substitutions, and dropping all hats, Eqs. (15) and (16) become 1 x¨ C x C "[ı1 x˙ C 1   2 C ˙ 2 ] 4 D "A cos(t) C O("2 ) ; 1 1 ¨ C  C "[ı2 ˙ C 2   x] D O("2 ) : 4 2

(18) (19)

We then look for solutions of the form  x D R1 (t) cos t C 1 (t) ;

 t  D R2 (t) cos C 2 (t) : 2 Making this substitution into (18), (19), and keeping only O(") terms we find

 1 2R˙ 1 D " ı1 R1  R22 sin( 1  2 2 ) A sin( 1 ) ; 4

 1 2R1 ˙ 1 D " 1 R1  R22 cos( 1  2 2 )  A cos 1 ; 4  2R˙ 2 D "  ı2 R2 C R1 R2 sin( 1  2 2 ) ;  2R2 ˙ 2 D " 2 R2  R1 R2 cos( 1  2 2 ) : Equilibrium solutions of these equations correspond to 4-periodic solutions that are valid close to the instability

tongue. Upon setting the left-hand sides to zero, a straightforward combination of the final two equations yields q R1 D 22 C ı22 ; (20) and a slightly longer calculation reveals that this leads to a unique (stable) solution for R2 provided A2 > (ı12 C 12 )(ı22 C 22 ). Equation (20) gives the amplitude of the quenched solution for the structure. Compare this with the amplitude R0 of the semi-trivial solution (17), which is the forced response of the structure without the pendulum-tuneddamper added. Written in the rescaled coordinates, this solution is R0 D

A2 : 12 C ı12

Here we make the important observation that, unlike the simple forced response R0 , the quenched amplitude R1 is independent of the forcing amplitude A, and does not blow up at the fundamental resonance 1 ! 0. In fact, at least to leading order in this asymptotic approximation, the amplitude of the quenched solution is independent of the frequency detuning and damping constant of the structure, and depends only on the frequency detuning and proportional damping of the pendulum! Tondl et al. [58] give a number of applications of autoparametric resonance. First, the above quenching effect means that analogues of the mass-spring-pendulum system can be used as tuned dampers, that are capable of suppressing certain undesirable response frequencies from structures or machines. Another important example of parametric resonance they cite is in flow-induced oscillation, where periodic flow features such as vortex shedding can cause a parametric response of a structure. The galloping of cables in strong winds is an example of just such an effect. Cable-stayed bridges are engineering structures that are particularly susceptible to autoparametric effects. Not only is there potential for resonant interaction between the cables and the wind, but, since the cables are all of different lengths, there is a good chance that there is at least one cable whose natural frequency is either half or twice that of a global vibration mode of the entire bridge. Models of deck-cable interaction have shown the propensity for undesirable effects, see e. g. [23,24]. For example, vibrations caused by traffic could cause particular cables to undergo large-amplitude vibration leading to potential problems with wear. Alternatively, large-amplitude cable vibrations (caused for example by fluid-structure interaction) could cause significant deck vibration leading to an uncomfortable ride for those crossing the bridge.

193

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Dynamics of Parametric Excitation

Combination Parametric Resonances Mailybayev and Seyranian [41] consider general systems of second-order equations of the form M y¨ C ıD y˙ C (A C ˇB(˝ t))y D 0 ;

y 2 Rn ;

(21)

where M, D and A are positive definite symmetric matrices, and B is a periodic matrix with period 2/˝. They derive conditions under which parametric resonance may occur. Suppose that when ı D 0 D ˇ D 0, the linear system has non-degenerate natural frequencies, !1 ; !2 ; : : : ! n . Then they showed that the only kinds of parametric resonance that are possible are when ˝D

2! j ; k

j D 1; : : : m ; k D 1; 2; : : : ;

(simple resonance) ;

!i ˙ ! j or ˝ D ; i; j D 1; : : : m ; i > j ; k k D 1; 2; : : : ; (combination resonance) :(22)

that the system with ˇ D 0 is written in diagonal form, and finally that time has been rescaled so that ˝ D 1. Hence we obtain a system of the form        x¨1 ˛1 0 b11 b12 C Cˇ cos(t) 0 ˛2 x¨2 b21 b22   x1 D 0 ; (23) x2 where ˛1 D !12 /˝ 2 , ˛2 D !22 /˝ 2 and the matrix Bˆ D fb i j g is the original constant matrix within B0 written in the transformed coordinates. Recalling, how in Subsect. “Floquet Analysis” we found the transition curves for the Mathieu equation using Floquet theory, we can look for solutions to (23) in the form x1 D c n

1 X

p 2

e i nt/

nD1 1 X

x2 D ˙c n

e

;

p si nt/ 2

(24) ;

nD1

The sign ‘+’ or ‘’ in (22) gives what are called sum or difference combination resonances respectively. There are many examples of such combination resonances in the literature. For example, the book by Nayfeh [45] considers many such cases, especially of mechanical systems where the two modes are derived from a Galerkin reduction of a continuum system such as a plate or beam. In all these examples, though, it is a sum combination resonance that is excited. That is, by forcing the system at the sum of the two individual frequencies ˝ D !1 C !2 , individual responses are found at frequency ! 1 or ! 2 . However, there do not seem to be any concrete examples of difference resonances in the literature. In fact, such a mechanical device might be somewhat strange. If we had !1  !2 , then ˝ D !1  !2 would be small, perhaps several orders of magnitude smaller. So a difference combination resonance would give a response at a high frequency from low frequency excitation. This would be like making a drum vibrate by sending it up and down in an elevator! In fact, for the case that B D B0 (t) is a constant matrix times a function of time, then it is shown in (Theorem 1 in [41]) that a given system can only excite either sum or difference combination resonances, but not both. To explain this result, it is sufficient to consider a simplification of (21) to the case n D 2. Consider (21) in the case ı D 0 and where B(t) is a constant matrix times a pure function of time with period ˝. Without loss of generality, let us suppose this function to be a pure cosine B D B0 cos(˝ t). Suppose that we change coordinates so

where s D ˙1. We find an infinite system of algebraic equations analogous to (10). It is straightforward to see that in the case ˇ D 0, there is a non-trivial solution with c n D 0 for all n ¤ k and c k ¤ 0, whenever ˛1 C s˛2 D k 2 . This would suggest that both sum and difference combination resonances are possible. However, when looking at the conditions for the bifurcation equations to have a real solution for nonzero ˇ, one finds the following condition sign(b12 b21 ) D s :

(25)

That is, to excite a sum resonance, the diagonal entries of B0 must be of the same sign, and to excite a difference resonance these diagonal entries must be of opposite sign. In particular, if B0 is a symmetric matrix (as is the case in many mechanical applications) then only sum combinations can be excited. In fact, it is argued in [68] that pure Hamiltonian systems cannot ever excite difference combination resonance; in other words, if difference parametric resonance is possible at all, then the matrix B(˝ t) must contain dissipative terms. One might wonder whether difference combination resonance can ever be excited in any physically derived system. To show that they can, consider the following example with a non-conservative parametric excitation force, a detailed analysis of which will appear elsewhere [59]. Figure 7 depicts a double pendulum (which we think of as lying in a horizontal plane so that gravity does not affect it) with stiff, damped joints and is loaded by an end

Dynamics of Parametric Excitation

Now, (26) is not in diagonal form, but a simple coordinate transformation can it in the form (21) with M the identity matrix,  DDAD  and

B0 D

0:1492 0 0:0466 0:1288

0 3:3508



0:3788 1:0466

 :

Hence the system has two natural frequencies !1 D p p 0:1492 D 0:3863 and !2 D 3:3508 D 1:8305, and the quantities b12 b21 D 0:3788  0:1288 D 0:0488 < 0. Hence, according to (25), a difference combination resonance should be possible by exciting the system at ˝ D !2  !1 D 1:4442, as verified in [59]. Upside-Down Stability Dynamics of Parametric Excitation, Figure 7 The partially follower-loaded elastic double pendulum, depicting a definition of the various physical quantities. In this study we take m1 D m2 D m and ˛ D 1

force – a so-called follower load – that is maintained in the direction of the axis of the second pendulum. Note that such a load is by its nature non-conservative because work (either positive or negative) has to be done to maintain the direction of the follower. In practice, such a force might be produced by a jet of fluid emerging from the end of the outer pendulum, for example if the pendula were actually pipes, although the equations of motion (and the consequent nonlinear dynamics) are rather different for that case [5,12,13]. Non-dimensional equations of motion for this system are derived in [56], where it was shown that the system can undergo quite violent chaotic dynamics for large deflections. We shall restrict ourselves here to small deflection theory and allow the follower load to be a harmonic function of time P D A cos ˝ t. After p nondimensionalization using ˇ D jAjl/k;pı D C/ kml 2 , and rescaling time according to tnew D k/m2 l 2 ) told , we obtain the equations  ¨  1 ¨2     ˙  2 1 1 2 1 1 Cı 1 1 2 1 1 ˙2    1 1 1 C p cos ˝ t D 0 : (26) 2 0 0



2 1

1 1



C

As mentioned in the Introduction, it has been known for 100 years [53] that the ‘negative gravity’ (˛ < 0) region of the Ince–Strutt diagram for the Mathieu equation implies that a simple pendulum can be stabilized in the inverted position by application of parametric excitation. More recently Acheson and Mullin [3] demonstrated experimentally that a double and even a triple pendulum can be similarly stabilized when started in the inverted position. Moreover, the stabilized equilibrium is quite robust, such that the system relaxes asymptotically back to its inverted state when given a moderate tap. At first sight, these observations seem remarkable. A multiple pendulum has several independent normal modes of oscillation and naively one might expect that single frequency excitation could at best stabilize only one of these modes at a time. However, Acheson [1] (see also the earlier analysis of Otterbein [46]) provided a simple one-line proof that, in fact, for sufficiently high frequency and sufficiently small amplitude sinusoidal excitation, in theory any ideal finite chain of N pendulum can be stabilized in the inverted position. To see why this is true, consider the multiple pendulum system represented in Fig. 8 but without the springs at the joints. Then, upon performing a modal analysis, we find that the equations of motion reduce to the study of N uncoupled Mathieu equations each with different parameters ˛ i > 0 and ˇ i > 0, for i D 1; : : : ; N. Hence the frequency can be chosen to be sufficiently high and the amplitude chosen to be sufficiently small for each ˛ i , ˇ i to lie in the thin wedge of stability for ˛ < 0 in Fig. 4. Thus, each mode is stable, and so the whole finite chain has been stabilized by parametric excitation.

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Dynamics of Parametric Excitation, Figure 8 a A multiple pendulum with stiff joints. b,c Ince–Strutt diagrams of dimensionless amplitude of vibration " against scaled elastic stiffness B for eight identical jointed pendulums with weak damping. b For dimensionless applied frequency ! D 10; c ! D 20. Shading represents the stability region of the upright equilibrium and digits in other regions give the number of unstable Floquet multipliers. After [22], to which the reader is referred to for the precise details; reproduced with permission from Elsevier

It has been suggested that the limit of such a multiple pendulum, in which the total length and mass of the system stays constant while the number of pendulums in the chain becomes infinite, could be used as a possible explanation of the so called ‘Indian rope trick’, a classic conjuring trick in which a rope is ‘charmed’ by a magician to appear vertically out of a basket like a snake. Unfortunately, as noted [1], in this limit, which is that of a piece of string, the stability region becomes vanishingly small and so this potential scientific explanation of the magic trick fails. However, a further experiment by Mullin announced in [4], demonstrated that a piece of ‘bendy curtain wire’, clamped at the bottom and free at the top, that is just too long to support its own weight can be stabilized by parametric excitation (see Fig. 9). In order to explain this result, Galán et al. [22] introduced the model depicted in Fig. 8 and studied the stability of the vertical equilibrium

position in the presence of both damping and stiffness in each of the joints. The normal modes of this problem lead to fully coupled equations and so Acheson’s argument does not apply. Using numerical bifurcation analysis Galán et al. were able to find stability boundaries and also to find an appropriate continuum limit; providing a good motivation for us to turn our attention to parametric resonance in continuous systems. Continuous Systems Parametric resonance is also of importance in continuum systems. Canonical effects we shall discuss here include the wire stiffening we have just introduced, the excitation of patterns at fluid interfaces, more general theories of flowinduced oscillations of structures, and the stabilizing effects of periodic modulation of optical fibers.

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 9 Experimental results on a piece of ‘bendy curtain wire’ which is just long enough to buckle under its own weight (a). For tiny vertical oscillations between about 15 and 35 Hz, the wire can be made to stand upright (b), and to be stable against even quite large perturbations (c). Beyond the upper frequency threshold, a violent dynamic stability sets in which involves the third vibration mode excited harmonically (d). After [44], reproduced with permission from the Royal Society

Structural Stiffening; the ‘Indian Rod Trick’ Thomsen [57] has shown that parametric excitation generally speaking has a stiffening effect on a structure. This is essentially because of the same effect that causes the upside down stability of the simple pendulum for high-enough frequencies and small enough amplitude. However, detailed experiments by Mullin and co-workers [44], reproduced in Fig. 9, showed that there can also be a destabilizing effect of parametric resonance. In particular, a piece of curtain wire (a thin helical spring surrounded by plastic) that was just too long (about 50 cm) to support its own weight, was clamped vertically upright and subjected to rapid, small-amplitude oscillations of about 1 cm peakto-peak. At around 15 Hz the rod was found to be stabilized, such that when released it remains erect (see Fig. 9b). This effect continues for higher frequencies, until about 30 Hz at which point a violent dynamic instability sets in that is commensurate with the driving frequency (harmonic, rather than subharmonic). The experimentally determined parameter values in which the upside-down stabilization can be achieved are represented in terms of dimensionless parameters in Fig. 10b. Champneys and Fraser [14,15,20], introduced a continuum theory to explain these experimental results, which produced the shaded region in Fig. 10b, that we now explain. Consider an intrinsically straight column of length `, with a uniform circular cross-section of radius a  ` and mass linear density m per unit length (see Fig. 10a). The column is assumed to be inextensible, unshearable and linearly elastic with bending stiffness Bˆ and material damping coefficient ˆ (where hatted variables are

dimensional). The lower end is clamped upright and is to vertical displacement  cos ! ˆt , whereas the upper end is free. In [15], it is shown that the stability of the vertical equilibrium position can be studied by analyzing solutions to the PDE Bussss C [(1  s)us ]s C  (u t t C usssst  "[(1  s)us ]s cos t) D 0 ;

(27)

for the scaled lateral displacement u(s; t) assumed without loss of generality to be in a fixed coordinate direction, where the scaled arc length s 2 (0; 1), the scaled time variable t D ! ˆt, and the boundary conditions are u D us D 0 ;

at s D 0 ;

(28)

uss D ussss D 0 ; at s D 1 : The dimensionless parameters appearing in (27) are

D

 `! ; mg

BD

Bˆ ; mg`3

D

! 2` ; g

"D

 ; (29) `

which represent respectively material damping, bending stiffness, the ratio of forcing frequency to the equivalent pendulum natural frequency, and forcing amplitude. If we momentarily consider the problem with no forcing or damping, " D D 0, then the peigenmodes  n and corresponding natural frequencies n / of free vibration of the rod satisfy B( n )ssss C [(1  s)( n )s ]s  n  n  n D 0 ; and form a complete basis of functions that satisfy the boundary conditions (28). Now the eigenmodes  n (s; B)

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Dynamics of Parametric Excitation, Figure 10 a Definition sketch of the parametrically excited flexible rod model. b Comparison between theory (solid line and shading) and the experimental results of [44] (dashed line), for the stability region as a function of the dimensionless parameters B and  using the experimental value " D 0:02, and the representative damping value 1 D 0:01 (which was not measured in the experiment) The theory is based on calculations close to the resonance tongue interaction between when B0;1 D B1;3 when  D c D 460:7. After [15], reproduced with permission from SIAM

are in general not expressible in closed form except at the special values of B at which n (B) D 0, in which case [20] there is a solution in terms of the Bessel function J1/3 . The same analysis shows that there are infinitely many such B-values, B0;n , for n D 1; 2; 3; 4; : : :, accumulating at B D 0, the first few values of which are B0;1 D 0:127594, B0;2 D 0:017864, B0;3 D 0:0067336 and B0;4 D 0:0003503. These correspond respectively to where each eigenvalue locus n (B), for n D 1; 2; 3; 4, crosses the B-axis, with the corresponding eigenmodes there having n  1 internal zeros. The lowering of B through each B0;n -value implies that the nth mode becomes linearly unstable. Hence for B > B c :D B0;1 the unforced rod is stable to self-weight buckling (a result known to Greenhill [26]). Moreover, it can be shown [14,15] that each eigenmode  n (s) retains a qualitatively similar mode shape for B > B0;n with the nth mode having n D 1 internal zeros, and that the corresponding loci n (B) are approximately straight lines as shown in the upper part of Fig. 11. The lower part of Fig. 11 shows the results of a parametric analysis of the Eq. (27) for D 0. Basically, each time one of the eigenvalue loci passes through  times the square of a half-integer ( j/2)2 we reach a B-value for which there is the root point of an instability tongue. Essentially we have a Mathieu-equation-like Ince–Strutt diagram in the (B; ") plane for each mode n. These diagrams are overlaid on top of each other to produce the overall stability

plot. However, note from Fig. 11 that the slope of each eigenvalue locus varies with the frequency ratio . Hence as  increases, the values of all the B j/2;n , j D 0; : : : ; 1 for a given n increase with the same fixed speed. But this speed varies with the mode number n. Hence the instability tongues slide over each other as the frequency varies. Note the schematic diagram in the lower panel of Fig. 11 indicates a region of the stability for B < B0;1 for small positive ". Such a region would indicate the existence of rod of (marginally) lower bending stiffness than that required for static stability, that can be stabilized by the introduction of the parametric excitation. Essentially, the critical bending stiffness for the buckling instability is reduced (or, equivalently since B / `3 , the critical length for buckling is increased). This effect has been dubbed the ‘Indian rod trick’ because the need for bending stiffness in the wire means that mathematically this is a rod, not a rope (or string); see, e. g. the account in [4]. To determine in theory the precise parameter values at which the rod trick works turns out to be an involved process. That the instability boundary emerging from B0;1 in the lower panel of Fig. 11 should usually bend back to the left can be argued geometrically [7]. However, for -values at which B0;1  B1;n for some n, it has been shown that there is actually a singularity in the coefficient of the quadratic part of this buckling curve causing the instability boundary to bend to the right [20]. A complete un-

Dynamics of Parametric Excitation

Dynamics of Parametric Excitation, Figure 11 Summarizing schematically the results of the parametric resonance analysis of the parametric excited vertical column in the absence of damping. The upper part shows eigenvalue loci n (B) and the definition of the points Bj/2;n , j D 0; 1; 2; 3; : : : In the lower part, instability tongues in the (B; ")-plane are shown to occur with root points Bj/2;n and to have width "j . The shaded regions correspond to the where the vertical solution is stable. Solid lines represent neutral stability curves with Floquet multiplier + 1 (where ˛ is an integer) and dashed lines to multipliers  1 (where ˛ is half an odd integer). Reproduced from [14] with permission from the Royal Society

folding of this so-called resonance tongue interaction was performed in [15], also allowing for the presence of material damping. The resulting asymptotic analysis needs to be carried out in the presence of four small parameters; ",

, B  B0;1 and  c where c is the frequency-ratio value at which the resonance-tongue interaction occurs. The results produce the shaded region shown in Fig. 10b, which can be seen to agree well with the experimental results. Faraday Waves and Pattern Formation When a vessel containing a liquid is vibrated vertically, it is well known that certain standing wave patterns form on the free surface. It was Faraday in 1831 [19] who first noted that the frequency of the liquid motion is half that of the vessel. These results were confirmed by Rayleigh [50] who suggested that what we now know as parametric resonance was the cause. This explanation was shown to be

correct by Benjamin and Ursell [6] who derived a system of uncoupled Mathieu equations from the Navier–Stokes equations describing the motion of an ideal (non-viscous) fluid. There is one Mathieu equation (4), for each wave number i that fits into the domain, each with its own parameters ˛ i and ˇ i . They argue that viscous effects would effectively introduce a linear damping term, as in (8) and they found good agreement between the broad principal (subharmonic) instability tongue and the results of a simple experiment. Later, Kumer and Tuckerman [36] produced a more accurate Ince–Strutt diagram (also known as a dispersion relation in this context) for the instabilities of free surfaces between two viscous fluids. Through numerical solution of appropriate coupled Mathieu equations that correctly take into account the boundary conditions, they showed that viscosity acts in a more complicated way than the simple velocity proportional damping term in (8). They produced the diagram shown in Fig. 12b, which showed a good match with the experimental results. Note that, compared with the idealized theory, Fig. 12a, the correct treatment of viscosity causes a small shift in the critical wavelength of each instability, a broadening of the higher-harmonic resonance tongues and a lowering of the amplitude required for the onset of the principal subharmonic resonance if damping is introduced to the idealized theory (see the inset to Fig. 12b). More interest comes when Faraday waves are excited in a vessel where two different wave modes have their subharmonic resonance at nearly the same applied frequency, or when the excitation contains two (or more) frequencies each being close to twice the natural frequency of a standing wave. See for example the book by Hoyle [30] and references therein, for a glimpse at the amazing complexity of the patterns of vibration that can result and for their explanation in terms of dynamical systems with symmetry. Recently a new phenomenon has also been established, namely that of localized spots of pattern that can form under parametric resonance, so-called oscillons see e. g. [61] for a review and [39] for the first steps at an explanation. Fluid-Structure Interaction Faraday waves are parametric effects induced by the gross mechanical excitation of a fluid. In Subsect. “Autoparametric Resonance” we briefly mentioned that fluid-structure interaction can go the other way, namely that external fluid flow effects such as vortex shedding can excite (auto)parametric resonances in simple mechanical systems. Such flow-induced parametric resonance can also occur for continuous structures, in which waves can be excited in the structure due to external parametric loading of

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Dynamics of Parametric Excitation, Figure 12 Stability boundaries for Faraday waves from: a Benjamin and Ursell’s theory for ideal fluids; b the results of full hydrodynamic stability analysis by Kumar and Tuckerman. The horizontal axis is the frequency and the vertical scale the amplitude of the excitation. The inset to panel b shows the difference between the true instability boundary for the principal subharmonic resonance (lower curve) and the idealized boundary in the presence of simple proportional damping. After [36], to which the reader is referred for the relevant parameter values; reproduced with permission from Cambridge University Press

the fluid; see for example the extensive review by Païdousis and Li [47]. They also treat cases where the flow is internal to the structure: we are all familiar with the noisy ducts and pipes, for example in central heating or rumbling digestive systems, and it could well be that many of these effects are a result of parametric resonance. Semler and Païdousis [52] consider in detail parametric resonance effects in the simple paradigm of a cantilevered (free at one end, clamped at the other) flexible pipe conveying fluid. Think of this as a hosepipe resting on a garden lawn. Many authors have considered such an apparatus as a paradigm for internal flow-induced oscillation

problems (see the extensive references in [47,52]), starting from Benjamin’s [5] pioneering derivation of equations of motion from the limit of a segmented rigid pipe. The equations of motion for such a situation are similar to those of the rod (27), but with significant nonlinear terms arising from inertial and fluid-structure interaction effects. In the absence of periodic fluctuations in the flow, then it is well known that there is a critical flow rate beyond which the pipe will become oscillatory, through a Hopf bifurcation (flutter instability). This effect can be seen in the way that hosepipes writhe around on a lawn, and indeed forms the basis for a popular garden toy. Another form of instability can occur when the mean flow rate is beneath the critical value for flutter, but the flow rate is subject to a periodic variation, see [52] and references therein. There, a parametric instability can occur when the pulse rate is twice the natural frequency of small amplitude vibrations of the pipe. In [52] this effect was observed in theory, in numerical simulation and in a careful experimental study, with good agreement between the three. Parametric resonance could also be found for super-critical flow rates. In this case, the effect was that the periodic oscillation born at the Hopf bifurcation was excited into a quasi-periodic response. Another form of fluid structure interaction can occur when the properties of an elastic body immersed in a fluid are subject to a temporal periodic variation. With a view to potential biological applications such as active hearing processes and heart muscles immersed in blood, Cortez et al. [16] consider a two-dimensional patch of viscous fluid containing a ring filament that is excited via periodic variation of its elastic modulus. Using a mixture of immersed boundary numerical methods and analytic reductions, they are able to find an effective Ince–Strutt diagram for this system and demonstrate the existence of parametric instability in which the excitation of the structure causes oscillations in the fluid. Taking a fixed wave number p of the ring in space, they find a sequence of harmonic and subharmonic instabilities as a parameter effectively like ˛ in the Mathieu equation is increased. Interestingly though, even for low viscosity, the principal subharmonic resonance tongue (corresponding to ˛ D 1/4) does not occur for low amplitude forcing, and the most easily excited instability is actually the fundamental, harmonic resonance (corresponding to ˛ D 1). Dispersion Managed Optical Solitons The stabilizing effect of high-frequency parametric excitation has in recent years seen a completely different application, in an area of high technological importance, namely

Dynamics of Parametric Excitation

optical communications. Pulses of light sent along optical fibers are a crucial part of global communication systems. A pulse represents a packet of linear waves, with a continuum of different frequencies. However, optical fibers are fundamentally dispersive, which means that waves with different wavelengths travel at different group velocities. In addition, nonlinearity, such as the Kerr effect of intensity-dependent refractive index change, and linear losses mean that there is a limit to how far a single optical signal can be sent without the need for some kind of amplifier. Depending on the construction of the fiber, dispersion can be either normal which means that lower frequencies travel faster, or anomalous in which case higher frequencies travel faster. One idea to overcome dispersive effects, to enable pulses to be travel over longer distances without breaking up, is to periodically vary the medium with which the fiber is constructed so that the dispersion is alternately normally and anomalously dispersive [38]. Another idea, first proposed by Hasegawa and Tappert [28], is to balance dispersion with nonlinearity. This works because the complex wave envelope A of the electric field of the pulse can be shown to leading order to satisfy the nonlinear Schrödinger (NLS) equation i

dA 1 d2 A C jAj2 A D 0 ; C dz 2 dt 2

(30)

in which time and space have been rescaled so that the anomalous dispersion coefficient (multiplying the secondderivative term) and the coefficient of the Kerr nonlinearity are both unity. Note the peculiarity in optics compared to other wave-bearing system in that time t plays the role of the spatial coordinate, and the propagation distance z is the evolution variable. Thus we think of pulses as being time-traces that propagate along the fiber. Now, the NLS equation is completely integrable and has the well-known soliton solution 2

A(z; t) D  sech(t  cz)e i z ;

(31)

which represents a smooth pulse with amplitude , which propagates at ‘speed’ c. Moreover, Eq. (30) is completely integrable [69] and so arbitrary initial conditions break up into solitons of different speed and amplitude. Optical solitons were first realized experimentally by Mollenauer et al. [43]. However, true optical fibers still suffer linear losses, and the necessary periodic amplification of the optical solitons to enable them to survive over long distances can cause them to jitter [25] and to interact with each other as they move about in time. The idea of combining nonlinearity with dispersion management in order to stabilize soliton propagation was

first introduced by Knox, Forysiak and Doran [35], was demonstrated experimentally by Suzuki et al. [54] and has shown surprisingly advantageous performance that is now being exploited technologically (see [60] for a review). Essentially, dispersion management introduces a periodic variation of the dispersion coefficient d(z), that can be used to compensate the effects of periodic amplification, which with the advent of modern erbium-doped optical amplifiers can be modeled as a periodic variation of the nonlinearity. As an envelope equation we arrive at the modified form of (30), i

dA d(z) d2 A C  (z)jAj2 A D 0 ; C dz 2 dt 2

(32)

where the period L of the dispersion map d(z) and the amplifier spacing (the period of  (z)) Z may not necessarily be the same. Two particular regimes are of interest. One, of application to long-haul, sub-sea transmission systems where the cable is already laid with fixed amplifiers, is to have L  Z. Here, one modifies an existing cable by applying bits of extra fiber with large dispersion of the opposite sign only occasionally. In this case, one can effectively look only at the long scale and average out the effect of the variation of  . Then the parametric term only multiplies the second derivative. Another interesting regime, of relevance perhaps when designing new terrestrial communication lines, is to have L D Z so that one adds loops of extra opposite-dispersion fiber at each amplification site. In both of these limits we have an evolution equation whose coefficients are periodic functions of the evolution variable z with period L. In either regime, the periodic (in z) parameter variation has a profound stabilizing influence, such that the effects of linear losses and jitter are substantially reduced. A particular observation is that soliton solutions appear to (32) breathe during each period; that is, their amplitude and phase undergo significant periodic modulation. Also, with increase in the amplitude of the oscillatory part of d, so the average shape of the breathing soliton becomes less like a sech-pulse and more like a Gaussian. An advantage is that the Gaussian has much greater energy the regular NLS soliton (31) for the same average values of d and  , hence there is less of a requirement for amplification. Surprisingly, it has been found that the dispersion managed solitons can propagate stably even if the average of d is slightly positive, i. e. in the normal dispersion region where there is no corresponding NLS soliton. This has a hugely advantageous consequence. The fact that the local dispersion can be chosen to be high, but the average dispersion can be chosen to be close to zero, is the key to enabling the jitter introduced by the amplifiers to be greatly suppressed

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compared with the regular solitons. Finally the large local modulation of phase that is a consequence of the breathing, means that neighboring solitons are less likely to interact with each other. A mathematical analysis of why these surprising effects arise is beyond the scope of this article, see e. g. [32,60] for reviews, but broadly-speaking the stable propagation of dispersion-managed solitons is due to the stabilizing effects of high frequency parametric excitation. Future Directions This article has represented a whistle-stop tour of a wide areas of science. One of the main simplifications we have taken is that we have assumed the applied excitation to the system to be periodic; in fact, in most examples it has been pure sinusoidal. There is a growing literature on quasiperiodic forcing, see e. g. [51], and it would be interesting to look at those effects that are unique to parametrically excited systems that contain two or more frequencies. The inclusion of noise in the excitation also needs to be investigated. What effect does small additive or multiplicative noise have on the shape of the resonance tongues? And what is the connection with the phenomenon of stochastic resonance [66]? There is a mature theory of nonlinear resonance in systems with two degrees of freedom which, when in an integrable limit, undergoes quasiperiodic motion with two independent frequencies. Delicate results, such as the KAM theorem, see e. g. [55], determine precisely which quasiperiodic motions (invariant tori) survive as a function of two parameters, the size of the perturbation from integrability and the frequency detuning between the two uncoupled periodic motions. For the tori that break up (inside so-called Arnol’d tongues) we get heteroclinic tangles and bands of chaotic motion. Parametric resonance tends to be slightly different though, since in the uncoupled limit, one of the two modes typically has zero amplitude (what we referred to earlier as a semi-trivial solution). It would seem that there is scope for a deeper connection to be established between the Arnol’d tongues of KAM theory and the resonance tongues that occur under parametric excitation. In terms of applications, a large area that remains to be explored further is the potential exploitation of parametric effects in fluid-structure interaction problems. As I write, the heating pipes in my office are buzzing at an extremely annoying frequency that seems to arise on warm days when many radiators have been switched off. Is this a parametric effect? It would also seem likely that nature uses parametric resonance in motion. Muscles contract

longitudinally, but motion can be in a transverse direction. Could phenomena such as the swishing of bacterial flagella, fish swimming mechanisms and the pumping of fluids through vessels, be understood in terms of exploitation of natural parametric resonances? How about motion of vocal chords, the Basilar membrane in the inner ear, or the global scale pumping of the heart? Does nature naturally try to tune tissue to find the primary subharmonic resonance tongue? Perhaps other effects of parametric excitation that we have uncovered here, such as the quenching of resonant vibrations, structural stiffening, and frequency tuned motion, are already being usefully exploited in nature. If so, there could be interesting consequences for nature inspired design in the engineered world. Finally, let us return to our opening paradigm of how children effectively excite a parametric instability in order to induce motion in a playground swing. In fact, it was carefully shown by Case and Swanson [11] that the mechanism the child uses when in the sitting position is far more accurately described as an example of direct forcing rather than parametric excitation. By pushing on the ropes of the swing during the backwards flight, and pulling on them during the forward flight, the child is predominately shifting his center of gravity to and fro, rather than up and down; effectively as in Fig. 1c rather than Fig. 1b. Later, Case [10] argued that, even in the standing position, the energy gained by direct forcing from leaning backwards and forwards is likely to greatly outweigh any gained from parametric up-and-down motion of the child. The popular myth that children swinging is a suitable playground science demonstrator of parametric resonance was finally put to bed by Post et al. [49]. They carried out experiments on human patients and analyzed the mechanisms they use to move, reaching the conclusion that even in the standing position, typically 95% of the energy comes from direct forcing, although parametric effects do make a slightly more significant contribution for larger amplitudes of swing.

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31. Ince E (1956) Ordinary Differential Equations. Dover, New York 32. Jones C (2003) Creating stability from instability. In: Nonlinear dynamics and chaos; where do we go from here? IOP Publishing, Bristol, chap 4, pp 73–90 33. Jordan DW, Smith P (1998) Nonlinear Ordinary Differential Equations, 3rd edn. Oxford University Press, Oxford 34. Kapitza P (1951) Dinamicheskaya ustoichivost mayatnika pri koleblyushcheisya tochke podvesa. Z Eksperimentalnoi I Teoreticheskoi Fiziki 21:588–597, in Russian 35. Knoz F, Forysiak W, Doran N (1995) 10 gbit/s soliton communication systems over standard fiber at 1.55m and the use of dispersion compensation. IEEE J Lightwave Technol 13:1960–1995 36. Kumar K, Tuckerman L (1994) Parametric instability of the interface between two fluids. J Fluid Mech 279:49–68 37. Kuznetsov YA (2004) Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York 38. Lin C, Kogelnik H, Cohen L (1980) Optical-pulse equilization of low-dispersion transmission in single fibers in the 1.3–1.7m spectral region. Opt Lett 5:476–480 39. Lloyd D, Sandstede B, Avitabilie D, Champneys A (2008) Localized hexagon patterns in the planar swift-hohenberg equation. To appear in SIAM Appl Dyn Sys 40. Magnus W, Winkler S (1979) Hill’s Equation. Dover, New York 41. Mailybayev A, Seyranian AP (2001) Parametric resonance in systems with small dissipation. PMM J App Math Mech 65:755–767 42. Mathieu E (1868) Mémoire sur le mouvement vibratoire d’une membrane de forme elliptique. J Math Pure Appl 13:137–203 43. Mollenauer L, Stolen R, Gordon J (1980) Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys Rev Lett 45:1095–1097 44. Mullin A, Champneys T, Fraser W, Galan J, Acheson D (2003) The ‘Indian wire trick’ via parametric excitation: a comparison between theory and experiment. Proc Roy Soc Lond A 459:539–546 45. Nayfeh A (2000) Nonlinear Interactions: Analytical, Computational and Experimental Methods. Wiley Interscience, New York 46. Otterbein S (1982) Stabilization of the N-pendulum and the Indian link trick. Arch Rat Mech Anal 78:381–393 47. Païdoussis MP, Li G (1993) Pipes conveying fluid: a model dynamical problem. J Fluids Structures 7:137–204 48. Piccoli B, Kulkarni J (2005) Pumping a swing by standing and squatting: Do children pump time optimally? IEEE Control Systems Magazine 25:48–56 49. Post A, de Groot G, Daffertshofer A, Beek P (2007) Pumping a playground swing. Motor Control 11:136–150 50. Rayliegh L (1883) On the crispations of fluid resting upon a vibrating support. Phil Mag 16:50 51. Romeiras F, Romeiras F, Bondeson A, Ott E, Antonsen TM, Grebogi C (1989) Quasiperiodic forcing and the observability of strange nonchaotic attractors. Phys Scr 40:442–444 52. Semler C, Païdoussis M (1996) Nonlinear analysis of the parametric resonances of a planar fluid-conveying cantilevered pipe. J Fluids Structures 10:787–825 53. Stephenson A (1908) On a new type of dynamical stability. Mem Proc Manch Lit Phil Soc 52:1–10 54. Suzuki M, Morita N, Edagawa I, Yamamoto S, Taga H, Akiba S (1995) Reduction of gordon-haas timing jitter by periodic

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Entropy in Ergodic Theory

Entropy in Ergodic Theory JONATHAN L. F. KING University of Florida, Gainesville, USA Article Outline Glossary Definition of the Subject Entropy example: How many questions? Distribution Entropy A Gander at Shannon’s Noisy Channel Theorem The Information Function Entropy of a Process Entropy of a Transformation Determinism and Zero-Entropy The Pinsker–Field and K-Automorphisms Ornstein Theory Topological Entropy Three Recent Results Exodos Bibliography Glossary Some of the following definitions refer to the “Notation” paragraph immediately below. Use mpt for ‘measure-preserving transformation’. Measure space A measure space (X; X ; ) is a set X, a field (that is, a -algebra) X of subsets of X, and a countably-additive measure  : X ! [0; 1]. (We often just write (X; ), with the field implicit.) For a collection C X , use Fld(C)  C for the smallest field including C. The number (B) is the “-mass of B”. Measure-preserving map A measure-preserving map : (X; X ; ) ! (Y; Y; ) is a map : X ! Y such that the inverse image of each B 2 Y is in X , and ( 1 (B)) D (B). A (measure-preserving) transformation is a measure-preserving map T : (X; X ; ) ! (X; X ; ). Condense this notation to (T : X; X ; ) or (T : X; ). Probability space A probability space is a measure space (X; ) with (X) D 1; this  is a probability measure. All our maps/transformations in this article are on probability spaces. Factor map A factor map : (T : X; X ; ) ! (S : Y; Y; ) is a measure-preserving map : X ! Y which intertwines the transformations, ıT D S ı . And is an

isomorphism if – after deleting a nullset (a mass-zero set) in each space – this is a bijection and 1 is also a factor map. Almost everywhere (a.e.) A measure-theoretic statement holds almost everywhere, abbreviated a.e., if it holds off of a nullset. (Eugene Gutkin once remarked to me that the problem with Measure Theory is . . . that you have to say “almost everywhere”, almost everywhere.) a:e: For example, B  A means that (B X A) is zero. The a.e. will usually be implicit. Probability vector A probability vector vE D (v1 ; v2 ; : : : ) is a list of non-negative reals whose sum is 1. We generally assume that probability vectors and partitions (see below) have finitely many components. Write “countable probability vector/partition”, when finitely or denumerably many components are considered. Partition A partition P D (A1 ; A2 ; : : : ) splits X into pairwise disjoint subsets A i 2 X so that the disjoint union F # i A i is all of X. Each Ai is an atom of P. Use jPj or P for the number of atoms. When P partitions a probability space, then it yields a probability vector vE, where v j :D (A j ). Lastly, use Phxi to denote the P-atom that owns x. Fonts We use the font H ; E ; I for distribution-entropy, entropy and the information function. In contrast, the script font ABC : : : will be used for collections of sets; usually subfields of X . Use E() for the (conditional) expectation operator. Notation Z D integers, ZC D positive integers, and N D natural numbers D 0; 1; 2; : : :. (Some wellmeaning folk use N for ZC , saying ‘Nothing could be more natural than the positive integers’. And this is why 0 2 N. Use de and bc for the ceiling and floor functions; bc is also called the “greatest-integer function”. For an interval J :D [a; b) [1; C1], let [a : : : b) denote the interval of integers J \ Z (with a similar convention for closed and open intervals). E. g., (e : : : ] D (e : : : ) D f3g. For subsets A and B of the same space, ˝, use A B for inclusion and A ¦ B for proper inclusion. The difference set B X A is f! 2 B j ! … Ag. Employ Ac for the complement ˝ X A. Since we work in a probability space, if we let x :D (A), then a convenient convention is to have xc

denote 1  x ;

since then (Ac ) equals xc . Use A4B for the symmetric difference [AXB][[BXA]. For a collection C D fE j g j of sets in ˝, let the disjoint F F S union j E j or (C) represent the union j E j and also assert that the sets are pairwise disjoint.

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Use “8large n” to mean: “9n0 such that 8n > n0 ”. To refer to left hand side of an Eq. (20), use LhS(20); do analogously for RhS(20), the right hand side.

Definition of the Subject The word ‘entropy’ (originally German, Entropie) was coined by Rudolf Julius Emanuel Clausius circa 1865 [2,3], taken from the Greek  o ˛, ‘a turning towards’. This article thus begins (Prolegomenon, “introduction”) and ends (Exodos1 , “the path out”) in Greek. Clausius, in his coinage, was referring to the thermodynamic notion in physics. Our focus in this article, however, will be the concept in measurable and topological dynamics. (Entropy in differentiable dynamics2 would require an article by itself.) Shannon’s 1948 paper [6] on Information Theory, then Kolmogorov’s [4] and Sinai’s [7] generalization to dynamical systems, will be our starting point. Our discussion will be of the one-dimensional case, where the acting-group is Z. “My greatest concern was what to call it. I thought of calling it ‘information’, but the word was overly used, so I decided to call it ‘uncertainty’. John von Neumann had a better idea, he told me, ‘You should call it entropy, for two reasons. In the first place your uncertainty function goes by that name in statistical mechanics. In the second place, and more important, nobody knows what entropy really is, so in a debate you will always have the advantage.’ ” (Shannon as quoted in [59]) Entropy example: How many questions? Imagine a dartboard, Fig. 1, split in five regions A; : : : ; E with known probabilities. Blindfolded, you throw a dart 1 This is

the Greek spelling. 2 For instance, see [15,18,24,25].

Entropy in Ergodic Theory, Figure 1 This dartboard is a probability space with a 5-set partition. The atoms have probabilities 12 ; 18 ; 18 ; 18 ; 18 . This probability distribution will be used later in Meshalkin’s example

at the board. What is the expected number V of Yes/No questions needed to ascertain the region in which the dart landed? Solve this by always dividing the remaining probability in half. ‘Is it A?’ – if Yes, then V D 1. Else: ‘Is it B or C?’ – if Yes, then ‘Is it B?’ – if No, then the dart landed in C, and V D 3 was the number of questions. Evidently V D 3 also for regions B; D; E. Using “log” to denote base-2 logarithm3 , the expected number of questions4 is thus 1 1 1 1 1 1C 3C 3C 3C 3 2 8 8 8 8   4 X 1 note D p j log D 2: pj

E(V ) D

(1)

jD0

  Letting vE :D 12 ; 18 ; 18 ; 18 ; 18 be the probability vector, we can write this expectation as X (x) : E(V ) D x2E v

Here,  : [0; 1] ! [0; 1) is the important function5 (x) :D x log(1/x) ;

so extending by continuity gives

(0) D 0 : (2) An interpretation of “(x)” is the number of questions needed to winnow down to an event of probability x. Distribution Entropy Given a probability vector vE, define its distribution entropy as X H (E v ) :D (x) : (3) x2E v

This article will use the term distropy for ‘distribution entropy’, reserving the word entropy for the corresponding 3 In this paper, unmarked logs will be to base-2. In entropy theory, it does not matter much what base is used, but base-2 is convenient for computing entropy for messages described in bits. When using the natural logarithm, some people refer to the unit of information as a nat. In this paper, I have picked bits rather than nats. 4 This is holds when each probability p is a reciprocal power of two. For general probabilities, the “expected number of questions” interpretation holds in a weaker sense: Throw N darts independently at N copies of the dartboard. Efficiently ask Yes/No questions to determine where all N darts  landed. Dividing by N, then sending N ! 1, will be the p log 1p sum of Eq. (1). 5 There does not seem to be a standard name for this function. I use , since an uppercase  looks like an H, which is the letter that Shannon used to denote what I am calling distribution-entropy.

Entropy in Ergodic Theory

dynamical concept, when there is a notion of time involved. Getting ahead of ourselves, the entropy of a stationary process is the asymptotic average value that its distropy decays to, as we look at larger and larger finite portions of the process.   An equi-probable vector vE :D K1 ; : K: :; K1 evidently has H (E v ) D log(K). On a probability space, the “distropy of partition P”, written H (P) or H (A1 ; A2 ; : : :) shall mean the distropy of probability vector j 7! (A j ). A (finite) partition necessarily has finite distropy. A countable partition can have finite distropy, e. g. H (1/2; 1/4; 1/8; : : : ) D 2. One could also have infinite distropy: Consider a piece B X of mass 1/2 N . Splitting B into 2 k many equal-mass atoms gives an -sum of 2 k (k C N)/ (2 k 2 N ). Setting k D k N :D 2 N  N makes this -sum F1 equal 1; so splitting the pieces of X D ND1 B N , with 1 (B N ) D 2N , yields an 1-distropy partition.

Entropy in Ergodic Theory, Figure 2 Using natural log, here are the graphs of: (x) in solid red, H (x; x c ) in dashed green, 1  x in dotted blue. Both (x) and H (x; x c ) are strictly convex-down. The 1  x line is tangent to (x) at x D 1

Function  The (x) D x log(1/x) function6 has vertical tangent at x D 0, maximum at 1/e and, when graphed in nats slope 1 at x D 1. Consider partitions P and Q on the same space (X; ). Their join, written P _ Q, has atoms A \ B, for each pair A 2 P and B 2 Q. They are independent, written P?Q if (A\ B) D (A)(B) for each A; B pair. We write P < Q, and say that “P refines Q”, if each P-atom is a subset of some Q-atom. Consequently, each Q-atom is a union of Patoms. Recall, for ı a real number, our convention that ı c means 1  ı, in analogy with (B c ) equaling 1  (B) on a probability space. Distropy Fact

Entropy in Ergodic Theory, Figure 3 Using natural log: The graph of H (x1 ; x2 ; x3 ) in barycentric coordinates; a slice has been removed, between z D 0:745 and z D 0:821. The three arches are copies of the distropy curve from Fig. 2.

For partitions P; Q; R on probability space (X; ): (a) H (P)  log(# P), with equality IFF P is an equi-mass partition. (b) H (Q _ R)   H (Q) C H (R), with equality IFF Q?R. (c) For ı 2 0; 12 , the function ı 7! H (ı; ı c ) is strictly increasing. a:e: (d) R 4 P implies H (R)  H (P), with equality IFF R D P. 6 Curiosity:

Just in this paragraph we compute distropy in nats, that is, using natural logarithm. Given a small probability p 2 [0; 1] and setting x :D 1/p, note that (p) D log(x)/x 1/(x), where (x) denotes the number of prime numbers less-equal x. (This approximation is a weak form of the Prime Number Theorem.) Is there any actual connection between the ‘approximate distropy’ function P H (E p) :D p2Ep 1/(1/p) and Number Theory, other than a coincidence of growth rate?

Proof Use the strict concavity of (), together with Jensen’s inequality.  Remark 1 Although we will not discuss it in this paper, most distropy statements remain true with ‘partition’ replaced by ‘countable partition of finite distropy’. Binomial Coefficients The dartboard gave an example where distropy arises in a natural way. Here is a second example. For a small   ı > 0, one might guess that the binomial coefficient ınn grows asymptotically (as n ! 1) like 2"n , for some small ". But what is the correct relation between " and ı?

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Well, Stirling’s formula n!  [n/e]n gives nn

n!  c [ın]![ı c n]! [ın][ı n] [ı c n][ı n] 1 (recall ı c D 1  ı) : D ın c ıc n [ı [ı ] ]   Thus n1 log ınn  H (ı; ı c ). But by means of the above distropy inequalities, we get an inequality true for all n, not just asymptotically.  Lemma 2 (Binomial Lemma) Fix a ı 2 0; 12 and let H :D H (ı; ı c ). Then for each n 2 ZC : ! X n  2Hn : (4) j j2[0:::ı n]

Proof Let X f0; 1gn be the set of xE with #

Noisy Channel Shannon’s theorem says that a noisy channel has a channel capacity. Transmitting above this speed, there is a minimum error-rate (depending how much “above”) that no error-correcting code can fix. Conversely, one can transmit below – but arbitrarily close to – the channel capacity, and encode the data so as to make the error-rate less than any given ". We use Corollary 3 to show the existence of such codes, in the simplest case where the noise7 is a binary independent-process (a “Bernoulli” process, in the language later in this article). We have a channel which can pass one bit per second. Alas, there is a fixed noise-probability  2 [0; 12 ) so that a bit in the channel is perturbed into the other value. Each perturbation is independent of all others. Let H :D H (;  c ). The value [1  H] bits-per-second is the channel capacity of this noise-afflicted channel.

fi 2 [1 : : : n] j x i D 1g  ın :

On X, let P1 ; P2 ; : : : be the coordinate partitions; e. g. P7 D (A7 ; Ac7 ), where A7 :D fE x j x7 D 1g. Weighting each point 1 , the uniform distribution on X, gives that (A7 )  by jXj ı. So H (P7 )  H, by (c) in Sect. “Distropy Fact”. Finally, the join P1 _    _ Pn separates the points of X. So log(# X) D H (P1 _    _ Pn )  H (P1 ) C : : : C H (P n )  Hn ; making use of (a),(b) in “Distropy Fact”. And # X equals LhS (4).  A Gander at Shannon’s Noisy Channel Theorem We can restate the Binomial lemma using the Hamming metric on f0; 1gn , Dist(E x ; yE) :D # fi 2 [1 : : : n] j x i ¤ y i g :

Encoding/Decoding Encode using an “k; n–blockcode”; an injective map F : f0; 1g k ! f0; 1gn . The source text is split into consecutive k-bit blocks. A block xE 2 f0; 1g k is encoded to F(E x ) 2 f0; 1gn and then sent through the channel, where it comes out perturbed to ˛E 2 f0; 1gn . The transmission rate is thus k/n bits per second. For this example, we fix a radius r > 0 to determine the decoding map, Dr : f0; 1gn ! fOopsg t f0; 1g k : We set Dr (E ˛) to zE if there is a unique zE with F(Ez) 2 Bal(E ˛; r); else, set Dr (E ˛) :D Oops. One can think of the noise as a f0; 1g-independentprocess, with Prob(1) D , which is added mod-2 to the signal-process. Suppose we can arrange that the set fF(E x )) j xE 2 f0; 1g k g of codewords, is a strongly r-separated-set. Then

Use Bal(E x ; r) for the open radius-r ball centered at xE, and Bal(E x ; r) :D f yE j Dist (E x ; yE)  rg for the closed ball. The above lemma can be interpreted as saying that x ; ın)j  2H (ı;ı jBal(E

c )n

for each xE 2 f0; 1gn : (5)  Corollary 3 Fix n 2 ZC and ı 2 0; 12 , and let ;

H :D H (ı; ı c ) : Then there is a set C f0; 1gn , with # C  2[1H]n , that is strongly ın-separated. I. e., Dist (E x ; yE) > ın for each distinct pair xE; yE 2 C.

The probability that a block is mis-decoded is the probability, flipping a -coin n times that we get more than r many Heads:

(6)  1 Theorem 4 (Shannon) Fix a noise-probability  2 0; 2 and let H :D H (;  c ). Consider a rate R < [1  H] and an " > 0. Then 8large n there exists a k and a code F : f0; 1g k ! f0; 1gn so that: The F-code transmits bits at faster than R bits-per-second, and with error-rate < ". 7 The noise-process is assumed to be independent of the signalprocess. In contrast, when the perturbation is highly dependent on the signal, then it is sometimes called distortion.

Entropy in Ergodic Theory

Proof Let H0 :D H (ı; ı c ), where ı >  was chosen so close to  that ı<

1 2

and 1  H0 > R :

(7)

The information function has been defined so that its expectation is the distropy of P, E(IP ) D

where

k :D b[1  H0 ]nc :

(8)

By Corollary 3, there is a strongly ın-separated-set C

0 f0; 1gn with # C  2[1H ]n . So C is big enough to permit an injection F : f0; 1g k ! C. Courtesy Eq. (6), the probability of a decoding error is that of getting more than ın many Heads in flipping a -coin n times. Since ı > , the Weak Law of Large Numbers guarantees – once n is large enough – that this probability is less than the given ".  The Information Function Agree to use P D (A1 ; : : :), Q D (B1 ; : : :), R D (C1 ; : : :) for partitions, and F ; G for fields. With C a (finite or infinite) family of subfields of X , W their join G2C G is the smallest field F such that G F , for each G 2 C. A partition Q can be interpreted also as a field; namely, the field of unions of its atoms. A join of denumerably many partitions will be interpreted as a field, but a join of finitely many, P1 _    _ PN , will be viewed as a partition or as a field, depending on context. Conditioning a partition P on a positive-mass set B, let PjB be the probability vector A 7! (A\B) (B) . Its distropy is   X 1 (A \ B) H (PjB) D log : (A \ B)/(B) (B) A2P So conditioning P on a partition Q gives conditional distropy X H (PjQ) D H (PjB)(B) B2Q

D

X A2P;B2Q

log

1 (A\B) (B)

Conditioning on a Field For a subfield F X , recall that each function g 2 L1 () has a conditional expectation E(gjF ) 2 L1 (). It is the unique F -measurable function with Z

8B 2 F :

E(gjF )d D B

Z gd : B

Returning to distropy ideas, use (AjF ) for the conditional probability function; it is the conditional expectation E(1A jF ). So the conditional information function is IPjF (x) :D

X A2P

 log

 1 1A (x) : (AjF (x))

(11)

Its integral H (PjF ) :D

Z

IPjF d ;

is the conditional distropy of P on F . When F is the field of unions of atoms from some partition Q, then the number H (PjF ) equals the H (PjQ) from Eq. (9). Write G j % F to indicate  fields G1 G2 : : : S1 that are nested, and that Fld 1 G j D F , a.e. The Martingale Convergence Theorem (p. 103 in [20]) gives (c) below. Conditional-Distropy Fact Consider partitions P; Q; R and fields F and G j . Then a:e:

! (A \ B) :

(9)

A “dartboard” interpretation of H (PjQ) is The expected number of questions to ascertain the P-atom that a random dart x 2 X fell in, given that we are told its Q-atom. For a set A X, use 1A : X ! f0; 1g for its indicator function; 1A (x) D 1 IFF x 2 A. The information function of partition P, a map IP : X ! [0; 1), is   X 1 IP :D log (10) 1A () : (A) A2P

IP ()d D H (P) : X

Pick a large n for which k >R; n

Z

(a) 0  H (PjF )  H (P), with equality IFF P F , respectively, P?F . (b) H (Q _ RjF )  H (QjF ) C H (RjF ). (c) Suppose G j % F . Then H (PjG j ) & H (PjF ). (d) H (Q _ R) D H (QjR) C H (R). (d’) H (Q _ R1 jR0 ) D H (QjR1 _ R0 ) C H (R1 jR0 ). Imagining our dartboard (Fig. 1) divided by superimposed partitions Q and R, equality (d) can interpreted as saying: ‘You can efficiently discover where the dart landed in both partitions, by first asking efficient questions about R, then – based on where it landed in R – asking intelligent questions about Q.’

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Entropy in Ergodic Theory

Entropy of a Process Consider a transformation (T : X; ) and partition P D (A1 ; A2 ; : : :). Each “time” n determines a partition Pn :D T n P, whose jth-atom is T n (A j ). The process T; P refers W to how T acts on the subfield 1 0 P n X . (An alternative view of a process is as a stationary sequence V0 ; V1 ; : : : of random variables Vn : X ! ZC , where Vn (x) :D j because x is in the jth-atom of Pn .) Write E (T; P) or E T (P) for the “entropy” of the T; P process”. It is the limit of the conditional-distropy-numbers c n :D H (P0 jP1 _ P2 _    _ Pn1 ) : This limit exists since H (P) D c1  c2      0. Define the average-distropy-number n1 h n , where h n :D H (P0 _ P1 _    _ Pn1 ) : Certainly h n D c n C H (P1 _    _ Pn1 ) D c n C h n1 , since T is measure preserving. Induction gives h n D Pn 1 jD1 c j . So the Cesàro averages n h n converge to the entropy. Theorem 5 The entropy of process (T; P : X; X ; ) equals

T(x) :D [n 7! xnC1 ] : The time-zero partition P separates points, under the action of the shift. This L-atom time-zero partition has Phxi D Phyi IFF x0 D y0 . So no matter what shift-invariant measure is put on X, the time-zero partition will generate under the action of T. Time Reversibility A transformation need not be isomorphic to its inverse. Nonetheless, the average-distropy-numbers show that E(T 1 ; P) D E(T; P); although this is not obvious from the conditioning-definition of entropy. Alternatively, ! ˇ_ ˇ n H P0 ˇˇ Pj 1

D H (P0 _    _ Pn )  H (P1 _    _ Pn ) D H (Pn _    _ P0 )  H (Pn _    _ P1 ) ! ˇ_ ˇ n (12) D H P0 ˇˇ P j :

1 H (P0 _    _ Pn1 ) n!1 n ! ˇ_ ˇ n ˇ D lim H P0 ˇ P j n!1 1 ˇ1 ! ˇ_ D H P0 ˇˇ P j : lim

1

Bernoulli Processes

1

Both limits are non-increasing. The entropy E T (P)  0, W T with equality IFF P 1 1 P j . And E (P)  H (P), with equality IFF T; P is an independent process. Generators We henceforth only discuss invertible mpts, that is, when T 1 is itself an mpt. Viewing the atoms of P as “letters”, then, each x 2 X has a T; P-name : : : x2 x1 x0 x1 x2 x3 : : : ; PhT n (x)i,

IFF P separates points. That is, after deleting a (T-invariant) nullset, distinct points of X have distinct T; P-names. A finite set [1 : : : L] of integers, interpreted as an alphabet, yields the shift space X :D [1 : : : L]Z of doubly-infinite sequences x D (: : : x1 x0 x1 : : : ). The shift T : X ! X acts on X by

T n (x).

where xn is the P-letter owning A partition P generates (the whole field) under (T : W n 8 X; ), if 1 1 T P D X . It turns out that P generates 8 I am now at liberty to reveal that our X has always been a Lebesgue space, that is, measure-isomorphic to an interval of R together with countably many point-atoms (points with positive mass).

A probability vector vE :D (v1 ; : : : ; v L ) can be viewed as a measure on alphabet [1 : : : L]. Let vE be the resulting product measure on X :D [1 : : : L]Z , with T the shift on X and P the time-zero partition. The independent process (T; P : X; vE ) is called, by ergodic theorists, a Bernoulli process. Not necessarily consistently, we tend to refer to the underlying  transformation as a Bernoulli  shift.  The 12 ; 12 -Bernoulli and the 13 ; 13 ; 13 -Bernoulli have different process-entropies, but perhaps their underlying transformations are isomorphic? Prior to the Kolmogorov–Sinai definition of entropy9 of a transformation, this question remained unanswered. The equivalence of generating and separating is a technical theorem, due to Rokhlin. Assuming  to be Lebesgue is not much of a limitation. For instance, if  is a finite measure on any Polish space, then  extends to a Lebesgue measure on the -completion of the Borel sets. To not mince words: All spaces are Lebesgue spaces unless you are actively looking for trouble. 9 This is sometimes called measure(-theoretic) entropy or (perhaps unfortunately) metric entropy, to distinguish it from topological

Entropy in Ergodic Theory

Entropy of a Transformation The Kolmogorov–Sinai definition of the entropy of an mpt is E(T) :D supfE T (Q) j Q a partition on Xg :

Certainly entropy is an isomorphism invariant – but is it useful? After all, the supremum of distropies of partitions is always infinite (on non-atomic spaces) and one might fear that the same holds for entropies. The key observation (restated in Lemma 8c and proved below) was this, from [4] and [7]. Theorem 6 (Kolmogorov–Sinai Theorem) If P generates under T, then E(T) D E(T; P).     Thereupon the 12 ; 12 and 13 ; 13 ; 13 Bernoulli-shifts are not isomorphic, since their respective entropies are log(2) ¤ log(3). Wolfgang Krieger later proved a converse to the Kolmogorov–Sinai theorem. Theorem 7 (Krieger Generator Theorem, 1970) Suppose T ergodic. If E(T) < 1, then T has a generating partition. Indeed, letting K be the smallest integer K > E(T), there is a K-atom generator.10 Proof See Rudolph [21], or § 5.1 in  Joinings in Ergodic Theory, where Krieger’s theorem is stated in terms of joinings.  Entropy Is Continuous

Then Lemma 8b says that process-entropy varies continuously with varying the partition. Lemma 8 Fix a mpt (T : X; ). For partitions P; Q; Q0 , define R :D Q4Q0 and let ı :D H (R). Then (a) jH (Q)  H (Q0 )j  ı. (Distropy varies continuously with the partition.) (b) jE T (Q)  E T (Q0 )j  ı. (Process-entropy varies continuously with the partition.) (c) For all partitions Q Fld(T; P) : E T (Q)  E T (P). Proof (of (a)) Evidently Q0 _ R D Q0 _ Q D Q _ R. So H (Q0 )  H (Q _ R)  H (Q) C ı.  Proof (of (b)) As above, H

N _

! Q0j

N _

H

1

! Qj C H

1

N _

! :

Rj

1

Sending N ! 1 gives E T (Q0 )  E T (Q) C E T (R). Finally, E T (R)  H (R) and so E T (Q0 )  E T (Q) C ı.  Proof (of (c)) Let K :D jQj. Then there is a sequence of WL P` . By above, K-set partitions Q(L) ! Q with Q(L) 4 L E T (Q(L) ) ! E T (Q), so showing that E

T

L _

!

?

P`  E T (P)

L

will suffice. Note that 0

(B01 ; : : :), 0

Given ordered partitions Q D (B1 ; : : :) and Q D extend the shorter by null-atoms until jQj D jQ j. Let F Fat :D j [B j \ B0j ]; this set should have mass close to 1 if Q and Q0 are almost the same partition. Define a new partition Q4Q0 :D fFatg t fB i \ B0j j with i ¤ jg : (In other words, take Q _ Q0 and coalesce, into a single atom, all the B k \ B0k sets.) Topologize the space of parti  tions by saying11 that Q(L) ! Q when H Q4Q(L) ! 0. entropy. Tools known prior to entropy, such as spectral properties, did not distinguish the two Bernoulli-shifts; see  Spectral Theory of Dynamical Systems for the definitions. 10 It is an easier result, undoubtedly known much earlier, that every ergodic T has a countable generating partition – possibly of 1-distropy. 11 On the set of ordered K-set partitions (with K fixed) this conver   gence is the same as: Q(L) ! Q when  Fat Q(L) ; Q ! 1. An alternative approach is the Rokhlin metric, Dist(P; Q) :D H (PjQ) C H (QjP), which has the advantage of working for unordered partitions.

h N :D H

N1 _

L _

Tn

1 NH

W

N1 Pj 0

0 D H@

P`

L

nD0

So N1 h N  N ! 1.

!!

N1CL _

1 PjA :

jDL



C

1 N 2LH (P).

Now send 

Entropy Is Not Continuous The most common topology placed on the space ˝ of mpts is the coarse topology12 that Halmos discusses in his “little red book” [14]. The Rokhlin lemma (see p. 33 in [21]) implies that the isomorphism-class of each ergodic mpt is dense in ˝, (e. g., see p. 77 in [14]) disclosing that the S 7! E(S) map is exorbitantly discontinuous. 12 i. e, S ! T IFF 8A 2 X : (S 1 (A)4T 1 (A)) ! 0; this is n n a metric-topology, since our probability space is countably generated. This can be restated in terms of the unitary operator U T on L2 (), where U T ( f ) :D f ı T. Namely, S n ! T in the coarse topology IFF U S n ! U T in the strong operator topology.

211

212

Entropy in Ergodic Theory

Indeed, the failure happens already for process-entropy with respect to a fixed partition. A Bernoulli process T; P has positive entropy. Take mpts S n ! T, each isomorphic to an irrational rotation. Then each E(S n ; P) is zero, as shown in the later section "Determinism and ZeroEntropy". Further Results When F is a T-invariant subfield, agree to use T F for “T restricted to F ”, which is a factor (see Glossary) of T. Transformations T and S are weakly isomorphic if each is isomorphic to a factor of the other. The foregoing entropy tools make short shrift of the following. Lemma 9 (Entropy Lemma) Consider T-invariant subfields G j and F . (a) Suppose G j % F . Then E T G j % E(T F ). In particular, G F implies that E(T G )  E(T F ), so entropy is an invariant of weak-isomorphism. P (b) E(T G1 _G2 _::: )  j E(T G j ). P And E(T; Q1 _ Q2 _ : : :)  j E(T; Q j ). P (c) For mpts (S j : Yj ;  j ): E(S1  S2    ) D j E(S j ). (d) E(T 1 ) D E(T). More generally, E(T n ) D jnj  E(T). Meshalkin’s Map In the wake of Kolmogorov’s 1958 entropy paper, for two Bernoulli-shifts to be isomorphic one now knew that they had to have equal entropies. Meshalkin provided the first non-trivial example in 1959 [45]. Let S : Y ! Y be the Bernoulli-shift over the “letter”  alphabet  fE; D; P; Ng, with probability distribution 14 ; 14 ; 14 ; 14 . The letters E; D; P; N stand for Even, oDd, Positive, Negative, and will be used to describe the code (isomorphism) between the processes. Use T : X ! X for the Bernoulli-shift over “digit” alphabet 1;  1 1f0;1 C1;  C2; 2g, with probability  1 1 1 distribu 1 1 1 tion ; ; ; ; . Both distributions 4 ; 4 ; 4 ; 4 and  1 1 12 18 18  8 8 2 ; 8 ; 8 ; 8 ; 8 have distropy log(4). After deleting invariant nullsets from X and Y, the construction will produce a measure-preserving isomorphism : X ! Y so that T ı D ı S. The Code

The leftmost 0 is linked to the rightmost +1, as indicated by the longest-overbar. The left/right-parentheses form a 12 ; 12 -random-walk. Since this random walk is recurrent, every position in x will be linked (except for a nullset of points x). Below each 0, write “P” or “N” as the 0 is linked to a positive or negative digit. And below the other digits, write “E” or “D” as the digit is even or odd. So the upper name in X is mapped to the lower name, a point y 2 Y. map : X !  Y carries  1 This   the upstairs 1 1 1 1 1 1 1 1 distribution to ; ; ; ; ; ; ; 2 8 8 8 8 4 4 4 4 , downstairs. It takes some arguing to show that independence is preserved. The inverse map 1 views D and E as right-parentheses, and P and N as left. Above D, write the odd digit C1 or 1, as this D is linked to Positive or Negative. Markov Shifts A Bernoulli process T; P has independence P(1:::0] ?P1 whereas a Markov process is a bit less aloof: The infinite Past P(1:::0] doesn’t provide any more information about Tomorrow than Today did. That is, the conditional distribution P1 jP(1:::0] equals P1 jP0 . Equivalently,   note H (P1 jP0 ) D H P1 jP(1:::0] D E(T; P) : (13) The simplest non-trivial Markov process (T; P : X; ) is over a two-letter alphabet fa; bg, and has transition graph Fig. 4, for some choice of transition probabilities s and c. The graph’s Markov matrix is

 s c ; M D [m i; j ] i; j D 1 0 where c D 1  s, and m i; j denotes the probability of going from state i to state j. If Today’s distribution on the two states is the probability-vector vE :D [pa pb ], then Tomorrow’s is the product vEM. So a stationary process needs vEM D vE. This equa1 c tion has the unique solution pa D 1Cc and pb D 1Cc .

In X, consider this point x:

: : : 0 0 0 1 0 0 +1 +2 1 +1 0 : : : Regard each 0 as a left-parenthesis, and each non-zero as a right-parenthesis. Link them according to the legal way of matching parentheses, as shown in the top row, below: 0 0 0 1 0 0 +1 +2 1 +1 0 P N N D P P D

E

D

D ?

Entropy in Ergodic Theory, Figure 4 Call the transition probabilities s :D Prob(a!a) for stay, and c :D Prob a!b for change. These are non-negative reals, and sCcD1

Entropy in Ergodic Theory

An example of computing the probability of a word or cylinder set; (see Sect. “The Carathéodory Construction” in  Measure Preserving Systems) in the process, is

approximately 2E(T;P)n ; this, after discarding small mass from the space. But the growth of n 7! 2n is sub-exponential and so, for our rotation, E(T; P) must be zero. Theorem 10 (Shannon–McMillan–Breiman Theorem (SMB-Theorem)) Set E :D E(T; P), where the tuple (T; P : X; ) is an ergodic process. Then the average information function

s (baaaba) D pb mba maa maa mab mba c D 1ssc1: 1Cc The subscript on s indicates the dependence on the transition probabilities; let’s also mark the mpt and call it T s . Using Eq. (13), the entropy of our Markov map is

1 n!1 IP (x) ! E ; n [0:::n)

for a.e.

x2X:

(15)

The functions f n :D IP[0:::n) converge to the constant function E both in the L1 -norm and in probability. 14

E(Ts ) D pa  H (s; c) C pb  H (1; 0)

„ ƒ‚ … D0

1 D  [s log(s) C c log(c)] : 1Cc

(14)

Determinism and Zero-Entropy Irrational rotations have zero-entropy; let’s reveal this in two different ways. Equip X :D [0; 1) with “length” (Lebesgue) measure and wrap it into a circle. With “˚” denoting addition mod-1, have T : X ! X be the rotation T(x) :D x ˚ ˛, where the rotation number ˛ is irrational. Pick distinct points y0 ; z0 2 X, and let P be the partition whose two atoms are the intervals [y0 ; z0 ) and [z0 ; y0 ), wrapping around the circle. The T-orbit of each point x is dense13 in X. In particular, y0 has dense orbit, so P separates points – hence ?

generates – under T. Our goal, thus, is E(T; P) D 0.

Proof See the texts of Karl Petersen (p. 261 in [20]), or Dan Rudolph (p. 77 in [21]).  Consequences Recall that P[0:::n) means P0 _ P1 _    _ Pn1 , where P j :D T j P. As usual, P[0:::n) hxi denotes the P[0:::n) -atom owning x. Having deleted a nullset, we can restate Eq. (15) to now say that 8"; 8x; 8large n:   1/2[EC"]n   P[0:::n) hxi  1/2[E"]n :

(16)

This has the following consequence. Fixing a number ı > 0, we consider any set with (B)  ı and count the number of n-names of points in B. The SMB-Thm implies 

8"; 8large n; 8B  ı : ˇ ˇ ˇfn-names in Bgˇ  2[E"]n :

(17)

Rotations are Deterministic

Rank-1 Has Zero-Entropy

The forward T-orbit of each point is dense. This is true for y0 , and so the backward T; P-name of each x actually W n tells us which point x is. I. e., P 1 1 T P, which is our definition of “process T; P is deterministic”. Our P being finite, this determinism implies that E(T; P) is zero, by Theorem 5.

There are several equivalent definitions for “rank-1 transformation”, several of which are discussed in the introduction of [28]. (See Chap. 6 in [13] as well as [51] and [27] for examples of stacking constructions.) A rank-1 transformation (T : X; ) admits a generating partition P and a sequence of Rokhlin stacks S n X, with heights going to 1, and with (S n ) ! 1. Moreover, each of these Rokhlin stacks is P-monochromatic, that is, each level of the stack lies entirely in some atom of P. Taking a stack of some height 2n, let B D B n be the union of the bottom n levels of the stack. There are at most n many length-n names starting in Bn , by monochromaticity. Finally, (B n ) is almost 12 , so is certainly larger than ı :D 13 . Thus Eq. (17) shows that our rank-1 T has zero entropy.

Counting Names in a Rotation The P0 _    _ Pn1 partition places n translates of points y0 and of z0 , cutting the circle into at most 2n intervals. Thus H (P0 _ : : : _ Pn1 )  log(2n). And n1 log(2n) ! 0. Alternatively, the below SMB-theorem implies, for an ergodic process T; P, that the number of length-n names is an " > 0 and an N > 1/". Points x; T(x); : : : ; T N (x) have some two at distance less than N1 ; say, Dist(T i (x); T j (x)) < ", for some 0  i < j  N. Since T is an isometry, " > Dist(x; T k (x)) > 0, where k :D j  i. So the T k -orbit of x is "-dense. 13 Fix

14 In engineering circles, this is called the Almost-everywhere equipartition theorem.

213

214

Entropy in Ergodic Theory

Cautions on Determinism’s Relation to Zero-Entropy

K-Processes

A finite-valued process T; P has zero-entropy iff P

W1 1 P j . Iterating gives

Kolmogorov introduced the notion of a K-process or Kolmogorov process, in which the present becomes asymptotically independent of the distant past. The asymptotic past of the T; P process is called its tail field, where

1 _ 0

Pj

1 _

Pj ;

1

i. e., the future is measurable with respect to the past. This was the case with the rotation, where a point’s past uniquely identified the point, thus telling us its future. While determinism and zero-entropy mean the same thing for finite-valued processes, this fails catastrophically for real-valued (i. e., continuum-valued) processes, as shown by an example of the author’s. A stationary realvalued process V D : : : V1 V0 V1 V2 : : : is constructed in [40] which is simultaneously strongly deterministic: The two values V0 ; V1 determine all of V, future and past. and non-consecutively independent. This latter means that for each bi-infinite increasing integer sequence fn j g1 jD1 with no consecutive pair (always 1 C n j < n jC1 ), then the list of random variables : : : Vn 1 Vn 0 Vn 1 Vn 2 : : : is an independent process. Restricting the random variables to be countablyvalued, how much of the example survives? Joint work with Kalikow [39] produced a countably-valued stationary V which is non-consecutively independent as well as deterministic. (Strong determinism is ruled out, due to cardinality considerations.) A side-effect of the construction is that V’s time-reversal n 7! Vn is not deterministic. The Pinsker–Field and K-Automorphisms Consider the collection of zero-entropy sets, Z D Z T :D fA 2 X j E(T; (A; Ac )) D 0g :

(18)

Courtesy of Lemma 9b, Z is a T-invariant field, and 8Q Z : E(T; Q) D 0 :

(19)

The Pinsker field15 of T is this Z. It is maximal with respect to Eq. (19). Unsurprisingly, the Pinsker factor T Z has zero entropy, that is, E(T Z ) D 0. A transformation T is said to have completely-positive entropy if it has no (nontrivial) zero-entropy factors. That is, its Pinsker field Z T is the trivial field, ; :D f¿; Xg. 15 Traditionally, this called the Pinsker algebra where, in this context, “algebra” is understood to mean “ -algebra”.

Tail(T; P) :D

1 \

M _

Pj :

MD1 jD1

This T; P is a K-process if Tail(T; P) D ;. This turns out to be equivalent to what we might a call a strong form of “sensitive dependence on initial conditions”: For each fixed length L, the distant future _

Pj

j2(G:::GCL]

becomes more and more independent of _

Pj ;

j2(1:::0]

as the gap G ! 1. A transformation T is a Kolmogorov automorphism if it possesses a generating partition P for which T; P is a K-process. Here is a theorem that relates the “asymptotic forgetfulness” of Tail(T; P) D ;, to the lack of determinism implied by having no zero-entropy factors (See Walters, [23], p. 113. Related results appear in Berg [44]). Theorem 11 (Pinsker-Algebra Theorem) Suppose P is a generating partition for an ergodic T. Then Tail(T; P) equals Z T . Since Z T does not depend on P, this means that all generating partitions have the same tail field, and therefore K-ness of T can be detected from any generator. Another non-evident fact follows from the above. The future field of T; P is defined to be Tail (T 1 P). It is not obvious that if the present is independent of the distant past, then it is automatically independent of the distant future. (Indeed, the precise definitions are important; witness the Cautions on determinism section.) But since the entropy of a process, E(T; (A; Ac )), equals the entropy of the time-reversed process E(T 1 ; (A; Ac )), it follows that Z T equals Z T 1 . Ornstein Theory In 1970, Don Ornstein [46] solved the long-standing problem of showing that entropy was a complete isomorphism-

Entropy in Ergodic Theory

invariant of Bernoulli transformations; that is, that two independent processes with same entropy necessarily have the same underlying transformation. (Earlier, Sinai [52]) had shown that two such Bernoulli maps were weakly isomorphic, that is, each isomorphic to a factor of the other.) Ornstein introduced the notion of a process being finitely determined, see [46] for a definition, proved that a transformation T was Bernoulli IFF it had a finitely-determined generator IFF every partition was finitely-determined with respect to T, and showed that entropy completely classified the finitely-determined processes up to isomorphism. This seminal result led to a vast machinery for proving transformations to be Bernoulli, as well as classification and structure theorems [47,51,53]. Showing that the class of K-automorphisms far exceeds the Bernoulli maps, Ornstein and Shields produced in [48] an uncountable family of non-isomorphic K-automorphisms all with the same entropy.

And the topological entropy of T is Etop (T) :D supV Etop (T; V ) ;

(21)

taken over all open covers V. Thus Etop counts, in some sense, the growth rate in the number of T-orbits of length n. Evidently, topological entropy is an isomorphism invariant. Two continuous maps T : X ! X and S : Y ! Y are topologically conjugate (as isomorphism is called in this category) if there exists a homeomorphism : X ! Y with T D S . Lemma 12 (Subadditive Lemma) Consider a sequence s D (s l )1 1 [1; 1] satisfying s kCl  s k C s l , for all k; l 2 Z. Then the following limit exists in [1; 1], and lim n!1 snn D inf n snn . Topological entropy, or “top ent” for short, satisfies many of the relations of measure-entropy. Lemma 13

Topological Entropy

(a) V 4 W implies

Adler, Konheim and McAndrew, in 1965, published the first definition of topological entropy in the eponymous article [33]. Here, T : X ! X is a continuous self-map of a compact topological space. The role of atoms is played by open sets. Instead of a finite partition, one uses a finite16 open-cover V D fU j gLjD1 , i. e. each patch U j is open, and S their union (V) D X. (Henceforth, ‘cover’ means “open cover”.) Let Card(V) be the minimum cardinality over all subcovers. [ ˚ (V0 ) D X ; Card(V) :D Min ] V0 j V0 V and and let H (V ) D Htop (V ) :D log(Card(V )) :

Analogous to the definitions for partitions, define V _ W :D fV \ W j V 2 V and W 2 W g ;

T V :D fT 1 (U) j U 2 Vg and V[0:::n) :D V0 _ V1 _    _ V n1 ; W < V ; if each W -patch is a subset of some V -patch:

The T; V-entropy is E T (V ) D E(T; V ) D Etop (T; V )

  1 :D lim sup Htop V[0:::n) : n!1 n

(20)

16 Because we only work on a compact space, we can omit “finite”. Some generalizations of topological entropy to non-compact spaces require that only finite open-covers be used [37].

H (V )  H (W )

and E(T; V )  E(T; W ) :

(b) H (V _ W )  H (V) C H (W ). (c) H (T(V))  H (V), with equality if T is surjective. Also, E(T; V )  H (V).   (d) In Eq. (20), the limn!1 n1 H V[0:::n) exists. (e) Suppose T is a homeomorphism. Then E(T 1 ; V ) D E(T; V ) ;

for each cover V. Consequently, Etop (T 1 ) D Etop (T). (f) Suppose C is a collection of covers such that: For each cover W , there exists a V 2 C with V < W . Then Etop (T) equals the supremum of Etop (T; V ), just taken over those V 2 C. (g) For all ` 2 N:   Etop T ` D `Etop (T) : Proof (of (c)) Let C 4 V be a min-cardinality subcover. Then T C is a subcover of T V. So Card T V  jT Cj D jCj. As for entropy, inequality (b) and the foregoing give H (V [0:::n) )  H (V )n.  Proof (of (d)) Set s n :D H (V[0:::n) ). Then    s kCl  s k C H T k V[0:::l )  s k C s l ; by (b) and (c), and so the Subadditive Lemma 12, applies. 

215

216

Entropy in Ergodic Theory

Proof (of (g)) WLOG, ` D 3. Given V a cover, triple it to b V :D V \ T V \ T 2 V ; so _  j   _ T3 b V D T i (V) : j2[0:::N)

i2[0:::3N)

  Thus H T 3 ; b V ; N D H (T; V ; 3N), extending notation. Part (d) and sending N ! 1, gives E(T 3 ; b V) D 3H (T; V). Lastly, take covers such that 









E T 3 ; C(k) ! Etop T 3



and

E T; D(k) ! Etop (T) ;

as k!1. Define V(k) :D C(k) _ D(k) . Apply the above to V (k) , then send k!1. 

You Take the High Road and I’ll Take the Low Road There are several routes to computing top-ent, some via maximization, others, minimization. Our foregoing discussion computed Etop (T) by a family of sizes f k (n) D f kT (n), depending on a parameter k which specifies the fineness of scale. (In Sect. “Metric Preliminaries”, this k is an integer; in the original definition, an open cover.) Define two numbers: 1 b L f (k) :D lim sup log f k (n) and n!1 n 1 L f (k) :D lim inf log f k (n) : (22) n!1 n L f (k). If the limit exists Finally, let E f (T) :D sup k b f in Eq. (22) then agree to write L (k) for the common value. The A-K-M definition used the size fV (n) :D Card(V[0:::n) ), where Card(W ) :DMinimum cardinality of a subcover from W :

Using a Metric From now on, our space is a compact metric space (X; d). Dinaburg [36] and Bowen [34,35], gave alternative, equivalent, definitions of topological entropy, in the compact metric-space case, that are often easier to work with than covers. Bowen gave a definition also when X is not compact17 (see [35] and Chap. 7 in [23]). Metric Preliminaries An "-ball-cover comprises finitely many balls, all of radius ". Since our space is compact, every cover V has a Lebesgue number " > 0. I. e., for each z 2 X, the Bal(z; ") lies entirely inside at least one V-patch. (In particular, there is an "-ball-cover which refines V.) Let LEB(V) be the supremum of the Lebesgue numbers. Courtesy of Lemma 13f we can Fix a “universal” list V(1) 4 V(2) 4 : : : , with V(k) a 1k -ball-cover. For every T : X ! X, then, the lim k E(T; V(k) ) computes Etop (T). An "-Microscope Three notions are useful in examining a metric space (X; m) at scale ". Subset A X is an "separated-set, if m(z; z0 )  " for all distinct z; z0 2 A. Subset F X is "-spanning if 8x 2 X; 9z 2 F with m(x; z) < ". Lastly, a cover V is "-small if Diam(U) < ", for each U 2 V. 17 When X is not compact, the definitions need not coincide; e. g. [37]. And topologically-equivalent metrics, but which are not uniformly equivalent, may give the same T different entropies (see p. 171 in [23]).

Here are three metric-space sizes f" (n): Sep(n; ") :DMaximum cardinality of a dn –"-separated set: Spn(n; ") :DMinimum cardinality of a dn –"-spanning set: Cov(n; ") :DMinimum cardinality of a dn –"-small cover: These use a list (d n )1 nD1 of progressively finer metrics on X, where   d N (x; y) :D Max j2[0:::N) d T j (x); T j (y) : Theorem 14 (All-Roads-Lead-to-Rome Theorem) Fix " and let W be any d-"-small cover. Then 8n : Cov(n; 2")  Spn(n; ")  Sep(n; ")  Card(W [0:::n) ) : (ii) Take a cover V and a ı < LEB(V). Then 8n: Card(V[0:::n) )  Cov(n; ı) : (iii) The limit LCov (") D lim n n1 log(Cov(n; ")) exists in [0 : : : 1) : (iv)

(i)

ESep (T) D ESpn (T) D ECov (T) D ECard (T) by defn

D Etop (T) :

Proof (of (i)) Take F X, a min-cardinality dn -"-spanS ning set. So z2F Dz D X, where note

Dz :D dn -Bal(z; ") D

n1 \ jD0

   T  j Bal T j z; " :

Entropy in Ergodic Theory

This D :D fDz gz is a cover, and it is dn –2"–small. Thus Cov(n; 2")  jDj D jFj. For any metric, a maximal "-separated-set is automatically "-spanning; adjoin a putative unspanned point to get a larger separated set. Let A be a max-cardinality dn -"-separated set. Take C, a min-cardinality subcover of W [0:::n) . For each z 2 A, pick a C-patch Cz 3 z. Could some pair x; y 2 A pick T j the same C? Well, write C D n1 jD0 T (Wj ), with each Wj 2 W . For every j 2 [0 : : : n), then, d(T j (x); T j (y))  Diam(Wj ) < " :

Proof (of (ii)) Choose a min-cardinality d n -ı-small cover C. For each C 2 C and j 2 [0 : : : n), the d-Diam(T j C) < ı. So there is a V-patch VC; j  T j (C). Hence note

n1 \

  T  j VC; j  C :

jD0

Thus V[0:::n) 4 C. So   Card V[0:::n)  Card(C)  jCj D Cov(n; ı) :



Proof (of (iii)) To upper-bound Cov(k C l; ") let V and W be min-cardinality "-small covers, respectively, for metrics d k and d l . Then V \ T l (W ) is a "-small for d kCl . Consequently Cov(kCl; ")  Cov(k; ")  Cov(l; "). Thus n 7! log(Cov(n; ")) is subadditive.  Proof (of (iv)) Pick a V from the list in Sect. “Metric Preliminaries”, choose some 2" < LEB(V) followed by an "-small W from Sect. “Metric Preliminaries”. Pushing n ! 1 gives LCard (V)  LCov (2") 

b LSep (") LSpn (")  b LSpn (")  LSep (")

1 log(Sep(n; ")) (24) n limit exists, in arguments that subsequently send "&0. Ditto for LSpn ("). LSep (") D lim

n!1

This will be used during the proof of the Variational Principle. But first, here are two entropy computations which illustrate the efficacy in having several characterizations of topological entropy. Etop (Isometry) D 0

Hence d n (x; y) < "; so x D y. Accordingly, the z 7! Cz map is injective, whence jAj  jCj. 

V [0:::n) 3

zero as "&0. Consequently, we can pretend that the

 LCard (W ) : (23)

Now send V and W along the list in Sect. “Metric Preliminaries”.  Pretension Topological entropy takes its values in [0; 1]. A useful corollary of Eq. (23) can be stated in terms of any Distance(; ) which topologizes [0; 1] as a compact interval. For each continuous T : X  ! X on a compact metric-space, the Distance b LSep ("); LSep (") goes to

Suppose (T : X; d) is a distancepreserving map of a compact metric-space. Fixing ", a set is dn -"-separated IFF it is d-"-separated. Thus Sep(n; ") does not grow with n. So each b LSep (") is zero. Topological Markov Shifts Imagine ourselves back in the days when computer data is stored on large reels of fast-moving magnetic tape. One strategy to maximize the density of binary data stored is to not put timing-marks (which take up space) on the tape. This has the defect that when the tape-writer writes, say, 577 consecutive 1-bits, then the tape-reader may erroneously count 578 copies of 1. We sidestep this flaw by first encoding our data so : :1 word, then writing to tape. as to avoid the 11:577 Generalize this to a finite alphabet Q and a finite list F of disallowed Q-words. Extend each word to a common length K C 1; now F QKC1 . The resulting “K-step TMS (topological Markov shift)” is the shift on the set of doubly-1 Q-names having no substring in F . In the above magnetic-tape example, K D 576. Making it more realistic, suppose that some string of zeros, say 00:574 : :0, is also forbidden18 . Extending to length 577, we get 23 D8 new disallowed words of form 00:574 : :0b1 b2 b3 . We recode to a 1-step TMS (just called a TMS or a subshift of finite type) over the alphabet P :D QK . Each outlawed Q-word w0 w1    w K engenders a length-2 forbidden P-word (w0 ; : : : ; w K1 )(w1 ; : : : ; w K ). The resulting TMS is topologically conjugate to the original K-step. The allowed length-2 words can be viewed as the edges in a directed-graph and the set of points x 2 X is the set of doubly-1 paths through the graph. Once trivialities removed, this X is a Cantor set and the shift T : X ! X is a homeomorphism. The Golden Shift As the simplest example, suppose our magnetic-tape is constrained by the Markov graph, Fig. 5 that we studied measure-theoretically in Fig. 4. 18 Perhaps the ;-bad-length, 574, is shorter than the 1-bad-length because, say, ;s take less tape-space than 1s and so – being written more densely – cause ambiguity sooner.

217

218

Entropy in Ergodic Theory

Entropy in Ergodic Theory, Figure 5 Ignoring the labels on the edges, for the moment, the Golden shift, T, acts on the space of doubly-infinite paths through this graph. The space can be represented as a subset XGold  fa; bgZ , namely, the set of sequences with no two consecutive b letters.

We want to store the text of The Declaration of Independence on our magnetic tape. Imagining that English is a stationary process, we’d like to encode English into this Golden TMS as efficiently as possible. We seek a shift-invariant measure  on XGold of maximum entropy, should such exist. View PDfa; bg as the time-zero partition on X Gold ; that is, name xD: : : x1 x0 x1 x2 : : :, is in atom b IFF letter x0 is “b”. Any measure  gives conditional probabilities (aja) D: s ; note

note

1 H (s; 1  s) 2s 1  slog(s) C (1  s)log(1  s) : D 2s

f (s) :D

(25)

Certainly f (0) D f (1) D 0, so f ’s maximum occurs at 0 s) the (it turns out) unique pointb ps where the derivative f (b equals zero. Thisb s D (1 C 5)/2. Plugging in, the maximum entropy supportable by the Golden Shift is p



 p 1 C 5 p  # 2 3 5 C log p : (26) 2 3 5 1 C 2

5

log

2

Exponentiating, the number of -typical n-names grows like Gn , where 2 G :D 4

2 1 C

p

p 1C p 5 5 5

5

1 ; 1 C jmj

(28)

0 . for the smallest jmj with x m ¤ x m

Lemma 15 Consider a subshift X. Then the 1 log(Names X (n)) n

exists in [0; 1], and equals Etop (X).

But recall, E(T) D H (P1 jP[1:::0) )  H (P1 jP0 ). So among all measures that make the conditional distribution Pja equal (s; c), the unique one maximizing entropy is the (s; c)-Markov-process. Its entropy, derived in Eq. (14), is

"

d(x; x 0 ) :D

n!1

(bjb) D 0 :

2 p MaxEnt D 5 5

Top-ent of the Golden Shift For a moment, let’s work more generally on an arbitrary subshift (a closed, shiftinvariant subset) X QZ , where Q is a finite alphabet. Here, the transformation is always the shift – but the space is varying – so agree to refer to the top-ent as Etop (X). Let ˚ Names X (n) be the number of distinct words in the set x [0:::n) j x 2 X . Note that a metric inducing the product-topology on QZ is

lim

(bja) D: c ;

(ajb) D 1 ;

This expression19 looks unpleasant to simplify – it isn’t even obviously an algebraic number – and yet topological entropy will reveal its familiar nature. This, because the Variational Principle (proved in the next section) says that the top-ent of a system is the supremum of measure-entropies supportable by the system.

32

p 3 3p5 2 5 5 54 5: p 3 5

(27)

Proof With " 2 (0; 1) fixed, two n-names are d n -"-separated IFF they are not the same name. Hence Sep(n; ") D  Names X (n). To compute Etop (X Gold ), declare that a word is “golden” if it appears in some x 2 XGold . Each [n C 1]-golden word ending in a has form wa, where w is n-golden. An [n C 1]golden word ending in b, must end in ab and so has form wab, where w is [n  1]-golden. Summing up, NamesXGold (n C 1) D NamesXGold (n) C Names XGold (n  1) : This is the Fibonacci recurrence, and indeed, these are the Fibonacci numbers, since NamesXGold (0) D 1 and Names XGold (1) D 2. Consequently, we have that NamesXGold (n) Const  n ; p

where  D 1C2 5 is the Golden Ratio. So the sesquipedalian number G from Eq. (27) is simply , and Etop (X Gold ) D log(). Since log()  0:694, each thousand bits written on tape (subject to the “no bb substrings” constraint) can carry at most 694 bits of information. 19 A popular computer-algebra-system was not, at least under my inexpert tutelage, able to simplify this. However, once top-ent gave the correct answer, the software was able to detect the equality.

Entropy in Ergodic Theory

Top-ent of a General TMS A (finite) digraph G engenders a TMS T : XG ! XG , as well as a f0; 1g-valued adjacency matrix ADAG , where a i; j is the number of directed-edges from state i to j. (Here, each a i; j is 0 or 1.) The (i; j)-entry in power An is automatically the number of length-n paths from i to j. Employing the matrix-norm P kMk :D i; j jm i; j j, then, kAkn D NamesX (n) : Happily Gelfand’s formula (see 10.13 in [58] or Spectral_ radius in [60]) applies: For an arbitrary (square) complex matrix, 1

lim kAn k n D SpecRad(A) :

n!1

(29)

This right hand side, the spectral radius of A, means the maximum of the absolute values of A’s eigenvalues. So the top-ent of a TMS is thus the Etop (X G ) D SpecRad(AG )

˚ ˇ :D Max jej ˇ e is an eigenvalue of AG : (30)

The (a; b)-adjacency matrix of Fig. 5 is

 1 1 ; 1 0 whose eigenvalues are  and

1  .

Labeling Edges Interpret (s; c; 1) simply as edge-labels in Fig. 5. The set of doubly-1 paths can also be viewed as a subset YGold fs; c; 1gZ , and it too is a TMS. The shift on YGold is conjugate (topologically isomorphic) to the shift on XGold , so they a fortiori have the same topent, log(). The (s; c; 1)-adjacency matrix is 2 3 1 1 0 40 0 15 : 1 1 0 Its j  j-largest eigenvalue is still , as it must. Now we make a new graph. We modify Fig. 5 by manufacturing a total of two s-edges, seven c-edges, and three edges 11 ; 12 ; 13 . Give these 2 C 7 C 3 edges twelve distinct labels. We could compute the resulting TMS-entropy from the corresponding 12 12 adjacency matrix. Alternatively, look at the (a; b)-adjacency matrix

 2 7 A :D : 3 0 p The roots of its characteristic polynomial arep1 ˙ 22. Hence Etop of this 12-symbol TMS is log(1 C 22).

The Variational Principle Let M :D M (X; d) be the set of Borel probability measures, and M (T) :D M (T : X; d) the set of T-invariant  2 M . Assign EntSup(T) :D sup fE (T) j  2 M (T)g : Theorem 16 (Variational EntSup(T) D Etop (T).

principle

(Goodson))

This says that top-ent is the top entropy – if there is a measure  which realizes the supremum. There doesn’t have to be. Choose a sequence of metric-systems (S k : Yk ; m k ) whose entropies strictly increase Etop (S k ) % L to some limit in (0; 1]. Let (S1 : Y1 ; m1 ) be the identity-map on a 1-point space. Define a new system (T : X; d), where F X :D k2[1:::1] Yk . Have T(x) :D S k (x), for the unique k with Yk 3 x. As for the metric, on Yk let d be a scaled version of m k , so that the d-Diam(Yk ) is less than 1/2 k . Finally, for points in distinct ˇcomponents, ˇ x 2 Yk and z 2 Y` , decree that d(x; z) :D ˇ2k  2` ˇ. Our T is continuous, and is a homeomorphism if each of the S k is. Certainly Etop (T) D L > Etop (S k ), for every k 2 [1 : : : 1]. If L is finite then there is no measure  of maximal entropy; for  must give mass to some Yk ; this pulls the entropy below L, since there are no compensatory components with entropy exceeding L. In contrast, when L D 1 then there is a maximalentropy measure (put mass 1/2 j on some component Yk j , where k j %1 swiftly); indeed, there are continuum-many maximal-entropy measures. But there is no 20 ergodic measure of maximal entropy. For a concrete L D 1 example, let S k be the shift on [1 : : : k]Z . Topology on M Let’s arrange our tools for establishing the Variational Principle. The argument will follow Misiurewicz’s proof, adapted from the presentations in [23] and [11]. Equip M with the weak- topology. 21 An A X is nice if its topological boundary @(A) is -null. And a partition is -nice if each atom is. 20 The ergodic measures are the extreme points of M(T); call them MErg (T). This M(T) is the set of barycenters obtained from Borel probability measures on MErg (T) (see Krein-Milman_ theorem, Choquet_theory in [60]). In this instance, what explains the failure to have an ergodic maximal-entropy measure? Let k be an invariant ergodic measure on Yk . These measures do converge to the one-point (ergodic) probability measure 1 on Y1 . But the map  7! E (T) is not continuousR at 1 . R 21 Measures ˛ !  IFF f d˛L ! f d, for each continuL ous f : X ! R. This metrizable topology makes M compact. Always, M(T) is a non-void compact subset (see  Measure Preserving Systems.)

219

220

Entropy in Ergodic Theory

Proposition 17 If ˛L !  and A X is -nice, then ˛L (A) ! (A). Proof Define operator U (D) :D lim supL ˛L (D). It suffices to show that U (A)  (A). For since Ac is -nice too, then U (Ac )  (Ac ). Thus limL ˛L (A) exists, and equals (A). Because C :D A is closed, the continuous functions f N & 1C pointwise, where f N (x) :D 1  Min(N  d(x; C); 1) : By the Monotone Convergence theorem, then, Z N f N d ! (C) : And (C) D (A), since AR is nice. Fixing N, then, it suffices to establish U (A)  f N d. But f N is continuous, so Z Z f N d D lim sup f N d˛L L!1 Z  lim sup 1A d˛L D U (A) :  L!1

Corollary 18 Suppose ˛L ! , and partition P is -nice. Then H˛L (P) ! H (P). The diameter of partition P is MaxA2P Diam(A). Proposition 19 Take  2 M and " > 0. Then there exists a -nice partition with Diam(P) < ". Proof Centered at an x, the uncountably many balls fBal(x; r) j r 2 (0; ")g have disjoint boundaries. So all but countably many are -nice; pick one and call it B x . Compactness gives a finite nice cover, say, fB1 ; : : : ; B7 g, at different centers. Then the partition P :D (A1 ; : : : ; A7 ) is S  nice, 22 where A k :D B k X k1 jD1 B j . Here is a consequence of Jensen’s inequality. Lemma 20 (Distropy-Averaging Lemma) For ;  2 M , a partition R, and a number t 2 [0; 1], t H (R) C t c H (R)  H tCt c  (R) : Strategy for EntSup(T)  Etop (T). Choose an " > 0. For L D 1; 2; 3; : : : , take a maximal (L; ")–separated–set FL X, then define 1 F D F" :D lim sup log(jFL j) : L!1 L 22 For any two sets B; B 0 X, the union @B [ @B 0 is a superset of the three boundaries @(B [ B0 ); @(B \ B0 ); @(B X B0 ).

Let 'L () be the equi-probable measure on FL ; each point has weight 1/jFL j. The desired invariant measure  will come from the Cesàro averages 1 X T ` 'L ; ˛L :D L `2[0:::L)

which get more and more invariant. Lemma 21 Let  be any weak- accumulation point of the above f˛L g1 1 . (Automatically,  is T-invariant.) Then E (T)  F. Indeed, if Q is any -nice partition with Diam(Q) < ", then E (T; Q)  F. Tactics As usual, Q[0:::N) means Q0 _Q1 _: : :_Q N1 . Our goal is 8N :

?

F

  1 H Q[0:::N) : N

(31)

Fix N and P :D Q[0:::N) , and a ı > 0. It suffices to verify: 8large L  N, ? 1 1 log(jFL j)  ı C H˛L (P) ; L N

(32)

since this and Corollary 17 will prove Eq. (31): Pushing L ! 1 along the sequence that produced  essentially sends LhS(32) to F, courtesy Eq. (24). And RhS(32) goes to ı C N1 H (P), by Corollary 17, since P is -nice. Descending ı & 0, hands us the needed Eq. (31). Remark 22 The idea in the following proof is to mostly fill interval [0::L) with N-blocks, starting with a offset K 2 [0::N). Averaging over the offset will create a Cesàro average over each N-block. Averaging over the N-blocks will allow us to compute distropy with respect to the averaged measure, ˛L . Proof (of Eq. (32)) Since L is frozen, agree to use ' for the ' L probability measure. Our dL -"-separated set F L has at most one point in any given atom of Q[0:::L) , thereupon   log(jFL j) D H' Q[0:::L) : Regardless of the “offset” K 2 [0 : : : N), we can always fit C :D b LN N c many N-blocks into [0 : : : L). Denote by G(K) :D [K : : : K C CN), this union of N-blocks, the good set of indices. Unsurprisingly, B(K) :D [0 : : : L) X G(K) is the bad index-set. Therefore, Bad(K)





‚  …„ _

H' Q[0:::L)  H'

j2B(K)

ƒ 

Good(K)

‚  …„ _

Q j C H'

ƒ  Qj

:

j2G(K)

(33)

Entropy in Ergodic Theory

this last inequality, when n is large. The upshot: E (T)  2 C Etop (T). Applied to a power T ` , this asserts that E (T ` )  2 C Etop (T ` ). Thus

Certainly Bad(K)  3Nlog(jQj). So 1 NL

X

Bad(K) 

K2[0:::N)

3N log(jQj) : L

This is less than ı, since L is large. Applying to Eq. (28) now produces 1 1 X Good(K) : log(jFL j)  ı C L NL

1 NL

P

K2[0:::N)

(34)

2 C Etop (T) ; ` using Lemma 9d and using Lemma 13g. Now coax ` ! 1.  E (T) 

K

Note _

j

T (Q) D

j2G(K)

Three Recent Results

_

:

c2[0:::C)T KCcN (P)

So Good(K) 

X

Having given an survey of older results in measure-theoretic entropy and in topological entropy, let us end this survey with a brief discussion of a few recent results, chosen from many.

H' (T KCc N P) :

c

P

This latter, by definition, equals c H T KCcN(') (P). We conclude that 1 X 1 XX Good(K)  H T KCcN ' (P) NL NL c K K 1 X H T ` ' (P) ;  NL `2[0:::L)

by adjoining a few translates of P; 1  H˛L (P) ; N by the Distropy-averaging lemma 18; P since ˛L is the average L1 ` T ` '. Thus Eq. (34) implies Eq. (32), our goal.  Proof (of EntSup(T)  Etop (T)) Fix a T-invariant . For partition Q D (B1 ; : : : ; B K ), choose a compact set A k B k with (B k X A k ) small. (This can be done, since F is automatically a regular measure [58].) Letting c and P :D (D; A1 ; : : : ; A K ), we can have D :D i Ai made H (PjQ) as small as desired. Courtesy of Lemma 8b, then, we only need consider partitions of the form that P has. Open-cover V D (U1 ; : : : ; U K ) has patches U k :D D[ A k . What atoms of, say, P[0:::3) , can the intersection U9 \ T 1 (U2 ) \ T 2 (U5 ) touch? Only the eight atoms (D or A9 ) \ T 1 (D or A2 ) \ T 2 (D or A5 ) : Thus ] P[0:::n)  2n  ] V[0:::n) . (Here, ] () counts the number of non-void atoms/patches.) So     1 1 H P[0:::n)  1 C log ] V [0:::n) n n  1 C 1 C Etop (T) ;

Ornstein–Weiss: Finitely-Observable Invariant In a landmark paper [10], Ornstein and Weiss show that all “finitely observable” properties of ergodic processes are secretly entropy; indeed, they are continuous functions of entropy. This was generalized by Gutman and Hochman [9]; some of the notation below is from their paper. Here is the setting. Consider an ergodic process, on a non-atomic space, taking on only finitely many values in N; let C be some family of such processes. An observation scheme is a metric space (˝; d) and a sequence of n functions S D (S n )1 1 , where Sn maps N  : : :  N into ˝. 1 On a point xE 2 N , the scheme converges if n 7! S n (x1 ; x2 ; : : : x n )

(35)

converges in ˝. And on a particular process X , say that S converges, if S converges on a.e. xE in X . A function J : C ! ˝ is isomorphism invariant if, whenever the underlying transformations of two processes X ; X 0 2 C are isomorphic, then J(X ) D J(X 0 ). Lastly, say that S “converges to J”, if for each X 2 C, scheme S converges to the value J(X ). The work of David Bailey [38], a student of Ornstein, produced an observation scheme for entropy. The Lempel-Ziv algorithm [43] was another entropy observer, with practical application. Ornstein and Weiss provided entropy schemes in [41] and [42]. Their recent paper “Entropy is the only finitelyobservable invariant” [10], gives a converse, a uniqueness result. Theorem 23 (Ornstein, Weiss) Suppose J is a finitely observable function, defined on all ergodic finite-valued processes. If J is an isomorphism invariant, then J is a continuous function of the entropy.

221

222

Entropy in Ergodic Theory

Gutman–Hochman: Finitely-Observable Extension Extending the Ornstein–Weiss result, Yonatan Gutman and Michael Hochman, in [9] proved that it holds even when the isomorphism invariant, J, is well-defined only on certain subclasses of the set of all ergodic processes. In particular they obtain the following result on three classes of zero-entropy transformations. Theorem 24 (Gutman, Hochman) Suppose J() is a finitely observable invariant on one of the following classes: (i)

The Kronecker systems; the class of systems with pure point spectrum. (ii) The zero-entropy mild mixing processes. (iii) The zero-entropy strong mixing processes. Then J() is constant.

Consider (G; G), a topological group and its Borel field (sigma-algebra). Let G  X be the field on G  X generated by the two coordinate-subfields. A map (36)

is measurable if 1

The paper introduces a new isomorphism invariant, the “f invariant”, and shows that, for Bernoulli actions, the f invariant agrees with entropy, that is, with the distropy of the independent generating partition. Exodos Ever since the pioneering work of Shannon, and of Kolmogorov and Sinai, entropy has been front and center as a major tool in Ergodic Theory. Simply mentioning all the substantial results in entropy theory would dwarf the length of this encyclopedia article many times over. And, as the above three results (cherry-picked out of many) show, Entropy shows no sign of fading away. . . Bibliography

Entropy of Actions of Free Groups

 : GxX ! X

Theorem 25 (Lewis Bowen) Let G be a finite-rank free group. Then two Bernoulli G-actions are isomorphic IFF they have the same entropy.

(X ) G  X :

Use g (x) for (g; x). This map in Eq. (36) is a (measure-preserving) group action if 8g; h 2 G: g ı h D g h , and each g : X ! X is measure preserving. This encyclopedia article has only discussed entropy for Z-actions, i. e., when G D Z. The ergodic theorem, our definition of entropy, and large parts of ergodic theory, involve taking averages (of some quantity of interest) over larger and larger “pieces of time”. In Z, we typically use the intervals I n :D [0 : : : n). When G is Z  Z, we might average over squares I n  I n . The amenable groups are those which possess, in a certain sense, larger and larger averaging sets. Parts of ergodic theory have been carried over to actions of amenable groups, e. g. [49] and [55]. Indeed, much of the Bernoulli theory was extended to certain amenable groups by Ornstein and Weiss, [50]. The stereotypical example of a non-amenable group, is a free group (on more than one generator). But recently, Lewis Bowen [8] succeeded in extending the definition of entropy to actions of finite-rank free groups.

Historical 1. Adler R, Weiss B (1967) Entropy, a complete metric invariant for automorphisms of the torus. Proc Natl Acad Sci USA 57:1573– 1576 2. Clausius R (1864) Abhandlungen ueber die mechanische Wärmetheorie, vol 1. Vieweg, Braunschweig 3. Clausius R (1867) Abhandlungen ueber die mechanische Wärmetheorie, vol 2. Vieweg, Braunschweig 4. Kolmogorov AN (1958) A new metric invariant of transitive automorphisms of Lebesgue spaces. Dokl Akad Nauk SSSR 119(5):861–864 5. McMillan B (1953) The basic theorems of information theory. Ann Math Stat 24:196–219 6. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423,623–656 7. Sinai Y (1959) On the Concept of Entropy of a Dynamical System, Dokl Akad Nauk SSSR 124:768–771

Recent Results 8. Bowen L (2008) A new measure-conjugacy invariant for actions of free groups. http://www.math.hawaii.edu/%7Elpbowen/ notes11.pdf 9. Gutman Y, Hochman M (2006) On processes which cannot be distinguished by finitary observation. http://arxiv.org/pdf/ math/0608310 10. Ornstein DS, Weiss B (2007) Entropy is the only finitely-observable invariant. J Mod Dyn 1:93–105; http://www.math.psu. edu/jmd

Ergodic Theory Books 11. Brin M, Stuck G (2002) Introduction to dynamical systems. Cambridge University Press, Cambridge 12. Cornfeld I, Fomin S, Sinai Y (1982) Ergodic theory. Grundlehren der Mathematischen Wissenschaften, vol 245. Springer, New York

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13. Friedman NA (1970) Introduction to ergodic theory. Van Nostrand Reinhold, New York 14. Halmos PR (1956) Lectures on ergodic theory. The Mathematical Society of Japan, Tokyo 15. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. (With a supplementary chapter by Katok and Leonardo Mendoza). Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge 16. Keller G, Greven A, Warnecke G (eds) (2003) Entropy. Princeton Series in Applied Mathematics. Princeton University Press, Princeton 17. Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge, 18. Mañé R (1987) Ergodic theory and differentiable dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete, ser 3, vol 8. Springer, Berlin 19. Parry W (1969) Entropy and generators in ergodic theory. Benjamin, New York 20. Petersen K (1983) Ergodic theory. Cambridge University Press, Cambridge 21. Rudolph DJ (1990) Fundamentals of measurable dynamics. Clarendon Press, Oxford 22. Sinai Y (1994) Topics in ergodic theory. Princeton Mathematical Series, vol 44. Princeton University Press, Princeton 23. Walters P (1982) An introduction to ergodic theory. Graduate Texts in Mathematics, vol 79. Springer, New York

Differentiable Entropy 24. Ledrappier F, Young L-S (1985) The metric entropy of diffeomorphisms. Ann Math 122:509–574 25. Pesin YB (1977) Characteristic Lyapunov exponents and smooth ergodic theory. Russ Math Surv 32:55–114 26. Young L-S (1982) Dimension, entropy and Lyapunov exponents. Ergod Theory Dyn Syst 2(1):109–124

Finite Rank 27. Ferenczi S (1997) Systems of finite rank. Colloq Math 73(1):35– 65 28. King JLF (1988) Joining-rank and the structure of finite rank mixing transformations. J Anal Math 51:182–227

Maximal-Entropy Measures 29. Buzzi J, Ruette S (2006) Large entropy implies existence of a maximal entropy measure for interval maps. Discret Contin Dyn Syst 14(4):673–688 30. Denker M (1976) Measures with maximal entropy. In: Conze J-P, Keane MS (eds) Théorie ergodique, Actes Journées Ergodiques, Rennes, 1973/1974. Lecture Notes in Mathematics, vol 532. Springer, Berlin, pp 70–112 31. Misiurewicz M (1973) Diffeomorphism without any measure with maximal entropy. Bull Acad Polon Sci Sér Sci Math Astron Phys 21:903–910

Topological Entropy 32. Adler R, Marcus B (1979) Topological entropy and equivalence of dynamical systems. Mem Amer Math Soc 20(219)

33. Adler RL, Konheim AG, McAndrew MH (1965) Topological entropy. Trans Am Math Soc 114(2):309–319 34. Bowen R (1971) Entropy for group endomorphisms and homogeneous spaces. Trans Am Math Soc 153:401–414. Errata 181:509–510 (1973) 35. Bowen R (1973) Topological entropy for noncompact sets. Trans Am Math Soc 184:125–136 36. Dinaburg EI (1970) The relation between topological entropy and metric entropy. Sov Math Dokl 11:13–16 37. Hasselblatt B, Nitecki Z, Propp J (2005) Topological entropy for non-uniformly continuous maps. http://www.citebase.org/ abstract?id=oai:arXiv.org:math/0511495

Determinism and Zero-Entropy, and Entropy Observation 38. Bailey D (1976) Sequential schemes for classifying and predicting ergodic processes. Ph D Dissertation, Stanford University 39. Kalikow S, King JLF (1994) A countably-valued sleeping stockbroker process. J Theor Probab 7(4):703–708 40. King JLF (1992) Dilemma of the sleeping stockbroker. Am Math Monthly 99(4):335–338 41. Ornstein DS, Weiss B (1990) How sampling reveals a process. Ann Probab 18(3):905–930 42. Ornstein DS, Weiss B (1993) Entropy and data compression schemes. IEEE Trans Inf Theory 39(1):78–83 43. Ziv J, Lempel A (1977) A universal algorithm for sequential data compression. IEEE Trans Inf Theory 23(3):337–343

Bernoulli Transformations, K-Automorphisms, Amenable Groups 44. Berg KR (1975) Independence and additive entropy. Proc Am Math Soc 51(2):366–370; http://www.jstor.org/stable/2040323 45. Meshalkin LD (1959) A case of isomorphism of Bernoulli schemes. Dokl Akad Nauk SSSR 128:41–44 46. Ornstein DS (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 5:337–352 47. Ornstein DS (1974) Ergodic theory randomness and dynamical systems, Yale Math Monographs, vol 5. Yale University Press, New Haven 48. Ornstein DS, Shields P (1973) An uncountable family of K-automorphisms. Adv Math 10:63–88 49. Ornstein DS, Weiss B (1983) The Shannon–McMillan–Briman theorem for a class of amenable groups. Isr J Math 44(3):53– 60 50. Ornstein DS, Weiss B (1987) Entropy and isomorphism theorems for actions of amenable groups. J Anal Math 48:1–141 51. Shields P (1973) The theory of Bernoulli shifts. University of Chicago Press, Chicago 52. Sinai YG (1962) A weak isomorphism of transformations having an invariant measure. Dokl Akad Nauk SSSR 147:797– 800 53. Thouvenot J-P (1975) Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli. Isr J Math 21:177– 207 54. Thouvenot J-P (1977) On the stability of the weak Pinsker property. Isr J Math 27:150–162

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Abramov Formula 55. Ward T, Zhang Q (1992) The Abramov–Rohlin entropy addition formula for amenable group actions. Monatshefte Math 114:317–329

Miscellaneous 56. Newhouse SE (1989) Continuity properties of entropy. Ann Math 129:215–235 57. http://www.isib.cnr.it/control/entropy/ 58. Rudin W (1973) Functional analysis. McGraw-Hill, New York 59. Tribus M, McIrvine EC (1971) Energy and information. Sci Am 224:178–184 60. Wikipedia, http://en.wikipedia.org/wiki/. pages: http://en.

wikipedia.org/wiki/Spectral_radius, wiki/Information_entropy

http://en.wikipedia.org/

Books and Reviews Boyle M, Downarowicz T (2004) The entropy theory of symbolic extensions. Invent Math 156(1):119–161 Downarowicz T, Serafin J (2003) Possible entropy functions. Isr J Math 135:221–250 Hassner M (1980) A non-probabilistic source and channel coding theory. Ph D Dissertation, UCLA Katok A, Sinai YG, Stepin AM (1977) Theory of dynamical systems and general transformation groups with an invariant measure. J Sov Math 7(6):974–1065

Ergodicity and Mixing Properties

Ergodicity and Mixing Properties ANTHONY QUAS Department of Mathematics and Statistics, University of Victoria, Victoria, Canada Article Outline Glossary Definition of the Subject Introduction Basics and Examples Ergodicity Ergodic Decomposition Mixing Hyperbolicity and Decay of Correlations Future Directions Bibliography Glossary Bernoulli shift Mathematical abstraction of the scenario in statistics or probability in which one performs repeated independent identical experiments. Markov chain A probability model describing a sequence of observations made at regularly spaced time intervals such that at each time, the probability distribution of the subsequent observation depends only on the current observation and not on prior observations. Measure-preserving transformation A map from a measure space to itself such that for each measurable subset of the space, it has the same measure as its inverse image under the map. Measure-theoretic entropy A non-negative (possibly infinite) real number describing the complexity of a measure-preserving transformation. Product transformation Given a pair of measure-preserving transformations: T of X and S of Y, the product transformation is the map of X  Y given by (T  S)(x; y) D (T(x); S(y)). Definition of the Subject Many physical phenomena in equilibrium can be modeled as measure-preserving transformations. Ergodic theory is the abstract study of these transformations, dealing in particular with their long term average behavior. One of the basic steps in analyzing a measure-preserving transformation is to break it down into its simplest possible components. These simplest components are its ergodic components, and on each of these components, the system enjoys the ergodic property: the long-term time

average of any measurement as the system evolves is equal to the average over the component. Ergodic decomposition gives a precise description of the manner in which a system can be split into ergodic components. A related (stronger) property of a measure-preserving transformation is mixing. Here one is investigating the correlation between the state of the system at different times. The system is mixing if the states are asymptotically independent: as the times between the measurements increase to infinity, the observed values of the measurements at those times become independent. Introduction The term ergodic was introduced by Boltzmann [8,9] in his work on statistical mechanics, where he was studying Hamiltonian systems with large numbers of particles. The system is described at any time by a point of phase space, a subset of R6N where N is the number of particles. The configuration describes the 3-dimensional position and velocity of each of the N particles. It has long been known that the Hamiltonian (i. e. the overall energy of the system) is invariant over time in these systems. Thus, given a starting configuration, all future configurations as the system evolves lie on the same energy surface as the initial one. Boltzmann’s ergodic hypothesis was that the trajectory of the configuration in phase space would fill out the entire energy surface. The term ergodic is thus an amalgamation of the Greek words for work and path. This hypothesis then allowed Boltzmann to conclude that the long-term average of a quantity as the system evolves would be equal to its average value over the phase space. Subsequently, it was realized that this hypothesis is rarely satisfied. The ergodic hypothesis was replaced in 1911 by the quasi-ergodic hypothesis of the Ehrenfests [17] which stated instead that each trajectory is dense in the energy surface, rather than filling out the entire energy surface. The modern notion of ergodicity (to be defined below) is due to Birkhoff and Smith [7]. Koopman [44] suggested studying a measure-preserving transformation by means of the associated isometry on Hilbert space, U T : L2 (X) ! L2 (X) defined by U T ( f ) D f ı T. This point of view was used by von Neumann [91] in his proof of the mean ergodic theorem. This was followed closely by Birkhoff [6] proving the pointwise ergodic theorem. An ergodic measure-preserving transformation enjoys the property that Boltzmann first intended to deduce from his hypothesis: that long-term averages of an observable quantity coincide with the integral of that quantity over the phase space.

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These theorems allow one to deduce a form of independence on the average: given two sets of configurations A and B, one can consider the volume of the phase space consisting of points that are in A at time 0 and in B at time t. In an ergodic measure-preserving transformation, if one computes the average of the volumes of these regions over time, the ergodic theorems mentioned above allow one to deduce that the limit is simply the product of the volume of A and the volume of B. This is the weakest mixing-type property. In this article, we will outline a rather full range of mixing properties with ergodicity at the weakest end and the Bernoulli property at the strongest end. We will set out in some detail the various mixing properties, basing our study on a number of concrete examples sitting at various points of this hierarchy. Many of the mixing properties may be characterized in terms of the Koopman operators operators mentioned above (i. e. they are spectral properties), but we will see that the strongest mixing properties are not spectral in nature. We shall also see that there are connections between the range of mixing properties that we discuss and measure-theoretic entropy. In measure-preserving transformations that arise in practice, there is a correlation between strong mixing properties and positive entropy, although many of these properties are logically independent. One important issue for which many questions remain open is that of higher-order mixing. Here, the issue is if instead of asking that the observations at two times separated by a large time T be approximately independent, one asks whether if one makes observations at more times, each pair suitably separated, the results can be expected to be approximately independent. This issue has an analogue in probability theory, where it is well-known that it is possible to have a collection of random variables that are pairwise independent, but not mutually independent. Basics and Examples In this article, except where otherwise stated, the measurepreserving transformations that we consider will be defined on probability spaces. More specifically, given a measurable space (X; B) and a probability measure  defined on B, a measure-preserving transformation of (X; B; ) is a B-measurable map T : X ! X such that (T 1 B) D (B) for all B 2 B. While this definition makes sense for arbitrary measures, not simply probability measures, most of the results and definitions below only make sense in the probability measure case. Sometimes it will be helpful to make the assumption that the underlying probability space is

a Lebesgue space (that is, the space together with its completed -algebra agrees up to a measure-preserving bijection with the unit interval with Lebesgue measure and the usual  -algebra of Lebesgue measurable sets). Although this sounds like a strong restriction, in practice it is barely a restriction at all, as almost all of the spaces that appear in the theory (and all of those that appear in this article) turn out to be Lebesgue spaces. For a detailed treatment of the theory of Lebesgue spaces, the reader is referred to Rudolph’s book [76]. The reader is referred also to the chapter on  Measure Preserving Systems. While many of the definitions that we shall present are valid for both invertible and non-invertible measure-preserving transformations, the strongest mixing conditions are most useful in the case of invertible transformations. It will be helpful to present a selection of simple examples, relative to which we will be able to explore ergodicity and the various notions of mixing. These examples and the lemmas necessary to show that they are measure-preserving transformations as claimed may be found in the books of Petersen [64], Rudolph [76] and Walters [92]. More details on these examples can also be found in the chapter on  Ergodic Theory: Basic Examples and Constructions. Example 1 (Rotation on the circle) Let ˛ 2 R. Let R˛ : [0; 1) ! [0; 1) be defined by R˛ (x) D x C ˛ mod 1. It is straightforward to verify that R˛ preserves the restriction of Lebesgue measure  to [0; 1) (it is sufficient to check that (R˛1 (J)) D (J) for an interval J). Example 2 (Doubling Map) Let M2 : [0; 1) ! [0; 1) be defined by M2 (x) D 2x mod 1. Again, Lebesgue measure is invariant under M 2 (to see this, one observes that for an interval J, M21 (J) consists of two intervals, each of half the length of J). This may be generalized in the obvious way to a map M k for any integer k  2. Example 3 (Interval Exchange Transformation) The class of interval exchange transformations was introduced by Sinai [85]. An interval exchange transformation is the map obtained by cutting the interval into a finite number of pieces and permuting them in such a way that the resulting map is invertible, and restricted to each interval is an order-preserving isometry. More formally, one takes a sequence of positive lengths `1 ; `2 ; : : : ; ` k summing to 1 and a permutation  P of f1; : : : ; kg and defines a i D j
Ergodicity and Mixing Properties

Example 4 (Bernoulli Shift) Let A be a finite set and fix a vector (p i ) i2A of positive numbers that sum to 1. Let AN denote the set of sequences of the form x0 x1 x2 : : : , where x n 2 A for each n 2 N and let AZ denote the set of biinfinite sequences of the form : : : x2 x1  x0 x1 x2 : : : (the  is a placeholder that allows us to distinguish (for example) between the sequences : : : 01010  10101 : : : and : : : 10101  01010 : : : ). We define a map (the shift map) S on AN by (S(x))n D x nC1 and define S on AZ by the same formula. Note that S is invertible as a transformation on AZ but non-invertible as a transformation on AN . We need to equip AN and AZ with measures. This is done by defining the measure of a preferred class of sets, checking certain consistency conditions and appealing to the Kolmogorov extension theorem. Here the preferred sets are the cylinder sets. Given m  n in the invertible case and a sequence a m : : : a n , we let [a m : : : a n ]nm denote fx 2 AZ : x m D a m ; : : : ; x n D a n g and define ([a m : : : a n ]nm ) D p a m p a mC1 : : : p a n . This is then shown to uniquely define a measure  on the -algebra of AZ generated by the cylinder sets. It is immediate to see that for any cylinder set C, (S 1 C) D (C), and it follows that S is a measure-preserving transformation of (AZ ; B; ). The construction is exactly analogous in the non-invertible case. See the chapter on  Measure Preserving Systems or the books of Walters [92] or Rudolph [76] for more details of defining measures in these systems.

of identical balls which move at constant velocity until two balls collide, whereupon they elastically swap momentum along the direction of contact. The phase space for this system is a region of R6N (with N 3-dimensional position vectors and N 3-dimensional velocity vectors). More abstractly, the system is equivalent to the motion of a single point particle in a region of R M  R M (with the first M-vector representing position and the second representing velocity). The system is constrained in that its position is required to lie in a bounded region S of R M with a piecewise smooth boundary. The system evolves by moving the position at a constant rate in the direction of the velocity vector until the point reaches @S, at which time the component of the velocity parallel to the normal to @S is reversed. This then defines a flow (i. e. a family of maps (Tt ) t2R satisfying TtCs D Tt ı Ts ) on the phase space. Since the magnitude of the velocity is conserved, it is convenient to restrict to flows with speed 1. This system is clearly the closest of the examples that we consider to the situation envisaged by Boltzmann. Perhaps not surprisingly, proofs of even the most basic properties for this system are much harder than those for the other examples that we consider.

The class of Bernoulli shifts will play a distinguished role in what follows.

1. g ı h and h ı g agree with the respective identity maps almost everywhere; 2. (g 1 F) D (F) and (h1 B) D (B) for all F 2 F and B 2 B; and 3. Sıg(x) D gıT(x) for -almost every x (or equivalently T ı h(y) D h ı S(y) for -almost every y).

Example 5 (Markov Shift) The spaces AN and AZ are exactly as above, as is the shift map. All that changes is the measure. To define a Markov shift, we need a stochastic matrix P (i. e. a matrix with non-negative entries whose rows sum to 1) with rows and columns indexed by A and a left eigenvector  for P with eigenvalue 1 with the property that the entries of  are non-negative and sum to 1. The existence of such an eigenvector is a consequence of the Perron–Frobenius theory of positive matrices. Provided that the matrix P is irreducible (for each a and a0 in A, there is n an n > 0 such that Pa;a 0 > 0), the eigenvector  is unique. Given the pair (P; ), one defines the measure of a cylinder set by ([a m : : : a n ]nm ) D  a m Pa m a mC1 : : : Pa n1 a n and extends  as before to a probability measure on AN or AZ . Example 6 (Hard Sphere Gases and Billiards) We wish to model the behavior of a gas in a bounded region. We make the assumption that the gas consists of a large number N

We will need to make use of the concept of measure-theoretic isomorphism. Two measure-preserving transformations T of (X; B; ) and S of (Y; F ; ) are measure-theoretically isomorphic (or just isomorphic) if there exist measurable maps g : X ! Y and h : Y ! X such that

Measure-theoretic isomorphism is the basic notion of ‘sameness’ in ergodic theory. It is in some sense quite weak, so that systems may be isomorphic that feel very different (for example, as we discuss later, the time one map of a geodesic flow is isomorphic to a Bernoulli shift). For comparison, the notion of sameness in topological dynamical systems (topological conjugacy) is far stronger. As an example of measure-theoretic isomorphism, it may be seen that the doubling map is isomorphic to the one-sided Bernoulli shift on f0; 1g with p0 D p1 D 1/2 (the map g takes an x 2 [0; 1) to the sequence of 0’s and 1’s in its binary expansion (choosing the sequence ending with 0’s, for example, if x is of the form p/2n ) and the inverse map h takes a sequence of 0’s and 1’s to the point in [0; 1) with that binary expansion.)

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Given a measure-preserving transformation T of a probability space (X; B; ), T is associated to an isometry of L2 (X; B; ) by U T ( f ) D f ı T. This operator is known as the Koopman Operator. In the case where T is invertible, the operator U T is unitary. Two measure-preserving transformations T and S of (X; B; ) and (Y; F ; ) are spectrally isomorphic if there is a Hilbert space isomorphism ‚ from L2 (X; B; ) to L2 (Y; F ; ) such that ‚ ı U T D U S ı ‚. As we shall see below, spectral isomorphism is a strictly weaker property than measure-theoretic isomorphism. Since in ergodic theory, measure-theoretic isomorphism is the basic notion of sameness, all properties that are used to describe measure-preserving systems are required to be invariant under measure-theoretic isomorphism (i. e. if two measure-preserving transformations are measure-theoretically isomorphic, the first has a given property if and only if the second does). On the other hand, we shall see that some mixing-type properties are invariant under spectral isomorphism, while others are not. If a property is invariant under spectral isomorphism, we say that it is a spectral property. There are a number of mixing type properties that occur in the probability literature (˛-mixing, ˇ-mixing, -mixing, -mixing etc.) (see Bradley’s survey [12] for a description of these conditions). Many of these are stronger than the Bernoulli property, and are therefore not preserved under measure-theoretic isomorphism. For this reason, these properties are not widely used in ergodic theory, although ˇ-mixing turns out to be equivalent to the so-called weak Bernoulli property (which turns out to be stronger than the Bernoulli property that we discuss in this article – see Smorodinsky’s paper [87]) and ˛-mixing is equivalent to strong-mixing. A basic construction (see the article on  Ergodic Theory: Basic Examples and Constructions) that we shall require in what follows is the product of a pair of measurepreserving transformations: given transformations T of (X; B; ) and S of (Y; F ; ), we define the product transformation T  S : (X  Y; B ˝ F ; ) by (T  S)(x; y) D (Tx; Sy). One issue that we face on occasion is that it is sometimes convenient to deal with invertible measure-preserving transformations. It turns out that given a non-invertible measure-preserving transformation, there is a natural way to uniquely associate an invertible measurepreserving transformation transformation sharing almost all of the ergodic properties of the original transformation. Specifically, given a non-invertible measure-preserving transformation T of (X; B; ), one lets X D f(x0 ; x1 ; : : : ) : x n 2 X and T(x n ) D x n1 for all ng, B

be the  -algebra generated by sets of the form A¯ n D ¯ A¯ n ) D (A) and T(x0 ; x1 ; : : : ) D fx¯ 2 X : x n 2 Ag, ( (T(x0 ); x0 ; x1 ; : : : ). The transformation T of (X; B; ) is called the natural extension of the transformation T of (X; B; ) (see the chapter on  Ergodic Theory: Basic Examples and Constructions for more details). In situations where one wants to use invertibility, it is often possible to pass to the natural extension, work there and then derive conclusions about the original non-invertible transformation. Ergodicity Given a measure-preserving transformation T : X ! X, if T 1 A D A, then T 1 Ac D Ac also. This allows us to decompose the transformation X into two pieces A and Ac and study the transformation T separately on each. In fact the same situation holds if T 1 A and A agree up to a set of measure 0. For this reason, we call a set A invariant if (T 1 AA) D 0. Returning to Boltzmann’s ergodic hypothesis, existence of an invariant set of measure between 0 and 1 would be a bad situation as his essential idea was that the orbit of a single point would ‘see’ all of X, whereas if X were decomposed in this way, the most that a point in A could see would be all of A, and similarly the most that a point in Ac could see would be all of Ac . A measure-preserving transformation will be called ergodic if it has no non-trivial decomposition of this form. More formally, let T be a measure-preserving transformation of a probability space (X; B; ). The transformation T is said to be ergodic if for all invariant sets, either the set or its complement has measure 0. Unlike the remaining concepts that we discuss in this article, this definition of ergodicity applies also to infinite measure-preserving transformations and even to certain non-measure-preserving transformations. See Aaronson’s book [1] for more information. The following lemma is often useful: Lemma 1 Let (X; B; ) be a probability space and let T : X ! X be a measure-preserving transformation. Then T is ergodic if and only if the only measurable functions f satisfying f ı T D f (up to sets of measure 0) are constant almost everywhere. For the straightforward proof, we notice that if the condition in the lemma holds and A is an invariant set, then 1A ı T D 1A almost everywhere, so that 1A is an a. e. constant function and so A or Ac is of measure 0. Conversely, if f is an invariant function, we see that for each ˛, fx : f (x) < ˛g is an invariant set and hence of measure 0

Ergodicity and Mixing Properties

or 1. It follows that f is constant almost everywhere. We remark for future use that it is sufficient to check that the bounded measurable invariant functions are constant. The following corollary of the lemma shows that ergodicity is a spectral property. Corollary 2 Let T be a measure-preserving transformation of the probability space (X; B; ). Then T is ergodic if and only if 1 is a simple eigenvalue of U T . The ergodic theorems mentioned earlier due to von Neumann and Birkhoff are the following (see also the chapter on  Ergodic Theorems). Theorem 3 (von Neumann Mean Ergodic Theorem [91]) Let T be a measure-preserving transformation of the probability space (X; B; ). For f 2 L2 (X; B; ), let A N f D 1/N( f C f ı T C    C f ı T N1 ). Then for all f 2 L2 (X; B; ), A N f converges in L2 to an invariant function f . Theorem 4 (Birkhoff Pointwise Ergodic Theorem [6]) Let T be a measure-preserving transformation of the probability space (X; B; ). Let f 2 L1 (X; B; ). Let A N f be as above. Then for -almost every x 2 X, (A N f (x)) is a convergent sequence. Of these two theorems, the pointwise ergodic theorem is the deeper result, and it is straightforward to deduce the mean ergodic theorem from the pointwise ergodic theorem. The mean ergodic theorem was reproved very concisely by Riesz [71] and it is this proof that is widely known now. Riesz’s proof is reproduced in Parry’s book [63]. There have been many different proofs given of the pointwise ergodic theorem. Notable amongst these are the argument due to Garsia [23] and a proof due to Katznelson and Weiss [40] based on work of Kamae [35], which appears in a simplified form in work of Keane and Petersen [42]. If the measure-preserving transformation T is ergodic, then by virtue of Lemma 1, the limit functions appearing in the ergodic theorems are constant. One sees that the constant is simply the integral of f with respect to , so R that in this situation A N f (x) converges to f d in norm and pointwise almost everywhere, thereby providing a justification of Boltzmann’s original claim: for ergodic measure-preserving transformations, time averages agree with spatial averages. In the case where T is not ergodic, it is also possible to identify the limit in the ergodic theorems: we have f  D E( f jI ), where I is the -algebra of T-invariant sets. Note that the set on which the almost everywhere convergence in Birkhoff’s theorem takes place depends on the L1 function f that one is considering. Straightforward considerations show that there is no single full measure set

that works simultaneously for all L1 functions. In the case where X is a compact metric space, it is well known that C(X), the space of continuous functions on X with the uniform norm has a countable dense set, ( f n )n1 say. If the invariant measure  is ergodic, then for each n, there is aR set Bn of measure 1 such that for all x 2 B n , A N f n (x) ! T f n d. Letting B D n B n , one obtains a full Rmeasure set such that for all n and all x 2 B, A N f n (x) ! f n d. A simple approximation argument thenRshows that for all x 2 B and all f 2 C(X), A N f (x) ! f d. A point x with this property is said to be generic for . The observations above show that for an ergodic invariant measure , we have fx : x is generic for g D 1. If T is ergodic, but T n is not ergodic for some n, then one can show that the space X splits up as A1 ; : : : ; A d for some djn in such a way that T(A i ) D A iC1 for i < d and T(A d ) D A1 with T n acting ergodically on each Ai . The transformation T is totally ergodic if T n is ergodic for all n 2 N. One can check that a non-invertible transformation T is ergodic if and only if its natural extension is ergodic. The following lemma gives an alternative characterization of ergodicity, which in particular relates it to mixing. Lemma 5 (Ergodicity as a Mixing Property) Let T be a measure-preserving transformation of the probability space (X; B; ). Then T is ergodic if and only if for all f and g in L2 , N 1 X h f ; g ı T n i ! h f ; 1ih1; gi: N nD0

P N1 In particular, if T is ergodic, then (1/N) nD0 (A \ T n B) ! (A)(B) for all measurable sets A and B. Proof Suppose that T is ergodic. Then the left-hand side P N1 of the equality is equal to h f ; (1/N) nD0 g ı T n i. The mean ergodic theorem shows that the second term conR verges in L2 to the constant function with value g d D hg; 1i, and the equality follows. Conversely, if the equation holds for all f and g in L2 , suppose that A is an invariant set. Let f D g D 1A . Then since g ı T n D 1A for all n, the left-hand side is h1A ; 1A i D (A). On the other hand, the right-hand side is (A)2 , so that the equation yields (A) D (A)2 , and (A) is either 0 or 1 as required. Taking f D 1A and g D 1B for measurable sets A and B gives the final statement.  We now examine the ergodicity of the examples presented above. Firstly, for the rotation of the circle, we claim that

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the transformation is ergodic if and only if the ‘angle’ ˛ is irrational. To see this, we argue as follows. If ˛ D p/q, then we see that f (x) D e 2 i qx is a non-constant R˛ invariant function, and hence R˛ is not ergodic. On the other hand, if ˛ is irrational, suppose f is a bounded measurable invariant function. Since f is bounded, it is an L2 function, and so f may be expressed in L2 as a Fourier seP 2 i nx . We then ries: f D n e n where e n (x) D e n2Z cP see that f ı R˛ D n2Z e 2 i n˛ c n e n . In order for f to be equal in L2 to f ı R˛ , they must have the same Fourier coefficients, so that c n D e 2 i n˛ c n for each n. Since ˛ is irrational, this forces c n D 0 for all n ¤ 0, so that f is constant as required. The doubling map and the Bernoulli shift are both ergodic, although we defer proof of this for the time being, since they in fact have the strong-mixing property. A Markov chain with matrix P and vector  is ergodic if and only if for all i and j in A with  i > 0 and  j > 0, there exists an n  0 with Pinj > 0. This follows from the ergodic theorem for Markov chains (which is derived from the Strong Law of Large Numbers) (see [18] for details). In particular, if the underlying Markov chain is irreducible, then the measure is ergodic. In the case of interval exchange transformations, there is a simple necessary condition on the permutation for irreducibility, namely for 1  j < k, we do not have f1; : : : ; jg D f1; : : : ; jg. Under this condition, Masur [49] and Veech [88] independently showed that for almost all values of the sequence of lengths (` i )1ik , the interval exchange transformation is ergodic. (In fact they showed the stronger condition of unique ergodicity: that the transformation has no other invariant measure than Lebesgue measure. This implies that Lebesgue measure is ergodic, because if there were a non-trivial invariant set, then the restriction of Lebesgue measure to that set would be another invariant measure). For the hard sphere systems, there are no results on ergodicity in full generality. Important special cases have been studied by Sinai [84], Sinai and Chernov [86], Krámli, Simányi and Szász [45], Simányi and Szász [81], Simnyi [79,80] and Young [95]. Ergodic Decomposition We already observed that if a transformation is not ergodic, then it may be decomposed into parts. Clearly if these parts are not ergodic, they may be further decomposed. It is natural to ask whether the transformation can be decomposed into ergodic parts, and if so what form does the decomposition take? In fact such a decomposition does exist, but rather than decompose the transforma-

tion, it is necessary to decompose the measure into ergodic pieces. This is known as ergodic decomposition. The set of invariant measures for a measurable map T of a measurable space (X; B) to itself forms a simplex. General functional analytic considerations (due to Choquet [14,15] – see also Phelps’ account [66] of this theory) mean that it is possible to write any member of the simplex as an integral-convex combination of the extreme points. Further, the extreme points of the simplex may be identified as precisely the ergodic invariant measures for T. It follows that any invariant probability measure  for T may be uniquely expressed in the form Z (A) D (A) dm() ; M erg (X;T)

where Merg (X; T) denotes the set of ergodic T-invariant measures on X and m is a measure on Merg (X; T). We will give a proof of this theorem in the special case of a continuous transformation of a compact space. Our proof is based on the Birkhoff ergodic theorem and the Riesz Representation Theorem identifying the dual space of the space of continuous functions on a compact space as the set of bounded signed measures on the space (see Rudin’s book [75] for details). We include it here because this special case covers many cases that arise in practice, and because few of the standard ergodic theory references include a proof of ergodic decomposition. An exception to this is Rudolph’s book [76] which gives a full proof in the case that X is a Lebesgue space. This is based on a detailed development of the theory of these spaces and builds measures using conditional exceptions. Kalikow’s notes [32] give a brief outline of a proof similar to that which follows. Oxtoby [62] also wrote a survey article containing much of the following (and much more besides). Theorem 6 Let X be a compact metric space, B be the Borel  -algebra,  be an invariant Borel probability measure and T be a continuous measure-preserving transformation of (X; B; ). Then for each x 2 X, there exists an invariant Borel measure x such that: R R R  1. For f 2 L1 (X; B; ), f d D f dx d(x); 2. Given f 2 L1 (X; R B; ), for -almost every x 2 X, one has A N f (x) ! f dx ; 3. The measure x is ergodic for -almost every x 2 X. Notice that conclusion (2) shows that x can be understood as the distribution on the phase space “seen” if one starts the system in an initial condition of x. This interpretation of the measures x corresponds closely with the ideas of Boltzmann and the Ehrenfests in the formulation

Ergodicity and Mixing Properties

of the ergodic and quasi-ergodic hypotheses, which can be seen as demanding that x is equal to  for (almost) all x. Proof The proof will be divided into 3 main steps: defining the measures x , proving measurability with respect to x and proving ergodicity of the measures. Step 1: Definition of x Given a function f 2 L1 (X; B; ), Birkhoff’s theorem states that for -almost every x 2 X, (A N f (x)) is convergent. It will be convenient to denote the limit by f˜(x). Let f1 ; f2 ; : : : be a sequence of continuous functions that is dense in C(X). For each k, there is a set Bk of x’s measure 1 for which (A n f k (x))1 nD1 is a convergent sequence. Intersecting these gives a set B of full measure such that for x 2 B, for each k  1, A n f k (x) is convergent. A simple approximation argument shows that for x 2 B and f an arbitrary continuous function, A n f (x) is convergent. Given x 2 B, define a map L x : C(X) ! R by L x ( f ) D f˜(x). This is a continuous linear functional on C(X), and hence by the Riesz Representation Theorem there exists a Borel R measure x such that f˜(x) D f dx for each f 2 C(X) and x 2 B. Since L x ( f )  0 when f is a non-negative function and L x (1) D 1, the measure x is a probability measure. Since L x ( f ı T) D L x ( f ) for f 2 C(X), one can check that x must be an invariant probability measure. For x 62 B, simply define x D . Since Bc is a set of measure 0, this will not affect any of the statements that we are trying to prove. Now for f continuous, we have A N f is a bounded seR to f d quence of functions withR A N f (x) converging x R almost everywhere and A N f d D f d since T is measure-preserving. It follows from the bounded convergence theorem that for f 2 C(X),  Z Z Z f d D f dx d(x) : (1) Step 2: Measurability of x 7! x (A) Lemma 7 Let C 2 B satisfy (C) D 0. Then x (C) D 0 for -almost every x 2 X. Proof Using regularity of Borel probability measures (see Rudin’s book [75] for details), there exist open sets U1  U2      C with (U k ) < 1/k. There exist continuous functions g k;m with (g k;m (x))1 mD1 increasing to 1U k everywhere R(e.Rg. g k;m (x) D min(1; m  d(x; U kc ))). By (1), we Rhave ( g k;m dx ) d(x) < 1/k for all k; m. Note that g k;m dx D lim n!1 A n g k;m (x) is a measurable function of x, so that using the monotone converR gence theorem (taking the limit in m), x ! 7 1 U k d x D R R x (U k ) is measurable and ( 1U k dx ) d(x)  1/k. T We now see that x 7! lim k!1 x (U k ) D x ( U k )

is measurable, and by monotone convergence we see R alsoT T x ( U k ) d(x) D 0. It follows that x ( U k ) D 0 T for -almost every x. Since U k  C, the lemma follows.  Given a set A 2 B, let f k be a sequence of continuous functions (uniformly bounded by 1) satisfying k f k  1A kL 1 () < 2n , so that in particular R f k (x) ! 1A (x) for -almost every x. For each k, x 7! f k dx D lim n!1 A n f k (x) is a measurable function. By Lemma 7, for -almost every x, f k ! 1A x -almost everywhere, so R that by the bounded convergence theorem limk!1 f k dx D x (A) for -almost every x. Since the limit of measurable functions is measurable, it follows that x 7! x (A) is measurable for any measurable set A 2 B. R This allows us to define a measure  by (A) D x (A) R d(x). RForR a bounded measurable function f , we have f d D ( f dx ) d(x). Since this agrees with R f d for continuous functions by (1), it follows that  D . Conclusion (1) of the theorem now follows easily. Given f 2 L1 (X), we let ( f k ) be a sequence of continuous functions such that k f k  f kL 1 () is summable. This implies that k f k  f kL 1 ( R for -almost evR x ) is summable ery x and in particular, f k dx ! f dx for almost every x. On the other hand, by the remark following the statement of Birkhoff ’s theorem, we have f˜k D E( f k jI ) so that k f˜  f˜k kL 1 () is summable and f˜k (x) ! f˜(x) for -almost every x. Combining these two statements, we see that for -almost every x, we have f˜(x) D lim f˜k (x) D lim k!1

k!1

Z

Z f k dx D

f dx :

This establishes conclusion (2) of the theorem. Step 3: Ergodicity of x We have shown how to disintegrate the invariant measure  as an integral combination of x ’s, and we have interpreted the x ’s as describing the average behavior starting from x. It remains to show that the x ’s are ergodic measures. Fix for now a continuous function f and a number 0 < < 1. Since A n f (x) ! f˜(x) -almost everywhere, there exists an N such that fx : jA N f (x) f˜(x)j > /2g < 3 /8. We now claim the following: R ˚  x : x fy : j f˜(y)  f dx j > g > < :

(2)

R To see this, note that fy : j f˜(y)  f dx j > g

fy : j f˜(y)A N f (y)j > /2g[fy : jA N f (y) f˜(x)j > /2g, R ˜ so that if x fy : j f (y)  f dx j > g > , then either

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x fy : j f˜(y) A N f (y)j > /2g > /2 or x fy : jA N f (y) f˜(x)j > /2g > /2. We show that the set of x’s satisfying each condition is small. 3 ˜ Firstly, R we have /8 > fy : j f (y)  A N f (y)j > /2g D x fy : j f˜(y)  A N f (y)j > /2g d(x), so that fx : x fy : j f˜(y)  A N f (y)j > /2g > /2g < 2 /4 < /2. For the second term, given c 2 R, let Fc (x) D jA N f (x)  cj and G(x) D F f˜(x) (x). Note that Z F f˜(x) (y) dx (y) D lim A n F f˜(x) (x) D lim A n G(x) n!1

n!1

(using the facts that y 7! F f˜(x) (y) is a continuous function and that since f˜(x) is R an invariant function, F f˜(x) (T k x) D G(T k x)). Since G(x) d(x) < 3 /8, it R follows that F f˜(x) (y) dx (y)  2 /4 except on a set of x’s of measure less than /2. Outside this bad set, we have x fy : jA N f (y)  f˜(x)j > /2g < /2 so that fx : x fy : jA N f (y)  f˜(x)j > /2g > /2g < /2 as required. This establishes our claim (2) above. Since > 0 is arbitrary, it follows that for each f 2 C(X), for R -almost every x, x -almost every y satisfies f˜(y) D f dx . As usual, taking a countable dense sequence ( f k ) in C(X), it ˜ is R the case that for all k and -almost every x, f k (y) D f k dx x -almost everywhere. Let the set of x’s with this property be D. We claim that for x 2 D, x is ergodic. Suppose not. Then let x 2 D and let J be an invariant set of x measure between ı and 1  ı for some ı > 0. Then by density of C(X) in L1 (x ), there exists an f k with k f k  1 J kL 1 (x ) < ı. Since 1 J is an invariant function, we have 1˜ J D 1 J . On the other hand, f˜k is a constant function. It follows that k f˜k  1˜ J kL 1 (x )  ı > k f k  1 J kL 1 (x ) . This contradicts the identification of the limit as a conditional expectation and concludes the proof of the theorem.  Mixing As mentioned above, ergodicity may be seen as an independence on average property. More specifically, one wants to know whether in some sense (A \ T n B) converges to (A)(B) as n ! 1. Ergodicity is the property that there is convergence in the Césaro sense. Weak-mixing is the property that there is convergence in the strong Césaro sense. That is, a measure-preserving transformation T is weak-mixing if N1 1 X j(A\ T n B)(A)(B)j ! 0 N nD0

as

N ! 1:

In order for T to be strong-mixing, we require simply (A \ T N B) ! (A)(B) as N ! 1. It is clear that strong-mixing implies weak-mixing and weak-mixing implies ergodicity. If T d is not ergodic (so that T d A D A for some A of measure strictly between 0 and 1), then j(T nd A \ A)  (A)2 j D (A)(1  (A)), so that T is not weak-mixing. An alternative characterization of weak-mixing is as follows: Lemma 8 The measure-preserving transformation T is weak-mixing if and only if for every pair of measurable sets A and B, there exists a subset J of N of density 1 (i. e. #(J \ f1; : : : ; Ng)/N ! 1) such that lim

n!1 n62 J

(A \ T n B) D (A)(B) :

(3)

By taking a countable family of measurable sets that are dense (with respect to the metric d(A; B) D (AB)) and taking a suitable intersection of the corresponding J sets, one shows that for a given weak-mixing measure-preserving transformation, there is a single set J N such that (3) holds for all measurable sets A and B (see Petersen [64] or Walters [92] for a proof). We show that an irrational rotation of the circle is not weak-mixing as follows: let ˛ 2 R n Q and let A be the interval [ 14 ; 34 ). There is a positive proportion of n’s in the natural numbers (in fact proportion 1/3) with the property that jT n ( 12 )  12 j < 16 . For these n’s (A \ T n A) > 13 , 1 . so that in particular j(A \ T n A)  (A)(A)j > 12 Clearly this precludes the required convergence to 0 in the definition of weak-mixing, so that an irrational rotation is ergodic but not weak-mixing. Since R˛n D R n˛ , the earlier argument shows that R˛n is ergodic, so that R˛ is totally ergodic. On the other hand, we show that any Bernoulli shift is strong-mixing. To see this, let A and B be arbitrary measurable sets. By standard measure-theoretic arguments, A and B may each be approximated arbitrarily closely by a finite union of cylinder sets. Since if A0 and B0 are finite unions of cylinder sets, we have that (A0 \ T n B0 ) is equal to (A0 )(B0 ) for large n, it is easy to deduce that (A \ T n B) ! (A)(B) as required. Since the doubling map is measure-theoretically isomorphic to a onesided Bernoulli shift, it follows that the doubling map is also strong-mixing. Similarly, if a Markov Chain is irreducible (i. e. for any states i and j, there exists an n  0 such that Pinj > 0) and aperiodic (there is a state i such that gcdfn : Pini > 0g D 1), then given any pair of cylinder sets A0 and B0 we have by standard theorems of Markov chains (A0 \ T n B0 ) !

Ergodicity and Mixing Properties

(A0 )(B0 ). The same argument as above then shows that an aperiodic irreducible Markov Chain is strong-mixing. On the other hand, if a Markov chain is periodic (d D gcdfn : Pini > 0g > 0), then letting A D B D fx : x0 D ig, we have that (A \ T n B) D 0 whenever d − n. It follows T d is not ergodic, so that T is not weak-mixing. Both weak- and strong-mixing have formulations in terms of functions: Lemma 9 Let T be a measure-preserving transformation of the probability space (X; B; ). 1. T is weak-mixing if and only if for every f ; g 2 L2 one has N1 1 X jh f ; g ı T n i  h f ; 1ih1; gij ! 0 as N ! 1 : N nD0

2. T is strong-mixing if and only if for every f ; g 2 L2 , one has h f ; g ı T N i ! h f ; 1ih1; gi as N ! 1 : Using this, one can see that both mixing conditions are spectral properties. Lemma 10 Weak- and strong-mixing are spectral properties. Proof Suppose S is a weak-mixing transformation of (Y; F ; ) and the transformation T of (X; B; ) is spectrally isomorphic to S by the Hilbert space isomorphism ‚. Then for f ; g 2 L2 (X; B; ), h f ; g ı T n i X  h f ; 1i X h1; gi X D h‚( f ); ‚(g) ı S n iY  h‚( f ); ‚(1)iY h‚(1); ‚(g)iY . Since 1 is an eigenfunction of U T with eigenvalue 1, ‚(1) is an eigenfunction of U S with an eigenvalue 1, so since S is ergodic, ‚(1) must be a constant function. Since ‚ preserves norms, ‚(1) must have a constant value of absolute value 1 and hence h f ; g ı T n i X  h f ; 1i X h1; gi X D h‚( f ); ‚(g) ı S n iY  h‚( f ); 1iY h1; ‚(g)iY . It follows from Lemma 9 that T is weak-mixing. A similar proof shows that strong-mixing is a spectral property.  Both weak- and strong-mixing properties are preserved by taking natural extensions. Recent work of Avila and Forni [4] shows that for interval exchange transformations of k  3 intervals with the underlying permutation satisfying the non-degeneracy condition above, almost all divisions of the interval (with respect to Lebesgue measure on the k1-dimensional simplex) lead to weak-mixing transformations. On the other

hand, work of Katok [36] shows that no interval exchange transformation is strong-mixing. It is of interest to understand the behavior of the ‘typical’ measure-preserving transformation. There are a number of Baire category results addressing this. In order to state them, one needs a set of measure-preserving transformations and a topology on them. As mentioned earlier, it is effectively no restriction to assume that a transformation is a Lebesgue-measurable map on the unit interval preserving Lebesgue measure. The classical category results are then on the collection of invertible Lebesgue-measure preserving transformations of the unit interval. One topology on these is the ‘weak’ topology, where a sub-base is given by sets of the form N(T; A; ) D fS : (S(A)T(A)) < g. With respect to this topology, Halmos [26] showed that a residual set (i. e. a dense Gı set) of invertible measurepreserving transformations is weak-mixing (see also work of Alpern [3]), while Rokhlin [72] showed that the set of strong-mixing transformations is meagre (i. e. a nowhere dense F  set), allowing one to conclude that with respect to this topology, the typical transformation is weak- but not strong-mixing. As often happens in these cases, even when a certain kind of behavior is typical, it may not be simple to exhibit concrete examples. In this case, a well-known example of a transformation that is weak-mixing but not strong-mixing was given by Chacon [13]. While on the face of it the formulation of weak-mixing is considerably less natural than that of strong-mixing, the notion of weak-mixing turns out to be extremely natural from a spectral point of view. Given a measure-preserving transformation T, let U T be the Koopman operator described above. Since this operator is an isometry, any eigenvalue must lie on the unit circle. The constant function 1 is always an eigenfunction with eigenvalue 1. If T is ergodic and g and h are eigenfunctions of U T with eigenvalue , then g h¯ is an eigenfunction with eigenvalue 1, hence invariant, so that g D Kh for some constant K. We see that for ergodic transformations, up to rescaling, there is at most one eigenfunction with any given eigenvalue. If U T has a non-constant eigenfunction f , then one has jhU Tn f ; f ij D k f k2 for each n, whereas by Cauchy– Schwartz, jh f ; 1ij2 < k f k2 . It follows that jhU Tn f ; f i  h f ; 1ih1; f ij  c for some positive constant c, so that using Lemma 9, T is not weak-mixing. Using the spectral theorem, the converse is shown to hold. Theorem 11 The measure-preserving transformation T is weak-mixing if and only U T has no non-constant eigenfunctions.

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Ergodicity and Mixing Properties

Of course this also shows that weak-mixing is a spectral property. Equivalently, this says that the transformation T is weak-mixing if and only if the apart from the constant eigenfunction, the operator U T has only continuous spectrum (that is, the operator has no other eigenfunctions). For a very nice and concise development of the part of spectral theory relevant to ergodic theory, the reader is referred to the Appendix in Parry’s book [63]. Using this theory, one can establish the following: Theorem 12 1. T is weak-mixing if and only if T  T is ergodic; 2. If T and S are ergodic, then T  S is ergodic if and only if U S and U T have no common eigenvalues other than 1. Proof The main factor in the proof is that the eigenvalues of U TS are precisely the set of ˛ˇ, where ˛ is an eigenvalue of U T and ˇ is an eigenvalue of U S . Further, the eigenfunctions of U TS with eigenvalue are spanned by eigenfunctions of the form f ˝ g, where f is an eigenfunction of U T , g is an eigenfunction of U S , and the product of the eigenvalues is . Suppose that T is weak-mixing. Then the only eigenfunction is the constant function, so that the only eigenfunction of U TT is the constant function, proving that T  T is ergodic. Conversely, if U T has an eigenvalue (so that f ı T D ˛T for some non-constant f ) then f ˝ f¯ is a non-constant invariant function of T  T so that T  T is not ergodic. For the second part, if U S and U T have a common eigenvalue other than 1 (say f ı T D ˛ f and g ı T D ˛g), then f ˝ g¯ is a non-constant invariant function. Conversely, if T  S has a non-constant invariant function h, then h can be decomposed into functions of the form f ˝g, where f and g are eigenfunctions of U T and U S respectively with eigenvalues ˛ and ˇ satisfying ˛ˇ D 1. Since the eigenvalues of S are closed under complex conjugation, we see that U T and U S have a common eigenvalue other than 1 as required.  For a measure-preserving transformation T, we let K be the subspace of L2 spanned by the eigenfunctions of U T . It is a remarkable fact that K may be identified as L2 (X; B0 ; ) where B0 is a sub--algebra of B. The space K is called the Kronecker factor of T. The terminology comes from the fact that any sub--algebra F of B gives rise to a factor mapping  : (X; B; ) ! (X; F ; ) with (x) D x. By construction L2 (X; B0 ; ) is the closed linear span of the eigenfunctions of T considered as a measure-preserving transformation of (X; B0 ; ). By the Discrete Spectrum Theorem of Halmos and von Neumann [27], T act-

ing on (X; B0 ; ) is measure-theoretically isomorphic to a rotation on a compact group. This allows one to split L2 (X; B; ) as L2 (X; B0 ; ) ˚ L2c (X; B; ), where, as mentioned above the first part is the discrete spectrum part, spanned by eigenfunctions, and the second part is the continuous spectrum part, consisting of functions whose spectral measure is continuous. Since we have split L2 into a discrete part and a continuous part, it is natural to ask whether the underlying transformation T can be split up in some way into a weak-mixing part and a discrete spectrum (compact group rotation) part, somewhat analogously to the ergodic decomposition. Unfortunately, there is nosuch decomposition available. However for some applications, for example to multiple recurrence (starting with the work of Furstenberg [20,21]), the decomposition of L2 (possibly into more complicated parts) plays a crucial role (see the chapters on  Ergodic Theory: Recurrence and  Ergodic Theory: Interactions with Combinatorics and Number Theory). For non-invertible measure-preserving transformations, the transformation is weak- or strong-mixing if and only if its natural extension has that property. The understanding of weak-mixing in terms of the discrete part of the spectrum of the operator also extends to total ergodicity. T n is ergodic if and only if T has no eigenvalues of the form e 2 i p/n other than 1. From this it follows that an ergodic measure-preserving transformation T is totally ergodic if and only if it has no rational spectrum (i. e. no eigenvalues of the form e 2 i p/q other than the simple eigenvalue 1). An intermediate mixing condition between strongand weak- mixing is that a measure-preserving transformation is mild-mixing if whenever f ı T n i ! f for an L2 function f and a sequence n i ! 1, then f is a.e. constant. Clearly mild-mixing is a spectral property. If a transformation has an eigenfunction f , then it is straightforward to find a sequence ni such that f ı T n i ! f , so we see that mild-mixing implies weak-mixing. To see that strong-mixing implies mild-mixing, suppose that T is and that f ı T n i ! f . Then we have R strong-mixing n ¯ i f ı T f ! k f k2 . On R the other hand, the strong mixing property implies that f ı T n i f¯ ! jh f ; 1ij2. The equality of these implies that f is a. e. constant. Mild-mixing has a useful reformulation in terms of ergodicity of general (not necessarily probability) measure-preserving transformations: A transformation T is mild-mixing if and only if for every conservative ergodic measure-preserving transformation S, T  S is ergodic. See Furstenberg and Weiss’ article [22] for further information on mild-mixing. The strongest spectral property that we consider is that of having countable Lebesgue spectrum. While we

Ergodicity and Mixing Properties

will avoid a detailed discussion of spectral theory in this article, this is a special case that can be described simply. Specifically, let T be an invertible measure-preserving transformation. Then T has countable Lebesgue spectrum if there is a sequence of functions f1 ; f2 ; : : : such that f1g [ fU Tn f j : n 2 Z; j 2 Ng forms an orthonormal basis for L2 (X). To see that this property is stronger than strongmixing, we simply observe that it implies that hU Tt U Tn f j ; U Tm f k i ! 0 as t ! 1. Then by approximating f and g by their expansions with respect to a finite part of the basis, we deduce that hU Tn f ; gi ! h f ; 1ih1; gi as required. Since already strong-mixing is atypical from the topological point of view, it follows that countable Lebesgue spectrum has to be atypical. In fact, Yuzvinskii [96] showed that the typical invertible measure-preserving transformation has simple singular spectrum. The property of countable Lebesgue spectrum is by definition a spectral property. Since it completely describes the transformation up to spectral isomorphism, there can be no stronger spectral properties. The remaining properties that we shall examine are invariant under measuretheoretic isomorphisms only. An invertible measure-preserving transformation T of (X; B; ) is said to be K (for Kolmogorov) if there is a sub-algebra F of B such that T n F is the trivial -algebra up to sets of mea1. 1 nD1 T sure 0 (i. e. the intersection consists only of null sets and sets of full measure). W n 2. 1 nD1 T F D B (i. e. the smallest -algebra containn ing T F for all n > 0 is B). The K property has a useful reformulation in terms of entropy as follows: T is K if and only if for every non-trivial partition P of X, the entropy of T with respect to the partition P is positive: T has completely positive entropy. See the chapter on  Entropy in Ergodic Theory for the relevant definitions. The equivalence of the K property and completely positive entropy was shown by Rokhlin and Sinai [74]. For a general transformation T, one can consider the collection of all subsets B of X such that with respect to the partition PB D fB; B c g, h(PB ) D 0. One can show that this is a -algebra. This -algebra is known as the Pinsker -algebra. The above reformulation allows us to say that a transformation is K if and only if it has a trivial Pinsker -algebra. The K property implies countable Lebesgue spectrum (see Parry’s book [63] for a proof). To see that K is not implied by countable Lebesgue spectrum, we point out that certain measure-preserving transformations derived from Gaussian systems (see for example the paper of Parry and

Newton [52]) have countable Lebesgue spectrum but zero entropy. The fact that (two-sided) Bernoulli shifts have the K property follows from Kolmogorov’s 0–1 law by taking W1 F D nD0 T n P , where P is the partition into cylinder sets (see Williams’s book [93] for details of the 0–1 law). Although the K property is explicitly an invertible property, it has a non-invertible counterpart, namely exactness. A transformation T of (X; B; ) is exact if T1 n B consists entirely of null sets and sets of meanD0 T sure 1. It is not hard to see that a non-invertible transformation is exact if and only if its natural extension is K. The final and strongest property in our list is that of being measure-theoretically isomorphic to a Bernoulli shift. If T is measure-theoretically isomorphic to a Bernoulli shift, we say that T has the Bernoulli property. While in principle this could apply to both invertible and non-invertible transformations, in practice the definition applies to a large class of invertible transformations, but occurs comparatively seldom for non-invertible transformations. For this reason, we will restrict ourselves to a discussion of the Bernoulli property for invertible transformations (see however work of Hoffman and Rudolph [29] and Heicklen and Hoffman [28] for work on the one-sided Bernoulli property). In the case of invertible Bernoulli shifts, Ornstein [53,58] developed in the early 1970s a powerful isomorphism theory, showing that two Bernoulli shifts are measure-theoretically isomorphic if and only if they have the same entropy. Entropy had already been identified as an invariant by Kolmogorov and Sinai [43,82], so this established that it was a complete invariant for Bernoulli shifts. Keane and Smorodinsky [41] gave a proof which showed that two Bernoulli shifts of the same entropy are isomorphic using a conjugating map that is continuous almost everywhere. With other authors, this theory was extended to show that the property of being isomorphic to a Bernoulli shift applied to a surprisingly large class of measure-preserving transformations (e. g. geodesic flows on manifolds of constant negative curvature (Ornstein and Weiss [60]), aperiodic irreducible Markov chains (Friedman and Ornstein [19]), toral automorphisms (Katznelson [39]) and more generally many Gibbs measures for hyperbolic dynamical systems (see the book of Bowen [11])). Initially, it was conjectured that the properties of being K and Bernoulli were the same, but since then a number of measure-preserving transformations that are K but not Bernoulli have been identified. The earliest was due to Ornstein [55]. Ornstein and Shields [59] then provided an uncountable family of non-isomorphic K automorphisms.

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Katok [37] gave an example of a smooth diffeomorphism that is K but not Bernoulli; and Kalikow [33] gave a very natural probabilistic example of a transformation that has this property (the T, T 1 process). While in systems that one regularly encounters there is a correlation between positive entropy and the stronger mixing properties that we have discussed, these properties are logically independent (for example taking the product of a Bernoulli shift and the identity transformation gives a positive entropy transformation that fails to be ergodic; also, the zero entropy Gaussian systems with countable Lebesgue spectrum mentioned above have relatively strong mixing properties but zero entropy). In many of the mixing criteria discussed above we have considered a pair of sets A and B and asked for asymptotic independence of A and B (so that for large n, A and T n B become independent). It is natural to ask, given a finite collection of sets A0 ; A1 ; : : : ; A k , under what conditions Q (A0 \T n 1 A1 \  \T n k A k ) converges to kjD0 (A j ). A measure-preserving transformation is said to be mixing of order k + 1 if for all measurable sets A0 ; : : : ; A k , lim

n 1 !1;n jC1n j !1

   A0 \ T n 1 A1 \    \ T n k A k D

k Y

(A j ) :

jD0

An outstanding open question asked by Rokhlin [73] appearing already in Halmos’ 1956 book [27] is to determine whether mixing (i. e. mixing of order 2) implies mixing of all orders. Kalikow [34] showed that mixing implies mixing of all orders for rank 1 transformations (existence of rank one mixing transformations having been previously established by Ornstein in [54]). Later Ryzhikov [77] used joining methods to establish the result for transformations with finite rank, and Host [30] also used joining methods to establish the result for measure-preserving transformations with singular spectrum, but the general question remains open. It is not hard to show using martingale arguments that K automorphisms and hence all Bernoulli measure-preserving transformations are mixing of all orders. For weak-mixing transformations, Furstenberg [21] has established the following weak-mixing of all orders statement: if a measure-preserving transformation T is weak-mixing, then given sets A0 ; : : : ; A k , there is a subsequence J of the integers of density 0 such that

lim

n!1 n62 J

k  Y   A0 \ T n A1 \    \ T kn A k D (A i ) : iD0

Bergelson [5] generalized this by showing that    A0 \ T p 1 (n) A1 \    \ T p k (n) A k n!1;n62 J k Y D (A i ) lim

iD0

whenever p1 (n); : : : ; p k (n) are non-constant integer-valued polynomials such that p i (n)  p j (n) is unbounded for i ¤ j. The method of proof of both of these results was a Hilbert space version of the van der Corput inequality of analytic number theory. Furstenberg’s proof played a key role in his ergodic proof [20] of Szemerédi’s theorem on the existence of arbitrarily long arithmetic progressions in a subset of the integers of positive density (see the chapter on  Ergodic Theory: Interactions with Combinatorics and Number Theory for more information about this direction of study). The conclusions that one draws here are much weaker than the requirement for mixing of all orders. For mixing of all orders, it was required that provided the gaps between 0; n1 ; : : : ; n k diverge to infinity, one achieves asymptotic independence, whereas for these weak-mixing results, the gaps are increasing along prescribed sequences with regular growth properties. It is interesting to note that the analogous question of whether mixing implies mixing of all orders is known to fail in higher-dimensional actions. Here, rather than a Z action, in which there is a single measure-preserving transformation (so that the integer n acts on a point x 2 X by mapping it to T n x), one takes a Zd action. For such an action, one has d commuting transformations T1 ; : : : ; Td and a vector (n1 ; : : : ; nd ) acts on a point x by sending it to n T1n 1    Td d x. Ledrappier [46] studied the following two2 dimensional action. Let X D fx 2 f0; 1gZ : xv C xvCe1 C xvCe2 D 0 (mod 2)g and let Ti (x)v D xvCe i . Since X is a compact Abelian group, it has a natural measure  invariant under the group operations (the Haar measure). It is not hard to show that this system is mixing (i. e. given any measurable sets A and B, (A \ T1n 1 T2n 2 B) ! (A)(B) as k(n1 ; n2 )k ! 1). Ledrappier showed that the system fails to be 3-mixing. Subsequently Masser [48] established necessary and sufficient conditions for similar higher-dimensional algebraic actions to be mixing of order k but not order k C 1 for any given k. Hyperbolicity and Decay of Correlations One class of systems in which the stronger mixing properties are often found is the class of smooth systems possessing uniform hyperbolicity (i. e. the tangent space to the manifold at each point splits into stable and unstable

Ergodicity and Mixing Properties

subspaces E s (x) and E u (x) such that the kDTj E s (x) k  a < 1 for all x and kDT 1 j E u (x) k  a and DT(Es (x)) D Es (T(x)) and DT(Eu (x)) D Eu (T(x))). In some cases similar conclusions are found in systems possessing non-uniform hyperbolicity. See Katok and Hasselblatt’s book [38] for an overview of hyperbolic dynamical systems, as well as the chapter in this volume on  Smooth Ergodic Theory. In the simple case of expanding piecewise continuous maps of the interval (that is, maps for which the absolute value of the derivative is uniformly bounded below by a constant greater than 1), it is known that if they are totally ergodic and topologically transitive (i. e. the forward images of any interval cover the entire interval), then provided that the map has sufficient smoothness (e. g. the map is C1 and the derivative satisfies a certain additional summability condition), the map has a unique absolutely continuous invariant measure which is exact and whose natural extension is Bernoulli (see the paper of Góra [25] for results of this type proved under some of the mildest hypotheses). These results were originally established for maps that were twice continuously differentiable, and the hypotheses were progressively weakened, approaching, but never meeting, C1 . Subsequent work of Quas [68,69] provided examples of C1 expanding maps of the interval for which Lebesgue measure was invariant, but respectively not ergodic and not weak-mixing. Some of the key tools in controlling mixing in one-dimensional expanding maps that are absent in the C1 case are bounded distortion estimates. Here, there is a constant 1  C < 1 such that given any interval I on which some power T n of T acts injectively and any sub-interval J of I, one has 1/C  (jT n Jj/jT n Ij)/(jJj/jIj)  C. An early place in which bounded distortion estimates appear is the work of Rényi [70]. One important class of results for expanding maps establishes an exponential decay of correlations. Here, one starts with f and g and one estiR a pair of smooth R functions R mates f  g ı T n d f d g d, where  is an absolutely continuous invariant measure. If  is mixing, we expect this to converge to 0. In fact though, in good cases this converges to 0 at an exponential rate for each pair of functions f and g belonging to a sufficiently smooth class. In this case, the measure-preserving transformation T is said to have exponential decay of correlations. See Liverani’s article [47] for an introduction to a method of establishing this based on cones. Exponential decay of correlations implies in particular that the natural extension is Bernoulli. Hu [31] has studied the situation of maps of the interval for which the derivative is bigger than 1 everywhere except at a fixed point, where the local behavior is of the form x 7! x C x 1C˛ for 0 < ˛ < 1. In this case, rather

than exhibiting exponential decay of correlations, the map has polynomial decay of correlations with a rate depending on ˛. In Young’s survey [94], a variety of techniques are outlined for understanding the strong ergodic properties of non-uniformly hyperbolic diffeomorphisms. In her article [95], methods are introduced for studying many classes of non-uniformly hyperbolic systems by looking at suitably high powers of the map, for which the power has strong hyperbolic behavior. The article shows how to understand the ergodic behavior of these systems. These methods are applied (for example) to billiards, one-dimensional quadratic maps and Hénon maps. Future Directions Problem 1 (Mixing of all orders) Does mixing imply mixing of all orders? Can the results of Kalikow, Ryzhikov and Host be extended to larger classes of measure-preserving transformations? Thouvenot observed that it is sufficient to establish the result for measure-preserving transformations of entropy 0. This observation (whose proof is based on the Pinsker  -algebra) was stated in Kalikow’s paper [34] and is reproduced as Proposition 3.2 in recent work of de la Rue [16] on the mixing of all orders problem. Problem 2 (Multiple weak-mixing) As mentioned above, Bergelson [5] showed that if T is a weak-mixing transformation, then there is a subset J of the integers of density 0 such that lim

n!1;n62 J

   A0 \ T p 1 (n) A1 \    \ T p k (n) A k D

k Y

(A i )

iD0

whenever p1 (n); : : : ; p k (n) are non-constant integer-valued polynomials such that p i (n)  p j (n) is unbounded for i ¤ j. It is natural to ask what is the most general class of times that can replace the sequences (p1 (n)); : : : ; (p k (n)). In unpublished notes, Bergelson and Håland considered as times the values taken by a family of integer-valued generalized polynomials (those functions of an integer variable that can be obtained by the operations of addition, multiplication, addition of or multiplication by apreal constantp and taking integer parts (e. g. g(n) D b 2b nc C b 3nc2 c)). They conjectured necessary and sufficient conditions for the analogue of Bergelson’s weakmixing polynomial ergodic theorem to hold, and proved the conjecture in certain cases.

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In a recent paper of McCutcheon and Quas [50], the analogous question was addressed in the case where T is a mild-mixing transformation. Problem 3 (Pascal adic transformation) Vershik [89,90] introduced a family of transformations known as the adic transformations. The underlying spaces for these transformations are certain spaces of paths on infinite graphs, and the transformations act by taking a path to its lexicographic neighbor. Amongst the adic transformations, the so-called Pascal adic transformation (so-called because the underlying graph resembles Pascal’s triangle) has been singled out for attention in work of Petersen and others [2,10,51,65]. In particular, it is unresolved whether this transformation is weak-mixing with respect to any of its ergodic measures. Weak-mixing has been shown by Petersen and Schmidt to follow from a number-theoretic condition on the binomial coefficients [2,10]. Problem 4 (Weak Pinsker Conjecture) Pinsker [67] conjectured that in a measure-preserving transformation with positive entropy, one could express the transformation as a product of a Bernoulli shift with a system with zero entropy. This conjecture (now known as the Strong Pinsker Conjecture) was shown to be false by Ornstein [56,57]. Shields and Thouvenot [78] showed that the collection of transformations that can be written as a product of a zero entropy transformation with a Bernoulli shift ¯ is closed in the so-called d-metric that lies at the heart of Ornstein’s theory. It is, however, the case that if T : X ! X has entropy h > 0, then for all h0  h, T has a factor S with entropy h0 (this was originally proved by Sinai [83] and reproved using the Ornstein machinery by Ornstein and Weiss in [61]). The Weak Pinsker Conjecture states that if a measure-preserving transformation T has entropy h > 0, then for all > 0, T may be expressed as a product of a Bernoulli shift and a measure-preserving transformation with entropy less than . Bibliography 1. Aaronson J (1997) An Introduction to Infinite Ergodic Theory. American Mathematical Society, Providence 2. Adams TM, Petersen K (1998) Binomial coefficient multiples of irrationals. Monatsh Math 125:269–278 3. Alpern S (1976) New proofs that weak mixing is generic. Invent Math 32:263–278 4. Avila A, Forni G (2007) Weak-mixing for interval exchange transformations and translation flows. Ann Math 165:637–664 5. Bergelson V (1987) Weakly mixing pet. Ergodic Theory Dynam Systems 7:337–349 6. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Nat Acad Sci 17:656–660

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57. Ornstein DS (1973) A mixing transformation for which Pinsker’s conjecture fails. Adv Math 10:103–123 58. Ornstein DS (1974) Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, Newhaven 59. Ornstein DS, Shields PC (1973) An uncountable family of K-automorphisms. Adv Math 10:89–102 60. Ornstein DS, Weiss B (1973) Geodesic flows are Bernoullian. Israel J Math 14:184–198 61. Ornstein DS, Weiss B (1975) Unilateral codings of Bernoulli systems. Israel J Math 21:159–166 62. Oxtoby JC (1952) Ergodic sets. Bull Amer Math Soc 58:116–136 63. Parry W (1981) Topics in Ergodic Theory. Cambridge, Cambridge 64. Petersen K (1983) Ergodic Theory. Cambridge, Cambridge 65. Petersen K, Schmidt K (1997) Symmetric Gibbs measures. Trans Amer Math Soc 349:2775–2811 66. Phelps R (1966) Lectures on Choquet’s Theorem. Van Nostrand, New York 67. Pinsker MS (1960) Dynamical systems with completely positive or zero entropy. Soviet Math Dokl 1:937–938 68. Quas A (1996) A C 1 expanding map of the circle which is not weak-mixing. Israel J Math 93:359–372 69. Quas A (1996) Non-ergodicity for C 1 expanding maps and g-measures. Ergodic Theory Dynam Systems 16:531–543 70. Rényi A (1957) Representations for real numbers and their ergodic properties. Acta Math Acad Sci Hungar 8:477–493 71. Riesz F (1938) Some mean ergodic theorems. J Lond Math Soc 13:274–278 72. Rokhlin VA (1948) A ‘general’ measure-preserving transformation is not mixing. Dokl Akad Nauk SSSR Ser Mat 60:349–351 73. Rokhlin VA (1949) On endomorphisms of compact commutative groups. Izvestiya Akad Nauk SSSR Ser Mat 13:329–340 74. Rokhlin VA, Sinai Y (1961) Construction and properties of invariant measurable partitions. Dokl Akad Nauk SSSR 141:1038– 1041 75. Rudin W (1966) Real and Complex Analysis. McGraw Hill, New York 76. Rudolph DJ (1990) Fundamentals of Measurable Dynamics. Oxford, Oxford University Press 77. Ryzhikov VV (1993) Joinings and multiple mixing of the actions of finite rank. Funct Anal Appl 27:128–140 78. Shields P, Thouvenot J-P (1975) Entropy zero × Bernoulli pro¯ cesses are closed in the d-metric. Ann Probab 3:732–736 79. Simányi N (2003) Proof of the Boltzmann–Sinai ergodic hypothesis for typical hard disk systems. Invent Math 154:123– 178 80. Simányi N (2004) Proof of the ergodic hypothesis for typical hard ball systems. Ann Henri Poincaré 5:203–233 81. Simányi N, Szász D (1999) Hard ball systems are completely hyperbolic. Ann Math 149:35–96 82. Sinai YG (1959) On the notion of entropy of a dynamical system. Dokl Russ Acad Sci 124:768–771 83. Sinai YG (1964) On a weak isomorphism of transformations with invariant measure. Mat Sb (NS) 63:23–42 84. Sinai YG (1970) Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat Nauk 25:141–192 85. Sinai Y (1976) Introduction to Ergodic Theory. Princeton, Princeton, translation of the 1973 Russian original

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Ergodic Theorems

Ergodic Theorems ANDRÉS DEL JUNCO Department of Mathematics, University of Toronto, Toronto, Canada

Article Outline Glossary Definition of the Subject Introduction Ergodic Theorems for Measure-Preserving Maps Generalizations to Continuous Time and Higher–Dimensional Time Pointwise Ergodic Theorems for Operators Subadditive and Multiplicative Ergodic Theorems Entropy and the Shannon–McMillan–Breiman Theorem Amenable Groups Subsequence and Weighted Theorems Ergodic Theorems and Multiple Recurrence Rates of Convergence Ergodic Theorems for Non-amenable Groups Future Directions Bibliography

Glossary Dynamical system in its broadest sense, any set X, with a map T : X ! X. The classical example is: X is a set whose points are the states of some physical system and the state x is succeeded by the state Tx after one unit of time. Iteration repeated applications of the map T above to arrive at the state of the system after n units of time. Orbit of x the forward images x; Tx; T 2 X : : : of x 2 X under iteration of T. When T is invertible one may consider the forward, backward or two-sided orbit of x. Automorphism a dynamical system T : X ! X, where X is a measure space and T is an invertible map preserving measure. Ergodic average if f is a function on X let A n f (x) D P i n1 n1 iD0 f (T x); the average of the values of f over the first n points in the orbit of x. Ergodic theorem an assertion that ergodic averages converge in some sense. Mean ergodic theorem an assertion that ergodic averages converge with respect to some norm on a space of functions.

Pointwise ergodic theorem an assertion that ergodic averages A n f (x) converge for some or all x 2 X, usually for a.e. x. Stationary process a sequence (X1 ; X2 ; : : :) of random variables (real or complex-valued measurable functions) on a probability space whose joint distributions are invariant under shifting (X1 ; X2 ; : : :) to (X2 ; X3 ; : : :). Uniform distribution a sequence fx n g in [0; 1] is uniformly distributed if for each interval I [0; 1], the time it spends in I is asymptotically proportional to the length of I. Maximal inequality an inequality which allows one to bound the pointwise oscillation of a sequence of functions. An essential tool for proving pointwise ergodic theorems. Operator any linear operator U on a vector space of functions on X, for example one arising from a dynamical system T by setting U f (x) D f (Tx). More generally any linear transformation on a real or complex vector space. Positive contraction an operator T on a space of functions endowed with a norm k  k such that T maps positive functions to positive functions and kT f k  j f j. Definition of the Subject Ergodic theorems are assertions about the long-term statistical behavior of a dynamical system. The subject arose out of Boltzmann’s ergodic hypothesis which sought to equate the spatial average of a function over the set of states in a physical system having a fixed energy with the time average of the function observed by starting with a particular state and following its evolution over a long time period. Introduction Suppose that (X; B; ) is a measure space and T : (X; B; ) ! (X; B; ) is a measurable and measurepreserving transformation, that is (T 1 E) D (E) for all E 2 B. One important motivation for studying such maps is that a Hamiltonian physical system (see the article by Petersen in this collection) gives rise to a one-parameter group fTt : t 2 Rg of maps in the phase space of the system which preserve Lebesgue measure. The ergodic theorem of Birkhoff asserts that for f 2 L1 () n1 1X f (T i x) n

(1)

iD0

converges a.e. and that if RT is ergodic (to be defined shortly) then the limit is f d. This may be viewed

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as a justification for Boltzmann’s ergodic hypothesis that “space averages equal time averages”. See Zund [214] for some history of the ergodic hypothesis. For physicists, then, the problem is reduced to showing that a given physical system is ergodic, which can be very difficult. However ergodic systems arise in many natural ways in mathematics. One example is rotation z 7! z of the unit circle if  is a complex number of modulus one which is not a root of unity. Another is the shift transformation on a sequence of i.i.d. random variables, for example a coin-tossing sequence. Another is an automorphism of a compact Abelian group. Often a transformation possesses an invariant measure which is not obvious at first sight. Knowledge of such a measure can be a very useful tool. See Petersen’s article for more examples. If (X; B; ) is a probability space and T is ergodic then Birkhoff’s ergodic theorem implies that if A is a measurable subset of X then for almost every x, the frequency with which x visits A is asymptotically equal to (A), a very satisfying justification of intuition. For example, applying this to the coin-tossing sequence one obtains the strong law of large numbers which asserts that almost every infinite sequence of coin tosses has tails occurring with asymptotic frequency 12 . One also obtains Borel’s theorem on normal numbers which asserts that for almost all x 2 [0; 1] each digit 0; 1; 2; : : : ; 9 occurs with limiting 1 frequency 10 . The so-called continued fraction transformation x 7! x 1 mod 1 on (0; 1) has a finite invariant dx . (Throughout this article x mod 1 denotes measure 1Cx the fractional part of x.) Applying Birkhoff’s theorem then gives precise information about the frequency of occurrence of any n 2 N in the continued fraction expansion of x, for a.e. x. See for example Billingsley [34]. These are the classical roots of the subject of ergodic theorems. The subject has evolved from these simple origins into a vast field in its own right, quite independent of physics or probability theory. Nonetheless it still has close ties to both these areas and has also forged new links with many other areas of mathematics. Our purpose here is to give a broad overview of the subject in a historical perspective. There are several excellent references, notably the books of Krengel [135] and Tempelman [194] which give a good picture of the state of the subject at the time they appeared. There has been tremendous progress since then. The time is ripe for a much more comprehensive survey of the field than is possible here. Many topics are necessarily absent and many are only glimpsed. For example this article will not touch on random ergodic theorems. See the articles [75,143] for some references on this topic.

I thank Mustafa Akcoglu, Ulrich Krengel, Michael Lin, Dan Rudolph and particularly Joe Rosenblatt and Vitaly Bergelson for many helpful comments and suggestions. I would like to dedicate this article to Mustafa Akcoglu who has been such an important contributor to the development of ergodic theorems over the past 40 years. He has also played a vital role in my mathematical development as well as of many other mathematicians. He remains a source of inspiration to me, and a valued friend. Ergodic Theorems for Measure-Preserving Maps Suppose (X; B; ) is a measure space. A (measure-preserving) endomorphism of (X; B; ) is a measurable mapping T : X ! X such that (T 1 E) D (E) for any measurable subset E X. If T has a measurable inverse then one says that it is an automorphism. In this article the unqualified terms “endomorphism” or “automorphism” will always mean measure-preserving. T is called ergodic if for all measurable E one has T 1 E D E ) (E) D 0

or (E c ) D 0 :

A very general class of examples comes from the notion of a stationary stochastic process in probability theory. A stochastic process is a sequence of measurable functions f1 ; f2 ; : : : on a probability space (X; B; ) taking values in a measurable space (Y; C ). The distribution of the process, a measure  on Y N , is defined as the image of  under the map ( f1 ; f2 ; : : :) : X ! Y N .  captures all the essential information about the process f f i g. In effect one may view any process f f i g as a probability measure on Y N . The process is said to be stationary if  is invariant under the left shift transformation S on Y N , S(y)(i) D y(i C 1), that is S is an endomorphism of the probability space (Y N ; ). From a probabilistic point of view the most natural examples of stationary stochastic processes are independent identically distributed processes, namely the case when  is a product measure N for some measure  on (Y; C ). More generally one can consider a stationary Markov process defined by transition probabilities on the state space Y and an invariant probability on Y. See for example Chap. 7 in [50], also Sect. “Pointwise Ergodic Theorems for Operators” below. The first and most fundamental result about endomorphisms is the celebrated recurrence theorem of Poincaré [173]). Theorem 1 Suppose  is finite, A 2 B and (A) > 0. Then for a.e. x 2 A there is an n > 0 such that T n x 2 A, in fact there are infinitely many such n. It may be viewed as an ergodic theorem, in that it is a qualitative statement about how x behaves under iteration of T.

Ergodic Theorems

For a proof, observe that if E A is the measurable set of points which never return to A then for each n > 0 the set T n E is disjoint from E. Applying T m one gets also T (nCm) E \ T m E D ;. Thus E; T 1 E; T 2 E; : : : is an infinite sequence of disjoint sets all having measure (E) so (E) D 0 since (X) < 1. Much later Kac [120] formulated the following quantitative version of Poincaré’s theorem. Theorem 2 Suppose that (X) D 1, T is ergodic and let r A x denote the time of first return to A, that is r A (x) is the least n > 0 such that T n x 2 A. Then Z 1 1 r A d D ; (2) (A) A (A) that is, the expected value of the return time to A is (A)1 . Koopman [131] made the observation that associated to an automorphism T there is a unitary operator U D U T defined on the Hilbert space L2 () by the formula U f D f ı T. This led von Neumann [151] to prove his mean ergodic theorem. Theorem 3 Suppose H is a Hilbert space, U is a unitary operator on H and let P denote the orthogonal projection on the subspace of U-invariant vectors. Then for any x 2 H one has  n1  1X    U i x  Px  ! 0 as n ! 1 : (3)  n  iD0

Von Neumann’s theorem is usually quoted as above but to be historically accurate he dealt with unitary operators indexed by a continuous time parameter. Inspired by von Neumann’s theorem Birkhoff very soon proved his pointwise ergodic theorem [35]. In spite of the later publication by von Neumann his result did come first. See [29,214] for an interesting discussion of the history of the two theorems and of the interaction between Birkhoff and von Neumann. Theorem 4 Suppose (X; B; ) is a measure space and T is an endomorphism of (X; B; ). Then for any f 2 L1 D L1 () there is a T-invariant function g 2 L1 such that A n f (x) D

n1 1X f (T i x) ! g(x) n

a.e.

(4)

iD0

Moreover if  is finite then the convergence alsoRholds with R respect to the L1 norm and one has E g d D E f d for all T-invariant subsets E.

Again this formulation of Birkhoff’s theorem is not historically accurate as he dealt with a smooth flow on a manifold. It was soon observed that the theorem, and its proof, remain valid for an abstract automorphism of a measure space although the realization that T need not be invertible seems to have taken a little longer. The notation A n f D A n (T) f as above will occur often in the sequel. Whenever T is an endomorphism one uses the notation T f D f ı T and with this notation P i A n (T) D n1 n1 iD0 T . When the scalars (R or C) for an L1 space are not specified the notation should be understood as referring to either possibility. In most of the theorems in this article the complex case follows easily from the real and any indications about proofs will refer to the real case. Although von Neumann originally used spectral theory to prove his result, there is a quick proof, attributed to Riesz by Hopf in his 1937 book [100], which uses only elementary properties of Hilbert space. Let I denote the (closed) subspace of U-invariant vectors and I 0 the (usually not closed) subspace of vectors of the form f  U f . It is easy to check that any vector orthogonal to I 0 must be in I, whence the subspace I C I 0 is dense in H . For any vector of the form x D y C y 0 ; y 2 I; y 0 2 I 0 it is clear P i that A n x D n1 n1 iD0 U x converges to y D Px, since 0 A n y D y and if y D z  Uz then the telescoping sum A n y 0 D n1 (z  U n z) converges to 0. This establishes the desired convergence for x 2 I C I 0 and it is easy to extend it to the closure of I C I 0 since the operators An are contractions of H (kA n k  1). Lorch [147] used a soft argument in a similar spirit to extend non Neumann’s theorem from the case of a unitary operator on a Hilbert space to that of an arbitrary linear contraction on any reflexive Banach space. Sine [188] gave a necessary and sufficient condition for the strong convergence of the ergodic averages of a contraction on an arbitrary Banach space. Birkhoff’s theorem has the distinction of being one of the most reproved theorems of twentieth century mathematics. One approach to the pointwise convergence, parallel to the argument to argument just seen, is to find a dense subspace E of L1 so that the convergence holds for all f 2 E and then try to extend the convergence to all f 2 L1 by an approximation argument. The first step is not too hard. For simplicity assume that  is a finite measure. As in the proof of von Neumann’s theorem the subspace E spanned by the T-invariant L1 functions together with functions of the form g  T g; g 2 L1 , is dense in L1 (). This can be seen by using the Hahn–Banach theorem and the duality of L1 and L1 . (Here one needs finiteness of  to know that L1 L1 .) The pointwise convergence of A n f for invariant f is trivial and for f D g  T g it follows from telescoping of the sum and the fact that n1 T n g ! 0 a.e. This

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last can be shown by using the Borel–Cantelli lemma. The second step, extending pointwise convergence, as opposed to norm convergence, for f in a dense subspace to all f in L1 is a delicate matter, requiring a maximal inequality. Roughly speaking a maximal inequality is an inequality which bounds the pointwise oscillation of A n f in terms of the norm of f . The now standard maximal inequality in the context of Birkhoff’s theorem is the following, due to Kakutani and Yosida [209]. Birkhoff’s proof of his theorem includes a weaker version of this result. Let S n f D nA n f , the nth partial sum of the iterates of f . S Theorem 5 Given any real f 2 L1 let A D n1 fS n f  0g. Then Z f d  0 : (5) A

Moreover if one sets M f D supn1 A n f then for any ˛ > 0 fM f > ˛g 

1 k f k1 ˛

(6)

A distributional inequality such as (6) will be referred to as a weak L1 inequality. Note that (6) follows easily from (5) by applying (5) to f ˛, at least in the case when  is finite. With the maximal inequality in hand it is straightforward to complete the proof of Birkhoff’s theorem. For a realvalued function f let Osc f D lim sup A n f  lim inf A n f :

(7)

Osc f D 0 a.e. if and only if A n f converges a.e.(to a possibly infinite limit). One has lim sup A n f  M f  Mj f j and by symmetry lim inf A n f  Mj f j, so Osc f  2Mj f j. To establish the convergence of A n f for a real-valued f 2 L1 let > 0 and write f D g C h with g 2 E (the subspace where convergence has already been established), h 2 L1 and khk < . Then since Osc g D 0 one has Osc f D Osc h. Thus for any fixed ˛ > 0, using (6) n ˛o fOsc f > ˛g D  fOsc h > ˛g   M h > 2 2khk1 2  < : (8) ˛ ˛ Since > 0 was arbitrary one concludes that fOsc f > ˛g D 0 and since ˛ > 0 is arbitrary it follows that fOsc f > 0g D 0, establishing the a.e. convergence. Moreover a simple application of Fatou’s lemma shows that the limiting function is in L1 , hence finite a.e. There are many proofs of (5). Two of particular interest are Garsia’s [89], perhaps the shortest and most mysterious, and the proof via the filling scheme of Chacón

and Ornstein [61], perhaps the most intuitive, which goes like this. Given a function g 2 L1 write g C D max(g; 0), g  D g C  g and let Ug D T g C  g  . Interpretation: the region between the graph of g C and the X-axis is a sheaf of vertical spaghetti sticks, the intervals [0; g C (x)], x in X, and g  is a hole. Now move the spaghetti (horizontally) by T and then let it drop (vertically) into the hole leaving a new hole and a new sheaf which are the negative and posS itive parts of Ug. Now let E 0 D n1 fU n f  0g, the set of points x at which the hole is eventually filled after finitely many iterations of U. The key point is that E D E 0 . Indeed if S n f (x)  0 for some n, then the total linear height of sticks over x; Tx; : : : T n1 x is greater than the total linear depth of holes at these points. The only way that spaghetti can escape from these points is by first filling the hole at x, which shows x 2 E 0 . Similar thinking shows that if x 2 E 0 and the hole at x is filled for the first time at time n then S n f (x)  0, so x 2 E, and that all the spaghetti that goes into the hole at x comes from points T i x which belong to E 0 . This shows that E D E 0 and that the part of the hole lying beneath E is eventually filled by spaghetti coming from E 0 D E. Thus the amount of spaghettiRover E is no less than the size of the hole under E, that is E f d  0. Most proofs of Birkhoff’s theorem use a maximal inequality in some form but a few avoid it altogether, for example [126,186]. It is also straightforward to deduce Birkhoff’s theorem directly from a maximal inequality, as Birkhoff does, without first establishing convergence on a dense subspace. However the technique of proving a pointwise convergence theorem by finding an appropriate dense subspace and a suitable maximal inequality has proved extremely useful, not only in ergodic theory. Indeed, in some sense maximal inequalities are unavoidable: this is the content of the following principle proved already in 1926 by Banach. The principle has many formulations; the following one is a slight simplification of the one to be found in [135]. Suppose B is a Banach space, (X; B; ) is a finite measure space and let E denote the space of -equivalence classes of measurable real-valued functions on X. A linear map T : B ! E is said to be continuous in measure if for each > 0 kx n  xk ! 0 ) fjTx n  Txj > g ! 0 : Suppose that T n is a sequence of linear maps from B to E which are continuous in measure and let Mx D sup n jTn xj. Of course if Tn x converges a.e. to a finite limit then Mx < 1 a.e. Theorem 6 (Banach principle) Suppose Mx < 1 a.e. for each x 2 X. Then there is a function C() such that

Ergodic Theorems

C() ! 0 as  ! 1 such that for all x 2 B and  > 0 one has fMx  kxkg  C() :

(9)

Moreover the set of x for which Tn x converges a.e. is closed in B. The first chapter of Garsia’s book [89] contains a nice introduction to the Banach principle. It should be noted that, for general integrable f , M f need not be in L1 . However if f 2 L p for p > 1 then M f does belong to Lp and one has the estimate kM f k p 

p k f kp : p1

(10)

This can be derived from (6) using also the (obvious) fact that kM f k1  k f k1 . Any such estimate on the norm of a maximal function will be called a strong Lp inequality. It also follows from (6) that if (X) is finite and f 2 L log L, R that is j f j logC j f jd < 1, then M f 2 L1 . In fact Ornstein [160] has shown that the converse of this last statement holds provided T is ergodic. There is a special setting where one has uniform convergence in the ergodic theorem. Suppose T is a homeomorphism of a compact metric space X. By a theorem of Krylov and Bogoliouboff [138] there is at least one probability measure on the Borel -algebra of X which is invariant under T. T is said to be uniquely ergodic if there is only one Borel probability measure, say , invariant under T. It is easy to see that when this is the case then T is an ergodic automorphism of (X; B; ). As an example, if ˛ is an irrational number then the rotation z 7! e2 i˛ z is a uniquely ergodic transformation of the circle fjzj D 1g. Equivalently x 7! x C ˛ mod 1 is a uniquely ergodic map on [0; 1]. A quick way to see this is to show that the ˆ Fourier co-efficients (n) of any invariant probability  are zero for n ¤ 0. The Jewett–Krieger theorem (see Jewett [109] and Krieger [137]) guarantees that unique ergodicity is ubiquitous in the sense that any automorphism of a probability space is measure-theoretically isomorphic to a uniquely ergodic homeomorphism. The following important result is due to Oxtoby [165]). Theorem 7 If T is uniquely ergodic,  is its unique invariant probability measure and f 2 C(X) R then the ergodic averages A n ( f ) converge uniformly to f d. This result can be proved along the same lines as the proof given above of von Neumann’s theorem. In a nutshell, one uses the fact that the dual of CR (X) is the space of finite signed measures on X and the unique ergodicity to show

that functions of the form f  f ı T together with the invariant functions (which are just constant functions) span a dense subspace of C(X). A sequence fx n g in the interval [0; 1]R is said to be uniP 1 formly distributed if n1 niD1 f (x n ) ! 0 f (x)dx for any f 2 C[0; 1] (equivalently for any Riemann integrable f or for any f D 1I where I is any subinterval of [0; 1]). As a simple application of Theorem 7 one obtains the linear case of the following result of Weyl [202]. Theorem 8 If ˛ is irrational and p(x) is any non-constant polynomial with integer coefficients then the sequence fp(n)˛mod1g is uniformly distributed. Furstenberg [84], see also [86], has shown that Weyl’s result, in full generality, can be deduced from the unique ergodicity of certain affine transformations of higher dimensional tori. For polynomials of degree k > 1 Weyl’s result is usually proved by inductively reducing to the case k D 1, using the following important lemma of van der Corput (see for example [139]). Theorem 9 Suppose that for each fixed h > 0 the sequence fx nCh  x n

mod 1gn

is uniformly distributed. Then fx n g is uniformly distributed. When  is infinite and T is ergodic the limiting function in Birkhoff’s theorem is 0 a.e. In 1937 Hopf [100] proved a generalization of Birkhoff’s theorem which is more meaningful in the case of an infinite invariant measure. It is a special case of a later theorem of Hurewicz [105], which we will discuss first. Suppose that (X; B; ) is a  -finite measure space. If  is another  -finite measure on B write    ( is absolutely continuous relative to ) if (E) D 0 implies (E) D 0 and one writes   if    and   . Consider a non-singular automorphism  : X ! X, meaning that  is measurable with a measurable inverse and that (E) D 0 if and only if ( E) D 0. In other words  D  ı  . By the Radon–Nikodym theorem there Ris a function 2 L1 () such that > 0 a.e. and (E) D E d for all measurable E. In order to obtain an associated operator T on L1 which is an (invertible) isometry one defines T f (x) D (x) f ( x) :

(11)

The dual operator on L1 is then given by T  g D g ı  1 . If  is a  -finite measure equivalent to  which is invariant under  then the ergodic theory of  can be reduced to the measure-preserving case using  . The interesting case is when there is no such . It was an open

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Ergodic Theorems

problem for some time whether there is always an equivalent invariant measure. In 1960 Ornstein [163] gave an example of a  which does not have an equivalent invariant measure. It is curious that, with hindsight, such examples were already known in the fifties to people studying von Neumann algebras. Pn i For f 2 L1 let S n f D iD0 T f .  is said to be conservative if there is no set E with (E) > 0 such that  i E; i D 0; 1; : : : are pairwise disjoint, that is if the Poincaré recurrence theorem remains valid. For example, the shift on Z is not conservative. Hurewicz [105] proved the following ratio ergodic theorem. Theorem 10 Suppose  is conservative, f ; g 2 L1 and a.e. to a -invarig(x) > 0 a.e. Then S n f /S n g converges R f d ant limit h. If  is ergodic then h D R gd . In the case when  is -invariant one has T f D f ı . If  is invariant and finite, taking g D 1 one recovers Birkhoff’s theorem. If  is invariant and -finite then Hurewicz’s theorem becomes the theorem of Hopf alluded to earlier. Wiener and Wintner [204] proved the following variant of Birkhoff’s theorem. Theorem 11 (Wiener–Wintner) Suppose T is an automorphism of a probability space (X; B; ). Then for any f 2 L1 there is a subset X 0 X of measure one such that for each x 2 X 0 and  2 C of modulus 1 the sequence 1 P n1 i i iD0  T f (x) converges. n It is an easy consequence of the ergodic theorem that one has a.e. convergence for a given f and  but the point here is that the set on which the convergence occurs is independent of . Generalizations to Continuous Time and Higher-Dimensional Time A (measure-preserving) flow is a one-parameter group fTt ; t 2 Rg of automorphisms of (X; B; ), that is TtCs D Tt Ts , such that Tt x is measurable as a function of (t; x). It will always be implicitly assumed that the map (t; x) 7! Tt x from R  X to X is measurable. Theorem 4 generalizes to flows by replacing sums with integrals and this generalization follows without difficulty from Theorem 4. (As already observed this observation reverses the historical record.) Theorem 4 may be viewed as a theorem about the “discrete flow” fTn D T n : n 2 Zg. Wiener was the first to generalize Birkhoff’s theorem to families of automorphisms fTg g indexed by groups more general than R or Z. A measure-preserving flow is an action of R while a single automorphism corresponds to an action of Z.

A (measure-preserving) action of a group G is a homomorphism T : g 7! Tg from G into the group of automorphisms of a measure space (X; B; ) (satisfying the appropriate joint measurability condition in case G is not discrete). Suppose now that G D R k or Z k and T is an action of G on (X; B; ). In the case of Z k an action amounts to an arbitrary choice of commuting maps T1 ; : : : Tk , Ti D T(e i ) where ei is the standard basis of Z k . In the case of R k one must specify k commuting flows. Let m denote counting measure on G in case G D Z k and Lebesgue measure in case G D R k . For any subset E of G with m(E) < 1 let Z 1 f (Tg x)dm(g) : (12) A E f (x) D m(E) E One may then ask whether A E f converges to a limit, either in the mean or pointwise, as E varies through some sequence of sets which “grow large” or, in case G D R k , “shrink to 0”. The second case is referred to as a local ergodic theorem. In the case of ergodic theorems at infinity the continuous and discrete theories are rather similar and often the continuous analogue of a discrete result can be deduced from the discrete result. In Wiener [203] proved the following result for actions of G D R k and ergodic averages over Euclidean balls Br D fx 2 R k : kxk2  rg. Theorem 12 Suppose T is an action of R k on (X; B; ) and f 2 L1 (). (a) For f 2 L1 limr!1 A B r f D g exists a.e. If  is finite the convergence also holds with respect to the L1 -norm, R R g is T-invariant and I gd D I f d for every T-invariant set I. (b) limr!0 A B r f D f a.e. The local aspect of Wiener’s theorem is closely related to the Lebesgue differentiation theorem, see for example Proposition 3.5.4 of [132], which, in its simplest form, states Rthat for f 2 L1 (R k ; m) one has a.e. convergence of 1 m(B r ) B r f (x C t)dt to f (x) as r ! 0. The local ergodic theorem implies Lebesgue’s theorem, simply by considering the action of R k on itself by translation. In fact the local ergodic theorem can also be deduced from Lebesgue’s theorem by a simple application of Fubini’s theorem (see for example [135], Chap. 1, Theorem 2.4 in the case k D 1). The key point in Wiener’s proof is the following weak L1 maximal inequality, similar to (6). Theorem 13 Let M f D supr>0 jA B r f j. Then one has fM f > ˛g 

C k f k1 ; ˛

(13)

Ergodic Theorems

where C is a constant depending only on the dimension d. (In fact one may take C D 3d .) In the case when T is the action of Rk on itself by translation (13) is the well-known maximal inequality for the Hardy–Littlewood maximal function ([132], Lemma 3.5.3). Wiener proves (13) by way of the following covering lemma. If B is a ball in R k let B0 denote the concentric ball with three times the radius. Rk

Theorem 14 Suppose a compact subset K of is covered by a (finite) collection U of open balls. Then there exist pairS wise disjoint B i 2 U; i D 1; : : : k such that i B0i covers K. To find the Bi it suffices to let B1 be the largest ball in U, then B2 the largest ball which does not intersect B1 and in general Bn the largest ball which does not intersect Sn1 0 iD1 B i . Then it is not hard to argue that the B i cover K. In general, a covering lemma is, roughly speaking, an assertion that, given a cover U of a set K in some space (often a group), one may find a subcollection U0 U which covers a substantial part of K and is in some sense efficient P in that U 2U0 1U  C, C some absolute constant. Covering lemmas play an important role in the proofs of many maximal inequalities. See Sect. 3.5 of [132] for a discussion of several of the best-known classical covering lemmas. As Wiener was likely aware, the same kind of covering argument easily leads to maximal inequalities and ergodic theorems for averages over “sufficiently regular” sets. For example, in the case of G D Z k use the standard total order  on G and for n 2 N k let S n D fm 2 Z k : 0  m  ng. Let e(n) denote the maximum value of n i /n j . Then one has the following result.

vergence if f 2 L p for some p > 1, provided  is finite. Let T1 ; : : : ; Tk be any k (possibly non-commuting!) automorphisms of a finite measure space (X; B; ). For n n D (n1 ; n2 ; : : : ; n k ) let Tn D T1n 1 : : : Tk k . (This is not an action of Z k unless the T i commute with each other.) As before when F Z k is a finite subset write A F f D 1 P n2F Tn f . Finally let P j f D lim n A n (T j ) f . jFj Theorem 16 For f 2 L p lim A S n f D P1 : : : Pk f

n!1

both a.e. and in Lp . The proof of Theorem 16 uses repeated applications of Birkhoff’s theorem and hinges on (10). Somewhat surprisingly Hurewicz’s theorem was not generalized to higher dimensions until the very recent work of Feldman [79]. In fact the theorem fails if one considers averages over the cube [0; n  1]d in Zd . However Feldman was able to prove a suitably formulated generalization of Hurewicz’s theorem for averages over symmetric cubes. For f 2 L1 (R) the classical Hilbert transform is defined for a.e. t by Z f (t  s) 1 lim ds (16) H f (t) D  !0C jsj> s R f (ts) Let H  f (t) D 1 sup >0 j jsj> s dsj, the corresponding maximal function. The proof that the limit (16) exists a.e. is based on the maximal inequality mfH f > g 

Theorem 15 For any e > 0 and any integrable f lim

n!1;e(n)<e

AS n f

exists a.e. and the limit is characterized by the same properties as in Theorem 12. Moreover there is a maximal inequality ( ) C  sup jA S n f j > ˛  k f k1 ; (14) ˛ e(n)<e

(15)

C k f k1 ; 

(17)

where C is an absolute constant. See Loomis [146] (1946) for a proof of (17) using real-variable methods. Cotlar [65] proved the existence of the following ergodic analogue of the Hilbert transform (actually for the n-dimensional Hilbert transform). Theorem 17 Suppose fTt g is a measure-preserving flow on a probability space (X; B; ) and f 2 L1 . Then Z f (Tt x) dt (18) lim !0
where C is a constant depending only on e and the dimension k.

exists for a.a. x.

If one does not impose a bound on e(n) the a.e. convergence of A S n f may fail for f 2 L1 , (although it does hold for any increasing sequence of n’s by Theorem 30 below). Nonetheless Dunford [72] and independently Zygmund [215] showed that one does have unrestricted con-

Motivated by Cotlar’s result Calderón [54]proved a general transfer principle which allows one to transfer maximal inequalities and convergence theorems for functions of a real or integer variable to the ergodic setting. Although stated for functions on R it applies equally well to R k

247

248

Ergodic Theorems

or Z k . This principle subsumes Birkhoff’s theorem, Wiener’s theorem and Cotlar’s result. For simplicity only a special case of Calderón’s result, in the discrete case, will be stated here. Let T be an automorphism of a probability space (X; B; ), let m denote counting measure on Z and suppose  is a probability measure on Z. For g 2 L1 (m) define P  (g)(n) D i2Z (i)g(nC i). For f 2 L1 () let (T) f D P i i2Z (i)T f , which is easily seen to converge a.e. and in L1 -norm. Given a fixed sequence  n of probabilities define M g D supn n g and M(T) f D supn n (T) f . Theorem 18 (Calderón) Suppose there is a constant C such that mfM g > g <

C kgk1 

for all g 2 L1 (m) :

(19)

Then one also has fM(T) f > g <

C k f k1 

for all T

and g 2 L1 () : (20)

In other words, in order to prove a maximal inequality for general T it suffices to prove it in case T is the shift map on the integers. The idea of transference could already be seen in Wiener’s proof of the Rn ergodic theorem. Transfer principles in various forms of have become an important tool in the study of ergodic theorems. See Bellow [18] for a very readable overview.

If  is a R -finite measure on X one may define the measure P D Px d(x). P is also meaningful if  is a finite signed measure. The case when P D  is the stochastic analogue of measure-preserving dynamics and the case when P   is the analogue of non-singular dynamics. It is easy to see that given any -finite measure  there is always a  such that    and P  . Let L˜ 1 denote the space of finite signed measures  such that   , which is identified with L1 D L1 (; R) via the Radon– Nikodym theorem. If P   then P maps L˜ 1 () into itself so the restriction of P is an operator T on L1 (). T is a positive contraction, that is kTk  1 and T maps nonnegative functions to non-negative functions. As proved in, for example, [153], every positive contraction arises in this way from a substochastic kernel under the assumption that (X; B) is standard. This simply means that there is some complete metric on X for which B is the -algebra of Borel sets. Virtually all measurable spaces encountered in analysis are standard so this should be viewed as a technicality only. See [153], [80] and [135] for more about the relation between kernels and positive contractions. The case when X is finite, P is stochastic and  is a a probability measure is classical in probability theory. P and  determine a probability measure  on X N characterized by its values on cylinder sets, namely for all x1 ; : : : ; x n 2 X fx 2 X N : x(i) D x i ; 1  i  n  1g D (x1 )

n1 Y

Px i (x iC1 ) :

(21)

iD1

Pointwise Ergodic Theorems for Operators Early in the history of ergodic theorems there were attempts to generalize the ergodic theorem to more general linear operators on Lp spaces, that is, operators which do not arise by composition with a mapping of X . In the case p D 1 the main motivation for this comes from the theory of Markov processes. If (X; B) is a measurable space a sub-stochastic kernel on X is a non-negative function P on X  B such that (a) for each x 2 XP(x; ) D Px is a measure on B such that Px (X)  1 and (b) P(; A) is a measurable function for each A 2 B. It is most intuitive to think about the stochastic case, namely when each Px is a probability measure. One then views P(x; A) as the probability that the point x moves into the set A in one unit of time, so one has stochastic dynamics as opposed to deterministic dynamics, namely the case when Px D ıT x for a map T. In this case the measures Px are called transition probabilities.

The co-ordinate functions X i (x) D x(i) on the space X N endowed with the probability  form a Markov process, which will be stationary if and only if  is P-invariant. For a general X the analogous construction is possible provided X is standard. Hopf [101] initiated the systematic study of positive L1 contractions and proved the following ergodic theorem. Theorem 19 Suppose (X; B; ) is a probability space and T is a positive contraction on L1 () satisfying T1 D 1 and P k T  1 D 1. Then limn!1 n1 n1 kD0 T f exists a.e. The importance of Hopf’s article lies less in this convergence result than in the methods he developed. He proved that the maximal inequality (5) generalizes to all positive L1 contractions T and used this to obtain the decomposition of X into its conservative and dissipative parts C and D, characterized by the fact that for any p 2 L1 such that P k p > 0 a.e. 1 kD0 T p is infinite a.e. on C and finite a.e. on D. These results are the cornerstone of much of the subsequent work on L1 contractions.

Ergodic Theorems

Theorem 19 contains Birkhoff’s theorem for an automorphism  of (X; B; ), simply by defining T f D f ı . In fact one can also deduce Theorem 19 from Birkhoff’s theorem (if one assumes only that X is standard). Indeed the hypotheses of Hopf’s theorem imply that the kernel P associated to T is stochastic (T  1 D 1) and and that  is P-invariant (T1 D 1). Hopf’s theorem then follows by applying Birkhoff’s theorem to the shift on the stationary Markov process associated to P and . In fact Kakutani [123] (see also Doob [71]) had already made essentially the same observation, except that his result assumes the stationary Markov process to be already given. In 1955 Dunford and Schwartz [73] made essential use of Hopf’s work to prove the following result. Theorem 20 Suppose (X) D 1 and T is a (not necessarily positive) contraction with respect to both the L1 and L1 norms. Then the conclusion of Hopf’s theorem remains valid. Note that the assumption that T contracts the L1 -norm is meaningful, as L1 L1 . The proof of the result is reduced to the positive case by defining a positive contraction jTj analogous to the total variation of a complex measure. Then in 1960 Chacón and Ornstein [61] proved a definitive ratio ergodic theorem for all positive contractions of L1 which generalizes both Hopf’s theorem and the Hurewicz theorem. Theorem 21 Suppose T is a positive contraction of L1 (), where  is -finite, f ; g 2 L1 and g  0. Then Pn Ti f PiD0 n i iD0 T g converges a.e. on the set f

(22) P1

iD0

T i g > 0g.

In 1963 Chacón [58] proved a very general theorem for non-positive operators which includes the Chacón–Ornstein theorem as well as the Dunford–Schwartz theorem. Theorem 22 Suppose T is a contraction of L1 and p n  0 is a sequence of measurable functions with the property that g 2 L1 ; jgj  p n ) jT gj  p nC1 :

(23)

Then Pn Ti f PiD0 n iD0 p i converges a.e. to a finite limit on the set f

(24) P1

iD0

p i > 0g.

If T is an L1 ; L1 -contraction and p n D 1 for all n then the hypotheses of this theorem are satisfied, so Theorem 22

reduces to the result of Dunford and Schwartz. See [59] for a concise overview of all of the above theorems in this section and the relations between them. The identification of the limit in the Chacón–Ornstein theorem on the conservative part C of X is a difficult problem. It was solved by Neveu [152] in case C D X, and in general by Chacón [57]. Chacón [60] has shown that there is a non-singular automorphism  of (X; B; ) such that for the associated invertible isometry T of L1 () given by (11) there is an f 2 L1 () such that lim sup A n f D 1 and lim inf A n f D 1. In 1975 Akcoglu [3] solved a major open problem when he proved the following celebrated theorem. Theorem 23 Suppose T : L p 7! L p is a positive contracP i tion. Then A n f D n1 n1 iD0 T f converges a.e. Moreover one has the strong Lp inequality     sup jA n f j     n

p

p k f kp : p1

(25)

As usual, the maximal inequality (25) is the key to the convergence. Note that it is identical in form to (10) the classical strong Lp inequality for automorphisms. (25) was proved by A. Ionesco–Tulcea (now Bellow) [106] in the case of positive invertible isometries of Lp . It is a result of Banach [16], see also [140], that in this case T arises from a non-singular automorphism  of (X; B; ) in the form T f D 1/p f ı . By a series of reductions Bellow was able to show that in this case (25) can be deduced from (10). Akcoglu’s brilliant idea was to consider a dilation S of T which is a positive invertible isometry on a larger Lp space L˜ p D L p (Y; C ; ). What this means is that there is a positive isometric injection D : L p ! L˜ p and a positive projection P on L˜ p whose range is D(L p ) such that DT n D PS n D for all n  0. Given the existence of such an S it is not hard to deduce (25) for T from (25) for S. In fact Akcoglu constructs a dilation only in the case when Lp is finite dimensional and shows how to reduce the proof of (25) to this case. In the finite dimensional case the construction is very concrete and P is a conditional expectation operator. Later Akcoglu and Kopp [7] gave a construction in the general case. It is noteworthy that the proof of Akcoglu’s theorem consists ultimately of a long string of reductions to the classical strong Lp inequality (10), which in turn is a consequence of (5). Subadditive and Multiplicative Ergodic Theorems Consider a family fX n;m g of real-valued random variables on a probability space indexed by the set of pairs (n; m) 2

249

250

Ergodic Theorems

Z2 such that 0  n < m. fX(n;m) g is called a (stationary) subadditive process if (a) the joint distribution of fX n;m g is the same as that of fX nC1;mC1 g (b) X n;m  X n;l C X l ;m whenever n < l < m. Denoting the index set by fn < mg Z2 the distribution of the process is a measure  on Rfn<mg which is invariant under the shift Tx(n; m) D x(n C 1; m C 1). Thus there is no loss of generality in assuming that there is an underlying endomorphism T such that X n;m ı T D X nC1;mC1 . In 1968 Kingman [129] proved the following generalizaR tion of Birkhoff’s theorem. D inf n1 X 0;n d is called the time constant of the process. Theorem 24 If the X n;m are integrable and > 1 then 1 n X 0;n converges a.e. and R in L1 -norm to a T -invariant limit ¯ D . X¯ 2 L1 () satisfying Xd It is easy to deduce from the above that if one assumes only C that X 0;1 is integrable then n1 X0;n still converges a.e. to a T-invariant limit X¯ taking values in [1; 1). Subadditive processes first arose in the work of Hammersley and Welsh [99] on percolation theory. Here is an example. Let G be the graph with vertex set Z2 and with edges joining every pair of nearest neighbors. Let E denote the edge set and let fTe : e 2 Eg be non-negative integrable i.i.d. random variables. To each finite path P in G P associate the “travel time” T(P) D e2E Te . For integers m > n  0 let X n;m be the infimum of T(P) over all paths P joining (0; R n) to (0; m). This is a subadditive process with 0  < Te d and it is not hard to see that the underlying endomorphism is ergodic. Thus Kingman’s theorem yields the result that n1 X0;n ! a.e. Suppose now that T is an ergodic automorphism of a probability space (X; B; ) and P is a function on X taking values in the space of d  d real matrices. Define Pn;m D P(T m1 x)P(T m2 x) : : : P(T n x) and let Pn;m (i; j) denote the i; j entry of Pn;m . Then X n;m D log(kPn;m k) (use any matrix norm) is a subadditive process so one obtains the first part of the following result of Furstenberg and Kesten [87] (1960) originally proved by more elaborate methods. The second part can also be deduced from the subadditive theorem with a little more work. See Kingman [130] for details and for some other applications of subadditive processes. Theorem 25 R (a) Suppose logC (jPj)d < 1. Then kP0;n k1/n converges a.e. to a finite limit. (b) Suppose that for each i; j P(i; j) is a strictly positive function such that log P(i; j) is integrable. Then the

1

limit p D lim(P0;n (i; j)) n exists a.e. and is independent of i and j. Partial results generalizing Kingman’s theorem to the multiparameter case were obtained by Smythe [189] and Nguyen [157]. In 1981 Akcoglu and Krengel [8] obtained a definitive multi-parameter subadditive theorem. They d consider an action fTm g of the semigroup G D Z0 by endomorphisms of a measure space (X; B; ). Using the standard total ordering  of G an interval in G is any set of the form fk 2 G : m  k  ng for any m  n 2 G. Let I denote the set of non-empty intervals. Reversing the direction of the inequality, they define a superadditive process as a collection of integrable functions FI ; I 2 I , such that (a) FI ı Tm D FICm , (b) FI  FI 1 C : : : C FI k whenever I is the disjoint union of I1 ; : : : ; I k and R (c) D supI2I jIj1 FI d < 1. A sequence fI n g of sets in I is called regular if there is an increasing sequence I 0n such that I n I 0n and jI 0n j  CjI n j for some constant C. Theorem 26 (Akcoglu–Krengel) Suppose F I is a superadditive process and fI n g is regular. Then jI1n j FI n converges a.e. fFI g is additive if the inequality in (b) is replaced by equalP ity. In this case FI D n2I f ı Tn where f is an integrable function. Thus in the additive case the Akcoglu–Krengel result is a theorem about ordinary multi-dimensional ergodic averages, which is in fact a special case of an earlier result of Tempelman [196] (see Sect. “Amenable Groups” below). Kingman’s proof of Theorem 24 hinged on the existence of a certain (typically non-unique) decomposition for subadditive processes. Akcoglu and Krengel’s proof of the multi-parameter result does not depend on a Kingman-type decomposition, in fact they show that there is no such decomposition in general. They prove a weak maximal inequality fsup jI n j1 FI n > g <

C

; 

(26)

where C is a constant depending only on the dimension, and show that this is sufficient to prove their result. In the case d D 1 the Akcoglu–Krengel argument provides a new and more natural proof of Kingman’s theorem, similar in spirit to Wiener’s arguments. Akcoglu and Sucheston [9] have proved a ratio ergodic theorem for subadditive processes with respect to

Ergodic Theorems

a positive L1 contraction, generalizing both the Chacón– Ornstein theorem and Kingman’s theorem. In 1968 Oseledec [164] proved his celebrated multiplicative ergodic theorem, which gives very precise information about the random matrix products studied by Furstenberg and Kesten. His theorem is an important tool for the study of Lyapunov exponents in differentiable dynamics, see notably Pesin [169]. If A is a d  d matrix let kAk D supfkAxk : kxk D 1g where kxk is the Euclidean norm on Rn . Theorem 27 Suppose T is an endomorphism of the probability space (X; B; ). Suppose P is a measurable function on XR whose values are d  d real matrices such that logC kPkd < 1 and let Pn (x) D P(T n1 x)P(T n2 x) : : : P(x). Then there is a T-invariant subset X 0 of X with measure 1 such that for x 2 X 0 the following hold. 1

(a) lim n!1 (Pn (x)Pn (x)) 2n D A(x) exists. (b) Let 0  exp 1 (x)  exp 2 (x)  : : :  exp r (x) be the distinct eigenvalues of A(x) (r D r(x) may depend on x and 1 may be 1) with multiplicities m1 (x); : : : mr (x). Let E i (x) be the eigenspace corresponding to exp( i (x)) and set F i (x) D E1 (x) C : : : C E i (x) : Then for each u 2 F i (x)nF i1 (x) lim

1 log kPn (x)uk D  i (x) : n

Entropy and the Shannon–McMillan–Breiman Theorem The notion of entropy was introduced by Shannon in his landmark work [185] which laid the foundations for a mathematical theory of information. Suppose (X; B; ) is a probability space, P is a finite measurable partition of X and T is an automorphism of (X; B; ). P(x) denotes the atom of P containing x. The entropy of P is h(P) D 

X Z

D

log((P(x)) d(x)  0 :

(d) If T is ergodic, det P(x) D 1 a.e. and lim sup n

1 n

Z log kPn kd > 0

then the i are constants, 1 < 0 and r > 0. Raghunathan [174] gave a much shorter proof of Oseledec’s theorem, valid for matrices with entries in a locally compact normed field. He showed that it could be reduced to the Furstenberg–Kesten theorem by considering the exterior powers of P. Ruelle [180] extended Oseledec’s theorem to the case where P takes values in the set of bounded operators on a Hilbert space. Walters [199] has given a proof (under slightly stronger hypotheses) which avoids the matrix calculations and tools from multilinear algebra used in other proofs.

(27)

 log((A)) may be viewed as a quantitative measure of the amount of information contained in the statement that a randomly chosen x 2 X happens to belong to A. So h(P) is the expected information if one is about to observe which atom of P a randomly chosen point falls in. See Billingsley [34] for more motivation of this concept. See also the article in this collection on entropy by J. King or any introductory book on ergodic theory, e. g. Petersen [171]. W If P and Q are partitions P Q denotes the common refinement which consists of all sets p\q, p 2 P; q 2 Q. It W is intuitive and not hard to show that h(P Q)  h(P) C W n i h(Q). Now let P0n D n1 iD0 T P and h n D h(P0 ). The subadditivity of entropy implies that h nCm  h n C h m , so by a well-known elementary lemma the limit h(P; T) D lim n

(c) The functions m i and i are T-invariant.

(p) log((p))

p2P

hn hn D inf 0 n n n

(28)

exists. If one thinks of P(x) as a measurement performed on the space X and Tx as the state succeeding x after one second has elapsed then h(P; T) is the expected information per second obtained by repeating the experiment every second for a very long time. See [177] for an alternative and very useful approach to h(P; T) via name-counting. The following result, which is known as the Shannon– McMillan–Breiman theorem, has proved to be of fundamental importance in ergodic theory, notably, for example, in the proof of Ornstein’s celebrated isomorphism theorem for Bernoulli shifts [159]. Theorem 28 If T is ergodic then 1 lim  log((P0n1 (x)) D h(P; T) n

n!1

a.e. and in L1 -norm.

(29)

251

252

Ergodic Theorems

In other words, the actual information obtained per second by observing x over time converges to the constant h(P; T), namely the limiting expected information per second. Shannon [185] formulated Theorem 28 and proved convergence in probability. McMillan [149] proved L1 convergence and Breiman [49] obtained the a.e. convergence. The original proofs of a.e. convergence used the martingale convergence theorem, were not very intuitive and did not generalize to Z n -actions, where the martingale theorem is not available. Ornstein and Weiss [161] (1983) found a beautiful and more natural argument which bypasses the martingale theorem and allows generalization to a class of groups which includes Z n . Amenable Groups Let G be any countable group and T D fTg g an action of G by automorphisms of a probability space. Suppose  is a complex measure on G, that is, f(g)g g2G is an absolutely summable sequence. Let T g act on functions via Tg f D f ı Tg , so T g is an isometry of Lp for every P 1  p  1. Let (T) D g2G (g)Tg . A very general framework for formulating ergodic theorems is to consider a sequence fn g and ask whether n (T) f converges, a.e. or in mean, for f in some Lp space. When n (T) f converges for all actions T and all f in Lp in p-norm or a.e. then one says that  n is mean or pointwise good in Lp . When the  n are probability measures it is natural to call such results weighted ergodic theorems and this terminology is retained for complex  as well. Birkhoff’s theorem says that if G D Z and  n is the normalized counting measure on f0; 1; : : : n  1g then fn g is pointwise good in L1 . This section will be concerned only with sequences fn g such that  n is normalized counting measure on a finite subset Fn G so one speaks of mean or pointwise good sequences fFn g. A natural condition to require of fFn g, which will ensure that the limiting function is invariant, is that it be asymptotically (left) invariant in the sense that 

jgFn Fn j !0 jFn j

8g 2 G :

(30)

Such a sequence is called a Følner sequence and a group G is amenable if it has a Følner sequence. As in most of this article G is restricted to be a discrete countable group for simplicity but most of the results to be seen actually hold for a general locally compact group. Amenability of G is equivalent to the existence of a finitely additive left invariant probability measure on G. It is not hard to see that any Abelian, and more generally

any solvable, group is amenable. On the other hand the free group F 2 on two generators is not amenable. See Paterson [167] for more information on amenable groups. The Følner property by itself is enough to give a mean ergodic theorem. Theorem 29 Any Følner sequence is mean good in Lp for 1  p < 1. The proof of this result is rather similar to the proof of Theorem 3. In fact Theorem 29 is only a special case of quite general results concerning amenable semi-groups acting on abstract Banach spaces. See the book of Paterson [167] for more on this. Turning to pointwise theorems, the Følner condition alone does not yield a pointwise theorem, even when G D Z and the F n are intervals. For example Akcoglu and del Junco [6] have shown that when G D Z and p Fn D [n; n C n] \ Z the pointwise ergodic theorem fails for any aperiodic T and for some characteristic function f . See also del Junco and Rosenblatt [119]. The following pointwise result of Tempelman [196] is often quoted. A Følner sequence fFn g is called regular if there is a constant C such that jFn1 Fn j  CjFn j and there is an increasing sequence Fn0 such that Fn Fn0 and jFn0 j  CjFn j. Theorem 30 Any regular Følner sequence is pointwise good in L1 . In case the F n are intervals in Z n this result can be proved by a variant of Wiener’s covering argument and in the general case by an abstraction thereof. The condition jFn1 Fn j  CjFn j captures the property of rectangles which is needed for the covering argument. Emerson [78] independently proved a very similar result. The work on ergodic theorems for abstract locally compact groups was pioneered by Calderón [53] who built on Wiener’s methods. The main result in this paper is somewhat technical but it already contains the germ of Tempelman’s theorem. Other ergodic theorems for amenable groups, whose main interest lies in the case of continuous groups, include Tempelman [195], Renaud [175], Greenleaf [93] and Greenleaf and Emerson[94]. The discrete versions of these results are all rather close to Tempelman’s theorem. Among pointwise theorems for discrete groups Tempelman’s result was essentially the best available for a long time. It was not known whether every amenable group had a Følner sequence which is pointwise good for some Lp . In 1988 Shulman [187] introduced the notion of a tempered

Ergodic Theorems

Følner sequence fFn g, namely one for which ˇ ˇ ˇ[ ˇ ˇ 1 ˇ F i Fn ˇ < CjFn j ; ˇ ˇ ˇ

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i
for some constant C. The advantage of the tempered condition is that any Følner sequence has a tempered subsequence, and in particular any amenable group has a tempered Følner sequence. Shulman proved a maximal inequality in L2 for such F n which implies that fFn g is pointwise good in L2 . An account of this work may be found in Section 5.6 of Tempelman’s book [194]. Lindenstrauss [145] was able to extend the result to L1 . Theorem 31 Any tempered Følner sequence is pointwise good in L1 . The key new idea in his proof is to use a probabilistic argument to establish a covering lemma. In the discrete case Ornstein and Weiss [201] have given a non-probabilistic proof of Lindenstrauss’s covering lemma. Lindenstrauss also generalizes the a.e. convergence in the Shannon–McMillan–Breiman theorem to this setting. L1 convergence was already established by Kieffer [128] in 1975. Subsequence and Weighted Theorems In this section G and T are as in the previous section and  n is a sequence of complex measures on G. This section will be concerned with conditions on  n that ensure that it is mean or pointwise good. For the most part G will be Z. Hopf’s ergodic theorem gives a class of examples for free. Choose any probability measure  on G, define the operator T D (T) and observe that (T )n D Tn , where n denotes the convolution power. Since T  satisfies the hypotheses of Hopf’s theorem it follows that the P i sequence n D n1 n1 iD0  is pointwise good in L1 . Another sort of example is given by choosing a seP quence g n in G and letting n D n1 n1 iD0 ı g n . If one has convergence for such a sequence one speaks of a subsequence ergodic theorem. This section will mainly focus on the case G D Z and sequences which are the increasing enumeration of a subset S N. Given S N write S;n for the corresponding probabilities  n and say S is good if S;n is good. Perhaps the first subsequence ergodic theorem is due to Blum and Hanson [38] who proved that an automorphism T of a probability space is strongly mixing if and P m i f converges in L norm for every only if n1 n1 2 iD0 T f 2 L2 and every increasing fm i g. Strong mixing means

that (T n A \ B) ! (A)(B). In 1969 Brunel and Keane [51] proved the first pointwise subsequence ergodic theorem Theorem 32 Suppose that T is a translation on a compact Abelian group G with Haar measure , g 2 G, and E G is any Borel set with (E) > 0 and (@E) D 0. Let S D fi > 0 : T i g 2 Eg. Then S is pointwise good in L1 . Krengel [133] constructed the first example of a sequence S N which is pointwise universally bad, in the strong sense that for any aperiodic T the a.e. convergence of S;n (T) f fails for some characteristic function f . Bellow [17] proved that any lacunary sequence (meaning a nC1 > ca n for some c > 1) is pointwise universally bad in L1 . Later Akcoglu at al. [1] were able to show that for lacunary sequences fS;n g is even strongly sweeping out. A sequence fn g of probability measures on Z is said to be strongly sweeping out if for any ergodic TRand for all ı > 0 there is a characteristic function f with f d < ı such that lim sup n (T) f D 1 a.e. It is not difficult to show that if fn g is strongly sweeping out then there are characteristic functions f such that lim inf n (T) f D 0 and lim sup n (T) f D 1. Thus for lacunary sequences the ergodic theorem fails in the worst possible way. Bellow and Losert [21] gave the first example of a sequence S Z of density 0 which is universally good for pointwise convergence, answering a question posed by Furstenberg. They construct an S which is pointwise good in L1 . This paper also contains a good overview of the progress on weighted and subsequence ergodic theorems at that time. Weyl’s theorem on uniform distribution (Theorem 9) suggests the possibility of an ergodic theorem for the sequence fn2 g. It is not hard to see that fn2 g is mean good in L2 . In fact the spectral theorem and the dominated convergence theorem show that it is enough to prove that theL1 P n 2 on the unit bounded sequence of functions n1 n1 iD0 z circle converges at each point z of the unit circle. When z is not a root of unity the sequence converges to 0 by Weyl’s result and when z is a root of unity the convergence is 2 trivial because fz n g is periodic. In 1987 Bourgain [39,43] proved his celebrated pointwise ergodic theorem for polynomial subsequences. Theorem 33 If p is any polynomial with rational coefficients taking integer values on the integers then S D fp(n)g is pointwise good in L2 . The first step in Bourgain’s argument is to reduce the problem of proving a maximal inequality to the case of the shift map on the integers, via Calderón’s transfer princi-

253

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Ergodic Theorems

ple. Then the problem is transferred to the circle by using Fourier transforms. At this point the problem becomes a very delicate question about exponential sums and a whole arsenal of tools is brought to bear. See Rosenblatt and Wierdl [176] and Quas and Wierdl [28](Appendix B) for nice expositions of Bourgain’s methods. Bourgain subsequently improved this to all Lp , p > 1 and also extended it to sequences f[q(n)]g where now q is an arbitrary real polynomial and [] denotes the greatest integer function. He also announced that his methods can be used to show that the sequence of primes is pointwise p 1C 3 good in Lp for any p > 2 . Wierdl [205] (1988) soon extended the result for primes to all p > 1. Theorem 34 The primes are pointwise good in Lp for p >1. It has remained a major open question for quite some time whether any of these results hold for p D 1. In 2005 there appeared a preprint of Mauldin and Buczolich [148], which remains unpublished, showing that polynomial sequences are L1 -universally bad. Another major result of Bourgain’s is the so-called return times theorem [44]. A simplification of Bourgain’s original proof was published jointly with Furstenberg, Katznelson and Ornstein as an appendix to an article [47] of Bourgain. To state it let us agree to say that a a sequence of complex numbers fa(n)gn0 has property P if the seP quence of complex measures n D n1 n1 iD0 a(i)ı i has property P, where ı i denotes the point mass at i. Theorem 35 (Bourgain) Suppose T is an automorphism of a probability space (X; B; ), 1  p; q  1 are conjugate exponents and f 2 L p (). Then for almost all x the sequence f f (T n x)g is pointwise good in Lq . Applying this to characteristic functions f D 1E one sees that the return time sequence fi > 0 : S i x 2 Eg is good for pointwise convergence in L1 . Theorem 32 is a very special case. It is also easy to see that Theorem 35 contains the Wiener–Wintner theorem. In 1998 Rudolph [179] proved a far-reaching generalization of the return times theorem using the technique of joinings. For an introduction to joinings see the article by de la Rue in this collection and also Thouvenot [198], Glasner [90] (2003) and Rudolph’s book [177]. Rudolph’s result concerns the convergence of multiple averages N1 k 1 XY f j (T jn x) N nD0

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jD1

where each T j is an automorphism of a probability space (X j ; B j ;  j ) and the f j are L1 functions. The point is that the convergence occurs whenever each x j 2 X 0j , sets of

measure one which may be chosen sequentially for j D 1; : : : ; k without knowing what T i or f i are for any i > j. He actually proves something stronger, namely he identifies an intrinsic property of a sequence fa i g, which he calls fully generic, such that the following hold. (a) The constant sequence {1} is fully generic. (b) If fa i g is fully generic then for any T and f 2 L1 the sequence a i f (T i x) is fully generic for almost all x. (c) Fully generic implies pointwise good in L1 . The definition of fully generic will not be quoted here as it is somewhat technical. For a proof of the basic return times theorem using joinings see Rudolph [178]. Assani, Lesigne and Rudolph [13] took a first step towards the multiple theorem, a Wiener–Wintner version of the return times theorem. Also Assani [11] independently gave a proof of Rudolph’s result in the case when all the T j are weakly mixing. Ornstein and Weiss [162] have proved the following version of the return times theorem for abstract discrete groups. As with Z, let us say that a sequence fa g g g2G of complex numbers has property P for fFn g if the sequence P n D jF1 j g2Fn a(g)ı g of complex measures has propn erty P. Theorem 36 Suppose that the increasing Følner sequence fFn g satisfies the Tempelman condition supn jFn1 Fn j/ S jFn j < 1 and Fn D G. If b 2 L1 then for a.a. x the sequence fb(Tg x)g is pointwise good in L1 for fFn g. Recently Demeter, Lacey, Tao and Thiele [67] have proved that the return times theorem remains valid for any 1 < p  1 and q  2. On the other hand Assani, Buczolich and Mauldin [14] (2005) showed that it fails for p D q D 1. Bellow, Jones and Rosenblatt have a series of papers [22,23,24,25] studying general weighted averages associated to a sequence  n of probability measures on Z, and, in some cases, more general groups. The following are a few of their results. [23] is concerned with Z-actions and moving block averages given by n D m I n , where the I n are finite intervals and mI denotes normalized counting measure on I. They resolve the problem completely, obtaining a checkable necessary and sufficient condition for such a sequence to be pointwise good in L1 . [24] gives sufficient conditions on a sequence  n for it to be pointwise good in Lp , p > 1, via properties of the Fourier transforms ˆ n . A particular consequence is P that if lim n!1 k2Z jn (k)  n (k  1)j D 0 then fn g has a subsequence which is pointwise good in Lp ,

Ergodic Theorems

p > 1. In [25] they obtain convergence results for sequences n D  n , the convolution powers of a probability measure . A consequence of one of their main results P is that if the expectation k2Z k(k) is zero, the second P moment k2Z k 2 (k) is finite and  is aperiodic (its support is not contained in any proper coset in Z) then  n is pointwise good in Lp for p > 1. Bellow and Calderón [19] later showed that this last result is valid also for p D 1. This is a consequence of the following sufficient condition for a sequence T to satisfy a weak L1 inequality. Given an automorphism of a probability space (X; B; ) let M f D sup jn (T) f (x)j be the maximal operator associated to fn g. Theorem 37 (Bellow and Calderón) Suppose there is an ˛ 2 (0; 1] and C > 0 such that for each n > 1 one has jn (x C y)  n (x)j  C

jyj˛ jxj1C˛

for all x; y 2 Z

Theorem 39 Suppose T is an automorphism of a probability space (X; B; ), (B) > 0 and k  1. Then there is T an n > 0 such that ( kiD1 T in B) > 0. In 1977, Furstenberg [85] proved the following ergodic theorem which implies the multiple recurrence theorem. He also established a general correspondence principle which puts the shaky analogy between the multiple recurrence theorem and Szemerédi’s theorem on a firm footing and allows each to be deduced from the other. Thus he obtained an ergodic theoretic proof of Szemerédi’s combinatorial result. Theorem 40 Suppose T is an automorphism of a probabilR ity space (X; B; ), f 2 L1 , f  0, f d > 0 and k  1. Then lim inf N

N1 Z k 1 X Y in T f d > 0 : N nD0

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iD1

such that 0 < 2jyj  jxj Then there is a constant D such that fM f > g 

D k f k1 

f 2 L1 ()

for all T;

and  > 0 :

Ergodic Theorems and Multiple Recurrence Suppose S N. The upper density of S is ¯ D lim sup jS \ [1; n]j : d(S) n n

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and the density d(S) is the limit of the same quantity, if it exists. In 1975 Szemerédi [190] proved the following celebrated theorem, answering an old question of Erd˝os and Turán. Theorem 38 Any subset of N with positive upper density contains an arithmetic progression of length k for each k  1. This result has a distinctly ergodic-theoretic flavor. Letting T denote the shift map on Z, it says that for each k there T is an n such that S 0 D kiD1 T i n S is non-empty. In fact ¯ 0 ) > 0. the result gives more: there is an n for which d(S In this light Szemerédi’s theorem becomes a multiple recurrence theorem for the shift map on N, equipped with ¯ Of course d¯ is not the invariant “measure-like” quantity d. even finitely additive so it is not a measure. d, however, is at least finitely additive, when defined, and d(N) D 1. This point of view suggests the following multiple recurrence theorem.

Furstenberg’s result opened the door to the study of socalled ergodic Ramsey theory which has yielded a vast array of deep results in combinatorics, many of which have no non-ergodic proof as yet. The focus of this article is not on this direction but the reader is referred to Furstenberg’s book [86] for an excellent introduction and to Bergelson [27,28] for surveys of later developments. There is also the article by Frantzikinakis and McCutcheon in this collection. Furstenberg’s proof relies on a deep structure theorem for a general automorphism which was also developed independently by Zimmer [213], [212] in a more general context. A factor of T is any sub- -algebra F B such that T(F ) D F . (It is more accurate to think of the factor as the action of T on the measure space (X; F ; jF ).) The structure theorem asserts that there is a transfinite increasing sequence of factors fF˛ g of T such that the following conditions hold. (a) F˛C1 is compact relative to F˛ . W (b) F˛ D ˇ <˛ Fˇ whenever ˛ is a limit ordinal. W (c) B is weakly mixing relative to F˛ . W ( denotes the  -algebra generated by a collection of  -alW gebras.) ˛ F˛ is called the maximal distal factor of T and W if F˛ D B then T is called distal. The definitions of the relative properties in (a) and (c) above are somewhat technical so only their absolute versions (i. e. relative to the trivial -algebra f;; Xg) will be described here. T is said to be compact if for each f 2 L2 the orbit fT i f : i 2 Zg is pre-compact in the norm topology of L2 . This turns out to be equivalent to the statement that T is

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Ergodic Theorems

a translation on a compact Abelian group endowed with its Haar measure. The property of weak mixing is a fundamental notion in ergodic theory which has many equivalent definitions. The most appropriate for our purposes is that T is weakly mixing if it has no compact factors. This turns out to be equivalent to the ergodicity of T  T acting on the product measure space (X; B; )  (X; B; ). The verification of 37 in the case of compact T is rather easy. In this case it is not hard to prove that for any f 2 L1 and > 0 the set fn 2 Z : kT n f  f k2 < g has bounded gaps and (34) follows easily. In the case when T is weakly mixing (34) is a consequence of the following theorem which Furstenberg proves in [85] (as a warm-up for it’s much harder relative version). Theorem 41 If T is weakly mixing and f1 ; f2 ; : : : ; f k are L1 functions then lim N

N1 Z k k Z Y 1 X Y in fi T f i d D N nD0 iD1

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iD1

Later Bergelson [26] showed that the result can be obtained easily by an induction argument using the following Hilbert space generalization of van der Corput’s lemma. Theorem 42 (Bergelson) Suppose fx n g is a bounded sequence of vectors in Hilbert space such that for each h > 0 P N1 one has N1 nD0 hx nCh ; x n i ! 0 as N ! 1. Then P N1 1 k N nD0 x n k ! 0. Ryzhikov has also given a beautiful short proof of Theorem 41 using joinings ([182]). Bergelson’s van der Corput lemma and variants of it have been a key tool in subsequent developments in ergodic Ramsey theory and in the convergence results to be be discussed in this section. Bergelson [26] used it to prove the following mean ergodic theorem for weakly mixing automorphisms. Theorem 43 Suppose T is weakly mixing, f1 ; : : : ; f k are L1 functions and p1 ; : : : ; p k are polynomials with rational coefficients taking integer values on the integers such that no p i  p j is constant for i ¤ j. Then   N1 k k Z  1 XY Y   p i (n) f i d D 0 : (36) T fi  lim   N!1  N nD0 iD1

iD1

Theorems 40 and 41 immediately raise the question of P N1 Q k in convergence of the multiple averages N1 nD0 iD1 T f i for a general T. Several authors obtained partial results on the question of mean convergence. It was finally resolved only recently by Host and Kra [104], who proved the following landmark theorem.

Theorem 44 Suppose f1 ; f2 ; : : : ; f k 2 L1 . Then there is a g 2 L1 such that   N1 k  1 XY   T in f i  g  D 0 : lim   N nD0 iD1

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2

Independently and somewhat later Ziegler [211]obtained the same result by somewhat different methods. Furstenberg had already established Theorem 44 for k D 2 in [85]. It was proved for k D 3 in the case of a totally ergodic T by Conze and Lesigne [63]and in general by Host and Kra [102]. It can also be obtained using the methods developed by Furstenberg and Weiss [88]. In this paper Furstenberg and Weiss proved a result for polynomial powers of T in the case k D 2. They also formalized the key notion of a characteristic factor. A factor C of T is said to be characteristic for the averages (37) if, roughly speaking, the L2 limiting behavior of the averages is unchanged when any one of the f i ’s is replaced by its conditional expectation on C . This means that the question of convergence of these averages may be reduced to the case when f i are all C -measurable. So the problem is to find the right (smallest) characteristic factor and prove convergence for that factor. The importance of characteristic factors was already apparent in Furstenberg’s original paper [85], where he showed that the maximal distal factor is characteristic for the averages (37). In fact he showed that for a given k a k-step distal factor is characteristic. (An automorphism is k-step distal if it is the top rung in a k-step ladder of factors as in the Furstenberg–Zimmer structure-theorem.) It turns out, though, that the right characteristic factor for (37) is considerably smaller. In their seminal paper [63] Conze and Lesigne identified the characteristic factor for k D 3, now called the Conze–Lesigne factor. As shown in [104], and [211], the characteristic factor for a general k is (isomorphic to) an inverse limit of k-step nilflows. A k-step nilflow is a compact homogeneous space N/ of a k-step nilpotent Lie group N, endowed with its unique left-invariant probability measure, on which T acts via left translation by an element of N. Ergodic properties of nilflows have been studied for some time in ergodic theory, for example in Parry [166]. In this way the problem of L2 -convergence of (37) is reduced to the case when T is a nilflow. In this case one has more: the averages converge pointwise by a result of Leibman [141] (See also Ziegler [210]). There have already been a good many generalizations of (37). Host and Kra [103], Frantzikinakis and Kra [82, 83], and Leibman [141] have proved results which replace

Ergodic Theorems

linear powers of T by polynomial powers. In increasing degrees of generality Conze and Lesigne [63], Frankzikinakis and Kra [81] and Tao [192] have obtained results which replace the maps T; T 2 ; : : : ; T k in (37) by commuting maps T1 ; : : : ; Tk . Bergelson and Leibman [30,31] have obtained results, both positive and negative, in the case of two noncommuting maps. In the direction of pointwise convergence the only general result is the following theorem of Bourgain [48] which asserts pointwise convergence in the case k D 2. Theorem 45 Suppose S and T are powers of a single autoPN f (T n x)g(S n x) morphism R and f ; g 2 L1 . Then N1 nD1 converges a.e. When T is a K-automorphism Derrien and Lesigne [68] have proved that the averages (35) converge pointwise to the product of the integrals, even with polynomial powers of T replacing the linear powers. Gowers [91] has given a new proof of Szemerédi’s theorem by purely finite methods using only harmonic analysis on Z n . His results give better quantitative estimates in the statement of finite versions of Szemerédi’s theorem. Although his proof contains no ergodic theory it is to some extent guided by Furstenberg’s approach. This section would be incomplete without mentioning the spectacular recent result of Green and Tao [92] on primes in arithmetic progression and the subsequent extensions of the Green-Tao theorem due to Tao [191] and Tao and Ziegler [193]. Rates of Convergence There are many results which say in various ways that, in general, there are no estimates for the rate of convergence of the averages A n f in Birkhoff’s theorem. For example there is the following result of Krengel [134]. Theorem 46 Suppose limn!1 c n D 1 and T is any ergodic automorphism of a probability space. Then there R is a bounded measurable f with f d D 0 such that lim sup c n A n f D 1 a.e. See Part 1 of Derriennic [69] for a selection of other results in this direction. In spite of these negative results one can obtain quantitative estimates by reformulating the ergodic theorem in various ways. Bishop [37] proved the following result which is purely finite and constructive in nature and evidently implies the a.e. convergence in Birkhoff’s theorem. If y D (y1 ; : : : ; y n ) is a finite sequence of real numbers and a < b, an upcrossing of y over [a; b] is a minimal integer interval [k; l] [1; n] satisfying y k < a and y l > b.

Theorem 47 Suppose T is an automorphism of the probability space (X; B; ). Let U(n; a; b; f ; x) be the number of upcrossings of the sequence A0 f (x); : : : ; A n f (x) over [a; b]. Then for every n fx : U(n; a; b; f ; x) > kg 

k f k1 : (b  a)k

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Ivanov [107] has obtained the following stronger upcrossing inequality for an arbitrary positive measurable f , which also implies Birkhoff’s theorem. Theorem 48 For any positive measurable f and 0 < a kg 

 a k b

:

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Note the exponential decay and the remarkable fact that the estimate does not depend on f . Ivanov has also obtained the following result (Theorem 23 in [121]) about fluctuations of A n f . An -fluctuation of a real sequence y D (y1 : : : y n ) is a minimal integer interval [k; l] satisfying jy k  y l j  . If f 2 L1 (R) let F( ; f ; x) be the number of -fluctuations of the sequence fA n f (x)g1 nD0 . Theorem 49 1

fx : F( ; f ; x)  kg  C

(log k) 2 k

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where C is a constant depending only on k f k1 / . If f 2 L1 then fx : F( ; f ; x)  kg  AeBk ;

(41)

where A and B are constants depending only on k f k1 / . See Kachurovskii’s survey [121] of results on rates of convergence in the ergodic theorem and also the article of Jones et al. [115] for more results on oscillation type inequalities. Under special assumptions on f and T it is possible to give more precise results on speed of convergence. If the sequence T i f is independent then there is a vast literature in probability theory giving very precise results, for example the central limit theorem and the law of the iterated logarithm (see, for example, [50]). See the surveys of Derrienic [69] (2006) and of Merlevède, Peligrad and Utev [150] for results on the central limit theorem for dynamical systems.

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Ergodic Theorems

Ergodic Theorems for Non-amenable Groups Guivarch [95] was the first to prove an ergodic theorem for a general pair of non-commuting unitary operators. Much work has been done in the last 15 years on mean and pointwise theorems for non-amenable groups. See the extensive survey by Nevo [155]. Here is just one result of Nevo [154] as a sample. Suppose G is a discrete group and fn g is a sequence of probability measures on G. Say that fn g is mean ergodic if for any unitary representation  of G on a Hilbert space H and any x 2 H one P has g2G n (g)(g)x converges in norm to the projection of x on the subspace of -invariant vectors. Say that fn g is pointwise ergodic if for any measure preserving action T of G on a probability space and f 2 L2 one has P g2G  n (g)Tg f converges a.e. to the projection of f on the subspace of T-invariant functions. Theorem 50 Let G be the free group on k generators, k  1. Let  n be the normalized counting measure on the set of elements whose word length (in terms of the generators and their inverses) is n. Let n D (n C nC1 )/2 and P n0 D n1 niD1  i . Then  n and n0 are mean and pointwise ergodic but  n is not mean ergodic. Future Directions This section will be devoted to a few open problems and some questions. Many of these have been suggested by my colleagues acknowledged in the introduction. The topic of convergence of Furstenberg’s multiple averages has seen some remarkable achievements in the last few years and will likely continue to be vital for some time to come. The question of pointwise convergence of multiple averages is completely open, beyond Bourgain’s result (Theorem 45). Even extending Bourgain’s result to three different powers of R or to any two commuting automorphisms S and T would be a very significant achievement. Another natural question is whether one has pointwise convergence of the averages of the sequence 2 f (T n (x))g(T n (x)), which are mean convergent by the result of Furstenberg and Weiss [88] (which is now subsumed in much more general results). A long-standing open problem relating to Akcoglu’s ergodic theorem (Theorem 23) for positive contractions of Lp is whether it can be extended to power-bounded operators T (this means that kTkn is bounded). It is also an open question whether it extends to non-positive contractions, excepting the case p D 2 where Burkholder [52] has shown that it fails. There are many natural questions in the area of subsequence and weighted ergodic theorems. For example,

which of Bourgain’s several pointwise theorems can be extended to L1 ? Are there other natural subsequences of an arithmetic character which have density 0 and for which an ergodic theorem is valid, either mean or pointwise, and in what Lp spaces might such theorems be valid? Are there higher dimensional analogues? Since lacunary sequences are bad, to have any sort of an pointwise theorem l n D log a nC1 must get close to 0 and for simplicity let us assume that lim l n D 0. How fast must the convergence be in order to get an ergodic theorem? Jones and Wierdl [113] have shown that if l n > 1 (log n) 2 C then the pointwise ergodic theorem fails in L2 while Jones, Lacey and Wierdl [116] have shown that an only slightly faster rate permits a sequence which is pointwise good in L2 . How well or badly does the ergodic theorem succeed or fail depending on the rate of convergence of ln ? In particular is there a (slow) rate which still guarantees strong sweeping out? [116] contains some interesting conjectures in this direction. There are also interesting questions concerning the mean and pointwise ergodic theorems for subsequences which are chosen randomly in some sense. See Bourgain [42] and [116] for some results in this direction. Again [116] contains some interesting conjectures along these lines. In a recent paper [32] Bergelson and Leibman prove some very interesting and surprising results about the distribution of generalized polynomials. A generalized polynomial is any function which can be built starting with polynomials in R[x] using the operations of addition, multiplication and taking the greatest integer. As a consequence they derive a generalization of von Neumann’s mean ergodic theorem to averages along generalized polynomial sequences. The following is a special case. Theorem 51 Suppose p is a generalized polynomial taking integer values on the integers and U is a unitary operator P p(i) x is norm convergent for all x 2 on H . Then n1 n1 iD0 U H. This begs the question: does one have pointwise convergence? If so, this would be a far-reaching generalization of Bourgain’s polynomial ergodic theorem. There are also lots of questions concerning the nature of Følner sequences fFn g in an amenable group which give a pointwise theorem. For example Lindenstrauss [145] has shown that in the lamplighter group, a semi-direct product L of Z with i2Z Z/2Z on which Z acts by the shift, there is no sequence satisfying Tempelman’s condition and that any fFn g satisfying the Shulman condition must grow super-exponentially. So, it is natural to ask for slower rates of growth. In particular, in any amenable group is there al-

Ergodic Theorems

ways a sequence fFn g which is pointwise good and grows at most exponentially? Can one do better either in general or in particular groups? Lindenstrauss’s theorem at least guarantees the existence of Følner sequences which are pointwise good in L1 but in particular groups there are often natural sequences L which one hopes might be good. For example in 1 iD1 Z one may take F n to be a cube based at 0 of sidelength ln and dimension dn (that is, all but the first dn co-ordinates are zero), where both sequences increase to 1. What conditions on ln and dn will give a good sequence? Note that no such sequence is regular in Tempelman’s sense. If d n D n then fl n g must be superexponential to ensure Shulman’s condition. Can one do better? What about l n D d n D n? Bibliography 1. Akcoglu M, Bellow A, Jones RL, Losert V, Reinhold–Larsson K, Wierdl M (1996) The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters. Ergodic Theory Dynam Systems 16(2):207–253 2. Akcoglu M, Jones RL, Rosenblatt JM (2000) The worst sums in ergodic theory. Michigan Math J 47(2):265–285 3. Akcoglu MA (1975) A pointwise ergodic theorem in Lp-spaces. Canad J Math 27(5):1075–1082 4. Akcoglu MA, Chacon RV (1965) A convexity theorem for positive operators. Z Wahrsch Verw Gebiete 3:328–332 (1965) 5. Akcoglu MA, Chacon RV (1970) A local ratio theorem. Canad J Math 22:545–552 6. Akcoglu MA, del Junco A (1975) Convergence of averages of point transformations. Proc Amer Math Soc 49:265–266 7. Akcoglu MA, Kopp PE (1977) Construction of dilations of positive Lp -contractions. Math Z 155(2):119–127 8. Akcoglu MA, Krengel U (1981) Ergodic theorems for superadditive processes. J Reine Angew Math 323:53–67 9. Akcoglu MA, Sucheston L (1978) A ratio ergodic theorem for superadditive processes. Z Wahrsch Verw Gebiete 44(4):269– 278 10. Alaoglu L, Birkhoff G (1939) General ergodic theorems. Proc Nat Acad Sci USA 25:628–630 11. Assani I (2000) Multiple return times theorems for weakly mixing systems. Ann Inst H Poincaré Probab Statist 36(2):153–165 12. Assani I (2003) Wiener–Wintner ergodic theorems. World Scientific Publishing Co. Inc., River Edge, NJ 13. Assani I, Lesigne E, Rudolph D (1995) Wiener–Wintner returntimes ergodic theorem. Israel J Math 92(1–3):375–395 14. Assani I, Buczolich Z, Mauldin RD (2005) An L1 counting problem in ergodic theory. J Anal Math 95:221–241 15. Auslander L, Green L, Hahn F (1963) Flows on homogeneous spaces. With the assistance of Markus L, Massey W and an appendix by Greenberg L Annals of Mathematics Studies, No 53. Princeton University Press, Princeton, NJ 16. Banach S (1993) Théorie des opérations linéaires. Éditions Jacques Gabay, Sceaux, reprint of the 1932 original 17. Bellow A (1983) On “bad universal” sequences in ergodic theory II. In: Belley JM, Dubois J, Morales P (eds) Measure theory and its applications. Lecture Notes in Math, vol 1033. Springer, Berlin, pp 74–78

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Ergodic Theory: Basic Examples and Constructions MATTHEW N ICOL1 , KARL PETERSEN2 Department of Mathematics, University of Houston, Houston, USA 2 Department of Mathematics, University of North Carolina, Chapel Hill, USA

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Article Outline Glossary Definition of the Subject Introduction Examples Constructions Future Directions Bibliography Glossary  A transformation T of a measure space (X; B; ) is measure-preserving if (T 1 A) D (A) for all measurable A 2 B.  A measure-preserving transformation (X; B; ; T) is ergodic if T 1 (A) D A (mod ) implies (A) D 0 or (Ac ) D 0 for each measurable set A 2 B.  A measure-preserving transformation (X; B; ; T) of a probability space is weak-mixing if lim n!1 n1 P n1 i iD0 j(T A \ B)  (A)(B)j D 0 for all measurable sets A; B 2 B. A measure-preserving transformation (X; B; ; T) of a probability space is strong-mixing if limn!1  (T n A\B) D (A)(B) for all measurable sets A; B 2 B.  A continuous transformation T of a compact metric space X is uniquely ergodic if there is only one T-invariant Borel probability measure on X. A continous transformation of a topological space X is topologically mixing for any two open sets U; V X there exists N > 0 such that T n (U) \ V ¤ ;, for each n  N.  Suppose (X; B; ) is a probability space. A finite partition P of X is a finite collection of disjoint (mod , i. e., up to sets of measure 0) measurable sets fP1 ; : : : ; Pn g such that X D [Pi (mod ). The entropy of P with reP spect to  is H(P ) D  i (Pi ) ln (Pi ) (other bases are sometimes used for the logarithm).  The metric (or measure-theoretic) entropy of T with respect to P is h (T; P ) D lim n!1 n1 H(P _    _ T nC1 (P )), where P _    _ T nC1 (P ) is the partition of X into sets of points with the same coding with







 









respect to P under T i , i D 0; : : : ; n  1. That is x, y are in the same set of the partition P _    _ T nC1 (P ) if and only if T i (x) and T i (y) lie in the same set of the partition P for i D 0; : : : ; n  1. The metric entropy h (T) of (X; B; ; T) is the supremum of h (T; P ) over all finite measurable partitions P . If T is a continuous transformation of a compact metric space X, then the topological entropy of T is the supremum of the metric entropies h (T), where the supremum is taken over all T-invariant Borel probability measures. A system (X; B; ; T) is loosely Bernoulli if it is isomorphic to the first-return system to a subset of positive measure of an irrational rotation or a (positive or infinite entropy) Bernoulli system. Two systems are spectrally isomorphic if the unitary operators that they induce on their L2 spaces are unitarily equivalent. A smooth dynamical system consists of a differentiable manifold M and a differentiable map f : M ! M. The degree of differentiability may be specified. Two submanifolds S1 , S2 of a manifold M intersect transversely at p 2 M if Tp (S1 ) C Tp (S2 ) D Tp (M). An ( -) small Cr perturbation of a Cr map f of a manifold M is a map g such that dC r ( f ; g) < i. e. the distance between f and g is less than in the Cr topology. A map T of an interval I D [a; b] is piecewise smooth (Ck for k  1) if there is a finite set of points a D x1 < x2 <    < x n D b such that Tj(x i ; x iC1 ) is Ck for each i. The degree of differentiability may be specified. A measure  on a measure space (X; B) is absolutely continuous with respect to a measure  on (X; B) if (A) D 0 implies (A) D 0 for all measurable A 2 B. A Borel measure  on a Riemannian manifold M is absolutely continuous if it is absolutely continuous with respect to the Riemannian volume on M. A measure  on a measure space (X; B) is equivalent to a measure  on (X; B) if  is absolutely continuous with respect to  and  is absolutely continuous with respect to .

Definition of the Subject Measure-preserving systems are a common model of processes which evolve in time and for which the rules governing the time evolution don’t change. For example, in Newtonian mechanics the planets in a solar system undergo motion according to Newton’s laws of motion: the planets

Ergodic Theory: Basic Examples and Constructions

move but the underlying rule governing the planets’ motion remains constant. The model adopted here is to consider the time-evolution as a transformation (either a map in discrete time or a flow in continuous time) on a probability space or more generally a measure space. This is the setting of the subject called ergodic theory. Applications of this point of view include the areas of statistical physics, classical mechanics, number theory, population dynamics, statistics, information theory and economics. The purpose of this chapter is to present a flavor of the diverse range of examples of measure-preserving transformations which have played a role in the development and application of ergodic theory and smooth dynamical systems theory. We also present common constructions involving measurepreserving systems. Such constructions may be considered a way of putting ‘building-block’ dynamical systems together to construct examples or decomposing a complicated system into simple ‘building-blocks’ to understand it better. Introduction In this chapter we collect a brief list of some important examples of measure-preserving dynamical systems, which we denote typically by (X; B; ; T) or (T; X; B; ) or slight variations. These examples have played a formative role in the development of dynamical systems theory, either because they occur often in applications in one guise or another or because they have been useful simple models to understand certain features of dynamical systems. There is a fundamental difference in the dynamical properties of those systems which display hyperbolicity: roughly speaking there is some exponential divergence of nearby orbits under iteration of the transformation. In differentiable systems this is associated with the derivative of the transformation possessing eigenvalues of modulus greater than one on a ‘dynamically significant’ subset of phase space. Hyperbolicity leads to complex dynamical behavior such as positive topological entropy, exponential divergence of nearby orbits (“sensitivity to initial conditions”) often coexisting with a dense set of periodic orbits. If ; are sufficiently regular functions on the phase space X of a hyperbolic measure-preserving transformation (T; X; ), then typically we have fast decay of correlations in the sense that ˇZ ˇ Z Z ˇ ˇ ˇ (T n x) (x) d   d ˇ  Ca(n) d ˇ ˇ X

where a(n) ! 0. If a(n) ! 0 at an exponential rate we say that the system has exponential decay of correlations. A theme in dynamical systems is that the time se-

ries formed by sufficiently regular observations on systems with some degree of hyperbolicity often behave statistically like independent identically distributed random variables. At this point it is appropriate to point out two pervasive differences between the usual probabilistic setting of a stationary stochastic process fX n g and the (smooth) dynamical systems setting of a time series of observations on a measure-preserving system f ı T n g. The most crucial is that for deterministic dynamical systems the time series is usually not an independent process, which is a common assumption in the strictly probabilistic setting. Even if some weak-mixing is assumed in the probabilistic setting it is usually a mixing condition on the  -algebras Fn D  (X1 ; : : : ; X n ) generated by successive random variables, a condition which is not natural (and usually very difficult to check) for dynamical systems. Mixing conditions on dynamical systems are given more naturally in terms of the mixing of the sets of the  -algebra B of the probability space (X; B; ) under the action of T and not by mixing properties of the  -algebras generated by the random variables f ı T n g. The other difference is that in the probabilistic setting, although fX n g satisfy moment conditions, usually no regularity properties, such as the Hölder property or smoothness, are assumed. In contrast in dynamical systems theory the transformation T is often a smooth or piecewise smooth transformation of a Riemannian manifold X and the observation  : X ! R is often assumed continuous or Hölder. The regularity of the observation  turns out to play a crucial role in proving properties such as rates of decay of correlation, central limit theorems and so on. An example of a hyperbolic transformation is an expanding map of the unit interval T(x) D (2x) (where (x) is x modulo the integers). Here the derivative has modulus 2 at all points in phase space. This map preserves Lebesgue measure, has positive topological entropy, Lebesgue almost every point x has a dense orbit and periodic points for the map are dense in [0; 1). Non-hyperbolic systems are of course also an important class of examples, and in contrast to hyperbolic systems they tend to model systems of ‘low complexity’, for example systems displaying quasiperiodic behavior. The simplest non-trivial example is perhaps an irrational rotation of the unit interval [0; 1) given by a map T(x) D (x C ˛), ˛ 2 R n Q. T preserves Lebesgue measure, every point has a dense orbit (there are no periodic orbits), yet the topological entropy is zero and nearby points stay the same distance from each other under iteration under T. There is a natural notion of equivalence for measurepreserving systems. We say that measure-preserving systems (T; X; B; ) and (S; Y; C ; ) are isomorphic if (possi-

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bly after deleting sets of measure 0 from X and Y) there is a one-to-one onto measurable map  : X ! Y with measurable inverse  1 such that  ı T D S ı   a.e. and ( 1 (A)) D (A) for all A 2 C . If X, Y are compact topological spaces we say that T is topologically conjugate to S if there exists a homeomorphism  : X ! Y such that  ı T D S ı . In this case we call  a conjugacy. If  is Cr for some r  1 we will call  a Cr -conjugacy and similarly for other degrees of regularity. We will consider X D [0; 1)(mod 1) as a representation of the unit circle S 1 D fz 2 C : jzj D 1g (under the map x ! e2 i x ) and similarly represent the k-dimensional torus T k D S 1      S 1 (k-times). If the -algebra is clear from the context we will write (T; X; ) instead of (T; X; B; ) when denoting a measure-preserving system. Examples Rigid Rotation of a Compact Group If G is a compact group equipped with Haar measure and a 2 G, then the transformation T(x) D ax preserves Haar measure and is called a rigid rotation of G. If G is abelian and the transformation is ergodic (in this setting transitivity implies ergodicity), then the transformation is uniquely ergodic. Such systems always have zero topological entropy. The simplest example of such a system is a circle rotation. Take X D [0; 1)(mod 1), with T(x) D (x C ˛) where ˛ 2 R : Then T preserves Lebesgue (Haar) measure and is ergodic (in fact uniquely ergodic) if and only if ˛ is irrational. Similarly, the map T(x1 ; : : : ; x k ) D (x1 C ˛1 ; : : : ; x k C ˛ k );

defined by  (k1  1; k2  1; : : : ) D (0; 0; : : : ) if each entry in the Z k i component is k i  1 , while  (k1  1; k2  1; : : : ; k n  1; x1 ; x2 ; : : : ) n times

‚ …„ ƒ D (0; 0; : : : ; 0; x1 C 1; x2 ; x3 ; : : : ) when x1 6D k nC1  1. The map  may be thought of as “add one and carry” and also as mapping each point to its successor in a certain order. See Sect. “Adic Transformations” for generalizations. If each k i D 2 then the system is called the dyadic (or von Neumann-Kakutani) adding machine or 2-odometer. Adding machines give examples of naturally occurring minimal systems of low orbit complexity in the sense that the topological entropy of an adding machine is zero. In fact if f is a continuous map of an interval with zero topological entropy and S is a closed, topologically transitive invariant set without periodic orbits, then the restriction of f to S is topologically conjugate to the dyadic adding machine (Theorem 11.3.13 in [46]). We say a non-empty set  is an attractor for a map T if there is an open set U containing  such that  D \n0 T n (U) (other definitions are found in the literature). The dyadic adding machine is topologically conjugate to the Feigenbaum attractor at the limit point of period doubling bifurcations (see Sect. “Unimodal Maps”). Furthermore, attractors for continuous unimodal maps of the interval are either periodic orbits, transitive cycles of intervals, or Cantor sets on which the dynamics is topologically conjugate to an adding machine [32].

where ˛1 ; : : : ; ˛ k 2 R ; preserves k-dimensional Lebesgue (Haar) measure and is ergodic (uniquely ergodic) if and only if there are no integers m1 ; : : : ; m k , not all 0, which satisfy m1 ˛1 C    C m k ˛ k 2 Z. Adding Machines Let fk i g i2N be a sequence of integers with k i  2. Equip each cyclic group Z k i with the discrete topology and form Q the product space ˙ D 1 iD1 Z k i equipped with the product topology. An adding machine corresponding to the seQ quence fk i g i2N is the topological space ˙ D 1 iD1 Z k i together with the map : ˙ !˙

Interval Exchange Maps A map T : [0; 1] ! [0; 1] is an interval exchange transformation if it is defined in the following way. Suppose that  is a permutation of f1; : : : ; ng and l i > 0, i D 1; : : : ; n, is a sequence of subintervals of I (open or closed) with P i l i D 1. Define ti by l i D t i  t i1 with t0 D 0. Suppose also that  is an n-vector with entries ˙1. T is defined by sending the interval t i1  x < t i1 of length li to the interval X X l( j)  x < l( j) ( j)<(i)

( j)>(i)

with orientation preserved if the ith entry of  is C1 and orientation reversed if the ith entry of  is -1. Thus on each

Ergodic Theory: Basic Examples and Constructions

interval li , T has the form T(x) D  i x Ca i , where  i is ˙1. If  i D 1 for each i, the transformation is called orientation preserving. The transformation T has finitely many discontinuities (at the endpoints of each li ), and modulo this set of discontinuities is smooth. T is also invertible (neglecting the finite set of discontinuities) and preserves Lebesgue measure. These maps have zero topological entropy and arise naturally in studies of polygonal billiards and more generally area-preserving flows. There are examples of minimal but non-ergodic interval exchange maps [56,69]. Full Shifts and Shifts of Finite Type Given a finite set (or alphabet) A D f0; : : : ; d  1g, take X D ˝ C (A) D AN (or X D AZ ) the sets of one-sided (two-sided) sequences, respectively, with entries from A. For example sequences in AN have the form x D x0 x1 : : : x n : : : . A cylinder set C(y n 1 ; : : : ; y n k ), y n i 2 A, of length k is a subset of X defined by fixing k entries; for example, C(y n 1 ; : : : ; y n k ) D fx : x n 1 D y n 1 ; : : : ; x n k D y n k g: We define the set Ak to consist of all cylinders C(y1 ; : : : ; y k ) determined by fixing the first k entries, i. e. an element of Ak is specified by fixing the first k entries of a sequence x0    x k by requiring x i D y i , i D 0; : : : ; k. Let p D (p0 ; : : : ; p d1 ) be a probability vector: all P p i  0 and d1 iD0 p i D 1. For any cylinder B D C(b1 ; : : : ; b k ) 2 Ak , define g k (B) D p b 1 : : : p b k :

(1)

It can be shown that these functions on Ak extend to a shift-invariant measure  p on AN (or AZ ) called product measure. (See the article on  Measure Preserving Systems.) The space AN or AZ may be given a metric by defining ( 1 if x0 6D y0 ; d(x; y) D 1 if x n 6D y n and x i D y i for jij < n : 2jnj The shift (:x0 x1    x n    ) D :x1 x2    x n    is ergodic with respect to  p . The measure-preserving system (˝; B; ; ) (with B the -algebra of Borel subsets of ˝(A), or its completion), is denoted by B(p) and is called the Bernoulli shift determined by p. This system models an infinite number of independent repetitions of an experiment with finitely many outcomes, the ith of which has probability pi on each trial. These systems are mixing of all orders (i. e.  n is mixing for all n  1) and have countable Lebesgue spectrum (hence are all spectrally isomorphic). Kolmogorov

and Sinai showed that two of them cannot be isomorphic unless they have the same entropy; Ornstein [81] showed the converse. B(1/2; 1/2) is isomorphic to the Lebesguemeasure-preserving transformation x ! 2x mod 1 on [0; 1]; similarly, B(1/3; 1/3; 1/3) is isomorphic to x ! 3x mod 1. Furstenberg asked whether the only nonatomic measure invariant for both x ! 2x mod 1 and x ! 3x mod 1 on [0; 1] is Lebesgue measure. Lyons [68] showed that if one of the actions is K; then the measure must be Lebesgue, and Rudolph [100] showed the same thing under the weaker hypothesis that one of the actions has positive entropy. For further work on this question, see [50,87]. This construction can be generalized to model onestep finite-state Markov stochastic processes as dynamical systems. Again let A D f0; : : : ; d1g, and let p D (p0 ; : : : ; p d1 ) be a probability vector. Let P be a d  d stochastic matrix with rows and columns indexed by A. This means that all entries of P are nonnegative, and the sum of the entries in each row is 1. We regard P as giving the transition probabilities between pairs of elements of A. Now we define for any cylinder B D C(b1 ; : : : ; b k ) 2 Ak  p;P (B) D p b 1 Pb 1 b 2 Pb 2 b 3 : : : Pb k1 b k :

(2)

It can be shown that  p;P extends to a measure on the Borel  -algebra of ˝ C (A), and its completion. (See the article on  Measure Preserving Systems.) The resulting stochastic process is a (one-step, finite-state) Markov process. If p and P also satisfy pP D p ;

(3)

then the Markov process is stationary. In this case we call the (one or two-sided) measure-preserving system the Markov shift determined by p and P. Aperiodic and irreducible Markov chains (those for which a power of the transition matrix P has all entries positive) are strongly mixing, in fact are isomorphic to Bernoulli shifts (usually by means of a complicated measure-preserving recoding). More generally we say a set  AZ is a subshift if it is compact and invariant under  . A subshift  is said to be of finite type (SFT) if there exists an d  d matrix M D (a i j ) such that all entries are 0 or 1 and x 2  if and only if a x i x iC1 D 1 for all i 2 Z. Shifts of finite type are also called topological Markov chains. There are many invariant measures for a non-trivial shift of finite type. For example the orbit of each periodic point is the support of an invariant measure. An important role in the theory, derived from motivations of statistical mechanics, is played by equilibrium measures (or equilibrium states) for continuous functions  :  ! R, i. e. those measures 

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R which maximize fh () C  dg over all shift-invariant probability measures, where h () is the measure-theoretic entropy of  with respect to . The study of full shifts or shifts of finite type has played a prominent role in the development of the hyperbolic theory of dynamical systems as physical systems with ‘chaotic’ dynamics ‘typically’ possess an invariant set with induced dynamics topologically conjugate to a shift of finite type (see the discussion by Smale in p. 147 in [108]). Dynamical systems in which there are transverse homoclinic connections are a common example (Theorem 5.3.5 in [45]). Furthermore in certain settings positive metric entropy implies the existence of shifts of finite type. One result along these lines is a theorem of Katok [53]. Let hto p ( f ) denote the topological entropy of a map f and h ( f ) denote metric entropy with respect to an invariant measure . Theorem 1 (Katok) Suppose T : M ! M is a C 1C diffeomorphism of a closed manifold and  is an invariant measure with positive metric entropy (i. e. h (T) > 0). Then for any 0 < < h (T) there exists an invariant set  topologically conjugate to a transitive shift of finite type with hto p (Tj) > h (T)  . More Examples of Subshifts We consider some further examples of systems that are given by the shift transformation on a subset of the set of (usually doubly-infinite) sequences on a finite alphabet, usually f0; 1g: Associated with each subshift is its language, the set of all finite blocks seen in all sequences in the subshift. These languages are extractive (or factorial ) (every subword of a word in the language is also in the language) and insertive (or extendable) (every word in the language extends on both sides to longer words in the language). In fact these two properties characterize the languages (subsets of the set of finite-length words on an alphabet) associated with subshifts. Prouhet–Thue–Morse An interesting (and often rediscovered) element of f0; 1gZC is produced as follows. Start with 0 and at each stage write down the opposite (00 D 1; 10 D 0) or mirror image of what is available so far. Or, repeatedly apply the substitution 0 ! 01; 1 ! 10: 0 01 0 1 10 0 1 10 0110 :: :

The nth entry is the sum, mod 2, of the digits in the dyadic expansion of n: Using Keane’s block multiplication [55] according to which if B is a block, B  0 D B; B  1 D B0 ; and B  (!1    ! n ) D (B  !1 )    (B  !n ); we may also obtain this sequence as 0  01  01  01     : The orbit closure of this sequence is uniquely ergodic (there is a unique shift-invariant Borel probability measure, which is then necessarily ergodic). It is isomorphic to a skew product (see Sect. “Skew Products”) over the von Neumann-Kakutani adding machine, or odometer (see Sect. “Adding Machines”). Generalized Morse systems, that is, orbit closures of sequences like 0001001001   ; are also isomorphic to skew products over compact group rotations. Chacon System This is the orbit closure of the sequence generated by the substitution 0 ! 0010; 1 ! 1: It is uniquely ergodic and is one of the first systems shown to be weakly mixing but not strongly mixing. It is prime (has no nontrivial factors) [30], and in fact has minimal self joinings [31]. It also has a nice description by means of cutting up the unit interval and stacking the pieces, using spacers (see Sect. “Cutting and Stacking”). This system has singular spectrum. It is not known whether or not its Cartesian square is loosely Bernoulli. Sturmian Systems Take the orbit closure of the sequence ! n D [1˛;1) (n˛); where ˛ is irrational. This is a uniquely ergodic system that is isomorphic to rotation by ˛ on the unit interval. These systems have minimal complexity in the sense that the number of n-blocks grows as slowly as possible (n C 1) [29]. Toeplitz Systems A bi-infinite sequence (x i ) is a Toeplitz sequence if the set of integers can be decomposed into arithmetic progressions such that each xi is constant on each arithmetic progression. A shift space X is a Toeplitz shift if it is the closure of the orbit of a Toeplitz sequence. It is possible to construct Toeplitz shifts which are uniquely ergodic and isomorphic to a rotation on a compact abelian group [34]. Sofic Systems These are images of SFT’s under continuous factor maps (finite codes, or block maps). They correspond to regular languages—languages whose words are recognizable by finite automata. These are the same as the languages defined by regular expressions—finite expressions built up from ; (empty set), (empty word), + (union of two languages),  (all concatenations of words

Ergodic Theory: Basic Examples and Constructions

from two languages), and  (all finite concatenations of elements). They also have the characteristic property that the family of all follower sets of all blocks seen in the system is a finite family; similarly for predecessor sets. These are also generated by phase-structure grammars which are linear, in the sense that every production is either of the form A ! Bw or A ! w; where A and B are variables and w is a string of terminals (symbols in the alphabet of the language). (A phase-structure grammar consists of alphabets V of variables and A of terminals, a set of productions, which is finite set of pairs of words (˛; w); usually written ˛ ! w; of words on V [ A; and a start symbol S. The associated language consists of all words on the alphabet A of terminals which can be made by starting with S and applying a finite sequence of productions.) Sofic systems typically support many invariant measures (for example they have many periodic points) but topologically transitive ones (those with a dense orbit) have a unique measures of maximal entropy (see [65]). Context-free Systems These are generated by phasestructure grammars in which all productions are of the form A ! w; where A is a variable and w is a string of variables and terminals. Coded Systems These are systems all of whose blocks are concatenations of some (finite or infinite) list of blocks. These are the same as the closures of increasing sequences of SFT’s [60]. Alternatively, they are the closures of the images under finite edge-labelings of irreducible countable-state topological Markov chains. They need not be context-free. Squarefree languages are not coded, in fact do not contain any coded systems of positive entropy. See [13,14,15]. Smooth Expanding Interval Maps Take X D [0; 1)(mod 1), m 2 N, m > 1 and define

If m D 2 the system is isomorphic to a model of tossing a fair coin, which is a common example of randomness. To see this let P D fP0 D [0; 1/2); P1 D [1/2; 1]g be a partition of [0; 1] into two subintervals. We code the orbit under T of any point x 2 [0; 1) by 0’s and 1’s by letting x k D i if T k x 2 Pi ; k D 0; 1; 2; : : : . The map  : X ! f0; 1gN which associates a point x to its itinerary in this way is a measure-preserving map from (X; ) to f0; 1gN equipped with the Bernoulli measure from p0 D p1 D 12 . The map  satisfies  ı T D  ı ,  a.e. and is invertible a.e., hence is an isomorphism. Furthermore, reading the binary expansion of x is equivalent to following the orbit of x under T and noting which element of the partition P is entered at each time. Borel’s theorem on normal numbers (base m) may be seen as a special case of the Birkhoff Ergodic Theorem in this setting. Piecewise C2 Expanding Maps The main statistical features of the examples in Sect. “Smooth Expanding Interval Maps” generalize to a broader class of expanding maps of the interval. For example: Let X D [0; 1] and let P D fI1 ; : : : ; I n g (n  2) be a partition of X into intervals (closed, half-open or open) such that I i \ I j D ; if i 6D j. Let I oi denote the interior of I i . Suppose T : X ! X satisfies: (a) For each i D 1; : : : ; n, TjI i has a C2 extension to the 0 closure I¯i of I i and jT (x)j  ˛ > 1 for all x 2 I oi . (b) T(I j ) D [ i2P j I i Lebesgue a.e. for some non-empty subset Pj f1; : : : ; ng. (c) For each I j there exists nj such that T n j (I j ) D [0; 1] Lebesgue a.e. Then T has an invariant measure  which is absolutely continuous with respect to Lebesgue measure m, and there d  C. Furthermore T is exists C > 0 such that C1  dm ergodic with respect to  and displays the same statistical properties listed above for the C2 expanding maps [20]. (See the “Folklore Theorem” in the article on  Measure Preserving Systems.)

T(x) D (mx): Then T preserves Lebesgue measure  (recall that T preserves  if (T 1 A) D (A) for all A 2 B). Furthermore it can be shown that T is ergodic. This simple map exemplifies many of the characteristics of systems with some degree of hyperbolicity. It is isomorphic to a Bernoulli shift. The map has positive topological entropy and exponential divergence of nearby orbits, and Hölder functions have exponential decay of correlations and satisfy the central limit theorem and other strong statistical properties [20].

More Interval Maps Continued Fraction Map This is the map T : [0; 1] ! [0; 1] given by Tx D 1/x mod 1, and it corresponds to the shift [0; a1 ; a2 ; : : : ] ! [0; a2 ; a3 ; : : : ] on the continued fraction expansions of points in the unit interval (a map on N N ): It preserves a unique finite measure equivalent to Lebesgue measure, the Gauss measure dx/(log 2)(1 C x): It is Bernoulli with entropy  2 /6 log 2 (in fact the natural partition into intervals is a weak Bernoulli generator, for the definition and details see [91]). By using the Ergodic

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Theorem, Khintchine and Lévy showed that

(a1    a n )1/n !

1

Y kD1

1 1C 2 k C 2k

there are good summaries by Bertrand-Mathis [9] and Blanchard [12]. p

k  log log 2

a.e. as n ! 1 ;

pn 1 2 if [0; a1 ; : : : ; a n ] D ; then log q n ! a.e. ; qn n 12 log 2 ˇ ˇ ˇ 1 p n (x) ˇˇ 2 log ˇˇx  a.e. ; ! ˇ n q n (x) 6 log 2 and if m is Lebesgue measure (or any equivalent measure) and  is Gauss measure, then for each interval I; m(T n I) ! (I); in fact exponentially fast, with a best constant 0:30366 : : : see [10,75]. The Farey Map This is the map U : [0; 1] ! [0; 1] given by Ux D x/(1  x) if 0  x  1/2; Ux D (1  x)/x if 1/2  x  1: It is ergodic for the -finite infinite measure dx/x (Rényi and Parry). It is also ergodic for the Minkowski measure d, which is a measure of maximal entropy. This map corresponds to the shift on the Farey tree of rational numbers which provide the intermediate convergence (best one-sided) as well as the continued fraction (best two-sided) rational approximations to irrational numbers. See [61,62]. f -Expansions Generalizing the continued fraction map, let f : [0; 1] ! [0; 1] and let fI n g be a finite or infinite partition of [0; 1] into subintervals. We study the map f by coding itineraries with respect to the partition fI n g: For many examples, absolutely continuous (with respect to Lebesgue measure) invariant measures can be found and their dynamical properties determined. See [104]. ˇ-Shifts This is the special case of f -expansions when f (x) D ˇx mod 1 for some fixed ˇ > 1: This map of the interval is called the ˇ-transformation. With a proper choice of partition, it is represented by the shift on a certain subshift of the set of all sequences on the alphabet D D f0; 1; : : : ; bˇcg, called the ˇ-shift. A point x is expanded as an infinite series in negative powers of ˇ with coefficients from this set; dˇ (x)n D bˇ f n (x)c: (By convention terminating expansions are replaced by eventually periodic ones.) A one-sided sequence on the alphabet D is in the ˇ-shift if and only if all of its shifts are lexicographically less than or equal to the expansion dˇ (1) of 1 base ˇ. A one-sided sequence on the alphabet D is the valid expansion of 1 for some ˇ if and only if it lexicographically dominates all its shifts. These were first studied by Bissinger [11], Everett [35], Rényi [93] and Parry [84,85];

For ˇ D 1C2 5 ; dˇ (1) D 10101010 : : : . For ˇ D 32 ; dˇ (1) D 101000001 : : : (not eventually periodic). Every ˇ-shift is coded. The topological entropy of a ˇ-shift is log ˇ: There is a unique measure of maximal entropy log ˇ. A ˇ-shift is a shift of finite type if and only if the ˇ-expansion of 1 is finite. It is sofic if and only if the expansion of 1 is eventually periodic. If ˇ is a Pisot–Vijayaragavhan number (algebraic integer all of whose conjugates have modulus less than 1), then the ˇ-shift is sofic. If the ˇ-shift is sofic, then ˇ is a Perron number (algebraic integer of maximum modulus among its conjugates). Theorem 2 (Parry [86]) Every strongly transitive (for every nonempty open set U, [n>0 T n U D X) piecewise monotonic map on [0; 1] is topologically conjugate to a ˇ-transformation. Gaussian Systems Consider a real-valued stationary process f f k :  1 < k < 1g on a probability space (˝; F ; P): The process (and the associated measure-preserving system consisting of the shift and a shift-invariant measure on RZ ) is called Gaussian if for each d  1, any d of the f k form an Rd valued Gaussian random variable on ˝ : this means that with E( f k ) D m for all k and Z Ai j D

˝

( f k i  m)( f k j  m)dP D C(k i  k j ) for i; j D 1; : : : ; d ;

where C() is a function, for each Borel set B R, Pf! : ( f k 1 (!); : : : ; f k d (!)) 2 Bg

Z 1 1 D p exp  (x  (m; : : : ; m)) tr d/2 2 2 det A B A1 (x  (m; : : : ; m)) dx1    dx d : where A is a matrix with entries (A i j ). The function C(k) is positive semidefinite and hence has an associated measure  on [0; 2) such that Z 2 e i k t d (t) : C(k) D 0

Theorem 3 (de la Rue [33]) The Gaussian system is ergodic if and only if the “spectral measure”  is continuous (i. e., nonatomic), in which case it is also weakly mixing. It

Ergodic Theory: Basic Examples and Constructions

is mixing if and only if C(k) ! 0 as jkj ! 1: If  is singular with respect to Lebesgue measure, then the entropy is 0; otherwise the entropy is infinite. For more details see [28]. Hamiltonian Systems (This paragraph is from the article on  Measure Preserving Systems.) Many systems that model physical situations can be studied by means of Hamilton’s equations. The state of the entire system at any time is specified by a vector (q; p) 2 R2n , the phase space, with q listing the coordinates of the positions of all of the particles, and p listing the coordinates of their momenta. We assume there is a time-independent Hamiltonian function H(q, p) such that the time development of the system satisfies Hamilton’s equations: dq i @H ; D dt @p i

dp i @H ; D dt @q i

i D 1; : : : ; n :

(4)

Often in applications the Hamiltonian function is the sum of kinetic and potential energy: H(q; p) D K(p) C U(q) :

(5)

Solving these equations with initial state (q, p) for the system produces a flow (q; p) ! Tt (q; p) in phase space which moves (q, p) to its position Tt (q; p)t units of time later. According to Liouville’s formula Theorem 3.2 in [69], this flow preserves Lebesgue measure on R2n . Calculating dH/dt by means of the Chain Rule  X  @H dp i dH @H dq i D C dt @p i dt @q i dt i

and using Hamilton’s equations shows that H is constant on orbits of the flow, and thus each set of constant energy X(H0 ) D f(q; p) : H(q; p) D H0 g is an invariant set. There is a natural invariant measure on a constant energy set X(H0 ) for the restricted flow, namely the measure given by rescaling the volume element dS on X(H0 ) by the factor 1/jjOHjj. Billiard Systems These form an important class of examples in ergodic theory and dynamical systems, motivated by natural questions in physics, particularly the behavior of gas models. Consider the motion of a particle inside a bounded region D in Rd with piecewise smooth (C1 at least) boundaries. In the case of planar billiards we have d D 2. The particle moves in a straight line with

constant speed until it hits the boundary, at which point it undergoes a perfectly elastic collision with the angle of incidence equal to the angle of reflection and continues in a straight line until it next hits the boundary. It is usual to normalize and consider unit speed, as we do in this discussion for convenience. We take coordinates (x, v) given by the Euclidean coordinates in x 2 D together with a direction vector v 2 S d1 . A flow  t is defined with respect to Lebesgue almost every (x, v) by translating x a distance t defined by the direction vector v, taking account of reflections at boundaries.  t preserves a measure absolutely continuous with respect to Riemannian volume on (x, v) coordinates. The flow we have described is called a billiard flow. The corresponding billiard map is formed by taking the Poincaré map corresponding to the cross-section given by the boundary @D. We will describe the planar billiard map; the higher dimensional generalization is clear. The billiard map is a map T : @D ! @D, where @D is coordinatized by (s; ), s 2 [0; L), where L is the length of @D and 2 (0; ) measures the angle that inward pointing vectors make with the tangent line to @D at s. Given a point (s; ), the angle defines an oriented line l(s; ) which intersects @D in two points s and s0 . Reflecting l in the tangent line to @D at the point s0 gives another oriented line passing through s0 with angle 0 (measured with respect to the angular coordinate system based at s0 ). The billiard map is the map T(s; ) D (s 0 ; 0 ). T preserves a measure  D sin ds  d . The billiard flow may be modeled as a suspension flow over the billiard map (see Sect. “Suspension Flows”). If the region D is a polygon in the plane (or polyhedron in Rd ), then @D consists of the faces of the polyhedron. The dynamical behavior of the billiard map or flow in regions with only flat (non-curved) boundaries is quite different to that of billiard flows or maps in regions D with strictly convex or strictly concave boundaries. The topological entropy of a flat polygonal billiard is zero. Research interest focuses on the existence and density of periodic or transitive orbits. It is known that if all the angles between sides are rational multiples of  then there are periodic orbits [17,74,112] and they are dense in the phase space [16]. It is also known that a residual set of polygonal billiards are topologically transitive and ergodic [58,117]. On the other hand, billiard maps in which @D has strictly convex components are physical examples of nonuniformly hyperbolic systems (with singularities). The meaning of concave or convex varies in the literature. We will consider a billiard flow inside a circle to be a system with a strictly concave boundary, while a billiard flow on the torus from which a circle has been excised to be a billiard flow with strictly convex boundary.

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Ergodic Theory: Basic Examples and Constructions

The class of billiards with some strictly convex boundary components, sometimes called dispersing billiards or Sinai billiards, was introduced by Sinai [106] who proved many of their fundamental properties. Lazutkin [63] proved that planar billiards with generic strictly concave boundary are not ergodic. Nevertheless Bunimovich [22,23] produced a large of billiard systems, Bunimovich billiards, with strictly concave boundary segments (perhaps with some flat boundaries as well) which were ergodic and non-uniformly hyperbolic. For more details see [26,54,66,109]. We will discuss possibly the simplest example of a dispersing billiard, namely a toral billiard with a single convex obstacle. Take the torus T 2 and consider a single strictly convex subdomain S with C 1 boundary. The domain of the billiard map is [0; L)(0; ), where L is the length of @S. The measure sin( )ds  d is preserved. If the curvature of @S is everywhere non-zero, then the billiard map T has positive topological entropy, periodic points are dense, and in fact the system is isomorphic to a Bernoulli shift [41]. KAM-Systems and Stably Non-Ergodic Behavior A celebrated theorem of Kolmogorov, Arnold and Moser (the KAM theorem) implies that the set of ergodic areapreserving diffeomorphisms of a compact surface without boundary is not dense in the Cr topology for r  4. This has important implications, in that there are natural systems in which ergodicity is not generic. The constraint of perturbing in the class of area-preserving diffeomorphisms is an appropriate imposition in many physical models. We will take the version of the KAM theorem as given in Theorem 5.1 in [69] (original references include [3,59,79]). An elliptic fixed point for an area-preserving diffeomorphism T of a surface M is called a nondegenerate elliptic fixed point if there is a local Cr , r  4, change of coordinates h so that in polar coordinates hT h1 (r; ) D (r; C ˛0 C ˛1 r) C F(r; ) ; where all derivatives of F up to order 3 vanish, ˛1 6D 0 and ˙2 ˛0 6D 0; ˙ 2 ; ; 3 . A map of the form (r; ) D (r; C ˛0 C ˛1 r) ; where ˛1 6D 0, is called a twist map. Note that a twist map leaves invariant the circle r D k, for any constant k, and rotates each invariant curve by a rigid rotation ˛1 r, the magnitude of the rotation depending upon r. With respect to two-dimensional Lebesgue measure a twist map is certainly not ergodic. Theorem 4 Suppose T is a volume-preserving diffeomorphism of class Cr , r  4, of a surface M. If x is a non-degenerate elliptic fixed point, then for every > 0 there exists

a neighborhood U of x and a set U0; U with the properties: (a) U0; is a union of T-invariant simple closed curves of class Cr1 containing x in their interior. (b) The restriction of T to each such invariant curve is topologically conjugate to an irrational rotation. (c) m(U U0; )  m(U ), where m is Lebesgue measure on M. It is possible to prove the existence of a Cr volume preserving diffeomorphism (r  4) with a non-degenerate elliptic fixed point and also show that if T possesses a non-degenerate elliptic fixed point then there is a neighborhood V of T in the Cr topology on volume-preserving diffeomorphisms such that each T 0 2 V possesses a non-degenerate elliptic fixed point Chapter II, Sect. 6 in [69]. As a corollary we have Corollary 1 Let M be a compact surface without boundary and Diffr (M) the space of Cr area-preserving diffeomorphisms with the Cr topology. Then the set of T 2 Diffr (M) which are ergodic with respect to the probability measure determined by normalized area is not dense in Diffr (M) for r  4. Smooth Uniformly Hyperbolic Diffeomorphisms and Flows Time series of measurements on deterministic dynamical systems sometimes display limit laws exhibited by independent identically distributed random variables, such as the central limit theorem, and also various mixing properties. The models of hyperbolicity we discuss in this section have played a key role in showing how this phenomenon of ‘chaotic behavior’ arises in deterministic dynamical systems. Hyperbolic sets and their associated dynamics have also been pivotal in studies of structural stability. A smooth system is Cr structurally stable if a small perturbation in the Cr topology gives rise to a system which is topologically conjugate to the original. When modeling a physical system it is desirable that slight changes in the modeling parameters do not greatly affect the qualitative or quantitative behavior of the ensemble of orbits considered as a whole. The orbit of a point may change drastically under perturbation (especially if the system has sensitive dependence on initial conditions) but the collection of all orbits should ideally be ‘similar’ to the original unperturbed system. In the latter case one would hope that statistical properties also vary only slightly under perturbation. Structural stability is one, quite strong, notion of stability. The conclusion of a body of work on structural

Ergodic Theory: Basic Examples and Constructions

stability is that a system is C1 structurally stable if and only if it is uniformly hyperbolic and satisfies a technical assumption called strong transversality (see below for details). Suppose M is a C1 compact Riemannian manifold equipped with metric d and tangent space TM with norm kk. Suppose also that U M is a non-empty open subset and T : U ! T(U) is a C1 diffeomorphism. A compact T invariant set  U is called a hyperbolic set if there is a splitting of the tangent space Tp M at each point p 2  into two invariant subspaces, Tp M D E u (p) ˚ E s (p), and a number 0 <  < 1 such that for n  0 kD p T n vk  Cn kvk for v 2 E s (p) ; kD p T n vk < Cn kvk for v 2 E u (p) : The subspace Eu is called the unstable or expanding subspace and the subspace Es the stable or contracting subspace. The stable and unstable subspaces may be integrated to produce stable and unstable manifolds W s (p) D fy : d(T n p; T n y) ! 0g u

W (p) D fy : d(T

n

p; T

n

as n ! 1 ;

y) ! 0g

as n ! 1 :

The stable and unstable manifolds are immersions of Euclidean spaces of the same dimension as E s (p) and E u (p), respectively, and are of the same differentiability as T. Moreover, Tp (W s (p)) D E s (p) and Tp (W u (p)) D E u (p). It is also useful to define local stable manifolds and local unstable manifolds by W s (p) D fy 2 W s (p) : d(T n p; T n y) < g for all n  0 ; W u (p) D fy 2 W u (p) : d(T n p; T n y) < g for all n  0 : Finally we discuss the notion of strong transversality. We say a point x is non-wandering if for each open neighborhood U of x there exists an n > 0 such that T n (U) \ U 6D ;. The NW set of non-wandering points is called the non-wandering set. We say a dynamical system has the strong transversal property if W s (x) intersects W u (y) transversely for each pair of points x; y 2 NW. In the Cr , r  1 topology Robbin [94], de Melo [78] and Robinson [95,96] proved that dynamical systems with the strong transversal property are structurally stable, and Robinson [97] in addition showed that strong transversality was also necessary. Mañé [70] showed that a C1 structurally stable diffeomorphism must be uniformly hyperbolic and Hayashi [47] extended this to flows. Thus a C1 diffeomorphism or flow on a compact manifold is structurally stable if and only it is uniformly hyperbolic and satisfies the strong transversality condition.

Geodesic Flow on Manifold of Negative Curvature The study of the geodesic flow on manifolds of negative sectional curvature by Hedlund and Hopf was pivotal to the development of the ergodic theory of hyperbolic systems. Suppose that M is a geodesically complete Riemannian manifold. Let p;v (t) be the geodesic with p;v (0) D p and ˙ p;v (0) D v, where ˙ p;v denotes the derivative with respect to time t. The geodesic flow is a flow  t on the tangent bundle TM of M,  t : R  TM ! TM, defined by  t (p; v) D ( p;v (t); ˙p;v (t)) : where (p; v) 2 TM. Since geodesics have constant speed, if kvk D 1 then k p;v (t)k D 1 for all t, and thus the unit tangent bundle T 1 M D f(p; v) 2 TM : kvk D 1g is preserved under the geodesic flow. The geodesic flow and its restriction to the unit tangent bundle both preserve a volume form, Liouville measure. In 1934 Hedlund [48] proved that the geodesic flow on the unit tangent bundle of a surface of strictly negative constant sectional curvature is ergodic, and in 1939 Hopf [49] extended this result to manifolds of arbitrary dimension and strictly negative (not necessarily constant) curvature. Hopf’s technique of proof of ergodicity (Hopf argument) was extremely influential and used the foliation of the tangent space into stable and unstable manifolds. For a clear exposition of this technique, and the property of absolute continuity of the foliations into stable and unstable manifolds, see [66]. The geodesic flow on manifolds of constant negative sectional curvature is an Anosov flow (see Sect. “Anosov Systems”). We remark that for surfaces sectional curvature is the same as Gaussian curvature. Recently the time-one map of the geodesic flow on the unit tangent bundle of a surface with constant negative curvature, which is a partially hyperbolic system (see Sect. “Partially Hyperbolic Dynamical Systems”), was shown to be stably ergodic [44], so the geodesic flow is still playing a major role in the development of ergodic theory. Horocycle Flow All surfaces endowed with a Riemannian metric of constant negative curvature are quotients of the upper half-plane H C :D fx C iy 2 C : y > 0g with dx 2 Cdy 2

, whose sectional curvature is the metric ds 2 D y2 1. The orientation-preserving isometries of this metric are exactly the linear fractional (also known as Möbius) transformations:   az C b a b 2 H C] : 2 SL(2; R); [z 2 H C 7! c d cz C d Since each matrix ˙I corresponds to the identity transformation, we consider matrices in PSL(2; R) :D SL(2; R)/f˙Ig.

273

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Ergodic Theory: Basic Examples and Constructions

The unit tangent bundle, S H C , of the upper halfplane can be identified with PSL(2; R). Then the geodesic flow corresponds to the transformations  t  e 0 t 2 R 7! 0 et seen as acting on PSL(2; R). The unstable foliation of an element A 2 PSL(2; R) Š S H C is given by   1 t A; t 2 R ; 0 1 and the flow along this foliation, given by   1 t ; t 2 R 7! 0 1 is called the horocycle flow. Similarly for the flow induced on the unit tangent bundle of each quotient of the upper half-plane by a discrete group of linear fractional transformations. The geodesic and horocycle flows acting on a (finitevolume) surface of constant negative curvature form the fundamental example of a transverse pair of actions. The geodesic flow often has many periodic orbits and many invariant measures, has positive entropy, and is in fact Bernoulli with respect to the natural measure [82], while the horocycle flow is often uniquely ergodic [39,72] and of entropy zero, although mixing of all orders [73]. See [46] for more details. Markov Partitions and Coding If (X; T; B; ) is a dynamical system then a finite partition of X induces a coding of the orbits and a semi-conjugacy with a subshift on a symbol space (it may not of course be a full conjugacy). For hyperbolic systems a special class of partitions, Markov partitions, induce a conjugacy for the invariant dynamics to a subshift of finite type. A Markov partition P for an invariant subset  of a diffeomorphism T of a compact manifold M is a finite collection of sets Ri , 1  i  n called rectangles. The rectangles have the property, for some > 0, if x; y 2 R i then W s (x) \ W u (y) 2 R i . This is sometimes described as being closed under local product structure. We let W u (x; R i ) denote W u (x) \ R i and W s (x; R i ) denote W s (x) \ R i . Furthermore we require for all i, j: (1) (2) (3) (4)

Each Ri is the closure of its interior.  [i R i R i \ R j D @R i \ @R j if i 6D j if x 2 R oi and T(x) 2 R oj then W u (T(x); R j )

T(W u (x; R i )) and W s (x; R i ) T 1 (W u (T(x); R j ))

Anosov Systems An Anosov diffeomorphism [2] is a uniformly hyperbolic system in which the entire manifold is a hyperbolic set. Thus an Anosov diffeomorphism is a C1 diffeomorphism T of M with a DT-invariant splitting (which is a continuous splitting) of the tangent space TM(x) at each point p into a disjoint sum Tp M D E u (p) ˚ E s (p) and there exist constants 0 <  < 1, constant C > 0 such that kDT n vk < Cn kvk for all v 2 E s (p) and kDT n wk  Cn kwk for all w 2 E u (p). A similar definition holds for Anosov flows  : R  M ! M. A flow is Anosov if there is a splitting of the tangent bundle into flow-invariant subspaces E u ; E s ; E c so D p  t E sp D Es t (p) , D p  t E up D Eu t (p) and D p  t E cp D Ec t (p) , and at each point p 2 M Tp M D E sp ˚ E up ˚ E cp k(D p  t )vk < C t kvk for v 2 E s (p) k(D p t )vk < C t kvk for v 2 E u (p) for some 0 <  < 1. The tangent to the flow direction E c (p) is a neutral direction: k(D p  t )vk D kvk for v 2 E c (p) : Anosov proved that Anosov flows and diffeomorphisms which preserve a volume form are ergodic [2] and are also structurally stable. Sinai [105] constructed Markov partitions for Anosov diffeomorphisms and hence coded trajectories via a subshift of finite type. Using ideas from statistical physics in [107] Sinai constructed Gibbs measures for Anosov systems. An SRB measure (see Sect. “Physically Relevant Measures and Strange Attractors”) is a type of Gibbs measure corresponding to the potential  log j det(DTj E u )j and is characterized by the property of absolutely continuous conditional measures on unstable manifolds. The simplest examples of Anosov diffeomorphisms are perhaps the two-dimensional hyperbolic toral automorphisms (the n > 2 generalization is clear). Suppose A is a 2  2 matrix with integer entries   a b c d such that det(A) D 1 and A has no eigenvalues of modulus 1. Then A defines a transformation of the two-dimensional torus T 2 D S 1  S 1 such that if v 2 T 2 ,   v vD 1 ; v2

Ergodic Theory: Basic Examples and Constructions

then   (av1 C bv2 ) : Av D (cv1 C dv2 ) A preserves Lebesgue (or Haar) measure and is ergodic. A prominent example of such a matrix is  2 1

1 1

 ;

which is sometimes called the Arnold Cat Map. Each point with rational coordinates (p1 /q1 ; p2 /q2 ) is periodic. p There are two eigenvalues 1/ < 1 <  D (3 C 5)/2 with orthogonal eigenvectors, and the projections of the eigenspaces from R2 to T 2 are the stable and unstable subspaces. Axiom A Systems In the case of Anosov diffeomorphisms the splitting into contracting and expanding bundles holds on the entire phase space M. A system T : M ! M is an Axiom A system if the non-wandering set NW is a hyperbolic set and periodic points are dense in the nonwandering set. NW M may have Lebesgue measure zero. A set  M is locally maximal if there exists an open set U such that  D \n2Z T n (U). The solenoid and horseshoe discussed below are examples of locally maximal sets. Bowen [18] constructed Markov partitions for Axiom A diffeomorphisms . Ruelle imported ideas from statistical physics, in particular the idea of an equilibrium state and the variational principle, to the study of Axiom A systems (see [101,102]) This work extended the notion of Gibbs measure and other ideas from statistical mechanics, introduced by Sinai for Anosov systems [107], into Axiom A systems. One achievement of the Axiom A program was the Smale Decomposition Theorem, which breaks the dynamics of Axiom A systems into locally maximal sets and describes the dynamics on each [18,19,108]. Theorem 5 (Spectral Decomposition Theorem) If T is Axiom A then there is a unique decomposition of the nonwandering set NW of T NW D 1 [    [  k as a disjoint union of closed, invariant, locally maximal hyperbolic sets  i such that T is transitive on each  i . Furthermore each  i may be further decomposed into a disj joint union of closed sets  i , j D 1; : : : ; n i such that T n i is j

topologically mixing on each  i and T cyclically permutes j

the  i .

Horseshoe Maps This type of map was introduced by Steven Smale in the 1960’s and has played a pivotal role in the development of dynamical systems theory. It is perhaps the canonical example of an Axiom A system [108] and is conjugate to a full shift on 2 symbols. Let S be a unit square in R2 and let T be a diffeomorphism of S onto its image such that S \ T(S) consists of two disjoint horizontal strips S0 and S1 . Think of stretching S uniformly in the horizontal direction and contracting uniformly in the vertical direction to form a long thin rectangle, and then bending the rectangle into the shape of a horseshoe and laying the straight legs of the horseshoe back on the unit square S. This transformation may be realized by a diffeomorphism and we may also require that T restricted to T 1 S i , i D 0; 1, acts as a linear map. The restriction of T i to the maximal invariant set H D \1 iD1 T S is a Smale horseshoe map. H is a Cantor set, the product of a Cantor set in the horizontal direction and a Cantor set in the vertical direction. The conjugacy with the shift on two symbols is realized by mapping x 2 H to its itinerary with respect to the sets S0 and S1 under powers of T (positive and negative powers). Solenoids The solenoid is defined on the solid torus X in R3 which we coordinatize as a circle of two-dimensional solid disks, so that X D f( ; z) : 2 [0; 1) and jzj  1; z 2 Cg The transformation T : X ! X is given by   1 1 T( ; z) D 2 (mod 1) ; z C e2 i 4 2 Geometrically the transformation stretches the torus to twice its length, shrinks its diameter by a factor of 4, then twists it and doubles it over, placing the resultant object without self-intersection back inside the original solid torus. T(X) intersects each disk D c D f( ; z) : D cg in two smaller disks of 14 the diameter. The transformation T contracts volume by a factor of 2 upon each application, yet there is expansion in the direction ( ! 2 ). The solenoid A D \n0 T n (X) has zero Lebesgue measure, is T-invariant and is (locally) topologically a line segment cross a two-dimensional Cantor set (A intersects each disk Dc in a Cantor set). The set A is an attractor, in that all points inside X limit under iteration by T upon A. T : A ! A is an Axiom A system. Partially Hyperbolic Dynamical Systems Partially hyperbolic dynamical systems are a generalization of uniformly hyperbolic systems in that an invariant central direction is allowed but the contraction in the

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central direction is strictly weaker than the contraction in the contracting direction and the expansion in the central direction is weaker than the expansion in the expanding direction. More precisely, suppose M is a C1 compact (adapted) Riemannian manifold equipped with metric d and tangent space TM with norm kk. A C1 diffeomorphism T of M is a partially hyperbolic diffeomorphism if there is a nontrivial continuous DT invariant splitting of the tangent space Tp M at each point p into a disjoint sum Tp M D E u (p) ˚ E c (p) ˚ E s (p) and continuous positive functions m; M; ˜ ; such that 

Es

is contracted: kD p T n v s k  m(p) < 1; if v s 2 E s (x) n f0g then kv sk  Eu is expanded: kD Tv u k if v u 2 E u (x) n f0g then kvp u k  M(p) > 1;  Ec is uniformly dominated by Eu and Es : ˜

(p) if v c 2 E c (x) n f0g then there are numbers (p); kD p Tv c k ˜ such that m(p) < (p)  kv c k  (p) < M(p) .

The notion of partial hyperbolicity was introduced by Brin and Pesin [21] who proved existence and properties, including absolute continuity, of invariant foliations in this setting. There has been intense recent interest in partially hyperbolic systems primarily because significant progress has been made in establishing that certain volume-preserving partially hyperbolic systems are ‘stably ergodic’—that is, they are ergodic and under small (Cr topology) volume-preserving perturbations remain ergodic. This phenomenon had hitherto been restricted to uniformly hyperbolic systems. For recent developments, and precise statements, on stable ergodicity of partially hyperbolic systems see [24,92]. Compact Group Extensions of Uniformly Hyperbolic Systems A natural example of a partially hyperbolic system is given by a compact group extension of an Anosov diffeomorphism. If the following terms are not familiar see Sect. “Constructions” on standard constructions. Suppose that (T; M; ) is an Anosov diffeomorphism, G is a compact connected Lie group and h : M ! G is a differentiable map. The skew product F : M  G ! M  G given by F(x; g) D (Tx; h(x)g) has a central direction in its tangent space corresponding to the Lie algebra LG of G (as a group element h acts isometrically on G so there is no expansion or contraction)

and uniformly expanding and contracting bundles corresponding to those of the tangent space of T : M ! M. Thus T(M  G) D E u ˚ LG ˚ E s . Time-One Maps of Anosov Flows Another natural context in which partial hyperbolicity arises is in time-one maps of uniformly hyperbolic flows. Suppose  t : R  M ! M is an Anosov flow. The diffeomorphism 1 : M ! M is a partially hyperbolic diffeomorphism with central direction given by the flow direction. There is no expansion or contraction in the central direction. Non-Uniformly Hyperbolic Systems The assumption of uniform hyperbolicity is quite restrictive and few ‘chaotic systems’ found in applications are likely to exhibit uniform hyperbolicity. A natural weakening of this assumption, and one that is non-trivial and greatly extends the applicability of the theory, is to require the hyperbolic splitting (no longer uniform) to hold only at almost every point of phase space. A systematic theory was built by Pesin [89,90] on the assumption that the system has non-zero Lyapunov exponents  almost everywhere where  is a Lebesgue equivalent invariant probability measure . Recall that a number  is a Lyapunov exponent for p 2 M if kD p T n vk en for some unit vector v 2 Tp M. Oseledet’s theorem [83] (see also p. 232 in [113]), which is also called the Multiplicative Ergodic Theorem, implies that if T is a C1 diffeomorphism of M then for any T-invariant ergodic measure  almost every point has well-defined Lyapunov exponents. One of the highlights of Pesin theory is the following structure theorem: If T : M ! M is a C 1C diffeomorphism with a T-invariant Lebesgue equivalent Borel measure  such that T has non-zero Lyapunov exponents with respect to  then T has at most a countable number of ergodic components fC i g on each of which the restriction of T is either Bernoulli or Bernoulli times a rotation (by which we mean the support of  i D jC i consists of a finite number ni of sets fS1i ; : : : S ni i g cyclically permuted and T n i is Bernoulli when restricted to each S ij ) [90,114]. This structure theorem has been generalized to SRB measures with non-zero Lyapunov exponents [64,90]. Physically Relevant Measures and Strange Attractors (This paragraph is from the article on  Measure Preserving Systems.) For Hamiltonian systems and other volumepreserving systems it is natural to consider ergodicity (and other statistical properties) of the system with respect to Lebesgue measure. In dissipative systems a measure equiv-

Ergodic Theory: Basic Examples and Constructions

alent to Lebesgue may not be invariant (for example the solenoid). Nevertheless Lebesgue measure has a distinguished role since sampling by experimenters is done with respect to Lebesgue measure. The idea of a physically relevant measure  is that it determines the statistical behavior of a positive Lebesgue measure set of orbits, even though the support of  may have zero Lebesgue measure. An example of such a situation in the uniformly hyperbolic setting is the solenoid , where the attracting set  has Lebesgue measure zero and is (locally) topologically the product of a two-dimensional Cantor set and a line segment. Nevertheless  determines the behavior of all points in a solid torus in R3 . More generally, suppose that T : M ! M is a diffeomorphism on a compact Riemannian manifold and that m is a version of Lebesgue measure on M, given by a smooth volume form. Although Lebesgue measure m is a distinguished physically relevant measure, m may not be invariant under T, and the system may even be volume contracting in the sense that m(T n A) ! 0 for all measurable sets A. Nevertheless an experimenter might observe long-term “chaotic” behavior whenever the state of the system gets close to some compact invariant set X which attracts a positive m-measure of orbits in the sense that these orbits limit on X. Possibly m(X) D 0, so that X is effectively invisible to the observer except through its effects on orbits not contained in X. The dynamics of T restricted to X can in fact be quite complicated—maybe a full shift, or a shift of finite type, or some other complicated topological dynamical system. Suppose there is a T-invariant measure  supported on X such that for all continuous functions  : M ! R Z n1 1X k  ı T (x) !  d ; n X

(6)

kD0

for a positive m-measure of points x 2 M. Then the longterm equilibrium dynamics of an observable set of points x 2 M (i. e. a set of points of positive m measure) is described by (X; T; ). In this situation  is described as a physical measure. There has been a great deal of research on the properties of systems with attractors supporting physical measures. In the dissipative non-uniformly hyperbolic setting the theory of ‘physically relevant’ measures is best developed in the theory of SRB (for Sinai, Ruelle and Bowen) measures. These dynamically invariant measures may be supported on a set of Lebesgue measure zero yet determine the asymptotic behavior of points in a set of positive Lebesgue measure. If T is a diffeomorphism of M and  is a T-invariant Borel probability measure with positive Lyapunov expo-

nents which may be integrated to unstable manifolds, then we call  an SRB measure if the conditional measure  induces on the unstable manifolds is absolutely continuous with respect to the Riemannian volume element on these manifolds. The reason for this definition is technical but is motivated by the following observations. Suppose that the diffeomorphism has no zero Lyapunov exponents with respect to . Since T is a diffeomorphism, this implies T has negative Lyapunov exponents as well as positive Lyapunov exponents and corresponding local stable manifolds as well as local unstable manifolds. Suppose that a T-invariant set A consists of a union of unstable manifolds and is the support of an ergodic SRB measure  and that  : M ! R is a continuous function. Since  has absolutely continuous conditional measures on unstable manifolds with respect to conditional Lebesgue measure on the unstable manifolds, almost every point x in the union of unstable manifolds U satisfies Z n1 1X lim  ı T j (x) D  d : (7) n!1 n jD0

If y 2 W s (x) for such an x 2 U then d(T n x; T n y) ! 1 and hence (7) implies Z n1 1X  ı T j (y) D  d : lim n!1 n jD0

Furthermore, if the holonomy between unstable manifolds defined by sliding along stable manifolds is absolutely continuous (takes sets of zero Lebesgue measure on W u to sets of zero Lebesgue measure on W u ), there is a positive Lebesgue measure of points (namely an unstable manifold and the union of stable manifolds through it) satisfying (7). Thus an SRB measure with absolutely continuous holonomy maps along stable manifolds is a physically relevant measure. If the stable foliation possesses this property it is called absolutely continuous. An Axiom A attractor for a C2 diffeomorphism is an example of an SRB attractor [19,101,102,107]. The examples we have given of SRB measures and attractors and measures have been uniformly hyperbolic. Recently much progress has been made in understanding the statistical properties of non-uniformly hyperbolic systems by using a tower (see Sect. “Induced Transformations”) to construct SRB measures. We refer to Young’s original papers [115,116], the book by Baladi [5] and to [114] for a recent survey on SRB measures in the nonuniformly hyperbolic setting. Unimodal Maps Unimodal Maps of an interval are simple examples of non-uniformly hyperbolic dynamical sys-

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Ergodic Theory: Basic Examples and Constructions

tems that have played an important role in the development of dynamical systems theory. Suppose I R is an interval; for simplicity we take I D [0; 1]. A unimodal map is a map T : [0; 1] ! [0; 1] such that there exists a point 0 < c < 1 and  T is C2 ;  T 0 (x) > 0 for x < c, T 0 (x) < 0 for x > c;  T 0 (c) D 0. Such a map is clearly not uniformly expanding, as jT 0 (x)j < 1 for points in a neighborhood of c. The family of maps T (x) D x(1  x), 0 <   4, is a family of unimodal maps with c D 1/2 and T2 (1/2) D 1/2, T4 (1/2) D 1. We could have taken the interval I to be [1; 1] or indeed any interval with an obvious modification of the definition above. A well-studied family of unimodal maps in this setting is the logistic family f a : [1; 1] ! [1; 1], f a (x) D 1  ax 2 , a 2 (0; 2]. The families f a and T are equivalent under a smooth coordinate change, so statements about one family may be translated into statements about the other. Unimodal maps are studied because of the insights they offer into transitions from regular or periodic to chaotic behavior as a parameter (e. g.  or a) is varied, the existence of absolutely continuous measures, and rates of decay of correlations of regular observations for non-uniformly hyperbolic systems. Results of Jakobson [52] and Benedicks and Carleson [6] implies that in the case of the logistic family there is a positive Lebesgue measure set of a such that f a has an absolutely continuous ergodic invariant measure  a . It has been shown by Keller and Nowicki [57] (see alsoYoung [116]) that if f a is mixing with respect to  a then the decay of correlations for Lipshitz observations on I is exponential. It is also known that the set of a such that f a is mixing with respect to  a has positive Lebesgue measure. There is a well-developed theory concerning bifurcations the maps T undergo as  varies [27]. We briefly describe the period-doubling route to chaos in the family T (x) D x(1  x). For a nice account see [46].p We let c denote the fixed point 1 . For  3 <   1 C 6, all points in [0; 1] except for 0; c and their preimages are attracted to a unique periodic orbit O(p ) of period 2. There is a monotone sequence of parameter values n ( 1 D 3) such that for n <   nC1 , T has a unique attracting periodic orbit O(n ) of period 2n and for each k D 1; 2; : : : ; n  1 a unique repelling orbit of period 2k . All points in the interval [0; 1] except for the repelling periodic orbits and their preimages are attracted to the attracting periodic orbit of period 2n . At

 D n the periodic orbit O(n ) undergoes a period-doubling bifurcation. Feigenbaum [36] found that the limit n1 ı D n  4:699 : : : exists and that in a wide class n nC1 of unimodal maps this period-doubling cascade occurs and the differences between successive bifurcation parameters give the same limiting ratio, an example of universality. At the end of the period-doubling cascade at a parameter 1 3:569 : : :, T1 has an invariant Cantor set C (the Feigenbaum attractor) which is topologically conjugate to the dyadic adding machine coexisting with isolated repelling orbits of period 2n , n D 0; 1; 2; : : : . There is a unique repelling orbit of period 2n for n  1 along with two fixed points. The Cantor set C is the !-limit set for all points that are not periodic or preimages of periodic orbits. C is the set of accumulation points of periodic orbits. Despite this picture of incredible complexity the topological entropy is zero for   1 . For  > 1 the map T has positive topological entropy and infinitely many periodic orbits whose periods are not powers of 2. For each   1 , T possesses an invariant Cantor set which is repelling for  > 1 . We say that T is hyperbolic if there is only one attracting periodic orbit and the only recurrent sets are the attracting periodic orbit, repelling periodic orbits and possibly a repelling invariant Cantor set. It is known that the set of  2 [0; 4] for which T is hyperbolic is open and dense [43]. Remarkably, by Jakobson’s result [52] there is also a positive Lebesgue measure set of parameters  for which T has an absolutely continuous invariant measure  with a positive Lyapunov exponent. Intermittent Maps Maps of the unit interval T : [0; 1] ! [0; 1] which are expanding except at the point x D 0, where they are locally of form T(x) x C x 1C˛ , ˛ > 0, have been extensively studied both for the insights they give into rates of decay of correlations for nonuniformly hyperbolic systems (hyperbolicity is lost at the point x D 0, where the derivative is 1) and for their use as models of intermittent behavior in turbulence [71]. A fixed point where the derivative is 1 is sometimes called an indifferent fixed point. It is a model of intermittency in the sense that orbits close to 1 will stay close for many iterates (since the expansion is very weak there) and hence a time series of observations will be quite uniform for long periods of time before displaying chaotic type behavior after moving away from the indifferent fixed into that part of the domain where the map is uniformly expanding. A particularly simple model [67] is provided by ( x(1 C 2˛ x ˛ ) if x 2 [0; 1/2) ; T(x) D 2x  1 if x 2 [1/2; 1] :

Ergodic Theory: Basic Examples and Constructions

For ˛ D 0 the map is uniformly expanding and Lebesgue measure is invariant. In this case the rate of decay of correlations for Hölder observations is exponential. For 0 < ˛ < 1 the map has an SRB measure ˛ with support the unit interval. For ˛  1 there are no absolutely continuous invariant probability measures though there are -finite absolutely continuous measures. Upper and lower polynomial bounds on the rate of decay of observations on such maps have been given as a function of 0 < ˛ < 1 and the regularity of the observable. For details see [51,67,103]. Hénon Diffeomorphisms The Henón family of diffeomorphisms was introduced and studied as Poincaré maps for the Lorenz system of equations. It is a two-parameter two-dimensional family which shares many characteristics with the logistic family and for small b > 0 may be considered a two-dimensional ‘perturbation’ of the logistic family. The parametrized mapping is defined as Ta;b (x; y) D (1  ax 2 C y; bx) ; so Ta;b : R2 ! R2 with 0 < a < 2 and b > 0. Benedicks and Carleson [7] showed that for a positivemeasure set of parameters (a, b), T a;b has a topologically transitive attractor  a;b . Benedicks and Young [8] later proved that for a positive-measure set of parameters (a, b), T a;b has a topologically transitive SRB attractor  a;b with SRB measure  a;b and that (Ta;b ;  a;b ;  a;b ) is isomorphic to a Bernoulli shift. Complex Dynamics Complex dynamics is concerned with the behavior of rational maps ˛1 z d C ˛2 z d1 C    ˛dC1 ˇ1 z d C ˇ2 z d1 C    ˇdC1 ¯ to itself, in which the of the extended complex plane C domain is C completed with the point at infinity (called the Riemann sphere). Recall that a family F of meromorphic functions is called normal on a domain D if every sequence possesses a subsequence that converges uniformly ¯ S 2 ) on compact subsets of (in the spherical metric C ¯ if it is normal on D. A family is normal at a point z 2 C ¯ of a rational a neighborhood of z. The Fatou set F(R) C ¯ ¯ ¯ map R : C ! C is the set of points z 2 C such that the family of forward iterates fR n gn0 is normal at z. The Julia set J(R) is the complement of the Fatou set F(R). The Fatou set is open and hence the Julia set is a closed set. Another characterization in the case d > 1 is that J(R) is the clo¯ ! C. ¯ sure of the set of all repelling periodic orbits of R : C

Both F(R) and J(R) are invariant under R. The dynamics of greatest interest is the restriction R : J(R) ! J(R). The Julia set often has a complicated fractal structure. In the case that R a (z) D z2  a, a 2 C, the Mandelbrot set is defined as the set of a for which the orbit of the origin 0 is bounded. The topology of the Mandelbrot set has been the subject of intense research. The study of complex dynamics is important because of the fascinating and complicated dynamics displayed and also because techniques and results in complex dynamics have direct implications for the behavior of one-dimensional maps. For more details see [25]. Infinite Ergodic Theory We may also consider a measure-preserving transformation (T; X; ) of a measure space such that (X) D 1. For example X could be the real line equipped with Lebesgue measure. This setting also arises with compact X in applications. For example, suppose T : [0; 1] ! [0; 1] is the simple model of intermittency given in Sect. “Intermittent Maps” and 2 (1; 2). Then T possesses an absolutely continuous invariant measure  with support [0; 1], but ([0; 1]) D 1. The Radon–Nikodym derivative of  with respect to Lebesgue measure m exists but is not in L1 (m). In this setting we say a measurable set A is a wandering set for T if fT n Ag1 nD0 are disjoint. Let D(T) be the measurable union of the collection of wandering sets for T. The transformation T is conservative with respect to  if (X n D(T)) D X (mod ) (see the article on Measure Preserving Systems). It is usually necessary to assume T conservative with respect to  to say anything interesting about its behavior. For example if T(x) D x C ˛, ˛ > 0, is a translation of the real line then D(T) D X. The definition of ergodicity in this setting remains the same: T is ergodic if A 2 B and T 1 A D A mod  implies that (A) D 0 or (Ac ) D 0. However the equivalence of ergodicity of T with respect to  and the equality of time and space averages for L1 () functions no longer holds. Thus in general  ergodic does not imply that lim

n!1

Z n1 1X  ı T j (x) D  d  a.e. x 2 X n X iD0

for all  2 L1 (). In the example of the intermittent map with 2 (1; 2) the orbit of Lebesgue almost every x 2 X is dense in X, yet the fraction of time spent near the indifferent fixed point x D 0 tends to one for Lebesgue almost every x 2 X. In fact it may be shown Sect. 2.4 in [1] that when (x) D 1 there are no constants a n > 0 such that Z n1 1 X lim  ı T j (x) D  d  a.e. x 2 X n!1 a n X iD0

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Nevertheless it is sometimes possible to obtain distributional limits, rather than almost sure limits, of Birkhoff sums under suitable normalization. We refer the reader to Aaronson’s book [1] for more details. Constructions We give examples of some of the standard constructions in dynamical systems. Often these constructions appear in modeling situations (for example skew products are often used to model systems which react to inputs from other systems, continuous time systems are often modeled as suspension flows over discrete-time dynamics) or to reduce systems to simpler components (often a factor system or induced system is simpler to study). Unless stated otherwise, in the sequel we will be discussing measure-preserving transformations on Lebesgue spaces (see the article on  Measure Preserving Systems). Products Given measure-preserving systems (X; B; ; T) and (Y; C ; ; S), their product consists of their completed product measure space with the transformation T S : X  Y ! X  Y defined by (T  S)(x; y) D (Tx; Sy) for all (x; y) 2 X  Y. Neither ergodicity nor transitivity is in general preserved by taking products; for example the product of an irrational rotation on the unit circle with itself is not ergodic. For a list of which mixing properties are preserved in various settings by forming a product see [113]. Given any countable family of measure-preserving transformations on probability spaces, their direct product is defined similarly. Factors We say that a measure-preserving system (Y; C ; ; S) is a factor of a measure-preserving system (X; B; ; T) if (possibly after deleting a set of measure 0 from X) there is a measurable onto map  : X ! Y such that  1 C B; T D S; T

1

and

(8)

D :

For Lebesgue spaces, there is a correspondence of factors of (X; B; ; T) and T-invariant complete sub- -algebras of B. According to Rokhlin’s theory of Lebesgue spaces [98] factors also correspond to certain partitions of X (see the article on  Measure Preserving Systems). A factor map  : X ! Y between Lebesgue spaces is an isomorphism if and only if it has a measurable inverse, or equivalently  1 C D B up to sets of measure 0.

Skew Products If (X; B; ; T) is a measure-preserving system, (Y; C ; ) is a measure-space, and fS x : x 2 Xg is a family of measurepreserving maps Y ! Y such that the map that takes (x, y) to S x y is jointly measurable in the two variables x and y, then we may define a skew product system consisting of the product measure space of X and Y equipped with product measure    together with the measure-preserving map T Ë S : X  Y ! X  Y defined by (T Ë S)(x; y) D (Tx; S x y) :

(9)

The space Y is called the fiber of the skew product and the space X the base. Sometimes in the literature the word skew product has a more general meaning and refers to the structure (T Ë S)(x; y) D (Tx; S x y) (without any assumption of measure-preservation), where the action of the map on the fiber Y is determined or ‘driven’ by the map T : X ! X. Some common examples of skew products include: Random Dynamical Systems Suppose that X indexes a collection of mappings S x : Y ! Y. We may have a transformation T : X ! X which is a full shift. Then the sequence of mappings fS T n x g may be considered a (random) choice of a mapping Y ! Y from the set fS x : x 2 Xg. The projection onto Y of the orbits of (Tx; S x y) give the orbits of a point y 2 Y under a random composition of maps S T n x ı  ıS T x ıS x . More generally we could consider the choice of maps Sx that are composed to come from any ergodic dynamical system, (T; X; ) to model the effect of perturbations by a stationary ergodic ‘noise’ process. Group Extensions of Dynamical Systems Suppose Y is a group,  is a measure on Y invariant under a left group action, and S x y :D g(x)y is given by a groupvalued function g : X ! Y. In this setting g is often called a cocycle, since upon defining g (n) (x) by (T Ë S)(n) (x; y) D (T n x; g (n) (x)y) we have a cocycle relation, namely g (mCn) (x) D g (m) (T n x)g (n) (x). Group extensions arise often in modeling systems with symmetry [37]. Common examples are provided by a random composition of matrices from a group of matrices (or more generally from a set of matrices which may form a group or not).

Induced Transformations Since by the Poincaré Recurrence Theorem (see [113]) a measure-preserving transformation (T; X; ; B) on a probability space is recurrent, given any set B of positive

Ergodic Theory: Basic Examples and Constructions

measure, the return-time function n B (x) D inffn  1 : T n x 2 Bg

(10)

is finite  a.e. We may define the first-return map by TB x D T n B (x) x :

(11)

Then (after perhaps discarding as usual a set of measure 0) TB : B ! B is a measurable transformation which preserves the probability measure B D /(B). The system (B; B \ B; B ; TB ) is called an induced, first-return or derived transformation. If (T; X; ; B) is ergodic then (B; B \ B; B ; TB ) is ergodic, but the converse is not in general true. The construction of the transformation T B allows us to represent the forward orbit of points in B via a tower or skyscraper over B. For each n D 1; 2; : : : , let B n D fx 2 B : n B (x) D ng :

(12)

Then fB1 ; B2 ; : : : g form a partition of B, which we think of as the bottom floor or base of the tower. The next floor is made up of TB2 ; TB3 ; : : : , which form a partition of TB n B, and so on. All these sets are disjoint. A column is a part of the tower of the form B n [ TB n [    [ T n1 B n for some n D 1; 2; : : : . The action of T on the entire tower is pictured as mapping each x not at the top of its column straight up to the point Tx above it on the next level, and mapping each point on the top level to T n B x 2 B. An equivalent way to describe the transformation on the tower is to write for each n and j < n, T j B n as f(x; j) : x 2 B n g, and then the transformation F on the tower is ( (x; l C 1) if l < n B (x)  1 ; F(x; l) D n (x) B x; 0) if l D n B (x)  1 : (T If T preserves a measure , then F preserves   dl, where l is counting measure. Sometimes the process of inducing yields an induced map which is easier to analyze (perhaps it has stronger hyperbolicity properties) than the original system. Sometimes it is possible to ‘lift’ ergodic or statistical properties from an induced system to the original system, so the process of inducing plays an important role in the study of statistical properties of dynamical systems [77]. It is possible to generalize the tower construction and relax the condition that nB (x) is the first-return time function. We may take a measurable set B X of positive  measure and define for almost every point x 2 B a height or ceiling function R : B ! N and take a countable partition fX n g of B into the sets on which R is constant. We

define the tower as the set  :D f(x; l) : x 2 B; 0  l < R(x)g and the tower map F :  !  by ( (x; l C 1) if l < R(x)  1 ; F(x; l) D R(x) x; 0) if l D R(x)  1 : (T R In this setting, if B R(x) d < 1, we may define an F-in  dl, where dl variant probability measure on  as C(R;B) is counting measure and C(R, B) is the normalizing conR stant C(R; B) D (B) B R(x)d. This viewpoint is connected with the construction of systems by cutting and stacking—see Sect. “Cutting and Stacking”. Suspension Flows The tower construction has an analogue in which the height function R takes values in R rather than N. Such towers are commonly used to model dynamical systems with continuous time parameter. Let (T; X; ) be a measure-preserving system and R : X ! (0; 1) a measurable “ceiling” function on X. The set X R D f(x; t) : 0  R(x) < tg ;

(13)

with measure  given locally by the product of  on X with Lebesgue measure m on R, is a measure space in a natural way. If  is a finite measure and R is integrable with respect to  then  is a finite measure. We define an action of R on X R by letting each point x flow at unit speed up the vertical lines f(x; t) : 0  t < R(x)g under the graph of R until it hits the ceiling, then jump to Tx, and so on. More precisely, defining R n (x) D R(x) C    C R(T n x), 8 (x; s C t) ˆ ˆ ˆ ˆ ˆ if 0  s C t < R(x); ˆ ˆ ˆ ˆ ˆ ˆ <(Tx; s C t  R(x)) Ts (x; t) D if R(x)  s C t < R(x) C R(Tx) ˆ ˆ ˆ ˆ ::: ˆ ˆ ˆ ˆ ˆ (T n x; s C t  [R(x) C    C R(T n1 x)]) ˆ ˆ : if R n1 (x)  s C t < R n (x) : (14) Ergodicity of (T; X; ) implies the ergodicity of (Ts ; X R ; ). Cutting and Stacking Several of the most interesting examples in ergodic theory have been constructed by this method; in fact, because of Rokhlin’s Lemma (see Sect. “Rokhlin’s Lemma”) every ergodic measure-preserving transformation on a Lebesgue

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space is isomorphic to one constructed by cutting and stacking. For example, the von Neumann-Kakutani adding machine (or 2-odometer) (Sect. “Adding Machines”), the Chacon weakly mixing but not strongly mixing system (Sect. “Chacon System”), Ornstein’s mixing rank one examples (see e. g. p. 160 ff. in [80]), and many more. We construct a Lebesgue measure-preserving transformation T on an interval X (bounded or maybe unbounded) by defining it as a translation on each of a pairwise disjoint countable collection of subintervals. The construction proceeds by stages, at each stage defining T on an additional part of X, until eventually T is defined a.e. At each stage X is represented as a tower, which is defined to be a disjoint union of columns. A column is defined to be a finite disjoint union of intervals of equal length, which are numbered from 0, for the “floor”, to the last one, for the “roof”, and which we picture as lying each above the preceding-numbered interval. T is defined on each level of a column (i. e. each interval in the column) except the roof by mapping it by translation to the next higher interval in the column. At stage 0, we have just one column, consisting of all of X as the floor, and T is not defined anywhere. To pass from one stage to the next, the columns are cut and stacked. This means that each column is divided, by vertical cuts, into a disjoint union of subcolumns of equal height (but maybe not equal width), and then some of these subcolumns are stacked above others (of the same width) so as to form a new tower. This allows the definition of T to be extended to some parts of X that were previously tops of towers, since they now may have levels above them. (Sometimes columns of height 1 are thought of as forming a reservoir for “spacers” to be inserted between subcolumns that are being stacked.) If the measure of the union of the tops of the columns tends to 0, eventually T becomes defined a.e. This description in words can be made precise with cumbersome notation, but the process can also be given a neater graphical description, which we sketch in the next section. Adic Transformations A.M. Vershik has introduced a family of models, called adic or Bratteli–Vershik transformations, into ergodic theory and dynamical systems. One begins with a graph which is arranged in levels, finitely many vertices on each level, with connections only from each level to the adjacent ones. The space X consists of the set of all infinite paths in this graph; it is a compact metric space in a natural way. We are given an order on the set of edges into each vertex,

and then X is partially ordered as follows: x and y are comparable if they agree from some point on, in which case we say that x < y if at the last level n where they traverse different edges, the edge xn of x is smaller than the edge yn of y: A map T is defined by letting Tx be the smallest y that is larger than x; if there is one. In nice situations, T is a homeomorphism after defining it and its inverse on perhaps countably many maximal and minimal elements. Invariant measures can sometimes be defined by assigning weights to edges, which are then multiplied to define the measure of each cylinder set. This is a nice combinatorial way to present the cutting and stacking method of constructing m.p.t.’s, allows for more convenient analysis of questions such as orbit equivalence, and leads to the construction of many interesting examples, such as those based on the Pascal or Euler graphs [4,38,76]. Odometers and generalizations are natural examples of adic systems. Vershik showed that in fact every ergodic measure-preserving transformation on a Lebesgue space is isomorphic to a uniquely ergodic adic transformation. See [111]. Rokhlin’s Lemma The following result is the fundamental starting point for many constructions in ergodic theory, from representing arbitrary systems in terms of cutting and stacking or adic systems, to constructing useful partitions and symbolic codings of abstract systems, to connecting convergence theorems in abstract ergodic theory with those in harmonic analysis. It allows us to picture arbitrarily long stretches of the action of a measure-preserving transformation as a translation within the set of integers. In the ergodic nonatomic case the statement follows readily from the construction of induced transformations. Lemma 1 (Rokhlin’s Lemma) Let T : X ! X be a measure-preserving transformation on a probability space (X; B; ). Suppose that (X; B; ) is nonatomic and T : X ! X is ergodic, or, more generally, (T; X; B; ) is aperiodic: that is to say, the set fx 2 X : there is n 2 N such that T n x D xg of periodic points has measure 0. Then given n 2 N and > 0, there is a measurable set B X such that the sets B; TB; : : : ; T n1 B are pairwise k disjoint and ([n1 kD0 T B) > 1  . Inverse Limits Suppose that for each i D 1; 2; : : : we have a Lebesgue probability space (X i ; B i ;  i ) and a measure-preserving transformation Ti : X i ! X i . Suppose also that for each i  j there is a factor map  ji : (T j ; X j ; B j ;  j ) ! (Ti ; X i ; B i ;  i ; ), such that each  j j is the identity on X j

Ergodic Theory: Basic Examples and Constructions

and  ji  k j D  ki whenever k  j  i. Let 1 X i :  ji x j D x i for all j  ig : X D fx 2 ˘ iD1

(15)

For each j, let  j : X ! X j be the projection defined by j x D x j. Let B be the smallest  -algebra of subsets of X which 1 contains all the  1 j B j . Define  on each  j B j by ( 1 j B) D  j (B)

for all B 2 B j :

(16)

Because  ji  j D  i for all j  i, the  1 j B j are increasing, and so their union is an algebra. The set function  can, with some difficulty, be shown to be countably additive on this algebra: since we are dealing with Lebesgue spaces, by means of measure-theoretic isomorphisms it is possible to replace the entire situation by compact metric spaces and continuous maps, then use regularity of the measures involved—see p. 137 ff. in [88]. Thus by Carathéodory’s Theorem (see the article on  Measure Preserving Systems)  extends to all of B. Define T : X ! X by T(x j ) D (T j x j ). Then (T; X; B; ) is a measure-preserving system such that any system which has all the (T j ; X j ; B j ;  j ) as factors, also has (T; X; B; ) a factor. Natural Extension The natural extension is a way to produce an invertible system from a non-invertible system. The original system is a factor of its natural extension and its orbit structure and ergodic properties are captured by the natural extension, as will be seen from its construction. Let (T; X; B; ) be a measure-preserving transformation of a Lebesgue probability space. Define ˝ : D f(x0 ; x1 ; x2 ; : : : ) : x n D T(x nC1 ); x n 2 X; n D 0; 1; 2; : : : g with  : ˝ ! ˝ defined by ((x0 ; x1 ; x2 ; : : : )) D (T(x0 ); x0 ; x1 ; : : : ). The map  is invertible on ˝. Given the invariant measure  we define the invariant measure ˜ on ˝ by defining it first on for the natural extension  cylinder sets C(A0 ; A1 ; : : : ; A k ) by ˜ (C(A 0 ; A1 ; : : : ; A k )) D (T k (A0 )\T kC1 (A1 )   \T kCi (A i )\  \A k ) and then extending it to ˝ using Kolmogorov’s extension theorem. We think of (x0 ; x1 ; x2 ; : : : ) as being an inverse branch of x0 2 X under the mapping T : X ! X. The ˜ if maps ;  1 : ˝ ! ˝ are ergodic with respect to  (T; X; B; ) is ergodic [113]. If  : ˝ ! X is projection onto the first component i. e. (x0 ; : : : ; x n ; : : : ) D x0 then

 ı  n (x0 ; : : : ; x n ; : : : ) D T n (x0 ) for all x0 and thus the natural extension yields information about the orbits of X under T. The natural extension is an inverse limit. Let (X; B; ) be a Lebesgue probability space and T : X ! X a map such that T 1 B B and T 1 D . For each i D 1; 2; : : : let (Ti ; X i ; B i ;  i ) D (T; X; B; ), and  ji D ˆ X; ˆ Bˆ ; ) ˆ T ji for each j > i. Then the inverse limit (T; of this system is an invertible measure-preserving system which is the natural extension of (T; X; B; ). We have Tˆ 1 (x1 ; x2 ; : : : ) D (x2 ; x3 ; : : : ) :

(17)

ˆ X; ˆ Bˆ ; ) ˆ The original system (T; X; B; ) is a factor of (T; (using any  i as the factor map), and any factor mapping from an invertible system onto (T; X; B; ) consists ˆ X; ˆ Bˆ ; ) ˆ followed by projecof a factor mapping onto (T; tion onto the first coordinate. Joinings Given measure-preserving systems (T; X; B; ) and (S; Y; C ; ), a joining of the two systems is a T  S-invariant measure P on their product measurable space that projects to  and , respectively, under the projections of X Y to X and Y, respectively. That is, if 1 : X Y ! X is the projection onto the first component i. e. 1 (x; y) D x then P(11 (A)) D (A) for all A 2 B and similarly for 2 : X  Y ! Y. This concept is the ergodic-theoretic version of the notion in probability theory of a coupling. The product measure  is always a joining of the two systems. If product measure is the only joining of the two systems, then we say that they are disjoint and write X ? Y [40]. If D is any family of systems, we write D? for the family of all measure-preserving systems which are disjoint from every system in D. Extensive recent accounts of the use of joinings in ergodic theory are in [42,99,110]. Future Directions The basic examples and constructions presented here are idealized, and many of the underlying assumptions (such as uniform hyperbolicity) are seldom satisfied in applications, yet they have given important insights into the behavior of real-world physical systems. Recent developments have improved our understanding of the ergodic properties of non-uniformly and partially hyperbolic systems. The ergodic properties of deterministic systems will continue to be an active research area for the foreseeable future. The directions will include, among others: establishing statistical and ergodic properties under weakened dependence assumptions; the study of systems which dis-

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play ‘anomalous statistics’; the study of the stability and typicality of ergodic behavior and mixing in dynamical systems; the ergodic theory of infinite-dimensional systems; advances in number theory (see the sections on Szemerédi and Ramsey theory); research into models with non-singular rather than invariant measures; and infinitemeasure systems. Other chapters in this Encyclopedia discuss in more detail these and other topics. Bibliography Primary Literature 1. Aaronson J (1997) An introduction to infinite ergodic theory, Mathematical Surveys and Monographs, vol 50. American Mathematical Society, Providence. MR 1450400 (99d:28025) 2. Anosov DV (1967) Geodesic flows on closed riemannian manifolds of negative curvature. Trudy Mat Inst Steklov 90:209. MR 0224110 (36 #7157) 3. Arnol0 d VI (1963) Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi Mat Nauk 18(6 (114)):91–192. MR 0170705 (30 #943) 4. Bailey S, Keane M, Petersen K, Salama IA (2006) Ergodicity of the adic transformation on the euler graph. Math Proc Cambridge Philos Soc 141(2):231–238. MR 2265871 (2007m:37010) 5. Baladi V (2000) Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics, vol 16. World Scientific Publishing Co Inc, River Edge. MR 1793194 (2001k:37035) 6. Benedicks M, Carleson L (1985) On iterations of 1  ax(X; B; )2 on (1; 1). Ann Math (2) 122(1):1–25. MR 799250 (87c:58058) 7. Benedicks M, Carleson L (1991) The dynamics of the hénon map. Ann Math (2) 133(1):73–169. MR 1087346 (92d:58116) 8. Benedicks M, Young LS (1993) Sina˘ı-bowen-ruelle measures for certain hénon maps. Invent Math 112(3):541–576. MR 1218323 (94e:58074) 9. Bertrand-Mathis A (1986) Développement en base ; répartition modulo un de la suite (x (X; B; )n)n0 ; langages codés et -shift. Bull Soc Math France 114(3):271–323. MR 878240 (88e:11067) 10. Billingsley P (1978) Ergodic Theory and Information. Robert E. Krieger Publishing Co, Huntington, N.Y., reprint of the 1965 original. MR 524567 (80b:28017) 11. Bissinger BH (1944) A generalization of continued fractions. Bull Amer Math Soc 50:868–876. MR 0011338 (6,150h) 12. Blanchard F (1989) ˇ -expansions and symbolic dynamics. Theoret Comput Sci 65(2):131–141. MR 1020481 (90j:54039) 13. Blanchard F, Hansel G (1986) Systèmes codés. Theoret Comput Sci 44(1):17–49. MR 858689 (88m:68029) 14. Blanchard F, Hansel G (1986) Systèmes codés et limites de systèmes sofiques. C R Acad Sci Paris Sér I Math 303(10):475–477. MR 865864 (87m:94009) 15. Blanchard F, Hansel G (1991) Sofic constant-to-one extensions of subshifts of finite type. Proc Amer Math Soc 112(1):259–265. MR 1050016 (91m:54050) 16. Boshernitzan M, Galperin G, Krüger T, Troubetzkoy S (1998) Periodic billiard orbits are dense in rational polygons. Trans Amer Math Soc 350(9):3523–3535. MR 1458298 (98k:58179)

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94. Robbin JW (1971) A structural stability theorem. Ann Math (2) 94:447–493. MR 0287580 (44 #4783) 95. Robinson C (1975) Errata to: “structural stability of vector fields” (ann. of math. (2) 99:154–175 (1974)). Ann Math (2) 101:368. MR 0365630 (51 #1882) 96. Robinson C (1976) Structural stability of c(X; B; )1 diffeomorphisms. J Differential Equations 22(1):28–73. MR 0474411 (57 #14051) 97. Robinson RC (1973) c(X; B; )r structural stability implies Kupka-Smale. In: Dynamical systems, (Proc Sympos, Univ Bahia, Salvador, 1971). Academic Press, New York, pp 443– 449. MR 0334282 (48 #12601) 98. Rohlin VA (1952) On the fundamental ideas of measure theory. Amer Math Soc Translation 1952(71):55. MR 0047744 (13,924e) 99. Rudolph DJ (1990) Fundamentals of Measurable Dynamics: Ergodic theory on Lebesgue spaces. Oxford Science Publications. The Clarendon Press Oxford University Press, New York. MR 1086631 (92e:28006) 100. Rudolph DJ (1990) 2 and 3 invariant measures and entropy. Ergodic Theory Dynam Systems 10(2):395–406. MR 1062766 (91g:28026) 101. Ruelle D (1976) A measure associated with axiom-a attractors. Amer J Math 98(3):619–654. MR 0415683 (54 #3763) 102. Ruelle D (1978) Thermodynamic formalism: the mathematical structures of classical equilibrium statistical mechanics. Encyclopedia of Mathematics and its Applications, vol 5. AddisonWesley Publishing Co, Reading, MA, with a foreword by Giovanni Gallavotti and Gian-Carlo Rota. MR 511655 (80g:82017) 103. Sarig O (2002) Subexponential decay of correlations. Invent Math 150(3):629–653. MR 1946554 (2004e:37010) 104. Schweiger F (1995) Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford Science Publications. The Clarendon Press Oxford University Press, New York. MR 1419320 (97h:11083) 105. Sina˘ı JG (1968) Markov partitions and u-diffeomorphisms. Funkcional Anal i Priložen 2(1):64–89. MR 0233038 (38 #1361) 106. Sina˘ı JG (1970) Dynamical systems with elastic reflections. Ergodic properties of dispersing billiards. Uspehi Mat Nauk 25(2 (152)):141–192. MR 0274721 (43 #481) 107. Sina˘ı JG (1972) Gibbs measures in ergodic theory. Uspehi Mat Nauk 27(4(166)):21–64. MR 0399421 (53 #3265) 108. Smale S (1980) The mathematics of time. Essays on dynamical systems, economic processes, and related topics. Springer, New York. MR 607330 (83a:01068) 109. Tabachnikov S (2005) Geometry and billiards, Student Mathematical Library, vol 30. American Mathematical Society, Providence, RI. MR 2168892 (2006h:51001) 110. Thouvenot JP (1995) Some properties and applications of joinings in ergodic theory. In: Ergodic theory and its connections with harmonic analysis (Alexandria, 1993). London Math Soc Lecture Note Ser, vol 205. Cambridge Univ Press, Cambridge, pp 207–235. MR 1325699 (96d:28017) 111. Vershik AM, Livshits AN (1992) Adic models of ergodic transformations, spectral theory, substitutions, and related topics. In: Representation theory and dynamical systems. Adv Soviet Math, vol 9. Amer Math Soc, Providence, RI, pp 185–204. MR 1166202 (93i:46131) 112. Vorobets YB, Gal0 perin GA, Stëpin AM (1992) Periodic billiard trajectories in polygons: generation mechanisms. Uspekhi Mat Nauk 47(3(285)):9–74, 207, (Russian with Russian

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Books and Reviews Baladi V (2000) Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics, vol 16. World Scientific Publishing Co Inc, River Edge. MR 1793194 (2001k:37035) Billingsley P (1978) Ergodic Theory and Information. Robert E. Krieger Publishing Co, Huntington, N.Y., pp xiii+194, reprint of the 1965 original. MR 524567 (80b:28017) Billingsley P (1995) Probability and Measure. Wiley Series in Probability and Mathematical Statistics, 3rd edn. Wiley, New York, pp xiv+593, A Wiley-Interscience Publication. MR 1324786 (95k:60001) Bonatti C, Díaz LJ, Viana M (2005) Dynamics beyond uniform hyperbolicity: A global geometric and probabilistic perspective; Mathematical Physics, III. In: Encyclopaedia of Mathematical Sciences,vol. 102. Springer, Berlin, pp xviii+384. MR 2105774 (2005g:37001) Boyarsky A, Góra P (1997) Laws of chaos: Invariant measures and dynamical systems in one dimension. Probability and its Applications. Birkhäuser, Boston. MR 1461536 (99a:58102) Brin M, Stuck G (2002) Introduction to dynamical systems. Cambridge University Press, Cambridge. MR 1963683 (2003m:37001) Carleson L, Gamelin TW (1993) Complex dynamics. Universitext: Tracts in Mathematics. Springer, New York. MR 1230383 (94h:30033) Chernov N, Markarian R (2006) Chaotic billiards, Mathematical Surveys and Monographs, vol 127. American Mathematical Society, Providence, RI. MR 2229799 (2007f:37050) Collet P, Eckmann JP (1980) Iterated maps on the interval as dynamical systems, Progress in Physics, vol 1. Birkhäuser, Boston. MR 613981 (82j:58078) Cornfeld IP, Fomin SV, Sina˘ı YG (1982) Ergodic Theory, Grundlehren der mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol 245. Springer, New York, translated from the Russian by A. B. Sosinski˘ı. MR 832433 (87f:28019) Denker M, Grillenberger C, Sigmund K (1976) Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, vol 527. Springer, Berlin. MR 0457675 (56 #15879)

Friedman NA (1970) Introduction to Ergodic Theory. Van Nostrand Reinhold Mathematical Studies, No 29. Van Nostrand Reinhold Co, New York. MR 0435350 (55 #8310) Glasner E (2003) Ergodic Theory via Joinings, Mathematical Surveys and Monographs, vol 101. American Mathematical Society, Providence, RI. MR 1958753 (2004c:37011) Halmos PR (1960) Lectures on Ergodic Theory. Chelsea Publishing Co, New York. MR 0111817 (22 #2677) Hasselblatt B, Katok A (2003) A First Course in Dynamics: With a panorama of recent developments. Cambridge University Press, Cambridge. MR 1995704 (2004f:37001) Hopf E (1937) Ergodentheorie, 1st edn. Ergebnisse der Mathematik und ihrer Grenzgebiete; 5. Bd, 2, J. Springer, Berlin Jacobs K (1965) Einige neuere Ergebnisse der Ergodentheorie. Jber Deutsch Math-Verein 67(Abt 1):143–182. MR 0186789 (32 #4244) Keller G (1998) Equilibrium States in Ergodic Theory. In: London Mathematical Society Student Texts, vol 42. Cambridge University Press, Cambridge, pp x+178. MR 1618769 (99e:28022) Mañé R (1987) Ergodic theory and differentiable dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) (Results in Mathematics and Related Areas (3)), vol 8. Springer, Berlin, translated from the Portuguese by Silvio Levy. MR 889254 (88c:58040) Nadkarni MG (1998) Spectral Theory of Dynamical Systems. Birkhäuser Advanced Texts: Basler Lehrbücher. (Birkhäuser Advanced Texts: Basel Textbooks), Birkhäuser, Basel. MR 1719722 (2001d:37001) Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge, With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374 (96c:58055) Parry W, Pollicott M (1990) Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque (187-188):268. MR 1085356 (92f:58141) Ornstein DS, Rudolph DJ, Weiss B (1982) Equivalence of measure preserving transformations. Mem Amer Math Soc 37(262):xii+116. MR 653094 (85e:28026) Petersen K (1989) Ergodic Theory, Cambridge Studies in Advanced Mathematics, vol 2. Cambridge University Press, Cambridge. corrected reprint of the 1983 original. MR 1073173 (92c:28010) Royden HL (1988) Real Analysis, 3rd edn. Macmillan Publishing Company, New York. MR 1013117 (90g:00004) Rudolph DJ (1990) Fundamentals of Measurable Dynamics: Ergodic theory on Lebesgue spaces. Oxford Science Publications. The Clarendon Press Oxford University Press, New York. MR 1086631 (92e:28006) Schweiger F (1995) Ergodic Theory of Fibred Systems and Metric Number Theory. Oxford Science Publications. The Clarendon Press Oxford University Press, New York. MR 1419320 (97h:11083) Thouvenot JP (1995) Some properties and applications of joinings in ergodic theory. In: Ergodic theory and its connections with harmonic analysis (Alexandria, 1993). London Math Soc Lecture Note Ser, vol 205. Cambridge Univ Press, Cambridge, pp 207–235. MR 1325699 (96d:28017) Walters P (1982) An Introduction to Ergodic Theory, Graduate Texts in Mathematics, vol 79. Springer, New York. MR 648108 (84e:28017)

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Ergodic Theory: Fractal Geometry JÖRG SCHMELING Center for Mathematical Sciences, Lund University, Lund, Sweden Article Outline Glossary Definition of the Subject Introduction Preliminaries Brief Tour Through Some Examples Dimension Theory of Low-Dimensional Dynamical Systems – Young’s Dimension Formula Dimension Theory of Higher-Dimensional Dynamical Systems Hyperbolic Measures General Theory Endomorphisms Multifractal Analysis Future Directions Bibliography Glossary Dynamical system A (discrete time) dynamical system describes the time evolution of a point in phase space. More precisely a space X is given and the time evolution is given by a map T : X ! X. The main interest is to describe the asymptotic behavior of the trajectories (orbits) T n (x), i. e. the evolution of an initial point x 2 X under the iterates of the map T. More generally one is interested in obtaining information on the geometrically complicated invariant sets or measures which describe the asymptotic behavior. Fractal geometry Many objects of interest (invariant sets, invariant measures etc.) exhibit a complicated structure that is far from being smooth or regular. The aim of fractal geometry is to study those objects. One of the main tools is the fractal dimension theory that helps to extract important properties of geometrically “irregular” sets. Definition of the Subject The connection between fractal geometry and dynamical system theory is very diverse. There is no unified approach and many of the ideas arose from significant examples. Also the dynamical system theory has been shown to have a strong impact on classical fractal geometry. In this article there are first presented some examples showing nontrivial results coming from the application of dimension the-

ory. Some of these examples require a deeper knowledge of the theory of smooth dynamical systems then can be provided here. Nevertheless, the flavor of these examples can be understood. Then there is a brief overview of some of the most developed parts of the application of fractal geometry to dynamical system theory. Of course a rigorous and complete treatment of the theory cannot be given. The cautious reader may wish to check the original papers. Finally, there is an outlook over the most recent developments. This article is by no means meant to be complete. It is intended to give some of the ideas and results from this field. Introduction In this section some of the aspects of fractal geometry in dynamical systems are pointed out. Some notions that are used will be defined later on and I intend only to give a flavor of the applications. The nonfamiliar reader will find the definitions in the corresponding sections and can return to this section later. The geometry of many invariant sets or invariant measures of dynamical systems (including attractors, measures defining the statistical properties) look very complicated at all scales, and their geometry is impossible to describe using standard geometric tools. For some important classes of dynamical systems, these complicated structures are intensively studied using notions of dimension. In many cases it becomes possible to relate these notions of dimension to other fundamental dynamical characteristics, such as Lyapunov exponents, entropies, pressure, etc. On the other hand tools from dynamical systems, especially from ergodic theory and thermodynamic formalism, are extremely useful to explore the fractal properties of the objects in question. This includes dimensions of limit sets of geometric constructions (the standard Cantor set being the most famous example), which a priori, are not related to dynamical systems [46,100]. Many dimension formulas for asymptotic sets of dynamical systems are obtained by means of Bowen-type formulas, i. e. as roots of some functionals arising from thermodynamic formalism. The dimension of a set is a subtle characteristic which measures the geometric complexity of the set at arbitrarily fine scales. There are many notions of dimension, and most definitions involve a measurement of geometric complexity at scale " (which ignores the irregularities of the set at size less than ") and then considers the limiting measurement as " ! 0. A priori (and in general) these different notions can be different. An important result is the affirmative solution of the Eckmann–Ruelle conjecture by Barreira, Pesin and Schmeling [17], which says that for smooth nonuniformly hyperbolic systems, the pointwise

Ergodic Theory: Fractal Geometry

dimension is almost everywhere constant with respect to a hyperbolic measure. This result implies that many dimension characteristics for the measure coincide. The deep connection between dynamical systems and dimension theory seems to have been first discovered by Billingsley [21] through several problems in number theory. Another link between dynamical systems and dimension theory is through Pesin’s theory of dimension-like characteristics. This general theory is a unification of many notions of dimension along with many fundamental quantities in dynamical system such as entropies, pressure, etc. However, there are numerous examples of dynamical systems exhibiting pathological behavior with respect to fractal geometrical characteristics. In particular higher-dimensional systems seem to be as complicated as general objects considered in geometric measure theory. Therefore, a clean and unified theory is still not available. The study of characteristic notions like entropy, exponents or dimensions is an essential issue in the theory of dynamical systems. In many cases it helps to classify or to understand the dynamics. Most of these characteristics were introduced for different questions and concepts. For example, entropy was introduced to distinguish nonisomorphic systems and appeared to be a complete invariant for Bernoulli systems (Ornstein). Later, the thermodynamic formalism (see [107]) introduced new quantities like the pressure. Bowen [27] and also Ruelle discovered a remarkable connection between the thermodynamic formalism and the dimension theory for invariant sets. Since then many efforts were taken to find the relations between all these different quantities. It occurred that the dimension of invariant sets or measures carries lots of information about the system, combining its combinatorial complexity with its geometric complexity. Unfortunately it is extremely difficult to compute the dimension in general. The general flavor is that local divergence of orbits and global recurrence cause complicated global behavior (chaos). It is impossible to study the exact (infinite) trajectory of all orbits. One way out is to study the statistical properties of “typical” orbits by means of an invariant measure. Although the underlying system might be smooth the invariant measures may often be singular. Preliminaries Throughout the article the following situation is considered. Let M be a compact Riemannian manifold without boundary. On M is acting a dynamical system generated by a C 1C˛ diffeomorphism T : M ! M. The presence of a dynamical system provides several important additional tools and methods for the theory of fractal dimensions.

Also the theory of fractal dimensions allows one to draw deep conclusions about the dynamical system. The importance and relevance of the study of fractal dimension will be explained in later sections. In the next sections some of the most important tools in the fractal theory of dynamical systems are considered. The definitions given here are not necessarily the original definitions but rather the ones which are closer to contemporary use. More details can be found in [95]. Some Ergodic Theory Ergodic theory is a powerful method to analyze statistical properties of dynamical systems. All the following facts can be found in standard books on ergodic theory like [103,124]. The main idea in ergodic theory is to relate global quantities to observations along single orbits. Let us consider an invariant measure: ( f 1 A) D (A) for all measurable sets A. Such a measure “selects” typical trajectories. It is important to note that the properties vary with the invariant measures. Any such invariant measure can be decomposed into elementary parts (ergodic components). An invariant measure is called ergodic if for any invariant set A D T 1 A one has (A)(M n A) D 0 (with the agreement 0  1 D 0), i. e. from the measure-theoretic point of view there are no nontrivial invariant subsets. The importance of ergodic probability measures (i. e. (M) D 1) lies in the following theorem of Birkhoff Theorem 1 (Birkhoff) Let  be an ergodic probability measure and ' 2 L1 (). Then Z n1 1X k '(T x) D ' d   a.e : lim n!1 n M kD0

Hausdorff Dimension With Z R N and s  0 one defines

X diam(B i )s : mH (s; Z) D lim inf ı!0 fB i g

i

sup diam(B i ) < " and i

[

 Bi  Z :

i

Note that this limit exists. mH (s; Z) is called the s-dimensional outer Hausdorff measure of Z. It is immediate that there exists a unique value s , called the Hausdorff dimension of Z, at which mH (s; Z) jumps from 1 to 0. In general it is very hard to find optimal coverings and hence it is often impossible to compute the Hausdorff dimension of a set. Therefore a simpler notion – the lower

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and upper Box dimension – was introduced. The difference to the Hausdorff dimension is that the covering balls are assumed to have the same radius ". Since then the limit as " ! 0 does not have to exist one arrives at the notion of the upper and lower dimension. Dimension of a Measure Definition 1 Let Z R N and let  be a probability measure supported on Z. Define the Hausdorff dimension of the measure  by dimH () 

inf

K Z : (K)D1

dimH (K):

Pointwise Dimension Most invariant sets or measures are not strongly self-similar, i. e. the local geometry at arbitrarily fine scales might look different from point to point. Therefore, the notion of pointwise dimension with respect to a Borel probability measure is defined. Let  be a Borel probability measure. By B(x; ") the ball with center x and radius " is denoted. The pointwise dimension of the measure  at the point x is defined as d (x) :D lim

"!0

log (B(x; ")) log "

if the above limit exists. If d (x) D d, then for small " the measure of small balls scales as (B(x; "))  "d . Proposition 1 Suppose  is a probability measure supported on Z R N . Then d (x) D d for  – almost all x 2 Z implies dimH () D d. One should not take the existence of a local dimension (even for good measures) for granted. Later on it will be seen that the spectrum of the pointwise dimension (dimension spectrum) is a main object of study in classical multifractal analysis. The dimension of a measure or a set of measures is its geometric complexity. However, under the presence of a dynamical system one also wants to measure the dynamical (combinatorial) complexity of the system. This leads to the notion of entropy. Dimension-Like Characteristics and Topological Entropy Pesin’s theory of dimension-like characteristics provides a unified treatment of dimensions and important dynamical quantities like entropies and pressure. The topological entropy of a continuous map f with respect to a subset Z

in a metric space (X; ) (in particular X D M – a Riemannian manifold) can be defined as a dimension-like characteristic. For each n 2 N and " > 0, define the Bowen ball B n (x; ") D fy 2 X : (T i (x); T i (y))  " for 0  i  ng. Then let m h (Z; ˛; e; n) :D ( ) X [ ˛n i e : n i > n; B n i (x; e)  Z : lim inf n!1

i

i

This gives rise to an outer measure that jumps from 1 to 0 at some value ˛  . This threshold value ˛  is called the topological entropy of Z (at scale e). However, in many situations this value does not depend on e. The topological entropy is denoted by htop (TjZ). If Z is f – invariant and compact, this definition of topological entropy coincides with the usual definition of topological entropy [124]. ˚The entropy h of a measure  is defined as h D inf htop (TjZ) : (Z) D 1 . For ergodic measures this definition coincides with the Kolmogorov–Sinai entropy (see [95]). One has to note that in the definition of entropy metric (“round”) balls are substituted by Bowen balls and the metric diameter by the “depth” of the Bowen ball. Therefore, the relation between entropy and dimension is determined by the relation of “round” balls to “oval” dynamical Bowen balls. If one understands how metric balls can be efficiently used to cover dynamical balls one can use the dynamical and relatively easy relation to compute notion of entropy to determine the dimension. However, in higher dimensions this relation is by far nontrivial. A heuristic argument for comparing “round” balls with dynamical balls is given in Subsect. “The Kaplan–Yorke Conjecture”. The Pressure Functional A useful tool in the dimension analysis of dynamical systems is the pressure functional. It was originally defined by means of statistical physics (thermodynamic formalism) as the free energy (or pressure) of a potential ' (see for example [107]). However, in this article a dimension-like definition (see [95]) is more suitable. Again an outer measure using Bowen balls will be used. Let ' : M ! R be a continuous function and (  X exp  ˛n i m P (Z; ˛; "; n; ') D lim inf n!1

C

i

sup

n X

x2B n i (x;") kD0

) '(T x) : k

Ergodic Theory: Fractal Geometry

This defines an outer measure that jumps from 1 to 0 as ˛ increases.The threshold value ˛  is called the topological pressure of the potential ' denoted by P('). In many situations it does not depend on ". There is also a third way of defining the pressure in terms of a variational principle (see [124]):   Z ' d P(') D sup h C invariant

M

Brief Tour Through Some Examples Before describing the fractal theory of dynamical systems in more detail some ideas about its role are presented. The application of dimension theory has many different aspects. At this point some (but by far not all) important examples are considered that should give the reader some feeling about the importance and wide use of dimension theory in dynamical system theory. Dimension of Conformal Repellers: Ruelle’s Pressure Formula Computing or estimating a dimension via a pressure formula is a fundamental technique. Explicit properties of pressure help to analyze subtle characteristics. For example, the smooth dependence of the Hausdorff dimension of basic sets for Axiom- A surface diffeomorphisms on the derivative of the map follows from smoothness of pressure. Ruelle proved the following pressure formula for the Hausdorff dimension of a conformal repeller. A conformal repeller J is an invariant set T(J) D J D fx 2 M : f n x 2 V8n 2 N and some neighborhood V of J} such that for any x 2 J the differential D x T D a(x) Isox where a(x) is a scalar with ja(x)j > 1 and Iso x an isometry of the tangent space Tx M. Theorem 2 ([108]) Let T : J ! J be a conformal repeller, and consider the function t ! P(t log jD x Tj), where P denotes pressure. Then P(s log jD x Tj) D 0

()

s D dimH (J)

Iterated Function Systems In fractal geometry one of the most-studied objects are iterated function systems, where there are given n contractions F1 ;    ; Fn of Rd . The unique compact set J fulfilling S J D i F i (J) is called the attractor of the iterated function system (see [42]). One question is to evaluate its Hausdorff dimension. Often (for example under the open set condition) the attractor J can be represented as the repeller of a piecewise expanding map T : Rd ! Rd where the F i are

the inverse branches of the map T. In general it is by far not trivial to determine the dimension of J or even the dimension of a measure sitting on J. The following example explains some of those difficulties. Let 1/2 <  < 1 and consider the maps F i : [0; 1] ! [0; 1] given by F1 (x) D x and F2 (x) D x C (1  ). Then the images of F1 ; F2 have an essential overlap and J D [0; 1]. If one randomizes this construction in the way that one applies both maps each with probability 1/2 a probability measure is induced on J. This measure might be absolute continuous with respect to Lebesgue measure or not. Already Erdös realized that for some special values of  (for example for the inverse of the golden mean) the induced measure is singular. In a breakthrough paper B. Solomyak ([119]) proved that for a.e.  the induced measure is absolutely continuous. A main ingredient in the proof is a transversality condition in the parameter space: the images of arbitrary two random samples of the (infinite) applications of the maps F i have to cross with nonzero speed when the parameter  changes. This is a general mechanism which allows one to handle more general situations. Homoclinic Bifurcations for Dissipative Surface Diffeomorphisms Homoclinic tangencies and their bifurcations play a fundamental role in the theory of dynamical systems [87,88,89]. Systems with homoclinic tangencies have a complicated and subtle quasi-local behavior. Newhouse showed that homoclinic tangencies can persist under small perturbations, and that horseshoes may co-exist with infinitely many sinks in a neighborhood of the homoclinic orbit and hence the system is not hyperbolic (Newhouse phenomenon). Let T : M 2 ! M 2 be a smooth parameter family of surface diffeomorphisms that exhibits for  D 0 an invariant hyperbolic set 0 (horseshoe) and undergoes a homoclinic bifurcation. The Hausdorff dimension of the hyperbolic set 0 for T 0 determines whether hyperbolicity is the typical dynamical phenomenon near T 0 or not. If dimH 0 < 1, then hyperbolicity is the prevalent dynamical phenomenon near f 0 . This is not the case if dimH 0 > 1. More precisely, let N W  denote the set of nonwandering points of T in an open neighborhood of 0 after the homoclinic bifurcation. Let ` denote Lebesgue measure. Theorem 3 ([87,88]) If dimH 0 < 1, then lim

0 !0

`f 2 [0; 0 ] : N W  is hyperbolicg D1: 0

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The hyperbolicity co-exists with the Newhouse phenomena for a residual set of parameter values. Theorem 4 (Palis–Yoccoz) If dimH 0 > 1, then `f 2 [0; 0 ] : N W  is hyperbolicg < 1: 0 !0 0 lim

Some Applications to Number Theory Sometimes dimension problems in number theory can be transferred to dynamical systems and attacked using tools from dynamics. Example Diadic expansion of numbers: Consider a real number expanded in base 2, i. e. x D P1 n nD1 x n /2 . Let ( Xp D

) n1 1X x : lim xk D p : n!1 n kD0

1

xD

1

a1 C

1

a2 C a3 C

1 : a4 C : :

D [a1 ; a2 ; a3 ; a4 ;    ] ; and the approximants p n (x)/q n (x) D [a1 ; a2 ; : : : ; a n ] (i. e. the approximation given by the finite continued fraction after step n). The set of numbers which admit a faster approximation by rational numbers is defined as follows. For   2, let F ˇ ˇ

ˇ pˇ 1 F D x 2 [0; 1] : ˇˇ x  ˇˇ   q q  for infinitely many p/q :

Borel showed that `(X 1/2 ) D 1, where ` denotes Lebesgue measure. This result is an easy consequence of the Birkhoff ergodic theorem applied to the characteristic function of the digit 1 (which in this simple case is the Strong Law of Large Numbers for i.i.d. processes).

It is well known that this set has zero measure for each   2. Jarnik [58] computed the Hausdorff dimension of F and showed that dimH (F ) D 2/. However, nowadays there are methods from dynamical systems which not only allow unified proofs but also can handle more subtle subsets of the reals defined by some properties of their continued fraction expansion (see for example [3,105]).

One can ask how large is the set X p in general. Eggleston [39] discovered the following wonderful dimension formula, which Billingsley [21] interpreted in terms of dynamics and reproved using tools from ergodic theory.

Infinite Iterated Function Systems and Parabolic Systems

Theorem 5 ([39]) The Hausdorff dimension of X p is given by dimH (X p ) D

1   p log p  (1  p) log(1  p) : log 2

The underlying dynamical system is E2 (x) D 2x mod 1 and the dimension is the dimension of the Bernoulli p measure. Other cases included that by Rényi who proposed generalization of the base d expansion from integer base d to noninteger base ˇ. In this case the underlying dynamical system is the (in general non-Markovian) beta shift ˇ(x) D ˇx mod 1 studied in [110]. There are many investigations concerning the approximation of real numbers by rationals by using dynamical methods. The underlying dynamical system in this case is the Gauss map T(x) D (1/x) mod 1. This map is uniformly expanding, but has infinitely many branches. Example Continued fraction approximation of numbers Consider the continued fraction expansion of x 2 [0; 1], i. e.,

In the previous section a system with infinitely many branches appeared. This can be regarded as an iterated function system with infinitely many maps F i . This situation is quite general. If one considers a (one-dimensional) system with a parabolic (indifferent) fixed point, i. e. there is a fixed point where the derivative has absolute value equal to 1, one often uses an induced system. For this one chooses nearby the parabolic point a higher iterate of the map in order to achieve uniform expansion away from the parabolic point. This leads to infinitely many branches since the number of iterates has to be increased the closer the parabolic point is. The main difference to the finite iterated function system is that the setting is no longer compact and many properties of the pressure functional are lost. Mauldin and Urba´nski and others (see for example [3,75,76,78]) developed a thermodynamic formalism adapted to the pressure functional for infinite iterated function systems. Besides noncompactness one of the main problems is the loss of analyticity (phase transitions) and convexity of the pressure functional for infinite iterated function systems.

Ergodic Theory: Fractal Geometry

Complex Dynamics Let T : C ! C be a polynomial and J its Julia set (repeller of this system). If this set is hyperbolic, i. e. the derivative at each point has absolute value larger than 1 the study of the dimension can be related to the study of a finite iterated function system. However, in the presence of a parabolic fixed point this leads to an infinite iterated function system. If one considers the coefficients of the polynomial as parameters one often sees a qualitative change in the asymptotic behavior. For example, the classical Mandelbrodt set for polynomials z2 C c is the locus of values c 2 C for which the orbit of the origin stays bounded. This set is well known to be fractal. However, its complete description is still not available. Embedology and Computational Aspects of Dimension Tools from dynamical systems are becoming increasingly important to study the time evolution of deterministic systems in engineering and the physical and biological sciences. One of the main ideas is to model a “real world” system by a smooth dynamical system which possesses a strange attractor with a natural ergodic invariant measure. When studying a complicated real world system, one can measure only a very small number of variables. The challenge is to reconstruct the underlying attractor from the time measurement of a scaler quantity. An idealized measurement is considered as a function h : M n ! R. The main tool researchers currently use to reconstruct the model system is called attractor reconstruction (see papers and references in [86]). This method is based on embedding with time delays (see the influential paper [33], where the authors attribute the idea of delay coordinates to Ruelle), where one attempts to reconstruct the attractor for the model using a single long trajectory. Then one considers the points in R pC1 defined by (x k ; x kC ; x kC2 ; : : : ; x kCp ). Takens [122] showed that for a smooth T : M n ! M n and for typical smooth h, the mapping ' : M n ! R2nC1 defined by x ! (h(x); h( f  (x));    ; h( f 2n (x)) is an embedding. Since the box dimension of the attractor  may be much less than the dimension of the ambient manifold n, an interesting mathematical question is whether there exists p < 2n C 1 such that the mapping on the attractor ' :  ! R p defined by x ! (h(x); h( f (x)); : : : ; h( f p (x)) is 1  1? It is known that for a typical smooth h the mapping ' is 1  1 for p > 2 dimB () [29]. Denjoy Systems This section will give some ideas indicating the principle difficulties that arise in systems with low complexity.

Contrary to hyperbolic systems (each vector in the tangent space is either contracted or expanded) finer mechanisms determine the local behavior of the scaling of balls. While in hyperbolic systems the dynamical scaling of small balls is exponential in a low-complexity system this scaling is subexponential and hence the linearization error is of the same magnitude. Up to now there is no general dimension theory for low complexity systems. A specific example presented here is considered in [67]. Poincaré showed that to each orientation preserving homeomorphism of the circle S 1 D R/Z is associated a unique real parameter ˛ 2 [0; 1), called the rotation number, so that the ordered orbit structure of T is the same as that of the rigid rotation R˛ , where R˛ (t) D (t C ˛) mod 1, provided that ˛ is irrational. Half a century later, Denjoy [35] constructed examples of C1 diffeomorphisms that are not conjugate (via a homeomorphism) to rotations. This was improved later on by Herman [55]. In these examples, the minimal set of T is necessarily a Cantor set ˝. The arithmetic properties of the rotation number have a strong effect on the properties of T. One area that has been well understood is the relation between the differentiability of T, the differentiability of the conjugation and the arithmetic properties of the rotation number. (See, for example, Herman [55]) Without stating any precise theorem, the results differ sharply for Diophantine and for Liouville rotation numbers (definition follows). Roughly speaking the conjugating map is always regular for Diophantine rotation numbers while it might be not smooth at all for Liuoville rotation numbers. Definition 2 An irrational number ˛ is of Diophantine class  D (˛) 2 RC if kq˛k <

1 q

has infinitely many solutions in integers q for  <  and at most finitely many for  >  where k  k denotes the distance to the nearest integer. In [67] the effect of the rotation number on the dimension of ˝ is studied. There the main result is Theorem 6 Assume that 0 < ı < 1 and that ˛ 2 (0; 1) is of Diophantine class  2 (0; 1). Then an orientation preserving C 1Cı diffeomorphism of the circle with rotation number ˛ and minimal set ˝˛ı satisfies dimH ˝˛ı 

ı : 

Furthermore, these results are sharp, i. e. the standard Denjoy examples attain the minimum.

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Return Times and Dimension Recently an interesting connection between the pointwise dimensions, multifractal analysis, and recurrence behavior of trajectories was discovered [1,11,23]. Roughly speaking, given an ergodic probability measure  the return time asymptotics (as the neighborhood of the point shrinks) of -a.e point is determined by the pointwise dimension of  at this point. The deeper understanding of this relation would help to get a unified approach to dimensions, exponents, entropies, recurrence times and correlation decay. Dimension Theory of Low-Dimensional Dynamical Systems – Young’s Dimension Formula In this section a remarkable extension of Ruelle’s dimension formula by Young [128] for the dimension of a measure is discussed. Theorem 7 Let T : M 2 ! M 2 be a C2 surface diffeomorphism and let  be an ergodic measure. Then   1 1 dimH () D h ( f )  ; 1 2 where 1  2 are the two Lyapunov exponents for .

2) Number theoretic properties of some scaling rates [104, 106] enter into dimension calculations in ways they do not in low dimensions (see Subsect. “Iterated Function Systems”). 3) The dimension theory of sets is often reduced to the theory of invariant measures. However, there is no invariant measure of full dimension in general and measure-theoretic considerations do not apply [79]. 4) The stable and unstable foliations for higher dimensional systems are typically not C1 [51,109]. Hence, to split the system into an expanding and a contracting part is far more subtle. Dimension Theory of Higher-Dimensional Dynamical Systems Here an example of a hyperbolic attractor in dimension 3 is considered to highlight some of the difficulties. Let 4 denote the unit disc in R2 . Let f : S 1  4 ! S 1  4 be of the form T(t; x; y) D ('(t);

Theorem 8 Let  be a basic set for a C2 Axiom-A surface diffeomorphism T : M 2 ! M 2 . Then dimH () D s1 C s2 , where s1 and s2 satisfy P(s log kDTx jE xu k) D 0 P(s log kDTx jE xs k) D 0 : where Es and Eu are the stable and unstable directions, respectively. Some Remarks on Dimension Theory for Low-Dimensional versus High-Dimensional Dynamical Systems Unlike lower dimensions (one, two, or conformal repellers), for higher-dimensional dynamical systems there are no general dimension formulas (besides the Ledrappier–Young formula), and in general dimension theory is much more difficult. This is due to several problems: 1) The geometry of the Bowen balls differs in a substantial way from round balls.

(t; x);

2

(t; y)) ;

with 0 < max

S 1 4

In [79], Manning and McCluskey prove the following dimension formula for a basic set (horseshoe) of an Axiom-A surface diffeomorphism which is a set-version of Young’s formula.

1

@ @x

1

(t; x) < min

S 1 4

@ @y

2

(t; y) <  < 1 :

The limit set \ T n (S 1  4)  :D n2N

is called the attractor or the solenoid. It is an example of a structurally stable basic set and is one of the fundamental examples of a uniformly hyperbolic attractor. The following result can be proved. Theorem 9 ([24,52]) For all t, the thermodynamic pressure ˇ ˇ  ˇ@ 2 ˇ s ˇ (t; y)ˇˇ D 0 : P dimH  t log ˇ @y In particular, the stable dimension is independent of the stable section. In this particular case the invariant axes for strong and weak contraction split the system smoothly and the difficulty is to show that the strong contraction is dominated by the weaker. In particular one has to ensure that effects as described in Subsect. “Iterated Function Systems” do not appear. In the general situation this is not the case and one lacks a similar theorem in the general situation. In particular, the unstable foliation is not better than Hölder and does not provide a “nice” coordinate.

Ergodic Theory: Fractal Geometry

Hyperbolic Measures

General Theory

Given x 2 M and a vector v 2 Tx M the Lyapunov exponent is defined as

In this section the dimension theory of higher-dimensional dynamical systems is investigated. Most developed is this theory for invariant measures. There is an important connection between Lyapunov exponents and the measure theoretic entropy and dimensions of measures that will be presented here. Let  be an ergodic invariant measure. The Oseledec and Pesin Theory guarantee that local stable manifolds exist at -a.e. point. As for the Kaplan–Yorke formula the idea is to consider the contributions to the entropy and to the dimension in the directions of i . Historically the first connections between exponents and entropy were discovered by Margulis and Ruelle. They proved that

(x; v) D lim

n!1

1 log kD x T n vk : n

provided this limit exists. For fixed x, the numbers (x; ) attain only finitely many values. By ergodicity they are a.e. constant, and the corresponding values are denoted by 1       p ; where p D dim M. Denote by s (s stands for stable) the largest index such that s < 0. Definition 3 The invariant ergodic measure  is said to be hyperbolic if  i ¤ 0 for every i D 1, : : :, p. The Kaplan–Yorke Conjecture In this section a heuristic argument for the dimension of an invariant ergodic measure will be given. This argument uses a specific cover of a dynamical ball by “round” balls. These ideas are essentially developed by Kaplan and Yorke [61] for the estimation of the dimension of an invariant set. Their estimates always provide an upper bound for the dimension. Kaplan and Yorke conjectured that in typical situations these estimates also provide a lower bound for the dimension of an attractor. Ledrappier and Young [72,73] showed that in case this holds for an invariant measure, then this measure has to be very special: an SBR-measure (after Sinai, Ruelle, and Bowen) (an SRBmeasure is a measure that describes the asymptotic behavior for a set of initial points of positive Lebesgue measure and has absolutely continuous conditional measures on unstable manifolds). Consider a small ball B in the phase space. The image T n B is almost an ellipsoid with axes of length e1 n ;    ; e p n : For 1  i  s, cover T n B by balls of radius e i n . Then approximately exp[ p n] exp[ iC1 n] ::: : exp[ i n] exp[ i n] balls are needed for covering. The dimension can be estimated from above by P j>i  j (1) C (p  i) :D dimLi : dimB   j i j This is the Kaplan–Yorke formula.

h ( f )  

s X

i

iD1

for a C1 diffeomorphism T. Pesin [91] showed that this inequality is actually an equality if the measure  is essentially the Riemannian volume on unstable manifolds. Ledrappier and Young [72] showed that this is indeed a necessary condition. They also provided an exact formula: Theorem 10 (Ledrappier–Young [72,73]) With d 0 D 0 for a C2 diffeomorphism holds h ( f ) D 

s X

   i d i  d i1

iD1

where di are the dimensions of the (conditional) measure on the ith unstable leaves. The proof of this theorem is difficult and uses the theory of nonuniform hyperbolic systems (Pesin theory). In dimension 1 and 2 the above theorem resembles Ruelle’s and Young’s theorems. The reader should note that the above theorem includes also the existence of the pointwise dimension along the stable and unstable direction. Here the question arises whether this implies the existence of the pointwise dimension itself. The Existence of the Pointwise Dimension for Hyperbolic Measure – the Eckmann–Ruelle Conjecture In [17], Barreira, Pesin, and Schmeling prove that every hyperbolic measure has an almost local product structure, i. e., the measure of a small ball can be approximated by the product of the stable conditional measure of the stable component and the unstable conditional measure of

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the unstable component, up to small exponential error. This was used to prove the existence of the pointwise dimension of every hyperbolic measure almost everywhere. Moreover, the pointwise dimension is the sum of the contributions from its stable and unstable part. This implies that most dimension-type characteristics of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide, and provides a rigorous mathematical justification of the concept of fractal dimension for hyperbolic measures. The existence of the pointwise dimension for hyperbolic measures had been conjectured long before by Eckmann and Ruelle. The hypotheses of this theorem are sharp. Ledrappier and Misiurewicz [70] constructed an example of a nonhyperbolic measure for which the pointwise dimension is not constant a.e. In [101], Pesin and Weiss present an example of a Hölder homeomorphism with Hölder constant arbitrarily close to one, where the pointwise dimension for the unique measure of maximal entropy does not exist a.e. There is also a one-dimensional example by Cutler [34].

Endomorphisms The previous section indicates that the dimensional properties of hyperbolic measures under invariant conditions for a diffeomorphism are understood. However, partial differential equations often generate only semi-flows and the corresponding dynamical system is noninvertible. Also, Poincaré sections sometimes introduce singularities. For such dynamical systems the theory of diffeomorphisms does not apply. However, the next theorem allows under some conditions application of this theory. It essentially rules out similar situations as considered in Subsect. “Iterated Function Systems”.

The Dynamical Characteristic View of Multifractal Analysis The aim of multifractal analysis is an attempt to understand the fine structure of the level sets of the fundamental asymptotic quantities in ergodic theory (e. g., Birkhoff averages, local entropy, Lyapunov exponents). For ergodic measures these quantities are a.e. constant, however may depend on the underlying ergodic measure. Important elements of multifractal analysis entail determining the range of values these characteristics attain, an analysis of the dimension of the level sets, and an understanding of the sets where the limits do not exist. A general concept of multifractal analysis was proposed by Barreira, Schmeling and Pesin [16]. An important field of applications of multifractal analysis is to describe sets of real numbers that have constraints on their digits or continued fraction expansion. General Multifractal Formalism In this section the abstract theory of multifractal analysis is described. Let X; Y be two measurable spaces and g : X n B ! Y be any measurable function where B is a measurable (possibly empty) subset of X (in the standard applications Y D R or Y D C). The associated multifractal decomposition of X is defined as [ g K˛ X DB[ ˛2Y

where g

K˛ :D fx 2 X : g(x) D ˛g For a given set function G : 2 X ! R the multifractal spectrum is defined by g

F (˛) :D G(K˛ ) :

Definition 4 A system (possibly with singularities) is almost surely invertible if it is invertible on a set of full measure. This implies that a full measure set of points has unique forward and backward trajectories.

At this point some classical and concrete examples of this general framework are considered.

Theorem 11 (Schmeling–Troubetzkoy [114]) A two-dimensional system with singularities is almost surely invertible (w.r.t. an SRB–measure) if and only if Young’s formula holds.

Let  be an ergodic invariant measure for T : X ! X. If one sets

The Entropy Spectrum

g E (x) :D h (x) 2 Y D R and

Multifractal Analysis A group of physicists [50] suggested the idea of a multifractal analysis.

G E (Z) D htop (TjZ) the associated multifractal spectrum fE (˛) is called the entropy spectrum.

Ergodic Theory: Fractal Geometry

The Dimension Spectrum This is the classical multifractal spectrum. Let  be an invariant ergodic measure on a complete separable metric space X. Set g D (x) :D d (x) 2 Y D R and G D (Z) D dimH Z : The associated multifractal spectrum fD (˛) is called the dimension spectrum. The Lyapunov Spectrum Let n1 1X g L (x) :D (x) D lim (T k x) 2 Y D R n!1 n kD0

where

(x) D log jD x Tj and

G D (Z) D dimH Z

or G E (Z) D htop (jZ) :

The associated multifractal spectra f LD (˛) and f LE (˛) are called Lyapunov spectra. It was observed by H. Weiss [126] and Barreira, Pesin and Schmeling [16] that for conformal repellers   log 2 f LD (˛) D fDh (2) ˛ where the dimension spectrum on the right-hand side is with respect to the measure of maximal entropy, and f LE (˛) D fED (dimH   ˛)

(3)

where the entropy spectrum on the right-hand side is with respect to the measure of maximal dimension. The following list summarizes the state of the art for the dynamical characteristic multifractal analysis of dynamical systems. The precise statements can be found in the original papers.  [102,126] For conformal repellers and Axiom-A surface diffeomorphisms, a complete multifractal analysis exists for the Lyapunov exponent.  [16,102,126] For mixing a subshift of finite type, a complete multifractal analysis exists for the Birkhoff average for a Hölder continuous potential and for the local entropy for a Gibbs measure with Hölder continuous potential.

 [12,97] There is a complete multifractal analysis for hyperbolic flows.  [123] There is a generalization of the multifractal analysis on subshifts with specification and continuous potentials.  [13,19] There is an analysis of “mixed” spectra like the dimension spectrum of local entropies and also an analysis of joint level sets determined by more than one (measurable) function.  [105] For the Gauss map (and a class of nonuniformly hyperbolic maps) a complete multifractal analysis exists for the Lyapunov exponent.  [57] A general approach to multifractal analysis for repellers with countably many branches is developed. It shows in contrary to finitely many branches features of nonanalytic behavior. In the first three statements the multifractal spectra are analytic concave functions that can be computed by means of the Legendre transform of the pressure functional with respect to a suitable chosen family of potentials. In the remaining items this is no longer the case. Analyticity and convexity properties of the pressure functional are lost. However, the authors succeeded to provide a satisfactory theory in these cases. Multifractal Analysis and Large Deviation Theory There are deep connections between large deviation theory and multifractal analysis. The variational formula for pressure is an important tool in the analysis, and can be viewed (and proven) as a large deviation result [41]. Some authors use large deviation theory as a tool to effect multifractal analysis. Future Directions The dimension theory is fast developing and of great importance in the theory of dynamical systems. In the most ideal situations (low dimensions and hyperbolicity) a generally far reaching and powerful theory has been developed. It uses ideas from statistical physics, fractal geometry, probability theory and other fields. Unfortunately, the richness of this theory does not carry over to higher-dimensional systems. However, recent developments have shown that it is possible to obtain a general theory for the dimension of measures. Part of this theory is the development of the analytic tools of nonuniformly hyperbolic systems. Therefore, the dimension theory of dynamical systems is far from complete. In particular, it is usually difficult to apply the general theory to concrete examples, for instance

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if one really wants to compute the dimension. The general theory does not provide a way to compute the dimension but gives rather connections to other characteristics. Moreover, in the presence of neutral directions (zero Lyapunov exponents) one encounters all the difficulties arising in low-complexity systems. Another important open problem is to understand the dimension theory of invariant sets in higher-dimensional spaces. One way would be to relate the dimension of sets to the dimension of measures. Such a connection is not clear. The reason is that most systems do not exhibit a measure whose dimension coincides with the dimension of its support (invariant set). But there are some reasons to conjecture that any compact invariant set of an expanding map in any dimension carries a measure of maximal dimension (see [48,63]). If this conjecture is true one obtains an invariant measure whose unstable dimension coincides with the unstable dimension of the invariant set. There is also a measure of maximal stable dimension. Combining these two measures one could establish an analogous theory for invariant sets as for invariant measures. Last but not least one has to mention the impact of the dimension theory of dynamical systems on other fields. This new point of view makes in many cases the posed problems more tractable. This is illustrated in examples from number theory, geometric limit constructions and others. The applications of the dimension theory of dynamical systems to other questions seem to be unlimited.

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79. McCluskey H, Manning A (1983) Hausdorff Dimension For Horseshoes. Erg Th Dyn Syst 3:251–260 80. Moran P (1946) Additive Functions Of Intervals and Hausdorff Dimension. Proceedings Of Cambridge Philosophical Society 42:15–23 81. Moreira C, Yoccoz J (2001) Stable Intersections of Cantor Sets with Large Hausdorff Dimension. Ann of Math (2) 154(1):45– 96 82. Mãné R (1981) On the Dimension of Compact Invariant Sets for Certain Nonlinear Maps. Lecture Notes in Mathematics, vol 898. Springer 83. Mãné R (1990) The Hausdorff Dimension of Horseshoes of Surfaces. Bol Soc Bras Math 20:1–24 84. Neunhäuserer J (1999) An analysis of dimensional theoretical properties of some affine dynamical systems. Thesis. Free University Berlin, Berlin 85. Oseledets V (1968) A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems. Trans Moscow Math Soc 19:197–221 86. Ott E, Sauer T, Yorke J (1994) Part I Background. In: Coping with chaos. Wiley Ser Nonlinear Sci, Wiley, New York, pp 1–62 87. Palis J, Takens F (1987) Hyperbolicity And The Creation Of Homoclinic Orbits. Ann Math 125:337–374 88. Palis J, Takens F (1993) Hyperbolicity And Sensitive Chaotic Dynamics At Homoclinic Bifurcations. Cambridge U Press, Cambridge 89. Palis J, Takens F (1994) Homoclinic Tangencies For Hyperbolic Sets Of Large Hausdorff Dimension. Acta Math 172:91–136 90. Palis J, Viana M (1988) On the continuity of Hausdorff dimension and limit capacity for horseshoes. Lecture Notes in Math, vol 1331. Springer 91. Pesin Y (1977) Characteristic exponents and smooth ergodic theory. Russian Math Surveys 32(4):55–114 92. Pesin Y (1992) Dynamical systems with generalized hyperbolic attractors: hyperbolic, ergodic and topological properties. Erg Th Dyn Syst 12:123–151 93. Pesin Y (1993) On Rigorous Mathematical Definition of Correlation Dimension and Generalized Spectrum for Dimensions. J Statist Phys 71(3–4):529–547 94. Pesin Y (1997) Dimension Theory In Dynamical Systems: Rigorous Results And Applications. Cambridge U Press, Cambridge 95. Pesin Y (1997) Dimension theory in dynamical systems: contemporary views and applications. In: Chicago Lectures in Mathematics. Chicago University Press, Chicago 96. Pesin Y, Pitskel’ B (1984) Topological pressure and the variational principle for noncompact sets. Funct Anal Appl 18:307– 318 97. Pesin Y, Sadovskaya V (2001) Multifractal analysis of conformal axiom A flows. Comm Math Phys 216(2):277–312 98. Pesin Y, Tempelman A (1995) Correlation Dimension of Measures Invariant Under Group Actions. Random Comput Dyn 3(3):137–156 99. Pesin Y, Weiss H (1994) On the Dimension of Deterministic and Random Cantor-like Sets. Math Res Lett 1:519–529 100. Pesin Y, Weiss H (1996) On The Dimension Of Deterministic and Random Cantor-like Sets, Symbolic Dynamics, And The Eckmann-Ruelle Conjecture. Comm Math Phys 182:105–153

101. Pesin Y, Weiss H (1997) A Multifractal Analysis of Equilibrium Measures For Conformal Expanding Maps and Moran-like Geometric Constructions. J Stat Phys 86:233–275 102. Pesin Y, Weiss H (1997) The Multifractal Analysis of Gibbs Measures: Motivation. Mathematical Foundation and Examples. Chaos 7:89–106 103. Petersen K (1983) Ergodic theory. Cambridge studies in advanced mathematics 2. Cambridge Univ Press, Cambridge 104. Pollicott M, Weiss H (1994) The Dimensions Of Some Self Affine Limit Sets In The Plane And Hyperbolic Sets. J Stat Phys 77:841–866 105. Pollicott M, Weiss H (1999) Multifractal Analysis for the Continued Fraction and Manneville-Pomeau Transformations and Applications to Diophantine Approximation. Comm Math Phys 207(1):145–171 ´ 106. Przytycki F, Urbanski M (1989) On Hausdorff Dimension of Some Fractal Sets. Studia Math 93:155–167 107. Ruelle D (1978) Thermodynamic Formalism. Addison-Wesley 108. Ruelle D (1982) Repellers For Real Analytic Maps. Erg Th Dyn Syst 2:99–107 109. Schmeling J (1994) Hölder Continuity of the Holonomy Maps for Hyperbolic basic Sets II. Math Nachr 170:211–225 110. Schmeling J (1997) Symbolic Dynamics for Beta-shifts and Self-Normal Numbers. Erg Th Dyn Syst 17:675–694 111. Schmeling J (1998) A dimension formula for endomorphisms – the Belykh family. Erg Th Dyn Syst 18:1283–1309 112. Schmeling J (1999) On the Completeness of Multifractal Spectra. Erg Th Dyn Syst 19:1–22 113. Schmeling J (2001) Entropy Preservation under Markov Codings. J Stat Phys 104(3–4):799–815 114. Schmeling J, Troubetzkoy S (1998) Dimension and invertibility of hyperbolic endomorphisms with singularities. Erg Th Dyn Syst 18:1257–1282 115. Schmeling J, Weiss H (2001) Dimension theory and dynamics. AMS Proceedings of Symposia in Pure Mathematics series 69:429–488 116. Series C (1985) The Modular Surface and Continued Fractions. J Lond Math Soc 31:69–80 117. Simon K (1997) The Hausdorff dimension of the general Smale–Williams solenoidwith different contraction coefficients. Proc Am Math Soc 125:1221–1228 118. Simpelaere D (1994) Dimension Spectrum of Axiom-A Diffeomorphisms, II. Gibbs Measures. J Stat Phys 76:1359–1375 P 119. Solomyak B (1995) On the random series ˙n (an Erdös problem). Ann of Math (2) 142(3):611–625 120. Solomyak B (2004) Notes on Bernoulli convolutions. In: Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Proc Sympos Pure Math, 72, Part 1. Amer Math Soc, Providence, pp 207–230 121. Stratmann B (1995) Fractal Dimensions for Jarnik Limit Sets of Geometrically Finite Kleinian Groups; The Semi-Classical Approach. Ark Mat 33:385–403 122. Takens F (1981) Detecting Strange Attractors in Turbulence. Lecture Notes in Mathematics, vol 898. Springer 123. Takens F, Verbitzki E (1999) Multifractal analysis of local entropies for expansive homeomorphisms with specification. Comm Math Phys 203:593–612 124. Walters P (1982) Introduction to Ergodic Theory. Springer 125. Weiss H (1992) Some Variational Formulas for Hausdorff Di-

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Books and Reviews Bowen R (1975) Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, vol 470. Springer Eckmann JP, Ruelle D (1985) Ergodic Theory Of Chaos And Strange Attractors. Rev Mod Phys 57:617–656 Federer H (1969) Geometric measure theory. Springer Hasselblatt B, Katok A (2002) Handbook of Dynamical Systems, vol 1, Survey 1. Principal Structures. Elsevier Katok A (1980) Lyapunov exponents, entropy and periodic or-

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Ergodic Theory on Homogeneous Spaces and Metric Number Theory DMITRY KLEINBOCK Department of Mathematics, Brandeis University, Waltham, USA Article Outline Glossary Definition of the Subject Introduction Basic Facts Connection with Dynamics on the Space of Lattices Diophantine Approximation with Dependent Quantities: The Set-Up Further Results Future Directions Acknowledgment Bibliography Glossary Diophantine approximation Diophantine approximation refers to approximation of real numbers by rational numbers, or more generally, finding integer points at which some (possibly vector-valued) functions attain values close to integers. Metric number theory Metric number theory (or, specifically, metric Diophantine approximation) refers to the study of sets of real numbers or vectors with prescribed Diophantine approximation properties. Homogeneous spaces A homogeneous space G/ of a group G by its subgroup  is the space of cosets fg g. When G is a Lie group and  is a discrete subgroup, the space G/ is a smooth manifold and locally looks like G itself. Lattice; unimodular lattice A lattice in a Lie group is a discrete subgroup of finite covolume; unimodular stands for covolume equal to 1. Ergodic theory The study of statistical properties of orbits in abstract models of dynamical systems. Hausdorff dimension A nonnegative number attached to a metric space and extending the notion of topological dimension of “sufficiently regular” sets, such as smooth submanifolds of real Euclidean spaces. Definition of the Subject The theory of Diophantine approximation, named after Diophantus of Alexandria, in its simplest set-up deals with the approximation of real numbers by rational num-

bers. Various higher-dimensional generalizations involve studying values of linear or polynomial maps at integer points. Often a certain “approximation property” is fixed, and one wants to characterize the set of numbers (vectors, matrices) which share this property, by means of certain measures (Lebesgue, or Hausdorff, or some other interesting measures). This is usually referred to as metric Diophantine approximation. The starting point for the theory is an elementary fact that Q, the set of rational numbers, is dense in R, the reals. In other words, every real number can be approximated by rationals: for any y 2 R and any " > 0 there exists p/q 2 Q with jy  p/qj < " :

(1)

To answer questions like “how well can various real numbers be approximated by rational numbers? i. e., how small can " in (1) be chosen for varying p/q 2 Q?”, a natural approach has been to compare the accuracy of the approximation of y by p/q to the “complexity” of the latter, which can be measured by the size of its denominator q in its reduced form. This seemingly simple set-up has led to introducing many important Diophantine approximation properties of numbers/vectors/matrices, which show up in various fields of mathematics and physics, such as differential equations, KAM theory, transcendental number theory. Introduction As the first example of refining the statement about the density of Q in R, consider a theorem by Kronecker stating that for any y 2 R and any c > 0, there exist infinitely many q 2 Z such that jy  p/qj < c/jqj

i. e. jqy  pj < c

(2)

for some p 2 Z. A comparison of (1) and (2) shows that it makes sense to multiply both sides of (1) by q, since in the right hand side of (2) one would still be able to get very small numbers. In other words, approximation of y by p/q translates into approximating integers by integer multiples of y. Also, if y is irrational, (p; q) can be chosen to be relatively prime, i. e. one gets infinitely many different rational numbers p/q satisfying (2). However if y 2 Q the latter is no longer true for small enough c. Thus it seems to be more convenient to talk about pairs (p; q) rather than p/q 2 Q, avoiding a necessity to consider the two cases separately. At this point it is convenient to introduce the following central definition: if is a function N ! RC and y 2 R, say that y is -approximable (notation: y 2 W ( )) if

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

there exist infinitely many q 2 N such that jqy  pj <

(q)

(3)

for some p 2 Z. Because of Kronecker’s Theorem, it is natural to assume that (x) ! 0 as x ! 1. Often will be assumed non-increasing, although many results do not require monotonicity of . One can similarly consider a higher-dimensional version of the above set-up. Note that y 2 R in the above formulas plays the role of a linear map from R to another copy of R, and one asks how close values of this map at integers are from integers. It is natural to generalize it by taking a linear operator Y from Rn to Rm for fixed m; n 2 N, that is, an m  n-matrix (interpreted as a system of m linear forms Y i on Rn ). We will denote by M m;n the space of m  n matrices with real coefficients. For as above, one says that Y 2 M m;n is -approximable (notation: Y 2 Wm;n ( )) if there are infinitely many q 2 Z n such that kYq C pk 

(kqk)

Recurrence by Frantzikinakis and McCutcheon and  Ergodic Theory: Interactions with Combinatorics and Number Theory by Ward, as well as in the survey papers [32,33, 50,58,66,70,71]. Here is a brief outline of the rest of the article. In the next section we survey basic results, some classical, some obtained relatively recently, in metric Diophantine approximation. Sect. “Connection with Dynamics on the Space of Lattices” is devoted to a description of the connection between Diophantine approximation and dynamics, specifically flows on the space of lattices. In Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up” and Sect. “Further Results”, we specialize to the set-up of Diophantine approximation on manifolds, or, more generally, approximation properties of vectors with respect to measures satisfying some natural conditions, and show how applications of homogeneous dynamics contributed to important recent developments in the field. Sect. “Future Directions” mentions several open questions and directions for further investigation.

(4)

for some p 2 Z m . Here k  k is the supremum norm on R k given by kyk D max1ik jy i j. (This definition is slightly different from the one used in [58], where powers of norms were considered). Traditionally, one of the main goals of metric Diophantine approximation has been to understand how big the sets Wm;n ( ) are for fixed m; n and various functions . Of course, (4) is not the only interesting condition that can be studied; various modifications of the approximation properties can also be considered. For example the Oppenheim Conjecture, now a theorem of Margulis [69] and a basis for many important recent developments [22,34,35], states that indefinite irrational quadratic forms can take arbitrary small values at integer points; Littlewood’s conjecture, see (18) below, deals with a similar statement about products of linear forms. See the article  Ergodic Theory: Rigidity by Nitica and surveys [32,70] for details. We remark that the standard tool for studying Diophantine approximation properties of real numbers (m D n D 1) is the continued fraction expansion, or, equivalently, the Gauss map x 7! 1/x mod 1 of the unit interval, see [49]. However the emphasis of this survey lies in higher-dimensional theory, and the dynamical system described below can be thought of as a replacement for the continued fraction technique applicable in the one-dimensional case. Additional details about interactions between ergodic theory and number theory can be found in the article by Nitica mentioned above, in  Ergodic Theory:

Basic Facts General references for this section: [17,80]. The simplest choice for functions happens to be the following: let us denote c;v (x) D cx v . It was shown by Dirichlet in 1842 that with the choice c D 1 and v D n/m, all Y 2 M m;n are -approximable. Moreover, Dirichlet’s Theorem states that for any Y 2 M m;n and for any t > 0 there exist q D (q1 ; : : : ; q n ) 2 Z n n f0g and p D (p1 ; : : : ; p m ) 2 Z m satisfying the following system of inequalities: kYq  pk < et/m

and kqk  e t/n :

(5)

From this it easily follows that Wm;n ( 1;n/m ) D M m;n . In fact, it is this paper of Dirichlet which gave rise to his box principle. Later another proof of the same result was given by Minkowski. The constant c D 1 is not optimal: p the smallest value of c for which W1;1 ( c;1 ) D R is 1/ 5, and the optimal constants are not known in higher dimensions, although some estimates can be given [80]. Systems of linear forms which do not belong to W m;n ( c;n/m ) for some positive c are called badly approximable; that is, we set def

BA m;n D M m;n n [c>0 Wm;n (

c;n/m )

:

Their existence in arbitrary dimensions was shown by Perron. Note that a real number y (m D n D 1) is badly approximable if and only if its continued fraction coefficients are uniformly bounded. It was proved by Jarnik [46] in the

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case m D n D 1 and by Schmidt in the general case [78] that badly approximable matrices form a set of full Hausdorff dimension: that is, dim(BA m;n ) D mn. On the other hand, it can be shown that each of the sets W m;n ( c;n/m ) for any c > 0 has full Lebesgue measure, and hence the complement BA m;n to their intersection has measure zero. This is a special case of a theorem due to Khintchine [48] in the case n D 1 and to Groshev [42] in full generality, which gives the precise condition on the function under which the set of -approximable matrices has full measure. Namely, if is non-increasing (this assumption can be removed in higher dimensions but not for n D 1, see [29]), then -almost no (resp. -almost every) Y 2 M m;n is -approximable, provided the sum 1 X

k n1 (k)m

(6)

kD1

converges (resp. diverges). (Here and hereafter  stands for Lebesgue measure). This statement is usually referred to as the Khintchine–Groshev Theorem. The convergence case of this theorem follows in a straightforward manner from the Borel–Cantelli Lemma, but the divergence case is harder. It was reproved and sharpened in 1960 by Schmidt [76], who showed that if the sum (6) diverges, then for almost all Y the number of solutions to (4) with kqk  N is asymptotic to the partial sum of the series (6) (up to a constant), and also gave an estimate for the error term. A special case of the convergence part of the theorem shows that Wm;n ( 1;v ) has measure zero whenever v > n/m. Y is said to be very well approximable if it belongs to W m;n ( 1;v ) for some v > n/m. That is, def

VWAm;n D [v>n/m Wm;n (

1;v )

:

More specifically, let us define the Diophantine exponent !(Y) of Y (sometimes called “the exact order” of Y) to be the supremum of v > 0 for which Y 2 Wm;n ( 1;v ). Then !(Y) is always not less than n/m, and is equal to n/m for Lebesgue-a.e. Y; in fact, VWA m;n D fY 2 M m;n : !(Y) > n/mg. The Hausdorff dimension of the null sets Wm;n ( 1;v ) was computed independently by Besicovitch [14] and Jarnik [45] in the one-dimensional case and by Dodson [26] in general: when v > n/m, one has  dim Wm;n (



1;v )

D (n  1)m C

mCn : vC1

(7)

See [27] for a nice exposition of ideas involved in the proof of both the aforementioned formula and the Khintchine– Groshev Theorem.

Note that it follows from (7) that the null set VWA m;n has full Hausdorff dimension. Matrices contained in the intersection \ W m;n ( 1;v ) D fY 2 M m;n : !(Y) D 1g v

are called Liouville and form a set of Hausdorff dimension (n  1)m, that is, to the dimension of Y for which Yq 2 Z for some q 2 Zn n f0g (the latter belong to Wm;n ( ) for any positive ). Note also that the aforementioned properties behave nicely with respect to transposition; this is described by the so-called Khintchine’s Transference Principle (Chap. V in [17]). For example, Y 2 BA m;n if and only if Y T 2 BA n;m , and Y 2 VWA m;n if and only if Y T 2 VWA n;m . In particular, many problems related to approximation properties of vectors (n D 1) and linear forms (m D 1) reduce to one another. We refer the readers to [43] and [8] for very detailed and comprehensive recent accounts of various further aspects of the theory. Connection with Dynamics on the Space of Lattices General references for this section: [4,86]. Interactions between Diophantine approximation and the theory of dynamical systems has a long history. Already in Kronecker’s Theorem one can see a connection. Indeed, the statement of the theorem can be rephrased as follows: the points on the orbit of 0 under the rotation of the circle R/Z by y approach the initial point 0 arbitrarily closely. This is a special case of the Poincare Recurrence Theorem in measurable dynamics. And, likewise, all the aforementioned properties of Y 2 M m;n can be restated in terms of recurrence properties of the Z n -action on the m-dimensional torus Rm /Z m given by x 7! Yx mod Z m . In other words, fixing Y gives rise to a dynamical system in which approximation properties of Y show up. However the theme of this section is a different dynamical system, whose phase space is (essentially) the space of parameters Y, and which can be used to read the properties of Y from the behavior of the associated trajectory. It has been known for a long time (see [81] for a historical account) that Diophantine properties of real numbers can be coded by the behavior of geodesics on the quotient of the hyperbolic plane by SL2 (Z). In fact, the latter flow can be viewed as the suspension flow of the Gauss map mentioned at the end of Sect. “Introduction”. There have been many attempts to construct a higher-dimensional analogue of the Gauss map so that it captures all the fea-

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

tures of simultaneous approximation, see [47,63,65] and references therein. On the other hand, it seems to be more natural and efficient to generalize the suspension flow itself, and this is where one needs higher rank homogeneous dynamics. As was mentioned above, in the basic set-up of simultaneous Diophantine approximation one takes a system of m linear forms Y1 ; : : : ; Ym on Rn and looks at the values of jYi (q) C p i j; p i 2 Z, when q D (q1 ; : : : ; q n ) 2 Z n is far from 0. The trick is to put together Y1 (q) C p1 ; : : : ; Ym (q) C p m

and

q1 ; : : : ; q n ;

fg t LY Z k : t 2 RC g

and consider the collection of vectors

 ˇ  Yq C p ˇˇ m n D LY Z k ; q 2 Z p 2 Z ˇ q where k D m C n and   Y def I LY D m ; Y 2 M m;n : 0 In

(8)

This collection is a unimodular lattice in R k , that is, a discrete subgroup of R k with covolume 1. Our goal is to keep track of vectors in such a lattice having small projections onto the first m components of R k and big projections onto the last n components. This is where dynamics comes into the picture. Denote by g t the one-parameter subgroup of SL k (R) given by g t D diag(et/m ; : : : ; e t/m ; et/n ; : : : ; et/n ) : „ ƒ‚ … „ ƒ‚ … mtimes

(9)

ntimes

The vectors in the lattice LY Z k are moved by the action of g t , t > 0, and a special role is played by the moment t when the “small” and “big” projections equalize. That is, one is led to consider a new dynamical system. Its phase space is the space of unimodular lattices in R k , which can be naturally identified with the homogeneous space def

˝ k D G/; where G D SL k (R) and  D SL k (Z) ;

Z k nf0gg such that g i (v i ) ! 0 as i ! 1. Equivalently, for " > 0 consider a subset K " of ˝ k consisting of lattices with no nonzero vectors of norm less than "; then all the sets K " are compact, and every compact subset of ˝ k is contained in one of them. Moreover, one can choose a metric on ˝ k such that dist(; Z k ) is, up to a uniform multiplicative constant, equal to  log minv2 nf0g kvk (see [25]); then the length of the smallest nonzero vector in a lattice  will determine how far away is this lattice in the “cusp” of ˝ k . Using Mahler’s Criterion, it is not hard to show that Y 2 BA m;n if and only if the trajectory

(10)

and the action is given by left multiplication by elements of the subgroup (9) of G, or perhaps other subgroups H G. Study of such systems has a rich history; for example, they are known to be ergodic and mixing whenever H is unbounded [74]. What is important in this particular case is that the space ˝ k happens to be noncompact, and its structure at infinity is described via Mahler’s Compactness Criterion, see Chap. V in [4]: a sequence of lattices g i Z k goes to infinity in ˝ k () there exists a sequence fv i 2

(11)

is bounded in ˝ k . This was proved by Dani [20] in 1985, and later generalized in [57] to produce a criterion for Y to be -approximable for any non-increasing function . An important special case is a criterion for a system of linear forms to be very well approximable: Y 2 VWAm;n if and only if the trajectory (11) has linear growth, that is, there exists a positive such that dist(g t LY Z k ; Z k ) > t for an unbounded set of t > 0. This correspondence allows one to link various Diophantine and dynamical phenomena. For example, from the results of [55] on abundance of bounded orbits on homogeneous spaces one can deduce the aforementioned theorem of Schmidt [78]: the set BAm;n has full Hausdorff dimension. And a dynamical Borel–Cantelli Lemma established in [57] can be used for an alternative proof of the Khintchine–Groshev Theorem; see also [87] for an earlier geometric approach. Note that both proofs are based on the following two properties of the g t -action: mixing, which forces points to return to compact subsets and makes preimages of cusp neighborhoods quasi-independent, and hyperbolicity, which implies that the behavior of points on unstable leaves is generic. The latter is important since the orbits of the group fLY Z k : Y 2 M m;n g are precisely the unstable leaves with respect to the g t -action. We note that other types of Diophantine problems, such as conjectures of Oppenheim and Littlewood mentioned in the previous section, can be reduced to statements involving ˝ k by means of the same principle: Mahler’s Criterion is used to relate small values of some function at integer points to excursions to infinity in ˝ k of orbit of the stabilizer of this function. Other important and useful recent applications of homogeneous dynamics to metric Diophantine approximation are related to the circle of ideas roughly called “Diophantine approximation with dependent quantities” (terminology borrowed from [84]), to be surveyed in the next two sections.

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Diophantine Approximation with Dependent Quantities: The Set-Up General references for this section: [12,84]. Here we restrict ourselves to Diophantine properties of vectors in Rn . In particular, we will look more closely at the set of very well approximable vectors, which we will simply denote by VWA, dropping the subscripts. In many cases it does not matter whether one works with row or column vectors, in view of the duality remark made at the end of Sect. “Basic Facts”. We begin with a non-example of an application of dynamics to Diophantine approximation: a celebrated and difficult theorem which currently, to the best of the author’s knowledge, has no dynamical proof. Suppose that y D (y1 ; : : : ; y n ) 2 Rn is such that each yi is algebraic and 1; y1 ; : : : ; y n are linearly independent over Q. It was established by Roth for n D 1 [75] and then generalized to arbitrary n by Schmidt [79], that y as above necessarily belongs to the complement of VWA. In other words, vectors with very special algebraic properties happen to follow the behavior of a generic vector in Rn . We would like to view the above example as a special case of a general class of problems. Namely, suppose we are given a Radon measure  on Rn . Let us say that  is extremal [85] if -a.e. y 2 Rn is not very well approximable. Further, define the Diophantine exponent !() of  to be the -essential supremum of the function !(); in other words, ˚   def !() D sup vj W ( 1;v ) > 0 : Clearly it only depends on the measure class of . If  is naturally associated with a subset M of Rn supporting  (for example, if M is a smooth submanifold of Rn and  is the measure class of the Riemannian volume on M, or, equivalently, the pushforward f  of  by a smooth map f parametrizing M), one defines the Diophantine exponent !(M) of M to be equal to that of , and says that M is extremal if f(x) is not very well approximable for -a.e. x. Then !()  n for any , and !() D !(Rn ) is equal to n. The latter justifies the use of the word “extremal”:  is extremal if !() is equal to n, i. e. attains the smallest possible value. The aforementioned results of Roth and Schmidt then can be interpreted as the extremality of atomic measures supported on algebraic vectors without rational dependence relations. Historically, the first measure (other than ) to be considered in the set-up described above was the pushforward of  by the map f(x) D (x; x 2 ; : : : ; x n ) :

(12)

The extremality of f  for f as above was conjectured in 1932 by K. Mahler [67] and proved in 1964 by Sprindžuk [82,83]. It was important for Mahler’s study of transcendental numbers: this result, roughly speaking, says that almost all transcendental numbers are “not very algebraic”. At about the same time Schmidt [77] proved the extremality of f  when f : I ! R2 , I R, is C3 and satisfies ˇ 0 ˇ ˇ f (x) f 0 (x) ˇ 2 ˇ 1 ˇ ˇ ˇ ¤ 0 for -a.e. x 2 I ; ˇ f 00 (x) f 00 (x)ˇ 1

2

in other words, the curve parametrized by f has nonzero curvature at almost all points. Since then, a lot of attention has been devoted to showing that measures f  are extremal for other smooth maps f. To describe a broader class of examples, recall the following definition. Let x 2 Rd and let f D ( f1 ; : : : ; f n ) be a Ck map from a neighborhood of x to Rn . Say that f is nondegenerate at x if Rn is spanned by partial derivatives of f at x up to some order. Say that f is nondegenerate if it is nondegenerate at -a.e. x. It was conjectured by Sprindžuk [84] in 1980 that f  for real analytic nondegenerate f are extremal. Many special cases were established since then (see [12] for a detailed exposition of the theory and many related results), but the general case stood open until the mid-1990s [56], when Sprindžuk’s conjecture was proved using the dynamical approach (later Beresnevich [6] succeeded in establishing and extending this result without use of dynamics). The proof in [56] uses the correspondence outlined in the previous section plus a measure estimate for flows on the space of lattices which is described below. In the subsequent work the method of [56] was adapted to a much broader class of measures. To define it we need to introduce some more notation and definitions. If x 2 Rd and r > 0, denote by B(x; r) the open ball of radius r centered at x. If B D B(x; r) and c > 0, cB will denote the ball B(x; cr). For B Rd and a real-valued function f on B, let def

k f kB D sup j f (x)j : x2B

If  is a measure on Rd such that (B) > 0, define k f k;B def Dk f kB \ supp  ; this is the same as the L1 ()-norm of f j B if f is continuous and B is open. If D > 0 and U Rd is an open subset, let us say that  is D-Federer on U if for any ball B U centered at supp  one has (3B)/(B) < D whenever 3B U. This condition is often called “doubling” in the literature. See [54,72] for examples and references.  is called Federer if for -a.e. x 2 Rd there exist a

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

neighborhood U of x and D > 0 such that  is D-Federer on U. Given C; ˛ > 0, open U Rd and a measure  on U, a function f : U ! R is called (C; ˛)-good on U with respect to  if for any ball B U centered in supp  and any " > 0 one has  ˛   "  fx 2 B : j f (x)j < "g  C (B) : (13) k f k;B This condition was formally introduced in [56] for  being Lebesgue measure, and in [54] for arbitrary . A basic example is given by polynomials, and the upshot of the above definition is the formalization of a property needed for the proof of several basic facts [19,21,68] about polynomial maps into the space of lattices. In [54] a strengthening of this property was considered: f was called absolutely (C; ˛)-good on U with respect to  if for B and " as above one has  ˛   "  fx 2 B : j f (x)j < "g  C (B) : (14) k f kB There is no difference between (13) and (14) when  has full support, but it turns out to be useful for describing measures supported on proper (e. g. fractal) subsets of Rd . Now suppose that we are given a measure  on Rd , an open U Rd with (U) > 0 and a map f D ( f1 ; : : : ; f n ) : Rd ! Rn . Following [62], say that a pair (f; ) is (absolutely) good on U if any linear combination of 1; f1 ; : : : ; f n is (absolutely) (C; ˛)-good on U with respect to . If for -a.e. X there exists a neighborhood U of X and C; ˛ > 0 such that  is (absolutely) (C; ˛)-good on U, we will say that the pair (f; ) is (absolutely) good. Another relevant notion is the nonplanarity of (f; ). Namely, (f; ) is said to be nonplanar if whenever B is a ball with (B) > 0, the restrictions of 1; f1 ; : : : ; f n to B \ supp  are linearly independent over R; in other words, f(B \ supp ) is not contained in any proper affine subspace of Rn . Note that absolutely good implies both good and nonplanar, but the converse is in general not true. Many examples of (absolutely) good and nonplanar pairs (f; ) can be found in the literature. Already the case n D d and f D Id is very interesting. A measure  on Rn is said to be friendly (resp., absolutely friendly) if and only if it is Federer and the pair (Id; ) is good and nonplanar (resp., absolutely good). See [54,88,89] for many examples. An important class of measures is given by limit measures of irreducible system of self-similar or self-conformal contractions satisfying the Open Set Condition [44]; those are shown to be absolutely friendly in [54]. The prime example is the middle-third Cantor set on the real line. The term

“friendly” was cooked up as a loose abbreviation for “Federer, nonplanar and decaying”, and later proved to be particularly friendly in dealing with problems arising in metric number theory, see e. g. [36]. Also let us say that a pair (f; ) is nondegenerate if f is nondegenerate at -a.e. X. When  is Lebesgue measure on Rd , it is proved in Proposition 3.4 in [56], that a nondegenerate (f; ) is good and nonplanar. The same conclusion is derived in Proposition 7.3 in [54], assuming that  is absolutely friendly. Thus volume measures on smooth nondegenerate manifolds are friendly, but not absolutely friendly. It turns out that all the aforementioned examples of measures can be proved to be extremal by a generalization of the argument from [56]. Specifically, let  be a Federer measure on Rd , U an open subset of Rd , and f : U ! Rn a continuous map such that the pair (f; ) is good and nonplanar; then f  is extremal. This can be derived from the Borel–Cantelli Lemma, the correspondence described in the previous section, and the following measure estimate: if , U and f are as above, then for -a.e. x0 2 U there ex˜ ˛ > 0 such that for ists a ball B U centered at x0 and C; any t 2 RC and any " > 0,  ˚ ˜ ˛: (15)  x 2 B : g t Lf(x) Z nC1 … K" < C" Here g t is as in (9) with m D 1 (assuming that the row vector viewpoint is adopted). This is a quantitative way of saying that for fixed t, the “flow” x 7! g t Lf(x) Z nC1 , B ! ˝nC1 , cannot diverge, and in fact must spend a big (uniformly in t) proportion of time inside compact sets K " . The inequality (15) is derived from a general “quantitative non-divergence” estimate, which can be thought of a substantial generalization of theorems of Margulis and Dani [19,21,68] on non-divergence of unipotent flows on homogeneous spaces. One of its most general versions [54] deals with a measure  on Rd , a continuous map h: e B ! G, where e B is a ball in Rd centered at supp  and G is as in (10). To describe the assumptions on h, one needs to employ the combinatorial structure of lattices in R k , and it will be convenient to use the following notation: if V is a nonzero rational subspace of R k and g 2 G, define `V (g) to be the covolume of g(V \ Z k ) in gV . Then, given positive constants C; D; ˛, there exists C1 D C1 (d; k; C; ˛; D) > 0 with the following property. Suppose  is D-Federer on e B, 0 <  1, and h is such that for each rational V R k (i) `V ı h is (C; ˛)-good on B˜ with respect to , and ˜ Then (ii) k`V ı hk;B  , where B D 3(k1) B. (iii) for any positive "  , one has  ˚  x 2 B : h(x)Z k … K"  C1 (" )˛ (B) : (16)

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Taking h(x) D g t Lf(x) and unwinding the definitions of good and nonplanar pairs, one can show that (i) and (ii) can be verified for some balls B centered at -almost every point, and derive (15) from (16). Further Results The approach to metric Diophantine approximation using quantitative non-divergence, that is, the implication (i) + (ii) ) (iii), is not omnipotent. In particular, it is difficult to use when more precise results are needed, such as for example computing/estimating the Hausdorff dimension of the set of 1;v -approximable vectors on a manifold. See [9,10] for such results. On the other hand, the dynamical approach can often treat much more general objects that its classical counterpart, and also can be perturbed in a lot of directions, producing many generalizations and modifications of the main theorems from the preceding section. One of the most important of them is the so-called multiplicative version of the set-up of Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up”. def Q def Namely, define functions ˘ (x) D i jx i j and ˘C (x) D Q : N ! RC , one i max(jx i j; 1) Then, given a function says that Y 2 M m;n is multiplicatively -approximable  ( )) if there are infinitely many (notation: Y 2 Wm;n q 2 Z n such that   ˘ (Yq C p)1/m  ˘C (q)1/n (17) for some p 2 Z m . Since ˘ (x)  ˘C (x)  kxk k for x 2 R k , any -approximable Y is multiplicatively -approximable; but the converse is in general not true, see e. g. [37]. However if one, as before, considers the family f 1;v g, the critical parameter for which the drop from full measure to measure zero occurs is again n/m. That is, if one defines the multiplicative Diophantine exponent def  ( !  (Y) of Y by !  (Y) D supfv : Y 2 Wm;n 1;v )g, then  clearly ! (Y)  !(Y) for all Y, and yet !  (Y) D n/m for -a.e. Y 2 M m;n . Now specialize to Rn (by the same duality principle as before, it does not matter whether to think in terms of row or column vectors, but we will adopt the row vector  set-up), and define the multiplicative˚ exponent   ! () of def n  a measure  on R by ! () D sup vj W ( 1;v ) > 0 ; then !  () D n. Following Sprindžuk [85], say that  is strongly extremal if !  () D n. It turns out that all the results mentioned in the previous section have their multiplicative analogues; that is, the measures described there happen to be strongly extremal. This was conjectured by A. Baker [1] for the curve (12), and then by Sprindžuk in 1980 [85] for analytic nondegenerate manifolds. (We re-

mark that only very few results in this set-up can be obtained by the standard methods, see e. g. [10]). The proof of this stronger statement is based on using the multi-parameter action of gt D diag(e t1 CCt n ; et1 ; : : : ; et n ) ; where t D (t1 ; : : : ; t n ) instead of g t considered in the previous section. One can show that the choice h(x) D gt Lf(x) allows one to verify n , and the proof is fin(i) and (ii) uniformly in t 2 RC ished by applying a multi-parameter version of the correspondence described in Sect. “Connection with Dynamics on the Space of Lattices”. Namely, one can show that k : y 2 VWA 1;n if and only if the trajectory fg t Ly Z n t 2 RC g grows linearly, that is, for some > 0 one has dist(gt LY Z nC1 ; Z nC1 ) > ktk for an unbounded n . A similar correspondence was recently set of t 2 RC used in [30] to prove that the set of exceptions to Littlewood’s Conjecture, which, using the terminology introduced above, can be called badly multiplicatively approximable vectors: def n [  BA W n;1 ( c;1/n ) n;1 D R n

D y:

c>0

 inf

q2Znf0g; p2Z n

jqj  ˘ (qy  p) > 0

; (18)

has Hausdorff dimension zero. This was done using a measure rigidity result for the action of the group of diagonal matrices on the space of lattices. See [18] for an implicit description of this correspondence and [32,66,70] for more detail. The dynamical approach also turned out to be fruitful in studying Diophantine properties of pairs (f; ) for which the nonplanarity condition fails. Note that obvious examples of non-extremal measures are provided by proper affine subspaces of Rn whose coefficients are rational or are well enough approximable by rational numbers. On the other hand, it is clear from a Fubini argument that almost all translates of any given subspace are extremal. In [51] the method of [56] was pushed further to produce criteria for the extremality, as well as the strong extremality, of arbitrary affine subspaces L of Rn . Further, it was shown that if L is extremal (resp. strongly extremal), then so is any smooth submanifold of L which is nondegenerate in L at a.e. point. (The latter property is a straightforward generalization of the definition of nondegeneracy in Rn : a map f is nondegenerate in L at x if the linear part of L is spanned by partial derivatives of f at x). In other words, extremality and strong extremality pass from affine subspaces to their nondegenerate submanifolds.

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

A more precise analysis makes it possible to study Diophantine exponents of measures with supports contained in arbitrary proper affine subspaces of Rn . Namely, in [53] it is shown how to compute !(L) for any L, and furthermore proved that if  is a Federer measure on Rd , U an open subset of Rd , and f : U ! Rn a continuous map such that the pair (f; ) is good and nonplanar in L, then !(f ) D !(L). Here we say, generalizing the definition from Sect.“Diophantine Approximation with Dependent Quantities: The Set-Up”, that (f; ) is nonplanar in L if for any ball B with (B) > 0, the f-image of B \ supp  is not contained in any proper affine subspace of L. (It is easy to see that for a smooth map f : U ! L, (f; ) is good and nonplanar in L whenever f is nondegenerate in L at a.e. point). It is worthwhile to point out that these new applications require a strengthening of the measure estimate described at the end of Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up”: it was shown in [53] that (i) and (ii) would still imply (iii) if in (ii) is replaced by dim V . Another application concerns badly approximable vectors. Using the dynamical description of the set BA

Rn due to Dani [20], it turns out to be possible to find badly approximable vectors inside supports of certain measures on Rn . Namely, if a subset K of Rn supports an absolutely friendly measure, then BA \ K has Hausdorff dimension not less than the Hausdorff dimension of this measure. In particular, it proves that limit measures of irreducible system of self-similar/self-conformal contractions satisfying the Open Set Condition, such as e. g. the middle-third Cantor set on the real line, contain subsets of full Hausdorff dimension consisting of badly approximable vectors. This was established in [60] and later independently in [64] using a different approach. See also [36] for a stronger result. The proof in [60] uses quantitative nondivergence estimates and an iterative procedure, which requires the measure in question to be absolutely friendly and not just friendly. A similar question for even the simplest not-absolutely friendly measures is completely open. For example, it is not known whether there exist uncountably many badly approximable pairs of the form (x; x 2 ). An analogous problem for atomic measures supported on algebraic numbers, that is, a “badly approximable” version of Roth’s Theorem, is currently beyond reach as well — there are no known badly approximable (or, for that matter, well approximable) algebraic numbers of degree bigger than two. It has been recently understood that the quantitative nondivergence method can be applied to the question of improvement to Dirichlet’s Theorem (see the beginning of Sect. “Basic Facts”). Given a positive " < 1, let us say

that Dirichlet’s Theorem can be "-improved for Y 2 M m;n , writing Y 2 DIm;n ("), if for every sufficiently large t the system kYq  pk < "et/m

and kqk < "e t/n

(19)

(that is, (5) with the right hand side terms multiplied by ") has a nontrivial integer solution (p; q). of  It is a theorem  Davenport and Schmidt [24] that  DIm;n (") D 0 for any " < 1; in other words, Dirichlet’s Theorem cannot be improved for Lebesgue-generic systems of linear forms. By a modification of the correspondence between dynamics and approximation, (19) is easily seen to be equivalent to g t LY Z k 2 K" , and since the complement to K " has nonempty interior for any " < 1, the result of Davenport and Schmidt follows from the ergodicity of the g t -action on ˝ k . Similar questions with  replaced by f  for some specific smooth maps f were considered in [2,3,15,23]. For example, [15], Theorem 7, provides an explicitly computable constant "0 D "0 (n) such that for f as in (12),   f  DI1;n (") D 0 for " < "0 : This had been previously done in [23] for n D 2 and in [2] for n D 3. In [62] this is extended to a much broader class of measures using estimates described in Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up”. In particular, almost every point of any nondegenerate smooth manifold is proved not to lie in DI(") for small enough " depending only on the manifold. Earlier this was done in [61] for the set of singular vectors, defined as the intersection of DI(") over all positive "; those correspond to divergent g t -trajectories. As before, the advantage of the method is allowing a multiplicative generalization of the Dirichlet-improvement set-up; see [62] for more detail. It is also worthwhile to mention that a generalization of the measure estimate discussed in Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up” was used in [13] to estimate the measure of the set of points x in a ball B Rd for which the system 8 ˆ <jf(x)  q C pj < " jf0 (x)  qj < ı ˆ : jq i j < Q i ; i D 1; : : : ; n; where f is a smooth nondegenerate map B ! Rn , has a nonzero integer solution. For that, Lf(x) as in (15) has to be replaced by the matrix 1 0 1 0 f(x) @0 1 f0 (x)A ; 0 0 In

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and therefore (i) and (ii) turn into more complicated conditions, which nevertheless can be checked when f is smooth and nondegenerate and  is Lebesgue measure. This has resulted in proving the convergence case of Khintchine–Groshev Theorem for nondegenerate manifolds [13], in both standard and multiplicative versions. The aforementioned estimate was also used in [7] for the proof of the divergence case, and in [38,40] for establishing the convergence case of Khintchine–Groshev theorem for affine hyperplanes and their nondegenerate submanifolds. This generalized results obtained by standard methods for the curve (12) by Bernik and Beresnevich [6,11]. Finally, let us note that in many of the problems mentioned above, the ground field R can be replaced by Q p , and in fact several fields can be taken simultaneously, thus giving rise to the S-arithmetic setting where S D fp1 ; : : : ; p s g is a finite set of normalized valuations of Q, which may or may not include the infinite valuation (cf. [83,90]). The space of lattices in RnC1 is replaced there by the space of lattices in QnC1 , where QS is the prodS uct of the fields R and Q p 1 ; : : : ; Q p s . This is the subject of the paper [59], where S-arithmetic analogues of many results reviewed in Sect. “Diophantine Approximation with Dependent Quantities: The Set-Up” have been established. Similarly one can consider versions of the above theorems over local fields of positive characteristic [39]. See also [52] where Sprindžuk’s solution [83] of the complex case of Mahler’s Conjecture has been generalized (the latter involves studying small values of linear forms with coefficients in C at real integer points), and [31] which establishes a p-adic analogue of the result of [30] on the set of exceptions to Littlewood’s Conjecture. Future Directions Interactions between ergodic theory and number theory have been rapidly expanding during the last two decades, and the author has no doubts that new applications of dynamics to Diophantine approximation will emerge in the near future. Specializing to the topics discussed in the present paper, it is fair to say that the list of “further results” contained in the previous section is by no means complete, and many even further results are currently in preparation. This includes: extending proofs of extremality and strong extremality of certain measures to the set-up of systems of linear forms (namely, with min(m; n) > 1; this was mentioned as work in progress in [56]); proving Khintchine-type theorems (both convergence and divergence parts) for p-adic and S-arithmetic nondegenerate manifolds, see [73] for results in this direction; extending [38,40] to establish Khintchine-type theo-

rems for submanifolds of arbitrary affine subspaces. The work of Dru¸tu [28], who used ergodic theory on homogeneous spaces to compute the Hausdorff dimension of the intersection of Wn;1 ( 1;v ), v > 1, with some rational quadratic hypersurfaces in Rn deserves a special mention; it is plausible that using this method one can treat more general situations. Several other interesting open directions are listed in [41], Section 9, in the final sections of papers [7,54], in the book [15], and in surveys by Frantzikinakis–McCutcheon, Nitica and Ward in this volume. Acknowledgment The work on this paper was supported in part by NSF Grant DMS-0239463. Bibliography 1. Baker A (1975) Transcendental number theory. Cambridge University Press, London 2. Baker RC (1976) Metric diophantine approximation on manifolds. J Lond Math Soc 14:43–48 3. Baker RC (1978) Dirichlet’s theorem on diophantine approximation. Math Proc Cambridge Phil Soc 83:37–59 4. Bekka M, Mayer M (2000) Ergodic theory and topological dynamics of group actions on homogeneous spaces. Cambridge University Press, Cambridge 5. Beresnevich V (1999) On approximation of real numbers by real algebraic numbers. Acta Arith 90:97–112 6. Beresnevich V (2002) A Groshev type theorem for convergence on manifolds. Acta Mathematica Hungarica 94:99–130 7. Beresnevich V, Bernik VI, Kleinbock D, Margulis GA (2002) Metric Diophantine approximation: the Khintchine–Groshev theorem for nondegenerate manifolds. Moscow Math J 2:203–225 8. Beresnevich V, Dickinson H, Velani S (2006) Measure theoretic laws for lim sup sets. Mem Amer Math Soc:179:1–91 9. Beresnevich V, Dickinson H, Velani S (2007) Diophantine approximation on planar curves and the distribution of rational points. Ann Math 166:367–426 10. Beresnevich V, Velani S (2007) A note on simultaneous Diophantine approximation on planar curves. Ann Math 337: 769–796 11. Bernik VI (1984) A proof of Baker’s conjecture in the metric theory of transcendental numbers. Dokl Akad Nauk SSSR 277:1036–1039 12. Bernik VI, Dodson MM (1999) Metric Diophantine approximation on manifolds. Cambridge University Press, Cambridge 13. Bernik VI, Kleinbock D, Margulis GA (2001) Khintchine-type theorems on manifolds: convergence case for standard and multiplicative versions. Int Math Res Notices 2001:453–486 14. Besicovitch AS (1929) On linear sets of points of fractional dimensions. Ann Math 101:161–193 15. Bugeaud Y (2002) Approximation by algebraic integers and Hausdorff dimension. J London Math Soc 65:547–559 16. Bugeaud Y (2004) Approximation by algebraic numbers. Cambridge University Press, Cambridge 17. Cassels JWS (1957) An introduction to Diophantine approximation. Cambridge University Press, New York

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41. Gorodnik A (2007) Open problems in dynamics and related fields. J Mod Dyn 1:1–35 42. Groshev AV (1938) Une théorème sur les systèmes des formes linéaires. Dokl Akad Nauk SSSR 9:151–152 43. Harman G (1998) Metric number theory. Clarendon Press, Oxford University Press, New York 44. Hutchinson JE (1981) Fractals and self-similarity. Indiana Univ Math J 30:713–747 45. Jarnik V (1928-9) Zur metrischen Theorie der diophantischen Approximationen. Prace Mat-Fiz 36:91–106 46. Jarnik V (1929) Diophantischen Approximationen und Hausdorffsches Mass. Mat Sb 36:371–382 47. Khanin K, Lopes-Dias L, Marklof J (2007) Multidimensional continued fractions, dynamical renormalization and KAM theory. Comm Math Phys 270:197–231 48. Khintchine A (1924) Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math Ann 92:115–125 49. Khintchine A (1963) Continued fractions. P Noordhoff Ltd, Groningen 50. Kleinbock D (2001) Some applications of homogeneous dynamics to number theory. In: Smooth ergodic theory and its applications. American Mathematical Society, Providence, pp 639–660 51. Kleinbock D (2003) Extremal subspaces and their submanifolds. Geom Funct Anal 13:437–466 52. Kleinbock D (2004) Baker–Sprindžuk conjectures for complex analytic manifolds. In: Algebraic groups and Arithmetic. Tata Inst Fund Res, Mumbai, pp 539–553 53. Kleinbock D (2008) An extension of quantitative nondivergence and applications to Diophantine exponents. Trans AMS, to appear 54. Kleinbock D, Lindenstrauss E, Weiss B (2004) On fractal measures and diophantine approximation. Selecta Math 10: 479–523 55. Kleinbock D, Margulis GA (1996) Bounded orbits of nonquasiunipotent flows on homogeneous spaces. In: Sina˘ı’s Moscow Seminar on Dynamical Systems. American Mathematical Society, Providence, pp 141–172 56. Kleinbock D, Margulis GA (1998) Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann Math 148:339–360 57. Kleinbock D, Margulis GA (1999) Logarithm laws for flows on homogeneous spaces. Invent Math 138:451–494 58. Kleinbock D, Shah N, Starkov A (2002) Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. In: Handbook on Dynamical Systems, vol 1A. Elsevier Science, North Holland, pp 813–930 59. Kleinbock D, Tomanov G (2007) Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation. Comm Math Helv 82:519–581 60. Kleinbock D, Weiss B (2005) Badly approximable vectors on fractals. Israel J Math 149:137–170 61. Kleinbock D, Weiss B (2005) Friendly measures, homogeneous flows and singular vectors. In: Algebraic and Topological Dynamics. American Mathematical Society, Providence, pp 281–292 62. Kleinbock D, Weiss B (2008) Dirichlet’s theorem on diophantine approximation and homogeneous flows. J Mod Dyn 2:43– 62

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63. Kontsevich M, Suhov Y (1999) Statistics of Klein polyhedra and multidimensional continued fractions. In: Pseudoperiodic topology. American Mathematical Society, Providence, pp 9–27 64. Kristensen S, Thorn R, Velani S (2006) Diophantine approximation and badly approximable sets. Adv Math 203:132–169 65. Lagarias JC (1994) Geodesic multidimensional continued fractions. Proc London Math Soc 69:464–488 66. Lindenstrauss E (2007) Some examples how to use measure classification in number theory. In: Equidistribution in number theory, an introduction. Springer, Dordrecht, pp 261– 303 67. Mahler K (1932) Über das Mass der Menge aller S-Zahlen. Math Ann 106:131–139 68. Margulis GA (1975) On the action of unipotent groups in the space of lattices. In: Lie groups and their representations (Budapest, 1971). Halsted, New York, pp 365–370 69. Margulis GA (1989) Discrete subgroups and ergodic theory. In: Number theory, trace formulas and discrete groups (Oslo, 1987). Academic Press, Boston, pp 377–398 70. Margulis GA (1997) Oppenheim conjecture. In: Fields Medalists’ lectures. World Sci Publishing, River Edge, pp 272– 327 71. Margulis GA (2002) Diophantine approximation, lattices and flows on homogeneous spaces. In: A panorama of number theory or the view from Baker’s garden. Cambridge University Press, Cambridge, pp 280–310 ´ 72. Mauldin D, Urbanski M (1996) Dimensions and measures in infinite iterated function systems. Proc London Math Soc 73:105–154 73. Mohammadi A, Salehi Golsefidy A (2008) S-Arithmetic Khintchine-Type Theorem. Preprint 74. Moore CC (1966) Ergodicity of flows on homogeneous spaces. Amer J Math 88:154–178 75. Roth KF (1955) Rational Approximations to Algebraic Numbers. Mathematika 2:1–20

76. Schmidt WM (1960) A metrical theorem in diophantine approximation. Canad J Math 12:619–631 77. Schmidt WM (1964) Metrische Sätze über simultane Approximation abhängiger Größen. Monatsh Math 63:154–166 78. Schmidt WM (1969) Badly approximable systems of linear forms. J Number Theory 1:139–154 79. Schmidt WM (1972) Norm form equations. Ann Math 96: 526–551 80. Schmidt WM (1980) Diophantine approximation. Springer, Berlin 81. Sheingorn M (1993) Continued fractions and congruence subgroup geodesics. In: Number theory with an emphasis on the Markoff spectrum (Provo, UT, 1991). Dekker, New York, pp 239–254 82. Sprindžuk VG (1964) More on Mahler’s conjecture. Dokl Akad Nauk SSSR 155:54–56 83. Sprindžuk VG (1969) Mahler’s problem in metric number theory. American Mathematical Society, Providence 84. Sprindžuk VG (1979) Metric theory of Diophantine approximations. VH Winston & Sons, Washington DC 85. Sprindžuk VG (1980) Achievements and problems of the theory of Diophantine approximations. Uspekhi Mat Nauk 35: 3–68 86. Starkov A (2000) Dynamical systems on homogeneous spaces. American Mathematical Society, Providence 87. Sullivan D (1982) Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math 149:215–237 88. Stratmann B, Urbanski M (2006) Diophantine extremality of the Patterson measure. Math Proc Cambridge Phil Soc 140:297– 304 89. Urbanski M (2005) Diophantine approximation of selfconformal measures. J Number Theory 110:219–235 90. Želudeviˇc F (1986) Simultane diophantische Approximationen abhängiger Grössen in mehreren Metriken. Acta Arith 46: 285–296

Ergodic Theory: Interactions with Combinatorics and Number Theory

Ergodic Theory: Interactions with Combinatorics and Number Theory TOM W ARD School of Mathematics, University of East Anglia, Norwich, UK Article Outline Glossary Definition of the Subject Introduction Ergodic Theory Frequency of Returns Ergodic Ramsey Theory and Recurrence Orbit-Counting as an Analogous Development Diophantine Analysis as a Toolbox Future Directions Bibliography Glossary Almost everywhere (abbreviated a. e.) A property that makes sense for each point x in a measure space (X; B; ) is said to hold almost everywhere (or a. e.) if the set N X on which it does not hold satisfies N 2 B and (N) D 0. ˇ Cech–Stone compactification of N, ˇN A compact Hausdorff space that contains N as a dense subset with the property that any map from N to a compact Hausdorff space K extends uniquely to a continuous map ˇN ! K. This property and the fact that ˇN is a compact Hausdorff space containing N characterizes ˇN up to homeomorphism. Curvature An intrinsic measure of the curvature of a Riemannian manifold depending only on the Riemannian metric; in the case of a surface it determines whether the surface is locally convex (positive curvature), locally saddle-shaped (negative) or locally flat (zero). Diophantine approximation Theory of the approximation of real numbers by rational numbers: how small can the distance from a given irrational real number to a rational number be made in terms of the denominator of the rational? Equidistributed A sequence is equidistributed if the asymptotic proportion of time it spends in an interval is proportional to the length of the interval. Ergodic A measure-preserving transformation is ergodic if the only invariant functions are equal to a constant a. e.; equivalently if the transformation exhibits the convergence in the quasi-ergodic hypothesis.

Ergodic theory The study of statistical properties of orbits in abstract models of dynamical systems; more generally properties of measure-preserving (semi-) group actions on measure spaces. Geodesic (flow) The shortest path between two points on a Riemannian manifold; such a geodesic path is uniquely determined by a starting point and the initial tangent vector to the path (that is, a point in the unit tangent bundle). The transformation on the unit tangent bundle defined by flowing along the geodesic defines the geodesic flow. Haar measure (on a compact group) If G is a compact topological group, the unique measure  defined on the Borel sets of G with the property that (A C g) D (A) for all g 2 G and (G) D 1. Measure-theoretic entropy A numerical invariant of measure-preserving systems that reflects the asymptotic growth in complexity of measurable partitions refined under iteration of the map. Mixing A measure-preserving system is mixing if measurable sets (events) become asymptotically independent as they are moved apart in time (under iteration). (Quasi) Ergodic hypothesis The assumption that, in a dynamical system evolving in time and preserving a natural measure, there are some reasonable conditions under which the ‘time average’ along orbits of an observable (that is, the average value of a function defined on the phase space) will converge to the ‘space average’ (that is, the integral of the function with respect to the preserved measure). Recurrence Return of an orbit in a dynamical system close to its starting point infinitely often. S-Unit theorems A circle of results stating that linear equations in fields of zero characteristic have only finitely many solutions taken from finitely-generated multiplicative subgroups of the multiplicative group of the field (apart from infinite families of solutions arising from vanishing sub-sums). Topological entropy A numerical invariant of topological dynamical systems that measures the asymptotic growth in the complexity of orbits under iteration. The variational principle states that the topological entropy of a topological dynamical system is the supremum over all invariant measures of the measure-theoretic entropies of the dynamical systems viewed as measurable dynamical systems. Definition of the Subject Number theory is a branch of pure mathematics concerned with the properties of numbers in general, and integers in particular. The areas of most relevance to this ar-

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ticle are Diophantine analysis (the study of how real numbers may be approximated by rational numbers, and the consequences for solutions of equations in integers); analytic number theory, and in particular asymptotic estimates for the number of primes smaller than X as a function of X; equidistribution, and questions about how the digits of real numbers are distributed. Combinatorics is concerned with identifying structures in discrete objects; of most interest here is that part of combinatorics connected with Ramsey theory, asserting that large subsets of highly structured objects must automatically contain large replicas of that structure. Ergodic theory is the study of asymptotic behavior of group actions preserving a probability measure; it has proved to be a powerful part of dynamical systems with wide applications. Introduction Ergodic theory, part of the mathematical study of dynamical systems, has pervasive connections with number theory and combinatorics. This article briefly surveys how these arise through a small sample of results. Unsurprisingly, many details are suppressed, and of course the selection of topics reflects the author’s interests far more than it does the full extent of the flow of ideas between ergodic theory and number theory. In addition the selection of topics has been chosen in part to be complementary to those in related articles in the Encyclopedia. A particularly enormous lacuna is the theory of arithmetic dynamical systems itself – the recent monograph by Silverman [94] gives a comprehensive overview. More sophisticated aspects of this connection – in particular the connections between ergodic theory on homogeneous spaces and Diophantine analysis – are covered in the articles  Ergodic Theory on Homogeneous Spaces and Metric Number Theory and  Ergodic Theory: Rigidity; more sophisticated overviews of the connections with combinatorics may be found in the article  Ergodic Theory: Recurrence. Ergodic Theory While the early origins of ergodic theory lie in the quasiergodic hypothesis of classical Hamiltonian dynamics, the mathematical study of ergodic theory concerns various properties of group actions on measure spaces, including but not limited to several special branches: 1. The classical study of single measure-preserving transformations. 2. Measure-preserving actions of Zd ; more generally of countable amenable groups.

3. Measure-preserving actions of Rd and more general amenable groups, called flows. 4. Measure-preserving actions of lattices in Lie groups. 5. Measure-preserving actions of Lie groups. The ideas and conditions surrounding the quasi-ergodic hypothesis were eventually placed on a firm mathematical footing by developments starting in 1890. For a single measure-preserving transformation T : X ! X of a probability space (X; B; ), Poincaré [74] showed a recurrence theorem: if E 2 B is any measurable set, then for a. e. x 2 E there is an infinite set of return times, 0 < n1 < n2 <    with T n j (x) 2 E (of course Poincaré noted this in a specific setting, concerned with a natural invariant measure for the “three-body” problem in planetary motion). Poincaré’s qualitative result was made quantitative in the 1930s, when von Neumann [105] used the approach of Koopman [52] to show the mean ergodic theorem: if f 2 L2 () then there is some f 2 L2 () for which  N1  1 X    f ı T n  f  ! 0 as  N  nD0

N ! 1;

2

clearly f then R R has the property that k f  f ı Tk2 D 0 and f d D f d. Around the same time, Birkhoff [13] showed the more delicate pointwise ergodic theorem: for any g 2 L1 () there is some g 2 L1 () for which N1 1 X g(T n x) ! g(x) a. e. ; N nD0

R again it is then clear that g(Tx) D g(x) a. e. and g d D R g d. The map T is called ergodic if the invariance condition forces the function (f or g) to be equal to a constant a. e. Thus an ergodic map has the property that the time or erP N1 godic average (1/N) nD0 f ı T j converges to the space R average d. An overview of ergodic theorems and their many extensions may be found in the article  Ergodic Theorems. Thus ergodic theory at its most basic level makes strong statements about the asymptotic behavior of orbits of a dynamical system as seen by observables (measurable functions on the space X). Applying the ergodic theorem to the indicator function of a measurable set A shows that ergodicity guarantees that a. e. orbit spends an asymptotic proportion of time in A equal to the volume (A) of that set (as measured by the invariant measure). This points to the start of the pervasive connections between ergodic theory and number theory – but as this and other articles relate, the connections extend far beyond this.

Ergodic Theory: Interactions with Combinatorics and Number Theory

Frequency of Returns

First Digits

In this section we illustrate the way in which a dynamical point of view may unify, explain and extend quite disparate results from number theory.

The astronomer Newcomb [67] noted that the first digits of large collections of numerical data that are not dimensionless have a specific and non-uniform distribution:

Normal Numbers Borel [15] showed (as a consequence of what became the Borel–Cantelli Lemma in probability) that a. e. real number (with respect to Lebesgue measure) is normal to every base: that is, has the property that any block of k digits in the base-r expansion appears with asymptotic frequency rk . Continued Fraction Digits Analogs of normality results for the continued fraction expansion of real numbers were found by Khinchin, Kuz’min, Lévy and others. Any irrational x 2 [0; 1] has a unique expansion as a continued fraction 1

a1 (x) C a2 (x) C

1 a3 (x) C   

and, just as in the case of the familiar base-r expansion, it turns out that the digits (a n (x)) obey precise statistical rules for a. e. x. Gauss conjectured that the appearance of individual digits would obey the law 1 jfk : 1  k  N; a k (x) D jgj N 2 log(1 C j)  log j  log(2 C j) ! : log 2

(1)

lim (a1 (x)a2 (x) : : : a n (x))1/n D

log n/ log 2 1  Y (n C 1)2 nD1

n(n C 2)

D 2:68545 : : :

n!1

1 jfk : 1  k  N; fa k g 2 [a; b]gj ! (b  a) N as N ! 1 ;

Z 1 N 1 X f (a k ) ! f (t)dt N 0

as

N !1

kD1

for all continuous functions f . Weyl showed that the sequence fn˛g is equidistributed if and only if ˛ is irrational. This result was refined and extended in many directions; for example, Hlawka [44] and others found rates for the convergence in terms of the discrepancy of the sequence, Weyl [110] proved equidistribution for fn2 ˛g, and Vinogradov for fp n ˛g where pn is the nth prime.

for a. e. x :

Lévy [60] showed that the denominator q n (x) of the nth convergent (p n (x))/(q n (x)) (the rational obtained by truncating the continued fraction expansion of x at the nth term) grows at a specific exponential rate, lim

Weyl [109] (and, separately, Bohl [14] and Sierpi´nski [91]) found an important instance of equidistribution. Writing fg for the fractional part, a sequence (a n ) of real numbers is said to be equidistributed modulo 1 if, for any interval [a; b] [0; 1),

equivalently if

This was eventually proved by Kuz’min [54] and Lévy [59], and the probability distribution of the digits is the Gauss– Kuz’min law. Khinchin [50] developed this further, showing for example that n!1

This is now known as Benford’s Law, following his popularization and possible rediscovery of the phenomenon [6]. In both cases, this was an empirical observation eventually made rigorous by Hill [42]. Arnold (App. 12 in [3]), pointed out the dynamics behind this phenomena in certain cases, best illustrated by the statistical behavior of the sequence 1; 2; 4; 8; 1; 3; 6; 1; : : : of first digits of powers of 2. Empirically, the digit 1 appears about 30% of the time, while the digit 9 appears about 5% of the time. Equidistribution

1

xD

“The law of probability of the occurrence of numbers is such that all mantissæ of their logarithms are equally probable.”

1 2 log q n (x) D n 12 log 2

for a. e. x :

The Ergodic Context All the results of this section are manifestations of various kinds of convergence of ergodic averages. Borel’s theorem on normal numbers is an immediate consequence of the fact that Lebesgue measure on [0; 1) is invariant and ergodic for the map x 7! bx modulo 1 with b > 2. The asymptotic properties of continued fraction digits are all

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a consequence of the fact that the Gauss measure defined by Z dx 1 (A) D for A [0; 1] log 2 A 1 C x ˚ is invariant and ergodic for the Gauss map x 7! x1 , and the orbit of an irrational number under the Gauss map determine the digits appearing in the continued fraction expansion much as the orbit under the map x 7! bx (mod 1) determines the digits in the base b expansion. The results on equidistribution and the frequency of first digits are related to ergodic averaging of a different sort. For example, writing R˛ (t) D t C ˛ modulo 1 for the circle rotation by ˛, the first digit of 2n is the digit j if and only if log10 j  Rlog10 (2) (0) < log10 ( j C 1) : Thus the asymptotic frequency of appearance concerns the orbit of a specific point. In order to see what this means, consider a continuous map T : X ! X of a compact metric space (X; d). The space M(T) of Borel probability measures on the Borel -algebra of (X; d) is a non-empty compact convex set in the weak*-topology, each extreme point is an ergodic measure for T, and these ergodic measures are mutually singular. If M(T) is not a singleton and 1 ; 2 2 M(T) are distinct ergodic measures, then for R R a continuous function f with X f d1 ¤ X f d2 it is P N1 f (T n x) must clear thatRthe ergodic averages N1 nD0 R converge to X f d1 a. e. with respect to 1 and to X f d2 a. e. with respect to 2 . Thus the presence of many invariant measures for a continuous map means that ergodic averages along the orbits of specific points need not converge to the space average with respect to a chosen invariant measure. In the extreme situation of unique ergodicity (a single invariant measure, which is necessarily an extreme point of M(T) and hence ergodic) the convergence of ergodic averages is much more uniform. Indeed, if T is uniquely ergodic with M(T) D fg then, for any continuous function f : X ! R, Z N1 1 X n f (T x) ! f d uniformly in x N nD0 X (see Oxtoby [69]). The circle rotation R˛ is uniquely ergodic for irrational ˛, leading to the equidistribution results. The ergodic viewpoint on equidistribution also places equidistribution results in a wider context. Weyl’s result that fn2 ˛g (indeed, the fractional part of any polynomial

with at least one irrational coefficient) is equidistributed for irrational ˛ was given another proof by Furstenberg [29] using the notion of unique ergodicity. These methods were then used in the study of nilsystems (translations on quotients of nilpotent Lie groups) by Auslander, Green and Hahn [4] and Parry [70], and these nilsystems play an essential role for polynomial (and other non-conventional) ergodic averaging (see Host and Kra [45] and Leibman [58] in the polynomial case; Host and Kra [46] in the multiple linear case). Remarkably, nilsystems are starting to play a role within combinatorics – an example is the work on the asymptotic number of 4-step arithmetic progressions in the primes by Green and Tao [40]. Pointwise ergodic theorems have also been found along sequences other than N; notably for integer-valued polynomials and along the primes for L2 functions by Bourgain [16,17]. For more details, see the survey paper of del Junco on ergodic theorems. Ergodic Ramsey Theory and Recurrence In 1927 van der Waerden proved a conjecture attributed to Baudet: if the natural numbers are written as a disjoint union of finitely many sets, N D C1 t C2 t    t C r ;

(2)

then there must be one set Cj that contains arbitrarily long arithmetic progressions. That is, there is some j 2 f1; : : : ; rg such that for any k > 1 there are a > 1 and n > 1 with a; a C n; a C 2n; : : : ; a C (k  1)n 2 C j : The original proof appears in van der Waerden’s paper [103], and there is a discussion of how he found the proof in [104]. Work of Furstenberg and Weiss [34] and others placed the theorem of van der Waerden in the context of topological dynamics, giving alternative proofs. Specifically, van der Waerden’s theorem is a consequence of topological multiple recurrence: the return of points under iteration in a topological dynamical system close to their starting point along finite sequences of times. The same approach readily gives dynamical proofs of Rado’s extension [76] of van der Waerden’s theorem, and of Hindman’s theorem [43]. The theorems of Rado and Hindman introduce a new theme: given a set A D fn1 ; n2 ; : : : g of natural numbers, write FS(A) for the set of numbers obtained as finite sums n i 1 C    C n i j with i1 < i2 <    < i j . Rado showed that for any large n there is some Cs containing some FS(A) for a set A of cardinality n. Hindman showed that there is some Cs containing some FS(A) for an infinite set A.

Ergodic Theory: Interactions with Combinatorics and Number Theory

In the theorem of van der Waerden, it is clear that for any reasonable notion of “proportion” or “density” one of the sets Cj must occupy a positive proportion of N. A set A N is said to have positive upper density if there are sequences (M i ) and (N i ) with N i  M i ! 1 as i ! 1 such that lim

i!1

1 jfa 2 A: M i < a < N i gj > 0 : Ni  Mi

Erd˝os and Turán [26] conjectured the stronger statement that any subset of N with positive upper density must contain arbitrary long arithmetic progressions. This statement was shown for arithmetic progressions of length 3 by Roth [79] in 1952, then for length 4 by Szemerédi [96] in 1969. The general result was eventually proved by Szemerédi [97] in 1975 in a lengthy and extremely difficult argument. Furstenberg saw that Szemerédi’s Theorem would follow from a deep extension of the Poincaré recurrence phenomena described in Sect. “Ergodic Theory” and proved that extension [30] (see also the survey article by Furstenberg, Katznelson and Ornstein [33]). The multiple recurrence result of Furstenberg says that for any measure-preserving system (X; B; ; T) and set A 2 B with (A) > 0, and for any k 2 N, lim inf

NM!1



N X

1 NMC1

   A \ T n A \ T 2n A \    \ T kn A > 0 :

nDM

An immediate consequence is that in the same setting there must be some n > 1 for which   (3)  A \ T n A \ T 2n A \    \ T kn A > 0 : A general correspondence principle, due to Furstenberg, shows that statements in combinatorics like Szemerédi’s Theorem are equivalent to statements in ergodic theory like (3). This opened up a significant new field of ergodic Ramsey theory, in which methods from dynamical systems and ergodic theory are used to produce new results in infinite combinatorics. For an overview, see the articles  Ergodic Theory on Homogeneous Spaces and Metric Number Theory,  Ergodic Theory: Rigidity,  Ergodic Theory: Recurrence and the survey articles of Bergelson [7,8,10]. The field is too large to give an overview here, but a few examples will give a flavor of some of the themes. Call a set R Z a set of recurrence if, for any finite measure-preserving invertible transformation T of a finite

measure space (X; B; ) and any set A 2 B with (A) > 0, there are infinitely many n 2 R for which (A \ T n A) > 0. Thus Poincaré recurrence is the statement that N is a set of recurrence. Furstenberg and Katznelson [31] showed that if T1 ; : : : ; Tk form a family of commuting measurepreserving transformations and A is a set of positive measure, then lim inf N!1

N1 1 X (T1n A \    \ Tkn A) > 0 : N nD0

This remarkable multiple recurrence implies a multi-dimensional form of Szemerédi’s theorem. Recently, Gowers has found a non-ergodic proof of this [38]. Furstenberg also gave an ergodic proof of Sárközy’s theorem [80]: if p 2 Q[t] is a polynomial with p(Z) Z and p(0) D 0, then fp(n)gn>0 is a set of recurrence. This was extended to multiple polynomial recurrence by Bergelson and Leibman [11]. Topology and Coloring Theorems ˇ The existence of idempotent ultrafilters in the Cech–Stone compactification ˇN gives rise to an algebraic approach to many questions in topological dynamics (this notion has its origins in the work of Ellis [24]). Using these methods, results like Hindman’s finite sums theorem find elegant proofs, and many new results in combinatorics have been found. For example, in the partition (2) there must be one set Cj containing a triple x; y; z solving x  y D z2 . A deeper application is to improve a strengthening of Kronecker’s theorem. To explain this, recall that a set S is called IP if there is a sequence (n i ) of natural numbers (which do not need to be distinct) with the property that S contains all the terms of the sequence and all finite sums of terms of the sequence with distinct indices. A set S is called IP if it has non-empty intersection with every IP set, and  if there is some t 2 Z for which S  t is a set S is called IPC  ) is an extreme form of ‘fatness’ IP . Thus being IP (or IPC for a set. Now let 1; ˛1 ; : : : ; ˛ k be numbers that are linearly independent over the rationals, and for any d 2 N and kd non-empty intervals I i j [0; 1] (1  i  d; 1  j  k), let D D fn 2 N : fn i ˛ j g 2 I i j for all i; jg : Kronecker showed that if d D 1 then D is non-empty; Hardy and Littlewood showed that D is infinite, and Weyl showed that D has positive density. Bergelson [9] uses these algebraic methods to improve the result by showing  set. that D is an IPC

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Polynomialization and IP-sets

Sets of Primes

As mentioned above, Bergelson and Leibman [11] extended multiple recurrence to a polynomial setting. For example, let

In a remarkable development, Szemerédi’s theorem and some of the ideas behind ergodic Ramsey theory joined results of Goldston and Yıldırım [37] in playing a part in Green and Tao’s proof [39] that the set of primes contains arbitrarily long arithmetic progressions. This profound result is surveyed from an ergodic point of view in the article of Kra [53]. As with Szemerédi’s theorem itself, this result has been extended to a polynomial setting by Tao and Ziegler [101]. Given integer-valued polynomials f1 ; : : : ; f k 2 Z[t] with

fp i; j : 1  i  k; 1  j  tg be a collection of polynomials with rational coefficients and p i; j (Z) Z, p i; j (0) D 0. Then if (A) > 0, we have 11 1 0 0 N k t X \ Y 1 p i; j (n) A AA > 0 : @ @ Tj lim inf N!1 N nD1 iD1

jD1

Using the Furstenberg correspondence principle, this gives a multi-dimensional polynomial Szemerédi theorem: If P : Zr ! Z` is a polynomial mapping with the property that P(0) D 0, and F Zr is a finite configuration, then any set S Z` of positive upper Banach density contains a set of the form u C P(nF) for some u 2 Z` and n 2 N. In a different direction, motivated in part by Hindman’s theorem, the multiple recurrence results generalize to IP-sets. Furstenberg and Katznelson [32] proved a linear IP-multiple recurrence theorem in which the recurrence is guaranteed to occur along an IP-set. A combinatorial proof of this result has been found by Nagle, Rödl and Schacht [66]. Bergelson and McCutcheon [12] extended these results by proving a polynomial IP-multiple recurrence theorem. To formulate this, make the following definitions. Write F for the family of non-empty finite subsets of N, so that a sequence indexed by F is an IP-set. More generally, an F -sequence (n˛ )˛2F taking values in an abelian group is called an IP-sequence if n˛[ˇ D n˛ C nˇ whenever ˛ \ ˇ D ;. An IP-ring is a set of the form S F (1) D f i2ˇ ˛ i : ˇ 2 F g where ˛1 < ˛2 <    is a sequence in F , and ˛ < ˇ means a < b for all a 2 ˛, b 2 ˇ; write F<m for the set of m-tuples (˛1 ; : : : ; ˛m ) from F m with ˛ i < ˛ j for i < j. Write PE(m; d) for the collection of all expressions of the form T(˛1 ; : : : ; ˛m ) D (b) p i ((n ˛ j )1bk; 1 jm ) Qr m, T ; (˛1 ; : : : ; ˛m ) 2 (F [ ;)< iD1 i where each pi is a polynomial in a k  m matrix of variables with integer coefficients and zero constant term with degree  d. Then for every m; t 2 N, there is an IP-ring F (1) , and an a D a(A; m; t; d) > 0, such that, for every set of polynomial expressions fS0 ; : : : ; S t g PE(m; d), ! t \ 1 S i (˛1 ; : : : ; ˛m ) A > 0 : IP  lim  m (˛1 ;:::;˛ m )2(F (1) )<

iD0

There are a large number of deep combinatorial consequences of this result, not all of which seem accessible by other means.

f1 (0) D    D f k (0) D 0 and any " > 0, Tao and Ziegler proved that there are infinitely many integers x; m with 1  m  x " for which x C f1 (m); : : : ; x C f k (m) are primes. Orbit-Counting as an Analogous Development Some of the connections between number theory and ergodic theory arise through developments that are analogous but not directly related. A remarkable instance of this concerns the long history of attempts to count prime numbers laid alongside the problem of counting closed orbits in dynamical systems. Counting Orbits and Geodesics Consider first the fundamental arithmetic function (X) D jfp  X : p is primegj. Tables of primes prepared by Felkel and Vega in the 18th century led Legendre to suggest that (X) is approximately x/(log(X)  1:08). Gauss, using both computational evidence and a heuristic argument, suggested that (X) is approximated by Z

X

li(X) D 2

dt : log t

Both of these suggestions imply the well-known asymptotic formula (X)

X : log X

(4)

Riemann brought the analytic ideas of Dirichlet and Chebyshev (who used the zeta function to find a weaker version of (4) with upper and lower bounds for the quan(X) log(X) ) to bear by proposing that the zeta function tity X (s) D

1 Y X 1 1 D ; 1  ps s n p nD1

(5)

Ergodic Theory: Interactions with Combinatorics and Number Theory

already studied by Euler, would connect properties of the primes to analytic methods. An essential step in these developments, due to Riemann, is the meromorphic extension of  from the region <(s) > 1 in (5) to the whole complex plane and a functional equation relating the value of the extension at s to the value at 1  s. Moreover, Riemann showed that the extension has readily understood real zeros, and that all the other zeros he could find were symmetric about <(s) D 12 . The Riemann hypothesis asserts that zeros in the region 0 < <(s) < 1 all lie on the line <(s) D 12 , and this remains open. Analytic properties of the Riemann zeta function were used by Hadamard and de la Vallée Poussin to prove (4), the Prime Number Theorem, in 1896. Tauberian methods developed by Wiener and Ikehara [111] later gave different approaches to the Prime Number Theorem. These ideas initiated the widespread use of zeta functions in several parts of mathematics, but it was not until the middle of the 20th century that Selberg [88] introduced a zeta function dealing directly with quantities arising in dynamical systems: the lengths of closed geodesics on surfaces of constant curvature 1. The geodesic flow acts on the unit tangent bundle to the surface by moving a point and unit tangent vector at that point along the unique geodesic they define at unit speed. Closed geodesics are then in one-to-one correspondence with periodic orbits of the associated geodesic flow on the unit tangent bundle, and it is in this sense that the quantities are dynamical. The function defined by Selberg takes the form Z(s) D

1  YY

 1  e(sCk)jj ;

 kD0

in which  runs over all the closed geodesics, and jj denotes the length of the geodesic. In a direct echo of the Riemann zeta function, Selberg found an analytic continuation to the complex plane, and showed that the zeros of Z lie on the real axis or on the line <(s) D 12 (the analogue of the Riemann hypothesis for Z; see also the paper of Hejhal [41]). The zeros of Z are closely connected to the eigenvalues for the Laplace–Beltrami operator, and thus give information about the lengths of closed geodesics via Selberg’s trace formula in the same paper. Huber [47] and others used this approach to give an analogue of the prime number theorem for closed geodesics – a prime orbit theorem. Sinai [92] considered closed geodesics on a manifold M with negative curvature bounded between R2 and r2 , and found the bounds (dim(M)  1) r  lim inf T!1

log (T) T

 lim sup T!1

log (T) T

 (dim(M)  1)R for the number (T) of closed geodesics of multiplicity one with length less than T, analogous to Chebyshev’s result. The essential dynamical feature behind the geodesic flow on a manifold of negative curvature is that it is an example of an Anosov flow [2]. These are smooth dynamical R-actions (equivalently, first-order differential equations on Riemannian manifolds) with the property that the tangent bundle has a continuously varying splitting into a direct sum E u ˚ E s ˚ E o and the action of the differential of the flow acts on Eu as an exponential expansion, on Es as an exponential contraction, Eo is the one-dimensional bundle of vectors that are tangent to orbits, and the expansion and contraction factors are bounded. In the setting of Anosov flows, the natural orbit counting function is (X) D jf :  a closed orbit of length jj  Xgj. Margulis [62,63] generalized the picture to weak-mixing Anosov flows by showing a prime orbit theorem of the form (X)

e htop X htop X

(6)

for the counting function (X) D jf :  a closed orbit of length jj  Xgj where as before htop denotes the topological entropy of the flow. Integral to Margulis’ work is a result on the spatial distribution of the closed geodesics reflected in a flow-invariant probability measure, now called the Margulis measure. Anosov also studied discrete dynamical systems with similar properties: diffeomorphisms of compact manifolds with a similar splitting of the tangent space (though in this setting Eo disappears). The archetypal Anosov diffeomorphism is a hyperbolic toral automorphism of the sort considered in Subsect. “Orbit Growth and Convergence”; for such automorphisms of the 2-torus Adler and Weiss [1] constructed Markov partitions, allowing the dynamics of the toral automorphism to be modeled by a topological Markov shift, and used this to determine when two such automorphisms are measurably isomorphic. Sinai [93], Ratner [77], Bowen [18,20] and others developed the construction of Markov partitions in general for Anosov diffeomorphisms and flows. Around the same time, Smale [95] introduced a more permissive hyperbolicity axiom for diffeomorphisms, Axiom A. Maps satisfying Axiom A are diffeomorphisms satisfying the same hypothesis as that of Anosov diffeomor-

319

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phisms, but only on the set of points that return arbitrarily close under the action of the flow (or iteration of the map). Thus Markov partitions, and with them associated transfer operators became a substitute for the geometrical Laplace–Beltrami operators of the setting considered by Selberg. Bowen [19] extended the uniform distribution result of Margulis to this setting and found an analogue of Chebychev’s theorem for closed orbits. Parry [71] (in a restricted case) and Parry and Pollicott [73] went on to prove the prime orbit theorem in this more general setting. The methods are an adaptation of the Ikehara–Wiener– Tauberian approach to the prime number theorem. Thus many facets of the prime number theorem story find their echoes in the study of closed orbits for hyperbolic flows: the role played by meromorphic extensions of suitable zeta functions, Tauberian methods, and so on. Moreover, related results from number theory have analogues in dynamics, for example Mertens’ theorem [65] in the work of Sharp [89] and Noorani [68] and Dirichlet’s theorem in work of Parry [72]. The “elementary” proof (not using analytic methods) of the prime number theorem by Erd˝os [25] and Selberg [87] (see the survey by Goldfeld [36] for the background to the results and the unfortunate priority dispute) has an echo in some approaches to orbit-counting problems from an elementary (non-Tauberian) perspective, including work of Lalley [56] on special flows and Everest, Miles, Stevens and the author [27] in the algebraic setting. In a different direction Lalley [55] found orbit asymptotics for closed orbits satisfying constraints in the Axiom A setting without using Tauberian theorems. His more direct approach is still analytic, using complex transfer operators (the same objects used to by Parry and Pollicott to study the dynamical zeta function at complex values) and indeed somewhat parallels a Tauberian argument. Further resonances with number theory arise here. For example, there are results on the distribution of closed orbits for group extensions (analogous to Chebotarev’s theorem) and for orbits with homological constraints (see Sharp [90], Katsuda and Sunada [49]). Of course the great diversity of dynamical systems subsumed in the phrase “prime orbit theorem” creates new problems and challenges, and in particular if there is not much geometry to work with then the reliance on Markov partitions and transfer operators makes it difficult to find higher-order asymptotics. Dolgopyat [22] has nonetheless managed to push the Markov methods to obtain uniform bounds on iterates of the associated transfer operators to the region <(s) > 0 with 0 < 1. This result has wide implications; an exam-

ple most relevant to the analogy with number theory is the work of Pollicott and Sharp [75] in which Dolgopyat’s result is used to show that for certain geodesic flows there is a two-term prime orbit theorem of the form     (X) D li e htop X C O e c X for some c < htop . For non-positive curvature manifolds less is known: Knieper [51] finds upper and lower bounds for the function counting closed geodesics on rank-1 manifolds of non-positive curvature of the form A

eh X  (X)  Be h X X

for constants A; B > 0. Counting Orbits for Group Endomorphisms A prism through which to view some of the deeper issues that arise in Subsect. “Counting Orbits and Geodesics” is provided by group endomorphisms. The price paid for having simple closed formulas for all the quantities involved is of course a severe loss of generality, but the diversity of examples illustrates many of the phenomena that may be expected in more general settings when hyperbolicity is lost. Consider an endomorphism T : X ! X of a compact group with the property that Fn (T) < 1 for all n > 1. The number of closed orbits of length n under T is then On (T) D

1X (n/d)Fd (T) : n

(7)

djn

In simple situations (hyperbolic toral automorphisms for example) it is straightforward to show that T (X) D jf :  a closed orbit under T of length  Xgj

e(XC1)htop (T) : X

(8)

Waddington [106] considered quasihyperbolic toral automorphisms, showing that the asymptotic (8) in this case is multiplied by an explicit almost-periodic function bounded away from zero and infinity. This result has been extended further into non-hyperbolic territory, which is most easily seen via the socalled connected S-integer dynamical systems introduced by Chothi, Everest and the author [21]. Fix an algebraic number field K with set of places P(K) and set of infinite places P1 (K), an element of infinite multiplicative order

Ergodic Theory: Interactions with Combinatorics and Number Theory

 2 K , and a finite set S P(K) n P1 (K) with the property that jjw  1 for all w … S [ P1 (K). The associated ring of S-integers is R S D fx 2 K : jxjw  1 for all w … S [ P1 (K)g : Let X be the compact character group of RS , and define the endomorphism T : X ! X to be the dual of the map x 7!  x on RS . Following Weil [108], write Kw for the completion at w, and for w finite, write rw for the maximal compact subring of Kw . Notice that if S D P then R S D K and Fn (T) D 1 for all n > 1 by the product formula for A-fields. As the set S shrinks, more and more periodic orbits come into being, and if S is as small as possible (given ) then the resulting system is (more or less) hyperbolic or quasi-hyperbolic. For S finite, it turns out that there are still sufficiently many periodic orbits to have the growth rate result (10), but the asymptotic (8) is modified in much the same way as Waddington observed for quasi-hyperbolic toral automorphisms: XT (X) >0 (9) e(XC1)htop (T) and there is an associated pair (X  ; a T ), where X  is a compact group and a T 2 X  , with the property that if Nj a T converges in X  as j ! 1, then there is convergence in (9). A simple special case will illustrate this. Taking K D Q,  D 2 and S D f3g gives a compact group endomorphism T with lim inf X!1

Fn (T) D (2n  1)j2n  1j3 : For this example the results of [21] are sharper: The expression in (9) converges along (X j ) if and only if 2 X j converges in the ring of 3-adic integers Z3 , the expression has uncountably many limit points, and the upper and lower limits are transcendental. Similarly, the dynamical analogue of Mertens’ theorem found by Sharp may be found for S-integer systems with S finite. Writing X 1 MT (N) D ; e h(T)jj

Diophantine Analysis as a Toolbox Many problems in ergodic theory and dynamical system exploit ideas and results from number theory in a direct way; we illustrate this by describing a selection of dynamical problems that call on particular parts of number theory in an essential way. The example of mixing in Subsect. “Mixing and Additive Relations in Fields” is particularly striking for two reasons: the results needed from number theory are relatively recent, and the ergodic application directly motivated a further development in number theory. Orbit Growth and Convergence The analysis of periodic orbits – how their number grows as the length grows and how they spread out through space – is of central importance in dynamics (see Katok [48] for example). An instance of this is that for many simple kinds of dynamical systems T : X ! X (where T is a continuous map of a compact metric space (X; d)) the logarithmic growth rate of the number of periodic points exists and coincides with the topological entropy h(T) (an invariant giving a quantitative measure of the average rate of growth in orbit complexity under T). That is, writing F n (T) D jfx 2 X : T n x D xgj ; we find 1 log Fn (T) ! htop (T) n

for many of the simplest dynamical systems. For example, if X D T r is the r-torus and T D TA is the automorphism of the torus corresponding to a matrix A in GLr (Z), then T A is ergodic with respect to Lebesgue measure if and only if no eigenvalue of A is a root of unity. Under this assumption, we have Fn (TA ) D

r Y

MT (N) D k T log N C C T C O (1/N) :

Without the restriction that K D Q, it is shown that there are constants k T 2 Q, CT and ı > 0 with MT (N) D k T log N C C T C O(N ı ) :

jni  1j

iD1

and

jjN

it is shown in [21] that for an ergodic S-integer map T with K D Q and S finite, there are constants k T 2 Q and CT such that

(10)

htop (TA ) D

r X

log maxf1; j i jg

(11)

iD1

where 1 ; : : : ; r are the eigenvalues of A. It follows that the convergence in (10) is clear under the assumption that T A is hyperbolic (that is, no eigenvalue has modulus one). Without this assumption the convergence is less clear: for r > 4 the automorphism T A may be ergodic without being hyperbolic. That is, while no eigenvalues are unit

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roots some may have unit modulus. As pointed out by Lind [61] in his study of these quasihyperbolic automorphisms, the convergence (10) does still hold for these systems, but this requires a significant Diophantine result (the theorem of Gel’fond [35] suffices; one may also use Baker’s theorem [5]). Even further from hyperbolicity lie the family of S-integer systems [21,107]; their orbit-growth properties are intimately tied up with Artin’s conjecture on primitive roots and prime divisors of linear recurrence sequences. Mixing and Additive Relations in Fields The problem of higher-order mixing for commuting group automorphisms provides a striking example of the dialogue between ergodic theory and number theory, in which deep results from number theory have been used to solve problems in ergodic theory, and questions arising in ergodic theory have motivated further developments in number theory. An action T of a countable group  on a probability space (X; B; ) is called k-fold mixing or mixing on (k C 1) sets if    A0 \ T g 1 A1 \    \ T g k A k ! (A0 )    (A k ) (12) as g i g 1 j ! 1 for i ¤ j with the convention that g0 D 1 , for any sets A0 ; : : : ; A k 2 B; g n ! 1 in  means that for any finite set F  there is an N with n > N H) g n … F. For k D 1 the property is called simply mixing. This notion for single transformations goes back to the foundational work of Rohlin [78], where he showed that ergodic group endomorphisms are mixing of all orders (and so the notion is not useful for distinguishing between group endomorphisms as measurable dynamical systems). He raised the (still open) question of whether any measure-preserving transformation can be mixing without being mixing of all orders. A class of group actions that are particularly easy to understand are the algebraic dynamical systems studied systematically by Schmidt [83]: here X is a compact abelian group, each T g is a continuous automorphism of X, and  is the Haar measure on X. Schmidt [82] related mixing properties of algebraic dynamical systems with  D Zd to statements in arithmetic, and showed that a mixing action on a connected group could only fail to mix in a certain way. Later Schmidt and the author [85] showed that for X

connected, mixing implies mixing of all orders. The proof proceeds by showing that the result is exactly equivalent to the following statement: if K is a field of characteristic zero, and G is a finitely generated subgroup of the multiplicative group K , then the equation a1 x1 C    C a n x n D 1

(13)

for fixed a1 ; : : : ; a n 2 K has a finite number of soluP tions x1 ; : : : ; x n 2 G for which no subsum i2I a i x i with I ¨ f1; : : : ; ng vanishes. The bound on the number of solutions to (13) follows from the profound extensions to W. Schmidt’s subspace theorem in Diophantine geometry [86] by Evertse and Schlickewei (see [28,81,102] for the details). The argument in [85] may be cast as follows: failure of k-fold mixing in a connected algebraic dynamical system implies (via duality) an infinite set of solutions to an equation of the shape (13) in some field of characteristic zero. The S-unit theorem means that this can only happen if there is some proper subsum that vanishes infinitely often. This infinite family of solutions to a homogeneous form of (13) with fewer terms can then be translated back via duality to show that the system fails to mix for some strictly lower order, proving that mixing implies mixing of all orders by induction. Mixing properties for algebraic dynamical systems without the assumption of connectedness are quite different, and in particular it is possible to have mixing actions that are not mixing of all orders. This is a simple consequence of the fact that the constituents of a disconnected algebraic dynamical system are associated with fields of positive characteristic, where the presence of the Frobenius automorphism can prevent higher-order mixing. Ledrappier [57] pointed this out via examples of the following shape. Let n 2 X D x 2 F2Z : x(aC1;b) C x(a;b) C x(a;bC1) D0

o (mod 2)

and define the Z2 -action T to be the natural shift action, (T (n;m) x)(a;b) D x(aCn;bCm) : It is readily seen that this action is mixing with respect to the Haar measure. The condition x(aC1;b) C x(a;b) C x(a;bC1) D 0 (mod 2) implies that, for any k > 1, x(0;2k )

! 2k X 2k D x( j;0) D x(0;0) Cx(2k ;0) j jD0

(mod 2) (14)

Ergodic Theory: Interactions with Combinatorics and Number Theory

since every entry in the 2 k th row of Pascal’s triangle is even apart from the first and the last. Now let A D fx 2 X : x(0;0) D 0g and let x 2 X be any element with x(0;0) D 1. Then X is the disjoint union of A and A C x , so (A) D (A C x ) D

1 2

stants b1 ; : : : ; b m 2 K and some (g1 ; : : : ; g m ) 2 G m with the following properties:  g j ¤ 1 for 1  j  m;  g i g 1 j ¤ 1 for 1  i < j  m;  there are infinitely many k for which

:

However, (14) shows that x 2 A \ T(2k ;0) A H) x 2 T(0;2k ) A ; so A \ T(2k ;0) A \ T(0;2k ) (A C x ) D ; for all k > 1, which shows that T cannot be mixing on three sets. The full picture of higher-order mixing properties on disconnected groups is rather involved; see Schmidt’s monograph [83]. A simple illustration is the construction by Einsiedler and the author [23] of systems with any prescribed order of mixing. When such systems fail to be mixing of all orders, they fail in a very specific way – along dilates of a specific shape (a finite subset of Zd ). In the example above, the shape that fails to mix is f(0; 0); (1; 0); (0; 1)g. This gives an order of mixing as detected by shapes; computing this is in principle an algebraic problem. On the other hand, there is a more natural definition of the order of mixing, namely the largest k for which (12) holds; computing this is in principle a Diophantine problem. A conjecture emerged (formulated explicitly by Schmidt [84]) that for any algebraic dynamical system, if every set of cardinality r > 2 is a mixing shape, then the system is mixing on r sets. This question motivated Masser [64] to prove an appropriate analogue of the S-unit theorem on the number of solutions to (13) in positive characteristic as follows. Let H be a multiplicative group and fix n 2 N. An infinite subset A H n is called broad if it has both of the following properties:  if h 2 H and 1  j  n, then there are at most finitely many (a1 ; : : : ; a n ) in A with a j D h;  if n > 2, h 2 H and 1  i < j  n then there are at most finitely many (a1 ; : : : ; a n ) 2 H with a i a1 j D h. Then Masser’s theorem says the following. Let K be a field of characteristic p > 0, let G be a finitely-generated subgroup of K and suppose that the equation a1 x1 C    C a n x n D 1 has a broad set of solutions (x1 ; : : : ; x n ) 2 G n for some constants a1 ; : : : ; a n 2 K . Then there is an m  n, con-

k b1 g1k C b2 g2k C    C b m g m D1:

The proof that shapes detect the order of mixing in algebraic dynamics then proceeds much as in the connected case. Future Directions The interaction between ergodic theory, number theory and combinatorics continues to expand rapidly, and many future directions of research are discussed in the articles  Ergodic Theory on Homogeneous Spaces and Metric Number Theory,  Ergodic Theory: Rigidity and  Ergodic Theory: Recurrence. Some of the directions most relevant to the examples discussed in this article include the following. The recent developments mentioned in Subsect. “Sets of Primes” clearly open many exciting prospects involving finding new structures in arithmetically significant sets (like the primes). The original conjecture of Erd˝os and P 1 Turán [26] asked if a2A N a D 1 is sufficient to force the set A to contain arbitrary long arithmetic progressions, and remains open. This would of course imply both Szemerédi’s theorem [97] and the result of Green and Tao [39] on arithmetic progressions in the primes. More generally, it is clear that there is still much to come from the dialogue subsuming the four parallel proofs of Szemerédi’s: one by purely combinatorial methods, one by ergodic theory, one by hypergraph theory, and one by Fourier analysis and additive combinatorics. For an overview, see the survey papers of Tao [98,99,100]. In the context of the orbit-counting results in Sect. “Orbit-Counting as an Analogous Development”, a natural problem is to on the one hand obtain finer asymptotics with better control of the error terms, and on the other to extend the situations that can be handled. In particular, relaxing the hypotheses related to hyperbolicity (or negative curvature) is a constant challenge. Bibliography Primary Literature 1. Adler RL, Weiss B (1970) Similarity of automorphisms of the torus. Memoirs of the American Mathematical Society, No. 98. American Mathematical Society, Providence

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2. Anosov DV (1967) Geodesic flows on closed Riemannian manifolds of negative curvature. Trudy Mat Inst Steklov 90:209 3. Arnol’d VI, Avez A (1968) Ergodic problems of classical mechanics. Translated from the French by A Avez. Benjamin Inc, New York 4. Auslander L, Green L, Hahn F (1963) Flows on homogeneous spaces. Annals of Mathematics Studies, No 53. Princeton University Press, Princeton 5. Baker A (1990) Transcendental number theory. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2nd edn 6. Benford F (1938) The law of anomalous numbers. Proc Amer Philos Soc 78:551–572 7. Bergelson V (1996) Ergodic Ramsey theory – an update. In: Ergodic theory of Zd actions (Warwick, 1993–1994), vol 228. London Math Soc Lecture Note Ser. Cambridge University Press, Cambridge, pp 1–61 8. Bergelson V (2000) Ergodic theory and Diophantine problems. In: Topics in symbolic dynamics and applications (Temuco, 1997), vol 279. London Math Soc Lecture Note Ser. Cambridge University Press, Cambridge, pp 167–205 9. Bergelson V (2003) Minimal idempotents and ergodic Ramsey theory. In: Topics in dynamics and ergodic theory, vol 310. London Math Soc, Lecture Note Series, Cambridge University Press, Cambridge, pp 8–39 10. Bergelson V (2006) Combinatorial and Diophantine applications of ergodic theory. In: Handbook of dynamical systems, vol 1B. Elsevier, Amsterdam, pp 745–869 11. Bergelson V, Leibman A (1996) Polynomial extensions of van der Waerden’s and Szemerédi’s theorems. J Amer Math Soc 9(3):725–753 12. Bergelson V, McCutcheon R (2000) An ergodic IP polynomial Szemerédi theorem. Mem Amer Math Soc 146(695):viii–106 13. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Natl Acad Sci USA 17:656–660 14. Bohl P (1909) Über ein in der Theorie der säkularen Störungen vorkommendes Problem. J Math 135:189–283 15. Borel E (1909) Les probabilités denombrables et leurs applications arithmetiques. Rend Circ Math Palermo 27:247–271 16. Bourgain J (1988) An approach to pointwise ergodic theorems. In: Geometric aspects of functional analysis (1986/87), vol 1317. Lecture Notes in Math, pp 204–223. Springer, Berlin 17. Bourgain J (1988) On the maximal ergodic theorem for certain subsets of the integers. Israel J Math 61(1):39–72 18. Bowen R (1970) Markov partitions for Axiom A diffeomorphisms. Amer J Math 92:725–747 19. Bowen R (1972) The equidistribution of closed geodesics. Amer J Math 94:413–423 20. Bowen R (1973) Symbolic dynamics for hyperbolic flows. Amer J Math 95:429–460 21. Chothi V, Everest G, Ward T (1997) S-integer dynamical systems: periodic points. J Reine Angew Math 489:99–132 22. Dolgopyat D (1998) On decay of correlations in Anosov flows. Ann Math 147(2):357–390 23. Einsiedler M, Ward T (2003) Asymptotic geometry of nonmixing sequences. Ergodic Theory Dyn Syst 23(1):75–85 24. Ellis R (1969) Lectures on topological dynamics. Benjamin Inc, New York

˝ P (1949) On a new method in elementary number the25. Erdos ory which leads to an elementary proof of the prime number theorem. Proc Natl Acad Sci USA 35:374–384 ˝ P, Turán P (1936) On some sequences of integers. J Lon26. Erdos don Math Soc 11:261–264 27. Everest G, Miles R, Stevens S, Ward T (2007) Orbit-counting in non-hyperbolic dynamical systems. J Reine Angew Math 608:155–182 28. Evertse JH, Schlickewei HP (1999) The absolute subspace theorem and linear equations with unknowns from a multiplicative group. In: Number theory in progress, vol 1 (ZakopaneKo´scielisko, 1997). de Gruyter, Berlin, pp 121–142 29. Furstenberg H (1961) Strict ergodicity and transformation of the torus. Amer J Math 83:573–601 30. Furstenberg H (1977) Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J Analyse Math 31:204–256 31. Furstenberg H, Katznelson Y (1979) An ergodic Szemerédi theorem for commuting transformations. J Analyse Math 34:275–291 32. Furstenberg H, Katznelson Y (1985) An ergodic Szemerédi theorem for IP-systems and combinatorial theory. J Analyse Math 45:117–168 33. Furstenberg H, Katznelson Y, Ornstein D (1982) The ergodic theoretical proof of Szemerédi’s theorem. Bull Amer Math Soc 7(3):527–552 34. Furstenberg H, Weiss B (1979) Topological dynamics and combinatorial number theory. J Analyse Math 34:61–85 35. Gel0 fond AO (1960) Transcendental and algebraic numbers. Translated from the first Russian edition by Leo F Boron. Dover Publications, New York 36. Goldfeld D (2004) The elementary proof of the prime number theorem: an historical perspective. In: Number theory (2003). Springer, New York, pp 179–192 37. Goldston DA, Yıldırım CY (2005) Small gaps between primes I. arXiv:math.NT/0504336 38. Gowers WT (2007) Hypergraph regularity and the multidimensional Szemerédi Theorem. Ann of Math 166:897–946 39. Green B, Tao T (2004) The primes contain arbitrarily long arithmetic progressions. arXiv:math.NT/0404188 40. Green B, Tao T (2006) Linear equations in primes. arXiv: math.NT/0606088 41. Hejhal DA (1976) The Selberg trace formula and the Riemann zeta function. Duke Math J 43(3):441–482 42. Hill TP (1995) Base-invariance implies Benford’s law. Proc Amer Math Soc 123(3):887–895 43. Hindman N (1974) Finite sums from sequences within cells of a partition of N. J Comb Theory Ser A 17:1–11 44. Hlawka E (1964) Discrepancy and uniform distribution of sequences. Compositio Math 16:83–91 45. Host B, Kra B (2005) Convergence of polynomial ergodic averages. Israel J Math 149:1–19, Probability in mathematics 46. Host B, Kra B (2005) Nonconventional ergodic averages and nilmanifolds. Ann Math 161(1):397–488 47. Huber H (1959) Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen. Math Ann 138:1–26 48. Katok A (1980) Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst Hautes Études Sci Publ Math (51):137–173 49. Katsuda A, Sunada T (1990) Closed orbits in homology classes. Inst Hautes Études Sci Publ Math (71):5–32

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50. Khinchin AI (1964) Continued fractions. The University of Chicago Press, Chicago 51. Knieper G (1997) On the asymptotic geometry of nonpositively curved manifolds. Geom Funct Anal 7(4):755–782 52. Koopman B (1931) Hamiltonian systems and transformations in Hilbert spaces. Proc Natl Acad Sci USA 17:315–318 53. Kra B (2006) The Green-Tao theorem on arithmetic progressions in the primes: an ergodic point of view. Bull Amer Math Soc 43(1):3–23 54. Kuz’min RO (1928) A problem of Gauss. Dokl Akad Nauk, pp 375–380 55. Lalley SP (1987) Distribution of periodic orbits of symbolic and Axiom A flows. Adv Appl Math 8(2):154–193 56. Lalley SP (1988) The “prime number theorem” for the periodic orbits of a Bernoulli flow. Amer Math Monthly 95(5):385–398 57. Ledrappier F (1978) Un champ markovien peut étre d’entropie nulle et mélangeant. C R Acad Sci Paris Sér A-B 287(7):A561–A563 58. Leibman A (2005) Convergence of multiple ergodic averages along polynomials of several variables. Israel J Math 146:303– 315 59. Levy P (1929) Sur les lois de probabilité dont dependent les quotients complets et incomplets d’une fraction continue. Bull Soc Math France 57:178–194 60. Levy P (1936) Sur quelques points de la théorie des probabilités dénombrables. Ann Inst H Poincaré 6(2):153–184 61. Lind DA (1982) Dynamical properties of quasihyperbolic toral automorphisms. Ergodic Theory Dyn Syst 2(1):49–68 62. Margulis GA (1969) Certain applications of ergodic theory to the investigation of manifolds of negative curvature. Funkcional Anal i Priložen 3(4):89–90 63. Margulis GA (2004) On some aspects of the theory of Anosov systems. Springer Monographs in Mathematics. Springer, Berlin. 64. Masser DW (2004) Mixing and linear equations over groups in positive characteristic. Israel J Math 142:189–204 65. Mertens F (1874) Ein Beitrag zur analytischen Zahlentheorie. J Reine Angew Math 78:46–62 66. Nagle B, Rödl V, Schacht M (2006) The counting lemma for regular k-uniform hypergraphs. Random Structures Algorithms 28(2):113–179 67. Newcomb S (1881) Note on the frequency of the use of digits in natural numbers. Amer J Math 4(1):39–40 68. Noorani MS (1999) Mertens’ theorem and closed orbits of ergodic toral automorphisms. Bull Malaysian Math Soc 22(2):127–133 69. Oxtoby JC (1952) Ergodic sets. Bull Amer Math Soc 58:116– 136 70. Parry W (1969) Ergodic properties of affine transformations and flows on nilmanifolds. Amer J Math 91:757–771 71. Parry W (1983) An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions. Israel J Math 45(1):41–52 72. Parry W (1984) Bowen’s equidistribution theory and the Dirichlet density theorem. Ergodic Theory Dyn Syst 4(1):117–134 73. Parry W, Pollicott M (1983) An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann Math 118(3):573–591 74. Poincaré H (1890) Sur le probléme des trois corps et les équations de la Dynamique. Acta Math 13:1–270

75. Pollicott M, Sharp R (1998) Exponential error terms for growth functions on negatively curved surfaces. Amer J Math 120(5):1019–1042 76. Rado R (1933) Studien zur Kombinatorik. Math Z 36(1):424– 470 77. Ratner M (1973) Markov partitions for Anosov flows on n-dimensional manifolds. Israel J Math 15:92–114 78. Rohlin VA (1949) On endomorphisms of compact commutative groups. Izvestiya Akad Nauk SSSR Ser Mat 13:329–340 79. Roth K (1952) Sur quelques ensembles d’entiers. C R Acad Sci Paris 234:388–390 80. Sárközy A (1978) On difference sets of sequences of integers. III. Acta Math Acad Sci Hungar 31(3-4):355–386 81. Schlickewei HP (1990) S-unit equations over number fields. Invent Math 102(1):95–107 82. Schmidt K (1989) Mixing automorphisms of compact groups and a theorem by Kurt Mahler. Pacific J Math 137(2):371–385 83. Schmidt K (1995) Dynamical systems of algebraic origin, vol 128. Progress in Mathematics. Birkhäuser, Basel 84. Schmidt K (2001) The dynamics of algebraic Zd -actions. In: European Congress of Mathematics, vol I (Barcelona, 2000), vol 201. Progress in Math. Birkhäuser, Basel, pp 543–553 85. Schmidt K, Ward T (1993) Mixing automorphisms of compact groups and a theorem of Schlickewei. Invent Math 111(1):69– 76 86. Schmidt WM (1972) Norm form equations. Ann Math 96(2):526–551 87. Selberg A (1949) An elementary proof of the prime-number theorem. Ann Math 50(2):305–313 88. Selberg A (1956) Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J Indian Math Soc 20:47–87 89. Sharp R (1991) An analogue of Mertens’ theorem for closed orbits of Axiom A flows. Bol Soc Brasil Mat 21(2):205–229 90. Sharp R (1993) Closed orbits in homology classes for Anosov flows. Ergodic Theory Dyn Syst 13(2):387–408 ´ W (1910) Sur la valeur asymptotique d’une certaine 91. Sierpinski somme. Bull Intl Acad Polonmaise des Sci et des Lettres (Cracovie), pp 9–11 92. Sina˘ı JG (1966) Asymptotic behavior of closed geodesics on compact manifolds with negative curvature. Izv Akad Nauk SSSR Ser Mat 30:1275–1296 93. Sina˘ı JG (1968) Construction of Markov partitionings. Funkcional Anal i Priložen 2(3):70–80 94. Silverman JH (2007) The Arithmetic of Dynamical Systems, vol 241. Graduate Texts in Mathematics. Springer, New York 95. Smale S (1967) Differentiable dynamical systems. Bull Amer Math Soc 73:747–817 96. Szemerédi E (1969) On sets of integers containing no four elements in arithmetic progression. Acta Math Acad Sci Hungar 20:89–104 97. Szemerédi E (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith 27:199–245 98. Tao T (2005) The dichotomy between structure and randomness, arithmetic progressions, and the primes. arXiv: math/0512114v2 99. Tao T (2006) Arithmetic progressions and the primes. Collect Math, vol extra, Barcelona, pp 37–88 100. Tao T (2007) What is good mathematics? arXiv:math/ 0702396v1

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101. Tao T, Ziegler T (2006) The primes contain arbitrarily long polynomial progressions. arXiv:math.NT/0610050 102. van der Poorten AJ, Schlickewei HP (1991) Additive relations in fields. J Austral Math Soc Ser A 51(1):154–170 103. van der Waerden BL (1927) Beweis einer Baudet’schen Vermutung. Nieuw Arch Wisk 15:212–216 104. van der Waerden BL (1971) How the proof of Baudet’s conjecture was found. In: Studies in Pure Mathematics (Presented to Richard Rado). Academic Press, London, pp 251–260 105. von Neumann J (1932) Proof of the quasi-ergodic hypothesis. Proc Natl Acad Sci USA 18:70–82 106. Waddington S (1991) The prime orbit theorem for quasihyperbolic toral automorphisms. Monatsh Math 112(3):235–248 107. Ward T (1998) Almost all S-integer dynamical systems have many periodic points. Ergodic Theory Dyn Syst 18(2):471–486 108. Weil A (1967) Basic number theory. Die Grundlehren der mathematischen Wissenschaften, Band 144. Springer, New York 109. Weyl H (1910) Über die Gibbssche Erscheinung und verwandte Konvergenzphänomene. Rendiconti del Circolo Matematico di Palermo 30:377–407 110. Weyl H (1916) Uber die Gleichverteilung von Zahlen mod Eins. Math Ann 77:313–352 111. Wiener N (1932) Tauberian theorems. Ann Math 33(1):1–100

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Ergodic Theory, Introduction to

Ergodic Theory, Introduction to BRYNA KRA Northwestern University, Evanston, USA Ergodic theory lies at the intersection of many areas of mathematics, including smooth dynamics, statistical mechanics, probability, harmonic analysis, and group actions. Problems, techniques, and results are related to many other areas of mathematics, and ergodic theory has had applications both within mathematics and to numerous other branches of science. Ergodic theory has particularly strong overlap with other branches of dynamical systems; to clarify what distinguishes it from other areas of dynamics, we start with a quick overview of dynamical systems. Dynamical systems is the study of systems that evolve with time. The evolution of a dynamical system is given by some fixed rule that determines the states of the system a short time into the future, given only the present states. Reflecting the origins of the subject in celestial mechanics, the set of states through which the system evolves with time is called an orbit. Many important concepts in dynamical systems are related to understanding the orbits in the system: Do the orbits fill out the entire space? Do orbits collapse? Do orbits return to themselves? What are statistical properties of the orbits? Are orbits stable under perturbation? For simple dynamical systems, knowing the individual orbits is often sufficient to answer such questions. However, in most dynamical systems it is impossible to write down explicit formulae for orbits, and even when one can, many systems are too complicated to be understood just in terms of individual orbits. The orbits may only be known approximately, some orbits may appear to be random while others exhibit regular behavior, and varying the parameters defining the system may give rise to qualitatively different behaviors. The various branches of dynamical systems have been developed for understanding long term properties of the orbits. To make the notion of a dynamical system more precise, let X denote the collection of all states of the system. The evolution of these states is given by some fixed rule T : X ! X, dictating where each state x 2 X is mapped. An application of the transformation T : X ! X corresponds to the passage of a unit of time and for a positive integer n, the map T n D T ı T ı : : : ı T denotes the composition of T with itself taken n times. Given a state x 2 X, the orbit of the point x under the transformation T is the collection of iterates x; Tx; T 2 x; : : : of the state x. Thus the single transformation T generates a semigroup of transformations acting on X, by considering the

powers T n . More generally, one can consider a family of transformations fTt : t 2 Rg with each Tt : X ! X. Assuming that T0 (x) D x and that TtCs (x) D Tt (Ts (x)) for all states x 2 X and all real t and s, this models the evolution of continuous time in a system. Autonomous differential equations are examples of such continuous time systems. In almost all cases of interest, the space X has some underlying structure which is preserved by the transformation T. Different underlying structures X and different properties of the transformation T give rise to different branches of dynamical systems. When X is a smooth manifold and T : X ! X is a differentiable mapping, one is in the framework of differentiable dynamics. When X is a topological space and T : X ! X is a continuous map, one is in the framework of topological dynamics. When X is a measure space and T : X ! X is a measure preserving map, one is in the framework of ergodic theory. These categories are not mutually exclusive, and the relations among them are deep and interesting. Some of these relations are explored in the articles  Topological Dynamics,  Symbolic Dynamics, and  Smooth Ergodic Theory. To further explain the role of ergodic theory, a few definitions are needed. The state space X is assumed to be a measure space, endowed with a  -algebra B of measurable sets and a measure . The measure  assigns each set B 2 B a non-negative number (its measure), and usually one assumes that (X) D 1 (thus  is a probability measure). The transformation T : X ! X is assumed to be a measurable and measure preserving map: For all B 2 B, (T 1 B) D (B). The quadruple (X; B; ; T) is called a measure preserving system. For the precise definitions and background on measure preserving transformations, see the article  Measure Preserving Systems. An extensive discussion of examples of measure preserving systems and basic constructions used in ergodic theory is given in  Ergodic Theory: Basic Examples and Constructions. The origins of ergodic theory are in the nineteenth century work of Boltzmann on the foundations of statistical mechanics. Boltzmann hypothesized that for large systems of interacting particles in equilibrium, the “time average” is equal to the “space average”. This question can be reformulated in the context of modern terminology. Assume that (X; B; ; T) is a measure preserving system and that f : X ! R is some measurement taken on the system. Thus if x 2 X is a state, evaluating the sequence f (x) ; f (Tx) ; f (T 2 x); : : : can be viewed as successive values of this measurement. Boltzmann’s question can be phrased as: Under what conditions is the time mean equal to the space mean? In short,

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when does lim

N!1

N1 1 X f (T n x) D N nD0

Z f d ? X

Boltzmann hypothesized that if orbits went “everywhere” in the space, then such a conclusion would hold. The study of the equality of space and time averages has been a major direction of research in ergodic theory. The long term behavior of the average lim

N!1

N1 1 X f (T n x) ; N nD0

and especially the existence of this limit, is a basic question. Roughly speaking, the ergodic theorem states that starting at almost any initial point, the distribution of its iterates obeys some asymptotic law. This, and more general convergence questions, are addressed in the article  Ergodic Theorems. Perhaps the earliest result in ergodic theory is the Poincaré Recurrence Theorem: In a finite measure space, some iterate of any set with positive measure intersects the set itself in a set of positive measure. More generally, the qualitative behavior of orbits is used to understand conditions under which the time average is equal to the space average of the system (see the article  Ergodic Theory: Recurrence). If the time average is equal almost everywhere to the space average, then the system is said to be ergodic. Ergodicity is a key notion, giving a simple expression for the time average of an arbitrary function. Moreover, using the Ergodic Decomposition Theorem, the study of arbitrary measure preserving systems can be reduced to ergodic ones. Ergodicity and related properties of a system are discussed in the article  Ergodicity and Mixing Properties. Another central problem in ergodic theory is the classification of measure preserving systems. There are various notions of equivalence, and a classical approach to checking if systems are equivalent is finding invariants that are preserved under the equivalence. This subject, including an introduction to Ornstein Theory, is covered in the article  Isomorphism Theory in Ergodic Theory. A map T : X ! X determines an associated unitary operator U D U T defined on L2 (X) by U T f (x) D f (Tx) : There are also numerical invariants that can be assigned to

a system, for example entropy, and this is discussed in the article  Entropy in Ergodic Theory. When two systems are not equivalent, one would like to understand what properties they do have in common. An essential tool in such a classification is the notion of joinings (see the article  Joinings in Ergodic Theory). Roughly speaking, a joining is a way of embedding two systems in the same space. When this can be done in a nontrivial manner, one obtains information on properties shared by the systems. Some systems have predictable behavior and can be classified according to the behavior of individual points and their iterates. Others have behavior that is too complex or unpredictable to be understood on the level of orbits. Ergodic theory provides a statistical understanding of such systems and this is discussed in the article  Chaos and Ergodic Theory. A prominent role in chaotic dynamical systems is played by one dimensional Gibbs measure and by equilibrium states and (see the article  Pressure and Equilibrium States in Ergodic Theory). Rigidity theory addresses the opposite case, studying what kinds of properties in a system are obstructions to general chaotic behavior. The role of ergodic theory in this area is discussed in the article  Ergodic Theory: Rigidity. Another important class of systems arises when one relaxes the condition that the transformation T preserves the measure of sets on X, only requiring that the transformation preserve the negligible sets. Such systems are discussed in  Joinings in Ergodic Theory. Ergodic theory has seen a burst of recent activity, and most of this activity comes from interaction with other fields. Historically, ergodic theory has interacted with numerous fields, including other areas of dynamics, probability, statistical mechanics, and harmonic analysis. More recently, ergodic theory and its techniques have been imported into number theory and combinatorics, proving new results that have yet to be proved by other methods, and in turn, combinatorial problems have given rise to new areas of research within ergodic theory itself. Problems related to Diophantine approximation are discussed in  Ergodic Theory on Homogeneous Spaces and Metric Number Theory and  Ergodic Theory: Rigidity and ones related to combinatorial problems are addressed in  Ergodic Theory: Interactions with Combinatorics and Number Theory and in  Ergodic Theory: Recurrence. Interaction with problems that are geometric in nature, in particular dimension theory, is discussed in  Ergodic Theory: Fractal Geometry.

Ergodic Theory: Non-singular Transformations

Ergodic Theory: Non-singular Transformations ALEXANDRE I. DANILENKO1, CESAR E. SILVA 2 1 Institute for Low Temperature Physics & Engineering, Ukrainian National Academy of Sciences, Kharkov, Ukraine 2 Department of Mathematics, Williams College, Williamstown, USA Article Outline Glossary Definition of the Subject Basic Results Panorama of Examples Mixing Notions and multiple recurrence Topological Group Aut(X, ) Orbit Theory Smooth Nonsingular Transformations Spectral Theory for Nonsingular Systems Entropy and Other Invariants Nonsingular Joinings and Factors Applications. Connections with Other Fields Concluding Remarks Bibliography Glossary Nonsingular dynamical system Let (X; B; ) be a standard Borel space equipped with a -finite measure. A Borel map T : X ! X is a nonsingular transformation of X if for any N 2 B, (T 1 N) D 0 if and only if (N) D 0. In this case the measure  is called quasi-invariant for T; and the quadruple (X; B; ; T) is called a nonsingular dynamical system. If (A) D (T 1 A) for all A 2 B then  is said to be invariant under T or, equivalently, T is measure-preserving. Conservativeness T is conservative if for all sets A of positive measure there exists an integer n > 0 such that (A \ T n A) > 0. Ergodicity T is ergodic if every measurable subset A of X that is invariant under T (i. e., T 1 A D A) is either null or -conull. Equivalently, every Borel function f : X ! R such that f ı T D f is constant a. e. Types II, II1 , II1 and III Suppose that  is non-atomic and T ergodic (and hence conservative). If there exists a -finite measure  on B which is equivalent to  and invariant under T then T is said to be of type II. It is easy to see that  is unique up to scaling. If  is finite then T is of type II 1 . If  is infinite then T is of type II1 . If T is not of type II then T is said to be of type III.

Definition of the Subject An abstract measurable dynamical system consists of a set X (phase space) with a transformation T : X ! X (evolution law or time) and a finite or -finite measure  on X that specifies a class of negligible subsets. Nonsingular ergodic theory studies systems where T respects  in a weak sense: the transformation preserves only the class of negligible subsets but it may not preserve . This survey is about dynamics and invariants of nonsingular systems. Such systems model ‘non-equilibrium’ situations in which events that are impossible at some time remain impossible at any other time. Of course, the first question that arises is whether it is possible to find an equivalent invariant measure, i. e. pass to a hidden equilibrium without changing the negligible subsets? It turns out that there exist systems which do not admit an equivalent invariant finite or even  -finite measure. They are of our primary interest here. In a way (Baire category) most of systems are like that. Nonsingular dynamical systems arise naturally in various fields of mathematics: topological and smooth dynamics, probability theory, random walks, theory of numbers, von Neumann algebras, unitary representations of groups, mathematical physics and so on. They also can appear in the study of probability preserving systems: some criteria of mild mixing and distality, a problem of Furstenberg on disjointness, etc. We briefly discuss this in Sect. “Applications. Connections with Other Fields”. Nonsingular ergodic theory studies all of them from a general point of view:    

What is the qualitative nature of the dynamics? What are the orbits? Which properties are typical withing a class of systems? How do we find computable invariants to compare or distinguish various systems?

Typically there are two kinds of results: some are extensions to nonsingular systems of theorems for finite measure-preserving transformations (for instance, the entire Sect. “Basic Results”) and the other are about new properly ‘nonsingular’ phenomena (see Sect. “Mixing Notions and Multiple Recurrence” or Sect. “Orbit Theory”). Philosophically speaking, the dynamics of nonsingular systems is more diverse comparatively with their finite measurepreserving counterparts. That is why it is usually easier to construct counterexamples than to develop a general theory. Because of shortage of space we concentrate only on invertible transformations, and we have not included as many references as we had wished. Nonsingular endomorphisms and general group or semigroup actions are practically not considered here (with some exceptions in

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Sect. “Applications. Connections with Other Fields” devoted to applications). A number of open problems are scattered through the entire text. We thank J. Aaronson, J.R. Choksi, V.Ya. Golodets, M. Lema´nczyk, F. Parreau, E. Roy for useful remarks. Basic Results This section includes the basic results involving conservativeness and ergodicity as well as some direct nonsingular counterparts of the basic machinery from classic ergodic theory: mean and pointwise ergodic theorems, Rokhlin lemma, ergodic decomposition, generators, Glimm–Effros theorem and special representation of nonsingular flows. The historically first example of a transformation of type III (due to Ornstein) is also given here with full proof. Nonsingular Transformations In this paper we will consider only invertible nonsingular transformations, i. e. those which are bijections when restricted to an invariant Borel subset of full measure. Thus when we refer to a nonsingular dynamical system (X; B; ; T) we shall assume that T is an invertible nonsingular transformation. Of course, each measure  on B which is equivalent to , i. e.  and  have the same null sets, is also quasi-invariant under T. In particular, since  is -finite, T admits an equivalent quasi-invariant probability measure. For each i 2 Z, we denote by ! i or ! i the Radon–Nikodym derivative d( ı T i )/d 2 L1 (X; ). The derivatives satisfy the cocycle equation ! iC j (x) D ! i (x) ! j (T i x) for a. e. x and all i; j 2 Z. Basic Properties of Conservativeness and Ergodicity A measurable set W is said to be wandering if for all i; j  0 with i ¤ j, T i W \ T  j W D ;. Clearly, if T has a wandering set of positive measure then it cannot be conservative. A nonsingular transformation T is incompressible if whenever T 1 C C, then (C n T 1 C) D 0. A set W of positive measure is said to be weakly wandering if there is a sequence n i ! 1 such that T n i W \ T n j W D ; for all i ¤ j. Clearly, a finite measure-preserving transformation cannot have a weakly wandering set. Hajian and Kakutani [83] showed that a nonsingular transformation T admits an equivalent finite invariant measure if and only if T does not have a weakly wandering set. Proposition 1 (see e. g. [123]) Let (X; B; ; T) be a nonsingular dynamical system. The following are equivalent: (i) T is conservative. S n A) D 0. (ii) For every measurable set A, (A n 1 nD1 T

(iii) T is incompressible. (iv) Every wandering set for T is null. Since any finite measure-preserving transformation is incompressible, we deduce that it is conservative. This is the statement of the classical Poincaré recurrence lemma. If T is a conservative nonsingular transformation of (X; B; ) and A 2 B a subset of positive measure, we can define an induced transformation T A of the space (A; B \ A;   A) by setting TA x :D T n x if n D n(x) is the smallest natural number such that T n x 2 A. T A is also conservative. As shownRin [179], if (X) D 1 and T is conservative and erP !(x) d(x) D 1, which is a nonsingular godic, A n(x)1 iD0 version of the well-known Kaçs formula. Theorem 2 (Hopf Decomposition, see e. g. [3]) Let T be a nonsingular transformation. Then there exist disjoint invariant sets C; D 2 B such that X D C t D, T reF1 n stricted to C is conservative, and D D nD1 T W, 1 where W is a wandering set. If f 2 L (X; ), f > 0, Pn1 i then C D fx : iD0 f (T x) ! i (x) D 1 a. e.g and D D P n1 i fx : iD0 f (T x) ! i (x) < 1 a. e.g. The set C is called the conservative part of T and D is called the dissipative part of T. If T is ergodic and  is non-atomic then T is automatically conservative. The translation by 1 on the group Z furnished with the counting measure is an example of an ergodic non-conservative (infinite measure-preserving) transformation. Proposition 4 Let (X; B; ; T) be a nonsingular dynamical system. The following are equivalent: (i) T is conservative and ergodic. (ii) For every set A of positive measure, (X n S1 n A) D 0. (In this case we will say A sweeps nD1 T out.) (iii) For every measurable set A of positive measure and for a. e. x 2 X there exists an integer n > 0 such that T n x 2 A. (iv) For all sets A and B of positive measure there exists an integer n > 0 such that (T n A \ B) > 0. (v) If A is such that T 1 A A, then (A) D 0 or (Ac ) D 0. This survey is mainly about systems of type III. For some time it was not quite obvious whether such systems exist at all. The historically first example was constructed by Ornstein in 1960. Example 5 (Ornstein [149]) Let A n D f0; 1; : : : ; ng, n (0) D 0:5 and n (i) D 1/(2n) for 0 < i  n and all n 2 N. Denote by (X; ) the infinite product probability

Ergodic Theory: Non-singular Transformations

N space 1 nD1 (A n ;  n ). Of course,  is non-atomic. A point of X is an infinite sequence x D (x n )1 nD1 with x n 2 A n for all n. Given a1 2 A1 ; : : : ; a n 2 A n , we denote the cylinder fx D (x i )1 iD1 2 X : x1 D a1 ; : : : ; x n D a n g by [a1 ; : : : ; a n ]. Define a Borel map T : X ! X by setting 8 if i < l(x) ˆ <0; (1) (Tx) i D x i C 1; if i D l(x) ˆ : if i > l(x) ; xi ; where l(x) is the smallest number l such that x l ¤ l. It is easy to verify that T is a nonsingular transformation of (X; ) and 1 Y n ((Tx)n ) d ı T (x) D d n (x n ) nD1 8 < (l(x)  1)! ; if x l (x) D 0 l(x) D : (l(x)  1)! ; if x l (x) ¤ 0 :

We prove that T is of type III by contradiction. Suppose that there exists a T-invariant -finite measure  equivalent to . Let ' :D d/d. Then 

! i (x) D '(x) '(T i x)1 for a. a. x 2 X and all i 2 Z: (2) Fix a real C > 1 such that the set EC :D ' 1 ([C 1 ; C])

X is of positive measure. By a standard approximation argument, for each sufficiently large n, there is a cylinder [a1 ; : : : ;a n ] such that (EC \ [a1 ; : : : ; a n ]) > 0:9([a1 ; : : : ; a n ]). Since nC1 (0) D 0:5, it follows that (EC \ [a1 ; : : : ; a n ; 0]) > 0:8([a1 ; : : : ; a n ; 0]). Moreover, by the pigeon hole principle there is 0 < i  n C 1 with (EC \ [a1 ; : : : ; a n ; i]) > 0:8([a1 ; : : : ; a n ; i]). Find N n > 0 such that T N n [a1 ; : : : ; a n ; 0] D [a1 ; : : : ; a n ; i]. Since ! Nn is constant on [a1 ; : : : ; a n ; 0], there is a subset E0 EC \ [a1 ; : : : ; a n ; 0] of positive measure such that T N n E0 EC \[a1 ; : : : ; a n ; i]. Moreover, ! Nn (x) D nC1 (i)/nC1 (0) D (n C 1)1 for a. a. x 2 [a1 ; : : : ; a n ; 0]. On the other hand,  we deduce from (2) that ! N n (x)  C 2 for all x 2 E0 , a contradiction. Mean and Pointwise Ergodic Theorems. Rokhlin Lemma Let (X; B; ; T) be a nonsingular dynamical system. Define a unitary operator U T of L2 (X; ) by setting s d( ı T)  f ıT: (3) U T f :D d We note that U T preserves the cone of positive functions L2C (X; ). Conversely, every positive unitary operator in

L2 (X; ) that preserves L2C (X; ) equals U T for a -nonsingular transformation T. Theorem 6 (von Neumann mean Ergodic Theorem, see e. g. [3]) If T has no -absolutely continuous T-invariant P i probability, then n1 n1 iD0 U T ! 0 in the strong operator topology. Denote by I the sub--algebra of T-invariant sets. Let E [:jI ] stand for the conditional expectation with Rrespect to I . Note that if T is ergodic, then E [ f jI ] D f d. Now we state a nonsingular analogue of Birkhoff’s pointwise ergodic theorem, due to Hurewicz [105] and in the form stated by Halmos [84]. Theorem 7 (Hurewicz pointwise Ergodic Theorem) If T is conservative, (X) D 1, f ; g 2 L1 (X; ) and g > 0, then n1 X iD0 n1 X

f (T i x) ! i (x) ! g(T i x) ! i (x)

E [ f jI ] as n ! 1 for a. e. x : E [gjI ]

iD0

A transformation T is aperiodic if the T-orbit of a. e. point from X is infinite. The following classical statement can be deduced easily from Proposition 1. Lemma 8 (Rokhlin’s lemma [161]) Let T be an aperiodic nonsingular transformation. For each " > 0 and integer N > 1 there exists a measurable set A such that the sets A; TA; : : : ; T N1 A are disjoint and (A [ TA [    [ T N1 A) > 1  ". This lemma was refined later (for ergodic transformations) by Lehrer and Weiss as follows. Theorem 9 ("-free Rokhlin lemma [132]) Let T be ergodic and  non-atomic. Then for a subset B X and any N S k N (X n B) D X, there is a set A such for which 1 kD0 T that the sets A; TA; : : : ; T N1 A are disjoint and A [ TA [    [ T N1 A  B. S k N (X n B) D X holds of course The condition 1 kD0 T for each B ¤ X if T is totally ergodic, i. e. T p is ergodic for any p, or if N is prime. Ergodic Decomposition A proof of the following theorem may be found in [3]. Theorem 10 (Ergodic Decomposition Theorem) Let T be a conservative nonsingular transformation on a standard probability space (X; B; ). There there exists a standard probability space (Y; ; A) and a family of probability measures y on (X; B), for y 2 Y, such that

331

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Ergodic Theory: Non-singular Transformations

(i)

For each A 2 B the map y 7!  y (A) is Borel and for each A 2 B Z (A) D  y (A) d(y) :

(ii) For y; y 0 2 Y the measures y and  y 0 are mutually singular. (iii) For each y 2 Y the transformation T is nonsingular and conservative, ergodic on (X; B;  y ). (iv) For each y 2 Y d y ı T d ı T  y -a. e : D d d y (v) (Uniqueness) If there exists another probability space (Y 0 ;  0 ; A0 ) and a family of probability measures 0y 0 on (X; B), for y 0 2 Y 0 , satisfying (i)–(iv), then there exists a measure-preserving isomorphism : Y ! Y 0 such that  y D 0 y for -a. e. y. It follows that if T preserves an equivalent -finite measure then the system (X; B;  y ; T) is of type II for a. a. y. The space (Y; ; A) is called the space of T-ergodic components. Generators It was shown in [157,162] that a nonsingular transformation T on a standard probability space (X; B; ) has a countable generator, i. e. a countable partition P so that W1 n nD1 T P generates the measurable sets. It was refined by Krengel [126]: if T is of type II1 or III then there exists a generator P consisting of two sets only. Moreover, given a sub--algebra F B such that F

S T F and k>0 T k F D B, the set fA 2 F j (A; X n A) is a generator of Tg is dense in F . It follows, in particular, that T is isomorphic to the shift on f0; 1gZ equipped with a quasi-invariant probability measure. The Glimm–Effros Theorem The classical Bogoliouboff–Krylov theorem states that each homeomorphism of a compact space admits an ergodic invariant probability measure [33]. The following statement by Glimm [76] and Effros [61] is a “nonsingular” analogue of that theorem. (We consider here only a particular case of Z-actions.) Theorem 11 Let X be a Polish space and T : X ! X an aperiodic homeomorphism. Then the following are equivalent: T has a recurrent point x, i. e. x D lim n!1 T n i x for a sequence n1 < n2 <    : (ii) There is an orbit of T which is not locally closed.

(i)

(iii) There is no a Borel set which intersects each orbit of T exactly once. (iv) There is a continuous probability Borel measure  on X such that (X; ; T) is an ergodic nonsingular system. A natural question arises: under the conditions of the theorem how many such  can exists? It turns out that there is a wealth of such measures. To state a corresponding result we first write an important definition. Definition 12 Two nonsingular systems (X; B; ; T) and (X; B0 ; 0 ; T 0 ) are called orbit equivalent if there is a oneto-one bi-measurable map ' : X ! X with 0 ı '  and such that ' maps the T-orbit of x onto the T 0 -orbit of '(x) for a. a. x 2 X. The following theorem was proved in [116,128] and [174]. Theorem 13 Let (X; T) be as in Theorem 11. Then for each ergodic dynamical system (Y; C ; ; S) of type II1 or III, there exist uncountably many mutually disjoint Borel measures  on X such that (X; T; B; ) is orbit equivalent to (Y; C ; ; S). On the other hand, T may not have any finite invariant measure. Indeed, let T be an irrational rotation on the circle T and X a non-empty T-invariant Gı subset of T of full Lebesgue measure. Let (X; T) contain a recurrent point. Then the unique ergodicity of (T ; T) implies that (X; T) has no finite invariant measures. Let T be an aperiodic Borel transformation of a standard Borel space X. Denote by M(T) the set of all ergodic T-nonsingular continuous measures on X. Given  2 M(T), let N() denote the family of all Borel -null subsets. Shelah and Weiss showed [178] T that 2M(T) N() coincides with the collection of all Borel T-wandering sets. Special Representations of Ergodic Flows Nonsingular flows (D R-actions) appear naturally in the study of orbit equivalence for systems of type III (see Sect. “Orbit Theory”). Here we record some basic notions related to nonsingular flows. Let (X; B; ) be a standard Borel space with a  -finite measure  on B. A nonsingular flow on (X; ) is a Borel map S : XR 3 (x; t) 7! S t x 2 X such that S t Ss D S tCs for all s; t 2 R and each St is a nonsingular transformation of (X; ). Conservativeness and ergodicity for flows are defined in a similar way as for transformations. A very useful example of a flow is a flow built under a function. Let (X; B; ; T) be a nonsingular dynamical system and f a positive Borel function on X such that P1 P1 i i iD0 f (T x) D iD0 f (T x) D 1 for all x 2 X. Set

Ergodic Theory: Non-singular Transformations

X f :D f(x; s) : x 2 X; 0  s < f (x)g. Define f to be the restriction of the product measure   Leb on X  R to X f and define, for t  0, ! n1 X f n i S t (x; s) :D T x; s C t  f (T x) ; iD0

where n is the unique integer that satisfies n1 X

f (T i x) < s C t 

iD0

n X

If x ¤ (0; 0; : : : ), we let l(x) be the smallest number l such that the lth coordinate of x is not m l  1. We define a Borel map T : X ! X by (1) if x ¤ (m1 ; m2 ; : : : ) and put Tx :D (0; 0; : : : ) if x D (m1 ; m2 ; : : : ). Of course, T is isomorphic to a rotation on a compact monothetic totally disconnected Abelian group. It is easy to check that T is -nonsingular and 1 Y d ı T n ((Tx)n ) (x) D d n (x n ) nD1

f (T i x) :

iD0

A similar definition applies when t < 0. In particular, f when 0 < s C t < '(x), S t (x; s) D (x; s C t), so that the flow moves the point (x; s) up t units, and when it reaches (x; '(x)) it is sent to (Tx; 0). It can be shown that f S f D (S t ) t2R is a free f -nonsingular flow and that it pref serves  if and only if T preserves  [148]. It is called the flow built under the function ' with the base transformation T. Of course, S f is conservative or ergodic if and only if so is T. Two flows S D (S t ) t2R on (X; B; ) and V D (Vt ) t2R on (Y; C ; ) are said to be isomorphic if there exist invariant co-null sets X 0 X and Y 0 Y and an invertible nonsingular map : X 0 ! Y 0 that interwines the actions of the flows: ı S t D Vt ı on X 0 for all t. The following nonsingular version of Ambrose–Kakutani representation theorem was proved by Krengel [120] and Kubo [130]. Theorem 14 Let S be a free nonsingular flow. Then it is isomorphic to a flow built under a function. Rudolph showed that in the Ambrose–Kakutani theorem one can choose the function ' to take two values. Krengel [122] showed that this can also be assumed in the nonsingular case. Panorama of Examples This section is devoted entirely to examples of nonsingular systems. We describe here the most popular (and simple) constructions of nonsingular systems: odometers, nonsingular Markov odometers, tower transformations, rankone and finite rank systems and nonsingular Bernoulli shifts. Nonsingular Odometers Given a sequence mn of natural numbers, we let A n :Df0; 1;: : : ;m n  1g. Let  n be a probability on An and n (a) > 0 for all a 2 A n . Consider now the infinite product N1 probability space (X; ) :D nD1 (A n ;  n ). Assume that Q1 maxf (a) j a 2 A g D 0. Then  is non-atomic. n n nD1 Given a1 2 A1 ; : : : ; a n 2 A n , we denote by [a1 ; : : : ; a n ] the cylinder x D (x i ) i>0 j x1 D a1 ; : : : ; x n D a n .

D

l (x)1 n (0)  l (x) (x l (x) C 1) Y  l (x) (x l (x) ) n (m n  1) nD1

for a. a. x D (x n )n>0 2 X. It is also easy to verify that T is ergodic. It is called the nonsingular odometer associated to (m n ; n )1 nD1 . We note that Ornstein’s transformation (Example 5) is a nonsingular odometer. Markov Odometers We define Markov odometers as in [54]. An ordered Bratteli diagram B [102] consists of (i)

a vertex set V which is a disjoint union of finite sets V (n) , n  0, V 0 is a singleton; (ii) an edge set E which is a disjoint union of finite sets E (n) , n > 0; (iii) source mappings s n : E (n) ! V (n1) and range mappings r n : E (n) ! V (n) such that s 1 n (v) ¤ ; for all (v) ¤ ; for all v 2 V (n) , n > 0; v 2 V (n1) and r1 n 0 (iv) a partial order on E so that e; e 2 E are comparable if and only if e; e 0 2 E (n) for some n and r n (e) D r n (e 0 ).

A Bratteli compactum X B of the diagram B is the space of infinite paths ˚ x D (x n )n>0 j x n 2 E (n) and r(x n ) D s(x nC1 ) on B. X B is equipped with the natural topology induced Q by the product topology on n>0 E (n) . We will assume always that the diagram is essentially simple, i. e. there is only one infinite path xmax D (x n )n>0 with xn maximal for all n and only one xmin D (x n )n>0 with xn minimal for all n. The Bratteli–Vershik map TB : X B ! X B is defined as follows: Txmax D xmin . If x D (x n )n>0 ¤ xmax then let k be the smallest number such that xk is not maximal. Let yk be a successor of xk . Let (y1 ; : : : ; y k ) be the unique path such that y1 ; : : : ; y k1 are all minimal. Then we let TB x :D (y1 ; : : : ; y k ; x kC1 ; x kC2 ; : : : ). It is easy to see that T B is a homeomorphism of X B . Suppose that we are given (n) a sequence P(n) D (P(v;e)2V n1 E (n) ) of stochastic matrices, i. e. (n) (i) Pv;e > 0 if and only if v D s n (e) and P (n) (n1) . (ii) f e2E (n) js n (e)Dv g Pv;e D 1 for each v 2 V

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For e1 2 E (1) ; : : : ; e n 2 E (n) , let [e1 ; : : : ; e n ] denote the cylinder fx D (x j ) j>0 j x1 D e1 ; : : : ; x n D e n g. Then we define a Markov measure on X B by setting P ([e1 ; : : : ; e n ]) D Ps11 (e 1 );e 1 Ps22 (e 2 );e 2    Psnn (e n );e n for each cylinder [e1 ; : : : ; e n ]. The dynamical system (X B ; P ; TB ) is called a Markov odometer. It is easy to see that every nonsingular odometer is a Markov odometer where the corresponding V (n) are all singletons. Tower Transformations This construction is a discrete analogue of flow under a function. Given a nonsingular dynamical system (X; ; T) and a measurable map f : X ! N, we define a new dynamical system (X f ;  f ; T f ) by setting X f :D f(x; i) 2 X  ZC j 0  i < f (x)g ; d f (x; i) :D d(x) and ( (x; i C 1); if i C 1 < f (x) f T (x; i) :D (Tx; 0); otherwise : Then T f is f -nonsingular and (d f ı T f /d f )(x; i) D (d ı T/d)(x) for a. a. (x; i) 2 X f . This transformation is called the (Kakutani) tower over T with height function f . It is easy to check that T f is conservative if and only if T is conservative; T f is ergodic if and only if T is ergodic; T f is of type III if and only if T is of type III. Moreover, the induced transformation (T f ) Xf0g is isomorphic to T. Given a subset A X of positive measure, T is the tower over the induced transformation T A with the first return time to A as the height function. Rank-One Transformations. Chacón Maps. Finite Rank The definition uses the process of “cutting and stacking”. We construct by induction a sequence of columns Cn . A column Cn consists of a finite sequence of bounded intervals (left-closed, right-open) C n D fI n;0 ; : : : ; I n;h n 1 g of height hn . A column Cn determines a column map TC n that sends each interval I n;i to the interval above it I n;iC1 by the unique orientation-preserving affine map between the intervals. TC n remains undefined on the top interval I n;h n 1 . Set C0 D f[0; 1)g and let fr n > 2g be a sequence of positive integers, let fs n g be a sequence of functions s n : f0; : : : ; r n  1g ! N0 , and let fw n g be a sequence of probability vectors on f0; : : : ; r n  1g. If Cn has been defined, column C nC1 is defined as follows. First “cut” (i. e., subdivide) each interval I n;i in Cn into rn subintervals I n;i [ j]; j D 0; : : : ; r n  1, whose lengths are in the proportions w n (0) : w n (1) :    : w n (r n  1). Next place, for each

j D 0; : : : ; r n  1, s n ( j) new subintervals above I n;h n 1 [ j], all of the same length as I n;h n 1 [ j]. Denote these intervals, called spacers, by S n;0 [ j]; : : : ; S n;s n ( j)1 [ j]. This yields, for each j 2 f0; : : : ; r n  1g, rn subcolumns each consisting of the subintervals I n;0 [ j]; : : : ; I n;h n 1 [ j] followed by the spacers S n;0 [ j]; : : : ; S n;s n ( j)1 [ j] : Finally each subcolumn is stacked from left to right so that the top subinterval in subcolumn j is sent to the bottom subinterval in subcolumn j C 1, for j D 0; : : : ; r n  2 (by the unique orientation-preserving affine map between the intervals). For example, S n;s n (0)1 [0] is sent to I n;0 [1]. This defines a new column C nC1 and new column map TC nC1 , which remains undefined on its top subinterval. Let X be the union of all intervals in all columns and let  be Lebesgue measure restricted to X. We assume that as n ! 1 the maximal length of the intervals in Cn converges to 0, so we may define a transformation T of (X; ) by Tx :D lim n!1 TC n x. One can verify that T is welldefined a. e. and that it is nonsingular and ergodic. T is said to be the rank-one transformation associated with (r n ; w n ; s n )1 nD1 . If all the probability vectors wn are uniform the resulting transformation is measure-preserving. The measure is infinite (-finite) if and only if the total mass of the spacers is infinite. In the case r n D 3 and s n (0) D s n (2) D 0, s n (1) D 1 for all n  0, the associated rank-one transformation is called a nonsingular Chacón map. It is easy to see that every nonsingular odometer is of rank-one (the corresponding maps sn are all trivial). Each rank-one map T is a tower over a nonsingular odometer (to obtain such an odometer reduce T to a column Cn ). A rank N transformation is defined in a similar way. A nonsingular transformation T is said to be of rank N or less if at each stage of its construction there exits N disjoint columns, the levels of the columns generate the  -algebra and the Radon–Nikodym derivative of T is constant on each non-top level of every column. T is said to be of rank N if it is of rank N or less and not of rank N  1 or less. A rank N transformation, N  2, need not be ergodic. Nonsingular Bernoulli Transformations – Hamachi’s Example A nonsingular Bernoulli transformation is a transformation T such that there exists a countable generator P (see Subsect. “Generators”) such that the partitions T n P , n 2 Z, are mutually independent and such that the Radon– Nikodym derivative ! 1 is measurable with respect to the W sub--algebra 0nD1 T n P .

Ergodic Theory: Non-singular Transformations

In [87], Hamachi constructed examples of conservative nonsingular Bernoulli transformations, hence ergodic (see Subsect. “Weak Mixing, Mixing, K-Property”), with a 2-set generating partition that are of type III. Krengel [121] asked if there are of type II1 examples of nonsingular Bernoulli automorphisms and the question remains open. Hamachi’s construction is the left-shift on Q the space X D 1 nD1 f0; 1g. The measure is a product Q1  D nD1 n where n D (1/2; 1/2) for n  0 and for n < 0 n is chosen carefully alternating on large blocks between the uniform measure and different non-uniform measures. Kakutani’s criterion for equivalence of infinite product measures is used to verify that  is nonsingular. Mixing Notions and multiple recurrence The study of mixing and multiple recurrence are central topics in classical ergodic theory [33,70]. Unfortunately, these notions are considerably less ‘smooth’ in the world of nonsingular systems. The very concepts of any kind of mixing and multiple recurrence are not well understood in view of their ambiguity. Below we discuss nonsingular systems possessing a surprising diversity of such properties that seem equivalent but are different indeed. Weak Mixing, Mixing, K-Property Let T be an ergodic conservative nonsingular transformation. A number  2 C is an L1 -eigenvalue for T if there exists a nonzero f 2 L1 so that f ı T D  f a. e. It follows that jj D 1 and f has constant modulus, which we assume to be 1. Denote by e(T) the set of all L1 -eigenvalues of T. T is said to be weakly mixing if e(T) D f1g. We refer to Theorem 2.7.1 in [3] for proof of the following Keane’s ergodic multiplier theorem: given an ergodic probability preserving transformation S, the product transformation T  S is ergodic if and only if S (e(T)) D 0, where  S denotes the measure of (reduced) maximal spectral type of the unitary U S (see (3)). It follows that T is weakly mixing if and only T  S is ergodic for every ergodic probability preserving S. While in the finite measure-preserving case this implies that T  T is ergodic, it was shown in [5] that there exits a weakly mixing nonsingular T with T  T not conservative, hence not ergodic. In [11], a weakly mixing T was constructed with T  T conservative but not ergodic. A nonsingular transformation T is said to be doubly ergodic if for all sets of positive measure A and B there exists an integer n > 0 such that (A \ T n A) > 0 and (A \ T n B) > 0. Furstenberg [70] showed that for finite measure-preserving transformations double ergodicity is equivalent to weak mixing. In [20] it is shown that for nonsingular transformations weak mixing does not imply

double ergodicity and double ergodicity does not imply that T  T is ergodic. T is said to have ergodic index k if the Cartesian product of k copies of T is ergodic but the product of k C 1 copies of T is not ergodic. If all finite Cartesian products of T are ergodic then T is said to have infinite ergodic index. Parry and Kakutani [113] constructed for each k 2 N [ f1g, an infinite Markov shift of ergodic index k. A stronger property is power weak mixing, which requires that for all nonzero integers k1 ; : : : ; kr the product T k 1      T k r is ergodic [47]. The following examples were constructed in [12,36,38]: (i) power weakly mixing rank-one transformations, (ii) non-power weakly mixing rank-one transformations with infinite ergodic index, (iii) non-power weakly mixing rank-one transformations with infinite ergodic index and such that T k 1      T k r are all conservative, k1 ; : : : ; kr 2 Z, of types II1 and III (and various subtypes of III, see Sect. “Orbit Theory”). Thus we have the following scale of properties (equivalent to weak mixing in the probability preserving case), where every next property is strictly stronger than the previous ones: T is weakly mixing ( T is doubly ergodic ( T  T is ergodic ( T  T  T is ergodic (  ( T has infinite ergodic index ( T is power weakly mixing : We also mention a recent example of a power weakly mixing transformation of type II1 which embeds into a flow [46]. We now consider several attempts to generalize the notion of (strong) mixing. Given a sequence of measurable sets fA n g let  k (fA n g) denote the  -algebra generated by A k ; A kC1 ; : : : . A sequence fA n g is said to be remotely trivT ial if 1 kD0  k (fA n g) D f;; Xg mod , and it is semi-remotely trivial if every subsequence contains a subsequence that is remotely trivial. Krengel and Sucheston [124] define a nonsingular transformation T of a  -finite measure space to be mixing if for every set A of finite measure the sequence fT n Ag is semi-remotely trivial, and completely mixing if fT n Ag is semi-remotely trivial for all measurable sets A. They show that T is completely mixing if and only if it is type II1 and mixing for the equivalent finite invariant measure. Thus there are no type III and II1 completely mixing nonsingular transformations on probability spaces. We note that this definition of mixing in infinite

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measure spaces depends on the choice of measure inside the equivalence class (but it is independent if we replace the measure by an equivalent measure with the same collection of sets of finite measure). Hajian and Kakutani showed [83] that an ergodic infinite measure-preserving transformation T is either of zero type: limn!1 (T n A \ A) D 0 for all sets A of finite measure, or of positive type: lim supn!1 (T n A \ A) > 0 for all sets A of finite positive measure. T is mixing if and only if it is of zero type [124]. For 0  ˛  1 Kakutani suggested a related definition of ˛-type: an infinite measure preserving transformation is of ˛-type if lim supn!1 (A \ T n A) D ˛(A) for every subset A of finite measure. In [153] examples of ergodic transformations of any ˛-type and a transformation of not any type were constructed. It may seem that mixing is stronger than any kind of nonsingular weak mixing considered above. However, it is not the case: if T is a weakly mixing infinite measure preserving transformation of zero type and S is an ergodic probability preserving transformation then T  S is ergodic and of zero type. On the other hand, the L1 -spectrum e(T  S) is nontrivial, i. e. T  S is not weakly mixing, whenever S is not weakly mixing. We also note that there exist rank-one infinite measure-preserving transformations T of zero type such that T  T is not conservative (hence not ergodic) [11]. In contrast to that, if T is of positive type all of its finite Cartesian products are conservative [7]. Another result that suggests that there is no good definition of mixing in the nonsingular case was proved recently in [110]. It is shown there that while the mixing finite measure-preserving transformations are measurably sensitive, there exists no infinite measure-preserving system that is measurably sensitive. (Measurable sensitivity is a measurable version of the strong sensitive dependence on initial conditions – a concept from topological theory of chaos.) A nonsingular transformation T of (X; B; ) is called K-automorphism [180] if there exists a sub--algebra F

T WC1 B such that T 1 F F , k0 T k F D f;; Xg, kD0 T k F D B and the Radon–Nikodym derivative d ı T/ d is F -measurable (see also [156] for the case when T is of type II1 ; the authors in [180] required T to be conservative). Evidently, a nonsingular Bernoulli transformation (see Subsect. “Nonsingular Bernoulli Transformations – Hamachi’s Example”) is a K-automorphism. Parry [156] showed that a type II1 K-automorphism is either dissipative or ergodic. Krengel [121] proved the same for a class of Bernoulli nonsingular transformations, and finally Silva and Thieullen extended this result to nonsingular K-automorphisms [180]. It is also shown in [180] that if T is

a nonsingular K-automorphism, for any ergodic nonsingular transformation S, if S  T is conservative, then it is ergodic. It follows that a conservative nonsingular K-automorphism is weakly mixing. However, it does not necessarily have infinite ergodic index [113]. Krengel and Sucheston [124] showed that an infinite measure-preserving conservative K-automorphism is mixing. Multiple and Polynomial Recurrence Let p be a positive integer. A nonsingular transformation T is called p-recurrent if for every subset B of positive measure there exists a positive integer k such that (B \ T k B \    \ T k p B) > 0 : If T is p-recurrent for any p > 0, then it is called multiply recurrent. It is easy to see that T is 1-recurrent if and only if it is conservative. T is called rigid if T n k ! Id for a sequence n k ! 1. Clearly, if T is rigid then it is multiply recurrent. Furstenberg showed [70] that every finite measure-preserving transformation is multiply recurrent. In contrast to that Eigen, Hajian and Halverson [64] constructed for any p 2 N [ f1g, a nonsingular odometer of type II1 which is p-recurrent but not (p C 1)-recurrent. Aaronson and Nakada showed in [7] that an infinite measure preserving Markov shift T is p-recurrent if and only if the product T      T (p times) is conservative. It follows from this and [5] that in the class of ergodic Markov shifts infinite ergodic index implies multiple recurrence. However, in general this is not true. It was shown in [12,45] and [82] that for each p 2 N [ f1g there exist (i) power weakly mixing rank-one transformations and (ii) non-power weakly mixing rank-one transformations with infinite ergodic index which are p-recurrent but not (p C 1)-recurrent (the latter holds when p ¤ 1, of course). A subset A is called p-wandering if (A \ T k A \    pk \T A) D 0 for each k. Aaronson and Nakada established in [7] a p-analogue of Hopf decomposition (see Theorem 2). Proposition 15 If (X; B; ; T) is conservative aperiodic nonsingular dynamical system and p 2 N then X D C p [ D p , where Cp and Dp are T-invariant disjoint subsets, Dp is a countable union of p-wandering sets, T  C p is p-reP k B \   \T d k B) D 1 for current and 1 kD1 (B \ T every B C p . Let T be an infinite measure-preserving transformation and let F be a  -finite factor (i. e., invariant subalgebra)

Ergodic Theory: Non-singular Transformations

of T. Inoue [106] showed that for each p > 0, if T  F is p-recurrent then so is T provided that the extension T ! T  F is isometric. It is unknown yet whether the latter assumption can be dropped. However, partial progress was recently achieved in [140]: if T  F is multiply recurrent then so is T. Let P :D fq 2 Q[t] j q(Z) Z and q(0) D 0g. An ergodic conservative nonsingular transformation T is called p-polynomially recurrent if for every q1 ; : : : ; q p 2 P and every subset B of positive measure there exists k 2 N with (B \ T q 1 (k) B \    \ T q p (k) B) > 0 : If T is p-polynomially recurrent for every p 2 N then it is called polynomially recurrent. Furstenberg’s theorem on multiple recurrence was significantly strengthened in [17], where it was shown that every finite measure-preserving transformation is polynomially recurrent. However, Danilenko and Silva [45] constructed (i)

nonsingular transformations T which are p-polynomially recurrent but not (p C 1)-polynomially recurrent (for each fixed p 2 N), (ii) polynomially recurrent transformations T of type II1 , (iii) rigid (and hence multiply recurrent) transformations T which are not polynomially recurrent. Moreover, such T can be chosen inside the class of rankone transformations with infinite ergodic index. Topological Group Aut(X, ) Let (X; B; ) be a standard probability space and let Aut(X; ) denote the group of all nonsingular transformations of X. Let  be a finite or -finite measure equivalent to ; the subgroup of the -preserving transformations is denoted by Aut0 (X; ). Then Aut(X; ) is a simple group [62] and it has no outer automorphisms [63]. Ryzhikov showed [169] that every element of this group is a product of three involutions (i. e. transformations of order 2). Moreover, a nonsingular transformation is a product of two involutions if and only if it is conjugate to its inverse by an involution. Inspired by [85], Ionescu Tulcea [107] and Chacon and Friedman [21] introduced the weak and the uniform topologies respectively on Aut(X; ). The weak one – we denote it by dw – is induced from the weak operator topology on the group of unitary operators in L2 (X; ) by the embedding T 7! U T (see Subsect. “Mean and Pointwise Ergodic Theorems. Rokhlin Lemma”). Then (Aut(X; ); dw ) is a Polish topological group and Aut0 (X; ) is a closed subgroup of Aut(X; ).

This topology will not be affected if we replace  with any equivalent measure. We note that T n weakly converges  to T if and only if (Tn1 A T 1 A) ! 0 for each A 2 B and d( ı Tn )/d ! d( ı T)/d in L1 (X; ). Danilenko showed in [34] that (Aut(X; ); dw ) is contractible. It follows easily from the Rokhlin lemma that periodic transformations are dense in Aut(X; ). For each p  1, one can also embed Aut(X; ) into the isometry group of L p (X; ) via a formula similar to (3) but with another power of the Radon–Nikodym derivative in it. The strong operator topology on the isometry group induces the very same weak topology on Aut(X; ) for all p  1 [24]. It is natural to ask which properties of nonsingular transformations are typical in the sense of Baire category. The following technical lemma (see see [24,68]) is an indispensable tool when considering such problems. Lemma 16 The conjugacy class of each aperiodic transformation T is dense in Aut(X; ) endowed with the weak topology. Using this lemma and the Hurewicz ergodic theorem Choksi and Kakutani [24] proved that the ergodic transformations form a dense Gı in Aut(X; ). The same holds for the subgroup Aut0 (X; ) [24,170]. Combined with [107] the above implies that the ergodic transformations of type III is a dense Gı in Aut(X; ). For further refinement of this statement we refer to Sect. “Orbit Theory”. Since the map T 7! T      T (p times) from Aut(X; ) to Aut(X p ; ˝p ) is continuous for each p > 0, we deduce that the set E1 of transformations with infinite ergodic index is a Gı in Aut(X; ). It is non-empty by [113]. Since this E1 is invariant under conjugacy, it is dense in Aut(X; ) by Lemma 16. Thus we obtain that E1 is a dense Gı . In a similar way one can show that E1 \ Aut0 (X; ) is a dense Gı in Aut0 (X; ) (see also [24,26,170] for original proofs of these claims). The rigid transformations form a dense Gı in Aut(X; ). It follows that the set of multiply recurrent nonsingular transformations is residual [13]. A finer result was established in [45]: the set of polynomially recurrent transformations is also residual. Given T 2 Aut(X; ), we denote the centralizer fS 2 Aut(X; ) j ST D T Sg of T by C(T). Of course, C(T) is a closed subgroup of Aut(X; ) and C(T)  fT n j n 2 Zg. The following problems solved recently (by the efforts of many authors) for probability preserving systems are still open for the nonsingular case. Are the properties: (i) T has square root; (ii) T embeds into a flow;

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(iii) T has non-trivial invariant sub--algebra; (iv) C(T) contains a torus of arbitrary dimension typical (residual) in Aut(X; )? The uniform topology on Aut(X; ), finer than dw , is defined by the metric du (T; S) D (fx : Tx ¤ Sxg) C (fx : T

1

x¤S

1

xg) :

This topology is also complete metric. It depends only on the measure class of . However the uniform topology is not separable and that is why it is of less importance in ergodic theory. We refer to [21,24,27] and [68] for the properties of du . Orbit Theory Orbit theory is, in a sense, the most complete part of nonsingular ergodic theory. We present here the seminal Krieger’s theorem on orbit classification of ergodic nonsingular transformations in terms of ratio sets and associated flows. Examples of transformations of various types III , 0    1 are also given here. Next, we consider the outer conjugacy problem for automorphisms of the orbit equivalence relations. This problem is solved in terms of a simple complete system of invariants. We discuss also a general theory of cocycles (of nonsingular systems) taking values in locally compact Polish groups and present an important orbit classification theorem for cocycles. This theorem is an analogue of the aforementioned result of Krieger. We complete the section by considering ITPFIsystems and their relation to AT-flows. Full Groups. Ratio Set and Types III , 0    1 Let T be a nonsingular transformation of a standard probability space (X; B; ). Denote by Orb T (x) the T-orbit of x, i. e. OrbT (x) D fT n x j n 2 Zg. The full group [T] of T consists of all transformations S 2 Aut(X; ) such that Sx 2 OrbT (x) for a. a. x. If T is ergodic then [T] is topologically simple (or even algebraically simple if T is not of type II1 ) [62]. It is easy to see that [T] endowed with the uniform topology du is a Polish group. If T is ergodic then ([T]; du ) is contractible [34]. The ratio set r(T) of T was defined by Krieger [126] and as we shall see below it is the key concept in the orbit classification (see Definition 1). The ratio set is a subset of [0; C1) defined as follows: t 2 r(T) if and only if for every A 2 B of positive measure and each > 0 there is a subset B A of positive measure and an integer k ¤ 0 such that  T k B A and j! k (x)  tj < for all x 2 B. It is easy to verify that r(T) depends only on the equivalence class of  and not on  itself. A basic fact is that 1 2 r(T) if and

only if T is conservative. Assume now T to be conservative and ergodic. Then r(T) \ (0; C1) is a closed subgroup of the multiplicative group (0; C1). Hence r(T) is one of the following sets: (i) f1g; (ii) f0; 1g; in this case we say that T is of type III 0 , (iii) fn j n 2 Zg [ f0g for 0 <  < 1; then we say that T is of type III  , (iv) [0; C1); then we say that T is of type III 1 . Krieger showed that r(T) D f1g if and only if T is of type II. Hence we obtain a further subdivision of type III into subtypes III0 , III , or III1 . Example 17 (i) Fix  2 (0; 1). Let n (0) :D 1/(1 C ) and n (1) :D /(1 C ) for all n D 1; 2; : : : . Let T be the nonsingular odometer associated with the sequence (2; n )1 nD1 (see Subsect. “Nonsingular Odometers”). We claim that T is of type III . Indeed, the group ˙ of finite permutations of N acts on X by ( x)n D x 1 (n) , for all n 2 N,  2 ˙ and x D (x n )1 nD1 2 X. This action preserves . Moreover, it is ergodic by the Hewitt–Savage 0–1 law. It remains to notice that (d ı T/d)(x) D  on the cylinder [0] which is of positive measure. (ii) Fix positive reals 1 and 2 such that log 1 and log 2 are rationally independent. Let n (0) :D 1/(1 C 1 C 2 ), n (1) :D 1 /(1 C 1 C 2 ) and n (2) :D 2 /(1 C 1 C

2 ) for all n D 1; 2; : : : . Then the nonsingular odometer associated with the sequence (3; n )1 nD1 is of type III1 . This can be shown in a similar way as (i). Non-singular odometer of type III0 will be constructed in Example 19 below. Maharam Extension, Associated Flow and Orbit Classification of Type III Systems On X  R with the  -finite measure   , where d(y) D exp (y)dy, consider the transformation   d ı T e T(x; y) :D Tx; y  log (x) : d We call it the Maharam extension of T (see [136], where these transformations were introduced). It is measure-preserving and it commutes with the flow S t (x; y) :D (x; y C t), t 2 R. It is conservative if and only if T is conservative [136]. However e T is not necessarily ergodic. Let (Z; ) denote the space of e T-ergodic components. Then (S t ) t2R acts nonsingularly on this space. The restriction of (S t ) t2R to (Z; ) is called the associated flow of T. The associated flow is ergodic whenever T is ergodic. It is easy to verify

Ergodic Theory: Non-singular Transformations

that the isomorphism class of the associated flow is an invariant of the orbit equivalence of the underlying system. Proposition 18 ([90]) (i)

T is of type II if and only if its associated flow is the translation on R, i. e. x 7! x C t, x; t 2 R, (ii) T is of type III , 0   < 1 if and only if its associated flow is the periodic flow on the interval [0;  log ), i. e. x 7! x C t mod ( log ), (iii) T is of type III1 if and only if its associated flow is the trivial flow on a singleton or, equivalently, e T is ergodic, (iv) T is of type III0 if and only if its associated flow is nontransitive.

We also note that every nontransitive ergodic flow can be realized as the associated flow of a type III0 transformation. However it is somewhat easier to construct a Z2 -action of type III0 whose associated flow is the given one. For this, we take an ergodic nonsingular transformation Q on a probability space (Z; B; ) and a measure-preserving transformation R of an infinite  -finite measure space (Y; F ; ) such that there is a continuous homomorphism  : R ! C(R) with (d ı (t)/d)(y) D exp (t) for a. a. y (for instance, take a type III1 transformation T and put R :D e T and (t) :D S t ). Let ' : Z ! R be a Borel map with inf Z ' > 0. Define two transformations R0 and Q0 of (Z  Y;   ) by setting: R0 (x; y) :D (x; Ry) ;

n

Example 19 Let A n D f0; 1; : : : ; 22 g and n (0) D 0:5 n and n (i) D 0:5  22 for all 0 < i  2n . Let T be the n nonsingular odometer associated with (22 C 1; n )1 nD0 . It is straightforward that the associated flow of T is the flow built under the constant function 1 with the probability preserving 2-adic odometer (associated with (2; n )1 nD1 , n (0) D n (1) D 0:5) as the base transformation. In particular, T is of type III0 . A natural problem arises: to compute Krieger’s type (or the ratio set) for the nonsingular odometers – the simplest class of nonsingular systems. Some partial progress was achieved in [56,141,152], etc. However in the general setting this problem remains open. The map  : Aut(X; ) 3 T 7! e T 2 Aut(X  R;  ) is a continuous group homomorphism. Since the set E of ergodic transformations on X  R is a Gı in Aut(X  R;   ) (see Sect. “Topological Group Aut(X, )”), the subset  1 (E ) of type III1 ergodic transformations on X is also Gı . The latter subset is non-empty in view of Example 17(ii). Since it is invariant under conjugacy, we deduce from Lemma 16 that the set of ergodic transformations of type III1 is a dense Gı in (Aut(X; ); dw ) [23,159]. Now we state the main result of this section – Krieger’s theorem on orbit classification for ergodic transformations of type III. It is a far reaching generalization of the basic result by H. Dye: any two ergodic probability preserving transformations are orbit equivalent [60]. Theorem 20 (Orbit equivalence for type III systems [125]–[129]) Two ergodic transformations of type III are orbit equivalent if and only if their associated flows are isomorphic. In particular, for a fixed 0 <   1, any two ergodic transformations of type III are orbit equivalent. The original proof of this theorem is rather complicated. Simpler treatment of it can be found in [90] and [117].

Q0 (x; y) D (Qx; U x y) ;

where U x D ('(x)  log(d ı Q/d)(x)). Notice that R0 and Q0 commute. The corresponding Z2 -action generated by these transformations is ergodic. Take any transformation V 2 Aut(Z  Y;   ) whose orbits coincide with the orbits of the Z2 -action. (According to [29], any ergodic nonsingular action of any countable amenable group is orbit equivalent to a single transformation.) Then V is of type III0 . It is now easy to verify that the associated flow of V is the special flow built under ' ı Q 1 with the base transformation Q 1 . Since Q and ' are arbitrary, we deduce the following from Theorem 14. Theorem 21 Every nontransitive ergodic flow is an associated flow of an ergodic transformation of type III0 . In [129] Krieger introduced a map ˚ as follows. Let T be an ergodic transformation of type III0 . Then the associated flow of T is a flow built under function with a base transformation ˚(T). We note that the orbit equivalence class of ˚(T) is well defined by the orbit equivalent class of T. If ˚ n (T) fails to be of type III0 for some 1  n < 1 then T is said to belong to Krieger’s hierarchy. For instance, the transformation constructed in Example 19 belongs to Krieger’s hierarchy. Connes gave in [28] an example of T such that ˚(T) is orbit equivalent to T (see also [73] and [90]). Hence T is not in Krieger’s hierarchy. Normalizer of the Full Group. Outer Conjugacy Problem Let N[T] D fR 2 Aut(X; ) j R[T]R1 D [T]g ; i. e. N[T] is the normalizer of the full group [T] in Aut(X; ). We note that a transformation R belongs to N[T] if and only if R(Orb T (x)) D OrbT (Rx) for a. a. x.

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Ergodic Theory: Non-singular Transformations

To define a topology on N[T] consider the T-orbit equivalence relation RT RX  X and a -finite measure R on P RT given by R T D X y2OrbT (x) ı(x;y) d(x). For R 2 N[T], we define a transformation i(R) 2 Aut(RT ; RT ) by setting i(R)(x; y) :D (Rx; Ry). Then the map R 7! i(R) is an embedding of N[T] into Aut(RT ; RT ). Denote by  the topology on N[T] induced by the weak topology on Aut(RT ; RT ) via i [34]. Then (N[T]; ) is a Polish group. A sequence Rn converges to R in N[T] if R n ! R weakly (in Aut(X; )) and R n TR1 ! RTR1 uniformly (in n [T]). Given R 2 N[T], denote by e R the Maharam extension of R. Then e R 2 N[e T] and it commutes with (S t ) t2R . Hence it defines a nonsingular transformation mod R on the space (Z; ) of the associated flow W D (Wt ) t2R of T. Moreover, mod R belongs to the centralizer C(W) of W in Aut(Z; ). Note that C(W) is a closed subgroup of (Aut(Z; ); dw ). Let T be of type II1 and let 0 be the invariant  -finite measure equivalent to . If R 2 N[T] then it is easy to see that the Radon–Nikodym derivative d0 ı R/d0 is invariant under T. Hence it is constant, say c. Then mod R D log c. Theorem 22 ([86,90]) If T is of type III then the map mod : N[T] ! C(W) is a continuous onto homomorphism. The kernel of this homomorphism is the -closure of  [T]. Hence the quotient group N[T]/[T] is (topologically)  isomorphic to C(W). In particular, [T] is co-compact in N[T] if and only if W is a finite measure-preserving flow with a pure point spectrum. The following theorem describes the homotopical structure of normalizers. Theorem 23 ([34]) Let T be of type II or III , 0   <  1. The group [T] is contractible. N[T] is homotopically equivalent to C(W). In particular, N[T] is contractible if T is of type II. If T is of type III with 0 <  < 1 then 1 (N[T]) D Z. The outer period p(R) of R 2 N[T] is the smallest positive integer n such that R n 2 [T]. We write p(R) D 0 if no such n exists. Two transformations R and R0 in N[T] are called outer conjugate if there are transformations V 2 N[T] and S 2 [T] such that V RV 1 D R0 S. The following theorem provides convenient (for verification) necessary and sufficient conditions for the outer conjugacy. Theorem 24 ([30] for type II and [18] for type III) Transformations R; R0 2 N[T] are outer conjugate if and only if p(R) D p(R0 ) and mod R is conjugate to mod R0 in the centralizer of the associated flow for T.

We note that in the case T is of type II, the second condition in the theorem is just mod R D mod R0 . It is always satisfied when T is of type II1 . Cocycles of Dynamical Systems. Weak Equivalence of Cocycles Let G be a locally compact Polish group and G a left Haar measure on G. A Borel map ' : X ! G is called a cocycle of T. Two cocycles ' and ' 0 are cohomologous if there is a Borel map b : X ! G such that ' 0 (x) D b(Tx)1 '(x)b(x) for a. a. x 2 X. A cocycle cohomologous to the trivial one is called a coboundary. Given a dense subgroup G 0 G, then every cocycle is cohomologous to a cocycle with values in G 0 [81]. Each cocycle ' extends to a (unique) map ˛' : RT ! G such that ˛' (Tx; x) D '(x) for a. a. x and ˛' (x; y)˛' (y; z) D ˛' (x; z) for a. a. (x; y); (y; z) 2 RT . ˛' is called the cocycle of RT generated by '. Moreover, ' and ' 0 are cohomologous via b as above if and only if ˛' and ˛' 0 are cohomologous via b, i. e. ˛' (x; y) D b(x)1 ˛' 0 (x; y)b(y) for RT -a. a. (x; y) 2 RT . The following notion was introduced by Golodets and Sinelshchikov [78,81]: two cocycles ' and ' 0 are weakly equivalent if there is a transformation R 2 N[T] such that the cocycles ˛' and ˛'0 ı (R  R) of RT are cohomologous. Let M(X; G) denote the set of Borel maps from X to G. It is a Polish group when endowed with the topology of convergence in measure. Since T is ergodic, it is easy to deduce from Rokhlin’s lemma that the cohomology class of any cocycle is dense in M(X; G). Given ' 2 M(X; G), we define the '-skew product extension T' of T acting on (X  G;   G ) by setting T' (x; g) :D (Tx; '(x)g). Thus Maharam extension is (isomorphic to) the Radon– Nikodym cocycle-skew product extension. We now specify some basic classes of cocycles [19,35,81,173]: (i) ' is called transient if T' is of type I. (ii) ' is called recurrent if T' is conservative (equivalently, T' is not transient). (iii) ' has dense range in G if T' is ergodic. (iv) ' is called regular if ' cobounds with dense range into a closed subgroup H of G (then H is defined up to conjugacy). These properties are invariant under the cohomology and the weak equivalence. The Radon–Nikodym cocycle ! 1 is a coboundary if and only if T is of type II. It is regular if and only if T is of type II or III , 0 <   1. It has dense range (in the multiplicative group RC ) if and only if T is of type III1 . Notice that ! 1 is never transient (since T is conservative).

Ergodic Theory: Non-singular Transformations

Schmidt introduced in [176] an invariant R(') :D fg 2 G j '  g is recurrentg. He showed in particular that (i) (ii) (iii) (iv)

R(') is a cohomology invariant, R(') is a Borel set in G, R(log !1 ) D f0g for each aperiodic conservative T, there are cocycles ' such that R(') and G n R(') are dense in G, (v) if (X) D 1, Rı T D  and ' : X ! R is integrable then R(') D f ' dg. We note that (v) follows from Atkinson theorem [15]. A nonsingular version of this theorem was established in [183]: if T is ergodic and -nonsingular and f 2 L1 () then ˇ ˇ n1 ˇ ˇX ˇ ˇ j f (T x)! j (x)ˇ D 0 for a. a. x lim inf ˇ n!1 ˇ ˇ jD0 R if and only if f d D 0. Since T' commutes with the action of G on X  G by inverted right translations along the second coordinate, this action induces an ergodic G-action W' D (W' (g)) g2G on the space (Z; ) of T' -ergodic components. It is called the Mackey range (or Poincaré flow) of ' [66,135,173,188]. We note that ' is regular (and cobounds with dense range into H G) if and only if W' is transitive (and H is the stabilizer of a point z 2 Z, i. e. H D fg 2 G j W' (g)z D zg). Hence every cocycle taking values in a compact group is regular. It is often useful to consider the double cocycle '0 :D '  !1 instead of '. It takes values in the group G  RC . Since T'0 is exactly the Maharam extension of T' , it follows from [136] that ' 0 is transient or recurrent if and only if ' is transient or recurrent respectively. Theorem 25 (Orbit classification of cocycles [81]) Let '; ' 0 : X ! G be two recurrent cocycles of an ergodic transformation T. They are weakly equivalent if and only if their Mackey ranges W'0 and W'00 are isomorphic. Another proof of this theorem was presented in [65]. Theorem 26 Let T be an ergodic nonsingular transformation. Then there is a cocycle of T with dense range in G if and only if G is amenable. It follows that if G is amenable then the subset of cocycles of T with dense range in G is a dense Gı in M(X; G) (just adapt the argument following Example 19). The ‘only if ’ part of Theorem 26 was established in [187]. The ‘if ’ part was considered by many authors in particular cases: G is compact [186], G is solvable or amenable almost connected [79], G is amenable unimodular [108], etc. The general case was proved in [78] and [100] (see also a recent treatment in [9]).

Theorem 21 is a particular case of the following result. Theorem 27 ([10,65,80]) Let G be amenable. Let V be an ergodic nonsingular action of G  RC . Then there is an ergodic nonsingular transformation T and a recurrent cocycle ' of T with values in G such that V is isomorphic to the Mackey range of the double cocycle ' 0 . Given a cocycle ' 2 M(X; G) of T, we say that a transformation R 2 N[T] is compatible with ' if the cocycles ˛' and ˛' ı (R  R) of RT are cohomologous. Denote by D(T; ') the group of all such R. It has a natural Polish topology which is stronger than  [41]. Since [T] is a normal subgroup in D(T; '), one can consider the outer conjugacy equivalence relation inside D(T; '). It is called '-outer conjugacy. Suppose that G is Abelian. Then an analogue of Theorem 24 for the '-outer conjugacy is established in [41]. Also, the cocycles ' with D(T; ') D N[T] are described there. ITPFI Transformations and AT-Flows A nonsingular transformation T is called ITPFI 1 if it is orbit equivalent to a nonsingular odometer (associated to a sequence (m n ; n )1 nD1 , see Subsect. “Nonsingular Odometers”). If the sequence mn can be chosen bounded then T is called ITPFI of bounded type. If m n D 2 for all n then T is called ITPFI2 . By [74], every ITPFI-transformation of bounded type is ITPFI2 . A remarkable characterization of ITPFI transformations in terms of their associated flows was obtained by Connes and Woods [31]. We first single out a class of ergodic flows. A nonsingular flow V D (Vt ) t2R on a space (˝; ) is called approximate transitive (AT) if given > 0 and f1 ; : : : ; f n 2 L1C (X; ), there exists f 2 L1C (X; ) and 1 ; : : : ; n 2 L1C (R; dt) such that ˇˇ ˇˇ Z ˇˇ ˇˇ d ı Vt ˇˇ f j  ˇˇ < f ı V (t)dt  t j ˇˇ ˇˇ d R 1 for all 1  j  n. A flow built under a constant ceiling function with a funny rank-one [67] probability preserving base transformation is AT [31]. In particular, each ergodic finite measure-preserving flow with a pure point spectrum is AT. Theorem 28 ([31]) An ergodic nonsingular transformation is ITPFI if and only if its associated flow is AT. The original proof of this theorem was given in the framework of von Neumann algebras theory. A simpler, purely 1 This abbreviates ‘infinite tensor product of factors of type I’ (came from the theory of von Neumann algebras).

341

342

Ergodic Theory: Non-singular Transformations

measure theoretical proof was given later in [96] (the ‘only if ’ part) and [88] (the ‘if ’ part). It follows from Theorem 28 that every ergodic flow with pure point spectrum is the associated flow of an ITPFI transformation. If the point spectrum of V is  , where  is a subgroup of Q and 2 R, then V is the associated flow of an ITPFT2 transformation [91]. Theorem 29 ([54]) Each ergodic nonsingular transformation is orbit equivalent to a Markov odometer (see Subsect. “Markov Odometers”). The existence of non-ITPFI transformations and ITPFI transformations of unbounded type was shown in [127]. In [55], an explicit example of a non-ITPFI Markov odometer was constructed. Smooth Nonsingular Transformations Diffeomorphisms of smooth manifolds equipped with smooth measures are commonly considered as physically natural examples of dynamical systems. Therefore the construction of smooth models for various dynamical properties is a well established problem of the modern (probability preserving) ergodic theory. Unfortunately, the corresponding ‘nonsingular’ counterpart of this problem is almost unexplored. We survey here several interesting facts related to the topic. r For r 2 N [ f1g, denote by DiffC (T ) the group of orientation preserving Cr -diffeomorphisms of the circle T . Endow this set with the natural Polish topology. Fix r T 2 DiffC (T ). Since T D R/Z, there exists a C1 -function f : R ! R such that T(x C Z) D f (x) C Z for all x 2 R. The rotation number (T) of T is the limit lim ( f ı    ı f )(x)(mod 1) : „ ƒ‚ …

n!1

n times

The limit exists and does not depend on the choice of x and f . It is obvious that T is nonsingular with respect to r (T ) and Lebesgue measure T . Moreover, if T 2 DiffC

(T) is irrational then the dynamical system (T ; T ; T) is ergodic [33]. It is interesting to ask: which Krieger’s type can such systems have? Katznelson showed in [114] that the subset of type III C 1 -diffeomorphisms and the subset of type II1 C 1 -dif1 (T ). Hawkins and feomorphisms are dense in DiffC Schmidt refined the idea of Katznelson from [114] to construct, for every irrational number ˛ 2 [0; 1) which is not of constant type (i. e. in whose continued fraction expansion the denominators are not bounded) a transformation 2 T 2 DiffC (T ) which is of type III1 and (T) D ˛ [97]. It should be mentioned that class C2 in the construction

is essential, since it follows from a remarkable result of 3 Herman that if T 2 DiffC (T ) then under some condition on ˛ (which determines a set of full Lebesgue measure), T is measure theoretically (and topologically) conjugate to a rotation by (T) [101]. Hence T is of type II1 . In [94], Hawkins shows that every smooth paracompact manifold of dimension  3 admits a type III diffeomorphism for every  2 [0; 1]. This extends a result of Herman [100] on the existence of type III1 diffeomorphisms in the same circumstances. It is also of interest to ask: which free ergodic flows are associated with smooth dynamical systems of type III0 ? Hawkins proved that any free ergodic C 1 -flow on a smooth, connected, paracompact manifold is the associated flow for a C 1 -diffeomorphism on another manifold (of higher dimension) [95]. 2 (T ) A nice result was obtained in [115]: if T 2 DiffC and the rotation number of T has unbounded continued fraction coefficients then (T ; T ; T) is ITPFI. Moreover, a converse also holds: given a nonsingular odometer R, the set of orientation-preserving C 1 -diffeomorphisms of the circle which are orbit equivalent to R is C 1 -dense in the Polish set of all C 1 -orientation-preserving diffeomorphisms with irrational rotation numbers. In contrast to that, Hawkins constructs in [93] a type III0 C 1 -diffeomorphism of the 4-dimensional torus which is not ITPFI. Spectral Theory for Nonsingular Systems While the spectral theory for probability preserving systems is developed in depth, the spectral theory of nonsingular systems is still in its infancy. We discuss below some problems related to L1 -spectrum which may be regarded as an analogue of the discrete spectrum. We also include results on computation of the maximal spectral type of the ‘nonsingular’ Koopman operator for rank-one nonsingular transformations. L1 -Spectrum and Groups of Quasi-Invariance Let T be an ergodic nonsingular transformation of (X; B; ). A number  2 T belongs to the L1 -spectrum e(T) of T if there is a function f 2 L1 (X; ) with f ı T D  f . f is called an L1 -eigenfunction of T corresponding to . Denote by E (T) the group of all L1 -eigenfunctions of absolute value 1. It is a Polish group when endowed with the topology of converges in measure. If T is of type II1 then the L1 -eigenfunctions are L2 (0 )-eigenfunctions of T, where 0 is an equivalent invariant probability measure. Hence e(T) is countable. Osikawa constructed in [151] the first examples of ergodic nonsingular transformations with uncountable e(T).

Ergodic Theory: Non-singular Transformations

We state now a nonsingular version of the von Neumann–Halmos discrete spectrum theorem. Let Q T be a countable infinite subgroup. Let K be a compact dual of Qd , where Qd denotes Q with the discrete topology. Let k0 2 K be the element defined by k0 (q) D q for all q 2 Q. Let R : K ! K be defined by Rk D k C k0 . The system (K; R) is called a compact group rotation. The following theorem was proved in [6]. Theorem 30 Assume that the L1 -eigenfunctions of T generate the entire -algebra B. Then T is isomorphic to a compact group rotation equipped with an ergodic quasi-invariant measure. A natural question arises: which subgroups of T can appear as e(T) for an ergodic T? Theorem 31 ([1,143]) e(T) is a Borel subset of T and carries a unique Polish topology which is stronger than the usual topology on T . The Borel structure of e(T) under this topology agrees with the Borel structure inherited from T . There is a Borel map : e(T) 3  7!  2 E (T) such that  ıT D   for each . Moreover, e(T) is of Lebesgue measure 0 and it can have an arbitrary Hausdorff dimension. A proper Borel subgroup E of T is called (i) weak Dirichlet if lim supn!1 b (n) D 1 for each finite complex measure  supported on E; (n)j  j(E)j for each fi(ii) saturated if lim supn!1 jb nite complex measure  on T , where b (n) denote the nth Fourier coefficient of . Every countable subgroup of T is saturated. Theorem 32 e(T) is -compact in the usual topology on T [104] and saturated [104,139]. It follows that e(T) is weak Dirichlet (this fact was established earlier in [175]). It is not known if every Polish group continuously embedded in T as a -compact saturated group is the eigenvalue group of some ergodic nonsingular transformation. This is the case for the so-called H 2 -groups and the groups of quasi-invariance of measures on T (see below). Given a sequence nj of positive integers and a sequence a j  0, P nj 2 the set of all z 2 T such that 1 jD1 a j j1  z j < 1 is a group. It is called an H 2 -group. Every H 2 -group is Polish in an intrinsic topology stronger than the usual circle topology. Theorem 33 ([104]) (i)

Every H 2 -group is a saturated (and hence weak Dirichlet) -compact subset of T .

P (ii) If 1 jD0 a j D C1 then the corresponding H 2 -group is a proper subgroup of T . P 2 (iii) If 1 jD0 a j (n j /n jC1 ) < 1 then the corresponding H 2 -group is uncountable. (iv) Any H 2 -group is e(T) for an ergodic nonsingular compact group rotation T. It is an open problem whether every eigenvalue group e(T) is an H 2 -group. It is known however that e(T) is close ‘to be an H 2 -group’: if a compact subset L T is disjoint from e(T) then there is an H 2 -group containing e(T) and disjoint from L. Example 34 ([6], see also [151]) Let (X; ; T) be the nonsingular odometer associated to a sequence (2;  j )1 jD1 . Let nj be a sequence of positive integers such that n j > P n for all j. For x 2 X, we put h(x) :D n l (x)  P i< j i j 0 then either H() is countable or  is equivalent to T [137]. Theorem 35 ([6]) Let  be an ergodic with respect to the H()-action by translations on T . Then there is a compact group rotation (K; R) and a finite measure on K quasiinvariant and ergodic under R such that e(R) D H(). Moreover, there is a continuous one-to-one homomorphism : e(R) ! E(R) such that  ı R D   for all  2 e(R). It was shown by Aaronson and Nadkarni [6] that if n1 D 1 and n j D a j a j1    a1 for positive integers a j  2 P 1 with 1 jD1 a j < 1 then the transformation S from Example 34 does not admit a continuous homomorphism : e(S) ! E(S) with  ı T D   for all  2 e(S). Hence e(S) ¤ H() for any measure  satisfying the conditions of Theorem 35.

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Ergodic Theory: Non-singular Transformations

Assume that T is an ergodic nonsingular compact group rotation. Let B0 be the -algebra generated by a subcollection of eigenfunctions. Then B0 is invariant under T and hence a factor (see Sect. “Nonsingular Joinings and Factors”) of T. It is not known if every factor of T is of this form. It is not even known whether every factor of T must have non-trivial eigenvalues. Unitary Operator Associated with a Nonsingular System Let (X; B; ; T) be a nonsingular dynamical system. In this subsection we consider spectral properties of the unitary operator U T defined by (3). First, we note that the spectrum of T is the entire circle T [147]. Next, if U T has an eigenvector then T is of type II1 . Indeed, if there are  2 T and 0 ¤ f 2 L2 (X; ) with U T f D  f then the measure , d(x) :D j f (x)j2 d(x), is finite, T-invariant and equivalent to . Hence if T is of type III or II1 then the maximal spectral type  T of U T is continuous. Another ‘restriction’ on  T was recently found in [166]: no Foïa¸s-Str˘atil˘a measure is absolutely continuous with respect to  T if T is of type II1 . We recall that a symmetric measure on T possesses Foïa¸s-Str˘atil˘a property if for each ergodic probability preserving system (Y; ; S) and f 2 L2 (Y; ), if  is the spectral measure of f then f is a Gaussian random variable [134]. For instance, measures supported on Kronecker sets possess this property. Mixing is an L2 -spectral property for type II1 transformations: T is mixing Rif and only if  T is a Rajchman measure, i. e. b  T (n) :D z n dT (z) ! 0 as jnj ! 1. P ki Also, T is mixing if and only if n1 n1 iD0 U T ! 0 in the strong operator topology for each strictly increasing sequence k1 < k2 <    [124]. This generalizes a well known theorem of Blum and Hanson for probability preserving maps. For comparison, we note that ergodicity is not an L2 -spectral property of infinite measure preserving systems. Now let T be a rank-one nonsingular transformation associated with a sequence (r n ; w n ; s n )1 nD1 as in Subsect. “Rank-One Transformations. Chacón Maps. Finite Rank”. Theorem 36 ([25,104]) The spectral multiplicity of U T is 1 and the maximal spectral type  T of U T (up to a discrete measure in the case T is of type II1 ) is the weak limit of the measures k defined as follows: d k (z) D

k Y

w j (0)jPj (z)j2 dz ;

jD1

where P j (z)

:D

1 C

p w j (1)/w j (0)zR1; j C    C

p

R

w j (m j  1)/w j (0)z r j 1; j , z 2 T , R i; j :D ih j1 C s j (0) C    C s j (i), 1  i  r k  1 and hj is the height of the jth column.

Thus the maximal spectral type of U T is given by a socalled generalized Riesz product. We refer the reader to [25,103,104,148] for a detailed study of Riesz products: their convergence, mutual singularity, singularity to T , etc. It was shown in [6] that H(T )  e(T) for any ergodic nonsingular transformation T. Moreover,  T is ergodic under the action of e(T) by translations if T is isomorphic to an ergodic nonsingular compact group rotation. However it is not known: (i) Whether H(T ) D e(T) for all ergodic T. (ii) Whether ergodicity of  T under e(T) implies that T is an ergodic compact group rotation. The first claim of Theorem 36 extends to the rank N nonsingular systems as follows: if T is an ergodic nonsingular transformation of rank N then the spectral multiplicity of U T is bounded by N (as in the finite measure-preserving case). It is not known whether this claim is true for a more general class of transformations which are defined as rank N but without the assumption that the Radon– Nikodym cocycle is constant on the tower levels. Entropy and Other Invariants Let T be an ergodic conservative nonsingular transformation of a standard probability space (X; B; ). If P is a finite partition of X, we define the entropy H(P ) of P P as H(P ) D  P2P (P) log (P). In the study of measure-preserving systems the classical (Kolmogorov–Sinai) entropy proved to be a very useful invariant for isomorphism [33]. The key fact of the theory is that if  ı T D  W then the limit limn!1 n1 H( niD1 T i P ) exists for every P . However if T does not preserve , the limit may no longer exist. Some efforts have been made to extend the use of entropy and similar invariants to the nonsingular domain. These include Krengel’s entropy of conservative measure-preserving maps and its extension to nonsingular maps, Parry’s entropy and Parry’s nonsingular version of Shannon–McMillan–Breiman theorem, critical dimension by Mortiss and Dooley, etc. Unfortunately, these invariants are less informative than their classical counterparts and they are more difficult to compute. Krengel’s and Parry’s Entropies Let S be a conservative measure-preserving transformation of a -finite measure space (Y; E ; ). The Krengel en-

Ergodic Theory: Non-singular Transformations

tropy [119] of S is defined by hKr (S) D sup f(E)h(S E ) j 0 < (E) < C1g ; where h(S E ) is the finite measure-preserving entropy of SE . It follows from Abramov’s formula for the entropy of induced transformation that hKr (S) D (E)h(S E ) whenS ever E sweeps out, i. e. i0 S i E D X. A generic transformation from Aut0 (X; ) has entropy 0. Krengel raised a question in [119]: does there exist a zero entropy infinite measure-preserving S and a zero entropy finite measurepreserving R such that hKr (S  R) > 0? This problem was recently solved in [44] (a special case was announced by Silva and Thieullen in an October 1995 AMS conference (unpublished)): (i) if hKr (S) D 0 and R is distal then hKr (S  R) D 0; (ii) if R is not distal then there is a rank-one transformation S with hKr (S  R) D 1. We also note that if a conservative S 2 Aut0 (X; ) commutes with another transformation R such that  ı R D c for a constant c ¤ 1 then hKr (S) is either 0 or 1 [180]. Now let T be a type III ergodic transformation of (X; B; ). Silva and Thieullen define an entropy h (T) T), where e T is the Maharam of T by setting h (T) :D hKr (e extension of T (see Subsect. “Maharam Extension, Associated Flow and Orbit Classification of Type III Systems”). Since e T commutes with transformations which ‘multiply’ e T-invariant measure, it follows that h (T) is either 0 or 1. Let T be the standard III -odometer from Example 17(i). Then h (T) D 0. The same is true for a socalled ternary odometer associated with the sequence (3; n )1 nD1 , where  n (0) D  n (2) D /(1 C 2) and  n (1) D /(1 C ) [180]. It is not known however whether every ergodic nonsingular odometer has zero entropy. On the other hand, it was shown in [180] that h (T) D 1 for every K-automorphism. The Parry entropy [158] of S is defined by ˚ hPa (S) :D H(S 1 FjF) j F is a -finite subalgebra of B such that F S 1 F : Parry showed [158] that hPa (S)  hKr (S). It is still an open question whether the two entropies coincide. This is the case when S is of rank one (since hKr (S) D 0) and when S is quasi-finite [158]. The transformation S is called quasifinite if there exists a subset of finite measure A Y such that the first return time partition (A n )n>0 of A has finite entropy. We recall that x 2 A n () n is the smallest positive integer such that T n x 2 A. An example of nonquasi-finite ergodic infinite measure preserving transformation was constructed recently in [8].

Parry’s Generalization of Shannon–MacMillan–Breiman Theorem Let T be an ergodic transformation of a standard nonatomic probability space (X; B; ). Suppose that f ı T 2 L1 (X; ) if and only if f 2 L1 (X; ). This means that there is K > 0 such that K 1 < (d ı T)/(d) (x) < K for a. a. x. Let P be a finite partition of X. Denote by C n (x) the W atom of niD0 T i P which contains x. We put !1 D 0. Parry shows in [155] that n X

  log  C n j (T j x) (! j (x)  ! j1 (x))

jD0 n X

! ! j (x)

iD0

! Z ˇ _ ˇ 1 1 d ı T T P  log E H P ˇˇ d X iD1

! ˇ _ ˇ 1 i ˇ T P d ˇ iD0

for a. a. x. Parry also shows that under the aforementioned conditions on T, 0 1@ n

n X jD0

0 H@

j _

1 T  j PA 

iD0

n1 X jD0

0 H@

jC1 _

11 T  j P AA

iD1

! ˇ 1 ˇ _ i ˇ T P : !H Pˇ iD1

Critical Dimension The critical dimension introduced by Mortiss [146] measures the order of growth for sums of Radon–Nikodym derivatives. Let (X; B; ; T) be an ergodic nonsingular dynamical system. Given ı > 0, let 8 9 n1 X ˆ > ˆ > ˆ > ! i (x) ˆ > ˆ ˇ > < = ˇ iD0 and (4) X ı :D x 2 X ˇˇ lim inf > 0 ı n!1 ˆ > n ˆ > ˆ > ˆ > ˆ > : ; 8 9 n1 X ˆ > ˆ > ˆ > ! i (x) ˆ > ˆ ˇ > < = ˇ iD0 ı ˇ : (5) X :D x 2 X ˇ lim inf D 0 n!1 ˆ > nı ˆ > ˆ > ˆ > ˆ > : ; Then Xı and X ı are T-invariant subsets. Definition 37 ([57,146]) The lower critical dimension ˛(T) of T is sup fı j (Xı ) D 1g. The upper critical dimension ˇ(T) of T is inf fı j (X ı ) D 1g.

345

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Ergodic Theory: Non-singular Transformations

It was shown in [57] that the lower and upper critical dimensions are invariants for isomorphism of nonsingular systems. Notice also that ! n X log ! i (x) ˛(T) D lim inf n!1

iD1

log n ! n X log ! i (x)

ˇ(T) D lim sup n!1

iD1

and

Theorem 40 For any > 0, there exists a Hamachi shift S with ˛(S) < and ˇ(S) > 1  . :

log n

Nonsingular Restricted Orbit Equivalence

Moreover, 0  ˛(T)  ˇ(T)  1. If T is of type II1 then ˛(T) D ˇ(T) D 1. If T is the standard III -odometer from Example 17 then ˛(T) D ˇ(T) D log(1 C ) /(1 C ) log . Theorem 38 (i) For every  2 [0; 1] and every c 2 [0; 1] there exists a nonsingular odometer of type III with critical dimension equal to c [145]. (ii) For every c 2 [0; 1] there exists a nonsingular odometer of type II1 with critical dimension equal to c [57]. Let T be the nonsingular odometer associated with a sequence (m n ; n )1 nD1 . Let s(n) D m1    m n and let H(P n ) denote the entropy of the partition of the first n coordinates with respect to . We now state a nonsingular version of Shannon–MacMillan–Breiman theorem for T from [57]. Theorem 39 Let mi be bounded from above. Then (i)  ˛(T) D lim inf inf

entropy (average coordinate). It also follows from Theorem 39 that if T is an odometer of bounded type then ˛(T 1 ) D ˛(T) and ˇ(T 1 ) D ˇ(T). In [58], Theorem 39 was extended to a subclass of Markov odometers. The critical dimensions for Hamachi shifts (see Subsect. “Nonsingular Bernoulli Transformations – Hamachi’s Example”) were investigated in [59]:

n X

log m i (x i )

iD1

log s(n) H(Pn ) ; D lim inf n!1 log s(n)

In [144] Mortiss initiated study of a nonsingular version of Rudolph’s restricted orbit equivalence [167]. This work is still in its early stages and does not yet deal with any form of entropy. However she introduced nonsingular orderings of orbits, defined sizes and showed that much of the basic machinery still works in the nonsingular setting. Nonsingular Joinings and Factors The theory of joinings is a powerful tool to study probability preserving systems and to construct striking counterexamples. It is interesting to study what part of this machinery can be extended to the nonsingular case. However, the definition of nonsingular joining is far from being obvious. Some progress was achieved in understanding 2-fold joinings and constructing prime systems of any Krieger type. As far as we know the higher-fold nonsingular joinings have not been considered so far. It turned out however that an alternative coding technique, predating joinings in studying the centralizer and factors of the classical measure-preserving Chacón maps, can be used as well to classify factors of Cartesian products of some nonsingular Chacón maps.

n!1

and (ii)  ˇ(T) D lim sup inf

n X

log m i (x i )

iD1

n!1

log s(n)

H(Pn ) D lim sup log s(n) n!1 for a. a. x D (x i ) i1 2 X. It follows that in the case when ˛(T) D ˇ(T), the critical dimension coincides with limn!1 H(Pn )/(log s(n)). In [145] this expression (when it exists) was called AC-

Joinings, Nonsingular MSJ and Simplicity In this section all measures are probability measures. A nonsingular joining of two nonsingular systems ˆ on the (X1 ; B1 ; 1 ; T1 ) and (X 2 ; B2 ; 2 ; T2 ) is a measure  product B1  B2 that is nonsingular for T1  T2 and satisˆ ˆ 1 B) D 2 (B) for all A 2 fies: (AX 2 ) D 1 (A) and (X B1 and B 2 B2 . Clearly, the product 1 2 is a nonsingular joining. Given a transformation S 2 C(T), the measure S given by S (A  B) :D (A \ S 1 B) is a nonsingular joining of (X; ; T) and (X; ıS 1 ; T). It is called a graphjoining since it is supported on the graph of S. Another important kind of joinings that we are going to define now is related to factors of dynamical systems. Recall that given a nonsingular system (X; B; ; T), a sub- -algebra A of B such that T 1 (A) D A mod  is called a factor of T.

Ergodic Theory: Non-singular Transformations

There is another, equivalent, definition. A nonsingular dynamical system (Y; C ; ; S) is called a factor of T if there exists a measure-preserving map ' : X ! Y, called a factor map, with 'T D S' a. e. (If ' is only nonsingular,  may be replaced with the equivalent measure  ı ' 1 , for which ' is measure-preserving.) Indeed, the sub--algebra ' 1 (C ) B is T-invariant and, conversely, any T-invariant sub--algebra of B defines a factor map by immanent properties of standard probability spaces, see e. g. [3]. If ' is a factor map as above,Rthen  has a disintegration with respect to ', i. e.,  D  y d(y) for a measurable map y 7!  y from Y to the probability measures on X so that  y (' 1 (y)) D 1, the measure S'(x) ı T is equivalent to '(x) and dS'(x) ı T d ı T d ı S (x) (x) D ('(x)) d d d'(x)

(6)

ˆ D for a. e. x 2 X. Define now theR relative product  ˆ D  y   y d(y). Then  '  on X  X by setting  ˆ is a nonsingular selfit is easy to deduce from (6) that  joining of T. We note however that the above definition of joining is not satisfactory since it does not reduce to the classical definition when we consider probability preserving systems. Indeed, the following result was proved in [168]. Theorem 41 Let (X 1 ; B1 ; 1 ; T1 ) and (X2 ; B2 ; 2 ; T2 ) be two finite measure-preserving systems such that T1  T2 is ergodic. Then for every ; 0 <  < 1, there exists a nonˆ of 1 and 2 such that (T1  T2 ; ) ˆ is singular joining  ergodic and of type III . ˆ can It is not known however if the nonsingular joining  be chosen in every orbit equivalence class. In view of the above, Rudolph and Silva [168] isolate an important subclass of joining. It is used in the definition of a nonsingular version of minimal self-joinings. Definition 42 ˆ of (X 1 ; 1 ; T1 ) and (i) A nonsingular joining  (X 2 ; 2 ; T2 ) is rational if there exit measurable functions c 1 : X1 ! RC and c 2 : X2 ! RC such that    !ˆ 1 (x1 ; x2 ) D !1 1 (x1 )!1 2 (x2 )c 1 (x1 ) ˆ

  ˆ a. e. D !1 1 (x1 )!1 2 (x2 )c 2 (x2 ) 

(ii) A nonsingular dynamical system (X; B; ; T) has minimal self-joinings (MSJ) over a class M of probability measures equivalent to , if for every 1 ; 2 2 M, ˆ of 1 ; 2 , a. e. ergodic for every rational joining  ˆ is either the product of its marginals component of  or is the graph-joining supported on T j for some j 2 Z.

Clearly, product measure, graph-joinings and the relative products are all rational joinings. Moreover, a rational joining of finite measure-preserving systems is measure-preserving and a rational joining of type II1 ’s is of type II1 [168]. Thus we obtain the finite measure-preserving theory as a special case. As for the definition of MSJ, it depends on a class M of equivalent measures. In the finite measure-preserving case M D fg. However, in the nonsingular case no particular measure is distinguished. We note also that Definition 42(ii) involves some restrictions on all rational joinings and not only ergodic ones as in the finite measure-preserving case. The reason is that an ergodic component of a nonsingular joining needs not be a joining of measures equivalent to the original ones [2]. For finite measure-preserving transformations, MSJ over fg is the same as the usual 2-fold MSJ [49]. A nonsingular transformation T on (X; B; ) is called prime if its only factors are B and fX; ;g mod . A (nonempty) class M of probability measures equivalent to  is said to be centralizer stable if for each S 2 C(T) and 1 2 M, the measure 1 ı S is in M. Theorem 43 ([168]) Let (X; B; ; T) be a ergodic nonatomic dynamical system such that T has MSJ over a class M that is centralizer stable. Then T is prime and the centralizer of T consists of the powers of T. A question that arises is whether if such nonsingular dynamical system (not of type II1 ) exist. Expanding on Ornstein’s original construction from [150], Rudolph and Silva construct in [168], for each 0    1, a nonsingular rank-one transformation T that is of type III and that has MSJ over a class M that is centralizer stable. Type II1 examples with analogues properties were also constructed there. In this connection it is worth to mention the example by Aaronson and Nadkarni [6] of II1 ergodic transformations that have no factor algebras on which the invariant measure is  -finite (except for the trivial and the entire ones); however these transformations are not prime. A more general notion than MSJ called graph self-joinings (GSJ), was introduced [181]: just replace the the words “on T j for some j 2 Z” in Definition 3(ii) with “on S for some element S 2 C(T)”. For finite measure-preserving transformations, GSJ over fg is the same as the usual 2fold simplicity [49]. The famous Veech theorem on factors of 2-fold simple maps (see [49]) was extended to nonsingular systems in [181] as follows: if a system (X; B; ; T) has GSJ then for every non-trivial factor A of T there exists a locally compact subgroup H in C(T) (equipped with the weak topology) which acts smoothly (i. e. the partition into H-orbits is measurable) and such that A D fB 2 B j (hB4B) D 0 for all h 2 Hg. It follows that there is a co-

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cycle ' from (X; A;   A) to H such that T is isomorphic to the '-skew product extension (T  A)' (see Subsect. “Cocycles of Dynamical Systems. Weak Equivalence of Cocycles”). Of course, the ergodic nonsingular odometers and, more generally, ergodic nonsingular compact group rotation (see Subsect. “L1 -Spectrum and Groups of Quasi-Invariance”) have GSJ. However, except for this trivial case (the Cartesian square is non-ergodic) plus the systems with MSJ from [168], no examples of type III systems with GSJ are known. In particular, no smooth examples have been constructed so far. This is in sharp contrast with the finite measure preserving case where abundance of simple (or close to simple) systems are known (see [39,40,49,182]). Nonsingular Coding and Factors of Cartesian Products of Nonsingular Maps As we have already noticed above, the nonsingular MSJ theory was developed in [168] only for 2-fold self-joinings. The reasons for this were technical problems with extending the notion of rational joinings form 2-fold to n-fold self-joinings. However while the 2-fold nonsingular MSJ or GSJ properties of T are sufficient to control the centralizer and the factors of T, it is not clear whether it implies anything about the factors or centralizer of T  T. Indeed, to control them one needs to know the 4-fold joinings of T. However even in the finite measure-preserving case it is a long standing open question whether 2-fold MSJ implies n-fold MSJ. That is why del Junco and Silva [51] apply an alternative – nonsingular coding – techniques to classify the factors of Cartesian products of nonsingular Chacón maps. The techniques were originally used in [48] to show that the classical Chacón map is prime and has trivial centralizer. They were extended to nonsingular systems in [50]. For each 0 <  < 1 we denote by T  the Chacón map (see Subsect. “Rank-One Transformations. Chacón Maps. Finite Rank”) corresponding the sequence of probability vectors w n D (/(1 C 2); 1/(1 C 2); /(1 C 2)) for all n > 0. One can verify that the maps T are of type III . (The classical Chacón map corresponds to  D 1.) All of these transformations are defined on the same standard Borel space (X; B). These transformations were shown to be power weakly mixing in [12]. The centralizer of any finite Cartesian product of nonsingular Chacón maps is computed in the following theorem. Theorem 44 ([51]) Let 0 < 1 < : : : <  k  1 and n1 ; : : : ; n k be positive integers. Then the centralizer of the ˝n 1 Cartesian product T˝n  : : :  T k is generated by maps 1 k of the form U1  : : :  U k , where each U i , acting on the

ni -dimensional product space X n i , is a Cartesian product of powers of T i or a co-ordinate permutation on X n i . Let  denote the permutation on X  X defined by (x; y) D (y; x) and let B2ˇ denote the symmetric factor, i. e. B2ˇ D fA 2 B ˝ B j (A) D Ag. The following theorem classifies the factors of the Cartesian product of any two nonsingular type III , 0 <  < 1, or type II1 Chacón maps. Theorem 45 ([51]) Let T1 and T2 be two nonsingular Chacón systems. Let F be a factor algebra of T1  T2 . If 1 ¤ 2 then F is equal mod 0 to one of the four algebras B ˝ B, B ˝ N , N ˝ B, or N ˝ N , where N D f;; Xg. (ii) If 1 D 2 then F is equal mod 0 to one of the following algebras B ˝ C , B ˝ N , N ˝ C , N ˝ N , or (T m  Id)B2ˇ for some integer m.

(i)

It is not hard to obtain type III1 examples of Chacón maps for which the previous two theorems hold. However the construction of type II1 and type III0 nonsingular Chacón transformations is more subtle as it needs the choice of ! n to vary with n. In [92], Hamachi and Silva construct type III0 and type II1 examples, however the only property proved for these maps is ergodicity of their Cartesian square. More recently, Danilenko [38] has shown that all of them (in fact, a wider class of nonsingular Chacón maps of all types) are power weakly mixing. In [22], Choksi, Eigen and Prasad asked whether there exists a zero entropy, finite measure-preserving mixing automorphism S, and a nonsingular type III automorphism T, such that T  S has no Bernoulli factors. Theorem 45 provides a partial answer (with a mildly mixing only instead of mixing) to this question: if S is the finite measure-preserving Chacón map and T is a nonsingular Chacón map as above, the factors of T  S are only the trivial ones, so T  S has no Bernoulli factors. Applications. Connections with Other Fields In this – final – section we shed light on numerous mathematical sources of nonsingular systems. They come from the theory of stochastic processes, random walks, locally compact Cantor systems, horocycle flows on hyperbolic surfaces, von Neumann algebras, statistical mechanics, representation theory for groups and anticommutation relations, etc. We also note that such systems sometimes appear in the context of probability preserving dynamics (see also a criterium of distality in Subsect. “Krengel’s and Parry’s Entropies”).

Ergodic Theory: Non-singular Transformations

Mild Mixing An ergodic finite measure-preserving dynamical system (X; B; ; T) is called mildly mixing if for each non-trivial factor algebra A B, the restriction T  A is not rigid. For equivalent definitions and extensions to actions of locally compact groups we refer to [3] and [177]. There is an interesting criterium of the mild mixing that involves nonsingular systems: T is mildly mixing if and only if for each ergodic nonsingular transformation S, the product T  S is ergodic [71]. Furthermore, T  S is then orbit equivalent to S [98]. Moreover, if R is a nonsingular transformation such that R  S is ergodic for any ergodic nonsingular S then R is of type II1 (and mildly mixing) [177]. Disjointness and Furstenberg’s Class W ? Two probability preserving systems (X; ; T) and (Y; ; S) are called disjoint if    is the only T  S-invariant probability measure on X  Y whose coordinate projections are  and  respectively. Furstenberg in [69] initiated studying the class W ? of transformations disjoint from all weakly mixing ones. Let D denote the class of distal transformations and M(W ? ) the class of multipliers of W ? (for the definitions see [75]). Then D M(W ? ) W ? . In [43] and [133] it was shown by constructing explicit examples that these inclusions are strict. We record this fact here because nonsingular ergodic theory was the key ingredient of the arguments in the two papers pertaining to the theory of probability preserving systems. The examples are of the form T';S (x; y) D (Tx; S'(x) y), where T is an ergodic rotation on (X; ), (S g ) g2G a mildly mixing action of a locally compact group G on Y and ' : X ! G a measurable map. Let W' denote the Mackey action of G associated with ' and let (Z; ) be the space of this action. The key observation is that there exists an affine isomorphism of the simplex of T';S -invariant probability measures whose pullback on X is  and the simplex of W'  S quasi-invariant probability measures whose pullback on Z is  and whose Radon–Nikodym cocycle is measurable with respect to Z. This is a far reaching generalization of Furstenberg theorem on relative unique ergodicity of ergodic compact group extensions. Symmetric Stable and Infinitely Divisible Stationary Processes Rosinsky in [163] established a remarkable connection between structural studies of stationary stochastic processes and ergodic theory of nonsingular transformations (and flows). For simplicity we consider only real processes in

discrete time. Let X D (X n )n2Z be a measurable stationary symmetric ˛-stable (S˛S) process, 0 < ˛ < 2. This P means that any linear combination nkD1 a k X j k , j k 2 Z, a k 2 R has an S˛S-distribution. (The case ˛ D 2 corresponds to Gaussian processes.) Then the process admits a spectral representation Z Xn D

f n (y) M(dy) ;

n2Z;

(7)

Y

where f n 2 L˛ (Y; ) for a standard  -finite measure space (Y;B;) and M is an independently scattered random measure on B such that E exp (iuM(A)) D exp (juj˛ (A)) for every A 2 B of finite measure. By [163], one can choose the kernel ( f n )n2Z in a special way: there are a -nonsingular transformation T and measurable maps ' : X ! f1; 1g and f 2 L˛ (Y; ) such that f n D U n f , n 2 Z, where U is the isometry of L˛ (X; ) given by Ug D '  (d ı T/d)1/˛  g ı T. If, in addition, the smallest T-invariant  -algebra containing f 1 (BR ) coincides with B and Supp f f ı T n : n 2 Zg D Y then the pair (T; ') is called minimal. It turns out that minimal pairs always exist. Moreover, two minimal pairs (T; ') and (T 0 ; ' 0 ) representing the same S˛S process are equivalent in some natural sense [163]. Then one can relate ergodic-theoretical properties of (T; ') to probabilistic properties of (X n )n2Z . For instance, let Y D C t D be the HopfR decomposition of Y (see Theorem 2). We R let X nD : D D f n (y) M(dy) and X nC :D C f n (y) M(dy). Then we obtain a unique (in distribution) decomposition of X into the sum X D C X C of two independent stationary S˛S-processes. Another kind of decomposition was considered in [171]. Let P be the largest invariant subset of Y such that T  P has a finite invariant measure. Partitioning Y into P and N :D Y n N and restricting the integration in (7) to P and N we obtain a unique (in distribution) decomposition of X into the sum X P C X N of two independent stationary S˛S-processes. Notice that the process X is ergodic if and only if (P) D 0. Recently, Roy considered a more general class of infinitely divisible (ID) stationary processes [165]. Using Maruyama’s representation of the characteristic function of an ID process X without Gaussian part he singled out the Lévy measure Q of X. Then Q is a shift invariant  -finite measure on RZ . Decomposing the dynamical system (RZ ; ; Q) in various natural ways (Hopf decomposition, 0-type and positive type, so-called ‘rigidity free’ part and its complement) he obtains corresponding decompositions for the process X. Here  stands for the shift on RZ .

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Ergodic Theory: Non-singular Transformations

Poisson Suspensions Poisson suspensions are widely used in statistical mechanics to model ideal gas, Lorentz gas, etc (see [33]). Let (X; B; ) be a standard -finite non-atomic measure space and (X) D 1. Denote by e X the space of unordered countable subsets of X. It is called the space of configurations. Fix t > 0. Let A 2 B have positive finite measure and let j 2 ZC . Denote by [A; j] the subset of all configurations e x2e X such that #(e x \ A) D j. Let e B be the -algebra generated by all [A; j]. We define a probability measure  et on e B by two conditions:

Boundaries of Random Walks

j

exp (t(A)); (i)  et ([A; j]) D (t(A)) j! (ii) if A1 ; : : : ; A p are pairwise disjoint Tp Qp et ([A k ; j k ]).  et ( kD1 [A k ; j k ]) D kD1 

singular transformation of a standard probability space (X; B; ) and let f : X ! Rn a measurable function. DeP n fine for m  1, Ym : X ! Rn by Ym :D m1 nD0 f ı T . In other words, (Ym )m1 is the random walk associated with the (non-stationary) process ( f ı T n )n0 . Let us call this random walk recurrent if the cocycle f of T is recurrent (see Subsect. “Cocycles of Dynamical Systems. Weak Equivalence of Cocycles”). It was shown in [176] that in the case  ı T D , i. e. the process is stationary, this definition is equivalent to the standard one.

then

If T is a -preserving transformation of X and e x D (x1 ; x2 ; : : : ) is a configuration then we set e T! :D T is a e -preserving (Tx1 ; Tx2 ; : : : ). It is easy to verify that e transformation of e X. The dynamical system (e X; e B; e ; e T) is called the Poisson suspension above (X; B; ; T). It is ergodic if and only if T has no invariant sets of finite positive measure. There is a canonical representation of L2 (e X; e ) as the Fock space over L2 (X; ) such that the unitary operator Ue of U T . Thus, the T is the ‘exponent’ P 1  n , where  is (n!) maximal spectral type of Ue n0 T is a measure of the maximal spectral type of U T . It is easy to see that a -finite factor of T corresponds to a factor (called Poissonian) of e T. Moreover, a -finite measurepreserving joining (with -finite projections) of two infinite measure-preserving transformations T 1 and T 2 gene2 [52,164]. e1 and T erates a joining (called Poissonian) of T Thus we see a similarity with the well studied theory of Gaussian dynamical systems [134]. However, the Poissonian case is less understood. There was a recent progress in this field. Parreau and Roy constructed Poisson suspensions whose ergodic self-joinings are all Poissonian [154]. In [111] partial solutions of the following (still open) problems are found: (i) whether the Pinsker factor of e T is Poissonian, (ii) what is the relationship between Krengel’s entropy of T, Parry’s entropy of T and Kolmogorov–Sinai entropy of e T.

Recurrence of Random Walks with Non-stationary Increments Using nonsingular ergodic theory one can introduce the notion of recurrence for random walks obtained from certain non-stationary processes. Let T be an ergodic non-

Boundaries of random walks on groups retain valuable information on the underlying groups (amenability, entropy, etc.) and enable one to obtain integral representation for harmonic functions of the random walk [112,186,187]. Let G be a locally compact group and  a probability measure on G. Let T denote the (one-sided) shift on the probability space (X; B X ; ) :D (G; BG ; )ZC and ' : X ! G a measurable map defined by (y0 ; y1 ; : : : ) 7! y0 . Let T' be the '-skew product extension of T acting on the space (X  G;   G ) (for noninvertible transformations the skew product extension is defined in the very same way as for invertible ones, see Subsect. “Cocycles of Dynamical Systems. Weak Equivalence of Cocycles”). Then T' is isomorphic to the homogeneous random walk on G with jump probability . Let I (T' ) denote the sub- -algebra of T' -invariant sets and T let F (T' ) :D n>0 T'n (B X ˝ BG ). The former is called the Poisson boundary of T' and the latter one is called the tail boundary of T' . Notice that a nonsingular action of G by inverted right translations along the second coordinate is well defined on each of the two boundaries. The two boundaries (or, more precisely, the G-actions on them) are ergodic. The Poisson boundary is the Mackey range of ' (as a cocycle of T). Hence the Poisson boundary is amenable [187]. If the support of  generates a dense subgroup of G then the corresponding Poisson boundary is weakly mixing [4]. As for the tail boundary, we first note that it can be defined for a wider family of non-homogeneous random walks. This means that the jump probability  is no longer fixed and a sequence (n )n>0 of probability measures on G is considered instead. Now let Q (X; B X ; ) :D n>0 (G; BG ; ). The one-sided shift on X may not be nonsingular now. Instead of it, we consider the tail equivalence relation R on X and a cocycle ˛ : R ! G given by ˛(x; y) D x1    x n y 1 n    y 1 , where x D (x i ) i>0 and y D (y i ) i>0 are R-equivalent and n in the smallest integer such that x i D y i for all i > n. The tail boundary of the random walk on G with time dependent jump

Ergodic Theory: Non-singular Transformations

probabilities (n )n>0 is the Mackey G-action associated with ˛. In the case of homogeneous random walks this definition is equivalent to the initial one. Connes and Woods showed [32] that the tail boundary is always amenable and AT. It is unknown whether the converse holds for general G. However it is true for G D R and G D Z: the class of AT-flows coincides with the class of tail boundaries of the random walks on R and a similar statement holds for Z [32]. Jaworsky showed [109] that if G is countable and a random walk is homogeneous then the tail boundary of the random walk possesses a so-called SAT-property (which is stronger than AT). Classifying  -Finite Ergodic Invariant Measures The description of ergodic finite invariant measures for topological (or, more generally, standard Borel) systems is a well established problem in the classical ergodic theory [33]. On the other hand, it seems impossible to obtain any useful information about the system by analyzing the set of all ergodic quasi-invariant (or just -finite invariant) measures because this set is wildly huge (see Subsect. “The Glimm–Effros Theorem”). The situation changes if we impose some restrictions on the measures. For instance, if the system under question is a homeomorphism (or a topological flow) defined on a locally compact Polish space then it is natural to consider the class of ( -finite) invariant Radon measures, i. e. measures taking finite values on the compact subsets. We give two examples. First, the seminal results of Giordano, Putnam and Skau on the topological orbit equivalence of compact Cantor minimal systems were extended to locally compact Cantor minimal (l.c.c.m.) systems in [37] and [138]. Given a l.c.c.m. system X, we denote by M(X) and M1 (X) the set of invariant Radon measures and the set of invariant probability measures on X. Notice that M1 (X) may be empty [37]. It was shown in [138] that two systems X and X 0 are topologically orbit equivalent if and only if there is a homeomorphism of X onto X 0 which maps bijectively M(X) onto M(X 0 ) and M1 (X) onto M1 (X 0 ). Thus M(X) retains an important information on the system – it is ‘responsible’ for the topological orbit equivalence of the underlying systems. Uniquely ergodic l.c.c.m. systems (with unique up to scaling infinite invariant Radon measure) were constructed in [37]. The second example is related to study of the smooth horocycle flows on tangent bundles of hyperbolic surfaces. Let D be the open disk equipped with the hyperbolic metric jdzj/(1  jzj2 ) and let Möb(D) denote the group of Möbius transformations of D. A hyperbolic surface can be written in the form M :D  nMöb(D), where  is a tor-

sion free discrete subgroup of Möb(D). Suppose that  is a nontrivial normal subgroup of a lattice  0 in Möb(D). Then M is a regular cover of the finite volume surface M0 : D 0 nMöb(D). The group of deck transformations G D 0 / is finitely generated. The horocycle flow (h t ) t2R and the geodesic flow (g t ) t2R defined on the unit tangent bundle T 1 (D) descend naturally to flows, say h and g, on T 1 (M). We consider the problem of classification of the h-invariant Radon measures on M. According to Ratner, h has no finite invariant measures on M if G is infinite (except for measures supported on closed orbits). However there are infinite invariant Radon measures, for instance the volume measure. In the case when G is free Abelian and  0 is co-compact, every homomorphism ' : G ! R determines a unique up to scaling ergodic invariant Radon measure (e.i.r.m.) m on T 1 (M) such that m ı dD D exp ('(D))m for all D 2 G [16] and every e.i.r.m. arises this way [172]. Moreover all these measures are quasi-invariant under g. In the general case, an interesting bijection is established in [131] between the e.i.r.m. which are quasi-invariant under g and the ‘nontrivial minimal’ positive eigenfunctions of the hyperbolic Laplacian on M. Von Neumann Algebras There is a fascinating and productive interplay between nonsingular ergodic theory and von Neumann algebras. The two theories alternately influenced development of each other. Let (X; B; ; T) be a nonsingular dynamical system. Given ' 2 L1 (X; ) and j 2 Z, we define operators A' and U j on the Hilbert space L2 (Z  Z;   ) by setting (A' f )(x; i) :D '(T i x) f (x; i) ; (U j f )(x; i) :D f (x; i  j) : Then U j A' U j D A'ıT j . Denote by M the von Neumann algebra (i. e. the weak closure of the -algebra) generated by A' , ' 2 L1 (X; ) and U j , j 2 Z. If T is ergodic and aperiodic then M is a factor, i. e. M \ M0 D C1, where M0 denotes the algebra of bounded operators commuting with M. It is called a Krieger’s factor. Murray–von Neumann–Connes’ type of M is exactly the Krieger’s type of T. The flow of weights of M is isomorphic to the associated flow of T. Two Krieger’s factors are isomorphic if and only if the underlying dynamical systems are orbit equivalent [129]. Moreover, a number of important problems in the theory of von Neumann algebras such as classification of subfactors, computation of the flow of weights and Connes’ invariants, outer conjugacy for

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automorphisms, etc. are intimately related to the corresponding problems in nonsingular orbit theory. We refer to [42,66,73,74,89,142] for details. Representations of CAR Representations of canonical anticommutation relations (CAR) is one of the most elegant and useful chapters of mathematical physics, providing a natural language for many body quantum physics and quantum field theory. By a representation of CAR we mean a sequence of bounded linear operators a1 ; a2 ; : : : in a separable Hilbert space K such that a j a k C a k a j D 0 and a j ak C a k a j D ı j;k . Consider f0; 1g as a group with addition mod 2. Then X D f0; 1gN is a compact Abelian group. Let  :D fx D (x1 ; x2 ; : : : ) : limn!1 x n D 0g. Then  is a dense countable subgroup of X. It is generated by elements k whose k-coordinate is 1 and the other ones are 0.  acts on X by translations. Let  be an ergodic  -quasi-invariant measure on X. Let (C k ) k1 be Borel maps from X to the group of unitary operators in a Hilbert space H satisfying C k (x) D C k (x C ı k ), C k (x)C l (x C ı l ) D C l (x)C k (x C ı k ), k ¤ l for a. a. x. In other words, (C k ) k1 f :D defines a cocycle of the  -action. We now put H 2 f L (X; ) ˝ H and define operators ak in H by setting (a k f )(x) D (1)x 1 CCx k1 (1  x k ) s d ı ı k (x) f (x C ı k ) ;  C k (x) d f and x D (x1 ; x2 ; where f : X ! H is an element of H : : : ) 2 X. It is easy to verify that a1 ; a2 ; : : : is a representation of CAR. The converse was established in [72] and [77]: every factor-representation (this means that the von Neumann algebra generated by all ak is a factor) of CAR can be represented as above for some ergodic measure , Hilbert space H and a  -cocycle (C k ) k1 . Moreover, using nonsingular ergodic theory Golodets [77] constructed for each k D 2; 3; : : : ; 1, an irreducible representation of CAR such that dim H D k. This answered a question of Gårding and Wightman [72] who considered only the case k D 1. Unitary Representations of Locally Compact Groups Nonsingular actions appear in a systematic way in the theory of unitary representations of groups. Let G be a locally compact second countable group and H a closed normal subgroup of G. Suppose that H is commutative (or, more generally, of type I, see [53]). Then the natural action of G by conjugation on H induces a Borel G-action, say ˛, on the dual space b H – the set of unitarily equivalent

classes of irreducible unitary representations of H. If now U D (U g ) g2G is a unitary representation of G in a separable Hilbert space then by applying Stone decomposition theorem to U  H one can deduce that ˛ is nonsingular with respect to a measure  of the ‘maximal spectral b Moreover, if U is irreducible then ˛ type’ for U  H on H. is ergodic. Whenever  is fixed, we obtain a one-to-one correspondence between the set of cohomology classes of irreducible cocycles for ˛ with values in the unitary group on a Hilbert space H and the subset of b G consisting of classes of those unitary representations V for which the measure associated to V  H is equivalent to . This correspondence is used in both directions. From information about cocycles we can deduce facts about representations and vise versa [53,118]. Concluding Remarks While some of the results that we have cited for nonsingular Z-actions extend to actions of locally compact Polish groups (or subclasses of Abelian or amenable ones), many natural questions remain open in the general setting. For instance: what is Rokhlin lemma, or the pointwise ergodic theorem, or the definition of entropy for nonsingular actions of general countable amenable groups? The theory of abstract nonsingular equivalence relations [66] or, more generally, nonsingular groupoids [160] and polymorphisms [184] is also a beautiful part of nonsingular ergodic theory that has nice applications: description of semifinite traces of AF-algebras, classification of factor representations of the infinite symmetric group [185], path groups [14], etc. Nonsingular ergodic theory is getting even more sophisticated when we pass from Z-actions to noninvertible endomorphisms or, more generally, semigroup actions (see [3] and references therein). However, due to restrictions of space we do not consider these issues in our survey. Bibliography 1. Aaronson J (1983) The eigenvalues of nonsingular transformations. Isr J Math 45:297–312 2. Aaronson J (1987) The intrinsic normalizing constants of transformations preserving infinite measures. J Analyse Math 49:239–270 3. Aaronson J (1997) An Introduction to Infinite Ergodic Theory. Amer Math Soc, Providence ´ 4. Aaronson J, Lemanczyk M (2005) Exactness of Rokhlin endomorphisms and weak mixing of Poisson boundaries, Algebraic and Topological Dynamics. Contemporary Mathematics, vol 385. Amer Math Soc, Providence 77–88 5. Aaronson J, Lin M, Weiss B (1979) Mixing properties of Markov operators and ergodic transformations, and ergodicity of cartesian products. Isr J Math 33:198–224

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Ergodic Theory: Recurrence

Ergodic Theory: Recurrence N IKOS FRANTZIKINAKIS, RANDALL MCCUTCHEON Department of Mathematics, University of Memphis, Memphis, USA Article Outline Glossary Definition of the Subject Introduction Quantitative Poincaré Recurrence Subsequence Recurrence Multiple Recurrence Connections with Combinatorics and Number Theory Future Directions Bibliography Glossary Almost every, essentially Given a Lebesgue measure space (X; B; ), a property P(x) predicated of elements of X is said to hold for almost every x 2 X, if the set X n fx : P(x) holdsg has zero measure. Two sets A; B 2 B are essentially disjoint if (A \ B) D 0. Conservative system Is an infinite measure preserving system such that for no set A 2 B with positive measure are A; T 1 A; T 2 A; : : : pairwise essentially disjoint. (cn )-Conservative system If (c n )n2N is a decreasing sequence of positive real numbers, a conservative ergodic measure preserving transformation T is (c n )conservative if for some non-negative function f 2 P n L1 (), 1 nD1 c n f (T x) D 1 a. e. Doubling map If T is the interval [0; 1] with its endpoints identified and addition performed modulo 1, the (non-invertible) transformation T : T ! T , defined by Tx D 2x mod 1, preserves Lebesgue measure, hence induces a measure preserving system on T . Ergodic system Is a measure preserving system (X; B; ; T) (finite or infinite) such that every A 2 B that is T-invariant (i. e. T 1 A D A) satisfies either (A) D 0 or (X n A) D 0. (One can check that the rotation R˛ is ergodic if and only if ˛ is irrational, and that the doubling map is ergodic). Ergodic decomposition Every measure preserving system (X; X ; ; T) can be expressed as an integral of ergodic systems; for example, one can write  D R  t d(t), where  is a probability measure on [0; 1] and  t are T-invariant probability measures on (X; X ) such that the systems (X; X ;  t ; T) are ergodic for t 2 [0; 1].

Ergodic theorem States that if (X; B; ; T) is a measure 2 preserving system and  1 PN  f 2 L (), then lim N!1 nf P   T D 0, where P f denotes the f L 2 () nD1 N orthogonal projection of the function f onto the subspace f f 2 L2 () : T f D f g. Hausdorff a-measure Let (X; B; ; T) be a measure preserving system endowed with a -compatible metric d. The Hausdorff a-measure H a (X) of X is an outer measure defined for all subsets of X as follows: First, for P a A X and " > 0 let H a;"(A) D inff 1 iD1 r i g, where the infimum is taken over all countable coverings of A by sets U i X with diameter r i < ". Then define H a (A) D lim sup"!0 H a;"(A). Infinite measure preserving system Same as measure preserving system, but (X) D 1. Invertible system Is a measure preserving system (X; B; ; T) (finite or infinite), with the property that there exists X0 2 X, with (X n X 0 ) D 0, and such that the transformation T : X0 ! X0 is bijective, with T 1 measurable. Measure preserving system Is a quadruple (X; B; ; T), where X is a set, B is a  -algebra of subsets of X (i. e. B is closed under countable unions and complementation),  is a probability measure (i. e. a countably additive function from B to [0; 1] with (X) D 1), and T : X ! X is measurable (i. e. T 1 A D fx 2 X : Tx 2 Ag 2 B for A 2 B), and -preserving (i. e. (T 1 A) D (A)). Moreover, throughout the discussion we assume that the measure space (X; B; ) is Lebesgue (see Sect. 1.0 in [2]). -Compatible metric Is a separable metric on X, where (X; B; ) is a probability space, having the property that open sets are measurable. Positive definite sequence Is a complex-valued sequence (a n )n2Z such that for any n1 ; : : : ; n k 2 Z and z1 ; : : : ; P z k 2 C, ki; jD1 a n i n j z i z j  0. Rotations on T If T is the interval [0; 1] with its endpoints identified and addition performed modulo 1, then for every ˛ 2 R the transformation R˛ : T ! T , defined by R˛ x D x C ˛mod1, preserves Lebesgue measure on T and hence induces a measure preserving system on T . Syndetic set Is a subset E Z having bounded gaps. If G is a general discrete group, a set E G is syndetic if G D FE for some finite set F G. Upper density Is the number d() D lim sup N!1 (j \ fN; : : : ; Ngj)/(2N C 1), where  Z (assuming the limit to exist). Alternatively for measurable E

Rm , D(E) D lim sup l (S)!1 (m(S \ E))/(m(S)), where S ranges over all cubes in Rm , and l(S) denotes the length of the shortest edge of S.

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Notation The following notation will be used throughout the article: T f D f ı T, fxg D x  [x], D- limn!1 (a n ) D a $ d fn : ja n  aj > "g D 0 for every " > 0.

a finite (or conservative) measure preserving system, every set of positive measure (or almost every point) comes back to itself infinitely many times under iteration. Despite the profound importance of these results, their proofs are extremely simple.

Definition of the Subject

Theorem 1 (Poincaré Recurrence for Sets) Let (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Then (A \ T n A) > 0 for infinitely many n 2 N.

The basic principle that lies behind several recurrence phenomena is that the typical trajectory of a system with finite volume comes back infinitely often to any neighborhood of its initial point. This principle was first exploited by Poincaré in his 1890 King Oscar prize-winning memoir that studied planetary motion. Using the prototype of an ergodic-theoretic argument, he showed that in any system of point masses having fixed total energy that restricts its dynamics to bounded subsets of its phase space, the typical state of motion (characterized by configurations and velocities) must recur to an arbitrary degree of approximation. Among the recurrence principle’s more spectacularly counterintuitive ramifications is that isolated ideal gas systems that do not lose energy will return arbitrarily closely to their initial states, even when such a return entails a decrease in entropy from equilibrium, in apparent contradiction to the second law of thermodynamics. Such concerns, previously canvassed by Poincaré himself, were more infamously expounded by Zermelo [74] in 1896. Subsequent clarifications by Boltzmann, Maxwell and others led to an improved understanding of the second law’s primarily statistical nature. (For an interesting historical/philosophical discussion, see [68]; also [10]. For a probabilistic analysis of the likelihood of observing second law violations in small systems over short time intervals, see [28]). These discoveries had a profound impact in dynamics, and the theory of measure preserving transformations (ergodic theory) evolved from these developments. Since then, the Poincaré recurrence principle has been applied to a variety of different fields in mathematics, physics, and information theory. In this article we survey the impact it has had in ergodic theory, especially as pertains to the field of ergodic Ramsey theory. (The heavy emphasis herein on the latter reflects authorial interest, and is not intended to transmit a proportionate image of the broader landscape of research relating to recurrence in ergodic theory.) Background information we assume in this article can be found in the books [35,63,71] ( Measure Preserving Systems). Introduction In this section we shall give several formulations of the Poincaré recurrence principle using the language of ergodic theory. Roughly speaking, the principle states that in

Proof Since T is measure preserving, the sets A; T 1 A; T 2 A; : : : have the same measure. These sets cannot be pairwise essentially disjoint, since then the union of finitely many of them would have measure greater than (X) D 1. Therefore, there exist m; n 2 N, with n > m, such that (T m A\T n A) > 0. Again since T is measure preserving, we conclude that (A \ T k A) > 0, where k D n  m > 0. Repeating this argument for the iterates A; T m A; T 2m A; : : :, for all m 2 N, we easily deduce  that (A \ T n A) > 0 for infinitely many n 2 N. We remark that the above argument actually shows that 1 (A \ T n A) > 0 for some n  [ (A) ] C 1. Theorem 2 (Poincaré Recurrence for Points) Let (X; B; ; T) be a measure preserving system and A 2 B. Then for almost every x 2 A we have that T n x 2 A for infinitely many n 2 N. Proof Let B be the set of x 2 A such that T n x … A for all S n 2 N. Notice that B D A n n2N T n A; in particular, B is measurable. Since the iterates B; T 1 B; T 2 B; : : : are pairwise essentially disjoint, we conclude (as in the proof of Theorem 1) that (B) D 0. This shows that for almost every x 2 A we have that T n x 2 A for some n 2 N. Repeating this argument for the transformation T m in place of T for all m 2 N, we easily deduce the advertised statement.  Next we give a variation of Poincaré recurrence for measure preserving systems endowed with a compatible metric: Theorem 3 (Poincaré Recurrence for Metric Systems) Let (X; B; ; T) be a measure preserving system, and suppose that X is endowed with a -compatible metric. Then for almost every x 2 X we have lim inf d(x; T n x) D 0 : n!1

The proof of this result is similar to the proof of Theorem 2 (see p. 61 in [35]). Applying this result to the doubling map Tx D 2x on T , we get that for almost every x 2 X, every string of zeros and ones in the dyadic expansion of x occurs infinitely often.

Ergodic Theory: Recurrence

n x 2 X the set R S x D fn 2 N : T x 2 Ag has well defined density and d(S x ) d(x) D (A). Furthermore, for ergodic measure preserving systems we have d(S x ) D (A) a. e.

We remark that all three formulations of the Poincaré Recurrence Theorem that we have given hold for conservative systems as well. See, e. g., [2] for details. This article is structured as follows. In Sect. “Quantitative Poincaré Recurrence” we give a few quantitative versions of the previously mentioned qualitative results. In Sects. “Subsequence Recurrence” and “Multiple Recurrence” we give several refinements of the Poincaré recurrence theorem, by restricting the scope of the return time n, and by considering multiple intersections (for simplicity we focus on Z-actions). In Sect. “Connections with Combinatorics and Number Theory” we give various implications of the recurrence results in combinatorics and number theory ( Ergodic Theory: Interactions with Combinatorics and Number Theory). Lastly, in Sect. “Future Directions” we give several open problems related to the material presented in Sects. “Subsequence Recurrence” to “Connections with Combinatorics and Number Theory”.

Theorem 6 (Kac [51]) Let (X; B; ; T) be an ergodic measure preserving system and A 2 B with (A) > 0. For x 2 X define R A (x) D minfn 2 N : T n x 2 Ag. RThen for x 2 A the expected value of R A (x) is 1/(A), i. e. A R A (x) d D 1.

Quantitative Poincaré Recurrence

More Recent Results

Early Results

As we mentioned in the previous section, if the space X is endowed with a -compatible metric d, then for almost every x 2 X we have that lim infn!1 d(x; T n x) D 0. A natural question is, how much iteration is needed to come back within a small distance of a given typical point? Under some additional hypothesis on the metric d we have the following answer:

For applications it is desirable to have quantitative versions of the results mentioned in the previous section. For example one would like to know how large (A \ T n A) can be made and for how many n. Theorem 4 (Khintchine [55]) Let (X; B; ; T) be a measure preserving system and A 2 B. Then for every " > 0 we have (A \ T n A) > (A)2  " for a set of n 2 N that has bounded gaps. By considering the doubling map Tx D 2x on T and letting A D 1[0;1/2) , it is easy to check that the lower bound of the previous result cannot be improved. We also remark that it is not possible to estimate the size of the gap by a function of (A) alone. One can see this by considering the rotations R k x D x C 1/k for k 2 N, defined on T , and letting A D 1[0;1/3] . Concerning the second version of the Poincaré recurrence theorem, it is natural to ask whether for almost every x 2 X the set of return times S x D fn 2 N : T n x 2 Ag has bounded gaps. This is not the case, as one can see by considering the doubling map Tx D 2x on T with the Lebesgue measure, and letting A D 1[0;1/2) . Since Lebesgue almost every x 2 T contains arbitrarily large blocks of ones in its dyadic expansion, the set S x has unbounded gaps. Nevertheless, as an easy consequence of the Birkhoff ergodic theorem [19], one has the following: Theorem 5 Let (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Then for almost every

Another question that arises naturally is, given a set A with positive measure and an x 2 A, how long should one wait until some iterate T n x of x hits A? By considering an irrational rotation R˛ on T , where ˛ is very near to, but not 1 , and letting A D 1[0;1/2] , one can see that less than, 100 the first return time is a member of the set f1; 50; 51g. So it may come as a surprise that the average first return time does not depend on the system (as long as it is ergodic), but only on the measure of the set A.

Theorem 7 (Boshernitzan [20]) Let (X; B; ; T) be a measure preserving system endowed with a -compatible metric d. Assume that the Hausdorff a-measure H a (X) of X is  -finite (i. e., X is a countable union of sets X i with H a (X i ) < 1). Then for almost every x 2 X, ˚ 1 lim inf n a  d(x; T n x) < 1 : n!1

Furthermore, if H a (X) D 0, then for almost every x 2 X, ˚ 1 lim inf n a  d(x; T n x) D 0 : n!1

One can see from rotations by “badly approximable” vectors ˛ 2 T k that the exponent 1/a in the previous theorem cannot be improved. Several applications of Theorem 7 to billiard flows, dyadic transformations, symbolic flows and interval exchange transformations are given in [20]. For a related result dealing with mean values of the limits in Theorem 7 see [67]. An interesting connection between rates of recurrence and entropy of an ergodic measure preserving system was established by Ornstein and Weiss [62], following earlier work of Wyner and Ziv [73]:

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Theorem 8 (Ornstein and Weiss [62]) Let (X; B; ; T) be an ergodic measure preserving system and P be a finite partition of X. Let Pn (x) be the element of the partition Wn1 i Tn1 i (i) : P(i) 2 P ; 0  i < ng iD0 T P D f iD0 T P that contains x. Then for almost every x 2 X, the first return time R n (x) of x to Pn (x) is asymptotically equivalent to e h(T;P )n , where h(T; P ) denotes the entropy of the system with respect to the partition P . More precisely, lim

n!1

log R n (x) D h(T; P ) : n

An extension of the above result to some classes of infinite measure preserving systems was given in [42]. Another connection of recurrence rates, this time with the local dimension of an invariant measure, is given by the next result: Theorem 9 (Barreira [4]) Let (X; B; ; T) be an ergodic measure preserving system. Define the upper and lower recurrence rates log r (x) and  log r log r (x) R(x) D lim sup ;  log r r!0

R(x) D lim inf r!0

where r (x) is the first return time of T k x to B(x; r), and the upper and lower pointwise dimensions log (B(x; r)) and log r log (B(x; r)) d  (x) D lim sup : log r r!0

d  (x) D lim inf r!0

Then for almost every x 2 X, we have R(x)  d  (x) and

R(x)  d  (x) :

Roughly speaking, this theorem asserts that for typical x 2 X and for small r, the first return time of x to B(x; r) is at most rd (x) . Since d  (x)  H a (X) for almost every x 2 X, we can conclude the first part of Theorem 7 from Theorem 9. For related results the interested reader should consult the survey [5] and the bibliography therein. We also remark that the previous results and related concepts have been applied to estimate the dimension of certain strange attractors (see [49] and the references therein) and the entropy of some Gibbsian systems [25]. We end this section with a result that connects “wandering rates” of sets in infinite measure preserving systems with their “recurrence rates”. The next theorem follows easily from a result about lower bounds on ergodic aver-

ages for measure preserving systems due to Leibman [57]; a weaker form for conservative, ergodic systems can be found in Aaronson [1]. Theorem 10 Let (X; B; ; T) be an infinite measure preserving system, and A 2 B with (A) < 1. Then for all N 2 N,  0 S N1 1 n A N1  T X nD0 1 @ (A \ T n A)A  ((A))2 :  N 2 nD0

Subsequence Recurrence In this section we discuss what restrictions one can impose on the set of return times in the various versions of the Poincaré recurrence theorem. We start with: Definition 11 Let R Z. Then R is a set of: (a) Recurrence if for any invertible measure preserving system (X; B; ; T), and A 2 B with (A) > 0, there is some nonzero n 2 R such that (A \ T n A) > 0. (b) Topological recurrence if for every compact metric space (X; d), continuous transformation T : X ! X and every " > 0, there are x 2 X and nonzero n 2 R such that d(x; T n x) < ". It is easy to check that the existence of a single n 2 R satisfying the previous recurrence conditions actually guarantees the existence of infinitely many n 2 R satisfying the same conditions. Moreover, if R is a set of recurrence then one can see from existence of some T-invariant measure  that R is also a set of topological recurrence. A (complicated) example showing that the converse is not true was given by Kriz [56]. Before giving a list of examples of sets of (topological) recurrence, we discuss some necessary conditions: A set of topological recurrence must contain infinitely many multiples of every positive integer, as one can see by considering rotations on Zd , d 2 N . Hence, the sets f2n C 1; n 2 Ng, fn2 C 1; n 2 Ng, fp C 2; p primeg are not good for (topological) recurrence. If (s n )n2N is a lacunary sequence (meaning lim infn!1 (s nC1 /s n ) D > 1), then one can construct an irrational number ˛ such that fs n ˛g 2 [ı; 1  ı] for all large n 2 N, where ı > 0 depends on (see [54], for example). As a consequence, the sequence (s n )n2N is not good for (topological) recurrence. Lastly, we mention that by considering product systems, one can immediately show that any set of (topological) recurrence R is partition regular, meaning that if R is partitioned into finitely many pieces then at least one of these pieces must still be a set of (topological) recur-

Ergodic Theory: Recurrence

rence. Using this observation, one concludes for example that any union of finitely many lacunary sequences is not a set of recurrence. We present now some examples of sets of recurrence: Theorem 12 The following are sets of recurrence: S (i) Any set of the form n2N fa n ; 2a n ; : : : ; na n g where a n 2 N. (ii) Any IP-set, meaning a set that consists of all finite sums of the members of some infinite set. (iii) Any difference set S  S, meaning a set that consists of all possible differences of the members of some some infinite set S. (iv) The set fp(n); n 2 Ng where p is any nonconstant integer polynomial with p(0) D 0 [35,66] (In fact we only have to assume that the range of the polynomial contains multiples of an arbitrary positive integer [53]). (v) The set fp(n); n 2 Sg, where p is an integer polynomial with p(0) D 0 and S is any IP-set [12]. (vi) The set of values of an admissible generalized polynomial (this class contains in particular the smallest function algebra G containing all integer polynomials having zero constant term and such that if g1 ; : : : ; g k 2 G and c1 ; : : : ; c k 2 R then P [[ kiD1 c i g i ]] 2 G, where [[x]] D [x C 12 ] denotes the integer nearest to x) [13]. (vii) The set of shifted primes fp  1; p primeg, and the set fp C 1; p primeg [66]. (viii) The set of values of a random non-lacunary sequence. (Pick n 2 N independently with probability b n where 0  b n  1 and limn!1 nb n D 1. The resulting set is almost surely a set of recurrence. If lim supn!1 nb n < 1 then the resulting set is almost surely a finite union of sets, each of which is the range of some lacunary sequence, hence is not a set of recurrence). Follows from [22]. Showing that the first three sets are good for recurrence is a straightforward modification of the argument used to prove Theorem 1. Examples (iv)  (viii) require more work. A criterion of Kamae and Mendés-France [53] provides a powerful tool that may be used in many instances to establish that a set R is a set of recurrence. We mention a variation of their result: Theorem 13 (Kamae and Mendés-France [53]) Suppose that R D fa1 < a2 < : : :g is a subset of N such that: (i) The sequence fa n ˛gn2N is uniformly distributed in T for every irrational ˛.

(ii) The set R m D fn 2 N : mja n g has positive density for every m 2 N. Then R is a set of recurrence. We sketch a proof for this result. First, recall Herglotz’s theorem: if (a n )n2Z is a positive definite sequence, then there isR a unique measure  on the torus T such that 2 i nt Ra n D e n d (t). The case of interest to us is a n D T f (x)  f (T x) d, where T is measure preserving and f 2 L1 (); (a n ) is positive definite, and we call  D  f the spectral measure of f . Let now (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Putting f D 1A , one has lim

N!1

N Z 1 X f (x)  f (T a n x) d N nD1

Z D

lim

T N!1

N 1 X 2 i a n t e N nD1

! d f (t) :

(1)

For t irrational the limit inside the integral is zero (by condition (i)), so the last integral can be taken over the rational points in T . Since the spectral measure of a function orthogonal to the subspace H D f f 2 L2 () : there exists k 2 N with T k f D f g

(2) has no rational point masses, we can easily deduce that when computing the first limit in (1), we can replace the function f by its orthogonal projection g onto the subspace H (g is again nonnegative and g ¤ 0). To complete the argument, we approximate g by a function g 0 such that T m g 0 D g 0 for some appropriately chosen m, and use condition (ii) to deduce that the limit of the average (1) is positive. In order to apply Theorem 13, one uses the standard machinery of uniform distribution. Recall Weyl’s criterion: a real-valued sequence (x n )n2N is uniformly distributed mod 1 if for every non-zero k 2 Z, lim

N!1

N 1 X 2 i kx n e D0: N nD1

This criterion becomes especially useful when paired with van der Corput’s so-called third principal property: if, for every h 2 N, (x nCh  x n )n2N is uniformly distributed mod 1, then (x n )n2N is uniformly distributed mod 1. Using the foregoing criteria and some standard (albeit nontrivial) exponential sum estimates, one can verify for example that the sets (iv) and (vii) in Theorem 12 are good for recurrence.

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In light of the connection elucidated above between uniform distribution mod 1 and recurrence, it is not surprising that van der Corput’s method has been adapted by modern ergodic theorists for use in establishing recurrence properties directly.

pute

N 1 X hx nCm ; x n i N!1 N nD1

lim

D lim

N!1

Theorem 14 (Bergelson [7]) Let (x n )n2N be a bounded sequence in a Hilbert space. If D-lim

m!1

! N 1 X hx nCm ; x n i D 0 ; lim N!1 N nD1

then   N 1 X    xn  D 0 : lim   N!1  N nD1

Let us illustrate how one uses this “van der Corput trick” by showing that S D fn2 : n 2 Ng is a set of recurrence. We will actually establish the following stronger fact: If (X; B; ; T) is a measure preserving system and f 2 L1 () is nonnegative and f ¤ 0 then lim inf N!1

1 N

N Z X

2

f (x)  f (T n x) d > 0 :

(3)

nD1

Then our result follows by setting f D 1A for some A 2 B with (A) > 0. The main idea is one that occurs frequently in ergodic theory; split the function f into two components, one of which contributes zero to the limit appearing in (3), and the other one being much easier to handle than f . To do this consider the T-invariant subspace of L2 (X) defined by H D f f 2 L2 () : there exists k 2 N with T k f D f g :

(4) Write f D g C h where g 2 H and h?H , and expand the average in (3) into a sum of four averages involving the functions g and h. Two of these averages vanish because iterates of g are orthogonal to iterates of h. So in order to show that the only contribution comes from the average that involves the function g alone, it suffices to establish that   N  1 X  n2  lim  T h D0: (5)  2 N!1  N nD1 L ()

To show this we will apply the Hilbert space van der Cor2 put lemma. For given h 2 N, we let x n D T n h and com-

N Z 1 X 2 2 2 T n C2nmCm h  T n h d N nD1

N Z 1 X 2 D lim T 2nm (T m h)  h d : N!1 N nD1

Applying the ergodic theorem to the transformation T 2m and using the fact that h?H , we get that the last limit is 0. This implies (5). Thus far we have shown that in order to compute the limit in (3) we can assume that f D g 2 H (g is also nonnegative and g ¤ 0). By the definition of H , given any " > 0, there exists a function f 0 2 H such that T k f 0 D f 0 for some k 2 N and k f  f 0 kL 2 ()  ". Then the limit in (3) is at least 1/k times the limit N Z 1 X 2 f (x)  f (T (kn) x) d: lim inf N!1 N nD1 Applying the triangle inequality twice we get that this is greater or equal than N Z 1 X 2 lim f 0 (x)  f 0 (T (kn) x) d  c  " N!1 N nD1 Z D ( f 0 (x))2 d  2" Z 

2 f 0 (x) d  c  ";

for some constant c that does not depend on " (we used that T k f 0 D f 0 and the Cauchy–Schwartz inequality). Choosing " small enough we conclude that the last quantity is positive, completing the proof. Multiple Recurrence Simultaneous multiple returns of positive measure sets to themselves were first considered by H. Furstenberg [34], who gave a new proof of Szemerédi’s theorem [69] on arithmetic progressions by deriving it from the following theorem: Theorem 15 (Furstenberg [34]) Let (X; B; ; T) be a measure preserving system and A 2 B with (A) > 0. Then for every k 2 N, there is some n 2 N such that (A \ T n A \    \ T kn A) > 0 :

(6)

Furstenberg’s proof came by means of a new structure theorem allowing one to decompose an arbitrary measure preserving system into component elements exhibiting

Ergodic Theory: Recurrence

one of two extreme types of behavior: compactness, characterized by regular, “almost periodic” trajectories, and weak mixing, characterized by irregular, “quasi-random” trajectories. On T , these types of behavior are exemplified by rotations and by the doubling map, respectively. To see the point, imagine trying to predict the initial digit of the dyadic expansion of T n x given knowledge of the initial digits of T i x, 1  i < n. We use the case k D 2 to illustrate the basic idea. It suffices to show that if f 2 L1 () is nonnegative and f ¤ 0, one has N Z 1 X f (x)  f (T n x)  f (T 2n x) d > 0 : (7) lim inf N!1 N nD1 An ergodic decomposition argument enables us to assume that our system is ergodic. As in the earlier case of the squares, we split f into “almost periodic” and “quasi-random” components. Let K be the closure in L2 of the subspace spanned by the eigenfunctions of T, i. e. the functions f 2 L2 () that satisfy f (Tx) D e 2 i˛ f (x) for some ˛ 2 R. We write f D g C h, where g 2 K and h?K . It can be shown that g; h 2 L1 () and g is again nonnegative with g ¤ 0. We expand the average in (7) into a sum of eight averages involving the functions g and h. In order to show that the only non-zero contribution to the limit comes from the term involving g alone, it suffices to establish that   N  1 X  n 2n  lim  T g  T h D0; (8)  2 N!1  N nD1

L ()

(and similarly with h and g interchanged, and with g D h, which is similar). To establish (8), we use the Hilbert space van der Corput lemma on x n D T n g  T 2n h. Some routine computations and a use of the ergodic theorem reduce the task to showing that  Z D-lim h(x)  h(T 2m x) d D 0 : m!1

But this is well known for h?K (in virtue of the fact that for h?K the spectral measure  h is continuous, for example). We are left with the average (7) when f D g 2 K . In this case f can be approximated arbitrarily well by a linear combination of eigenfunctions, which easily implies that given " > 0 one has kT n f  f kL 2 ()  " for a set of n 2 N with bounded gaps. Using this fact and the triangle inequality, one finds that for a set of n 2 N with bounded gaps, Z 3 Z f (x)  f (T n x)  f (T 2n x) d  f d  c  "

for a constant c that is independent of ". Choosing " small enough, we get (7). The new techniques developed for the proof of Theorem 15 have led to a number of extensions, many of which have to date only ergodic proofs. To expedite discussion of some of these developments, we introduce a definition: Definition 16 Let R Z and k 2 N. Then R is a set of k-recurrence if for every invertible measure preserving system (X; B; ; T) and A 2 B with (A) > 0, there is some nonzero n 2 R such that (A \ T n A \    \ T kn A) > 0 : The notions of k-recurrence are distinct for different values of k. An example of a difference set that is a set of 1-recurrence but not a set of 2-recurrence was given in [34]; sets of k-recurrence that are not sets of (k C 1)recurrencepfor general k were given in [31] (R k D fn 2 N : fn kC1 2g 2 [1/4; 3/4]g is such). Aside from difference sets, the sets of (1-)recurrence given in Theorem 12 may well be sets of k-recurrence for every k 2 N, though this has not been verified in all cases. Let us summarize the current state of knowledge. The following are sets of k-recurrence for every k: Sets of S the form n2N fa n ; 2a n ; : : : ; na n g where a n 2 N (this follows from a uniform version of Theorem 15 that can be found in [15]). Every IP-set [37]. The set fp(n); n 2 Ng where p is any nonconstant integer polynomial with p(0) D 0 [16], and more generally, when the range of the polynomial contains multiples of an arbitrary integer [33]. The set fp(n); n 2 Sg where p is an integer polynomial with p(0) D 0 and S is any IP-set [17]. The set of values of an admissible generalized polynomial [60]. Moreover, the set of shifted primes fp  1; p primeg, and the set fp C 1; p primeg are sets of 2-recurrence [32]. More generally, one would like to know for which sequences of integers a1 (n); : : : ; a k (n) it is the case that for every invertible measure preserving system (X; B; ; T) and A 2 B with (A) > 0, there is some nonzero n 2 N such that (A \ T a 1 (n) A \    \ T a k (n) A) > 0 :

(9)

Unfortunately, a criterion analogous to the one given in Theorem 13 for 1-recurrence is not yet available for k-recurrence when k > 1. Nevertheless, there have been some notable positive results, such as the following: Theorem 17 (Bergelson and Leibman [16])

Let (X;

B; ; T) be an invertible measure preserving system and

p1 (n); : : : ; p k (n) be integer polynomials with zero constant term. Then for every A 2 B with (A) > 0, there is some

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n 2 N such that (A \ T p 1 (n) A \    \ T p k (n) A) > 0 :

(10)

Furthermore, it has been shown that the n in (10) can be chosen from any IP set [17], and the polynomials p1 ; : : : ; p k can be chosen to belong to the more general class of admissible generalized polynomials [60]. Very recently, a new boost in the area of multiple recurrence was given by a breakthrough of Host and Kra [50]. Building on work of Conze and Lesigne [26,27] and Furstenberg and Weiss [41] (see also the excellent survey [52], exploring close parallels with [45] and the seminal paper of Gowers [43]), they isolated the structured component (or factor) of a measure preserving system that one needs to analyze in order to prove various multiple recurrence and convergence results. This allowed them, in particular, to prove existence of L2 limits for the so-called PN Qk in “Furstenberg ergodic averages” N1 nD1 iD0 f (T x), which had been a major open problem since the original ergodic proof of Szemerédi’s theorem. Subsequently Ziegler in [75] gave a new proof of the aforementioned limit theorem and established minimality of the factor in question. It turns out that this minimal component admits of a purely algebraic characterization; it is a nilsystem, i. e. a rotation on a homogeneous space of a nilpotent Lie group. This fact, coupled with some recent results about nilsystems (see [58,59] for example), makes the analysis of some otherwise intractable multiple recurrence problems much more manageable. For example, these developments have made it possible to estimate the size of the multiple intersection in (6) for k D 2; 3 (the case k D 1 is Theorem 4): Theorem 18 (Bergelson, Host and Kra [14]) Let (X; B; ; T) be an ergodic measure preserving system and A 2 B. Then for k D 2; 3 and for every " > 0, (A \ T n A \    \ T kn A) >  kC1 (A)  "

(11)

for a set of n 2 N with bounded gaps. Based on work of Ruzsa that appears as an appendix to the paper, it is also shown in [14] that a similar estimate fails for ergodic systems (with any power of (A) on the right hand side) when k  4. Moreover, when the system is nonergodic it also fails for k D 2; 3, as can be seen with the help of an example in [6]. Again considering the doubling map Tx D 2x and the set A D [0; 1/2], one sees that the positive results for k  3 are sharp. When the polynomials n; 2n; : : : ; kn are replaced by linearly independent polynomials p1 ; p2 ; : : : ; p k with zero constant term, similar lower bounds hold for every k 2 N without assuming

ergodicity [30]. The case where the polynomials n; 2n; 3n are replaced with general polynomials p1 ; p2 ; p3 with zero constant term is treated in [33]. Connections with Combinatorics and Number Theory The combinatorial ramifications of ergodic-theoretic recurrence were first observed by Furstenberg, who perceived a correspondence between recurrence properties of measure preserving systems and the existence of structures in sets of integers having positive upper density. This gave rise to the field of ergodic Ramsey theory, in which problems in combinatorial number theory are treated using techniques from ergodic theory. The following formulation is from [8]. Theorem 19 Let  be a subset of the integers. There exists an invertible measure preserving system (X; B; ; T) and a set A 2 B with (A) D d() such that d( \ (  n1 ) \ : : : \ (  n k ))  (A \ T n 1 A \    \ T n k A) ;

(12)

for all k 2 N and n1 ; : : : ; n k 2 Z. Proof The space X will be taken to be the sequence space f0; 1gZ , B is the Borel  -algebra, while T is the shift map defined by (Tx)(n) D x(n C 1) for x 2 f0; 1gZ , and A is the set of sequences x with x(0) D 1. So the only thing that depends on  is the measure  which we now define. For m 2 N set 0 D Z n  and 1 D . Using a diagonal argument we can find an increasing sequence of integers (N m )m2N such that limm!1 j \ [1; N m ]j/N m D d() and such that ˇ ˇ ˇ ( i 1  n1 ) \ ( i 2  n2 ) \    ˇ ˇ ˇ ˇ ˇ \( i r  nr ) \ [1; N m ] (13) lim m!1 Nm exists for every n1 ; : : : ; nr 2 Z, and i1 ; : : : ; i r 2 f0; 1g. For n1 ; n2 ; : : : ; nr 2 Z, and i1 ; i2 ; : : : ; i r 2 f0; 1g, we define the measure  of the cylinder set fx(n1 ) D i1 ; x(n2 ) D i2 ; : : : ; x(nr ) D i r g to be the limit (13). Thus defined,  extends to a premeasure on the algebra of sets generated by cylinder sets and hence by Carathéodory’s extension theorem [24] to a probability measure on B. It is easy to check that (A) D d(), the shift transformation T preserves the measure  and (12) holds.  Using this principle for k D 1, one may check that any set of recurrence is intersective, that is intersects E  E for every set E of positive density. Using it for n1 D n; n2 D

Ergodic Theory: Recurrence

2n; : : : ; n k D kn, together with Theorem 15, one gets an ergodic proof of Szemerédi’s theorem [69], stating that every subset of the integers with positive upper density contains arbitrarily long arithmetic progressions (conversely, one can easily deduce Theorem 15 from Szemerédi’s theorem, and that intersective sets are sets of recurrence). Making the choice n1 D n2 and using part (iv) of Theorem 13, we get an ergodic proof of the surprising result of Sárközy [66] stating that every subset of the integers with positive upper density contains two elements whose difference is a perfect square. More generally, using Theorem 19, one can translate all of the recurrence results of the previous two sections to results in combinatorics. (This is not straightforward for Theorem 18 because of the ergodicity assumption made there. We refer the reader to [14] for the combinatorial consequence of this result). We mention explicitly only the combinatorial consequence of Theorem 17: Theorem 20 (Bergelson and Leibman [16]) Let  Z with d() > 0, and p1 ; : : : ; p k be integer polynomials with zero constant term. Then  contains infinitely many configurations of the form fx; x C p1 (n); : : : ; x C p k (n)g. The ergodic proof is the only one known for this result, even for patterns of the form fx; xCn2 ; xC2n2 g or fx; xC n; x C n2 g. Ergodic-theoretic contributions to the field of geometric Ramsey theory were made by Furstenberg, Katznelson, and Weiss [40], who showed that if E is a positive upper density subset of R2 then: (i) E contains points with any large enough distance (see also [21] and [29]), (ii) Every ı-neighborhood of E contains three points forming a triangle congruent to any given large enough dilation of a given triangle (in [21] it is shown that if the three points lie on a straight line one cannot always find three points with this property in E itself). Recently, a generalization of property (ii) to arbitrary finite configurations of Rm was obtained by Ziegler [76]. It is also worth mentioning some recent exciting connections of multiple recurrence with some structural properties of the set of prime numbers. The first one is in the work of Green and Tao [45], where the existence of arbitrarily long arithmetic progressions of primes was demonstrated, the authors, in addition to using Szemerédi’s theorem outright, use several ideas from its ergodic-theoretic proofs, as appearing in [34] and [39]. The second one is in the recent work of Tao and Ziegler [70], where a quantitative version of Theorem 17 was used to prove that the primes contain arbitrarily long polynomial progressions. Furthermore, several recent results in ergodic theory, related to the structure of the minimal characteristic fac-

tors of certain multiple ergodic averages, play an important role in the ongoing attempts of Green and Tao to get asymptotic formulas for the number of k-term arithmetic progressions of primes up to x (see for example [46] and [47]). This project has been completed for k D 3, thus verifying an interesting special case of the Hardy– Littlewood k-tuple conjecture predicting the asymptotic growth rate of N a 1 ;:::;a k (x) D the number of configurations of primes having the form fp; p C a1 ; : : : ; p C a k g with p  x. Finally, we remark that in this article we have restricted attention to multiple recurrence and Furstenberg correspondence for Z actions, while in fact there is a wealth of literature on extensions of these results to general commutative, amenable and even non-amenable groups. For an excellent exposition of these and other recent developments the reader is referred to the surveys [9] and [11]. Here, we give just one notable combinatorial corollary to some work of this kind, a density version of the classical Hales–Jewett coloring theorem [48]. Theorem 21 (Furstenberg and Katznelson [38]) Let Wn (A) denote the set of words of length n with letters in the alphabet A D fa1 ; : : : ; a k g. For every " > 0 there exists N0 D N0 ("; k) such that if n  N0 then any subset S of Wn (A) with jSj  "k n contains a combinatorial line, i. e., a set consisting of k n-letter words, having fixed letters in l positions, for some 0  l < n, the remaining n  l positions being occupied by a variable letter x, for x D a1 ; : : : ; a k . (For example, in W4 (A) the sets f(a1 ; x; a2 ; x) : x 2 Ag and f(x; x; x; x); : x 2 Ag are combinatorial lines). At first glance, the uninitiated reader may not appreciate the importance of this “master” density result, so it is instructive to derive at least one of its immediate consequences. Let A D f0; 1; : : : ; k  1g and interpret Wn (A) as integers in base k having at most n digits. Then a combinatorial line in Wn (A) is an arithmetic progression of length k – for example, the line f(a1 ; x; a2 ; x) : x 2 Ag corresponds to the progression fm; m C n; m C 2n; m C 3ng, where m D a1 C a2 d 2 and n D d C d 3 . This allows one to deduce Szemerédi’s theorem. Similarly, one can deduce from Theorem 21 multidimensional and IP extensions of Szemerédi’s theorem [36,37], and some related results about vector spaces over finite fields [37]. Again, the only known proof for the density version of the Hales– Jewett theorem relies heavily on ergodic theory. Future Directions In this section we formulate a few open problems relating to the material in the previous three sections. It should be

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noted that this selection reflects the authors’ interests, and does not strive for completeness. We start with an intriguing question of Katznelson [54] about sets of topological recurrence. A set S N is a set of Bohr recurrence if for every ˛1 ; : : : ; ˛ k 2 R and " > 0 there exists s 2 S such that fs˛ i g 2 [0; "] [ [1  "; 1) for i D 1; : : : ; k. Problem 1 Is every set of Bohr recurrence a set of topological recurrence? Background for this problem and evidence for a positive answer can be found in [54,72]. As we mentioned in Sect. “Subsequence Recurrence”, there exists a set of topological recurrence (and hence Bohr recurrence) that is not a set of recurrence. Problem 2 Is the set S D fl!2m 3n : l; m; n 2 Ng a set of recurrence? Is it a set of k-recurrence for every k 2 N? It can be shown that S is a set of Bohr recurrence. Theorem 13 cannot be applied since the uniform distribution condition fails for some irrational numbers ˛. A relevant question was asked by Bergelson in [9]: “Is the set S D f2m 3n : m; n 2 Ng good for single recurrence for weakly mixing systems?” As we mentioned in Sect. “Multiple Recurrence”, the set of primes shifted by 1 (or  1) is a set of 2-recurrence [32]. Problem 3 Show that the sets P  1 and P C 1, where P is the set of primes, are sets of k-recurrence for every k 2 N. As remarked in [32], a positive answer to this question will follow if some uniformity conjectures of Green and Tao [47] are verified. We mentioned in Sect. “Subsequence Recurrence” that random non-lacunary sequences (see definition there) are almost surely sets of recurrence. Problem 4 Show that random non-lacunary sequences are almost surely sets of k-recurrence for every k 2 N. The answer is not known even for k D 2, though, in unpublished work, Wierdl and Lesigne have shown that the answer is positive for random sequences with at most quadratic growth. We refer the reader to the survey [65] for a nice exposition of the argument used by Bourgain [22] to handle the case k D 1. It was shown in [31] that if S is a set of 2-recurrence then the set of its squares is a set of recurrence for circle rotations. The same method shows that it is actually a set of Bohr recurrence. Problem 5 If S Z is a set of 2-recurrence, is it true that S 2 D fs 2 : s 2 Sg is a set of recurrence?

A similar question was asked in [23]: “If S is a set of k-recurrence for every k, is the same true of S2 ?”. One would like to find a criterion that would allow one to deduce that a sequence is good for double (or higher order) recurrence from some uniform distribution properties of this sequence. Problem 6 Find necessary conditions for double recurrence similar to the one given in Theorem 13. It is now well understood that such a criterion should involve uniform distribution properties of some generalized polynomials or 2-step nilsequences. We mentioned in Sect. “Connections with Combinatorics and Number Theory” that every positive density subset of R2 contains points with any large enough distance. Bourgain [21] constructed a positive density subset E of R2 , a triangle T, and numbers t n ! 1, such that E does not contain congruent copies of all t n -dilations of T. But the triangle T used in this construction is degenerate, which leaves the following question open: Problem 7 Is it true that every positive density subset of R2 contains a triangle congruent to any large enough dilation of a given non-degenerate triangle? For further discussion on this question the reader can consult the survey [44]. The following question of Aaronson and Nakada [1] is related to a classical question of Erd˝os concerning whether P every K N such that n2K 1/n D 1 contains arbitrarily long arithmetic progressions: Problem 8 Suppose that (X; B; ; T) is a f1/ng-conservative ergodic measure preserving system. Is it true that for every A 2 B with (A) > 0 and k 2 N we have (A \ T n A \    \ T kn A) > 0 for some n 2 N? The answer is positive for the class of Markov shifts, and it is remarked in [1] that if the Erd˝os conjecture is true then the answer will be positive in general. The converse is not known to be true. For a related result showing that multiple recurrence is preserved by extensions of infinite measure preserving systems see [61]. Our next problem is motivated by the question whether Theorem 21 has a polynomial version (for a precise formulation of the general conjecture see [9]). Not even this most special consequence of it is known to hold. Problem 9 Let " > 0. Does there exist N D N(") having the property that every family P of subsets of f1; : : : ; Ng2 2 satisfying jPj  "2 N contains a configuration fA; A [ ( 

)g, where A f1; : : : ; Ng2 and f1; : : : ; Ng with A \ (  ) D ;?

Ergodic Theory: Recurrence

A measure preserving action of a general countably infinite group G is a function g ! Tg from G into the space of measure preserving transformations of a probability space X such that Tg h D Tg Th . It is easy to show that a version of Khintchine’s recurrence theorem holds for such actions: if (A) > 0 and " > 0 then fg : (A \ Tg A) > ((A))2  "g is syndetic. However it is unknown whether the following ergodic version of Roth’s theorem holds. Problem 10 Let (Tg ) and (S g ) be measure preserving G-actions of a probability space X that commute in the sense that Tg S h D S h Tg for all g; h 2 G. Is it true that for all positive measure sets A, the set of g such that (A \ Tg A \ S g A) > 0 is syndetic? We remark that for general (possibly amenable) groups G not containing arbitrarily large finite subgroups nor elements of infinite order, it is not known whether one can find a single such g ¤ e. On the other hand, the answer is known to be positive for general G in case (Tg1 S g ) is a G-action [18]; even under such strictures, however, it is unknown whether a triple recurrence theorem holds. Bibliography 1. Aaronson J (1981) The asymptotic distribution behavior of transformations preserving infinite measures. J Analyse Math 39:203–234 2. Aaronson J (1997) An introduction to infinite ergodic theory. Mathematical Surveys and Monographs 50. American Mathematical Society, Providence 3. Aaronson J, Nakada H (2000) Multiple recurrence of Markov shifts and other infinite measure preserving transformations. Israel J Math 117:285–310 4. Barreira L (2001) Hausdorff dimension of measures via Poincaré recurrence. Comm Math Phys 219:443–463 5. Barreira L (2005) Poincaré recurrence: old and new. XIVth International Congress on Mathematical Physics, World Sci Publ Hackensack, NJ, pp 415–422 6. Behrend F (1946) On sets of integers which contain no three in arithmetic progression. Proc Nat Acad Sci 23:331–332 7. Bergelson V (1987) Weakly mixing PET. Ergod Theory Dynam Syst 7:337–349 8. Bergelson V (1987) Ergodic Ramsey Theory. In: Simpson S (ed) Logic and Combinatorics. Contemporary Mathematics 65. American Math Soc, Providence, pp 63–87 9. Bergelson V (1996) Ergodic Ramsey Theory – an update. In: Pollicot M, Schmidt K (eds) Ergodic theory of Zd -actions. Lecture Note Series 228. London Math Soc, London, pp 1–61 10. Bergelson V (2000) The multifarious Poincaré recurrence theorem. In: Foreman M, Kechris A, Louveau A, Weiss B (eds) Descriptive set theory and dynamical systems. Lecture Note Series 277. London Math Soc, London, pp 31–57 11. Bergelson V (2005) Combinatorial and diophantine applications of ergodic theory. In: Hasselblatt B, Katok A (eds) Handbook of dynamical systems, vol 1B. Elsevier, pp 745–841 12. Bergelson V, Furstenberg H, McCutcheon R (1996) IP-sets and polynomial recurrence. Ergod Theory Dynam Syst 16:963–974

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57. Leibman A (2002) Lower bounds for ergodic averages. Ergod Theory Dynam. Syst 22:863–872 58. Leibman A (2005) Pointwise convergence of ergodic averages for polynomial sequences of rotations of a nilmanifold. Ergod Theory Dynam Syst 25:201–213 59. Leibman A (2005) Pointwise convergence of ergodic averages for polynomial actions of Zd by translations on a nilmanifold. Ergod Theory Dynam Syst 25:215–225 60. McCutcheon R (2005) FVIP systems and multiple recurrence. Israel J Math 146:157–188 61. Meyerovitch T (2007) Extensions and multiple recurrence of infinite measure preserving systems. Preprint. Available at http://arxiv.org/abs/math/0703914 62. Ornstein D, Weiss B (1993) Entropy and data compression schemes. IEEE Trans Inform Theory 39:78–83 63. Petersen K (1989) Ergodic theory. Cambridge Studies in Advanced Mathematics 2. Cambridge University Press, Cambridge 64. Poincaré H (1890) Sur le problème des trois corps et les équations de la dynamique. Acta Math 13:1–270 65. Rosenblatt J, Wierdl M (1995) Pointwise ergodic theorems via harmonic analysis. Ergodic theory and its connections with harmonic analysis (Alexandria, 1993). London Math Soc, Lecture Note Series 205, Cambridge University Press, Cambridge, pp 3–151 66. Sárközy A (1978) On difference sets of integers III. Acta Math Acad Sci Hungar 31:125–149 67. Shkredov I (2002) Recurrence in the mean. Mat Zametki 72(4):625–632; translation in Math Notes 72(3–4):576–582 68. Sklar L (2004) Philosophy of Statistical Mechanics. In: Zalta EN (ed) The Stanford Encyclopedia of Philosophy (Summer 2004 edn), http://plato.stanford.edu/archives/sum2004/ entries/statphys-statmech/ 69. Szemerédi E (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith 27:299–345 70. Tao T, Ziegler T () The primes contain arbitrarily long polynomial progressions. Acta Math (to appear). Available at http:// www.arxiv.org/abs/math.DS/0610050 71. Walters P (1982) An introduction to ergodic theory. Graduate Texts in Mathematics, vol 79. Springer, Berlin 72. Weiss B (2000) Single orbit dynamics. CBMS Regional Conference Series in Mathematics 95. American Mathematical Society, Providence 73. Wyner A, Ziv J (1989) Some asymptotic properties of the entropy of a stationary ergodic data source with applications to data compression. IEEE Trans Inform Theory 35:1250–1258 74. Zermelo E (1896) Über einen Satz der Dynamik und die mechanische Wärmetheorie. Annalen der Physik 57:485–94; English translation, On a theorem of dynamics and the mechanical theory of heat. In: Brush SG (ed) Kinetic Theory. Oxford, 1966, II, pp 208–17 75. Ziegler T (2007) Universal characteristic factors and Furstenberg averages. J Amer Math Soc 20:53–97 76. Ziegler T (2006) Nilfactors of Rm -actions and configurations in sets of positive upper density in Rm . J Anal Math 99:249– 266

Ergodic Theory: Rigidity

Ergodic Theory: Rigidity ˘ 1,2 VIOREL N I TIC ¸ A 1 West Chester University, West Chester, USA 2 Institute of Mathematics, Bucharest, Romania

Article Outline Glossary Definition of the Subject Introduction Basic Definitions and Examples Differentiable Rigidity Local Rigidity Global Rigidity Measure Rigidity Future Directions Acknowledgment Bibliography Glossary Differentiable rigidity Differentiable rigidity refers to finding invariants to the differentiable conjugacy of dynamical systems, and, more general, group actions. Local rigidity Local rigidity refers to the study of perturbations of homomorphisms from discrete or continuous groups into diffeomorphism groups. Global rigidity Global rigidity refers to the classification of all group actions on manifolds satisfying certain conditions. Measure rigidity Measure rigidity refers to the study of invariant measures for actions of abelian groups and semigroups. Lattice A lattice in a Lie group is a discrete subgroup of finite covolume. Conjugacy Two elements g1 ; g2 in a group G are said to be conjugated if there exists an element h 2 G such that g1 D h1 g2 h. The element h is called conjugacy. Ck Conjugacy Two diffeomorphisms 1 ; 2 acting on the same manifold M are said to be Ck -conjugated if there exists a Ck diffeomorphism h of M such that 1 D h1 ı 2 ı h. The diffeomorphism h is called Ck conjugacy. Definition of the Subject As one can see from this volume, chaotic behavior of complex dynamical systems is prevalent in nature and in large classes of transformations. Rigidity theory can be viewed as the counterpart to the generic theory of dynamical systems which often investigates chaotic dynamics for a typical transformation belonging to a large class. In rigid-

ity one is interested in finding obstructions to chaotic, or generic, behavior. This often leads to rather unexpected classification results. As such, rigidity in dynamics and ergodic theory is difficult to define precisely and the best approach to this subject is to study various results and themes that developed so far. A classification is offered below in local, global, differentiable and measurable rigidity. One should note that all branches are strongly intertwined and, at this stage of the development of the subject, it is difficult to separate them. Rigidity is a well developed and prominent topic in modern mathematics. Historically, rigidity has two main origins, one coming from the study of lattices in semi-simple Lie groups, and one coming from the theory of hyperbolic dynamical systems. From the start, ergodic theory was an important tool used to prove rigidity results, and a strong interdependence developed between these fields. Many times a result in rigidity is obtained by combining techniques from the theory of lattices in Lie groups with techniques from hyperbolic dynamical systems and ergodic theory. Among other mathematical disciplines using results and contributing to this field one can mention representation theory, smooth, continuous and measurable dynamics, harmonic and spectral analysis, partial differential equations, differential geometry, and number theory. Additional details about the appearance of rigidity in ergodic theory as well as definitions for some terminology used in the sequel can be found in the articles  Ergodic Theory on Homogeneous Spaces and Metric Number Theory by Kleinbock,  Ergodic Theory: Recurrence by Frantzikinakis, McCutcheon, and  Ergodic Theory: Interactions with Combinatorics and Number Theory by Ward. The theory of hyperbolic dynamics is presented in the article  Hyperbolic Dynamical Systems by Viana and in the article  Smooth Ergodic Theory by Wilkinson. Introduction The first results about classification of lattices in semi-simple Lie groups were local, aimed at trying to understand the space of small perturbations of a given linear representation. A major contributor was Weil [110,111,112], who proved local rigidity of linear representations for large classes of groups, in particular lattices. Another breakthrough was the contribution of Kazhdan [66], who introduced property (T), allowing to show that large classes of lattices are finitely generated. Rigidity theory matured due to the remarkable global rigidity results obtained by Mostow [90] and Margulis [82], leading to a complete classification of lattices in large classes of semi-simple Lie groups.

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Briefly, a hyperbolic (Anosov) dynamical system is one that exhibits strong expansion and contraction along complementary directions. An early contribution introducing this class of objects is the paper of Smale [106], in which basic examples and techniques are introduced. A breakthrough came with the results of Anosov [1], who proved structural stability of the Anosov systems and ergodicity of the geodesic flow on a manifold of negative curvature. Motivated by questions arising in mathematical physics, chaos theory and other areas, hyperbolic dynamics emerged as one of the major fields of contemporary mathematics. From the beginning, a major unsolved problem in the field was the classification of Anosov diffeomorphisms and flows. In the 80s a change in philosophy occurred, partially motivated by a program introduced by Zimmer [114]. The goal of the program was to classify the smooth actions of higher rank semi-simple Lie groups and of their (irreducible) lattices on compact manifolds. It was expected that any such lattice action that preserves a smooth volume form and is ergodic can be reduced to one of the following standard models: isometric actions, linear actions on infranilmanifolds, and left translations on compact homogeneous spaces. This original conjecture was disproved by Katok, Lewis (see [56]): by blowing up a linear nilmanifold-action at some fixed points they exhibit real-analytic, volume preserving, ergodic lattice actions on manifolds with complicated topology. Nevertheless, imposing extra assumptions on a higher rank action, for example the existence of some hyperbolicity, allows local and global classification results. The concept of Anosov action, that is, an action that contains at least one Anosov diffeomorphism, was introduced for general groups by Pugh, Shub [99]. The significant differences between the classical Z and R cases and those of higher rank lattices, or at a more basic level, of higher rank abelian groups, went unnoticed for a while. The surge of activity in the 80s allowed these differences to surface: for lattices in the work of Hurder, Katok, Lewis, Zimmer (see [43,55,56,63]); and for higher rank abelian groups in the work of Katok, Lewis [55] and Katok, Spatzier [59]. As observed in these papers, local and global rigidity are typical for such Anosov actions. This generated additional research which is summarized in Sects. “Local Rigidity” and “Global Rigidity”. Differentiable rigidity is covered in Sect. “Differentiable Rigidity”. An interesting problem is to find moduli for the Ck conjugacy, k  1, of Anosov diffeomorphisms and flows. This was tackled so far only for low dimensional cases (n D 2 for diffeomorphisms and n D 3 for flows). Another direction that can be included here refers to find-

ing obstructions for higher transverse regularity of the stable/unstable foliation of a hyperbolic system. A spin-off of the research done so far, which is of high interest by itself, and has applications to local and global rigidity, consists of results lifting the regularity of solutions of cohomological equations over hyperbolic systems. In turn, these results motivated a more careful study of analytic questions about lifting the regularity of real valued continuous functions that enjoy higher regularity along webs of foliations. We also include in this section rigidity results for cocycles over higher rank abelian actions. These are crucial to the proof of local rigidity of higher rank abelian group actions. A more detailed presentation of the material relevant to differentiable rigidity can be found in the forthcoming monograph [58]. Measure rigidity refers to the study of invariant measures under actions of abelian groups and semigroups. If the actions are hyperbolic, higher-rank, and satisfy natural algebraic and irreducibility assumptions, one expects the invariant measures to be rare. This direction was started by a question of Furstenberg, asking if any nonatomic probability measure on the circle, invariant and ergodic under multiplications by 2 and 3, is the Lebesgue measure. An early contribution is that of Rudolph [103], who answered positively if the action has an element of strictly positive entropy. Katok, Spatzier [61] extended the question to more general higher rank abelian actions, such as actions by linear automorphisms of tori and Weyl chamber flows. A related direction is the study of the invariant sets and measures under the action of horocycle flows, where important progress was made by Ratner [100,101] and earlier by Margulis [16,81,83]. An application of these results present in the last papers is the proof of the long standing Oppenheim’s conjecture, about the density of the values of the quadratic forms at integer points. Recent developments due to and Einsiedler, Katok, Lindenstrauss [20] give a partial answer to another outstanding conjecture in number theory, Littlewood’s conjecture, and emphasize measure rigidity as one of a more promising directions in rigidity. More details are shown in Sect. “Measure Rigidity”. Four other recent surveys of rigidity theory, each one with a fair amount of overlap but also complementary in part to the present one, that discuss various aspects of the field and its significance are written by Fisher [23], Lindenstrauss [68], Ni¸tic˘a, Török [95], and Spatzier [107]. Among these, [23] concentrates mostly on local and global rigidity, [95] on differentiable rigidity, [68] on measure rigidity, and [107] gives a general overview. Here is a word of caution for the reader. Many times, instead of the most general results, we present an example

Ergodic Theory: Rigidity

that contains the essence of what is available. Also, several important facts that should have been included, are left out. This is because stating complete results would require more space than allocated to this material. The limited knowledge of the author also plays a role here. He apologizes for any obvious omissions and hopes that the bibliography will help fill the gaps. Basic Definitions and Examples A detailed introduction to the theory of Anosov systems and hyperbolic dynamics is given in the monograph [51]. The proofs of the basic results for diffeomorphisms stated below can be found there. The proofs for flows are similar. Surveys about hyperbolic dynamics in this volume are the article  Hyperbolic Dynamical Systems by Viana and the article  Smooth Ergodic Theory by Wilkinson. Consider a compact differentiable manifold M and f : M ! M a C1 diffeomorphism. Let TM be the tangent bundle of M, and D f : TM ! TM be the derivative of f . The map f is said to be an Anosov diffeomorphism if there is a smooth Riemannian metric k  k on M, which induces a metric dM called adapted, a number  2 (0; 1), and a continuous Df invariant splitting TM D E s ˚ E u such that kD f vk  kvk ; v 2 E s ;

kD f 1 vk  kvk ; v 2 E u :

For each x 2 M there is a pair of embedded C1 discs s u (x), Wloc (x), called the local stable manifold and the Wloc local unstable manifold at x, respectively, such that: s (x) D E s (x) ; T W u (x) D E u (x); 1. Tx Wloc x loc s s ( f x) ; f 1 (W u (x)) W u ( f 1 x); 2. f (Wloc (x)) Wloc loc loc 3. For any  2 (; 1), there exists a constant C > 0 such that for all n 2 N,

d M ( f n x; f n y)  Cn d M (x; y) ;

for

s y 2 Wloc (x) ;

d M ( f n x; f n y)  Cn d M (x; y) ;

for

u y 2 Wloc (x) :

The local stable (unstable) manifolds can be extended to global stable (unstable) manifolds W s (x) and W u (x) which are well defined and smoothly injectively immersed. These global manifolds are the leaves of global foliations W s and W u of M. In general, these foliations are only continuous, but their leaves are differentiable. Let  : R  M ! M be a C1 flow. The flow  is said to be an Anosov flow if there is a Riemannian metric k  k on M, a constant 0 <  < 1, and a continuous Df invariant splitting TM D E s ˚ E 0 ˚ E u such that for all x 2 M and t > 0: 1.

d t dt j tD0 

2 E xc n f0g ; dim E xc D 1,

2. kD t vk   t kvk ; v 2 E s , 3. kD t vk   t kvk ; v 2 E u . For each x 2 M there is a pair of embedded C1 discs s (x) ; W u (x), called the local (strong) stable manifold Wloc loc and the local (strong) unstable manifold at x, respectively, such that: s (x) D E s (x) ; T W u (x) D E u (x); 1. Tx Wloc x loc s s (x)) Wloc ( t x) ; 2.  t (Wloc u (x)) W u ( t x) for t > 0;  t (Wloc loc 3. For any  2 (; 1), there exists a constant C > 0 such that for all n 2 N,

d M ( t x ;  t y)  C t d M (x; y) ; for d M (

t

x ;

t

s y 2 Wloc (x) ; t > 0 ;

t

y)  C d M (x; y) ; for

u y 2 Wloc (x) ; t > 0 :

The local stable (unstable) manifolds can be extended to global stable (unstable) manifolds W s (x) and W u (x). These global manifolds are the leaves of global foliations W s and W u of M. One can also define weak stable and weak unstable foliations with leaves given by W cs (x) D [ t2R (W s (x)) and W cu (x) D [ t2R (W u (x)), which have as tangent distributions E cs D E c ˚E s and E cu D E c ˚E s . In general, all these foliations are only continuous, but their leaves are differentiable. Any Anosov diffeomorphism is structurally stable, that is, any C1 diffeomorphism that is C1 close to an Anosov diffeomorphism is topologically conjugate to the unperturbed one via a Hölder homeomorphism. An Anosov flow is structurally stable in the orbit equivalence sense: any C1 small perturbation of an Anosov flow has the orbit foliation topologically conjugate via a Hölder homeomorphism to the orbit foliation of the unperturbed flow. Let SL(n; R) be the group of all n-dimensional square matrices with real valued entries of determinant 1. Let SL(n; Z) SL(n; R) be the subgroup with integer entries. Basic examples of Anosov diffeomorphisms are automorphisms of the n-torus T n D Rn /Z n induced by hyperbolic matrices in SL(n; Z). A hyperbolic matrix is one that has only nonzero eigenvalues, all away in absolute value from 1. A specific example of such matrix in SL(2; Z) is   2 1 : 1 1 Basic examples of Anosov flows are given by the geodesic flows of surfaces of constant negative curvature. The unitary bundle of such a surface can be realized as M D  n PSL(2; R), where PSL(2; R) D SL(2; R)/f˙1g and  is a cocompact lattice in PSL(2; R). The action of

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the geodesic flow on M is induced by right multiplication with elements in the diagonal one-parameter subgroup  

 t/2 0 e ; t 2 R : 0 et/2 A related transformation, which is not hyperbolic, but will be of interest in this presentation, is the horocycle flow induced by right multiplication on M by elements in the one parameter subgroup

   1 t ;t 2 R : 0 1 Of interest in this survey are also actions of more general groups than Z and R. Typical examples of higher rank Z k Anosov actions are constructed on tori using groups of units in number fields. See [65] for more details about this construction. A particular example of Anosov Z2 -action on T 3 is induced by the hyperbolic matrices: 0 1 0 1 0 1 0 2 1 0 A D @0 0 1A ; B D @0 2 1A : 1 8 2 1 8 4 One can check, by looking at the eigenvalues, that A and B are not multiples of the same matrix. Moreover, A and B commute. Typical examples of higher rank Anosov R k -actions are given by Weyl chamber flows, which we now describe using some notions from the theory of Lie groups. A good reference for the background in Lie group theory necessary here is the book of Helgason [40]. Note that for a hyperbolic element of such an action the center distribution is k-dimensional and coincides with the tangent distribution to the orbit foliation of R k . Let G be a semi-simple connected real Lie group of the noncompact type, with Lie algebra g. Let K G be a maximal compact subgroup that gives a Cartan decomposition g D k C p, where k is the Lie algebra of K and p is the orthogonal complement of k with respect to the Killing form of g. Let a p be a maximal abelian subalgebra and A D exp a be the corresponding subgroup. The simultaneous diagonalization of adg (a) gives the decomposition X gDgC g ; g0 D a C m ;

cones in a called Weyl chambers. The faces of the Weyl chambers are called Weyl chamber walls. Let M be the centralizer of A in K. Suppose  is an irreducible torsion-free cocompact lattice in G. Since A commutes with M, the action of A by right translations on  n G descends to an A-action on N :D  n G/M. This action is called a Weyl chamber flow. Any Weyl chamber flow is an Anosov action, that is, has an element that acts hyperbolically transversally to the orbit foliation of A. Note that all maximal connected R diagonalizable subgroups of G are conjugate and their common dimension is called the R-rank of G. If the R-rank k of G is higher than 2, then the Weyl chamber flow is a higher rank hyperbolic R k -action. An example of semi-simple Lie group is SL(n; R). Let A be the diagonal subgroup of matrices with positive entries in SL(n; R). An example of Weyl chamber flow that will be discussed in the sequel is the action of A by right translations on  nSL(n; R), where  is a cocompact lattice. In this case the centralizer M is trivial. The rank of this action is n  1. The picture of the Weyl chambers for n D 3 is shown in Fig. 1. The signs that appear in each chamber are the signs of half of the Lyapunov exponents of a regular element from the chamber with respect to a certain fixed basis. For this action, the Lyapunov exponents appear in pairs of opposite signs. An example of higher rank lattice Anosov action that will be discussed in the sequel is the standard action of SL(n; Z) on the torus T n , (A; x) 7! Ax; A 2 SL(n; Z); x 2 T n : SL(n; Z) is a (noncocompact!) lattice in SL(n; R). As shown in [55], this action is generated by Anosov diffeomorphisms. We describe now a class of dynamical systems more general than the hyperbolic one. A C1 diffeomorphism f of a compact differentiable manifold M is called partially

2

where  is the set of restricted roots. The spaces g are called root spaces. A point H 2 a is called regular if (H) ¤ 0 for all  2 . Otherwise it is called singular. The set of regular elements consists of the complement of a union of finitely many hyperplanes. Its components are

Ergodic Theory: Rigidity, Figure 1 Weyl chambers for SL(3; R)

Ergodic Theory: Rigidity

hyperbolic if there exists a continuous invariant splitting of the tangent bundle TM D E s ˚ E 0 ˚ E u such the the derivative of f expands Eu much more than E0 , and contracts Es much more than E0 . See [9,41] and [10] for the theory of partially hyperbolic diffeomorphisms. Es and Eu are called stable, respectively unstable distributions, and are integrable. E0 is called center distribution and, in general, it is not integrable. A structural stability result proved by Hirsch, Pugh, Shub [41], that is a frequently used tool in rigidity, shows that, if E0 is integrable to a smooth foliation, then any perturbation f¯ of f is partially hyperbolic and has an integrable center foliation. Moreover, the center foliations of f¯ of f are mapped one into the other by a homeomorphism that conjugates the maps induced on the factor spaces of the center foliations by f¯ of f respectively. We review now basic facts about cocycles. These basic definitions refer to several regularity classes: measurable, continuous, or differentiable. Let G be a group acting on a set M, and denote the action G  M ! M by (g; x) 7! gx; thus (g1 g2 )x D g1 (g2 x). Let  be a group with unit 1 . M is usually endowed with a measurable, continuous, or differentiable structure. A cocycle ˇ over the action is a function ˇ : G  M !  such that ˇ(g1 g2 ; x) D ˇ(g1 ; g2 x)ˇ(g2 ; x) ;

(1)

for all g1 ; g2 2 G; x 2 M. Note that any group representation  : G !  defines a cocycle called constant cocycle. The trivial representation defines the trivial cocycle. A natural equivalence relation for the set of cocycles is given by cohomology. Two cocycles ˇ1 ; ˇ2 : G  M !  are cohomologous if there exists a map P : M !  , called transfer map, such that ˇ1 (g; x) D P(gx)ˇ2 (g; x)P(x)1 ;

(2)

for all g 2 G; x 2 M. Differentiable Rigidity We start by reviewing cohomological results. Several basic notions are already defined in Sect. “Basic Definitions and Examples”. In this section we assume that the cocycles are at least continuous. A cocycle ˇ : G  M !  over an action (g; x) 7! gx; g 2 G; x 2 M, is said to satisfy closing conditions if for any g 2 G and x 2 M such that gx D x, one has ˇ(g; x) D 1 . Note that closing conditions are necessary in order for a cocycle to be cohomologous to the trivial one. Since a Z-cocycle is determined by a function ˇ : M ! ; ˇ(x) :D ˇ(1; x), the closing conditions become f nx D x

implies ˇ( f n1 x) : : : ˇ(x) D 1 ;

(3)

where f : M ! M is the function that implements the Zaction. The first cohomological results for hyperbolic systems were obtained by Livshits [70,71]. Let M be a compact Riemannian manifold with a Z-action implemented by a topologically transitive Anosov C1 diffeomorphism f . Then an ˛-Hölder function ˇ : M ! R determines a cocycle cohomologous to a trivial cocycle if and only if ˇ satisfies the closing conditions (3). The transfer map is ˛-Hölder. Moreover, for each Hölder class ˛ and each finite dimensional Lie group  , there is a neighborhood U of the identity in  such that an ˛-Hölder function ˇ : M !  determines a cocycle cohomologous to the trivial cocycle if and only if ˇ satisfies the closing conditions (3). The transfer map is again ˛-Hölder. Similar results are true for Anosov flows. Using Fourier analysis, Veech [108] extended Livshits’s result to real valued cocycles over Z-actions induced by ergodic endomorphisms of an n-dimensional torus, not necessarily hyperbolic. For cocycles with values in abelian groups, the question of two arbitrary cocycles being cohomologous reduces to the question of an arbitrary cocycle being cohomologous to a trivial one. This is not the case for cocycles with values in nonabelian groups. Parry [98] extended Livshits’s criteria to one for cohomology of two arbitrary cocycles with values in compact Lie groups. Parry’s result was generalized by Schmidt [104] to cocycles with values in Lie groups that, in addition, satisfy a center bunching condition. Ni¸tic˘a, Török [92] extended Livshits’s result to cocycles with values in the group Diff k (M) of Ck diffeomorphism of a compact manifold M with stably trivial bundle, k  3. Examples of such manifolds are the tori and the spheres. In this case, the transfer map takes values in Diff k3 (M), and it is Hölder with respect to a natural metric on Diff k3 (M). In [92] one can also find a generalization of Livshits’s result to generic Anosov actions, that is, actions generated by families of Anosov diffeomorphisms that do not interchange the stable and unstable directions of elements in the family. An example of such an action is the standard action of SL(n; Z) on the n-dimensional torus. A question of interest is the following: if two Ck cocycles, 1  k  !, over a hyperbolic action, are cohomologous through a continuous/measurable transfer map P, what can be said about the higher regularity of P? For real valued cocycles the question can be reduced to one about cohomologically trivial cocycles. Livshits showed that for a real valued C1 cocycle cohomologous to a constant the transfer map is C1 . He also obtained C 1 regularity results if the action is given by hyperbolic automorphisms of a torus. After preliminary results by Guillemin and Kazh-

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dan for geodesic flows on surfaces of negative curvature, for general hyperbolic systems the question was answered positively by de la Llave, Marco, Moriyon [76] in the C 1 case and by de la Llave [74] in the real analytic case. Ni¸tic˘a, Török [93] considered the lift of regularity for a transfer map between two cohomologous cocycles with values in a Lie group or a diffeomorphism group. In contrast to the case of cocycles cohomologous to trivial ones, here one needs to require for the transfer map a certain amount of Hölder regularity that depends on the ratio between the expansion/contraction that appears in the base and the expansion/contraction introduced by the cocycle in the fiber. This assumption is essential, as follows from a counterexample found by de la Llave’s [73]. Useful tools in this development have been results from analysis that lift the regularity of a continuous real valued function which is assumed to have higher regularity along pairs of transverse Hölder foliations. Many times the foliations are the stable and unstable ones associated to a hyperbolic system. Journé [46] proved the C n;˛ regularity of a continuous function that is C n;˛ along two transverse continuous foliations with C n;˛ leaves. If one is interested only in C 1 regularity, a convenient alternative is a result of Hurder, Katok [44]. This has a simpler proof and can be applied to the more general situation in which the function is regular along a web of transverse foliations. A real analytic regularity result along these lines belongs to de la Llave [74]. In certain problems, for example when working with Weyl chamber flows, it is difficult to control the regularity in enough directions to span the whole tangent space. Nevertheless, the tangent space can be generated if one consider higher brackets of good directions. A C 1 regularity result for this case belongs to Katok, Spatzier [60]. In order to apply this result, the foliations need to be C 1 not only along the leaves, but also transversally. An application of the above regularity results is to questions about transverse regularity of the stable and unstable foliations of the geodesic flow on certain C 1 surfaces of nonpositive curvature. For compact negatively curved C 1 surfaces, E. Hopf showed that these foliations are C1 , and it follows from the work of Anosov that individual leaves are C 1 . Hurder, Katok [44] showed that once the weak-stable and weak-unstable foliations of a volume-preserving Anosov flow on a compact 3-manifold are C2 , they are C 1 . Another application of the regularity results is to the study of invariants for Ck conjugacy of hyperbolic systems. By structural stability, a small C1 perturbation of a hyperbolic system is C0 conjugate to the unperturbed one. The conjugacy, in general, is only Hölder. If the conjugacy is

C1 then it preserves the eigenvalues of the derivative at the periodic orbit. The following two results describe the invariants of smooth and real analytic conjugacy of low dimensional hyperbolic systems. They are proved in a series of papers written in various combinations by de la Llave, Marco, Moryion [72,74,75,79,80]. Let X; Y be two C 1 (C ! ) transitive Anosov vector fields on a compact three-dimensional manifold. If they are C0 conjugate and the eigenvalues of the derivative at the corresponding periodic orbits are the same, then the conjugating homeomorphism is C 1 (C! ). In particular, any C1 conjugacy is C 1 (C! ). Assume now that f ; g are two C 1 (C! ) Anosov diffeomorphisms on a compact two dimensional manifold. If they are C0 conjugate and the eigenvalues of the derivative at the corresponding periodic orbits are the same, then the conjugating diffeomorphism is C 1 (C! ). In particular, any C1 conjugacy is C 1 (C! ). An important direction was initiated by Katok, Spatzier [59] who studied cohomological results over hyperbolic Z k or R k -actions, k  2. They show that real valued smooth/Hölder cocycles over typical classes of hyperbolic Z k or R k , k  2, actions are smoothly/Hölder cohomologous to constants. These results cover, in particular, actions by hyperbolic automorphisms of a torus, and Weyl chamber flows. The proofs rely on harmonic analysis techniques, such as Fourier transform and group representations for semi-simple Lie groups. A geometric method for cocycle rigidity was developed in [64]. One constructs a differentiable form using invariant structures along stable/unstable foliations, and the commutativity of the action. The form is exact if and only if the cocycle is cohomologous to a constant one. The method covers actions on nilmanifolds satisfying a condition called TNS (Totally Non-Symplectic). This condition means that the action is higher rank abelian hyperbolic, and that the tangent space is a direct sum of invariant distributions, with each pair of these included in the stable distribution of a hyperbolic element of the action. The method was also applied to small (i. e. close to identity on a set of generators) Lie group valued cocycles. A related paper is [96] which contains rigidity results for cocycles over (TNS) actions with values in compact Lie groups. In this situation the number of cohomology classes is finite. An example of (TNS) action is given by the action of a maximal diagonalizable subgroup of SL(n; Z) on T n . Recently Damjanovi´c, Katok [14] developed a new method that was applied to the action of the matrix diagonal group on  n SL(n; R). They use techniques from [54], where one finds cohomology invariants for cocycles over partially hyperbolic actions that satisfy accessibility prop-

Ergodic Theory: Rigidity

erty. Accessibility means that one can connect any two points from the manifold supporting the partially hyperbolic dynamical system by transverse piecewise smooth paths included in stable/unstable leaves. This notion was introduced by Brin, Pesin [9] and it is playing a crucial role in the recent surge of activity in the field of partially hyperbolic diffeomorphisms. See [10] for a recent survey of the subject. The cohomology invariants described in [54] are heights of the cocycle over cycles constructed in the base out of pieces inside stable/unstable leaves. They provide a complete set of obstructions for solving the cohomology equation. A new tool introduced in [14] is algebraic K-theory [88]. The method can be extended to cocycles with non-abelian range. In [57] one finds related results for small cocycles with values in a Lie group or the diffeomorphism group of a compact manifold. The equivalent of the Livshits theorem in the higherrank setting appears to be a description of the highest cohomology rather than the first cohomology. Indeed, for higher rank partially hyperbolic actions of the torus, the intermediate cohomologies are trivial, while for the highest one the closing conditions characterize the cohomology classes. This behavior provides a generalization of Veech cohomological result and of Katok, Spatzier cohomological result for toral automorphisms, and was discovered by A. Katok, S. Katok [52,53]. Flaminio, Forni [27] studied the cohomological equation over the horocycle flow. It is shown that there are infinitely many obstructions to the existence of a smooth solution. Moreover, if these obstructions vanish, then one can solve the cohomological equation. In [28] it is shown a similar result for cocycles over area preserving flows on compact higher-genus surfaces under certain assumptions that hold generically. Mieczkowski [87] extended these techniques and studied the cohomology of parabolic higher rank abelian actions. All these results rely on noncommutative Fourier analysis, more specifically representation theory of SL(2; R) and SL(2; C). Local Rigidity Let  be a finitely (or compactly) generated group, G a topological group, and  :  ! G a homomorphism. The target of local rigidity theory is to understand the space of perturbations of various homomorphisms . Trivial perturbations of a homomorphism arise from conjugation by an arbitrary element of G. In order to rule them out, one says that  is locally rigid if any nearby homomorphism  0 , (that is,  0 close to  on a finite or compact set of generators of  ), is conjugate to  by an element g 2 G, that is, ( ) D g 0 ( )g 1 for all 2  . If G

is path-wise connected, one can also consider deformation rigidity, meaning that any nearby continuous path of homomorphisms is conjugated to the initial one via a continuous path of elements in G that has an end in the identity. Initial results on local rigidity are about embeddings of lattices into semi-simple Lie groups. The main results belong to Weil [110,111,112]. He showed that if G is a semisimple Lie group that is not locally isomorphic to SL(2; R) and if  G is an irreducible cocompact lattice, then the natural embedding of  into G is locally rigid. Earlier results were obtained by Selberg [105], Calabi, Vesentini [12], and Calabi [11]. Selberg proved the local rigidity of the natural embedding of cocompact lattices into SL(n; R). His proof used the dynamics of iterates of matrices, in particular the existence of singular directions, or walls of Weyl chambers, in the maximal diagonalizable subgroups of SL(n; R). Selberg’s approach inspired Mostow [90] to use the boundaries at infinity in his proof of strong rigidity of lattices, which in turn was crucial to the development of superrigidity due to Margulis [82]. See Sect. “Global Rigidity” for more details. Recall that hyperbolic dynamical systems are structurally stable. Thus they are, in a certain sense, locally rigid. We introduce now a precise definition of local rigidity in the infinite-dimensional setup. The fact that for general group actions one needs to consider different regularities for the actions, perturbations and conjugacies is apparent from the description of structural stability for Anosov systems. A Ck action ˛ of a finitely generated discrete group  on a manifold M, that is, a homomorphism ˛ :  ! Diff k (M), is said to be C k;l ;p;r locally rigid if any Cl perturbation ˛˜ which is Cp close to ˛ on a family of generators, is Cr conjugate to ˛, i.e there exists a Cr diffeomorphism H : M ! M which conjugates ˛˜ to ˛, that is, ˜ H ı ˛(g) D ˛(g) ı H for all g 2  . Note that for Anosov Z-actions, C 1;1;1;0 rigidity is known as structural stability. One can also introduce the notion of deformation rigidity if the initial action and the perturbation are conjugate by a continuous path of diffeomorphisms that has an end coinciding to the identity. A weaker notion of local rigidity can be defined in the presence of invariant foliations for the initial group action and for the perturbation. The map H is now required to preserve the leaves of the foliations and to conjugate only after factorization by the invariant foliations. The importance of this notion is apparent from the leafwise conjugacy structural stability theorem of Hirsch, Pugh, Shub [41]. See Sect. “Basic Definitions and Examples”. Moreover, for Anosov flows this is the natural notion of structural stability, and appears by taking the invariant fo-

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liation to be the one-dimensional orbit foliation. For more general actions, of lattices or higher rank abelian groups, this property is often used in combination to cocycle rigidity in order to show local rigidity. We discuss more about this when we review local rigidity results for partially hyperbolic actions. We summarize now several developments in local rigidity that emerged in the 80s. Initial results [67,115] were about infinitesimal rigidity, that is, a weaker version of local rigidity suitable for discrete groups representations in infinite-dimensional spaces of smooth vector fields. Then Hurder [43] proved C 1;1;1;1 deformation rigidity and Katok, Lewis, Zimmer [55,56,63] proved C 1;1;1;1 local rigidity of the standard action of SL(n; Z); n  3, on the n-dimensional torus. In these results crucial use was made of the presence of an Anosov element in the action. Due to the uniqueness of the conjugacy coming from structural stability, one has a continuous candidate for the conjugacy between the actions. Margulis, Qian [85] used the existence of a spanning family of directions that are hyperbolic for certain elements of the action to show local rigidity of partially hyperbolic actions that are not hyperbolic. Another important tool present in many proofs is Margulis and Zimmer superrigidity. These results allow one to produce a measurable conjugacy for the perturbation. Then one shows that the conjugacy has higher regularity using the presence of hyperbolicity. Having enough directions to span the whole tangent space is essential to lift the regularity. A cocycle to which superrigidity can be applied is the derivative cocycle. The study of local rigidity of partially hyperbolic actions that contain a compact factor was initiated by Ni¸tic˘a, Török [92,94]. Let n  3 and d  1. Let be the action of SL(n; Z) on T nCd D T n  T d given by (A)(x; y) D (Ax; y) ; x 2 T n ; y 2 T d ; A 2 SL(n; Z). Then, for K  1, [92] shows that is C 1;1;5;K1 deformation rigid. The proof is based on three results in hyperbolic dynamics: the generalization of Livshits’s cohomological results to cocycles with values in diffeomorphism groups, the extension of Livshits’s result to general Anosov actions, and a version of the Hirsch, Pugh, Shub structural stability theorem improving the regularity of the conjugacy. Assume now n  3 and K  1. If is the action of SL(n; Z) on T nC1 D T n  T given by (A)(x; y) D (Ax; y) ; x 2 T n ; y 2 T ; A 2 SL(n; Z), [94] shows that the action is C 1;1;2;K1 locally rigid. Ingredients in the proof are two rigidity results, one about TNS actions, and one about actions of property (T) groups. A locally compact group has property (T) if the trivial representation is isolated in the Fell topology. This means that if G acts on a Hilbert space unitarily and it has almost invari-

ant vectors, then it has invariant vectors. Hirsch–Pugh– Shub theorem implies that perturbations of abelian partially hyperbolic actions of product type are conjugated to skew-products of abelian Anosov actions via cocycles with values in diffeomorphism groups. In addition, the TNS property implies that the sum of the stable and unstable distributions of any regular element of the perturbation is integrable. The leaves of the integral foliation are closed, covering the base simply. Thus one obtains a conjugacy between the perturbation and a product action. Property (T) is used to show that the conjugacy reduces the perturbed action to a family of perturbations of hyperbolic actions. But the last ones are already known to be conjugate to the hyperbolic action in the base. Recent important progress in the question of local rigidity of lattice actions was made by Fisher, Margulis [24, 25,26]. Their proofs are modeled along the proof of Weil’s local rigidity result [112] and use an analog of Hamilton’s [39] hard implicit function theorem. Let G be a connected semi-simple Lie group with all simple factors of rank at least two, and  G a lattice. The main result shows that a volume preserving affine action of G or  on a compact smooth manifold X is C 1;1;1;1 locally rigid. Lower regularity results are also available. A component of the proof shows that if  is a group with property (T), X a compact smooth manifold, and a smooth action of  on X by Riemannian isometries, then is C 1;1;1;1 locally rigid. An earlier local rigidity result for this type of actions by cocompact lattices was obtained by Benveniste [5]. Many lattices act naturally on “boundaries” of type G/P, where G is a semi-simple algebraic Lie group and P is a parabolic subgroup. An example is given by G D SL(2; R) and P the subgroup in G consisting of upper triangular matrices. Local rigidity results for this type of actions were found by Ghys [34], Kanai [50] and Katok, Spatzier [62]. Starting with the work of Katok and Lewis, a related direction was the study of local rigidity for higher rank abelian actions. They prove in [55] the C 1;1;1;1 local rigidity of the action of a Zn maximal diagonalizable (over R) subgroup of SL(n C 1; Z); n  2, acting on the torus T nC1 . These type of results were later pushed forward by Katok, Spatzier [62]. Using the theory of nonstationary normal forms developed in [38] and [37] by Katok, Guysinsky, they proved several local rigidity results. The first one assumes that G is a semisimple Lie group with all simple factors of rank at least two,  a lattice in G, N a nilpotent Lie group and  a lattice in N. Then any Anosov affine action of  on N/ is C 1;1;1;1 locally rigid. Second, let Zd be a group of affine transformations

Ergodic Theory: Rigidity

of N/ for which the derivatives are simultaneously diagonalizable over R with no eigenvalues on the unit circle. Then the Zd -action on N/ is C 1;1;1;1 locally rigid. A related result for continuous groups is the C 1;1;1;1 local rigidity (after factorization by the orbit foliation) of the action of a maximal abelian R-split subgroup in an R-split semi-simple Lie group of real rank at least two on G/, where  is a cocompact lattice in G. One can also study rigidity of higher rank abelian partially hyperbolic actions that are not hyperbolic. Natural examples appear as automorphisms of tori and as variants of Weyl chamber flows. For the case of ergodic actions by automorphisms of a torus, this was investigated using a version of the KAM (Kolmogorov, Arnold, Moser) method by Damianovi´c, Katok [15]. As usual in the KAM method, one starts with a linearization of the conjugacy equation. At each step of the iterative KAM scheme, some twisted cohomological equations are solved. The existence of the solutions is forced by the ergodicity of the action and the higher rank assumptions. Diophantine conditions present in this case allow to control the fixed loss of regularity which is necessary for the convergence of these solutions to a conjugacy. Global Rigidity The first remarkable result in global rigidity belongs to Mostow [90]. For G a connected non-compact semi-simple Lie group not locally isomorphic to SL(2; R), and two irreducible cocompact lattices 1 ; 2 G, Mostow showed that any isomorphism from  1 into  2 extends to an isomorphism of G into itself. G has an involution  whose fixed set is a maximal compact subgroup K. One constructs the symmetric Riemannian space X D G/K. To each chamber of X corresponds a parabolic group and these parabolic groups are endowed with a Tits geometry similar to the projective geometry of lines, planes etc. formed in the classical case when G D PGL(n; R). The proof of Mostow’s result starts by building a -equivariant pseudo-isometric map  : G/K1 ! G/K2 . The map  induces an incidence preserving -equivariant isomorphism  0 of the Tits geometries. By Tits’ generalized fundamental theorem of projective geometry,  0 is induced by an isomorphism of G. Finally, ( ) D 0   01 gives the desired conclusion. The next remarkable result in global rigidity is Margulis’ superrigidity theorem. An account of this development can be found in the monograph [82]. For large classes of irreducible lattices in semi-simple Lie groups, this result classifies all finite dimensional representations. Let G be a semi-simple simply connected Lie group of rank

higher than two and  < G an irreducible lattice. Then any linear representation  of  is almost the restriction of a linear representation of G. That is, there exists a linear representation  1 of G and a bounded image representation  2 of  such that  D 1 2 . The possible representations  2 are also classified by Margulis up to some facts concerning finite image representations. As in the case of Mostow’s result, the proof involved the analysis of a map defined on the boundary at infinity. In this case the map is studied using deep results from dynamics like the multiplicativity ergodic theorem of Oseledec [97] or the theory of random walks on groups developed by Furstenberg [31]. An important consequence of Margulis superrigidity result is the arithmeticity of irreducible lattices in connected semi-simple Lie groups of rank higher than two. A basic example of arithmetic lattice can be obtained by taking the integer points in a semi-simple Lie group that is a matrix group, like taking SL(n; Z) inside SL(n; R). Special cases of superrigidity theorems were proved by Corlette [13] and Gromov, Schoen [36] for the rank one groups Sp(1; n) and respectively F 4 using the theory of harmonic maps. A consequence is the arithmeticity of lattices in these groups. Some of these results are put into differential geometric setting in [89]. Margulis supperrigidity result was extended to cocycles by Zimmer. A detailed exposition, including a self contained presentation of several rigidity results of Margulis, can be found in the monograph [113]. We mention here a version of this result that can be found in [24]. Let M be a compact manifold, H a matrix group, P D M  H, and  a lattice in a simply connected, semi-simple Lie group with all factors of rank higher that two. Assume that  acts on M and H in a way that makes the projection from P to M equivariant. Moreover, the action of  on P is measure preserving and ergodic. Then there exists a measurable map s : M ! H, a representation  : G ! H, a compact subgroup K < H which commute with (G) and a measurable map k :   M ! K such that  s(m) D k( ; m)  ( )  s(  m). One can easily check from the last equation that k is a cocycle. So, up to a measurable change of coordinates given by the map s, the action of  on P is a compact extension via a cocycle of a linear representation of G. Developing further the method of Mostow for studying the Tits building associated to a symmetric space of non-positive curvature led Ballman, Brin, Eberlein, Spatzier [2,3] to a number of characterizations of symmetric spaces. In particular, they showed that if M is a complete Riemannian manifold of non-positive curvature, finite volume, with simply connected cover, irreducible and of rank at least two, then M is isometric to a symmetric

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space with the connected component of Isom(M) having no compact factors. A topological rigidity theorem has been proved by Farrell, Jones [21]. They showed that if N is a complete connected Riemannian manifold whose sectional curvature lies in a closed interval included in (1; 0], and M is a topological manifold of dimension greater than 5, then any proper homotopy equivalence f : M ! N is properly homotopic to a homeomorphism. In particular, if M and N are both compact connected negatively curved Riemannian manifolds with isomorphic fundamental groups, then M and N are homeomorphic. Likewise to the case of local rigidity, a source of inspiration for results in global rigidity was the theory of hyperbolic systems, in particular their classification. The only known examples of Anosov diffeomorphisms are hyperbolic automorphisms of infranilmanifolds. Moreover, any Anosov diffeomorphism on an infranilmanifold is topologically conjugate to a hyperbolic automorphism [29,78]. It is conjectured that any Anosov diffeomorphism is topologically conjugate to a hyperbolic automorphism of an infranilmanifold. Partial results are obtained in [91], where the conjecture is proved for Anosov diffeomorphisms with codimension one stable/unstable foliation. The proof of the general conjecture eluded efforts done so far. It is not even known if any Anosov diffeomorphism is topologically transitive, that is, if it has a dense orbit. A few positive results are available. Let M be a C 1 compact manifold endowed with a C 1 affine connection. Let f be a topologically transitive Anosov diffeomorphism preserving the connection and such that the stable and unstable distributions are C 1 . Then Benoist, Labourie [4] proved that f is C 1 conjugate to a hyperbolic automorphism of an infranilmanifold. The situation for Anosov flows is somehow different. As shown in [30], there exist Anosov flows that are not topologically transitive, so a general analog of the conjecture is false. Nevertheless, for the case of codimension one stable or unstable foliation, it is conjectured in [109] that any Anosov flow on a manifold of dimension greater than three admits a global cross-section. This would imply that the flow is topologically conjugate to the suspension of a linear automorphism of a torus. For actions of groups larger than Z, or R, global classification results are more abundant. A useful strategy in these type of results, which are quite technical, is to start by obtaining a measurable description of the action, most of the time using Margulis–Zimmer superrigidity results, and then use extra assumptions on the action, such as the presence of a hyperbolic element, or the presence of an invariant geometric structure, or both, in order to show that

the measurable model is actually continuous or even differentiable. For actions of higher rank Lie groups and their lattices some representative papers are by Katok, Lewis, Zimmer [63] and Goetze, Spatzier [35]. For actions of higher rank abelian groups see Kalinin, Spatzier [49]. Measure Rigidity Measure rigidity is the study of invariants measures for actions of one parameter and multiparameter abelian groups and semigroups acting on manifolds. Typical situations when interesting rigidity phenomena appear are for one parameter unipotent actions and higher rank hyperbolic actions, discrete or continuous. A unipotent matrix is one all of whose eigenvalues are one. An important case where the action of a unipotent flow appears is that of the horocycle flow. The invariant measures for it were studied by Furstenberg [33], who showed that the horocycle flow on a compact surface is uniquely ergodic, that is, the ergodic measure is unique. Dani and Smillie [17] extended this result to the case of non-compact surfaces, with the only other ergodic measures appearing being those supported on compact horocycles. An important breakthrough is the work of Margulis [81], who solved a long standing question in number theory, Oppenheim’s conjecture. The conjecture is about the density properties of the values of an indefinite quadratic form in three or more variables, provided the form is not proportional to a rational form. The proof of the conjecture is based on the study of the orbits for unipotent flows acting by translations on the homogenous space SL(n; Z) n SL(n; R). All these results were special cases of the Raghunathan conjecture about the structure of the orbits of the actions of unipotent flows on homogenous spaces. Raghunathan’s conjecture was proved in full generality by Ratner [100,101]. Borel, Prasad [8] raised the question of an analog of Raghunathan’s conjecture for S-algebraic groups. S-algebraic groups are products of real and p-adic algebraic groups. This was answered independently by Margulis, Tomanov [86] and Ratner [102]. A basic example of higher rank abelian hyperbolic action is given by the action of S m;n , the multiplicative semigroup of endomorphisms generated by the multiplication by m and n, two nontrivial integers, on the one dimensional torus T 1 . In a pioneering paper [32] Furstenberg showed that for m; n that are not powers of the same integer the action of S m;n has a unique closed, infinite invariant set, namely T 1 itself. Since there are many closed, infinite invariant sets for multiplication by m, and by n, this result shows a remarkable rigidity property of the joint action. Furstenberg’s result was generalized later by Berend

Ergodic Theory: Rigidity

for other group actions on higher dimensional tori and on other compact abelian groups in [6] and [7]. Furstenberg further opened the field by raising the following question: Conjecture 1 Let  be a S m;n -invariant and ergodic probability measure on T 1 . Then  is either an atomic measure supported on a finite union of (rational) periodic orbits or  is the Lebesgue measure. While the statement appears to be simple, proving it has been elusive. The first partial result was given by Lyons [77] under the strong additional assumption that the measure makes one of the endomorphisms generating the action exact. Later Rudolph [103] and Johnson [45] weaken the exactness assumption and proved that  must be the Lebesgue measure provided that multiplication by m (or multiplication by n) has positive entropy with respect to . Their results have been proved again using slightly different methods by Feldman [22]. A further extension was given by Host [42]. Katok proposed another example for which measure rigidity can be fruitfully tested, the Z2 -action induced by two commuting hyperbolic automorphisms on the torus T 3 . An example of such action is shown in Sect. “Basic Definitions and Examples”. One can also consider the action induced by a Z n1 maximal abelian group of hyperbolic automorphisms acting on the torus T n . Katok, Spatzier [61] developed a more geometric technique allowing to prove measure rigidity for these actions if they have an element acting with positive entropy. Their technique can be applied in higher generality if the action is irreducible in a strong sense and, in addition, it has individual ergodic elements or it is TNS. See also [47]. This method is based on the study of conditional measures induced by the invariant measure on certain invariant foliations that appear naturally in the presence of a hyperbolic action. Besides stable and unstable foliations, one can also consider various intersections of them. Einsiedler, Lindenstrauss [19] were able to eliminate the ergodicity and TNS assumptions. Yet another interesting example of higher rank abelian action is given by Weyl chamber flows. These do not satisfy the TNS condition. Einsiedler, Katok [18] proved that if G is SL(n; R);  G is a lattice, H is the subgroup of positive diagonal matrices in G, and  a H-invariant and ergodic measure on G/ such that the entropy of  with respect to all one parameter subgroups of H is positive, then  is the G invariant measure on G/ . These results are useful in the investigation of several deep questions in number theory. Define X D SL(3; Z) n SL(3; R) the diagonal subgroup of matrices with posi-

tive entries in SL(3; R). This space is not compact but is endowed with a unique translation invariant probability measure. The diagonal subgroup 80 s 9 1 0 0 < e = H D @ 0 et 0 A : s; t 2 R : ; 0 0 est acts naturally on X by right translations. It was conjectured by Margulis [83] that any compact H-invariant subset of X is a union of compact H-orbits. A positive solution to this conjecture implies a long standing conjecture of Littlewood: Conjecture 2 Let kxk denote the distance between the real number x and the closest integer. Then lim inf nkn˛kknˇk D 0 n!1

(4)

for any real numbers ˛ and ˇ. A partial result was obtained by Einsiedler, Katok, Lindenstrauss [20] who proved that the set of pairs (˛; ˇ) 2 R2 for which (4) is not satisfied has Hausdorff dimension zero. Applications of these techniques to questions in quantum ergodicity were found by Lindenstrauss [69]. A current direction in measure rigidity is attempting to classify the invariant measures under rather weak assumptions about the higher rank abelian action, like the homotopical data for the action. Kalinin and Katok [48] proved that any Z k ; k  2, smooth action ˛ on a k C 1dimensional torus whose elements are homotopic to the corresponding elements of an action by hyperbolic automorphisms preserves an absolutely continuous measure. Future Directions An important open problem in differential rigidity is to find invariants for the Ck conjugacy of the perturbations of higher dimensional hyperbolic systems. For Anosov diffeomorphisms, de la Llave counterexample [73] shows that this extension is not possible for a four dimensional example that appears as a direct product of two dimensional Anosov diffeomorphisms. Indeed, there are C 1 perturbations of the product that are only Ck conjugate to the unperturbed system for any k  1. In the positive direction, Katok conjectured that generalizations are possible for the diffeomorphism induced by an irreducible hyperbolic automorphism of a torus. One can also investigate this question for Anosov flows. Examples of higher rank Lie groups can be obtained by taking products of rank one Lie groups. Many actions of irreducible lattices in these type of groups are believed

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to be locally rigid, but the techniques available so far cannot be applied. A related problem is to study local rigidity in low regularity classes, for example the local rigidity of homomorphisms from higher rank lattices into homeomorphism groups. More problems related to local rigidity can be found in [23]. An important problem in global rigidity, emphasized by Katok and Spatzier, is to show that, up to differentiable conjugacy, any higher rank Anosov or partially hyperbolic action is algebraic under the assumption that it is sufficiently irreducible. The irreducibility assumption is needed in order to exclude actions obtained by successive application of products, extensions, restrictions and time changes from basic ingredients which include some rank one actions. Another problem of current research in measure rigidity is to develop a counterpart of Ratner’s theory for the case of actions by hyperbolic higher rank abelian groups on homogenous spaces. It was conjectured by Katok, Spatzier [61] that the invariant measures for such actions given by toral automorphisms or Weyl chamber flows are essentially algebraic, that is, supported on closed orbits of connected subgroups. Margulis in [84] extended this conjecture to a rather general setup addressing both the topological and measurable aspects of the problem. More details about actions on homogenous spaces, as well as connections to diophantine approximation, can be found in the article  Ergodic Theory on Homogeneous Spaces and Metric Number Theory by Kleinbock. Acknowledgment This research was supported in part by NSF Grant DMS0500832. Bibliography Primary Literature 1. Anosov DV (1967) Geodesic flows on closed Riemannian manifolds with negative curvature. Proc Stek Inst 90:1–235 2. Ballman W, Brin M, Eberlein P (1985) Structure of manifolds of non-negative curvature. I. Ann Math 122:171–203 3. Ballman W, Brin M, Spatzier R (1985) Structure of manifolds of non-negative curvature. II. Ann Math 122:205–235 4. Benoist Y, Labourie F (1993) Flots d’Anosov á distribuitions stable et instable différentiables. Invent Math 111:285–308 5. Benveniste EJ (2000) Rigidity of isometric lattice actions on compact Riemannian manifolds. Geom Func Anal 10:516–542 6. Berend D (1983) Multi-invariant sets on tori. Trans AMS 280:509–532 7. Berend D (1984) Multi-invariant sets on compact abelian groups. Trans AMS 286:505–535 8. Borel A, Prasad G (1992) Values of isotropic quadratic forms at S-integral points. Compositio Math 83:347–372

9. Brin MI, Pesin YA (1974) Partially hyperbolic dynamical systems. Izvestia 38:170–212 10. Burns K, Pugh C, Shub M, Wilkinson A (2001) Recent results about stable ergodicity. In: Katok A, Pesin Y, de la Llave R, Weiss H (eds) Smooth ergodic theory and its applications, Seattle, 1999. Proc Symp Pure Math, vol 69. AMS, Providence, pp 327–366 11. Calabi E (1961) On compact Riemannian manifolds with constant curvature. I. Proc Symp Pure Math, vol 3. AMS, Providence, pp 155–180 12. Calabi E, Vesentini E (1960) On compact, locally symmetric Kähler manifolds. Ann Math 17:472–507 13. Corlette K (1992) Archimedian superrigidity and hyperbolic geometry. Ann Math 135:165–182 14. Damianovi´c D, Katok A (2005) Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic actions. Disc Cont Dynam Syst 13:985–1005 15. Damianovi´c D, Katok A (2007) Local rigidity of partially hyperbolic actions KAM, I.method and Zk -actions on the torus. Preprint available at http://www.math.psu.edu/katok_a 16. Dani SG, Margulis GA (1990) Values of quadratic forms at integer points: an elementary approach. Enseign Math 36:143– 174 17. Dani SG, Smillie J (1984) Uniform distributions of horocycle orbits for Fuchsian groups. Duke Math J 51:185–194 18. Einsiedler M, Katok A (2003) Invariant measures on G/ for split simple Lie groups. Comm Pure Appl Math 56:1184–1221 19. Einsiedler M, Lindenstrauss E (2003) Rigidity properties of Zd actions on tori and solenoids. Elec Res Ann AMS 9:99–110 20. Einsiedler M, Katok A, Lindenstrauss E (2006) Invariant measures and the set of exceptions to Littlewood conjecture. Ann Math 164:513–560 21. Farrell FT, Jones LE (1989) A topological analog of Mostow’s rigidity theorem. J AMS 2:237–370 22. Feldman J (1993) A generalization of a result of R Lyons about measures on [0,1). Isr J Math 81:281–287 23. Fisher D (2006) Local rigidity of group actions: past, present, future. In: Dynamics, Ergodic Theory and Geometry (2007). Cambridge University Press 24. Fisher D, Margulis GA (2003) Local rigidity for cocycles. In: Surv Diff Geom VIII. International Press, Cambridge, pp 191– 234 25. Fisher D, Margulis GA (2004) Local rigidity of affine actions of higher rank Lie groups and their lattices. 2003 26. Fisher D, Margulis GA (2005) Almost isometric actions, property T, and local rigidity. Invent Math 162:19–80 27. Flaminio L, Forni G (2003) Invariant distributions and time averages for horocycle flows. Duke Math J 119:465–526 28. Forni G (1997) Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann Math 146:295–344 29. Franks J (1970) Anosov diffeomorphisms. In: Chern SS, Smale S (eds) Global Analysis (Proc Symp Pure Math, XIV, Berkeley 1968). AMS, Providence, pp 61–93 30. Franks J, Williams R (1980) Anomalous Anosov flows. In: Global theory of dynamical systems, Proc Inter Conf Evanston, 1979. Lecture Notes in Mathematics, vol 819. Springer, Berlin, pp 158–174 31. Furstenberg H (1963) A Poisson formula for semi-simple Lie groups. Ann Math 77:335–386

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32. Furstenberg H (1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math Syst Theor 1:1–49 33. Furstenberg H (1973) The unique ergodicity of the horocycle flow. In: Recent advances in topological dynamics, Proc Conf Yale Univ, New Haven 1972. Lecture Notes in Mathematics, vol 318. Springer, Berlin, pp 95–115 34. Ghys E (1985) Actions localement libres du groupe affine. Invent Math 82:479–526 35. Goetze E, Spatzier R (1999) Smooth classification of Cartan actions of higher rank semi-simple Lie groups and their lattices. Ann Math 150:743–773 36. Gromov M, Schoen R (1992) Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one. Publ Math IHES 76:165–246 37. Guysinsky M (2002) The theory of nonstationary normal forms. Erg Th Dyn Syst 22:845–862 38. Guysinsky M, Katok A (1998) Normal forms and invariant geometric structures for dynamical systems with invariant contracting foliations. Math Res Lett 5:149–163 39. Hamilton R (1982) The inverse function theorem of Nash and Moser. Bull AMS 7:65–222 40. Helgason S (1978) Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, New York 41. Hirsch M, Pugh C, Shub M (1977) Invariant Manifolds. Lecture Notes in Mathematics, vol 583. Springer, Berlin 42. Host B (1995) Nombres normaux, entropie, translations. Isr J Math 91:419–428 43. Hurder S (1992) Rigidity of Anosov actions of higher rank lattices. Ann Math 135:361–410 44. Hurder S, Katok A (1990) Differentiability, rigidity and Godbillon-Vey classes for Anosov flows. Publ Math IHES 72:5–61 45. Johnson AS (1992) Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of integers. Isr J Math 77:211–240 46. Journé JL (1988) A regularity lemma for functions of several variables. Rev Mat Iberoam 4:187–193 47. Kalinin B, Katok A (2002) Measurable rigidity and disjointness for Zk -actions by toral automorphisms. Erg Theor Dyn Syst 22:507–523 48. Kalinin B, Katok A (2007) Measure rigidity beyond uniform hyperbolicity: invariant measures for Cartan actions on tori. J Modern Dyn 1:123–146 49. Kalinin B, Spatzier R (2007) On the classification of Cartan actions. Geom Func Anal 17:468–490 50. Kanai M (1996) A new approach to the rigidity of discrete group actions. Geom Func Anal 6:943–1056 51. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications 54. Cambridge University Press, Cambridge 52. Katok A, Katok S (1995) Higher cohomology for abelian groups of toral automorphisms. Erg Theor Dyn Syst 15:569– 592 53. Katok A, Katok S (2005) Higher cohomology for abelian groups of toral automorphisms. II. The partially hyperbolic case, and corrigendum. Erg Theor Dyn Syst 25:1909–1917 54. Katok A, Kononenko A (1996) Cocycles’ stability for partially hyperbolic systems. Math Res Lett 3:191–210 55. Katok A, Lewis J (1991) Local rigidity for certain groups of toral automorphisms. Isr J Math 75:203–241

56. Katok A, Lewis J (1996) Global rigidity results for lattice actions on tori and new examples of vol preserving actions. Isr J Math 93:253–280 57. Katok A, Ni¸tic˘a V (2007) Rigidity of higher rank abelian cocycles with values in diffeomorphism groups. Geometriae Dedicata 124:109–131 58. Katok A, Ni¸tic˘a V () Differentiable rigidity of abelian group actions. Cambridge University Press (to appear) 59. Katok A, Spatzier R (1994) First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publ Math IHES 79:131–156 60. Katok A, Spatzier R (1994) Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions. Math Res Lett 1:193–202 61. Katok A, Spatzier R (1996) Invariant measures for higherrank abelian actions. Erg Theor Dyn Syst 16:751–778; Katok A, Spatzier R (1998) Corrections to: Invariant measures for higher-rank abelian actions.; (1996) Erg Theor Dyn Syst 16:751–778; Erg Theor Dyn Syst 18:503–507 62. Katok A, Spatzier R (1997) Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions. Trudy Mat Inst Stek 216:292–319 63. Katok A, Lewis J, Zimmer R (1996) Cocycle superrigidity and rigidity for lattice actions on tori. Topology 35:27–38 64. Katok A, Ni¸tic˘a V, Török A (2000) Non-abelian cohomology of abelian Anosov actions. Erg Theor Dyn Syst 2:259–288 65. Katok A, Katok S, Schmidt K (2002) Rigidity of measurable structure for Zd -actions by automorphisms of a torus. Comm Math Helv 77:718–745 66. Kazhdan DA (1967) On the connection of a dual space of a group with the structure of its closed subgroups. Funkc Anal Prilozen 1:71–74 67. Lewis J (1991) Infinitezimal rigidity for the action of SLn (Z) on T n . Trans AMS 324:421–445 68. Lindenstrauss E (2005) Rigidity of multiparameter actions. Isr Math J 149:199–226 69. Lindenstrauss E (2006) Invariant measures and arithmetic quantum unique ergodicity. Ann Math 163:165–219 70. Livshits A (1971) Homology properties of Y-systems. Math Zametki 10:758–763 71. Livshits A (1972) Cohomology of dynamical systems. Izvestia 6:1278–1301 he Livšic cohomology equation. Ann Math 123:537–611 72. de la Llave R (1987) Invariants for smooth conjugacy of hyperbolic dynamical systems. I. Comm Math Phys 109:369–378 73. de la Llave R (1992) Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic dynamical systems. Comm Math Phys 150:289–320 74. de la Llave R (1997) Analytic regularity of solutions of Livshits’s cohomology equation and some applications to analytic conjugacy of hyperbolic dynamical systems. Erg Theor Dyn Syst 17:649–662 75. de la Llave R, Moriyon R (1988) Invariants for smooth conjugacy of hyperbolic dynamical systems. IV. Comm Math Phys 116:185–192 76. de la Llave R, Marco JM, Moriyon R (1986) Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation. Ann Math 123:537–611 77. Lyons R (1988) On measures simultaneously 2- and 3-invariant. Isr J Math 61:219–224

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78. Manning A (1974) There are no new Anosov diffeomorphisms on tori. Amer J Math 96:422–429 79. Marco JM, Moriyon R (1987) Invariants for smooth conjugacy of hyperbolic dynamical systems. I. Comm Math Phys 109:681–689 80. Marco JM, Moriyon R (1987) Invariants for smooth conjugacy of hyperbolic dynamical systems. III. Comm Math Phys 112:317–333 81. Margulis GA (1989) Discrete subgroups and ergodic theory. In: Number theory, trace formulas and discrete groups, Oslo, 1987. Academic Press, Boston, pp 277–298 82. Margulis GA (1991) Discrete subgroups of semi-simple Lie groups. Springer, Berlin 83. Margulis GA (1997) Oppenheim conjecture. In: Fields Medalists Lectures, vol 5. World Sci Ser 20th Century Math. World Sci Publ, River Edge, pp 272–327 84. Margulis GA (2000) Problems and conjectures in rigidity theory. In: Mathematics: frontiers and perspectives. AMS, Providence, pp 161–174 85. Margulis GA, Qian N (2001) Local rigidity of weakly hyperbolic actions of higher rank real Lie groups and their lattices. Erg Theor Dyn Syst 21:121–164 86. Margulis GA, Tomanov G (1994) Invariant measures for actions of unipotent groups over local fields of homogenous spaces. Invent Math 116:347–392 87. Mieczkowski D (2007) The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. J Modern Dyn 1:61–92 88. Milnor J (1971) Introduction to algebraic K-theory. Princeton University Press, Princeton 89. Mok N, Siu YT, Yeung SK (1993) Geometric superrigidity. Invent Math 113:57–83 90. Mostow GD (1973) Strong rigidity of locally symmetric spaces. Ann Math Studies 78. Princeton University Press, Princeton 91. Newhouse SE (1970) On codimension one Anosov diffeomorphisms. Amer J Math 92:761–770 92. Ni¸tic˘a V, Török A (1995) Cohomology of dynamical systems and rigidity of partially hyperbolic actions of higher rank lattices. Duke Math J 79:751–810 93. Ni¸tic˘a V, Török A (1998) Regularity of the transfer map for cohomologous cocycles. Erg Theor Dyn Syst 18:1187– 1209 94. Ni¸tic˘a V, Török A (2001) Local rigidity of certain partially hyperbolic actions of product type. Erg Theor Dyn Syst 21:1213– 1237 95. Ni¸tic˘a V, Török A (2002) On the cohomology of Anosov actions. In: Rigidity in dynamics and geometry, Cambridge, 2000. Springer, Berlin, pp 345–361 96. Ni¸tic˘a V, Török A (2003) Cocycles over abelian TNS actions. Geom Ded 102:65–90 97. Oseledec VI (1968) A multiplicative ergodic theorem. Characteristic Lyapunov, exponents of dynamical systems. Trudy Mosk Mat Obsc 19:179–210 98. Parry W (1999) The Livšic periodic point theorem for nonabelian cocycles. Erg Theor Dyn Syst 19:687–701 99. Pugh C, Shub M (1972) Ergodicity of Anosov actions. Invent Math 15:1–23 100. Ratner M (1991) On Ragunathan’s measure conjecture. Ann Math 134:545–607

101. Ratner M (1991) Ragunathan’s topological conjecture and distributions of unipotent flows. Duke Math J 63:235–280 102. Ratner M (1995) Raghunathan’s conjecture for Cartesians products of real and p-adic Lie groups. Duke Math J 77:275–382 103. Rudolph D (1990) × 2 and × 3 invariant measures and entropy. Erg Theor Dyn Syst 10:395–406 104. Schmidt K (1999) Remarks on Livšic’ theory for nonabelian cocycles. Erg Theor Dyn Syst 19:703–721 105. Selberg A (1960) On discontinuous groups in higher-dimensional symmetric spaces. In: Contributions to function theory. Inter Colloq Function Theory, Bombay. Tata Institute of Fundamental Research, pp 147–164 106. Smale S (1967) Differentiable dynamical systems. Bull AMS 73:747–817 107. Spatzier R (1995) Harmonic analysis in rigidity theory. In: Ergodic theory and its connections with harmonic analysis. Alexandria, 1993. London Math Soc Lect Notes Ser, vol 205. Cambridge University Press, Cambridge, pp 153–205 108. Veech WA (1986) Periodic points and invariant pseudomeasures for toral endomorphisms. Erg Theor Dyn Syst 6:449–473 109. Verjovsky A (1974) Codimension one Anosov flows. Bul Soc Math Mex 19:49–77 110. Weil A (1960) On discrete subgroups of Lie groups. I. Ann Math 72:369–384 111. Weil A (1962) On discrete subgroups of Lie groups. II. Ann Math 75:578–602 112. Weil A (1964) Remarks on the cohomology of groups. Ann Math 80:149–157 113. Zimmer R (1984) Ergodic theory and semi-simple groups. Birhhäuser, Boston 114. Zimmer R (1987) Actions of semi-simple groups and discrete subgroups. Proc Inter Congress of Math (1986). AMS, Providence, pp 1247–1258 115. Zimmer R (1990) Infinitesimal rigidity of smooth actions of discrete subgroups of Lie groups. J Diff Geom 31:301–322

Books and Reviews de la Harpe P, Valette A (1989) La propriété (T) de Kazhdan pour les groupes localement compacts. Astérisque 175 Feres R (1998) Dynamical systems and semi-simple groups: An introduction. Cambridge Tracts in Mathematics, vol 126. Cambridge University Press, Cambridge Feres R, Katok A (2002) Ergodic theory and dynamics of G-spaces. In: Handbook in Dynamical Systems, 1A. Elsevier, Amsterdam, pp 665–763 Gromov M (1988) Rigid transformation groups. In: Bernard D, Choquet-Bruhat Y (eds) Géométrie Différentielle (Paris, 1986). Hermann, Paris, pp 65–139; Travaux en Cours. 33 Kleinbock D, Shah N, Starkov A (2002) Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. In: Handbook in Dynamical Systems, 1A. Elsevier, Amsterdam, pp 813–930 Knapp A (2002) Lie groups beyond an introduction, 2nd edn. Progress in Mathematics, 140. Birkhäuser, Boston Raghunathan MS (1972) Discrete subgroups of Lie groups. Springer, Berlin Witte MD (2005) Ratner’s theorems on unipotent flows. Chicago Lectures in Mathematics. University of Chicago Press, Chicago

Existence and Uniqueness of Solutions of Initial Value Problems

Existence and Uniqueness of Solutions of Initial Value Problems GIANNE DERKS Department of Mathematics, University of Surrey, Guildford, UK Article Outline Glossary Definition of the Subject Introduction Existence Uniqueness Continuous Dependence on Initial Conditions Extended Concept of Differential Equation Further Directions Bibliography Glossary Ordinary differential equation An ordinary differential equation is a relation between a (vector) function u : I ! Rm , (I an interval in R, m 2 N) and its derivatives. A function u, which satisfies this relation, is called a solution. Initial value problem An initial value problem is an ordinary differential equation with a prescribed value for the solution at one instance of its variable, often called the initial time. The initial value is the pair consisting of the initial time and the prescribed value of the solution. Vector field For an ordinary differential equation of the m ! Rm form du dt D f (t; u), the function f : R  R is called the vector field. Thus, at a solution, the vector field gives the tangent to the solution. Flow map The flow map describes the solution of an ordinary differential equation for varying initial values. Hence, for an ordinary differential equation du dt D f (t; u), the flow map is a function ˚ : R  R  Rm ! Rm such that ˚(t; t0 ; u0 ) is a solution of the ordinary differential equation, starting at t D t0 in u D u0 . Functions: bounded, continuous, uniformly Lipschitz continuous Let D be a connected set in R k with k 2 N and let f : D ! Rm be a function on D:  The function f is bounded if there is some M > 0 such that j f (x)j  M for all x 2 D.  The function f is continuous on D if lim x!x 0 f (x) D f (x0 ), for every x0 2 D. A continuous function on a bounded and closed set D is bounded. The function f is equicontinuous or

uniform continuous if the convergence of the limits in the definition of continuity is uniform for all x0 2 D, i. e., for every " > 0, there is a ı > 0, such that for all x0 2 D and all x 2 D with jx  x0 j < ı it holds that j f (x)  f (x0 )j < ". A continuous function on a compact interval is equicontinuous.  Let D R  Rm . The function f : D ! Rm is uniformly Lipschitz continuous on D with respect to its second variable, if f is continuous on D and there exists some constant L > 0 such that j f (t; u)  f (t; u)j  Lju  vj ; for all (t; u); (t; v) 2 D : The constant L is called the Lipschitz constant. Pointwise and uniform convergence A sequence of functions fu n g with u n : I ! Rm is Pointwise convergent if lim n!1 u n (t) exists for every t 2 I. The sequence of functions fu n g is uniform convergent with limit function u : I ! Rm if lim n!1 supfju  u n j j t 2 Ig D 0. A sequence of pointwise convergent, equicontinuous functions is uniform convergent and the limit function is equicontinuous. Notation u˙ Derivative of u, i. e., u(k)

du dt

The kth derivative of u, i. e.,

dk u dt k

I a (t0 ) The closed interval [t0 ; t0 C a] Bb (u0 ) The closed ball with radius b about u0 in Rm , i. e., Bb (u0 ) :D fu 2 Rm j ju  u0 j  bg kuk1 The supremum norm for a bounded function u : I ! Rm , i. e., kuk1 D supfju(t)j j t 2 Ig Definition of the Subject Many problems in physics, engineering, biology, economics, etc., can be modeled as relations between observables or states and their derivatives, hence as differential equations. When only derivatives with respect to one variable play a role, the differential equation is called an ordinary differential equation. The field of differential equations has a long history, starting with Newton and Leibniz in the seventeenth century. In the beginning of the study of differential equations, the focus is on finding explicit solutions as the emphasis is on solving the underlying physical problems. But soon one starts to wonder: If a starting point for a solution of a differential equation is given, does the solution always exist? And if such a solution exists, how long does it exist and is there only one such solution? These are the questions of existence and uniqueness of solutions of initial value problems. The first existence

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result is given in the middle of the nineteenth century by Cauchy. At the end of the nineteenth century and the beginning of the twentieth century, substantial progress is made on the existence and uniqueness of solutions of initial value problems and currently the heart of the topic is quite well understood. But there are many open questions as soon as one considers delay equations, functional differential equations, partial differential equations or stochastic differential equations. Another area of intensive current research, which uses the existence and uniqueness of differential equations, is the area of finite- and infinitedimensional dynamical systems. Introduction As indicated before, an initial value problem is the problem of finding a solution of an ordinary differential equation with a given initial condition. To be precise, let D be an open, connected set in R  Rm . Given are a function f : D ! Rm and a point (t0 ; u0 ) 2 D. The initial value problem is the problem of finding an interval I R and a function u : I ! Rm such that t0 2 I, f(t; u(t)) j t 2 Ig D and du D f (t; u(t)) ; dt

t2I;

with u(t0 ) D u0 :

(1)

The function f is called the vector field and the point (t0 ; u0 ) the initial value. One often writes the deriva˙ tive as du dt D u. If f is continuous, then the initial value problem (1) is equivalent to finding a function u : I ! Rm such that Z

t

u(t) D u0 C

f (; u())d ;

for t 2 I :

(2)

t0

If f is continuous and a solution exists, then clearly this solution is continuously differentiable on I. It might seem restrictive to only consider first-order systems; however, most higher-order equations can be written as a system of first-order equations. Consider for example the nth order differential equation for v : I ! R   v (n) (t) D F t; v; v˙; : : : ; v (n1) ; with initial conditions v (k) (t0 ) D v k ;

k D 0; : : : ; n  1 :

This can be written as a first-order system by using the vector function u : I ! Rm defined as u D (v; v˙; : : : ; v (n1) ).

The equivalent initial value problem for u is 0 B B u˙ D f (t; u) D B @

u2 :: : un F(t; u1 ; u2 ; : : : ; u n )

1 C C C; A 0

B B with u(t0 ) D B @

v0 v1 :: :

1 C C C: A

v n1 Obviously, this system is not a unique representation, there are many other ways of obtaining first-order systems from the nth-order problem. In the beginning of the study of differential equations, the focus is on finding explicit solutions. The first existence theorem is by Cauchy [3] in the middle of the nineteenth century and an initial value problem is also called a Cauchy problem. At the end of the nineteenth century substantial progress is made on the existence of solutions of an initial value problem when Peano [20] shows that if f is continuous, then a solution exists near the initial value (t0 ; u0 ), i. e., there is local existence of solutions. Global existence of solutions of initial value problems needs more than smoothness, as is illustrated by the following example. Example 1 Consider the initial value problem on D D R  R given as u˙ D u2

and

u(0) D u0

for some u0 > 0. As can be verified easily, the solution is u(t) D u0 /(1  tu0 ), for t 2 (1; 1/u0 ). This solution cannot be extended for t  1/u0 , even though the vector field f (t; u) D u2 is infinitely differentiable on the full domain. As can be seen from later theorems, the lack of global existence is related to the fact that the vector field f is unbounded. Once existence of solutions of initial value problems is known, the next question is if such a solution is unique. As can be seen from the following example, continuity of f is not sufficient for uniqueness of solutions. Example 2 Consider the following initial value problem on D D R  R: u˙ D juj˛

and u(0) D 0 :

If 0 < ˛ < 1, then there is an infinite number of solutions. Two obvious solutions for t 2 [0; 1) are u(t) D 0 and

Existence and Uniqueness of Solutions of Initial Value Problems

1

ˆ D ((1  ˛)t) 1˛ . But these solutions are members of u(t) a large family of solutions. For any c  0, the following functions are solutions for I D R: ( 0; t
in more general differential equations and the use of existence and uniqueness of solutions of initial value problems in dynamical systems. Existence Cauchy [3] seems to have been the first one to publish an existence result for differential equations, using an approximation of solutions by joined line segments. This work was extended considerably by Peano [20] in 1890. Theorem 1 (Cauchy–Peano Theorem) Let a; b > 0, define I a (t0 ) :D [t0 ; t0 C a] and Bb (u0 ) :D fu 2 Rm j ju  u0 j  bg and assume that the cylinder S :D I a (t0 )  Bb (u0 ) D. Let the vector field f be a continuous function on the cylinder S. This implies that f is bounded on S, say, M D maxfj f (t; u)j j (t; u) 2 Sg.  

b . Then there exDefine the parameter D min a; M ists a solution u(t), with t0  t  t0 C , which solves the initial value problem (1).

The theorem can also be phrased for an interval [t0  ; t0 ] or [t0  ; t0 C ]. The parameter gives a time interval such that that the solution (t; u(t)) is guaranteed to be within S. From the integral formulation of the initial value problem (2), it follows that Z t ju(t)  u0 j  j f (; u())jd  M(t  t0 ) : t0

To guarantee that u(t) is within S, the condition t  t0  is sufficient as it gives M(t  t0 )  b and t  t0  a. In Fig. 1, this is illustrated in the case D R  R.

Existence and Uniqueness of Solutions of Initial Value Problems, Figure 1 The parameter D min( Mb ; a) guarantees that the solution is within the shaded area S, on the left in the case a < if a > Mb

b M

and on the right

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Existence and Uniqueness of Solutions of Initial Value Problems

To prove the Cauchy–Peano existence theorem, a construction which goes back to Cauchy is used. This so-called Cauchy–Euler construction of approximate solutions uses joined line segments which are such that the tangent of each line segment is given by the vector field evaluated at the start of the line segment. To be specific, for any N 2 N, the interval I (t0 ) D [t0 ; t0 C ] is divided in N equal parts with intermediate points t k :D t0 C Nk , k D 1; : : : ; N. The approximate solution is the function u N : I (t0 ) ! R, with u N (t0 ) D u0 ; u N (t) D u N (t k1 ) C f (t k1 ; u N (t k1 )) (t  t k1 ) ; (3) t k1 < t  t k ;

k D 1; : : : ; N ;

see also Fig. 2. One sees immediately that function uN is continuous and piecewise differentiable and, using the bound on the vector field f , ju N (t)  u N (s)j  Mjt  sj ;

for all t; s 2 I (t0 ) : (4)

This estimate implies that the sequence of functions fu N g is uniformly bounded and that the functions uN are equicontinuous. Indeed, the estimate above gives that for any t 2 I (t0 ) and any N 2 N, ju N (t)j  ju N (t0 )j C Mjt  t0 j  ju0 j C M : And equicontinuity follows immediately from (4) as M does not depend on N. Arzela–Ascoli’s lemma states that a sequence of functions which is uniformly bounded and equicontinuous on a compact set has a subsequence which is uniformly convergent on this compact set [5]. As the sequence fu N g is uniformly bounded and equicontinuous on the com-

pact set I (t0 ), it follows immediately that it has a convergent subsequence. This convergent subsequence is denoted by u Nk and its limit by u. To prove the Cauchy– Peano theorem, we will show that this limit function u satisfies the integral equation (2) and hence is a solution of the initial value problem (1). Proof First it will be shown that if N is sufficiently large, then the functions uN are close to solutions of the differential equation in the following sense: for all " > 0, there is some N0 such that for all N > N0 ju˙ N (t)  f (t; u N (t))j < " ; t 2 I (t0 ) ;

t ¤ tk ;

k D 0; : : : ; N :

(5)

Let " > 0. As f is a continuous function on the compact set S, it follows that f is uniform continuous. Thus, there is some ı" such that for all (t; u); (s; v) 2 S with jt  sj C ju  vj < ı" j f (t; u)  f (s; v)j < " :

(6)

m l and let N > N0 . Then for Now define N0 D (MC1)

ın any t 2 I (t0 ), t ¤ t k , k D 0; : : : ; N, we have u˙ N (t) D f (t l ; u N (t l )), where l is such that t l < t < t l C1 . Hence, (6) gives ju˙ N (t)  f (t; u N (t))j D j f (t l ; u N (t l ))  f (t; u N (t))j < " as (4) shows that jt l  tj C ju N (t l )  u N (t)j < (1 C M)jt l  tj < (1CM)

 (1CM)

 ı" . N N0 Next it will be shown that (5) implies that the functions uN almost satisfy the integral equation (2) for N sufficiently large. From (5), it follows that f (t; u N (t))  " < u˙ N (t) < f (t; u N (t)) C " for all N > N0 and all t 2 I (t0 ), except at the special points tk , k D 0; : : : ; N. Hence, for any k D 1; : : : ; N and all t k1 < t  t k , this gives Z

t

Z

t k1 t



u N (t) < u N (t k1 ) C  u N (t k1 ) C

 f (; u N ()) C " d

f (; u N ())d C t k1

" : N

Thus, R t k also for any k D" 1; : : : ; N, u N (t k )  u N (t k1 ) < t k1 f (; u N ())d C N and hence Existence and Uniqueness of Solutions of Initial Value Problems, Figure 2 The approximate solution uN , with N D 5, and the solution u(t). The dotted lines show the tangent given by the vector field f(t; u) at the points (ti ; u) for various values of u and i D 0; : : : ; 5

u N (t k )  u N (t0 ) D

k X  u N (t j )  u N (t j1 ) jD1

Z

tk

<

f (; u N ())d C " : t0

Existence and Uniqueness of Solutions of Initial Value Problems

Combination of the last two results gives that for any t 2 I (t0 ) Z t   u N (t) < u N (t0 ) C f (; u N ()) C " d C " : t0

In R t a similar way it can be shown that u N (t) > u N (t0 ) C t 0 f (; u N ())d  " and hence for all N > N0 , the following holds: ˇ ˇ Z t ˇ ˇ ˇ < " ; ˇu N (t)  u N (t0 )  f (; u ()d N ˇ ˇ t0

for all t 2 I (t0 ) :

(7)

Thus, the function uN satisfies the integral equation (2) up to an order " error if N is sufficiently large. As the subsequence u N k converges R t uniformly to u and f is continuous, this implies that t0 f (; u N k ()) ! Rt t 0 f (; u())d. Thus, from (7), it can be concluded that u satisfies the integral equation (2) exactly.  Note that the full sequence fu N g does not necessarily converge. A counterexample can be found in exercise 12 in Chap. 1 of [5]. It is based on the initial value problem 1 u˙ D juj 4 sgn(u)Ct sin(/t) with u(0) D 0, and shows that on a small interval near t0 D 0, the even approximations are bounded below by a strictly positive constant, while the odd approximations are bounded above by a strictly negative constant. However, if it is known that the solution of the initial value problem is unique, then the full sequence of approximate solutions fu N g must converge to the solution. This can be seen by a contradiction argument. If there is a unique solution, but the sequence fu N g is not convergent, then there is a convergent subsequence and a remaining set of functions which does not converge to the unique solution. But Arzela–Ascoli’s lemma can be applied to the remaining set as well; thus, there exists a convergent subsequence in the remaining set, which must converge to a different solution. This is not possible as the solution is unique. In case of the initial value problem as presented in Example 2, the sequence of Cauchy–Euler approximate solutions fu N g converges to the zero function; hence, no subsequence is required to obtain a solution. As there are many solutions of this initial value problem, this implies that not all solutions of the initial value problem in Example 2 solutions can be approximated by the Cauchy–Euler construction of approximate solutions as defined in (3). As indicated in exercise 2.2 in Chap. 2 of [13], it is possible to get any solution by using appropriate Lipschitz continuous approximations of the vector field f , instead of the

original vector field itself, in the Cauchy–Euler construction. In the example p below this is illustrated for the vector field f (t; u) D juj. p Example 3 Let f : R  R be defined as f (t; u) D juj. First we define the approximate functions fˆn : [0; 2]  [0; 1) ! [0; 1) as (p u; u > n1 ˆf n (t; u) D   p p1  1 n 1  u ; u  1 2 n n n and the definition of the Euler–Cauchy approximate functions is modified to u N (t) D u N (t k1 ) C fˆN (t k1 ; u N (t k1 )) (t  t k1 ); t k1 < t  t k ;

k D 1; : : : ; N :

This sequence converges to the solution u(t) D 4t 2 for t 2 [0; 2]. Next define the approximate functions f˜n : [0; 2]  [0; 1) ! [0; 1) as f˜n (t; u) 8p u; ˆ ˆ ˆ1 1  ˆ ˆ u ˆ < 2n  2n 1 2 D Ca2 2n  u ˆ 1 3 ˆ ˆ ˆ ˆ Ca3 n  u ; ˆ : u;

t 1; u >

t 1;

1 2n

1 n

;


t 1; u 

1 2n

1 n

;

;

p p with a2 D 4n( n1) and a3 D 12n3 (3 n2), f˜n (t; u) D fˆn (t; u), for t  1 C n1 and and a smooth connection between those two components of the approximation for a < t < 1 C n1 . Then the related approximate solutions will converge to the solution u(t) D 4(t  1)2 , 1  t  2 and u(t) D 0, 0  t < 1. In a similar way all other solutions presented in Example 2 can be obtained. Example 2 gives a connected and closed family of solutions to the initial value problem. The following theorem shows this is typical. Theorem 2 ([15]) Assume that the assumptions of Theorem 1 are satisfied. Let t0 < c  t0 C and define Ac to be the set of points that can be reached at time t D c by some solution of the initial value problem. Then Ac is closed, bounded and connected. If the initial value problem is one-dimensional, i. e., u 2 R, then this implies that the set of points reached by possible solutions at time t D c is the empty set, a point or a closed interval. A proof of this theorem, based on [19], can be found in Theorem 4.1 in Chap. 2 of [13].

387

388

Existence and Uniqueness of Solutions of Initial Value Problems

The existence theorems presented so far give existence of solutions near the initial value (t0 ; u0 ). But this local result can be extended to a global one. Theorem 3 (Extension Theorem) Let f be continuous on D. If u(t) is a solution of the initial value problem (1) on some interval, then the solution can be extended to a maximal interval of existence (t ; tC ) such that (t; u(t)) converges to the boundary of D as t " tC or t # t (where t˙ D ˙1 is possible). Proof Let Dn be open subsets of D such that [n2N D n D D, the closures D n are compact and D n D nC1 (e. g., D n D f(t; u) 2 R j j(t; u)j < n; dist((t; u); @D) > 1/ng, see [13]). First it will be shown that for all n 2 N, there is some n such that for each (t0 ; u0 ) 2 D n , the initial value problem has a solution on the interval [t0  n ; t0 C n ]. Indeed, for each n 2 N, the function f is continuous on the compact set D n . Hence, there is some M n > 0 such that j f (t; u)j  M n for all (t; u) 2 D n . Furthermore, for each n 2 N, the distance d n :D dist(D n ; D nC1 ) > 0. Thus, the Cauchy–Peano theorem implies that for each (t0 ; u0 ) 2 D n , the solution of the initial value problem exists on the interval [t0  n ; t0 C n ], with n D 1/2 min (d n /M nC1 ; d n ). If the solution u(t) is defined on an interval I, which is not a right maximal interval, then the argument above shows that the right endpoint of I can be included in the interval of existence. So it can be assumed that the interval I is of the form [a1 ; a2 ]. The continuity of the solution u gives that the set f(t; u(t)) j t 2 [a1 ; a2 ]g is a compact set in the open set D; hence, there is some n 2 N such that f(t; u(t)) j t 2 [a1 ; a2 ]g D n D n . But this implies that the solution can be extended to the interval [a1 ; 12 C n ]. And if f(t; u(t)) j t 2 [a1 ; a2 C n ]g D n , then this can be repeated. Since D n is compact, there is some k 2 N such that f(t; u(t)) j t 2 [a1 ; a2 C k n ]g 6 D n ; hence, there exists some (t n ; u(t n )) such that (t n ; u(t n )) 2 D nC1 nD n . This argument can be repeated for each n 2 N to give sequence (t n ; u(t n )) 2 D nC1 nD n , with t n < t nC1 . Thus, the sequence tn is monotone increasing. Either it is unbounded, which implies that the solution exists on an interval [a1 ; 1) and hence tC D 1 or it is bounded and hence is convergent to some limit tC . Similarly, the sequence (t n ; u(t n )) is either an unbounded sequence in D or has a cluster point on the boundary of D. In either case it is clear that the solution cannot be extended outside the interval [a1 ; tC ). A similar argument can be used for the left endpoint and it can be concluded that there exists a maximal interval of existence of the form (t ; tC ). Finally it will be shown that the solution converges to the boundary of D by a contradiction argument. Assume

that the solution does not converge to the boundary if t " tC . Then there is some "0 > 0 and a sequence fn g with tC  n < n1 and d(n ; u(n )); D) > "0 . This implies that there is some N 2 N such that the sequence f(n ; u(n ))g D N . With the arguments above, this implies that the solution can be extend to an interval including tC , which contradicts tC being the maximal right point.  Example 1 gives an example of a solution for which u(t) ! 1 if t ! u10 , the boundary of its interval of existence. Example 2 gives an example of a solution with an unbounded interval of existence. Example 4 shows that the endpoint can also be related to the failure of continuity of f . Example 4 Consider the initial value problem with D D R  (0; 1) and 1 u˙ D  ; and u(0) D 2 : u p Then the solution is u(t) D 4  2t for t 2 (1; 2). If t ! 2, then (t; u(t)) ! (2; 0), hence the boundary of D and the point where the continuity of f fails. If the initial value problem has a nonunique solution, then the maximal interval of extension in the extension theorem will in general depend on the initial solution u(t). Uniqueness Uniqueness follows if the vector field f is Lipschitz continuous with respect to its second variable u as shown first by Picard [21] and Lindelöf [18]. Their proof is based again on approximate solutions, but the approximate solutions are smoother than the Cauchy–Euler ones. The approximate solutions are obtained by successive iterations and are defined as u0 (t) :D u0

and Z (t) :D u C u nC1 0

t

f (; u n ())d ;

n2N; t2I:

t0

(8) Sometimes this iteration is called Picard iteration. Clearly, this iteration is based on the integral formula (2). The iteration process fails if u n () 62 D for some  2 I. If f is continuous, then there is some interval I for which the sequence is well-defined, as follows from the proof below. Theorem 4 (Picard–Lindelöf Theorem) Let a; b > 0 be such that the cylinder S :D I a (t0 )  Bb (u0 ) D. Let the vector field f be a uniformly Lipschitz continuous function on the cylinder S with respect to its second variable u and let M be the upper bound on f , i. e., M D maxf f (t; u) j (t; u) 2 Sg.

Existence and Uniqueness of Solutions of Initial Value Problems

  b Define the parameter D min a; M . For t 2 [t0 ; t0 C ], there exists a unique solution u(t) of the initial value problem (1). Successive iterations play an important role in proving existence for more general differential equations. Thus, although the existence of solutions already follows from the Cauchy–Peano theorem, we will prove again existence by using the successive iterations instead of the Cauchy–Euler approximations as the ideas in the proof can be extended to more general differential equations. Proof By using the bound on the vector field f , we will show that if (t; u n (t)) 2 S for all t 2 [t0 ; t0 C 0 ], then (t; u nC1 (t)) 2 S for all t 2 [t0 ; t0 C 0 ]. Indeed, for any t 2 [t0 ; t0 C 0 ], Z t ju nC1 (t)  u0 j  j f (; u n ())jd

Z

t

ju nC2 (t)  u nC1 (t)j 

L ju nC1 ()  u n ()jd t0

 L2 ku n  u n1 k1

Z

t

(  t0 )d t0

D

L2 ku n  u n1 k1 (t  t0 )2 : 2

Repeating this process, it follows for any k 2 N that Lk ku n  u n1 k1 (t  t0 ) k k! Lk k  ku n  u n1 k1 : k!

ju nCkC1 (t)  u nCk (t)j 

By using the triangle inequality repeatedly this gives for any n; k; 2 N and t 2 [t0 ; t0 C 0 ], ju nCkC1 (t)  u n (t)j 

t0 Z t

k X

ju nCiC1 (t)  u nCi (t)j

iD0

Md  M(t  t0 )  b ;



(9)



t0

k X L nCi nCi ku1  u0 k1 (n C i)! iD0

thus, ju nC1 (t)  u0 j  M  b and therefore u nC1 (t) 2 B b (u0 ). The boundedness of f also ensures that the functions fu n g are equicontinuous as for any n 2 N and any t0  t1 < t2  C t0 : ju n (t1 )  u n (t2 )j  Z t2 j f (; u n ())jd  Mjt2  t1 j : t1

Up to this point, we have only used the boundedness of f (which follows from the continuity of f ). But for the next part, in which it will be shown that for any t 2 [t0 ; t0 C

0 ] the sequence fu n (t)g is a Cauchy sequence in Rm , we will need the Lipschitz continuity. Let L be the Lipschitz constant of f on S, then for any n 2 N and t 2 [t0 ; t0 C 0 ], we have Z t ju nC1 (t)  u n (t)j  j f (; u n ())  f (; u n1 ())jd Z

t0 t

L ju n ()  u n1 ()jd

 t0

Z

t

 L ku n  u n1 k1

d t0

D L ku n  u n1 k1 (t  t0 ) ; where ku n  u n1 k1 D supfju n ()  u n1 ()j j t0    t0 C g. This implies that



k X Li i (L )n ku1  u0 k1 n! i! iD0

(L )n b eL :  n! Thus, for any t 2 [t0 ; t0 C 0 ], the sequence fu n (t)g is a Cauchy sequence in Rm ; hence, the sequence has a limit, which will be denoted by u(t). In other words, the sequence of functions fu n g is pointwise convergent in I (t0 ). We have already seen that the functions un are equicontinuous; hence, the convergence is uniform and the limit function u(t) is equicontinuous [23]. To see that u satisfies the integral equation (2), observe that for any t 2 [t0 ; t0 C 0 ] ˇ ˇ Z t ˇ ˇ ˇu(t)  u0  f (; u())d ˇˇ ˇ t0

 ju(t)  u nC1 (t)j ˇZ t ˇ ˇ ˇ ˇ Cˇ j f (; u n ())  f (; u())jd ˇˇ t0

 ju(t)  u nC1 (t)j C Lku  u n k1 (t  t0 )  (1 C L ) (ku  u nC1 k1 C ku  u n k1 ) for any n 2 N. As the sequence fu n g is uniformly convergent, this implies that u(t) satisfies the integral equation (2). Finally the uniqueness will be proved using techniques similar to those in “Existence.” Assume that there are two

389

390

Existence and Uniqueness of Solutions of Initial Value Problems

solutions u and v of the integral equation (2). Hence, for any t0  t  t0 C , Z t j f (; u())  f (; v())jd ju(t)  v(t)j  t0

 Lku  vk1 (t  t0 ) : Thus, for k D 1, it holds that ju(t)  v(t)j 

L k ku  vk1 (t  t0 ) k ; k! for any t0  t  t0 C :

(10)

For the induction step, assume that (10) holds for some k 2 N. Then for any t0  t  t0 C , Z t k L ku  vk1 (  t0 ) k d ju(t)  v(t)j  L k! t0 D

L kC1 ku  vk1 (t  t0 ) kC1 : (k C 1)!

By using the principle of induction, it follows that (10) holds for any k 2 N. This implies that for any   t0kC1 satisfy ju(t)  v(t)j  (L ) t  t0 C ,the solutions ju  vk1 / (k C 1)! for any k 2 N. Since the expression on the right converges to 0 for k ! 1, this implies that u(t) D v(t) for all t 2 [t0 ; t0 C ] and hence the solution is unique.  Implicitly, the proof shows that the integral equation gives rise to a contraction on the space of continuous functions with the supremum norm and this could also have been used to show existence and uniqueness; see [11,25], where Schauder’s fixed-point theorem is used to prove the Picard–Lindelöf theorem. With Schauder’s fixed-point theorem, existence and uniqueness can be shown for much more general differential equations than ordinary differential equations. The iteration process in (8) does not necessarily have to start with the initial condition. It can start with any continuously differentiable function u0 (t) and the interation process will still converge. In specific problems, there are often obvious choices for the initial function u0 which give a faster convergence or are more efficient, for instance, when proving unboundedness. Techniques similar to those in the proof can be used to derive a bound on the difference between the successive approximations and the solution, showing that ju n (t)  u(t)j 

ML n (t  t0 )n ; (n C 1)!

t 2 [t0 ; t0 C ] ;

see [13]. This illustrates that the convergence is fast for t near t0 , but might be slow further away: see Fig. 3 for an example.

Existence and Uniqueness of Solutions of Initial Value Problems, Figure 3 The first four iterations for initial value problem with the vector field f(t; u) D t cos u C sin t and the initial value u(0) D 0. Note that the convergence in the interval [0; 1] is very fast (two steps seem sufficient for graphical purposes), but near t D 2, the convergence has not yet been reached in three steps

Continuous Dependence on Initial Conditions In this section, the vector field f is uniformly Lipschitz continuous on D with respect to its second variable u and its Lipschitz constant is denoted by L. By combining the Picard–Lindelöf theorem (Theorem 4) and the extension theorem (Theorem 3), it follows that for every (t0 ; u0 ) 2 D, there exists a unique solution of (1) passing through (t0 ; u0 ) with a maximal interval of existence (t (t0 ; u0 ); tC (t0 ; u0 )). The trajectory through (t0 ; u0 ) is the set of points (t; u(t)), where u(t) solves (1) and t (t0 ; u0 ) < t < tC (t0 ; u0 ). Now define the set E RnC2 as E :D f (t; t0 ; u0 ) j t (t0 ; u0 ) < t < tC (t0 ; u0 ); (t0 ; u0 ) 2 D g : The flow map ˚ : E ! Rm is a mapping which describes the solution of the initial value problem with the initial condition varying through D, i. e., d ˚(t; t0 ; u0 ) D f (t; ˚ (t; t0 ; u0 )) and dt

Existence and Uniqueness of Solutions of Initial Value Problems

˚(t0 ; t0 ; u0 ) D u0 ;

for (t; t0 ; u0 ) 2 E :

From the integral equation (2), it follows that the flow map satisfies Z

Z

t

(t)  K C

()()d ;

for all a  t  b :

a

Then

t

˚(t; t0 ; u0 ) D u0 C

such that

f (; ˚ (; t0 ; u0 ))d ; t0

Z (t)  K exp

()d

Lemma 5 (Gronwall’s Lemma) Let : [a; b] ! R and  : [a; b] ! R be nonnegative continuous functions on an interval [a; b]. Let K  0 be some nonnegative constant

;

for all a  t  b :

a

for (t; t0 ; u0 ) 2 E : Furthermore, the uniqueness implies for the flow map that starting at the point (t0 ; u0 ), flowing to (t; ˚ (t; t0 ; u0 )) is the same as starting at the point (t0 ; u0 ) and flowing to the intermediate point (s; ˚ (s; t0 ; u0 )) and continuing to the point (t; ˚ (t; t0 ; u0 )). This implies that the flow map has a group structure: for any (t0 ; u0 ) 2 D and any s; t 2 (t (t0 ; u0 ); tC (t0 ; u0 )), the flow map satisfies ˚(t; s; ˚ (s; t0 ; u0 )) D ˚(t; t0 ; u0 ); the identity map is given by ˚(t0 ; t0 ; u0 ) D u0 ; and by combining the last two results, it follows that ˚ (t0 ; t; ˚ (t; t0 ; u0 )) D ˚ (t0 ; t0 ; u0 ) D u0 , and hence there is an inverse. See also Fig. 4 for a sketch of the flow map for D R  R. To show that the flow map is continuous in all its variables, Gronwall’s lemma [9] will be very useful. There are many versions of Gronwall’s lemma; the one presented here follows [1].



t

Rt Proof Define F(t) D K C a ()()d, for a  t  b. Then the assumption in the lemma gives (t)  F(t) for a  t  b. Furthermore, F is differentiable, with F˙ D  ; hence, for a  t  b  

Z t d ()d F(t) exp  dt a

Z t  D (t)(t) exp  ()d a

Z t   F(t) (t) exp  ()d  0 a

as (t)  F(t). Integrating the left-hand side gives 

Z t ()d  F(a) D K ; F(t) exp  a

which implies for  that (t)  F(t)  K exp

hR

t a

i ()d . 

Gronwall’s lemma will be used to show that small variations in the initial conditions give locally small variations in the solution, or Theorem 6 The flow map ˚ : E ! RnC2 is continuous in E.

Existence and Uniqueness of Solutions of Initial Value Problems, Figure 4 The flow map ˚(t; t0 ; u0 ). Note the group property ˚(t; s; ˚(s; t0 ; u0 )) D ˚(t; t0 ; u0 ). The lightly shaded area is the tube Uı1 and the darker shaded area is the tube Uı as used in the proof of Theorem 6. If a solution starts from within Uı , then it will stay within the tube Uı1 for t1  t  t2 , as the solution ˚(t; s0 ; v0 ) demonstrates

Proof Let (t; t0 ; u0 ) 2 E. As t; t0 2 (t (t0 ; u0 ); tC (t0 ; u0 )), there is some t1 < t2 such that t; t0 2 (t1 ; t2 ), [t1 ; t2 ]

(t (t0 ; u0 ); tC (t0 ; u0 )) and the solution ˚(s; t0 ; u0 ) exists for any s 2 [t1 ; t2 ]. First we will show that there is some tube around the solution curve f(s; ˚ (s; t0 ; u0 )) j t1  s  t2 g such that for any (s0 ; v0 ) in this tube, the solution ˚(s; s0 ; v0 ) exists for t1  s  t2 . As D is open, there is some ı1 > 0 such that the closed tube Uı1 :D f(s; v) j j˚(s; t0 ; u0 )vj  ı1 ; t1  s  t2 g D. As f is Lipschitz continuous, hence continuous, there is some M > 0 such that j f (s; v)j  M for all (s; v) 2 Uı1 . Recall that the Lipschitz constant of f on D is denoted by L. Now define ı :D ı1 eL(t2 t1 ) < ı1 and the open tube Uı D f(s; v) j j˚(s; t0 ; u0 )  vj < ı; t1 < s < t2 g ; thus, Uı Uı1 D. Thus, for every (s0 ; v0 ) 2 Uı , the solution ˚(s; s0 ; v0 ) exists for s in some closed interval in

391

392

Existence and Uniqueness of Solutions of Initial Value Problems

[t1 ; t2 ] around s0 and the solution in this interval satisfies Z s ˚(s; s0 ; v0 ) D v0 C f (; ˚ (; s0 ; v0 ))d ; s0

see also Fig. 4. Furthermore, for any s 2 [t1 ; t2 ], we have ˚ (s; t0 ; u0 ) D ˚(s; s0 ; ˚ (s0 ; t0 ; u0 )) Z s f (; ˚ (; s0 ; ˚ (s0 ; t0 ; u0 )))d D ˚(s0 ; t0 ; u0 ) C s Z 0s f (; ˚ (; t0 ; u0 ))d : D ˚ (s0 ; t0 ; u0 ) C s0

Subtracting these expressions gives j˚(s; s0 ; v0 )  ˚(s; t0 ; u0 )j  jv0  ˚(s0 ; t0 ; u0 )j Z s C j f (; ˚ (; s0 ; v0 ))  f (; ˚ (; t0 ; u0 ))j d s0 Z s L j˚(; s0 ; v0 )  ˚(; t0 ; u0 )jd ; ıC s0

so Gronwall’s lemma implies that j˚(s; s0 ; v0 )  ˚ (s; t0 ; u0 )j  ıe L(ss 0 )  ı1 . Hence, for any s, ˚ (s; s0 ; v0 ) 2 Uı1 D, and hence this solution can be extended to its maximal interval of existence, which contains the interval [t1 ; t2 ]. So it can be concluded that for any (s0 ; v0 ) 2 Uı , the solution ˚(s; s0 ; v0 ) exists for any s 2 [t1 ; t2 ]. Next we will show continuity of the flow map in its last two variables, i. e., in the initial value. For any (s0 ; v0 ) 2 Uı and t1  t  t2 , we have j˚(t; s0 ; v0 )  ˚(t; t0 ; u0 )j  jv0  u0 j ˇZ t ˇ Z t ˇ ˇ ˇ C ˇ f (; ˚ (; s0 ; v0 ))d  f (; ˚ (; t0 ; u0 ))dˇˇ s0 t0 ˇZ t 0 ˇ ˇ ˇ ˇ j f (; ˚ (; s0 ; v0 ))jd ˇˇ  jv0  u0 j C ˇ s0 ˇ ˇZ t ˇ ˇ ˇ j f (; ˚ (; t0 ; u0 ))  f (; ˚ (; s0 ; v0 ))jd ˇˇ Cˇ t0

 jv0  u0 j C Mjt0  s0 j ˇ ˇZ t ˇ ˇ ˇ Lj˚(; t0 ; u0 )  ˚(; s0 ; v0 )jd ˇˇ : Cˇ t0

Thus, Gronwall’s lemma implies that j˚(t; s0 ; v0 )  ˚(t; t0 ; u0 )j  (jv0  u0 j C Mjt0  s0 j) e Ljtt0 j  (jv0  u0 j C Mjt0  s0 j) e Ljt2 t1 j ;

and it follows that ˚ is continuous in its last two arguments. The continuity of ˚ in its first argument follows immediately from the fact that the solution of the initial value problem is continuous.  If the vector field f is smooth, then the flow map ˚ is smooth as well. Theorem 7 Let f 2 C 1 (D; R m ). Then the flow map ˚ 2 C 1 (E; R m ) and det(Du 0 ˚(t; t0 ; u0 )) Z t    D exp tr Du f (; ˚ (; t0 ; u0 )) d t0

for any (t; t0 ; u0 ) 2 E. In this theorem, tr(Du f (; ˚ (; t0 ; u0 ))) stands for the trace of the matrix Du f (; ˚ (; t0 ; u0 )). For second-order linear systems, this identity is known as Abel’s identity and det(Du 0 ˚(t; t0 ; u0 )) is the Wronskian. The proof of Theorem 7 uses the fact that Du 0 ˚(t; t0 ; u0 ) satisfies the linear differential equation d Du ˚(t; t0 ; u0 ) D Du f (t; ˚ (t; t0 ; u0 )) Du 0 ˚(t; t0 ; u0 ); dt 0 with the initial condition Du 0 ˚(t0 ; t0 ; u0 )) D I. This last fact follows immediately from ˚ (t0 ; t0 ; u0 ) D u0 . And the linear differential equation follows by differentiating the differential equation for ˚(t; t0 ; u0 ) with respect to u0 . The full details of the proof can be found, for example, in [5,11]. Extended Concept of Differential Equation Until now, we have looked for continuously differentiable functions u(t), which satisfy the initial value problem (1). But the initial value problem can be phrased for less smooth functions as well. For example, one can define a solution as an absolute continuous function u(t) which satisfies (1). A function u : I ! Rm is absolutely continuous on I if for every positive number ", no matter how small, there is a positive number ı small enough so that whenever a sequence of pairwise disjoint subintervals P [s k ; t k ] of I, k D 1; 2; : : : ; N satisfies N kD1 jt k  s k j < ı then n X

d (u(t k ); u(s k )) < " :

kD1

An absolute continuous function has a derivative almost everywhere and is uniformly continuous, thus continuous [24]. Thus, the initial value problem for an absolute continuous function can be stated almost everywhere.

Existence and Uniqueness of Solutions of Initial Value Problems

For the existence of absolute continuous solutions, the continuity condition for the vector field in the Cauchy– Peano theorem (Theorem 1) is replaced by the so-called Carathéodory conditions on the vector field. A function f : D ! Rm satisfies the Carathéodory conditions if [2]  f (t; u) is Lebesgue measurable in t for each fixed u;  f (t; u) is continuous in u for each fixed t;  for each compact set A D, there is a measurable function m A (t) such that j f (t; u)j  m A (t) for any (t; u) 2 A. Similar theorems to the ones in the previous sections about existence and uniqueness can be stated in this case. Theorem 8 (Carathéodory) If D is an open set in RnC1 and f : D ! Rm satisfies the Carathéodory conditions on D, then for every (t0 ; u0 ) 2 D there is a solution of the initial value problem (1) through (t0 ; u0 ). Theorem 9 If D is an open set in RnC1 , f : D ! Rm satisfies the Carathéodory conditions on D and u(t) satisfies the initial value problem (1) on some interval, then there exists a maximal open interval of existence. Furthermore, if (t ; tC ) denotes the maximal interval of existence, then the solution u(t) tends to the boundary of D if t # t or t " tC . Theorem 10 If D is an open set in RnC1 , f : D ! Rm satisfies the Carathéodory conditions on D and for every compact set U D there is an integrable function LU (t) such that j f (t; u) f (t; v)j  LU juvj;

for all (t; u); (t; v) 2 U ;

then for every (t0 ; u0 ) 2 D, there is a unique solution ˚ (t; t0 ; u0 ) of the initial value problem (1). The domain E of the flow map ˚ is open and ˚ is continuous in E. Proofs of those theorems can be found, for example, in Sect. I.5 in [11]. Many other generalizations are possible, for example, to vector fields which are discontinuous in u. Such vector fields play an important role in control theory. More details can be found in [6,7,17].

similar to the ones for ordinary differential equations as presented in the earlier sections and can be proved by using Schauder’s fixed-point theorem. But not all solutions can be extended as in the extension theorem (Theorem 3); see Chap. 2 in [12]. If one considers differential equations with more variables, i. e., partial differential equations, then there are no straightforward general existence and uniqueness theorems. For partial differential equations, the existence and uniqueness depends very much on the details of the partial differential equation. One possible theorem is the Cauchy– Kowaleskaya theorem, which applies to partial differential 2 equations of the form @m t u D F(t; u; @x u; @ t u; @ tx u; : : : ), where the total number of derivatives of u in F should be less than or equal to m [22]. But there are still many open questions as well. A famous open question is the existence and uniqueness of solutions of the three-dimensional Navier–Stokes equations, a system of equations which describes fluid motion. Existence and uniqueness is known for a fluid in a two-dimensional domain, but not in a threedimensional domain. The question is one of the seven “Millennium Problems,” stated as prize problems at the beginning of the third millennium by the Clay Mathematics Institute [26]. Existence and uniqueness results are also used in dynamical systems. An area of dynamical systems is bifurcation theory and for bifurcation theory the smooth dependence on parameters is crucial. The following theorem gives sufficient conditions for the smooth dependence parameters; see Theorem 3.3 in Chap. 1 of [11]. Theorem 11 If the vector field depends on parameters  2 R k , i. e., f : D  R k ! Rm and f 2 C 1 (D  R k ; Rm ), then the flow map is continuously differentiable with respect to its parameters . Furthermore, D ˚ satisfies an inhomogeneous linear equation d D ˚(t; t0 ; u0 ; ) dt D Du f (t; ˚ (t; t0 ; u0 ; ); ) D ˚(t; t0 ; u0 ; ) C D f (t; ˚ (t; t0 ; u0 ; ); ) with initial condition D ˚(t0 ; t0 ; u0 ; ) D 0.

Further Directions As follows from the previous sections, the theory of initial value problems for ordinary differential equations is quite well understood, at least for continuous vector fields. For more general differential equations, the situation is quite different. Consider for example a retarded differential equation, hence a differential equation with delay effects [12]. For retarded differential equations, there are local existence and uniqueness theorems which are quite

Results and overviews of bifurcation theory can be found, for example, in [4,8,10,14,16]. Another area of dynamical systems is stability theory. Roughly speaking, a solution is called stable if other solutions which start near this solution stay near it for all time. Note that the local continuity result in Theorem 6 is not a stability result as it only states that nearby solutions will stay nearby for a short time. For stability results, long-time existence of nearby solutions is a crucial property [10,14].

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Bibliography Primary Literature 1. Bellman R (1943) The stability of solutions of linear differential equations. Duke Math J 10:643–647 2. Carathéodory C (1918) Vorlesungen über Reelle Funktionen. Teubner, Leipzig (reprinted: (1948) Chelsea Publishing Company, New York) 3. Cauchy AL (1888) Oevres complètes (1) 6. Gauthiers-Villars, Paris 4. Chow S-N, Hale JK (1982) Methods of bifurcation theory. Springer, New York 5. Coddington EA, Levinson N (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York 6. Filippov AF (1988) Differential equations with discontinuous righthand sides. Kluwer, Dordrecht 7. Flugge-Lotz I (1953) Discontinuous automatic control. Princeton University Press, Princeton 8. Golubitsky M, Stewart I, Schaeffer DG (1985–1988) Singularities and groups in bifurcation theory, vol 1 and 2. Springer, New York 9. Gronwall TH (1919) Note on the derivative with respect to a parameter of the solutions of a system of differential equations. Ann Math 20:292–296 10. Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York 11. Hale JK (1969) Ordinary Differential Equations. Wiley, New York 12. Hale JK, Verduyn Lunel SM (1993) Introduction to Functional Differential Equations. Springer, New York 13. Hartman P (1973) Ordinary Differential Equations. Wiley, Baltimore 14. Iooss G, Joseph DD (1980) Elementary stability and bifurcation theory. Springer, New York 15. Kneser H (1923) Ueber die Lösungen eines Systems gewöhnlicher Differentialgleichungen das der Lipschitzschen Bedingung nicht genügt. S-B Preuss Akad Wiss Phys-Math Kl 171– 174 16. Kuznetsov YA (1995) Elements of applied bifurcation analysis. Springer, New York 17. Lee B, Markus L (1967) Optimal Control Theory. Wiley, New York

18. Lindelöf ME (1894) Sur l’application de la méthode des approximations successives aux équations différentielles ordinaires du premier ordre. Comptes rendus hebdomadaires des séances de l’Académie des sciences 114:454–457 19. Müller M (1928) Beweis eines Satzes des Herrn H. Kneser über die Gesamtheit der Lösungen, die ein System gewöhnlicher Differentialgleichungen durch einen Punkt schickt. Math Zeit 28:349–355 20. Peano G (1890) Démonstration de l’integrabilité des équations differentielles ordinaires. Math Ann 37:182–228 21. Picard É (1890) Mémoire sur la théorie de équations aux dérivées partielles et la méthode des approximations successives. J Math, ser 4, 6:145–210 22. Rauch J (1991) Partial Differential Equations. Springer, New York 23. Rudin W (1976) Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York 24. Rudin W (1987) Real and Complex Analysis, 3rd edn. McGrawHill, New York 25. Zeidler E (1995) Applied functional analysis. Vol 1 Applications to mathematical physics. Springer, New York 26. Clay Mathematics Institute (2000) The Millenium Problems: Navier Stokes equation. http://www.claymath.org/ millennium/Navier-Stokes_Equations/

Books and Reviews Arnold VI (1992) Ordinary Differential Equations. Springer, Berlin Arrowsmith DK, Place CM (1990) An Introduction to Dynamical Systems. Cambridge University Press, Cambridge Braun M (1993) Differential Equations and their Applications. Springer, Berlin Brock WA, Malliaris AG (1989) Differential Equations, stability and chaos in Dynamic Economics. Elsevier, Amsterdam Grimshaw R (1993) Nonlinear Ordinary Differential Equations. CRC Press, Boca Raton Ince EL (1927) Ordinary differential equations. Longman, Green, New York Jordan DW, Smith P (1987) Nonlinear Differential Equations. Oxford University Press, Oxford Werner H, Arndt H (1986) Gewöhnliche Differentialgleichungen: eine Einführung in Theorie und Praxis. Springer, Berlin

Finite Dimensional Controllability

Finite Dimensional Controllability LIONEL ROSIER Institut Elie Cartan, Vandoeuvre-lès-Nancy, France Article Outline Glossary Definition of the Subject Introduction Control Systems Linear Systems Linearization Principle High Order Tests Controllability and Observability Controllability and Stabilizability Flatness Future Directions Bibliography Glossary Control system A control system is a dynamical system incorporating a control input designed to achieve a control objective. It is finite dimensional if the phase space (e. g. a vector space or a manifold) is of finite dimension. A continuous-time control system takes the form dx/dt D f (x; u), x 2 X, u 2 U and t 2 R denoting respectively the state, the input, and the continuous time. A discrete-time system assumes the form x kC1 D f (x k ; u k ), where k 2 Z is the discrete time. Open/closed loop A control system is said to be in open loop form when the input u is any function of time, and in closed loop form when the input u is a function of the state only, i. e., it takes the more restrictive form u D h(x(t)), where h : X ! U is a given function called a feedback law. Controllability A control system is controllable if any pair of states may be connected by a trajectory of the system corresponding to an appropriate choice of the control input. Stabilizability A control system is asymptotically stabilizable around an equilibrium point if there exists a feedback law such that the corresponding closed loop system is asymptotically stable at the equilibrium point. Output function An output function is any function of the state. Observability A control system given together with an output function is said to be observable if two different

states give rise to two different outputs for a convenient choice of the input function. Flatness An output function is said to be flat if the state and the input can be expressed as functions of the output and of a finite number of its derivatives. Definition of the Subject A control system is controllable if any state can be steered to another one in the phase space by an appropriate choice of the control input. While the stabilization issue has been addressed since the XIXth century (Watt’s steam engine governor providing a famous instance of a stabilization mechanism at the beginning of the English Industrial Revolution), the controllability issue has been addressed for the first time by Kalman in the 1960s [20]. The controllability is a basic mathematical property which characterizes the degrees of freedom available when we try to control a system. It is strongly connected to other control concepts: optimal control, observability, and stabilizability. While the controllability of linear finite-dimensional systems is well understood since Kalman’s seminal papers, the situation is more tricky for nonlinear systems. For the later, concepts borrowed from differential geometry (e. g. Lie brackets, holonomy, . . . ) come into play, and the study of their controllability is still a field of active research. The controllability of finite dimensional systems is a basic concept in control theory, as well as a notion involved in many applications, such as spatial dynamics (with e. g. spatial rendezvous), airplane autopilot, industrial robots, quantic chemistry. Introduction A very familiar example of a controllable finite dimensional system is given by a car that one attempts to park at some place in a parking. The phase space is roughly the three dimensional Euclidean space R3 , a state being composed of the two coordinates of the center of mass together with the angle formed by some axis linked to the car with the (fixed) abscissa axis. The driver may act on the angle of the wheels and on their velocity, which may thus be taken as control inputs. In general, the presence of obstacles (e. g. other cars) impose to change the phase space R3 to a subset of it. The controllability issue is roughly how to combine changes of direction and of velocity to drive the car from a position to another one. Note that the system is controllable, even if the number of control inputs (2) is less than the number of independent coordinates (3). This is an important property resting upon the many connections between the coordinates of the state. While in nature each motion is generally controlled by an input (think of

395

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Finite Dimensional Controllability

the muscles in an arm), the control theory focuses on the study of systems in which an input living in a space of low dimension (typically, one) is sufficient to control the coordinates of a state living in a space of high dimension. The article is outlined as follows. In Sect. “Control Systems”, we introduce the basic concepts (controllability, stabilizability) used thereafter. In the next section, we review the linear theory, recalling the Kalman and Hautus tests for the controllability of a time invariant system, and the Gramian test for a time dependent system. Section “Linearization Principle” is devoted to the linearization principle, which allows to deduce the controllability of a nonlinear system from the controllability of its linearization along a trajectory. The focus in Sect. “High Order Tests” is on nonlinear systems for which the linear test fails, i. e., the linearized system fails to be controllable. High order conditions based upon Lie brackets ensuring controllability will be given, first for systems without drift, and next for systems with a drift. Section “Controllability and Observability” explores the connections between controllability and observability, while Sect. “Controllability and Stabilizability” shows how to derive stabilization results from the controllability property. A final section on the flatness, a new theory used in many applications to design explicit control inputs, is followed by some thoughts on future directions. Control Systems A finite dimensional (continuous-time) control system is a differential equation of the form x˙ D f (x; u)

(1)

where x 2 X is the state, u 2 U is the input, f : X  U ! U is a smooth (typically real analytic) nonlinear function, x˙ D dx/dt, and X and U denote finite dimensional manifolds. For the sake of simplicity, we shall assume here that X Rn and U Rm are open sets. Sometimes, we impose U to be bounded (or to be a compact set) to force the control input to be bounded. Given some control input u 2 L1 (I; U), i. e. a measurable essentially bounded function u : I ! U, a solution of (1) is a locally Lipschitz continuous function x() : J ! X, where J I, such that x˙ (t) D f (x(t); u(t))

for almost every t 2 J :

(2)

Note that J I only, that is, x needs not exist on all I, as it may escape to infinity in finite time. In general, u is piecewise smooth, so that (2) holds actually for all t except for finitely many values. The basic problem of the controllability is the issue whether, given an initial state x0 2 X,

a terminal state x T 2 X, and a control time T > 0, one may design a control input u 2 L1 ([0; T]; U) such that the solution of the system (

x˙ (t) D f (x(t) ; u(t)) x(0) Dx0

(3)

satisfies x(T) D x T . An equilibrium position for (1) is a point x 2 X such that there exists a value u 2 U (typically, 0) such that f (x; u) D 0. An asymptotically stabilizing feedback law is a function k : X ! U with k(x) D u, such that the closed loop system x˙ D f (x; k(x))

(4)

obtained by plugging the control input u(t) :D k(x(t))

(5)

into (1), is locally asymptotically stable at x. Recall that it means that (i) the equilibrium point is stable: for any " > 0, there exists some ı > 0 such that any solution of (4) starting from a point x0 such that jx0  xj < ı at t D 0, is defined on RC and satisfies jx(t)j  " for all t  0; (ii) the equilibrium point is attractive. For some ı > 0 as in (i), we have also that x(t) ! 0 as t ! 1 whenever jx0 j < ı. The feedback laws considered here will be continuous, and we shall mean by a solution of (4) any function x(t) satisfying (4) for all t. Notice that the solutions of the Cauchy problems exist (locally in time) by virtue of Peano theorem. A control system is asymptotically stabilizable around an equilibrium position if an asymptotically stabilizing feedback law as above does exist. In the following, we shall also consider time-varying systems, i. e. systems of the form x˙ D f (x; t; u)

(6)

where f : X  I  U ! X is smooth and I R denotes some interval. The controllability and stabilizability concepts extend in a natural way to that setting. Time-varying feedback laws u(t) D k(x(t); t)

(7)

where k : X  R ! U is a smooth (generally time periodic) function, prove to be useful in situations where the classical static stabilization defined above fails.

Finite Dimensional Controllability

and let r D rank R(A; B) < n. It may be seen that the reachable space from the origin, that is the set (   R D x T 2 Rn ; 9T > 0; 9u 2 L2 [0; T] ; Rn ;

Linear Systems Time Invariant Linear Systems A time invariant linear system is a system of the form x˙ D Ax C Bu

(8)

B 2 denote some time invariwhere A 2 ant matrices. This corresponds to the situation where the function f in (1) is linear in both the state and the input. Here, x 2 Rn and u 2 Rm , and we consider square integrable inputs u 2 L2 ([0; T]; Rm ). We say that (8) is controllable in time T if for any pair of states x0 ; x T 2 Rn , one may construct an input u 2 L2 ([0; T]; Rm ) that steers (8) from x0 to xT . Recall that the solution of (8) emanating from x0 at t D 0 is given by Duhamel formula x(t) D e tA x0 C

Z

t

e(ts)A Bu(s) ds :

0

Let us introduce the nnm matrix R(A; B) D (BjABjA2 Bj    jAn1 B) obtained by gathering together the matrices B; AB; A2 B; : : : ; An1 B. Then we have the following rank condition due to Kalman [20] for the controllability of a linear system. Theorem 1 (Kalman) The linear system (8) is controllable in time T if and only if rank R(A; B) D n. We notice that the controllability of a linear system does not depend of the control time T, and that the control input may actually be chosen very smooth (e. g. in C 1 ([0; T]; Rn )). Example 1 Consider a pendulum to which is applied a torque as a control input. A simplified model is then given by the following linear system x˙1 D x2 x˙2 D x1 C u

Z

(9) (10)

where x1 ; x2 and u stand respectively for the angle with the vertical, the angular velocity, and the torque. Here, n D 2, m D 1, and     0 1 0 AD and B D : (11) 1 0 1 As rank(B; AB) D 2, we infer from Kalman rank test that (9)–(10) is controllable. When (8) fails to be controllable, it may be important (e. g. when studying the stabilizability) to identify the uncontrollable part of (8). Assume that (8) is not controllable,

)

T

(Ts)A

e

Rnm

Rnn ,

Bu(s) ds D x T

(12) ;

0

coincides with the space spanned by the columns of the matrix R(A; B): R D fR(A; B)V ; V 2 Rnm g

(13)

In particular, dim R D rank R(A; B) D r < n. Let e D fe1 ; : : : ; e n g be the canonical basis of Rn , and let f f1 ; : : : ; f r g be a basis of R, that we complete in a basis f D ˜ denotes the vector of the f f1 ; : : : ; f n g of Rn . If x (resp. x) coordinates of a point in the basis e (resp. f), then x D T x˜ where T D ( f1 ; f2 ; : : : ; f n ) 2 Rnn . In the new coordi˜ (8) may be written nates x, ˜x C Bu ˜ x˙˜ D A˜

(14)

where A˜ :D T 1 AT and B˜ :D T 1 B read     A˜1 A˜ 2 B˜ 1 ˜D A˜ D B 0 A˜ 3 0

(15)

with A˜ 1 2 Rrr , B˜ 1 2 Rrm . Writing x˜ D (x˜1 ; x˜2 ) 2 Rr  Rnr , we have x˙˜ 1 D A˜ 1 x˜1 C A˜ 2 x˜2 C B˜ 1 u

(16)

x˙˜ 2 D A˜ 3 x˜2

(17)

and it may be proved that rank R(A˜1 ; B˜ 1 ) D r :

(18)

This is the Kalman controllability decomposition. By (18), the dynamics of x˜1 is well controlled. Actually, a solution of (18) evaluated at t D T assumes the form Z T ˜ ˜ x˜1 (T) D e T A 1 x˜1 (0) C e(Ts)A1 A˜ 2 x˜2 (s) ds 0

Z

T

C ˜

˜

e(Ts)A1 B˜ 1 u(s)ds

0

D eT A1 x 1 C

Z

T

˜

e(Ts)A1 B˜ 1 u(s)ds

0

RT ˜ if we set x 1 D x˜1 (0) C 0 es A 1 A˜ 2 x˜2 (s)ds. Hence x˜1 (T) r may be given any value in R . On the other hand, no control input is present in (17). Thus x˜2 stands for the uncontrolled part of the dynamics of (8).

397

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Finite Dimensional Controllability

Another test based upon a spectral analysis has been furnished by Hautus in [16]. Theorem 2 (Hautus) The control system (8) is controllable in time T if and only if rank(I  A; B) D n for all  2 C. Notice that in Hautus test we may restrict ourselves to the complex numbers  which are eigenvalues of A, for otherwise rank(I  A) D n.

Note that the Gramian test provides a third criterion to test whether a time invariant linear system (8) is controllable or not. Corollary 4 (8) is controllable in time T if and only if the Gramian Z T  GD e(Tt)A BB e(Tt)A dt 0

is invertible. Time-Varying Linear Systems Let us now turn to the controllability issue for a time-varying linear system x˙ (t) D A(t)x(t) C B(t)u(t)

(19)

where A 2 L1 ([0; T] ; Rnn ), B 2 L1 ([0; T] ; Rnm ) denote time-varying matrices. Such a system arises in a natural way when linearizing a control system (1) along a trajectory. The input u(t) is any function in L1 ([0; T] ; Rm ) and a solution of (19) is any locally Lipschitz continuous function satisfying (19) almost everywhere. We define the fundamental solution  associated with A as follows. Pick any s 2 [0; T], and let M : [0; T] ! Rnn denote the solution of the system ˙ M(t) D A(t)M(t)

(20)

M(s) D I

Z

t

x(t) D (t; t0 )x0 C

(t; s)B(s)u(s)ds : t0

The controllability Gramian of (20) is the matrix T





(T; t)B(t)B (t) (T; t)dt

GD

u(t) D B (t)  (T; t)G 1 (x T  (T; 0)x0 ) :

(21)

A remarkable property of the control input u is that u minimizes the control cost Z T ju(t)j2 dt E(u) D 0

among all the control inputs u 2 L1 ([0; T] ; Rm ) (or u 2 L2 ([0; T]; Rm )) steering (19) from x0 to xT . Actually a little more can be said. Proposition 5 If u 2 L2 ([0; T] ; Rm ) is such that the solution x of (19) emanating from x0 at t D 0 reaches xT at time T, and if u ¤ u, then E(u) < E(u):

Then (t; s) :D M(t). Notice that (t; s) D e(ts)A when A is constant. The solution x of (20) starting from x0 at time t0 reads then

Z

As the value of the control time T plays no role according to Kalman test, it follows that the Gramian G is invertible for all T > 0 whenever it is invertible for one T > 0. If (19) is controllable, then an explicit control input steering (19) from x0 to xT is given by

0

where  denotes transpose. Note that G 2 Rnn , and that G is a nonnegative symmetric matrix. Then the following result holds. Theorem 3 (Gramian test) The system (19) is controllable on [0; T] if and only if the Gramian G is invertible.

The main drawback of the Gramian test is that the knowledge of the fundamental solution (t; s) and the computation of an integral term are both required. In the situation where A(t) and B(t) are smooth functions of time, a criterion based only upon derivatives in time is also available. Assume that A 2 C 1 ([0; T] ; Rnn ) and that B 2 C 1 ([0; T] ; Rnm ), and define a sequence of functions B i 2 C 1 ([0; T] ; Rnm ) by induction on i by B0 D B B i D AB i1 

dB i1 : dt

(22)

Then the following result holds (see, e. g., [11]). Theorem 6 Assume that there exists a time t 2 [0; T] such that span(B i (t)v ; v 2 Rm ; i  0) D Rn : Then (19) is controllable.

(23)

Finite Dimensional Controllability

The converse of Theorem 6 is true when A and B are real analytic functions of time. More precisely, we have the following result.

 Hautus test admits the following extension due to Liu [25]: (24) is (exactly) controllable in time T if and only if there exists some constant ı > 0 such that

Theorem 7 If A and B are real analytic on [0; T], then (20) is controllable if and only if for all t 2 [0; T] span(B i (t)v ; v 2 R ; i  0) D R : m

jj(A  I)zjj2 C jjB zjj2  ıjjzjj2 8z 2 D(A ) ; 8 2 C :

(25)

n

Clearly, Theorem 7 is not valid when A and B are merely of class C 1 . (Take n D m D 1, A(t) D 0, B(t) D exp(t 1 ) and t D 0.) Linear Control Systems in Infinite Dimension Let us end this section with some comments concerning the extensions of the above controllability tests to control systems in infinite dimension (see [30] for more details). Let us consider a control system of the form x˙ D Ax C Bu

(24)

where A: D(A) X ! X is an (unbounded) operator generating a strongly continuous semigroup (S(t)) t0 on a (complex) Hilbert space X, and B : U ! X is a bounded operator, U denoting another Hilbert space. Definition 8 We shall say that (24) is  Exactly controllable in time T if for any x0 ; x T 2 X there exists u 2 L2 ([0; T]; U) such that the solution x of (24) emanating from x0 at t D 0 satisfies x(T) D x T ;  Null controllable in time T if for any x0 2 X there exists u 2 L2 ([0; T] ; U) such that the solution x of (24) emanating from x0 at t D 0 satisfies x(T) D 0;  Approximatively controllable in time T if for any x0 ; x T 2 X and any " > 0 there exists u 2 L2 ([0; T]; U) such that the solution x of (24) emanating from x0 at t D 0 satisfies jjx(T)  x T jj < " (jj  jj denoting the norm in X). This setting is convenient for a partial differential equation with an internal control Bu :D gu(t), where g D g(x) is such that gU X. Let us review the above controllability tests.  Kalman rank test, which is based on a computation of the dimension of the reachable space, possesses some extension giving the approximate (not exact!) controllability of (24) (See [13], Theorem 3.16). More interestingly, a Kalman-type condition has been introduced in [23] to investigate the null controllability of a system of coupled parabolic equations.

In (25), A (resp. B ) denotes the adjoint of the operator A (resp. B), and jj  jj denotes the norm in X.  The Gramian test admits the following extension, due to Dolecky–Russell [14] and J.L. Lions [24]: (24) is exactly controllable in time T if and only if there exists some constant ı > 0 such that Z T jjB S  (t)x0 jj2 dt  ıjjx0jj2 8x0 2 X : 0

Linearization Principle Assume given a smooth nonlinear control system x˙ D f (x; u)

(26)

where f : X  U ! Rn is a smooth map (i. e. of class C 1 ) and X Rn , U Rm denote some open sets. Assume also given a reference trajectory (x; u) : [0; T] ! X  U where u 2 L1 ([0; T] ; U) is the control input and x solves (26) for u  u. We introduce the linearized system along the reference trajectory defined as y˙ D A(t)y C B(t)v

(27)

where @f (x(t); u(t)) 2 Rnn ; @x @f (x(t); u(t)) 2 Rnm B(t) D @u

A(t) D

(28)

and y 2 Rn , v 2 Rm . (27) is formally derived from (26) by letting x D x C y, u D u C v and observing that y˙ D x˙  x˙ D f (x C y; u C v)  f (x; u) 

@f @f (x; u)y C (x; u)v : @x @u

Notice that if (x 0 ; u0 ) is an equilibrium point of f (i. e. f (x 0 ; u 0 ) D 0) and x(t)  x0 , u(t)  u0 , then A(t) D @f @f A D @x (x 0 ; u 0 ) and B(t) D B D @u (x 0 ; u 0 ) take constant values. Equation (27) is in that case the time invariant linear system y˙ D Ay C Bv :

(29)

399

400

Finite Dimensional Controllability

Let x0 ; x T 2 X. We seek for a trajectory x of (26) connecting x0 to xT when x0 (resp. xT ) is close to x(0) (resp. x(T)). In addition, we will impose that the trajectory (x, u) be uniformly close to the reference trajectory (x; u). We are led to the following Definition 9 The system (26) is said to be controllable along (x; u) if for each " > 0, there exists some ı > 0 such that for each x0 ; x T 2 X with jjx0  x(0)jj < ı, jjx T  x(T)jj < ı, there exists a control input u 2 L1 ([0; T] ; U) such that the solution of (26) starting from x0 at t D 0 satisfies x(T) D x T and   sup jx(t)  x(t)j C ju(t)  u(t)j  " :

tory such that the linearization of (30)–(32) along it is controllable. Pick any time T > 0 and let u 1 (t) D cos( t/T), u 2 (t) D 0 for t 2 [0; T] and x0 D 0. Let x denote the corresponding solution of (30)–(32). Notice that x 1 (t) D (T/) sin( t/T) (hence x 1 (T) D 0), and x 2 D x 3  0. The linearization of (30)–(32) along (x; u) is (27) with 0 1 0 0 0 A(t) D @ 0 0 0 A; 0  cos( t/T) 0 (33) 0 1 1 0 A: B(t) D @ 0 1 0 (T/) sin( t/T)

t2[0;T]

We are in a position to state the linearization principle. Theorem 10 Let (x; u) be a trajectory of (26). If the linearized system (27) along (x; u) is controllable, then the system (26) is controllable along the trajectory (x; u). When the reference trajectory is stationary, we obtain the following result. Corollary 11 Let (x0 ; u0 ) be such that f (x0 ; u0 ) D 0. If the linearized system (29) is controllable, then the system (26) is controllable along the stationary trajectory (x; u) D (x0 ; u0 ). Notice that the converse of Corollary 11 is not true. (Consider the system x˙ D u3 with n D m D 1, x0 D u0 D 0, which is controllable at the origin as it may be seen by performing the change of inputs v D u3 .) Often, the nonlinear part of f plays a crucial role in the controllability of (26). This will be explained in the next section using Lie algebraic techniques. Another way to use the nonlinear contribution in f is to consider a linearization of (26) along a convenient (not stationary) trajectory. We consider a system introduced by Brockett in [6] to exhibit an obstruction to stabilizability. Example 2 The Brockett’s system reads

Notice that A and B are real analytic, so that we may apply Theorem 8 to check whether (27) is controllable or not on [0; T]. Simple computations give 0 1 0 0 A: B1 (t) D @ 0 0 0 2 cos( t/T) Clearly,  span B0 (0)

1 0



 ; B0 (0)

0 1



 ; B1 (0)

0 1

!

D R3

hence (27) is controllable. We infer from Theorem 10 that (30)–(32) is controllable along (x; u). Notice that if we want to prove the controllability around an equilibrium point as above for Brockett’s system, we have to design a reference control input u so that x0 D x(0) D x(T) :

(34)

When f is odd with respect to the control, i. e. f (x; u) D  f (x; u) ;

(35)

then (34) is automatically satisfied whenever u fulfills

x˙1 D u1

(30)

x˙2 D u2

(31)

x˙3 D x1 u2  x2 u1 :

(32)

u(t) D u(T  t) 8t 2 [0; T] :

(36)

Indeed, it follows from (35)–(36) that the solution x to (26) starting from x0 at t D 0 satisfies

Its linearization along (x0 ; u0 ) D (0; 0), which reads y˙1 D v1 y˙2 D v2 y˙3 D 0 ; is not controllable, by virtue of the Kalman rank condition. We may however construct a smooth closed trajec-

x(t) D x(T  t) 8t 2 [0; T] :

(37)

In particular, x(T) D x(0) D x0 . Of course, the control inputs of interest are those for which the linearized system (27) is controllable, and the latter property is “generically” satisfied for a controllable system. The above construction of the reference trajectory is due to J.-M. Coron,

Finite Dimensional Controllability

and is referred to as the return method. A precursor to that method is [34]. Beside giving interesting results for the stabilization of finite dimensional systems (see below Sect. “Controllability and Stabilizability”), the return method has also been successfully applied for the control of some important partial differential equations arising in Fluid Mechanics (see [11]). High Order Tests In this section, we shall derive new controllability tests for systems for which the linearization principle is inconclusive; that is, the linearization at an equilibrium point fails to be controllable. To simplify the exposition, we shall limit ourselves to systems affine in the control, i. e. systems of the form x˙ D f0 (x) C u1 f1 (x) C    C u m f m (x) ; x 2 Rn ; juj1  ı

(38)

where juj1 :D sup1im ju i j and ı > 0 denotes a fixed number. We assume that f0 (0) D 0 and that f i 2 C 1 (Rn ; Rn ) for each i 2 f0; : : : ; mg. To state the results we need to introduce a few notations. Let v D (v1 ; : : : ; v n ) and w D (w1 ; : : : ; w n ) be two vector fields of class C 1 on Rn . The Lie bracket of v and w, denoted by [v; w], is the vector field [v; w] D

@v @w v w @x @x

where @w/@x is the Jacobian matrix (@w i /@x j ) i; jD1;:::;n , and the vector v(x) !D (v1 (x); : : : ; v n (x)) is identified v 1 (x)

to the column

: : : v n (x)

. As [v; w] is still a smooth vector

field, we may bracket it with v, or w, etc. Vector fields like [v; [v; w]], [[v; w]; [v; [v; w]]], etc. are termed iterated Lie brackets of v, w. The Lie bracketing of vector fields is an operation satisfying the two following properties (easy to check) 1. Anticommutativity: [w; v] D [v; w]; 2. Jacobi identity: [ f ; [g; h]]C[g; [h; f ]]C[h; [ f ; g]] D 0. The Lie algebra generated by f 1 , . . . ,f m , denoted Lie( f1 ; : : : ; f m ), is the smallest vector subspace v of C 1 (Rn ; Rn ) which contains f 1 , . . . ,f m and which is closed under Lie bracketing (i. e. v; w 2 V ) [v; w] 2 V ). It is easily to see that Lie( f1 ; : : : ; f m ) is the vector space spanned by all the iterated Lie brackets of f 1 ,. . . ,f m . (Actually, using the anticommutativity and Jacobi identity, we may restrict ourselves to iterated Lie brackets of the form

[ f i 1 ; [ f i 2 ; [: : : [ f i p1 ; f i p ] : : : ]]] to span Lie( f1 ; : : : ; f m ).) For any x 2 Rn , we set Lie( f1 ; : : : ; f m )(x) D fg(x); g 2 Lie( f1 ; : : : ; f m )g Rn : Example 3 Let us consider the system 0

1 0 1 1 0 x˙ D u1 f1 (x) C u2 f2 (x) D u1 @ 0 A C u2 @ 1 A (39) 0 x1 where x D (x1 ; x2 ; x3 ) 2 R3 . Clearly, the linearized system at (x0 ; u0 ) D (0; 0), which reduces to x˙ D u1 f1 (0) C u2 f2 (0), is not controllable. However, we shall see later that the system (39) is controllable. First, each point (˙"; 0; 0) (resp. (0; ˙"; 0)) may be reached from the origin by letting u(t) D (˙1; 0) (resp. u(t) D (0; ˙1)) on the time interval [0; "]. More interestingly, any point (0; 0; "2 ) may also be reached from the origin in a time T D O("), even though (0; 0; 1) 62 span( f1 (0); f2 (0)). To prove the last claim, let us introduce for i D 1; 2 the flow map  it defined by  it (x0 ) D x(t), where x() solves x˙ D f i (x) ;

x(0) D x0 :

Then, it is easy to see that 2" 1" 2" 1" (0) D (0; 0; "2 ) : It means that the control 8 (1; 0) if ˆ ˆ < (0; 1) if u(t) D (1; 0) if ˆ ˆ : (0; 1) if

0t<" "  t < 2" 2"  t < 3" 3"  t < 4"

steers the solution of (39) from x0 D (0; 0; 0) at t D 0 to x T D (0; 0; "2 ) at T D 4". (See Fig. 1.) More generally, if f 1 and f 2 denote arbitrary smooth vector fields and 1t , 2t denote the corresponding flow maps, we have that 2" 1" 2" 1" (0) D "2 [ f1 ; f2 ](0) C O("3 ) :

(40)

Thus, it is possible to reach points in the direction of the Lie bracket [ f1 ; f2 ](0). However, the process is not very efficient: in the direction of the Lie bracket [ f1 ; f2 ](0), only points located at a distance from the origin of order "2 may be reached in a time of order ". Let us proceed to the controllability properties of affine systems. We first review the results for systems without drift (i. e. f0 D 0), and next consider the general (only partially understood) situation where a drift exists.

401

402

Finite Dimensional Controllability

Example 2 continued Brockett’s system is also controllable, since [ f1 ; f2 ] D (0; 0; 2) gives (43). Notice that Theorem 13 is almost sharp. Indeed, it has been proved by Sussmann–Jurdjevic in [36] that (43) has to be satisfied for a controllable system (41) with real analytic vector fields (i. e. f i 2 C ! (Rn ; Rn ) for each i). Affine Systems with Drifts We consider now a control system with a drift x˙ D f0 (x) C u1 f1 (x) C    C u m f m (x)

where f i 2 C 1 (Rn ; Rn ) for any i 2 f0; : : : ; mg, and f0 (0) D 0 but f0 6 0. As no control is associated with the vector field f 0 , one expects that the latter will occupy a singular place in the controllability tests. Let > 0 be a given number. We assume that the control input u is a measurable function taking its values in the compact set U D [ ; ]m . To emphasize the dependence in u, we shall sometimes denote by x(t, u) the solution of (45) such that x(0; u) D 0. The attainable set at time T > 0 is the set

Finite Dimensional Controllability, Figure 1 Trajectory from x0 D (0; 0; 0) to xT D (0; 0; "2 )

Affine Systems Without Drift We assume that f0 D 0, so that (38) takes the form x˙ D u1 f1 (x) C    C u m f m (x) :

(41)

When m < n, the linearized system at any stationary trajectory (x; 0), which reads y˙ D ( f1 (x)    f m (x))v ;

(42)

cannot be controllable. High order tests are therefore very useful in this setting. We adopt the following Definition 12 The system (41) is said to be controllable if for any T > 0 and any x0 ; x T 2 Rn , there exists a control u 2 L1 ([0; T] ; Rm ) such that the solution x(t) of (41) emanating from x0 at t D 0 reaches xT at time t D T. Note that the controllability may be supplemented with the words: local, with small controls, in small time, in large time, etc. An important instance, the small-time local controllability, is introduced below in Definition 14. The following result has been obtained by Rashevski [28] and Chow [8]. Theorem 13 (Rashevski–Chow) Assume that f i 2 C 1 (Rn ; Rn ) for all i 2 f1; : : : ; mg. If Lie( f1 ; : : : ; f m )(x) D Rn

8x 2 Rn ;

(43)

then (41) is controllable. Example 3 continued (39) is controllable, since [ f1 ; f2 ] D (0; 0; 1) gives span( f1 (x); f2 (x); [ f1 ; f2 ](x)) D R3

(45)

8x 2 R3 : (44)

˚



A (T) D x(T; u); u(t) 2 U 8t 2 [0; T] :

Definition 14 We shall say that the control system (45) is  accessible from the origin if the interior of A (T) is nonempty for any T > 0;  small time locally controllable (STLC) at the origin if for any T > 0 there exists a number ı > 0 such that for any pair of states (x0 ; x T ) with jx0 j < ı; jx T j < ı, there exists a control input u 2 L1 ([0; T] ; U ) such that the solution x of x˙ D f0 (x)Cu1 f1 (x)C    Cu m f m (x);

x(0) D x0 (46)

satisfies x(T) D x T . (Notice that in the definition of the small time local controllability, certain authors assume x0 D 0 in above definition, or require the existence of ı > 0 for any T > 0 and any > 0.) The accessibility property turns out to be easy to characterize. As a STLC system is clearly accessible, the accessibility property is often considered as the first property to test before investigating the controllability of a given affine system. The following result provides an accessibility test based upon the rank at the origin of the Lie algebra generated by all the vectors fields involved in the control system.

Finite Dimensional Controllability

Theorem 15 (Hermann–Nagano) If dim Lie( f0 ; f1 ; : : : ; f m )(0) D n

(47)

then the system (45) is accessible from the origin. Conversely, if (45) is accessible from the origin and the vector fields f 0 , f 1 , . . . , f m are real analytic, then (47) has to be satisfied. Example 4 Pick any k 2 N  and consider as in [22] the system

x˙1 D u; juj  1 ; (48) x˙2 D (x1 ) k so that n D 2, m D 1, and f0 (x) D (0; (x1 ) k ), f1 (x) D (1; 0). Setting

h D [[ f2 ; [ f2 ; f0 ]]; [ f2 ; f3 ]] : Finally, we set X  (h) D h : 2S m

With h ˚ as in (49), and S3 D idf1;2;3g ; (1 2 3); (1 3 2); (1 2); (1 3); (2 3) , we obtain  (h) D [[ f1 ; [ f1 ; f0 ]]; [ f1 ; f2 ]] C [[ f2 ; [ f2 ; f0 ]]; [ f2 ; f3 ]]

(ad f1 ; f0 ) D [ f1 ; f0 ] (ad iC1 f1 ; f0 ) D [ f1 ; (ad i f1 ; f0 )]

Let Sm denote the usual symmetric group, i. e. Sm is the group of all the permutations of the set f1; : : : ; mg. If 2 S m and h is an iterated Lie bracket of f0 ; f1 ; : : : ; f m , we denote by h the iterated Lie bracket obtained by replacing, in the definition of h, f i by f (i) for each i 2 f1; : : : ; mg. For instance, if D (1 2 3) and h is as in (49), then

C [[ f3 ; [ f3 ; f0 ]]; [ f3 ; f1 ]] C [[ f2 ; [ f2 ; f0 ]]; [ f2 ; f1 ]]

8i  1

C [[ f3 ; [ f3 ; f0 ]]; [ f3 ; f2 ]] C [[ f1 ; [ f1 ; f0 ]]; [ f1 ; f3 ]] :

we obtain at once that

We need the following

k

(ad f1 ; f0 )(x) D (0; k !)

Definition 16 Let 2 [0; 1]. We say that the system (45) satisfies the condition S( ) if, for any iterated Lie bracket h with ı0 (h) odd and ı i (h) even for all i 2 f1; : : : ; mg, the vector  (h)(0) belongs to the vector space spanned by the vectors g(0) where g is an iterated Lie bracket satisfying

hence Lie( f0 ; f1 )(0) D R2 : It follows from Theorem 15 that (48) is accessible from the origin. On the other hand, it is clear that (48) is not STLC at the origin when k is even, since x˙2 D (x1 ) k  0, hence x2 is nondecreasing. Using a controllability test due to Hermes [17], it may be shown that (48) is STLC at the origin if and only if k is odd. Note that the linearized system at the origin fails to be controllable whenever k  2. Let us consider some affine system (45) which is accessible from the origin. We exclude the trivial situation when the linearization principle may be applied, and seek for a Lie algebraic condition ensuring that (45) is STLC. If h is a given iterated Lie bracket of f0 ; f1 ; : : : ; f m , we let ı i (h) denote the number of occurrences of f i in the definition of h. For instance, if m D 3 and h D [[ f1 ; [ f1 ; f0 ]]; [ f1 ; f2 ]]

(49)

then ı0 (h) D 1 ; ı1 (h) D 3 ; ı2 (h) D 1 ; ı3 (h) D 0 : Notice that the fields f0 ; f1 ; : : : ; f m are considered as indeterminates when computing the ı i (h)’s, their effective values as vector fields being ignored.

ı0 (g) C

m X

ı i (g) < ı0 (h) C

iD1

m X

ı i (h) :

iD1

Then we have the following result proved in [35]. Theorem 17 (Sussmann) If the condition S( ) is satisfied for some 2 [0; 1], then the system (45) is small time locally controllable at the origin. When D 0, Sussmann’s theorem is nothing else than Hermes’ theorem. Sussmann’s theorem, which is in itself very useful in Engineering (see, e. g., [5]), has been extended in [1,4,18]. Controllability and Observability Assume given a control system x˙ D f (x; u)

(50)

together with an output function y D h(x)

(51)

where f : X  U ! Rn and h : X ! Y are smooth functions, and X Rn , U Rm and Y R p denote some

403

404

Finite Dimensional Controllability

open sets. y typically stands for a (partial) measurement of the state, e. g. the p first coordinates, where p < n. Often, only y is available, and for the stabilization of (50) we should consider an output feedback law of the form u D k(y). We shall say that (50)–(51) is observable on the interval [0; T] if, for any pair x0 ; x˜0 of points in X, one may find a control input u 2 L1 ([0; T] ; U) such that if x (resp. x˜ ) denotes the solution of (50) emanating from x0 (resp. x˜0 ) at time t D 0, and y (resp. y˜) denotes the corresponding output function, we have y(t) ¤ y˜(t) for some t 2 [0; T] :

(52)

For a time invariant linear control system and a linear output function x˙ D Ax C Bu

(53)

y D Cx

(54)

with A 2 Rnn , B 2 Rnm , C 2 R pn , the output is found to be Z t y(t) D Cx(t) D Ce tA x0 C C e(ts)A Bu(s)ds : (55) 0

In particular, y(t)  y˜(t) D Ce tA (x0  x˜0 )

Notice that the observability of the adjoint system is easily shown to be equivalent to the existence of a constant ı > 0 such that Z T  jjB e tA x0 jj2 dt  ıjjx0 jj2 8x0 2 Rn : (59) 0

As it has been pointed out in Sect. “Linear Systems”, the equivalence between the controllability of a system and the observability of its adjoint, expressed in the form (59), is still true in infinite dimension, and provides a very useful way to investigate the controllability of a partial differential equation. Finally, using the duality principle, we see that any controllability test gives rise to an observability test, and vice-versa. Controllability and Stabilizability In this section we shall explore the connections between the controllability and the stabilizability of a system. Let us begin with a linear control system x˙ D Ax C Bu :

(56)

and B does not play any role. Therefore, (53)–(54) is observable in time T if and only if the only state x 2 Rn such that Ce tA x D 0 for any t 2 [0; T] is x D 0. Differentiating in time and applying Cayley–Hamilton theorem, we obtain the following result. Theorem 18 System (53)–(54) is observable in time T if and only if rank O(A; C) D n, where the observability matrix O(A; C) 2 Rn pn is defined by 1 0 C B CA C C B O(A; C) D B C :: A @ : n1 CA Noticing that R(A; B) D O(A ; B )

and introducing the adjoint system (57)

y˜ D B x˜

(58)

(60)

Performing a linear change of coordinates if needed, we may assume that (60) has the block structure given by the Kalman controllability decomposition        x˙1 A1 A2 x1 B1 u (61) D C x˙2 0 A3 x2 0 where A1 2 Rrr , B1 2 Rrm , and where rank R (A1 ; B1 ) D r. For any square matrix M 2 C nn we denote by  (M) its spectrum, i. e.  (M) D f 2 C ; det(M  I) D 0g : Let us note C :D f 2 C ; Re  < 0g. Then the asymptotic stabilizability of (60) may be characterized as follows. Theorem 20 There exists a (continuous) asymptotically stabilizing feedback law u D k(x) with k(0) D 0 for (60) if and only if  (A3 ) C . If it is the case, then for any family S D ( i )1ir of elements of C invariant by conjugation, there exists a linear asymptotically stabilizing feedback law u(x) D Kx, with K 2 Rmn , such that  (A C BK) D  (A3 ) [ S :

x˙˜ D A x˜

we arrive to the following duality principle.

Theorem 19 A time invariant linear system is controllable if and only if its adjoint system is observable.

(62)

The property (62), which shows to what extend the spectrum of the closed loop system can be assigned, is referred to as the pole shifting theorem.

Finite Dimensional Controllability

As a direct consequence of Theorem 20, we obtain the following result. Corollary 21 A time invariant linear system which is controllable is also asymptotically stabilizable. It is natural to ask whether Corollary 21 is still true for a nonlinear system, i. e. if a controllable system is necessarily asymptotically stabilizable. A general result cannot be obtained, as is shown by Brockett’s system. Example 2 continued Brockett’s system (30)–(32) may be written x˙ D u1 f1 (x) C u2 f2 (x) D f (x; u), and it follows from Theorem 13 that (30)–(32) is controllable around (x0 ; u0 ) D (0; 0). Let us now recall a necessary condition for stabilizability due to R. Brockett [6]. Brockett third condition for stabilizability. Let f 2 C(Rn  Rm ; Rn ) with f (x0 ; u0 ) D 0. If the control system x˙ D f (x; u) can be locally asymptotically stabilized at x0 by means of a continuous feedback law u satisfying u(x0 ) D u0 , then the image by f of any open neighborhood of (x0 ; u0 ) 2 Rn  Rm contains an open neighborhood of 0 2 Rn . Here, for any neighborhood V of (x0 ; u0 ) in R5 , f (V) does not cover an open neighborhood of 0 in R3 , since (0; 0; ") 62 f (V) for any " 2 R. According to Brockett’s condition, the system (30)–(32) is not asymptotically stabilizable at the origin. Thus a controllable system may fail to be asymptotically stabilizable by a continuous feedback law u D k(x) due to topological obstructions. It turns out that this phenomenon does not occur when the phase space is the plane, as it has been demonstrated by Kawski in [21]. Theorem 22 (Kawski) Let f0 ; f1 be real analytic vector fields on R2 with f0 (0) D 0 and f1 (0) ¤ 0. Assume that the system x˙ D f0 (x) C u f1 (x) ;

u2R

(63)

is small time locally controllable at the origin. Then it is also asymptotically stabilizable at the origin by a Hölder continuous feedback law u D k(x). In larger dimension (n  3), a way to go round the topological obstruction is to consider a time-varying feedback law u D k(x; t). It has been first observed by Sontag and Sussmann in [33] that for one-dimensional state and input (n D m D 1), the controllability implies the asymptotic stabilizability by means of a time-varying feedback law. This kind of stabilizability was later established by Samson in [31] for Brockett’s system (n D 3 and m D 2). Finally, using the return method, Coron proved that the implication

Controllability ) Asymptotic Stabilizability by TimeVarying Feedback was a principle verified in most cases. To state precise results, we consider affine systems x˙ D f0 (x) C

m X

u i f i (x) ;

x 2 Rn

iD1

and distinguish again two cases: (1) f0  0 (no drift); (2) f0 6 0 (a drift). (1) System without drift Theorem 23 (Coron [9]) Assume that (43) holds for the system (41). Pick any number T > 0. Then there exists a feedback law u D (u1 ; : : : ; u m ) 2 C 1 (Rn  R; Rm ) such that u(0; t) D 0

8t 2 R

u(x; t C T) D u(x; t)

(64) 8(x; t) 2 Rn  R

(65)

and such that 0 is globally asymptotically stable for the system x˙ D

m X

u i (x; t) f i (x) :

iD1

(2) System with a drift Theorem 24 (Coron [10]) Assume that the system (45) satisfies the condition S( ) for some 2 [0; 1]. Assume also that n 62 f2; 3g and that Lie( f0 ; f1 ; : : : ; f m )(0) D Rn : Pick any T > 0. Then there exists a feedback law u D (u1 ; : : : ; u m ) 2 C 0 (Rn R; Rm ) with u 2 C 1 ((Rn nf0g) R; Rm ) such that (64)–(65) hold and such that 0 is locally asymptotically stable for the system x˙ D f0 (x) C

m X

u i (x; t) f i (x) :

iD1

Flatness While powerful criterions enable us to decide whether a control system is controllable or not, most of them do not provide any indication on how to design an explicit control input steering the system from a point to another one. Fortunately, there exists a large class of systems, the so-called flat systems, for which explicit control inputs may easily be found. The flatness theory has been introduced by Fliess, Levine, Martin, and Rouchon in [15], and since then it has attracted the interest of many researchers

405

406

Finite Dimensional Controllability

thanks to its numerous applications in Engineering. Here, we only sketch the main ideas, referring the interested reader to [27] for a comprehensive introduction to the subject. Let us consider a smooth control system x 2 Rn ; u 2 Rm

x˙ D f (x; u) ;

given together with an output y 2 Rm depending on x, u, and a finite number of derivatives of u ˙ : : : ; u(r) ) : y D h(x; u; u;

Example 6 Consider now the nonlinear system x˙1 D u1

(72)

x˙2 D u2

(73)

x˙3 D x2 u1 :

(74)

Eliminating the input u1 in (72)–(74) yields x˙3 D x2 x˙1 , so that x2 may be expressed as a function of x1 ; x3 and their derivatives. The same is true for u2 , thanks to (73). Let us pick y D (y1 ; y2 ) D (x1 ; x3 ) :

Following [15], we shall say that y is a flat output if the components of y are differentially independent, and both x and u may be expressed as functions of y and of a finite number of its derivatives   (66) x D k y; y˙; : : : ; y (p)   u D l y; y˙; : : : ; y (q) : (67) In (66)–(67), p and q denote some nonnegative integers, and k and l denote some smooth functions. Since the state x and the input u are parameterized by the flat output y, to solve the controllability problem

We claim that y is a flat output. Indeed,   y˙2 (x1 ; x2 ; x3 ) D y1 ; ; y2 ; y˙1   y¨2 y˙1  y˙2 y¨1 : (u1 ; u2 ) D y˙1 ; ( y˙1 )2

(75) (76)

Pick x0 D (0; 0; 0) and x T D (0; 0; 1). Notice that, by the mean value theorem, y˙1 has to vanish somewhere, say at t D ¯t . We shall construct y1 in such a way that y˙1 vanishes only at t D ¯t . For x2 not to be singular at ¯t , we have to impose the condition y˙2 ( ¯t ) D 0 :

x˙ D f (x; u) x(0) D x0 ;

(68) x(T) D x T

(69)

it is sufficient to pick any function y 2 C max(p;q) ([0; T]; Rm ) such that k(y; y˙; : : : ; y (p) )(0) D x(0) D x0 k(y; y˙; : : : ; y

(p)

)(T) D x(T) D x T :

(70)

If y1 and y2 are both analytic near ¯t , we notice that x2 D y˙2 / y˙1 is analytic near ¯t , hence u2 D x˙2 is well-defined (and analytic) near t¯. To steer the solution of (72)–(74) from x0 D (0; 0; 0) to x T D (0; 0; 1), it is then sufficient to pick a function y D (y1 ; y2 ) 2 C ! ([0; T]; R2 ) such that y1 (0) D y2 (0) D 0 ; y1 (T) D 0 ;

The constraints (70)–(71) are generally very easy to satisfy. The desired control input u is then given by (67). Let us show how this program may be carried out on two simple examples.

y˙1

y2 (T) D 1 ;

  T D0; 2

Example 5 For the simple integrator x˙1 Dx2

and y˙1 (0) ¤ 0 ;

y˙2 (T) D 0 ;

and y˙1 (T) ¤ 0 :     T T y˙2 D 0 ; y¨1 ¤0; 2 2 T and y˙1 (t) ¤ 0 for t ¤ : 2

Clearly, 

x˙2 Du the output y D x1 is flat, for x2 DR y˙ and u D y¨. The output z D x2 is not flat, as x1 D z(t) dt. To steer the system from x0 D (0; 0) to x T D (1; 0) in time T, we only have to pick a function y 2 C 2 ([0; T]; R) such that y(0) D 0 ;

y˙2 (0) D 0 ;

(71)

y˙(0) D 0 ;

y(T) D 1 ; and y˙(T) D 0 :

Clearly, y(t) D t 2 (2T  t)2 /T 4 is convenient.

t 4 Tt 3 T 2 t2 C (y1 (t); y2 (t)) D t(T  t);  4 2 4



is convenient. Future Directions Lie algebraic techniques have been used to provide powerful controllability tests for affine systems. However, there is still an important gap between the known necessary con-

Finite Dimensional Controllability

ditions (e. g. the Legendre Clebsh condition or its extensions) and the sufficient conditions (e. g. the S( ) condition) for the small time local controllability. On the other hand, it has been noticed by Kawski in ([22], Example 6.1) that certain systems can be controlled on small time intervals [0; T] only by using faster switching control variations, the number of switchings tending to infinity as T tends to zero. As for switched systems [2], it is not clear whether purely algebraic conditions be sufficient to characterize the controllability of systems with drift. Of great interest for applications is the development of methods providing explicit control inputs, or control inputs computed in real time with the aid of some numerical schemes, both for the motion planning and the stabilization issue. The flatness theory seems to be a very promising method, and it has been successfully applied in Engineering. An active research is devoted to filling the gap between necessary and sufficient conditions for the existence of flat outputs, and to extending the theory to partial differential equations. The control of nonlinear partial differential equations may sometimes be reduced to the control of a family of finite-dimensional systems by means of the Galerkin method [3]. On the other hand, the spatial discretization of partial differential equations by means of finite differences, finite elements, or spectral methods leads in a natural way to the control of finite dimensional systems (see, e. g., [38]). Of great importance is the uniform boundedness with respect to the discretization parameter of the L2 (0; T)-norms of the control inputs associated with the finite dimensional approximations. Geometric ideas borrowed from the control of finite dimensional systems (e. g. the return method, the power series expansion, the quasistatic deformation) have been applied to prove the controllability of certain nonlinear partial differential equations whose linearization fails to be controllable. (See [11] for a survey of these techniques.) The Korteweg–de Vries equation provides an interesting example of a partial differential equation whose linearization fails to be controllable for certain lengths of the space domain [29]. However, for these critical lengths the reachable space proves to be of finite codimension, and it may be proved that the full equation is controllable by using the nonlinear term in order to reach the missing directions [7,12]. Bibliography Primary Literature 1. Agrachev AA, Gamkrelidze RV (1993) Local controllability and semigroups of diffeomorphisms. Acta Appl Math 32:1–57

2. Agrachev A, Liberzon D (2001) Lie-algebraic stability criteria for switched systems. SIAM J Control Optim 40(1):253–269 3. Agrachev A, Sarychev A (2005) Navier–Stokes equations: controllability by means of low modes forcing. J Math Fluid Mech 7(1):108–152 4. Bianchini RM, Stefani G (1986) Sufficient conditions of local controllability. In: Proc. 25th IEEE–Conf. Decision & Control. Athens 5. Bonnard B (1992) Contrôle de l’altitude d’un satellite rigide. RAIRO Autom Syst Anal Control 16:85–93 6. Brockett RW (1983) Asymptotic stability and feedback stabilization. In: Brockett RW, Millman RS, Sussmann HJ (eds) Differential Geometric Control Theory. Progr. Math., vol 27. Birkhäuser, Basel, pp 181–191 7. Cerpa E, Crépeau E (2007) Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain 8. Chow WL (1940) Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math Ann 117:98–105 9. Coron JM (1992) Global asymptotic stabilization for controllable systems without drift. Math Control Signals Syst 5: 295–312 10. Coron JM (1995) Stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws. SIAM J Control Optim 33:804–833 11. Coron JM (2007) Control and nonlinearity, mathematical surveys and monographs. American Mathematical Society, Providence 12. Coron JM, Crépeau E (2004) Exact boundary controllability of a nonlinear KdV equation with critical lengths. J Eur Math Soc 6(3):367–398 13. Curtain RF, Pritchard AJ (1978) Infinite Dimensional Linear Systems Theory. Springer, New York 14. Dolecky S, Russell DL (1977) A general theory of observation and control. SIAM J Control Optim 15:185–220 15. Fliess M, Lévine J, Martin PH, Rouchon P (1995) Flatness and defect of nonlinear systems: introductory theory and examples. Intern J Control 31:1327–1361 16. Hautus MLJ (1969) Controllability and observability conditions for linear autonomous systems. Ned Akad Wetenschappen Proc Ser A 72:443–448 17. Hermes H (1982) Control systems which generate decomposable Lie algebras. J Differ Equ 44:166–187 18. Hermes H, Kawski M (1986) Local controllability of a singleinput, affine system. In: Proc. 7th Int. Conf. Nonlinear Analysis. Dallas 19. Hirschorn R, Lewis AD (2004) High-order variations for families of vector fields. SIAM J Control Optim 43:301–324 20. Kalman RE (1960) Contribution to the theory of optimal control. Bol Soc Mat Mex 5:102–119 21. Kawski M (1989) Stabilization of nonlinear systems in the plane. Syst Control Lett 12:169–175 22. Kawski M (1990) High-order small time local controllability. In: Sussmann HJ (ed) Nonlinear controllability and optimal control. Textbooks Pure Appl. Math., vol 113. Dekker, New York, pp 431–467 23. Khodja FA, Benabdallah A, Dupaix C, Kostin I (2005) Nullcontrollability of some systems of parabolic type by one control force. ESAIM Control Optim Calc Var 11(3):426–448 24. Lions JL (1988) Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. In: Recherches en Mathématiques Appliquées, Tome 1, vol 8. Masson, Paris

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25. Liu K (1997) Locally distributed control and damping for the conservative systems. SIAM J Control Optim 35:1574– 1590 26. Lobry C (1970) Contrôlabilité des systèmes non linéaires. SIAM J Control 8:573–605 27. Martin PH, Murray RM, Rouchon P (2002) Flat systems. Mathematical Control Theory, Part 1,2 (Trieste 2001). In: ICTP Lect. Notes, vol VIII. Abdus Salam Int Cent Theoret Phys, Trieste 28. Rashevski PK (1938) About connecting two points of complete nonholonomic space by admissible curve. Uch Zapiski ped inst Libknexta 2:83–94 29. Rosier L (1997) Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain. ESAIM: Control Optim Calc Var 2:33–55 30. Rosier L (2007) A Survey of controllability and stabilization results for partial differential equations. J Eur Syst Autom (JESA) 41(3–4):365–412 31. Samson C (1991) Velocity and torque feedback control of a nonholonomic cart. In: de Witt C (ed) Proceedings of the International workshop on nonlinear and adaptative control: Issues in robotics, Grenoble, France nov. 1990. Lecture Notes in Control and Information Sciences, vol 162. Springer, Berlin, pp 125–151 32. Silverman LM, Meadows HE (1967) Controllability and observability in time-variable linear systems. SIAM J Control 5: 64–73 33. Sontag ED, Sussmannn H (1980) Remarks on continuous feedbacks. In: Proc. IEEE Conf. Decision and Control, (Albuquerque 1980). pp 916–921

34. Stefani G (1985) Local controllability of nonlinear systems: An example. Syst Control Lett 6:123–125 35. Sussmann H (1987) A general theorem on local controllability. SIAM J Control Optim 25:158–194 36. Sussmann H, Jurdjevic V (1972) Controllability of nonlinear systems. J Differ Equ 12:95–116 37. Tret’yak AI (1991) On odd-order necessary conditions for optimality in a time-optimal control problem for systems linear in the control. Math USSR Sbornik 79:47–63 38. Zuazua E (2005) Propagation, observation and control of waves approximated by finite difference methods. SIAM Rev 47(2):197–243

Books and Reviews Agrachev AA, Sachkov YL (2004) Control theory from the geometric viewpoint. In: Encyclopaedia of Mathematical Sciences, vol 87. Springer, Berlin Bacciotti A (1992) Local stabilization of nonlinear control systems. In: Ser. Adv. Math. Appl. Sci, vol 8. World Scientific, River Edge Bacciotti A, Rosier L (2005) Liapunov functions and stability in control theory. Springer, Berlin Isidori A (1989) Nonlinear control systems. Springer, Berlin Nijmeijer H, van der Schaft AJ (1990) Nonlinear dynamical control systems. Springer, New York Sastry S (1999) Nonlinear systems. Springer, New York Sontag E (1990) Mathematical control theory. Springer, New York Zabczyk J (1992) Mathematical control theory: An introduction. Birkhäuser, Boston

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to ARMIN BUNDE1 , SHLOMO HAVLIN2 1 Institut für Theoretische Physik, Gießen, Germany 2 Institute of Theoretical Physics, Bar-Ilan-University, Ramat Gan, Israel Article Outline Definition of the Subject Deterministic Fractals Random Fractal Models How to Measure the Fractal Dimension Self-Affine Fractals Long-Term Correlated Records Long-Term Correlations in Financial Markets and Seismic Activity Multifractal Records Acknowledgments Bibliography Definition of the Subject In this chapter we present some definitions related to the fractal concept as well as several methods for calculating the fractal dimension and other relevant exponents. The purpose is to introduce the reader to the basic properties of fractals and self-affine structures so that this book will be self contained. We do not give references to most of the original works, but, we refer mostly to books and reviews on fractal geometry where the original references can be found. Fractal geometry is a mathematical tool for dealing with complex systems that have no characteristic length scale. A well-known example is the shape of a coastline. When we see two pictures of a coastline on two different scales, with 1 cm corresponding for example to 0.1 km or 10 km, we cannot tell which scale belongs to which picture: both look the same, and this features characterizes also many other geographical patterns like rivers, cracks, mountains, and clouds. This means that the coastline is scale invariant or, equivalently, has no characteristic length scale. Another example are financial records. When looking at a daily, monthly or annual record, one cannot tell the difference. They all look the same. Scale-invariant systems are usually characterized by noninteger (“fractal”) dimensions. The notion of noninteger dimensions and several basic properties of fractal objects were studied as long ago as the last century by Georg Cantor, Giuseppe Peano, and David Hilbert, and in

Fractal Geometry, A Brief Introduction to, Figure 1 The Dürer pentagon after five iterations. For the generating rule, see Fig. 8. The Dürer pentagon is in blue, its external perimeter is in red, Courtesy of M. Meyer

the beginning of this century by Helge von Koch, Waclaw Sierpinski, Gaston Julia, and Felix Hausdorff. Even earlier traces of this concept can be found in the study of arithmetic-geometric averages by Carl Friedrich Gauss about 200 years ago and in the artwork of Albrecht Dürer (see Fig. 1) about 500 years ago. Georg Friedrich Lichtenberg discovered, about 230 years ago, fractal discharge patterns. He was the first to describe the observed self-similarity of the patterns: A part looks like the whole. Benoit Mandelbrot [1] showed the relevance of fractal geometry to many systems in nature and presented many important features of fractals. For further books and reviews on fractals see [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. Before introducing the concept of fractal dimension, we should like to remind the reader of the concept of dimension in regular systems. It is well known that in regular systems (with uniform density) such as long wires, large thin plates, or large filled cubes, the dimension d characterizes how the mass M(L) changes with the linear size L of the system. If we consider a smaller part of the system of linear size bL (b < 1), then M(bL) is decreased by a factor of bd , i. e., M(bL) D b d M(L) :

(1)

The solution of the functional equation (1) is simply M(L) D ALd . For the long wire the mass changes linearly with b, i. e., d D 1. For the thin plates we obtain d D 2, and for the cubes d D 3; see Fig. 2.

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Fractal Geometry, A Brief Introduction to, Figure 2 Examples of regular systems with dimensions d D 1, d D 2, and dD3

Next we consider fractal objects. Here we distinguish between deterministic and random fractals. Deterministic fractals are generated iteratively in a deterministic way, while random fractals are generated using a stochastic process. Although fractal structures in nature are random, it is useful to study deterministic fractals where the fractal properties can be determined exactly. By studying deterministic fractals one can gain also insight into the fractal properties of random fractals, which usually cannot be treated rigorously. Deterministic Fractals In this section, we describe several examples of deterministic fractals and use them to introduce useful fractal concepts such as fractal and chemical dimension, self similarity, ramification, and fractal substructures (minimum path, external perimeter, backbone, and red bonds).

One of the most common deterministic fractals is the Koch curve. Figure 3 shows the first n D 4 iterations of this fractal curve. By each iteration the length of the curve is increased by a factor of 4/3. The mathematical fractal is defined in the limit of infinite iterations, n ! 1, where the total length of the curve approaches infinity. The dimension of the curve can be obtained as for regular objects. From Fig. 3 we notice that, if we decrease the linear size by a factor of b D 1/3, the total length (mass) of the curve decreases by a factor of 1/4, i. e., 1  3 L D

1 4

M(L) :

d D log 4/ log 3. For such non-integer dimensions Mandelbrot coined the name “fractal dimension” and those objects described by a fractal dimension are called fractals. Thus, to include fractal structures, Eq. (1) is generalized by M(bL) D b d f M(L) ;

(2)

This feature is very different from regular curves, where the length of the object decreases proportional to the linear scale. In order to satisfy Eqs. (1) and (2) we are led to introduce a noninteger dimension, satisfying 1/4 D (1/3)d , i. e.,

(3)

and M(L) D ALd f ;

The Koch Curve

M

Fractal Geometry, A Brief Introduction to, Figure 3 The first iterations of the Koch curve. The fractal dimension of the Koch curve is df D log 4/ log 3

(4)

where df is the fractal dimension. When generating the Koch curve and calculating df , we observe the striking property of fractals – the property of self-similarity. If we examine the Koch curve, we notice that there is a central object in the figure that is reminiscent of a snowman. To the right and left of this central snowman there are two other snowmen, each being an exact reproduction, only smaller by a factor of 1/3. Each of the smaller snowmen has again still smaller copies (by 1/3) of itself to the right and to the left, etc. Now, if we take any such triplet of snowmen (consisting of 1/3m of the curve), for any m, and magnify it by 3m , we will obtain exactly the original Koch curve. This property of self-similarity or scale invariance is the basic feature of all deterministic and random fractals: if we take a part of a fractal and magnify it by the same magnification factor in all directions, the magnified picture cannot be distinguished from the original.

Fractal Geometry, A Brief Introduction to

For the Koch curve as well as for all deterministic fractals generated iteratively, Eqs. (3) and (4) are of course valid only for length scales L below the linear size L0 of the whole curve (see Fig. 3). If the number of iterations n is finite, then Eqs. (3) and (4) are valid only above a lower cut off length Lmin , Lmin D L0 /3n for the Koch curve. Hence, for a finite number of iterations, there exist two cut-off length scales in the system, an upper cut-off Lmax D L0 representing the total linear size of the fractal, and a lower cut-off Lmin . This feature of having two characteristic cutoff lengths is shared by all fractals in nature. An interesting modification of the Koch curve is shown in Fig. 4, which demonstrates that the chemical distance is an important concept for describing structural properties of fractals (for a review see, for example, [16] and Chap. 2 in [13]). The chemical distance ` is defined as shortest path on the fractal between two sites of the fractal. In analogy to the fractal dimension df that characterizes how the mass of a fractal scales with (air) distance L, we introduce the chemical dimension d` in order to characterize how the mass scales with the chemical distance `, M(b`) D b d ` M(`) ;

or M(`) D B`d ` :

(5)

From Fig. 4 we see that if we reduce ` by a factor of 5, the mass of the fractal within the reduced chemical distance is reduced by a factor of 7, i. e., M(1/5 `)D 1/7 M(`), yielding d` D log 7/ log 5 Š 1:209. Note that the chemical dimension is smaller than the fractal dimension df D log 7/ log 4 Š 1:404, which follows from M(1/4 L) D 1/7 M(L).

Fractal Geometry, A Brief Introduction to, Figure 4 The first iterations of a modified Koch curve, which has a nontrivial chemical distance metric

The structure of the shortest path between two sites represents an interesting fractal by itself. By definition, the length of the path is the chemical distance `, and the fractal dimension of the shortest path, dmin , characterizes how ` scales with (air) distance L. Using Eqs. (4) and (5), we obtain ` Ld f /d `  Ld min ;

(6)

from which follows dmin D df /d` . For our example we find that dmin D log 5/ log 4 Š 1:161. For the Koch curve, as well as for any linear fractal, one simply has d` D 1 and hence dmin D df . Since, by definition, dmin  1, it follows that d`  df for all fractals. The Sierpinski Gasket, Carpet, and Sponge Next we discuss the Sierpinski fractal family: the “gasket”, the “carpet”, and the “sponge”. The Sierpinski Gasket The Sierpinski gasket is generated by dividing a full triangle into four smaller triangles and removing the central triangle (see Fig. 5). In the following iterations, this procedure is repeated by dividing each of the remaining triangles into four smaller triangles and removing the central triangles. To obtain the fractal dimension, we consider the mass of the gasket within a linear size L and compare it with the mass within 1/2 L. Since M(1/2 L) D 1/3 M(L), we have df D log 3/ log 2 Š 1:585. It is easy to see that d` D df and dmin D 1.

Fractal Geometry, A Brief Introduction to, Figure 5 The Sierpinski gasket. The fractal dimension of the Sierpinski gasket is df D log 3/ log 2

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Fractal Geometry, A Brief Introduction to, Figure 6 A Sierpinski carpet with n D 5 and k D 9. The fractal dimension of this structure is df D log 16/ log 5

The Sierpinski Carpet The Sierpinski carpet is generated in close analogy to the Sierpinski gasket. Instead of starting with a full triangle, we start with a full square, which we divide into n2 equal squares. Out of these squares we choose k squares and remove them. In the next iteration, we repeat this procedure by dividing each of the small squares left into n2 smaller squares and removing those k squares that are located at the same positions as in the first iteration. This procedure is repeated again and again. Figure 6 shows the Sierpinski carpet for n D 5 and the specific choice of k D 9. It is clear that the k squares can be chosen in many different ways, and the fractal structures will all look very different. However, since M(1/n L) D 1/(n2  k) M(L) it follows that df D log(n2  k)/ log n, irrespective of the way the k squares are chosen. Similarly to the gasket, we have d` D df and hence dmin D 1. In contrast, the external perimeter (“hull”, see also Fig. 1) of the carpet and its fractal dimension dh depend strongly on the way the squares are chosen. The hull consists of those sites of the cluster, which are adjacent to empty sites and are connected with infinity via empty sites. In our example, see Fig. 6, the hull is a fractal with the fractal dimension dh D log 9/ log 5 Š 1:365. On the other hand, if a Sierpinski gasket is constructed with the k D 9 squares chosen from the center, the external perimeter stays smooth and dh D 1. Although the rules for generating the Sierpinski gasket and carpet are quite similar, the resulting fractal structures belong to two different classes, to finitely ramified and infinitely ramified fractals. A fractal is called finitely ramified if any bounded subset of the fractal can be isolated by cutting a finite number of bonds or sites. The Sierpinski

Fractal Geometry, A Brief Introduction to, Figure 7 The Sierpinski sponge (third iteration). The fractal dimension of the Sierpinski sponge is df D log 20/ log 3

gasket and the Koch curve are finitely ramified, while the Sierpinski carpet is infinitely ramified. For finitely ramified fractals like the Sierpinski gasket many physical properties, such as conductivity and vibrational excitations, can be calculated exactly. These exact solutions help to provide insight onto the anomalous behavior of physical properties on fractals, as was shown in Chap. 3 in [13]. The Sierpinski Sponge The Sierpinski sponge shown in Fig. 7 is constructed by starting from a cube, subdividing it into 3  3  3 D 27 smaller cubes, and taking out the central small cube and its six nearest neighbor cubes. Each of the remaining 20 small cubes is processed in the same way, and the whole procedure is iterated ad infinitum. After each iteration, the volume of the sponge is reduced by a factor of 20/27, while the total surface area increases. In the limit of infinite iterations, the surface area is infinite, while the volume vanishes. Since M(1/3 L) D 1/20 M(L), the fractal dimension is df D log 20/ log 3 Š 2:727. We leave it to the reader to prove that both the fractal dimension dh of the external surface and the chemical dimension d` is the same as the fractal dimension df . Modification of the Sierpinski sponge, in analogy to the modifications of the carpet can lead to fractals, where the fractal dimension of the hull, dh , differs from df . The Dürer Pentagon Five-hundred years ago the artist Albrecht Dürer designed a fractal based on regular pentagons, where in each iteration each pentagon is divided into six smaller pentagons and five isosceles triangles, and the triangles are removed (see Fig. 8). In each triangle, the ratio of the larger side to

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 8 The first iterations of the Dürer pentagon. The fractal dimension of the Dürer pentagon is df D log 6/ log(1 C g) Fractal Geometry, A Brief Introduction to, Figure 10 The first iterations of the triadic Cantor set. The fractal dimension of this Cantor set is df D log 2/ log 3

Fig. 10). We divide a unit interval [0; 1] into three equal intervals and remove the central one. In each following iteration, each of the remaining intervals is treated in this way. In the limit of n D 1 iterations one obtains a set of points. Since M(1/3 L) D 1/2 M(L), we have df D log 2/ log 3 Š 0:631, which is smaller than one. In chaotic systems, strange fractal attractors occur. The simplest strange attractor is the Cantor set. It occurs, for example, when considering the one-dimensional logistic map x tC1 D x t (1  x t ) :

Fractal Geometry, A Brief Introduction to, Figure 9 The first iterations of the David fractal. The fractal dimension of the David fractal is df D log 6/ log 3

the smaller side is the famous proportio divina or golden p ratio, g  1/(2 cos 72ı )  (1 C 5)/2. Hence, in each iteration the sides of the pentagons are reduced by 1 C g. Since M(L/(1 C g)) D 1/6 M(L), the fractal dimension of the Dürer pentagon is df D log 6/ log(1 C g) Š 1:862. The external perimeter of the fractal (see Fig. 1) forms a fractal curve with dh D log 4/ log(1 C g). A nice modification of the Dürer pentagon is a fractal based on regular hexagons, where in each iteration one hexagon is divided into six smaller hexagons, six equilateral triangles, and a David-star in the center, and the triangles and the David-star are removed (see Fig. 9). We leave it as an exercise to the reader to show that df D log 6/ log 3 and dh D log 4/ log 3. The Cantor Set Cantor sets are examples of disconnected fractals (fractal dust). The simplest set is the triadic Cantor set (see

(7)

The index t D 0; 1; 2; : : : represents a discrete time. For 0    4 and x0 between 0 and 1, the trajectories xt are bounded between 0 and 1. The dynamical behavior of xt for t ! 1 depends on the parameter . Below 1 D 3, only one stable fixed-point exists to which xt is attracted. At 1 , this fixed-point becomes unstable and bifurcates into two new stable fixed-points. At large times, the trajectories move alternately between both fixed-points, p and the motion is periodic with period 2. At 2 D 1C 6 Š 3:449 each of the two fixed-points bifurcates into two new stable fix points and the motion becomes periodic with period 4. As  is increased, further bifurcation points n occur, with periods of 2n between n and nC1 . For large n, the differences between nC1 and n become smaller and smaller, according to the law nC1  n D (n  n1 )/ı, where ı Š 4:6692 is the socalled Feigenbaum constant. The Feigenbaum constant is “universal”, since it applies to all nonlinear “single-hump” maps with a quadratic maximum [17]. At 1 Š 3:569 945 6, an infinite period occurs, where the trajectories xt move in a “chaotic” way between the infinite attractor points. These attractor points define the strange attractor, which forms a Cantor set with a fractal dimension df Š 0:538 [18]. For a further discussion of strange attractors and chaotic dynamics we refer to [3,8,9].

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Fractal Geometry, A Brief Introduction to, Figure 11 Three generations of the Mandelbrot–Given fractal. The fractal dimension of the Mandelbrot–Given fractal is df D log 8/ log 3

The Mandelbrot–Given Fractal This fractal was suggested as a model for percolation clusters and its substructures (see Sect. 3.4 and Chap. 2 in [13]). Figure 11 shows the first three generations of the Mandelbrot–Given fractal [19]. At each generation, each segment of length a is replaced by 8 segments of length a/3. Accordingly, the fractal dimension is df D log 8/ log 3 Š 1:893, which is very close to df D 91/46 Š 1:896 for percolation in two dimensions. It is easy to verify that d` D df , and therefore dmin D 1. The structure contains loops, branches, and dangling ends of all length scales. Imagine applying a voltage difference between two sites at opposite edges of a metallic Mandelbrot–Given fractal: the backbone of the fractal consists of those bonds which carry the electric current. The dangling ends are those parts of the cluster which carry no current and are connected to the backbone by a single bond only. The red bonds (or singly connected bonds) are those bonds that carry the total current; when they are cut the current flow stops. The blobs, finally, are those parts of the backbone that remain after the red bonds have been removed. The backbone of this fractal can be obtained easily by eliminating the dangling ends when generating the fractal (see Fig. 12). It is easy to see that the fractal dimension of the backbone is dB D log 6/ log 3 Š 1:63. The red bonds are all located along the x axis of the figure and form a Cantor set with the fractal dimension dred D log 2/ log 3 Š 0:63. Julia Sets and the Mandelbrot Set A complex version of the logistic map (7) is z tC1 D z2t C c ;

(8)

Fractal Geometry, A Brief Introduction to, Figure 12 The backbone of the Mandelbrot–Given fractal, with the red bonds shown in bold

where both the trajectories zt and the constant c are complex numbers. The question is: if a certain c-value is given, for example c D 1:5652  i1:03225, for which initial values z0 are the trajectories zt bounded? The set of those values forms the filled-in Julia set, and the boundary points of them form the Julia set. To clarify these definitions, consider the simple case c D 0. For jz0 j > 1, zt tends to infinity, while for jz0 j < 1, zt tends to zero. Accordingly, the filled-in Julia set is the set of all points jz0 j  1, the Julia set is the set of all points jz0 j D 1. In general, points on the Julia set form a chaotic motion on the set, while points outside the Julia set move away from the set. Accordingly, the Julia set can be regarded as a “repeller” with respect to Eq. (8). To generate the Julia set, it is thus practical to use the inverted transformation p z t D ˙ z tC1  c ; (9) start with an arbitrarily large value for t C 1, and go back-

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 13 Julia sets for a c D i and b c D 0:11031  i0:67037. After [9]

ward in time. By going backward in time, even points far away from the Julia set are attracted by the Julia set. For obtaining the Julia set for a given value of c, one starts with some arbitrary value for z tC1 , for example, z tC1 D 2. To obtain zt , we use Eq. (9), and determine the sign randomly. This procedure is continued to obtain z t1 , z t2 , etc. By disregarding the initial points, e. g., the first 1000 points, one obtains a good approximation of the Julia set. The Julia sets can be connected (Fig. 13a) or disconnected (Fig. 13b) like the Cantor sets. The self-similarity of the pictures is easy to see. The set of c values that yield connected Julia sets forms the famous Mandelbrot set. It has been shown by Douady and Hubbard [20] that the Mandelbrot set is identical to that set of c values for which zt converges starting from the initial point z0 D 0. For a detailed discussion with beautiful pictures see [10] and Chaps. 13 and 14 in [3]. Random Fractal Models In this section we present several random fractal models that are widely used to mimic fractal systems in nature. We begin with perhaps the simplest fractal model, the random walk. Random Walks Imagine a random walker on a square lattice or a simple cubic lattice. In one unit of time, the random walker advances one step of length a to a randomly chosen nearest neighbor site. Let us assume that the walker is unwinding a wire, which he connects to each site along his way. The length (mass) M of the wire that connects the random walker with his starting point is proportional to the number of steps n (Fig. 14) performed by the walker. Since for a random walk in any d-dimensional space the mean end-to-end distance R is proportional to n1/2 (for a simple derivation see e. g., Chap. 3 in [13], it follows that

Fractal Geometry, A Brief Introduction to, Figure 14 a A normal random walk with loops. b A random walk without loops

M R2 . Thus Eq. (4) implies that the fractal dimension of the structure formed by this wire is df D 2, for all lattices. The resulting structure has loops, since the walker can return to the same site. We expect the chemical dimension d` to be 2 in d D 2 and to decrease with increasing d, since. Loops become less relevant. For d  4 we have d` D 1. If we assume, however, that there is no contact between sections of the wire connected to the same site (Fig. 14b), the structure is by definition linear, i. e., d` D 1 for all d. For more details on random walks and its relation to Brownian motion, see Chap. 5 in [15] and [21]. Self-Avoiding Walks Self-avoiding walks (SAWs) are defined as the subset of all nonintersecting random walk configurations. An example is shown in Fig. 15a. As was found by Flory in 1944 [22], the end-to-end distance of SAWs scales with the number of steps n as R n ;

(10)

with  D 3/(d C 2) for d  4 and  D 1/2 for d > 4. Since n is proportional to the mass of the chain, it follows from Eq. (4) that df D 1/. Self-avoiding walks serve as models for polymers in solution, see [23]. Subsets of SAWs do not necessarily have the same fractal dimension. Examples are the kinetic growth walk (KGW) [24] and the smart growth walk (SGW) [25], sometimes also called the “true” or “intelligent” self-avoiding walk. In the KGW, a random walker can only step on those sites that have not been visited before. Asymptotically, after many steps n, the KGW has the same fractal dimension as SAWs. In d D 2, however, the asymptotic regime is difficult to reach numerically, since the random walker can be trapped with high probability (see Fig. 15b). A related structure is the hull of a random walk in d D 2. It has been conjectured by Mandelbrot [1] that the fractal dimension of the hull is dh D 4/3, see also [26].

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Fractal Geometry, A Brief Introduction to, Figure 16 a Generation of a DLA cluster. The inner release radius is usually a little larger than the maximum distance of a cluster site from the center, the outer absorbing radius is typically 10 times this distance. b A typical off-lattice DLA cluster of 10,000 particles

Fractal Geometry, A Brief Introduction to, Figure 15 a A typical self-avoiding walk. b A kinetic growth walk after 8 steps. The available sites are marked by crosses. c A smart growth walk after 19 steps. The only available site is marked by “yes”

In the SGW, the random walker avoids traps by stepping only at those sites from which he can reach infinity. The structure formed by the SGW is more compact and characterized by df D 7/4 in d D 2 [25]. Related structures with the same fractal dimension are the hull of percolation clusters (see also Sect. “Percolation”) and diffusion fronts (for a detailed discussion of both systems see also Chaps. 2 and 7 in [13]). Kinetic Aggregation The simplest model of a fractal generated by diffusion of particles is the diffusion-limited aggregation (DLA) model, which was introduced by Witten and Sander in 1981 [27]. In the lattice version of the model, a seed particle is fixed at the origin of a given lattice and a second particle is released from a circle around the origin. This particle performs a random walk on the lattice. When it comes to a nearest neighbor site of the seed, it sticks and a cluster (aggregate) of two particles is formed. Next, a third particle is released from the circle and performs a random walk. When it reaches a neighboring site of the aggregate, it sticks and becomes part of the cluster. This procedure is repeated many times until a cluster of the desired number of sites is generated. For saving computational time it is convenient to eliminate particles that have diffused too far away from the cluster (see Fig. 16).

In the continuum (off-lattice) version of the model, the particles have a certain radius a and are not restricted to diffusing on lattice sites. At each time step, the length ( a) and the direction of the step are chosen randomly. The diffusing particle sticks to the cluster, when its center comes within a distance a of the cluster perimeter. It was found numerically that for off-lattice DLA, df D 1:71 ˙ 0:01 in d D 2 and df D 2:5 ˙ 0:1 in d D 3 [28,29]. These results may be compared with the mean field result df D (d 2 C 1)/(d C 1) [30]. For a renormalization group approach, see [31] and references therein. The chemical dimension d` is found to be equal to df [32]. Diffusion-limited aggregation serves as an archetype for a large number of fractal realizations in nature, including viscous fingering, dielectric breakdown, chemical dissolution, electrodeposition, dendritic and snowflake growth, and the growth of bacterial colonies. For a detailed discussion of the applications of DLA we refer to [5,13], and [29]. Models for the complex structure of DLA have been developed by Mandelbrot [33] and Schwarzer et al. [34]. A somewhat related model for aggregation is the cluster-cluster aggregation (CCA) [35]. In CCA, one starts from a very low concentration of particles diffusing on a lattice. When two particles meet, they form a cluster of two, which can also diffuse. When the cluster meets another particle or another cluster, a larger cluster is formed. In this way, larger and larger aggregates are formed. The structures are less compact than DLA, with df Š 1:4 in d D 2 and df Š 1:8 in d D 3. CCA seems to be a good model for smoke aggregates in air and for gold colloids. For a discussion see Chap. 8 in [13]. Percolation Consider a square lattice, where each site is occupied randomly with probability p or empty with probability 1  p.

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 17 Square lattice of size 20  20. Sites have been randomly occupied with probability p (p D 0:20, 0.59, 0.80). Sites belonging to finite clusters are marked by full circles, while sites on the infinite cluster are marked by open circles

At low concentration p, the occupied sites are either isolated or form small clusters (Fig. 17a). Two occupied sites belong to the same cluster, if they are connected by a path of nearest neighbor occupied sites. When p is increased, the average size of the clusters increases. At a critical concentration pc (also called the percolation threshold) a large cluster appears which connects opposite edges of the lattice (Fig. 17b). This cluster is called the infinite cluster, since its size diverges when the size of the lattice is increased to infinity. When p is increased further, the density of the infinite cluster increases, since more and more sites become part of the infinite cluster, and the average size of the finite clusters decreases (Fig. 17c). The percolation transition is characterized by the geometrical properties of the clusters near pc . The probability P1 that a site belongs to the infinite cluster is zero below pc and increases above pc as P1 (p  pc )ˇ :

(11)

The linear size of the finite clusters, below and above pc , is characterized by the correlation length . The correlation length is defined as the mean distance between two sites on the same finite cluster and represents the characteristic length scale in percolation. When p approaches pc ,  increases as  jp  pc j ;

(12)

with the same exponent  below and above the threshold. While pc depends explicitly on the type of the lattice (e. g., pc Š 0:593 for the square lattice and 1/2 for the triangular lattice), the critical exponents ˇ and  are universal and depend only on the dimension d of the lattice, but not on the type of the lattice. Near pc , on length scales smaller than , both the infinite cluster and the finite clusters are self-similar. Above pc , on length scales larger than , the infinite cluster can be regarded as an homogeneous system which is composed of

Fractal Geometry, A Brief Introduction to, Figure 18 A large percolation cluster in d D 3. The colors mark the topological distance from an arbitrary center of the cluster in the middle of the page. Courtesy of M. Meyer

many unit cells of size . Mathematically, this can be summarized as ( r df ; r   ; M(r) d (13) r ; r: The fractal dimension df can be related to ˇ and : df D d 

ˇ : 

(14)

Since ˇ and  are universal exponents, df is also universal. One obtains df D 91/48 in d D 2 and df Š 2:5 in d D 3. The chemical dimension d` is smaller than df , d` Š 1:15 in d D 2 and d` Š 1:33 in d D 3. A large percolation cluster in d D 3 is shown in Fig. 18. Interestingly, a percolation cluster is composed of several fractal sub-structures such as the backbone, dangling ends, blobs, External perimeter, and the red bonds, which are all described by different fractal dimensions. The percolation model has found applications in physics, chemistry, and biology, where occupied and empty sites may represent very different physical, chemical, or biological properties. Examples are the physics of two component systems (the random resistor, magnetic or superconducting networks), the polymerization process in chemistry, and the spreading of epidemics and forest fires. For reviews with a comprehensive list of references, see Chaps. 2 and 3 of [13] and [36,37,38].

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How to Measure the Fractal Dimension One of the most important “practical” problems is to determine the fractal dimension df of either a computer generated fractal or a digitized fractal picture. Here we sketch the two most useful methods: the “sandbox” method and the “box counting” method. The Sandbox Method To determine df , we first choose one site (or one pixel) of the fractal as the origin for n circles of radii R1 < R2 <    < R n , where Rn is smaller than the radius R of the fractal, and count the number of points (pixels) M1 (R i ) within each circle i. (Sometimes, it is more convenient to choose n squares of side length L1 . . . Ln instead of the circles.) We repeat this procedure by choosing randomly many other (altogether m) pixels as origins for the n circles and determine the corresponding number of points M j (R i ), j D 2; 3; : : : ; m within each circle (see Fig. 19a). We obtain the mean number of points M(R i ) within each P circle by averaging, M(R i ) D 1/m m jD1 M j (R i ), and plot M(R i ) versus Ri in a double logarithmic plot. The slope of the curve, for large values of Ri , determines the fractal dimension. In order to avoid boundary effects, the radii must be smaller than the radius of the fractal, and the centers of the circles must be chosen well inside the fractal, so that the largest circles will be well within the fractal. In order to obtain good statistics, one has either to take a very large fractal cluster with many centers of circles or many realizations of the same fractal. The Box Counting Method We draw a grid on the fractal that consists of N12 squares, and determine the number of squares S(N1 ) that are

needed to cover the fractal (see Fig. 19b). Next we choose finer and finer grids with N12 < N22 < N32 <    < 2 squares and calculate the corresponding numbers of Nm squares S(N1 ) . . . S(N m ) needed to cover the fractal. Since S(N) scales as S(N) N d f ;

we obtain the fractal dimension by plotting S(N) versus 1/N in a double logarithmic plot. The asymptotic slope, for large N, gives df . Of course, the finest grid size must be larger than the pixel size, so that many pixels can fall into the smallest square. To improve statistics, one should average S(N) over many realizations of the fractal. For applying this method to identify self-similarity in real networks, see Song et al. [39]. Self-Affine Fractals The fractal structures we have discussed in the previous sections are self-similar: if we cut a small piece out of a fractal and magnify it isotropically to the size of the original, both the original and the magnification look the same. By magnifying isotropically, we have rescaled the x, y, and z axis by the same factor. There exist, however, systems that are invariant only under anisotropic magnifications. These systems are called self-affine [1]. A simple model for a self-affine fractal is shown in Fig. 20. The structure is invariant under the anisotropic magnification x ! 4x, y ! 2y. If we cut a small piece out of the original picture (in the limit of n ! 1 iterations), and rescale the x axis by a factor of four and the y axis by a factor of two, we will obtain exactly the original structure. In other words, if we describe the form of the curve in Fig. 20 by the function F(x), this function satisfies the equation F(4x) D 2F(x) D 41/2 F(x). In general, if a self-affine curve is scale invariant under the transformation x ! bx, y ! ay, we have F(bx) D aF(x)  b H F(x) ;

Fractal Geometry, A Brief Introduction to, Figure 19 Illustrations for determining the fractal dimension: a the sandbox method, b the box counting method

(15)

(16)

where the exponent H D log a/ log b is called the Hurst exponent [1]. The solution of the functional equation (16) is simply F(x) D Ax H . In the example of Fig. 20, H D 1/2. Next we consider random self-affine structures, which are used as models for random surfaces. The simplest structure is generated by a one-dimensional random walk, where the abscissa is the time axis and the ordinate is the P displacement Z(t) D tiD1 e i of the walker from its starting point. Here, e i D ˙1 is the unit step made by the random walker at time t. Since different steps of the random walker are uncorrelated, he i e j i D ı i j , it follows that the

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 20 A simple deterministic model of a self-affine fractal

root mean square displacement F(t)  hZ 2 (t)i1/2 D t 1/2 , and the Hurst exponent of the structure is H D 1/2. Next we assume that different steps i and j are correlated in such a way that he i e j i D bji  jj , 1 >  0. To see how the Hurst exponent depends on , we have to Pt evaluate again hZ 2 (t)i D i; j he i e j i. For calculating the double sum it is convenient to introduce the Fourier transP form of ei , e! D (1/˝)1/2 ˝ l D1 e l exp(i! l), where ˝ is the number of sites in the system. It is easy to verify that hZ 2 (t)i can be expressed in terms of the power spectrum he! e! i [40]: hZ 2 (t)i D

1 X he! e! ij f (!; t)j2 ; ˝ !

(17a)

where f (!; t) 

ei!(tC1)  1 : ei!  1

Since the power spectrum scales as he! e! i ! (1 ) ;

(17b)

the integration of (17a) yields, for large t, hZ 2 (t)i t 2 :

(17c)

Therefore, the Hurst exponent is H D (2 )/2. According to Eq. (17c), for 0 < < 1, hZ 2 (t)i increases faster in time than the uncorrelated random walk. The long-range correlated random walks were called fractional Brownian motion by Mandelbrot [1].

There exist several methods to generate correlated random surfaces. We shall describe the successive random additions method [41], which iteratively generates the selfaffine function Z(x) in the unit interval 0  x  1. An alternative method that is detailed in the chapter of Jan Kantelhardt is the Fourier-filtering technique and its variants. In the n D 0 iteration, we start at the edges x D 0 and x D 1 of the unit interval and choose the values of Z(0) and Z(1) from a distribution with zero mean and variance 02 D 1 (see Fig. 21). In the n D 1 iteration, we choose the midpoint x D 1/2 and determine Z(1/2) by linear interpolation, i. e., Z(1/2) D (Z(0) C Z(1))/2, and add to all so-far calculated Z values (Z(0), Z(1/2), and Z(1)) random displacements from the same distribution as before, but with a variance 1 D (1/2)H (see Fig. 21). In the n D 2 iteration we again first choose the midpoints (x D 1/4 and x D 3/4), determine their Z values by linear interpolation, and add to all so-far calculated Z values random displacements from the same distribution as before, but with a variance 2 D (1/2)2H . In general, in the nth iteration, one first interpolates the Z values of the midpoints and then adds random displacements to all existing Z values, with variance n D (1/2)nH . The procedure is repeated until the required resolution of the surface is obtained. Figure 22 shows the graphs of three random surfaces generated this way, with H D 0:2, H D 0:5, and H D 0:8. The generalization of the successive random addition method to two dimensions is straightforward (see Fig. 21). We consider a function Z(x; y) on the unit square 0 

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Fractal Geometry, A Brief Introduction to, Figure 21 Illustration of the successive random addition method in d D 1 and d D 2. The circles mark those points that have been considered already in the earlier iterations, the crosses mark the new midpoints added at the present iteration. At each iteration n, first the Z values of the midpoints are determined by linear interpolation from the neighboring points, and then random displacements of variance n are added to all Z values

x; y  1. In the n D 0 iteration, we start with the four corners (x; y) D (0; 0); (1; 0); (1; 1); (0; 1) of the unit square and choose their Z values from a distribution with zero mean and variance 02 D 1 (see Fig. 21). In the n D 1 iteration, we choose the midpoint at (x; y) D (1/2; 1/2) and determine Z(1/2; 1/2) by linear interpolation, i. e., Z(1/2; 1/2) D (Z(0; 0)CZ(0; 1)CZ(1; 1)CZ(1; 0))/4. Then we add to all so far calculated Z-values (Z(0; 0), Z(0; 1), Z(1; 0), Z(1; 1) and Z(1/2; 1/2)) random displacements from the same p distribution as before, but with a variance 1 D (1/ 2)H (see Fig. 21). In the n D 2 iteration we again choose the midpoints of the five sites (0; 1/2), (1/2; 0), (1/2; 1) and (1; 1/2), determine their Z value by linear interpolation, and add to all so far calculated Z values random displacements from the same p distribution as before, but with a variance 2 D (1/ 2)2H . This procedure is repeated again and again, until the required resolution of the surface is obtained. At the end of this section we like to note that selfsimilar or self-affine fractal structures with features similar to those fractal models discussed above can be found in nature on all, astronomic as well as microscopic, length scales. Examples include clusters of galaxies (the fractal dimension of the mass distribution is about 1.2 [42]), the crater landscape of the moon, the distribution of earthquakes (see Chap. 2 in [15]), and the structure of coastlines, rivers, mountains, and clouds. Fractal cracks (see, for example, Chap. 5 in [13]) occur on length scales ranging from 103 km (like the San Andreas fault) to micrometers (like fractures in solid materials) [44]. Many naturally growing plants show fractal structures, examples range from trees and the roots of trees to cauliflower and broccoli. The patterns of blood vessels in the human body, the kidney, the lung, and some types of nerve cells have fractal features (see Chap. 3 in [15]). In materials sciences, fractals appear in polymers, gels, ionic glasses, aggregates, electro-deposition, rough interfaces and surfaces (see [13] and Chaps. 4 and 6 in [15]), as well as in fine particle systems [43]. In all these structures there is no characteristic length scale in the system besides the physical upper and lower cut-offs. The occurrence of self-similar or self-affine fractals is not limited to structures in real space as we will discuss in the next section. Long-Term Correlated Records

Fractal Geometry, A Brief Introduction to, Figure 22 Correlated random walks with H D 0:2, 0.5, and 0.8, generated by the successive random addition method in d D 1

Long-range dependencies as described in the previous section do not only occur in surfaces. Of great interest is longterm memory in climate, physiology and financial markets, the examples range from river floods [45,46,47,48,

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 23 Comparison of an uncorrelated and a long-term correlated record with  D 0:4. The full line is the moving average over 30 data points

49], temperatures [50,51,52,53,54], and wind fields [55] to market volatilities [56], heart-beat intervals [57,58] and internet traffic [59]. Consider a record xi of discrete numbers, where the index i runs from 1 to N. xi may be daily or annual temperatures, daily or annual river flows, or any other set of data consisting of N successive data points. We are interested in the fluctuations of the data around their (sometimes seasonal) mean value. Without loss of generality, we assume that the mean of the data is zero and the variance equal to one. In analogy to the previous section, we call the data long-term correlated, when the corresponding autocorrelation function C x (s) decays by a power law, C x (s) D hx i x iCs i 

1 N s

Ns X

x1 x iCs s  ;

(18)

iD1

where denotes the correlation exponent, 0 < < 1. Such correlations are Rnamed ‘long-term’ since the mean 1 correlation time T D 0 C x (s) ds diverges in the limit of infinitely long series where N ! 1. If the xi are uncorrelated, C x (s) D 0 for s > 0. More generally, if correlations exist up to a certain correlation time sx , then C(s) > 0 for s < s x and C(s) D 0 for s > s x . Figure 23 shows parts of an uncorrelated (left) and a long-term correlated (right) record, with D 0:4; both series have been generated by the computer. The red line is the moving average over 30 data points. For the uncorrelated data, the moving average is close to zero, while for the long-term correlated data set, the moving average can have large deviations from the mean, forming some kind of mountain valley structure. This structure is a consequence of the power-law persistence. The mountains and valleys in Fig. 23b look as if they had been generated by external trends, and one might be inclined to draw a trend-

line and to extrapolate the line into the near future for some kind of prognosis. But since the data are trend-free, only a short-term prognosis utilizing the persistence can be made, and not a longer-term prognosis, which often is the aim of such a regression analysis. Alternatively, in analogy to what we described above for self-affine surfaces, one can divide the data set in K s equidistant windows of length s and determine in each window  the squared sum !2 s X 2 F (s) D xi (19a) iD1

and detect how the average of this quantity over all winPK s F2 (s), scales with the window dows, F 2 (s) D 1/K s D1 size s. For long-term correlated data one can show that F 2 (s) scales as hZ 2 (t)i in the previous section, i. e. F 2 (s) s 2˛ ;

(19b)

where ˛ D 1  /2. This relation represents an alternative way to determine the correlation exponent . Since trends resemble long-term correlations and vice versa, there is a general problem to distinguish between trends and long-term persistence. In recent years, several methods have been developed, mostly based on the hierarchical detrended fluctuation analysis (DFAn) where longterm correlations in the presence of smooth polynomial trends of order n  1 can be detected [57,58,60] (see also  Fractal and Multifractal Time Series). In DFAn, one considers the cumulated sum (“profile”) of the xi and divides the N data points of the profile into equidistant windows of fixed length s. Then one determines, in each window, the best fit of the profile by an nth order polynomial and determines the variance around the fit. Finally, one av2 and erages these variances to obtain the mean variance F(n)

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Fractal Geometry, A Brief Introduction to, Figure 24 DFAn analysis of six temperature records, one precipitation record and two run-off records. The black curves are the DFA0 results, while the upper red curves refer to DFA1 and the lower red curves to DFA2. The blue numbers denote the asymptotic slopes of the curves

the corresponding mean standard deviation (mean fluctuation) F(n) (s). One can show that for long-term correlated trend-free data F(n) (s) scales with the window size s as F(s) in Eq. (19b), i. e., F(n) (s) s ˛ , with ˛ D 1  /2, irrespective of the order of the polynomial n. For short-term correlated records (including the case  1), the exponent is 1/2 for s above sx . It is easy to verify that trends of order k  1 in the original data are eliminated in F(k) (s) but contribute to F(k1) ; F(k2) etc., and this allows one to determine the correlation exponent in the presence of trends. For example, in the case of a linear trend, DFA0 and DFA1 (where F(0) (s) and F(1) (s) are determined) are affected by the trend and will exaggerate the asymptotic exponents ˛, while DFA2, DFA3 etc. (where F(2) (s) and F(3) (s) etc. is determined) are not affected by the trend and will show, in a double logarithmic plot, the same value of ˛, which then gives immediately the correlation exponent . When is known this way, one can try to detect the trend, but there is no unique treatment available. In recent papers [61,62,63], different kinds of analysis have been elaborated and applied to estimate trends in the temperature records of the Northern Hemisphere and Siberian locations.

Climate Records Figure 24 shows representative results of the DFAn analysis, for temperature, precipitation and run-off data. For continental temperatures, the exponent ˛ is around 0.65, while for island stations and sea surface temperatures the exponent is considerably higher. There is no crossover towards uncorrelated behavior at larger time scales. For the precipitation data, the exponent is close to 0.55, not being significantly larger than for uncorrelated records. Figure 25 shows a summary of the exponent ˛ for a large number of climate records. It is interesting to note that while the distribution of ˛-values is quite broad for run-off, sea-surface temperature, and precipitation records, the distribution is quite narrow, located around ˛ D 0:65 for continental atmospheric temperature records. For the island records, the exponent is larger. The quite universal exponent ˛ D 0:65 for continental stations can be used as an efficient test bed for climate models [62, 64,65]. The time window accessible by DFAn is typically 1/4 of the length of the record. For instrumental records, the time window is thus restricted to about 50 years. For ex-

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 25 Distribution of fluctuation exponents ˛ for several kinds of climate records (from [53,66,67])

tending this limit, one has to take reconstructed records or model data, which range up to 2000y. Both have, of course, large uncertainties, but it is remarkable that exactly the same kind of long-term correlations can be found in these data, thus extending the time scale where long-term memory exists to at least 500y [61,62]. Clustering of Extreme Events Next we consider the consequences of long-term memory on the occurrence of rare events. Understanding (and predicting) the occurrence of extreme events is one of the major challenges in science (see, e. g., [68]). An important quantity here is the time interval between successive extreme events (see Fig. 26), and by understanding the statistics of these return intervals one aims to better understand the occurrence of extreme events. Since extreme events are, by definition, very rare and the statistics of their return intervals poor, one usually studies also the return intervals between less extreme events, where the data are above some threshold q and where the statistics is better, and hopes to find some general “scaling” relations between the return intervals at low and high thresholds, which then allows one to extrapolate the results to very large, extreme thresholds (see Fig. 26).

Fractal Geometry, A Brief Introduction to, Figure 26 Illustration of the return intervals for three equidistant threshold values q1 ; q2 ; q3 for the water levels of the Nile at Roda (near Cairo, Egypt). One return interval for each threshold (quantile) q is indicated by arrows

For uncorrelated data, the return intervals are independent of each other and their probability density function (pdf) is a simple exponential, Pq (r) D (1/R q )  exp(r/R q ). In this case, all relevant quantities can be derived from the knowledge of the mean return interval Rq . Since the return intervals are uncorrelated, a sequential ordering cannot occur. There are many cases, however, where some kind of ordering has been observed where the

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hazardous events cluster, for example in the floods in Central Europe during the middle ages or in the historic water levels of the Nile river which are shown in Fig. 26 for 663y. Even by eye one can realize that the events are not distributed randomly but are arranged in clusters. A similar clustering was observed for extreme floods, winter storms, and avalanches in Central Europe (see, e. g., Figs. 4.4, 4.7, 4.10, and 4.13 in [69], Fig. 66 in [70], and Fig. 2 in [71]). The reason for this clustering is the long-term memory. Figure 27 shows Pq (r) for long-term correlated records with ˛ D 0:4 (corresponding to D 0:8), for three values of the mean return interval Rq (which is easily obtained from the threshold q and independent of the correlations). The pdf is plotted in a scaled way, i. e., R q Pq (r) as a function of r/R q . The figure shows that all three curves collapse. Accordingly, when we know the functional form of the pdf for one value of Rq , we can easily deduce its functional form also for very large Rq values which due to its poor statistics cannot be obtained directly from the data. This scaling is a very important property, since it allows one to make predictions also for rare events which otherwise are not accessible with meaningful statistics. When the data are shuffled, the long-term correlations are destroyed and the pdf becomes a simple exponential. The functional form of the pdf is a quite natural extension of the uncorrelated case. The figure suggests that ln Pq (r) (r/R q )

(20)

i. e. simple stretched exponential behavior [72,73]. For

approaching 1, the long-term correlations tend to vanish

Fractal Geometry, A Brief Introduction to, Figure 27 Probability density function of the return intervals in long-term correlated data, for three different return periods Rq , plotted in a scaled way. The full line is a stretched exponential, with exponent  D 0:4 (after [73])

and we obtain the simple exponential behavior characteristic for uncorrelated processes. For r well below Rq , however, there are deviations from the pure stretched exponential behavior. Closer inspection of the data shows that for r/R q  the decay of the pdf is characterized by a power law, with the exponent  1. This overall behavior does not depend crucially on the way the original data are distributed. In the cases shown here, the data had a Gaussian distribution, but similar results have been obtained also for exponential, power-law and log-normal distributions [74]. Indeed, the characteristic stretched exponential behavior of the pdf can also be seen in long historic and reconstructed records [73]. The form of the pdf indicates that return intervals both well below and well above their average value are considerably more frequent for long-term correlated data than for uncorrelated data. The distribution does not quantify, however, if the return intervals themselves are arranged in a correlated or in an uncorrelated fashion, and if clustering of rare events may be induced by long-term correlations. To study this question, [73] and [74] have evaluated the autocorrelation function of the return intervals in synthetic long-term correlated records. They found that also the return intervals are arranged in a long-term correlated fashion, with the same exponent as the original data. Accordingly, a large return interval is more likely to be followed by a large one than by a short one, and a small return interval is more likely to be followed by a small one than by a large one, and this leads to clustering of events above some threshold q, including extreme events. As a consequence of the long-term memory, the probability of finding a certain return interval depends on the preceding interval. This effect can be easily seen in synthetic data sets generated numerically, but not so well in climate records where the statistics is comparatively poor. To improve the statistics, we now only distinguish between two kinds of return intervals, “small” ones (below the median) and “large” ones (above the median), and determine  the mean RC q and R q of those return intervals following a large (C) or a small () return interval. Due to scal ing, RC q /R q and R q /R q are independent of q. Figure 28 shows both quantities (calculated numerically for longterm correlated Gaussian data) as a function of the correlation exponent . The lower dashed line is R q /R q , the /R . In the limit of vanishing longupper dashed line is RC q q term memory, for D 1, both quantities coincide, as ex pected. Figure 28 also shows RC q /R q and R q /R q for five climate records with different values of . One can see that the data agree very well, within the error bars, with the theoretical curves.

Fractal Geometry, A Brief Introduction to

Fractal Geometry, A Brief Introduction to, Figure 28 Mean of the (conditional) return intervals that either follow a return interval below the median (lower dashed line) or above the median (upper dashed line), as a function of the correlation exponent , for five long reconstructed and natural climate records. The theoretical curves are compared with the corresponding values of the climate records (from right to left): The reconstructed run-offs of the Sacramento River, the reconstructed temperatures of Baffin Island, the reconstructed precipitation record of New Mexico, the historic water levels of the Nile and one of the reconstructed temperature records of the Northern hemisphere (Mann record) (after [73])

Long-Term Correlations in Financial Markets and Seismic Activity The characteristic behavior of the return intervals, i. e. long-term correlations and stretched exponential decay, can also be observed in financial markets and seismic activity. It is well known (see, e. g. [56]) that the volatility of stocks and exchange rates is long-term correlated. Figure 29 shows that, as expected from the foregoing, also the return intervals between daily volatilities are long-term

correlated, with roughly the same exponent as the original data [75]. It has further been shown [75] that also the pdfs show the characteristic behavior predicted above. A further example where long-term correlations seem to play an important role, are earthquakes in certain bounded areas (e. g. California) in time regimes where the seismic activity is (quasi) stationary. It has been discovered recently by [76] that the magnitudes of earthquakes in Northern and Southern California, from 1995 until 1998, are long-term correlated with an exponent around

D 0:4, and that also the return intervals between the earthquakes are long-term correlated with the same exponent. For the given exponential distribution of the earthquake magnitudes (following the Gutenberg–Richter law), the long-term correlations lead to a characteristic dependence on the scaled variable r/R q which can explain, without any fit parameter, the previous results on the pdf of the return intervals by [77]. Multifractal Records Many records do not exhibit a simple monofractal scaling behavior, which can be accounted for by a single scaling exponent. In some cases, there exist crossover (time-) scales sx separating regimes with different scaling exponents, e. g. long-term correlations on small scales below sx and another type of correlations or uncorrelated behavior on larger scales above sx . In other cases, the scaling behavior is more complicated, and different scaling exponents are required for different parts of the series. In even more complicated cases, such different scaling behavior can be observed for many interwoven fractal subsets of the time series. In this case a multitude of scaling exponents is required for a full description of the scaling behavior, and

Fractal Geometry, A Brief Introduction to, Figure 29 Long-term correlation exponent  for the daily volatility (left) and the corresponding return intervals (right). The studied commodities are (from left to right), the S&P 500 index, six stocks (IBM, DuPont, AT&T, Kodak, General Electric, Coca-Cola) and seven currency exchange rates (US$ vs. Japanese Yen, British Pound vs. Swiss Franc, US$ vs. Swedish Krona, Danish Krone vs. Australian $, Danish Krone vs. Norwegian Krone, US$ vs. Canadian $ and US$ vs. South African $). Courtesy of Lev Muchnik

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a multifractal analysis must be applied (see, e. g., [78,79] and literature therein). To see this, it is meaningful to extend Eqs. (19a) and (19b) by considering the more general average F q (s) D

1 Ks

Ks X D1



q/2 F2 (s)

(21)

with q between 1 and C1. For q  1 the small fluctuations will dominate the sum, while for q  1 the large fluctuations are dominant. It is reasonable to assume that the q-dependent average scales with s as F q (s) s qˇ (q) ;

(22)

with ˇ(2) D ˛. Equation (22) generalizes Eq. (19b). If ˇ(q) is independent of q, we have (F q (s))1/q s ˛ , independent of q, and both large and small fluctuations scale the same. In this case, a single exponent is sufficient to characterize the record, which then is referred to as monofractal. If ˇ(q) is not identical to ˛, we have a multifractal [1,4,12]. In this case, the dependence of ˇ(q) on q characterizes the record. Instead of ˇ(q) one considers frequently the spectrum f (!) that one obtains by Legendre transform of qˇ(q), ! D d(qˇ(q))/dq, f (!) D q!  qˇ(q) C 1. In the monofractal limit we have f (!) D 1. For generating multifractal data sets, one considers mostly multiplicative random cascade processes, described, e. g., in [3,4]. In this process, the data set is obtained in an iterative way, where the length of the record doubles in each iteration. It is possible to generate random cascades with vanishing autocorrelation function (Cx (s) D 0 for s  1) or with algebraically decaying autocorrelation functions (C x (s) s ). Here we focus on a multiplicative random cascade with vanishing autocorrelation function, which is particularly interesting since it can be used as a model for the arithmetic returns (Pi  Pi1 )/Pi of daily stock closing prices Pi [80]. In the zeroth iteration n D 0, the data set (x i ) consists of one value, x1(nD0) D 1. In the nth iteration, the data x (n) i consist of 2n values that are obtained from (n) (n1) (n) x2l m2l 1 1 D x l

and (n) x2l D x (n1) m(n) l 2l ;

(23)

where the multipliers m are independent and identically distributed (i.i.d.) random numbers with zero mean and unit variance. The resulting pdf is symmetric with lognormal tails, with vanishing correlation function C x (s) for s  1.

It has been shown that in this case, the pdf of the return intervals decays by a power-law  Pq (r)

r Rq

ı(q) ;

(24)

where the exponent ı depends explicitly on Rq and seems to converge to a limiting curve for large data sets. Despite of the vanishing autocorrelation function of the original record, the autocorrelation function of the return intervals decays by a power law with a threshold-dependent exponent [80]. Obviously, these long-term correlations have been induced by the nonlinear correlations in the multifractal data set. Extracting the return interval sequence from a data set is a nonlinear operation, and thus the return intervals are influenced by the nonlinear correlations in the original data set. Accordingly, the return intervals in data sets without linear correlations are sensitive indicators for nonlinear correlations in the data records. The power-law dependence of Pq (r) can be used for an improved risk estimation. Both power-law dependencies can be observed in economic and physiological records that are known to be multifractal [81]. Acknowledgments We like to thank all our coworkers in this field, in particular Eva Koscielny-Bunde, Mikhail Bogachev, Jan Kantelhardt, Jan Eichner, Diego Rybski, Sabine Lennartz, Lev Muchnik, Kazuko Yamasaki, John Schellnhuber and Hans von Storch. Bibliography 1. Mandelbrot BB (1977) Fractals: Form, chance and dimension. Freeman, San Francisco; Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San Francisco 2. Jones H (1991) Part 1: 7 chapters on fractal geometry including applications to growth, image synthesis, and neutral net. In: Crilly T, Earschaw RA, Jones H (eds) Fractals and chaos. Springer, New York 3. Peitgen H-O, Jürgens H, Saupe D (1992) Chaos and fractals. Springer, New York 4. Feder J (1988) Fractals. Plenum, New York 5. Vicsek T (1989) Fractal growth phenomena. World Scientific, Singapore 6. Avnir D (1992) The fractal approach to heterogeneous chemistry. Wiley, New York 7. Barnsley M (1988) Fractals everywhere. Academic Press, San Diego 8. Takayasu H (1990) Fractals in the physical sciences. Manchester University Press, Manchester 9. Schuster HG (1984) Deterministic chaos – An introduction. Physik Verlag, Weinheim 10. Peitgen H-O, Richter PH (1986) The beauty of fractals. Springer, Heidelberg

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32. Meakin P, Majid I, Havlin S, Stanley HE (1984) Topological properties of diffusion limited aggregation and cluster-cluster aggregation. Physica A 17:L975 33. Mandelbrot BB (1992) Plane DLA is not self-similar; is it a fractal that becomes increasingly compact as it grows? Physica A 191:95; see also: Mandelbrot BB, Vicsek T (1989) Directed recursive models for fractal growth. J Phys A 22:L377 34. Schwarzer S, Lee J, Bunde A, Havlin S, Roman HE, Stanley HE (1990) Minimum growth probability of diffusion-limited aggregates. Phys Rev Lett 65:603 35. Meakin P (1983) Formation of fractal clusters and networks by irreversible diffusion-limited aggregation. Phys Rev Lett 51:1119; Kolb M (1984) Unified description of static and dynamic scaling for kinetic cluster formation. Phys Rev Lett 53:1653 36. Stauffer D, Aharony A (1992) Introduction to percolation theory. Taylor and Francis, London 37. Kesten H (1982) Percolation theory for mathematicians. Birkhauser, Boston 38. Grimmet GR (1989) Percolation. Springer, New York 39. Song C, Havlin S, Makse H (2005) Self-similarity of complex networks. Nature 433:392 40. Havlin S, Blumberg-Selinger R, Schwartz M, Stanley HE, Bunde A (1988) Random multiplicative processes and transport in structures with correlated spatial disorder. Phys Rev Lett 61:1438 41. Voss RF (1985) In: Earshaw RA (ed) Fundamental algorithms in computer graphics. Springer, Berlin, p 805 42. Coleman PH, Pietronero L (1992) The fractal structure of the universe. Phys Rep 213:311 43. Kaye BH (1989) A random walk through fractal dimensions. Verlag Chemie, Weinheim 44. Turcotte DL (1997) Fractals and chaos in geology and geophysics. Cambridge University Press, Cambridge 45. Hurst HE, Black RP, Simaika YM (1965) Long-term storage: An experimental study. Constable, London 46. Mandelbrot BB, Wallis JR (1969) Some long-run properties of geophysical records. Wat Resour Res 5:321–340 47. Koscielny-Bunde E, Kantelhardt JW, Braun P, Bunde A, Havlin S (2006) Long-term persistence and multifractality of river runoff records: Detrended fluctuation studies. Hydrol J 322:120–137 48. Mudelsee M (2007) Long memory of rivers from spatial aggregation. Wat Resour Res 43:W01202 49. Livina VL, Ashkenazy Y, Braun P, Monetti A, Bunde A, Havlin S (2003) Nonlinear volatility of river flux fluctuations. Phys Rev E 67:042101 50. Koscielny-Bunde E, Bunde A, Havlin S, Roman HE, Goldreich Y, Schellnhuber H-J (1998) Indication of a universal persistence law governing athmospheric variability. Phys Rev Lett 81: 729–732 51. Pelletier JD, Turcotte DL (1999) Self-affine time series: Application and models. Adv Geophys 40:91 52. Talkner P, Weber RO (2000) Power spectrum and detrended fluctuation analysis: Application to daily temperatures. Phys Rev E 62:150–160 53. Eichner JF, Koscielny-Bunde E, Bunde A, Havlin S, Schellnhuber H-J (2003) Power-law persistence and trends in the atmosphere: A detailed study of long temperature records. Phys Rev E 68:046133 54. Király A, Bartos I, Jánosi IM (2006) Correlation properties of daily temperature anormalies over land. Tellus 58A(5):593–600

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55. Santhanam MS, Kantz H (2005) Long-range correlations and rare events in boundary layer wind fields. Physica A 345: 713–721 56. Liu YH, Cizeau P, Meyer M, Peng C-K, Stanley HE (1997) Correlations in economic time series. Physica A 245:437; Liu YH, Gopikrishnan P, Cizeau P, Meyer M, Peng C-K, Stanley HE (1999) Statistical properties of the volatility of price fluctuations. Phys Rev E 60:1390 57. Peng C-K, Mietus J, Hausdorff JM, Havlin S, Stanley HE, Goldberger AL (1993) Long-range anticorrelations and non-gaussian behavior of the heartbeat. Phys Rev Lett 70:1343–1346 58. Bunde A, Havlin S, Kantelhardt JW, Penzel T, Peter J-H, Voigt K (2000) Correlated and uncorrelated regions in heart-rate fluctuations during sleep. Phys Rev Lett 85:3736 59. Leland WE, Taqqu MS, Willinger W, Wilson DV (1994) On the self-similar nature of Ethernet traffic. IEEE/Transactions ACM Netw 2:1–15 60. Kantelhardt JW, Koscielny-Bunde E, Rego HA, Bunde A, Havlin S (2001) Detecting long-range correlations with detrended fluctuation analysis. Physica A 295:441 61. Rybski D, Bunde A, Havlin S, Von Storch H (2006) Long-term persistence in climate and the detection problem. Geophys Res Lett 33(6):L06718 62. Rybski D, Bunde A (2008) On the detection of trends in longterm correlated records. Physica A 63. Giese E, Mossig I, Rybski D, Bunde A (2007) Long-term analysis of air temperature trends in Central Asia. Erdkunde 61(2): 186–202 64. Govindan RB, Vjushin D, Brenner S, Bunde A, Havlin S, Schellnhuber H-J (2002) Global climate models violate scaling of the observed atmospheric variability. Phys Rev Lett 89:028501 65. Vjushin D, Zhidkov I, Brenner S, Havlin S, Bunde A (2004) Volcanic forcing improves atmosphere-ocean coupled general circulation model scaling performance. Geophys Res Lett 31:L10206 66. Monetti A, Havlin S, Bunde A (2003) Long-term persistence in the sea surface temperature fluctuations. Physica A 320: 581–589 67. Kantelhardt JW, Koscielny-Bunde E, Rybski D, Braun P, Bunde A, Havlin S (2006) Long-term persistence and multifractality of precipitation and river runoff records. Geophys J Res Atmosph 111:1106

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Fractal Growth Processes

Fractal Growth Processes LEONARD M. SANDER Physics Department and Michigan Center for Theoretical Physics, The University of Michigan, Ann Arbor, USA Article Outline Glossary Definition of the Subject Introduction Fractals and Multifractals Aggregation Models Conformal Mapping Harmonic Measure Scaling Theories Future Directions Bibliography Glossary Fractal A fractal is a geometric object which is self-similar and characterized by an effective dimension which is not an integer. Multifractal A multifractal measure is a non-negative real function) defined on a support (a geometric region) which has a spectrum of scaling exponents. Diffusion-limited aggregation Diffusion-limited aggregation (DLA) is a discrete model for the irreversible growth of a cluster. The rules of the model involve a sequence of random walkers that are incorporated into a growing aggregate when they wander into contact with one of the previously aggregated walkers. Dielectric breakdown model The dielectric breakdown model (DBM) is a generalization of DLA where the probability to grow is proportional to a power of the diffusive flux onto the aggregate. If the power is unity, the model is equivalent to DLA: in this version it is called Laplacian growth. Harmonic measure If a geometric object is thought of as an isolated grounded conductor of fixed charge, the distribution of electric field on its surface is the harmonic measure. The harmonic measure of a DLA cluster is the distribution of growth probability on the surface. Definition of the Subject Fractal growth processes are a class of phenomena which produce self-similar, disordered objects in the course of development far from equilibrium. The most famous of these processes involve growth which can be modeled on

the large scale by the diffusion-limited aggregation algorithm of Witten and Sander [1]. DLA is a paradigm for pattern formation modeling which has been very influential. The algorithm describes growth limited by diffusion: many natural processes fall in this category, and the salient characteristics of clusters produced by the DLA algorithm are observed in a large number of systems such as electrodeposition clusters, viscous fingering patterns, colonies of bacteria, dielectric breakdown patterns, and patterns of blood vessels. At the same time the DLA algorithm poses a very rich problem in mathematical physics. A full “solution” of the DLA problem, in the sense of a scaling theory that can predict the important features of computer simulations is still lacking. However, the problem shows many features that resemble thermal continuous phase transitions, and a number of partially successful approaches have been given. There are deep connections between DLA in two dimensions and theories such as Loewner evolution that use conformal maps. Introduction In the 1970s Benoit Mandelbrot [2] developed the idea of fractal geometry to unify a number of earlier studies of irregular shapes and natural processes. Mandelbrot focused on a particular set of such objects and shapes, those that are self-similar, i. e., where a part of the object is identical (either in shape, or for an ensemble of shapes, in distribution) to a larger piece. He dubbed these fractals. He noted the surprising ubiquity of self-similar shapes in nature. Of particular interest to Mandelbrot and his collaborators were incipient percolation clusters [3]. These are the shapes generated when, for example, a lattice is diluted by cutting bonds at random until a cluster of connected bonds just reaches across a large lattice. This model has obvious applications in physical processes such as transport in random media. The model has a non-trivial mapping to a statistical model [4] and can be treated by the methods of equilibrium statistical physics. It is likely that percolation processes account for quite a few observations of fractals in nature. In 1981 Tom Witten and the present author observed that a completely different type of process surprisingly appears to make fractals [1,5]. These are unstable, irreversible, growth processes which we called diffusion-limited aggregation (DLA). DLA is a kinetic process which is not related to equilibrium statistical physics, but rather defined by growth rules. The rules idealize growth limited by diffusion: in the model there are random walking “par-

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ticles” which attach irreversibly to a single cluster made up of previously aggregated particles. As we will see, quite a few natural processes can be described by DLA rules, and DLA-like clusters are reasonably common in nature. The Witten–Sander paper and subsequent developments unleashed a large amount of activity – the original work has been cited more than 2700 times as of this writing. The literature in this area up to 1998 was reviewed in a very comprehensive manner by T. Vicsek [6] and P. Meakin [7]. See also the chapter by the present author in [8]. For non-technical reviews see [9,10,11]. There are three major areas where self-similar shapes arising from non-equilibrium processes have been studied. The first is the related to the original DLA algorithm. The model may be seen as an idealization of solidification of an amorphous substance. The study of this simple-seeming model is quite unexpectedly rich, and quite difficult to treat theoretically. It will be our focus in this article. We will review the early work, but emphasize developments since [7]. Meakin [12] and Kolb, et al. [13] generalized DLA to consider cluster-cluster or colloid aggregation. In this process particles aggregate when they come into contact, but the clusters so formed are mobile, and themselves aggregate by attaching to each other. This is an idealization of colloid or coagulated aerosol formation. This model also produces fractals but this is not really a surprise: at each stage, clusters of similar size are aggregating, and the result is an approximately hierarchical object. This model is quite important in applications in colloid science. The interested reader should consult [6,14]. A third line of work arose from studies of ballistic aggregation [15] and the Eden model [16]. In the former case particles attach to an aggregate after moving in straight paths, and in the latter, particles are simply added to the surface of an aggregate at any available site, with equal probability. These models give rise to non-fractal clusters with random rough surfaces. The surfaces have scaling properties which are often characterized at the continuum level by a stochastic partial differential equation proposed by Kardar, Parisi, and Zhang [17]. For accounts of this area the reader can consult [18,19,20]. The remainder of this article is organized as follows: we first briefly review fractals and multifractals. Then we give details about DLA and related models, numerical methods, and applications of the models. The major development that has fueled a remarkable revival of interest in this area was the work of Hastings and Levitov [21] who related two-dimensional DLA to a certain class of conformal maps. Developments of this theme is the subject of the subsequent section. Then we discuss the ques-

tion of the distribution of growth probabilities on the cluster surface, and finally turn to scaling theories of DLA. Fractals and Multifractals The kind of object that we deal with in this article is highly irregular; see Fig. 1. We think of the object as being made up of a large number, N, of units, and to be of overall linear dimension R. In growth processes the objects are formed by adding the units according to some dynamics. We will call the units “particles”, and take their size to be a. Such patterns need not be merely random, but can have well-defined scaling properties in the regime a  R. The picture is of a geometric object in two dimensions, but the concepts we introduce here are often applied to more abstract spaces. For example, a strange attractor in phase space often has the kind of scaling properties described here. In order to characterize the geometry of such complex objects, we first cover the points in question with a set of n(l) “boxes” of fixed size, l such that a  l  R. Clearly, for a smooth curve the product ln(l) approaches a limit (the length of the curve) as l ! 0. This is a number of order R. For a planar region with smooth boundaries l 2 n(l) approaches the area, of order R2 . The objects of ordinary geometry in d dimensions have measures given by the limit of l d n(l). For an object with many scales (a fractal), in general none of these relations hold. Rather, the product l D n(l) approaches a limit with D not necessarily an integer; D is called the (similarity) fractal dimension.

Fractal Growth Processes, Figure 1 A partial covering of a pattern with boxes. Smaller boxes reveal smaller scales of the pattern. For a pattern with many scales (like this one) there is a non-trivial scaling between the box size, l and the number of boxes

Fractal Growth Processes

Said another way, we define the fractal dimension by: D

n(l) / (R/l) :

(1)

For many simple examples of mathematical objects with non-integer values of D see [2,6]. For an infinite fractal there are no characteristic lengths. For a finite size object there is a characteristic length, the overall scale, R. This can be taken to be any measure of the size such as the radius of gyration or the extremal radius – all such lengths must be proportional. It is useful to generalize this definition in two ways. First we consider not only a geometric object, but also a measure, that is a non-negative R function  defined on the points of the object such that d D 1. For the geometry, we take the measure to be uniform on the points. However, for the case of growing fractals, we could also consider the growth probability at a point. As we will see, this is very non-uniform for DLA. Second, we define a sequence of generalized dimensions. If we are interested in geometry, we denote the mass of the object covered by box i by pi . For an arbitrary measure, we define: Z p i D d ; (2) where the integral is over the box labeled i. Then, following [22,23] we define a partition function for the pi : (q) D

n X

q

pi ;

(3)

points can be found by focusing on any point, and drawing a d-dimensional disk of radius r around it. The number of other points in the disk will scale as r D 2 . For DLA, it is common to simply count the number of points within radius r of the origin, or, alternatively, the dependence of some mean radius, R on the total number of aggregated particles, N, that is N / R D 2 This method is closely related to the Grassberger–Procaccia correlation integral [24]. For a simple fractal all of the Dq are the same, and we use the symbol D. If the generalized dimensions differ, then we have a multifractal. Multifractals were introduced by Mandelbrot [25] in the context of turbulence. In the context of fractal growth processes, the clusters themselves are usually simple fractals. They are the support for a multifractal measure, the growth probability. We need another bit of formalism. It is useful to look at the fractal measure and note how the pi scale with l/R. Suppose we assume a power-law form, p i / (l/R)˛ , where there are different values of ˛ for different parts of the measure. Also, suppose that we make a histogram of the ˛, and look at the parts of the support on which we have the same scaling. It is natural to adopt a form like Eq. (1) for the size of these sets, (l/R) f (˛) . (It is natural to think of f (˛) as the fractal dimension of the set on which the measure has exponent ˛, but this is not quite right because f can be negative due to ensemble averaging.) Then it is not hard to show [23] that ˛; f (˛) are related to q; (q) by a Legendre transform:

iD1

where q is a real number. For an object with well-defined scaling properties we often find that  scales with l in the following way as l/R ! 0: (q) / (R/l)(q)  (R/l)(q1)D q ; (q) D (q  1)D q :

(4)

Objects with this property are called fractals if all the Dq are the same. Otherwise they are multifractals. Some of the Dq have special significance. The similarity (or box-counting) dimension mentioned above is D0 since in this case  D n. If we take the limit q ! 1 we have the information dimension of dynamical systems theory: ˇ X ln p i d ˇˇ D pi : (5) D1 D ˇ dq qD1 ln l i

D2 is called the correlation dimension since p2i measures the probability that two points are close together, i. e. the number of pairs within distance l. This interpretation gives rise to a popular way to measure D2 . If we suppose that the structure is homogeneous, then the number of pairs of

f (˛) D q

d  ; dq

˛D

d : dq

(6)

Aggregation Models In this section we review aggregation models of the DLA type, and their relationship to certain continuum processes. We look at practical methods of simulation, and exhibit a few applications to physical and biological processes. DLA The original DLA algorithm [1,5] was quite simple: on a lattice, declare that the point at the origin is the first member of the cluster. Then launch a random walker from a distant point and allow it to wander until it arrives at a neighboring site to the origin, and attach it to the cluster, i. e., freeze its position. Then launch another walker and let it attach to one of the two previous points, and so on. The name, diffusion-limited aggregation, refers to the fact that random walkers, i. e., diffusing particles, control the growth. DLA is a simplified view of a common physical process, growth limited by diffusion.

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Fractal Growth Processes, Figure 2 a A DLA cluster of 50,000,000 particles produced with an offlattice algorithm by E. Somfai. b A three-dimensional off-lattice cluster. Figure courtesy of R. Ball

It became evident that for large clusters the overall shape is dependent on the lattice type [26,27], that is, DLA clusters are deformed by lattice anisotropy. This is an interesting subject [26,28,29,30,31] but most modern work is on DLA clusters without anisotropy, off-lattice clusters. The off-lattice algorithm is similar to the original one: instead of a random walk on a lattice the particle is considered to have radius a. For each step of the walk the particle moves its center from the current position to a random point on its perimeter. If it overlaps a particle of the current cluster, it is backed up until it just touches the cluster, and frozen at that point. Then another walker is launched. A reasonably large cluster grown in two dimensions is shown in Fig. 2, along with a smaller three-dimensional example. Most of the work on DLA has been for two dimensions, but dimensions up to 8 have been considered [32]. Patterns like the one in Fig. 2 have been analyzed for fractal properties. Careful measurements of both D0 and D2 (using the method, mentioned above, of fitting n(r) to r D 2 ) give the same fractal dimension, D=1.71 [32,33,34]; see Fig. 3. There is some disagreement about the next digit. There have been suggestions that DLA is a mass multifractal [35], but most authors now agree that all of the Dq are the same for the mass distribution. For three dimensions D  2:5 [31,32], and for four dimensions D  3:4 [32]. However, some authors [36,37] have claimed that plane DLA is not a self-similar fractal at all, and that the fractal dimension will drift towards 2 as the number of particles increases. More recent work based on conformal maps [38] has cast doubt on this. We will return to this point below. Plane DLA clusters can be grown in restricted geometries, see Fig. 4. The shape of such clusters is an interesting problem in pattern formation [39,40,41,42]. It was long thought that DLA grown in a channel had a different fractal dimension than in radial geometry [43,44,45].

Fractal Growth Processes, Figure 3 The number of particles inside of radius r for a large DLA cluster. This plot gives an estimate of D2

However, more careful work has shown that the dimensions are the same in the two cases [34]. Laplacian Growth and DBM Suppose we ask how the cluster of Fig. 2 gets to be rough. A simple answer is that if we already have a rough shape, it is quite difficult for a random walker to penetrate a narrow channel. (Just how difficult is treated in the Sect. “Harmonic Measure”, below) The channels don’t fill in, and the shape might be preserved. But we can also ask why a a smooth outline, e. g. a disk, does not continue to grow smoothly. In fact, it is easy to test that any initial condition is soon forgotten in the growth [5]. If we start with a smooth shape it roughens immediately because of a growth instability intrinsic to diffusion-limited growth. This instability was discovered by Mullins and Sekerka [46] who used a statement of the problem of diffusion-limited growth in continuum terms: this is known as the Stefan problem (see [47,48]), and is the standard way to idealize crystallization in the diffusion-limited case. The Stefan problem goes as follows: suppose that we have a density (r; t) of particles that diffuse until they reach the growing cluster where they deposit. Then we have: @ D r 2  @t

(7)

Fractal Growth Processes

Fractal Growth Processes, Figure 4 DLA grown in a channel (a) and a wedge (b). The boxes in a are the hierarchical maps of the Sect. “Numerical Methods”. Figures due to E. Somfai

@ / vn @n

(8)

That is,  should obey the diffusion equation;  is the diffusion constant. The normal growth velocity, vn , of the interface is proportional to the flux onto the surface, @/@n. However the term @/@t is of order v@/@x, where v is a typical growth velocity. Now jr 2 j  (v/D)j@/@nj. In the DLA case we launch one particle at a time, so that the velocity goes to zero. Hence Eq. (7) reduces to the Laplace equation, r 2 D 0

(9)

Since the cluster absorbs the particles, we should think of it as having  D 0 on the surface. We are to solve an electrostatics problem: the cluster is a grounded conductor with fixed electric flux far away. We grow by an amount proportional to the electric field at each point on the surface. This is called the quasi-static or Laplacian growth regime for deterministic growth. A linear stability analysis of these equations gives the Mullins–Sekerka instability [46]. The qualitative reason for the instability is that near the tips of the cluster the contours of  are compressed so that @/@n,

the growth rate, is large. Thus tips grow unstably. We expect DLA to have a growth instability. However, we can turn the argument, and use these observations to give a restatement of the DLA algorithm in continuum terms: we calculate the electric field on the surface of the aggregate, and interpret Eq. (8) as giving the distribution of the growth probability, p, at a point on the surface. We add a particle with this probability distribution, recalculate the potential using Eq. (9) and continue. This is called Laplacian growth. Simulations of Laplacian growth yield the same sort of clusters as the original discrete algorithm. (Some authors use the term Laplacian growth in a different way, to denote deterministic growth according to the Stefan model without surface tension [49].) DLA is thus closely related to one of the classic problems of mathematical physics, dendritic crystal growth in the quasistatic regime. However, it is not quite the same for several reasons: DLA is dominated by noise, whereas the Stefan problem is deterministic. Also, the boundary conditions are different [48]: for a crystal, if we interpret u as T  Tm , where T is the temperature, and T m the melting temperature, we have  D 0 only on a flat surface. On a curved surface we need  /  where is the surface stiffness, and  is the curvature. The surface tension acts

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Fractal Growth Processes

Fractal Growth Processes, Figure 5 DBM patterns for 1000 particles on a triangular lattice. a  D 0:5 b  D 1. This is a small DLA cluster. c  D 2. d  D 3

as a regularization which prevents the Mullins–Sekerka instability from producing sharp cusps [50]. In DLA the regularization is provided by the finite particle size. And, of course, crystals have anisotropy in the surface tension. There is another classic problem very similar to this, that of viscous fingering in Hele–Shaw flow [48]. This is the description of the displacement of an incompressible viscous fluid by an inviscid one: the “bubble” of inviscid fluid plays the role of the cluster,  is the difference of pressures in the viscous fluid and the bubble, and the Laplace equation is the direct result of incompressibility, r  v D 0 and D’Arcy’s law, v D kr where k is the permeability, and v the fluid velocity [51]. These considerations led Niemeyer, Pietronero, and Weismann [52] to a clever generalization of Laplacian growth. They were interested in dielectric breakdown with  representing a real electrostatic potential. This is known to be a threshold process so that we expect that the breakdown probability is non-linear in @/@n. To generalize they chose:  p/

@ @n

 ;

(10)

where  is a non-negative real number. There are some interesting special cases for this model. For  D 0 each growth site is equally likely to be used. This is the Eden model [16]. For  D 1 we have the Laplacian growth version of DLA, and for larger  we get a higher probability to grow at the tips so that the aggregates are more spread out (as in a real dielectric breakdown pattern like atmospheric lightning). There is a remarkable fact which was suggested numerically in [53] and confirmed more recently [54,55]: for  > 4 the aggregate prefers growth at tips so much that it becomes essentially linear and non-fractal. DBM patterns for small numbers of particles are shown in Fig. 5.

A large number of variants of the basic model have been proposed such as having the random walkers perform Lévy flights, having a variable particle size, imposing a drift on the random walkers, imposing anisotropy in attachment, etc. For references on these and other variants, see [7]. There are two qualitative results from these studies that are worth mentioning here: In the presence of drift, DLA clusters cross over to (non-fractal) Eden-like clusters [56,57]. And, anisotropy deforms the shape of the clusters on the large scale, as we mentioned above. Numerical Methods In the foregoing we have talked about the algorithm for DLA, but we have not described what is actually done to compute the positions of 50,000,000 particles. This is a daunting computational task, and considerable ingenuity has been devoted to making efficient algorithms. The techniques are quite interesting in their own right, and have been applied to other types of simulation. The first observation to make is that the algorithm is an idealization of growth due to particles wandering into contact with the aggregate from far away. However, it is not necessary to start the particles far away: they arrive at the aggregate with uniform probability on a circle which just circumscribes all of the presently aggregated particles; thus we can start the particles there. As the cluster grows, the starting circle grows. This was already done in the original simulations [1,5]. However, particles can wander in and then out of the starting circle without attaching. These walkers must be followed, and could take a long time to find the cluster again. However, since there is no matter outside, it is possible to speed up the algorithm by noting that if the random walker takes large steps it will still have the same probability distribution, provided that it cannot encounter any matter. For example, if it walks onto the circumference of

Fractal Growth Processes

Fractal Growth Processes, Figure 6 A cluster is surrounded by a starting circle (gray). A random walker outside the circle can safely take steps of length of the black circle. If the walker is at any point, it can walk on a circle equal in size to the distance to the nearest point of the cluster. However, finding that circle appears to require a time-consuming search

a circle that just reaches the starting circle, it will have the correct distribution. The radius of this circle is easy to find. This observation, due to P. Meakin, is the key to what follows: see Fig. 6. We should note that the most efficient way to deal with particles that wander away is to return the particle to the starting circle in one step using the Green’s function for a point charge outside an absorbing circle. A useful algorithm to do this is given in [58]. However, the idea of taking big steps is still a good one because there is a good deal of empty space inside the starting circle. If we could take steps in this empty space (see Fig. 6) we could again speed up the algorithm. The trick is to efficiently find the largest circle centered on the random walker that has no point of the aggregate within it. One could imagine simply doing a spiral search from the current walker position. This technique has actually been used in a completely different setting, that of Kinetic Monte Carlo simulations of surface growth in materials science [59]. For the case of DLA an even more efficient method, the method of hierarchical maps, was devised [26]. It was extended and applied to higher dimensions and off-lattice in [32], and is now the standard method. One version of the idea is illustrated in Fig. 4a for the case of growth in a channel. What is shown is an adaptively refined square mesh. The cluster is covered with a square – a map. The square is subdivided into four smaller squares, and each is further divided, but only if the cluster is closer to it than half of the side of the square. The subdivision continues only up to a predefined maximum depth so that the smallest maps are a few particle diameters. All particles of the cluster will be in one of the smallest maps: a list of the particles is attached to these maps.

As the cluster grows the maps are updated. Each time a particle is added to a previously empty smallest map, the neighboring maps (on all levels) are checked to see whether they satisfy the rule. If not, they are subdivided until they do. When a walker lands, we find the smallest map containing the point. If this map is not at the maximum depth, then the particle is far away from any matter, and half the side of the map is a lower estimate of the walker’s distance from the cluster. If, on the other hand, the particle lands in a map of maximum depth, then it is close to the cluster. The particle lists of the map and of the neighboring smallest size maps can be checked to calculate the exact distance from the cluster. Either way, the particle is enclosed in an empty circle of known radius, and can be brought to the perimeter of the circle in one step. Note that updating the map means that there is only a search for the cluster if we are in the smallest map. Empirically, the computational time, T for an N particle cluster obeys T N 1:1 , and the memory is linear in N. A more recent version of the algorithm for three-dimensional growth uses a covering with balls rather than cubes [31]. For simulations of the Laplacian growth version of the situation is quite different, and simulations are much slower. The reason is that a literal interpretation of the algorithm requires that the Laplace equation be solved each time a particle is added, for example, by the relaxation or boundary-integral method. This is how corresponding simulations for viscous fingering are done [60]. For DBM clusters the growth step requires taking a power of the electric field at the surface. It is possible to make DBM clusters using random walkers [61], but the method is rather subtle. It involves estimating the growth probability at a point by figuring out the “age” of a site, i. e., the time between attachments. This can be converted into an estimate of the growth probability. This algorithm is quite fast. There is another class of methods involving conformal maps. These are slow methods: the practical limit of the size of clusters that can be grown this way is about 50,000. However, this method calculates the growth probability as it goes along, and thus is of great interest. Conformal mapping is discussed in the Sect. “Loewner Evolution and the Hastings–Levitov Scheme”, below. Selected Applications Probably the most important impact of the DLA model has been the realization that diffusion-limited growth naturally gives rise to tenuous, branched objects. Of course, in the context of crystallization, this was understood in the centuries-long quest to understand the shape of

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Fractal Growth Processes, Figure 7 a A scanning tunneling microscope picture of Rh metal islands. The color scale indicates height, and the figure is about 500 Å across. Figure courtesy of R. Clarke. b A bacteria colony on a Petri dish. The figure is a few centimeters across. Figure courtesy of E. Ben-Jacob

snowflakes. However, applications of DLA gave rise to a unification of this with the study of many other physical applications. Broad surveys are given in [6,7,10]. Here we will concentrate on a few illustrative examples. Rapid crystallization in a random environment is the most direct application of DLA scheme. One particularly accessible example is the growth of islands on surfaces in molecular beam epitaxy experiments. In the proper growth conditions it is easy to see patterns dominated by the Mullins–Sekerka instability. These are often referred to with the unlovely phrase “fractal-like”. An example is given in Fig. 7a. There are many examples of this type, e. g. [62]. A related example is the electrodeposition of metal ions from solution. For overpotential situations DLA patterns are often observed [44,63,64,65]. There are also examples of Laplacian growth. A case of this type was discovered by Matsushita and collaborators [66], and exploited in a series of very interesting studies by the group of Ben-Jacob [67] and others. This is the growth of colonies of bacteria on hard agar plates in conditions of low nutrient supply. In this case, bacteria movement is suppressed (by the hard agar) and the limiting step in colony growth is the diffusion of nutrients to the colony. Thus we have an almost literal realization of Laplacian growth, and, indeed, colonies do look like DLA clusters in these conditions; see Fig. 7b. The detailed study of this system has led to very interesting insights into the biophysics of bacteria: these are far from our subject here, and the reader should consult [67]. We remarked above that noisy viscous fingering patterns are similar to DLA clusters, but not the same in detail: in viscous fingering the surface tension boundary con-

dition is different from that of DLA which involves discrete particles. We should note that for viscous fingering patterns in a channel, the asymptotic state is not a disorderly cluster, but rather a single finger that fills half the channel [48] because surface tension smooths the finger. In radial growth this is not true: the Mullins–Sekerka instability gives rise to a disorderly pattern [51] which looks very much like a DLA cluster, see Fig. 8. Even for growth in a channel, if there is sufficient noise, patterns look rather like those in Fig. 4a. In fact, Tang [68] used a version of DLA which allowed him to reduce the noise (see below) and introduce surface tension to do simulations of viscous fingering.

Fractal Growth Processes, Figure 8 A radial viscous fingering pattern. Courtesy of M. Moore and E. Sharon

Fractal Growth Processes

We should ask if this resemblance is more than a mere coincidence. It is the case that the measured fractal dimension of patterns like the one in Fig. 8 is close to 1.71, as for DLA. On this basis there are several claims in the literature that the large-scale structure of the patterns is identical [9,51,69]. Many authors have disagreed and given ingenious arguments about how to verify this. For example, some have claimed that viscous patterns become two-dimensional at large scales [70], or that viscous fingering patterns are most like DBM patterns with   1:2 [71]. Most recently a measurement of the growth probability of a large viscous fingering pattern was found to agree with that of DLA [72]. On this basis, these authors claim that DLA and viscous fingering are in the same universality class. In our opinion, this subject is still open. Conformal Mapping For pattern formation in two dimensions the use of analytic function theory and conformal mapping methods allows a new look at growth processes. The idea is to think of a pattern in the z plane as the image of a simple reference shape, e. g. the unit circle in the w plane, under a time-dependent analytic function, z D F t (w). More precisely, we think of the region outside of the pattern as the image of the region outside of the reference shape. By the Riemann mapping theorem the map exists and is unique if we set a boundary condition such as F(w) ! r0 w as w ! 1. We will also use the inverse map, w D G(z) D F 1 (z). For Laplacian growth processes this idea is particularly interesting since the growth rules depend on solving the Laplace equation outside the cluster. We recall that the Laplace equation is conformally invariant; that is, it retains its form under a conformal transformation. Thus we can solve in the w plane, and transform the solution: if r 2 (w) D 0, and  D 0 on the unit circle, and we take  to be the real part of a complex potential ˚ in the w plane, then Re ˚ (G(z)) solves the Laplace equation in the z plane with Re ˚ D 0 on the cluster boundary. Thus we can solve for the potential outside the unit circle in w-space (which is easy): ˚ D ln(w). Then if we map to z space we have the solution outside of the cluster: ˚(z) D ln G(z) :

(11)

Note that the constant r0 has the interpretation of a mean radius since ˚ ! ln(z/r0 ) as z ! 1. In fact, r0 is the radius of the disk that gives the same potential as the cluster far away. This has another consequence: the electric field (i. e. the growth probability) is uniform around the unit circle in the w plane. This means that equal intervals on the unit

circle in w space map into equal regions of growth probability in the z plane. The map contains not only the shape of the cluster (the image of jwj D 1) but also information about the growth probability: using Eq. (11) we have: jr˚j D jG 0 j D

1 : jF 0 j

(12)

The problem remains to construct the maps G or F for a given cluster. Somfai, et al. gave a direct method [38]: for a given cluster release a large number, M, of random walkers and record where they hit, say at points zj . We know from the previous paragraph that the images of these points are spaced roughly equally on the unit circle in the w plane. That is, if we start somewhere on the cluster and number the landing positions sequentially around the cluster surface, the images in the w-plane, w j D r j e i j , are given by j D 2 j/M; r j D 1. Thus we have the boundary values of the map and by analytic continuation we can construct the entire map. In fact, if we represent F by a Laurent series: F(w) D r0 w C A0 C

1 X Aj ; wj

(13)

jD1

it is easy to see that the Fourier coefficients of the function F( j ) are the Aj . Unfortunately, for DLA this method only gives a map of the highly probable points on the surface of the cluster. The inner, frozen, regions are very badly sampled by random walkers, so the generated map will not represent these regions. Loewner Evolution and the Hastings–Levitov Scheme A more useful approach to finding F is to impose a dynamics on the map which gives rise to the growth of the cluster. This is very closely related to Loewner evolution, where a curve in the z plane is generated by a map that obeys an equation of motion: 2 dG t (z) D : dt G t (z)  (t)

(14)

The map is to the upper half plane from the upper half plane minus the set of singularities, G D . (For an elementary discussion and references see [73].) If (t) is a stochastic process then many interesting statistical objects such as percolation clusters can be generated. For DLA a similar approach was presented by Hastings and Levitov [21,30]. In this case the evolution is discrete, and is designed to represent the addition of particles to a cluster. The process is iterative: suppose we know the

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Fractal Growth Processes

Fractal Growth Processes, Figure 9 The Hastings–Levitov scheme for fractal growth. At each stage a “bump” of the proper size is added to the unit circle. The composition of the bump map, , with the map at stage N gives the map at stage N C 1

map for N particles, F N . Then we want to add a “bump” corresponding to a new particle. This is accomplished by adding the bump of area  in the w-plane on the surface of the unit circle at angle . There are various explicit functions that generate bumps; for the most popular example see [21]. This is a function that depends on a parameter which gives the aspect ratio of the bump. Let us call the resulting transformation f; . If we use F N to transform the unit circle with a bump we get a cluster with an extra bump in the z-plane: that is, we have added a particle to the cluster and F NC1 D F N ı f . The scheme is represented in Fig. 9. There are two matters that need to be dealt with. First, we need to pick . Since the probability to grow is uniform on the circle in w-space, we take to be a random variable uniformly distributed between 0 and 2. Also, we need the bump in z-space to have a fixed area, 0 . That means that the bump in w-space needs to be adjusted because conformal transformations stretch lengths by jF 0 j. A first guess for the area  is:  NC1 D

0 0 jF N (e i NC1 ) j2

:

(15)

However, this is just a first guess. The stretching of lengths varies over the cluster, and there can be some regions where the approximation of Eq. (15) is not adequate. In this case an iterative procedure is necessary to get the area right [30,34]. The transformation itself is given by: F N D f1 ; 1 ı f2 ; 2 ı    ı fN ; N :

(16)

Fractal Growth Processes, Figure 10 A DLA cluster made by iterated conformal maps. Courtesy of E. Somfai

All of the information needed to specify the mapping is contained in the list  j ; j ; 1  j  N. An example of a cluster made this way is shown in Fig. 10. If we choose uniformly in w-space, we have chosen points with the harmonic measure, jF 0 j1 and we make DLA clusters. To simulate DBM clusters with  ¤ 1 we must choose the angles non-uniformly. Hastings [54] has shown how to do this: since a uniform distribution gives a distribution according to jF 0 j1 (see Eq. (12)) we have to pick angles with probability jF 0 j1 in order to grow with probability jrj . This can be done with a Metropolis algorithm with p( NC1 ) D jF N0 (e i )j1 playing the role of a Boltzmann factor. Applications of the Hastings–Levitov Method: Scaling and Crossovers The Hastings–Levitov method is a new numerical algorithm for DLA, but not a very efficient one. Constructing the composed map takes of order N steps so that the algorithm is of order N 2 . One may wonder what has been gained. The essential point is that new understanding arises from considering the objects that are produced in the course of the computation. For example, Davidovitch and collaborators [74] showed that averages of  over the cluster boundary are related to the generalized dimensions of the growth probability measure, cf. Eq. (4). Further, the Laurent coefficients of the map, Eq. (13), are meaningful in themselves and have interesting scaling properties. We

Fractal Growth Processes

have seen above that r0 is the radius of a disk with the same capacitance as the cluster. It scales as N 1/D , as we expect. The constant, A0 gives the wandering of the center of the cluster from its original position. Its scaling can be worked out in terms of D and the generalized dimension, D2 . The other coefficients, Aj , are related to the moments of the charge density on the cluster surface. We expect them to scale in the same way as r0 for the following reason: there is an elementary theorem in complex analysis [75] for any univalent map that says, in our notation:  r02 D S N C 

X

kjA k j2 :

(17)

kD1

Here SN is the area of the cluster. However, this is just the area of N particles, and is linear in N. Therefore, in leading order, the sum must cancel the N 2/D dependence of r02 . The simplest way this can occur is if every term in the sum goes as N 2/D . This seemed to be incorrect according to the data in [74]: they found that for for the first few k the scaling exponents of the hjA k j2 i were smaller than 2/D. However, later work [38] showed that the asymptotic scaling of all of the coefficients is the same. The apparent difference in the exponents is due to a slow crossover. This effect also seems to resolve a long-standing controversy about the asymptotic scaling behavior of DLA clusters, namely the anomalous behavior of the penetration depth of random walkers as the clusters grow. The anomaly was pointed out numerically by Plischke and Racz [76] soon after the DLA algorithm was introduced. These authors showed numerically that the width of the region where deposition occurred, , (a measure of penetration of random walkers into the cluster) seemed to grow more slowly with N than the mean radius of deposition, Rdep . However for a simple fractal all of the characteristic lengths must scale in the same way, R / N 1/D . Mandelbrot and collaborators [36,37,77] used this and other numerical evidence to suggest that DLA would not be a simple fractal for large N. However, Meakin and Sander [78] gave numerical evidence that the anomaly in the scaling of  is not due to a different exponent, but is a crossover. The controversy is resolved [38] by noting that the penetration depth can be estimated from the Laurent coefficients of f : 1X jA k j2 : (18) 2 D 2 kD1

It is easy to see [38] that this version of the penetration depth can be interpreted as the rms deviation of the position of the end of a field line from those for a disk of radius r0 . Thus an anomaly in the scaling of the Ak is related

to the effect discovered in [76]. Further, a slow crossover in Ak for small k is reasonable geometrically since this is a slow crossover in low moments of the charge. These low moments might be expected to have intrinsically slow dynamics. In [38,79,80] strong numerical evidence was given for the crossover and for universal asymptotic scaling of the Ak . Indeed, many quantities with the interpretation of a length fit an expression of the form:   C 1/D N 1C  ; (19) N where the subdominant correction to scaling is characterized by an exponent  D 0:33˙0:06. This observation verifies the crossover of . The asymptotic value of the penetration depth is measured to be /Rdep ! 0:12. The same value is found in three dimensions [81]. In fact, the crossover probably accounts for the anomalies reported in [36,37,77]. Further, a careful analysis of numerical data shows that the reported multiscaling [82] (an apparent dependence of D on the distance from the center of the cluster) is also a crossover effect. These insights lead to the view that DLA clusters are simple fractals up to a slow crossover. The origin of the crossover exponent,   1/3, is not understood. In three dimensions a similar crossover exponent was found by different techniques [81] with a value of 0:22 ˙ 0:03. Conformal Mapping for Other Discrete Growth Processes The Hastings–Levitov scheme depends on the conformal invariance of the Laplace equation. Bazant and collaborators [83,84] pointed out that the same techniques can be used for more general growth processes. For example, consider growth of an aggregate in a flowing fluid. Now there are two relevant fields, the particle concentration, c, and the velocity of the fluid, v D r (for potential flow). The current associated with c can be written j D cv  rc, where  is a mobility and  a diffusion coefficient. For steady incompressible flow we have, after rescaling: Per

 rc D r 2 c ;

r2

D0:

(20)

Here Pe is the Péclet number, U L/, where U is the value of the velocity far from the aggregate and L its initial size. These equations are conformally invariant, and the problem of flow past a disk in the w plane is easily solved. In [83] growth was accomplished by choosing bumps with probability distribution @c/@n using the method described above. This scheme solves the flow equation past the complex growing aggregate and adds particles according to

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Fractal Growth Processes, Figure 11 An aggregate grown by the advection-diffusion mechanism, Eq. (20) for Pe=10. Courtesy of M. Bazant

their flux at the surface. An example of a pattern of this type is given in Fig. 11. It has long been known that it is possible to treat quasi-static fracture as a growth process similar to DLA [85,86,87]. This is due to the fact that the Lamé equation of elasticity is similar in form to the Laplace equation and the condition for breaking a region of the order of a process zone is related to the stress at the current crack, i. e., boundary values of derivatives of the displacement field. Recent work has exploited this similarity to give yet another application of conformal mapping. For example, for Mode III fracture the quasi-static elastic equation reduces to the Laplace equation, and it is only necessary to replace the growth probability and the boundary conditions in order to use the method of iterated conformal maps [88]. For Mode I and Mode II fracture it is necessary to solve for two analytic functions, but this can also be done [89]. Harmonic Measure The distribution of the boundary values of the normal derivatives of a potential on a electrode of complex shape is called the problem of the harmonic measure. For DLA it is equivalent to the distribution of growth probabilities on the surface of the cluster, or, in other terms, the penetration of random walkers into the cluster. For other complex shapes the problem is still interesting. Its practical significance is that of the study of electrodes [90] or catalytic surfaces. In some cases the harmonic measure has deep relationships to conformal field theory. For the case of the harmonic measure we can interpret the variable ˛ of Eq. (6) as the singularity strength of the electric field near a sharp tip. This is seen as follows: it is well known [91] that near the apex of a wedge-shaped conductor the electric field diverges as r/ˇ 1 where r is the

distance from the tip and the exterior angle of the wedge is ˇ. For example, near a square corner with ˇ D 3/2 there is a r1/3 divergence. Now the quantity pi is the integral over a box of size l. Thus a sequence of boxes centered on the tip will give a power-law l /ˇ D l ˛ . Smaller ˛ means stronger divergence. For fractals that can be treated with the methods of conformal field theory, a good deal is known about the harmonic measure. For example, for a percolation cluster at pc the harmonic measure is completely understood [92]. The Dq are given by a formula: Dq D

1 5 ; Cp 2 24q C 1 C 5

q

1 : 24

(21)

The f (˛) spectrum is easy to compute from this. This formula in good accord with the numerical results of Meakin and collaborators [93] who sampled the measure by firing many random walkers at a cluster. There is an interesting feature of this formula: D0 D 4/3 is the dimension of the support of the measure. This is less than the dimension of a percolation hull, 7/4. There is a large part of the surface of a percolation cluster which is inaccessible to random walkers – the interior surfaces are cut off from the exterior by narrow necks whose widths vanish in the scaling regime. For DLA the harmonic measure is much less well understood. Some results are known. For example, Makarov [94] proved that D1 D 1 under very general circumstances for a two-dimensional harmonic measure. Halsey [95] used Green’s functions for electrostatics to prove the following for DBM models with parameter : ( C 2)  () D D ;

(22)

Here D is the fractal dimension of the cluster. For DLA  D 1, and (1) D 0. Thus (3) D D D 1:71. We introduced the function f (˛) as the Legendre transform of D(q). There is an argument due to Turkevich and Sher [96] which allows us to see a feature of f (˛) directly by giving an estimate of the singularity associated with the most active tip of the growing cluster. Note that the growth rate of the extremal radius of the cluster is related to the fractal dimension because Rext / N 1/D . Suppose we imagine adding one particle per unit time and recall that ptip / (l/Rext )˛tip . Then: dRext dRext dN ˛ D / Rext tip dt dN dt D D 1 C ˛tip  1 C ˛min :

(23)

Since the singularity at the tip is close to being the most active one, we have an estimate of the minimum value of ˛.

Fractal Growth Processes

Fractal Growth Processes, Figure 12 A sketch of the f(˛) curve. Some authors find a maximum slope at a position like that marked by the dot so that the curve ends there. The curve is extended to the real axis with a straight line. This is referred to as a phase transition

There have been a very large number of numerical investigations of D(q); f (˛) for two dimensional DLA; see [7] for a comprehensive list. They proceed either by launching many random walkers, e. g. [93] or by solving the Laplace equation [97], or, most recently, by using the Hastings–Levitov method [98]. The general features of the results are shown in Fig. 12. There is fairly good agreement about the left hand side of the curve which corresponds to large probabilities. The intercept for small ˛ is close to the Turkevich–Sher relation above. The maximum of the curve corresponds to d f /d˛ D q D 0; thus it is the dimension of the support of the measure. This seems to be quite close to D so that the whole surface is accessible to random walkers. There is very little agreement about the right hand side of the curve. It arises from regions with small probabilities which are very hard to estimate. The most reliable current method is that of [98] where conformal maps are manipulated to get at probabilities as small as 1070 , quite beyond the reach of other techniques. Unfortunately, these computations are for rather small clusters (N  50; 000) which are the largest ones that can be made by the Hastings–Levitov method. A particularly active controversy relates to the value of ˛max , if it exists at all; that is, the question is whether there is a maximum value of the slope d/dq. The authors of [98] find ˛max  20. Scaling Theories Our understanding of non-equilibrium fractal growth processes is not very satisfactory compared to that of equi-

librium processes. A long-term goal of many groups has been to find a “theory of DLA” which has the nice features of the renormalization theory of critical phenomena. The result of a good deal of work is that we have theories that give the general features of DLA, but they do not explain things in satisfactory detail. We should note that a mere estimate of the fractal dimension, D, is not what we have in mind. There are several ad hoc estimates in the literature that give reasonable values of D [41,99]. We seek, rather, a theory that allows a good understanding of the fixed point of fractal growth with a description of relevant and irrelevant operators, crossovers, etc. There have been an number of such attempts. In the first section below we describe a semi-numerical scheme which sheds light on the fixed point. Then we look at several attempts at ab initio theory. Scaling of Noise The first question one might ask about DLA growth is the role of noise. It might seem that for a very large cluster the noise of individual particle arrivals should average out. However, this is not the case. Clusters fluctuate on all scales for large N. Further, the noise-free case seems to be unstable. We can see this by asking about the related question of the growth of viscous fingers in two dimensions. As was remarked long ago [51] this sort of growth is always unstable. Numerical simulations of viscous fingering [69] show that any asymmetric initial condition develops into a pattern with a fractal dimension close to that of DLA. DLA organizes its noise to a fixed point exactly as turbulence does. Several authors [79,81,100] have looked at the idea that the noise (measured in some way) flows to a fixed value as clusters grow large. For example, we could characterize the shape fluctuations by measuring the scaled variance of the number of particles necessary to grow to a fixed extremal radius: ˇ p ıN ˇˇ D A : (24) ˇ N Rext This is easy to measure for large clusters. In two dimensions A D :0036 [79]. In [100] it was argued that DLA will be at its fixed point if if one unit of growth acts as a coarse graining of DLA on finer length scales. This amounts to a kind of real-space renormalization. For the original DLA model the scaled noise to grow one unit of length is of order unity. We can reduce it to the fixed point value by noise reduction. The slow approach to the fixed point governed by the exponent  can be in-

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Fractal Growth Processes

Fractal Growth Processes, Figure 13 A small noise-reduced cluster and a larger one with no noise reduction. Noise reduction of the proper size accelerates the approach to the fixed point. From [79]

terpreted as the drift of the noise to its proper value as N grows. Noise reduction was introduced into lattice models by Tang [68]. He kept a counter on each site and only grew there if a certain number of random walkers, m, hit that point. For off-lattice clusters a similar reduction is obtained if shallow bumps of height A are added to the cluster by letting the particles overlap as they stick. We must add m D 1/A particles to advance the growth by one particle diameter [79]. For the Hastings–Levitov method it is equivalent to use a bump map with a small aspect ratio [30]. In either case if we examine a number of sites p whose mean advance is one unit, we will find ıN/N D A. We should expect that if we tune the input value of A to A we will get to asymptotic behavior quickly. This is what is observed in two dimensions [79] and in three dimensions [81]. The amplitude of the crossover, the parameter C in Eq. (19), is smallest for A near the fixed point. In Fig. 13 we show two clusters, a small one grown with noise reduction, and a much larger one with A D 1. They both have the same value of /Rdep , near the asymptotic value of 0.12. Attempts at Theoretical Description The last section described a semi-empirical approach to DLA. The reader may wonder why the techniques of phase transition theory are not simply applicable here. One problem is that one of the favorite methods in equilibrium theory, the "-expansion, cannot be used because DLA has no upper critical dimension [5,101]. To be more precise, in other cluster theories such as percolation there is a dimension, dc such that if d > d c the fractal dimension of the cluster does not change. For example, percolation clusters are 4 dimensional for all di-

mensions above 6. For DLA this is not true because if d is much bigger than D then random walkers will penetrate and fill up the cluster so that its dimension would increase. This results [5] from the fact that for a random walker the number of sites visited in radius R is Nw / R2 . In the same region there are RD sites of the cluster, so the density of cluster sites goes as R Dd . The mean number of intersections is R Dd R2 . Therefore if D C 2 < d there are a vanishing number of intersections, and the cluster will fill up. Thus we must have D  d  2. A related argument [101] sharpens the bound to D  d  1: the fractal dimension increases without limit as d increases. Halsey and collaborators have given a theoretical description based on branch competition [102,103]. This method has been used to give an estimate of Dq for positive q [104]. The idea is to think of two branches that are born from a single site. Then the probability to stick to the first or second is called p1 ; p2 where p1 C p2 D p b is the total probability to stick to that branch. Similarly there are numbers of particles, n1 C n2 D nb . Define x D p1 /p b ; y D n1 /nb . The two variables x; y regulate the branch competition. On the average p1 D dn1 /dnb so we have: dy nb Dxy: (25) dnb The equation of motion for x is assumed to be of similar form: dx nb D g(x; y) : (26) dnb Thus the competition is reduced to a two-dimensional dynamical system with an unstable fixed point at x D y D 1/2 corresponding to the point when the two branches start with equal numbers. The fixed point is unstable because one branch will eventually screen the other. If g is known, then the unstable manifold of this fixed point describes the growth of the dominant branch, and it turns out, by counting the number of particles that attach to the dominant branch, that the eigenvalue of the unstable manifold is 1/D. The starting conditions for the growth are taken to result from microscopic processes that distribute points randomly near the fixed point. The problem remains to find g. This was done several ways: numerically, by doing a large number of simulations of branches that start equally, or in terms of a complicated self-consistent equation [103]. The result is a fractal dimension of 1.66, and a multifractal spectrum that agrees pretty well with direct simulations [104]. Another approach is due to Pietronero and collaborators [105]. It is called the method of fixed scale transformations. It is a real-space method where a small system

Fractal Growth Processes

at one scale is solved essentially exactly, and the behavior at the next coarse-grained scale estimated by assuming that there is a scale-invariant dynamics and estimating the parameters from the fixed-scale solution. The method is much more general than a theory of DLA: in [105] it is applied to directed percolation, the Eden model, sandpile models, and DBM. For DLA the fractal dimension calculated is about 1.6. The rescaled noise (cf. the previous section) comes out to be of order unity rather than the small value, 0.0036, quoted above [106] . The most recent attempt at a fundamental theory is due to Ball and Somfai [71,107]. The idea depends on a mapping from DLA to an instance of the DBM which has different boundary conditions on the growing tip. The scaling of the noise and the multifractal spectrum (for small ˛) are successfully predicted.

Future Directions The DLA model is 27 years old as of this writing. Every year (including last year) there have been about 100 references to the paper. Needless to say, this author has only read a small fraction of them. Space and time prevented presenting here even the interesting ones that I am familiar with. For example, there is a remarkable literature associated with the viscous-fingering problem without surface tension which seems, on the one hand, to describe some facets of experiments [108] and on the other to have deep relationships with the theory of 2d quantum gravity [109,110]. Where this line of work will lead is a fascinating question. There are other examples: I hope that those whose work I have not covered will not feel slighted. There is simply too much going on. A direction that should be pursued is to use the ingenious techniques that have been developed for the DLA problem for problems in different areas; [59,83] are examples of this. It is clear that this field is as lively as ever after 27 years, and will certainly hold more surprises.

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94. Makarov NG (1985) On the distortion of boundary sets under conformal-mappings. P Lond Math Soc 51:369–384 95. Halsey TC (1987) Some consequences of an equation of motion for diffusive growth. Phys Rev Lett 59:2067–2070 96. Turkevich LA, Scher H (1985) Occupancy-probability scaling in diffusion-limited aggregation. Phys Rev Lett 55(9):1026–1029 97. Ball RC, Spivack OR (1990) The interpretation and measurement of the f(alpha) spectrum of a multifractal measure. J Phys A 23:5295–5307 98. Jensen MH, Levermann A, Mathiesen J, Procaccia I (2002) Multifractal structure of the harmonic measure of diffusion-limited aggregates. Phys Rev E 65:046109 99. Ball RC (1986) Diffusion limited aggregation and its response to anisotropy. Physica A: Stat Theor Phys 140(1–2):62–69 100. Barker PW, Ball RC (1990) Real-space renormalization of diffusion-limited aggregation. Phys Rev A 42(10):6289–6292 101. Ball RC, Witten TA (1984) Causality bound on the density of aggregates. Phys Rev A 29(5):2966–2967 102. Halsey TC, Leibig M (1992) Theory of branched growth. Phys Rev A 46:7793–7809 103. Halsey TC (1994) Diffusion-limited aggregation as branched growth. Phys Rev Lett 72(8):1228–1231 104. Halsey TC, Duplantier B, Honda K (1997) Multifractal dimensions and their fluctuations in diffusion-limited aggregation. Phys Rev Lett 78(9):1719–1722 105. Erzan A, Pietronero L, Vespignani A (1995) The fixed-scale transformation approach to fractal growth. Rev Mod Phys 67(3):545–604 106. Cafiero R, Pietronero L, Vespignani A (1993) Persistence of screening and self-criticality in the scale-invariant dynamics of diffusion-limited aggregation. Phys Rev Lett 70(25):3939–3942 107. Ball RC, Somfai E (2003) Diffusion-controlled growth: Theory and closure approximations. Phys Rev E 67(2):021401, part 1 108. Ristroph L, Thrasher M, Mineev-Weinstein MB, Swinney HL (2006) Fjords in viscous fingering: Selection of width and opening angle. Phys Rev E 74(1):015201, part 2 109. Mineev-Weinstein MB, Wiegmann PB, Zabrodin A (2000) Integrable structure of interface dynamics. Phys Rev Lett 84(22):5106–5109 110. Abanov A, Mineev-Weinstein MB, Zabrodin A (2007) Self-similarity in Laplacian growth. Physica D – Nonlinear Phenomena 235(1–2):62–71

Books and Reviews Vicsek T (1992) Fractal Growth Phenomena, 2nd edn. World Scientific, Singapore Meakin P (1998) Fractals, scaling, and growth far from equilibrium. Cambridge University Press, Cambridge Godreche G (1991) Solids far from equilibrium. Cambridge, Cambridge, New York Sander LM (2000) Diffusion limited aggregation, a kinetic critical phenomenon? Contemporary Physics 41:203–218 Halsey TC (2000) Diffusion-limited aggregation: A model for pattern formation. Physics Today 53(4)

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks SIDNEY REDNER Center for Polymer Studies and Department of Physics, Boston University, Boston, USA Article Outline Glossary Definition of the Subject Introduction Solving Resistor Networks Conduction Near the Percolation Threshold Voltage Distribution in Random Networks Random Walks and Resistor Networks Future Directions Bibliography Glossary Conductance (G) The relation between the current I in an electrical network and the applied voltage V: I D GV . Conductance exponent (t) The relation between the conductance G and the resistor (or conductor) concentration p near the percolation threshold: G (p p c ) t . Effective medium theory (EMT) A theory to calculate the conductance of a heterogeneous system that is based on a homogenization procedure. Fractal A geometrical object that is invariant at any scale of magnification or reduction. Multifractal A generalization of a fractal in which different subsets of an object have different scaling behaviors. Percolation Connectivity of a random porous network. Percolation threshold pc The transition between a connected and disconnected network as the density of links is varied. Random resistor network A percolation network in which the connections consist of electrical resistors that are present with probability p and absent with probability 1  p. Definition of the Subject Consider an arbitrary network of nodes connected by links, each of which is a resistor with a specified electrical resistance. Suppose that this network is connected to the leads of a battery. Two natural scenarios are: (a) the “busbar geometry” (Fig. 1), in which the network is connected

to two parallel lines (in two dimensions), plates (in three dimensions), etc., and the battery is connected across the two plates, and (b) the “two-point geometry”, in which a battery is connected to two distinct nodes, so that a current I injected at a one node and the same current withdrawn from the other node. In both cases, a basic question is: what is the nature of the current flow through the network? There are many reasons why current flows in resistor networks have been the focus of more than a century of research. First, understanding currents in networks is one of the earliest subjects in electrical engineering. Second, the development of this topic has been characterized by beautiful mathematical advancements, such as Kirchhoff’s formal solution for current flows in networks in terms of tree matrices [52], symmetry arguments to determine the electrical conductance of continuous two-component media [10,29,48,69,74], clever geometrical methods to simplify networks [33,35,61,62], and the use of integral transform methods to solve node voltages on regular networks [6,16,94,95]. Third, the nodes voltages of a network through which a steady electrical current flows are harmonic [26]; that is, the voltage at a given node is a suitably-weighted average of the voltages at neighboring nodes. This same harmonicity also occurs in the probability distribution of random walks. Consequently, there are deep connections between the probability distribution of random walks on a given network and the node voltages on the same network [26]. Another important theme in the subject of resistor networks is the essential role played by randomness on current-carrying properties. When the randomness is weak, effective medium theory [10,53,54,55,57,74] is appropriate to characterize how the randomness affects the conductance. When the randomness is strong, as embodied by a network consisting of a random mixture of resistors and insulators, this random resistor network undergoes a transition between a conducting phase and an insulating phase when the resistor concentration passes through a percolation threshold [54]. The feature underlying this phase change is that for a small density of resistors, the network consists of disconnected clusters. However, when the resistor density passes through the percolation threshold, a macroscopic cluster of resistors spans the system through which current can flow. Percolation phenomenology has motivated theoretical developments, such as scaling, critical point exponents, and multifractals that have advanced our understanding of electrical conduction in random resistor networks. This article begins with an introduction to electrical current flows in networks. Next, we briefly discuss ana-

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 1 Resistor networks in a the bus-bar geometry, and b the two-point geometry

lytical methods to solve the conductance of an arbitrary resistor network. We then turn to basic results related to percolation: namely, the conduction properties of a large random resistor network as the fraction of resistors is varied. We will focus on how the conductance of such a network vanishes as the percolation threshold is approached from above. Next, we investigate the more microscopic current distribution within each resistor of a large network. At the percolation threshold, this distribution is multifractal in that all moments of this distribution have independent scaling properties. We will discuss the meaning of multifractal scaling and its implications for current flows in networks, especially the largest current in the network. Finally, we discuss the relation between resistor networks and random walks and show how the classic phenomena of recurrence and transience of random walks are simply related to the conductance of a corresponding electrical network. The subject of current flows on resistor networks is a vast subject, with extensive literature in physics, mathematics, and engineering journals. This review has the modest goal of providing an overview, from my own myopic perspective, on some of the basic properties of random resistor networks near the percolation threshold. Thus many important topics are simply not mentioned and the reference list is incomplete because of space limitations. The reader is encouraged to consult the review articles listed in the reference list to obtain a more complete perspective. Introduction In an elementary electromagnetism course, the following classic problem has been assigned to many generations

of physics and engineering students: consider an infinite square lattice in which each bond is a 1 ohm resistor; equivalently, the conductance of each resistor (the inverse resistance) also equals 1. There are perfect electrical connections at all vertices where four resistors meet. A current I is injected at one point and the same current I is extracted at a nearest-neighbor lattice point. What is the electrical resistance between the input and output? A more challenging question is: what is the resistance between two diagonal points, or between two arbitrary points? As we shall discuss, the latter questions can be solved elegantly using Fourier transform methods. For the resistance between neighboring points, superposition provides a simple solution. Decompose the current source and sink into its two constituents. For a current source I, symmetry tells us that a current I/4 flows from the source along each resistor joined to this input. Similarly, for a current sink I, a current I/4 flows into the sink along each adjoining resistor. For the source/sink combination, superposition tells us that a current I/2 flows along the resistor directly between the source and sink. Since the total current is I, a current of I/2 flows indirectly from source to sink via the rest of the lattice. Because the direct and indirect currents between the input and output points are the same, the resistance of the direct resistor and the resistance of rest of the lattice are the same, and thus both equal to 1. Finally, since these two elements are connected in parallel, the resistance of the infinite lattice between the source and the sink equals 1/2 (conductance 2). As we shall see in Sect. “Effective Medium Theory”, this argument is the basis for constructing an effective medium theory for the conductance of a random network. More generally, suppose that currents I i are injected at each node of a lattice network (normally many of these

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currents are zero and there would be both positive and negative currents in the steady state). Let V i denote the voltage at node i. Then by Kirchhoff’s law, the currents and voltages are related by Ii D

X

P P Fi D j gi j F j/ j g i j ! We then exploit this connection to provide insights about random walks in terms of known results about resistor networks and vice versa. Solving Resistor Networks

g i j (Vi  Vj ) ;

(1)

j

where g ij is the conductance of link ij, and the sum runs over all links ij. This equation simply states that the current flowing into a node by an external current source equals the current flowing out of the node along the adjoining resistors. The right-hand side of Eq. (1) is a discrete Laplacian operator. Partly for this reason, Kirchhoff’s law has a natural connection to random walks. At nodes where the external current is zero, the node voltages in Eq. (1) satisfy P 1X j g i j Vj ! Vj : (2) Vi D P z j gi j j

The last step applies if all the conductances are identical; here z is the coordination number of the network. Thus for steady current flow, the voltage at each unforced node equals the weighted average of the voltages at the neighboring sites. This condition defines V i as a harmonic function with respect to the weight function g ij . An important general question is the role of spatial disorder on current flows in networks. One important example is the random resistor network, where the resistors of a lattice are either present with probability p or absent with probability 1  p [54]. Here the analysis tools for regular lattice networks are no longer applicable, and one must turn to qualitative and numerical approaches to understand the current-carrying properties of the system. A major goal of this article is to outline the essential role that spatial disorder has on the current-carrying properties of a resistor network by such approaches. A final issue that we will discuss is the deep relation between resistor networks and random walks [26,63]. Consider a resistor network in which the positive terminal of a battery (voltage V D 1) is connected to a set of boundary nodes, defined to be BC ), and where a disjoint set of boundary nodes B are at V D 0. Now suppose that a random walk hops between nodes of the same geometrical network in which the probability of hopping from P node i to node j in a single step is g i j / k g i k , where k is one of the neighbors of i and the boundary sets are absorbing. For this random walk, we can ask: what is the probability F i for a walk to eventually be absorbed on BC when it starts at node i? We shall show in Sect. “Random Walks and Resistor Networks” that F i satisfies Eq. (2):

Fourier Transform The translational invariance of an infinite lattice resistor network with identical bond conductances g i j D 1 cries out for applying Fourier transform methods to determine node voltages. Let’s study the problem mentioned previously: what is the voltage at any node of the network when a unit current enters at some point? Our discussion is specifically for the square lattice; the extension to other lattices is straightforward. For the square lattice, we label each site i by its x; y coordinates. When a unit current is injected at r0 D (x0 ; y0 ), Eq. (1) becomes  ıx;x 0 ı y;y0 D V(x C 1; y) C V(x  1; y) C V(x; y C 1) C V(x; y  1)  4V(x; y) ;

(3)

which clearly exposes the second difference operator of the discrete Laplacian. To find the node voltages, we deP fine V(k) D r V(r) e ikr and then we Fourier transform Eq. (3) to convert this infinite set of difference equations into the single algebraic equation V(k) D

e ikr0 : 4  2(cos k x C cos k y )

(4)

Now we calculate V(r) by inverting the Fourier transform 1 V(r) D (2)2

Z Z  

eik(rr0 ) dk : 4  2(cos k x C cos k y )

(5)

Formally, at least, the solution is trivial. However, the integral in the inverse Fourier transform, known as a Watson integral [96], is non-trivial, but considerable understanding has gradually been developed for evaluating this type of integral [6,16,94,95,96]. For a unit input current at the origin and a unit sink of current at r0 , the resistance between these two points is V (0)  V(r0 ), and Eq. (5) gives R D V(0)  V(r0 ) Z Z (1  e ikr0 ) 1 dk : D (2)2 4  2(cos k x C cos k y )

(6)

 

Tables for the values of R for a set of closely-separated input and output points are given in [6,94]. As some specific examples, for r0 D (1; 0), R D 1/2, thus reproducing

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

the symmetry argument result. For two points separated by a diagonal, r0 D (1; 1), R D 2/. For r0 D (2; 0), R D 24/. Finally, for two points separated by a knight’s move, r0 D (2; 1), R D 4/  1/2.

method is prohibitively inefficient for larger networks because the number of spanning trees grows exponentially with network size. Potts Model Connection

Direct Matrix Solution Another way to solve Eq. (1), is to recast Kirchhoff’s law as the matrix equation Ii D

N X

G i j Vj ;

i D 1; 2; : : : ; N

(7)

jD1

where the elements of the conductance matrix are: (P k¤i g i k ; i D j Gi j D i ¤ j: g i j ; The conductance matrix is an example of a tree matrix, as G has the property that the sum of any row or any column equals zero. An important consequence of this tree property is that all cofactors of G are identical and are equal to the spanning tree polynomial [43]. This polynomial is obtained by enumerating all possible tree graphs (graphs with no closed loops) on the original electrical network that includes each node of the network. The weight of each spanning tree is simply the product of the conductances for each bond in the tree. Inverting Eq. (7), one obtains the voltage V i at each node i in terms of the external currents I j ( j D 1; 2; : : : ; N) and the conductances gij . Thus the two-point resistance Rij between two arbitrary (not necessarily connected) nodes i and j is then given by R i j D (Vi  Vj ) / I, where the network is subject to a specified external current; for example, for the two-point geometry, I i D 1, I j D 1, and I k D 0 for k ¤ i; j. Formally, the two-point resistance can be written as [99] jG (i j) j ; Ri j D jG ( j) j

(8)

where jG ( j) j is the determinant of the conductance matrix with the jth row and column removed and jG (i j) j is the determinant with the ith and jth rows and columns removed. There is a simple geometric interpretation for this conductance matrix inversion. The numerator is just the spanning tree polynomial for the original network, while the denominator is the spanning tree polynomial for the network with the additional constraint that nodes i and j are identified as a single point. This result provides a concrete prescription to compute the conductance of an arbitrary network. While useful for small networks, this

The matrix solution of the resistance has an alternative and elegant formulation in terms of the spin correlation function of the q-state Potts model of ferromagnetism in the q ! 0 limit [88,99]. This connection between a statistical mechanical model in a seemingly unphysical limit and an enumerative geometrical problem is one of the unexpected charms of statistical physics. Another such example is the n-vector model, in which ferromagnetically interacting spins “live” in an n-dimensional spin space. In the limit n ! 0 [20], the spin correlation functions of this model are directly related to all self-avoiding walk configurations. In the q-state Potts model, each site i of a lattice is occupied by a spin si that can assume one of q discrete values. The Hamiltonian of the system is X H D J ıs i ;s j ; i; j

where the sum is over all nearest-neighbor interacting spin pairs, and ıs i ;s j is the Kronecker delta function (ıs i ;s j D 1 if s i D s j and ıs i ;s j D 0 otherwise). Neighboring aligned spin pairs have energy J, while spin pairs in different states have energy zero. One can view the spins as pointing from the center to a vertex of a q-simplex, and the interaction energy is proportional to the dot product of two interacting spins. The partition function of a system of N spins is X ˇ P Jı ZN D e i; j s i ;s j ; (9) fsg

where the sum is over all 2N spin states fsg. To make the connection to resistor networks, notice that: (i) the exponential factor associated with each link ij in the partition function takes the values 1 or eˇ J , and (ii) the exponential of the sum can be written as the product XY ZN D (1 C vıs i ;s j ) ; (10) fs i g i; j

with v D tanh ˇJ. We now make a high-temperature (small-v) expansion by multiplying out the product in (10) to generate all possible graphs on the lattice, in which each bond carries a weight vıs i ;s j . Summing over all states, the spins in each disjoint cluster must be in the same state, and the last sum over the common state of all spins leads to

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each cluster being weighted by a factor of q. The partition function then becomes X ZN D qN c v N b ; (11) graphs

where N c is the number of distinct clusters and N b is the total number of bonds in the graph. It was shown by Kasteleyn and Fortuin [47] that the limit q D 1 corresponds to the percolation problem when one chooses v D p / (1  p), where p is the bond occupation probability in percolation. Even more striking [34], if one chooses v D ˛q1/2 , where ˛ is a constant, then lim q!0 Z N /q(NC1)/2 D ˛ N1 TN , where T N is again the spanning tree polynomial; in the case where all interactions between neighboring spins have the same strength, then the polynomial reduces to the number of spanning trees on the lattice. It is because of this connection to spanning trees that the resistor network and Potts model are intimately connected [99]. In a similar vein, one can show that the correlation function between two spins at nodes i and j in the Potts model is simply related to the conductance between these same two nodes when the interactions J ij between the spins at nodes i and j are equal to the conductances g ij between these same two nodes in the corresponding resistor network [99]. -Y and Y- Transforms In elementary courses on circuit theory, one learns how to combine resistors in series and parallel to reduce the complexity of an electrical circuit. For two resistors with resistances R1 and R2 in series, the net resistance is R D R1 C R2 , while for resistors in parallel, the net resistance  1 is R D R11 C R22 . These rules provide the resistance of a network that contains only series and parallel connections. What happens if the network is more complicated? One useful way to simplify such a network is by the Y and Y- transforms that was apparently first discovered by Kennelly in 1899 [50] and applied extensively since then [33,35,61,62,81]. The basic idea of the -Y transform is illustrated in Fig. 2. Any triangular arrangement of resistors R12 , R13 , and R23 within a larger circuit can be replaced by a star, with resistances R1 , R2 , and R3 , such that all resistances between any two points among the three vertices in the triangle and the star are the same. The conditions that all two-point resistances are the same are: (R1 C R2 ) D



1 R12

1 1

C (R13 C R23 )  a12

C cyclic permutations :

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 2 Illustration of the -Y and Y- transforms

Solving for R1 ; R2 , and R3 gives R1 D 12 (a12  a23 C a13 ) + cyclic permutations; the explicit result in terms of the Rij is: R1 D

R12 R13 R12 C R13 C R23

C cyclic permutations; (12)

as well as the companion result for the conductances G i D R1 i : G1 D

G12 G13 C G12 G23 C G13 G23 Ccyclic permutations: G23

These relations allow one to replace any triangle by a star to reduce an electrical network. However, sometimes we need to replace a star by a triangle to simplify a network. To construct the inverse Y- transform, notice that the -Y transform gives the resistance ratios R1 /R2 D R13 /R23 + cyclic permutations, from which R13 D R12 (R3 /R2 ) and R23 D R12 (R3 /R1 ). Substituting these last two results in Eq. (12), we eliminate R13 and R23 and thus solve for R12 in terms of the Ri : R12 D

R1 R2 C R1 R3 C R2 R3 Ccyclic permutations; (13) R3

and similarly for G i j D R1 i j . To appreciate the utility of the -Y and Y- transforms, the reader is invited to apply them on the Wheatstone bridge. When employed judiciously and repeatedly, these transforms systematically reduce planar lattice circuits to a single bond, and thus provide a powerful approach to calculate the conductance of large networks near the percolation threshold. We will return to this aspect of the problem in Sect. “Conductance Exponent”. Effective Medium Theory Effective medium theory (EMT) determines the macroscopic conductance of a random resistor network by a homogenization procedure [10,53,54,55,57,74] that is reminiscent of the Curie–Weiss effective field theory of mag-

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

netism. The basic idea in EMT is to replace the random network by an effective homogeneous medium in which the conductance of each resistor is determined self-consistently to optimally match the conductances of the original and homogenized systems. EMT is quite versatile and has been applied, for example, to estimate the dielectric constant of dielectric composites and the conductance of conducting composites. Here we focus on the conductance of random resistor networks, in which each resistor (with conductance g 0 ) is present with probability p and absent with probability 1  p. The goal is to determine the conductance as a function of p. To implement EMT, we first replace the random network by an effective homogeneous medium in which each bond has the same conductance g m (Fig. 3). If a voltage is applied across this effective medium, there will be a potential drop V m and a current I m D g m Vm across each bond. The next step in EMT is to assign one bond in the effective medium a conductance g and adjust the external voltage to maintain a fixed total current I passing through the network. Now an additional current ıi passes through the conductor g. Consequently, a current ıi must flow through one terminal of g to the other terminal via the remainder of the network (Fig. 3). This current perturbation leads to an additional voltage drop ıV across g. Thus the current-voltage relations for the marked bond and the remainder of the network are I m C ıi D g(Vm C ıV ) ıi D G ab ıV ;

(14)

where Gab is the conductance of the rest of the lattice between the terminals of the conductor g. The last step in EMT is to require that the mean value ıV averaged over the probability distribution of individual bond conductances is zero. Thus the effective medium “matches” the current-carrying properties of the original network. Solving Eq. (14) for ıV , and using the probability distribution P(g) D pı(gg0 )C(1p)ı(g) appropriate

for the random resistor network, we obtain

 (g m  g0 )p g m (1  p) C D0: hıVi D Vm (G ab C g0 ) G ab

It is now convenient to write Gab D ˛g m , where ˛ is a lattice-dependent constant of the order of one. With this definition, Eq. (15) simplifies to g m D go

p(1 C ˛)  1 : ˛

(16)

The value of ˛ – the proportionality constant for the conductance of the initial lattice with a single bond removed – can usually be determined by a symmetry argument of the type presented in Sect. “Introduction to Current Flows”. For example, for the triangular lattice (coordination number 6), the conductance Gab D 2g m and ˛ D 2. For the hypercubic lattice in d dimensions (coordination number z D 2d ), Gab D ((z  2)/2)g m . The main features of the effective conductance g m that arises from EMT are: (i) the conductance vanishes at a lattice-dependent percolation threshold p c D 1/(1 C ˛); for the hypercubic lattice ˛ D (z  2)/2 and the percolation threshold p c D 2/z D 21d (fortuitously reproducing the exact percolation threshold in two dimensions); (ii) the conductance varies linearly with p and vanishes linearly in p  p c as p approaches pc from above. The linearity of the effective conductance away from the percolation threshold accords with numerical and experimental results. However, EMT fails near the percolation threshold, where large fluctuations arise that invalidate the underlying assumptions of EMT. In this regime, alternative methods are needed to estimate the conductance. Conduction Near the Percolation Threshold Scaling Behavior EMT provides a qualitative but crude picture of the current-carrying properties of a random resistor network. While EMT accounts for the existence of a percolation transition, it also predicts a linear dependence of the conductance on p. However, near the percolation threshold it is well known that the conductance varies non-linearly in p  p c near pc [85]. This non-linearity defines the conductance exponent t by G (p  p c ) t

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 3 Illustration of EMT. (left) The homogenized network with conductances gm and one bond with conductance g. (right) The equivalent circuit to the lattice

(15)

p # pc ;

(17)

and much research on random resistor networks [85] has been performed to determine this exponent. The conductance exponent generically depends only on the spatial dimension of the network and not on any other details (a notable exception, however, is when link resistances are

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 4 (left) Realization of bond percolation on a 25  25 square lattice at p=0.505. (right) Schematic picture of the nodes (shaded circles), links and blobs picture of percolation for p & pc

broadly distributed, see [40,93]). This universality is one of the central tenets of the theory of critical phenomena [64, 83]. For percolation, the mechanism underlying universality is the absence of a characteristic length scale; as illustrated in Fig. 4, clusters on all length scales exist when a network is close to the percolation threshold. The scale of the largest cluster defines the correlation length  by  (p c  p) as p ! p c . The divergence in  also applies for p > p c by defining the correlation length as the typical size of finite clusters only (Fig. 4), thus eliminating the infinite percolating cluster from consideration. At the percolation threshold, clusters on all length scales exist, and the absence of a characteristic length implies that the singularity in the conductance should not depend on microscopic variables. The only parameter remaining upon which the conductance exponent t can depend upon is the spatial dimension d [64,83]. As typifies critical phenomena, the conductance exponent has a constant value in all spatial dimensions d > d c , where dc is the upper critical dimension which equals 6 for percolation [22]. Above this critical dimension, mean-field theory (not to be confused with EMT) gives the correct values of critical exponents. While there does not yet exist a complete theory for the dimension dependence of the conductance exponent below the critical dimension, a crude but useful nodes, links, and blobs picture of the infinite cluster [21,82,84] provides partial information. The basic idea of this picture is that for p & p c , a large system has an irregular network-like

topology that consists of quasi-linear chains that are separated by the correlation length  (Fig. 4). For a macroscopic sample of linear dimension L with a bus bar-geometry, the percolating cluster above pc then consists of (L/)d1 statistically identical chains in parallel, in which each chain consists of L/ macrolinks in series, and the macrolinks consists of nested blob-like structures. The conductance of a macrolink is expected to vanish as (p  p c ) , with  a new unknown exponent. Although a theory for the conductance of a single macrolink, and even a precise definition of a macrolink, is still lacking, the nodes, links, and blobs picture provides a starting point for understanding the dimension dependence of the conductance exponent. Using the rules for combining parallel and series conductances, the conductance of a large resistor network of linear dimension L is then G(p; L)

 d1 L (p  p c )  L/

Ld2 (p  p c )(d2)C :

(18)

In the limit of large spatial dimension, we expect that a macrolink is merely a random walk between nodes. Since the spatial separation between nodes is , the number of bonds in the macrolink, and hence its resistance, scales as  2 [92]. Using the mean-field result  (p  p c )1/2 , the resistance of the macrolink scales as (p  p c )1 and thus the exponent  D 1. Using the mean-field exponents

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

 D 1/2 and  D 1 at the upper critical dimension of d c D 6, we then infer the mean-field value of the conductance exponent t D 3 [22,91,92]. Scaling also determines the conductance of a finitesize system of linear dimension L exactly at the percolation threshold. Although the correlation length formally diverges when p  p c D 0,  is limited by L in a finite system of linear dimension L. Thus the only variable upon which the conductance can depend is L itself. Equivalently, deviations in p  p c that are smaller than L1/ cannot influence critical behavior because  can never exceed L. Thus to determine the dependence of a singular observable for a finite-size system at pc , we may replace (pp c ) by L1/ . By this prescription, the conductance at pc of a large finite-size system of linear dimension L becomes G(p c ; L) Ld2 (L1/ )(d2)C L/ :

(19)

In this finite-size scaling [85], we fix the occupation probability to be exactly at pc and study the dependence of an observable on L to determine percolation exponents. This approach provides a convenient and more accurate method to determine the conductance exponent compared to studying the dependence of the conductance of a large system as a function of p  p c . Conductance Exponent In percolation and in the random resistor network, much effort has been devoted to computing the exponents that characterize basic physical observables – such as the correlation length  and the conductance G – to high precision. There are several reasons for this focus on exponents. First, because of the universality hypothesis, exponents are a meaningful quantifier of phase transitions. Second, various observables near a phase transition can sometimes be related by a scaling argument that leads to a corresponding exponent relation. Such relations may provide a decisive test of a theory that can be checked numerically. Finally, there is the intellectual challenge of developing accurate numerical methods to determine critical exponents. The best such methods have become quite sophisticated in their execution. A seminal contribution was the “theorists’ experiment” of Last and Thouless [58] in which they punched holes at random in a conducting sheet of paper and measured the conductance of the sheet as a function of the area fraction of conducting material. They found that the conductance vanished faster than linearly with (p  p c ); here p corresponds to the area fraction of the conductor. Until this experiment, there was a sentiment that the conductance should be related to the fraction of material in

the percolating cluster [30] – the percolation probability P(p) – a quantity that vanished slower than linearly with (p  p c ). The reason for this disparity is that in a resistor network, much of the percolating cluster consists of dangling ends – bonds that carry no current – and thus make no contribution to the conductance. A natural geometrical quantity that ought to be related to the conductance is the fraction of bonds B(p) in the conducting backbone – the subset of the percolating cluster without dangling ends. However, a clear relation between the conductivity and a geometrical property of the backbone has not yet been established. Analytically, there are primary two methods that have been developed to compute the conductance exponent: the renormalization group [44,86,87,89] and low-density series expansions [1,2,32]. In the real-space version of the renormalization group, the evolution of conductance distribution under length rescaling is determined, while the momentum-space version involves a diagrammatic implementation of this length rescaling in momentum space. The latter is a perturbative approach away from mean-field theory in the variable 6  d that become exact as d ! 6. Considerable effort has been devoted to determining the conductance exponent by numerical and algorithmic methods. Typically, the conductance is computed for networks of various linear dimensions L at p D p c , and the conductance exponent is extracted from the L dependence of the conductance, which should vanish as L/ . An exact approach, but computationally impractical for large networks, is Gauss elimination to invert the conductance matrix [79]. A simple approximate method is Gauss relaxation [59,68,80,90,97] (and its more efficient variant of Gauss–Seidel relaxation [71]). This method uses Eq. (2) as the basis for an iteration scheme, in which the voltage V i at node i at the nth update step is computed from (2) using the values of V j at the (n  1)st update in the right-hand side of this equation. However, one can do much better by the conjugate gradient algorithm [27] and speeding up this method still further by Fourier acceleration methods [7]. Another computational approach is based on the node elimination method, in which the -Y and Y- transforms are used to successively eliminate bonds from the network and ultimately reduce a large network to a single bond [33,35,62]. In a different vein, the transfer matrix method has proved to be extremely accurate and efficient [24,25,70,100]. The method is based on building up the network one bond at a time and immediately calculating the conductance of the network after each bond addition. This method is most useful when applied to very long strips of transverse dimension L so that a single realization gives an accurate value for the conductance.

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

As a result of these investigations, as well as by series expansions for the conductance, the following exponents have been found. For d D 2, where most of the computational effort has been applied, the best estimate [70] for the exponent t (using  D t in d D 2 only) is t D 1:299˙0:002. One reason for the focus on two dimensions is that early estimates for t were tantalizingly close to the correlation length exponent  that is now known to exactly equal 4/3 [23]. Another such connection was the Alexander–Orbach conjecture [5], which predicted t D 91/72 D 1:2638 : : :, but again is incompatible with the best numerical estimate for t. In d D 3, the best available numerical estimate for t appears to be t D 2:003 ˙ 0:047 [12,

36], while the low concentration series method gives an equally precise result of t D 2:02 ˙ 0:05 [1,2]. These estimates are just compatible with the rigorous bound that t  2 in d D 3 [37,38]. In greater than three dimensions, these series expansions give t D 2:40 ˙ 0:03 for d D 4 and t D 2:74 ˙ 0:03 for d D 5, and the dimension dependence is consistent with t D 3 when d reaches 6. Voltage Distribution in Random Networks Multifractal Scaling While much research has been devoted to understanding the critical behavior of the conductance, it was re-

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 5 The voltage distribution on an L  L square lattice random resistor network at the percolation threshold for a L D 4 (exact), b L D 10, and c L D 130. The latter two plots are based on simulation data. For L D 4, a number of peaks, that correspond to simple rational fractions of the unit potential drop, are indicated. Also shown are the average voltage over all realizations, Vav , the most probable voltage, Vmp , and the average of the minimum voltage in each realization, hVmin i. [Reprinted from Ref. [19]]

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

alized that the distribution of voltages across each resistor of the network was quite rich and exhibited multifractal scaling [17,19,72,73]. Multifractality is a generalization of fractal scaling in which the distribution of an observable is sufficiently broad that different moments of the distribution scale independently. Such multifractal scaling arises in phenomena as diverse as turbulence [45,66], localization [13], and diffusion-limited aggregation [41,42]. All these diverse examples showed scaling properties that were much richer than first anticipated. To make the discussion of multifractality concrete, consider the example of the Maxwell–Boltzmann velocity distribution of a one-dimensional ideal gas r m 2 1 2 2 P(v) D ev /2v th ; emv /2k B T  q 2 kB T 2 2v th where kB is Boltzmann’s constant,pm is the particle mass, T is the temperature, and v th D kB T/m is the characteristic thermal velocity. The even integer moments of the velocity distribution are  n   p(n) h(v 2 )n i / v th2  v th2 : Thus a single velocity scale, vth , characterizes all positive moments of the velocity distribution. Alternatively, the exponent p(n) is linear in n. This linear dependence of successive moment exponents characterizes single-parameter scaling. The new feature of multifractal scaling is that a wide range of scales characterizes the voltage distribution (Fig. 5). As a consequence, the moment exponent p(n) is a non-linear function of n. One motivation for studying the voltage distribution is its relation to basic aspects of electrical conduction. If a voltage V D 1 is applied across a resistor network, then the conductance G and the total current flow I are equal: I D G. Consider now the power dissipated through the network P D IV D GV 2 ! G. We may also compute the dissipated power by adding up these losses in each resistor to give X X X PDGD g i j Vi2j ! Vi2j D V 2 N(V) : (20) ij

ij

tribution, we define the family of exponents p(k) for the scaling dependence of the voltage distribution at p D p c by X M(k)  N(V)V k Lp(k)/ : (21) V

Since M(2) is just the network conductance, p(2) D . Other moments of the voltage distribution also have simple interpretations. For example, hV 4 i is related to the magnitude of the noise in the network [9,73], while hV k i for k ! 1 weights the bonds with the highest currents, or the “hottest” bonds of the network, most strongly, and they help understand the dynamics of fuse networks of failure [4,18]. On the other hand, negative moments weight low-current bonds more strongly and emphasize the low-voltage tail of the distribution. For example, M(1) characterizes hydrodynamic dispersion [56], in which passive tracer particles disperse in a network due to a multiplicity of network paths. In hydrodynamics dispersion, the transit time across each bond is proportional to the inverse of the current in the bond, while the probability for tracer to enter a bond is proportional to the entering current. As a result, the kth moment of the transit time distribution varies as M(k C 1), so that the quantity that quantifies dispersion, ht 2 i  hti2 , scales as M(1). A simple fractal model [3,11,67] of the conducting backbone (Fig. 6) illustrates the multifractal scaling of the voltage distribution near the percolation threshold [17]. To obtain the Nth-order structure, each bond in the (N  1)st iteration is replaced by the first-order structure. The resulting fractal has a hierarchical embedding of links and blobs that captures the basic geometry of the percolating backbone. Between successive generations, the length scale changes by a factor of 3, while the number of bonds

V

Here g i j D 1 is the conductance of resistor ij, and V ij is the corresponding voltage drop across this bond. In the last equality, N(V) is the number of resistors with a voltage drop V. Thus the conductance is just the second moment of the distribution of voltage drops across each bond in the network. From the statistical physics perspective it is natural to study other moments of the voltage distribution and the voltage distribution itself. Analogous to the velocity dis-

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 6 The first few iterations of a hierarchical model

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

changes by a factor of 4. Defining the fractal dimension df as the scaling relation between mass (M D 4 N ) and the length scale (` D 3 N ) via M `d f , gives a fractal dimension d f D ln 4/ ln 3. Now let’s determine the distribution of voltage drops across the bonds. If a unit voltage is applied at the opposite ends of a first-order structure (N D 1) and each bond is a 1 ohm resistor, then the two resistors in the central bubble each have a voltage drop of 1/5, while the two resistors at the ends have a voltage drop 2/5. In an Nth-order hierarchy, the voltage of any resistor is the product of these two factors, with number of times each factor occurs dependent on the level of embedding of a resistor within the blobs. It is a simple exercise to show that the voltage distribution is [17] ! N N N(V( j)) D 2 ; (22) j where the voltage V(j) can take the values 2 j /5 N (with j D 0; 1; : : : ; N). Because j varies logarithmically in V, the voltage distribution is log binomial [75]. Using this distribution in Eq. (21), the moments of the voltage distribution are " #N 2(1 C 2 k ) M(k) D : (23) 5k In particular, the average voltage, M(1)/M(0)  Vav equals ((3/2)/5) N , which ispvery different from the most probable voltage, Vmp D ( 2/5) N as N ! 1. The underlying multiplicativity of the bond voltages is the ultimate source of the large disparity between the average and most probable values. To calculate the moment exponent p(k), we first need to relate the iteration index N to a physical length scale. For percolation, the appropriate relation is based on Coniglio’s theorem [15], which is a simple but profound statement about the structure of the percolating cluster. This theorem states that the number of singly-connected bonds in a system of linear dimension L, Ns , varies as L1/ . Singly-connected bonds are those that would disconnect the network if they were cut. An equivalent form of the theorem is Ns D @p0 /@p, where p0 is the probability that a spanning cluster exists in the system. This relation reflects the fact that when p is decreased slightly, p0 changes only if a singly-connected bond happens to be deleted. In the Nth-order hierarchy, the number of such singlyconnected links is simply 2N . Equating these two gives an effective linear dimension, L D 2 N . Using this relation in (23), the moment exponent p(k) is iı h p(k) D k  1 C k ln (5/4)  ln(1 C 2k ) ln 2 : (24)

Because each p(k) is independent, the moments of the voltage distribution are characterized by an infinite set of exponents. Equation (24) is in excellent agreement with numerical data for the voltage distribution in two-dimensional random resistor networks at the percolation threshold [19]. A similar multifractal behavior was also found for the voltage distribution of the resistor network at the percolation threshold in three dimensions [8]. Maximum Voltage An important aspect of the voltage distribution, both because of its peculiar scaling properties [27] and its application to breakdown problems [18,27], is the maximum voltage in a network. The salient features of this maximum voltage are: (i) logarithmic scaling as a function of system size [14,27,28,60,65], and (ii) non-monotonic dependence on the resistor concentration p [46]. The former property is a consequence of the expected size of the largest defect in the network that gives maximal local currents. Here, we use the terms maximum local voltage and maximum local current interchangeably because they are equivalent. To find the maximal current, we first need to identify the optimal defects that lead to large local currents. A natural candidate is an ellipse [27,28] with major and minor axes a and b (continuum), or its discrete analog of a linear crack (hyperplanar crack in greater than two dimensions) in which n resistors are missing (Fig. 7). Because current has to detour around the defect, the local current at the ends of the defect is magnified. For the continuum problem, the current at the tip of the ellipse is Itip D I0 (1Ca / b), where I 0 is the current in the unperturbed system [27]. For the maximum current in the lattice system, one must integrate the continuum current over a one lattice spacing and identify a/b with n [60]. This approach gives the maximal current at the tip of a crack Imax / (1 C n1/2 ) in two dimensions and as Imax / (1 C n1/2(d1) ) in d dimensions. Next, we need to find the size of the largest defect, which is an extreme-value statistics exercise [39]. For a linear crack, each broken bond occurs with probability 1  p, so that the probability for a crack of length n is (1  p)n  ean , with a D  ln(1  p). In a network of d , we estimate the size of the largest defect by volume R 1 Lan d L n max e dx D 1; that is, there exists of the order of one defect of size nmax or larger in the network [39]. This estimate gives nmax varying as ln L. Combining this result with the current at the tip of a crack of length n, the largest current in a system of linear dimension L scales as (ln L)1/2(d1) . A more thorough analysis shows, however, that a single crack is not quite optimal. For a continuum two-com-

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 7 Defect configurations in two dimensions. a An ellipse and its square lattice counterpart, b a funnel, with the region of good conductor shown shaded, and a 2-slit configuration on the square lattice

ponent network with conductors of resistance 1 with probability p and with resistance r > 1 with probability 1  p, the configuration that maximizes the local current is a funnel [14,65]. For a funnel of linear dimension `, the maximum current at the apex of the funnel is proportional to `1 , where  D (4/) tan1 (r1/2 ) [14,65]. The probability to find a funnel of linear dimension ` now scales as 2 eb` (exponentially in its area), with b a constant. By the same extreme statistics reasoning given above, the size of the largest funnel in a system of linear dimension L then scales as (ln L)1/2 , and the largest expected current correspondingly scales as (ln L)(1)/2 . In the limit r ! 1, where one component is an insulator, the optimal discrete configuration in two dimensions becomes two parallel slits, each of length n, between which a single resistor remains [60]. For this two-slit configuration, the maximum current is proportional to n in two dimensions, rather than n1/2 for the single crack. Thus the maximal current in a system of linear dimension L scales as ln L rather than as a fractional power of ln L. The p dependence of the maximum voltage is intriguing because it is non-monotonic. As p, the fraction of occupied bonds, decreases from 1, less total current flows (for a fixed overall voltage drop) because the conductance is decreasing, while local current in a funnel is enhanced because such defects grow larger. The competition between these two effects leads to Vmax attaining its

peak at ppeak above the percolation threshold that only slowly approaches pc as L ! 1. An experimental manifestation of this non-monotonicity in Vmax occurred in a resistor-diode network [77], where the network reproducibly burned (solder connections melting and smoking) when p ' 0:77, compared to a percolation threshold of p c ' 0:58. Although the directionality constraint imposed by diodes enhances funneling, similar behavior should occur in a random resistor network. The non-monotonic p dependence of Vmax can be understood within the quasi-one-dimensional “bubble” model [46] that captures the interplay between local funneling and overall current reduction as p decreases (Fig. 8). Although this system looks one-dimensional, it can be engineered to reproduce the percolation properties of a system in greater than one dimension by choosing the length L to scale exponentially with the width w. The probability for a spanning path in this structure is L  p0 D 1  (1  p)w ! exp[L epw ] L; w ! 1 ;

(25)

which suddenly changes from 0 to 1 – indicative of percolation – at a threshold that lies strictly within (0,1) as L ! 1 and L ew . In what follows, we take L D 2w , which gives p c D 1/2. To determine the effect of bottlenecking, we appeal to the statement of Coniglio’s theorem [15], @p0 /@p equals the average number of singly-connected bonds in the system. Evaluating @p0 /@p in Eq. (25) at the percolation threshold of p c D 1/2 gives @p0 D w C O(ew ) ln L : @p

(26)

Thus at pc there are w ln L bottlenecks. However, current focusing due to bottlenecks is substantially diluted because the conductance, and hence the total current through the network, is small at pc . What is needed is a single bottleneck of width 1. One such bottleneck ensures the total current flow is still substantial, while the narrowing to width 1 endures that the focusing effect of the bottleneck is maximally effective.

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 8 The bubble model: a chain of L bubbles in series, each consisting of w bonds in parallel. Each bond is independently present with probability p

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

Clearly, a single bottleneck of width 1 occurs above the percolation threshold. Thus let’s determine when a such an isolated bottleneck of width 1 first appears as a function of p. The probability that a single non-empty bubble contains at least two bonds is (1 qw w pqw1 ) / (1 qw ), where q D 1 p. Then the probability P1 (p) that the width of the narrowest bottleneck has width 1 in a chain of L bubbles is  L w pqw1 P1 (p) D 1  1  1  qw   p pw : (27) 1  exp Lw e q(1  qw ) The subtracted term is the probability that L non-empty bubbles contain at least two bonds, and then P1 (p) is the complement of this quantity. As p decreases from 1, P1 (p) sharply increases from 0 to 1 when the argument of the outer exponential becomes of the order of 1; this change occurs at pˆ p c C O(ln(ln L) / ln L). At this point, a bottleneck of width 1 first appears and therefore Vmax also occurs for this value of p. Random Walks and Resistor Networks The Basic Relation We now discuss how the voltages at each node in a resistor network and the resistance of the network are directly related to first-passage properties of random walks [31,63,76, 98]. To develop this connection, consider a random walk on a finite network that can hop between nearest-neighbor sites i to j with probability pij in a single step. We divide the boundary points of the network into two disjoint classes, BC and B , that we are free to choose; a typical situation is the geometry shown in Fig. 9. We now ask: starting at an arbitrary point i, what is the probability that the walk eventually reaches the boundary set BC without first reaching any node in B ? This quantity is termed the exit probability EC (i) (with an analogous definition for the exit probability E (i) D 1  EC (i) to B ). We obtain the exit probability EC (i) by summing the probabilities for all walk trajectories that start at i and reach a site in BC without touching any site in B (and similarly for E (i)). Thus X E˙ (i) D P p ˙ (i) ; (28)

E˙ (i) D

X

p i j E˙ ( j) :

(29)

j

Thus E˙ (i) is a harmonic function because it equals a weighted average of E˙ at neighboring points, with weighting function pij . This is exactly the same relation obeyed by the node voltages in Eq. (2) for the corresponding resistor network when we identify the single-step hopP ping probabilities pij with the conductances g i j / j g i j . We thus have the following equivalence:  Let the boundary sets BC and B in a resistor network be fixed at voltages 1 and 0 respectively, with g ij the conductance of the bond between sites i and j. Then the voltage at any interior site i coincides with the probability for a random walk, which starts at i, to reach BC before reaching B , when the hopping probability from P i to j is p i j D g i j / j g i j . If all the bond conductances are the same – corresponding to single - step hopping probabilities in the equivalent random walk being identical – then Eq. (29) is just the discrete Laplace equation. We can then exploit this correspondence between conductances and hopping probabilities to infer non-trivial results about random walks and about resistor networks from basic electrostatics. This correspondence can also be extended in a natural way to general random walks with a spatially-varying bias and diffusion coefficient, and to continuous media. The consequences of this equivalence between random walks and resistor networks is profound. As an example [76], consider a diffusing particle that is initially at distance r0 from the center of a sphere of radius a < r0 in otherwise empty d-dimensional space. By the correspondence with electrostatics, the probability that this particle eventually hits the sphere is simply the electrostatic potential at r0 ; E (r0 ) D (a/r0 )d2 !

p˙

where P p ˙ (i) denotes the probability of a path from i to B˙ that avoids B . The sum over all these restricted paths can be decomposed into the outcome after one step, when the walk reaches some intermediate site j, and the sum over all path remainders from j to B˙ . This decomposition gives

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 9 a Lattice network with boundary sites BC or B . b Corresponding resistor network in which each rectangle is a 1 ohm resistor. The sites in BC are all fixed at potential V D 1, and sites in B are all grounded

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

fact, Eq. (31) gives the fundamental result

Network Resistance and Pólya’s Theorem An important extension of the relation between exit probability and node voltages is to infinite resistor networks. This extension provides a simple connection between the classic recurrence/transience transition of random walks on a given network [31,63,76,98] and the electrical resistance of this same network [26]. Consider a symmetric random walk on a regular lattice in d spatial dimensions. Suppose that the walk starts at the origin at t D 0. What is the probability that the walk eventually returns to its starting point? The answer is strikingly simple:  For d  2, a random walk is certain to eventually return to the origin. This property is known as recurrence.  For d > 2, there is a non-zero probability that the random walk will never return to the origin. This property is known as transience. Let’s now derive the transience and recurrence properties of random walks in terms of the equivalent resistor network problem. Suppose that the voltage V at the boundary sites BC is set to one. Then by Kirchhoff’s law, the total current entering the network is ID

X X X (1  Vj )gC j D (1  Vj )pC j gCk : (30) j

j

k

Here g C j is the conductance of the resistor between BC P and a neighboring site j, and pC j D gC j / j gC j . Because the voltage V j also equals the probability for the corresponding random walk to reach BC without reaching B , the term Vj pC j is just the probability that a random walk starts at BC , makes a single step to one of the sites j adjacent to BC (with hopping probability pij ), and then returns to BC without reaching B . We therefore deduce that X ID (1  Vj )gC j j

D

X k

D

X

gCk

escape probability  Pescape D P

G : k g Ck

(32)

Suppose now that a current I is injected at a single point of an infinite network, with outflow at infinity (Fig. 10). Thus the probability for a random walk to never return to its starting point, is simply proportional to the conductance G from this starting point to infinity of the same network. Thus a subtle feature of random walks, namely, the escape probability, is directly related to currents and voltages in an equivalent resistor network. Part of the reason why this connection is so useful is that the conductance of the infinite network for various spatial dimensions can be easily determined, while a direct calculation of the return probability for a random walk is more difficult. In one dimension, the conductance of an infinitely long chain of identical resistors is clearly zero. Thus Pescape D 0 or, equivalently, Preturn D 1. Thus a random walk in one dimension is recurrent. As alluded to at the outset of Sect. “Introduction to Current Flows”, the conductance between one point and infinity in an infinite resistor lattice in general spatial dimension is somewhat challenging. However, to merely determine the recurrence or transience of a random walk, we only need to know if the return probability is zero or greater than zero. Such a simple question can be answered by a crude physical estimate of the network conductance. To estimate the conductance from one point to infinity, we replace the discrete lattice by a continuum medium

X (1  Vj )pC j j

gCk  (1  return probability )

k

D

X

gCk  escape probability :

(31)

k

Here “escape” means that the random walk reaches the set B without returning to a node in BC . On the other hand, the current and the voltage drop across the network are related to the conductance G between the two boundary sets by I D GV D G. From this

Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks, Figure 10 Decomposition of a conducting medium into concentric shells, each of which consists of fixed-conductance blocks. A current I is injected at the origin and flows radially outward through the medium

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Fractal and Multifractal Scaling of Electrical Conduction in Random Resistor Networks

of constant conductance. We then estimate the conductance of the infinite medium by decomposing it into a series of concentric shells of fixed thickness dr. A shell at radius r can be regarded as a parallel array of r d1 volume elements, each of which has a fixed conductance. The conductance of one such shell is proportional to its surface area, and the overall resistance is the sum of these shell resistances. This reasoning gives Z1 R

Rshell (r)dr Z1



dr r d1

( D

1 (Pescape

P

for d  2 j

gC j

)1

for d > 2 :

(33)

The above estimate gives an easy solution to the recurrence/transience transition of random walks. For d  2, the conductance to infinity is zero because there are an insufficient number of independent paths from the origin to infinity. Correspondingly, the escape probability is zero and the random walk is recurrent. The case d D 2 is more delicate because the integral in Eq. (33) diverges only logarithmically at the upper limit. Nevertheless, the conductance to infinity is still zero and the corresponding random walk is recurrent (but just barely). For d > 2, the conductance between a single point and infinity in an infinite homogeneous resistor network is non zero and therefore the escape probability of the corresponding random walk is also non zero – the walk is now transient. There are many amusing ramifications of the recurrence of random walks and we mention two such properties. First, for d  2, even though a random walk eventually returns to its starting point, the mean time for this event is infinite! This divergence stems from a power-law tail in the time dependence of the first-passage probability [31,76], namely, the probability that a random walk returns to the origin for the first time. Another striking aspect of recurrence is that because a random walk returns to its starting point with certainty, it necessarily returns an infinite number of times. Future Directions There is a good general understanding of the conductance of resistor networks, both far from the percolation threshold, where effective medium theory applies, and close to percolation, where the conductance G vanishes as (p  p c ) t . Many advancements in numerical techniques have been developed to determine the conductance accurately and thereby obtain precise values for the conductance exponent, especially in two dimensions. In spite of this progress, we still do not yet have the right way, if it

exists at all, to link the geometry of the percolation cluster or the conducting backbone to the conductivity itself. Furthermore, many exponents of two-dimensional percolation are known exactly. Is it possible that the exact approaches developed to determine percolation exponents can be extended to give the exact conductance exponent? Finally, there are aspects about conduction in random networks that are worth highlighting. The first falls under the rubric of directed percolation [51]. Here each link in a network has an intrinsic directionality that allows current to flow in one direction only – a resistor and diode in series. Links are also globally oriented; on the square lattice for example, current can flow rightward and upward. A qualitative understanding of directed percolation and directed conduction has been achieved that parallels that of isotropic percolation. However, there is one facet of directed conduction that is barely explored. Namely, the state of the network (the bonds that are forward biased) must be determined self consistently from the current flows. This type of non-linearity is much more serious when the circuit elements are randomly oriented. These questions about the coupling between the state of the network and its conductance are central when the circuit elements are intrinsically non-linear [49,78]. This is a topic that seems ripe for new developments. Bibliography 1. Adler J (1985) Conductance Exponents From the Analysis of Series Expansions for Random Resistor Networks. J Phys A Math Gen 18:307–314 2. Adler J, Meir Y, Aharony A, Harris AB, Klein L (1990) Low-Concentration Series in General Dimension. J Stat Phys 58:511– 538 3. Aharony A, Feder J (eds) (1989) Fractals in Physics. Phys D 38:1–398 4. Alava MJ, Nukala PKVV, Zapperi S (2006) Statistical Models for Fracture. Adv Phys 55:349–476 5. Alexander S, Orbach R (1982) Density of States of Fractals: Fractons. J Phys Lett 43:L625–L631 6. Atkinson D, van Steenwijk FJ (1999) Infinite Resistive Lattice. Am J Phys 67:486–492 7. Batrouni GG, Hansen A, Nelkin M (1986) Fourier Acceleration of Relaxation Processes in Disordered Systems. Phys Rev Lett 57:1336–1339 8. Batrouni GG, Hansen A, Larson B (1996) Current Distribution in the Three-Dimensional Random Resistor Network at the Percolation Threshold. Phys Rev E 53:2292–2297 9. Blumenfeld R, Meir Y, Aharony A, Harris AB (1987) Resistance Fluctuations in Randomly Diluted Networks. Phys Rev B 35:3524–3535 10. Bruggeman DAG (1935) Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Ann Phys (Leipzig) 24:636–679. [Engl Trans: Computation of Different Physical Constants of Hetero-

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Fractal and Multifractal Time Series JAN W. KANTELHARDT Institute of Physics, Martin-Luther-University Halle-Wittenberg, Halle, Germany Article Outline Glossary Definition of the Subject Introduction Fractal and Multifractal Time Series Methods for Stationary Fractal Time Series Analysis Methods for Non-stationary Fractal Time Series Analysis Methods for Multifractal Time Series Analysis Statistics of Extreme Events in Fractal Time Series Simple Models for Fractal and Multifractal Time Series Future Directions Acknowledgment Bibliography Glossary Time series One dimensional array of numbers (x i ); i D 1; : : : ; N, representing values of an observable x usually measured equidistant (or nearly equidistant) in time. Complex system A system consisting of many non-linearly interacting components. It cannot be split into simpler sub-systems without tampering with the dynamical properties. Scaling law A power law with a scaling exponent (e. g. ˛) describing the behavior of a quantity F (e. g., fluctuation, spectral power) as function of a scale parameter s (e. g., time scale, frequency) at least asymptotically: F(s) s˛ . The power law should be valid for a large range of s values, e. g., at least for one order of magnitude. Fractal system A system characterized by a scaling law with a fractal, i. e., non-integer exponent. Fractal systems are self-similar, i. e., a magnification of a small part is statistically equivalent to the whole. Self-affine system Generalization of a fractal system, where different magnifications s and s 0 D s H have to be used for different directions in order to obtain a statistically equivalent magnification. The exponent H is called Hurst exponent. Self-affine time series and time series becoming self-affine upon integration are commonly denoted as fractal using a less strict terminology.

Multifractal system A system characterized by scaling laws with an infinite number of different fractal exponents. The scaling laws must be valid for the same range of the scale parameter. Crossover Change point in a scaling law, where one scaling exponent applies for small scale parameters and another scaling exponent applies for large scale parameters. The center of the crossover is denoted by its characteristic scale parameter s in this article. Persistence In a persistent time series, a large value is usually (i. e., with high statistical preference) followed by a large value and a small value is followed by a small value. A fractal scaling law holds at least for a limited range of scales. Short-term correlations Correlations that decay sufficiently fast that they can be described by a characteristic correlation time scale; e. g., exponentially decaying correlations. A crossover to uncorrelated behavior is observed on larger scales. Long-term correlations Correlations that decay sufficiently slow that a characteristic correlation time scale cannot be defined; e. g., power-law correlations with an exponent between 0 and 1. Power-law scaling is observed on large time scales and asymptotically. The term long-range correlations should be used if the data is not a time series. Non-stationarities If the mean or the standard deviation of the data values change with time, the weak definition of stationarity is violated. The strong definition of stationarity requires that all moments remain constant, i. e., the distribution density of the values does not change with time. Non-stationarities like monotonous, periodic, or step-like trends are often caused by external effects. In a more general sense, changes in the dynamics of the system also represent non-stationarities. Definition of the Subject Data series generated by complex systems exhibit fluctuations on a wide range of time scales and/or broad distributions of the values. In both equilibrium and nonequilibrium situations, the natural fluctuations are often found to follow a scaling relation over several orders of magnitude. Such scaling laws allow for a characterization of the data and the generating complex system by fractal (or multifractal) scaling exponents, which can serve as characteristic fingerprints of the systems in comparisons with other systems and with models. Fractal scaling behavior has been observed, e. g., in many data series from experimental physics, geophysics, medicine, physiology, and

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even social sciences. Although the underlying causes of the observed fractal scaling are often not known in detail, the fractal or multifractal characterization can be used for generating surrogate (test) data, modeling the time series, and deriving predictions regarding extreme events or future behavior. The main application, however, is still the characterization of different states or phases of the complex system based on the observed scaling behavior. For example, the health status and different physiological states of the human cardiovascular system are represented by the fractal scaling behavior of the time series of intervals between successive heartbeats, and the coarsening dynamics in metal alloys are represented by the fractal scaling of the time-dependent speckle intensities observed in coherent X-ray spectroscopy. In order to observe fractal and multifractal scaling behavior in time series, several tools have been developed. Besides older techniques assuming stationary data, there are more recently established methods differentiating truly fractal dynamics from fake scaling behavior caused by non-stationarities in the data. In addition, short-term and long-term correlations have to be clearly distinguished to show fractal scaling behavior unambiguously. This article describes several methods originating from statistical physics and applied mathematics, which have been used for fractal and multifractal time series analysis in stationary and non-stationary data. Introduction The characterization and understanding of complex systems is a difficult task, since they cannot be split into simpler subsystems without tampering with the dynamical properties. One approach in studying such systems is the recording of long time series of several selected variables (observables), which reflect the state of the system in a dimensionally reduced representation. Some systems are characterized by periodic or nearly periodic behavior, which might be caused by oscillatory components or closed-loop regulation chains. However, in truly complex systems such periodic components are usually not limited to one or two characteristic frequencies or frequency bands. They rather extend over a wide spectrum, and fluctuations on many time scales as well as broad distributions of the values are found. Often no specific lower frequency limit – or, equivalently, upper characteristic time scale – can be observed. In these cases, the dynamics can be characterized by scaling laws which are valid over a wide (possibly even unlimited) range of time scales or frequencies; at least over orders of magnitude. Such dynamics are usually denoted as fractal or multifractal, depending on the

question if they are characterized by one scaling exponent or by a multitude of scaling exponents. The first scientist who applied fractal analysis to natural time series was Benoit B. Mandelbrot [1,2,3], who included early approaches by H.E. Hurst regarding hydrological systems [4,5]. For extensive introductions describing fractal scaling in complex systems, we refer to [6,7,8, 9,10,11,12,13]. In the last decade, fractal and multifractal scaling behavior has been reported in many natural time series generated by complex systems, including  Geophysics time series (recordings of temperature, precipitation, water runoff, ozone levels, wind speed, seismic events, vegetational patterns, and climate dynamics),  Medical and physiological time series (recordings of heartbeat, respiration, blood pressure, blood flow, nerve spike intervals, human gait, glucose levels, and gene expression data),  DNA sequences (they are not actually time series),  Astrophysical time series (X-ray light sources and sunspot numbers),  Technical time series (internet traffic, highway traffic, and neutronic power from a reactor),  Social time series (finance and economy, language characteristics, fatalities in conflicts), as well as  Physics data (also going beyond time series), e. g., surface roughness, chaotic spectra of atoms, and photon correlation spectroscopy recordings. If one finds that a complex system is characterized by fractal (or multifractal) dynamics with particular scaling exponents, this finding will help in obtaining predictions on the future behavior of the system and on its reaction to external perturbations or changes in the boundary conditions. Phase transitions in the regulation behavior of a complex system are often associated with changes in their fractal dynamics, allowing for a detection of such transitions (or the corresponding states) by fractal analysis. One example for a successful application of this approach is the human cardiovascular system, where the fractality of heartbeat interval time series was shown to reflect certain cardiac impairments as well as sleep stages [14,15]. In addition, one can test and iteratively improve models of the system until they reproduce the observed scaling behavior. One example for such an approach is climate modeling, where the models were shown to need input from volcanos and solar radiation in order to reproduce the long-term correlated (fractal) scaling behavior [16] previously found in observational temperature data [17]. Fractal (or multifractal) scaling behavior certainly cannot be assumed a priori, but has to be established. Hence,

Fractal and Multifractal Time Series

there is a need for refined analysis techniques, which help to differentiate truly fractal dynamics from fake scaling behavior caused, e. g., by non-stationarities in the data. If conventional statistical methods are applied for the analysis of time series representing the dynamics of a complex system [18,19], there are two major problems. (i) The number of data series and their durations (lengths) are usually very limited, making it difficult to extract significant information on the dynamics of the system in a reliable way. (ii) If the length of the data is extended using computer-based recording techniques or historical (proxy) data, non-stationarities in the signals tend to be superimposed upon the intrinsic fluctuation properties and measurement noise. Non-stationarities are caused by external or internal effects that lead to either continuous or sudden changes in the average values, standard deviations or regulation mechanism. They are a major problem for the characterization of the dynamics, in particular for finding the scaling properties of given data. Fractal and Multifractal Time Series Fractality, Self-Affinity, and Scaling The topic of this article is the fractality (and/or multifractality) of time series. Since fractals and multifractals in general are discussed in many other articles of the encyclopedia, the concept is not thoroughly explained here. In particular, we refer to the articles  Fractal Geometry, A Brief Introduction to and  Fractals and Multifractals, Introduction to for the formalism describing fractal and multifractal structures, respectively. In a strict sense, most time series are one dimensional, since the values of the considered observable are measured in homogeneous time intervals. Hence, unless there are missing values, the fractal dimension of the support is D(0) D 1. However, there are rare cases where most of the values of a time series are very small or even zero, causing a dimension D(0) < 1 of the support. In these cases, one has to be very careful in selecting appropriate analysis techniques, since many of the methods presented in this article are not accurate for such data; the wavelet transform modulus maxima technique (see Subsect. “Wavelet Transform Modulus Maxima (WTMM) Method”) is the most advanced applicable method. Even if the fractal dimension of support is one, the information dimension D(1) and the correlation dimension D(2) can be studied. As we will see in Subsect. “The Structure Function Approach and Singularity Spectra”, D(2) is in fact explicitly related to all exponents studied in monofractal time series analysis. However, usually a slightly different approach is employed based on the no-

tion of self-affinity instead of (multi-) fractality. Here, one takes into account that the time axis and the axis of the measured values x(t) are not equivalent. Hence, a rescaling of time t by a factor a may require rescaling of the series values x(t) by a different factor aH in order to obtain a statistically similar (i. e., self-similar) picture. In this case the scaling relation x(t) ! a H x(at)

(1)

holds for an arbitrary factor a, describing the data as selfaffine (see, e. g., [6]). The Hurst exponent H (after the water engineer H.E. Hurst [4]) characterizes the type of self affinity. Figure 1a shows several examples of self-affine time series with different H. The trace of a random walk (Brownian motion, third line in Fig. 1a), for example, is characterized by H D 0:5, implying that the position axis must be rescaled by a factor of two if the time axis is rescaled by a factor of four. Note that self-affine series are often denoted as fractal even though they are not fractal in the strict sense. In this article the term “fractal” will be used in the more general sense including all data, where a Hurst exponent H can be reasonably defined. The scaling behavior of self-affine data can also be characterized by looking at their mean-square displacement. Since the mean-square displacement of a random walker is known to increase linearly in time, hx 2 (t)i t, deviations from this law will indicate the presence of selfaffine scaling. As we will see in Subsect. “Fluctuation Analysis (FA)”, one can thus retrieve the Hurst (or self-affinity) exponent H by studying the scaling behavior of the mean-square displacement, or the mean-square fluctuations hx 2 (t)i t 2H . Persistence, Long- and Short-Term Correlations Self-affine data are persistent in the sense that a large value is usually (i. e., with high statistical preference) followed by a large value and a small value is followed by a small value. For the trace of a random walk, persistence on all time scales is trivial, since a later position is just a former one plus some random increment(s). The persistence holds for all time scales, where the self-affinity relation (1) holds. However, the degree of persistence can also vary on different time scales. Weather is a typical example: while the weather tomorrow or in one week is probably similar to the weather today (due to a stable general weather condition), persistence is much harder to be seen on longer time scales. Considering the increments x i D x i  x i1 of a selfaffine series, (x i ), i D 1; : : : ; N with N values measured equidistant in time, one finds that the x i can be either

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Fractal and Multifractal Time Series, Figure 1 a Examples of self-affine series x i characterized by different Hurst exponents H = 0.9, 0.7, 0.5, 0.3 (from top to bottom). The data has been generated by Fourier filtering using the same seed for the random number generator. b Differentiated series xi of the data from a; the xi are characterized by positive long-term correlations (persistence) with  = 0.2 and 0.6 (first and second line), uncorrelated behavior (third line), and anti-correlations (bottom line), respectively

persistent, independent, or anti-persistent. Examples for all cases are shown in Fig. 1b. In our example of the random walk with H D 0:5 (third line in the figure), the increments (steps) are fully independent of each other. Persistent and anti-persistent increments, where a positive increment is likely to be followed by another positive or negative increment, respectively, are also leading to persistent P integrated series x i D ijD1 x j . For stationary data with constant mean and standard deviation the auto-covariance function of the increments, ˛ ˝ C(s) D x i x iCs D

Ns 1 X x i x iCs Ns

(2)

iD1

can be studied to determine the degree of persistence. If C(s) is divided by the variance h(x i )2 i, it becomes the auto-correlation function; both are identical if the data are normalized with unit variance. If the x i are uncorrelated (as for the random walk), C(s) is zero for s > 0. Shortrange correlations of the increments x i are usually described by C(s) declining exponentially, C(s) exp(s/t )

(3)

with a characteristic decay time t . Such behavior is typical for increments generated by an auto-regressive (AR) process x i D cx i1 C " i

(4)

with random uncorrelated offsets "i and c D exp(1/t ). Figure 2a shows the auto-correlation function for one configuration of an AR process with t D 48.

Fractal and Multifractal Time Series, Figure 2 Comparison of the autocorrelation functions C(s) (decreasing functions) and fluctuation functions F2 (s) (increasing functions) for short-term correlated data (top panel) and long-term correlated data ( D 0:4, bottom panel). The asymptotic slope H  ˛ D 0:5 of F2 (s) clearly indicates missing long-term correlations, while H  ˛ D 1  /2 > 0:5 indicates long-term correlations. The difference is much harder to observe in C(s), where statistical fluctuations and negative values start occurring above s  100. The data have been generated by an AR process Eq. (4) and Fourier filtering, respectively. The dashed lines indicate the theoretical curves

R1 For so-called long-range correlations 0 C(s)ds diverges in the limit of infinitely long series (N ! 1). In practice, this means that t cannot be defined because it increases with increasing N. For example, C(s) declines as a power-law

Fractal and Multifractal Time Series

C(s) / s 

(5)

with an exponent 0 < < 1. Figure 2b shows C(s) for one configuration with D 0:4. This type of behavior can be modeled by the Fourier filtering technique (see Subsect. “Fourier Filtering”). Long-term correlated, i. e. persistent, behavior of the x i leads to self-affine scaling behavior of the xi , characterized by H D 1  /2, as will be shown below. Crossovers and Non-Stationarities in Time Series Short-term correlated increments x i characterized by a finite characteristic correlation decay time t lead to a crossover in the scaling behavior of the integrated sePi ries x i D jD1 x j , see Fig. 2a for an example. Since the position of the crossover might be numerically different from t , we denote it by s here. Time series with a crossover are not self-affine and there is no unique Hurst exponent H characterizing them. While H > 0:5 is observed on small time scales (indicating correlations in the increments), the asymptotic behavior (for large time scales s  t and  s ) is always characterized by H D 0:5, since all correlations have decayed. Many natural recordings are characterized by pronounced short-term correlations in addition to scaling long-term correlations. For example, there are short-term correlations due to particular general weather situations in temperature data and due to respirational effects in heartbeat data. Crossovers in the scaling behavior of complex time series can also be caused by different regulation mechanisms on fast and slow time scales. Fluctuations of river runoff, for example, show different scaling behavior on time scales below and above approximately one year. Non-stationarities can also cause crossovers in the scaling behavior of data if they are not properly taken into account. In the most strict sense, non-stationarities are variations in the mean or the standard deviation of the data (violating weak stationarity) or the distribution of the data values (violating strong stationarity). Non-stationarities like monotonous, periodic or step-like trends are often caused by external effects, e. g., by the greenhouse warming and seasonal variations for temperature records, different levels of activity in long-term physiological data, or unstable light sources in photon correlation spectroscopy. Another example for non-stationary data is a record consisting of segments with strong fluctuations alternating with segments with weak fluctuations. Such behavior will cause a crossover in scaling at the time scale corresponding to the typical duration of the homogeneous segments. Different mechanisms of regulation during different time segments – like, e. g., different heartbeat reg-

ulation during different sleep stages at night – can also cause crossovers; they are regarded as non-stationarities here, too. Hence, if crossovers in the scaling behavior of data are observed, more detailed studies are needed to find out the cause of the crossovers. One can try to obtain homogenous data by splitting the original series and employing methods that are at least insensitive to monotonous (polynomially shaped) trends. To characterize a complex system based on time series, trends and fluctuations are usually studied separately (see, e. g., [20] for a discussion). Strong trends in data can lead to a false detection of long-range statistical persistence if only one (non-detrending) method is used or if the results are not carefully interpreted. Using several advanced techniques of scaling time series analysis (as described in Sect. “Methods for Non-stationary Fractal Time Series Analysis”) crossovers due to trends can be distinguished from crossovers due to different regulation mechanisms on fast and slow time scales. The techniques can thus assist in gaining insight into the scaling behavior of the natural variability as well as into the kind of trends of the considered time series. It has to be stressed that crossovers in scaling behavior must not be confused with multifractality. Even though several scaling exponents are needed, they are not applicable for the same regime (i. e., the same range of time scales). Real multifractality, on the other hand, is characterized by different scaling behavior of different moments over the full range of time scales (see next section). Multifractal Time Series Many records do not exhibit a simple monofractal scaling behavior, which can be accounted for by a single scaling exponent. As discussed in the previous section, there might exist crossover (time-) scales s separating regimes with different scaling exponents. In other cases, the scaling behavior is more complicated, and different scaling exponents are required for different parts of the series. In even more complicated cases, such different scaling behavior can be observed for many interwoven fractal subsets of the time series. In this case a multitude of scaling exponents is required for a full description of the scaling behavior in the same range of time scales, and a multifractal analysis must be applied. Two general types of multifractality in time series can be distinguished: (i) Multifractality due to a broad probability distribution (density function) for the values of the time series, e. g. a Levy distribution. In this case the multifractality cannot be removed by shuffling the series. (ii) Multifractality due to different long-term correlations

467

468

Fractal and Multifractal Time Series

of the small and large fluctuations. In this case the probability density function of the values can be a regular distribution with finite moments, e. g., a Gaussian distribution. The corresponding shuffled series will exhibit nonmultifractal scaling, since all long-range correlations are destroyed by the shuffling procedure. Randomly shuffling the order of the values in the time series is the easiest way of generating surrogate data; however, there are more advanced alternatives (see Sect. “Simple Models for Fractal and Multifractal Time Series”). If both kinds of multifractality are present, the shuffled series will show weaker multifractality than the original series. A multifractal analysis of time series will also reveal higher order correlations. Multifractal scaling can be observed if, e. g., three or four-point correlations scale differently from the standard two-point correlations studied by classical autocorrelation analysis (Eq. (2)). In addition, multifractal scaling is observed if the scaling behavior of small and large fluctuations is different. For example, extreme events might be more or less correlated than typical events.

C(s) / s  with a correlation exponent 0 < < 1 (see Eqs. (3) and (5), respectively). As illustrated by the two examples shown in Fig. 2, a direct calculation of C(s) is usually not appropriate due to noise superimposed on the data x˜ i and due to underlying non-stationarities of unknown origin. Non-stationarities make the definition of C(s) problematic, because the average hxi is not welldefined. Furthermore, C(s) strongly fluctuates around zero on large scales s (see Fig. 2b), making it impossible to find the correct correlation exponent . Thus, one has to determine the value of indirectly.

Methods for Stationary Fractal Time Series Analysis

The spectral exponent ˇ and the correlation exponent

can thus be obtained by fitting a power-law to a double logarithmic plot of the power spectrum S( f ). An example is shown in Fig. 3. The relation (6) can be derived from the Wiener–Khinchin theorem (see, e. g., [25]). If, instead of x˜ i D x i the integrated runoff time series is Fourier P transformed, i. e., x˜ i D x i X D ijD1 x j , the resulting power spectrum scales as S( f ) f 2ˇ .

In this section we describe four traditional approaches for the fractal analysis of stationary time series, see [21,22,23] for comparative studies. The main focus is on the determination of the scaling exponents H or , defined in Eqs. (1) and (5), respectively, and linked by H D 1  /2 in long-term persistent data. Methods taking non-stationarities into account will be discussed in the next chapter.

Spectral Analysis If the time series is stationary, we can apply standard spectral analysis techniques (Fourier transform) and calculate the power spectrum S( f ) of the time series (x˜ i ) as a function of the frequency f to determine self-affine scaling behavior [24]. For long-term correlated data characterized by the correlation exponent , we have S( f ) f ˇ

with ˇ D 1  :

(6)

Autocorrelation Function Analysis We consider a record (x i ) of i D 1; : : : ; N equidistant measurements. In most applications, the index i will correspond to the time of the measurements. We are interested in the correlation of the values xi and x iCs for different time lags, i. e. correlations over different time scales s. In order to remove a constant offset in the data, the mean PN ˜ i  x i  hxi. Alhxi D N1 iD1 x i is usually subtracted, x ternatively, the correlation properties of increments x˜ i D x i D x i  x i1 of the original series can be studied (see also Subsect. “Persistence, Long- and Short-Term Correlations”). Quantitatively, correlations between x˜ -values separated by s steps are defined by the (auto-) covariance function C(s) D hx˜ i x˜ iCs i or the (auto-) correlation function C(s)/hx˜ 2i i, see also Eq. (2). As already mentioned in Subsect. “Persistence, Longand Short-Term Correlations”, the x˜ i are short-term correlated if C(s) declines exponentially, C(s) exp(s/t ), and long-term correlated if C(s) declines as a power-law

Fractal and Multifractal Time Series, Figure 3 Spectral analysis of a fractal time series characterized by longterm correlations with  D 0:4 (ˇ D 0:6). The expected scaling behavior (dashed line indicating the slope –ˇ) is observed only after binning of the spectrum (circles). The data has been generated by Fourier filtering

Fractal and Multifractal Time Series

Spectral analysis, however, does not yield more reliable results than auto-correlation analysis unless a logarithmic binning procedure is applied to the double logarithmic plot of S( f ) [21], see also Fig. 3. I. e., the average of log S( f ) is calculated in successive, logarithmically wide bands from a n f0 to a nC1 f0 , where f 0 is the minimum frequency, a > 1 is a factor (e. g., a D 1:1), and the index n is counting the bins. Spectral analysis also requires stationarity of the data. Hurst’s Rescaled-Range Analysis The first method for the analysis of long-term persistence in time series based on random walk theory has been proposed by the water construction engineer Harold Edwin Hurst (1880–1978), who developed it while working in Egypt. His so-called rescaled range analysis (R/S analysis) [1,2,4,5,6] begins with splitting of the time series (x˜ i ) into non-overlapping segments  of size (time scale) s (first step), yielding Ns D int(N/s) segments altogether. In the second step, the profile (integrated data) is calculated in each segment  D 0; : : : ; Ns  1, Y ( j) D

j X

(x˜sCi  hx˜sCi is )

iD1

D

j X

x˜sCi 

iD1

s jX x˜sCi : s

(7)

By the subtraction of the local averages, piecewise constant trends in the data are eliminated. In the third step, the differences between minimum and maximum value (ranges) R (s) and the standard deviations S (s) in each segment are calculated,

(8)

jD1

Finally, the rescaled range is averaged over all segments to obtain the fluctuation function F(s), FRS (s) D

N s 1 R (s) 1 X sH N s D0 S (s)

for s  1 ;

Fluctuation Analysis (FA) The standard fluctuation analysis (FA) [8,26] is also based on random walk theory. For a time series (x˜ i ), i D 1; : : : ; N, with zero mean, we consider the global profile, i. e., the cumulative sum (cf. Eq. (7)) Y( j) D

j X

x˜ i ;

j D 0; 1; 2; : : : ; N ;

(10)

iD1

iD1

R (s) D maxsjD1 Y ( j)  minsjD1 Y ( j); v u X u1 s Y2 ( j) : S (s) D t s

that H actually characterizes the self-affinity of the profile function (7), while ˇ and refer to the original data. The values of H, that can be obtained by Hurst’s rescaled range analysis, are limited to 0 < H < 2, and significant inaccuracies are to be expected close to the bounds. Since H can be increased or decreased by one if Pj ˜ i ) or differentiated the data is integrated (x˜ j ! iD1 x (x˜ i ! x˜ i  x˜ i1 ), respectively, one can always find a way to calculate H by rescaled range analysis provided the data is stationary. While values H < 1/2 indicate long-term anticorrelated behavior of the data x˜ i , H > 1/2 indicates longterm positively correlated behavior. For power-law correlations decaying faster than 1/s, we have H D 1/2 for large s values, like for uncorrelated data. Compared with spectral analysis, Hurst’s rescaled range analysis yields smoother curves with less effort (no binning procedure is necessary) and works also for data with piecewise constant trends.

(9)

where H is the Hurst exponent already introduced in Eq. (1). One can show [1,24] that H is related to ˇ and

by 2H  1 C ˇ D 2  (see also Eqs. (6) and (14)). Note that 0 < < 1, so that the right part of the equation does not hold unless 0:5 < H < 1. The relationship does not hold in general for multifractal data. Note also

and study how the fluctuations of the profile, in a given time window of size s, increase with s. The procedure is illustrated in Fig. 4 for two values of s. We can consider the profile Y( j) as the position of a random walker on a linear chain after j steps. The random walker starts at the origin and performs, in the ith step, a jump of length x˜ i to the bottom, if x˜ i is positive, and to the top, if x˜ i is negative. To find how the square-fluctuations of the profile scale with s, we first divide the record of N elements into N s D int(N/s) non-overlapping segments of size s starting from the beginning (see Fig. 4) and another N s non-overlapping segments of size s starting from the end of the considered series. This way neither data at the end nor at the beginning of the record is neglected. Then we determine the fluctuations in each segment . In the standard FA, we obtain the fluctuations just from the values of the profile at both endpoints of each segment  D 1; : : : ; Ns , 2 (; s) D [Y(s)  Y((1)s)]2 ; FFA

(11)

(see Fig. 4) and analogous for  D N sC1 ; : : : ; 2N s , 2 (; s) D [Y(N(N s )s)Y(N(1N s )s)]2 : (12) FFA

469

470

Fractal and Multifractal Time Series

differentiation of the data, the same rules apply as listed for H in the previous subsection. The results of FA become statistically unreliable for scales s larger than one tenth of the length of the data, i. e. the analysis should be limited by s < N/10. Methods for Non-stationary Fractal Time Series Analysis Wavelet Analysis

Fractal and Multifractal Time Series, Figure 4 Illustration of the fluctuation analysis (FA) and the detrended fluctuation analysis (DFA). For two segment durations (time scales) s D 100 (a) and 200 (b), the profiles Y( j) (blue lines; defined in Eq. (11), the values used for fluctuation analysis in Eq. (12) (green circles), and least-square quadratic fits to the profiles (red lines) are shown

The origins of wavelet analysis come from signal theory, where frequency decompositions of time series were studied [27,28]. Like the Fourier transform, the wavelet transform of a signal x(t) is a convolution integral to be replaced by a summation in case of a discrete time series (x˜ i ); i D 1; : : : ; N, Z 1 1 L (; s) D x(t) [(t  )/s] dt s 1 N

2 (; s) over all subsequences to obtain Then we average FFA the mean fluctuation F2 (s),

" F2 (s) D

#1/2 2N 1 Xs 2 F (; s) s˛ : 2N s D1 FA

(13)

By definition, F2 (s) can be viewed as the root-mean-square displacement of the random walker on the chain, after s steps (the reason for the index 2 will become clear later). For uncorrelated xi values, we obtain Fick’s diffusion law F2 (s) s1/2 . For the relevant case of long-term correlations, in which C(s) follows the power-law behavior of Eq. (5), F2 (s) increases by a power law, F2 (s) s ˛

with ˛  H ;

(14)

where the fluctuation exponent ˛ is identical with the Hurst exponent H for mono-fractal data and related to

and ˇ by 2˛ D 1 C ˇ D 2  :

(15)

The typical behavior of F2 (s) for short-term correlated and long-term correlated data is illustrated in Fig. 2. The relation (15) can be derived straightforwardly by inserting Eqs. (10), (2), and (5) into Eq. (11) and separating sums over products x˜ i x˜ j with identical and different i and j, respectively. The range of the ˛ values that can be studied by standard FA is limited to 0 < ˛ < 1, again with significant inaccuracies close to the bounds. Regarding integration or

D

1X x˜ i s

[(i  )/s] :

(16)

iD1

Here, (t) is a so-called mother wavelet, from which all daughter wavelets ;s (t) D ((t  )/s) evolve by shifting and stretching of the time axis. The wavelet coefficients L (; s) thus depend on both time position  and scale s. Hence, the local frequency decomposition of the signal is described with a time resolution appropriate for the considered frequency f D 1/s (i. e., inverse time scale). All wavelets (t) must have zero mean. They are often chosen to be orthogonal to polynomial trends, so that the analysis method becomes insensitive to possible trends in the data. Simple examples are derivatives of a Gaus(n) sian, Gauss (t) D dn /(dt n ) exp(x 2 /2), like the Mexican (2) (1) hat wavelet  Gauss and the Haar wavelet, Haar (t) D C1 if 0  t < 1, 1 if 1  t < 2, and 0 otherwise. It is straightforward to construct higher order Haar wavelets that are orthogonal to linear, quadratic and cubic trends, (2) e. g., Haar (t) D 1 for t 2 [0; 1) [ [2; 3), 2 for t 2 [1; 2), (3) (t) D 1 for t 2 [0; 1), 3 for and 0 otherwise, or Haar t 2 [1; 2), +3 for t 2 [2; 3), 1 for t 2 [3; 4), and 0 otherwise. Discrete Wavelet Transform (WT) Approach A detrending fractal analysis of time series can be easily implemented by considering Haar wavelet coefficients of the profile Y( j), Eq. (10) [17,30]. In this case the convolution (16) corresponds to the addition and subtraction of mean values of Y( j) within segments of size s. Hence, P defining Y¯ (s) D 1s sjD1 Y(s C j), the coefficients can

Fractal and Multifractal Time Series

be written as FWT1 (; s)  L FWT2 (; s)  L

(0) Haar

(s; s) D Y¯ (s)  Y¯C1 (s) ;

(1) Haar

(17)

(s; s)

D Y¯ (s)  2Y¯C1 (s) C Y¯C2 (s) ;

Detrended Fluctuation Analysis (DFA) (18)

and FWT3 (; s)  L

(2) Haar

(s; s)

D Y¯ (s)  3Y¯C1 (s) C 3Y¯C2 (s)  Y¯C3 (s)

results become statistically unreliable for scales s larger than one tenth of the length of the data, just as for FA.

(19)

for constant, linear and quadratic detrending, respectively. The generalization for higher orders of detrending is ob2 vious. The resulting mean-square fluctuations FWTn (; s) are averaged over all  to obtain the mean fluctuation F2 (s), see Eq. (13). Figure 5 shows typical results for WT analysis of long-term correlated, short-term correlated and uncorrelated data. Regarding trend-elimination, wavelet transform WT0 corresponds to standard FA (see Subsect. “Fluctuation Analysis (FA)”), and only constant trends in the profile are eliminated. WT1 is similar to Hurst’s rescaled range analysis (see Subsect. “Hurst’s Rescaled-Range Analysis”): linear trends in the profile and constant trends in the data are eliminated, and the range of the fluctuation exponent ˛  H is up to 2. In general, WTn determines the fluctuations from the nth derivative, this way eliminating trends described by (n  1)st-order polynomials in the data. The

In the last 14 years Detrended Fluctuation Analysis (DFA), originally introduced by Peng et al. [31], has been established as an important method to reliably detect longrange (auto-) correlations in non-stationary time series. The method is based on random walk theory and basically represents a linear detrending version of FA (see Subsect. “Fluctuation Analysis (FA)”). DFA was later generalized for higher order detrending [15], separate analysis of sign and magnitude series [32] (see Subsect. “Sign and Magnitude (Volatility) DFA”), multifractal analysis [33] (see Subsect. “Multifractal Detrended Fluctuation Analysis (MFDFA)”), and data with more than one dimension [34]. Its features have been studied in many articles [35,36,37,38,39,40]. In addition, several comparisons of DFA with other methods for stationary and nonstationary time-series analysis have been published, see, e. g., [21,23,41,42] and in particular [22], where DFA is compared with many other established methods for short data sets, and [43], where it is compared with recently suggested improved methods. Altogether, there are about 600 papers applying DFA (till September 2008). In most cases positive auto-correlations were reported leaving only a few exceptions with anti-correlations, see, e. g., [44,45,46]. Like in the FA method, one first calculates the global profile according to Eq. (10) and divides the profile into N s D int(N/s) non-overlapping segments of size s starting from the beginning and another N s segments starting from the end of the considered series. DFA explicitly deals with monotonous trends in a detrending procedure. This m ( j) within is done by estimating a polynomial trend y;s each segment  by least-square fitting and subtracting this trend from the original profile (‘detrending’), m ( j) : Y˜s ( j) D Y( j)  y;s

Fractal and Multifractal Time Series, Figure 5 Application of discrete wavelet transform (WT) analysis on uncorrelated data (black circles), long-term correlated data ( D 0:8; ˛ D 0:6, red squares), and short-term correlated data (summation of three AR processes, green diamonds). Averages of F2 (s) averaged over 20 series with N D 216 points and divided by s1/2 are shown, so that a horizontal line corresponds to uncorrelated behavior. The blue open triangles show the result for one selected extreme configuration, where it is hard to decide about the existence of long-term correlations (figure after [29])

(20)

The degree of the polynomial can be varied in order to eliminate constant (m D 0), linear (m D 1), quadratic (m D 2) or higher order trends of the profile function [15]. Conventionally the DFA is named after the order of the fitting polynomial (DFA0, DFA1, DFA2, . . . ). In DFAm, trends of order m in the profile Y( j) and of order m  1 in the original record x˜ i are eliminated. The variance of the detrended profile Y˜s ( j) in each segment  yields the meansquare fluctuations, s

2 (; s) D FDFAm

1 X ˜2 Ys ( j) : s jD1

(21)

471

472

Fractal and Multifractal Time Series

Fractal and Multifractal Time Series, Figure 6 Application of Detrended Fluctuation Analysis (DFA) on the data already studied in Fig. 5 (figure after [29])

2 As for FA and discrete wavelet analysis, the FDFAm (; s) are averaged over all segments  to obtain the mean fluctuations F2 (s), see Eq. (13). Calculating F2 (s) for many s, the fluctuation scaling exponent ˛ can be determined just as with FA, see Eq. (14). Figure 6 shows typical results for DFA of the same long-term correlated, short-term correlated and uncorrelated data studied already in Fig. 5. We note that in studies that include averaging over many records (or one record cut into many separate pieces by the elimination of some unreliable intermediate data points) the averaging procedure (13) must be performed for all data. Taking the square root is always the final step after all averaging is finished. It is not appropriate to calculate F2 (s) for parts of the data and then average the F2 (s) values, since such a procedure will bias the results towards smaller scaling exponents on large time scales. If F2 (s) increases for increasing s by F2 (s) s ˛ with 0:5 < ˛ < 1, one finds that the scaling exponent ˛  H is related to the correlation exponent by ˛ D 1  /2 (see Eq. (15)). A value of ˛ D 0:5 thus indicates that there are no (or only short-range) correlations. If ˛ > 0:5 for all scales s, the data are long-term correlated. The higher ˛, the stronger the correlations in the signal are. ˛ > 1 indicates a non-stationary local average of the data; in this case, FA fails and yields only ˛ D 1. The case ˛ < 0:5 corresponds to long-term anti-correlations, meaning that large values are most likely to be followed by small values and vice versa. ˛ values below 0 are not possible. Since the maximum value for ˛ in DFAm is mC1, higher detrending orders should be used for very non-stationary data with large ˛. Like in FA and Hurst’s analysis, ˛ will decrease or increase by one upon additional differentiation or integration of the data, respectively.

Small deviations from the scaling law (14), i. e. deviations from a straight line in a double logarithmic plot, occur for small scales s, in particular for DFAm with large detrending order m. These deviations are intrinsic to the usual DFA method, since the scaling behavior is only approached asymptotically. The deviations limit the capability of DFA to determine the correct correlation behavior in very short records and in the regime of small s. DFA6, e. g., is only defined for s  8, and significant deviations from the scaling law F2 (s) s ˛ occur even up to s  30. They will lead to an over-estimation of the fluctuation exponent ˛, if the regime of small s is used in a fitting procedure. An approach for correction of this systematic artefact in DFA is described in [35]. The number of independent segments of length s is larger in DFA than in WT, and the fluctuations in FA are larger than in DFA. Hence, the analysis has to be based on s values lower than smax D N/4 for DFA compared with smax D N/10 for FA and WT. The accuracy of scaling exponents ˛ determined by DFA was recently studied as a function of the length N of the data [43] (fitting range s 2 [10; N/2] was used). The results show that statistical standard errors of ˛ (one standard deviation) are approximately 0.1 for N D 500, 0.05 for N D 3000, and reach 0.03 for N D 10,000. Findings of long-term correlations with ˛ D 0:6 in data with only 500 points are thus not significant. A generalization of DFA for two-dimensional data (or even higher dimensions d) was recently suggested [34]. The generalization works well when tested with synthetic surfaces including fractional Brownian surfaces and multifractal surfaces. In the 2D procedure, a double cumulative sum (profile) is calculated by summing over both directional indices analogous with Eq. (10), Y(k; l) D Pk Pl ˜ i; j . This surface is partitioned into squares iD1 jD1 x of size s  s with indices  and , in which polynomials 2 like y;;s (i; j) D ai 2 C b j2 C ci j C di C e j C f are fitted. The fluctuation function F2 (s) is again obtained by calculating the variance of the profile from the fits. Detection of Trends and Crossovers with DFA Frequently, the correlations of recorded data do not follow the same scaling law for all time scales s, but one or sometimes even more crossovers between different scaling regimes are observed (see Subsect. “Crossovers and Non-stationarities in Time Series”). Time series with a well-defined crossover at s and vanishing correlations above s are most easily generated by Fourier filtering (see Subsect. “Fourier Filtering”). The power spectrum S( f ) of an uncorrelated random series is multiplied by ( f / f )ˇ

Fractal and Multifractal Time Series

with ˇ D 2˛  1 for frequencies f > f D 1/s only. The series obtained by inverse Fourier transform of this modified power spectrum exhibits power-law correlations on time scales s < s only, while the behavior becomes uncorrelated on larger time scales s > s . The crossover from F2 (s) s ˛ to F2 (s) s 1/2 is clearly visible in double logarithmic plots of the DFA fluctuation function for such short-term correlated data. However, it (m) occurs at times s that are different from the original s used for the generation of the data and that depend on the detrending order m. This systematic deviation is most significant in the DFAm with higher m. Extensive numerical simulations (see Fig. 3 in [35]) show that the ratios of (m) /s are 1.6, 2.6, 3.6, 4.5, and 5.4 for DFA1, DFA2, . . . , s DFA5, with an error bar of approximately 0.1. Note, however, that the precise value of this ratio will depend on the (m) method used for fitting the crossover times s (and the method used for generating the data if generated data is analyzed). If results for different orders of DFA shall be (m) compared, an observed crossover s can be systematically corrected dividing by the ratio for the corresponding DFAm. If several orders of DFA are used in the procedure, several estimates for the real s will be obtained, which can be checked for consistency or used for an error approximation. A real crossover can thus be well distinguished from the effects of non-stationarities in the data, which lead to a different dependence of an apparent crossover on m. The procedure is also required if the characteristic time scale of short-term correlations shall be studied with DFA. If consistent (corrected) s values are obtained based on DFAm with different m, the existence of a real characteristic correlation time scale is positively confirmed. Note that lower detrending orders are advantageous in this case, (m) might besince the observed crossover time scale s come quite large and nearly reach one forth of the total series length (N/4), where the results become statistically inaccurate. We would like to note that studies showing scaling long-term correlations should not be based on DFA or variants of this method alone in most applications. In particular, if it is not clear whether a given time series is indeed long-term correlated or just short-term correlated with a fairly large crossover time scale, results of DFA should be compared with other methods. For example, one can employ wavelet methods (see, e. g., Subsect. “Discrete Wavelet Transform (WT) Approach”). Another option is to remove short-term correlations by considering averaged series for comparison. For a time series with daily observations and possible short-term correlations up to two years, for example, one might consider the series of two-

year averages and apply DFA together with FA, binned power spectra analysis, and/or wavelet analysis. Only if these methods still indicate long-term correlations, one can be sure that the data are indeed long-term correlated. As discussed in Subsect. “Crossovers and Non-stationarities in Time Series”, records from real measurements are often affected by non-stationarities, and in particular by trends. They have to be well distinguished from the intrinsic fluctuations of the system. To investigate the effect of trends on the DFAm fluctuation functions, one can generate artificial series (x˜ i ) with smooth monotonous trends by adding polynomials of different power p to the original record (x i ), x˜ i D x i C Ax p

with

x D i/N :

(22)

For the DFAm, such trends in the data can lead to an artificial crossover in the scaling behavior of F2 (s), i. e., the slope ˛ is strongly increased for large time scales s. The position of this artificial crossover depends on the strength A and the power p of the trend. Evidently, no artificial crossover is observed, if the detrending order m is larger than p and p is integer. The order p of the trends in the data can be determined easily by applying the different DFAm. If p is larger than m or p is not an integer, an artificial crossover is observed, the slope ˛trend in the large s regime strongly depends on m, and the position of the artificial crossover also depends strongly on m. The artificial crossover can thus be clearly distinguished from real crossovers in the correlation behavior, which result in identical slopes ˛ and rather similar crossover positions for all detrending orders m. For more extensive studies of trends with non-integer powers we refer to [35,36]. The effects of periodic trends are also studied in [35]. If the functional form of the trend in given data is not known a priori, the fluctuation function F2 (s) should be calculated for several orders m of the fitting polynomial. If m is too low, F2 (s) will show a pronounced crossover to a regime with larger slope for large scales s [35,36]. The maximum slope of log F2 (s) versus log s is m C 1. The crossover will move to larger scales s or disappear when m is increased, unless it is a real crossover not due to trends. Hence, one can find m such that detrending is sufficient. However, m should not be larger than necessary, because shifts of the observed crossover time scales and deviations on short scales s increase with increasing m. Sign and Magnitude (Volatility) DFA To study the origin of long-term fractal correlations in a time series, the series can be split into two parts which are analyzed separately. It is particularly useful to split the

473

474

Fractal and Multifractal Time Series

series of increments, x i D x i  x i1 , i D 1; : : : ; N, into a series of signs x˜ i D s i D signx i and a series of magnitudes x˜ i D m i D jx i j [32,47,48]. There is an extensive interest in the magnitude time series in economics [49,50]. These data, usually called volatility, represent the absolute variations in stock (or commodity) prices and are used as a measure quantifying the risk of investments. While the actual prices are only short-term correlated, long-term correlations have been observed in volatility series [49,50]. Time series having identical distributions and longrange correlation properties can exhibit quite different temporal organizations of the magnitude and sign sub-series. The DFA method can be applied independently to both of these series. Since in particular the signs are often rather strongly anti-correlated and DFA will give incorrect results if ˛ is too close to zero, one often studies integrated sign and magnitude series. As mentioned above, integraP tion x˜ i ! ijD1 x˜ j increases ˛ by one. Most published results report short-term anti-correlations and no long-term correlations in the sign series, i. e., ˛sign < 1/2 for the non-integrated signs si (or ˛sign < 3/2 for the integrated signs) on low time scales and ˛sign ! 1/2 asymptotically for large s. The magnitude series, on the other hand, are usually either uncorrelated ˛magn D 1/2 (or 3/2) or positively long-term correlated ˛magn > 1/2 (or 3/2). It has been suggested that findings of ˛magn > 1/2 are related with nonlinear properties of the data and in particular multifractality [32,47,48], if ˛ < 1:5 in standard DFA. Specifically, the results suggest that the correlation exponent of the magnitude series is a monotonically increasing function of the multifractal spectrum (i. e., the singularity spectrum) width of the original series (see Subsect. “The Structure Function Approach and Singularity Spectra”). On the other hand, the sign series mainly relates to linear properties of the original series. At small time scales s < 16 the standard ˛ is approximately the average of ˛sign and ˛magn , if integrated sign and magnitude series are analyzed. For ˛ > 1:5 in the original series, the integrated magnitude and sign series have approximately the same two-point scaling exponents [47]. An analytical treatment is presented in [48]. Further Detrending Approaches A possible drawback of the DFA method is the occurrence of abrupt jumps in the detrended profile Y˜s ( j) (Eq. (20)) at the boundaries between the segments, since the fitting polynomials in neighboring segments are not related. A possible way to avoid these jumps would be the calculation of F2 (s) based on polynomial fits in overlapping windows. However, this is rather time consuming due to the polynomial fit in each segment and is consequently not

done in most applications. To overcome the problem of jumps several modifications and extensions of the FA and DFA methods have been suggested in recent years. These methods include  The detrended moving average technique [51,52,53], which we denote by the backward moving average (BMA) technique (following [54]),  The centered moving average (CMA) method [54], an essentially improved version of BMA,  The modified detrended fluctuation analysis (MDFA) [55], which is essentially a mixture of old FA and DFA,  The continuous DFA (CDFA) technique [56,57], which is particularly useful for the detection of crossovers,  The Fourier DFA [58],  A variant of DFA based on empirical mode decomposition (EMD) [59],  A variant of DFA based on singular value decomposition (SVD) [60,61], and  A variant of DFA based on high-pass filtering [62]. Detrended moving average techniques will be thoroughly described and discussed in the next section. A study comparing DFA with CMA and MDFA can be found in [43]. For studies comparing DFA and BMA, see [63,64]; note that [64] also discusses CMA. The method we denote as modified detrended fluctuation analysis (MDFA) [55], eliminates trends similar to the DFA method. A polynomial is fitted to the profile function Y( j) in each segment  and the deviation between the profile function and the polynomial fit is calculated, p Y˜s ( j) D Y( j)  y;s ( j) (Eq. (20)). To estimate correlations in the data, this method uses a derivative of Y˜s ( j), obtained for each segment , by Y˜s ( j) D Y˜s ( jCs/2)Y˜s ( j). Hence, the fluctuation function (compare with Eqs. (13) and (21)) is calculated as follows: 31/2 2 N X 2  1 (23) Y˜s ( j C s/2)  Y˜s ( j) 5 : F2 (s) D 4 N jD1

As in case of DFA, MDFA can easily be generalized to remove higher order trends in the data. Since the fitting polynomials in adjacent segments are not related, Y˜s ( j) shows abrupt jumps on their boundaries as well. This leads to fluctuations of F2 (s) for large segment sizes s and limits the maximum usable scale to s < N/4 as for DFA. The detection of crossovers in the data, however, is more exact with MDFA (compared with DFA), since no correction of the estimated crossover time scales seems to be needed [43]. The Fourier-detrended fluctuation analysis [58] aims to eliminate slow oscillatory trends which are found espe-

Fractal and Multifractal Time Series

cially in weather and climate series due to seasonal influences. The character of these trends can be rather periodic and regular or irregular, and their influence on the detection of long-range correlations by means of DFA was systematically studied previously [35]. Among other things it has been shown that slowly varying periodic trends disturb the scaling behavior of the results much stronger than quickly oscillating trends and thus have to be removed prior to the analysis. In the case of periodic and regular oscillations, e. g., in temperature fluctuations, one simply removes the low frequency seasonal trend by subtracting the daily mean temperatures from the data. Another way, which the Fourier-detrended fluctuation analysis suggests, is to filter out the relevant frequencies in the signals’ Fourier spectrum before applying DFA to the filtered signal. Nevertheless, this method faces several difficulties especially its limitation to periodic and regular trends and the need for a priori knowledge of the interfering frequency band. To study correlations in data with quasi-periodic or irregular oscillating trends, empirical mode decomposition (EMD) was suggested [59]. The EMD algorithm breaks down the signal into its intrinsic mode functions (IMFs) which can be used to distinguish between fluctuations and background. The background, estimated by a quasi-periodic fit containing the dominating frequencies of a sufficiently large number of IMFs, is subtracted from the data, yielding a slightly better scaling behavior in the DFA curves. However, we believe that the method might be too complicated for wide-spread applications. Another method which was shown to minimize the effect of periodic and quasi-periodic trends is based on singular value decomposition (SVD) [60,61]. In this approach, one first embeds the original signal in a matrix whose dimension has to be much larger than the number of frequency components of the periodic or quasi-periodic trends obtained in the power spectrum. Applying SVD yields a diagonal matrix which can be manipulated by setting the dominant eigenvalues (associated with the trends) to zero. The filtered matrix finally leads to the filtered data, and it has been shown that subsequent application of DFA determines the expected scaling behavior if the embedding dimension is sufficiently large. None the less, the performance of this rather complex method seems to decrease for larger values of the scaling exponent. Furthermore SVD-DFA assumes that trends are deterministic and narrow banded. The detrending procedure in DFA (Eq. (20)) can be regarded as a scale-dependent high-pass filter since (low-frequency) fluctuations exceeding a specific scale s are eliminated. Therefore, it has been suggested to obtain the de-

trended profile Y˜s ( j) for each scale s directly by applying digital high-pass filters [62]. In particular, Butterworth, Chebyshev-I, Chebyshev-II, and an elliptical filter were suggested. While the elliptical filter showed the best performance in detecting long-range correlations in artificial data, the Chebyshev-II filter was found to be problematic. Additionally, in order to avoid a time shift between filtered and original profile, the average of the directly filtered signal and the time reversed filtered signal is considered. The effects of these complicated filters on the scaling behavior are, however, not fully understood. Finally, a continuous DFA method has been suggested in the context of studying heartbeat data during sleep [56, 57]. The method compares unnormalized fluctuation functions F2 (s) for increasing length of the data. I. e., one starts with a very short recording and subsequently adds more points of data. The method is particularly suitable for the detection of change points in the data, e. g., physiological transitions between different activity or sleep stages. Since the main objective of the method is not the study of scaling behavior, we do not discuss it in detail here. Centered Moving Average (CMA) Analysis Particular attractive modifications of DFA are the detrended moving average (DMA) methods, where running averages replace the polynomial fits. The first suggested version, the backward moving average (BMA) method [51,52,53], however, suffers from severe problems, because an artificial time shift of s between the original signal and the moving average is introduced. This time shift leads to an additional contribution to the detrended profile Y˜s ( j), which causes a larger fluctuation function F2 (s) in particular for small scales in the case of long-term correlated data. Hence, the scaling exponent ˛ is systematically underestimated [63]. In addition, the BMA method preforms even worse for data with trends [64], and its slope is limited by ˛ < 1 just as for the non-detrending method FA. It was soon recognized that the intrinsic error of BMA can be overcome by eliminating the artificial time shift. This leads to the centered moving average (CMA) method [54], where Y˜s ( j) is calculated as 1 Y˜s ( j) D Y( j)  s

(s1)/2 X

Y( j C i) ;

(24)

iD(s1)/2

replacing Eq. (20) while Eq. (21) and the rest of the DFA procedure described in Subsect. “Detrended Fluctuation Analysis (DFA)” stay the same. Unlike DFA, the CMA method cannot easily be generalized to remove linear and higher order trends in the data.

475

476

Fractal and Multifractal Time Series

It was recently proposed [43] that the scaling behavior of the CMA method is more stable than for DFA1 and MDFA1, suggesting that CMA could be used for reliable computation of ˛ even for scales s < 10 (without correction of any systematic deviations needed in DFA for this regime) and up to smax D N/2. The standard errors in determining the scaling exponent ˛ by fitting straight lines to the double logarithmic plots of F2 (s) have been studied in [43]; they are comparable with DFA1 (see end of Subsect. “Detrended Fluctuation Analysis (DFA)”). Regarding the determination of crossovers, CMA is comparable to DFA1. Ultimately, the CMA seems to be a good alternative to DFA1 when analyzing the scaling properties in short data sets without trends. Nevertheless for data with possible unknown trends we recommend the application of standard DFA with several different detrending polynomial orders in order to distinguish real crossovers from artificial crossovers due to trends. In addition, an independent approach (e. g., wavelet analysis) should be used to confirm findings of long-term correlations (see also Subsect. “Detection of Trends and Crossovers with DFA”). Methods for Multifractal Time Series Analysis This section describes the multifractal characterization of time series, for an introduction, see Subsect. “Multifractal Time Series”. The simplest type of multifractal analysis is based upon the standard partition function multifractal formalism, which has been developed for the multifractal characterization of normalized, stationary measures [6,12,65,66]. Unfortunately, this standard formalism does not give correct results for non-stationary time series that are affected by trends or that cannot be normalized. Thus, in the early 1990s an improved multifractal formalism has been developed, the wavelet transform modulus maxima (WTMM) method [67,68,69,70,71], which is based on wavelet analysis and involves tracing the maxima lines in the continuous wavelet transform over all scales. An important alternative is the multifractal DFA (MFDFA) algorithm [33], which does not require the modulus maxima procedure, and hence involves little more effort in programming than the conventional DFA. For studies comparing methods for detrending multifractal analysis (MFDFA and WTMM, see [33,72,73]. The Structure Function Approach and Singularity Spectra In the general multifractal formalism, one considers a normalized measure (t), t 2 [0; 1], and defines the box R tCs/2 ˜ s (t) D ts/2 (t 0 ) dt 0 in neighborhoods of probabilities 

(scale) length s  1 around t. The multifractal approach is then introduced by the partition function Z q (s) D

1/s1 X

˜ s [( C 1/2)s] s (q)  q

for s  1 ; (25)

D0

where (q) is the Renyi scaling exponent and q is a real parameter that can take positive as well as negative values. Note that (q) is sometimes defined with opposite sign (see, e. g., [6]). A record is called monofractal (or selfaffine), when the Renyi scaling exponent (q) depends linearly on q; otherwise it is called multifractal. The generalized multifractal dimensions D(q) (see also Subsect. “Multifractal Time Series”) are related to (q) by D(q) D (q)/(q  1), such that the fractal dimension of the support is D(0) D (0) and the correlation dimension is D(2) D (2). In time series, a discrete version has to be used, and the considered data (x i ), i D 1; : : : ; N may usually include negative values. Hence, setting N s D int(N/s) and P X(; s) D siD1 xsCi for  D 0; : : : ; N s  1 we can define [6,12], NX s 1

Z q (s) D

jX(; s)j q s (q)

for s > 1 :

(26)

D0

Inserting the profile Y( j) and FFA (; s) from Eqs. (10) and (11), respectively, we obtain Z q (s) D

NX s 1

˚

[Y(( C 1)s)  Y(s)]2

q/2

D0

D

Ns X

jFFA (; s)j :

(27)

D1

Comparing Eq. (27) with (13), we see that this multifractal approach can be considered as a generalized version of the fluctuation analysis (FA) method, where the exponent 2 is replaced by q. In particular we find (disregarding the summation over the second partition of the time series)

F2 (s)

1/2 1 Z2 (s) s [1C(2)]/2 Ns ) 2˛ D 1 C (2) D 1 C D(2):

(28)

We thus see that all methods for (mono-)fractal time analysis (discussed in Sect. “Methods for Stationary Fractal Time Series Analysis”and Sect. “Methods for Non-stationary Fractal Time Series Analysis”) in fact study the correlation dimension D(2) D 2˛ 1 D ˇ D 1 (see Eq. (15)).

Fractal and Multifractal Time Series

It is straightforward to define a generalized (multifractal) Hurst exponent h(q) for the scaling behavior of the qth moments of the fluctuations [65,66],

1/q 1 Fq (s) D Z2 (s) s [1C(q)]/q D s h(q) Ns 1 C (q) ) h(q) D q

d (q) dq

and

f (˛) D q˛  (q) :

(29)

(30)

Here, ˛ is the singularity strength or Hölder exponent (see also  Fractals and Multifractals, Introduction to in the encyclopedia), while f (˛) denotes the dimension of the subset of the series that is characterized by ˛. Note that ˛ is not the fluctuation scaling exponent in this section, although the same letter is traditionally used for both. Using Eq. (29), we can directly relate ˛ and f (˛) to h(q), ˛ D h(q)C qh0 (q)

and

Z(q; s) D

j max X

jL ( j ; s)j q :

(32)

jD1

with h(2) D ˛  H. In the following, we will use only h(2) for the standard fluctuation exponent (denoted by ˛ in the previous chapters), and reserve the letter ˛ for the Hölder exponent. Another way to characterize a multifractal series is the singularity spectrum f (˛), that is related to (q) via a Legendre transform [6,12], ˛D

up the qth power of the maxima,

The reason for the maxima procedure is that the absolute wavelet coefficients jL (; s)j can become arbitrarily small. The analyzing wavelet (x) must always have positive values for some x and negative values for other x, since it has to be orthogonal to possible constant trends. Hence there are always positive and negative terms in the sum (16), and these terms might cancel. If that happens, jL (; s)j can become close to zero. Since such small terms would spoil the calculation of negative moments in Eq. (32), they have to be eliminated by the maxima procedure. In fluctuation analysis, on the other hand, the calculation of the variances F 2 (; s), e. g. in Eq. (11), involves only positive terms under the summation. The variances cannot become arbitrarily small, and hence no maximum procedure is required for series with compact support. In addition, the variances will increase if the segment length s is increased, because the fit will usually be worse for a longer segment. In the WTMM method, in contrast, the absolute

f (˛) D q[˛  h(q)]C1: (31)

Wavelet Transform Modulus Maxima (WTMM) Method The wavelet transform modulus maxima (WTMM) method [67,68,69,70,71] is a well-known method to investigate the multifractal scaling properties of fractal and selfaffine objects in the presence of non-stationarities. For applications, see e. g. [74,75]. It is based upon the wavelet transform with continuous basis functions as defined in Subsect. “Wavelet Analysis”, Eq. (16). Note that in this case the series x˜ i are analyzed directly instead of the profile Y( j) defined in Eq. (10). Using wavelets orthogonal to mth order polynomials, the corresponding trends are eliminated. Instead of averaging over all wavelet coefficients L (; s), one averages, within the modulo-maxima method, only the local maxima of jL (; s)j. First, one determines for a given scale s, the positions  j of the local maxima of jW(; s)j as a function of , so that jL ( j  1; s)j < jL ( j ; s)j  jL ( j C1; s)j for j D 1; : : : ; jmax . This maxima procedure is demonstrated in Fig. 7. Then one sums

Fractal and Multifractal Time Series, Figure 7 Example of the wavelet transform modulus maxima (WTMM) method, showing the original data (top), its continuous wavelet transform (gray scale coded amplitude of wavelet coefficients, middle), and the extracted maxima lines (bottom) (figure taken from [68])

477

478

Fractal and Multifractal Time Series

wavelet coefficients jL (; s)j need not increase with increasing scale s, even if only the local maxima are considered. The values jL (; s)j might become smaller for increasing s since just more (positive and negative) terms are included in the summation (16), and these might cancel even better. Thus, an additional supremum procedure has been introduced in the WTMM method in order to keep the dependence of Z(q; s) on s monotonous. If, for a given scale s, a maximum at a certain position  j happens to be smaller than a maximum at  j0   j for a lower scale s 0 < s, then L ( j ; s) is replaced by L ( j0 ; s 0 ) in Eq. (32). Often, scaling behavior is observed for Z(q; s), and scaling exponents ˆ (q) can be defined that describe how Z(q; s) scales with s, Z(q; s) sˆ (q) :

(33)

The exponents ˆ (q) characterize the multifractal properties of the series under investigation, and theoretically they are identical with the (q) defined in Eq. (26) [67,68,69,71] and related to h(q) by Eq. (29). Multifractal Detrended Fluctuation Analysis (MFDFA) The multifractal DFA (MFDFA) procedure consists of five steps [33]. The first three steps are essentially identical to the conventional DFA procedure (see Subsect. “Detrended Fluctuation Analysis (DFA)” and Fig. 4). Let us assume that (x˜ i ) is a series of length N, and that this series is of compact support. The support can be defined as the set of the indices j with nonzero values x˜ j , and it is compact if x˜ j D 0 for an insignificant fraction of the series only. The value of x˜ j D 0 is interpreted as having no value at this j. Note that we are not discussing the fractal or multifractal features of the plot of the time series in a two-dimensional graph (see also the discussion in Subsect. “Fractality, SelfAffinity, and Scaling”), but analyzing time series as onedimensional structures with values assigned to each point. Since real time series always have finite length N, we explicitly want to determine the multifractality of finite series, and we are not discussing the limit for N ! 1 here (see also Subsect. “The Structure Function Approach and Singularity Spectra”). Step 1 Calculate the profile Y( j), Eq. (10), by integrating the time series. Step 2 Divide the profile Y( j) into N s D int(N/s) nonoverlapping segments of equal length s. Since the length N of the series is often not a multiple of the considered time scale s, the same procedure can be repeated starting from the opposite end. Thereby, 2N s segments are obtained altogether.

Step 3 Calculate the local trend for each of the 2N s segments by a least-square fit of the profile. Then determine the variance by Eqs. (20) and (21) for each segment  D 1; : : : ; 2Ns . Again, linear, quadratic, cubic, or higher order polynomials can be used in the fitting procedure, and the corresponding methods are thus called MFDFA1, MFDFA2, MFDFA3, . . . ) [33]. In (MF-)DFAm [mth order (MF-)DFA] trends of order m in the profile (or, equivalently, of order m  1 in the original series) are eliminated. Thus a comparison of the results for different orders of DFA allows one to estimate the type of the polynomial trend in the time series [35,36]. Step 4 Average over all segments to obtain the qth order fluctuation function ( ) 1/q 2N q/2 1 Xs  2 F (; s) : (34) Fq (s) D 2N s D1 DFAm This is the generalization of Eq. (13) suggested by the relations derived in Subsect. “The Structure Function Approach and Singularity Spectra”. For q D 2, the standard DFA procedure is retrieved. One is interested in how the generalized q dependent fluctuation functions Fq (s) depend on the time scale s for different values of q. Hence, we must repeat steps 2 to 4 for several time scales s. It is apparent that Fq (s) will increase with increasing s. Of course, Fq (s) depends on the order m. By construction, Fq (s) is only defined for s  m C 2. Step 5 Determine the scaling behavior of the fluctuation functions by analyzing log-log plots Fq (s) versus s for each value of q. If the series x˜ i are long-range powerlaw correlated, Fq (s) increases, for large values of s, as a power-law, Fq (s) s h(q)

with

h(q) D

1 C (q) : q

(35)

For very large scales, s > N/4, Fq (s) becomes statistically unreliable because the number of segments N s for the averaging procedure in step 4 becomes very small. Thus, scales s > N/4 should be excluded from the fitting procedure determining h(q). Besides that, systematic deviations from the scaling behavior in Eq. (35), which can be corrected, occur for small scales s  10. The value of h(0), which corresponds to the limit h(q) for q ! 0, cannot be determined directly using the averaging procedure in Eq. (34) because of the diverging exponent. Instead, a logarithmic averaging procedure has to be employed, ( ) 2N 1 Xs  2 F0 (s) D exp ln F (; s) s h(0) : (36) 4N s D1

Fractal and Multifractal Time Series

Note that h(0) cannot be defined for time series with fractal support, where h(q) diverges for q ! 0. For monofractal time series with compact support, h(q) is independent of q, since the scaling behavior of the 2 variances FDFAm (; s) is identical for all segments , and the averaging procedure in Eq. (34) will give just this identical scaling behavior for all values of q. Only if small and large fluctuations scale differently, there will be a significant dependence of h(q) on q. If we consider positive values of q, the segments  with large variance F 2 (; s) (i. e. large deviations from the corresponding fit) will dominate the average Fq (s). Thus, for positive values of q, h(q) describes the scaling behavior of the segments with large fluctuations. On the contrary, for negative values of q, the 2 segments  with small variance FDFAm (; s) will dominate the average Fq (s). Hence, for negative values of q, h(q) describes the scaling behavior of the segments with small fluctuations. Figure 8 shows typical results obtained for Fq (s) in the MFDFA procedure. Usually the large fluctuations are characterized by a smaller scaling exponent h(q) for multifractal series than the small fluctuations. This can be understood from the following arguments. For the maximum scale s D N the fluctuation function Fq (s) is independent of q, since the sum in Eq. (34) runs over only two identical segments. For smaller scales s  N the averaging procedure runs over several segments, and the average value Fq (s) will be dominated by the F 2 (; s) from the segments with small (large) fluctuations if q < 0 (q > 0). Thus, for s  N, Fq (s) with

Fractal and Multifractal Time Series, Figure 8 Multifractal detrended fluctuation analysis (MFDFA) of data from the binomial multifractal model (see Subsect. “The Extended Binomial Multifractal Model”) with a D 0:75. Fq (s) is plotted versus s for the q values given in the legend; the slopes of the curves correspond to the values of h(q). The dashed lines have the slopes of the theoretical slopes h(˙1) from Eq. (42). 100 configurations have been averaged

q < 0 will be smaller than Fq (s) with q > 0, while both become equal for s D N. Hence, if we assume an homogeneous scaling behavior of Fq (s) following Eq. (35), the slope h(q) in a log-log plot of Fq (s) with q < 0 versus s must be larger than the corresponding slope for Fq (s) with q > 0. Thus, h(q) for q < 0 will usually be larger than h(q) for q > 0. However, the MFDFA method can only determine positive generalized Hurst exponents h(q), and it already becomes inaccurate for strongly anti-correlated signals when h(q) is close to zero. In such cases, a modified (MF-)DFA technique has to be used. The most simple way to analyze such data is to integrate the time series before the MFDFA procedure. Following the MFDFA procedure as described above, we obtain a generalized fluctuation ˜ functions described by a scaling law with h(q) D h(q) C 1. The scaling behavior can thus be accurately determined even for h(q) which are smaller than zero for some values of q. The accuracy of h(q) determined by MFDFA certainly depends on the length N of the data. For q D ˙10 and data with N D 10,000 and 100,000, systematic and statistical error bars (standard deviations) up to h(q)  ˙0:1 and  ˙0:05 should be expected, respectively [33]. A difference of h(10)  h(C10) D 0:2, corresponding to an even larger width ˛ of the singularity spectrum f (˛) defined in Eq. (30) is thus not significant unless the data was longer than N D 10,000 points. Hence, one has to be very careful when concluding multifractal properties from differences in h(q). As already mentioned in the introduction, two types of multifractality in time series can be distinguished. Both of them require a multitude of scaling exponents for small and large fluctuations: (i) Multifractality of a time series can be due to a broad probability density function for the values of the time series, and (ii) multifractality can also be due to different long-range correlations for small and large fluctuations. The most easy way to distinguish between these two types is by analyzing also the corresponding randomly shuffled series [33]. In the shuffling procedure the values are put into random order, and thus all correlations are destroyed. Hence the shuffled series from multifractals of type (ii) will exhibit simple random behavior, hshuf (q) D 0:5, i. e. non-multifractal scaling. For multifractals of type (i), on the contrary, the original h(q) dependence is not changed, h(q) D hshuf (q), since the multifractality is due to the probability density, which is not affected by the shuffling procedure. If both kinds of multifractality are present in a given series, the shuffled series will show weaker multifractality than the original one.

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Fractal and Multifractal Time Series, Figure 9 Illustration for the definition of return intervals rq between extreme events above two quantiles (thresholds) q1 and q2 (figure by Jan Eichner)

Comparison of WTMM and MFDFA The MFDFA results turn out to be slightly more reliable than the WTMM results [33,72,73]. In particular, the MFDFA has slight advantages for negative q values and short series. In the other cases the results of the two methods are rather equivalent. Besides that, the main advantage of the MFDFA method compared with the WTMM method lies in the simplicity of the MFDFA method. However, contrary to WTMM, MFDFA is restricted to studies of data with full one-dimensional support, while WTMM is not. Both WTMM and MFDFA have been generalized for higher dimensional data, see [34] for higher dimensional MFDFA and, e. g., [71] for higher dimensional WTMM. Studies of other generalizations of detrending methods like the discrete WT approach (see Subsect. “Discrete Wavelet Transform (WT) Approach”) and the CMA method (see Subsect. “Centered Moving Average (CMA) Analysis”) are currently under investigation [76]. Statistics of Extreme Events in Fractal Time Series The statistics of return intervals between well defined extremal events is a powerful tool to characterize the temporal scaling properties of observed time series and to derive quantities for the estimation of the risk for hazardous events like floods, very high temperatures, or earthquakes. It was shown recently that long-term correlations represent a natural mechanism for the clustering of the hazardous events [77]. In this section we will discuss the most important consequences of long-term correlations and fractal scaling of time series upon the statistics of extreme events [77,78,79,80,81]. Corresponding work regarding multifractal data [82] is not discussed here. Return Intervals Between Extreme Events To study the statistics of return intervals we consider again a time series (x i ), i D 1; : : : ; N with fractal scaling behavior, sampled homogeneously and normalized to zero mean

and unit variance. For describing the reoccurrence of rare events exceeding a certain threshold q, we investigate the return intervals rq between these events, see Fig. 9. The average return interval R q D hr q i increases as a function of the threshold q (see, e. g. [83]). It is known that for uncorrelated records (‘white noise’), the return intervals are also uncorrelated and distributed according to the Poisson distribution, Pq (r) D R1q exp(r/R q ). For fractal (long-term correlated) data with auto-correlations following Eq. (5), we obtain a stretched exponential [77,78,79,80,84], Pq (r) D

 a exp b (r/R q ) : Rq

(37)

This behavior is shown in Fig. 10. The exponent is the correlation exponent from C(s), and the parameters a and b are independent of q. They can be determined from R (r), i. e., P (r) dr D1 the normalization conditions for P q q R and rPq (r) dr D R q . The form of the distribution (37) indicates that return intervals both well below and well above their average value Rq (which is independent of ) are considerably more frequent for long-term correlated than for uncorrelated data. It has to be noted that there are deviations from the stretched exponential law (37) for very small r (discretization effects and an additional power-law regime) and for very large r (finite size effects), see Fig. 10. The extent of the deviations from Eq. (37) depends on the distribution of the values xi of the time series. For a discussion of these effects, see [80]. Equation (37) does not quantify, however, if the return intervals themselves are arranged in a correlated or in an uncorrelated fashion, and if clustering of rare events may be induced by long-term correlations. To study this question, one has to evaluate the auto-covariance function Cr (s) D hr q (l)r q (l C s)i  R2q of the return intervals. The results for model data suggests that also the return intervals are long-term power-law correlated, with the same exponent as the original record. Accordingly, large and small return intervals are not arranged in a random fash-

Fractal and Multifractal Time Series

Fractal and Multifractal Time Series, Figure 10 Normalized rescaled distribution density functions Rq Pq (r) of r values with Rq D 100 as a function of r/Rq for long-term correlated data with  D 0:4 (open symbols) and  D 0:2 (filled symbols; we multiplied the data for the filled symbols by a factor 100 to avoid overlapping curves). In a the original data were Gaussian distributed, in b exponentially distributed, in c power-law distributed with power 5.5, and in d log-normally distributed. All four figures follow quite well stretched exponential curves (solid lines) over several decades. For small r/Rq values a power-law regime seems to dominate, while on large scales deviations from the stretched exponential behavior are due to finite-size effects (figure by Jan Eichner)

ion but are expected to form clusters. As a consequence, the probability of finding a certain return interval r depends on the value of the preceding interval r0 , and this effect has to be taken into account in predictions and risk estimations [77,80]. The conditional distribution function Pq (rjr0 ) is a basic quantity, from which the relevant quantities in risk estimations can be derived [83]. For example, the first moment of Pq (rjr0 ) is the average value R q (r0 ) of those return intervals that directly follow r0 . By definition, R q (r0 ) is the expected waiting time to the next event, when the two events before were separated by r0 . The more general quantity is the expected waiting time q (xjr0 ) to the next event, when the time x has elapsed. For x D 0, q (0jr0 ) is identical to R q (r0 ). In general, q (xjr0 ) is related to Pq (rjr0 ) by Z 1 (r  x)Pq (rjr0 )dr : (38) q (xjr0 ) D x Z 1 Pq (rjr0 )dr

Distribution of Extreme Events In this section we describe how the presence of fractal long-term correlations affects the statistics of the extreme events, i. e., maxima within time segments of fixed duration R, see Fig. 11 for illustration. By definition, extreme events are rare occurrences of extraordinary nature, such as, e. g. floods, very high temperatures, or earthquakes. In hydrological engineering such conventional extreme value statistics are commonly applied to decide what building projects are required to protect river-side areas against typical floods that occur, for example, once in 100 years. Most of these results are based on statistically independent values xi and hold only in the limit R ! 1. However, both of these assumptions are not strictly fulfilled for correlated fractal scaling data. In classical extreme value statistics one assumes that records (x i ) consist of i.i.d. data, described by density distributions P(x), which can be, e. g., a Gaussian or an exponential distribution. One is interested in the distribution

x

For uncorrelated records, q (xjr0 )/R q D 1 (except for discreteness effects that lead to q (xjr0 )/R q > 1 for x > 0, see [85]). Due to the scaling of Pq (rjr0 ), also q (xjr0 )/R q scales with r0 /R q and x/R q . Small and large return intervals are more likely to be followed by small and large ones, respectively, and hence q (0jr0 )/R q D R q (r0 )/R q is well below (above) one for r0 /R q well below (above) one. With increasing x, the expected residual time to the next event increases. Note that only for an infinite long-term correlated record, the value of q (xjr0 ) will increase indefinitely with x and r0 . For real (finite) records, there exists a maximum return interval which limits the values of x, r0 and q (xjr0 ).

Fractal and Multifractal Time Series, Figure 11 Illustration for the definition of maxima mR within periods of R D 365 values (figure by Jan Eichner)

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density function PR (m) of the maxima (m j ) determined in segments of length R in the original series (x i ), see Fig. 11. Note that all maxima are also elements of the original data. The corresponding integrated maxima distribution G R (m) is defined as Z m G R (m) D 1  E R (m) D PR (m0 )dm0 : (39)

previous maximum [81]. The last item implies that conditional mean maxima and conditional maxima distributions should be considered for improved extreme event predictions. Simple Models for Fractal and Multifractal Time Series

1

Fourier Filtering Since G R (m) is the probability of finding a maximum smaller than m, E R (m) denotes the probability of finding a maximum that exceeds m. One of the main results of traditional extreme value statistics states that for independently and identically distributed (i.i.d.) data (x i ) with Gaussian or exponential distribution density function P(x) the integrated distribution G R (m) converges to a double exponential (Fisher–Tippet–Gumbel) distribution (often labeled as Type I) [86,87,88,89,90], i. e., G R (m) ! G

m  u ˛

h  m  u i D exp  exp  (40) ˛

for R ! 1, where ˛ is the scale parameter and u the location parameter. By the method of moments those pap rameters are given by ˛ D 6/ R and u D m R  n e ˛ with the Euler constant n e D 0:577216 [89,91,92,93]. Here mR and  R denote the (R-dependent) mean maximum and the standard deviation, respectively. Note that different asymptotics will be reached for broader distributions of data (x i ) that belong to other domains of attraction [89]. For example, for data following a power-law distribution (or Pareto distribution), P(x) D (x/x0 )k , G R (m) converges to a Fréchet distribution, often labeled as Type II. For data following a distribution with finite upper endpoint, for example the uniform distribution P(x) D 1 for 0  x  1, G R (m) converges to a Weibull distribution, often labeled as Type III. We do not consider the latter two types of asymptotics here. Numerical studies of fractal model data have recently shown that the distribution P(x) of the original data has a much stronger effect upon the convergence towards the Gumbel distribution than the long-term correlations in the data. Long-term correlations just slightly delay the convergence of G R (m) towards the Gumbel distribution (40). This can be observed very clearly in a plot of the integrated and scaled distribution G R (m) on logarithmic scale [81]. Furthermore, it was found numerically that (i) the maxima series (m j ) exhibit long-term correlations similar to those of the original data (x i ), and most notably (ii) the maxima distribution as well as the mean maxima significantly depend on the history, in particular on the

Fractal scaling with long-term correlations can be introduced most easily into time series by the Fourier-filtering technique, see, e. g., [94,95,96]. The Fourier filtering technique is not limited to the generation of long-term correlated data characterized by a power-law auto-correlation function C(s) x  with 0 < < 1. All values of the scaling exponents ˛ D h(2)  H or ˇ D 2˛  1 can be obtained, even those that cannot be found directly by the fractal analysis techniques described in Sect. “Methods for Stationary Fractal Time Series Analysis” and Sect. “Methods for Non-stationary Fractal Time Series Analysis” (e. g. ˛ < 0). Note, however, that Fourier filtering will always yield Gaussian distributed data values and that no nonlinear or multifractal properties can be achieved (see also Subsect. “Multifractal Time Series”, Subsect. “Sign and Magnitude (Volatility) DFA”, and Sect. “Methods for Multifractal Time Series Analysis”). In Subsect. “Detection of Trends and Crossovers with DFA”, we have briefly described a modification of Fourier filtering for obtaining reliable short-term correlated data. For the generation of data characterized by fractal scaling with ˇ D 2˛  1 [94,95] we start with uncorrelated Gaussian distributed random numbers xi from an i.i.d. generator. Transforming a series of such numbers into frequency space with discrete Fourier transform or FFT (fast Fourier transform, for suitable series lengths N) yields a flat power spectrum, since random numbers correspond to white noise. Multiplying the (complex) Fourier coefficients by f ˇ /2 , where f / 1/s is the frequency, will rescale the power spectrum S( f ) to follow Eq. (6), as expected for time series with fractal scaling. After transforming back to the time domain (using inverse Fourier transform or inverse FFT) we will thus obtain the desired longterm correlated data x˜i . The final step is the normalization of this data. The Fourier filtering method can be improved using modified Bessel functions instead of the simple factors f ˇ /2 in modifying the Fourier coefficients [96]. This way problems with the divergence of the autocorrelation function C(s) at s D 0 can be avoided. An alternative method to the Fourier filtering technique, the random midpoint displacement method, is

Fractal and Multifractal Time Series

based on the construction of self-affine surfaces by an iterative procedure, see, e. g. [6]. Starting with one interval with constant values, the intervals are iterative split in the middle and the midpoint is displaced by a random offset. The amplitude of this offset is scaled according to the length of the interval. Since the method generates a selfaffine surface xi characterized by a Hurst exponent H, the differentiated series x i can be used as long-term correlated or anti-correlated random numbers. Note, however, that the correlations do not persist for the whole length of the data generated this way. Another option is the use of wavelet synthesis, the reverse of wavelet analysis described in Subsect. “Wavelet Analysis”. In that method, the scaling law is introduced by setting the magnitudes of the wavelet coefficients according to the corresponding time scale s. The Schmitz–Schreiber Method When long-term correlations in random numbers are introduced by the Fourier-filtering technique (see previous section), the original distribution P(x) of the time series values xi is always modified such that it becomes closer to a Gaussian. Hence, no series (x i ) with broad distributions of the values and fractal scaling can be generated. In these cases an iterative algorithm introduced by Schreiber and Schmitz [98,99] must be applied. The algorithm consists of the following steps: First one creates a Gaussian distributed long-term correlated data set with the desired correlation exponent by standard Fourier-filtering [96]. The power spectrum SG ( f ) D FG ( f )FG ( f ) of this data set is considered as reference spectrum (where f denotes the frequency in Fourier space and the FG ( f ) are the complex Fourier coefficients). Next one creates an uncorrelated sequence of random numbers (x ref i ), following a desired distribution P(x). The (complex) Fourier transform F( f ) of the (x ref i ) is now divided by its absolute value and multiplied by the square root of the reference spectrum, p F( f ) SG ( f ) : (41) Fnew( f ) D jF( f )j After the Fourier back-transformation of Fnew ( f ), the new sequence (x new i ) has the desired correlations (i. e. the desired ), but the shape of the distribution has changed towards a (more or less) Gaussian distribution. In order to enforce the desired distribution, we exchange the (x new i ) by the (x ref ), such that the largest value of the new set is i replaced by the largest value of the reference set, the second largest of the new set by the second largest of the reference set and so on. After this the new sequence has the desired distribution and is clearly correlated. However, due

to the exchange algorithm the perfect long-term correlations of the new data sequence were slightly altered again. So the procedure is repeated: the new sequence is Fourier transformed followed by spectrum adjustment, and the exchange algorithm is applied to the Fourier back-transformed data set. These steps are repeated several times, until the desired quality (or the best possible quality) of the spectrum of the new data series is achieved. The Extended Binomial Multifractal Model The multifractal cascade model [6,33,65] is a standard model for multifractal data, which is often applied, e. g., in hydrology [97]. In the model, a record xi of length N D 2n max is constructed recursively as follows. In generation n D 0, the record elements are constant, i. e. x i D 1 for all i D 1; : : : ; N. In the first step of the cascade (generation n D 1), the first half of the series is multiplied by a factor a and the second half of the series is multiplied by a factor b. This yields x i D a for i D 1; : : : ; N/2 and x i D b for i D N/2 C 1; : : : ; N. The parameters a and b are between zero and one, 0 < a < b < 1. One need not restrict the model to b D 1  a as is often done in the literature [6]. In the second step (generation n D 2), we apply the process of step 1 to the two subseries, yielding x i D a2 for i D 1; : : : ; N/4, x i D ab for i D N/4 C 1; : : : ; N/2, x i D ba D ab for i D N/2 C 1; : : : ; 3N/4, and x i D b2 for i D 3N/4 C 1; : : : ; N. In general, in step n C 1, each subseries of step n is divided into two subseries of equal length, and the first half of the xi is multiplied by a while the second half is multiplied by b. For example, in generation n D 3 the values in the eight subseries are a3 ; a2 b; a2 b; ab2 ; a2 b; ab2 ; ab2 ; b3 . After nmax steps, the final generation has been reached, where all subseries have length 1 and no more splitting is possible. We note that the final record can be written as x i D a n max n(i1) b n(i1) , where n(i) is the number of digits 1 in the binary representation of the index i, e. g. n(13) D 3, since 13 corresponds to binary 1101. For this multiplicative cascade model, the formula for (q) has been derived earlier [6,33,65]. The result is (q) D [ ln(a q C b q ) C q ln(a C b)]/ ln 2 or h(q) D

1 ln(a q C b q ) ln(a C b)  C : q q ln 2 ln 2

(42)

It is easy to see that h(1) D 1 for all values of a and b. Thus, in this form the model is limited to cases where h(1), which is the exponent Hurst defined originally in the R/S method, is equal to one. In order to generalize this multifractal cascade process such that any value of h(1) is possible, one can sub-

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tract the offset h D ln(a C b)/ ln(2) from h(q) [100]. The constant offset h corresponds to additional longterm correlations incorporated in the multiplicative cascade model. For generating records without this offset, we rescale the power spectrum. First, we transform (FFT) the simple multiplicative cascade data into the frequency domain. Then, we multiply all Fourier coefficients by f h , where f is the frequency. This way, the slope ˇ of the power spectra S( f ) f ˇ is decreased from ˇ D 2h(2)  1 D [2 ln(a C b)  ln(a2 C b2 )]/ ln 2 into ˇ 0 D 2[h(2)  h]  1 D  ln(a 2 C b2 )/ ln 2. Finally, backward FFT is employed to transform the signal back into the time domain. The Bi-fractal Model In some cases a simple bi-fractal model is already sufficient for modeling apparently multifractal data [101]. For bifractal records the Renyi exponents (q) are characterized by two distinct slopes ˛ 1 and ˛ 2 , ( q  q q˛1  1 (43) (q) D q˛2 C q (˛1  ˛2 )  1 q > q or ( (q) D

q˛1 C q (˛2  ˛1 )  1

q  q

q˛2  1

q > q

:

(44)

If this behavior is translated into the h(q) picture using Eq. (29), we obtain that h(q) exhibits a plateau from q D 1 up to a certain q and decays hyperbolically for q > q , ( q  q ˛1 ; (45) h(q) D 1 q (˛1  ˛2 ) q C ˛2 q > q or vice versa, ( q (˛2  ˛1 ) 1q C ˛1 h(q) D ˛2

q  q q > q

:

(46)

Both versions of this bi-fractal model require three parameters. The multifractal spectrum is degenerated to two single points, thus its width can be defined as ˛ D ˛1  ˛2 . Future Directions The most straightforward future direction is to analyze more types of time series from other complex systems than those listed in Sect. “Introduction” to check for the presence of fractal scaling and in particular long-term correlations. Such applications may include (i) data that are not

recorded as a function of time but as a function of another parameter and (ii) higher dimensional data. In particular, the inter-relationship between fractal time series and spatially fractal structures can be studied. Studies of fields with fractal scaling in time and space have already been performed in Geophysics. In some cases studying new types of data will require dealing with more difficult types of non-stationarities and transient behavior, making further development of the methods necessary. In many studies, detrending methods have not been applied yet. However, discovering fractal scaling in more and more systems cannot be an aim on its own. Up to now, the reasons for observed fractal or multifractal scaling are not clear in most applications. It is thus highly desirable to study causes for fractal and multifractal correlations in time series, which is a difficult task, of course. One approach might be based on modeling and comparing the fractal aspects of real and modeled time series by applying the methods described in this article. The fractal or multifractal characterization can thus be helpful in improving the models. For many applications, practically usable models which display fractal or transient fractal scaling still have to be developed. One example for a model explaining fractal scaling might be a precipitation, storage and runoff model, in which the fractal scaling of runoff time series could be explained by fractional integration of rainfall in soil, groundwater reservoirs, or river networks characterized by a fractal structure. Also studies regarding the inter-relationship between fractal scaling and complex networks, representing the structure of a complex system, are desirable. This way one could gain an interpretation of the causes for fractal behavior. Another direction of future research is regarding the linear and especially non-linear inter-relationships between several time series. There is great need for improved methods characterizing cross-correlations and similar statistical inter-relationships between several non-stationary time series. Most methods available so far are reserved to stationary data, which is, however, hardly found in natural recordings. An even more ambitious aim is the (timedependent) characterization of a larger network of signals. In such a network, the signals themselves would represent the nodes, while the (possibly directed) inter-relationships between each pair represent the links (or bonds) between the nodes. The properties of both nodes and links can vary with time or change abruptly, when the represented complex system goes through a phase transition. Finally, more work will have to be invested in studying the practical consequences of fractal scaling in time series. Studies should particularly focus on predictions of future values and behavior of time series and whole complex

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systems. This is very relevant, not only in hydrology and climate research, where a clear distinguishing of trends and natural fluctuations is crucial, but also for predicting dangerous medical events on-line in patients based on the continuous recording of time series. Acknowledgment We thank Ronny Bartsch, Amir Bashan, Mikhail Bogachev, Armin Bunde, Jan Eichner, Shlomo Havlin, Diego Rybski, Aicko Schumann, and Stephan Zschiegner for helpful discussions and contribution. This work has been supported by the Deutsche Forschungsgemeinschaft (grant KA 1676/3) and the European Union (STREP project DAPHNet, grant 018474-2). Bibliography 1. Mandelbrot BB, van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Review 10: 422 2. Mandelbrot BB, Wallis JR (1969) Some long-run properties of geophysical records. Water Resour Res 5:321–340 3. Mandelbrot BB (1999) Multifractals and 1/f noise: wild selfaffinity in physics. Springer, Berlin 4. Hurst HE (1951) Long-term storage capacity of reservoirs. Tran Amer Soc Civ Eng 116:770 5. Hurst HE, Black RP, Simaika YM (1965) Long-term storage: an experimental study. Constable, London 6. Feder J (1988) Fractals. Plenum Press, New York 7. Barnsley MF (1993) Fractals everywhere. Academic Press, San Diego 8. Bunde A, Havlin S (1994) Fractals in science. Springer, Berlin 9. Jorgenssen PET (2000) Analysis and probability: Wavelets, signals, fractals. Springer, Berlin 10. Bunde A, Kropp J, Schellnhuber HJ (2002) The science of disasters – climate disruptions, heart attacks, and market crashes. Springer, Berlin 11. Kantz H, Schreiber T (2003) Nonlinear time series analysis. Cambridge University Press, Cambridge 12. Peitgen HO, Jürgens H, Saupe D (2004) Chaos and fractals. Springer, Berlin 13. Sornette D (2004) Critical phenomena in natural sciences. Springer, Berlin 14. Peng CK, Mietus J, Hausdorff JM, Havlin S, Stanley HE, Goldberger AL (1993) Long-range anti-correlations and non-Gaussian behaviour of the heartbeat. Phys Rev Lett 70: 1343 15. Bunde A, Havlin S, Kantelhardt JW, Penzel T, Peter JH, Voigt K (2000) Correlated and uncorrelated regions in heart-rate fluctuations during sleep. Phys Rev Lett 85:3736 16. Vyushin D, Zhidkov I, Havlin S, Bunde A, Brenner S (2004) Volcanic forcing improves atmosphere-ocean coupled general circulation model scaling performance. Geophys Res Lett 31:L10206 17. Koscielny-Bunde E, Bunde A, Havlin S, Roman HE, Goldreich Y, Schellnhuber HJ (1998) Indication of a universal persistence law governing atmospheric variability. Phys Rev Lett 81:729

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Fractals in Biology SERGEY V. BULDYREV Department of Physics, Yeshiva University, New York, USA

Article Outline Glossary Definition of the Subject Introduction Self-similar Branching Structures Fractal Metabolic Rates Physical Models of Biological Fractals Diffusion Limited Aggregation and Bacterial Colonies Measuring Fractal Dimension of Real Biological Fractals Percolation and Forest Fires Critical Point and Long-Range Correlations Lévy Flight Foraging Dynamic Fractals Fractals and Time Series SOC and Biological Evolution Fractal Features of DNA Sequences Future Directions Bibliography Glossary Allometric laws An allometric law describes the relationship between two attributes of living organisms y and x, and is usually expressed as a power-law: y x ˛ , where ˛ is the scaling exponent of the law. For example, x can represent total body mass M and y can represent the mass of a brain mb . In this case mb M 3/4 . Another example of an allometric law: B M 3/4 where B is metabolic rate and M is body mass. Allometric laws can be also found in ecology: the number of different species N found in a habitat of area A scales as N A1/4 . Radial distribution function Radial distribution function g(r) describes how the average density of points of a set behaves as function of distance r from a point of this set. For an empirical set of N data points, the distances between all pair of points are computed and the number of pairs Np (r) such that their distance is less then r is found. Then M(r) D 2Np (r)/N gives the average number of the neighbors (mass) of the set within a distance r. For a certain distance bin r1 < r < r2 , we define g[(r2 C r1 )/2] D [M(r2 )  M(r1 )]/[Vd (r2 )  Vd (r1 )], where

Vd (r) D 2 d/2 r d /[d (d/2)] is the volume/area/length of a d-dimensional sphere/circle/interval of radius r. Fractal set We define a fractal set with the fractal dimension 0 < df < d as a set for which M(r) r d f for r ! 1. Accordingly, for such a set g(r) decreases as a power law of the distance g(r) r , where  D d  df . Correlation function For a superposition of a fractal set and a set with a finite density defined as D limr!1 M(r)/Vd (r), the correlation function is defined as h(r)  g(r)  . Long-range power law correlations The set of points has long-range power law correlations (LRPLC) if h(r) r for r ! 1 with 0 <  < d. LRPLC indicate the presence of a fractal set with fractal dimension df D d   superposed with a uniform set. Critical point Critical point is defined as a point in the system parameter space (e. g. temperature, T D Tc , and pressure P D Pc ), near which the system acquires LRPLC h(r)

1 r d2C

exp(r/) ;

(1)

where  is the correlation length which diverges near the critical point as jT  Tc j . Here  > 0 and  > 0 are critical exponents which depend on the few system characteristics such as dimensionality of space. Accordingly, the system is characterized by fractal density fluctuations with df D 2  . Self-organized criticality Self-organized criticality (SOC) is a term which describes a system for which the critical behavior characterized by a large correlation length is achieved for a wide range of parameters and thus does not require special tuning. This usually occurs when a critical point corresponds to an infinite value of a system parameter, such as a ratio of the characteristic time of the stress build up and a characteristic time of the stress release. Morphogenesis Morphogenesis is a branch of developmental biology concerned with the shapes of organs and the entire organisms. Several types of molecules are particularly important during morphogenesis. Morphogens are soluble molecules that can diffuse and carry signals that control cell differentiation decisions in a concentration-dependent fashion. Morphogens typically act through binding to specific protein receptors. An important class of molecules involved in morphogenesis are transcription factor proteins that determine the fate of cells by interacting with DNA. The morphogenesis of the branching fractal-like structures such as lungs involves a dozen of morpho-

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genes. The mechanism for keeping self-similarity of the branches at different levels of branching hierarchy is not yet fully understood. The experiments with transgenic mice with certain genes knocked-out produce mice without limbs and lungs or without terminal buds.

for each model and therefore may shed light on the origin of a particular biological phenomenon. In Sect. “Diffusion Limited Aggregation and Bacterial Colonies”, we discuss the techniques for measuring fractal dimension and their limitations. Introduction

Definition of the Subject Fractals occur in a wide range of biological applications: 1) In morphology when the shape of an organism (tree) or an organ (vertebrate lung) has a self-similar brunching structure which can be approximated by a fractal set (Sect. “Self-Similar Branching Structures”). 2) In allometry when the allometric power laws can be deduced from the fractal nature of the circulatory system (Sect. “Fractal Metabolic Rates”). 3) In ecology when a colony or a habitat acquire fractal shapes due to some SOC processes such as diffusion limited aggregation (DLA) or percolation which describes forest fires (Sects. “Physical Models of Biological Fractals”–“Percolation and Forest Fires”). 4) In epidemiology when some of the features of the epidemics is described by percolation which in turn leads to fractal behavior (Sect. “Percolation and Forest Fires”). 5) In behavioral sciences, when a trajectory of foraging animal acquires fractal features (Sect. “Lévy Flight Foraging”). 6) In population dynamics, when the population size fluctuates chaotically (Sect. “Dynamic Fractals”). 7) In physiology, when time series have LRPLC (Sect. “Fractals and Time Series”). 8) In evolution theory, which may have some features described by SOC (Sect. “SOC and Biological Evolution”). 9) In bioinformatics when a DNA sequence has a LRPLC or a network describing protein interactions has a selfsimilar fractal behavior (Sect. “Fractal Features of DNA Sequences”). Fractal geometry along with Euclidian geometry became a part of general culture which any scientist must be familiar with. Fractals often originate in the theory of complex systems describing the behavior of many interacting elements and therefore have a great number of biological applications. Complex systems have a general tendency for self-organization and complex pattern formation. Some of these patterns have certain nontrivial symmetries, for example fractals are characterized by scale invariance i. e. they look similarly on different magnification. Fractals are characterized by their fractal dimension, which is specific

The fact that simple objects of Euclidian geometry such as straight lines, circles, cubes, and spheres are not sufficient to describe complex biological shapes has been known for centuries. Physicists were always accused by biologists for introducing a “spherical cow”. Nevertheless people from antiquity to our days were fascinated by finding simple mathematical regularities which can describe the anatomy and physiology of leaving creatures. Five centuries ago, Leonardo da Vinci observed that “the branches of a tree at every stage of its height when put together are equal in thickness to the trunk” [107]. Another famous but still poorly understood phenomenon is the emergence of the Fibonacci numbers in certain types of pine cones and composite flowers [31,134]. In the middle of the seventies a new concept of fractal geometry was introduced by Mandelbrot [79]. This concept was readily accepted for analysis of the complex shapes in the biological world. However, after initial splash of enthusiasm, [14,40,41,74,75,76,88,122] the application of fractals in biology significantly dwindled and today the general consensus is that a “fractal cow” is often not much better than a “spherical cow”. Nature is always more complex than mathematical abstractions. Strictly speaking the fractal is an object whose mass, M, grows as a fractional power law of its linear dimension, L, M Ldf ;

(2)

where df is a non-integer quantity called fractal dimension. Simple examples of fractal objects of various fractal dimensions are given by iterative self-similar constructs such as a Cantor set, a Koch curve, a Sierpinski gasket, and a Menger sponge [94]. In all this constructs a next iteration of the object is created by arranging p exact copies of the previous iteration of the object (Fig. 1) in such a way that the linear size of the next iteration is q times larger than the linear size of the previous iteration. Thus the mass, M n , and the length, Ln , of the nth iteration scale as M n D p n M0 L n D q n L0 ;

(3)

where M 0 and L0 are mass and length of the zero order

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Fractals in Biology, Figure 1 a Cantor set (p D 2, q D 3, df D ln 2/ ln 3  0:63) is an example of fractal “dust” with fractal dimension less then 1; b Fractal tree (p D 3, q D 2, df D ln 3/ ln 2  1:58) with branches removed so that only the terminal points (leaves) can be seen. c The same tree with the trunks added. Both trees are produced by recursive combining of the three smaller trees: one serving as the top of the tree and the other two rotated by 90ı clockwise and counter-clockwise serving as two brunches joined with the top at the middle. In c a vertical segment representing a trunk of the length equal to the diagonal of the branches is added at each recursive step. Mathematically, the fractal dimensions of sets b and c are the same, because the mass of the tree with trunks (number of black pixels) for the system of linear size 2n grows as 3nC1  2nC1 , while for the tree without trunks mass scales simply as 3n . In the limit n ! 1 this leads to the same fractal dimension ln 3/ ln 2. However, visual inspection suggests that the tree with branches has a larger fractal dimension. This is in accord with the box counting method, which produces a higher value of the slope for the finite tree with the trunks. The slope slowly converges to the theoretical value as the number of recursive steps increases. d Sierpinski gasket has the same fractal dimension as the fractal tree (p D 3, q D 2) but has totally different topology and visual appearance than the tree

iteration. Excluding n from Eq. (3), we get  d f Ln ; M n D M0 L0

Self-similar Branching Structures (4)

where df D ln p/ ln q

(5)

can be identified as fractal dimension. The above-described objects have the property of self-similarity, the previous iteration magnified q times looks exactly like the next iteration once we neglect coarse-graining on the lowest iteration level, which can be assumed to be infinitely small. An interesting feature of such fractals is the power law distribution of their parts. For example, the cumulative distribution of distances L between the points of Cantor set and the branch lengths of the fractal tree (Fig. 1) follows a power law: P(L > x) x d f :

(6)

Thus the emergence of power law distributions and other power law dependencies is often associated with fractals and often these power law regularities not necessarily related to geometrical fractals are loosely referred as fractal properties.

Perfect deterministic fractals described above are never observed in nature, where all shapes are subjects to random variations. The natural examples closest to the deterministic fractals are structures of trees, certain plants such as a cauliflower, lungs and a cardiovascular system. The control of branching morphogenesis [85,114,138] involves determining when and where a branch will occur, how long the tube grows before branching again, and at what angle the branch will form. The development of different organs (such as salivary gland, mammary gland, kidney, and lung) creates branching patterns easily distinguished from each other. Moreover, during the development of a particular organ, the form of branching often changes, depending on the place or time when the branching occurs. Morphogenesis is controlled by complex molecular interactions of morphogenes. Let us illustrate the idea of calculating fractal dimension of a branching object such as a cauliflower [51,64]. If we assume (which is not quite correct, see Fig. 2) that each branch of a cauliflower give rise to exactly p branches of the next generation exactly q times smaller than the original branch applying Eq. (5), we get df D ln q/ ln p. Note

Fractals in Biology

Fractals in Biology, Figure 2 a Cauliflower anatomy. A complete head of a cauliflower (left) and after it is taken apart (right). We remove 17 branches until the diameter of the remaining part becomes half of the diameter of the original head. b Diameters of the branches presented in a. c Estimation of the fractal dimension for asymmetric lungs ( D 3) and trees ( D 2) with branching parameter r using Eq. (7). For r > 0:855 or r < 0:145 the fractal dimension is not defined due to the “infrared catastrophe”: the number of large branches diverge in the limit of an infinite tree

that this is not the fractal dimension of the cauliflower itself, but of its skeleton in which each brunch is represented by the elements of its daughter branches, with an addition of a straight line connecting the daughter branches as in the example of a fractal tree (Fig. 1b, c). Because this addition does not change the fractal dimension formula, the fractal dimension of the skeleton is equal to the

fractal dimension of the surface of a cauliflower, which can be represented as the set of the terminal branches. As a physical object, the cauliflower is not a fractal but a three-dimensional body so that the mass of a branch of length L scales as L3 . This is because the diameter of each branch is proportional to the length of the branch.

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The distribution of lengths of the collection of all the branches of a cauliflower is also a power law given by P k n a simple idea that there are N n D n1 kD0 p  p /(p  1) branches of length larger than L n D L0 qn . Excluding n d gives N n (L > L n ) L n f . In reality, however (Fig. 2a, b), the branching structure of a cauliflower is more complex. Simple measurements show that there are about p D 18 branches of sizes L k D L0  L0 [0:5  0:2(k  1)] for k D 1; 2; : : :; p, where L0 is the diameter of the complete head of the cauliflower. (We keep removing branches until the diameter of the central remaining part is equal to the half diameter of the complete head and we assume that it is similar to the rest of the branches). Subsequent generations (we count at least eight) obey similar rules, however p has a tendency to decrease with the generation number. To find fractal dimension of a cauliflower we will count the number of branches N(L) larger than certain size L. As in case of equal branches, this number scales as Ld f . Calculations, similar to those presented in [77] show that in this case the fractal dimension is equal to 0s 1  @ 2 2 1  ln p 2  1A ; (7) df D  2   where  and  are the mean and the standard deviation of ln q k  ln(L0 /L k ). For the particular cauliflower shown in Fig. 2a the measurements presented in Fig. 2b give df D 2:75, which is very close to the estimate df D 2:8, of [64]. The physiological reason for such a peculiar branching pattern of cauliflower is not known. It is probably designed to store energy for a quick production of a dense inflorescence. For a tree, [142] the sum of the cross-section areas of the two daughter branches according to Leonardo is equal to the cross-section area of the mother branch. This can be understood because the number of capillary bundles going from the mother to the daughters is conserved. Accordingly, we have a relation between the daughter branch diameters d1 and d2 and the mother branch diameter d0 , 





d1 C d2 D d0 ;

(8)

where  D 2. We can assume that the branch diameter of the largest daughter branch scales as d1 D rd0 , where r is asymmetry ratio maintaining for each generation of branches. In case of a symmetric tree d1 D d2 , r D 1/2. If we assume that the branch length L D sd, where s is a constant which is the same for any branch of the tree, then we can relate our tree to a fractal model with p D 2 and q D 21/ . Using Eq. (5) we get df D  D 2, i. e. the tree skeleton or the surface of the terminal branches is

an object of fractal dimension two embedded in the threedimensional space. This is quite natural because all the leaves whose number is proportional to the number of terminal branches must be exposed to the sunlight and the easiest way to achieve this is to place all the leaves on the surface of a sphere which is a two-dimensional object. For an asymmetric tree the fractal dimension can be computed using Eq. (7) with  D j ln[r(1  r)]/2j

(9)

and  D j ln[r/(1  r)]/2j :

(10)

This value is slightly larger than 2 for a wide range of r (Fig. 2c). This property may be related to a tendency of a tree to maximize the surface of it leaves, which must not be necessarily exposed to the direct sunlight but can suffice on the light reflected by the outer leaves. For a lung [56,66,77,102,117], the flow is not a capillary but can be assumed viscous. According to flow conservation, the sum of the air flows of the daughter branches must be equal to the flow of the mother branch: Q1 C Q2 D Q0 :

(11)

For the Poiseuille flow, Q Pd 4 /L, where P is the pressure drop, which is supposed to be the same for the airways of all sizes. Assuming that the lung maintains the ratio s D L/d in all generations of branches, we conclude that the diameters of the daughter and mother branches must satisfy Eq. (8) with  D 4  1 D 3. Accordingly, for a symmetrically branching lung df D  D 3, which means that the surface of the alveoli which is proportional to the number of terminal airways scales as L3 , i. e. it is a space filling object. Again this prediction is quite reasonable because nature tends to maximize the gas exchange area so that it completely fills the volume of the lung. In reality, the flow in the large airways is turbulent, and the parameter  of lungs of different species varies between 2 and 3 [77]. Also the lungs are known to be asymmetric and the ratio r D Q1 /Q0 ¤ 1/2 changes from one generation of the airways to the next [77]. However, Eq. (7) with  and  given by Eqs. (9) and (10) shows that the fractal dimension of an asymmetric tree remains very close to 3 for a wide range of 0:146 < r < 0:854 (Fig. 2c). The fact that the estimated fractal dimension is slightly larger than 3 does not contradict common sense because the branching stops when the airway diameter becomes smaller than some critical cutoff. Other implications of the lung asymmetry are discussed in [77]. An interesting idea

Fractals in Biology

was proposed in [117], according to which the dependence of the branching pattern on the generation of the airways can be derived from optimization principles and give rise to the complex value of the fractal dimension. An interesting property of the crackle sound produced by diseased lungs is that during inflation the resistance to airflow of the small airways decreases in discrete jumps. Airways do not open individually, but in a sequence of bursts or avalanches involving many airways; both the size of these jumps and the time intervals between jumps follow power-law distributions [128]. These avalanches are not related to the SOC as one might expect, but their power law distributions directly follow from the branching structure of lungs (see [3] and references therein).

Fractal Metabolic Rates An important question in biology is the scaling of the metabolic rate with respect to the body mass (Kleiber’s Law). It turns out that for almost all species of animals, which differ in terms of mass by 21 orders of magnitudes, the metabolic rate B scales as B M 3/4 [28,112,140]. The scatter plot ln B vs ln M is a narrow cloud concentrated around the straight line with a slope 3/4 (Fig. 3). This is one of the many examples of the allometric laws which describe the dependence of various biological parameters on the body mass or population size [1,9,87,105,106]. A simple argument based on the idea that the thermal energy loss and hence the metabolic rate should be propor-

Fractals in Biology, Figure 3 Dependence of metabolic rate on body mass for different types of organisms (Kleiber’s Law, after [112]). The least square fit lines have slopes 0:76 ˙ 0:01

tional to the surface area predicts however that B L2 M 2/3 . Postulating that the metabolic rate is proportional to some effective metabolic area leads to a strange prediction that this area should have a fractal dimension of 9/4. Many attempts has been made to explain this interesting fact [10,32,142,143]. One particular attempt [142] was based on the ideas of energy optimization and the fractal organization of the cardiovascular system. However, the arguments were crucially dependent on such details as turbulent vs laminar flow, the elasticity of the blood vessels and the pulsatory nature of the cardiovascular system. The fact that this derivation does not work for species with different types of circulatory systems suggests that there might be a completely different and quite general explanation of this phenomenon. Recently [10], as the theory of networks has became a hot subject, a general explanation of the metabolic rates was provided using a mathematical theorem that the total flow Q (e. g. of the blood) in the most efficient supply network must scale as Q BL/u, where L is the linear size of the network, B is the total consumption rate (e. g. metabolic rate), and u is the linear size of each consumption unit (e. g. cell). The authors argue that the total flow is proportional to the total amount of the liquid in the network, which must scale as the body mass M. On the other hand, L M 1/3 . If one assumes that u is independent of the body mass, then B M 2/3 , which is identical to the simple but incorrect prediction based on the surface area. In order to make ends meet, the authors postulate that some combination of parameters must be independent of the body size, from where it follows that u M 1/12 . If one identifies a consumption unit with the cell, it seems that the cell size must scale as L1/4 . Thus the cells of a whale (L D 30 m) must be about 12 times larger than those of a C. elegance (L D 1 mm), which more or less consistent with the empirical data [110] for slowly dividing cells such as neurons in mammals. However, the issue of the metabolic scaling is still not fully resolved [11,33,144]. It is likely that the universal explanation of Kleiber’s law is impossible. Moreover, recently it was observed that Kleiber’s law does not hold for plants [105]. An interesting example of an allometric law which may have some fractal implications is the scaling of the brain size with the body mass Mb M 3/4 [1]. Assuming that the mass of the brain is proportional to the mass of the neurons in the body, we can conclude that if the average mass of a neuron does not depend on the body mass, neurons must form a fractal set with the fractal dimension 9/4. However, if we assume that the mass of a neuron must scale with the body mass as M 1/12 , as it follows from [10,110], we can conclude that the number of neu-

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rons in the body scales simply as M 2/3 L2 , which means that the number of neurons is proportional to the surface area of the body. The latter conclusion is physiologically more plausible than the former, since it is obvious that the neurons are more likely to be located near the surface of an organism. From the point of view of comparative zoology, the universal scaling of the brain mass is not as important as the deviations from it. It is useful [1] to characterize an organism by the ratio of its actual brain mass to its expected brain mass which is defined as Eb D AM 3/4 , where A is a constant measured by the intercept of the log–log graph of brain mass versus body mass. Homo sapiens have the largest Mb /Eb D 8, several times larger than those of gorillas and chimpanzees. Physical Models of Biological Fractals The fractals discussed in Sect. “Self-Similar Branching Structures”, which are based on the explicit rules of construction and self-similarity, are useful concepts for analyzing branching structures in the living organisms whose development and growth are programmed according to these rules. However, in nature, there are many instances when fractal shapes emerge just from general physical principles without a special fractal “blueprint” [26,34, 58,124,125,131]. One of such examples particularly relevant to biology is the diffusion limited aggregation (DLA) [145]. Another phenomenon, described by the same equations as DLA and thus producing similar shapes is viscous fingering [17,26,131]. Other examples are a random walk (RW) [42,58,104,141], a self-avoiding walk (SAW) [34,58], and a percolation cluster [26,58,127]. DLA explains the growth of ramified inorganic aggregates which sometimes can be found in rock cracks. The aggregates grow because certain ions or molecules deposit on the surface of the aggregate. These ions come to the surface due to diffusion which is equivalent on the microscopic level to Brownian motion of individual ions. The Brownian trajectories can be modeled as random walks in which the direction of each next step is independent of the previous steps. The Brownian trajectory itself has fractal properties expressed by the Einstein formula: r2 D 2dtD ;

(12)

where r2 is the average square displacement of the Brownian particle during time t, D is the diffusion coefficient and d is the dimensionality of the embedding space. The number of steps, n, of the random walk is proportional to time of the Brownian motion t D n, where  is the

Fractals in Biology, Figure 4 A random walk of n D 216 steps is surrounded by its p D p 4 parts of m D n/p D 214 steps magnified by factor of q D 4 D 2. One can see that the shapes and sizes of each of the magnified parts are similar to the shape and size of the whole

average duration of the random walk step. The diffusion coefficient can be expressed as D D r2 (m)/(2dm), where r2 (m) is the average square displacement during m time steps. Accordingly, p r(n) D pr(m) ; (13) which shows that the random walk of n steps consists of p D n/m copies of the one-step walks arranged in space in such a way pthat the average linear size of this arrangement is q D n/m times larger than the size of the m-step walk (Fig. 4). Applying Eq. (5), we get df D 2 which does not depend on the embedding space. It is the same in one-dimensional, two-dimensional, and three-dimensional space. Note that Brownian trajectory is self-similar only in a statistical sense: each of the n concatenated copies of a m-step trajectory are different from each other, but their average square displacements are the same. Our eye can easily catch this statistical self-similarity. A random walk is a cloudy object of an elongated shape, its inertia ellipsoid is characterized by the average ratios of the squares of its axis: 12.1/2.71/1 [18]. The Brownian trajectory itself has a relevance in biology, since it may describe the changing of the electrical potential of a neuron [46], spreading of a colony [68], the foraging trajectory of a bacteria or an animal as well as the motion of proteins and other molecules in the cell. The self-avoiding walk (SAW) which has a smaller fractal dimension (df,SAW D 4/3 in d D 2 and df,SAW  1:7 in

Fractals in Biology

Fractals in Biology, Figure 5 A self-avoiding walk (a) of n D 104 in comparison with a random walk (b) of the same number of steps, both in d D 3. Both walks are produced by molecular dynamic simulations of the bead-on-a-string model of polymers. The hard core diameter of monomers in the SAW is equal to the bond length `, while for the RW it is zero. Comparing their fractal dimensions df,SAW  1:7, and df,RW D 2, one can predict that the average size (radius of inertia) of SAW must be n1/df,SAW 1/df,RW  2:3 times larger than that of RW. Indeed the average radius of inertia of SAW and RW are 98` and 40`, respectively. The fact that their complex shapes resemble leaving creatures was noticed by P. G. de Gennes in a cartoon published in his book “Scaling Concepts in Polymer Physics” [34]

d D 3) is a model of a polymer (Fig. 5) in a good solvent, such as for example a random coil conformation of a protein. Thus a SAW provides another example of a fractal object which has certain relevance in molecular biology. The fractal properties of SAWs are well established in the works of Nobel Laureates Flory and de Gennes [34,44,63]. Diffusion Limited Aggregation and Bacterial Colonies The properties of a Brownian trajectory explain the ramified structure of the DLA cluster. Since the Brownian trajectory is not straight, it is very difficult for it to penetrate deep into the fiords of a DLA cluster. With a much higher probability it hits the tips of the branches. Thus DLA cluster (Fig. 6a) has a tree-like structure with usually 5 main branches in 2 dimensions. The analytical determination of its fractal dimension is one of the most challenging questions in modern mathematics. In computer simulations it is defined by measuring the number of aggregated particles versus the radius of gyration of the aggregate (Fig. 6c). The fractal dimension thus found is approximately 1.71

Fractals in Biology, Figure 6 a A DLA cluster of n D 214 particles produced by the aggregation of the random walks on the square lattice (top). A comparison of a small BD cluster (b) and a DLA cluster (c). The color in b and c indicates the deposition time of a particle. The slopes of the loglog graphs of the mass of the growing aggregates versus their gyration radius give the values of their fractal dimensions in the limit of large mass

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Fractals in Biology, Figure 7 A typical DLA-like colony grown in a Petri dish with a highly viscous substrate (top). A change in morphology from DLA-like colony growth to a swimming chiral pattern presumably due to a cell morphotype transition (bottom) (from [15])

Fractals in Biology

in two dimensions and 2.50 in three dimensions [34]. It seems that as the aggregate grows larger the fractal dimension slightly increases. Note that if the deposition is made by particles moving along straight lines (ballistic deposition) the aggregate changes its morphology and becomes almost compact and circular with small fluctuation on the boundaries and small holes in the interior (Fig. 6b). The fractal dimension of the ballistic aggregate coincides with the dimension of embedding space. Ballistic deposition (BD) belongs to the same universality class as the Eden model, the first model to describe the growth of cancer. In this model, each cell on the surface of the cluster can produce an offspring with equal probability. This cell does not diffuse but occupies one of the empty spaces neighboring to the parent cell [13,83]. DLA is supposed to be a good model for growth of bacterial colonies in the regime when nutrients are coming via diffusion in a viscous media from the exterior. Under different conditions bacteria colonies observe various transition in their morphology. If the nutrient supply is greater, the colonies increase their fractal dimension and start to resemble BD. An interesting phenomenon is observed upon changing the viscosity of the substrate [15]. If the viscosity is small it is profitable for bacteria to swim in order to get to the regions with high nutrient concentrations. If the viscosity is large it is more profitable to grow in a static colony which is supplied by the diffusion of the nutrient from the outside. It seems that the way how the colony grows is inherited in the bacteria gene expression. When a bacteria grown at high viscosity is planted into a low viscosity substrate, its descendants continue to grow in a DLA pattern until a transition in a gene-expression happens in a sufficiently large number of neighboring bacteria which change their morphotype to a swimming one (Fig. 7). All the descendants of these bacteria start to swim and thus quickly take over the slower growing morphotype. Conversely, when a swimming bacteria is planted into a viscous media its descendants continue to swim until a reverse transition of a morphotype happens and the descendants of the bacteria with this new morphotype start a DLA-like growing colony. The morphotype transition can be also induced by fungi. Thus, it is likely that although bacteria are unicellular organisms, they exchange chemical signals similarly to the cells in the multicellular organisms, which undergo a complex process of cell differentiation during organism development (morphogenesis). There is evidence that the tree roots also follow the DLA pattern, growing in the direction of diffusing nutrients [43]. Coral reefs [80] whose life depends on the diffu-

sion of oxygen and nutrients also may to a certain degree follow DLA or BD patterns. Another interesting conjecture is that neuronal dendrites grow in vivo obeying the same mechanism [29]. In this case, they follow signaling chemicals released by other cells. It was also conjectured that even fingers of vertebrate organisms may branch in a DLA controlled fashion, resembling the pattern of viscous fingering which are observed when a liquid of low viscosity is pushed into a liquid of higher viscosity. Measuring Fractal Dimension of Real Biological Fractals There were attempts to measure fractal dimension of growing neurons [29] and other biological objects from photographs using a box-counting method and a circle method developed for empirical determination of fractal dimension. The circle method is the simplified version of the radial distribution function analysis described in the definition section. It consists of placing a center of a circle or a sphere of a variable radius R at each point of an object and counting the average number of other points of this object M(R) found inside these circles or spheres. The slope of ln M(R) vs. ln R gives an estimate of the fractal dimension. The box counting method consists of placing a square grid of a given spacing ` on a photograph and counting number of boxes n(`) needed to cover all the points belonging to a fractal set under investigation. Each box containing at least one point of a fractal set is likely to contain on average N(`) D `d f other points of this set, and n(`)N(`) D N, where N is the total number of the points. Thus for a fractal image n(`) N/`d f `d f . Accordingly the fractal dimension of the image can be estimated by the slope of a graph of ln n(`) vs. ln `. The problem is that real biological objects do not have many orders of self-similarity, and also, as we see above, only a skeleton or a perimeter of a real object has fractal properties. The box counting method usually produces curvy lines on a log–log paper with at best one decade of approximately constant slope (Fig. 8). What is most disappointing is that almost any image analyzed in this fashion produces a graph of similar quality. For example, an Einstein cartoon presented in [93] has the same fractal dimension as some of the growing neurons. Therefore, experimental determination of the fractal dimension from the photographs is no longer in use in scientific publications. However, in the 1980s and early 1990s when the computer scanning of images became popular and the fractals were at the highest point of their career, these exercises were frequent.

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Fractals in Biology, Figure 8 An image of a neuron and its box-counting analysis (a) together with the analogous analysis of an Einstein cartoon (b). The log–log plot of the box-counting data has a shape of a curve with the changing slope. Bars represent the local slope of this graph, which may be interpreted as some effective fractal dimension. However, in both graphs the slopes change dramatically from 1 to almost 1.7 and it is obvious that any comparison to the DLA growth mechanism based on these graphs is invalid (from [93])

For example, R. Voss [137] analyzed fractal dimensions of Chinese graphics of various historical periods. He found that the fractal dimension of the drawings fluctuated from century to century and was about 1.3 at the time of the highest achievements of this technique. It was argued that the perimeters of many natural objects such as perimeters of the percolation clusters and random walks have similar fractal dimension; hence the works of the artists who were able to reproduce this feature were the most pleasant for our eyes. Recently, the box counting method was used to analyze complex networks such as protein interaction networks (PIN). In this case the size of the box was defined as the maximal number of edges needed to connect any two nodes belonging to the box. The box-counting method appears to be a powerful tool for analyzing self-similarity of the network structure [120,121].

Percolation and Forest Fires It was also conjectured that biological habitats of certain species may have fractal properties. However it is not clear whether we have a true self-similarity or just an apparent mosaic pattern which is due to complex topography and geology of the area. If there is no reasonable theoretical model of an ecological process, which displays true fractal properties the assertions of the fractality of a habitat are pointless. One such model, which produces fractal clusters is percolation [26,127]. Suppose a lightning hits a random tree in the forest. If the forest is sufficiently dry, and the inflammable trees are close enough the fire will spread from a tree to a tree and can burn the entire forest. If the inflammable trees form a connected cluster, defined as such that its any two trees are connected with a path along which the fire can spread, all the trees in this cluster will

Fractals in Biology

be burnt down. Percolation theory guarantees that there is a critical density pc of inflammable trees below which all the connected clusters are finite and the fire will naturally extinguish burning only an infinitesimally small portion of the forest. In contrast, above this threshold there exists a giant cluster which constitutes a finite portion of the forest so that the fire started by a random lightning will on average destroy a finite portion of the forest. Exactly at the critical threshold, the giant cluster is a fractal with a fractal dimension df D 91/48  1:89. The probability to burn a finite cluster of mass S follows a power law P(S) S d/d f . Above and below the percolation threshold, this distribution is truncated by an exponential factor which specifies that clusters of linear dimensions exceeding a characteristic size are exponentially rare. The structure of the clusters which do not exceed this characteristic scale is also fractal, so if their mass S is plotted versus their linear size R (e. g. radius of inertia) it follows a power law S Rdf In the natural environment the thunderstorms happen regularly and they produce fractal patterns of burned down patches. However, if the density of trees is small, the patches are finite and the fires are confined. As the density the trees reaches the critical threshold, the next thunderstorm is likely to destroy the giant cluster of the forest, which will produce a bare patch of a fractal shape spreading over the entire forest. No major forest fire will happen until the new forest covers this patch creating a new giant cluster, because the remaining disconnected groves are of finite size. There is an evidence that the forest is the system which drives itself to a critical point [30,78,111] (Fig. 9). Suppose that each year there is certain number of lightning strokes per unit area nl . The average number of trees in a cluster is hSi D ajp  pc j , where D 43/18 is one of the critical exponents describing percolation, a is a constant, and p is the number of inflammable trees per unit area. Thus the number of trees destroyed annually per unit area is nl ajpc  p] . On the other hand, since the trees are growing, the number of trees per unit area increases annually by nt , which is the parameter of the ecosystem. At equilibrium, nl ajpc  p] D nt . Accordingly, the equilibrium density of trees must reach pe D pc  (anl /nt )1/ . For very low nl , pe will be almost equal to pc , and the chance that in the next forest fire a giant cluster which spans the entire ecosystem will be burned is very high. It can be shown that for a given chance c of hitting such a cluster in a random lightning stroke, the density of trees must reach p D pc  f (c)L1/ , where L is the linear size of the forest,  D 4/3 is the correlation critical exponent and f (c) > 0 is a logarithmically growing function of c.

Accordingly, if nt /nl > bL / , where b is some constant, the chance of getting a devastating forest fire is close to 100%. We have here a paradoxical situation: the more frequent are the forest fires, the least dangerous they are. This implies that the fire fighters should not extinguish small forest fires which will be contained by themselves. Rather they should annually cut a certain fraction of trees to decrease nt . As we see, the forest fires can be regarded as a self-organized critical (SOC) system which drive themselves towards criticality. As in many SOC systems, here there are two processes one is by several orders of magnitudes faster than the other. In this case they are the tree growing process and the lightning striking process. The model reaches the critical point if a tuning parameter nt /nl ! 1 and in addition nt ! 0 and L ! 1, which are quite reasonable assumptions. In a regular critical system a tuning parameter (e. g. temperature) must be in the vicinity of a specific finite value. In a SOC system the tuning parameter must be just large enough. There is an evidence, that the forest fires follow a power law distribution [109]. One can also speculate that the areas burned down by the previous fires shape fractal habitats for light loving and fire-resistant trees such as pines. The attempts to measure a fractal dimension of such habitats from aerial photographs are dubious due to the limitations discussed above and also because by adjusting a color threshold one can produce fractal-like clusters in almost any image. This artifact is itself a trivial consequence of the percolation theory. The frequently reported Zipf’s law for the sizes of colonies of various species including the areas and populations of the cities [146] usually arises not from fractality of the habitats but from preferential attachment growth in which the old colonies grow in proportion to their present population, while the new colonies may form with a small probability. The preferential attachment model is a very simple mechanism which can create a power-law distribution of colony sizes P(S) S 2 , where  is a small correction which is proportional to the probability of formation of new colonies [25]. Other examples of power laws in biology [67], such as distributions of clusters in the metabolic networks, the distribution of families of proteins, etc. are also most likely come from the preferential attachment or also, in some cases, can arise as artifacts of specifically selected similarity threshold, which brings a network under consideration to a critical point of the percolation theory. The epidemic spreading can also be described by a percolation model. In this case the contagious disease spreads from a person to a person as the forest fire spreads from

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Fractals in Biology, Figure 9 A sequence of frames of a forest fire model. Each tree occupies a site on 500  500 square lattice. At each time step a tree (a colored site) is planted at a randomly chosen empty site (black). Each 10,000 time steps a lightning strikes a randomly chosen tree and the forest fire eliminates a connected cluster of trees. The frames are separated by 60,000 time steps. The color code indicates the age of the trees from blue (young) to red (old). The initial state of the system is an empty lattice. As the concentration of trees reaches percolation threshold (frame 3) a small finite cluster is burned. However it does not sufficiently decrease the concentration of trees and it continues to build up until a devastating forest fire occurs between frames 3 and 4, with only few green groves left. Between frames 4 and 5 several lightnings hit these groves and they are burned down, while surrounding patch of the old fire continues to be populated by the new trees. Between frame 5 and 6 a new devastating forest fire occurs. At the end of the movie huge intermittent forest fires produce gigantic patches of dense and rare groves of various ages

a tree to a tree. Analogously to the forest fire model, a person who caught the disease dies or recovers and becomes immune to the disease, so he or she cannot catch it again. This model is called susceptible-infective-removed (SIR) [62]. The main difference is that the epidemics spread not on a two-dimensional plane but on the network describing contacts among people [61,86]. This network is usually supposed to be scale free, i. e. the number of contacts different people have (degree) is distributed according to a an inverse power law [2,71]. As the epidemic spreads the number of connections in the susceptible population depletes [115] so the susceptible population comes to a percolation threshold after which the epidemic stops.

This model explains for example why the Black Death epidemic stopped in the late 14th century after killing about one third of the total European population. Critical Point and Long-Range Correlations Percolation critical threshold [26,127] is an example of critical phenomena the most well known of which is the liquid-gas critical point [124]. As one heats a sample of any liquid occupying a certain fraction of a closed rigid container, part of the liquid evaporates and the pressure in the container increases so that the sample of liquid remains at equilibrium with its vapor. However, at certain tempera-

Fractals in Biology

ture the visible boundary between the liquid at the bottom and the gas on the top becomes fuzzy and eventually the system becomes completely nontransparent as milk. This phenomena is called critical opalescence and the temperature, pressure, and density at which it happens is called a critical point of this liquid. For water, the critical point is Tc D 374 Cı , Pc D 220 atm and c D 330 kg/m3 . As the temperature goes above the critical point the system becomes transparent again, but the phase boundary disappears: liquid and gas cannot coexist above the critical temperature. They form a single phase, supercritical fluid, which has certain properties of both gas (high compressibility) and liquid (high density and slow diffusivity). At the critical point the system consists of regions of high density (liquid like) and low density (gas like) of all sizes from the size of a single molecule to several microns across which are larger than the wave length of visible light  0:5 m. These giant density fluctuations scatter visible light causing the critical opalescence. The characteristic linear size of the density fluctuations is called the correlation length  which diverges at critical temperature as  jT  Tc j . The shapes of these density fluctuations are self-similar, fractal-like. The system can be represented as a superposition of a uniform set with the density corresponding to the average density of the system and a fractal set with the fractal dimension df . The density correlation function h(r) decreases for r ! 1 as r exp(r/), where   d  2 C  D d  df and d D 1; 2; 3; : : : is the dimension of space in which the critical behavior is observed. The exponential cutoff sets the upper limit of the fractal density fluctuations to be equal to the correlation length. The lower limit of the fractal behavior of the density fluctuations is one molecule. As the system approaches the critical point, the range of fractal behavior increases and can reach at least three orders of magnitude in the narrow vicinity of the critical point. The intensity of light scattered by density fluctuations at certain angle can be expressed via the Fourier transform of the density R correlation function S(f) h(r) exp(ir  f)dr, where the integral is taken over d-dimensional space. This experimentally observed quantity S(f ) is called the structure factor. It is a powerful tool to study the fractal properties of matter since it can uncover the presence of a fractal set even if it is superposed with a set of uniform density. If the density correlation function has a power law behavior h(r) r , the structure factor also has a power law behavior S( f ) f d . For the true fractals with df < d, the average density over the entire embedding space is zero, so the correlation function coincides with the average density of points of the set at certain distance from a given point, h(r) rdC1 dM(r)/dr r d f d , where M(r) r d f is

the mass of the fractal within the radius r from a given point. Thus, for a fractal set of points with fractal dimension df , the structure factor has a simple power law form S( f ) f d f . Therefore whenever S( f ) f ˇ , ˇ < d is identified with the fractal dimension and the system is said to have fractal density fluctuations even if the average density of the system is not zero. There are several methods of detecting spatial correlations in ecology [45,108] including box-counting [101]. In addition one can study the density correlation function h(r) of certain species on the surface of the Earth defined as h(r) D N(r)/2 rr  N/A, where N(r) is the average number of the representatives of a species within a circular rim of radius r and width r from a given representative, N is the total population, and A is the total area. For certain species, the correlation function may follow about one order of magnitude of a fractal (power-law) behavior [9,54,101]. One can speculate that there is some effective attraction between the individuals such as cooperation (mimicking the van der Walls forces between molecules) and a tendency to spread over the larger territory (mimicking the thermal motion of the particles). The interplay of these two tendencies may produce a fractal behavior like in the vicinity of the critical point. However, there is no reason of why the ecological system although complex and chaotic [81] should drive itself to a critical point. Whether or not there is a fractal behavior, the density correlation function is a useful way to describe the spatial distribution of the population. It can be applied not only in ecology but also in physiology and anatomy to describe the distribution of cells, for example, neurons in the cortex [22]. Lévy Flight Foraging A different set of mathematical models which produce fractal patterns and may be relevant in ecology is the Lévy flight and Lévy walk models [118,147]. The Lévy flight model is a generalization of a random walk, in which the distribution of the steps (flights) of length ` follows a power law P(`) ` with  < 3. Such distributions do not have a finite variance. The probability density of the landing points of the Lévy flight converges not to a Gaussian as for a normal random walk with a finite step variance but to a Lévy stable distribution with the parameter ˛    1. The landing points of a Lévy flight form fractal dust similar to a Cantor set with the fractal dimension df D ˛. It was conjectured that certain animals may follow Lévy flights during foraging [103,116,132,133]. It is a mathematical theorem [23,24] that in case when targets

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are scarce but there is a high probability that a new target could be discovered in the vicinity of the previously found target, the optimal strategy for a forager is to perform a Lévy flight with  D 2 C , where  is a small correction which depends on the density of the target sites, the radius of sight of the forager and the probability to find a new target in the vicinity of the old one. In this type of “inverse square” foraging strategy a balance is reached between finding new rich random target areas and returning back to the previously visited area which although depleted, may still provide some food necessary for survival. The original report [132] of the distribution of the flight times of wandering albatross was in agreement with the theory. However, recent work [39] using much longer flight time records and more reliable analysis showed that the distribution of flight times is better described by a Poisson distribution corresponding to regular random walk rather then by a power law. Several other reports of the Lévy flight foraging are also found dubious. The theory [23] however predicts that the inverse square law of foraging is optimal only in case of scarce sources distribution. Thus if the harvest is good and the food is plenty there is no reason to discover the new target locations and the regular random walk strategy becomes the most efficient. Subsequent observations show that power laws exist in some other marine predator search behavior [119]. In order to find a definitive answer, the study must be repeated for the course of many successive years characterized by different harvests. Once the miniature electronic devices for tracing animals are becoming cheaper and more efficient, a new branch of electronic ecology is emerging with the goal of quantifying migratory patterns and foraging strategies of various species in various environments. Whether or not the foraging patterns are fractal, this novel approach will help to establish better conservational policy with scientifically sound borders of wild-life reservations. Recent observations indicate that human mobility patterns might also possess Lévy flight properties [49].

pose that there is a population N t of a certain species enclosed in a finite habitat (e. g. island) at time t. Here t is an integer index which may denote year. Suppose that at time t C 1 the population becomes

Dynamic Fractals

Fractals and Time Series

It is not necessary that a fractal is a real geometrical object embedded in the regular three-dimensional space. Fractals can be found in the properties of the time series describing the behavior of the biological objects. One classical example of a biological phenomenon which under certain condition may display fractal properties is the famous logistic map [38,82,95,123] based on the ideas of a great British economist and demographer Robert Malthus. Sup-

As we can see in the previous section, even simple systems characterized by nonlinear feedbacks may display complex temporal behavior, which often becomes chaotic and sometimes to fractal. Obviously, such features must be present in the behavior of the nervous system, in particular in the human brain which is probably the most complex system known to contemporary science. Nevertheless, the source of fractal behavior can be sometimes trivial with-

N tC1 D bN t  dN t2 ; where b is the natural birth rate and d is the death rate caused by the competition for limited resources. In the most primitive model, the animals are treated as particles randomly distributed over area A annihilating at each time step if the distance between them is less than r. In this case d D  r2 /A. The normalized population x t D dN t /b obeys a recursive relation with a single parameter b: x tC1 D bx t (1  x t ) : The behavior of the population is quite different for different b. For b  1 the population dies out as t ! 1. For 1 < b  b0 D 3 the population converges to a stable size. If b n1 < b  b n , the population repetitively visits 2n values 0 < x1 ; : : : ; x2n < 1 called attractors as t ! 1. The bifurcation points 3 D b0 < b1 < b2 <    < b n < b1  3:569945672 converge to a critical value b1 , at which the set of population sizes becomes a fractal with fractal dimension df  0:52 [50]. This fractal set resembles a Cantor set confined between 0 and 1. For b1 < b < 4, the behavior is extremely complex. At certain values b chaos emerges and the behavior of the population become unpredictable, i. e. exponentially sensitive to the initial conditions. At some intervals of parameter b, the predictable behavior with a finite attractor set is restored. For b > 4, the population inevitably dies out. Although the set of attractors becomes truly fractal only at certain values of the birth rate, and the particular value of the fractal dimension does not have any biological meaning, the logistic map has a paradigmatic value in the studies of population dynamics with an essentially Malthusian take home massage: excessive birth rate leads to disastrous consequences such as famines coming at unpredictable times, and in the case of Homo sapiens to devastating wars and revolutions.

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out evidence of any cognitive ability. An a example of such a trivial fractal behavior is the random firing of a neuron [46] which integrates through its dendritic synapses inhibitory and excitatory signals from its neighbors. The action potential of such a neuron can be viewed as performing a one-dimensional random walk going down if an inhibitory signal comes from a synapse or going up if an excitatory signal comes form a different synapse. As soon as the action potential reaches a firing threshold the neuron fires, its action potential drops to an original value and the random walk starts again. Thus the time intervals between the firing spikes of such a neuron are distributed as the returning times of a one-dimensional random walk to an origin. It is well known [42,58,104,141], that the probability density P(t) of the random walk returns scales as (t) with  D 3/2. Accordingly, the spikes on the time axis form a fractal dust with the fractal dimension df D   1 D 1/2. A useful way to study the correlations in the time series is to compute its autocorrelation function and its Fourier transform which is called power spectrum, analogous to the structure factor for the spatial correlation analysis. Due to the property of the Fourier transform to convert a convolution into a product, the power spectrum is also equal to the square of the Fourier transform of the original time series. Accordingly it has a simple physical meaning telling how much energy is carried in a certain frequency range. In case of a completely uncorrelated signal, the power spectrum is completely flat, which means that all frequencies carry the same energy as in white light which is the mixture of all the rainbow colors of different frequencies. Accordingly a signal which has a flat power spectrum is called white noise. If the autocorrelation function C(t) decays for t ! 1 as C(t) t  , where 0 <  < 1, the power spectrum S(f ) of this time series diverges as f ˇ , with ˇ D 1  . Thus the LRPLC in the time series can be detected by studying the power spectrum. There are alternative ways of detecting LRPLC in time series, such as Hurst analysis [41], detrended fluctuation analysis (DFA) [19,98] and wavelet analysis [4]. These methods are useful for studying short time series for which the power spectrum is too noisy. They measure the Hurst exponent ˛ D (1 C ˇ)/2 D 1  /2. It can be shown [123] that for a time series which is equal to zero everywhere except at points t1 ; t2 ; : : : ; t n ; : : : , at which it is equal to unity and these points form a fractal set with fractal dimension df , the time autocorrelation function C(t) decreases for t ! 1 as C(t) t  , where  D 1  df . Therefore the power spectrum S(f ) of this time series diverges for f ! 0 as f ˇ , with ˇ D 1   D df . Accordingly, for the random

walk model of neuron firing, the power spectrum is characterized by ˇ D 1/2. In the more general case, the distribution of the intervals t n D t n  t n1 can decay as a power law P(t) (t) , with 1 <  < 3. For  < 2, the set t1 ; t2 ; : : : ; t n is a fractal, and the power spectrum decreases as power law S( f ) f d f D f C1 . For 2 <  < 3, the set t1 ; t2 ; : : : ; t n has a finite density, with df D D D 1. However, the power spectrum and the correlation function maintain their power law behavior for  < 3. This behavior indicates that although the time series itself is uniform, the temporal fluctuations remain fractal. In this case, the exponent ˇ characterizing the low frequency limit is given by ˇ D 3  . The maximal value of ˇ D 1 is achieved when  D 2. This type of signal is called 1/ f noise or “red” noise. If   3, ˇ D 0 in the limit of low frequencies and we again recover white noise. The physical meaning of 1/ f noise is that the temporal correlations are of infinite range. For the majority of the processes in nature, temporal correlations decay exponentially C(t) exp(t/), where the characteristic memory span  is called relaxation or correlation time. For a time series with a finite correlation time , the power spectrum has a Lorentzian shape, namely it stays constant for f < 1/ and decreases as 1/ f 2 for f > 1/. The signal in which S( f ) 1/ f 2 is called brown noise, because it describes the time behavior of the one-dimensional Brownian motion. Thus for the majority of the natural processes, the time spectrum has a crossover from a white noise at low frequencies to a brown noise at high frequencies. The relatively unusual case when S( f ) f ˇ , where 0 < ˇ < 1 is called fractal noise because as we see above it describes the behavior of the fractal time series. 1/ f -noise is the special type of the fractal noise corresponding to the maximal value of ˇ, which can be achieved in the limit of low frequencies. R. Voss and J. Clarke [135,136] had analyzed the music written by different composers and found that it follows 1/ f noise over at least three orders of magnitude. It means that music does not have a characteristic time-scale. There is an evidence that physiological process such as heart-beat, gate, breath and sleeping patterns as well as certain types of human activity such as sending e-mails has certain fractal features [6,16,20,27,47,55,57, 72,73,97,99,100,113,126,129]. A. Goldberger [99,126] suggested that music is pleasant for us because it mimics the fractal features of our physiology. It has to be pointed out that in all these physiological time series there is no clear power law behavior expanding over many orders of magnitude. There is also no simple clear explanation of the origins of fractality. One possible mechanism could be due

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the distribution of the return times of a random walk, which has been used to explain the sleeping patterns and response to the e-mails. SOC and Biological Evolution Self-organized criticality (SOC) [8,92] describes the behavior of the systems far form equilibrium, the general feature of which is a slow increase of strain which is interrupted by an avalanche-like stress release. These avalanches are distributed in a power law fashion. The power spectrum of an activity at a given spatial site is described by a fractal noise. One of the most successful application of SOC is the explanation of the power-law distribution of the magnitudes of the earthquakes (Gutenberg–Richter’s law) [89,130]. The simple physical models of self-organized criticality are invasion percolation [26,127], sand-pile model [8], and Bak–Sneppen model of biological evolution [7]. In the one-dimensional Bak–Sneppen model, an ecosystem is represented by a linear chain of the pray-predator relationships in with each species is represented by a site on a straight line surrounded by its predator (a site to the right) and a pray (a site to the left). Each site is characterized by its fitness f which at the beginning is uniformly distributed between 0 and 1. At each time step, a site with the lowest fitness becomes extinct and is replaced by a mutated species with a new fitness randomly taken from a uniform distribution between 0 and 1. The fitnesses of its two neighbors (predator and prey) are also changed at random. After certain equilibration time, the fitness of almost all the species except a few becomes larger than a certain critical value fc . These few active species with low fitness which can spontaneously mutate form a fractal set on the pray-predator line. The activity of each site can be represented by a time series of mutations shown as spikes corresponding to the times of individual mutations at this site. The power spectrum of this time series indicates the presence of the fractal noise. At a steady state the minimal fitness value which spontaneously mutates fluctuates below fc with a small probability P( ) comes into the interval between fc  and fc . The distribution of the first return times t to a given -vicinity of fc follows a power law with an -dependent exponential cut-off. Since each time step corresponds to a mutation, the time interval for which f stays below fc  corresponds to an avalanche of mutations caused by a mutation of a very stable species with f > fc  . Accordingly one can speculate, that evolution goes as a punctuated equilibrium so that an extinction of a stable species causes a gigantic extinction of many other species which hitherto have been well adopted. The problem with this SOC model is the def-

inition of a time step. In order to develop a realistic model of evolution one needs to assume that the real time needed for a spontaneous mutation of species with different fitness dramatically increases with f , going for example as exp( f A), where A is a large value. Unfortunately, it is impossible to verify the predictions of this model because paleontological records do not provide us with a sufficient statistics. Fractal Features of DNA Sequences DNA molecules are probably the largest molecules in nature [139]. Each strand of DNA in large human chromosomes consist of about 108 monomers or base-pairs (bp) which are adenine (A), cytosine (C), guanine (G), and thymine (T). The length of this molecule if stretched would reach several centimeters. The geometrical packing of DNA in a cell resembles a self-similar structure with at least 6 levels of packing: a turn of the double helix (10 bp), a nucleosome (200 bp), a unit of 30 nm fiber (6 nucleosomes), a loop domain ( 100 units of 30 nm fiber), a turn of a metaphase chromosome ( 100 loop domains), a metaphase chromosome ( 100 turns). The packing principle is quite similar to the organization of the information in the library: letters form lines, lines form pages, pages form books, books are placed on the shelves, shelves form bookcases, bookcases form rows, and rows are placed in different rooms. This structure however is not a rigorous fractal, because packing of the units on different levels follows different organization principles. The DNA sequence treated as a sequence of letters also has certain fractal properties [4,5,19,69,70,96]. This sequence can be transformed into a numerical sequence by several mapping rules: for example A rule, in which A is replaced by 1 and C, T, G are replaced by 0, or SW rule in which strongly bonded bp (C and G) are replaced by 1 and weakly bonded bp (A and T) are replaced by  1. Purine-Pyrimidine (RY) mapping rule (A,G: +1; C,T:1) and KM mapping rule (A,C:+1; G,T -1) are also possible. Power spectra of such sequences display large regions of approximate power law behavior in the range from f D 102 to f D 108 . For SW mapping rule we has almost perfect 1/ f noise in the region of low frequencies (Fig. 10). This is not surprising because chromosomes are organized in a large CG rich patches followed by AT rich patches called isochors which extend over millions of bp. The changing slope of the power spectra for different frequency ranges clearly indicates that DNA sequences are also not rigorous fractals but rather a mosaic structures with different organization principles on different length-scales [60,98]. A possible relation between the frac-

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Fractals in Biology, Figure 10 a Power spectra for seven different mapping rules computed for the Homo sapiens chromosome XIV, genomic contig NT_026437. The result is obtained by averaging 1330 power spectra computed by fast Fourier transform for non-overlapping segments of length N D 216 D 65536. b Power spectra for SW, RY, and KM mapping rules for the same contig extended to the low frequency region characterizing extremely long range correlations. The extension is obtained by extracting low frequencies from the power spectra computed by FFT with N D 224  16  106 bp. Three distinct correlation regimes can be identified. High frequency regime (f < 0:003) is characterized by small sharp peaks. Medium frequency regime (0:5  105 < f < 0:003) is characterized by approximate power-law behavior for RY and SW mapping rules with exponent ˇM D 0:57. Low frequency regime (f < 0:5  105 ) is characterized by ˇ D 1:00 for SW rule. The high frequency regime for RY rule can be approximated by ˇH D 0:16 in agreement with the data of Fig. 11. c RY Power spectra for the entire genome of E. coli (bacteria), S. cerevisae (yeast) chromosome IV, H. sapiens (human) chromosome XIV and the largest contig (NT_032977.6) on the chromosome I; and C. elegans (worm) chromosome X. It can be clearly seen that the high frequency peaks for the two different human chromosomes are exactly the same, while they are totally different from the high frequency peaks for other organisms. These high frequency peaks are associated with the interspersed repeats. One can also notice the presence of enormous peaks for f D 1/3 in E. coli and yeast, indicating that their genomes do not have introns, so that the lengths of coding segments are very large. The C. elegans data can be very well approximated by power law correlations S(f) f 0:28 for 104 < f < 102 . d Log–log plot of the RY power spectrum for E. coli with subtracted white noise level versus jf 1/3j. It shows a typical behavior for a signal with finite correlation length, indicating that the distribution of the coding segments in E. coli has finite average square length of approximately 3  103 bp

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Fractals in Biology, Figure 11 RY Power spectra averaged over all eukaryotic sequences longer than 512 bp, obtained by FFT with window size 512. Upper curve is average over 29,453 coding sequences; lower curve is average over 33,301 noncoding sequences. For clarity, the power spectra are shifted vertically by arbitrary quantities. The straight lines are least squares fits for second decade (Region M). The values of ˇM for coding and noncoding DNA obtained from the slopes of the fits are 0.03 and 0.21, respectively (from [21])

tal correlations of DNA sequences and packing of DNA molecules was suggested in [52,53]. An intriguing property of the DNA of multicellular organisms is that an overwhelming portion of it (97% in case of humans) is not used for coding proteins. It is interesting that the percent of non-coding DNA increases with the complexity of an organism. Bacteria practically do not have non-coding DNA, and Yeast has only 30% of it. The coding sequences form genes ( 104 bp) which carry information for one protein. The genes of multicellular organisms are broken by many noncoding intervening sequences (introns). Only exons which are short ( 102 bp) coding sequences located between introns ( 103 bp) are eventually translated into a protein. The genes themselves are separated by very long intergenic sequences  105 bp. Thus the coding structure of DNA resembles a Cantor set. The purpose and properties of coding DNA are well understood. Each three consequent bp form a codon, which is translated into one amino acid of the protein sequence. Accordingly, the power spectrum computed for the coding DNA has a characteristic peak at f D 1/3 corresponding to the inverse codon length (Figs. 11 and 12). Coding DNA is highly conserved and the power spectra of coding DNA of different organisms are very similar and in case of mammals are indistinguishable (Fig. 12). The properties of noncoding DNA are very different. Non coding DNA contains a lot of useful information in-

Fractals in Biology, Figure 12 Comparison of the correlation properties of coding and non-coding DNA of different mammals. Shown RY power spectra averaged over all complimentary DNA sequences of Homo sapiens (HS-C) and Mus musculus (mouse) (MM-C). The complimentary DNA sequences are obtained from messenger RNA by reverse transcriptase and thus lack non-coding elements. They are characterized by huge peaks at f D 1/3, corresponding to the inverse codon length 3 bp. The power spectra for human and mouse are almost indistinguishable. Also shown RY power spectra of large continuously sequenced segments of chromosomes (contigs) of about 107 bp long for mouse (MM), human (HS) and chimpanzee (PT, Pan troglodytes). Their power spectra have different high frequency peaks absent in the coding DNA power spectra: a peak at f D 1/2, corresponding to simple repeats and several large peaks in the range from f D 1/3 to f D 1/100 corresponding to interspersed repeats. Notice that the magnitudes of these peaks are similar for humans and chimpanzees (although for humans they are slightly larger, especially the peak at 80 bp corresponding to the long interspersed repeats) and much larger than those of mouse. This means that mouse has much smaller number of the interspersed repeats than primates. On the other hand, mouse has much larger fraction of dimeric simple repeats indicated by the peak at f D 1/2

cluding protein binding sites controlling gene transcription and expression and other regulatory sequences. However its overwhelming fraction lacks any known purpose. This “junk” DNA is full of simple repeats such as CACACACA. . . as well as the interspersed repeats or retroposons which are virus-like sequences inserting themselves in great number of copies into the intergenic DNA. It is a great challenge of molecular biology and genetics to understand the meaning of non-coding DNA and even to learn how to manipulate it. It would be very interesting to create transgenic animals without non-coding DNA and test if their phenotype will differ from that of the wild type species. The non-coding DNA even for very closely

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related species can significantly differ (Fig. 12). The length of simple repeats varies even for close relatives. That is why simple repeats are used in forensic studies. The power spectra of non-coding DNA significantly differ from those of coding DNA. Non-coding DNA does not have a peak at f D 1/3. The presence of simple repeats make non-coding DNA more correlated than coding DNA on a scale from 10 to 100 bp [21] (Fig. 11). This difference observed in 1992 [70,96] lead to the hypothesis that noncoding DNA is governed by some mutation-duplication stochastic process which creates long-range (fractal) correlations, while the coding DNA lacks long-range correlations because it is highly conserved. Almost any mutation which happens in coding DNA alter the sequence of the corresponding protein and thus may negatively affect its function and lead to a non-viable or less fit organism. Researchers have proposed using the difference in the long-range correlations to find the coding sequences in the sea of non-coding DNA [91]. However this method appears to be unreliable and today the non-coding sequences are found with much greater accuracy by the bioinformatics methods based on sequence similarity to the known proteins. Bioinformatics has developed powerful methods for comparing DNA of different species. Even a few hundred bp stretch of the mitochondrial DNA of a Neanderthal man can tell that Neanderthals diverged from humans about 500 000 years ago [65]. The power spectra and other correlation methods such as DFA or wavelet analysis does not have such an accuracy. Never the less, power spectra of the large stretches of DNA carry important information on the evolutionary processes in the DNA. They are similar for different chromosomes of the same organism, but differ even for closely related species (Fig. 12). In this sense they can play a role similar to the role played by X-ray or Raman spectra for chemical substances. A quick look at them can tell a human and a mouse and even a human and a monkey apart. Especially interesting is the difference in peak heights produced by different interspersed repeats. The height of these peaks is proportional to the number of the copies of the interspersed repeats. According to this simple criterion, the main difference between humans and chimps is the insertion of hundreds of thousands of extra copies of interspersed repeats into human DNA [59]. Future Directions All the above examples show that there are no rigorous fractals in biology. First of all, there are always a lower and an upper cutoff of the fractal behavior. For example, a polymer in a good solvent which is probably the most

rigorous of all real fractals in biology has the lower cutoff corresponding to the persistence length comprised of a few monomers and the upper cutoff corresponding to the length of the entire polymer or to the size of the compartment in which it is confined. For the hierarchical structures such as trees and lungs, in addition to the obvious lower and upper cutoffs, the branching pattern changes from one generation to the next. In the DNA, the packing principles employ different mechanisms on the different levels of packing. In bacterial colonies, the lower cutoff is due to some tendency of bacteria to clump together analogous to surface tension, and the upper cutoff is due to the finite concentration of the nutrients, which originally are uniformly distributed on the Petri dish. In the ideal DLA case, the concentration of the diffusing particles is infinitesimally small, so at any given moment of time there is only one particle in the vicinity of the aggregate. The temporal physiological series are also not exactly self-similar, but are strongly affected by the daily schedule with a characteristic frequency of 24 h and shorter overtones. Some of these signals can be described with help of multifractals. Fractals in ecology are limited by the topographical features of the land which may be also fractal due to complex geological processes, so it is very difficult to distinguish whether certain features are caused by biology or geology. The measurements of the fractal dimension are hampered by the lack of statistics, the noise in the image or signal, and by the crossovers due to intrinsic features of the system which is differently organized on different length scales. So very often the mosaic organization of the system with patches of several fixed length scales can be mistakenly identified with the fractal behavior. Thus the use of fractal dimension or Hurst exponent for diagnostics or distinguishing some parts of the system from one another has a limited value. After all, the fractal dimension is only one number which is usually obtained from a slope of a least square linear fit of a log–log graph over a subjectively identified range of values. Other features of the power spectrum, such as peaks at certain characteristic frequencies may have more biological information than fractal dimension. Moreover, the presence of certain fractal behavior may originate from simple physical principles, while the deviation from it may indicate the presence of a nontrivial biological phenomenon. On the other hand, fractal geometry is an important concept which can be used for qualitative understanding of the mechanisms behind certain biological processes. The use of similar organization principles on different length scales is a remarkable feature, which is certainly em-

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ployed by nature to design the shape and behavior of living organisms. Though fractals themselves may have a limited value in biology, the theory of complex systems in which they often emerge continues to be a leading approach in understanding life. One of the most impelling challenges of modern interdisciplinary science which involves biology, chemistry, physics, mathematics, computer science, and bioinformatics is to build a comprehensive theory of morphogenesis. Such a great mind as de Gennes turned to this subject in his late years [35,36,37]. In these studies the theory of complex networks [2,12,48,67,90] which describe interactions of biomolecules with many complex positive and negative feedbacks will take a leading part. Challenges of the same magnitude face researches in neuroscience, behavioral science, and ecology. Only complex interdisciplinary approach involving the specialists in the theory of complex systems may lead to new breakthroughs in this field.

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Fractals and Economics MISAKO TAKAYASU1 , HIDEKI TAKAYASU2 1 Tokyo Institute of Technology, Tokyo, Japan 2 Sony Computer Science Laboratories Inc, Tokyo, Japan Article Outline Glossary Definition of the Subject Introduction Examples in Economics Basic Models of Power Laws Market Models Income Distribution Models Future Directions Bibliography Glossary Fractal An adjective or a noun representing complex configurations having scale-free characteristics or selfsimilar properties. Mathematically, any fractal can be characterized by a power law distribution. Power law distribution For this distribution the probability density is given by a power law, p(r) D c  r˛1 , where c and ˛ are positive constants. Foreign exchange market A free market of currencies, exchanging money in one currency for other, such as purchasing a United States dollar (USD) with Japanese yen (JPY). The major banks of the world are trading 24 hours and it is the largest market in the world. Definition of the Subject Market price fluctuation was the very first example of fractals, and since then many examples of fractals have been found in the field of Economics. Fractals are everywhere in economics. In this article the main attention is focused on real world examples of fractals in the field of economics, especially market properties, income distributions, money flow, sales data and network structures. Basic mathematics and physics models of power law distributions are reviewed so that readers can start reading without any special knowledge. Introduction Fractal is the scientific word coined by B.B. Mandelbrot in 1975 from the Latin word fractus, meaning “fractured” [25]. However, fractal does not directly mean fracture itself. As an image of a fractal Fig. 1 shows a photo of

Fractals and Economics, Figure 1 Fractured pieces of plaster fallen on a hard floor (provided by H. Inaoka)

fractured pieces of plaster fallen on a hard floor. There are several large pieces, many middle size pieces and countless fine pieces. If you have a microscope and observe a part of floor carefully then you will find in your vision several large pieces, many small pieces and countless fine pieces, again in the microscopic world. Such scale-invariant nature is the heart of the fractal. There is no explicit definition on the word fractal, it generally means a complicated scale-invariant configuration. Scale-invariance can be defined mathematically [42]. Let P( r) denote the probability that the diameter of a randomly chosen fractured piece is larger than r, then this distribution is called scale-invariant if this function satisfies the following proportional relation for any positive scale factor  in a considering scale range: P( r) / P( r) :

(1)

The proportional factor should be a function of , so we can re-write Eq. (1) as P( r) D C()P( r) :

(2)

Assuming that P( r) is a differentiable function, and differentiate Eq. (2) by , and then let  D 1. rP 0 ( r) D C 0 (1)P( r)

(3)

As C 0 (1) is a constant this differential equation is readily integrated as P( r) D c0 r C

0 (1)

:

(4)

P( r) is a cumulative distribution and it is a non-increasing function in general, the exponent C 0 (1) can be replaced by ˛ where ˛ is a positive constant. Namely, from

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the scale-invariance with the assumption of differentiability we have the following power law: P( r) D c0 r˛ :

(5)

The reversed logic also holds, namely for any power law distribution there is a fractal configuration or a scale-invariant state. In the case of real impact fracture, the size distribution of pieces is experimentally obtained by repeating sieves of various sizes, and it is empirically well-known that a fractured piece’s diameter follows a power law with the exponent about ˛ D 2 independent of the details about the material or the way of impact [14]. This law is one of the most stubborn physical laws in nature as it is known to hold from 106 m to 105 m, from glass pieces around us to asteroids. From theoretical viewpoint this phenomenon is known to be described by a scale-free dynamics of crack propagation and the universal properties of the exponent value are well understood [19]. Usually fractal is considered geometric concept introducing the quantity fractal dimension or the concept of self-similarity. However, in economics there are very few geometric objects, so, the concept of fractals in economics are mostly used in the sense of power law distributions. It should be noted that any geometrical fractal object accompanies a power law distribution even a deterministic fractal such as Sierpinski gasket. Figure 2 shows Sierpinski gasket which is usually characterized by the fractal dimension D given by DD

log 3 : log 2

(6)

Paying attention to the distribution of length r of white triangles in this figure, it is easy to show that the probability that a randomly chosen white triangle’s side is larger than r, P( r), follows the power law, P( r) / r ˛ ;

˛DDD

Fractals and Economics, Figure 2 Sierpinski gasket

log 3 : log 2

(7)

Here, the power law exponent of distribution equals to the fractal dimension; however, such coincidence occurs only when the considering distribution is for a length distribution. For example, in Sierpinski gasket the area s of white triangles follow the power law, P( s) / s ˛ ;

˛D

log 3 : log 4

(8)

The fractal dimension is applicable only for geometric fractals, however, power law distributions are applicable for any fractal phenomena including shapeless quantities. In such cases the power law exponent is the most important quantity for quantitative characterization of fractals. According to Mandelbrot’s own review on his life the concept of fractal was inspired when he was studying economics data [26]. At that time he found two basic properties in the time series data of daily prices of New York cotton market [24]: (A) Geometrical similarity between large scale chart and an expanded chart. (B) Power law distribution of price changes in a unit time interval, which is independent of the time scale of the unit. He thought such scale invariance in both shape and distribution is a quite general property, not only in price charts but also in nature at large. His inspiration was correct and the concept of fractals spread over physics first and then over almost all fields of science. In the history of science it is a rare event that a concept originally born in economics has been spread widely to all area of sciences. Basic mathematical properties of cumulative distribution can be summarized as follows (here we consider distribution of non-negative quantity for simplicity): 1. P( 0) D 1 , P( 1) D 0. 2. P( r) is a non-increasing function of r. 3. The probability density is given as p(r)   drd P( r). As for power law distributions there are three peculiar characteristics: 4. Difficulty in normalization. Assuming that P( r) D c0 r˛ for all in the range 0  r < 1, then the normalization factor c0 must be 0 considering the limit of r ! 0. To avoid this difficulty it is generally assumed that the power law does not hold in the vicinity of r D 0. In the case of observing distribution from real data there are naturally lower and upper bounds, so this difficulty should be necessary only for theoretical treatment. 5. Divergence R 1of moments. As for moments defined by hr n i  0 r n p(r)dr, hr n i D 1 for n  ˛. In the

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special case of 2  ˛ > 0 the basic statistical quantity, the variance, diverges,  2  hr2 i  hri2 D 1. In the case of 1  ˛ > 0 even the average can not be defined as hri D 1. 6. Stationary or non-stationary? In view of the data analysis, the above characteristics of diverging moments is likely to cause a wrong conclusion that the phenomenon is non-stationary by observing its averaged value. For example, assume that we observe k samples fr1 ; r2 ; : : : ; r k g independently from the power law distribution with the exponent, 1  ˛ > 0. Then, the sample average, hri k  1k fr1 C r2 C    C r k g, is shown to diverge as, hri k / k 1/˛ . Such tendency of monotonic increase of averaged quantity might be regarded as a result of non-stationarity, however, this is simply a general property of a power law distribution. The best way to avoid such confusion is to observe the distribution directly from the data. Other than the power law distribution there is another important statistical quantity in the study of fractals, that is, the autocorrelation. For given time series, fx(t)g, the autocorrelation is defined as, C(T) 

hx(t C T)x(t)i  hx(t)i2 ; hx(t)2 i  hx(t)i2

(9)

where h   i denotes an average over realizations. The autocorrelation can be defined only for stationary time series with finite variance, in which any statistical quantities do not depend on the location of the origin of time axis. For any case, the autocorrelation satisfies the following basic properties, 1. 2. 3.

C(0) D 1 and C(1) D 0 jC(T)j  1 for any T  0. The R 1 Wiener–Khinchin theorem holds, C(T) D 0 S( f ) cos 2 f d f , where S(f ) is the power spectrum defined by S( f R)  hb x( f )b x( f )i, with the Fourier transform, b x( f )  x(t)e2 i f t dt.

In the case that the autocorrelation function is characterized by a power law, C(T) / T ˇ , ˇ > 0, then the time series fx(t)g is said to have a fractal property, in the sense that the autocorrelation function is scale-independent for any scale-factor,  > 0, C(T) / C(T). In the case 1 > ˇ > 0 the corresponding power spectrum is given as S( f ) / f 1Cˇ . The power spectrum can be applied to any time series including non-stationary situations. A simple way of telling non-stationary situation is to check the power law exponent of S( f ) / f 1Cˇ in the vicinity of f D 0, for 0 > ˇ the time series is non-stationary.

Three basic examples of fractal time series are the followings: 1. White noise. In the case that fx(t)g is a stationary independent noise, the autocorrelation is given by the Kronecker’s function, C(T) D ıT , where ( 1; T D 0 ıT D 0; T ¤ 0: The corresponding power spectrum is S( f ) / f 0 . This case is called white noise from an analogy that superposition of all frequency lights with the same amplitude make a colorless white light. White noise is a plausible model of random phenomena in general including economic activities. 2. Random walk. This is defined by summation of a white Pt x(s), and the power specnoise, X(t) D X(0) C sD0 trum is given by S( f ) / f 2 . In this case the autocorrelation function can not be defined because the data is non-stationary. Random walks are quite generic models widely used from Brownian motions of colloid to market prices. The graph of a random walk has a fractal property such that an expansion of any part of the graph looks similar to the whole graph. 3. The 1/f noise. The boundary of stationary and nonstationary states is given by the so-called 1/f noise, S( f ) / f 1 . This type of power spectrum is also widely observed in various fields of sciences from electrical circuit noise [16] to information traffics in the Internet [53]. The graph of this 1/f noise also has the fractal property. Examples in Economics In this chapter fractals observed in real economic activities are reviewed. Mathematical models derived from these empirical findings will be summarized in the next chapter. As mentioned in the previous chapter the very first example of a fractal was the price fluctuation of the New York cotton market analyzed by Mandelbrot with the daily data for a period of more than a hundred years [24]. This research attracted much attention at that time, however, there was no other good market data available for scientific analysis, and no intensive follow-up research was done until the 1990s. Instead of earnest scientific data analysis artificial mathematical models of market prices based on random walk theory became popular by the name of Financial Technology during the years 1960–1980. Fractal properties of market prices are confirmed with huge amount of high resolution market data since the 1990s [26,43,44]. This is due to informationization of fi-

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nancial markets in which transaction orders are processed by computers and detail information is recorded automatically, while until the 1980s many people gathered at a market and prices are determined by shouting and screaming which could not be recorded. Now there are more than 100 financial market providers in the world and the number of transacted items exceeds one million. Namely, millions of prices in financial markets are changing with time scale in seconds, and you can access any market price at real time if you have a financial provider’s terminal on your desk via the Internet. Among these millions of items one of the most representative financial markets is the US Dollar-Japanese Yen (USD-JPY) market. In this market Dollar and Yen are exchanged among dealers of major international banks. Unlike the case of stock markets there is no physical trading place, but major international banks are linked by computer networks and orders are emitted from each dealer’s terminal and transactions are done at an electronic broking system. Such a broking system and the computer networks are provided by financial provider companies like Reuters. The foreign exchange markets are open 24 hours and deals are done whenever buy- and sell-orders meet. The minimum unit of a deal is one million USD (called a bar), and about three million bars are traded everyday in the whole foreign exchange markets in which more than 100 kinds of currencies are exchanged continuously. The total amount of money flow is about 100 times bigger than the total amount of daily world trade, so it is believed that most of deals are done not for the real world’s needs, but they are based on speculative strategy or risk hedge, that is, to get profit by buying at a low price and selling at a high price, or to avoid financial loss by selling decreasing currency. In Fig. 3 the price of one US Dollar paid by Japanese Yen in the foreign exchange markets is shown for 13 years [30]. The total number of data points is about 20 million, that is, about 10 thousand per day or the averaged transaction interval is seven seconds. A magnified part of the top figure for one year is shown in the second figure. The third figure is the enlargement of one month in the second figure. The bottom figure is again a part of the third figure, here the width is one day. It seems that at least the top three figures look quite similar. This is one of the fractal properties of market price (A) introduced in the previous chapter. This geometrical fractal property can be found in any market, so that this is a very universal market property. However, it should be noted that this geometrical fractal property breaks down for very short time scale as typi-

Fractals and Economics, Figure 3 Dollar-Yen rate for 13 years (Top). Dark areas are enlarged in the following figure [30]

Fractals and Economics, Figure 4 Market price changes in 10 minutes

cally shown in Fig. 4. In this figure the abscissa is 10 minutes range and we can observe each transaction separately. Obviously the price up down is more zigzag and more discrete than the large scale continuous market fluctuations

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Fractals and Economics, Figure 5 Log–log plot of cumulative distribution of rate change [30]

shown in Fig. 3. In the case of USD-JPY market the time scale that this breakdown of scale invariance occurs typically at time scale of several hours. The distribution of rate change in a unit time (one minute) is shown in Fig. 5. Here, there are two plots of cumulative distributions, P(> x) for positive rate changes and P(> jxj) for negative rate changes, which are almost identical meaning that the up-down symmetry of rate changes is nearly perfect. In this log–log plot the estimated power law distribution’s exponent is 2.5. In the original finding of Mandelbrot, (B) in the previous chapter, the reported exponent value is about 1.7 for cotton prices. In the case of stock markets power laws are confirmed universally for all items, however, the power exponents are not universal, taking value from near one to near five, typically around three [15]. Also the exponent values change in time year by year. In order to demonstrate the importance of large fluctuations, Fig. 6 shows a comparison of three market prices. The top figure is the original rate changes for a week. The middle figure is produced from the same data, but it is consisted of rate changes of which absolute values are larger than 2, that is, about 5 % of all the data. In the bottom curve such large rate changes are omitted and the residue of 95 % of small changes makes the fluctuations. As known from these figures the middle figure is much closer to the original market price changes. Namely, the contribution from the power law tails of price change distribution is very large for macro-scale market prices. Power law distribution of market price changes is also a quite general property which can be confirmed for any market. Up-down symmetry also holds universally in short time scale in general, however, for larger unit time the distribution of price changes gradually deforms and for very large unit time the distribution becomes closer to

Fractals and Economics, Figure 6 USD-JPY exchange rate for a week (top) Rate changes smaller than 2 are neglected (middle) Rate changes larger than 2 are neglected (bottom)

a Gaussian distribution. It should be noted that in special cases of market crashes or bubbles or hyper-inflations the up-down symmetry breaks down and the power law distribution is also likely to be deformed. The autocorrelation of the time sequence of price changes generally decays quickly to zero, sometimes accompanied by a negative correlation in a very short time. This result implies that the market price changes are apparently approximated by white noise, and market prices are known to follow nearly a random walk as a result. However, market price is not a simple random walk. In Fig. 7 the autocorrelation of volatility, which is defined by the square of price change, is shown in log–log scale. In the case of a simple random walk this autocorrelation should also decay quickly. The actual volatility autocorrelation nearly satisfies a power law implying that the volatility time series has a fractal clustering property. (See also Fig. 31 representing an example of price change clustering.) Another fractal nature of markets can be found in the intervals of transactions. As shown in Fig. 8 the transaction intervals fluctuate a lot in very short time scale. It is known that the intervals make clusters, namely, shorter intervals tend to gather. To characterize such clustering effect we can make a time sequence consisted of 0 and 1, where 0 denotes no deal was done at that time, and 1 denotes a deal was done. The corresponding power spectrum follows a 1/f power spectrum as shown in Fig. 9 [50].

Fractals and Economics

Fractals and Economics, Figure 7 Autocorrelation of volatility [30]

Fractals and Economics, Figure 10 Income distribution of companies in Japan

Fractals and Economics, Figure 8 Clustering of transaction intervals

Fractal properties are found not only in financial markets. Company’s income distribution is known to follow also a power law [35]. A company’s income is roughly given by subtraction of incoming money flow minus outgoing money flow, which can take both positive and negative values. There are about six million companies in Japan and Fig. 10 shows the cumulative distribution of annual income of these companies. Clearly we have a power law distribution of income I with the exponent very close to 1 in the middle size range, so-called the Zipf’s law, P(> I) / I ˇ ;

Fractals and Economics, Figure 9 Power spectrum of transaction intervals [50]

ˇ D 1:

(10)

Although in each year every company’s income fluctuates, and some percentage of companies disappear or are newly born, this power law is known to hold for more than 30 years. Similar power laws are confirmed in various countries, the case of France is plotted in Fig. 11 [13]. Observing more details by categorizing the companies, it is found that the income distribution in each job category follows nearly a power law with the exponent depending on the job category as shown in Fig. 12 [29]. The implication of this phenomenon will be discussed in Sect. “Income Distribution Models”. A company’s size can also be viewed by the amount of whole sale or the number of employee. In Figs. 13 and 14 distributions of these quantities are plotted [34]. In both cases clear power laws are confirmed. The size distribution of debts of bankrupted companies is also known to follow a power law as shown Fig. 15 [12].

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Fractals and Economics, Figure 11 Income distribution of companies in France [13]

Fractals and Economics, Figure 14 The distribution of employee numbers [34]

Fractals and Economics, Figure 12 Income distribution of companies in each category [29]

Fractals and Economics, Figure 15 The size distribution of debts of bankrupted companies [12]

Fractals and Economics, Figure 13 The distribution of whole sales [34]

Fractals and Economics

Fractals and Economics, Figure 16 Personal income distribution in Japan [1]

Fractals and Economics, Figure 17 The distribution of the amount of transferred money [21]

A power law distribution can also be found in personal income. Figure 16 shows the personal income distribution in Japan in a log–log plot [1]. The distribution is clearly separated into two parts. The majority of people’s incomes are well approximated by a log-normal distribution (the left top part of the graph), and the top few percent of people’s income distribution is nicely characterized by a power law (the linear line in the left part of the graph). The majority of people are getting salaries from companies. This type of composite of two distributions is well-known from the pioneering study by Pareto about 100 years ago and it holds in various countries [8,22]. A typical value of the power exponent is about two, significantly larger than the income distribution of companies. However, the exponent of the power law seems to be not universal and the value changes county by country or year by year. There is a tendency that the exponent is smaller, meaning more rich people, when the economy is improving [40]. Another fractal in economics can be found in a network of economic agents such as banks’ money transfer network. As a daily activity banks transfer money to other banks for various reasons. In Japan all of these interbank money transfers are done via a special computer network provided by the Bank of Japan. Detailed data of actual money transfer among banks are recorded and analyzed for the basic study. The total amount of money flow among banks in a day is about 30  1012 yen with the number of transactions about 10 000. Figure 17 shows the distribution of the amount of money at a transaction. The range is not wide enough but we can find a power law with an exponent about 1.3 [20].

The number of banks is about 600, so the daily transaction number is only a few percent of the theoretically possible combinations. It is confirmed that there are many pairs of banks which never transact directly. We can define active links between banks for pairs with the averaged number of transaction larger than one per day. By this criterion the number of links becomes about 2000, that is, about 0.5 percent of all possible link numbers. Compared with the complete network, the actual network topologies are much more sparse. In Fig. 18 the number distribution of active links per site are plotted in log–log plot [21]. As is known from this graph, there is an intermediate range in which the link number distribution follows a power law. In the terminology of recent complex network study, this property is called the scale-free network [5]. The scale-free network structure among these intermediate banks is shown in Fig. 19. There are about 10 banks with large link numbers which deviate from the power law, also small link number banks with link number less than four are out of the power law. Such small banks are known to make a satellite structure that many banks linked to one large link number banks. It is yet to clarify why intermediate banks make fractal network, and also to clarify the role of large banks and small banks which are out of the fractal configuration. In relation with the banks, there are fractal properties other than cash flow and the transaction network. The distribution of the whole amount of deposit of Japanese bank is approximated by a power law as shown in Fig. 20 [57]. In recent years large banks merged making a few mega banks and the distribution is a little deformed. Historically there were more than 6000 banks in Japan, however,

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Fractals and Economics, Figure 20 Distribution of total deposit for Japanese banks [57] Power law breaks down from 1999 Fractals and Economics, Figure 18 The number distribution of active links per site [20]

Fractals and Economics, Figure 19 Scale-free network of intermediate banks [20]

now we have about 600 as mentioned. It is very rare that a bank disappears, instead banks are merged or absorbed. The number distribution of banks which are historically behind a present bank is plotted in Fig. 21, again a power law can be confirmed. Other than the example of the bank network, network structures are very important generally in economics. In production process from materials, through various parts to final products the network structure is recently studied in view of complex network analysis [18]. Trade networks among companies can also be described by network terminology. Recently, network characterization quantities such as link numbers (Fig. 22), degrees of authority, and Pageranks are found to follow power laws from real trade data for nearly a million of companies in Japan [34]. Still more power laws in economics can be found in sales data. A recent study on the distribution of expen-

Fractals and Economics, Figure 21 Distribution of bank numbers historically behind a present bank [57]

diture at convenience stores in one shopping trip shows a clear power law distribution with the exponent close to two as shown in Fig. 23 [33]. Also, book sales, movie hits, news paper sales are known to be approximated by power laws [39]. Viewing all these data in economics, we may say that fractals are everywhere in economics. In order to understand why fractals appear so frequently, we firstly need to make simple toy models of fractals which can be analyzed completely, and then, based on such basic models we can make more realistic models which can be directly comparable with real data. At that level of study we will be able to predict or control the complex real world economy.

Fractals and Economics

a power law, P(> y) D y ˛ for y  1. This is a useful transformation in case of numerical simulation using random variable following power laws. 2. Let x be a stochastic variable following an exponential distribution, P(> x) D ex , for positive x, then, y  e x/˛ satisfies a power law, P(> y) / y ˛ . As exponential distributions occur frequently in random process such as the Poisson process, or energy distribution in thermal equilibrium, this simple exponential variable transformation can make it a power law. Superposition of Basic Distributions

Fractals and Economics, Figure 22 Distribution of in-degrees and out-degrees in Japanese company network [34]

A power law distribution can also be easily produced by superposition of basic distributions. Let x be a Gaussian distribution with the probability density given by p R 2 R p R (x) D p e 2 x ; (11) 2 and R be a 2 distribution with degrees of freedom ˛,  1 ˛/2 w(R) D

2



˛ 2

˛

R

 R 2 1 e 2 :

(12)

Then, the superposition of Gaussian distribution, Eq. (11), with the weight given by Eq. (12) becomes the T-distribution having power law tails: Z1 p(x) D

W(R)p R (x)dR 0

   ˛C1 1 2  / jxj˛1 ; Dp ˛  2 (1 C x 2 ) ˛C1 2 Fractals and Economics, Figure 23 Distribution of expenditure in one shopping trip [33]

Basic Models of Power Laws In this chapter we introduce general mathematical and physical models which produce power law distributions. By solving these simple and basic cases we can deepen our understanding of the underlying mechanism of fractals or power law distributions in economics. Transformation of Basic Distributions A power law distribution can be easily produced by variable transformation from basic distributions. 1. Let x be a stochastic variable following a uniform distribution in the range (0, 1], then, y  x 1/˛ satisfies

(13)

which is P(> jxj) / jxj˛ in cumulative distribution. In the special case that R, the inverse of variance of the normal distribution, distributes exponentially, the value of ˛ is 2. Similar super-position can be considered for any basic distributions and power law distributions can be produced by such superposition. Stable Distributions Assume that stochastic variables, x1 ; x2 ; : : : ; x n , are independent and follow the same distribution, p(x), then consider the following normalized summation; Xn 

x1 C x2 C    C x n  n : n1/˛

(14)

If there exists ˛ > 0 and n , such that the distribution of X n is identical to p(x), then, the distribution belongs to

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one of the Levy stable distributions [10]. The parameter ˛ is called the characteristic exponent which takes a value in the range (0, 2]. The stable distribution is characterized by four continuous parameters, the characteristic exponent, an asymmetry parameter which takes a value in [1, 1], the scale factor which takes a positive value and the location parameter which takes any real number. Here, we introduce just a simple case of symmetric distribution around the origin with the unit scale factor. The probability density is then given as p(x; ˛) D

1 2

Z1

˛

ei x ej j d :

(15)

the power law is obtained,  P( x) D

1  Mlog x

0

:

(18)

In other words, a power law distribution maximizes the entropy in the situation where products are conserved. To be more precise, consider two time dependent random variables interacting each other satisfying the relation, x1 (t)  x2 (t) D x1 (t 0 )  x2 (t 0 ), then the equilibrium distribution follows a power law. Another entropy approach to the power laws is to generalize the entropy by the following form [56],

1

For large jxj the cumulative distribution follows the power law, P(> x; ˛) / jxj˛ except the case of ˛ D 2. The stable distribution with ˛ D 2 is the Gaussian distribution. The most important property of the stable distribution is the generalized central limit theorem: If the distribution of sum of any independent identically distributed random variables like X n in Eq. (14) converges in the limit of n ! 1 for some value of ˛, then the limit distribution is a stable distribution with the characteristic exponent ˛. For any distribution with finite variance, the ordinary central limit theory holds, that is, the special case of ˛ D 2. For any infinite variance distribution the limit distribution is ˛ ¤ 2 with a power law tail. Namely, a power law realizes simply by summing up infinitely many stochastic variables with diverging variance.

x x0

1 Sq 

1 R

p(x)q dx

x0

q1

;

(19)

where q is a real number. This function is called the q-entropy and the ordinary entropy, Eq. (15), recovers in the limit of q ! 1. Maximizing the q-entropy keeping the variance constant, so-called a q-Gaussian distribution is obtained, which has the same functional form with the T-distribution, Eq. (12), with the exponent ˛ given by ˛D

q3 : 1q

(20)

This generalized entropy formulation is often applied to nonlinear systems having long correlations, in which power law distributions play the central role.

Entropy Approaches Let x0 be a positive constant and consider a probability density p(x) defined in the interval [x0 ; 1), the entropy of this distribution is given by Z1

Here, we find a distribution that maximizes the entropy with a constraint such that the expectation of logarithm of x is a constant, hlog xi D M. Then, applying the variational principle to the following function,

L x0

01 1 Z p(x) log p(x)dx  1 @ p(x)dx  1A 0 C 2 @

x0

Z1

x0

(21)

(16)

x0

Z1

Stochastic time evolution described by the following formulation is called the multiplicative process, x(t C 1) D b(t)x(t) C f (t) ;

p(x) log p(x)dx :

S

Random Multiplicative Process

P(> x) / jxj˛ ;

1

p(x) log xdx  M A

where b(t) and f (t) are both independent random variables [17]. In the case that b(t) is a constant, the distribution of x(t) depends on the distribution of f (t), for example, if f (t) follows a Gaussian distribution, then the distribution of x(t) is also a Gaussian. However, in the case that b(t) fluctuates randomly, the resulting distribution of x(t) is known to follows a power law independent of f (t),

(17)

(22)

where the exponent ˛ is determined by solving the following equation [48], hjb(t)j˛ i D 1 :

(23)

Fractals and Economics

This steady distribution exists when hlog jb(t)ji < 0 and f (t) is not identically 0. As a special case that b(t) D 0 with a finite probability, then a steady state exists. It is proved rigorously that there exists only one steady state, and starting from any initial distribution the system converges to the power law steady state. In the case hlog jb(t)ji  0 there is no statistically steady state, intuitively the value of jb(t)j is so large that x(t) is likely to diverge. Also in the case f (t) is identically 0 there is no steady state as known from Eq. (21) that log jx(t)j follows a simple random walk with random noise term, log jb(t)j. The reason why this random multiplicative process produces a power law can be understood easily by considering a special case that b(t) D b > 1 with probability 0.5 and b(t) D 0 otherwise, with a constant value of f (t) D 1. In such a situation the value of x(t) is 1 C b C b2 C   C b K with probability (0:5)K . From this we can directly evaluate the distribution of x(t),   b KC1  1 D 2KC1 i: e: P  b1 (24) log 2 P( x) D 4(1 C (b  1)x)˛ ; ˛D : log b As is known from this discussion, the mechanism of this power law is deeply related to the above mentioned transformation of exponential distribution in Sect. “Transformation of Basic Distributions”. The power law distribution of a random multiplicative process can also be confirmed experimentally by an electrical circuit in which resistivity fluctuates randomly [38]. In an ordinary electrical circuit the voltage fluctuations in thermal equilibrium is nearly Gaussian, however, for a circuit with random resistivity a power law distribution holds. Aggregation with Injection Assume the situation that many particles are moving randomly and when two particles collide they coalesce making a particle with mass conserved. Without any injection of particles the system converges to the trivial state that only one particle remains. In the presence of continuous injection of small mass particles there exists a non-trivial statistically steady state in which mass distribution follows a power law [41]. Actually, the mass distribution of aerosol in the atmosphere is known to follow a power law in general [11]. The above system of aggregation with injection can be described by the following model. Let j be the discrete space, and x j (t) be the mass on site j at time t, then choose

one site and let the particle move to another site and particles on the visited site merge, then add small mass particles to all sites, this process can be mathematically given as, 8 ˆ <x j (t) C x k (t) C f j (t) ; x j (tC1) D x j (t) C f j (t) ; ˆ : f j (t) ;

prob D 1/N prob D (N  2)/N prob D 1/N (25)

where N is the total number of sites and f j (t) is the injected mass to the site j. The characteristic function, Z( ; t)  he x j (t) i, which is the Laplace transform of the probability density, satisfies the following equation by assuming uniformity, Z( ; t C 1)

 N 2 1 1 D Z( ; t)2 C Z( ; t) C he f j (t) i : N N N (26) The steady state solution in the vicinity of D 0 is obtained as p (27) Z( ) D 1  h f i 1/2 C    : From this behavior the following power law steady distribution is obtained. P( x) / x ˛ ;

˛D

1 : 2

(28)

By introducing a collision coefficient depending on the size of particles power laws with various values of exponents realized in the steady state of such aggregation with injection system [46]. Critical Point of a Branching Process Consider the situation that a branch grows and splits with probability q or stops growing with probability 1  q as shown in Fig. 24. What is the size distribution of the branch? This problem can be solved in the following way. Let p(r) be the probability of finding a branch of size r, then the next relation holds. p(r C 1) D q

r1 X

p(s)p(r  s) :

(29)

sD1

Multiplying y rC1 and summing up by r from 0 to 1, a closed equation of the generating function, M(y), is obtained, M(y)1Cq D q y  M(y)2 ;

M(y) 

1 X rD0

y r p(r) : (30)

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tributed in N-sites. At each time step choose one site randomly, and transport a finite portion, x j (t), to another randomly chosen site, where is a parameter in the range [0, 1]. x j (t C 1) D (1  )x j (t) ; x k (t C 1) D x k (t) C x j (t) :

Fractals and Economics, Figure 24 Branching process (from left to right)

Solving this quadratic equation and expanding in terms of y, we have the probability density, p(r) / r3/2 eQ(q)r ;

Q(q)  log 4q(1  q) :

(31)

For q < 0:5 the probability decays exponentially for large r, in this case all branches has a finite size. At q D 0:5 the branch size follows the power law, P( r) / r1/2 , and the average size of branch becomes infinity. For q > 0:5 there is a finite probability that a branch grows infinitely. The probability of having an infinite branch, p(1) D 1M(1), is given as, p 2q  1 C 1  4q(1  q) p(1) D ; (32) 2q which grows monotonically from zero to one in the range q D [0:5; 1]. It should be noted that the power law distribution realizes at the critical point between the finite-size phase and the infinite-size phase [42]. Compared with the preceding model of aggregation with injection, Eq. (28), the mass distribution is the same as the branch size distribution at the critical point in Eq. (31). This coincidence is not an accident, but it is known that aggregation with injection automatically chooses the critical point parameter. Aggregation and branching are reversed process and the steady occurrence of aggregation implies that branching numbers keep a constant value on average and this requires the critical point condition. This type of critical behaviors is called the self-organized criticality and examples are found in various fields [4].

It is known that for small positive the statistically steady distribution x is well approximated by a Gaussian like the case of thermal fluctuations. For close to 1 the fluctuation of x is very large and its distribution is close to a power law. In the limit goes to 1 and the distribution converges to Eq. (28), the aggregation with injection case. For intermediate values of the distribution accompanies a fat tail between Gaussian and a power law [49]. Fractal Tiling A fractal tiling is introduced as the final basic model. Figure 25 shows an example of fractal tiling of a plane by squares. Like this case Euclidean space is covered by various sizes of simple shapes like squares, triangles, circles etc. The area size distribution of squares in Fig. 25 follows the power law, P( x) / x ˛ ;

˛D 1/2 :

(34)

Generalizing this model in d-dimensional space, the distribution of d-dimensional volume x is characterized by a power law distribution with an exponent, ˛ D (d  1)/d, therefore, the Zipf’s law which is the case of ˛ D 1 realizes in the limit of d D 1. The fracture size distribution measured in mass introduced in the beginning of this article corresponds to the case of d D 3.

Finite Portion Transport Here, a kind of mixture of aggregation and branching is considered. Assume that conserved quantities are dis-

(33)

Fractals and Economics, Figure 25 An example of fractal tiling

Fractals and Economics

ence as, x(t) D  (t) f (t) ;

(35)

where f (t) is a random variable following a Gaussian distribution with 0 mean and variance unity, the local variance  (t) is given as  (t)2 D c0 C

k X

c k (x(t  k))2 ;

(36)

jD1

Fractals and Economics, Figure 26 Fractal tiling by river patterns [45]

A classical example of fractal tiling is the Apollonian gasket, that is, a plane is covered totally by infinite number of circles which are tangent each other. For a given river pattern like Fig. 26 the basin area distribution follows a power law with exponent about ˛ D 0:4 [45]. Although these are very simple geometric models, simple models may sometimes help our intuitive understanding of fractal phenomena in economics.

Market Models In this chapter market price models are reviewed in view of fractals. There are two approaches for construction of market models. One is modeling the time sequences directly by some stochastic model, and the other is modeling markets by agent models which are artificial markets in computer consisted of programmed dealers. The first market price model was proposed by Bachelier in 1900 written as his Ph.D thesis [3], that is, five years before the model of Einstein’s random walk model of colloid particles. His idea was forgotten for nearly 50 years. In 1950’s Markowitz developed the portfolio theory based on a random walk model of market prices [28]. The theory of option prices by Black and Scholes was introduced in the 1970s, which is also based on random walk model of market prices, or to be more precise a logarithm of market prices in continuum description [7]. In 1982 Engle introduced a modification of the simple random walk model, the ARCH model, which is the abbreviation of auto-regressive conditional heteroscedasticity [9]. This model is formulated for market price differ-

with adjustable positive parameters, fc0 ; c1 ; : : : ; c k g. By the effect of this modulation on variance, the distribution of price difference becomes superposition of Gaussian distribution with various values of variance, and the distribution becomes closer to a power law. Also, volatility clustering occurs automatically so that the volatility autocorrelation becomes longer. There are many variants of ARCH models, such as GARCH and IGARCH, but all of them are based on purely probabilistic modeling, and the probability of prices going up and that of going down are identical. Another type of market price model has been proposed from physics view point [53]. The model is called the PUCK model, an abbreviation of potentials of unbalanced complex kinetics, which assumes the existence of market’s time-dependent potential force, UM (x; t), and the time evolution of market price is given by the following set of equations; ˇ ˇ d C f (t) ; (37) x(tC1)x(t) D  UM (x; t)ˇˇ dx xDx(t)x M (t) b(t) x 2 ; (38) UM (x; t)  M1 2 where M is the number of moving average needed to define the center of potential force, xM (t) 

M1 1 X x(t  k) : M

(39)

kD0

In this model f (t) is the external noise and b(t) is the curvature of quadratic potential which changes with time. When b(t) D 0 the model is identical to the simple random walk model. When b(t) > 0 the market prices are attracted to the moving averaged price, xM (t), the market is stable, and when b(t) < 0 prices are repelled from xM (t) so that the price fluctuation is large and the market is unstable. For b(t) < 2 the price motion becomes an exponential function of time, which can describe singular behavior such as bubbles and crashes very nicely.

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In the simplest case of M D 2 the time evolution equation becomes, x(t C 1) D 

b(t) x(t) C f (t) : 2

(40)

As is known from this functional form in the case b(t) fluctuates randomly, the distribution of price difference follows a power law as mentioned in the previous Sect. “Random Multiplicative Process”, Random multiplicative process. Especially the PUCK model derives the ARCH model by introducing a random nonlinear potential function [54]. The value of b(t) can be estimated from the data and most of known empirical statistical laws including fractal properties are fulfilled as a result [55]. The peculiar difference of this model compared with financial technology models is that directional prediction is possible in some sense. Actually, from the data it is known that b(t) changes slowly in time, and for non-zero b(t) the autocorrelation is not zero implying that the up-down statistics in the near future is not symmetric. Moreover in the case of b(t) < 2 the price motion show an exponential dynamical growth hence predictable. As introduced in Sect. “Examples in Economics” the tick interval fluctuations can be characterized by the 1/f power spectrum. This power law can be explained by a model called the self-modulation model [52]. Let t j be the jth tick interval, and we assume that the tick interval can be approximated by the following random process, t jC1 D  j

K1 1 X t jk C g j ; K

Fractals and Economics, Figure 27 Tick intervals of Poisson process (top) and the self-modulation process (bottom) [52]

(41)

kD0

where j is a positive random number following an exponential distribution with the mean value 1, and K is an integer which means the number of moving average, g j is a positive random variable. Due to the moving average term in Eq. (41) the tick interval automatically make clusters as shown in Fig. 27, and the corresponding power spectrum is proved to be proportional to 1/ f as typically represented in Fig. 28. The market data of tick intervals are tested whether Eq. (41) really works or not. In Fig. 29 the cumulative probability of estimated value of j from market data is plotted where the moving average size is determined by the physical time of 150 seconds and 400 seconds. As known from this figure, the distribution fits very nicely with the exponential distribution when the moving average size is 150 seconds. This result implies that dealers in the market are mostly paying attention to the latest transaction for about a few minutes only. And the dealers’ clocks in their minds move quicker if the market becomes busier. By this

Fractals and Economics, Figure 28 The power spectrum of the self-modulation process [52]

Fractals and Economics, Figure 29 The distribution of normalized time interval [50]

Fractals and Economics

self-modulation effect transactions in markets automatically make a fractal configuration. Next, we introduce a dealer model approach to the market [47]. In any financial market dealers’ final goal is to gain profit from the market. To this end dealers try to buy at the lowest price and to sell at the highest price. Assume that there are N dealers at a market, and let the jth dealer’s buying and selling prices in their mind B j (t) and S j (t). For each dealer the inequality, B j (t) < S j (t), always holds. We pay attention to the maximum price of fB j (t)g called the best bid, and to the minimum price of fS j (t)g called the best ask. Transactions occur in the market if there exists a pair of dealers, j and k, who give the best bid and best ask respectively, and they fulfill the following condition, B j (t)  S k (t) :

(42)

Fractals and Economics, Figure 30 Price evolution of a market with deterministic three dealers

In the model the market price is given by the mean value of these two prices. As a simple situation we consider a deterministic time evolution rule for these dealers. For all dealers the spread, S j (t) B j (t), is set to be a constant L. Each dealer has a position, either a seller or a buyer. When the jth dealer’s position is a seller the selling price in mind, S j (t), decreases every time step until he can actually sell. Similar dynamics is applied to a buyer with the opposite direction of motion. In addition we assume that all dealers shift their prices in mind proportional to a market price change. When this proportional coefficient is positive, the dealer is categorized as a trend-follower. If this coefficient is negative, the dealer is called a contrarian. These rules are summarized by the following time evolution equations. B j (t C 1) D B j (t) C a j S j C b j x(t) ;

(43)

where Sj takes either +1 or 1 meaning the buyer position or seller position, respectively, x(t) gives the latest market price change, fa j g are positive numbers given initially, fb j g are also parameters given initially. Figure 30 shows an example of market price evolution in the case of three dealers. It should be noted that although the system is deterministic, namely, the future price is determined uniquely by the set of initial values, resulting market price fluctuates almost randomly even in the minimum case of three dealers. The case of N D 2 gives only periodic time evolution as expected, while for N  3 the system can produce market price fluctuations similar to the real market price fluctuations, for example, the fractal properties of price chart and the power law distribution of price difference are realized. In the case that the value of fb j g are identical for all dealers, b, then the distribution of market price difference follows a power law where the exponent is controllable by

Fractals and Economics, Figure 31 Cumulative distribution of a dealer model for different values of b. For weaker trend-follow the slope is steeper [38]

this trend-follow parameter, b as shown in Fig. 31 [37]. The volatility clustering is also observed automatically for large dealer number case as shown in Fig. 32 (bottom) which looks quite similar to a real price difference time series Fig. 32 (top). By adding a few features to this basic dealer model it is now possible to reproduce almost all statistical characteristics of market, such as tick-interval fluctuations, abnormal diffusions etc. [58]. In this sense the study of market behaviors are now available by computer simulations based on the dealer model. Experiments on the market is either impossible or very difficult for a real market, however, in an artificial market we can repeat occurrence of bubbles and crashes any times, so that we might be able to find a way to avoid catastrophic market behaviors by numerical simulation.

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This mathematical answer implies that even X is one million dollars this lottery is generous enough and you should buy because expectation is infinity. But, would you dare to buy this lottery, in which you will win only one dollar with probability 0.5, and two dollars with probability 0.25, . . . ? Bernoulli’s answer to this paradox is to introduce the human feeling of value, or utility, which is proportional to the logarithm of price, for example. Based on this expected utility hypothesis the fair value of X is given as follows, XD

1 1 X X U(2n ) log(2n ) D D 1 C log 2 nC1 2 2nC1 nD0 nD0

 1:69 ;

Fractals and Economics, Figure 32 Price difference time series for a real market (top) and a dealer model (bottom)

Income Distribution Models Let us start with a famous historical problem, the St. Petersburg Paradox, as a model of income. This paradox was named after Daniel Bernoulli’s paper written when he was staying in the Russian city, Saint Petersburg, in 1738 [6]. This paradox treats a simple lottery as described in the following, which is deeply related to the infinite expected value problem in probability theory and also it has been attracting a lot of economists’ interest in relation with the essential concept in economics, the utility [2]. Assume that you enjoy a game of chance, you pay a fixed fee, X dollars, to enter, and then you toss a fair coin repeatedly until a tail firstly appears. You win 2n dollars where n is the number of heads. What is the fair price of the entrance fee, X? Mathematically a fair price should be equal to the expectation value, therefore, it should be given as, 1 X 1 XD 2n  nC1 D 1 : (44) 2 nD0

(45)

where the utility function, U(x) D 1Clog x, is normalized to satisfy U(1) D 1. This result implies that the appropriate entry fee X should be about two dollars. The idea of utility was highly developed in economics for description of human behavior, in the way that human preference is determined by maximal point of utility function, the physics concept of the variational principle applied to human action. Recently, in the field of behavioral finance which emerged from psychology the actual observation of human behaviors about money is the main task and the St. Petersburg paradox is attracting attention [36]. Although Bernoulli’s solution may explain the human behavior, the fee X D 2 is obviously so small that the bookmaker of this lottery will bankrupt immediately if the entrance fee is actually fixed as two dollars and if a lot of people actually buy it. The paradox is still a paradox. To clarify what is the problem we calculate the distribution of income of a gambler. As an income is 2 n with probability 2n1 , the cumulative distribution of income is readily obtained as, P( x) /

1 : x

(46)

This is the power law which we observed for income distribution of companies in Sect. “Examples in Economics”. The key of this lottery is the mechanism that the prize money doubles at each time a head appears and the coin toss stops when a tail appears. By denoting the number of coin toss by t, we can introduce a stochastic process or a new lottery which is very much related to the St. Petersburg lottery. x(t C 1) D b(t)x(t) C 1 ;

(47)

where b(t) is 2 with probability 0.5 and is 0 otherwise. As introduced in Sect. “Random Multiplicative Process”, this problem is solved easily and it is confirmed that the steady

Fractals and Economics

state cumulative distribution of x(t) also follows Eq. (46). The difference between the St. Petersburg lottery and the new lottery Eq. (47) is the way of payment of entrance fee. In the case of St. Petersburg lottery the entrance fee X is paid in advance, while in the case of new lottery you have to add one dollar each time you toss a coin. This new lottery is fair from both the gambler side and the bookmaker side because the expectation of income is given by hx(t)i D t and the amount of paid fee is also t. Now we introduce a company’s income model by generalizing this new fair lottery in the following way, I(t C 1) D b(t)I(t) C f (t) ;

(48)

where I(t) denotes the annual income of a company, b(t) represents the growth rate which is given randomly from a growth rate distribution g(b), and f (t) is a random noise. Readily from the results of Sect. “Random Multiplicative Process”, we have a condition to satisfy the empirical relation, Eq. (10), Z hb(t)i D bg(b) D 1 : (49)

An implication of this result is that if a job category is expanding, namely, hb(t)i > 1, then the power law exponent determined by Eq. (50) is smaller than 1. On the other hand if a job category is shrinking, we have an exponent that is larger than 1. This type of company’s income model can be generalized to take into account the effect of company’s size dependence on the distribution of growth rate. Also, the magnitude of the random force term can be estimated from the probability of occurrence of negative income. Then, assuming that the present growth rate distribution continues we can perform a numerical simulation of company’s income distribution starting from a uniform distribution as shown in Fig. 34 for Japan and in Fig. 35 for USA. It is shown that in the case of Japan, the company size distribution converges to the power law with exponent 1 in 20 years, while in the case of USA the steady power law’s slope is about 0.7 and it takes about 100 years to converge [31]. According to this result extremely large com-

This relation is confirmed to hold approximately in actual company data [32]. In order to explain the job category dependence of the company’s income distribution already shown in Fig. 12, we plot the comparison of exponents in Fig. 33. Empirically estimated exponents are plotted in the ordinate and the solutions of the following equation calculated in each job category are plotted in the abscissa, hb(t)ˇ i D 1 :

(50)

The data points are roughly on a straight line demonstrating that the simple growth model of Eq. (48) seems to be meaningful.

Fractals and Economics, Figure 33 Theoretical predicted exponent value vs. observed value [29]

Fractals and Economics, Figure 34 Numerical simulation of income distribution evolution of Japanese companies [32]

Fractals and Economics, Figure 35 Numerical simulation of income distribution evolution of USA companies [32]

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panies with size about 10 times bigger than the present biggest company will appear in USA in this century. Of course the growth rate distribution will change faster than this prediction, however, this model can tell the qualitative direction and the speed of change in very macroscopic economical conditions. Other than this simple random multiplicative model approach there are various approaches to explain empirical facts about company’s statistics assuming a hierarchical structure of organization, for example [23]. Future Directions Fractal properties generally appear in almost any huge data in economics. As for financial market models, empirical fractal laws are reproduced and the frontier of study is now at the level of practical applications. However, there are more than a million markets in the world and little is known about their interaction. More research on market interaction will be promising. Company data so far analyzed show various fractal properties as introduced in Sect. “Examples in Economics”, however, they are just a few cross-sections of global economics. Especially, companies’ interaction data are inevitable to analyze the underlying network structures. Not only money flow data it will be very important to observe material flow data in manufacturing and consumption processes. From the viewpoint of environmental study, such material flow network will be of special importance in the near future. Detail sales data analysis is a new topic and progress is expected. Bibliography Primary Literature 1. Aoyama H, Nagahara Y, Okazaki MP, Souma W, Takayasu H, Takayasu M (2000) Pareto’s law for income of individuals and debt of bankrupt companies. Fractals 8:293–300 2. Aumann RJ (1977) The St. Petersburg paradox: A discussion of some recent comments. J Econ Theory 14:443–445 http://en. wikipedia.org/wiki/Robert_Aumann 3. Bachelier L (1900) Theory of Speculation. In: Cootner PH (ed) The Random Character of Stock Market Prices. MIT Press, Cambridge (translated in English) 4. Bak P (1996) How Nature Works. In: The Science of Self-Organized Criticality. Springer, New York 5. Barabási AL, Réka A (1999) Emergence of scaling in random networks. Science 286:509–512; http://arxiv.org/abs/ cond-mat/9910332 6. Bernoulli D (1738) Exposition of a New Theory on the Measurement of Risk; Translation in: (1954) Econometrica 22:22–36 7. Black F, Scholes M (1973) The Pricing of Options and Corporate Liabilities. J Political Econ 81:637–654

8. Brenner YS, Kaelble H, Thomas M (1991) Income Distribution in Historical Perspective. Cambridge University Press, Cambridge 9. Engle RF (1982) Autoregressive Conditional Heteroskedasticity With Estimates of the Variance of UK Inflation. Econometrica 50:987–1008 10. Feller W (1971) An Introduction to Probability Theory and Its Applications, 2nd edn, vol 2. Wiley, New York 11. Friedlander SK (1977) Smoke, dust and haze: Fundamentals of aerosol behavior. Wiley-Interscience, New York 12. Fujiwara Y (2004) Zipf Law in Firms Bankruptcy. Phys A 337:219–230 13. Fujiwara Y, Guilmi CD, Aoyama H, Gallegati M, Souma W (2004) Do Pareto-Zipf and Gibrat laws hold true? An analysis with European firms. Phys A 335:197–216 14. Gilvarry JJ, Bergstrom BH (1961) Fracture of Brittle Solids. II Distribution Function for Fragment Size in Single Fracture (Experimental). J Appl Phys 32:400–410 15. Gopikrishnan P, Meyer M, Amaral LAN, Stanley HE (1998) Inverse Cubic Law for the Distribution of Stock Price Variations. Eur Phys J B 3:139–143 16. Handel PH (1975) 1/f Noise-An “Infrared” Phenomenon. Phys Rev Lett 34:1492–1495 17. Havlin S, Selinger RB, Schwartz M, Stanley HE, Bunde A (1988) Random Multiplicative Processes and Transport in Structures with Correlated Spatial Disorder. Phys Rev Lett 61: 1438–1441 18. Hidalgo CA, Klinger RB, Barabasi AL, Hausmann R (2007) The Product Space Conditions the Development of Nations. Science 317:482–487 19. Inaoka H, Toyosawa E, Takayasu H (1997) Aspect Ratio Dependence of Impact Fragmentation. Phys Rev Lett 78:3455–3458 20. Inaoka H, Ninomiya T, Taniguchi K, Shimizu T, Takayasu H (2004) Fractal Network derived from banking transaction – An analysis of network structures formed by financial institutions. Bank of Japan Working Paper. http://www.boj.or.jp/en/ ronbun/04/data/wp04e04.pdf 21. Inaoka H, Takayasu H, Shimizu T, Ninomiya T, Taniguchi K (2004) Self-similarity of bank banking network. Phys A 339: 621–634 22. Klass OS, Biham O, Levy M, Malcai O, Solomon S (2006) The Forbes 400 and the Pareto wealth distribution. Econ Lett 90:290–295 23. Lee Y, Amaral LAN, Canning D, Meyer M, Stanley HE (1998) Universal Features in the Growth Dynamics of Complex Organizations. Phys Rev Lett 81:3275–3278 24. Mandelbrot BB (1963) The variation of certain speculative prices. J Bus 36:394–419 25. Mandelbrot BB (1982) The Fractal Geometry of Nature. W.H. Freeman, New York 26. Mandelbrot BB (2004) The (mis)behavior of markets. Basic Books, New York 27. Mantegna RN, Stanley HE (2000) An Introduction to Economics: Correlations and Complexity in Finance. Cambridge Univ Press, Cambridge 28. Markowitz HM (1952) Portfolio Selection. J Finance 7:77–91 29. Mizuno T, Katori M, Takayasu H, Takayasu M (2001) Statistical laws in the income of Japanese companies. In: Empirical Science of Financial Fluctuations. Springer, Tokyo, pp 321–330 30. Mizuno T, Kurihara S, Takayasu M, Takayasu H (2003) Analysis of high-resolution foreign exchange data of USD-JPY for 13 years. Phys A 324:296–302

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31. Mizuno T, Kurihara S, Takayasu M, Takayasu H (2003) Investment strategy based on a company growth model. In: Takayasu H (ed) Application of Econophysics. Springer, Tokyo, pp 256–261 32. Mizuno T, Takayasu M, Takayasu H (2004) The mean-field approximation model of company’s income growth. Phys A 332:403–411 33. Mizuno T, Toriyama M, Terano T, Takayasu M (2008) Pareto law of the expenditure of a person in convenience stores. Phys A 387:3931–3935 34. Ohnishi T, Takayasu H, Takayasu M (in preparation) 35. Okuyama K, Takayasu M, Takayasu H (1999) Zipf’s law in income distribution of companies. Phys A 269:125– 131. http://www.ingentaconnect.com/content/els/03784371; jsessionid=5e5wq937wfsqu.victoria 36. Rieger MO, Wang M (2006) Cumulative prospect theory and the St. Petersburg paradox. Econ Theory 28:665–679 37. Sato AH, Takayasu H (1998) Dynamic numerical models of stock market price: from microscopic determinism to macroscopic randomness. Phys A 250:231–252 38. Sato AH, Takayasu H, Sawada Y (2000) Power law fluctuation generator based on analog electrical circuit. Fractals 8: 219–225 39. Sinha S, Pan RK (2008) How a “Hit” is Born: The Emergence of Popularity from the Dynamics of Collective Choice. http:// arxiv.org/PS_cache/arxiv/pdf/0704/0704.2955v1.pdf 40. Souma W (2001) Universal structure of the personal income distribution. Fractals 9:463–470; http://www.nslij-genetics. org/j/fractals.html 41. Takayasu H (1989) Steady-state distribution of generalized aggregation system with injection. Phys Rev Lett 63: 2563–2566 42. Takayasu H (1990) Fractals in the physical sciences. Manchester University Press, Manchester 43. Takayasu H (ed) (2002) Empirical Science of Financial Fluctuations–The Advent of Econophysics. Springer, Tokyo 44. Takayasu H (ed) (2003) Application of Econophysics. Springer, Tokyo 45. Takayasu H, Inaoka H (1992) New type of self-organized criticality in a model of erosion. Phys Rev Lett 68:966–969

46. Takayasu H, Takayasu M, Provata A, Huber G (1991) Statistical properties of aggregation with injection. J Stat Phys 65: 725–745 47. Takayasu H, Miura H, Hirabayashi T, Hamada K (1992) Statistical properties of deterministic threshold elements–The case of market price. Phys A 184:127–134 48. Takayasu H, Sato AH, Takayasu M (1997) Stable infinite variance fluctuations in randomly amplified Langevin systems. Phys Rev Lett 79:966–969 49. Takayasu M, Taguchi Y, Takayasu H (1994) Non-Gaussian distribution in random transport dynamics. Inter J Mod Phys B 8:3887–3961 50. Takayasu M (2003) Self-modulation processes in financial market. In: Takayasu H (ed) Application of Econophysics. Springer, Tokyo, pp 155–160 51. Takayasu M (2005) Dynamics Complexity in Internet Traffic. In: Kocarev K, Vatty G (eds) Complex Dynamics in Communication Networks. Springer, New York, pp 329–359 52. Takayasu M, Takayasu H (2003) Self-modulation processes and resulting generic 1/f fluctuations. Phys A 324:101–107 53. Takayasu M, Mizuno T, Takayasu H (2006) Potentials force observed in market dynamics. Phys A 370:91–97 54. Takayasu M, Mizuno T, Takayasu H (2007) Theoretical analysis of potential forces in markets. Phys A 383:115–119 55. Takayasu M, Mizuno T, Watanabe K, Takayasu H (preprint) 56. Tsallis C (1988) Possible generalization of Boltzmann–Gibbs statistics. J Stat Phys 52:479–487; http://en.wikipedia.org/wiki/ Boltzmann_entropy 57. Ueno H, Mizuno T, Takayasu M (2007) Analysis of Japanese bank’s historical network. Phys A 383:164–168 58. Yamada K, Takayasu H, Takayasu M (2007) Characterization of foreign exchange market using the threshold-dealer-model. Phys A 382:340–346

Books and Reviews Takayasu H (2006) Practical Fruits of Econophysics. Springer, Tokyo Chatterjee A, Chakrabarti BK (2007) Econophysics of Markets and Business Networks (New Economic Windows). Springer, New York

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Fractals in Geology and Geophysics DONALD L. TURCOTTE Department of Geology, University of California, Davis, USA Article Outline Glossary Definition of the Subject Introduction Drainage Networks Fragmentation Earthquakes Volcanic Eruptions Landslides Floods Self-Affine Fractals Topography Earth’s Magnetic Field Future Directions Bibliography Glossary Fractal A collection of objects that have a power-law dependence of number on size. Fractal dimension The power-law exponent in a fractal distribution. Definition of the Subject The scale invariance of geological phenomena is one of the first concepts taught to a student of geology. When a photograph of a geological feature is taken, it is essential to include an object that defines the scale, for example, a coin or a person. It was in this context that Mandelbrot [7] introduced the concept of fractals. The length of a rocky coastline is obtained using a measuring rod with a specified length. Because of scale invariance, the length of the coastline increases as the length of the measuring rod decreases according to a power law. It is not possible to obtain a specific value for the length of a coastline due to small indentations down to a scale of millimeters or less. A fractal distribution requires that the number of objects N with a linear size greater than r has an inverse power-law dependence on r so that C (1) rD where C is a constant and the power D is the fractal dimension. This power-law scaling is the only distribution that is ND

scale invariant. However, the power-law dependence cannot be used to define a statistical distribution because the integral of the distribution diverges to infinity either for large values or small values of r. Thus fractal distributions never appear in compilations of statistical distributions. A variety of statistical distributions have power-law behavior either at large scales or small scales, but not both. An example is the Pareto distribution. Many geological phenomena are scale invariant. Examples include the frequency-size distributions of fragments, faults, earthquakes, volcanic eruptions, and landslides. Stream networks and landforms exhibit scale invariance. In terms of these applications there must always be upper and lower cutoffs to the applicability of a fractal distribution. As a specific application consider earthquakes on the Earth. The number of earthquakes has a power-law dependence on the size of the rupture over a wide range of sizes. But the largest earthquake cannot exceed the size of the Earth, say 104 km. Also, the smallest earthquake cannot be smaller than the grain size of rocks, say 1 mm. But this range of scales is 1010 . Actual earthquakes appear to satisfy fractal scaling over the range 1 m to 103 km. An example of fractal scaling is the number-area distribution of lakes [10], this example is illustrated in Fig. 1. Excellent agreement with the fractal relation given in Eq. (1) is obtained taking D D 1:90. The linear dimension r is taken to be the square root of the area A and the power-law (fractal) scaling extends from r D 100 m to r D 300 km. Introduction Fractal scaling evolved primarily as an empirical means of correlating data. A number of examples are given below. More recently a theoretical basis has evolved for the applicability of fractal distributions. The foundation of this basis is the concept of self-organized criticality. A number of simple computational models have been shown to yield fractal distributions. Examples include the sand-pile model, the forest-fire model, and the slider-block model. Drainage Networks Drainage networks are a universal feature of landscapes on the Earth. Small streams merge to form larger streams, large streams merge to form rivers, and so forth. Strahler [16] quantified stream networks by introducing an ordering system. When two like-order streams of order i merge they form a stream of order i C 1. Thus two i D 1 streams merge to form a i D 2 stream, two i D 2 streams merge to from a i D 3 stream and so forth. A bifurcation

Fractals in Geology and Geophysics

self-similar, side-branching topology was developed. Applications to drainage networks have been summarized by Peckham [11] and Pelletier [13]. Fragmentation

Fractals in Geology and Geophysics, Figure 1 Dependence of the cumulative number of lakes N with areas greater than A as a function of A. Also shown is the linear dimension r which is taken to be the square root of A. The straight-line correlation is with Eq. (1) taking the fractal dimension D D 1:90

ratio Rb is defined by Rb D

Ni N iC1

(2)

where N i is the number of streams of order i. A length order ratio Rr is defined by Rr D

r iC1 ri

(3)

where ri is the mean length of streams of order i. Empirically both Rb and Rr are found to be nearly constant for a range of stream orders in a drainage basin. From Eq. (1) the fractal dimension of a drainage basin DD

ln(N i /N iC1 ) ln Rb D ln(r iC1 /r i ) ln Rr

(4)

Typically Rb D 4:6, Rr D 2:2, and the corresponding fractal dimension is D D 1:9. This scale invariant scaling of drainage networks was recognized some 20 years before the concept of fractals was introduced in 1967. A major advance in the quantification of stream networks was made by Tokunaga [17]. This author was the first to recognize the importance of side branching, that is some i D 1 streams intersect i D 2, i D 3, and all higher-order streams. Similarly, i D 2 streams intersect i D 3 and higher- order streams and so forth. A fully

An important application of power-law (fractal) scaling is to fragmentation. In many examples the frequency-mass distributions of fragments are fractal. Explosive fragmentation of rocks (for example in mining) give fractal distributions. At the largest scale the frequency size distribution of the tectonic plates of plate tectonics are reasonably well approximated by a power-law distribution. Fault gouge is generated by the grinding process due to earthquakes on a fault. The frequency-mass distribution of the gouge fragments is fractal. Grinding (comminution) processes are common in tectonics. Thus it is not surprising that fractal distributions are ubiquitous in geology. As a specific example consider the frequency-mass distribution of asteroids. Direct measurements give a fractal distribution. Since asteroids are responsible for the impact craters on the moon, it is not surprising that the frequencyarea distribution of lunar craters is also fractal. Using evidence from the moon and a fractal extrapolation it is estimated that on average, a 1m diameter meteorite impacts the earth every year, that a 100 m diameter meteorite impacts every 10,000 years, and that a 10 km diameter meteorite impacts the earth every 100,000,000 years. The classic impact crater is Meteor Crater in Arizona, it is over 1 km wide and 200 m deep. Meteor Crater formed about 50,000 years ago and it is estimated that the impacting meteorite had a diameter of 30 m. The largest impact to occur in the 20th century was the June 30, 1908 Tunguska event in central Siberia. The impact was observed globally and destroyed over 1000 km2 of forest. It is believed that this event was the result of a 30 m diameter meteorite that exploded in the atmosphere. One of the major global extinctions occurred at the Cretaceous/Tertiary boundary 65 million years ago. Some 65% of the existing species were destroyed including dinosaurs. This extinction is attributed to a massive impact at the Chicxulub site on the Yucatan Peninsula, Mexico. It is estimated that the impacting meteorite had a 10 km diameter. In addition to the damage done by impacts there is evidence that impacts on the oceans have created massive tsunamis. The fractal power-law scaling can be used to quantify the risk of future impacts. Earthquakes Earthquakes universally satisfy several scaling laws. The most famous of these is Gutenberg–Richter frequency-

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magnitude scaling. The magnitude M of an earthquake is an empirical measure of the size of an earthquake. If the magnitude is increased by one unit it is observed that the cumulative number of earthquakes greater than the specified magnitude is reduced by a factor of 10. For the entire earth, on average, there is 1 magnitude 8 earthquake per year, 10 magnitude 7 earthquakes per year, and 100 magnitude 6 earthquakes per year. When magnitude is converted to rupture area a fractal relation is obtained. The numbers of earthquakes that occur in a specified region and time interval have a power-law dependence on the rupture area. The validity of this fractal scaling has important implications for probabilistic seismic risk assessment. The number of small earthquakes that occur in a region can be extrapolated to estimate the risk of larger earthquakes [1]. As an example consider southern California. On average there are 30 magnitude 4 or larger earthquakes per year. Using the fractal scaling it is estimated that the expected intervals between magnitude 6 earthquakes will be 3 years, between magnitude 7 earthquakes will be 30 years, and between magnitude 8 earthquakes will be 300 years. The fractal scaling of earthquakes illustrate a useful aspect of fractal distributions. The fractal distribution requires two parameters. The first parameter, the fractal dimension D (known as the b-value in seismology), gives the dependence of number on size (magnitude). For earthquakes the fractal dimension is almost constant independent of the tectonic setting. The second parameter gives the level of activity. For example, this can be the number of earthquakes greater than a specified magnitude in a region. This level of activity varies widely and is an accepted measure of seismic risk. The level is essentially zero in states like Minnesota and is a maximum in California. Volcanic Eruptions There is good evidence that the frequency-volume statistics of volcanic eruptions are also fractal [9]. Although it is difficult to quantify the volumes of magma and ash associated with older eruptions, the observations suggest that an eruption with a volume of 1 km3 would be expected each 10 years, 10 km3 each 100 years, and 100 km3 each 1000 years. For example, the 1991 Mount Pinatubo, Philippines eruption had an estimated volume of about 5 km3 . The most violent eruption in the last 200 years was the 1815 Tambora, Indonesia eruption with an estimated volume of 150 km3 . This eruption influenced the global climate in 1816 which was known as the year without a summer. It is estimated that the Long Valley, California eruption with an age of about 760,000 years had a volume of about

600 km3 and the Yellowstone eruptions of about 600,000 years ago had a volume of about 2000 km3 . Although the validity of the power-law (fractal) extrapolation of volcanic eruption volumes to long periods in the past can be questioned, the extrapolation does give some indication of the risk of future eruptions to global climate. There is no doubt that the large eruptions that are known to have occurred on time scales of 105 to 106 years would have a catastrophic impact on global agricultural production. Landslides Landslides are a complex natural phenomenon that constitutes a serious natural hazard in many countries. Landslides also play a major role in the evolution of landforms. Landslides are generally associated with a trigger, such as an earthquake, a rapid snowmelt, or a large storm. The landslide event can include a single landslide or many thousands. The frequency-area distribution of a landslide event quantifies the number of landslides that occur at different sizes. It is generally accepted that the number of large landslides with area A has a power-law dependence on A with an exponent in the range 1.3 to 1.5 [5]. Unlike earthquakes, a complete statistical distribution can be defined for landslides. A universal fit to an inversegamma distribution has been found for a number of event inventories. This distribution has a power-law (fractal) behavior for large landslides and an exponential cut-off for small landslides. The most probable landslides have areas of about 40 m2 . Very few small landslides are generated. As a specific example we consider the 11,111 landslides generated by the magnitude 6.7 Northridge (California) earthquake on January 17, 1994. The total area of the landslides was 23.8 km2 and the area of the largest landslide was 0.26 km2 . The inventory of landslide areas had a good power-law dependence on area for areas greater than 103 m2 (103 km2 ). The number of landslides generated by earthquakes have a strong dependence on earthquake magnitude. Typically earthquakes with magnitudes M less than 4 do not generate any landslides [6]. Floods Floods are a major hazard to many cities and estimates of flood hazards have serious economic implications. The standard measure of the flood hazard is the 100-year flood. This is quantified as the river discharge Q100 expected during a 100 year period. Since there is seldom a long enough history to establish Q100 directly, it is necessary to extrapolate smaller floods that occur more often.

Fractals in Geology and Geophysics

One extrapolation approach is to assume that flood discharges are fractal (power-law) [3,19]. This scale invariant distribution can be expressed in terms of the ratio F of the peak discharge over a 10 year interval to the peak discharge over a 1 year interval, F D Q10 /Q1 . With self-similarity the parameter F is also the ratio of the 100 year peak discharge to the 10 year peak discharge, F D Q100 /Q10 . Values of F have a strong dependence on climate. In temperate climates such as the northeastern and northwestern US values are typically in the range F D 2–3. In arid and tropical climates such as the southwestern and southeastern US values are typically in the range F D 4–6. The applicability of fractal concepts to flood forecasting is certainly controversial. In 1982, the US government adopted the log-Pearson type 3 (LP3) distribution for the legal definition of the flood hazard [20]. The LP3 is a thintailed distribution relative to the thicker tailed power-law (fractal) distribution. Thus the forecast 100 year flood using LP3 is considerably smaller than the forecast using the fractal approach. This difference is illustrated by considering the great 1993 Mississippi River flood. Considering data at the Keukuk, Iowa gauging station [4] this flood was found to be a typical 100 year flood using the power-law (fractal) analysis and a 1000 to 10,000 year flood using the federal LP3 formulation. Concepts of self-similarity argue for the applicability of fractal concepts for flood-frequency forecasting. This applicability also has important implications for erosion. Erosion will be dominated by the very largest floods.

Adjacent values are not correlated with each other. In this case the spectrum is flat and the power spectral density coefficients are not a function of frequency, ˇ D 0. The classic example of a self-affine fractal is a Brownian walk. A Brownian walk is obtained by taking the running sum of a Gaussian white noise. In this case we have ˇ D 2. Another important self-affine time series is a red (or pink) noise with power spectral density coefficients proportional to 1/ f , that is ˇ D 1. We will see that the variability in the Earth’s magnetic field is well approximated by a 1/ f noise. Self-affine fractal time series in the range ˇ D 0 to 1 are known as fractional Gaussian noises. These noises are stationary and the standard deviation is a constant independent of the length of the time series. Self-affine time series with ˇ larger than 1 are known as fractional Brownian walk. These motions are not stationary and the standard deviation increases as a power of the length of the time series, there is a drift. For a Brownian walk the standard deviation increases with the square root of the length of the time series.

Self-Affine Fractals

Spectral expansions of global topography have been carried out, an example [15] is given in Fig. 2. Excellent agreement with the fractal relation given in Eq. (6) is obtained taking ˇ D 2, topography is well approximated by a Brownian walk. It has also shown that this fractal behavior of topography is found for the moon, Venus, and Mars [18].

Mandelbrot and Van Ness [8] extended the concept of fractals to time series. Examples of time series in geology and geophysics include global temperature, the strength of the Earth’s magnetic field, and the discharge rate in a river. After periodicities and trends have been removed, the remaining values are the stochastic (noise) component of the time series. The standard approach to quantifying the noise component is to carry out a Fourier transform on the time series [2]. The power-spectral density coefficients Si are proportional to the squares of the Fourier coefficients. The time series is a self-affine fractal if the power-spectral density coefficients have an inverse power-law dependence on frequency f i , that is Si D

C ˇ

(5)

fi

where C is a constant and ˇ is the power-law exponent. For a Gaussian white noise the values in the time series are selected randomly from a Gaussian distribution.

Topography The height of topography along linear tracks can be considered to be a continuous time series. In this case we consider the wave number ki (1/wave length) instead of frequency. Topography is a self-affine fractal if Si D

C k ˇi

(6)

Earth’s Magnetic Field Paleomagnetic studies have given the strength and polarity of the Earth’s magnetic field as a function of time over millions of years. These studies have also shown that the field has experienced a sequence of reversals. Spectral studies of the absolute amplitude of the field have been shown that it is a self-affine fractal [12,14]. The power-spectral density is proportional to one over the frequency, it is a 1/ f noise. When the fluctuations of the 1/ f noise take the magnitude to zero the polarity of the field reverses. The predicted distribution of polarity intervals is fractal and is in good agreement with the observed polarity intervals.

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Fractals in Geology and Geophysics, Figure 2 Power spectral density S as a function of wave number k for a spherical harmonic expansion of the Earth’s topography (degree l). The straight-line correlation is with Eq. (6) taking ˇ D 2, a Brownian walk

Future Directions There is no question that fractals are a useful empirical tool. They provide a rational means for the extrapolation and interpolation of observations. The wide applicability of power-law (fractal) distributions is generally accepted, but does this applicability have a more fundamental basis? Fractality appears to be fundamentally related to chaotic behavior and to numerical simulations exhibiting self-organized criticality. The entire area of fractals, chaos, selforganized criticality, and complexity remains extremely active, and it is impossible to predict with certainty what the future holds. Bibliography

4. Malamud BD, Turcotte DL, Barton CC (1996) The 1993 Mississippi river flood: A one hundred or a one thousand year event? Env Eng Geosci 2:479 5. Malamud BD, Turcotte DL, Guzzetti F, Reichenbach P (2004) Landslide inventories and their statistical properties. Earth Surf Process Landf 29:687 6. Malamud BD, Turcotte DL, Guzzetti F, Reichenbach P (2004) Landslides, earthquakes, and erosion. Earth Planet Sci Lett 229:45 7. Mandelbrot BB (1967) How long is the coast of Britain? Statistical self-similarity and fractional dimension. Science 156:636 8. Mandelbrot BB, Van Ness JW (1968) Fractional Brownian motions, fractional noises and applications. SIAM Rev 10:422 9. McClelland L et al (1989) Global Volcanism 1975-1985. Prentice-Hall, Englewood Cliffs 10. Meybeck M (1995) Global distribution of lakes. In: Lerman A, Imboden DM, Gat JR (eds) Physics and Chemistry of Lakes, 2nd edn. Springer, Berlin, pp 1-35 11. Peckham SD (1989) New results for self-similar trees with applications to river networks. Water Resour Res 31:1023 12. Pelletier JD (1999) Paleointensity variations of Earth’s magnetic field and their relationship with polarity reversals. Phys Earth Planet Int 110:115 13. Pelletier JD (1999) Self-organization and scaling relationships of evolving river networks. J Geophys Res 104:7259 14. Pelletier JD, Turcotte DL (1999) Self-affine time series: II. Applications and models. Adv Geophys 40:91 15. Rapp RH (1989) The decay of the spectrum of the gravitational potential and the topography of the Earth. Geophys J Int 99:449 16. Strahler AN (1957) Quantitative analysis of watershed geomorphology. Trans Am Geophys Un 38:913 17. Tokunaga E (1978) Consideration on the composition of drainage networks and their evolution. Geogr Rep Tokyo Metro Univ 13:1 18. Turcotte DL (1987) A fractal interpretation of topography and geoid spectra on the earth, moon, Venus, and Mars. J Geophys Res 92:E597 19. Turcotte DL (1994) Fractal theory and the estimation of extreme floods. J Res Natl Inst Stand Technol 99:377 20. US Water Resources Council (1982) Guidelines for Determining Flood Flow Frequency. Bulletin 17B. US Geological Survey, Reston

Primary Literature 1. Kossobokov VG, Keilis-Borok VI, Turcotte DL, Malamud BD (2000) Implications of a statistical physics approach for earthquake hazard assessment and forecasting. Pure Appl Geophys 157:2323 2. Malamud BD, Turcotte DL (1999) Self-affine time series: I. Generation and analyses. Adv Geophys 40:1 3. Malamud BD, Turcotte DL (2006) The applicability of powerlaw frequency statistics to floods. J Hydrol 332:168

Books and Reviews Feder J (1988) Fractals. Plenum Press, New York Korvin G (1992) Fractal Models in the Earth Sciences. Elsevier, Amsterdam Mandelbrot BB (1982) The Fractal Geometry of Nature. Freeman, San Francisco Turcotte DL (1997) Fractals and Chaos in Geology and Geophysics, 2nd edn. Cambridge University Press, Cambridge

Fractals Meet Chaos

Fractals Meet Chaos TONY CRILLY Middlesex University, London, UK Article Outline Glossary Definition of the Subject Introduction Dynamical Systems Curves and Dimension Chaos Comes of Age The Advent of Fractals The Merger Future Directions Bibliography Glossary Dimension The traditional meaning of dimension in modern mathematics is “topological dimension” and is an extension of the classical Greek meaning. In modern concepts of this dimension can be defined in terms of a separable metric space. For the practicalities of Fractals and Chaos the notion of dimension can be limited to subsets of Euclidean n-space where n is an integer. The newly arrived “fractal dimension” is metrically based and can take on fractional values. Just as for topological dimension. As for topological dimension itself there is a profusion of different (but related) concepts of metrical dimension. These are widely used in the study of fractals, the ones of principal interest being:  Hausdorff dimension (more fully, Hausdorff–Besicovitch dimension),  Box dimension (often referred to as Minkowski– Bouligand dimension),  Correlation dimension (due to A. Rényi, P. Grassberger and I. Procaccia). Other types of metric dimension are also possible. There is “divider dimension” (based on ideas of an English mathematician/meteorologist L. F. Richardson in the 1920s); the “Kaplan–Yorke dimension” (1979) derived from Lyapunov exponents, known also as the “Lyapunov dimension”; “packing dimension” introduced by Tricot (1982). In addition there is an overall general dimension due to A. Rényi (1970) which admits box dimension, correlation dimension and information dimension as special cases. With many

of the concepts of dimension there are upper and lower refinements, for example, the separate upper and lower box dimensions. Key references to the vast (and highly technical) subject of mathematical dimension include [31,32,33,60,73,92,93]. Hausdorff dimension (Hausdorff–Besicovitch dimension). In the study of fractals, the most sophisticated concept of dimension is Hausdorff dimension, developed in the 1920s. The following definition of Hausdorff dimension is given for a subset A of the real number line. This is readily generalized to subsets of the plane, Euclidean 3-space and Euclidean n-space, and more abstractly to separable metric spaces by taking neighborhoods as disks instead of intervals. Let fU i g be an r-covering of A, (a covering of A where the width of all intervals U i , satisfies w(U i )  r). The measure mr is defined by mr (A) D inf

1 X

! w(U i )

;

iD1

where the infimum (or greatest of the minimum values) is taken over all r-coverings of A. The Hausdorff dimension DH of A is: DH D lim mr (A) ; r!0

provided the limit exists. The subset E D f1/n : n D 1; 2; ; 3; : : :g of the unit interval has DH D 0 (the Hausdorff dimension of a countable set is always zero). The Hausdorff dimension is the basis of “fractal dimension” but because it takes into account intervals of unequal widths it may be difficult to calculate in practice. Box dimension (or Minkowski–Bouligand dimension, known also as capacity dimension, cover dimension, grid dimension). The box counting dimension is a more direct and practical method for computing dimension in the case of fractals. To define it, we again confine our attention to the real number line in the knowledge that box dimension is readily extended to subsets of more general spaces. As before, let fU i g be an r-covering of A, and let N r (A) be the least number of sets in such a covering. The box dimension DB of A is defined by: log N r (A) : r!0 log 1/r

DB D lim

The box dimension of the subset E D f1/n : n D 1; 2; 3; : : :g of the unit interval can be calculated to give DB D 0:5.

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In general Hausdorff and Box dimensions are related to each other by the inequality DB  DH , as happens in the above example. The relationship between DH and DB is investigated in [49]. For compact, selfsimilar fractal sets DB D DH but there are fractal sets for which DB > DH [76]. Though Hausdorff dimension and Box dimension have similar properties, Box dimension is only finitely additive, while Hausdorff dimension is countably additive. Correlation dimension For a set of n points A on the real number line, let P(r) be the probability that two different points of A chosen at random are closer than r apart. For a large number of points n, the graph of v D log P(r) against u D log r is approximated to its slope for small values of r, and theoretically, a straight line. The correlation dimension DC is defined as its slope for small values of r, that is, dv : r!0 du

DC D lim

The correlation dimension involves the separation of points into “boxes”, whereas the box dimension merely counts the boxes that cover A. If Pi is the probability of a point of A being in box i (approximately n i /n where ni is the number of points in box i and n is the totality of points in A) then an alternative definition of correlation dimension is P log Pi2 i : DC D lim r!0 log r Attractor A point set in phase space which “attracts” trajectories in its vicinity. More formally, a bounded set A in phase space is called an attractor for the solution x(t) of a differential equation if  x(0) 2 A ) x(t) 2 A f or al l t. Thus, an attractor is invariant under the dynamics (trajectories which start in A remain in A).  There is a neighborhood U  A such that any trajectory starting in U is attracted to A (the trajectory gets closer and closer to A).  If B A and if B satisfies the above two properties then B D A. An attractor is therefore the minimal set of points A which attracts all orbits starting at some point in a neighborhood of A. Orbit is a sequence of points fx i g D x0 ; x1 ; x2 ; : : : defined by an iteration x n D f n (x0 ). If n is a positive it is called a forwards orbit, and if n is negative a backwards orbit. If x0 D x n for some finite value n, the orbit is periodic. In this case, the smallest value of n for

which this is true is called the period of the orbit. For an invertible function f , a point x is homoclinic to a if lim f n (x) D lim f n (x) D a

as n ! 1 :

and in this case the orbit f f n (x0 )g is called a homoclinic orbit – the orbit which converges to the same saddle point a forwards or backwards. This term was introduced by Poincaré. The terminology “orbit” may be regarded as applied to the solution of a difference equation, in a similar way the solution of a differential equation x(t) is termed a trajectory. Orbit is the term used for discrete dynamical system and trajectory for the continuous time case. Basin of attraction If a is an attractive fixed point of a function f , its basin of attraction B(a) is the subset of points defined by B(a) D fx : f k (x) ! a ; as k ! 1g : It is the subset containing all the initial points of orbits attracted to a. The basins of attraction may have a complicated structure. An important example applies to the case where a is a point in the complex plane C. Julia set A set J f is the boundary between the basins of attraction of a function f . For example, in the case where z D ˙1 are attracting points (solutions of z2  1 D 0), the Julia set of the “Newton–Fourier” function f (z) D z((z2 1)/2z) is the set of complex numbers which lie along the imaginary axis x D 0 (as proved by Schröder and Cayley in the 1870s). The case of the Julia set involved with the solutions of z3  1 D 0 was beyond these pioneers and is fractal in nature. An alternative definition for a Julia set is the closure of the subset of the complex plane whose orbits of f tend to infinity. Definition of the Subject Though “Chaos” and “Fractals” are yoked together to form “Fractals and Chaos” they have had separate lines of development. And though the names are modern, the mathematical ideas which lie behind them have taken more than a century to gain the prominence they enjoy today. Chaos carries an applied connotation and is linked to differential equations which model physical phenomena. Fractals is directly linked to subsets of Euclidean space which have a fractional dimension, which may be obtained by the iteration of functions. This brief survey seeks to highlight some of the significant points in the history of both of these subjects. There are brief academic histories of the field. A history of

Fractals Meet Chaos

Chaos has been shown attention [4,46], while an account of the early history of the iteration of complex functions (up to Julia and Fatou) is given in [3]. A broad survey of the whole field of fractals is given in [18,50]. Accounts of Chaos and Fractals tend to concentrate on one side at the expense of the other. Indeed, it is only quite recently that the two subjects have been seen to have a substantial bearing on each other [72]. This account treats a “prehistory” and a “modern period”. In the prehistory period, before about 1960, topics which contributed to the modern theory are not so prominent. There is a lack of impetus to create a new field. In the modern period there is a greater sense of continuity, driven on by the popular interest in the subject. The modern theory coincided with a rapid increase in the power of computers unavailable to the early workers in the prehistory period. Scientists, mathematicians, and a wider public could now “see” the beautiful geometrical shapes displayed before them [25]. The whole theory of “fractals and chaos” necessarily involves nonlinearity. It is a mathematical theory based on the properties of processes which are assumed to be modeled by nonlinear differential equations and nonlinear functions. Chaos shows itself when solutions to these differential equations become unstable. The study of stability is rooted in the mathematics of the nineteenth century. Fractals are derived from the geometric study of curves and sets of points generally, and from abstract iterative schemes. The modern theory of fractals is the outcome of explorations by mathematicians and scientists in the 1960s and 1970s, though, as we shall see, it too has an extensive prehistory. Recently, there has been an explosion of published work in Fractals and Chaos. Just in Chaos theory alone a bibliography of six hundred articles and books compiled by 1982, grew to over seven thousand by the end of 1990 [97]. This is most likely an excessive underestimate. Fractals and Chaos has since grown into a wide-ranging and variegated theory which is rapidly developing. It has widespread applications in such areas as Astronomy the motions of planets and galaxies Biology population dynamics chemistry chemical reactions Economics time series, analysis of financial markets Engineering capsize of ships, analysis of road traffic flow Geography measurement of coastlines, growth of cities, weather forecasting Graphic art analysis of early Chinese Landscape Paintings, fractal geometry of music Medicine dynamics of the brain, psychology, heart rhythms.

There are many works which address the application of Chaos and Fractals to a broad sweep of subjects. In particular, see [13,19,27,59,63,64,87,96]. Introduction “Fractals and Chaos” is the popular name for the subject which burst onto the scientific scene in the 1970s and 1980s and drew together practical scientists and mathematicians. The subject reached the popular ear and perhaps most importantly, its eye. Computers were becoming very powerful and capable of producing remarkable visual images. Quite elementary mathematics was capable of creating beautiful pictures that the world had never seen before. The popular ear was captivated – momentarily at least – by the neologisms created for the theory. “Chaos” was one, but “fractals”, the “horseshoe”, and the superb “strange attractors” became an essential part of the scientific vocabulary. In addition, these were being passed around by those who dwelled far from the scientific front. It seemed all could take part, from scientific and mathematical researchers to lay scientists and amateur mathematicians. All could at least experience the excitement the new subject offered. Fractals and Chaos also carried implications for the philosophy of science. The widespread interest in the subject owes much to articles and books written for the popular market. J. Gleick’s Chaos was at the top of the heap in this respect and became a best seller. He advised his readership that chaos theory was one of the great discoveries of the twentieth century and quoted scientists who placed chaos theory alongside the revolutions of Relativity and Quantum Mechanics. Gleick claimed that “chaos [theory] eliminates the Laplacean fantasy of deterministic predictability”. In a somewhat speculative flourish, he reasoned: “Of the three theories, the revolution in chaos applies to the universe we see and touch, to objects at human scale. Everyday experience and real pictures of the world become legitimate targets for inquiry. There has long been a feeling, not always expressed openly, that theoretical physics has strayed far from human intuition about the world [38].” More properly chaos is “deterministic chaos”. The equations and functions used to model a dynamical system are stated exactly. The contradictory nature of chaos is that the mathematical solutions appeared to be random. On the one hand the situation is deterministic, but on the other there did not seem to be any order in the solutions. Chaos theory resolves this difficulty by conceptualizing the notion of orbits, trajectories phase space, attractors, and fractals. What appeared paradoxical seen through the

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old lenses offered explanation through a more embracing mathematical theory. In the scientific literature the modern Chaos is presented through “dynamical systems”, and this terminology gives us a clue to its antecedents. Dynamical systems conveys the idea of a physical system. It is clear that mechanical problems, for example, those involving motion are genuine dynamical systems because they evolve in space through time. Thus the pendulum swings backwards and forwards in time, planets trace out orbits, and the beat of a heart occurs in time. The differential equations which describe physical dynamical system give rise to chaos, but how do fractals enter the scene? In short, the trajectories in the phase space which describes the physical system through a process of spreading and folding pass through neighborhoods of the attractor set infinitely often, and on magnification reveal them running closer and closer together. This is the fine structure of the attractor. Measuring the metric dimension of this set results in a fraction and it is there that the connection with fractals is made. But this is running ahead of the story. Dynamical Systems The source of “Chaos” lies in the analysis of physical systems and goes back to the eighteenth century and the work of the Swiss mathematician Leonhard Euler. The most prolific mathematician of all time, Euler (whose birth tercentenary occurred in 2007) was amongst natural philosophers who made a study of differential equations in order to solve practical mechanical and astronomical problems. Chief among the problems is the problem of fluid flow which occurs in hydrodynamics. Sensitive Initial Conditions The key characteristic of “chaotic solutions” is their sensitivity to initial conditions: two sets of initial conditions close together can generate very different solution trajectories, which after a long time has elapsed will bear very little relation to each other. Twins growing up in the same household will have a similar life for the childhood years but their lives may diverge completely in the fullness of time. Another image used in conjunction with chaos is the so-called “butterfly effect” – the metaphor that the difference between a butterfly flapping its wings in the southern hemisphere (or not) is the difference between fine weather and hurricanes in Europe. The butterfly effect notion most likely got its name from the lecture E. Lorenz gave in Washington in 1972 entitled “Predictability: Does the Flap of a Butterfly’s wings in Brazil Set off a Tornado in

Texas?” [54]. An implication of chaos theory is that prediction in the long term is impossible for we can never know for certain whether the “causal” butterfly really did flap its wings. The sensitivity of a system to initial conditions, the hallmark of what makes a chaotic solution to a differential equation is derived from the stability of a system. Writing in 1873, the mathematical physicist James Clerk Maxwell alluded to this sensitivity in a letter to man of science Francis Galton: Much light may be thrown on some of these questions [of mechanical systems] by the consideration of stability and instability. When the state of things is such that an infinitely small variation of the present state will alter only by an infinitely small quantity the state at some future time, the condition of the system, whether at rest or in motion, is said to be stable; but when an infinitely small variation in the present state may bring about a finite difference in the state of the system in a finite time, the condition is said to be unstable [44]. H. Poincaré too, was well aware that small divergences in initial conditions could result in great differences in future outcomes, and said so in his discourse on Science and Method: “It may happen that slight differences in the initial conditions produce very great differences in the final phenomena; a slight error in the former would make an enormous error in the latter. Prediction becomes impossible.” As an example of this he looked at the task of weather forecasting: Why have the meteorologists such difficulty in predicting the weather with any certainty? Why do the rains, the tempests seem to us to come by chance, so that many persons find it quite natural to pray for rain or shine, when they would think it ridiculous to pray for an eclipse? We see that great perturbations generally happen in regions where the atmosphere is in unstable equilibrium. . . . one tenth of a degree more or less at any point, and the cyclone bursts here and not there, and spreads its ravages over countries it would have spared. . . . Here again we find the same contrast between a very slight cause, unappreciable to the observer, and important effects, which are sometimes tremendous disasters [75]. So here is the answer to one conundrum – Poincaré is the true author of the butterfly effect! Weather forecasting is ultimately a problem of fluid flow in this case air flow.

Fractals Meet Chaos

Fluid Motion and Turbulence The study of the motion of fluids predates Poincaré and Lorenz. Euler published “General Principles of the Motion of Fluids” in 1755 in which he wrote down a set of partial differential equations to describe the motion of non-viscous fluids. These were improved by the French engineer C. Navier when in 1821 he published a paper which took viscosity into account. From a different starting point in 1845 the British mathematical physicist G. G. Stokes derived the same equations in a paper entitled “On the Theories of the Internal Friction of Fluids in Motion.” These are the Navier–Stokes equations, a set of nonlinear partial differential equations which generally apply to fluid motion and which are studied by the mathematical physicist. In modern vector notation (in the case of unforced incompressible flow) they are @v 1  C v:rv D  r p C r 2 v ; @t

where v is velocity, p is pressure, and and  are density and viscosity constants. It is the nonlinearity of the Navier–Stokes equations which makes them intractable, the nonlinearity manifested by the “products of terms” like v:rv which occur in them. They have been studied intensively since the nineteenth century particularly in special forms obtained by making simplifying assumptions. L. F. Richardson, who enters the subject of Fractals and Chaos at different points made attempts to solve nonlinear differential equations by numerical methods. In the 1920s, Richardson adapted words from Gulliver’s Travels in one of his well-known refrains on turbulence: “big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity – in the molecular sense” [79]. This numerical work was hampered by the lack of computing power. The only computers available in the 1920s were human ones, and the traditions of using paid “human computers” was still in operation. Richardson visualized an orchestra of human computers harmoniously carrying out the vast array of calculations under the baton of a mathematician. There were glimmers of all this changing, and during the 1920s the appearance of electrical devices gave an impetus to the mathematical study of both numerical analysis and the study of nonlinear differential equations. John von Neumann for one saw the need for the electronic devices as an aid to mathematics. Notwithstanding this development, the problem of fluid flow posed intrinsic mathematical difficulties. In 1932, Horace Lamb addressed the British Association for the Advancement of Science, with a prophetic

statement dashed with his impish touch of humor: “I am an old man now, and when I die and go to Heaven there are two matters on which I hope for enlightenment. One is quantum electro-dynamics, and the other is the turbulent motion of fluids. And about the former I am really rather optimistic.” Forty years on Werner Heisenberg continued in the same vein. He certainly knew about quantum theory, having invented it, but chose relativity as the competing theory with turbulence for his own lamb’s tale. On his death bed it is said he compared quantum theory with turbulence and is reputed to have singled out turbulence as of the greater difficulty. In 1941, A. Kolmogorov, the many-sided Russian mathematician published two papers on problems of turbulence caused by the jet engine and astronomy. Kolmogorov also made contributions to probability and topology though these two papers are the ones rated highly by fluid dynamicists and physicists. In 1946 he was appointed to head the Turbulence Laboratory of the Academic Institute of Theoretical Geophysics. With Kolmolgov, an influential school of mathematicians that included L. S. Pontryagin, A. A. Andronov, D. V. Anosov and V. I. Arnol’d became active in Russia in the field of dynamical systems [24]. Qualitative Differential Equations and Topology Poincaré’s qualitative study of differential equations pioneered the idea of viewing the solution of differential equations as curves rather than functions and replacing the local with the global. Poincaré’s viewpoint was revolutionary. As he explained it in Science and Method: “formerly an equation was considered solved only when its solution had been expressed by aid of a finite number of known functions; but that is possible scarcely once in a hundred times. What we always can do, or rather what we should always seek to do, is to solve the problem qualitatively [his italics] so to speak; that is to say; seek to know the general form of the curve [trajectory] which represents the unknown function.” For the case of the plane, m D 2, for instance, what do the solutions to differential equations look like across the whole plane, viewing them as trajectories x(t) starting at an initial point? This contrasts with the traditional view of solving differential equations whereby specific functions are sought which satisfy initial and boundary conditions. Poincaré’s attack on the three body problem (the motion of the moon, earth, sun, is an example) was stimulated by a prize offered in 1885 to commemorate the sixtieth birthday of King Oscar II of Sweden. The problem set was for an n-body problem but Poincaré’s essay on the re-

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stricted three-body problem was judged so significant that he was awarded the prize in January 1889. Before publication, Poincaré went over his working and found a significant error. It proved a profound error. According to Barrow-Green his description of doubly asymptotic trajectories is the first mathematical description of chaotic motion in a dynamical system [6,7]. Poincaré introduced the “Poincaré section” the surface section that transversely intersects the trajectory in phase space and defines a sequence of points on it. In the case of the damped pendulum, for example, a sequence of points is obtained as the spiral trajectory in phase space hits the section. This idea brings a physical dynamical systems problem in conjunction with topology, a theme developed further by Birkhoff in 1927 [1,10]. Poincaré also introduced the idea of phase space but his study mainly revolves around periodic trajectories and not the non periodic ones typical of chaos. Poincaré said of periodic orbits, that they were “the only gap through which may attempt to penetrate, into a place where up to now was reputed to be unreachable”. The concentration on periodic trajectories has a parallel in nearby pure mathematics – where irregular curves designated as “monsters” were ignored in favor of “normal curves”. G. D. Birkhoff learned from Poincaré. It was said that, apart from J. Hadamard, no other mathematician knew Poincaré’s work as well as Birkhoff did. Like his mentor, Birkhoff adopted phase space as his template, and emphasised periodic solutions and treated conservative systems and not dissipative systems (such as the damped pendulum which loses energy). Birkhoff spent the major part of his career, between 1912 and 1945 contributing to the theory of dynamical systems. His aim was to provide a qualitative theory which characterized equilibrium points in connection with their stability. A dynamical system was defined by a set of n differential equations dx i /dt D F i (x1 ; : : : ; x n ) defined locally. An interpretation of these could be a situation in chemistry where x1 (t); : : : ; x n (t) might be the concentrations of n chemical reactants at time t where the initial values x1 (0); : : : ; x n (0) are given, though applications were not Birkhoff’s main concern. In 1941, towards the end of his career, Birkhoff reappraised the field in “Some unsolved problems of theoretical dynamics” Later M. Morse discussed these and pointed out that Birkhoff listed the same problems he had considered in 1920. In 1920 Birkhoff had written about dynamical systems in terms of the “general analysis” propounded by E. H. Moore in Chicago where he had been a student. In the essay of 1941 modern topological language was used:

As was first realized about fifty years ago by the great French mathematician, Henri Poincaré, the study of dynamical systems (such as the solar system) leads directly to extraordinary diverse and important problems in point-set theory, topology and the theory of functions of real variables. The idea was to describe the phase space by an abstract topological space. In his talk of 1941, Birkhoff continued: The kind of abstract space which it seems best to employ is a compact metric space. The individual points represent “states of motion”, and each curve of motion represents a complete motion of the abstract dynamical system [11]. Using Birkhoff’s reappraisal, Morse set out future goals: “ ‘Conservative flows’ are to be studied both in the topological and the statistical sense, and abstract variational theory is to enter. There is no doubt about the challenge of the field, and the need for a powerful and varied attack [68].” Duffing, Van der Pol and Radar It is significant that chaos theory was first derived from practical problems. Poincaré was an early pioneer with a problem in astronomy but other applications shortly arrived. C. Duffing (1918) introduced a second order nonlinear differential equation which described a mechanical oscillating device. In a simple form of it (with zero forcing term): d2 x/dt 2 C ˛dx/dt C (ˇx 3 C x) D 0 ; Duffing’s equation exhibits chaotic solutions. B. van der Pol working at the Radio Scientific Research group at the Philips Laboratory at Eindhoven, described “irregular noise” in an electronic diode. Van der Pol’s equations of the form (with right hand side forcing term): d2 x/dt 2 C k(x 2  1)dx/dt C x D A cos ˝ t described “relaxational” oscillations or arrhythmic beats of an electrical circuit. Such “relaxational” oscillations are of the type which also occur in the beating of the heart. Richardson noted the transition from periodic solutions to van der Pol’s equation (1926) to unstable solutions. Both Duffing’s equation and Van der Pol’s equation play an important part in chaos theory. In 1938 the English mathematician Mary Cartwright answered a call for help from the British Department of Scientific and Industrial Research. A solution to the differential equations connected with the new radar technology

Fractals Meet Chaos

was wanted, and van der Pol’s equation was relevant. It was in the connection this equation that she collaborated with J. E. Littlewood in the 1940s and made discoveries which presaged modern Chaos. The mathematical difficulty was caused by the equation being nonlinear but otherwise the equation appear nondescript. Yet in mathematics, the nondescript can yield surprises. A corner of mathematics was discovered that Cartwright described as “a curious branch of mathematics developed by different people from different standpoints – straight mechanics, radio oscillations, pure mathematics and servo-mechanisms of automatic control theory [65]” The mathematician and physicist Freeman Dyson wrote of this work and its practical significance: Cartwright had been working with Littlewood on the solutions of the equation, which describe the output of a nonlinear radio amplifier when the input is a pure sine-wave. The whole development of radio in World War II depended on high power amplifiers, and it was a matter of life and death to have amplifiers that did what they were supposed to do. The soldiers were plagued with amplifiers that misbehaved, and blamed the manufacturers for their erratic behavior. Cartwright and Littlewood discovered that the manufacturers were not to blame. The equation itself was to blame. They discovered that as you raise the gain [the ratio of output to input] of the amplifier, the solutions of the equation become more and more irregular. At low power the solution has the same period as the input, but as the power increases you see solutions with double the period, and finally you have solutions that are not periodic at all [30].

Crinkly Curves In 1872, K. Weierstrass introduced the famous function defined by a convergent series: f (x) D

1 X

b k cos(a k  x) (a > 1; 0 < b < 1; ab > 1);

kD0

which was continuous everywhere but differentiable nowhere. It was the original “crinkly curve” but in 1904, H. von Koch produced a much simpler one based only on elementary geometry. While the idea of a function being continuous but not differentiable could be traced back to A-M Ampère at the beginning of the nineteenth century, von Koch’s construction has similarities with examples produced by B. Bolzano. Von Koch’s curve has become a classic half-way between a “monster curves” and regularity – an example of a curve of infinite length which encloses a finite area, as well as being an iconic fractal. In retrospect G. Cantor’s middle third set (also discovered by H.J.S. Smith (1875) a professor at Oxford), is also a fractal. It is a totally disconnected and uncountable set, with the curiosity that after subtractions of the middlethird segments from the unit interval the ultimate Cantor set has same cardinality as the original unit interval. At the beginning of the twentieth century searching questions about the theory of curves were being asked. The very basic question “what is a curve” had been brought to life by G. Peano by his space filling curve which is defined in accordance with Jordan’s definition of a curve but fills out a “two-dimensional square.” Clearly the theory of dimension needed serious attention for how could an ostensibly two-dimensional “filled-in square” be a curve? The Iteration of Functions

The story is now familiar: there is the phenomenon of period doubling of solutions followed by chaotic solutions as the gain of the amplifier is raised still higher. A further contribution to the theory of this equation was made by N. Levinson of the Massachusetts Institute of Technology in the United States.

Curves and Dimension The subject of “Fractals” is a more recent development. First inklings of them appeared in “Analysis situs” in the latter part of the nineteenth century when the young subject of topology was gaining ground. Questions were being asked about the properties of sets of points in Euclidean spaces, the nature of curves, and the meaning of dimension itself.

A principal source of fractals is obtained by the iteration of functions, what is now called a discrete dynamical system, or a system of symbolic dynamics [15,23]. One of the earliest forays in this field was made by the English mathematician A. Cayley, but he was not the first as D. S. Alexander has pointed out. F.W. K. E. (Ernst) Schröder anticipated him, and may even have been the inspiration for Cayley’s attraction to the problem. Is there a link between the two mathematicians? Schröder studied at Heidelberg and where L. O. Hesse was his doctoral adviser. Hesse who contributed to algebra, geometry, and invariant theory and was known to Cayley. Nowadays Schröder is known for his contributions to logic but in 1871 he published “Ueber iterite Functionen” in the newly founded Mathematische Annalen. The principal objective of this journal was the publication of articles

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on invariant theory then a preoccupation of the English and German schools of mathematics, and Cayley was at the forefront of this research. So did Cayley know of Schröder’s work? Cayley had an appetite for all things (pure) mathematical and had acquired encyclopedic knowledge of the field, just about possible in the 1870s. In 1871 Cayley published an article (“Note on the theory of invariants”) in the same volume and issue of the Mathematische Annalen in which Schröder’s article appeared, and actually published three articles in the volume. In this period of his life, when he was in his fifties he allowed his interests full rein and became a veritable magpie of mathematics. He covered a wide range of topics, too many to list here, but his contributions were invariably short. Moreover he dropped invariant theory at this point and only resumed in 1878 when J. J. Sylvester reawakened his interest. There was time to rediscover the four color problem and perhaps Schröder’s work on the iteration of functions. It was a field where Schroeder had previously “encountered very few collaborators.” The custom of English mathematicians citing previous work of others began in the 1880s and Cayley was one of the first to do this. The fact that Cayley did not cite Schröder is not significant. In February 1879, Cayley wrote to Sir William Thomson (the later Lord Kelvin) about the Newton–Fourier method of finding the root of an equation, named after I. Newton (c.1669) and J. Fourier (1818) that dealt with the real variable version of the method. This achieved a degree of significance in Cayley’s mind, for a few days later he wrote to another colleague about it: I have a beautiful question which is bothering me – the extension of the Newton–Fourier method of approximation to imaginary values: it is very easy and pretty for a quadric equation, but I do not yet see how it comes out for a cubic. The general notion is that the plane must be divided into regions; such that starting with a point P in one of these say the A-region . . . [his ellipsis], the whole series of derived points P1 ; P2 ; P3 ; : : : up is P1 (which will be the point A) lies in this [planar] region; . . . and so for the B. and C. regions. But I do not yet see how to find the bounding curves [of these regions] [21]. So Cayley’s regions are the modern basins of attraction for the point A, the bounding curves now known as Julia sets. He tried out the idea before the Cambridge Philosophical Society, and by the beginning of March had done enough to send the problem for publication: In connexion herewith, throwing aside the restrictions as to reality, we have what I call the Newton–

Fourier Imaginary Problem, as follows. Take f (u) a given rational and integral function [a polynomial] of u, with real or imaginary coefficients; z, a given real or imaginary value, and from this derive z1 by the formula z1 D z 

f (z) f 0 (z)

and thence z1 ; z2 ; z3 ; : : : each from the preceding one by the like formula. . . . The solution is easy and elegant in the case of a quadric equation: but the next succeeding case of the cubic equation appears to present considerable difficulty [17]. Cayley’s connection with German mathematicians was close in the 1870s, and later on in 1879 (and in 1880) he went on tours of Germany, and visited mathematicians. No doubt he took the problem with him, and it was one he returned to periodically. Both Cayley and Schröder solved this problem for the roots of z2 D 1 but their methods differ. Cayley’s is geometrical whereas Schröder’s was analytical. There are only two fixed points z D ˙1 and the boundary (Julia set) between the two basins of attraction of the root finding function is the “y-axis.” The algorithm for finding the complex roots of the cubic equation z3 D 1 has three stable fixed points at z D 1; z D exp(2 i/3); z D exp(2 i/3) and with the iteration z ! z  (z3  1)/3z2 , three domains, basins of attraction exist with highly interlaced boundaries. The problem of determining the Julia set for the quadratic was straightforward but the cubic seemed impossible [42]. For good reason, with the aid of modern computing machinery the Julia set in the case of the cubic is an intricately laced trefoil. But some progress was made before computers entered the field. After WWI G. Julia and P. Fatou published lengthy papers on iteration and later C. L. Siegel studied the field [35,48,85]. Topological Dimension The concept of dimension has been an enduring study for millennia, and fractals has prolonged the centrality of this concept in mathematics. The ordinary meaning of dimensions one, two, three dimensions applied to line, plane, solid of the Greeks, was initially extended to n-dimensions. Since the 1870s mathematicians ascribed newer meanings to the term. At the beginning of the twentieth century, Topology was in its infancy, and emerging from “analysis situs.” An impetus was the Hausdorff’s definition of a topological space in terms of neighborhoods in 1914. The evolution of topological dimension was developed in the hands

Fractals Meet Chaos

of such mathematicians as L. E. J. Brouwer, H. Poincaré, K. Menger, P. Urysohn, F. Hausdorff but in the new surroundings, the concepts of curve and dimension proved elusive. Poincaré made several attempts to define dimension. One was in terms of group theory, one he described as a “dynamical theory.” This was unsatisfactory for why should dimension depend on the idea of a group. He gave another definition that he described as a “statical theory.” Accordingly in papers of 1903, and 1912, a topological notion of n-dimensions (where n a natural number) was based on the notion of a cut, and Poincaré wrote: If to divide a continuum it suffices to consider as cuts a certain number of elements all distinguishable from one another, we say this continuum is of one dimension; if, on the contrary, to divide a continuum it is necessary to consider as cuts a system of elements themselves forming one or several continua, we shall say that this continuum is of several dimensions. If to divide a continuum C, cuts which form one or several continua of one dimension suffice, we shall say that C is a continuum of two dimensions; if cuts which form one or several continua of at most two dimensions suffice, we shall say that C is a continuum of three dimensions; and so on [74]. To illustrate the pitfalls of this game of cat and mouse where “definition” attempts to capture the right notion, we see this definition yielded some curious results – the dimension of a double cone ostensibly of two dimensions turns out to be of one dimension, since one can delete the zero dimensional point where the two ends of the cone meet. Curves were equally difficult to pin down. Menger used a physical metaphor to get at the pure notion of a curve: We can think of a curve as being represented by fine wires, surfaces as produced from thin metal sheets, bodies as if they were made of wood. Then we see that in order to separate a point in the surface from points in a neighborhood or from other surfaces, we have to cut the surfaces along continuous lines with a scissors. In order to extract a point in a body from its neighborhood we have to saw our way through whole surfaces. On the other hand in order to excise a point in a curve from its neighborhood irrespective of how twisted or tangled the curve may be, it suffices to pinch at discrete points with tweezers. This fact, that is independent of the particular form of curves or surfaces we consider, equips us with a strong conceptual description [22].

Menger set out his ideas about basic notions in mathematical papers and in the books Dimensionstheorie (1928) and Kurventheorie (1932). In Dimensionstheorie he gave an inductive definition of dimension, on the implicit understanding that dimension only made sense for n an integer ( 1): A space is called at most n-dimensional, if every point is contained in arbitrarily small neighborhoods with an most (n1)-dimensional boundaries. A space that is not at most (n  1)-dimensional we call at least n-dimensional. . . . A Space is called n-dimensional, if it is both at most n-dimensional and also at least n-dimensional, in other words, if every point is contained in arbitrarily small neighborhoods with at most (n 1)-dimensional boundaries, but at least one point is not contained in arbitrarily small boundaries with less than (n  1)-dimensional boundaries. . . . The empty set and only this is (–1)-dimensional (and at most (–1)-dimensional. A space that for no natural number n is n-dimensional we call infinite dimensional. Different notions of topological dimension defined in the 1920s and 1930s, were ind X (the small inductive dimension), Ind X (the large inductive dimension), and dim X (the Lebesgue covering dimension). Researchers investigated the various inequalities between these and the properties of abstract topological spaces which ensured all these notions coincided. By 1950, the theory of topological dimension was still in its infancy [20,47,66]. Metric Dimension For the purposes of fractals, it is the metric definitions of dimension which are fundamental. For instance, how can we define the dimension of the Cantor’s “middle third” set which takes into account its metric structure? What about the iconic fractal known as the von Koch curve snow flake curve (1904)? These sets pose interesting questions. Pictorially von Koch’s curve is made up of small 1 dimensional lines and we might be persuaded that it too should be 1 dimensional. But the real von Koch curve is defined as a limiting curve and for this there are differences between it and a line. Between any two points on a 1 dimensional line there is a finite distance but between any two points on the Koch curve the distance is infinite. This suggests its dimensionality is greater than 1 while at the same time it does not fill out a 2 dimensional region. The Sierpinski curve (or gasket) is another example, but perhaps more spectacular is the “sponge curve”. In Kurventheorie (1932) Menger gave what is now known as

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the Menger Sponge, the set obtained from the unit cube by successively deleting sub-blocks in the same way as Cantor’s middle third set is obtained from the unit interval. A colorful approach to defining metric dimension is provided by L. F. Richardson. Richardson (1951) took on the practical measurement of coastlines from maps. To measure the coastline, Richardson used dividers set a disP tance l apart. The length of the coastline will be L D l after walking the dividers around the coast. Richardson was a practical man, and he found by plotting L against l, the purely empirical result that L / l ˛ for a constant ˛ that depends on the chosen coastline. So, for a given country we get an approximate length L D cl ˛ but the smaller the dividers are set apart the longer becomes the coastline! The phenomenon of the coastline length being “divider dependent” explains the discrepancy of 227 kilometers in the measurement by of the border between Spain and Portugal (987 km stated by Spain, and 1214 km stated by Portugal). The length of the Australian coastline turns out to be 9400 km if one divider space represents 2000 km, and is 14,420 km if the divider space represent 100 km. Richardson cautions “to speak simply of the “length” of a coast is therefore to make an unwarranted assumption. When a man says that he “walked 10 miles along the coast,” he usually means that he walked 10 miles [on a smooth curve] near the coast.” Richardson goes into incredible numerical data and his work is of an empirical nature, but it paid off, and he was able to combine two branches of science, the empirical and the mathematical. Moreover here we have a link with chaos: Richardson described the “sensitivity to initial conditions” where a small difference in the setting of the dividers can result in a large difference in the “length” of the coastline. The constant ˛ is a characteristic of the coastline or frontier whose value is dependent on its place in the range between smoothness and jaggedness. So if the frontier is a straight line, ˛ would be zero, and increases the more irregular the coast line. In the case of Australia, ˛ was found to be about 0.13 and for the very irregular west coast of Britain, ˛ was found to be about 0.25. The value 1 C ˛ anticipates the mathematical concept known as “divider dimension”. So, for example, the divider dimension of the west coast of Britain would be 1.25. The divider dimension idea is fruitful – it can be applied to Brownian motion, a mathematical theory set out by Norbert Wiener in the 1920s but it is not the main one when applied to fractal dimension. “Fractal dimension” means Hausdorff dimension, or as we already noted, the Hausdorff–Besicovitch dimension. Generally Hausdorff dimension is difficult to calculate, but for the self-similar sets and curves such as Can-

tor’s middle-third set, the Sierpinski curve, the von Koch curve, Menger’s sponge, it is equivalent to calculating the box dimension or Minkowski–Bouligand dimension (see also [61]). The middle third set has fractal dimension DH D log 2/ log 3 D 0:63 : : :, and the Sierpinski curve has DH D log 3/ log 2 D 1:58 : : :, the von Koch curve DH D log 4/ log 3 D 1:26 : : :, and Menger’s sponge embedded in three-dimensional Euclidean space has Hausdorff dimension DH D log 20/ log 3 D 2:72 : : :. Chaos Comes of Age The modern theory of Chaos occurred around the end of the 1950s. S. Smale, Y. Ueda, E. N. Lorenz made discoveries which ushered in the new age. The subject became extremely popular in the early 1970s and the “avant-garde” looked back to this period as the beginning of chaos theory. They contributed some of the most often cited papers in further developments. Physical Systems In 1961 Y. Ueda, a third year undergraduate student in Japan discovered a curious phenomenon in connection with the single “Van der Pol” type nonlinear equation d2 x/dt 2 C k( x 2  1)dx/dt C x 3 D ˇ cos t ; another example of an equation used to model an oscillator. With the parameter values set at k D 0:2; D 8; ˇ D 0:35 Ueda found the dynamics was “chaotic” – though of course he did not use that term. With an analogue computer, of a type then used to solve differential equations, the attractor in phase space appeared as a “shattered egg.” Solutions for many other values of the parameters revealed orderly behavior, but what was special about the values 0.2, 8, 0.35? Ueda had no idea he had stumbled on a major discovery. Forty years later he reminisced on the higher principles of the scientific quest: “but while I was toiling alone in my laboratory,” he said, “I was never trying to pursue such a grandiose dream as making a revolutionary new discovery, not did I ever anticipate writing a memoir about it. I was simply trying to find an answer to a persistent question, faithfully trying to follow the lead of my own perception of a problem [2].” In 1963, E. N. Lorenz published a landmark paper (with at least 5000 citations to date) but it was one published in a journal off the beaten track for mathematicians. Lorenz investigated a simple model for atmospheric convection along the lines of the Rayleigh–Bernard convection model. To see what Lorenz actually did we quote the abstract to his original paper:

Fractals Meet Chaos

Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamical flow. Solutions of these equations can be identified with trajectories in phase space. For those solutions with bounded solutions, it is found that non periodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are non periodic [53]. The oft quoted equations (with simplifying assumptions, and a truncated version of the Navier–Stokes equations) used to model Rayleigh–Bernard convection are the nonlinear equations [89]: dx/dt D (y  x) dy/dt D rx  y  xz dz/dt D x y  bz ; where  is called the Prandtl number, r is the Rayleigh number and b is a geometrically determined parameter. From the qualitative viewpoint, the solution of these equations, (x1 (t); x2 (t); x3 (t)) with initial values (x1 (0); x2 (0); x3 (0)) traces out a trajectory in 3-dimensional phase space. Lorenz solved the equations numerically by a forward difference procedure based on the time honored Runge–Kutta method which set up an iterative scheme of difference equations. Lorenz discovered that the case where the parameters have specific values of  D 10; r D 28; b D 8/3 gave chaotic trajectories which wind around the famous Lorenz attractor in the phase space – and by accident, he discovered the butterfly effect. The significance of Lorenz’s work was the discovery of Chaos in low dimensional systems – the equations described dynamical behavior of a kind seen by only a few – including Ueda in Japan. One can imagine his surprise and delight, for though working in Meteorology, Lorenz had been a graduate student of G. D. Birkhoff at Harvard. Once the chink in the mathematical fabric was made, the same kind of mathematical behavior of chaos was discovered in other systems of differential equations. The Lorenz attractor became the subject of intensive research. Once it was discovered by mathematicians – not many mathematicians read the meteorology journals – it excited much attention and helped to launch the chaos craze. But one outstanding problem remained: did the

Lorenz attractor actually exist or was its presence due to the accumulation of numerical errors in the approximate methods used to solve the differential equations. Computing power was in its infancy and Lorenz made his calculations on a fairly primitive string and sealing-wax Royal McBee LGP-30 machine. Some believed that the numerical evidence was sufficient for the existence of a Lorenz attractor but this did not satisfy everyone. This question of actual existence resisted all attempts at its solution because there were no mathematical tools to solve the equations explicitly. The problem was eventually cracked by W. Tucker in 1999 then a postgraduate student at the University of Uppsala [94]. Tucker’s proof that the Lorenz attractor actually exists involved a rigorous computer algorithm in conjunction with a rigorous set of bounds on the possible numerical errors which could occur. It is very technical [95]. Other sets of differential equations which exemplified the chaos phenomena, one even more basic than Lorenz’s is due to O. Rössler, equations which modeled chemical reactions: dx/dt D y  z dy/dt D x C ˛ y dz/dt D ˛  z C xz : Rössler discovered chaos for the parameter values ˛ D 0:2 and  D 5:7 [77]. This is the simplest system yet found, for it has only one nonlinear term xz compared with two in the case of Lorenz’s equations. Other examples of chaotic solutions are found in the dripping faucet experiment caried out by Shaw (1984) [84]. Strange Attractors An attractor is a set of points in phase space with the property that a trajectory with an initial point near it will be attracted to it – and if the trajectory touches the attractor it is trapped to stay within it. But what makes an attractor “strange”? The simple pendulum swings to and fro. This is a dissipative system as the loss of energy causes the bob to come to rest. Whatever the initial point stating point of the bob, the trajectory in phase space will spiral down to the origin. The attractor in this case is simply the point at the origin; the rest point where displacement is nil and velocity is nil. This point is said to attract all solutions of the differential equations which models the pendulum. This single point attractor is hardly strange. If the pendulum is configured to swing to and fro with a fixed amplitude, the attractor in phase space will be a circle. In this case the system conserves energy and the whole

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system is called a conservative system. If the pendulum is slightly perturbed it will return to the previous amplitude, and in this sense the circle or a limit cycle will have attracted the displaced trajectory. Neither is the circle strange. The case of the driven pendulum is different. In the driven pendulum the anchor point of the pendulum oscillates with a constant amplitude a and constant drive frequency f . The equation of motion of the angular displacement from the vertical with damping parameter q can be described as a second order nonlinear (because of the presence of the sin term in the differential equation): d2 /dt 2 C (1/q) d /dt C sin D a cos f t : Alternatively this motion can be described by three simultaneous equations: dw/dt D (1/q)w  sin C a cos  d /dt D w d/dt D f ; where  is the phase of the drive term. The three variables (w; ; ) describe the motion of the driven pendulum in three–dimensional phase space and its shape will depend on the values of the parameters (a; f ; q). For some values, just as for the Lorenz attractor, the motion will be chaotic [5]. For an attractor to be strange, it should be fractal in structure. The notion of strange attractor is due to D. Ruelle and F. Takens in papers of 1971 [82]. In their work there is no mention of fractals, simply because fractals had not risen to prominence at that time. Strange attractors for Ruelle and Takens were infinite sets of points in phase space corresponding to points of a physical dynamic system which appeared to have a complicated evolution – they were just very weird sets of points. But the principle was gained. Dynamical systems, like the gusts in wind turbulence, are in principle modeled by deterministic differential equations but now their solution trajectories seemed random. In classical physics mechanical processes were supposed to be as uncomplicated as the pendulum where the attractor was a single point or a circular limit cycle. The seemingly random processes that now appeared offered the mathematician and physicist a considerable challenge. The name “strange attractor” caught on and quickly captured the scientific and popular imagination. Ruelle asked Takens if he had dreamed up the name, and he replied: “Did you ever ask God whether he created this damned universe? . . . I don’t remember anything . . . I often create without remembering it . . . ” and Ruelle wrote

the “creation of strange attractors thus seems to be surrounded by clouds and thunder. Anyway, the name is beautiful, and I well suited to these astonishing objects, of which we understand so little [80].” Ruelle wrote “These systems of curves [arising in the study of turbulence], these clouds of points, sometimes evoke galaxies or fireworks, other times quite weird and disturbing blossomings. There is a whole world of forms still to be explored, and harmonies still to be discovered [45,91].” Initially a strange attractor was a set which was extraordinary in some sense and there were naturally attempts to pin this down. In particular distinctions between “strange attractors” and “chaotic behavior” were drawn out [14]. Accordingly, a strange attractor is an attractor which is not (i) a finite set of points (ii) a closed curve (iii) a smooth or piecewise smooth surface or a volume bounded by such a surface. “Chaotic behavior” relates to the behavior of trajectories on the points of the attractor where nearby orbits around the attractor diverge with time. A strange nonchaotic attractor is one (with the above exclusions) where the trajectories are not chaotic, that is, there is an absence of sensitivity to initial conditions. Pure Dynamical Systems Chaos had its origins in real physical problems and the consequent differential equations, but the way had been set by Poincaré and Birkhoff for the entry of pure mathematicians. S. Smale gained his doctorate from the University of Michigan in 1956 supervised by the topologist R. Bott, stepped into this role. In seeking out fresh fields for research he surveyed Birkhoff’s Collected Papers and read the papers of Levinson on the van der Pol equation.. On a visit to Rio de Janeiro in late 1959, Smale focused on general dynamical systems. What better place to do research than on the beach: My work was mostly scribbling down ideas and trying to see how arguments could be sequenced. Also I would sketch crude diagrams of geometric objects flowing through space and try to link the pictures with formal deductions. Deeply involved in this kind of thinking and writing on a pad of paper, the distraction of the beach did not bother me. Moreover, one could take time off from the research to swim [88]. He got into hotter water to the north when it was brought to the attention of government officials that national research grants were being spent at the seaside. They seemed unaware that a mathematical researcher is always working. Smale certainly had a capacity for hard concentrated work.

Fractals Meet Chaos

Receiving a letter from N. Levinson, about the Van der Pol equation, he reported: I worked day and night to try to resolve the challenge to my beliefs that letter posed. It was necessary to translate Levinson’s analytic argument into my own geometric way of thinking. At least in my own case, understanding mathematics doesn’t come from reading or even listening. It comes from rethinking what I see or hear. I must redo the mathematics in the context of my particular background. . . . In any case I eventually convinced myself that Levinson was correct, and that my conjecture was wrong. Chaos was already implicit in the analyzes of Cartwright and Littlewood! The paradox was resolved, I had guessed wrongly. But while learning that, I discovered the horseshoe [8]. The famous “horseshoe” mapping allowed him to construct a proof of the Poincaré conjecture in dimensions greater than or equal to five, and for this, he was awarded a Field’s medal in 1966. Iteration provides an example of a dynamical system, of a kind involving discrete time intervals. These were the systems suggested by Schröder and Cayley, and take on a pure mathematical guise with no physical experimenting to act as a guide. In the late 1950s, a research group in Toulouse led by I. Gumowski and his student C. Mira pursued an exploration of nonlinear systems from this point of view. Their approach was inductive, and started with the simplest example of nonlinear maps of the plane which were then iterated. Functions chosen for exploration were the family, x ! y  F(x) y ! x C F(y  F(x)) ; for various rational functions F(x). The pictures in the plane were noted for their spectacular “chaos esthétique.” The connection of chaos with the iteration of functions was pioneered by Gumowski and Mira, O. M. Sharkovsky (in 1964), Smale (in 1967) [86], and N. Metropolis, M. Stein, P. Stein (in 1973) [67]. M. Feigenbaum (in the late 1970s) studied the quadratic logistic map x ! x(1  x) and its iterates and discovered period doubling and the connection with physical phenomena. For the value  D 3:5699456 : : : the orbit is non periodic and the attractor is the Cantor set so it is a strange attractor. Feigenbaum studied one dimensional iterative maps of a general type x !  f (x) for a general f and discovered properties independent of the form of the recursion function. The Feigenbaum number, the bifurcation rate ı with

value ı D 4:6692 : : : is the limit of the parameter intervals in which period doubling takes place before the onset of chaos. The intervals are shortened at each period doubling by the inverse of the Feigenbaum number, a universal value for many functions [36,37]. In the case of fluid flow, it was discovered that the transition from smooth flow to turbulence is linked with period doubling. The iteration of functions, a very simple process, was brought to the attention of the wider scientific public by R. May. In his oft quoted piece on the one dimension quadratic mapping, he stressed the importance it held for mathematical education: I would therefore urge that people be introduced to, say, equation x ! x(1  x) early in their mathematical education. This equation can be studied phenomenologically by iterating it on a calculator, or even by hand. Its study does not involve as much conceptual sophistication as does elementary calculus. Such study would greatly enrich the student’s intuition about nonlinear systems. Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties [62]. The example of x ! x(1  x) provides another link with the mathematical past, which amply demonstrates that Fractals and Chaos is not merely a child of the sixties but is joined to the mainstream of mathematics which slowly evolves over centuries. The Schwarzian derivative y D f (x) defined by dx d3 y 3 S(y) D  dy dx 3 2

dx d2 y dy dx 2

!2 ;

is connected with complex analysis and also introduced into invariant theory (where it was called a differentiant) of a hundred years previously. It was introduced by D. Singer in 1978 in the context of one dimensional dynamical systems [26]. While the quadratic map is one dimensional, we can go into two dimensions with pairs of nonlinear difference equations. A beautiful example is the one (x; y) ! (y  x 2 ; a C bx) or in the equivalent form (x; y) ! (1Cyax 2 ; bx). In this case, for some values of the parameters ellipses are generated, but in others we gain strange attractors. This is the Hénon schemeb D 0:3, and a near 1.4 the attractor has an infinity of attracting points, and being a fractal set it is a strange attractor [81]. A Poincaré

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section of this attractor is a Cantor set, further evidence for the set being a strange attractor. The Hénon map is one of the earliest examples illustrating chaos. It has Jacobian (which measures area) b for all points in the plane, so if jbj < 1 the Hénon mapping of the plane contracts area by a constant factor b for any point in the plane [41]. The effect of iterating this mapping is to stretch and fold its image in the manner of the Smale horseshoe mapping. The Hénon attractor is the prototype strange attractor. It arises in the case of a diffeomorphism of the plane which stretches and folds an open set U with ¯ U. the property that f (U) The Advent of Fractals The term was coined by B. B. Mandelbrot in the 1970s and now Mandelbrot is regarded as the “father of fractals” [55,56]. His first forays were to study fractals which were invariant under linear transformations. His uncle (Szolem Mandelbrojt, a professor of Mathematics and Mechanics at the Collège de France and later at the French Académie des Sciences) encouraged him to read the original papers by G. Julia and P. Fatou as a young man. Euclid’s Elements Fractals are revolutionary because they challenge one of the sacred texts of mathematics. Euclid’s Elements had reigned over mathematics for well over two thousand years and enjoys the distinction of still be referred to by working mathematicians. Up to the nineteenth century it was the essential fare of mathematics taught in schools. In Britain the Elements exerted the same authority as the Bible which was the other book committed to memory by generations of school pupils. Towards the end of the nineteenth century its central position in the curriculum was challenged, and it was much for the same reasons that Mandelbrot challenged it in the 1980s. Praised for its logical structure, learning Euclid’s deductive proofs by heart was simply not the way to teach an understanding of geometry. In France it had been displaced at the beginning of the century but in Britain and many other countries the attack on its centrality came at the end of the century. Ironically one of the main upholders of the sovereignty of Euclid and a staunch defender of the sacred book was Cayley. He was just the man who a few years before had shown himself to be more in the Mandelbrot mode, who threw aside “the restrictions as to reality” in his investigation of primitive symbolic dynamic systems. And there is a second irony, for Cayley was a man who loved nature and likened the beauty of the natural world to the terrain of mathematics itself – and in prac-

tical terms used his mountaineering exploits as a way of discovering new ideas in the subject. A century later Mandelbrot’s criticism came but from a different direction but containing enough phrases which would have gained a nod of agreement from those who mounted their opposition all those years before. Mandelbrot’s opening paragraph launched the attack: Why is geometry often described as “cold” and “dry”? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are nor spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line [57]. Mandelbrot was setting out an ambitious claim, and behind it was the way he used for conducting mathematical research. He advocated a different methodology from the Euclidean style updated by the Bourbakian mantra of “definition, theorem, proof”. In fact he describes himself as an exile, driven from France by the Bourbaki school. Jean Dieudonné, a leading Bourbakian was at the opposite end of the spectrum in his mathematical style. Dieudonné’s objective, as set out in a graduate text Foundations of Modern Analysis (1960) was to “train the student in the use of the most fundamental mathematical tool of our time – the axiomatic method with which he [sic] will have had very little contact, if any at all, during his undergraduate years”. Echoing the attitude of the nineteenth century geometer Jacob Steiner, who eschewed diagrams of any kind as inimical to the property development of the subject, Dieudonné appealed only to axiomatic methods, and to make the point in his book, he deliberately avoided introducing any diagrams and appeal to “geometric intuition”[28]. What Mandelbrot advocated was the methodology of the physician’s and lawyer’s “casebook.” He noted “this term has no counterpart in science, and I suggest we appropriate it”. This was rather novel and presents a counterpoint to the image that the Bourbakian ideal of great theorems with nicely turned out proofs are what is required above all else. If mathematics is presented as a completed house built by a great mathematician, as the Bourkadians could suggest, questions are only to be found in the higher reaches of the subjects, that demand years of study to reach them. In his autobiography, Mandelbrot claimed “in pure mathematics, my main contribution has not been to provide proofs, but to ask new questions – usually very hard ones – suggested by physics and pictures”. His questions spring from elementary objects in mathematics, but ones

Fractals Meet Chaos

which start off in the world. He summarizes his long career in his autobiography: My whole career became one long, ardent pursuit of the concept of roughness. The roughness of clusters in the physics of disorder, of turbulent flows, of exotic noises, of chaotic dynamical systems, of the distribution of galaxies, of coastlines, of stock-prize charts, and of mathematical constructions [58]. The “monster” examples which previously existed as instances of known theorems (and therefore were of little interest in themselves from the novelty point of view) or put aside because they were not examples, were now brought into the limelight and put on to the dissecting table to be investigated. What is a Fractal? In his essay (1975) Mandelbrot said “I stopped short of giving a mathematical definition, because I felt this notion – like a good wine – demanded a bit of ageing before being ‘bottled’.” [71]. He knew of Hausdorff dimension and developed an intuitive understanding for it, but postponed a definition. A little later he adopted a working definition, and in his Fractal Geometry of Nature (1982) he gave his manifesto for fractals. What is Mandelbrot’s subsequent working definition of a fractal? It is simply a point set for which the Hausdorff dimension exceeds its topological dimension, DH > DT . Examples are Cantor middle third set. where DH D log 2/ log 3 D 0:63 : : : with DT D 0, and the von Koch Curve where DH D log 4/ log 3 D 1:26 : : : with DT D 1. Mandelbrot’s definition gives rise to a programme of research similar in nature to one carried out by Menger and Urysohn in the 1920s. Just as they asked “what is a curve?” and “what is dimension?” it could now be asked “what is a fractal exactly?” and “what is fractal dimension?” The whole game of topological dimension was to be played over for metric dimension. Mandelbrot’s definition of a fractal fired a first shot in answering one question but, just as there were exceptions to C. Jordan’s definitions of curve like Peano’s famous space filling curve of the 1890s, exceptions were found to Mandelbrot’s preliminary definition. It was a regret, for example, that the example of the “devil’s staircase” (characterized in terms of a continuous weight function defined on the Cantor middle third set) which looked like a fractal did not conform to the working definition of a fractal since in this case DH D DT D 1. There are, however, many examples of obvious fractals, in nature and in mathematics. The pathological curves of the late nineteenth century which suffered this fate of

not conforming to definitions of “normal curves” were brought back to life. The pathologies, such as those invented by Brouwer and Cantor had been put in a cupboard and locked away. An instance of this is Brouwer’s decomposition of the plane into three “countries” so that the boundary points touch each other (not just meeting at one point). Brouwer’s decomposition is not unlike the shape of a fractal. Cantor’s examples were equally pathological. Mandelbrot made his own discoveries, but when he opened the cupboard of “monster curves” he liked what he saw. Otto Rössler, with justification called Mandelbrot “Cantor’s time-displaced younger friend”. As Freeman Dyson wrote: “Now, as Mandelbrot points out, . . . , Nature has played a joke on the mathematicians. The nineteenth century mathematicians may have been lacking in imagination [in limiting themselves to Euclid and Newton], but nature has not. The same pathological structures [the gallery of monster curves] that the mathematicians invented to break loose from nineteenth century naturalism turn out to be inherent in familiar objects all around us [38].” The Mandelbrot Set Just how did Mandelbrot arrive at the famous Mandelbrot set? With the freedom of being funded by an IBM fellowship, he went further into the study of p c (z) D z2 C c and experimented. The basement of the Thomas J. Watson Research Center in the early 1980s held a brand new VAX main frame computer and a Tektronix cathode ray tube for viewing results. The quadratic map p c (z) D z2 C c is the simplest of all the nonlinear rational transformations. The key element is to consider z; c as complex numbers. The Mandelbrot set M in the plane is defined as M D fc 2 C : the Julia set J c is connectedg or, alternatively, in terms of the sequence fp kc (0)g M D fc 2 C : fp kc (0)g is boundedg : It was an exciting event when Mandelbrot first glimpsed the set, which was initially called an M-set. First observed on the Tektronix cathode ray tube, was the messy outline a fault in the primitive equipment? He and his colleagues tried again with a more powerful computer but found “the mess had failed to vanish” and even that the image showed signs of being more systematic. Mandebrot reported “we promptly took a much closer look. Many specks of dirt duly vanished after we zoomed in. But some

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specks failed to vanish; in fact, they proved to resolve into complex structures endowed with ‘sprouts’ very similar to those of the whole set M. Peter Moldave and I could not contain our excitement.” Yet concrete mathematical properties were hard to come by. One immediate result did follow quickly, when the M-set, now termed the Mandelbrot set was proved to be a connected set [29].

As an example, the fractal dimension of the original Lorenz attractor derived from the meteorology differential equations was found to be approximately 2.05 [34].

The Merger

Newer Concepts of Dimension

When chaos theory was being discussed in the 1970s, strange attractors were regarded as weird subsets of Euclidean space. To be sure their importance was recognized but their nature was only vaguely understood. Towards the end of the 1970s, prompted by the advent of fractals, attractors could be viewed in a new light. But Chaos which arose in physical situations did not give rise to ordinary self-similar fractals, like the Koch curve, but to more complicated sets. The outcome of Chaos is apparent randomness while the key properties of ordinary fractals is regularity.

Just as for metric free topological dimension itself, there are a myriad of different concepts of metrically based dimension. What was wanted were measures which could be used in practice. What “dimensions” would shine light on the onset of chaotic solutions to a set of differential equations ? One was the Lyapunov dimension drawing on the work of the Russian mathematician Alexander Lyapunov, and investigated by J. L. Kaplan and J. A. Yorke (1979). The Lyapunov dimension of an attractor A embedded in an Euclidean space of dimension n is defined by:

Measuring Strange Attractors Once strange attractors were brought to light, the next stage was to measure them. This perspective was put by practitioners in the 1980s: Nonlinear physics presents us with a perplexing variety of complicated fractal objects and strange sets. Notable examples include strange attractors for chaotic dynamical systems. . . . Naturally one wishes to characterize the objects and described the events occurring on them. For example, in dynamical systems theory one is often interested in a strange attractor . . . [43]. At the same time other researchers put a similar point of view, and an explanation for making the calculations. The way to characterize strange attractors was by the natural measure of a fractal – its fractal dimension – and this became the way to proceed: The determination of the fractal dimension d of strange attractors has become a standard diagnostic tool in the analysis of dynamical systems. The dimension roughly speaking measures the number of degrees of freedom that are relevant to the dynamics. Most work on dimension has concerned maps, such as the Hénon map, or systems given by a few coupled ordinary differential equations, such as the Lorenz and Rössler models. For any chaotic system described by differential equations, d must be

greater than 2, but d could be much larger for a system described by partial differential equations, such as the Navier–Stokes equations [12].

DL D k C

1 C 2 C    C  k ; j kC1 j

where 1 ; 2 ; : : : ;  k are Lyapunov exponents and k is P the maximum integer for which kiD1  i  0. In the case where k D 1 which occurs for two-dimensional chaotic maps, for instance, DL D 1 

1 2

and because in this case 2 < 0 the value of the Lyapunov dimension will be sandwiched between 1 and the topological dimension which is 2. In the case of the Hénon mapping of the plane, the Lyapunov dimension of the attractor is approximately 1.25. In a different direction, a generalized dimension Dq was introduced by A. Rényi in the 1950s [78]. Working in probability and information theory, he defined a spectrum of dimensions: P q log Pi 1 i D q D lim r!0 q  1 log r depending on the value of q. The special cases are worth noting:  D0 D DB the box dimension  D1 D “information dimension“ related to C. Shannon’s entropy [39,40]  D2 D DC the correlation dimension.

Fractals Meet Chaos

The correlation dimension is a very practical measure and can be applied to experimental data which has been presented visually, as well as other shapes, such as photographs where the calculation involves counting points. It has been used for applications in hydrodynamics, business, lasers, astronomy, signal analysis. It can be calculated directly for time series where the “phase space” points are derived from lags or time-delays. In two dimensions this would be a sequence of points (x(n); x(n C l)) where l is the chosen lag. A successful computation depends on the number of points chosen. For example, it is possible to estimate the fractal dimension of the Hénon attractor to within 6% of its supposed value with 500 data points [70]. There is a sequence of inequalities between the various dimensions of a set A from the embedding space of topological dimension D E D n to the topological dimension DT of A [90]: DT  : : :  D2  D1  DH  D0  n : Historically these questions of inequalities between the various types of dimension are of the same type as had been asked about topological dimension fifty years before. Just as it was shown that the “coincidence theorem” ind X D Ind X D dim X holds for well behaved spaces, such as compact metric spaces, so it is true that DL D DB D D1 for ordinary point sets such as single points and limit cycles. For the Lorenz attractor DL agrees with the value of DB and that of D2 [69]. But each of the equalities fails for some fractal sets; the study of fractal dimension of attractors is covered in [34,83]. Multifractals An ordinary fractal is one where the generalized Rényi dimension Dq is independent of q and returns a single value for the whole fractal. This occurs for the standard fractals such as the von Koch curve. A multifractal is a set in which Dq is dependent on q and a continuous spectrum of values results. While an ordinary self-similar fractal are useful for expository purposes (and are beautifully symmetric shapes) the attractors found in physics are not uniformly self-similar. The resulting object is called a multifractal because it is multi-dimensional. The values of Dq is called its spectrum of the multifractal. Practical examples usually require their strange attractors to be modeled by multifractals.

Future Directions The range of written material on Fractals and Chaos has exploded since the 1970s. A vast number of expository articles have been lodged in such journals as Nature and New Scientist and technical articles in Physica D and Nonlinearity [9]. Complexity Theory and Chaos Theory are relatively new sciences that can revolutionize the way we see our world. Stephen Hawking has said, “Complexity will be the science of the 21st century.” There is even a relationship between fractals with the yet unproved Riemann hypothesis [51]. Many problems remain. Here it is sufficient to mention one representative of the genre. Fluid dynamicists are broadly in agreement that fluids are accurately modeled by the nonlinear Navier–Stokes equations. These are based on Newtonian principles and are deterministic, but theoretically the solution of these equations is largely unknown territory. A proof of the global regularity of the solutions represents a formidable challenge. The current view is that there is a dichotomy between laminar flow (like flow in the upper atmosphere) being smooth while turbulent flow (like flow near the earth’s surface) is violent. Yet even this “turbulent” flow could be regular but of a complicated kind. A substantial advance in the theory will be rewarded with one of the Clay Institute’s million dollar prizes. One expert is not optimistic the prize will be claimed in the near future [52]. The implication of Chaos dependent as it is on the sensitivity of initial conditions, suggests that forecasting some physical processes is theoretically impossible. Long range weather forecasting falls into this mould since it is predicated on knowing the weather conditions exactly at some point in time. There will inevitably be inaccuracies so exactness appears to be an impossibility. No doubt “Chaos and (multi)Fractals” is here to stay. Rössler wrote of Chaos as the key to understanding Nature: “hairs and noodles and spaghettis and dough and taffy form an irresistible, disentanglable mess. The world of causality is thereby caricatured and, paradoxically, faithfully represented [2]”. Meanwhile, the challenge for scientists and mathematicians remains [16].

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26. Devaney RL (2003) An Introduction to Chaotic Dynamical Systems, 2nd edn. Westview Press, Boulder 27. Diacu F, Holmes P (1996) Celestial Encounters: the Origins of Chaos and Stability. Princeton University Press, Princeton 28. Dieudonné J (1960) Foundations of Modern Analysis. Academic Press, New York 29. Douady A, Hubbard J (1982) Iteration des polynomes quadratiques complexes. Comptes Rendus Acad Sci Paris Sér 1 Math 294:123–126 30. Dyson F (1997) ‘Nature’s Numbers’ by Ian Stewart. Math Intell 19(2):65–67 31. Edgar G (1993) Classics on Fractals. Addison-Wesley, Reading 32. Falconer K (1990) Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester 33. Falconer K (1997) Techniques in Fractal Geometry. Wiley, Chichester 34. Farmer JD, Ott E, Yorke JA (1983) The Dimension of Chaotic Attractors. Physica D 7:153–180 35. Fatou P (1919/20) Sur les equations fonctionelles. Bull Soc Math France 47:161–271; 48:33–94, 208–314 36. Feigenbaum MJ (1978) Quantitative Universality for a Class of Nonlinear Transformations. J Stat Phys 19:25–52 37. Feigenbaum MJ (1979) The Universal Metric Properties of Nonlinear Transformations. J Stat Phys 21:669–706 38. Gleick J (1988) Chaos. Sphere Books, London 39. Grassberger P (1983) Generalised dimensions of strange attractors. Phys Lett A 97:227–230 40. Grassberger P, Procaccia I (1983) Measuring the strangeness of strange attractors. Physica D 9:189–208 41. Hénon M (1976) A Two dimensional Mapping with a Strange Attractor. Commun Math Phys 50:69–77 42. Haeseler F, Peitgen H-O, Saupe D (1984) Cayley’s Problem and Julia Sets. Math Intell 6:11–20 43. Halsey TC, Jensen MH, Kadanoff LP, Procaccia Shraiman I (1986) Fractal measures and their singularities: the characterization of strange sets. Phys Rev A 33:1141–1151 44. Harman PM (1998) The Natural Philosophy of James Clerk Maxwell. Cambridge University Press, Cambridge 45. Hofstadter DR (1985) Metamathematical Themas: questing for the essence of mind and pattern. Penguin, London 46. Holmes P (2005) Ninety plus years of nonlinear dynamics: More is different and less is more. Int J Bifurc Chaos Appl Sci Eng 15(9):2703–2716 47. Hurewicz W, Wallman H (1941) Dimension Theory. Princeton University Press, Princeton 48. Julia G (1918) Sur l’iteration des functions rationnelles. J Math Pures Appl 8:47–245 49. Kazim Z (2002) The Hausdorff and box dimension of fractals with disjoint projections in two dimensions. Glasgow Math J 44:117–123 50. Lapidus ML, van Frankenhuijsen M (2004) Fractal geometry and Applications: A Jubilee of Benoît Mandelbrot, Parts, 1, 2. American Mathematical Society, Providence 51. Lapidus ML, van Frankenhuysen (2000) Fractal geometry and Number Theory: Complex dimensions of fractal strings and zeros of zeta functions. Birkhäuser, Boston 52. Li YC (2007) On the True Nature of Turbulence. Math Intell 29(1):45–48 53. Lorenz EN (1963) Deterministic Nonperiodic Flow. J Atmospheric Sci 20:130–141

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54. Lorenz EN (1993) The Essence of Chaos. University of Washington Press, Seattle 55. Mandelbrot BB (1975) Les objets fractals, forme, hazard et dimension. Flammarion, Paris 56. Mandelbrot BB (1980) Fractal aspects of the iteration of z ! (1  z) for complex ; z. Ann New York Acad Sci 357:249–259 57. Mandelbrot BB (1982) The Fractal Geometry of Nature. Freeman, San Francisco 58. Mandelbrot BB (2002) A maverick’s apprenticeship. Imperial College Press, London 59. Mandelbrot BB, Hudson RL (2004) The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward. Basic Books, New York 60. Mattila P (1995) Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge 61. Mauldin RD, Williams SC (1986) On the Hausdorff dimension of some graphs. Trans Am Math Soc 298:793–803 62. May RM (1976) Simple Mathematical models with very complicated dynamics. Nature 261:459–467 63. May RM (1987) Chaos and the dynamics of biological populations. Proc Royal Soc Ser A 413(1844):27–44 64. May RM (2001) Stability and Complexity in Model Ecosystems, 2nd edn, with new introduction. Princeton University Press, Princeton 65. McMurran S, Tattersall J (1999) Mary Cartwright 1900–1998. Notices Am Math Soc 46(2):214–220 66. Menger K (1943) What is dimension? Am Math Mon 50:2–7 67. Metropolis, Stein ML, Stein P (1973) On finite limit sets for transformations on the unit interval. J Comb Theory 15:25–44 68. Morse M (1946) George David Birkhoff and his mathematical work. Bull Am Math Soc 52(5, Part 1):357–391 69. Nese JM, Dutton JA, Wells R (1987) Calculated Attractor Dimensions for low-Order Spectral Models. J Atmospheric Sci 44(15):1950–1972 70. Ott E, Sauer T, Yorke JA (1994) Coping with Chaos. Wiley Interscience, New York 71. Peitgen H-O, Richter PH (1986) The Beauty of Fractals. Springer, Berlin 72. Peitgen H-O, Jürgens H, Saupe D (1992) Chaos and fractals. Springer, New York 73. Pesin YB (1997) Dimension Theory in Dynamical Systems: contemporary views and applications. University of Chicago Press, Chicago 74. Poincaré H (1903) L’Espace et ses trois dimensions. Rev Métaphys Morale 11:281–301 75. Poincaré H, Halsted GB (tr) (1946) Science and Method. In: Cattell JM (ed) Foundations of Science. The Science Press, Lancaster 76. Przytycki F, Urbañski M (1989) On Hausdorff dimension of some fractal sets. Studia Mathematica 93:155–186 77. Rössler OE (1976) An Equation for Continuous Chaos. Phys Lett A 57:397–398 78. Rényi A (1970) Probability Theory. North Holland, Amsterdam 79. Richardson LF (1993) Collected Papers, 2 vols. Cambridge University Press, Cambridge 80. Ruelle D (1980) Strange Attractors. Math Intell 2:126–137 81. Ruelle D (2006) What is a Strange Attractor? Notices Am Math Soc 53(7):764–765 82. Ruelle D, Takens F (1971) On the nature of turbulence. Commun Math Phys 20:167–192; 23:343–344

83. Russell DA, Hanson JD, Ott E (1980) Dimension of strange attractors. Phys Rev Lett 45:1175–1178 84. Shaw R (1984) The Dripping Faucet as a Model Chaotic System. Aerial Press, Santa Cruz 85. Siegel CL (1942) Iteration of Analytic Functions. Ann Math 43:607–612 86. Smale S (1967) Differentiable Dynamical Systems. Bull Am Math Soc 73:747–817 87. Smale S (1980) The Mathematics of Time: essays on dynamical systems, economic processes, and related topics. Springer, New York 88. Smale S (1998) Chaos: Finding a horseshoe on the Beaches of Rio. Math Intell 20:39–44 89. Sparrow C (1982) The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer, New York 90. Sprott JC (2003) Chaos and Time-Series Analysis. Oxford University Press, Oxford 91. Takens F (1980) Detecting Strange Attractors in Turbulence. In: Rand DA, Young L-S (eds) Dynamical Systems and Turbulence. Springer Lecture Notes in Mathematics, vol 898. Springer, New York, pp 366–381 92. Theiler J (1990) Estimating fractal dimensions. J Opt Soc Am A 7(6):1055–1073 93. Tricot C (1982) Two definitions of fractional dimension. Math Proc Cambridge Philos Soc 91:57–74 94. Tucker W (1999) The Lorenz Attractor exists. Comptes Rendus Acad Sci Paris Sér 1, Mathematique 328:1197–1202 95. Viana M (2000) What’s New on Lorenz Strange Attractors. Math Intell 22(3):6–19 96. Winfree AT (1983) Sudden Cardiac death- a problem for topology. Sci Am 248:118–131 97. Zhang S-Y (compiler) (1991) Bibliography on Chaos. World Scientific, Singapore

Books and Reviews Abraham RH, Shaw CD (1992) Dynamics, The Geometry of Behavior. Addison-Wesley, Redwood City Barnsley MF, Devaney R, Mandelbrot BB, Peitgen H-O, Saupe D, Voss R (1988) The Science of Fractal Images. Springer, Berlin Barnsley MF, Rising H (1993) Fractals Everywhere. Academic Press, Boston Çambel AB (1993) Applied Complexity Theory: A Paradigm for Complexity. Academic Press, San Diego Crilly AJ, Earnshaw RA, Jones H (eds) (1991) Fractals and Chaos. Springer, Berlin Cvitanovic P (1989) Universality in Chaos, 2nd edn. Adam Hilger, Bristol Elliott EW, Kiel LD (eds) (1997) Chaos Theory in the Social Sciences: foundations and applications. University of Michigan Press, Ann Arbor Gilmore R, Lefranc M (2002) The Topology of Chaos: Alice in Stretch and Squeezeland. Wiley Interscience, New York Glass L, MacKey MM (1988) From Clocks to Chaos. Princeton University Press, Princeton Holden A (ed) (1986) Chaos. Manchester University Press, Manchester Kellert SH (1993) In the Wake of Chaos. University of Chicago Press, Chicago Lauwerier H (1991) Fractals: Endlessly Repeated Geometrical Figures. Princeton University Press, Princeton

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Mullin T (ed) (1994) The Nature of Chaos. Oxford University Press, Oxford Ott E (2002) Chaos in Dynamical Systems. Cambridge University Press, Cambridge Parker B (1996) Chaos in the Cosmos: The Stunning Complexity of the Universe. Plenum, New York, London Peitgen H-O, Jürgens, Saupe D, Zahlten C (1990) Fractals: An animated discussion with Edward Lorenz and Benoit Mandelbrot. A VHS film in color (63 mins). Freeman, New York

Prigogine I, Stengers I (1985) Order out of Chaos: man’s new dialogue with Nature. Fontana, London Ruelle D (1993) Chance and Chaos. Penguin, London Schroeder M (1991) Fractals, Chaos, Power Laws. Freeman, New York Smith P (1998) Explaining Chaos. Cambridge University Press, Cambridge Thompson JMT, Stewart HB (1988) Nonlinear Dynamics and Chaos. Wiley, New York

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Fractals and Percolation

Fractals and Percolation YAKOV M. STRELNIKER1 , SHLOMO HAVLIN1 , ARMIN BUNDE2 1 Department of Physics, Bar–Ilan University, Ramat–Gan, Israel 2 Institut für Theoretische Physik III, Justus-Liebig-Universität, Giessen, Germany

Definition of the Subject Percolation theory is useful for characterizing many disordered systems. Percolation is a pure random process of choosing sites to be randomly occupied or empty with certain probabilities. However, the topology obtained in such processes has a rich structure related to fractals. The structural properties of percolation clusters have become clearer thanks to the development of fractal geometry since the 1980s.

Article Outline Glossary Definition of the Subject Introduction Percolation Percolation Clusters as Fractals Anomalous Transport on Percolation Clusters: Diffusion and Conductivity Networks Summary and Future Directions Bibliography Glossary Percolation In the traditional meaning, percolation concerns the movement and filtering of fluids through porous materials. In this chapter, percolation is the subject of physical and mathematical models of porous media that describe the formation of a long-range connectivity in random systems and phase transitions. The most common percolation model is a lattice, where each site is occupied randomly with a probability p or empty with probability 1  p. At low p values, there is no connectivity between the edges of the lattice. Above some concentration pc , the percolation threshold, connectivity appears between the edges. Percolation represents a geometric critical phenomena where p is the analogue of temperature in thermal phase transitions. Fractal A fractal is a structure which can be subdivided into parts, where the shape of each part is similar to that of the original structure. This property of fractals is called self-similarity, and it was first recognized by G.C. Lichtenberg more than 200 years ago. Random fractals represent models for a large variety of structures in nature, among them porous media, colloids, aggregates, flashes, etc. The concepts of self-similarity and fractal dimensions are used to characterize percolation clusters. Self-similarity is strongly related to renormalization properties used in critical phenomena, in general, and in percolation phase transition properties.

Introduction Percolation represents the simplest model of a phase transition [1,8,13,14,26,27,30,48,49,61,64,65,68]. Assume a regular lattice (grid) where each site (or bond) is occupied with probability p or empty with probability 1  p. At a critical threshold, pc , a long-range connectivity first appears: pc is called the percolation threshold (see Fig. 1). Occupied and empty sites (or bonds) may stand for very different physical properties. For example, occupied sites may represent electrical conductors, empty sites may represent insulators, and electrical current may flow only through nearest-neighbor conducting sites. Below pc , the grid represents an isolator since there is no conducting path between two adjacent bars of the lattice, while above pc , conducting paths start to occur and the grid becomes a conductor. One can also consider percolation as a model for liquid filtration (i. e., invasion percolation (see Fig. 2), which is the source of this terminology) through porous media. A possible application of bond percolation in chemistry is the polymerization process [25,31,44], where small branching molecules can form large molecules by activating more and more bonds between them. If the activation probability p is above the critical concentration, a network of chemical bonds spanning the whole system can be formed, while below pc only macromolecules of finite size can be generated. This process is called a sol-gel transition. An example of this gelation process is the boiling of an egg, which at room temperature is liquid but, upon heating, becomes a solid-like gel. An example from biology concerns the spreading of an epidemic [35]. In its simplest form, an epidemic starts with one sick individual which can infect its nearest neighbors with probability p in one time step. After one time step, it dies, and the infected neighbors in turn can infect their (so far) uninfected neighbors, and the process is continued. Here the critical concentration separates a phase at low p where the epidemic always dies out after a finite number of time steps, from a phase where the epidemic can continue forever. The same process can be used as a model for

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Fractals and Percolation, Figure 1 Square lattice of size 20  20. Sites have been randomly occupied with probability p (p D 0:20, 0.59, 0.80). Sites belonging to finite clusters are marked by full circles, while sites on the infinite cluster are marked by open circles

Fractals and Percolation, Figure 2 Invasion percolation through porous media

forest fires [52,60,64,71], with the infection probability replaced by the probability that a burning tree can ignite its nearest-neighbor trees in the next time step. In addition to these simple examples, percolation concepts have been found useful for describing a large number of disordered systems in physics and chemistry. The first study introducing the concept of percolation is attributable to Flory and Stockmayer about 65 years ago, when studying the gelation process [32]. The name percolation was proposed by Broadbent and Hammersley in 1957 when they were studying the spreading of fluids in random media [15]. They also introduced relevant geometrical and probabilistic concepts. The developments of phase transition theory in the following years, in particular the series expansion method by Domb [27] and renormalization group theory by Wilson, Fisher and Kadanoff [51,65], very much stimulated research activities into the geometric percolation transition. At the percolation threshold, the conducting (as well as insulating) clusters are self-similar (see Fig. 3) and, therefore, can be described by fractal geometry [53], where various fractal dimensions are introduced to quantify the clusters and their physical properties.

Fractals and Percolation, Figure 3 Self-similarity of the random percolation cluster at the critical concentration; courtesy of M. Meyer

Percolation As above (see Sect. “Introduction”), consider a square lattice, where each site is occupied randomly with probability p (see Fig. 1). For simplicity, let us assume that the occupied sites are electrical conductors and the empty sites represent insulators. At low concentration p, the occupied sites either are isolated or form small clusters (Fig. 1a). Two occupied sites belong to the same cluster if they are connected by a path of nearest-neighbor occupied sites and a current can flow between them. When p is increased, the average size of the clusters increases. At a critical concentration pc (also called the percolation threshold), a large cluster appears which connects opposite edges of the lattice (Fig. 1). This cluster is called the infinite cluster, since its size diverges when the size of the lattice is increased to infinity. When p is increased further, the density of the infinite cluster increases, since more and more sites become

Fractals and Percolation

part of the infinite cluster, and the average size of the finite clusters decreases (Fig. 1c). The percolation threshold separates two different phases and, therefore, the percolation transition is a geometrical phase transition, which is characterized by the geometric features of large clusters in the neighborhood of pc . At low values of p, only small clusters of occupied sites exist. When the concentration p is increased, the average size of the clusters increases. At the critical concentration pc , a large cluster appears which connects opposite edges of the lattice. Accordingly, the average size of the finite clusters which do not belong to the infinite cluster decreases. At p D 1, trivially, all sites belong to the infinite cluster. Similar to site percolation, it is possible to consider bond percolation when the bonds between sites are randomly occupied. An example of bond percolation in physics is a random resistor network, where the metallic wires in a regular network are cut at random. If sites are occupied with probability p and bonds are occupied with probability q, we speak of site-bond percolation. Two occupied sites belong to the same cluster if they are connected by a path of nearest-neighbor occupied sites with occupied bonds in between. The definitions of site and bond percolation on a square lattice can easily be generalized to any lattice in d-dimensions. In general, in a given lattice, a bond has more nearest neighbors than a site. Thus, large clusters of bonds can be formed more effectively than large clusters of sites, and therefore, on a given lattice, the percolation threshold for bonds is smaller than the percolation threshold for sites (see Table 1). A natural example of percolation, is continuum percolation, where the positions of the two components of Fractals and Percolation, Table 1 Percolation thresholds for the Cayley tree and several two- and three-dimensional lattices (see Refs. [8,14,41,64,75] and references therein)

Lattice Triangular Square Honeycomb Face Centered Cubic Body Centered Cubic Simple Cubic (1st nn) Simple Cubic (2nd nn) Simple Cubic (3rd nn) Cayley Tree

Percolation of Sites Bonds 1/2 2 sin(/18) 0.5927460 1/2 0.6962 1  2 sin(/18) 0.198 0.119 0.245 0.1803 0.31161 0.248814 0.137 – 0.097 – 1/(z  1) 1/(z  1)

Fractals and Percolation, Figure 4 Continuum percolation: Swiss cheese model

a random mixture are not restricted to the discrete sites of a regular lattice [9,73]. As a simple example, consider a sheet of conductive material, with circular holes punched randomly in it (Swiss cheese model, see Fig. 4). The relevant quantity now is the fraction p of remaining conductive material. Hopping Percolation Above, we have discussed traditional percolation with only two values of local conductivities, 0 and 1 (insulator– conductor) or 1 and 1 (superconductor–normal conductor). However, quantum systems should be treated by hopping conductivity, which can be described by an exponential function representing the local conductivities (between ith and jth sites):  i j exp(x i j ). Here  can be interpreted as the dimensionless mean hopping distance or as the degree of disorder (the smaller the density of the deposited grains, the larger  becomes), and xij is a random number taken from a uniform distribution in the range (0,1) [70]. In contrast to the traditional bond (or site) percolation model, in which the system is either a metal or an insulator, in the hopping percolation model the system always conducts some current. However, there are two regimes of such percolation [70]. A regime with many conducting paths which is not sensitive to the removal of a single bond (weak disorder L/  > 1, where L is size of the system and  is percolation critical exponent) and a regime with a single or only a few dominating conducting paths which is very sensitive to the removal of a specific single bond with the highest current (strong disorder L/   1). In the strong disorder regime, the trajectories along which

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P1 can be identified as the order parameter similar to magnetization, m(T) (Tc  T)ˇ , in magnetic materials. With decreasing temperature, T, more elementary magnetic moments (spins) become aligned in the same direction, and the system becomes more ordered. The linear size of the finite clusters, below and above pc , is characterized by correlation length . Correlation length is defined as the mean distance between two sites on the same finite cluster. When p approaches pc ,  increases as  jp  pc j ;

(2)

with the same exponent  below and above the threshold. The mean number of sites (mass) of a finite cluster also diverges, S jp  pc j ;

Fractals and Percolation, Figure 5 A color density plot of the current distribution in a bond-percolating lattice for which voltage is applied in the vertical direction for strong disorder with D 10. The current between the sites is shown by the different colors (orange corresponds to the highest value, green to the lowest). a The location of the resistor, on which the value of the local current is maximal, is shown by a circle. b The current distribution after removing the above resistor. This removal results in a significant change of the current trajectories

(3)

again with the same exponent above and below pc . Analogous to S in magnetic systems is the susceptibility  (see Fig. 6 and Table 2). The exponents ˇ, , and describe the critical behavior of typical quantities associated with the percolation transition, and are called critical exponents. The exponents are universal and do not depend on the structural details of the lattice (e. g., square or triangular) nor on the type of percolation (site, bond, or continuum), but depend only on the dimension d of the lattice.

the highest current flows (analogous to the spanning cluster at criticality in the traditional percolation network, see Fig. 5) can be distinguished and a single bond can determine the transport properties of the entire macroscopic system. Percolation as a Critical Phenomenon In percolation, the concentration p of occupied sites plays the same role as temperature in thermal phase transitions. The percolation transition is a geometrical phase transition where the critical concentration pc separates a phase of finite clusters (p < pc ) from a phase where an infinite cluster is present (p > pc ). An important quantity is the probability P1 that a site (or a bond) belongs to the infinite cluster. For p < pc , only finite clusters exist, and P1 D 0. For p > pc , P1 increases with p by a power law P1 (p  pc )ˇ :

(1)

Fractals and Percolation, Figure 6 P1 and S compared with magnetization M and susceptibility  Fractals and Percolation, Table 2 Exact and best estimate values for the critical exponents for percolation (see Refs. [8,14,41,64] and references therein) Percolation Order parameter P1 : ˇ Correlation length  :  Mean cluster size S: 

dD2 5/36 4/3 43/18

dD3 0:417 ˙ 0:003 0:875 ˙ 0:008 1:795 ˙ 0:005

d6 1 1/2 1

Fractals and Percolation

This universality property is a general feature of phase transitions, where the order parameter vanishes continuously at the critical point (second order phase transition). In Table 2, the values of the critical exponents ˇ, , and for percolation in two, three, and six dimensions. The exponents considered here describe the geometrical properties of the percolation transition. The physical properties associated with this transition also show power-law behavior near pc and are characterized by critical exponents. Examples include the conductivity in a random resistor or random superconducting network and the spreading velocity of an epidemic disease near the critical infection probability. It is believed that the “dynamical” exponents cannot be generally related to the geometric exponents discussed above. Note that all quantities described above are defined in the thermodynamic limit of large systems. In a finite system, for example, P1 , is not strictly zero below pc .

Fractals and Percolation, Figure 7 Examples of regular systems with dimensions d D 1, d D 2, and dD3

fractals. Thus, to include fractal structures, (4) we can generalize M(bL) D b d f M(L) ; and M(L) D ALd f ;

Percolation Clusters as Fractals As first noticed by Stanley [66], the structure of percolation clusters (when the length scale is smaller than ) can be well described by the fractal concept [53]. Fractal geometry is a mathematical tool for dealing with complex structures that have no characteristic length scale. Scaleinvariant systems are usually characterized by noninteger (“fractal”) dimensions. This terminology is associated with B. Mandelbrot [53] (though some notion of noninteger dimensions and several basic properties of fractal objects were studied earlier by G. Cantor, G. Peano, D. Hilbert, H. von Koch, W. Sierpinski, G. Julia, F. Hausdorff, C. F. Gauss, and A. Dürer).

Sierpinski Gasket This fractal is generated by dividing a full triangle into four smaller triangles and removing the central triangle (see Fig. 8). In subsequent iterations, this procedure is repeated by dividing each of the remaining triangles into four smaller triangles and removing the central triangles. To obtain the fractal dimension, we consider

In regular systems (with uniform density) such as long wires, large thin plates, or large filled cubes, the dimension d characterizes how the mass M(L) changes with the linear size L of the system. If we consider a smaller part of a system of linear size bL (b < 1), then M(bL) is decreased by a factor of bd , i. e., (4)

The solution of the functional Eq. (4) is simply M(L) D ALd . For a long wire, mass changes linearly with b, i. e., d D 1. For the thin plates, we obtain d D 2, and for cubes d D 3; see Fig. 7. Mandelbrot coined the name “fractal dimension”, and those objects described by a fractal dimension are called

(6)

where df is the fractal dimension and can be a noninteger. Below, we present two examples of dealing with df : (i) the deterministic Sierpinski Gasket and (ii) random percolation clusters and criticality.

Fractal Dimension df

M(bL) D b d M(L) :

(5)

Fractals and Percolation, Figure 8 2D Sierpinski gasket. Generation and self-similarity

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the mass of the gasket within a linear size L and compare it with the mass within 12 L. Since M( 21 L) D 13 M(L), we have d f D log 3/ log 2 Š 1:585. Percolation Fractal We assume that at pc ( D 1) the clusters are fractals. Thus for p > pc , we expect length scales smaller than  to have critical properties and therefore a fractal structure. For length scales larger than , one expects a homogeneous system which is composed of many unit cells of size : ( rd f ; r   ; M(r) d (7) r ; r:

Shortest Path Dimensions, dmin and d` The fractal dimension, however, is not sufficient to fully characterize a percolation cluster, since two clusters with very different topologies may have the same fractal dimension df . As an additional characterization of a fractal, one can consider, e. g., the shortest path between two arbitrary sites A and B on the cluster (see Figs. 10, 11) [3,16,35, 42,55,58]. The structure formed by the sites of this path is also self-similar and is described by a fractal dimension dmin [46,67]. Accordingly, the length ` of the path, which is often called the “chemical distance”, scales with the “Eu-

For a demonstration of this feature see Fig. 9. One can relate the fractal dimension df of percolation clusters to the exponents ˇ and . The probability that an arbitrary site within a circle of radius r smaller than  belongs to the infinite cluster, is the ratio between the number of sites on the infinite cluster and the total number of sites, P1 r d f /r d ;

r<:

(8)

This equation is certainly correct for r D a, where a is an arbitrary constant smaller than 1. Substituting r D a in (8) yields P1  d f / d . Both sides are powers of p  pc . Substituting Eqs. (1) and (2) into the latter one obtains [8,28,41,49,64], d f D d  ˇ/ :

Fractals and Percolation, Figure 10 Shortest path between two sites A and B on a percolation cluster

(9)

Thus, the fractal dimension of the infinite cluster at pc is not a new independent exponent but depends on ˇ and . Since ˇ and  are universal exponents, df is also universal.

Fractals and Percolation, Figure 9 Lattice composed of Sierpinski gasket cells of size 

Fractals and Percolation, Figure 11 Percolation system at critical concentration in a 510510 square lattice. The finite clusters are in yellow. Substructures of the infinite percolation cluster are shown in different colors: the shortest path between two points at opposite sites of the system is shown in white, the single connected sites (“red” sites) in red, the loops in blue and the dangling ends in green; courtesy of S. Schwarzer

Fractals and Percolation

clidean distance” r between A and B as ` r d min :

(10)

The inverse relation r `1/d min  `˜

(11)

tells how r scales with `. Closely related to dmin and df is the “chemical” dimension d` , which describes how the cluster mass M within the chemical distance ` from a given site scales with `, M(`) `d ` :

(12)

While the fractal dimension df characterizes how the mass of a cluster scales with the Euclidean distance r, the graph dimension d` characterizes how the mass scales with the chemical distance `. Combining Eqs. (7), (10) and (12) we obtain the relation between dmin , d` and df d` D d f /dmin :

(13)

To measure df , an arbitrary site is chosen on the cluster and one determines the number M(r) of sites within a distance r from this site. To measure d` , an arbitrary site is chosen on the cluster at criticality and one determines the number M(`) of sites which are connected to this site by a shortest path with length less than or equal to `. Finally, to measure dmin , two arbitrary sites are chosen on the cluster and one determines the length `(r) of the shortest path connecting them. As for M(r), averages must be performed for M(`) and r(`) over many realizations. In regular Euclidean lattices, both d` and df coincide with the Euclidean space dimension d and dmin D 1. The chemical dimension d` (or dmin D 1/˜ ) is an important tool for distinguishing between different fractal structures which may have a similar fractal dimension. In d D 3, for example, DLA (diffusion limited aggregation) clusters and percolation clusters have approximately the same fractal dimension d f Š 2:5, but have different ˜ : ˜ D 1 for DLA [54] but ˜ Š 0:73 for percolation [36,46,63,69]. While df has been related to the (known) critical exponents, (9), no such relation has been found for dmin or d` . The values of d` or dmin are known only from approximate methods, mainly numerical simulations (see also Refs. [17,37,57,76]). Fractal Substructures The fractal dimensions df and d` are not the only exponents characterizing a percolation cluster at pc . A percola-

tion cluster is composed of several fractal sub-structures, which are described by other exponents. Imagine applying a voltage difference between two sites at opposite edges of a metallic percolation cluster: The backbone of the cluster consists of those sites (or bonds) which carry the electric current. The dangling ends are those parts of the cluster which carry no current and are connected to the backbone by a single site only. The red bonds (or singly connected bonds) [22,66] are those bonds that carry the total current; when they are cut the current flow stops. In analogy to red bonds we can define anti-red bonds [34]. If an antired bond is added to a nonconducting percolation system below pc , the current will be able to flow in the system. The blobs, finally, are those parts of the backbone that remain after the red bonds have been removed. Further fractal substructures of the cluster are the external perimeter (which is also called the hull), the skeleton and the elastic backbone. The hull consists of those sites of the cluster which are adjacent to empty sites and are connected with infinity via empty sites. In contrast, the total perimeter also includes the holes in the cluster. The external perimeter is an important model for random fractal interfaces. The skeleton is defined as the union of all shortest paths from a given site to all sites at a chemical distance ` [43]. The elastic backbone is the union of all shortest paths between two sites [47]. The fractal dimension dB of the backbone is smaller than the fractal dimension df of the cluster (see Table 3). This reflects the fact that most of the mass of the cluster is concentrated in the dangling ends, which is seen clearly in Fig. 12. The value of the fractal dimension of the backbone is known only from numerical simulations [45,59]. Note, also, that the graph dimension d`B of the backbone is smaller than that of percolation. In contrast, ˜ is the same for both backbone and percolation cluster, indicating the more universal nature of ˜ . This can be understood by recalling that every two sites on a percolation cluster are located on the corresponding backbone. The fractal dimensions of the red bonds dred and the hull dh are known from exact analytical arguments. It has been proven by Coniglio [22,23] that the mean number of red bonds varies with p as nred (p  pc )1  1/ r1/ ;

(14)

and the fractal dimension of the red bonds is therefore dred D 1/. The fractal dimension of the skeleton is very close to dmin D 1/˜ , supporting the assumption that percolation clusters at criticality are finitely ramified [43]. The hull of the cluster in d D 2 has the fractal dimension d h D 7/4, which was first found numerically by

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Anomalous Transport on Percolation Clusters: Diffusion and Conductivity Due to self-similarity, transport quantities are significantly modified on fractal substrates. This can be seen in two representative examples: (1) total resistance or the conductivity, (2) mean square displacement and the probability density of random walks. Consider a metallic network of size Ld . At opposite faces of the network are metallic bars with a voltage difference between them. If we vary the linear size L of the system, the total resistance R varies as R  1 L/Ld1 ; Fractals and Percolation, Figure 12 Percolation cluster at the critical concentration in a simple cubic lattice. The backbone between two (green) cluster sites is shown in red, the gray sites represent the dangling ends; courtesy of M. Porto

Sapoval, Rosso, and Gouyet [63] and proven rigorously by Saleur and Duplantier [62]. If the hull is defined slightly differently and next-nearest neighbors of the perimeter are regarded as connected, many “fjords” are removed from the hull. According to Grossmann and Aharony [38], the fractal dimension of this modified hull is close to 4/3, the fractal dimension of self-avoiding random walks in d D 2. In three dimensions, in contrast, the mass of the hull seems to be proportional to the mass of the cluster, and both have the same fractal dimension. In Table 3 we summarize the values of the fractal dimension df and the graph dimension d` of the percolation cluster and its fractal substructures.

Fractals and Percolation, Table 3 Fractal dimensions of the substructures composing percolation clusters (see Ref. [8,14,41,59,64] and references therein). For fractal dimensions in d D 4 and d D 5 see Ref. [57] Fractal Space dimension dimensions d D 2 dD3 df 91/48 2:524 ˙ 0:008 d` 1:678 ˙ 0:005 1:84 ˙ 0:02 dmin 1:13 ˙ 0:004 1:374 ˙ 0:004 dred 3/4 1:143 ˙ 0:01 dh 7/4 2:548 ˙ 0:014 dB 1:64 ˙ 0:02 1:87 ˙ 0:04 d`B 1:43 ˙ 0:02 1:34 ˙ 0:03

d6 4 2 2 2 4 2 1

(15)

where  L0 D const is the conductivity (inverse to the resistivity,  D 1 ) of the metal. Since  does not depend on L, (15) states that the total resistance of the network depends on its linear size L via the power law R L2d  ˜ ˜ here ˜ D 2  L , which defines the resistance exponent , d. The idea that transport properties of percolation systems can be efficiently studied by means of diffusion was suggested by de Gennes [24] (see also Kopelman [50]). The diffusion process can be modeled by random walkers, which can jump randomly between nearest-neighbor occupied sites in the lattice. For such a random walker moving in a disordered environment, including bottlenecks, loops, and dead ends, de Gennes coined the term ant in the labyrinth. By calculating the mean square displacement of the walker, one obtains the diffusion constant, which according to Einstein is proportional to dc conductivity. Not only are the conductivity and diffusion exponents above pc related; also related are the exponents characterizing the size dependence of the dc conductivity and the time dependence of the mean square displacement of the random walker. Since it is numerically more efficient to calculate the relevant transport quantities by simulating random walks than to determine the conductivity directly from Kirchhoff’s equations, the study of random walks has improved our knowledge not only of diffusion but also of the transport process in percolation in general [5,6,10,12, 18,39,40,56,74]. Due to the presence of large holes, bottlenecks, and dangling ends in the fractal, the motion of a random walker is slowed down. Fick’s law for the mean square displacement (hr2 (t)i D a 2 t) is no longer valid. Instead, the mean square displacement is described by a more general

Fractals and Percolation

power law, hr2 (t)i t 2/d w ;

(16)

where the new exponent dw (“diffusion exponent” or “fractal dimension of the random walk”) is always greater than 2. Both the resistance exponent ˜ and the exponent dw can be related by the Einstein equation  D (e2 n/kB T)D ;

(17)

which relates the dc conductivity  of the system to the diffusion constant D D lim t!1 hr2 (t)i/2dt of the random walk. In Eq. (17), e and n denote charge and density of the mobile particles, respectively. Simple scaling arguments can now be used to relate ˜ Since n is proportional to the density of dw to ˜ and . the substrate, n Ld f d , the right-hand side of Eq. (17) is proportional to Ld f d t 2/d w 1 . The left-hand side of Eq. (17) is proportional to L˜ . Since the time a random walker takes to travel a distance L scales as Ld w , we find L˜ Ld f dC2d w , from which the Einstein relation [4] ˜ D d f C ˜ dw D d f  d C 2 C 

(18)

follows. For example, for the Sierpinski gasket d f D log 3/ log 2, and ˜ D log(5/3)/ log(2/4), therefore dw D log 5/ log 2. In general, determining dw for random fractals is not easy. An exception is topologically linear fractal structures (d` D 1), which can be considered as nonintersecting paths. Along a path (in `-space), diffusion is normal and h`2 (t)i D t. Since ` r d f , the mean square displacement in r-space scales as hr2 i t 1/d f , leading to dw D 2d f in this case. In percolation, dw cannot be calculated exactly, but upper and lower bounds can be derived which are very close to each other in d  3 dimensions. A good estimate is dw Š 3d f /2 (Alexander–Orbach conjecture [4]). The long-term behavior of the mean square displacement of a random walker on an infinite percolation cluster is characterized by the diffusion constant D. It is easy to see that D is related to the diffusion constant D0 of the whole percolation system: above pc , the dc conductivity of the percolation system increases as  (p  pc ) , so due to the Einstein relation, Eq. (17), the diffusion constant D0 must also increase in this way. The mean square displacement (and hence D0 ) is obtained by averaging over all possible starting points of a particle in the percolation system. It is clear that only those particles which start on the infinite cluster can travel from one side of the system to the other, and thus contribute to D0 . Particles that start

on a finite cluster cannot leave the cluster, and thus do not contribute to D0 . Hence D0 is related to D by D0 D DP1 , implying D (p  pc )ˇ  (ˇ )/ :

(19)

Combining (16) and (19), the mean square displacement on the infinite cluster can be written as [7,33,72] ( if t  t ; t 2/d w 2 hr (t)i (20) ˇ t if t  t ; (p  pc ) where t  d w

(21)

describes the time scale the random walker needs, on average, to explore the fractal regime in the cluster. As  (p  pc ) is the only length scale here, t is the only relevant time scale, and we can bridge the short time regime and the long time regime by a scaling function f (t/t ), hr2 (t)i D t 2/d w f (t/t ) :

(22)

To satisfy (20)–(21), we require f (x) x 0 for x  1 and f (x) x 12/d w for x  1. The first relation trivially satisfies (20)–(21). The second relation gives D D lim t!1 hr2 (t)i/2dt t2/d w 1 , which in connection with (19) and (21) yields a relation between dw and  [7, 33,72], dw D 2 C (  ˇ)/ :

(23)

˜ Comparing (18) and (23) one can express the exponent  by , ˜ D / : 

(24)

Networks Networks are defined as nodes connected by links, called graphs in mathematics. Many real world system can be describe as networks. Perhaps the best known example of a network is the Internet, where computers (nodes) around the globe are connected by cables (links) in such a way that an email message can travel from one computer to another by traveling along only a few links. Social relations between people can be represented by a social network [2]; nodes represent people and links represent their relations. One important property of a network is the “small world” phenomena. The shortest path (minimum number of hops) between any two nodes is very small, of the order of log N or smaller, where N is the number of nodes in the network [11,19,29]. The lattices discussed in

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Sect. “Percolation” are also networks, where the sites of the lattice are the nodes and the bonds represent the links. In this case, the number of links per node is fixed but, in general, the number of links per node can be taken from any degree distribution, P(k). In lattices, due to spatial constraints, the distances between nodes is large and scales as N 1/d , where d is the dimension of the lattice. Since many networks have no spatial constraints, it follows that such networks can be regarded as embedded in infinite dimension, d D 1, justifying a very small distance, of order log N. In networks which are not constrained to geographical space, there is no Euclidean distance and the distance metric is only the shortest path ` defined in Sect. “Percolation Clusters as Fractals”. We will show in this chapter that ideas from percolation and fractals can be applied to obtain useful results in networks which are not embedded in space. The main difference compared to lattices is that the condition for percolation is no longer the spanning property, but rather the property of having a cluster containing number of nodes of order N, where N is the total original number of nodes in the network. Such a component, if it exists, is termed the giant component. The condition for the existence of a giant component above the percolation threshold, and its absence below the threshold, applies also to lattices, and therefore can be regarded as more general than the spanning property. An interesting property of percolation, called universality, is that the behavior at and close to the critical point depends only on the dimensionality of the lattice, and not on the microscopic connection details of the lattice. This behavior is characterized by a set of critical exponents that are the same for all two-dimensional lattices, square, triangular or hexagonal, and for either site or bond percolation. However, a different set of critical exponents will be obtained for a lattice of another dimension. Furthermore, above some critical dimension (dc D 6 for percolation in d-dimensional lattices), known as the upper critical dimension, the critical behavior remains the same for all d  dc . This is due to the insignificance of loops in high dimensions, and thus usually allows for easy determination of the critical exponents for high dimensions, using the “infinite dimensional” or “mean field” approach. Erd˝os and Rényi (ER) studied an ensemble of networks with N nodes and 2M links that randomly connect pairs of nodes. They found that pc D 1/hki D N/2M. Percolation on ER networks or on infinite dimensional lattices, as well as on Cayley trees, has the same critical exponents, due to the fact that their topology is the same and no spatial constraint is imposed on the networks. For ER networks, as for lattices, in d  6 the size S of the percolation cluster at pc , scales with N as, S N 2/3 [11,20].

The value of 2/3 can be related to the upper critical dimension, dc D 6, and to the fractal dimension of percolation clusters d f D 4 for d D 6. Since N D L6 and S D L4 , it follows that S N 2/3 . In recent years it was realized [2] that P(k) for many real networks is very broad and, in many cases, is best represented by a power law, P(k) k  . Networks with a power law degree distribution are called scale free (SF) network. Heterogeneity of the degrees may affect critical behavior, even above the upper critical dimension. The heterogeneity of the degrees can be regarded as a breakdown of translational symmetry that exists in lattices, ER networks and Cayley trees. In these cases, each node has a typical number of neighbors, while in scale free networks the variation between node degrees is very large. A general result for pc for any random network with a given degree distribution is [21] pc D

1 ;  1



hk 2 i : hki

This result yields that for < 3, hk 2 i ! 1 and therefore pc ! 0. That is, no finite percolation threshold exists. Thus, even if most of the nodes of the Internet are removed, those which are left can still communicate. This explains the puzzle of why viruses and worms stay a long time in the Internet even if many people use antivirus software. It also explains why in order to effectively immunize populations, one needs to immunize most of the people. The percolation critical exponents for SF networks are still mean-field or infinite-dimensional in the sense of the insignificance of loops. However, they are different from those of the standard mean field percolation. Indeed, for scale free networks [2], the size of the spanning percolation cluster is [20], ( N ( 2)/( 1) ; 3 < < 4 S (25)

>4 N 2/3 ; As shown above, for < 3, there is no percolation threshold, and therefore no spanning percolation cluster. Note that SF networks can be regarded as a generalization of ER networks, since for > 4 one obtains the ER network results. Summary and Future Directions The percolation problem and its numerous modifications can be useful in describing several physical, chemical, and biological processes, such as the spreading of epidemics or forest fires, gelation processes, and the invasion of water into oil in porous media, which is relevant for the process of recovering oil from porous rocks. In some cases, modi-

Fractals and Percolation

fication changes the universality class of a percolation. We begin with an example in which the universality class does not change. We showed that a random process such as percolation can lead naturally to fractal structures. This may be one of the reasons why fractals occur frequently in nature. Bibliography Primary Literature 1. Aharony A (1986) In: Grinstein G, Mazenko G (eds) Directions in condensed matter physics. World Scientific, Singapore 2. Albert R, Barabasi A-L (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47 3. Alexandrowicz Z (1980) Phys Lett A 80:284 4. Alexander S, Orbach R (1982) J Phys Lett 43:L625 5. Alexander S, Bernasconi J, Schneider WR, Orbach R (1981) Rev Mod Phys 53:175; Alexander S (1983) In: Deutscher G, Zallen R, Adler J (eds) Percolation Structures and Processes. Adam Hilger, Bristol, p 149 6. Avnir D (ed) (1989) The fractal approach to heterogeneos chemistry. Wiley, Chichester 7. Ben-Avraham D, Havlin S (1982) J Phys A 15:L691; Havlin S, BenAvraham D, Sompolinsky H (1983) Phys Rev A 27: 1730 8. Ben-Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge 9. Benguigui L (1984) Phys Rev Lett 53:2028 10. Blumen A, Klafter J, Zumofen G (1986) In: Zschokke I (ed) Optical spectroscopy of glasses. Reidel, Dordrecht, pp 199–265 11. Bollobás B (1985) Random graphs. Academic Press, London 12. Bouchaud JP, Georges A (1990) Phys Rep 195:127 13. Bunde A (1986) Adv Solid State Phys 26:113 14. Bunde A, Havlin S (eds) (1996) Fractals and disordered systems, 2nd edn. Springer, Berlin; Bunde A, Havlin S (eds) (1995) Fractals in Science, 2nd edn. Springer, Berlin 15. Broadbent SR, Hammersley JM (1957) Proc Camb Phil Soc 53:629 16. Cardey JL, Grassberger P (1985) J Phys A 18:L267 17. Cardy J (1998) J Phys A 31:L105 18. Clerc JP, Giraud G, Laugier JM, Luck JM (1990) Adv Phys 39:191 19. Cohen R, Havlin S (2003) Scale-free networks are ultrasmall. Phys Rev Lett 90:058701 20. Cohen R, Havlin S (2008) Complex networks: Structure, stability and function. Cambridge University Press, Cambridge 21. Cohen R, Erez K, Ben-Avraham D, Havlin S (2000) Resilience of the internet to random breakdowns. Phys Rev Lett 85:4626 22. Coniglio A (1982) J Phys A 15:3829 23. Coniglio A (1982) Phys Rev Lett 46:250 24. de Gennes PG (1976) La Recherche 7:919 25. de Gennes PG (1979) Scaling concepts in polymer physics. Cornell University Press, Ithaca 26. Deutscher G, Zallen R, Adler J (eds) (1983) A collection of review articles: percolation structures and processes. Adam Hilger, Bristol 27. Domb C (1983) In: Deutscher G, Zallen R, Adler J (eds) Percolation structures and processes. Adam Hilger, Bristol; Domb C, Stoll E, Schneider T (1980) Contemp Phys 21: 577 28. Elam WT, Kerstein AR, Rehr JJ (1984) Phys Rev Lett 52:1515

˝ P, Rényi A (1959) On random graphs. Publicationes 29. Erdos Mathematicae 6:290; (1960) Publ Math Inst Hung Acad Sci 5:17 30. Essam JW (1980) Rep Prog Phys 43:843 31. Family F, Landau D (eds) (1984) Kinetics of aggregation and gelation. North Holland, Amsterdam; For a review on gelation see: Kolb M, Axelos MAV (1990) In: Stanley HE, Ostrowsky N (eds) Correlatios and Connectivity: Geometric Aspects of Physics, Chemistry and Biology. Kluwer, Dordrecht, p 225 32. Flory PJ (1971) Principles of polymer chemistry. Cornell University, New York; Flory PJ (1941) J Am Chem Soc 63:3083–3091– 3096; Stockmayer WH (1943) J Chem Phys 11:45 33. Gefen Y, Aharony A, Alexander S (1983) Phys Rev Lett 50:77 34. Gouyet JF (1992) Phys A 191:301 35. Grassberger P (1986) Math Biosci 62:157; (1985) J Phys A 18:L215; (1986) J Phys A 19:1681 36. Grassberger P (1992) J Phys A 25:5867 37. Grassberger P (1999) J Phys A 32:6233 38. Grossman T, Aharony A (1987) J Phys A 20:L1193 39. Haus JW, Kehr KW (1987) Phys Rep 150:263 40. Havlin S, Ben-Avraham D (1987) Adv Phys 36:695 41. Havlin S, Ben-Avraham D (1987) Diffusion in random media. Adv Phys 36:659 42. Havlin S, Nossal R (1984) Topological properties of percolation clusters. J Phys A 17:L427 43. Havlin S, Nossal R, Trus B, Weiss GH (1984) J Stat Phys A 17:L957 44. Herrmann HJ (1986) Phys Rep 136:153 45. Herrmann HJ, Stanley HE (1984) Phys Rev Lett 53:1121; Hong DC, Stanley HE (1984) J Phys A 16:L475 46. Herrmann HJ, Stanley HE (1988) J Phys A 21:L829 47. Herrmann HJ, Hong DC, Stanley HE (1984) J Phys A 17:L261 48. Kesten H (1982) Percolation theory for mathematicians. Birkhauser, Boston (A mathematical approach); Grimmet GR (1989) Percolation. Springer, New York 49. Kirkpatrick S (1979) In: Maynard R, Toulouse G (eds) Le Houches Summer School on Ill Condensed Matter. North Holland, Amsterdam 50. Kopelman R (1976) In: Fong FK (ed) Topics in applied physics, vol 15. Springer, Heidelberg 51. Ma SK (1976) Modern theory of critical phenomena. Benjamin, Reading 52. Mackay G, Jan N (1984) J Phys A 17:L757 53. Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San Francisco; Mandelbrot BB (1977) Fractals: Form, Chance and Dimension. Freeman, San Francisco 54. Meakin P, Majid I, Havlin S, Stanley HE (1984) J Phys A 17:L975 55. Middlemiss KM, Whittington SG, Gaunt DC (1980) J Phys A 13:1835 56. Montroll EW, Shlesinger MF (1984) In: Lebowitz JL, Montroll EW (eds) Nonequilibrium phenomena II: from stochastics to hydrodynamics. Studies in Statistical Mechanics, vol 2. NorthHolland, Amsterdam 57. Paul G, Ziff RM, Stanley HE (2001) Phys Rev E 64:26115 58. Pike R, Stanley HE (1981) J Phys A 14:L169 59. Porto M, Bunde A, Havlin S, Roman HE (1997) Phys Rev E 56:1667 60. Ritzenberg AL, Cohen RI (1984) Phys Rev B 30:4036 61. Sahimi M (1993) Application of percolation theory. Taylor Francis, London 62. Saleur H, Duplantier B (1987) Phys Rev Lett 58:2325 63. Sapoval B, Rosso M, Gouyet JF (1985) J Phys Lett 46:L149

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Books and Reviews Bak P (1996) How nature works: The science of self organized criticality. Copernicus, New York Barabási A-L (2003) Linked: How everything is connected to every-

thing else and what it means for business, science and everyday life. Plume Bergman DJ, Stroud D (1992) Solid State Phys 46:147–269 Dorogovtsev SN, Mendes JFF (2003) Evolution of networks: From biological nets to the internet and www (physics). Oxford University Press, Oxford Eglash R (1999) African fractals: Modern computing and indigenous design. Rutgers University Press, New Brunswick, NJ Feder J (1988) Fractals. Plenum, New York Gleick J (1997) Chaos. Penguin Books, New York Gould H, Tobochnik J (1988) An introduction to computer simulation methods. In: Application to physical systems. AddisonWesley, Reading, MA Meakin P (1998) Fractals, scaling and growth far from equilibrium. Cambridge University Press, Cambridge Pastor-Satorras R, Vespignani A (2004) Evolution and structure of the internet: A statistical physics approach. Cambridge University Press, Cambridge Peitgen HO, Jurgens H, Saupe D (1992) Chaos and fractals. Springer, New York Peng G, Decheng T (1990) The fractal nature of a fracture surface. J Physics A 14:3257–3261 Pikovsky A, Rosenblum M, Kurths J, Chirikov B, Cvitanovic P, Moss F, Swinney H (2003) Synchronization: A universal concept in nonlinear sciences. Cambridge University Press, Cambridge Vicsek T (1992) Fractal growth phenomena. World Scientific, Singapore

Fractals in the Quantum Theory of Spacetime

Fractals in the Quantum Theory of Spacetime LAURENT N OTTALE CNRS, Paris Observatory and Paris Diderot University, Paris, France Article Outline Glossary Definition of the Subject Introduction Foundations of Scale Relativity Theory Scale Laws From Fractal Space to Nonrelativistic Quantum Mechanics From Fractal Space-Time to Relativistic Quantum Mechanics Gauge Fields as Manifestations of Fractal Geometry Future Directions Bibliography Glossary Fractality In the context of the present article, the geometric property of being structured over all (or many) scales, involving explicit scale dependence which may go up to scale divergence. Spacetime Inter-relational level of description of the set of all positions and instants (events) and of their transformations. The events are defined with respect to a given reference system (i. e., in a relative way), but a spacetime is characterized by invariant relations which are valid in all reference systems, such as, e. g., the metric invariant. In the generalization to a fractal space-time, the events become explicitly dependent on resolution. Relativity The property of physical quantities according to which they can be defined only in terms of relationships, not in an abolute way. These quantities depend on the state of the reference system, itself defined in a relative way, i. e., with respect to other coordinate systems. Covariance Invariance of the form of equations under general coordinate transformations. Geodesics Curves in a space (more generally in a spacetime) which minimize the proper time. In a geometric spacetime theory, the motion equation is given by a geodesic equation. Quantum Mechanics Fundamental axiomatic theory of elementary particle, nuclear, atomic, molecular, etc. physical phenomena, according to which the state of

a physical system is described by a wave function whose square modulus yields the probability density of the variables, and which is solution of a Schrödinger equation constructed from a correspondence principle (among other postulates). Definition of the Subject The question of the foundation of quantum mechanics from first principles remains one of the main open problems of modern physics. In its current form, it is an axiomatic theory of an algebraic nature founded upon a set of postulates, rules and derived principles. This is to be compared with Einstein’s theory of gravitation, which is founded on the principle of relativity and, as such, is of an essentially geometric nature. In its framework, gravitation is understood as a very manifestation of the curvature of a Riemannian space-time. It is therefore relevant to question the nature of the quantum space-time and to ask for a possible refoundation of the quantum theory upon geometric first principles. In this context, it has been suggested that the quantum laws and properties could actually be manifestations of a fractal and nondifferentiable geometry of space-time [52,69,71], coming under the principle of scale relativity [53,54]. This principle extends, to scale transformations of the reference system, the theories of relativity (which have been, up to now, applied to transformations of position, orientation and motion). Such an approach allows one to recover the main tools and equations of standard quantum mechanics, but also to suggest generalizations, in particular toward high energies, since it leads to the conclusion that the Planck length scale could be a minimum scale in nature, unreachable and invariant under dilations [53]. But it has another radical consequence. Namely, it allows the possibility of a new form of macroscopic quantum-type behavior for a large class of complex systems, namely those whose behavior is characterized by Newtonian dynamics, fractal stochastic fluctuations over a large range of scales, and small scale irreversibility. In this case the equation of motion may take the form of a Schrödinger equation, which yields peaks of probability density according to the symmetry, field and limit conditions. These peaks may be interpreted as a tendency for the system to form structures [57], in terms of a macroscopic constant which is no longer ¯, therefore possibly leading to a new theory of self-organization. It is remarkable that, under such a relativistic view, the question of complexity may be posed in an original way. Namely, in a fully relativistic theory there is no intrinsic complexity, since the various physical properties of an ‘ob-

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ject’ are expected to vanish in the proper system of coordinates linked to the object. Therefore, in such a framework the apparent complexity of a system comes from the complexity of the change of reference frames from the proper frame to the observer (or measurement apparatus) reference frame. This does not mean that the complexity can always be reduced, since this change can itself be infinitely complex, as it is the case in the situation described here of a fractal and nondifferentiable space-time. Introduction There have been many attempts during the 20th century at understanding the quantum behavior in terms of differentiable manifolds. The failure of these attempts indicates that a possible ‘quantum geometry’ should be of a completely new nature. Moreover, following the lessons of Einstein’s construction of a geometric theory of gravitation, it seems clear that any geometric property to be attributed to space-time itself, and not only to particular objects or systems embedded in space-time, must necessarily be universal. Fortunately, the founders of quantum theory have brought to light a universal and fundamental behavior of the quantum realm, in opposition to the classical world; namely, the explicit dependence of the measurement results on the apparatus resolution described by the Heisenberg uncertainty relations. This leads one to contemplate the possibility that the space-time underlying the quantum mechanical laws can be explicitly dependent on the scale of observation [52,69,71]. Now the concept of a scale-dependent geometry (at the level of objects and media) has already been introduced and developed by Benoit Mandelbrot, who coined the word ‘fractal’ in 1975 to describe it. But here we consider a complementary program that uses fractal geometry, not only for describing ‘objects’ (that remain embedded in an Euclidean space), but also for intrinsically describing the geometry of space-time itself. A preliminary work toward such a goal may consist of introducing the fractal geometry in Einstein’s equations of general relativity at the level of the source terms. This would amount to giving a better description of the density of matter in the Universe accounting for its hierarchical organization and fractality over many scales (although possibly not all scales), then to solve Einstein’s field equations for such a scale dependent momentum-energy tensor. A full implementation of this approach remains a challenge to cosmology. But a more direct connection of the fractal geometry with fundamental physics comes from its use in describing not only the distribution of matter in space, but also the ge-

ometry of space-time itself. Such a goal may be considered as the continuation of Einstein’s program of generalization of the geometric description of space-time. In the new fractal space-time theory, [27,52,54,69,71], the essence of quantum physics is a manifestation of the nondifferentiable and fractal geometry of space-time. Another line of thought leading to the same suggestion comes, not from relativity and space-time theories, but from quantum mechanics itself. Indeed, it has been discovered by Feynman [30] that the typical quantum mechanical paths (i. e., those that contribute in a main way to the path integral) are nondifferentiable and fractal. Namely, Feynman has proved that, although a mean velocity can be defined for them, no mean-square velocity exists at any point, since it is given by hv 2 i / ıt 1 . One now recognizes in this expression the behavior of a curve of fractal dimension DF D 2 [1]. Based on these premises, the reverse proposal, according to which the laws of quantum physics find their very origin in the fractal geometry of space-time, has been developed along three different and complementary approaches. Ord and co-workers [50,71,72,73], extending the Feynman chessboard model, have worked in terms of probabilistic models in the framework of the statistical mechanics of binary random walks. El Naschie has suggested to give up not only the differentiability, but also the continuity of space-time. This leads him to work in terms of a ‘Cantorian’ spacetime [27,28,29], and to therefore use in a preferential way the mathematical tool of number theory (see a more detailed review of these two approaches in Ref. [45]). The scale relativity approach [52,54,56,63,69] which is the subject of the present article, is, on the contrary, founded on a fundamentally continuous geometry of space-time which therefore includes the differentiable and nondifferentiable cases, constrained by the principle of relativity applied to both motion and scale. Other applications of fractals to the quantum theory of space-time have been proposed in the framework of a possible quantum gravity theory. They are of another nature than those considered in the present article, since they are applicable only in the framework of the quantum theory instead of deriving it from the geometry, and they concern only very small scales on the magnitude of the Planck scale. We send the interested reader to Kröger’s review paper on “Fractal geometry in quantum mechanics, field theory and spin systems”, and to references therein [45]. In the present article, we summarize the steps by which one recovers, in the scale relativity and fractal space-time framework, the main tools and postulates of quantum me-

Fractals in the Quantum Theory of Spacetime

chanics and of gauge field theories. A more detailed account can be found in Refs. [22,23,54,56,63,67,68], including possible applications of the theory to various sciences. Foundations of Scale Relativity Theory The theory of scale relativity is based on giving up the hypothesis of manifold differentiability. In this framework, the coordinate transformations are continuous but can be nondifferentiable. This has several consequences [54], leading to the following preliminary steps of construction of the theory: (1) One can prove the following theorem [5,17,18,54, 56]: a continuous and nondifferentiable curve is fractal in a general meaning, namely, its length is explicitly dependent on a scale variable ", i. e., L D L("), and it diverges, L ! 1, when " ! 0. This theorem can be readily extended to a continuous and nondifferentiable manifold, which is therefore fractal not as an hypothesis, but as a consequence of giving up an hypothesis (that of differentiability). (2) The fractality of space-time [52,54,69,71] involves the scale dependence of the reference frames. One therefore adds a new variable " which characterizes the ‘state of scale’ to the usual variables defining the coordinate system. In particular, the coordinates themselves become functions of these scale variables, i. e., X D X("). (3) The scale variables " can never be defined in an absolute way, but only in a relative way. Namely, only their ratio D "0 /" has a physical meaning. In experimental situations, these scale variables amount to the resolution of the measurement apparatus (it may be defined as standard errors, intervals, pixel size, etc.). In a theoretical analysis, they are the space and time differential elements themselves. This universal behavior extends to the principle of relativity in such a way that it also applies to the transformations (dilations and contractions) of these resolution variables [52,53,54]. Scale Laws Fractal Coordinate and Differential Dilation Operator Consider a variable length measured on a fractal curve, and (more generally) a non-differentiable (fractal) curvilinear coordinate L(s; "), that depends on some parameter s which characterizes the position on the curve (it may be, e. g., a time coordinate), and on the resolution ". Such a coordinate generalizes the concept of curvilinear coordinates introduced for curved Riemannian space-times in Einstein’s general relativity [54] to nondifferentiable and fractal space-times.

Such a scale-dependent fractal length L(s; ") remains finite and differentiable when " ¤ 0; namely, one can define a slope for any resolution ", being aware that this slope is itself a scale-dependent fractal function. It is only at the limit " ! 0 that the length is infinite and the slope undefined, i. e., that nondifferentiability manifests itself. Therefore the laws of dependence of this length upon position and scale may be written in terms of a double differential calculus, i. e., it can be the solution of differential equations involving the derivatives of L with respect to both s and ". As a preliminary step, one needs to establish the relevant form of the scale variables and the way they intervene in scale differential equations. For this purpose, let us apply an infinitesimal dilation d to the resolution, which is therefore transformed as " ! "0 D "(1 C d ). The dependence on position is omitted at this stage in order to simplify the notation. By applying this transformation to a fractal coordinate L, one obtains, to the first order in the differential element, L("0 ) D L(" C " d ) D L(") C

@L(") " d @"

˜ L(") ; D (1 C Dd )

(1)

˜ is, by definition, the dilation operator. where D Since d"/" D d ln ", the identification of the two last members of Eq. (1) yields ˜ D" @ D @ : D @" @ ln "

(2)

This form of the infinitesimal dilation operator shows that the natural variable for the resolution is ln ", and that the expected new differential equations will indeed involve quantities such as @L(s; ")/@ ln ". This theoretical result agrees and explains the current knowledge according to which most measurement devices (of light, sound, etc.), including their physiological counterparts (eye, ear, etc.) respond according to the logarithm of the intensity (e. g., magnitudes, decibels, etc.). Self-similar Fractals as Solutions of a First Order Scale Differential Equation Let us start by writing the simplest possible differential equation of scale, and then solve it. We shall subsequently verify that the solutions obtained comply with the principle of relativity. As we shall see, this very simple approach already yields a fundamental result: it gives a foundation and an understanding from first principles for self-similar fractal laws, which have been shown by Mandelbrot and many others to be a general description of a large

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Fractals in the Quantum Theory of Spacetime, Figure 1 Scale dependence of the length (left) and of the effective fractal dimension DF D F C 1 (right) in the case of “inertial” scale laws (which are solutions of the simplest, first order scale differential equation). Toward the small scale one gets a scale-invariant law with constant fractal dimension, while the explicit scale-dependence is lost at scales larger than a transition scale 

number of natural phenomena, in particular biological ones (see, e. g., [48,49,70], other volumes of these series and references therein). In addition, the obtained laws, which combine fractal and scale-independent behaviours, are the equivalent for scales of what inertial laws are for motion [49]. Since they serve as a fundamental basis of description for all the subsequent theoretical constructions, we shall now describe their derivation in detail. The simplest differential equation of explicit scale dependence which one can write is of first order and states that the variation of L under an infinitesimal scale transformation d ln " depends only on L itself. Basing ourselves on the previous derivation of the form of the dilation operator, we thus write @L(s; ") D ˇ(L) : @ ln "

(3)

The function ˇ is a priori unknown. However, still looking for the simplest form of such an equation, we expand ˇ(L) in powers of L, namely we write ˇ(L) D a C bL C : : : . Disregarding for the moment the s dependence, we obtain, to the first order, the following linear equation in which a and b are constants: dL D a C bL : d ln "

(4)

In order to find the solution of this equation, let us change the names of the constants as F D b and L0 D a/F , so that a C bL D F (L  L0 ). We obtain the equation dL L  L0

D F d ln " :

(5)

Its solution is (see Fig. 1)

 F   ; L(") D L0 1 C "

(6)

where  is an integration constant. This solution corresponds to a length measured on a fractal curve up to a given point. One can now generalize it to a variable length that also depends on the position characterized by the parameter s. One obtains

 F   L(s; ") D L0 (s) 1 C (s) ; (7) " in which, in the most general case, the exponent F may itself be a variable depending on the position. The same kind of result is obtained for the projections on a given axis of such a fractal length [54]. Let X(s; ") be one of these projections, it reads

 F   : (8) X(s; ") D x(s) 1 C x (s) " In this case x (s) becomes a highly fluctuating function which may be described by a stochastic variable, as can be seen in Fig. 2. The important point here and for what follows is that the solution obtained is the sum of two terms (a classicallike, “differentiable part” and a nondifferentiable “fractal part”) which is explicitly scale-dependent and tends to infinity when " ! 0 [22,54]. By differentiating these two parts in the above projection, we obtain the differential formulation of this essential result, dX D dx C d ;

(9)

Fractals in the Quantum Theory of Spacetime

that this result has been derived from a theoretical analysis based on first principles, instead of being postulated or deduced from a fit of observational data. It should be noted that in the above expressions, the resolution is a length interval, " D ıX defined along the fractal curve (or one of its projected coordinate). But one may also travel on the curve and measure its length on constant time intervals, then change the time scale. In this case the resolution " is a time interval, " D ıt. Since they are related by the fundamental relation (see Fig. 2) ıX D F ıt ; the fractal length depends on the time resolution as  11/D F T : X(s; ıt) D X0 (s)  ıt Fractals in the Quantum Theory of Spacetime, Figure 2 A fractal function. An example of such a fractal function is given by the projections of a fractal curve on Cartesian coordinates, as a function of a continuous and monotonous parameter (here the time t) which marks the position on the curve. The figure also exhibits the relation between space and time differential elements for such a fractal function, and compares the differentiable part ıx and nondifferentiable part ı of the space elementary displacement ıX D ıx C ı. While the differentiable coordinate variation ıx D hıXi is of the same order as the time differential ıt, the fractal fluctuation becomes much larger than ıt when ıt T, where T is a transition time scale, and it depends on the fractal dimension DF as ı / ıt1/DF . Therefore the two contributions to the full differential displacement are related by the fractal law ı DF / ıx, since ıx and ıt are differential elements of the same order

where dx is a classical differential element, while d is a differential element of fractional order (see Fig. 2, in which the parameter s that characterizes the position on the fractal curve has been taken to be the time t). This relation plays a fundamental role in the subsequent developments of the theory. Consider the case when F is constant. In the asymptotic small scale regime, "  , one obtains a power-law dependence on resolution which reads  F  L(s; ") D L0 (s) : (10) " In this expression we recognize the standard form of a selfsimilar fractal behavior with constant fractal dimension DF D 1 C F , which has already been found to yield a fair description of many physical and biological systems [49]. Here the topological dimension is DT D 1, since we deal with a length, but this can be easily generalized to surfaces (DT D 2), volumes (DT D 3), etc., according to the general relation DF D DT C F . The new feature here is

(11)

(12)

An example of the use of such a relation is Feynman’s result according to which the mean square value of the velocity of a quantum mechanical particle is proportional to ıt 1 (see p. 176 in [30]), which corresponds to a fractal dimension DF D 2, as later recovered by Abbott and Wise [1] by using a space resolution. More generally (in the usual case when " D ıX), following Mandelbrot, the scale exponent F D DF  DT can be defined as the slope of the (ln "; ln L) curve, namely F D

d ln L : d ln(/")

(13)

For a self-similar fractal such as that described by the fractal part of the above solution, this definition yields a constant value which is the exponent in Eq. (10). However, one can anticipate on the following, and use this definition to compute an “effective” or “local” fractal dimension, now variable, from the complete solution that includes the differentiable and the nondifferentiable parts, and therefore a transition to effective scale independence. Differentiating the logarithm of Eq. (16) yields an effective exponent given by F eff D : (14) 1 C ("/)F The effective fractal dimension DF D 1 C F therefore jumps from the nonfractal value DF D DT D 1 to its constant asymptotic value at the transition scale  (see right part of Fig. 1). Galilean Relativity of Scales We can now check that the fractal part of such a law is compatible with the principle of relativity extended to scale transformations of the resolutions (i. e., with the principle of scale relativity). It reads L D L0 (/")F

575

576

Fractals in the Quantum Theory of Spacetime

(Eq. 10), and it is therefore a law involving two variables (ln L and F ) as a function of one parameter (") which, according to the relativistic view, characterizes the state of scale of the system (its relativity is apparent in the fact that we need another scale  to define it by their ratio). More generally, all the following statements remain true for the complete scale law including the transition to scale-independence, by making the replacement of L by L  L0 . Note that, to be complete, we anticipate on what follows and consider a priori F to be a variable, even if, in the simple law first considered here, it takes a constant value. Let us take the logarithm of Eq. (10). It yields ln(L/L0 ) D F ln(/"). The two quantities ln L and F then transform under a finite scale transformation " ! "0 D " as ln

L("0 ) L0

D ln

L(") L0

 F ln ;

(15)

and to be complete, F0 D F :

(16)

These transformations have exactly the same mathematical structure as the Galilean group of motion transformation (applied here to scale rather than motion), which reads x 0 D x  t v;

t0 D t :

(17)

This is confirmed by the dilation composition law, " ! "0 ! "00 , which reads "00 "0 "00 (18) D ln C ln 0 ; " " " and is therefore similar to the law of composition of velocities between three reference systems K, K 0 and K 00 , ln

V 00 (K 00 /K) D V(K 0 /K) C V 0 (K 00 /K 0 ) :

(19)

Since the Galileo group of motion transformations is known to be the simplest group that implements the principle of relativity, the same is true for scale transformations. It is important to realize that this is more than a simple analogy: the same physical problem is set in both cases, and is therefore solved under similar mathematical structures (since the logarithm transforms what would have been a multiplicative group into an additive group). Indeed, in both cases, it is equivalent to finding the transformation law of a position variable (X for motion in a Cartesian system of coordinates, ln L for scales in a fractal system of coordinates) under a change of the state of the coordinate system (change of velocity V for motion and of resolution ln for scale), knowing that these state variables are defined only in a relative way. Namely, V is the

relative velocity between the reference systems K and K 0 , and is the relative scale: note that " and "0 have indeed disappeared in the transformation law, only their ratio remains. This remark establishes the status of resolutions as (relative) “scale velocities” and of the scale exponent F as a “scale time”. Recall finally that since the Galilean group of motion is only a limiting case of the more general Lorentz group, a similar generalization is expected in the case of scale transformations, which we shall briefly consider in Sect. “Special Scale-Relativity”. Breaking of Scale Invariance The standard self-similar fractal laws can be derived from the scale relativity approach. However, it is important to note that Eq. (16) provides us with another fundamental result, as shown in Fig. 1. Namely, it also contains a spontaneous breaking of the scale symmetry. Indeed, it is characterized by the existence of a transition from a fractal to a non-fractal behavior at scales larger than some transition scale . The existence of such a breaking of scale invariance is also a fundamental feature of many natural systems, which remains, in most cases, misunderstood. The advantage of the way it is derived here is that it appears as a natural, spontaneous, but only effective symmetry breaking, since it does not affect the underlying scale symmetry. Indeed, the obtained solution is the sum of two terms, the scale-independent contribution (differentiable part), and the explicitly scale-dependent and divergent contribution (fractal part). At large scales the scaling part becomes dominated by the classical part, but it is still underlying even though it is hidden. There is therefore an apparent symmetry breaking (see Fig. 1), though the underlying scale symmetry actually remains unbroken. The origin of this transition is, once again, to be found in relativity (namely, in the relativity of position and motion). Indeed, if one starts from a strictly scale-invariant law without any transition, L D L0 (/")F , then adds a translation in standard position space (L ! L C L1 ), one obtains  F

 F   1 L0 D L1 C L0 D L1 1 C : (20) " " Therefore one recovers the broken solution (that corresponds to the constant a ¤ 0 in the initial scale differential equation). This solution is now asymptotically scaledependent (in a scale-invariant way) only at small scales, and becomes independent of scale at large scales, beyond some relative transition 1 which is partly determined by the translation itself.

Fractals in the Quantum Theory of Spacetime

Multiple Scale Transitions Multiple transitions can be obtained by a simple generalization of the above result [58]. Still considering a perturbative approach and taking the Taylor expansion of the differential equation dL/d ln " D ˇ(L), but now to the second order of the expansion, one obtains the equation dL D a C bL C c L2 C    : d ln "

(21)

One of its solutions, which generalizes that of Eq. (5), describes a scaling behavior which is broken toward both the small and large scales, as observed in most real fractal systems,   1 C (0 /")F L D L0 : (22) 1 C (1 /")F Due to the non-linearity of the ˇ function, there are now two transition scales in such a law. Indeed,  when " < 1 < 0 , one has (0 /")  1 and (1 /")  1, so that L D L0 (0 /1 )F  cst, independent of scale;  when 1 < " < 0 , one has (0 /")  1 but (1 /")  1, so that the denominator disappears, and one recovers the previous pure scaling law L D L0 (0 /")F ;  when 1 < 0 < ", one has (0 /")  1 and (1 /")  1, so that L D L0 D cst, independent of scale. Scale Relativity Versus Scale Invariance Let us briefly be more specific about the way in which the scale relativity viewpoint differs from scaling or simple scale invariance. In the standard concept of scale invariance, one considers scale transformations of the coordinate, X ! X0 D q  X ;

(23)

then one looks for the effect of such a transformation on some function f (X). It is scaling when f (qX) D q˛  f (X) :

(24)

The scale relativity approach involves a more profound level of description, since the coordinate X is now explicitly resolution-dependent, i. e. X D X("). Therefore we now look for a scale transformation of the resolution, " ! "0 D " ;

(25)

which implies a scale transformation of the position variable that takes in the self-similar case the form X( ") D F X(") :

(26)

But now the scale factor on the variable has a physical meaning which goes beyond a trivial change of units. It corresponds to a coordinate measured at two different resolutions on a fractal curve of fractal dimension DF D 1 C F , and one can obtain a scaling function of a fractal coordinate, f ( F X) D ˛F  f (X) :

(27)

In other words, there are now three levels of transformation in the scale relativity framework (the resolution, the variable, and its function) instead of only two in the usual conception of scaling. Generalized Scale Laws Discrete Scale Invariance, Complex Dimension, and Log-Periodic Behavior Fluctuations with respect to pure scale invariance are potentially important, namely the log-periodic correction to power laws that is provided, e. g., by complex exponents or complex fractal dimensions. It has been shown that such a behavior provides a very satisfactory and possibly predictive model of the time evolution of many critical systems, including earthquakes and market crashes ([79] and references therein). More recently, it has been applied to the analysis of major event chronology of the evolutionary tree of life [14,65,66], of human development [11] and of the main economic crisis of western and precolumbian civilizations [36,37,40,65]. One can recover log-periodic corrections to self-similar power laws through the requirement of covariance (i. e., of form invariance of equations) applied to scale differential equations [58]. Consider a scale-dependent function L("). In the applications to temporal evolution quoted above, the scale variable is identified with the time interval jt  t c j, where t c is the date of a crisis. Assume that L satisfies a first order differential equation, dL  L D 0 ; d ln "

(28)

whose solution is a pure power law L(") / " (cf Sect. “Self-Similar Fractals as Solutions of a First Order Scale Differential Equation”). Now looking for corrections to this law, one remarks that simply incorporating a complex value of the exponent  would lead to large log-periodic fluctuations rather than to a controllable correction to the power law. So let us assume that the right-hand side of Eq. (28) actually differs from zero: dL  L D  : d ln "

(29)

577

578

Fractals in the Quantum Theory of Spacetime

Fractals in the Quantum Theory of Spacetime, Figure 3 Scale dependence of the length L of a fractal curve and of the effective “scale time” F D DF  DT (fractal dimension minus topological dimension) in the case of a log-periodic behavior with fractal/non-fractal transition at scale , which reads L(") D L0 f1 C (/") exp[b cos(! ln("/))]g

We can now apply the scale covariance principle and require that the new function  be a solution of an equation which keeps the same form as the initial equation, d  0 D 0 : d ln "

(30)

Setting  0 D  C , we find that L must be a solution of a second-order equation: d2 L dL  (2 C ) C ( C )L D 0 : (d ln ")2 d ln "

(31)

The solution reads L(") D a" (1 C b" ), and finally, the choice of an imaginary exponent  D i! yields a solution whose real part includes a log-periodic correction: L(") D a" [1 C b cos(! ln ")] :

(32)

As previously recalled in Sect. “Breaking of Scale Invariance,” adding a constant term (a translation) provides a transition to scale independence at large scales (see Fig. 3). Lagrangian Approach to Scale Laws In order to obtain physically relevant generalizations of the above simplest (scale-invariant) laws, a Lagrangian approach can be used in scale space analogously to using it to derive the laws of motion, leading to reversal of the definition and meaning of the variables [58]. This reversal is analogous to that achieved by Galileo concerning the laws of motion. Indeed, from the Aristotle viewpoint, “time is the measure of motion”. In the same way, the fractal dimension, in its standard (Mandelbrot’s)

acception, is defined from the topological measure of the fractal object (length of a curve, area of a surface, etc.) and resolution, namely (see Eq. 13) tD

x v

$

F D DF  D T D

d ln L : d ln(/")

(33)

In the case mainly considered here, when L represents a length (i. e., more generally, a fractal coordinate), the topological dimension is DT D 1 so that F D DF  1. With Galileo, time becomes a primary variable, and the velocity is deduced from space and time, which are therefore treated on the same footing, in terms of a space-time (even though the Galilean space-time remains degenerate because of the implicitly assumed infinite velocity of light). In analogy, the scale exponent F D DF 1 becomes in this new representation a primary variable that plays, for scale laws, the same role as played by time in motion laws (it is called “djinn” in some publications which therefore introduce a five-dimensional ‘space-time-djinn’ combining the four fractal fluctuations and the scale time). Carrying on the analogy, in the same way that the velocity is the derivative of position with respect to time, v D dx/dt, we expect the derivative of ln L with respect to scale time F to be a “scale velocity”. Consider as reference the self-similar case, that reads ln L D F ln(/"). Deriving with respect to F , now considered as a variable, yields d ln L/dF D ln(/"), i. e., the logarithm of resolution. By extension, one assumes that this scale velocity provides a new general definition of resolution even in more general situations, namely,    d ln L V D ln : (34) D " dF

Fractals in the Quantum Theory of Spacetime

One can now introduce a scale Lagrange function e L(ln L; V ; F ), from which a scale action is constructed: Z 2 e e SD (35) L(ln L; V ; F )dF : 1

The application of the action principle yields a scale Euler– Lagrange equation which reads L d @e @e L : D dF @V @ ln L

(36)

One can now verify that in the free case, i. e., in the ab˜ ln L D 0), one recovsence of any “scale force” (i. e., @L/@ ers the standard fractal laws derived hereabove. Indeed, in this case the Euler–Lagrange equation becomes ˜ @L/@V D const ) V D const:

(37)

which is the equivalent for scale of what inertia is for motion. Still in analogy with motion laws, the simplest possible form for the Lagrange function is a quadratic dependence on the scale velocity, (i. e., L˜ / V 2 ). The constancy of V D ln(/") means that it is independent of the scale time F . Equation (34) can therefore be integrated to give the usual power law behavior, L D L0 (/")F , as expected. But this reversed viewpoint also has several advantages which allow a full implementation of the principle of scale relativity: The scale time F is given the status of a fifth dimension and the logarithm of the resolution V D ln(/"), is given the status of a scale velocity (see Eq. 34). This is in accordance with its scale-relativistic definition, in which it characterizes the state of scale of the reference system, in the same way as the velocity v D dx/dt characterizes its state of motion. (ii) This allows one to generalize the formalism to the case of four independent space-time resolutions, V  D ln( /" ) D d ln L /dF . (iii) Scale laws more general than the simplest self-similar ones can be derived from more general scale Lagrangians [57,58] involving “scale accelerations” I% D d2 ln L/dF2 D d ln(/")/dF , as we shall see in what follows. (i)

Note however that there is also a shortcoming in this approach. Contrary to the case of motion laws, in which time is always flowing toward the future (except possibly in elementary particle physics at very small time scales), the variation of the scale time may be non-monotonic, as exemplified by the previous case of log-periodicity. Therefore this Lagrangian approach is restricted to monotonous variations of the fractal dimension, or, more generally, to scale intervals on which it varies in a monotonous way.

Scale Dynamics The previous discussion indicates that the scale invariant behavior corresponds to freedom (i. e. scale force-free behavior) in the framework of a scale physics. However, in the same way as there are forces in nature that imply departure from inertial, rectilinear uniform motion, we expect most natural fractal systems to also present distortions in their scale behavior with respect to pure scale invariance. This implies taking non-linearity in the scale space into account. Such distorsions may be, as a first step, attributed to the effect of the dynamics of scale (“scale dynamics”), i. e., of a “scale field”, but it must be clear from the very beginning of the description that they are of geometric nature (in analogy with the Newtonian interpretation of gravitation as the result of a force, which has later been understood from Einstein’s general relativity theory as a manifestation of the curved geometry of space-time). In this case the Lagrange scale-equation takes the form of Newton’s equation of dynamics, FD

d2 ln L ; dF2

(38)

where  is a “scale mass”, which measures how the system resists to the scale force, and where I% D d2 ln L/dF2 D d ln(/")/dF is the scale acceleration. In this framework one can therefore attempt to define generic, scale-dynamical behaviours which could be common to very different systems, as corresponding to a given form of the scale force. Constant Scale Force A typical example is the case of a constant scale force. Setting G D F/, the potential reads ' D G ln L, analogous to the potential of a constant force f in space, which is ' D  f x, since the force is @'/@x D f . The scale differential equation is d2 ln L DG: dF2

(39)

It can be easily integrated. A first integration yields d ln L/dF D GF CV0 , where V0 is a constant. Then a second integration yields a parabolic solution (which is the equivalent for scale laws of parabolic motion in a constant field), 1 V D V0 C GF ; ln L D ln L0 C V0 F C G F2 ; (40) 2 where V D d ln L/dF D ln(/"). However the physical meaning of this result is not clear under this form. This is due to the fact that, while in the case of motion laws we search for the evolution of the system with time, in the case of scale laws we search for the

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Fractals in the Quantum Theory of Spacetime

Fractals in the Quantum Theory of Spacetime, Figure 4 Scale dependence of the length of a fractal curve ln L and of its effective fractal dimension (DF D DT CF , where DT is the topological dimension) in the case of a constant scale force, with an additional fractal to non-fractal transition

dependence of the system on resolution, which is the directly measured observable. Since the reference scale  is arbitrary, the variables can be re-defined in such a way that V0 D 0, i. e.,  D 0 . Indeed, from Eq. (40) one gets F D (V  V0 )/G D [ln(/")  ln(/0 )]/G D ln(0 /")/G. Then one obtains       0 L 1 1 2 0 F D ln ; ln ln : (41) D G " L0 2G " The scale time F becomes a linear function of resolution (the same being true, as a consequence, of the fractal dimension DF D 1 C F ), and the (ln L; ln ") relation is now parabolic instead of linear (see Fig. 4 and compare to Fig. 1). Note that, as in previous cases, we have considered here only the small scale asymptotic behavior, and that we can once again easily generalize this result by including a transition to scale-independence at large scale. This is simply achieved by replacing L by (L  L0 ) in every equations. There are several physical situations where, after careful examination of the data, the power-law models were clearly rejected since no constant slope could be defined in the (log L; log ") plane. In the several cases where a clear curvature appears in this plane, e. g., turbulence [26], sandpiles [9], fractured surfaces in solid mechanics [10], the physics could come under such a scale-dynamical description. In these cases it might be of interest to identify and study the scale force responsible for the scale distortion (i. e., for the deviation from standard scaling). Special Scale-Relativity Let us close this section about the derivation of scale laws of increasing complexity by coming back to the question of

finding the general laws of scale transformations that meet the principle of scale relativity [53]. It has been shown in Sect. “Galilean Relativity of Scales” that the standard selfsimilar fractal laws come under a Galilean group of scale transformations. However, the Galilean relativity group is known, for motion laws, to be only a degenerate form of the Lorentz group. It has been proved that a similar result holds for scale laws [53,54]. The problem of finding the linear transformation laws of fields in a scale transformation V D ln (" ! "0 ) amounts to finding four quantities, a(V ); b(V ); c(V ), and d(V ), such that ln

L0 L0

D a(V ) ln

F 0 D c(V ) ln

L L0 L L0

C b(V ) F ; (42) C d(V )F :

Set in this way, it immediately appears that the current ‘scale-invariant’ scale transformation law of the standard form (Eq. 8), given by a D 1; b D V ; c D 0 and d D 1, corresponds to a Galilean group. This is also clear from the law of composition of dilatations, " ! "0 ! "00 , which has a simple additive form, V 00 D V C V 0 :

(43)

However the general solution to the ‘special relativity problem’ (namely, find a; b; c and d from the principle of relativity) is the Lorentz group [47,53]. This result has led to the suggestion of replacing the standard law of dilatation, " ! "0 D %" by a new Lorentzian relation, namely, for " < 0 and "0 < 0 ln

"0 ln("/0 ) C ln % D : 0 1 C ln % ln("/0 )/ ln2 (/0 )

(44)

Fractals in the Quantum Theory of Spacetime

Fractals in the Quantum Theory of Spacetime, Figure 5 Scale dependence of the logarithm of the length and of the effective fractal dimension, DF D 1 C F , in the case of scale-relativistic Lorentzian scale laws including a transition to scale independence toward large scales. The constant C has been taken to be C D 4 2  39:478, which is a fundamental value for scale ratios in elementary particle physics in the scale relativity framework [53,54,63], while the effective fractal dimension jumps from DF D 1 to DF D 2 at the transition, then increases without any limit toward small scales

This relation introduces a fundamental length scale , which is naturally identified, towards the small scales, with the Planck length (currently 1:6160(11)  1035 m) [53],  D lP D („G/c 3 )1/2 ;

(45)

and toward the large scales (for " > 0 and "0 > 0 ) with the scale of the cosmological constant, L D 1/2 (see Chap. 7.1 in [54]). As one can see from Eq. (44), if one starts from the scale " D  and apply any dilatation or contraction %, one obtains again the scale "0 D , whatever the initial value of 0 . In other words,  can be interpreted as a limiting lower (or upper) length-scale, which is impassable and invariant under dilatations and contractions. Concerning the length measured along a fractal coordinate which was previously scale-dependent as ln(L/L0 ) D 0 ln(0 /") for " < 0 , it becomes in the new framework, in the simplified case when one starts from the reference scale L0 (see Fig. 5) ln

L L0

0 ln(0 /") : D p 1  ln2 (0 /")/ ln2 (0 /)

(46)

The main new feature of scale relativity with respect to the previous fractal or scale-invariant approaches is that the scale exponent F and the fractal dimension DF D 1 C F , which were previously constant (DF D 2; F D 1), are now explicitly varying with scale (see Fig. 5), following the law (given once again in the simplified case when we start from the reference scale L0 ): 0 F (") D p : (47) 2 1  ln (0 /")/ ln2 (0 /) Under this form, the scale covariance is explicit, since one keeps a power law form for the length variation, L D

L0 (/")F (") , but now in terms of a variable fractal dimen-

sion. For a more complete development of special relativity, including its implications regarding new conservative quantities and applications in elementary particle physics and cosmology, see [53,54,56,63]. The question of the nature of space-time geometry at the Planck scale is a subject of intense work (see, e. g., [3,46] and references therein). This is a central question for practically all theoretical attempts, including noncommutative geometry [15,16], supersymmetry, and superstring theories [35,75] – for which the compactification scale is close to the Planck scale – and particularly for the theory of quantum gravity. Indeed, the development of loop quantum gravity by Rovelli and Smolin [76] led to the conclusion that the Planck scale could be a quantized minimal scale in Nature, involving also a quantization of surfaces and volumes [77]. Over the last years, there has also been significant research effort aimed at the development of a ‘Doubly-Special-Relativity’ [2] (see a review in [3]), according to which the laws of physics involve a fundamental velocity scale c and a fundamental minimum length scale Lp , identified with the Planck length. The concept of a new relativity in which the Planck length-scale would become a minimum invariant length is exactly the founding idea of the special scale relativity theory [53], which has been incorporated in other attempts of extended relativity theories [12,13]. But, despite the similarity of aim and analysis, the main difference between the ‘Doubly-Special-Relativity’ approach and the scale relativity one is that the question of defining an invariant lengthscale is considered in the scale relativity/fractal space-time theory as coming under a relativity of scales. Therefore

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Fractals in the Quantum Theory of Spacetime

the new group to be constructed is a multiplicative group, that becomes additive only when working with the logarithms of scale ratios, which are definitely the physically relevant scale variables, as one can show by applying the Gell-Mann-Levy method to the construction of the dilation operator (see Sect. “Fractal Coordinate and Differential Dilation Operator”).

relations and the explicit dependence of measurement results on the resolution of the measurement apparatus. Let us start from the result of the previous section, according to which the solution of a first order scale differential equation reads for DF D 2, after differentiation and reintroduction of the indices, p (48) dX  D dx  C d  D v  ds C   c ds ;

From Fractal Space to Nonrelativistic Quantum Mechanics

where c is a length scale which must be introduced for dimensional reasons and which, as we shall see, generalizes the Compton length. The   are dimensionless highly fluctuating functions. Due to their highly erratic character, we can replace them by stochastic variables such that h  i D 0, h( 0 )2 i D 1 and h( k )2 i D 1 (k D1 to 3). The mean is taken here on a purely mathematic probability law which can be fully general, since the final result does not depend on its choice.

The first step in the construction of a theory of the quantum space-time from fractal and nondifferentiable geometry, which has been described in the previous sections, has consisted of finding the laws of explicit scale dependence at a given “point” or “instant” (under their new fractal definition). The next step, which will now be considered, amounts to writing the equation of motion in such a fractal space(time) in terms of a geodesic equation. As we shall see, after integration this equation takes the form of a Schrödinger equation (and of the Klein-Gordon and Dirac equations in the relativistic case). This result, first obtained in Ref. [54], has later been confirmed by many subsequent physical [22,25,56,57] and mathematical works, in particular by Cresson and Ben Adda [6,7,17,19] and Jumarie [41,42,43,44], including attempts of generalizations using the tool of the fractional integro-differential calculus [7,21,44]. In what follows, we consider only the simplest case of fractal laws, namely, those characterized by a constant fractal dimension. The various generalized scale laws considered in the previous section lead to new possible generalizations of quantum mechanics [56,63]. Critical Fractal Dimension 2 Moreover, we simplify again the description by considering only the case DF D 2. Indeed, the nondifferentiability and fractality of space implies that the paths are random walks of the Markovian type, which corresponds to such a fractal dimension. This choice is also justified by Feynman’s result [30], according to which the typical paths of quantum particles (those which contribute mainly to the path integral) are nondifferentiable and of fractal dimension DF D 2 [1]. The case DF ¤ 2, which yields generalizations to standard quantum mechanics has also been studied in detail (see [56,63] and references therein). This study shows that DF D 2 plays a critical role in the theory, since it suppresses the explicit scale dependence in the motion (Schrödinger) equation – but this dependence remains hidden and reappears through, e. g., the Heisenberg

Metric of a Fractal Space-Time Now one can also write the fractal fluctuations in terms p of the coordinate differentials, d  D    dx  . The identification of this expression with that of Eq. (3) leads one to recover the Einstein-de Broglie length and time scales, x D

c „ ; D dx/ds px

D

c „ D : dt/ds E

(49)

Let us now assume that the large scale (classical) behavior is given by the Riemannian metric potentials g (x; y; z; t). The invariant proper time dS along a geodesic in terms of the complete differential elements dX  D dx  C d  dS 2 D g dX  dX  D g (dx  Cd  )(dx  Cd  ): (50) Now replacing the d’s by their expression, one obtains a fractal metric [54,68]. Its two-dimensional and diagonal expression, neglecting the terms of the zero mean (in order to simplify its writing) reads  F  2 2 c dt dS 2 Dg00 (x; t) 1 C 02 dt   x  g11 (x; t) 1 C 12 (51) dx 2 : dx We therefore obtain generalized fractal metric potentials which are divergent and explicitly dependent on the coordinate differential elements [52,54]. Another equivalent way to understand this metric consists in remarking that it is no longer only quadratic in the space-time differental elements, but that it also contains them in a linear way.

Fractals in the Quantum Theory of Spacetime

As a consequence, the curvature is also explicitly scaledependent and divergent when the scale intervals tend to zero. This property ensures the fundamentally non-Riemannian character of a fractal space-time, as well as the possibility to characterize it in an intrinsic way. Indeed, such a characterization, which is a necessary condition for defining a space in a genuine way, can be easily made by measuring the curvature at smaller and smaller scales. While the curvature vanishes by definition toward the small scales in Gauss-Riemann geometry, a fractal space can be characterized from the interior by the verification of the divergence toward small scales of curvature, and therefore of physical quantities like energy and momentum. Now the expression of this divergence is nothing but the Heisenberg relations themselves, which therefore acquire in this framework the status of a fundamental geometric test of the fractality of space-time [52,54,69]. Geodesics of a Fractal Space-Time The next step in such a geometric approach consists of the identifying the wave-particles with fractal space-time geodesics. Any measurement is interpreted as a selection of the geodesics bundle linked to the interaction with the measurement apparatus (that depends on its resolution) and/or to the information known about it (for example, the which-way-information in a two-slit experiment [56]. The three main consequences of nondifferentiability are: (i) The number of fractal geodesics is infinite. This leads one to adopt a generalized statistical fluid-like description where the velocity V  (s) is replaced by a scaledependent velocity field V  [X  (s; ds); s; ds]. (ii) There is a breaking of the reflexion invariance of the differential element ds. Indeed, in terms of fractal functions f (s; ds), two derivatives are defined, X(s C ds; ds)  X(s; ds) ; ds X(s; ds)  X(s  ds; ds) 0 (s; ds) D ; X ds

0 XC (s; ds) D

(52)

which transform into each other under the reflection (ds $ ds), and which have a priori no reason to be equal. This leads to a fundamental two-valuedness of the velocity field. (iii) The geodesics are themselves fractal curves of fractal dimension DF D 2 [30]. This means that one defines two divergent fractal velocity fields, VC [x(s; ds); s; ds] D vC [x(s); s] C wC [x(s; ds); s; ds] and V [x(s; ds); s; ds] D v [x(s); s] C w [x(s; ds); s; ds], which can be decomposed in terms of

differentiable parts vC and v , and of fractal parts wC and w . Note that, contrary to other attempts such as Nelson’s stochastic quantum mechanics which introduces forward and backward velocities [51] (and which has been later disproved [34,80]), the two velocities are here both forward, since they do not correspond to a reversal of the time coordinate, but of the time differential element now considered as an independent variable. More generally, we define two differentiable parts of derivatives dC /ds and d /ds, which, when they are applied to x  , yield the differential parts of the velocity fields,   D d x  /ds. vC D dC x  /ds and v  Covariant Total Derivative Let us first consider the non-relativistic case. It corresponds to a three-dimensional fractal space, without fractal time, in which the invariant ds is therefore identified with the time differential element dt. One describes the elementary displacements dX k , where k D 1; 2; 3, on the geodesics of a nondifferentiable fractal space in terms of the sum of the two terms (omitting the indices for simplicity) dX˙ D d˙ x C d˙ , where dx represents the differentiable part and d the fractal (nondifferentiable) part, defined as p d˙ x D v˙ dt ; d˙ D ˙ 2D dt 1/2 : (53) Here ˙ are stochastic dimensionless variables such that 2 i D 1, and D is a parameter that generh˙ i D 0 and h˙ alizes the Compton scale (namely, D D „/2m in the case of standard quantum mechanics) up to the fundamental constant c/2. The two time derivatives are then combined in terms of a complex total time derivative operator [54],     b d 1 dC d i dC d D C   : (54) dt 2 dt dt 2 dt dt Applying this operator to the differentiable part of the position vector yields a complex velocity V D

b d vC C v vC  v x(t) D V  iU D i : (55) dt 2 2

In order to find the expression for the complex time derivative operator, let us first calculate the derivative of a scalar function f . Since the fractal dimension is 2, one needs to go to the second order of expansion. For one variable it reads @f @ f dX 1 @2 f dX 2 df D C C : dt @t @X dt 2 @X 2 dt

(56)

Generalizing this process to three dimensions is straightforward.

583

584

Fractals in the Quantum Theory of Spacetime

Let us now take the stochastic mean of this expression, i. e., we take the mean on the stochastic variables ˙ which appear in the definition of the fractal fluctuation d˙ . By definition, since dX D dx C d and hdi D 0, we have hdXi D dx, so that the second term is reduced (in 3 dimensions) to v:r f . Now concerning the term dX 2 /dt, it is infinitesimal and therefore it would not be taken into account in the standard differentiable case. But in the nondifferentiable case considered here, the mean square fluctuation is non-vanishing and of order dt, namely, hd 2 i D 2Ddt, so that the last term of Eq. (56) amounts in three dimensions to a Laplacian operator. One obtains, respectively for the (+) and () processes,   @ d˙ f (57) D C v˙ :r ˙ D f : dt @t Finally, by combining these two derivatives in terms of the complex derivative of Eq. (54), it reads [54] b @ d D C V :r  i D: dt @t

b d @L @L  D0: dt @V @x

(62)

From the homogeneity of space and Noether’s theorem, one defines a generalized complex momentum given by the same form as in classical mechanics as PD

@L : @V

(63)

If the action is now considered as a function of the upper limit of integration in Eq. (61), the variation of the action from a trajectory to another nearby trajectory yields a generalization of another well-known relation of classical mechanics, P D rS :

(64)

(58) Motion Equation

Under this form, this expression is not fully covariant [74], since it involves derivatives of the second order, so that its Leibniz rule is a linear combination of the first and second order Leibniz rules. By introducing the velocity operator [61] b D V  i Dr; V

Generalized Euler–Lagrange equations that keep their standard form in terms of the new complex variables can be derived from this action [22,54], namely

(59)

As an example, consider the case of a single particle in an external scalar field with potential energy  (but the method can be applied to any situation described by a Lagrange function). The Lagrange function, L D 12 mv 2  , is generalized as L(x; V ; t) D 12 mV 2  . The Euler– Lagrange equations then keep the form of Newton’s fundamental equation of dynamics F D m dv/dt, namely,

it may be given a fully covariant expression, b d @ b :r : D CV dt @t

(60)

Under this form it satisfies the first order Leibniz rule for partial derivatives. We shall now see that b d/dt plays the role of a “covariant derivative operator” (in analogy with the covariant derivative of general relativity). Namely, one may write in its terms the equation of physics in a nondifferentiable space under a strongly covariant form identical to the differentiable case. Complex Action and Momentum The steps of construction of classical mechanics can now be followed, but in terms of complex and scale dependent quantities. One defines a Lagrange function that keeps its usual form, L(x; V ; t), but which is now complex. Then one defines a generalized complex action Z t2 SD L(x; V ; t)dt : (61) t1

m

b d V D r ; dt

(65)

which is now written in terms of complex variables and complex operators. In the case when there is no external field ( D 0), the covariance is explicit, since Eq. (65) takes the free form of the equation of inertial motion, i. e., of a geodesic equation, b d V D0: dt

(66)

This is analogous to Einstein’s general relativity, where the equivalence principle leads to the covariant equation of the motion of a free particle in the form of an inertial motion (geodesic) equation Du /ds D 0, written in terms of the general-relativistic covariant derivative D, of the four-vector u and of the proper time differential ds. The covariance induced by the effects of the nondifferentiable geometry leads to an analogous transformation of the equation of motions, which, as we show below, become after integration the Schrödinger equation, which

Fractals in the Quantum Theory of Spacetime

can therefore be considered as the integral of a geodesic equation in a fractal space. In the one-particle case the complex momentum P reads P D mV ;

(67)

so that from Eq. (64) the complex velocity V appears as a gradient, namely the gradient of the complex action V D r S/m :

(68)

Wave Function Up to now the various concepts and variables used were of a classical type (space, geodesics, velocity fields), even if they were generalized to the fractal and nondifferentiable, explicitly scale-dependent case whose essence is fundamentally not classical. We shall now make essential changes of variable, that transform this apparently classical-like tool to quantum mechanical tools (without any hidden parameter or new degree of freedom). The complex wave function is introduced as simply another expression for the complex action S, by making the transformation D e i S/S 0 :

(69)

Note that, despite its apparent form, this expression involves a phase and a modulus since S is complex. The factor S0 has the dimension of an action (i. e., an angular momentum) and must be introduced because S is dimensioned while the phase should be dimensionless. When this formalism is applied to standard quantum mechanics, S0 is nothing but the fundamental constant ¯. As a consequence, since S D i S0 ln

;

(70)

one finds that the function is related to the complex velocity appearing in Eq. (68) as follows V D i

S0 r ln m

:

(71)

This expression is the fundamental relation that connects the two description tools while giving the meaning of the wave function in the new framework. Namely, it is defined here as a velocity potential for the velocity field of the infinite family of geodesics of the fractal space. Because of nondifferentiability, the set of geodesics that defines a ‘particle’ in this framework is fundamentally non-local. It can easily be generalized to a multiple particle situation (in particular to entangled states) which are described by

a single wave function , from which the various velocity fields of the subsets of the geodesic bundle are derived as V k D i (S0 /m k ) r k ln , where k is an index for each particle. The indistinguishability of identical particles naturally follows from the fact that the ‘particles’ are identified with the geodesics themselves, i. e., with an infinite ensemble of purely geometric curves. In this description there is no longer any point-mass with ‘internal’ properties which would follow a ‘trajectory’, since the various properties of the particle – energy, momentum, mass, spin, charge (see next sections) – can be derived from the geometric properties of the geodesic fluid itself. Correspondence Principle Since we have P D i S0 r ln tain the equality [54] P

D i S0 (r )/ , we ob-

D i„r

(72)

in the standard quantum mechanical case S0 D „, which establishes a correspondence between the classical momentum p, which is the real part of the complex momentum in the classical limit, and the operator i„r. This result is generalizable to other variables, in particular to the Hamiltonian. Indeed, a strongly covariant form of the Hamiltonian can be obtained by using the fully covariant form of the covariant derivative operator given by Eq. (60). With this tool, the expression of the relation between the complex action and the complex Lagrange function reads LD

b dS @S b :r S : D CV dt dt

(73)

Since P D r S and H D @S/@t, one obtains for the generalized complex Hamilton function the same form it has in classical mechanics, namely [63,67], b :P  L : H DV

(74)

After expanding the velocity operator, one obtains H D V :P i Dr:P  L, which includes an additional term [74], whose origin is now understood as an expression of nondifferentiability and strong covariance. Schrödinger Equation and Compton Relation The next step of the construction amounts to writing the fundamental equation of dynamics Eq. (65) in terms of the function . It takes the form iS0

b d (r ln ) D r : dt

(75)

585

586

Fractals in the Quantum Theory of Spacetime

As we shall now see, this equation can be integrated in a general way in the form of a Schrödinger equation. Replacing b d/dt and V by their expressions yields

r˚ D iS0

@ r ln @t

i

D2 

S0 (r ln :r)(r ln ) m  C D(r ln ) : (76)

This equation may be simplified thanks to the identity [54],  r



 D 2(r ln :r)(r ln ) C (r ln ) : (77)

We recognize, in the right-hand side of Eq. (77), the two terms of Eq. (76), which were respectively a factor of S0 /m and D. This leads to the definition of the wave function as D e i S/2mD ;

(78)

which means that the arbitrary parameter S0 (which is identified with the constant ¯ in standard QM) is now linked to the fractal fluctuation parameter by the relation S0 D 2mD :

(79)

This relation (which can actually be proved instead of simply being set as a simplifying choice, see [62,67]) is actually a generalization of the Compton relation, since the geometric parameter D D hd 2 i/2dt can be written in terms of a length scale as D D c/2, so that, when S0 D „, it becomes  D „/mc. But a geometric meaning is now given to the Compton length (and therefore to the inertial mass of the particle) in the fractal space-time framework. The fundamental equation of dynamics now reads r D 2imD



@ r ln @t

˚  i 2D(r ln :r)(r ln )  C D(r ln ) : (80)

Using the above remarkable identity and the fact that @/@t and r commute, it becomes

 r  @  D 2Dr i ln C D : (81) m @t The full equation becomes a gradient,

r

  2D r m



and it can be easily integrated to finally obtain a generalized Schrödinger equation [54]

i @ /@t C D

 D0

(82)

C iD

@ @t



 2m

D0;

(83)

up to an arbitrary phase factor which may be set to zero by a suitable choice of the phase. One recovers the standard Schrödinger equation of quantum mechanics for the particular case when D D „/2m. Von Neumann’s and Born’s Postulates In the framework described here, “particles” are identified with the various geometric properties of fractal space(time) geodesics. In such an interpretation, a measurement (and more generally any knowledge about the system) amounts to selecting the sub-set of the geodesics family which only contains the geodesics with the geometric properties corresponding to the measurement result. Therefore, just after the measurement, the system is in the state given by the measurement result, which is precisely the von Neumann postulate of quantum mechanics. The Born postulate can also be inferred from the scalerelativity construction [22,62,67]. Indeed, the probability for the particle to be found at a given position must be proportional to the density of the geodesics fluid at this point. The velocity and the density of the fluid are expected to be solutions of a Euler and continuity system of four equations, with four unknowns, ( ; Vx ; Vy ; Vz ). Now, by separating the real and imaginary parts of the p Schrödinger equation, setting D P  e i and using a mixed representation (P; V ), where V D fVx ; Vy ; Vz g, one precisely obtains such a standard system of fluid dynamics equations, namely, p !   @ 2 P ; C V  r V D r   2D p @t P (84) @P C div(PV ) D 0 : @t This allows one to unequivocally identify P D j j2 with the probability density of the geodesics and therefore with the probability of presence of the ‘particle’. Moreover, p 2 P (85) Q D 2D p P can be interpreted as the new potential which is expected to emerge from the fractal geometry, in analogy with the identification of the gravitational field as a manifestation of the curved geometry in Einstein’s general relativity. This result is supported by numerical simulations, in which the

Fractals in the Quantum Theory of Spacetime

probability density is obtained directly from the distribution of geodesics without writing the Schrödinger equation [39,63]. Nondifferentiable Wave Function In more recent works, instead of taking only the differentiable part of the velocity field into account, one constructs the covariant derivative and the wave function in terms of the full velocity field, including its divergent nondifferentiable part of zero mean [59,62]. As we shall briefly see now, this still leads to the standard form of the Schrödinger equation. This means that, in the scale relativity framework, one expects the Schrödinger equation to have fractal and nondifferentiable solutions. This result agrees with a similar conclusion by Berry [8] and Hall [38], but it is considered here as a direct manifestation of the nondifferentiability of space itself. Consider the full complex velocity field, including its differentiable and nondifferentiable parts, 

 vC C v vC  v i 2 2   wC C w wC  w C i : 2 2

˜ DV CW D V

(86)

It is related to a full complex action S˜ and a full wave function ˜ as ˜ D V C W D r S˜/m D 2i Dr ln ˜ : V

(87)

Under the standard point of view, the complex fluctuation W is infinite and therefore r ln ˜ is undefined, so that

Eq. (87) would be meaningless. In the scale relativity approach, on the contrary, this equation keeps a mathematical and physical meaning, in terms of fractal functions, which are explicitly dependent on the scale interval dt and divergent when dt ! 0. After some calculations [59,62], one finds that the covariant derivative built from the total process (including the differentiable and nondifferentiable divergent terms) is finally b d @ D C V˜ :r  i D : dt @t

(88)

The subsequent steps of the derivation of the Schrödinger equation are unchanged (now in terms of scale-dependent fractal functions), so that one obtains D2  ˜ C i D

@˜  ˜  D0; @t 2m

(89)

where ˜ can now be a nondifferentiable and fractal function. The research of such a behavior in laboratory experiments is an interesting new challenge for quantum physics. One may finally stress the fact that this result is obtained by accounting for all the terms, differentiable (dx) and nondifferentiable (d), in the description of the elementary displacements in a nondifferentiable space, and that it does not depend at all on the probability distribution of the stochastic variables d, about which no hypothesis is needed. This means that the description is fully general, and that the effect on motion of a nondifferentiable space(-time) amounts to a fundamental indeterminism, i. e., to a total loss of information about the past path which will in all cases lead to the QM description. From Fractal Space-Time to Relativistic Quantum Mechanics All these results can be generalized to relativistic quantum mechanics, which corresponds in the scale relativity framework to a full fractal space-time. This yields, as a first step, the Klein–Gordon equation [22,55,56]. Then an account of the new two-valuedness of the velocity allows one to suggest a geometric origin for the spin and to obtain the Dirac equation [22]. Indeed, the total derivative of a physical quantity also involves partial derivatives with respect to the space variables, @/@x  . From the very definition of derivatives, the discrete symmetry under the reflection dx  $ dx  is also broken. Since, at this level of description, one should also account for parity as in the standard quantum theory, this leads to introduce a bi-quaternionic velocity field [22], in terms of which the Dirac bispinor wave function can be constructed. The successive steps that lead to the Dirac equation naturally generalize the Schrödinger case. One introduces a biquaternionic generalization of the covariant derivative that keeps the same form as in the complex case, namely, b d  D V  @ C i @ @ ; ds 2

(90)

where  D 2D/c. The biquaternionic velocity field is related to the biquaternionic (i. e., bispinorial) wave function, by V D i

S0 m

1

@

:

(91)

This is the relativistic expression of the fundamental relation between the scale relativity tools and the quantum

587

588

Fractals in the Quantum Theory of Spacetime

mechanical tools of description. It gives a geometric interpretation to the wave function, which is, in this framework, a manifestation of the geodesic fluid and of its associated fractal velocity field. The covariance principle allows us to write the equation of motion under the form of a geodesic differential equation, b d V D0: ds

(92)

After some calculations, this equation can be integrated and factorized, and one finally derives the Dirac equation [22], @ mc 1@ D ˛ k k  i ˇ c @t „ @x

:

(93)

Finally it is easy to recover the Pauli equation and Pauli spinors as nonrelativistic approximations of the Dirac equation and Dirac bispinors [23]. Gauge Fields as Manifestations of Fractal Geometry General Scale Transformations and Gauge Fields Finally, let us briefly recall the main steps of applying of the scale relativity principles to the foundation of gauge theories, in the Abelian [55,56] and non-Abelian [63,68] cases. This application is based on a general description of the internal fractal structures of the “particle” (identified with the geodesics of a nondifferentiable space-time) in terms of scale variables ˛ˇ (x; y; z; t) D %˛ˇ "˛ "ˇ whose true nature is tensorial, since it involves resolutions that may be different for the four space-time coordinates and may be correlated. This resolution tensor (similar to a covariance error matrix) generalizes the single resolution variable ". Moreover, one considers a more profound level of description in which the scale variables may now be functions of the coordinates. Namely, the internal structures of the geodesics may vary from place to place and during the time evolution, in agreement with the nonabsolute character of the scale space. This generalization amounts to the construction of a ‘general scale relativity’ theory. We assume, for simplicity of the writing, that the two tensorial indices can be gathered under one common index. We therefore write the scale variables under the simplified form ˛1 ˛2 D ˛ , ˛ D 1 to N D n(n C 1)/2, where n is the number of space-time dimensions (n D 3; N D 6 for fractal space, n D 4; N D 10 for fractal spacetime and n D 5; N D 15 in the special scale relativity case where one treats the djinn (scale-time F ) as a fifth dimension [53]).

Let us consider infinitesimal scale transformations. The transformation law on the ˛ can be written in a linear way as 0˛ D ˛ C ı˛ D (ı˛ˇ C ı ˛ˇ ) ˇ ;

(94)

where ı˛ˇ is the Kronecker symbol. Let us now assume that the ˛ ’s are functions of the standard space-time coordinates. This leads one to define a new scale-covariant derivative by writing the total variation of the resolution variables as the sum of the inertial variation, described by the covariant derivative, and of the new geometric contribution, namely, 

d˛ D D˛  ˇ ı ˛ˇ D D˛  ˇ W˛ˇ dx :

(95)

This covariant derivative is similar to that of general relativity, i. e., it amounts to the subtraction of the new geometric part in order to only keep the inertial part (for which the motion equation will therefore take a geodesical, free-like form). This is different from the case of the previous quantum-covariant derivative, which includes the effects of nondifferentiability by adding new terms in the total derivative. In this new situation we are led to introduce “gauge  field potentials” W˛ˇ that enter naturally in the geometrical framework of Eq. (95). These potentials are linked to the scale transformations as follows:  ı ˛ˇ D W˛ˇ dx :

(96)

But one should keep in mind, when using this expression, that these potentials find their origin in a covariant derivative process and are therefore not gradients. Generalized Charges After having written the transformation law of the basic variables (the ˛ ’s), one now needs to describe how various physical quantities transform under these ˛ transformations. These new transformation laws are expected to depend on the nature of the objects they transform (e. g., vectors, tensors, spinors, etc.), which implies a jump to group representations. We anticipate the existence of charges (which are fully constructed herebelow) by generalizing the relation (91) to multiplets between the velocity field and the wave function. In this case the multivalued velocity becomes a biquaternionic matrix, 

V jk D i

1  j @

k

:

(97)

The biquaternionic, and therefore non-commutative, nature of the wave function [15], which is equivalent to Dirac

Fractals in the Quantum Theory of Spacetime

bispinors, plays an essential role here. Indeed, the general structure of Yang–Mills theories and the correct construction of non-Abelian charges can be obtained thanks to this result [68]. The action also becomes a tensorial biquaternionic quantity,    dS jk D dS jk x  ; V jk ; ˛ ;

(98)

and, in the absence of a field (free particle) it is linked to the generalized velocity (and therefore to the spinor multiplet) by the relation 

@ S jk D mc V jk D i„

1  j @

k

:

(99)

Now, in the presence of a field (i. e., when the secondorder effects of the fractal geometry appearing in the right hand side of Eq. (95) are included), using the complete expression for @ ˛ , 

@ ˛ D D ˛  W˛ˇ ˇ ;

(100)

one obtains a non-Abelian relation, @ S jk D D S jk  ˇ

@S jk  W : @˛ ˛ˇ

(101)

This finally leads to the definition of a general group of scale transformations whose generators are T ˛ˇ D ˇ @˛

In this contribution, we have recalled the main steps that lead to a new foundation of quantum mechanics and of gauge fields on the principle of relativity itself (once it includes scale transformations of the reference system), and on the generalized geometry of space-time which is naturally associated with such a principle, namely, nondifferentiability and fractality. For this purpose, two covariant derivatives have been constructed, which account for the nondifferentiable and fractal geometry of space-time, and which allow one to write the equations of motion as geodesics equations. After a change of variable, these equations finally take the form of the quantum mechanical and quantum field equations. Let us conclude by listing some original features of the scale relativity approach which could lead to experimental tests of the theory and/or to new experiments in the future [63,67]: (i) nondifferentiable and fractal solutions of the Schrödinger equation; (ii) zero particle interference in a Young slit experiment; (iii) possible breaking of the Born postulate for a possible effective kinetic energy operator b T ¤ („2 /2m) [67]; (iv) underlying quantum behavior in the classical domain, at scales far larger than the de Broglie scale [67]; (v) macroscopic systems described by a Schrödinger-type mechanics based on a generalized macroscopic parameter D ¤ „/2m (see Chap. 7.2 in [54] and [24,57]); (vi) applications to cosmology [60]; (vii) applications to life sciences and other sciences [4,64,65], etc.

(102)

(where we use the compact notation @˛ D @/@˛ ), yielding the generalized charges, @S jk g˜ ˛ˇ : t D ˇ c jk @˛

Future Directions

(103)

This unified group is submitted to a unitarity condition,  must since, when it is applied to the wave functions, be conserved. Knowing that ˛; ˇ represent two indices each, this is a large group – at least SO(10) – that contains the electroweak theory [33,78,81] and the standard model U(1)  SU(2)  SU(3) and its simplest grand unified extension SU(5) [31,32] as a subset (see [53,54] for solutions in the special scale relativity framework to the problems encountered by SU(5) GUTs). As it is shown in more detail in Ref. [68], the various ingredients of Yang–Mills theories (gauge covariant derivative, gauge invariance, charges, potentials, fields, etc.) may subsequently be recovered in such a framework, but they now have a first principle and geometric scalerelativistic foundation.

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57. Nottale L (1997) Astron Astrophys 327:867 58. Nottale L (1997) In: Scale invariance and beyond, Proceedings of Les Houches school, Dubrulle B, Graner F, Sornette D (eds) EDP Sciences, Les Ullis/Springer, Berlin, New York, p 249 59. Nottale L (1999) Chaos Solit Fractals 10:459 60. Nottale L (2003) Chaos Solit Fractals 16:539 61. Nottale L (2004) American Institute of Physics Conference Proceedings 718:68 62. Nottale L (2008) Proceedings of 7th International Colloquium on Clifford Algebra and their applications, 19–29 May 2005, Toulouse, Advances in Applied Clifford Algebras (in press) 63. Nottale L (2008) The theory of scale relativity. (submitted) 64. Nottale L, Auffray C (2007) Progr Biophys Mol Bio 97:115 65. Nottale L, Chaline J, Grou P (2000) Les arbres de l’évolution: Univers, Vie, Sociétés. Hachette, Paris, 379 pp 66. Nottale L, Chaline J, Grou P (2002) In: Fractals in biology and medicine, vol 3. Proceedings of Fractal (2000) Third International Symposium, Losa G, Merlini D, Nonnenmacher T, Weibel E (eds), Birkhäuser, p 247 67. Nottale L, Célérier MN (2008) J Phys A 40:14471 68. Nottale L, Célérier MN, Lehner T (2006) J Math Phys 47:032303 69. Nottale L, Schneider J (1984) J Math Phys 25:1296 70. Novak M (ed) (1998) Fractals and beyond: Complexities in the sciences, Proceedings of the Fractal 98 conference, World Scientific 71. Ord GN (1983) J Phys A: Math Gen 16:1869 72. Ord GN (1996) Ann Phys 250:51 73. Ord GN, Galtieri JA (2002) Phys Rev Lett 1989:250403 74. Pissondes JC (1999) J Phys A: Math Gen 32:2871 75. Polchinski J (1998) String theories. Cambridge University Press, Cambridge 76. Rovelli C, Smolin L (1988) Phys Rev Lett 61:1155 77. Rovelli C, Smolin L (1995) Phys Rev D(52):5743 78. Salam A (1968) Elementary particle theory. Svartholm N (ed). Almquist & Wiksells, Stockholm 79. Sornette D (1998) Phys Rep 297:239 80. Wang MS, Liang WK (1993) Phys Rev D(48):1875 81. Weinberg S (1967) Phys Rev Lett 19:1264

Books and Reviews Georgi H (1999) Lie Algebras in particle physics. Perseus books, Reading, Massachusetts Landau L, Lifchitz E (1970) Theoretical physics, 10 volumes, Mir, Moscow Lichtenberg AJ, Lieberman MA (1983) Regular and stochastic motion. Springer, New York Lorentz HA, Einstein A, Minkowski H, Weyl H (1923) The principle of relativity. Dover, New York Mandelbrot B (1975) Les objets fractals. Flammarion, Paris Misner CW, Thorne KS, Wheeler JA (1973) Gravitation. Freeman, San Francisco Peebles J (1980) The large-scale structure of the universe. Princeton University Press, Princeton Rovelli C (2004) Quantum gravity. Cambridge Universty Press, Cambridge Weinberg S (1972) Gravitation and cosmology. Wiley, New York

Fractal Structures in Condensed Matter Physics

Fractal Structures in Condensed Matter Physics TSUNEYOSHI N AKAYAMA Toyota Physical and Chemical Research Institute, Nagakute, Japan Article Outline Glossary Definition of the Subject Introduction Determining Fractal Dimensions Polymer Chains in Solvents Aggregates and Flocs Aerogels Dynamical Properties of Fractal Structures Spectral Density of States and Spectral Dimensions Future Directions Bibliography Glossary Anomalous diffusion It is well known that the meansquare displacement hr2 (t)i of a diffusing particle on a uniform system is proportional to the time t such as hr2 (t)i t. This is called normal diffusion. Particles on fractal networks diffuse more slowly compared with the case of normal diffusion. This slow diffusion called anomalous diffusion follows the relation given by hr2 (t)i t a , where the condition 0 < a < 1 always holds. Brownian motion Einstein published the important paper in 1905 opening the way to investigate the movement of small particles suspended in a stationary liquid, the so-called Brownian motion, which stimulated J. Perrin in 1909 to pursue his experimental work confirming the atomic nature of matter. The trail of a random walker provides an instructive example for understanding the meaning of random fractal structures. Fractons Fractons, excitations on fractal elastic-networks, were named by S. Alexander and R. Orbach in 1982. Fractons manifest not only static properties of fractal structures but also their dynamic properties. These modes show unique characteristics such as strongly localized nature with the localization length of the order of wavelength. Spectral density of states The spectral density of states of ordinary elastic networks are expressed by the Debye spectral density of states given by D(!) ! d1 , where d is the Euclidean dimensionality. The spec-

tral density of states of fractal networks is given by D(!) ! d s 1 , where ds is called the spectral or fracton dimension of the system. Spectral dimension This exponent characterizes the spectral density of states for vibrational modes excited on fractal networks. The spectral dimension constitutes the dynamic exponent of fractal networks together with the conductivity exponent and the exponent of anomalous diffusion. Definition of the Subject The idea of fractals is based on self-similarity, which is a symmetry property of a system characterized by invariance under an isotropic scale-transformation on certain length scales. The term scale-invariance has the implication that objects look the same on different scales of observations. While the underlying concept of fractals is quite simple, the concept is used for an extremely broad range of topics, providing a simple description of highly complex structures found in nature. The term fractal was first introduced by Benoit B. Mandelbrot in 1975, who gave a definition on fractals in a simple manner “A fractal is a shape made of parts similar to the whole in some way”. Thus far, the concept of fractals has been extensively used to understand the behaviors of many complex systems or has been applied from physics, chemistry, and biology for applied sciences and technological purposes. Examples of fractal structures in condensed matter physics are numerous such as polymers, colloidal aggregations, porous media, rough surfaces, crystal growth, spin configurations of diluted magnets, and others. The critical phenomena of phase transitions are another example where self-similarity plays a crucial role. Several books have been published on fractals and reviews concerned with special topics on fractals have appeared. Length, area, and volume are special cases of ordinary Euclidean measures. For example, length is the measure of a one-dimensional (1d) object, area the measure of a two-dimensional (2d) object, and volume the measure of a three-dimensional (3d) object. Let us employ a physical quantity (observable) as the measure to define dimensions for Euclidean systems, for example, a total mass M(r) of a fractal object of the size r. For this, the following relation should hold r / M(r)1/d ;

(1)

where d is the Euclidean dimensionality. Note that Euclidean spaces are the simplest scale-invariant systems. We extend this idea to introduce dimensions for selfsimilar fractal structures. Consider a set of particles with

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unit mass m randomly distributed on a d-dimensional Euclidean space called the embedding space of the system. Draw a sphere of radius r and denote the total mass of particles included in the sphere by M(r). Provided that the following relation holds in the meaning of statistical average such as r / hM(r)i1/D f ;

(2)

where h: : :i denotes the ensemble-average over different spheres of radius r, we call Df the similarity dimension. It is necessary, of course, that Df is smaller than the embedding Euclidean dimension d. The definition of dimension as a statistical quantity is quite useful to specify the characteristic of a self-similar object if we could choose a suitable measure. There are many definitions to allocate dimensions. Sometimes these take the same value as each other and sometimes not. The capacity dimension is based on the coverage procedure. As an example, the length of a curved line L is given by the product of the number N of straightline segment of length r needed to step along the curve from one end to the other such as L(r) D N(r)r. While, the area S(r) or the volume V(r) of arbitrary objects can be measured by covering it with squares or cubes of linear size r. The identical relation, M(r) / N(r)r d

(3)

should hold for the total mass M(r) as measure, for example. If this relation does not change as r ! 0, we have the relation N(r) / rd . We can extend the idea to define the dimensions of fractal structures such as N(r) / rD f ;

(4)

from which the capacity dimension Df is given by Df :D lim

r!0

ln N(r) : ln(1/r)

(5)

The definition of Df can be rendered in the following implicit form lim N(r)r D f D const :

r!0

(6)

Equation (5) brings out a key property of the Hausdorff dimension [10], where the product N(r)r D f remains finite as r ! 0. If Df is altered even by an infinitesimal amount, this product will diverge either to zero or to infinity. The Hausdorff dimension coincides with the capacity dimension for many fractal structures, although the Hausdorff dimension is defined less than or equal to the capacity dimension. Hereafter, we refer to the capacity dimension or the Hausdorff dimension mentioned above as the fractal dimension. Introduction Fractal structures are classified into two categories; deterministic fractals and random fractals. In condensed matter physics, we encounter many examples of random fractals. The most important characteristic of random fractals is the spatial and/or sample-to-sample fluctuations in their properties. We must discuss their characteristics by averaging over a large ensemble. The nature of deterministic fractals can be easily understood from some examples. An instructive example is the Mandelbrot–Given fractal [12], which can be constructed by starting with a structure with eight line segments as shown in Fig. 1a (the first stage of the Mandelbrot–Given fractal). In the second stage, each line segment of the initial structure is replaced by the initial structure itself (Fig. 1b). This process is repeated indef-

Fractal Structures in Condensed Matter Physics, Figure 1 Mandelbrot–Given fractal. a The initial structure with eight line segments, b the object obtained by replacing each line segment of the initial structure by the initial structure itself (the second stage), and c the third stage of the Mandelbrot–Given fractal obtained by replacing each line segment of the second-sage structure by the initial structure

Fractal Structures in Condensed Matter Physics

Fractal Structures in Condensed Matter Physics, Figure 2 a Homogeneous random structure in which particles are randomly but homogeneously distributed, and b the distribution functions of local densities , where (l) is the average mass density independent of l

initely. The Mandelbrot–Given fractal possesses an obvious dilatational symmetry, as seen from Fig. 1c, i. e., when we magnify a part of the structure, the enlarged portion looks just like the original one. Let us apply (5) to determine Df of the Mandelbrot–Given fractal. The Mandelbrot–Given fractal is composed of 8 parts of size 1/3, hence, N(1/3) D 8, N((1/3)2 ) D 82 , and so on. We thus have a relation of the form N(r) / r ln3 8 , which gives the fractal dimension Df D ln3 8 D 1:89278 : : :. The Mandelbrot–Given fractal has many analogous features with percolation networks (see Sect. “Dynamical Properties of Fractal Structures”), a typical random fractal, such that the fractal dimension of a 2d percolation network is Df D 91/48 D 1:895833: : :, which is very close to that of the Mandelbrot–Given fractal. The geometric characteristics of random fractals can be understood by considering two extreme cases of random structures. Figure 2a represents the case in which particles are randomly but homogeneously distributed in a d-dimensional box of size L, where d represents ordinary Euclidean dimensionality of the embedding space. If we divide this box into smaller boxes of size l, the mass density of the ith box is

i (l) D

M i (l) ; ld

(7)

where M i (l) represents the total mass (measure) inside box i. Since this quantity depends on the box i, we plot the distribution function P( ), from which curves like those in Fig. 2b may be obtained for two box sizes l1 and (l2 < l2 ). We see that the central peak position of the distribution function P( ) is the same for each case. This means that the average mass density yields

(l) D

hM i (l)i i ld

becomes constant, indicating that hM i (l)i i / l d . The above is equivalent to

D

m ad

;

(8)

where a is the average distance (characteristic lengthscale) between particles and the mass of a single particle. This indicates that there exists a single length scale a characterizing the random system given in Fig. 2a. The other type of random structure is shown in Fig. 3a, where particle positions are correlated with each other and i (l) greatly fluctuates from box to box, as shown in Fig. 3b. The relation hM i (l)i i / l d may not hold at all for this type of structure. Assuming the fractality for this system, namely, if the power law hM i (l)i i / lfD holds, the average mass density becomes ¯ D

(l)

hM i (l)i i / l D f d ; ld

(9)

¯ dewhere i (l) D 0 is excluded. In the case Df < d, (l) pends on l and decreases with increasing l. Thus, there is no characteristic length scale for the type of random structure shown in Fig. 3a. If (9) holds with Df < d, so that hM i (l)i i is proportional to lfD , the structure is said to be fractal. It is important to note that there is no characteristic length scale for the type of random fractal structure shown in Fig. 3b. Thus, we can extend the idea of self-similarity not only for deterministic self-similar structures, but also for random and disordered structures, the so-called random fractals, in the meaning of statistical average. The percolation network made by putting particles or bonds on a lattice with the probability p is a typical example of random fractals. The theory of percolation was initiated in 1957 by S.R. Broadbent and J.M. Hammersley [5] in connection with the diffusion of gases through porous

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Fractal Structures in Condensed Matter Physics, Figure 3 a Correlated random fractal structure in which particles are randomly distributed, but correlated with each other, and b the distribution functions of local densities with finite values, where the average mass densities depend on l

media. Since their work, it has been widely accepted that the percolation theory describes a large number of physical and chemical phenomena such as gelation processes, transport in amorphous materials, hopping conduction in doped semiconductors, the quantum Hall effect, and many other applications. In addition, it forms the basis for studies of the flow of liquids or gases through porous media. Percolating networks thus serve as a model which helps us to understand physical properties of complex fractal structures. For both deterministic and random fractals, it is remarkable that no characteristic length scale exists, and this is a key feature of fractal structures. In other words, fractals are defined to be objects invariant under isotropic scale transformations, i. e., uniform dilatation of the system in every spatial direction. In contrast, there exist systems which are invariant under anisotropic transformations. These are called self-affine fractals. Determining Fractal Dimensions There are several methods to determine fractal dimensions Df of complex structures encountered in condensed matter physics. The following methods for obtaining the fractal dimension Df are known to be quite efficient. Coverage Method The idea of coverage in the definition of the capacity dimension (see (5)) can be applied to obtain the fractal dimension Df of material surfaces. An example is the fractality of rough surfaces or inner surfaces of porous media. The fractal nature is probed by changing the sizes of adsorbed molecules on solid surfaces. Power laws are verified by plotting the total number of adsorbed molecules versus

their size r. The area of a surface can be estimated with the aid of molecules weakly adsorbed by van der Waals forces. Gas molecules are adsorbed on empty sites until the surface is uniformly covered with a layer one molecule thick. Provided that the radius r of one adsorbed molecule and the number of adsorbed molecules N(r) are known, the surface area S obtained by molecules is given by S(r) / N(r)r2 :

(10)

If the surface of the adsorbate is perfectly smooth, we expect the measured area to be independent of the radius r of the probe molecules, which indicates the power law N(r) / r2 :

(11)

However, if the surface of the adsorbate is rough or contains pores that are small compared with r, less of the surface area S is accessible with increasing size r. For a fractal surface with fractal dimension Df , (11) gives the relation N(r) / rD f :

(12)

Box-Counting Method Consider as an example a set of particles distributed in a space. First, we divide the space into small boxes of size r and count the number of boxes containing more than one particle, which we denote by N(r). From the definition of the capacity dimension (4), the number of particle N(r) / rD f :

(13)

For homogeneous objects distributed in a d-dimensional space, the number of boxes of size r becomes, of course N(r) / rd :

Fractal Structures in Condensed Matter Physics

Fractal Structures in Condensed Matter Physics, Figure 4 a 2d site-percolation network and circles with different radii. b The power law relation holds between r and the number of particles in the sphere of radius r, indicating the fractal dimension of the 2d network is Df D 1:89 : : : D 91/48

Correlation Function The fractal dimension Df can be obtained via the correlation function, which is the fundamental statistical quantity observed by means of X-ray, light, and neutron scattering experiments. These techniques are available to bulk materials (not surface), and is widely used in condensed matter physics. Let (r) be the number density of atoms at position r. The density-density correlation function G(r; r 0 ) is defined by G(r; r 0 ) D h (r) (r 0 )i ;

(14)

where h: : :i denotes an ensemble average. This gives the correlation of the number-density fluctuation. Provided that the distribution is isotropic, the correlation function becomes a function of only one variable, the radial distance r D jr  r 0 j, which is defined in spherical coordinates. Because of the translational invariance of the system on average, r 0 can be fixed at the coordinate origin r 0 D 0. We can write the correlation function as G(r) D h (r) (0)i :

(15)

The quantity h (r) (0)i is proportional to the probability that a particle exists at a distance r from another particle. This probability is proportional to the particle density (r) within a sphere of radius r. Since (r) / r D f d for a fractal distribution, the correlation function becomes G(r) / r D f d ;

(16)

where Df and d are the fractal and the embedded Euclidean dimensions, respectively. This relation is often used directly to determine Df for random fractal structures. The scattering intensity in an actual experiment is proportional to the structure factor S(q), which is the Fourier transform of the correlation function G(r). The structure factor is calculated from (16) as Z 1 S(q) D G(r)e i qr dr / qD f (17) V V where V is the volume of the system. Here dr is the d-dimensional volume element. Using this relation, we can determine the fractal dimension Df from the data obtained by scattering experiments. When applying these methods to obtain the fractal dimension Df , we need to take care over the following point. Any fractal structures found in nature must have upper and lower length-limits for their fractality. There usually exists a crossover from homogeneous to fractal. Fractal properties should be observed only between these limits. We describe in the succeeding Sections several examples of fractal structures encountered in condensed matter physics. Polymer Chains in Solvents Since the concept of fractal was coined by B.B. Mandelbrot in 1975, scientists reinterpreted random complex structures found in condensed matter physics in terms of fractals. They found that a lot of objects are classified as fractal

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structures. We show at first from polymer physics an instructive example exhibiting the fractal structure. That is an early work by P.J. Flory in 1949 on the relationship between the mean-square end-to-end distance of a polymer chain hr2 i and the degree of polymerization N. Consider a dilute solution of separate coils in a solvent, where the total length of a flexible polymer chain with a monomer length a is Na. The simplest idealization views the polymer chain in analogy with a Brownian motion of a random walker. The walk is made by a succession of N steps from the origin r D 0 to the end point r. According to the central limit theorem of the probability theory, the probability to find a walker at r after N steps (N  1) follows the diffusion equation and we have the expression for the probability to find a particle after N steps at r PN (r) D (2 Na 2/3)3/2 exp(3r2 /2Na2) ;

(18)

where the prefactor arises from the normalization of PN (r). The mean squared distance calculated from PN (r) becomes Z hr2 i D r2 PN (r)d3 r D Na2 : (19) Then, the mean-average end-to-end distance of a polymer chain yields R D hr2 i1/2 D N 1/2 a. Since the number of polymerization N corresponds to the total mass M of a polymer chain, the use of (19) leads to the relation such as M(R) R2 . The mass M(R) can be considered as a measure of a polymer chain, the fractal dimension of this ideal chain as well as the trace of Brown motion becomes Df D 2 for any d-dimensional embedding space. The entropy of the idealized chain of the length L D Na is obtained from (18) as S(r) D S(0) 

3r2 ; 2R2

(20)

from which the free energy Fel D U  T S is obtained as Fel (r) D Fel(0) C

3k B Tr2 : 2R2

(21)

Here U is assumed to be independent of distinct configurations of polymer chains. This is an elastic energy of an ideal chain due to entropy where Fel decreases as N ! large. P.J. Flory added the repulsive energy term due to monomer-monomer interactions, the so-called excluded volume effect. This has an analogy with self-avoiding random walk. The contribution to the free energy is obtained by the virial expansion into the power series on the concentration cint D N/r d . According to the mean field

2 , we have the total theory on the repulsive term Fint / cint free-energy F such as

F v(T)N 2 3r2 C ; D kB T 2Na2 rd

(22)

where v(T) is the excluded volume parameter. We can obtain a minimum of F(r) at r D R by differentiating F(r) with respect to r such that M(R) / R

dC2 3

:

(23)

Here the number of polymerization N corresponds to the total mass M(R) of a polymer chain. Thus, we have the fractal dimension Df D (d C 2)/3, in particular, Df D 5/3 D 1:666 : : : for a polymer chain in a solvent. Aggregates and Flocs The structures of a wide variety of flocculated colloids in suspension (called aggregates or flocs) can be described in terms of fractals. A colloidal suspension is a fluid containing small charged particles that are kept apart by Coulomb repulsion and kept afloat by Brownian motion. A change in the particle-particle interaction can be induced by varying the chemical composition of the solution and in this manner an aggregation process can be initiated. Aggregation processes are classified into two simple types: diffusion-limited aggregation (DLA) and diffusion-limited cluster-cluster aggregation (DLCA), where a DLA is due to the cluster-particle coalescence and a DLCA to the cluster-cluster flocculation. In most cases, actual aggregates involve a complex interplay between a variety of flocculation processes. The pioneering work was done by M.V. Smoluchowski in 1906, who formulated a kinetic theory for the irreversible aggregation of particles into clusters and further clusters combining with clusters. The inclusion of cluster-cluster aggregation makes this process distinct from the DLA process due to particle-cluster interaction. There are two distinct limiting regimes of the irreversible colloidal aggregation process: the diffusion-limited CCA (DLCA) in dilute solutions and the reaction-limited CCA (RLCA) in dense solutions. The DLCA is due to the fast process determined by the time for the clusters to encounter each other by diffusion, and the RLCA is due to the slow process since the cluster-cluster repulsion has to dominate thermal activation. Much of our understanding on the mechanism forming aggregates or flocs has been mainly due to computer simulations. The first simulation was carried out by Vold in 1963 [23], who used the ballistic aggregation model and found that the number of particles N(r) within a distance r measured from the first seed particle is given by

Fractal Structures in Condensed Matter Physics

N(r) r2:3 . Though this relation surely exhibits the scaling form of (2), the applicability of this model for real systems was doubted in later years. The researches on fractal aggregates has been developed from a simulation model on DLA introduced by T.A. Witten and L.M. Sander in 1981 [26] and on the DLCA model proposed by P. Meakin in 1983 [14] and M. Kolb et al. in 1983 [11], independently. The DLA has been used to describe diverse phenomena forming fractal patterns such as electro-depositions, surface corrosions and dielectric breakdowns. In the simplest version of the DLA model for irreversible colloidal aggregation, a particle is located at an initial site r D 0 as a seed for cluster formation. Another particle starts a random walk from a randomly chosen site in the spherical shell of radius r with width dr( r) and center r D 0. As a first step, a random walk is continued until the particle contacts the seed. The cluster composed of two particles is then formed. Note that the finite-size of particles is the very reason of dendrite structures of DLA. This procedure is repeated many times, in each of which the radius r of the starting spherical shell should be much larger than the gyration radius of the cluster. If the number of particles contained in the DLA cluster is huge (typically 104 108 ), the cluster generated by this process is highly branched, and forms fractal structures in the meaning of statistical average. The fractality arises from the fact that the faster growing parts of the cluster shield the other parts, which therefore become less accessible to incoming particles. An arriving random walker is far more likely to attach to one of the tips of the cluster. Thus, the essence of the fractal-pattern formation arises surely from nonlinear process. Figure 5 illustrates a simulated result for a 2d DLA cluster obtained by the procedure mentioned above. The number of particles N inside a sphere of radius L ( the gyration radius of the cluster) follows the scaling law given by N / L Df ;

(24)

where the fractal dimension takes a value of Df  1:71 for the 2d DLA cluster and Df  2:5 for the 3d DLA cluster without an underlying lattice. Note that these fractal dimensions are sensitive to the embedding lattice structure. The reason for this open structure is that a wandering molecule will settle preferentially near one of the tips of the fractal, rather than inside a cluster. Thus, different sites have different growth probabilities, which are high near the tips and decrease with increasing depth inside a cluster. One of the most extensively studied DLA processes is the growth of metallic forms by electrochemical deposition. The scaling properties of electrodeposited metals were pointed out by R.M. Brady and R.C. Ball in 1984 for

Fractal Structures in Condensed Matter Physics, Figure 5 Simulated results of a 2d diffusion-limited aggregation (DLA). The number of particles contained in this DLA cluster is 104

copper electrodepositions. The confirmation of the fractality for zinc metal leaves was made by M. Matsushita et al. in 1984. In their experiments [13], zinc metal leaves are grown two-dimensionally by electrodeposition. The structures clearly recover the pattern obtained by computer simulations for the DLA model proposed by T.A. Witten and L.M. Sander in 1981. Figure 6 shows a typical zinc dendrite that was deposited on the cathode in one of these experiments. The fractal dimensionality Df D 1:66 ˙ 0:33 was obtained by computing the density-density correlation function G(r) for patterns grown at applied voltages of less than 8V. The fractality of uniformly sized gold-colloid aggregates according to the DLCA was experimentally demonstrated by D.A. Weitz in 1984 [25]. They used transmission-electron micrographs to determine the fractal dimension of this systems to be Df D 1:75. They also performed quasi-elastic light-scattering experiments to investigate the dynamic characteristics of DLCA of aqueous gold colloids. They confirmed the scaling behaviors for the dependence of the mean cluster size on both time and initial concentration. These works were performed consciously to examine the fractality of aggregates. There had been earlier works exhibiting the mass-size scaling relationship for actual aggregates. J.M. Beeckmans [2] pointed out in 1963 the power law behaviors by analyzing the data for aerosol and

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Fractal Structures in Condensed Matter Physics, Figure 6 The fractal structures of zinc metal leaves grown by electrodeposition. Photographs a–d were taken 3,5,9, and 15 min after initiating the electrolysis, respectively. After [13]

precipitated smokes in the literature (1922–1961). He used in his paper the term “aggregates-within-aggregates”, implying the fractality of aggregates. However, the data available at that stage were not adequate and scattered. Therefore, this work did not provide decisive results on the fractal dimensions of aggregates. There were smarter experiments by N. Tambo and Y. Watanabe in 1967 [20], which precisely determined fractal dimensions of flocs formed in an aqueous solution. These were performed without being aware of the concept of fractals. Original works were published in Japanese. The English versions of these works were published in 1979 [21]. We discuss these works below. Flocs generated in aqueous solutions have been the subject of numerous studies ranging from basic to applied sciences. In particular, the settling process of flocs formed in water incorporating kaolin colloids is relevant to water and wastewater treatment. The papers by N. Tambo and Y. Watanabe pioneered the discussion on the socalled fractal approach to floc structures; they performed their own settling experiments to clarifying the size dependences of mass densities for clay-aluminum flocs by using Stokes’ law ur /  (r)r2 where  is the difference between the densities of water and flocs taking so-small values  0:01–0:001 g/cm3 . Thus, the settling velocities ur are very slow of the order of 0:001 m/sec for flocs of sizes r 0:1 mm, which enabled them to perform precise

measurements. Since flocs are very fragile aggregates, they made the settling experiments with special cautions on convection and turbulence, and by careful and intensive experiments of flocculation conditions. They confirmed from thousands of pieces of data the scaling relationship between settling velocities ur and sizes of aggregates such as ur / r b . From the analysis of these data, they found the scaling relation between effective mass densities and sizes of flocs such as  (r) / rc , where the exponents c were found to take values from 1.25 to 1.00 depending on the aluminum-ion concentration, showing that the fractal dimensions become Df D 1:75 to 2.00 with increasing aluminum-ion concentration. This is because the repulsive force between charged clay-particles is screened, and van der Waals attractive force dominates between the pair of particles. It is remarkable that these fractal dimensions Df show excellent agreement with those determined for actual DLCA and RLCA clusters in the 1980s by using various experimental and computer simulation methods. Thus, they had found that the size dependences of mass densities of flocs are controlled by the aluminum-ion concentration dosed/suspended particle concentration, which they named the ALT ratio. These correspond to the transition from DLCA (established now taking the value of Df  1:78 from computer simulations) process to the RLCA one (established at present from computer simulations as Df  2:11). The ALT ratio has since the publica-

Fractal Structures in Condensed Matter Physics

of the network is preserved and decimeter-size monolithic blocks with a range of densities from 50 to 500 kg/m3 can be obtained. As a consequence, aerogels exhibit unusual physical properties, making them suitable for a number of practical applications, such as Cerenkov radiation detectors, supports for catalysis, or thermal insulators. Silica aerogels possess two different length scales. One is the radius r of primary particles. The other length is the correlation length of the gel. At intermediate length scales, lying between these two length scales, the clusters possess a fractal structure and at larger length scales the gel is a homogeneous porous glass. Aerogels have a very low thermal conductivity, solid-like elasticity, and very large internal surfaces. In elastic neutron scattering experiments, the scattering differential cross-section measures the Fourier components of spatial fluctuations in the mass density. For aerogels, the differential cross-section is the product of three factors, and is expressed by d D Af 2 (q)S(q)C(q) C B : d˝

Fractal Structures in Condensed Matter Physics, Figure 7 Observed scaling relations between floc densities and their diameters where aluminum chloride is used as coagulants. After [21]

tion of the paper been used in practice as a criterion for the coagulation to produce flocs with better settling properties and less sludge volume. We show their experimental data in Fig. 7, which demonstrate clearly that flocs (aggregates) are fractal. Aerogels Silica aerogels are extremely light materials with porosities as high as 98% and take fractal structures. The initial step in the preparation is the hydrolysis of an alkoxysilane Si(OR)4 , where R is CH3 or C2 H5 . The hydrolysis produces silicon hydroxide Si(OH)4 groups which polycondense into siloxane bonds –Si–O–Si–, and small particles start to grow in the solution. These particles bind to each other by diffusion-limited cluster-cluster aggregation (DLCA) (see Sect. “Aggregates and Flocs”) until eventually they produce a disordered network filling the reaction volume. After suitable aging, if the solvent is extracted above the critical point, the open porous structure

(25)

Here A is a coefficient proportional to the particle concentration and f (q) is the primary-particle form factor. The structure factor S(q) describes the correlation between particles in a cluster and C(q) accounts for cluster-cluster correlations. The incoherent background is expressed by B. The structure factor S(q) is proportional to the spatial Fourier transform of the density-density correlation function defined by (16), and is given by (17). Since the structure of the aerogel is fractal up to the correlation length  of the system and homogeneous for larger scales, the correlation function G(r) is expressed by (25) for r   and G(r) D Const. for r  . Corresponding to this, the structure factor S(q) is given by (17) for q  1, while S(q) is independent of q for q  1. The wavenumber regime for which S(q) becomes a constant is called the Guinier regime. The value of Df can be deduced from the slope of the observed intensity versus momentum transfer (q  1) in a double logarithmic plot. For very large q, there exists a regime called the Porod regime in which the scattering intensity is proportional to q4 . The results in Fig. 8 by R. Vacher et al. [22] are from small-angle neutron scattering experiments on silica aerogels. The various curves are labeled by the macroscopic density of the corresponding sample in Fig. 8. For example, 95 refers to a neutrally reacted sample with D 95 kg/m3 . Solid lines represent best fits. They are presented even in the particle regime q > 0:15 Å-1 to emphasize that the fits do not apply in the region, particularly for the denser samples. Remarkably, Df is independent of sample

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and the mean-square displacement follows the power law hr2 (t)i / t 2/d w ;

(26)

where dw is termed the exponent of anomalous diffusion. The exponents is expressed as dw D 2 C with a positive > 0 (see (31)), implying that the diffusion becomes slower compared with the case of normal diffusion. This is because the inequality 2/dw < 1 always holds. This slow diffusions on fractal supports are called anomalous diffusion. The scaling relation between the length-scale and the time-scale can be easily extended to the problem of atomic vibrations of elastic fractal-networks. This is because various types of equations governing dynamics can be mapped onto the diffusion equation. This implies that both equations are governed by the same eigenvalue problem, namely, the replacement of eigenvalues ! ! ! 2 between the diffusion equation and the equation of atomic vibrations is justified. Thus, the basic properties of vibrations of fractal networks, such as the density of states, the dispersion relation and the localization/delocalization property, can be derived from the same arguments for diffusion on fractal networks. The dispersion relation between the frequency ! and the wavelength (!) is obtained from (26) by using the reciprocal relation t ! ! 2 (here the diffusion problem is mapped onto the vibrational one) and hr2 (t)i ! (!)2 . Thus we obtain the dispersion relation for vibrational excitations on fractal networks such as ! / (!)d w /2 : Fractal Structures in Condensed Matter Physics, Figure 8 Scattered intensities for eight neutrally reacted samples. Curves are labeled with in kg/m3 . After [22]

density to within experimental accuracy: Df D 2:40 ˙ 0:03 for samples 95 to 360. The departure of S(q) from the qD f dependence at large q indicates the presence of particles with gyration radii of a few Å. Dynamical Properties of Fractal Structures The dynamics of fractal objects is deeply related to the time-scale problems such as diffusion, vibration and transport on fractal support. For the diffusion of a particle on any d-dimensional ordinary Euclidean space, it is well known that the mean-square displacement hr2 (t)i is proportional to the time such as hr2 (t)i / t for any Euclidean dimension d (see also (19)). This is called normal diffusion. While, on fractal supports, a particle more slowly diffuses,

(27)

If dw D 2, we have the ordinary dispersion relation ! / (!) for elastic waves excited on homogeneous systems. Consider the diffusion of a random walker on a percolating fractal network. How does hr2 (t)i behave in the case of fractal percolating networks? For this, P.G. de Gennes in 1976 [7] posed the problem called an ant in the labyrinth. Y. Gefen et al. in 1983 [9] gave a fundamental description of this problem in terms of a scaling argument. D. BenAvraham and S. Havlin in 1982 [3] investigated this problem in terms of Monte Carlo simulations. The work by Y. Gefen [9] triggered further developments in the dynamics of fractal systems, where the spectral (or fracton) dimension ds is a key dimension for describing the dynamics of fractal networks, in addition to the fractal dimension Df . The fractal dimension Df characterizes how the geometrical distribution of a static structure depends on its length scale, whereas the spectral dimension ds plays a central role in characterizing dynamic quantities on fractal networks. These dynamical properties are described in a unified way by introducing a new dynamic exponent called

Fractal Structures in Condensed Matter Physics

the spectral or fracton dimension defined by ds D

2Df : dw

(28)

The term fracton, coined by S. Alexander and R. Orbach in 1982 [1], denotes vibrational modes peculiar to fractal structures. The characteristics of fracton modes cover a rich variety of physical implications. These modes are strongly localized in space and their localization length is of the order of their wavelengths. We give below the explicit form of the exponent of anomalous diffusion dw by illustrating percolation fractal networks. The mean-square displacement hr2 (t)i after a sufficiently long time t should follow the anomalous diffusion described by (26). For a finite network with a size , the mean-square distance at sufficiently large time becomes hr2 (t)i   2 , so we have the diffusion coefficient for anomalous diffusion from (26) such as D /  2d w :

(29)

For percolating networks, the diffusion constant D in the vicinity of the critical percolation density pc behaves D / (p  pc ) tˇ /  (tˇ )/ ;

(30)

where t is called the conductivity exponent defined by dc (p  pc ) t , ˇ the exponent for the percolation order parameter defined by S(p) / (p  pc )ˇ , and  the exponent for the correlation length defined by  / jp  pc j , respectively. Comparing (29) and (30), we have the relation between exponents such as dw D 2 C

tˇ D2C : 

(31)

Due to the condition t > ˇ, and hence > 0, implying that the diffusion becomes slow compared with the case of normal diffusion. This slow diffusion is called anomalous diffusion. Spectral Density of States and Spectral Dimensions The spectral density of states of atomic vibrations is the most fundamental quantity describing the dynamic properties of homogeneous or fractal systems such as specific heats, heat transport, scattering of waves and others. The simplest derivation of the spectral density of states (abbreviated, SDOS) of a homogeneous elastic system is given below. The density of states at ! is defined as the number of modes per particle, which is expressed by D(!) D

1 ; !Ld

(32)

where ! is the frequency interval between adjacent eigenfrequencies close to ! and L is the linear size of the system. In the lowest frequency region, ! is the lowest eigenfrequency which depends on the size L. The relation between the frequency ! and L is obtained from the well-known linear dispersion relationship ! D ( k, where ( is the velocity of phonons (quantized elastic waves) such that ! D

2( 1 / :  L

(33)

The substitution of (33) into (32) yields D(!) / ! d1 :

(34)

Since this relation holds for any length scale L due to the scale-invariance property of homogeneous systems, we can replace the frequency ! by an arbitrary !. Therefore, we obtain the conventional Debye density of states as D(!) / ! d1 :

(35)

It should be noted that this derivation is based on the scale invariance of the system, suggesting that we can derive the SDOS for fractal networks in the same line with this treatment. Consider the SDOS of a fractal structure of size L with fractal dimension Df . The density of states per particle at the lowest frequency ! for this system is, as in the case of (32), written as D(!) /

1 : LfD !

(36)

Assuming that the dispersion relation for ! corresponding to (33) is ! / Lz ;

(37)

we can eliminate L from (36) and obtain D(!) / ! D f /z1 :

(38)

The exponent z of the dispersion relation (37) is evaluated from the exponent of anomalous diffusion dw . Considering the mapping correspondence between diffusion and atomic vibrations, we can replace hr2 (t)i and t by L2 and 1/! 2 , respectively. Equation (26) can then be read as L / ! 2/d w :

(39)

The comparison of (28),(37) and (39) leads to zD

dw Df : D 2 ds

(40)

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Fractal Structures in Condensed Matter Physics, Figure 9 a Spectral densities of states (SDOS) per atom for 2d, 3d, and 4d BP networks at p D pc . The angular frequency ! is defined with mass units m D 1 and force constant Kij D 1. The networks are formed on 1100  1100 (2d), 100  100  100 (3d), and 30  30  30  30 (4d) lattices with periodic boundary conditions, respectively. b Integrated densities of states for the same

Since the system has a scale-invariant fractal (self-similar) structure !, can be replaced by an arbitrary frequency !. Hence, from (38) and (40) the SDOS for fractal networks is found to be D(!) / ! d s 1 ;

(41)

and the dispersion relation (39) becomes ! / L(!)D f /d s :

(42)

For percolating networks, the spectral dimension is obtained from (40) ds D

2Df 2( Df D : 2C 2( C   ˇ

(43)

This exponent ds is called the fracton dimension after S. Alexander and R. Orbach [1] or the spectral dimension after R. Rammal and G. Toulouse [17], hereafter we use the term spectral dimension for ds . S. Alexander and R. Orbach [1] estimated the values of ds for percolating networks on d-dimensional Euclidean lattices from the known values of the exponents Df ; (;  and ˇ. They pointed out that, while these exponents depend largely on d, the spectral dimension (fracton) dimension ds does not. The spectral dimension ds can be obtained from the value of the conductivity exponent t or vice versa. In the case of percolating networks, the conductivity exponent t is related to ds through (43), which means that the con-

ductivity dc (p  pc ) t is also characterized by the spectral dimension ds . In this sense, the spectral dimension ds is an intrinsic exponent related to the dynamics of fractal systems. We can determine the precise values of ds from the numerical calculations of the spectral density of states of percolation fractal networks. The fracton SDOS for 2d, 3d, and 4d bond percolation networks at the percolation threshold p D pc are given in Fig. 9a and b, which were calculated by K. Yakubo and T. Nakayama in 1989. These were obtained by large-scale computer simulations [27]. At p D pc , the correlation length diverges as  / jp  pc j and the network has a fractal structure at any length scale. Therefore, fracton SDOS should be recovered in the wide frequency range !L  !  !D , where !D is the Debye cutoff frequency and !L is the lower cutoff determined by the system size. The SDOSs and the integrated SDOSs per atom are shown by the filled squares for a 2d bond percolation (abbreviated, BP) network at pc D 0:5. The lowest frequency !L is quite small (! 105 for the 2d systems) as seen from the results in Fig. 9 because of the large sizes of the systems. The spectral dimension ds is obtained as ds D 1:33 ˙ 0:11 from Fig. 9a, whereas data in Fig. 9b give the more precise value ds D 1:325 ˙ 0:002. The SDOS and the integrated SDOS for 3d BP networks at pc D 0:249 are given in Fig. 9a and b by the filled triangles (middle). The spectral dimension ds is obtained as ds D 1:31 ˙ 0:02 from Fig. 9a and ds D 1:317 ˙ 0:003 from Fig. 9b. The SDOS and the integrated SDOS of 4d BP networks at pc D 0:160.

Fractal Structures in Condensed Matter Physics

Fractal Structures in Condensed Matter Physics, Figure 10 a Typical fracton mode (! D 0:04997) on a 2d network. Bright region represents the large amplitude portion of the mode. b Crosssection of the fracton mode shown in a along the white line. The four figures are snapshots at different times. After [28]

A typical mode pattern of a fracton on a 2d percolation network is shown in Fig. 10a, where the eigenmode belongs to the angular frequency ! D 0:04997. To bring out the details more clearly, Fig. 10b by K. Yakubo and T. Nakayama [28] shows cross-sections of this fracton mode along the line drawn in Fig. 10a. Filled and open circles represent occupied and vacant sites in the percolation network, respectively. We see that the fracton core (the largest amplitude) possesses very clear boundaries for the edges of the excitation, with an almost step-like character and a long tail in the direction of the weak segments. It should be noted that displacements of atoms in dead ends (weakly connected portions in the percolation network) move in phase, and fall off sharply at their edges. The spectral dimension can be obtained exactly for deterministic fractals. In the case of the d-dimensional Sierpinski gasket, the spectral dimension is given by [17] ds D

2 log(d C 1) : log(d C 3)

We see from this that the upper bound for a Sierpinski gasket is ds D 2 as d ! 1. The spectral dimension for the Mandelbrot–Given fractal depicted is also calculated ana-

lytically as ds D

2 log 8 D 1:345 : : : : log 22

This value is close to those for percolating networks mentioned above, in addition to the fact that the fractal dimension ds D log 8/ log 3 of the Mandelbrot–Given fractal is close to Df D 91/48 for 2d percolating networks and that the Mandelbrot–Given fractal has a structure with nodes, links, and blobs as in the case of percolating networks. For real systems, E. Courtens et al. in 1988 [6] observed fracton excitations in aerogels by means of inelastic light scattering. Future Directions The significance of fractal researches in sciences is that the very idea of fractals opposes reductionism. Modern physics has developed by making efforts to elucidate the physical mechanisms of smaller and smaller structures such as molecules, atoms, and elementary particles. An example in condensed matter physics is the band theory of electrons in solids. Energy spectra of electrons can be obtained by incorporating group theory based on the translational and

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rotational symmetry of the systems. The use of this mathematical tool greatly simplifies the treatment of systems composed of 1022 atoms. If the energy spectrum of a unit cell molecule is solved, the whole energy spectrum of the solid can be computed by applying the group theory. In this context, the problem of an ordered solid is reduced to that of a unit cell. Weakly disordered systems can be handled by regarding impurities as a small perturbation to the corresponding ordered systems. However, a different approach is required for elucidating the physical properties of strongly disordered/complex systems with correlations, or of medium-scale objects, for which it is difficult to find an easily identifiable small parameter that would allow a perturbative analysis. For such systems, the concept of fractals plays an important role in building pictures of the realm of nature. Our established knowledge on fractals is mainly due to experimental observations or computer simulations. The researches are at the phenomenological level stage, not at the intrinsic level, except for a few examples. Concerning future directions of the researches on fractals in condensed matter physics apart from such a question as “What kinds of fractal structures are involved in condensed matter?”, we should consider two directions: one is the very basic aspect such as the problem “Why are there numerous examples showing fractal structures in nature/condensed matter?” However, this type of question is hard. The kinetic growth-mechanisms of fractal systems have a rich variety of applications from the basic to applied sciences and attract much attention as one of the important subjects in non-equilibrium statistical physics and nonlinear physics. Network formation in society is one example where the kinetic growth is relevant. However, many aspects related to the mechanisms of network formations remain puzzling because arguments are at the phenomenological stage. If we compare with researches on Brownian motion as an example, the DLA researches need to advance to the stage of Einstein’s intrinsic theory [8], or that of Smoluchowski [18] and H. Nyquist [15]. It is notable that the DLA is a stochastic version of the Hele–Shaw problem, the flow in composite fluids with high and low viscosities: the particles diffuse in the DLA, while the fluid pressure diffuses in Hele–Shaw flow [19]. These are deeply related to each other and involve many open questions for basic physics and mathematical physics. Concerning the opposite direction, one of the important issues in fractal research is to explore practical uses of fractal structures. In fact, the characteristics of fractals are applied to many cases such as the formation of tailormade nano-scale fractal structures, fractal-shaped antennae with much reduced sizes compared with those of ordi-

nary antennae, and fractal molecules sensitive to frequencies in the infrared region of light. Deep insights into fractal physics in condensed matter will open the door to new sciences and its application to technologies in the near future.

Bibliography Primary Literature 1. Alexander S, Orbach R (1982) Density of states: Fractons. J Phys Lett 43:L625–631 2. Beeckmans JM (1963) The density of aggregated solid aerosol particles. Ann Occup Hyg 7:299–305 3. Ben-Avraham D, Havlin S (1982) Diffusion on percolation at criticality. J Phys A 15:L691–697 4. Brady RM, Ball RC (1984) Fractal growth of copper electro-deposits. Nature 309:225–229 5. Broadbent SR, Hammersley JM (1957) Percolation processes I: Crystals and mazes. Proc Cambridge Philos Soc 53:629–641 6. Courtens E, Vacher R, Pelous J, Woignier T (1988) Observation of fractons in silica aerogels. Europhys Lett 6:L691–697 7. de Gennes PG (1976) La percolation: un concept unificateur. Recherche 7:919–927 8. Einstein A (1905) Über die von der molekularkinetischen Theorie der Waerme geforderte Bewegung von in Ruhenden Fluessigkeiten Suspendierten Teilchen. Ann Phys 17:549–560 9. Gefen Y, Aharony A, Alexander S (1983) Anomalous diffusion on percolating clusters. Phys Rev Lett 50:70–73 10. Hausdorff F (1919) Dimension und Aeusseres Mass. Math Ann 79:157–179 11. Kolb M, Botel R, Jullien R (1983) Scaling of kinetically growing clusters. Phys Rev Lett 51:1123–1126 12. Mandelbrot BB, Given JA (1984) Physical properties of a new fractal model of percolation clusters. Phys Rev Lett 52: 1853–1856 13. Matsushita M, et al (1984) Fractal structures of zinc metal leaves grown by electro-deposition. Phys Rev Lett 53:286–289 14. Meakin P (1983) Formation of fractal clusters and networks by irreversible diffusion-limited aggregation. Phys Rev Lett 51:1119–1122 15. Nyquist H (1928) Termal agitation nof electric charge in conductors. Phys Rev 32:110–113 16. Perrin J (1909) Movement brownien et realite moleculaire. Ann Chim Phys 19:5–104 17. Rammal R, Toulouze G (1983) Random walks on fractal structures and percolation clusters. J Phys Lett 44:L13–L22 18. Smoluchowski MV (1906) Zur Kinematischen Theorie der Brownschen Molekular Bewegung und der Suspensionen. Ann Phys 21:756–780 19. Saffman PG, Taylor GI (1959) The penetration of a fluid into a porous medium or hele-shaw cell containing a more viscous fluid. Proc Roy Soc Lond Ser A 245:312–329 20. Tambo N, Watanabe Y (1967) Study on the density profiles of aluminium flocs I (in japanese). Suidou Kyoukai Zasshi 397:2– 10; ibid (1968) Study on the density profiles of aluminium flocs II (in japanese) 410:14–17 21. Tambo N, Watanabe Y (1979) Physical characteristics of flocs

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Books and Reviews Barabasi AL, Stanley HE (1995) Fractal concepts in surface growth. Cambridge University Press, Cambridge Ben-Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge Bunde A, Havlin S (1996) Fractals and disordered systems. Springer, New York de Gennes PG (1979) Scaling concepts in polymer physics. Cornell University Press, Itahca

Falconer KJ (1989) Fractal geometry: Mathematical foundations and applications. Wiley, New York Feder J (1988) Fractals. Plenum, New York Flory PJ (1969) Statistical mechanics of chain molecules. Inter Science, New York Halsey TC (2000) Diffusion-limited aggregation: A model for pattern formation. Phys Today 11:36–41 Kadanoff LP (1976) Domb C, Green MS (eds) Phase transitions and critical phenomena 5A. Academic Press, New York Kirkpatrick S (1973) Percolation and conduction. Rev Mod Phys 45:574–588 Mandelbrot BB (1979) Fractals: Form, chance and dimension. Freeman, San Francisco Mandelbrot BB (1982) The fractal geometry of nature. Freeman, San Francisco Meakin P (1988) Fractal aggregates. Adv Colloid Interface Sci 28:249–331 Meakin P (1998) Fractals, scaling and growth far from equilibrium. Cambridge University Press, Cambridge Nakayama T, Yakubo K, Orbach R (1994) Dynamical properties of fractal networks: Scaling, numerical simulations, and physical realizations. Rev Mod Phys 66:381–443 Nakayama T, Yakubo K (2003) Fractal concepts in condensed matter. Springer, Heidelberg Sahimi M (1994) Applications of percolation theory. Taylor and Francis, London Schroeder M (1991) Fractals, chaos, power laws. W.H. Freeman, New York Stauffer D, Aharony A (1992) Introduction to percolation theory, 2nd edn. Taylor and Francis, London Vicsek T (1992) Fractal growth phenomena, 2nd edn. World Scientific, Singapore

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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

Fractals and Wavelets: What Can We Learn on Transcription and Replication from Wavelet-Based Multifractal Analysis of DNA Sequences? ALAIN ARNEODO1 , BENJAMIN AUDIT1 , EDWARD-BENEDICT BRODIE OF BRODIE1 , SAMUEL N ICOLAY2 , MARIE TOUCHON3,5 , YVES D’AUBENTON-CARAFA 4 , MAXIME HUVET4 , CLAUDE THERMES4 1 Laboratoire Joliot–Curie and Laboratoire de Physique, ENS-Lyon CNRS, Lyon Cedex, France 2 Institut de Mathématique, Université de Liège, Liège, Belgium 3 Génétique des Génomes Bactériens, Institut Pasteur, CNRS, Paris, France 4 Centre de Génétique Moléculaire, CNRS, Gif-sur-Yvette, France 5 Atelier de Bioinformatique, Université Pierre et Marie Curie, Paris, France Article Outline Glossary Definition of the Subject Introduction A Wavelet-Based Multifractal Formalism: The Wavelet Transform Modulus Maxima Method Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription From the Detection of Relication Origins Using the Wavelet Transform Microscope to the Modeling of Replication in Mammalian Genomes A Wavelet-Based Methodology to Disentangle Transcription- and Replication-Associated Strand Asymmetries Reveals a Remarkable Gene Organization in the Human Genome Future Directions Acknowledgments Bibliography Glossary Fractal Fractals are complex mathematical objects that are invariant with respect to dilations (self-similarity) and therefore do not possess a characteristic length scale. Fractal objects display scale-invariance properties that can either fluctuate from point to point (multifractal) or be homogeneous (monofractal). Mathematically, these properties should hold over all scales.

However, in the real world, there are necessarily lower and upper bounds over which self-similarity applies. Wavelet transform The continuous wavelet transform (WT) is a mathematical technique introduced in the early 1980s to perform time-frequency analysis. The WT has been early recognized as a mathematical microscope that is well adapted to characterize the scaleinvariance properties of fractal objects and to reveal the hierarchy that governs the spatial distribution of the singularities of multifractal measures and functions. More specifically, the WT is a space-scale analysis which consists in expanding signals in terms of wavelets that are constructed from a single function, the analyzing wavelet, by means of translations and dilations. Wavelet transform modulus maxima method The WTMM method provides a unified statistical (thermodynamic) description of multifractal distributions including measures and functions. This method relies on the computation of partition functions from the wavelet transform skeleton defined by the wavelet transform modulus maxima (WTMM). This skeleton provides an adaptive space-scale partition of the fractal distribution under study, from which one can extract the D(h) singularity spectrum as the equivalent of a thermodynamic potential (entropy). With some appropriate choice of the analyzing wavelet, one can show that the WTMM method provides a natural generalization of the classical box-counting and structure function techniques. Compositional strand asymmetry The DNA double helix is made of two strands that are maintained together by hydrogen bonds involved in the base-pairing between Adenine (resp. Guanine) on one strand and Thymine (resp. Cytosine) on the other strand. Under no-strand bias conditions, i. e. when mutation rates are identical on the two strands, in other words when the two strands are strictly equivalent, one expects equimolarities of adenine and thymine and of guanine and cytosine on each DNA strand, a property named Chargaff’s second parity rule. Compositional strand asymmetry refers to deviations from this rule which can be assessed by measuring departure from intrastrand equimolarities. Note that two major biological processes, transcription and replication, both requiring the opening of the double helix, actually break the symmetry between the two DNA strands and can thus be at the origin of compositional strand asymmetries. Eukaryote Organisms whose cells contain a nucleus, the structure containing the genetic material arranged into

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

chromosomes. Eukaryotes constitute one of the three domains of life, the two others, called prokaryotes (without nucleus), being the eubacteria and the archaebacteria. Transcription Transcription is the process whereby the DNA sequence of a gene is enzymatically copied into a complementary messenger RNA. In a following step, translation takes place where each messenger RNA serves as a template to the biosynthesis of a specific protein. Replication DNA replication is the process of making an identical copy of a double-stranded DNA molecule. DNA replication is an essential cellular function responsible for the accurate transmission of genetic information though successive cell generations. This process starts with the binding of initiating proteins to a DNA locus called origin of replication. The recruitment of additional factors initiates the bi-directional progression of two replication forks along the chromosome. In eukaryotic cells, this binding event happens at a multitude of replication origins along each chromosome from which replication propagates until two converging forks collide at a terminus of replication. Chromatin Chromatin is the compound of DNA and proteins that forms the chromosomes in living cells. In eukaryotic cells, chromatin is located in the nucleus. Histones Histones are a major family of proteins found in eukaryotic chromatin. The wrapping of DNA around a core of 8 histones forms a nucleosome, the first step of eukaryotic DNA compaction. Definition of the Subject The continuous wavelet transform (WT) is a mathematical technique introduced in signal analysis in the early 1980s [1,2]. Since then, it has been the subject of considerable theoretical developments and practical applications in a wide variety of fields. The WT has been early recognized as a mathematical microscope that is well adapted to reveal the hierarchy that governs the spatial distribution of singularities of multifractal measures [3,4,5]. What makes the WT of fundamental use in the present study is that its singularity scanning ability equally applies to singular functions than to singular measures [3,4,5,6,7, 8,9,10,11]. This has led Alain Arneodo and his collaborators [12,13,14,15,16] to elaborate a unified thermodynamic description of multifractal distributions including measures and functions, the so-called Wavelet Transform Modulus Maxima (WTMM) method. By using wavelets instead of boxes, one can take advantage of the freedom in the choice of these “generalized oscillating boxes” to get

rid of possible (smooth) polynomial behavior that might either mask singularities or perturb the estimation of their strength h (Hölder exponent), remedying in this way for one of the main failures of the classical multifractal methods (e. g. the box-counting algorithms in the case of measures and the structure function method in the case of functions [12,13,15,16]). The other fundamental advantage of using wavelets is that the skeleton defined by the WTMM [10,11], provides an adaptative space-scale partitioning from which one can extract the D(h) singularity spectrum via the Legendre transform of the scaling exponents (q) (q real, positive as well as negative) of some partition functions defined from the WT skeleton. We refer the reader to Bacry et al. [13], Jaffard [17,18] for rigorous mathematical results and to Hentschel [19] for the theoretical treatment of random multifractal functions. Applications of the WTMM method to 1D signals have already provided insights into a wide variety of problems [20], e. g., the validation of the log-normal cascade phenomenology of fully developed turbulence [21,22,23, 24] and of high-resolution temporal rainfall [25,26], the characterization and the understanding of long-range correlations in DNA sequences [27,28,29,30], the demonstration of the existence of causal cascade of information from large to small scales in financial time series [31,32], the use of the multifractal formalism to discriminate between healthy and sick heartbeat dynamics [33,34], the discovery of a Fibonacci structural ordering in 1D cuts of diffusion limited aggregates (DLA) [35,36,37,38]. The canonical WTMM method has been further generalized from 1D to 2D with the specific goal to achieve multifractal analysis of rough surfaces with fractal dimensions D F anywhere between 2 and 3 [39,40,41]. The 2D WTMM method has been successfully applied to characterize the intermittent nature of satellite images of the cloud structure [42,43], to perform a morphological analysis of the anisotropic structure of atomic hydrogen (H I ) density in Galactic spiral arms [44] and to assist in the diagnosis in digitized mammograms [45]. We refer the reader to Arneodo et al. [46] for a review of the 2D WTMM methodology, from the theoretical concepts to experimental applications. In a recent work, Kestener and Arneodo [47] have further extended the WTMM method to 3D analysis. After some convincing test applications to synthetic 3D monofractal Brownian fields and to 3D multifractal realizations of singular cascade measures as well as their random function counterpart obtained by fractional integration, the 3D WTMM method has been applied to dissipation and enstrophy 3D numerical data issued from direct numerical simulations (DNS) of isotropic turbulence. The results so-obtained have revealed that the multifractal spatial structure of both

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dissipation and enstrophy fields are likely to be well described by a multiplicative cascade process clearly nonconservative. This contrasts with the conclusions of previous box-counting analysis [48] that failed to estimate correctly the corresponding multifractal spectra because of their intrinsic inability to master non-conservative singular cascade measures [47]. For many years, the multifractal description has been mainly devoted to scalar measures and functions. However, in physics as well as in other fundamental and applied sciences, fractals appear not only as deterministic or random scalar fields but also as vector-valued deterministic or random fields. Very recently, Kestener and Arneodo [49,50] have combined singular value decomposition techniques and WT analysis to generalize the multifractal formalism to vector-valued random fields. The so-called Tensorial Wavelet Transform Modulus Maxima (TWTMM) method has been applied to turbulent velocity and vorticity fields generated in (256)3 DNS of the incompressible Navier–Stokes equations. This study reveals the existence of an intimate relationship Dv (h C 1) D D! (h) between the singularity spectra of these two vector fields that are found significantly more intermittent that previously estimated from longitudinal and transverse velocity increment statistics. Furthermore, thanks to the singular value decomposition, the TWTMM method looks very promising for future simultaneous multifractal and structural (vorticity sheets, vorticity filaments) analysis of turbulent flows [49,50]. Introduction The possible relevance of scale invariance and fractal concepts to the structural complexity of genomic sequences has been the subject of considerable increasing interest [20,51,52]. During the past fifteen years or so, there has been intense discussion about the existence, the nature and the origin of the long-range correlations (LRC) observed in DNA sequences. Different techniques including mutual information functions [53,54], auto-correlation functions [55,56], power-spectra [54,57,58], “DNA walk” representation [52,59], Zipf analysis [60,61] and entropies [62,63], were used for the statistical analysis of DNA sequences. For years there has been some permanent debate on rather struggling questions like the fact that the reported LRC might be just an artifact of the compositional heterogeneity of the genome organization [20,27,52, 55,56,64,65,66,67]. Another controversial issue is whether or not LRC properties are different for protein-coding (exonic) and non-coding (intronic, intergenic) sequences [20, 27,52,54,55,56,57,58,59,61,68]. Actually, there were many

objective reasons for this somehow controversial situation. Most of the pioneering investigations of LRC in DNA sequences were performed using different techniques that all consisted in measuring power-law behavior of some characteristic quantity, e. g., the fractal dimension of the DNA walk, the scaling exponent of the correlation function or the power-law exponent of the power spectrum. Therefore, in practice, they all faced the same difficulties, namely finite-size effects due to the finiteness of the sequence [69, 70,71] and statistical convergence issue that required some precautions when averaging over many sequences [52, 65]. But beyond these practical problems, there was also a more fundamental restriction since the measurement of a unique exponent characterizing the global scaling properties of a sequence failed to resolve multifractality [27], and thus provided very poor information upon the nature of the underlying LRC (if they were any). Actually, it can be shown that for a homogeneous (monofractal) DNA sequence, the scaling exponents estimated with the techniques previously mentioned, can all be expressed as a function of the so-called Hurst or roughness exponent H of the corresponding DNA walk landscape [20, 27,52]. H D 1/2 corresponds to classical Brownian, i. e. uncorrelated random walk. For any other value of H, the steps (increments) are either positively correlated (H > 1/2: Persistent random walk) or anti-correlated (H < 1/2: Anti-persistent random walk). One of the main obstacles to LRC analysis in DNA sequences is the genuine mosaic structure of these sequences which are well known to be formed of “patches” of different underlying composition [72,73,74]. When using the “DNA walk” representation, these patches appear as trends in the DNA walk landscapes that are likely to break scale-invariance [20,52,59,64,65,66,67,75,76]. Most of the techniques, e. g. the variance method, used for characterizing the presence of LRC are not well adapted to study non-stationary sequences. There have been some phenomenological attempts to differentiate local patchiness from LRC using ad hoc methods such as the so-called “min-max method” [59] and the “detrended fluctuation analysis” [77]. In previous works [27,28], the WT has been emphasized as a well suited technique to overcome this difficulty. By considering analyzing wavelets that make the WT microscope blind to low-frequency trends, any bias in the DNA walk can be removed and the existence of powerlaw correlations with specific scale invariance properties can be revealed accurately. In [78], from a systematic WT analysis of human exons, CDSs and introns, LRC were found in non-coding sequences as well as in coding regions somehow hidden in their inner codon structure. These results made rather questionable the model based

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

on genome plasticity proposed at that time to account for the reported absence of LRC in coding sequences [27,28, 52,54,59,68]. More recently, some structural interpretation of these LRC has emerged from a comparative multifractal analysis of DNA sequences using structural coding tables based on nucleosome positioning data [29,30]. The application of the WTMM method has revealed that the corresponding DNA chain bending profiles are monofractal (homogeneous) and that there exists two LRC regimes. In the 10–200 bp range, LRC are observed for eukaryotic sequences as quantified by a Hurst exponent value H ' 0:6 (but not for eubacterial sequences for which H D 0:5) as the signature of the nucleosomal structure. These LRC were shown to favor the autonomous formation of small (a few hundred bps) 2D DNA loops and in turn the propensity of eukaryotic DNA to interact with histones to form nucleosomes [79,80]. In addition, these LRC might induce some local hyperdiffusion of these loops which would be a very attractive interpretation of the nucleosomal repositioning dynamics. Over larger distances (& 200 bp), stronger LRC with H ' 0:8 seem to exist in any sequence [29,30]. These LRC are actually observed in the S. cerevisiae nucleosome positioning data [81] suggesting that they are involved in the nucleosome organization in the so-called 30 nm chromatin fiber [82]. The fact that this second regime of LRC is also present in eubacterial sequences shows that it is likely to be a possible key to the understanding of the structure and dynamics of both eukaryotic and prokaryotic chromatin fibers. In regards to their potential role in regulating the hierarchical structure and dynamics of chromatin, the recent report [83] of sequence-induced LRC effects on the conformations of naked DNA molecules deposited onto mica surface under 2D thermodynamic equilibrium observed by Atomic Force Microscopy (AFM) is a definite experimental breakthrough. Our purpose here is to take advantage of the availability of fully sequenced genomes to generalize the application of the WTMM method to genome-wide multifractal sequence analysis when using codings that have a clear functional meaning. According to the second parity rule [84,85], under no strand-bias conditions, each genomic DNA strand should present equimolarities of adenines A and thymines T and of guanines G and cytosines C [86,87]. Deviations from intrastrand equimolarities have been extensively studied during the past decade and the observed skews have been attributed to asymmetries intrinsic to the replication and transcription processes that both require the opening of the double helix. Actually, during these processes mutational events can affect the two strands differently and an asymmetry can

result if one strand undergoes different mutations, or is repaired differently than the other strand. The existence of transcription and/or replication associated strand asymmetries has been mainly established for prokaryote, organelle and virus genomes [88,89,90,91,92,93,94]. For a long time the existence of compositional biases in eukaryotic genomes has been unclear and it is only recently that (i) the statistical analysis of eukaryotic gene introns have revealed the presence of transcription-coupled strand asymmetries [95,96,97] and (ii) the genome wide multiscale analysis of mammalian genomes has clearly shown some departure from intrastrand equimolarities in intergenic regions and further confirmed the existence of replication-associated strand asymmetries [98,99,100]. In this manuscript, we will review recent results obtained when using the WT microscope to explore the scale invariance properties of the TA and GC skew profiles in the 22 human autosomes [98,99,100]. These results will enlighten the richness of information that can be extracted from these functional codings of DNA sequences including the prediction of 1012 putative human replication origins. In particular, this study will reveal a remarkable human gene organization driven by the coordination of transcription and replication [101]. A Wavelet-Based Multifractal Formalism The Continuous Wavelet Transform The WT is a space-scale analysis which consists in expanding signals in terms of wavelets which are constructed from a single function, the analyzing wavelet , by means of translations and dilations. The WT of a real-valued function f is defined as [1,2]: T [ f ] (x0 ; a) D

1 a

C1 Z

f (x)

x  x 

1

0

a

dx ;

(1)

where x0 is the space parameter and a (> 0) the scale parameter. The analyzing wavelet is generally chosen to be well localized in both space and frequency. Usually is required to be of zero mean for the WT to be invertible. But for the particular purpose of singularity tracking that is of interest here, we will further require to be orthogonal to low-order polynomials in Fig. 1 [7,8,9,10,11,12,13, 14,15,16]: C1 Z

x m (x)dx ;

0m
(2)

1

As originally pointed out by Mallat and collaborators [10,11], for the specific purpose of analyzing the reg-

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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 1 Set of analyzing wavelets (x) that can be used in Eq. (1). a g(1) and b g(2) as defined in Eq. (10). c R as defined in Eq. (12), that will be used in Sect. “A Wavelet-Based Methodology to Disentangle Transcription- and Replication-Associated Strand Asymmetries Reveals a Remarkable Gene Organization in the Human Genome” to detect replication domains. d Box function T that will be used in Sect. “A Wavelet-Based Methodology to Disentangle Transcription- and Replication-Associated Strand Asymmetries Reveals a Remarkable Gene Organization in the Human Genome” to model step-like skew profiles induced by transcription

ularity of a function, one can get rid of the redundancy of the WT by concentrating on the WT skeleton defined by its modulus maxima only. These maxima are defined, at each scale a, as the local maxima of jT [ f ](x; a)j considered as a function of x. As illustrated in Figs. 2e, 2f, these WTMM are disposed on connected curves in the spacescale (or time-scale) half-plane, called maxima lines. Let us define L(a0 ) as the set of all the maxima lines that exist at the scale a0 and which contain maxima at any scale a  a0 . An important feature of these maxima lines, when analyzing singular functions, is that there is at least one maxima line pointing towards each singularity [10,11,16]. Scanning Singularities with the Wavelet Transform Modulus Maxima The strength of the singularity of a function f at point x0 is given by the Hölder exponent, i. e., the largest exponent such that there exists a polynomial Pn (x  x0 ) of order

n < h(x0 ) and a constant C > 0, so that for any point x in a neighborhood of x0 , one has [7,8,9,10,11,13,16]: j f (x)  Pn (x  x0 )j  C jx  x0 j h :

(3)

If f is n times continuously differentiable at the point x0 , then one can use for the polynomial Pn (x  x0 ), the ordern Taylor series of f at x0 and thus prove that h(x0 ) > n. Thus h(x0 ) measures how irregular the function f is at the point x0 . The higher the exponent h(x0 ), the more regular the function f . The main interest in using the WT for analyzing the regularity of a function lies in its ability to be blind to polynomial behavior by an appropriate choice of the analyzing wavelet . Indeed, let us assume that according to Eq. (3), f has, at the point x0 , a local scaling (Hölder) exponent h(x0 ); then, assuming that the singularity is not oscillating [11,102,103], one can easily prove that the local behavior of f is mirrored by the WT which locally behaves

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 2 WT of monofractal and multifractal stochastic signals. Fractional Brownian motion: a a realization of B1/3 (L D 65 536); c WT of B1/3 as coded, independently at each scale a, using 256 colors from black (jT j D 0) to red (maxb jT j); e WT skeleton defined by the set of all the maxima lines. Log-normal random W -cascades: b a realization of the log-normal W -cascade model (L D 65 536) with the following parameter values m D 0:355 ln 2 and 2 D 0:02 ln 2 (see [116]); d WT of the realization in b represented with the same color coding as in c; f WT skeleton. The analyzing wavelet is g(1) (see Fig. 1a)

like [7,8,9,10,11,12,13,14,15,16,17,18]: T [ f ](x0 ; a) a h(x 0 ) ;

a ! 0C ;

(4)

provided n > h(x0 ), where n is the number of vanishing moments of (Eq. (2)). Therefore one can extract the exponent h(x0 ) as the slope of a log-log plot of the WT amplitude versus the scale a. On the contrary, if one chooses

n < h(x0 ), the WT still behaves as a power-law but with a scaling exponent which is n : T [ f ](x0 ; a) a n ;

a ! 0C :

(5)

Thus, around a given point x0 , the faster the WT decreases when the scale goes to zero, the more regular f is around

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that point. In particular, if f 2 C 1 at x0 (h(x0 ) D C1), then the WT scaling exponent is given by n , i. e. a value which is dependent on the shape of the analyzing wavelet. According to this observation, one can hope to detect the points where f is smooth by just checking the scaling behavior of the WT when increasing the order n of the analyzing wavelet [12,13,14,15,16]. Remark 1 A very important point (at least for practical purpose) raised by Mallat and Hwang [10] is that the local scaling exponent h(x0 ) can be equally estimated by looking at the value of the WT modulus along a maxima line converging towards the point x0 . Indeed one can prove that both Eqs. (4) and (5) still hold when following a maxima line from large down to small scales [10,11]. The Wavelet Transform Modulus Maxima Method As originally defined by Parisi and Frisch [104], the multifractal formalism of multi-affine functions amounts to compute the so-called singularity spectrum D(h) defined as the Hausdorff dimension of the set where the Hölder exponent is equal to h [12,13,16]: D(h) D dimH fx; h(x) D hg ;

(6)

where h can take, a priori, positive as well as negative real values (e. g., the Dirac distribution ı(x) corresponds to the Hölder exponent h(0) D 1) [17]. A natural way of performing a multifractal analysis of fractal functions consists in generalizing the “classical” multifractal formalism [105,106,107,108,109] using wavelets instead of boxes. By taking advantage of the freedom in the choice of the “generalized oscillating boxes” that are the wavelets, one can hope to get rid of possible smooth behavior that could mask singularities or perturb the estimation of their strength h. But the major difficulty with respect to box-counting techniques [48,106,110,111, 112] for singular measures, consists in defining a covering of the support of the singular part of the function with our set of wavelets of different sizes. As emphasized in [12,13, 14,15,16], the branching structure of the WT skeletons of fractal functions in the (x; a) half-plane enlightens the hierarchical organization of their singularities (Figs. 2e, 2f). The WT skeleton can thus be used as a guide to position, at a considered scale a, the oscillating boxes in order to obtain a partition of the singularities of f . The wavelet transform modulus maxima (WTMM) method amounts to compute the following partition function in terms of WTMM coefficients [12,13,14,15,16]: X  ˇ q ˇ (7) Z(q; a) D sup ˇT [ f ](x; a0 )ˇ ; l 2L(a)

(x;a 0 )2l a 0 a

where q 2 R and the sup can be regarded as a way to define a scale adaptative “Hausdorff-like” partition. Now from the deep analogy that links the multifractal formalism to thermodynamics [12,113], one can define the exponent (q) from the power-law behavior of the partition function: Z(q; a) a(q) ;

a ! 0C ;

(8)

where q and (q) play respectively the role of the inverse temperature and the free energy. The main result of this wavelet-based multifractal formalism is that in place of the energy and the entropy (i. e. the variables conjugated to q and ), one has h, the Hölder exponent, and D(h), the singularity spectrum. This means that the singularity spectrum of f can be determined from the Legendre transform of the partition function scaling exponent (q) [13,17, 18]: D(h) D min(qh  (q)) :

(9)

q

From the properties of the Legendre transform, it is easy to see that homogeneous monofractal functions that involve singularities of unique Hölder exponent h D @/@q, are characterized by a (q) spectrum which is a linear function of q. On the contrary, a nonlinear (q) curve is the signature of nonhomogeneous functions that exhibit multifractal properties, in the sense that the Hölder exponent h(x) is a fluctuating quantity that depends upon the spatial position x. Defining Our Battery of Analyzing Wavelets There are almost as many analyzing wavelets as applications of the continuous WT [3,4,5,12,13,14,15,16]. In the present work, we will mainly used the class of analyzing wavelets defined by the successive derivatives of the Gaussian function: g (N) (x) D

d N x 2 /2 e ; dx N

(10)

for which n D N and more specifically g (1) and g (2) that are illustrated in Figs. 1a, 1b. Remark 2 The WT of a signal f with g (N) (Eq. (10)) takes the following simple expression: 1 Tg (N) [ f ](x; a) D a D

C1 Z

f (y)g (N)

1 dN aN N dx

y  x a

Tg (0) [ f ](x; a) :

dy ; (11)

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

Equation (11) shows that the WT computed with g (N) at scale a is nothing but the Nth derivative of the signal f (x) smoothed by a dilated version g (0) (x/a) of the Gaussian function. This property is at the heart of various applications of the WT microscope as a very efficient multi-scale singularity tracking technique [20]. With the specific goal of disentangling the contributions to the nucleotide composition strand asymmetry coming respectively from transcription and replication processes, we will use in Sect. “A Wavelet-Based Methodology to Disentangle Transcription- and Replication-Associated Strand Asymmetries Reveals a Remarkable Gene Organization in the Human Genome”, an adapted analyzing wavelet of the following form (Fig. 1c) [101,114]:  

 1 1 1  R (x) D  x  ; for x 2  ; 2 2 2 (12) D 0 elsewhere : By performing multi-scale pattern recognition in the (space, scale) half-plane with this analyzing wavelet, we will be able to define replication domains bordered by putative replication origins in the human genome and more generally in mammalian genomes [101,114]. Test Applications of the WTMM Method on Monofractal and Multifractal Synthetic Random Signals This section is devoted to test applications of the WTMM method to random functions generated either by additive models like fractional Brownian motions [115] or by multiplicative models like random W -cascades on wavelet dyadic trees [21,22,116,117]. For each model, we first wavelet transform 1000 realizations of length L D 65 536 with the first order (n D 1) analyzing wavelet g (1) . From the WT skeletons defined by the WTMM, we compute the mean partition function (Eq. (7)) from which we extract the annealed (q) (Eq. (8)) and, in turn, D(h) (Eq. (9)) multifractal spectra. We systematically test the robustness of our estimates with respect to some change of the shape of the analyzing wavelet, in particular when increasing the number n of zero moments, going from g (1) to g (2) (Eq. (10)). Fractional Brownian Signals Since its introduction by Mandelbrot and van Ness [115], the fractional Brownian motion (fBm) B H has become a very popular model in signal and image processing [16,20,39]. In 1D, fBm has proved useful for modeling various physical phenomena with long-range dependence, e. g., “1/ f ” noises. The fBm exhibits a power spectral density S(k) 1/k ˇ , where the

spectral exponent ˇ D 2H C 1 is related to the Hurst exponent H. fBm has been extensively used as test stochastic signals for Hurst exponent measurements. In Figs. 2, 3 and 4, we report the results of a statistical analysis of fBm’s using the WTMM method [12,13,14,15,16]. We mainly concentrate on B1/3 since it has a k 5/3 power-spectrum similar to the spectrum of the multifractal stochastic signal we will study next. Actually, our goal is to demonstrate that, where the power spectrum analysis fails, the WTMM method succeeds in discriminating unambiguously between these two fractal signals. The numerical signals were generated by filtering uniformly generated pseudo-random noise in Fourier space in order to have the required k 5/3 spectral density. A B1/3 fractional Brownian trail is shown in Fig. 2a. Figure 2c illustrates the WT coded, independently at each scale a, using 256 colors. The analyzing wavelet is g (1) (n D 1). Figure 3a displays some plots of log2 Z(q; a) versus log2 (a) for different values of q, where the partition function Z(q; a) has been computed on the WTMM skeleton shown in Fig. 2e, according to the definition (Eq. (7)). Using a linear regression fit, we then obtain the slopes (q) of these graphs. As shown in Fig. 3c, when plotted versus q, the data for the exponents (q) consistently fall on a straight line that is remarkably fitted by the theoretical prediction: (q) D qH  1 ;

(13)

with H D 1/3. From the Legendre transform of this linear (q) (Eq. (9)), one gets a D(h) singularity spectrum that reduces to a single point: D(h) D 1

if h D H ;

D 1 if h ¤ H :

(14)

Thus, as expected theoretically [16,115], one finds that the fBm B1/3 is a nowhere differentiable homogeneous fractal signal with a unique Hölder exponent h D H D 1/3. Note that similar good estimates are obtained when using analyzing wavelets of different order (e. g. g (2) ), and this whatever the value of the index H of the fBm [12,13,14,15, 16]. Within the perspective of confirming the monofractality of fBm’s, we have studied the probability density function (pdf) of wavelet coefficient values a (Tg (1) (:; a)), as computed at a fixed scale a in the fractal scaling range. According to the monofractal scaling properties, one expects these pdfs to satisfy the self-similarity relationship [20, 27,28]: a H a (a H T) D (T) ;

(15)

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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 3 Determination of the (q) and D(h) multifractal spectra of fBm B1/3 (red circles) and log-normal random W -cascades (green dots) using the WTMM method. a log2 Z(q; a) vs. log2 a: B1/3 . b log2 Z(q; a) vs. log2 a: Log-normal W -cascades with the same parameters as in Fig. 2b. c (q) vs. q; the solid lines correspond respectively to the theoretical spectra (13) and (16). d D(h) vs. h; the solid lines correspond respectively to the theoretical predictions (14) and (17). The analyzing wavelet is g(1) . The reported results correspond to annealed averaging over 1000 realizations of L D 65 536

where (T) is a “universal” pdf (actually the pdf obtained at scale a D 1) that does not depend on the scale parameter a. As shown in Figs. 4a, 4a0 for B1/3 , when plotting a H a (a H T) vs. T, all the a curves corresponding to different scales (Fig. 4a) remarkably collapse on a unique curve when using a unique exponent H D 1/3 (Fig. 4a0 ). Furthermore the so-obtained universal curve cannot be distinguished from a parabola in semi-log representation

as the signature of the monofractal Gaussian statistics of fBm fluctuations [16,20,27]. Random W -Cascades Multiplicative cascade models have enjoyed increasing interest in recent years as the paradigm of multifractal objects [16,19,48,105,107, 108,118]. The notion of cascade actually refers to a selfsimilar process whose properties are defined multiplica-

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 4 Probability distribution functions of wavelet coefficient values of fBm B1/3 (open symbols) and log-normal random W -cascades (filled symbols) with the same parameters as in Fig. 2b. a a vs. Tg(1) for the set of scales a D 10 (4), 50 (), 100 (), 1000 (˙), 9000 (O); a0 aH a (aH Tg(1) ) vs. Tg(1) with H D 1/3; The symbols have the same meaning as in a. b a vs. Tg(1) for the set of scales a D 10 (▲), 50 (■), 100 (●), 1000 (◆), 9000 (▼); (b0 ) aH a (aH Tg(1) ) vs. Tg(1) with H D m/ ln 2 D 0:355. The analyzing wavelet is g(1) (Fig. 1a)

tively from coarse to fine scales. In that respect, it occupies a central place in the statistical theory of turbulence [48, 104]. Originally, the concept of self-similar cascades was introduced to model multifractal measures (e. g. dissipation or enstrophy) [48]. It has been recently generalized to the construction of scale-invariant signals (e. g. longitudinal velocity, pressure, temperature) using orthogonal wavelet basis [116,119]. Instead of redistributing the measure over sub-intervals with multiplicative weights, one allocates the wavelet coefficients in a multiplicative way on the dyadic grid. This method has been implemented to generate multifractal functions (with weights W) from a given deterministic or probabilistic multiplicative process. Along the line of the modeling of fully developed turbulent signals by log-infinitely divisible multiplicative pro-

cesses [120,121], we will mainly concentrate here on the log-normal W -cascades in order to calibrate the WTMM method. If m and  2 are respectively the mean and the variance of ln W (where W is a multiplicative random variable with log-normal probability distribution), then, as shown in [116], a straightforward computation leads to the following (q) spectrum: (q) D  log2 hW q i  1 ; D

2 2 ln 2

q2 

8q 2 R

m q1; ln 2

(16)

where h: : :i means ensemble average. The corresponding D(h) singularity spectrum is obtained by Legendre

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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

transforming (q) (Eq. (9)): D(h) D 

(h C m/ ln 2)2 C1: 2 2 / ln 2

Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription (17)

According to the convergence criteria established in [116], m andp 2 have to satisfy the conditions: m < 0 and jmj/ > 2 ln 2. Moreover, by solving D(h) D 0, one gets the following bounds for the supportpof thepD(h) singularity spectrum: hmin m/ ln 2  ( 2)/ ln 2 p D p and hmax D m/ ln 2 C ( 2)/ ln 2. In Fig. 2b is illustrated a realization of a log-normal W -cascade for the parameter values m D 0:355 ln 2 and  2 D 0:02 ln 2. The corresponding WT and WT skeleton as computed with g (1) are shown in Figs. 2d and 2f respectively. The results of the application of the WTMM method are reported in Fig. 3. As shown in Fig. 3b, when plotted versus the scale parameter a in a logarithmic representation, the annealed average of the partition functions Z(q; a) displays a well defined scaling behavior over a range of scales of about 5 octaves. Note that scaling of quite good quality is found for a rather wide range of q values: 5  q  10. When processing to a linear regression fit of the data over the first four octaves, one gets the (q) spectrum shown in Fig. 3c. This spectrum is clearly a nonlinear function of q, the hallmark of multifractal scaling. Moreover, the numerical data are in remarkable agreement with the theoretical quadratic prediction (Eq. (16)). Similar quantitative agreement is observed on the D(h) singularity spectrum in Fig. 3d which displays a single humped parabola shape that characterizes intermittent fluctuations corresponding to Hölder exponents values ranging from hmin D 0:155 to hmax D 0:555. Unfortunately, to capture the strongest and the weakest singularities, one needs to compute the (q) spectrum for very large values of jqj. This requires the processing of many more realizations of the considered log-normal random W -cascade. The multifractal nature of log-normal W -cascade realizations is confirmed in Figs. 4b, 4b0 where the self-similarity relationship (Eq. (15)) is shown not to apply. Actually there does not exist a H value allowing to superimpose onto a single curve the WT pdfs computed at different scales. The test applications reported in this section demonstrate the ability of the WTMM method to resolve multifractal scaling of 1D signals, a hopeless task for classical power spectrum analysis. They were used on purpose to calibrate and to test the reliability of our methodology, and of the corresponding numerical tools, with respect to finite-size effects and statistical convergence.

During genome evolution, mutations do not occur at random as illustrated by the diversity of the nucleotide substitution rate values [122,123,124,125]. This non-randomness is considered as a by-product of the various DNA mutation and repair processes that can affect each of the two DNA strands differently. Asymmetries of substitution rates coupled to transcription have been mainly observed in prokaryotes [88,89,91], with only preliminary results in eukaryotes. In the human genome, excess of T was observed in a set of gene introns [126] and some large-scale asymmetry was observed in human sequences but they were attributed to replication [127]. Only recently, a comparative analysis of mammalian sequences demonstrated a transcription-coupled excess of G+T over A+C in the coding strand [95,96,97]. In contrast to the substitution biases observed in bacteria presenting an excess of C!T transitions, these asymmetries are characterized by an excess of purine (A!G) transitions relatively to pyrimidine (T!C) transitions. These might be a by-product of the transcription-coupled repair mechanism acting on uncorrected substitution errors during replication [128]. In this section, we report the results of a genome-wide multifractal analysis of strand-asymmetry DNA walk profiles in the human genome [129]. This study is based on the computation of the TA and GC skews in non-overlapping 1 kbp windows: nT  nA nG  nC STA D ; SGC D ; (18) nT C nA nG C nC where nA , nC , nG and nT are respectively the numbers of A, C, G and T in the windows. Because of the observed correlation between the TA and GC skews, we also considered the total skew S D STA C SGC :

(19)

From the skews STA (n), SGC (n) and S(n), obtained along the sequences, where n is the position (in kbp units) from the origin, we also computed the cumulative skew profiles (or skew walk profiles): ˙TA (n) D

n X

STA ( j) ;

jD1

˙GC (n) D

n X

SGC ( j) ; (20)

jD1

and ˙ (n) D

n X

S( j) :

(21)

jD1

Our goal is to show that the skew DNA walks of the 22 human autosomes display an unexpected (with respect

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

to previous monofractal diagnosis [27,28,29,30]) bifractal scaling behavior in the range 10 to 40 kbp as the signature of the presence of transcription-induced jumps in the LRC noisy S profiles. Sequences and gene annotation data (“refGene”) were retrieved from the UCSC Genome Browser (May 2004). We used RepeatMasker to exclude repetitive elements that might have been inserted recently and would not reflect long-term evolutionary patterns. Revealing the Bifractality of Human Skew DNA Walks with the WTMM Method As an illustration of our wavelet-based methodology, we show in Fig. 5 the S skew profile of a fragment of human chromosome 6 (Fig. 5a), the corresponding skew DNA walk (Fig. 5b) and its space-scale wavelet decomposition using the Mexican hat analyzing wavelet g (2) (Fig. 1b). When computing Z(q; a) (Eq. (7)) from the WT skeletons of the skew DNA walks ˙ of the 22 human autosomes, we get convincing power-law behavior for 1:5  q  3 (data not shown). In Fig. 6a are reported the (q) exponents obtained using a linear regression fit of ln Z(q; a) vs. ln a over the range of scales 10 kbp  a  40 kbp. All the data points remarkably fall on two straight lines 1 (q) D 0:78q  1 and 2 (q) D q  1 which strongly suggests the presence of two types of singularities h1 D 0:78 and h2 D 1, respectively on two sets S1 and S2 with the same Haussdorf dimension D D 1 (0) D 2 (0) D 1, as confirmed when computing the D(h) singularity spectrum in Fig. 6b. This observation means that Z(q; a) can be split in two parts [12,16]: Z(q; a) D C1 (q)a q h1 1 C C2 (q)a q h2 1 ;

(22)

where C1 (q) and C2 (q) are prefactors that depend on q. Since h1 < h2 , in the limit a 7! 0C , the partition function is expected to behave like Z(q; a) C1 (q)a q h1 1 for q > 0 and like Z(q; a) C2 (q)a q h2 1 for q < 0, with a so-called phase transition [12,16] at the critical value q c D 0. Surprisingly, it is the contribution of the weakest singularities h2 D 1 that controls the scaling behavior of Z(q; a) for q > 0 while the strongest ones h1 D 0:78 actually dominate for q < 0 (Fig. 6a). This inverted behavior originates from finite (1 kbp) resolution which prevents the observation of the predicted scaling behavior in the limit a 7! 0C . The prefactors C1 (q) and C2 (q) in Eq. (22) are sensitive to (i) the number of maxima lines in the WT skeleton along which the WTMM behave as a h1 or a h2 and (ii) the relative amplitude of these WTMM. Over the range of scales used to estimate (q), the WTMM along the maxima lines pointing (at small scale) to h2 D 1 singularities are significantly larger than those along the maxima lines associated

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 5 a Skew profile S(n) (Eq. (19)) of a repeat-masked fragment of human chromosome 6; red (resp. blue) 1 kbp window points correspond to (+) genes (resp. () genes) lying on the Watson (resp. Crick) strand; black points to intergenic regions. b Cumulated skew profile ˙ (n) (Eq. (21)). c WT of ˙ ; Tg(2) (n; a) is coded from black (min) to red (max); the WT skeleton defined by the maxima lines is shown in solid (resp. dashed) lines corresponding to positive (resp. negative) WT values. For illustration yellow solid (resp. dashed) maxima lines are shown to point to the positions of 2 upward (resp. 2 downward) jumps in S (vertical dashed lines in a and b) that coincide with gene transcription starts (resp. ends). In green are shown maxima lines that persist above a  200 kbp and that point to sharp upward jumps in S (vertical solid lines in a and b) that are likely to be the locations of putative replication origins (see Sect. “From the Detection of Relication Origins Using the Wavelet Transform Microscope to the Modeling of Replication in Mammalian Genomes”) [98,100]; note that 3 out of those 4 jumps are co-located with transcription start sites [129]

to h1 D 0:78 (see Figs. 6c, 6d). This implies that the larger q > 0, the stronger the inequality C2 (q)  C1 (q) and the more pronounced the relative contribution of the second term in the r.h.s. of Eq. (22). On the opposite for q < 0, C1 (q)  C2 (q) which explains that the strongest singularities h1 D 0:78 now control the scaling behavior of Z(q; a) over the explored range of scales. In Figs. 6c, 6d are shown the WTMM pdfs computed at scales a D 10, 20 and 40 kbp after rescaling by a h1 and a h2 respectively. We note that there does not exist a value of H such that all the pdfs collapse on a single curve as expected from Eq. (15) for monofractal DNA walks. Consistently with the (q) data in Fig. 6a and with the inverted

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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 6 Multifractal analysis of ˙ (n) of the 22 human (filled symbols) and 19 mouse (open circle) autosomes using the WTMM method with g(2) over the range 10 kbp  a  40 kbp [129]. a (q) vs. q. b D(h) vs. h. c WTMM pdf: is plotted versus jTj/aH where H D h1 D 0:78, in semi-log representation; the inset is an enlargement of the pdf central part in linear representation. d Same as in c but with H D h2 D 1. In c and d, the symbols correspond to scales a D 10 (●), 20 (■) and 40 kbp (▲)

scaling behavior discussed above, when using the two exponents h1 D 0:78 and h2 D 1, one succeeds in superimposing respectively the central (bump) part (Fig. 6c) and the tail (Fig. 6d) of the rescaled WTMM pdfs. This corroborates the bifractal nature of the skew DNA walks that display two competing scale-invariant components of Hölder exponents: (i) h1 D 0:78 corresponds to LRC homogeneous fluctuations previously observed over the range 200 bp . a . 20 kbp in DNA walks generated with structural codings [29,30] and (ii) h2 D 1 is associated to convex _ and concave ^ shapes in the DNA walks ˙ indicating the presence of discontinuities in the derivative of ˙ , i. e., of jumps in S (Figs. 5a, 5b). At a given scale a, according to Eq. (11), a large value of the WTMM in Fig. 5c corresponds to a strong derivative of the smoothed S profile and the maxima line to which it belongs is likely to point to a jump location in S. This is particularly the case for the

colored maxima lines in Fig. 5c: Upward (resp. downward) jumps (Fig. 5a) are so-identified by the maxima lines corresponding to positive (resp. negative) values of the WT. Transcription-Induced Step-like Skew Profiles in the Human Genome In order to identify the origin of the jumps observed in the skew profiles, we have performed a systematic investigation of the skews observed along 14 854 intron containing genes [96,97]. In Fig. 7 are reported the mean values of STA and SGC skews for all genes as a function of the distance to the 50 - or 30 - end. At the 50 gene extremities (Fig. 7a), a sharp transition of both skews is observed from about zero values in the intergenic regions to finite positive values in transcribed regions ranging between 4 and 6% for S¯TA and between 3 and 5% for S¯GC . At the

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 7 TA (●) and GC (green ●) skew profiles in the regions surrounding 50 and 30 gene extremities [96]. STA and SGC were calculated in 1 kbp windows starting from each gene extremities in both directions. In abscissa is reported the distance (n) of each 1 kbp window to the indicated gene extremity; zero values of abscissa correspond to 50 - (a) or 30 - (b) gene extremities. In ordinate is reported the mean value of the skews over our set of 14 854 intron-containing genes for all 1 kbp windows at the corresponding abscissa. Error bars represent the standard error of the means

gene 30 - extremities (Fig. 7b), the TA and GC skews also exhibit transitions from significantly large values in transcribed regions to very small values in untranscribed regions. However, in comparison to the steep transitions observed at 50 - ends, the 30 - end profiles present a slightly smoother transition pattern extending over 5 kbp and including regions downstream of the 30 - end likely reflecting the fact that transcription continues to some extent downstream of the polyadenylation site. In pluricellular organisms, mutations responsible for the observed biases are expected to have mostly occurred in germ-line cells. It could happen that gene 30 - ends annotated in the databank differ from the poly-A sites effectively used in the germline cells. Such differences would then lead to some broadening of the skew profiles. From Skew Multifractal Analysis to Gene Detection In Fig. 8 are reported the results of a statistical analysis of the jump amplitudes in human S profiles [129]. For maxima lines that extend above a  D 10 kbp in the WT skeleton (see Fig. 5c), the histograms obtained for upward and downward variations are quite similar, especially their tails that are likely to correspond to jumps in the S profiles (Fig. 8a). When computing the distance between upward or downward jumps (jSj  0:1) to the closest transcription start (TSS) or end (TES) sites (Fig. 8b), we reveal that the number of upward jumps in close proximity (jnj . 3 kpb) to TSS over-exceeds the number of such jumps close to TES. Similarly, downward jumps are preferentially located at TES. These observations are consistent with the step-like shape of skew profiles induced

by transcription: S > 0 (resp. S < 0) is constant along a (+) (resp. ()) genes and S D 0 in the intergenic regions (Fig. 7) [96]. Since a step-like pattern is edged by one upward and one downward jump, the set of human genes that are significantly biased is expected to contribute to an even number of S > 0 and S < 0 jumps when exploring the range of scales 10 . a . 40 kbp, typical of human gene size. Note that in Fig. 8a, the number of sharp upward jumps actually slightly exceeds the number of sharp downward jumps, consistently with the experimental observation that whereas TSS are well defined, TES may extend over 5 kbp resulting in smoother downward skew transitions (Fig. 7b). This TES particularity also explains the excess of upward jumps found close to TSS as compared to the number of downward jumps close to TES (Fig. 8b). In Fig. 9a, we report the analysis of the distance of TSS to the closest upward jump [129]. For a given upward jump amplitude, the number of TSS with a jump within jnj increases faster than expected (as compared to the number found for randomized jump positions) up to jnj ' 2 kbp. This indicates that the probability to find an upward jump within a gene promoter region is significantly larger than elsewhere. For example, out of 20 023 TSS, 36% (7228) are delineated within 2 kbp by a jump with S > 0:1. This provides a very reasonable estimate for the number of genes expressed in germline cells as compared to the 31.9% recently experimentally found to be bound to Pol II in human embryonic stem cells [130]. Combining the previous results presented in Figs. 8b and 9a, we report in Fig. 9b an estimate of the efficiency/coverage relationship by plotting the proportion of upward jumps (S > S  ) lying in TSS proximity as

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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 8 Statistical analysis of skew variations at the singularity positions determined at scale 1 kbp from the maxima lines that exist at scales a  10 kbp in the WT skeletons of the 22 human autosomes [129]. For each singularity, we computed the variation amplitudes ¯ 0 ) over two adjacent 5 kbp windows, respectively in the 30 and 50 directions and the distances n to the closest ¯ 0 )  S(5 S D S(3 TSS (resp. TES). a Histograms N(jSj) for upward (S > 0, red) and downward (S < 0, black) skew variations. b Histograms of the distances n of upward (red) or downward (black) jumps with jSj  0:1 to the closest TSS (●, red ●) and TES (, red )

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 9 a Number of TSS with an upward jump within jnj (abscissa) for jump amplitudes S > 0.1 (black), 0.15 (dark gray) and 0.2 (light gray). Solid lines correspond to true jump positions while dashed lines to the same analysis when jump positions were randomly drawn along each chromosome [129]. b Among the Ntot (S ) upward jumps of amplitude larger than some threshold S , we plot the proportion of those that are found within 1 kbp (●), 2 kbp (■) or 4 kbp (▲) of the closest TSS vs. the number NTSS of the sodelineated TSS. Curves were obtained by varying S from 0.1 to 0.3 (from right to left). Open symbols correspond to similar analyzes performed on random upward jump and TSS positions

a function of the number of so-delineated TSS [129]. For a given proximity threshold jnj, increasing S  results in a decrease of the number of delineated TSS, characteristic of the right tail of the gene bias pdf. Concomitant to this decrease, we observe an increase of the efficiency up to a maximal value corresponding to some optimal value for S  . For jnj < 2 kbp, we reach a maximal efficiency of 60% for S  D 0:225; 1403 out of 2342 up-

ward jumps delineate a TSS. Given the fact that the actual number of human genes is estimated to be significantly larger ( 30 000) than the number provided by refGene, a large part of the the 40% (939) of upward jumps that have not been associated to a refGene could be explained by this limited coverage. In other words, jumps with sufficiently high amplitude are very good candidates for the location of highly-biased gene promoters. Let us point that out of

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

the above 1403 (resp. 2342) upward jumps, 496 (resp. 624) jumps are still observed at scale a D 200 kbp. We will see in the next section that these jumps are likely to also correspond to replication origins underlying the fact that large upward jumps actually result from the cooperative contributions of both transcription- and replication- associated biases [98,99,100,101]. The observation that 80% (496/624) of the predicted replication origins are co-located with TSS enlightens the existence of a remarkable gene organization at replication origins [101]. To summarize, we have demonstrated the bifractal character of skew DNA walks in the human genome. When using the WT microscope to explore (repeatmasked) scales ranging from 10 to 40 kbp, we have identified two competing homogeneous scale-invariant components characterized by Hölder exponents h1 D 0:78 and h2 D 1 that respectively correspond to LRC colored noise and sharp jumps in the original DNA compositional asymmetry profiles. Remarkably, the so-identified upward (resp. downward) jumps are mainly found at the TSS (resp. TES) of human genes with high transcription bias and thus very likely highly expressed. As illustrated in Fig. 6a, similar bifractal properties are also observed when investigating the 19 mouse autosomes. This suggests that the results reported in this section are general features of mammalian genomes [129]. From the Detection of Relication Origins Using the Wavelet Transform Microscope to the Modeling of Replication in Mammalian Genomes DNA replication is an essential genomic function responsible for the accurate transmission of genetic information through successive cell generations. According to the so-called “replicon” paradigm derived from prokaryotes [131], this process starts with the binding of some “initiator” protein to a specific “replicator” DNA sequence called origin of replication. The recruitment of additional factors initiate the bi-directional progression of two divergent replication forks along the chromosome. One strand is replicated continuously (leading strand), while the other strand is replicated in discrete steps towards the origin (lagging strand). In eukaryotic cells, this event is initiated at a number of replication origins and propagates until two converging forks collide at a terminus of replication [132]. The initiation of different replication origins is coupled to the cell cycle but there is a definite flexibility in the usage of the replication origins at different developmental stages [133,134,135,136, 137]. Also, it can be strongly influenced by the distance and timing of activation of neighboring replication ori-

gins, by the transcriptional activity and by the local chromatin structure [133,134,135,137]. Actually, sequence requirements for a replication origin vary significantly between different eukaryotic organisms. In the unicellular eukaryote Saccharomyces cerevisiae, the replication origins spread over 100–150 bp and present some highly conserved motifs [132]. However, among eukaryotes, S. cerevisiae seems to be the exception that remains faithful to the replicon model. In the fission yeast Schizosaccharomyces pombe, there is no clear consensus sequence and the replication origins spread over at least 800 to 1000 bp [132]. In multicellular organisms, the nature of initiation sites of DNA replication is even more complex. Metazoan replication origins are rather poorly defined and initiation may occur at multiple sites distributed over a thousand of base pairs [138]. The initiation of replication at random and closely spaced sites was repeatedly observed in Drosophila and Xenopus early embryo cells, presumably to allow for extremely rapid S phase, suggesting that any DNA sequence can function as a replicator [136,139,140]. A developmental change occurs around midblastula transition that coincides with some remodeling of the chromatin structure, transcription ability and selection of preferential initiation sites [136,140]. Thus, although it is clear that some sites consistently act as replication origins in most eukaryotic cells, the mechanisms that select these sites and the sequences that determine their location remain elusive in many cell types [141,142]. As recently proposed by many authors [143,144,145], the need to fulfill specific requirements that result from cell diversification may have led multicellular eukaryotes to develop various epigenetic controls over the replication origin selection rather than to conserve specific replication sequence. This might explain that only very few replication origins have been identified so far in multicellular eukaryotes, namely around 20 in metazoa and only about 10 in human [146]. Along the line of this epigenetic interpretation, one might wonder what can be learned about eukaryotic DNA replication from DNA sequence analysis. Replication Induced Factory-Roof Skew Profiles in Mammalian Genomes The existence of replication associated strand asymmetries has been mainly established in bacterial genomes [87,90, 92,93,94]. SGC and STA skews abruptly switch sign (over few kbp) from negative to positive values at the replication origin and in the opposite direction from positive to negative values at the replication terminus. This step-like profile is characteristic of the replicon model [131] (see Fig. 13, left panel). In eukaryotes, the existence of compo-

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sitional biases is unclear and most attempts to detect the replication origins from strand compositional asymmetry have been inconclusive. Several studies have failed to show compositional biases related to replication, and analysis of nucleotide substitutions in the region of the ˇ-globin replication origin in primates does not support the existence of mutational bias between the leading and the lagging strands [92,147,148]. Other studies have led to rather opposite results. For instance, strand asymmetries associated with replication have been observed in the subtelomeric regions of Saccharomyces cerevisiae chromosomes, supporting the existence of replication-coupled asymmetric mutational pressure in this organism [149]. As shown in Fig. 10a for the TOP1 replication origin [146], most of the known replication origins in the human genome correspond to rather sharp (over several kbp) transitions from negative to positive S (STA as well as SGC ) skew values that clearly emerge from the noisy background. But when examining the behavior of the skews at larger distances from the origin, one does not observe a step-like pattern with upward and downward jumps at the origin and termination positions, respectively, as expected for the bacterial replicon model (Fig. 13, left panel). Surprisingly, on both sides of the upward jump, the noisy S profile decreases steadily in the 50 to 30 direction without clear evidence of pronounced downward jumps. As shown in Figs. 10b–10d, sharp upward jumps of amplitude S & 15%, similar to the ones observed for the known replication origins (Fig. 10a), seem to exist also at many other locations along the human chromosomes. But the most striking feature is the fact that in between two neighboring major upward jumps, not only the noisy S profile does not present any comparable downward sharp transition, but it displays a remarkable decreasing linear behavior. At chromosome scale, we thus get jagged S profiles that have the aspect of “factory roofs” [98,100,146]. Note that the jagged S profiles shown in Figs. 10a–10d look somehow disordered because of the extreme variability in the distance between two successive upward jumps, from spacing 50–100 kbp ( 100–200 kbp for the native sequences) mainly in GC rich regions (Fig. 10d), up to 1–2 Mbp ( 2–3 Mbp for native sequences) (Fig. 10c) in agreement with recent experimental studies [150] that have shown that mammalian replicons are heterogeneous in size with an average size 500 kbp, the largest ones being as large as a few Mbp. But what is important to notice is that some of these segments between two successive skew upward jumps are entirely intergenic (Figs. 10a, 10c), clearly illustrating the particular profile of a strand bias resulting solely from replication [98,100,146]. In most other cases, we observe the superimposition of this replication

profile and of the step-like profiles of (+) and () genes (Fig. 7), appearing as upward and downward blocks standing out from the replication pattern (Fig. 10c). Importantly, as illustrated in Figs. 10e, 10f, the factory-roof pattern is not specific to human sequences but is also observed in numerous regions of the mouse and dog genomes [100]. Hence, the presence of strand asymmetry in regions that have strongly diverged during evolution further supports the existence of compostional bias associated with replication in mammalian germ-line cells [98,100,146]. Detecting Replication Origins from the Skew WT Skeleton We have shown in Fig. 10a that experimentally determined human replication origins coincide with large-amplitude upward transitions in noisy skew profiles. The corresponding S ranges between 14% and 38%, owing to possible different replication initiation efficiencies and/or different contributions of transcriptional biases (Sect. “Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription”). Along the line of the jump detection methodology described in Sect. “Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription”, we have checked that upward jumps observed in the skew S at these known replication origins correspond to maxima lines in the WT skeleton that extend to rather large scales a > a D 200 kbp. This observation has led us to select the maxima lines that exist above a  D 200 kbp, i. e. a scale which is smaller than the typical replicon size and larger than the typical gene size [98,100]. In this way, we not only reduce the effect of the noise but we also reduce the contribution of the upward (50 extremity) and backward (30 extremity) jumps associated to the step-like skew pattern induced by transcription only (Sect. “Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription”), to the benefit of maintaining a good sensitivity to replication induced jumps. The detected jump locations are estimated as the positions at scale 20 kbp of the so-selected maxima lines. According to Eq. (11), upward (resp. downward) jumps are identified by the maxima lines corresponding to positive (resp. negative) values of the WT as illustrated in Fig. 5c by the green solid (resp. dashed) maxima lines. When applying this methodology to the total skew S along the repeat-masked DNA sequences of the 22 human autosomal chromosomes, 2415 upward jumps are detected and, as expected, a similar number (namely 2686) of downward jumps. In Fig. 11a are reported the histograms of the amplitude jSj of the so-identified upward (S > 0) and downward (S < 0) jumps respec-

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 10 S profiles along mammalian genome fragments [100,146]. a Fragment of human chromosome 20 including the TOP1 origin (red vertical line). b and c Human chromosome 4 and chromosome 9 fragments, respectively, with low GC content (36%). d Human chromosome 22 fragment with larger GC content (48%). In a and b, vertical lines correspond to selected putative origins (see Subsect. “Detecting Replication Origins from the Skew WT Skeleton”); yellow lines are linear fits of the S values between successive putative origins. Black intergenic regions; red, (+) genes; blue, () genes. Note the fully intergenic regions upstream of TOP1 in a and from positions 5290–6850 kbp in c. e Fragment of mouse chromosome 4 homologous to the human fragment shown in c. f Fragment of dog chromosome 5 syntenic to the human fragment shown in c. In e and f, genes are not represented

tively. These histograms no longer superimpose as previously observed at smaller scales in Fig. 8a, the former being significantly shifted to larger jSj values. When plotting N(jSj > S  ) versus S  in Fig. 11b, we can see that the number of large amplitude upward jumps overexceeds the number of large amplitude downward jumps.

These results confirm that most of the sharp upward transitions in the S profiles in Fig. 10 have no sharp downward transition counterpart [98,100]. This excess likely results from the fact that, contrasting with the prokaryote replicon model (Fig. 13, left panel) where downward jumps result from precisely positioned replication terminations,

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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 11 Statistical analysis of the sharp jumps detected in the S profiles of the 22 human autosomal chromosomes by the WT microscope at ¯ 0 )  S(5 ¯ 0 )j, where the averages were computed over the two scale a D 200 kbp for repeat-masked sequences [98,100]. jSj D jS(3 adjacent 20 kbp windows, respectively, in the 30 and 50 direction from the detected jump location. a Histograms N(jSj) of jSj values. b N(jSj > S ) vs. S . In a and b, the black (resp. red) line corresponds to downward S < 0 (resp. upward S > 0) jumps. R D 3 corresponds to the ratio of upward over downward jumps presenting an amplitude jSj  12:5% (see text)

in mammals termination appears not to occur at specific positions but to be randomly distributed. Accordingly the small number of downward jumps with large jSj is likely to result from transcription (Fig. 5) and not from replication. These jumps are probably due to highly biased genes that also generate a small number of large-amplitude upward jumps, giving rise to false-positive candidate replication origins. In that respect, the number of large downward jumps can be taken as an estimation of the number of false positives. In a first step, we have retained as acceptable a proportion of 33% of false positives. As shown in Fig. 11b, this value results from the selection of upward and downward jumps of amplitude jSj  12:5%, corresponding to a ratio of upward over downward jumps R D 3. Let us notice that the value of this ratio is highly variable along the chromosome [146] and significantly larger than 1 for GCC . 42%. In a final step, we have decided [98,100,146] to retain as putative replication origins upward jumps with jSj  12:5% detected in regions with G+C  42%. This selection leads to a set of 1012 candidates among which our estimate of the proportion of true replication origins is 79% (R D 4:76). In Fig. 12 is shown the mean skew profile calculated in intergenic windows on both sides of the 1012 putative replication origins [100]. This mean skew profile presents a rather sharp transition from negative to positive values when crossing the origin position. To avoid any bias in the skew values that could result from incompletely annotated gene extremities (e. g. 50 and 30 UTRs), we have removed 10-kbp sequences at both ends of all annotated transcripts. As shown in Fig. 12, the removal of

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 12 Mean skew profile of intergenic regions around putative replication origins [100]. The skew S was calculated in 1 kbp windows (Watson strand) around the position (˙300 kbp without repeats) of the 1012 detected upward jumps; 50 and 30 transcript extremities were extended by 0.5 and 2 kbp, respectively (●), or by 10 kbp at both ends (). The abscissa represents the distance (in kbp) to the corresponding origin; the ordinate represents the skews calculated for the windows situated in intergenic regions (mean values for all discontinuities and for 10 consecutive 1 kbp window positions). The skews are given in percent (vertical bars, SEM). The lines correspond to linear fits of the values of the skew () for n < 100 kbp and n > 100 kbp

these intergenic sequences does not significantly modifies the mean skew profile, indicating that the observed values do not result from transcription. On both sides of the jump, we observe a linear decrease of the bias with some

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 13 Model of replication termination [98,100]. Schematic representation of the skew profiles associated with three replication origins O1 , O2 , and O3 ; we suppose that these replication origins are adjacent, bidirectional origins with similar replication efficiency. The abscissa represents the sequence position; the ordinate represents the S value (arbitrary units). Upward (or downward) steps correspond to origin (or termination) positions. For convenience, the termination sites are symmetric relative to O2 . (Left) Three different termination positions T i , T j , and T k , leading to elementary skew profiles Si , Sj , and Sk as predicted by the replicon model [146]. (Center) Superposition of these three profiles. (Right) Superposition of a large number of elementary profiles leading to the final factoryroof pattern. In the simple model, termination occurs with equal probability on both sides of the origins, leading to the linear profile (thick line). In the alternative model, replication termination is more likely to occur at lower rates close to the origins, leading to a flattening of the profile (gray line)

flattening of the profile close to the transition point. Note that, due to (i) the potential presence of signals implicated in replication initiation and (ii) the possible existence of dispersed origins [151], one might question the meaningfulness of this flattening that leads to a significant underestimate of the jump amplitude. Furthermore, according to our detection methodology, the numerical uncertainty on the putative origin position estimate may also contribute to this flattening. As illustrated in Fig. 12, when extrapolating the linear behavior observed at distances > 100 kbp from the jump, one gets a skew of 5.3%, i. e. a value consistent with the skew measured in intergenic regions around the six experimentally known replication origins namely 7:0 ˙ 0:5%. Overall, the detection of sharp upward jumps in the skew profiles with characteristics similar to those of experimentally determined replication origins and with no downward counterpart further supports the existence, in human chromosomes, of replication-associated strand asymmetries, leading to the identification of numerous putative replication origins active in germ-line cells. A Model of Replication in Mammalian Genomes Following the observation of jagged skew profiles similar to factory roofs in Subsect. “Replication Induced Factory-Roof Skew Profiles in Mammalian Genomes”, and the quantitative confirmation of the existence of such (piecewise linear) profiles in the neighborhood of 1012 putative origins in Fig. 12, we have proposed, in Tou-

chon et al. [100] and Brodie of Brodie et al. [98], a rather crude model for replication in the human genome that relies on the hypothesis that the replication origins are quite well positioned while the terminations are randomly distributed. Although some replication terminations have been found at specific sites in S. cerevisiae and to some extent in Schizosaccharomyces pombe [152], they occur randomly between active origins in Xenopus egg extracts [153, 154]. Our results indicate that this property can be extended to replication in human germ-line cells. As illustrated in Fig. 13, replication termination is likely to rely on the existence of numerous potential termination sites distributed along the sequence. For each termination site (used in a small proportion of cell cycles), strand asymmetries associated with replication will generate a steplike skew profile with a downward jump at the position of termination and upward jumps at the positions of the adjacent origins (as in bacteria). Various termination positions will thus correspond to classical replicon-like skew profiles (Fig. 13, left panel). Addition of these profiles will generate the intermediate profile (Fig. 13, central panel). In a simple picture, we can reasonably suppose that termination occurs with constant probability at any position on the sequence. This behavior can, for example, result from the binding of some termination factor at any position between successive origins, leading to a homogeneous distribution of termination sites during successive cell cycles. The final skew profile is then a linear segment decreasing between successive origins (Fig. 13, right panel).

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Let us point out that firing of replication origins during time interval of the S phase [155] might result in some flattening of the skew profile at the origins as sketched in Fig. 13 (right panel, gray curve). In the present state, our results [98,100,146] support the hypothesis of random replication termination in human, and more generally in mammalian cells (Fig. 10), but further analyzes will be necessary to determine what scenario is precisely at work. A Wavelet-Based Methodology to Disentangle Transcription- and Replication-Associated Strand Asymmetries Reveals a Remarkable Gene Organization in the Human Genome During the duplication of eukaryotic genomes that occurs during the S phase of the cell cycle, the different replication origins are not all activated simultaneously [132,135, 138,150,155,156]. Recent technical developments in genomic clone microarrays have led to a novel way of detecting the temporal order of DNA replication [155,156]. The arrays are used to estimate replication timing ratios i. e. ratios between the average amount of DNA in the S phase at a locus along the genome and the usual amount of DNA present in the G1 phase for that locus. These ratios should vary between 2 (throughout the S phase, the amount of DNA for the earliest replicating regions is twice the amount during G1 phase) and 1 (the latest replicating regions are not duplicated until the end of S phase). This approach has been successfully used to generate genomewide maps of replication timing for S. cerevisiae [157], Drosophila melanogaster [137] and human [158]. Very recently, two new analyzes of human chromosome 6 [156] and 22 [155] have improved replication timing resolution from 1 Mbp down to 100 kbp using arrays of overlapping tile path clones. In this section, we report on a very promising first step towards the experimental confirmation of the thousand putative replication origins described in Sect. “From the Detection of Relication Origins Using the Wavelet Transform Microscope to the Modeling of Replication in Mammalian Genomes”. The strategy will consist in mapping them on the recent high-resolution timing data [156] and in checking that these regions replicate earlier than their surrounding [114]. But to provide a convincing experimental test, we need as a prerequisite to extract the contribution of the compositional skew specific to replication. Disentangling Transcription- and Replication-Associated Strand Asymmetries The first step to detect putative replication domains consists in developing a multi-scale pattern recognition

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 14 Wavelet-based analysis of genomic sequences. a Skew profile S of a 9 Mbp repeat-masked fragment of human chromosome 21. b WT of S using 'R (Fig. 1c); TR [S](n; a) is color-coded from dark-blue (min; negative values) to red (max; positive values) through green (null values). Light-blue and purple lines illustrate the detection of two replication domains of significantly different sizes. Note that in b, blue cone-shape areas signing upward jumps point at small scale (top) towards the putative replication origins and that the vertical positions of the WT maxima (red areas) corresponding to the two indicated replication domains match the distance between the putative replication origins (1.6 Mbp and 470 kbp respectively)

methodology based on the WT of the strand compositional asymmetry S using as analyzing wavelet  R (x) (Eq. (12)) that is adapted to perform an objective segmentation of factory-roof skew profiles (Fig. 1c). As illustrated in Fig. 14, the space-scale location of significant maxima values in the 2D WT decomposition (red areas in Fig. 14b) indicates the middle position (spatial location) of candidate replication domains whose size is given by the scale location. In order to avoid false positives, we then check that there does exist a well-defined upward jump at each domain extremity. These jumps appear in Fig. 14b as blue cone-shape areas pointing at small scale to the jumps positions where are located the putative replication origins. Note that because the analyzing wavelet is of zero mean (Eq. (2)), the WT decomposition is insensitive to (global) asymmetry offset. But as discussed in Sect. “Bifractality of Human DNA Strand-Asymmetry Profiles Results from Transcription”, the overall observed skew S also contains some contribution induced by transcription that generates step-like

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

blocks corresponding to (+) and () genes [96,97,129]. Hence, when superimposing the replication serrated and transcription step-like skew profiles, we get the following theoretical skew profile in a replication domain [114]: S(x 0 ) D S R (x 0 ) C S T (x 0 )   X   1 D 2ı  x 0  cg g x0 ; C 2 gene

(23)

where position x 0 within the domain has been rescaled between 0 and 1, ı > 0 is the replication bias, g is the characteristic function for the g th gene (1 when x 0 points within the gene and 0 elsewhere) and cg is its transcriptional bias calculated on the Watson strand (likely to be positive for (+) genes and negative for () genes). The objective is thus to detect human replication domains by delineating, in the noisy S profile obtained at 1 kbp resolution (Fig. 15a), all chromosomal loci where S is well fitted by the theoretical skew profile Eq. (23). In order to enforce strong compatibility with the mammalian replicon model (Subsect. “A Model of Replication in Mammalian Genomes”), we will only retain the domains the most likely to be bordered by putative replication origins, namely those that are delimited by upward jumps corresponding to a transition from a negative S value < 3% to a positive S value > C3%. Also, for each domain so-identified, we will use a least-square fitting procedure to estimate the replication bias ı, and each of the gene transcription bias cg . The resulting 2 value will then be used to select the candidate domains where the noisy S profile is well described by Eq. (23). As illustrated in Fig. 15 for a fragment of human chromosome 6 that contains 4 adjacent replication domains (Fig. 15a), this method provides a very efficient way of disentangling the step-like transcription skew component (Fig. 15b) from the serrated component induced by replication (Fig. 15c). Applying this procedure to the 22 human autosomes, we delineated 678 replication domains of mean length hLi D 1:2 ˙ 0:6 Mbp, spanning 28.3% of the genome and predicted 1060 replication origins. DNA Replication Timing Data Corroborate in silico Human Replication Origin Predictions Chromosome 22 being rather atypical in gene and GC contents, we mainly report here on the correlation analysis [114] between nucleotide compositional skew and timing data for chromosome 6 which is more representative of the whole human genome. Note that timing data for clones completely included in another clone have been removed after checking for timing ratio value consistency leaving

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 15 a Skew profile S of a 4.3 Mbp repeat-masked fragment of human chromosome 6 [114]; each point corresponds to a 1 kbp window: Red, (+) genes; blue, () genes; black, intergenic regions (the color was defined by majority rule); the estimated skew profile (Eq. (23)) is shown in green; vertical lines correspond to the locations of 5 putative replication origins that delimit 4 adjacent domains identified by the wavelet-based methodology. b Transcription-associated skew ST obtained by subtracting the estimated replication-associated profile (green lines in c) from the original S profile in a; the estimated transcription step-like profile (second term on the rhs of Eq. (23)) is shown in green. c Replication-associated skew SR obtained by subtracting the estimated transcription step-like profile (green lines in b) from the original S profile in a; the estimated replication serrated profile (first term in the rhs of Eq. (23)) is shown in green; the light-blue dots correspond to high-resolution tr data

1648 data points. The timing ratio value at each point has been chosen as the median over the 4 closest data points to remove noisy fluctuations resulting from clone heterogeneity (clone length 100 ˙ 51 kbp and distance between successive clone mid-points 104 ˙ 89 kbp), so that the spatial resolution is rather inhomogeneous 300 kbp. Note that using asynchronous cells also results in some smoothing of the data, possibly masking local maxima. Our wavelet-based methodology has identified 54 replication domains in human chromosome 6 [114]; these domains are bordered by 83 putative replication origins among which 25 are common to two adjacent domains. Four of these contiguous domains are shown in Fig. 15. In Fig. 15c, on top of the replication skew profile SR , are

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Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 16 a Average replication timing ratio (˙SEM) determined around the 83 putative replication origins (●), 20 origins with well-defined local maxima () and 10 late replicating origins (4). x is the native distance to the origins in Mbp units [114]. b Histogram of Pearson’s correlation coefficient values between tr and the absolute value of SR over the 38 predicted domains of length L  1 Mbp. The dotted line corresponds to the expected histogram computed with the correlation coefficients between tr and jSj profiles over independent windows randomly positioned along chromosome 6 and with the same length distribution as the 38 detected domains

reported for comparison the high-resolution timing ratio tr data from [156]. The histogram of tr values obtained at the 83 putative origin locations displays a maximum at tr ' htr i ' 1:5 (data not shown) and confirms what is observed in Fig. 15c, namely that a majority of the predicted origins are rather early replicating with tr & 1:4. This contrasts with the rather low tr ('1.2) values observed in domain central regions (Fig. 15c). But there is an even more striking feature in the replication timing profile in Fig. 15c: 4 among the 5 predicted origins correspond, relatively to the experimental resolution, to local maxima of the tr profile. As shown in Fig. 16a, the average tr profile around the 83 putative replication origins decreases regularly on both sides of the origins over a few (4– 6) hundreds kbp confirming statistically that domain borders replicate earlier than their left and right surroundings which is consistent with these regions being true replication origins mostly active early in S phase. In fact, when averaging over the top 20 origins with a well-defined local maximum in the tr profile, htr i displays a faster decrease on both sides of the origin and a higher maximum value 1.55 corresponding to the earliest replicating origins. On the opposite, when averaging tr profiles over the top 10 late replicating origins, we get, as expected, a rather flat mean profile (tr 1:2) (Fig. 16a). Interestingly, these origins are located in rather wide regions of very low GC

content (. 34%, not shown) correlating with chromosomal G banding patterns predominantly composed of GCpoor isochores [159,160]. This illustrates how the statistical contribution of rather flat profiles observed around late replicating origins may significantly affect the overall mean tr profile. Individual inspection of the 38 replication domains with L  1 Mbp shows that, in those domains that are bordered by early replicating origins (tr & 1:4 1:5), the replication timing ratio tr and the absolute value of the replication skew jS R j turn out to be strongly correlated. This is quantified in Fig. 16b by the histogram of the Pearson’s correlation coefficient values that is clearly shifted towards positive values with a maximum at 0.4. Altogether the results of this comparative analysis provide the first experimental verification of in silico replication origins predictions: The detected putative replication domains are bordered by replication origins mostly active in the early S phase, whereas the central regions replicate more likely in late S phase. Gene Organization in the Detected Replication Domains Most of the 1060 putative replication origins that border the detected replication domains are intergenic (77%) and are located near to a gene promoter more often than

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . .

Fractals and Wavelets: What Can We Learn on Transcription and Replication . . . , Figure 17 Analysis of the genes located in the identified replication domains [101]. a Arrows indicate the R+ orientation, i. e. the same orientation as the most frequent direction of putative replication fork progression; R orientation (opposed direction); red, (+) genes; blue, () genes. b Gene density. The density is defined as the number of 50 ends (for (+) genes) or of 30 ends (for () genes) in 50-kbp adjacent windows, divided by the number of corresponding domains. In abscissa, the distance, d, in Mbp, to the closest domain extremity. c Mean gene length. Genes are ranked by their distance, d, from the closest domain extremity, grouped by sets of 150 genes, and the mean length (kbp) is computed for each set. d Relative number of base pairs transcribed in the + direction (red),  direction (blue) and non-transcribed (black) determined in 10-kbp adjacent sequence windows. e Mean expression breadth using EST data [101]

would be expected by chance (data not shown) [101]. The replication domains contain approximately equal numbers of genes oriented in each direction (1511 (+) genes and 1507 () genes). Gene distributions in the 50 halves of domains contain more (+) genes than () genes, regardless of the total number of genes located in the halfdomains (Fig. 17b). Symmetrically, the 30 halves contain more () genes than (+) genes (Fig. 17b). 32.7% of halfdomains contain one gene, and 50.9% contain more than one gene. For convenience, (+) genes in the 50 halves and () genes in the 30 halves are defined as R+ genes (Fig. 17a): their transcription is, in most cases, oriented in the same direction as the putative replication fork progression (genes transcribed in the opposite direction are defined as R genes). The 678 replication domains contain significantly more R+ genes (2041) than R genes (977). Within 50 kbp of putative replication origins, the mean density of R+ genes is 8.2 times greater than that of R genes. This asymmetry weakens progressively with the distance from the putative origins, up to 250 kbp (Fig. 17b). A similar asymmetric pattern is observed when the do-

mains containing duplicated genes are eliminated from the analysis, whereas control domains obtained after randomization of domain positions present similar R+ and R gene density distributions (Supplementary in [101]). The mean length of the R+ genes near the putative origins is significantly greater ( 160 kbp) than that of the R genes ( 50 kbp), however both tend towards similar values ( 70 kbp) at the center of the domain (Fig. 17c). Within 50 kbp of the putative origins, the ratio between the numbers of base pairs transcribed in the R+ and R directions is 23.7; this ratio falls to 1 at the domain centers (Fig. 17d). In Fig. 17e are reported the results of the analysis of the breadth of expression, N t (number of tissues in which a gene is expressed) of genes located within the detected domains [101]. As measured by EST data (similar results are obtained by SAGE or microarray data [101]), N t is found to decrease significantly from the extremities to the center in a symmetrical manner in the 50 and 30 halfdomains (Fig. 17e). Thus, genes located near the putative replications origins tend to be widely expressed whereas those located far from them are mostly tissue-specific.

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To summarize, the results reported in this section provide the first demonstration of quantitative relationships in the human genome between gene expression, orientation and distance from putative replication origins [101]. A possible key to the understanding of this complex architecture is the coordination between replication and transcription [101]. The putative replication origins would mostly be active early in the S phase in most tissues. Their activity could result from particular genomic context involving transcription factor binding sites and/or from the transcription of their neighboring housekeeping genes. This activity could also be associated with an open chromatin structure, permissive to early replication and gene expression in most tissues [161,162,163,164]. This open conformation could extend along the first gene, possibly promoting the expression of further genes. This effect would progressively weaken with the distance from the putative replication origin, leading to the observed decrease in expression breadth. This model is consistent with a number of data showing that in metazoans, ORC and RNA polymerase II colocalize at transcriptional promoter regions [165], and that replication origins are determined by epigenetic information such as transcription factor binding sites and/or transcription [166,167,168,169]. It is also consistent with studies in Drosophila and humans that report correlation between early replication timing and increased probability of expression [137,155,156,165,170]. Furthermore, near the putative origins bordering the replication domains, transcription is preferentially oriented in the same direction as replication fork progression. This coorientation is likely to reduce head-on collisions between the replication and transcription machineries, which may induce deleterious recombination events either directly or via stalling of the replication fork [171,172]. In bacteria, co-orientation of transcription and replication has been observed for essential genes, and has been associated with a reduction in head-on collisions between DNA and RNA polymerases [173]. It is noteworthy that in human replication domains such co-orientation usually occurs in widely-expressed genes located near putative replication origins. Near domain centers, head-on collisions may occur in 50% of replication cycles, regardless of the transcription orientation, since there is no preferential orientation of the replication fork progression in these regions. However, in most cell types, there should be few head-on collisions due to the low density and expression breadth of the corresponding genes. Selective pressure to reduce head-on collisions may thus have contributed to the simultaneous and coordinated organization of gene orientation and expression breadth along the detected replication domains [101].

Future Directions From a statistical multifractal analysis of nucleotide strand asymmetries in mammalian genomes, we have revealed the existence of jumps in the noisy skew profiles resulting from asymmetries intrinsic to the transcription and replication processes [98,100]. This discovery has led us to extend our 1D WTMM methodology to an adapted multi-scale pattern recognition strategy in order to detect putative replication domains bordered by replication origins [101,114]. The results reported in this manuscript show that directly from the DNA sequence, we have been able to reveal the existence in the human genome (and very likely in all mammalian genomes), of regions bordered by early replicating origins in which gene position, orientation and expression breadth present a high level of organization, possibly mediated by the chromatin structure. These results open new perspectives in DNA sequence analysis, chromatin modeling as well as in experiment. From a bioinformatic and modeling point of view, we plan to study the lexical and structural characteristics of our set of putative origins. In particular we will search for conserved sequence motifs in these replication initiation zones. Using a sequence-dependent model of DNA-histones interactions, we will develop physical studies of nucleosome formation and diffusion along the DNA fiber around the putative replication origins. These bioinformatic and physical studies, performed for the first time on a large number of replication origins, should shed light on the processes at work during the recognition of the replication initiation zone by the replication machinery. From an experimental point of view, our study raises new opportunities for future experiments. The first one concerns the experimental validation of the predicted replication origins (e. g. by molecular combing of DNA molecules [174]), which will allow us to determine precisely the existence of replication origins in given genome regions. Large scale study of all candidate origins is in current progress in the laboratory of O. Hyrien (École Normale Supérieure, Paris). The second experimental project consists in using Atomic Force Microscopy (AFM) [175] and Surface Plasmon Resonance Microscopy (SPRM) [176] to visualize and study the structural and mechanical properties of the DNA double helix, the nucleosomal string and the 30 nm chromatin fiber around the predicted replication origins. This work is in current progress in the experimental group of F. Argoul at the Laboratoire Joliot– Curie (ENS, Lyon) [83]. Finally the third experimental perspective concerns in situ studies of replication origins. Using fluorescence techniques (FISH chromosome painting [177]), we plan to study the distributions and dynam-

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ics of origins in the cell nucleus, as well as chromosome domains potentially associated with territories and their possible relation to nuclear matrix attachment sites. This study is likely to provide evidence of chromatin rosette patterns as suggested in [146]. This study is under progress in the molecular biology experimental group of F. Mongelard at the Laboratoire Joliot–Curie. Acknowledgments We thank O. Hyrien, F. Mongelard and C. Moskalenko for interesting discussions. This work was supported by the Action Concertée Incitative Informatique, Mathématiques, Physique en Biologie Moléculaire 2004 under the project “ReplicOr”, the Agence Nationale de la Recherche under the project “HUGOREP” and the program “Emergence” of the Conseil Régional Rhônes-Alpes and by the Programme d’Actions Intégrées Tournesol. Bibliography Primary Literature 1. Goupillaud P, Grossmann A, Morlet J (1984) Cycle-octave and related transforms in seismic signal analysis. Geoexploration 23:85–102 2. Grossmann A, Morlet J (1984) Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J Math Anal 15:723–736 3. Arneodo A, Argoul F, Bacry E, Elezgaray J, Freysz E, Grasseau G, Muzy J-F, Pouligny B (1992) Wavelet transform of fractals. In: Meyer Y (ed) Wavelets and applications. Springer, Berlin, pp 286–352 4. Arneodo A, Argoul F, Elezgaray J, Grasseau G (1989) Wavelet transform analysis of fractals: Application to nonequilibrium phase transitions. In: Turchetti G (ed) Nonlinear dynamics. World Scientific, Singapore, pp 130–180 5. Arneodo A, Grasseau G, Holschneider M (1988) Wavelet transform of multifractals. Phys Rev Lett 61:2281–2284 6. Holschneider M (1988) On the wavelet transform of fractal objects. J Stat Phys 50:963–993 7. Holschneider M, Tchamitchian P (1990) Régularité locale de la fonction non-différentiable de Riemann. In: Lemarié PG (ed) Les ondelettes en 1989. Springer, Berlin, pp 102–124 8. Jaffard S (1989) Hölder exponents at given points and wavelet coefficients. C R Acad Sci Paris Sér. I 308:79–81 9. Jaffard S (1991) Pointwise smoothness, two-microlocalization and wavelet coefficients. Publ Mat 35:155–168 10. Mallat S, Hwang W (1992) Singularity detection and processing with wavelets. IEEE Trans Info Theory 38:617–643 11. Mallat S, Zhong S (1992) Characterization of signals from multiscale edges. IEEE Trans Patt Recog Mach Intell 14:710–732 12. Arneodo A, Bacry E, Muzy J-F (1995) The thermodynamics of fractals revisited with wavelets. Physica A 213:232–275 13. Bacry E, Muzy J-F, Arneodo A (1993) Singularity spectrum of fractal signals from wavelet analysis: Exact results. J Stat Phys 70:635–674

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Wavelets Abry P (1997) Ondelettes et turbulences. Diderot Éditeur, Art et Sciences, Paris Arneodo A, Argoul F, Bacry E, Elezgaray J, Muzy J-F (1995) Ondelettes, multifractales et turbulences: de l’ADN aux croissances cristallines. Diderot Éditeur, Art et Sciences, Paris Chui CK (1992) An introduction to wavelets. Academic Press, Boston Combes J-M, Grossmann A, Tchamitchian P (eds) (1989) Wavelets. Springer, Berlin Daubechies I (1992) Ten lectures on wavelets. SIAM, Philadelphia Erlebacher G, Hussaini MY, Jameson LM (eds) (1996) Wavelets: Theory and applications. Oxford University Press, Oxford Farge M, Hunt JCR, Vassilicos JC (eds) (1993) Wavelets, fractals and Fourier. Clarendon Press, Oxford Flandrin P (1993) Temps-Fréquence. Hermès, Paris Holschneider M (1996) Wavelets: An analysis tool. Oxford University Press, Oxford Jaffard S, Meyer Y, Ryan RD (eds) (2001) Wavelets: Tools for science and technology. SIAM, Philadelphia Lemarie PG (ed) (1990) Les ondelettes en 1989. Springer, Berlin

DNA and Chromatin Alberts B, Watson J (1994) Molecular biology of the cell, 3rd edn. Garland Publishing, New-York Calladine CR, Drew HR (1999) Understanding DNA. Academic Press, San Diego Graur D, Li WH (1999) Fundamentals of molecular evolution. Sinauer Associates, Sunderland Hartl DL, Jones EW (2001) Genetics: Analysis of genes and genomes. Jones and Bartlett, Sudbury Kolchanov NA, Lim HA (1994) Computer analysis of genetic macromolecules: Structure, function and evolution. World Scientific, Singapore Kornberg A, Baker TA (1992) DNA Replication. WH Freeman, NewYork Lewin B (1994) Genes V. Oxford University Press, Oxford Sudbery P (1998) Human molecular genetics. Addison Wesley, Singapore Van Holde, KE (1988) Chromatin. Springer, New-York Watson JD, Gilman M, Witkowski J, Zoller M (1992) Recombinant DNA. Freeman, New-York Wolfe AP (1998) Chromatin structure and function, 3rd edn. Academic Press, London

Fractal and Transfractal Scale-Free Networks

Fractal and Transfractal Scale-Free Networks HERNÁN D. ROZENFELD, LAZAROS K. GALLOS, CHAOMING SONG, HERNÁN A. MAKSE Levich Institute and Physics Department, City College of New York, New York, USA Article Outline Glossary Definition of the Subject Introduction Fractality in Real-World Networks Models: Deterministic Fractal and Transfractal Networks Properties of Fractal and Transfractal Networks Future Directions Acknowledgments Appendix: The Box Covering Algorithms Bibliography Glossary Degree of a node Number of edges incident to the node. Scale-free network Network that exhibits a wide (usually power-law) distribution of the degrees. Small-world network Network for which the diameter increases logarithmically with the number of nodes. Distance The length (measured in number of links) of the shortest path between two nodes. Box Group of nodes. In a connected box there exists a path within the box between any pair of nodes. Otherwise, the box is disconnected. Box diameter The longest distance in a box. Definition of the Subject The explosion in the study of complex networks during the last decade has offered a unique view in the structure and behavior of a wide range of systems, spanning many different disciplines [1]. The importance of complex networks lies mainly in their simplicity, since they can represent practically any system with interactions in a unified way by stripping complicated details and retaining the main features of the system. The resulting networks include only nodes, representing the interacting agents and links, representing interactions. The term ‘interactions’ is used loosely to describe any possible way that causes two nodes to form a link. Examples can be real physical links, such as the wires connecting computers in the Internet or roads connecting cities, or alternatively they may be virtual

links, such as links in WWW homepages or acquaintances in societies, where there is no physical medium actually connecting the nodes. The field was pioneered by the famous mathematician P. Erd˝os many decades ago, when he greatly advanced graph theory [27]. The theory of networks would have perhaps remained a problem of mathematical beauty, if it was not for the discovery that a huge number of everyday life systems share many common features and can thus be described through a unified theory. The remarkable diversity of these systems incorporates artificially or man-made technological networks such as the Internet and the World Wide Web (WWW), social networks such as social acquaintances or sexual contacts, biological networks of natural origin, such as the network of protein interactions of Yeast [1,36], and a rich variety of other systems, such as proximity of words in literature [48], items that are bought by the same people [16] or the way modules are connected to create a piece of software, among many others. The advances in our understanding of networks, combined with the increasing availability of many databases, allows us to analyze and gain deeper insight into the main characteristics of these complex systems. A large number of complex networks share the scale-free property [1,28], indicating the presence of few highly connected nodes (usually called hubs) and a large number of nodes with small degree. This feature alone has a great impact on the analysis of complex networks and has introduced a new way of understanding these systems. This property carries important implications in many everyday life problems, such as the way a disease spreads in communities of individuals, or the resilience and tolerance of networks under random and intentional attacks [19,20,21,31,59]. Although the scale-free property holds an undisputed importance, it has been shown to not completely determine the global structure of networks [6]. In fact, two networks that obey the same distribution of the degrees may dramatically differ in other fundamental structural properties, such as in correlations between degrees or in the average distance between nodes. Another fundamental property, which is the focus of this article, is the presence of self-similarity or fractality. In simpler terms, we want to know whether a subsection of the network looks much the same as the whole [8,14,29,66]. In spite of the fact that in regular fractal objects the distinction between self-similarity and fractality is absent, in network theory we can distinguish the two terms: in a fractal network the number of boxes of a given size that are needed to completely cover the network scales with the box size as a power law, while a self-similar network is defined as a network whose degree distribution remains invariant under renormaliza-

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tion of the network (details on the renormalization process will be provided later). This essential result allows us to better understand the origin of important structural properties of networks such as the power-law degree distribution [35,62,63]. Introduction Self-similarity is a property of fractal structures, a concept introduced by Mandelbrot and one of the fundamental mathematical results of the 20th century [29,45,66]. The importance of fractal geometry stems from the fact that these structures were recognized in numerous examples in Nature, from the coexistence of liquid/gas at the critical point of evaporation of water [11,39,65], to snowflakes, to the tortuous coastline of the Norwegian fjords, to the behavior of many complex systems such as economic data, or the complex patterns of human agglomeration [29,66]. Typically, real world scale-free networks exhibit the small-world property [1], which implies that the number of nodes increases exponentially with the diameter of the network, rather than the power-law behavior expected for self-similar structures. For this reason complex networks were believed to not be length-scale invariant or self-similar. In 2005, C. Song, S. Havlin and H. Makse presented an approach to analyze complex networks, that reveals their self-similarity [62]. This result is achieved by the application of a renormalization procedure which coarse-grains the system into boxes containing nodes within a given size [62,64]. As a result, a power-law relation between the number of boxes needed to cover the network and the size of the box is found, defining a finite self-similar exponent. These fundamental properties, which are shown for the WWW, cellular and protein-protein interaction networks, help to understand the emergence of the scale-free property in complex networks. They suggest a common self-organization dynamics of diverse networks at different scales into a critical state and in turn bring together previously unrelated fields: the statistical physics of complex networks with renormalization group, fractals and critical phenomena. Fractality in Real-World Networks The study of real complex networks has revealed that many of them share some fundamental common properties. Of great importance is the form of the degree distribution for these networks, which is unexpectedly wide. This means that the degree of a node may assume values that span many decades. Thus, although the majority of nodes have a relatively small degree, there is a finite probability

that a few nodes will have degree of the order of thousands or even millions. Networks that exhibit such a wide distribution P(k) are known as scale-free networks, where the term refers to the absence of a characteristic scale in the degree k. This distribution very often obeys a powerlaw form with a degree exponent , usually in the range 2 < < 4 [2], P(k) k  :

(1)

A more generic property, that is usually inherent in scalefree networks but applies equally well to other types of networks, such as in Erd˝os–Rényi random graphs, is the small-world feature. Originally discovered in sociological studies [47], it is the generalization of the famous ‘six degrees of separation’ and refers to the very small network diameter. Indeed, in small-world networks a very small number of steps is required to reach a given node starting from any other node. Mathematically this is expressed by the slow (logarithmic) increase of the average diame¯ with the total number of nodes N, ter of the network, `, ¯ ` ln N, where ` is the shortest distance between two nodes and defines the distance metric in complex networks [2,12,27,67], namely, ¯

N e`/`0 ;

(2)

where `0 is a characteristic length. These network characteristics have been shown to apply in many empirical studies of diverse systems [1,2,28]. The simple knowledge that a network has the scale-free and/or small-world property already enables us to qualitatively recognize many of its basic properties. However, structures that have the same degree exponents may still differ in other aspects [6]. For example, a question of fundamental importance is whether scale-free networks are also self-similar or fractals. The illustrations of scale-free networks (see, e. g,. Figs. 1 and 2b) seem to resemble traditional fractal objects. Despite this similarity, Eq. (2) definitely appears to contradict a basic property of fractality: fast increase of the diameter with the system size. Moreover, a fractal object should be self-similar or invariant under a scale transformation, which is again not clear in the case of scale-free networks where the scale has necessarily limited range. So, how is it even possible that fractal scalefree networks exist? In the following, we will see how these seemingly contradictory aspects can be reconciled. Fractality and Self-Similarity The classical theory of self-similarity requires a power-law relation between the number of nodes N and the diameter of a fractal object ` [8,14]. The fractal dimension can

Fractal and Transfractal Scale-Free Networks

Fractal and Transfractal Scale-Free Networks, Figure 1 Representation of the Protein Interaction Network of Yeast. The colors show different subgroups of proteins that participate in different functionality classes [36]

be calculated using either box-counting or cluster-growing techniques [66]. In the first method the network is covered with NB boxes of linear size `B . The fractal dimension or box dimension dB is then given by [29]: B : NB `d B

(3)

In the second method, instead of covering the network with boxes, a random seed node is chosen and nodes centered at the seed are grown so that they are separated by a maximum distance `. The procedure is then repeated by choosing many seed nodes at random and the average “mass” of the resulting clusters, hMc i (defined as the number of nodes in the cluster) is calculated as a function of ` to obtain the following scaling: hMc i `d f ;

(4)

defining the fractal cluster dimension df [29]. If we use Eq. (4) for a small-world network, then Eq. (2) readily implies that df D 1. In other words, these networks cannot be characterized by a finite fractal dimension, and should be regarded as infinite-dimensional objects. If this were true, though, local properties in a part of the network would not be able to represent the whole system. Still, it is also well established that the scale-free nature is simi-

lar in different parts of the network. Moreover, a graphical representation of real-world networks allows us to see that those systems seem to be built by attaching (following some rule) copies of itself. The answer lies in the inherent inhomogeneity of the network. In the classical case of a homogeneous system (such as a fractal percolation cluster) the degree distribution is very narrow and the two methods described above are fully equivalent, because of this local neighborhood invariance. Indeed, all boxes in the box-covering method are statistically similar with each other as well as with the boxes grown when using the cluster-growing technique, so that Eq. (4) can be derived from Eq. (3) and dB D df . In inhomogeneous systems, though, the local environment can vary significantly. In this case, Eqs. (3) and (4) are no longer equivalent. If we focus on the box-covering technique then we want to cover the entire network with the minimum possible number of boxes NB (`B ), where the distance between any two nodes that belong in a box is smaller than `B . An example is shown in Fig. 2a using a simple 8-node network. After we repeat this procedure for different values of `B we can plot NB vs. `B . When the box-covering method is applied to real large-scale networks, such as the WWW [2] (http://www. nd.edu/~networks), the network of protein interaction

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Fractal and Transfractal Scale-Free Networks

of H. sapiens and E. coli [25,68] and several cellular networks [38,52], then they follow Eq. (3) with a clear power-law, indicating the fractal nature of these systems (Figs. 3a,b,c). On the other hand when the method is applied to other real world networks such as the Internet [24] or the Barabási–Albert network [7], they do not satisfy Eq. (3), which manifests that these networks are not fractal. The reason behind the discrepancy in the fractality of homogeneous and inhomogeneous systems can be better clarified studying the mass of the boxes. For a given `B value, the average mass of a box hMB (`B )i is hMB (`B )i 

N `Bd B ; NB (`B )

(5)

as also verified in Fig. 3 for several real-world networks. On the other hand, the average performed in the cluster growing method (averaging over single boxes without tiling the system) gives rise to an exponential growth of the mass hMc (`)i e`/`1 ;

Fractal and Transfractal Scale-Free Networks, Figure 2 The renormalization procedure for complex networks. a Demonstration of the method for different `B and different stages in a network demo. The first column depicts the original network. The system is tiled with boxes of size `B (different colors correspond to different boxes). All nodes in a box are connected by a minimum distance smaller than the given `B . For instance, in the case of `B D 2, one identifies four boxes which contain the nodes depicted with colors red, orange, white, and blue, each containing 3, 2, 1, and 2 nodes, respectively. Then each box is replaced by a single node; two renormalized nodes are connected if there is at least one link between the unrenormalized boxes. Thus we obtain the network shown in the second column. The resulting number of boxes needed to tile the network, NB (`B ), is plotted in Fig. 3 vs. `B to obtain dB as in Eq. (3). The renormalization procedure is applied again and repeated until the network is reduced to a single node (third and fourth columns for different `B ). b Three stages in the renormalization scheme applied to the entire WWW. We fix the box size to `B D 3 and apply the renormalization for four stages. This corresponds, for instance, to the sequence for the network demo depicted in the second row in part a of this figure. We color the nodes in the web according to the boxes to which they belong

(6)

in accordance with the small-world effect, Eq. (2). Correspondingly, the probability distribution of the mass of the boxes MB using box-covering is very broad, while the cluster-growing technique leads to a narrow probability distribution of Mc . The topology of scale-free networks is dominated by several highly connected hubs—the nodes with the largest degree—implying that most of the nodes are connected to the hubs via one or very few steps. Therefore, the average performed in the cluster growing method is biased; the hubs are overrepresented in Eq. (6) since almost every node is a neighbor of a hub, and there is always a very large probability of including the same hubs in all clusters. On the other hand, the box covering method is a global tiling of the system providing a flat average over all the nodes, i. e. each part of the network is covered with an equal probability. Once a hub (or any node) is covered, it cannot be covered again. In conclusion, we can state that the two dominant methods that are routinely used for calculations of fractality and give rise to Eqs. (3) and (4) are not equivalent in scale-free networks, but rather highlight different aspects: box covering reveals the self-similarity, while cluster growth reveals the small-world effect. The apparent contradiction is due to the hubs being used many times in the latter method. Scale-free networks can be classified into three groups: (i) pure fractal, (ii) pure small-world and (iii) a mixture

Fractal and Transfractal Scale-Free Networks

Fractal and Transfractal Scale-Free Networks, Figure 3 Self-similar scaling in complex networks. a Upper panel: Log-log plot of the NB vs. `B revealing the self-similarity of the WWW according to Eq. (3). Lower panel: The scaling of s(`B ) vs. `B according to Eq. (9). b Same as a but for two protein interaction networks: H. sapiens and E. coli. Results are analogous to b but with different scaling exponents. c Same as a for the cellular networks of A. fulgidus, E. coli and C. elegans. d Internet. Log-log plot of NB (`B ). The solid line shows that the internet [24] is not a fractal network since it does not follow the power-law relation of Eq. (5). e Same as d for the Barabási–Albert model network [7] with m D 3 and m D 5

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between fractal and small-world. (i) A fractal network satisfies Eq. (3) at all scales, meaning that for any value of `B , the number of boxes always follows a power-law (examples are shown in Fig. 3a,b,c). (ii) When a network is a pure small-world, it never satisfies Eq. (3). Instead, NB follows an exponential decay with `B and the network cannot be regarded as fractal. Figures 3d and 3e show two examples of pure small-world networks. (iii) In the case of mixture between fractal and small-world, Eq. (3) is satisfied up to some cut-off value of `B , above which the fractality breaks down and the small-world property emerges. The smallworld property is reflected in the plot of NB vs. `B as an exponential cut-off for large `B . We can also understand the coexistence of the smallworld property and the fractality through a more intuitive approach. In a pure fractal network the length of a path between any pair of nodes scales as a power-law with the number of nodes in the network. Therefore, the diameter L also follows a power-law, L N 1/d B . If one adds a few shortcuts (links between randomly chosen nodes), many paths in the network are drastically shortened and the small-world property emerges as L LogN. In spite of this fact, for shorter scales, `B  L, the network still behaves as a fractal. In this sense, we can say that globally the network is small-world, but locally (for short scales) the network behaves as a fractal. As more shortcuts are added, the cut-off in a plot of NB vs. `B appears for smaller `B , until the network becomes a pure small-world for which all paths lengths increase logarithmically with N. The reasons why certain networks have evolved towards a fractal or non-fractal structure will be described later, together with models and examples that provide additional insight into the processes involved.

The method works as follows. Start by fixing the value of `B and applying the box-covering algorithm in order to cover the entire network with boxes (see Appendix). In the renormalized network each box is replaced by a single node and two nodes are connected if there existed at least one connection between the two corresponding boxes in the original network. The resulting structure represents the first stage of the renormalized network. We can apply the same procedure to this new network, as well, resulting in the second renormalization stage network, and so on until we are left with a single node. The second column of the panels in Fig. 2a shows this step in the renormalization procedure for the schematic network, while Fig. 2b shows the results for the same procedure applied to the entire WWW for `B D 3. The renormalized network gives rise to a new probability distribution of links, P(k 0 ) (we use a prime 0 to denote quantities in the renormalized network). This distribution remains invariant under the renormalization: P(k) ! P(k 0 ) (k 0 ) :

(7)

Fig. 4 supports the validity of this scale transformation by showing a data collapse of all distributions with the same

according to (7) for the WWW. Here, we present the basic scaling relations that characterize renormalizable networks. The degree k 0 of each node in the renormalized network can be seen to scale with

Renormalization Renormalization is one of the most important techniques in modern Statistical Physics [17,39,58]. The idea behind this procedure is to continuously create smaller replicas of a given object, retaining at the same time the essential structural features, and hoping that the coarse-grained copies will be more amenable to analytic treatment. The idea for renormalizing the network emerges naturally from the concept of fractality described above. If a network is self-similar, then it will look more or less the same under different scales. The way to observe these different length-scales is based on renormalization principles, while the criterion to decide on whether a renormalized structure retains its form is the invariance of the main structural features, expressed mainly through the degree distribution.

Fractal and Transfractal Scale-Free Networks, Figure 4 Invariance of the degree distribution of the WWW under the renormalization for different box sizes, `B . We show the data collapse of the degree distributions demonstrating the self-similarity at different scales. The inset shows the scaling of k 0 D s(`B )k for different `B , from where we obtain the scaling factor s(`B ). Moreover, renormalization for a fixed box size (`B D 3) is applied, until the network is reduced to a few nodes. It was found that P(k) is invariant under these multiple renormalizations procedures

Fractal and Transfractal Scale-Free Networks

the largest degree k in the corresponding original box as 0

k ! k D s(`B )k :

(8)

This equation defines the scaling transformation in the connectivity distribution. Empirically, it was found that the scaling factor s(< 1) scales with `B with a new expod nent, dk , as s(`B ) `B k , so that d k

k 0 `B

k;

(9)

This scaling is verified for many networks, as shown in Fig. 3. The exponents , dB , and d k are not all independent from each other. The proof starts from the density balance equation n(k)dk D n0 (k 0 )dk 0 , where n(k) D N P(k) is the number of nodes with degree k and n0 (k 0 ) D N 0 P(k 0 ) is the number of nodes with degree k 0 after the renormalization (N 0 is the total number of nodes in the renormalized network). Substituting Eq. (8) leads to N 0 D s  1 N. Since the total number of nodes in the renormalized network is the number of boxes needed to cover the unrenormalized network at any given `B we have the identity N 0 D NB (`B ). Finally, from Eqs. (3) and (9) one obtains the relation between the three indexes dB : (10)

D1C dk The use of Eq. (10) yields the same exponent as that obtained in the direct calculation of the degree distribution. The significance of this result is that the scale-free properties characterized by can be related to a more fundamental length-scale invariant property, characterized by the two new indexes dB and dk . We have seen, thus, that concepts introduced originally for the study of critical phenomena in statistical physics, are also valid in the characterization of a different class of phenomena: the topology of complex networks. A large number of scale-free networks are fractals and an even larger number remain invariant under a scale-transformation. The influence of these features on the network properties will be delayed until the sixth chapter, after we introduce some algorithms for efficient numerical calculations and two theoretical models that give rise to fractal networks.

the q ! 1 limit of the Potts model. Unfortunately, in those days the importance of the power-law degree distribution and the concept of fractal and non-fractal complex networks were not known. Much work has been done on these types of hierarchical networks. For example, in 1984, M. Kaufman and R. Griffiths made use of Berker and Ostlund’s model to study the percolation phase transition and its percolation exponents [22,37,40,41]. Since the late 90s, when the importance of the powerlaw degree distribution was first shown [1] and after the finding of C. Song, S. Havlin and H. Makse [62], many hierarchical networks that describe fractality in complex networks have been proposed. These artificial models are of great importance since they provide insight into the origins and fundamental properties that give rise to the fractality and non-fractality of networks.

The Song–Havlin–Makse Model The correlations between degrees in a network [46,49, 50,54] are quantified through the probability P(k1 ; k2 ) that a node of degree k1 is connected to another node of degree k2 . In Fig. 5 we can see the degree correlation profile R(k1 ; k2 ) D P(k1 ; k2 )/Pr (k1 ; k2 ) of the cellular metabolic network of E. coli [38] (known to be a fractal network) and for the Internet at the router level [15] (a non-fractal network), where Pr (k1 ; k2 ) is obtained by randomly swapping the links without modifying the degree distribution. Figure 5 shows a dramatic difference between the two networks. The network of E. coli, that is a fractal network, presents an anti-correlation of the degrees (or dissasortativity [49,50]), which means that mostly high degree nodes are linked to low degree nodes. This property leads to fractal networks. On the other hand, the Internet exhibits a high correlation between degrees leading to a non-fractal network. With this idea in mind, in 2006 C. Song, S. Havlin and H. Makse presented a model that elucidates the way

Models: Deterministic Fractal and Transfractal Networks The first model of a scale-free fractal network was presented in 1979 when N. Berker and S. Ostlund [9] proposed a hierarchical network that served as an exotic example where renormalization group techniques yield exact results, including the percolation phase transition and

Fractal and Transfractal Scale-Free Networks, Figure 5 Degree correlation profile for a the cellular metabolic network of E. coli, and b the Internet at the router level

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Fractal and Transfractal Scale-Free Networks

Fractal and Transfractal Scale-Free Networks, Figure 6 The model grows from a small network, usually two nodes connected to each other. During each step and for every link in the system, each endpoint of a link produces m offspring nodes (in this drawing m D 3). In this case, with probability e D 1 the original link is removed and x new links between randomly selected nodes of the new generation are added. Notice that the case of x D 1 results in a tree structure, while loops appear for x > 1

new nodes must be connected to the old ones in order to build a fractal, a non-fractal network, or a mixture between fractal and non-fractal network [63]. This model shows that, indeed, the correlations between degrees of the nodes are a determinant factor for the fractality of a network. This model was later extended [32] to allow loops in the network, while preserving the self-similarity and fractality properties. The algorithm is as follows (see Fig. 6): In generation n D 0, start with two nodes connected by one link. Then, generation n C 1 is obtained recursively by attaching m new nodes to the endpoints of each link l of generation n. In addition, with probability e remove link l and add x new links connecting pairs of new nodes attached to the endpoints of l. The degree distribution, diameter and fractal dimension can be easily calculated. For example, if e D 1 (pure fractal network), the degree distribution follows a powerlaw P(k) k  with exponent D 1Clog(2mC x)/logm and the fractal dimension is dB D log(2m C x)/logm. The diameter L scales, in this case, as power of the number of nodes as L N 1/d B [63,64]. Later, in Sect. “Properties of Fractal and Transfractal Networks”, several topological properties are shown for this model network.

Fractal and Transfractal Scale-Free Networks, Figure 7 (u; v)-flowers with u C v D 4( D 3). a u D 1 (dotted line) and v D 3 (broken line). b u D 2 and v D 2. The graphs may also be iterated by joining four replicas of generation n at the hubs A and B, for a, or A and C, for b

(u; v)-Flowers In 2006, H. Rozenfeld, S. Havlin and D. ben-Avraham proposed a new family of recursive deterministic scalefree networks, the (u; v)-flowers, that generalize both, the original scale-free model of Berker and Ostlund [9] and the pseudo-fractal network of Dorogovstev, Goltsev and Mendes [26] and that, by appropriately varying its two parameters u and v, leads to either fractal networks or non-fractal networks [56,57]. The algorithm to build the (u; v)-flowers is the following: In generation n D 1 one starts with a cycle graph (a ring) consisting of u C v  w links and nodes (other choices are possible). Then, generation n C 1 is obtained recursively by replacing each link by two parallel paths of u and v links long. Without loss of generality, u  v. Examples of (1; 3)- and (2; 2)-flowers are shown in Fig. 7. The DGM network corresponds to the special case of u D 1 and v D 2 and the Berker and Ostlund model corresponds to u D 2 and v D 2. An essential property of the (u; v)-flowers is that they are self-similar, as evident from an equivalent method of construction: to produce generation nC1, make w D uCv

Fractal and Transfractal Scale-Free Networks

copies of the net in generation n and join them at the hubs. The number of links of a (u; v)-flower of generation n is M n D (u C v)n D w n ;

(11)

and the number of nodes is Nn D

w  2  w  wn C : w1 w1

(12)

The degree distribution of the (u; v)-flowers can also be easily obtained since by construction, (u; v)-flowers have only nodes of degree k D 2m , m D 1; 2; : : : ; n. As in the DGM case, (u; v)-flowers follow a scale-free degree distribution, P(k) k  , of degree exponent

D1C

ln(u C v) : ln 2

(13)

Recursive scale-free trees may be defined in analogy to the flower nets. If v is even, one obtains generation n C 1 of a (u; v)-tree by replacing every link in generation n with a chain of u links, and attaching to each of its endpoints chains of v/2 links. Figure 8 shows how this works for the (1; 2)-tree. If v is odd, attach to the endpoints (of the chain of u links) chains of length (v ˙ 1)/2. The trees may be also constructed by successively joining w replicas at the appropriate hubs, and they too are self-similar. They share many of the fundamental scaling properties with (u; v)-flowers: Their degree distribution is also scale-free, with the same degree exponent as (u; v)-flowers. The self-similarity of (u; v)-flowers, coupled with the fact that different replicas meet at a single node, makes them amenable to exact analysis by renormalization techniques. The lack of loops, in the case of (u; v)-trees, further simplifies their analysis [9,13,56,57]. Dimensionality of the (u; v)-Flowers There is a vast difference between (u; v)-nets with u D 1 and u > 1. If u D 1 the diameter Ln of the nth generation flower scales linearly with n. For example, L n for the (1; 2)-flower [26] and L n D 2n for the (1; 3)-flower. It is easy to see that the diameter of the (1; v)-flower, for v odd, is L n D (v  1)n C (3  v)/2, and, in general one can show that L n (v  1)n. For u > 1, however, the diameter grows as a power of n. For example, for the (2; 2)-flower we find L n D 2n , and, more generally, the diameter satisfies L n u n . To summarize, ( Ln

(v  1)n

u D1;

un

u >1;

flowers :

(14)

Fractal and Transfractal Scale-Free Networks, Figure 8 The (1; 2)-tree. a Each link in generation n is replaced by a chain of u D 1 links, to which ends one attaches chains of v/2 D 1 links. b Alternative method of construction highlighting selfsimilarity: u C v D 3 replicas of generation n are joined at the hubs. c Generations n D 1; 2; 3

Similar results are quite obvious for the case of (u; v)-trees, where ( Ln

vn

u D1;

un

u >1;

trees :

(15)

Since N n (u C v)n (Eq. (12)), we can recast these relations as ( L

ln N

u D1;

N ln u/ ln(uCv)

u > 1:

(16)

Thus, (u; v)-nets are small world only in the case of u D 1. For u > 1, the diameter increases as a power of N, just as in finite-dimensional objects, and the nets are in fact fractal. For u > 1, the change of mass upon the rescaling of length by a factor b is N(bL) D b d B N(L) ;

(17)

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Fractal and Transfractal Scale-Free Networks

where dB is the fractal dimension [8]. In this case, N(uL) D (u C v)N(L), so dB D

ln(u C v) ; ln u

u >1:

(18)

Transfinite Fractals Small world nets, such as (1; v)nets, are infinite-dimensional. Indeed, their mass (N, or M) increases faster than any power (dimension) of their diameter. Also, note that a naive application of (4) to u ! 1 yields df ! 1. In the case of (1; v)-nets one can use their weak self-similarity to define a new measure of dimensionality, d˜f , characterizing how mass scales with diameter: `d˜f

N(L C `) D e

N(L) :

(19)

Instead of a multiplicative rescaling of length, L 7! bL, a slower additive mapping, L 7! L C `, that reflects the small world property is considered. Because the exponent d˜f usefully distinguishes between different graphs of infinite dimensionality, d˜f has been termed the transfinite fractal dimension of the network. Accordingly, objects that are self-similar and have infinite dimension (but finite transfinite dimension), such as the (1; v)-nets, are termed transfinite fractals, or transfractals, for short. For (1; v)-nets, we see that upon ‘zooming in’ one generation level the mass increases by a factor of w D 1 C v, while the diameter grows from L to L C v  1 (for flowers), or to L C v (trees). Hence their transfractal dimension is ( ln(1Cv) (1; v)-trees ; v ˜ df D ln(1Cv) (20) (1; v)-flowers : v1 There is some arbitrariness in the selection of e as the base of the exponential in the definition (19). However the base is inconsequential for the sake of comparison between dimensionalities of different objects. Also, scaling relations between various transfinite exponents hold, irrespective of the choice of base: consider the scaling relation of Eq. (10) valid for fractal scale-free nets of degree exponent [62,63]. For example, in the fractal (u; v)-nets (with u > 1) renormalization reduces lengths by a factor b D u and all degrees are reduced by a factor of 2, so b d k D 2. Thus dk D ln 2/ ln u, and since dB D ln(u C v)/ ln u and

D 1Cln(uCv)/ ln 2, as discussed above, the relation (10) is indeed satisfied. For transfractals, renormalization reduces distances by an additive length, `, and we express the self-similarity manifest in the degree distribution as ˜

˜

P0 (k) D e`d k P(e`d k k) ;

(21)

where d˜k is the transfinite exponent analogous to dk . Renormalization of the transfractal (1; v)-nets reduces the

link lengths by ` D v  1 (for flowers), or ` D v (trees), while all degrees are halved. Thus, ( ln 2 (1; v)-trees ; ˜ dk D lnv 2 (1; v)-flowers : v1 Along with (20), this result confirms that the scaling relation

D1C

d˜f d˜k

(22)

is valid also for transfractals, and regardless of the choice of base. A general proof of this relation is practically identical to the proof of (10) [62], merely replacing fractal with transfractal scaling throughout the argument. For scale-free transfractals, following m D L/` renormalizations the diameter and mass reduce to order one, ˜ and the scaling (19) implies L m`, N em`d f , so that 1 L ˜ ln N ; df in accordance with their small world property. At the same ˜ ˜ ˜ time the scaling (21) implies K e m`d k , or K N d k /d f . 1/( Using the scaling relation (22), we rederive K N 1) , which is indeed valid for scale-free nets in general, be they fractal or transfractal. Properties of Fractal and Transfractal Networks The existence of fractality in complex networks immediately calls for the question of what is the importance of such a structure in terms of network properties. In general, most of the relevant applications seem to be modified to a larger or lesser extent, so that fractal networks can be considered to form a separate network sub-class, sharing the main properties resulting from the wide distribution of regular scale-free networks, but at the same time bearing novel properties. Moreover, from a practical point of view a fractal network can be usually more amenable to analytic treatment. In this section we summarize some of the applications that seem to distinguish fractal from non-fractal networks. Modularity Modularity is a property closely related to fractality. Although this term does not have a unique well-defined definition we can claim that modularity refers to the existence of areas in the network where groups of nodes share some common characteristics, such as preferentially connecting within this area (the ‘module’) rather than to the rest of

Fractal and Transfractal Scale-Free Networks

the network. The isolation of modules into distinct areas is a complicated task and in most cases there are many possible ways (and algorithms) to partition a network into modules. Although networks with significant degree of modularity are not necessarily fractals, practically all fractal networks are highly modular in structure. Modularity naturally emerges from the effective ‘repulsion’ between hubs. Since the hubs are not directly connected to each other, they usually dominate their neighborhood and can be considered as the ‘center of mass’ for a given module. The nodes surrounding hubs are usually assigned to this module. The renormalization property of self-similar networks is very useful for estimating how modular a given network is, and especially for how this property is modified under varying scales of observation. We can use a simple definition for modularity M, based on the idea that the number of links connecting nodes within a module, Lin i , is higher than the number of link connecting nodes in different modules, Lout i . For this purpose, the boxes that result from the boxcovering method at a given length-scale `B are identified as the network modules for this scale. This partitioning assumes that the minimization of the number of boxes corresponds to an increase of modularity, taking advantage of the idea that all nodes within a box can reach each other within less than `B steps. This constraint tends to assign the largest possible number of nodes in a given neighborhood within the same box, resulting in an optimized modularity function. A definition of the modularity function M that takes advantage of the special features of the renormalization process is, thus, the following [32]: M(`B ) D

NB Lin 1 X i ; NB Lout i

(23)

iD1

where the sum is over all the boxes. The value of M through Eq. (23) for a given `B value is of small usefulness on its own, though. We can gather more information on the network structure if we measure M for different values of `B . If the dependence of M on `B has the form of a power-law, as if often the case in practice, then we can define the modularity exponent d M through M(`B ) `Bd M :

(24)

The exponent d M carries the important information of how modularity scales with the length, and separates modular from non-modular networks. The value of d M is easy

to compute in a d-dimensional lattice, since the number of d links within any module scales with its bulk, as Lin i `B and the number of links outside the module scale with the length of its interface, i. e. Lout `Bd1 . So, the resulting i scaling is M `B i. e. d M D 1. This is also the borderline value that separates non-modular structures (d M < 1) from modular ones (d M > 1). For the Song–Havlin–Makse fractal model introduced in the previous section, a module can be identified as the neighborhood around a central hub. In the simplest version with x D 1, the network is a tree, with well-defined modules. Larger values of x mean that a larger number of links are connecting different modules, creating more loops and ‘blurring’ the discreteness of the modules, so that we can vary the degree of modularity in the network. For this model, it is also possible to analytically calculate the value of the exponent d M . During the growth process at step t, the diameter in the network model increases multiplicatively as L(t C 1) D 3L(t). The number of links within a module grows with 2m C x (each node on the side of one link gives rise to m new links and x extra links connect the new nodes), while the number of links pointing out of a module is by definition proportional to x. Thus, the modularity M(`B ) of a network is proportional to (2m C x)/x. Equation (24) can then be used to calculate d M for the model: 2m C x 3d M ; x

(25)

which finally yields

dM

  ln 2 mx C 1 D : ln 3

(26)

So, in this model the important quantity that determines the degree of modularity in the system is the ratio of the growth parameters m/x. Most of the real-life networks that have been measured display some sort of modular character, i. e. d M > 1, although many of them have values very close to 1. Only in a few cases we have observed exponents d M < 1. Most interesting is, though, the case of d M values much larger than 1, where a large degree of modularity is observed and this trend is more pronounced for larger length-scales. The importance of modularity as described above can be demonstrated in biological networks. There, it has been suggested that the boxes may correspond to functional modules and in protein interaction networks, for example, there may be an evolution drive of the system behind the development of its modular structure.

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Robustness Shortly after the discovery of the scale-free property, the first important application of their structure was perhaps their extreme resilience to removal of random nodes [3,10,19,21,57,60,61]. At the same time such a network was found to be quite vulnerable to an intentional attack, where nodes are removed according to decreasing order of their degree [20,30]. The resilience of a network is usually quantified through the size of the largest remaining connected cluster Smax (p), when a fraction p of the nodes has been removed according to a given strategy. At a critical point pc where this size becomes equal to Smax (pc ) ' 0, we consider that the network has been completely disintegrated. For the random removal case, this threshold is pc ' 1, i. e. practically all nodes need to be destroyed. In striking contrast, for intentional attacks pc is in general of the order of only a few percent, although the exact value depends on the system details. Fractality in networks considerably strengthens the robustness against intentional attacks, compared to nonfractal networks with the same degree exponent . In Fig. 9 the comparison between two such networks clearly shows that the critical fraction pc increases almost 4 times from pc ' 0:02 (non-fractal topology) to pc ' 0:09 (fractal topology). These networks have the same exponent, the same number of links, number of nodes, number of loops and the same clustering coefficient, differing only in whether hubs are directly connected to each other. The fractal property, thus, provides a way of increasing resistance against the network collapse, in the case of a targeted attack. The main reason behind this behavior is the dispersion of hubs in the network. A hub is usually a central node that helps other nodes to connect to the main body of the system. When the hubs are directly connected to each other, this central core is easy to destroy in a targeted attack leading to a rapid collapse of the network. On the contrary, isolating the hubs into different areas helps the network to retain connectivity for longer time, since destroying the hubs now is not similarly catastrophic, with most of the nodes finding alternative paths through other connections. The advantage of increased robustness derived from the combination of modular and fractal network character, may provide valuable hints on why most biological networks have evolved towards a fractal architecture (better chance of survival against lethal attacks). Degree Correlations We have already mentioned the importance of hub-hub correlations or anti-correlations in fractality. Generalizing

Fractal and Transfractal Scale-Free Networks, Figure 9 Vulnerability under intentional attack of a non-fractal Song– Makse–Havlin network (for e D 0) and a fractal Song–Makse– Havlin network (for e D 1). The plot shows the relative size of the largest cluster, S, and the average size of the remaining isolated clusters, hsi as a function of the removal fraction f of the largest hubs for both networks

this idea to nodes of any degree, we can ask what is the joint degree probability P(k1 ; k2 ) that a randomly chosen link connects two nodes with degree k1 and k2 , respectively. Obviously, this is a meaningful question only for networks with a wide degree distribution, otherwise the answer is more or less trivial with all nodes having similar degrees. A similar and perhaps more useful quantity is the conditional degree probability P(k1 jk2 ), defined as the probability that a random link from a node having degree k2 points to a node with degree k1 . In general, the following balance condition is satisfied k2 P(k1 jk2 )P(k2 ) D k1 P(k2 jk1 )P(k1 ) :

(27)

It is quite straightforward to calculate P(k1 jk2 ) for completely uncorrelated networks. In this case, P(k1 jk2 ) does not depend on k2 , and the probability to chose a node with degree k1 becomes simply P(k1 jk2 ) D k1 P(k1 )/hk1 i. In the case where degree-degree correlations are present, though, the calculation of this function is very difficult, even when restricting ourselves to a direct numerical evaluation, due to the emergence of huge fluctuations. We can still estimate this function, though, using again the self-similarity principle. If we consider that the function P(k1 ; k2 ) remains invariant under the network renormalization scheme described above, then it is possible to

Fractal and Transfractal Scale-Free Networks

show that [33] ( 1)  k2

P(k1 ; k2 ) k1

(k1 > k2 ) ;

(28)

and similarly P(k1 jk2 ) k1( 1) k2(  C1) (k1 > k2 ) ;

(29)

In the above equations we have also introduced the correlation exponent , which characterizes the degree of correlations in a network. For example, the case of uncorrelated networks is described by the value D  1. The exponent can be measured quite accurately using an appropriate quantity. For this purpose, we can introduce a measure such as R1 P(kjk2 )dk2 Eb (k)  bkR 1 ; (30) bk P(k)dk which estimates the probability that a node with degree k has neighbors with degree larger than bk, and b is an arbitrary parameter that has been shown not to influence the results. It is easy to show that Eb (k)

k 1 D k (  ) : k 1

(31)

This relation allows us to estimate for a given network, after calculating the quantity Eb (k) as a function of k. The above discussion can be equally applied to both fractal and non-fractal networks. If we restrict ourselves to fractal networks, then we can develop our theory a bit further. If we consider the probability E(`B ) that the largest degree node in each box is connected directly with the other largest degree nodes in other boxes (after optimally covering the network), then this quantity scales as a power law with `B : d e

E(`B ) `B

;

(32)

where d e is a new exponent describing the probability of hub-hub connection [63]. The exponent , which describes correlations over any degree, is related to d e , which refers to correlations between hubs only. The resulting relation is D2C

de de D 2 C (  1) : dk dB

(33)

For an infinite fractal dimension dB ! 1, which is the onset of non-fractal networks that cannot be described by the above arguments, we have the limiting case of D 2. This value separates fractal from non-fractal networks, so that fractality is indicated by > 2. Also, we have seen

that the line D 1 describes networks for which correlations are minimal. Measurements of many real-life networks have verified the above statements, where networks with > 2 having been clearly characterized as fractals with alternate methods. All non-fractal networks have values of < 2 and the distance from the D  1 line determines how much stronger or weaker the correlations are, compared to the uncorrelated case. In short, using the self-similarity principle makes it possible to gain a lot of insight on network correlations, a notoriously difficult task otherwise. Furthermore, the study of correlations can be reduced to the calculation of a single exponent , which is though capable of delivering a wealth of information on the network topological properties. Diffusion and Resistance Scale-free networks have been described as objects of infinite dimensionality. For a regular structure this statement would suggest that one can simply use the known diffusion laws for d D 1. Diffusion on scale-free structures, however, is much harder to study, mainly due to the lack of translational symmetry in the system and different local environments. Although exact results are still not available, the scaling theory on fractal networks provides the tools to better understand processes, such as diffusion and electric resistance. In the following, we describe diffusion through the average first-passage time TAB , which is the average time for a diffusing particle to travel from node A to node B. At the same time, assuming that each link in the network has an electrical resistance of 1 unit, we can describe the electrical properties through the resistance between the two nodes A and B, RAB . The connection between diffusion (first-passage time) and electric networks has long been established in homogeneous systems. This connection is usually expressed through the Einstein relation [8]. The Einstein relation is of great importance because it connects a static quantity RAB with a dynamic quantity TAB . In other words, the behavior of a diffusing particle can be inferred by simply having knowledge of a static topological property of the network. In any renormalizable network the scaling of T and R follows the form: T0 R0  w ; D `d D `B ; B T R

(34)

where T 0 (R0 ) and T(R) are the first-passage time (resistance) for the renormalized and original networks, respectively. The dynamical exponents dw and  characterize the

649

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Fractal and Transfractal Scale-Free Networks

scaling in any lattice or network that remains invariant under renormalization. The Einstein relation relates these two exponents through the dimensionality of the substrate dB , according to: dw D  C dB :

(35)

The validity of this relation in inhomogeneous complex networks, however, is not yet clear. Still, in fractal and transfractal networks there are many cases where this relation has been proved to be valid, hinting towards a wider applicability. For example, in [13,56] it has been shown that the Einstein Relation [8] in (u; v)-flowers and (u; v)-trees is valid for any u and v, that is for both fractal and transfractal networks. In general, in terms of the scaling theory we can study diffusion and resistance (or conductance) in a similar manner [32]. Because of the highly inhomogeneous character of the structure, though, we are interested in how these quantities behave as a function of the end-node degrees k1 and k2 when they are separated by a given distance `. Thus, we are looking for the full dependence of T(`; k1 ; k2 ) and R(`; k1 ; k2 ). Obviously, for lattices or networks with narrow degree distribution there is no degree dependence and those results should be a function of ` only. For self-similar networks, we can rewrite Eq. (34) above as  0 d w /d B  0 /d B R0 N N T0 ; ; (36) D D T N R N where we have taken into account Eq. (3). This approach offers the practical advantage that the variation of N 0 /N is larger than the variation of `B , so that the exponents calculation can be more accurate. To calculate these exponents, we fix the box size `B and we measure the diffusion time T and resistance R between any two points in a network before and after renormalization. If for every such pair we plot the corresponding times and resistances in T 0 vs. T and R0 vs. R plots, as shown in Fig. 10, then all these points fall in a narrow area, suggesting a constant value for the ratio T 0 /T over the entire network. Repeating this procedure for different `B values yields other ratio values. The plot of these ratios vs. N 0 /N (Fig. 11) finally exhibits a power-law dependence, verifying Eq. (36). We can then easily calculate the exponents dw and  from the slopes in the plot, since the dB exponent is already known through the standard box-covering methods. It has been shown that the results for many different networks are consistent, within statistical error, with the Einstein relation [32,56]. The dependence on the degrees k1 , k2 and the distance ` can also be calculated in a scaling form using the

Fractal and Transfractal Scale-Free Networks, Figure 10 Typical behavior of the probability distributions for the resistance R vs. R0 and the diffusion time T vs. T 0 , respectively, for a given `B value. Similar plots for other `B values verify that the ratios of these quantities during a renormalization stage are roughly constant for all pairs of nodes in a given biological network

Fractal and Transfractal Scale-Free Networks, Figure 11 Average value of the ratio of resistances R/R0 and diffusion times T/T 0 , as measured for different `B values (each point corresponds to a different value of `B ). Results are presented for both biological networks, and two fractal network models with different dM values. The slopes of the curves correspond to the exponents /dB (top panel) and dw /dB (bottom panel)

self-similarity properties of fractal networks. After renormalization, a node with degree k in a given network, will d have a degree k 0 D `B k k according to Eq. (9). At the same time all distances ` are scaled down according to `0 D `/`B . This means that Eqs. (36) can be written as 

R0 (`0 ; k10 ; k20 ) D `B R(`; k1 ; k2 )

(37)

w T 0 (`0 ; k10 ; k20 ) D `d T(`; k1 ; k2 ) : B

(38)

Substituting the renormalized quantities we get: d k

R0 (`1 B `; `B

d k

k 1 ; `B



k2 ) D `B R(`; k1 ; k2 ) :

(39)

Fractal and Transfractal Scale-Free Networks

The above equation holds for all values of `B , so we can 1/d select this quantity to be `B D k2 k . This constraint allows us to reduce the number of variables in the equation, with the final result: 0 1  ` k 1 d R @ 1 ; ; 1A D k2 k R(`B ; k1 ; k2 ) : (40) k2 dk k2 This equation suggests a scaling for the resistance R: 0 1  ` k d 1 (41) R(`; k1 ; k2 ) D k2 k f R @ 1 ; A ; k2 dk k2 where f R () is an undetermined function. All the above arguments can be repeated for the diffusion time, with a similar expression: 0 1 dw ` k dk 1 T(`; k1 ; k2 ) D k2 f T @ 1 ; A ; (42) k2 dk k2 where the form of the right-hand function may be different. The final result for the scaling form is Eqs. (41) and (42), which is also supported by the numerical data collapse in Fig. 12. Notice that in the case of homogeneous networks, where there is almost no k-dependence, the unknown functions in the rhs reduce to the forms f R (x; 1) D x  , f T (x; 1) D x d w , leading to the well-established classical relations R ` and T `d w . Future Directions Fractal networks combine features met in fractal geometry and in network theory. As such, they present many unique aspects. Many of their properties have been well-studied and understood, but there is still a great amount of open and unexplored questions remaining to be studied. Concerning the structural aspects of fractal networks, we have described that in most networks the degree distribution P(k), the joint degree distribution P(k1 ; k2 ) and a number of other quantities remain invariant under renormalization. Are there any quantities that are not invariable, and what would their importance be? Of central importance is the relation of topological features with functionality. The optimal network covering leads to the partitioning of the network into boxes. Do these boxes carry a message other than nodes proximity? For example, the boxes could be used as an alternative definition for separated communities, and fractal methods could be used as a novel method for community detection in networks [4,5,18,51,53].

Fractal and Transfractal Scale-Free Networks, Figure 12 Rescaling of a the resistance and b the diffusion time according to Eqs. (41) and (42) for the protein interaction network of yeast (upper symbols) and the Song–Havlin–Makse model for e D 1 (lower filled symbols). The data for PIN have been vertically shifted upwards by one decade for clarity. Each symbol corresponds to a fixed ratio k1 /k2 and the different colors denote a different value for k 1 . Inset: Resistance R as a function of distance `, before rescaling, for constant ratio k1 /k2 D 1 and different k 1 values

The networks that we have presented are all static, with no temporal component, and time evolution has been ignored in all our discussions above. Clearly, biological networks, the WWW, and other networks have grown (and continue to grow) from some earlier simpler state to their present fractal form. Has fractality always been there or has it emerged as an intermediate stage obeying certain evolutionary drive forces? Is fractality a stable condition or growing networks will eventually fall into a non-fractal form? Finally, we want to know what is the inherent reason behind fractality. Of course, we have already described how hub-hub anti-correlations can give rise to fractal networks. However, can this be directly related to some underlying mechanism, so that we gain some information on the process? In general, in Biology we already have some idea on the advantages of adopting a fractal structure. Still, the question remains: why fractality exists in certain networks and not in others? Why both fractal and non-fractal networks are needed? It seems that we will be able to increase our knowledge for the network evolutionary mechanisms through fractality studies. In conclusion, a deeper understanding of the self-similarity, fractality and transfractality of complex networks will help us analyze and better understand many fundamental properties of real-world networks.

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Fractal and Transfractal Scale-Free Networks

Acknowledgments We acknowledge support from the National Science Foundation. Appendix: The Box Covering Algorithms The estimation of the fractal dimension and the self-similar features in networks have become standard properties in the study of real-world systems. For this reason, in the last three years many box covering algorithms have been proposed [64,69]. This section presents four of the main algorithms, along with a brief discussion on the advantages and disadvantages that they offer. Recalling the original definition of box covering by Hausdorff [14,29,55], for a given network G and box size `B , a box is a set of nodes where all distances ` i j between any two nodes i and j in the box are smaller than `B . The minimum number of boxes required to cover the entire network G is denoted by NB . For `B D 1, each box encloses only 1 node and therefore, NB is equal to the size of the network N. On the other hand, NB D 1 for `B  `max B , is the diameter of the network plus one. where `max B The ultimate goal of a box-covering algorithm is to find the minimum number of boxes NB (`B ) for any `B . It has been shown that this problem belongs to the family of NP-hard problems [34], which means that the solution cannot be achieved in polynomial time. In other words, for a relatively large network size, there is no algorithm that can provide an exact solution in a reasonably short amount of time. This limitation requires treating the box covering problem with approximations, using for example optimization algorithms. The Greedy Coloring Algorithm The box-covering problem can be mapped into another NP-hard problem [34]: the graph coloring problem. An algorithm that approximates well the optimal solution of this problem was presented in [64]. For an arbitrary value of `B , first construct a dual network G 0 , in which two nodes are connected if the distance between them in G (the original network) is greater or equal than `B . Figure 13 shows an example of a network G which yields such a dual network G 0 for `B D 3 (upper row of Fig. 13). Vertex coloring is a well-known procedure, where labels (or colors) are assigned to each vertex of a network, so that no edge connects two identically colored vertices. It is clear that such a coloring in G 0 gives rise to a natural box covering in the original network G, in the sense that vertices of the same color will necessarily form a box since the distance between them must be less than `B . Accordingly, the minimum number of boxes NB (G) is equal

Fractal and Transfractal Scale-Free Networks, Figure 13 Illustration of the solution for the network covering problem via mapping to the graph coloring problem. Starting from G (upper left panel) we construct the dual network G0 (upper right panel) for a given box size (here `B D 3), where two nodes are connected if they are at a distance `  `B . We use a greedy algorithm for vertex coloring in G0 , which is then used to determine the box covering in G, as shown in the plot

to the minimum required number of colors (or the chromatic number) in the dual network G 0 , (G 0 ). In simpler terms, (a) if the distance between two nodes in G is greater than `B these two neighbors cannot belong in the same box. According to the construction of G 0 , these two nodes will be connected in G 0 and thus they cannot have the same color. Since they have a different color they will not belong in the same box in G. (b) On the contrary, if the distance between two nodes in G is less than `B it is possible that these nodes belong in the same box. In G 0 these two nodes will not be connected and it is allowed for these two nodes to carry the same color, i. e. they may belong to the same box in G, (whether these nodes will actually be connected depends on the exact implementation of the coloring algorithm). The algorithm that follows both constructs the dual network G 0 and assigns the proper node colors for all `B values in one go. For this implementation a two-dimenis needed, whose values sional matrix c i` of size N  `max B represent the color of node i for a given box size ` D `B . 1. Assign a unique id from 1 to N to all network nodes, without assigning any colors yet. 2. For all `B values, assign a color value 0 to the node with id=1, i. e. c1` D 0. 3. Set the id value i D 2. Repeat the following until i D N. (a) Calculate the distance ` i j from i to all the nodes in the network with id j less than i.

Fractal and Transfractal Scale-Free Networks

(b) Set `B D 1 (c) Select one of the unused colors c j` i j from all nodes j < i for which ` i j  `B . This is the color c i`B of node i for the given `B value. (d) Increase `B by one and repeat (c) until `B D `max B . (e) Increase i by 1. The results of the greedy algorithm may depend on the original coloring sequence. The quality of this algorithm was investigated by randomly reshuffling the coloring sequence and applying the greedy algorithm several times and in different models [64]. The result was that the probability distribution of the number of boxes NB (for all box sizes `B ) is a narrow Gaussian distribution, which indicates that almost any implementation of the algorithm yields a solution close to the optimal. Strictly speaking, the calculation of the fractal dimenB sion dB through the relation NB `d is valid only for B the minimum possible value of NB , for any given `B value, so any box covering algorithm must aim to find this minimum NB . Although there is no rule to determine when this minimum value has been actually reached (since this would require an exact solution of the NP-hard coloring problem) it has been shown [23] that the greedy coloring algorithm can, in many cases, identify a coloring sequence which yields the optimal solution. Burning Algorithms This section presents three box covering algorithms based on more traditional breadth-first search algorithm. A box is defined as compact when it includes the maximum possible number of nodes, i. e. when there do not exist any other network nodes that could be included in this box. A connected box means that any node in the box can be reached from any other node in this box, without having to leave this box. Equivalently, a disconnected box denotes a box where certain nodes can be reached by other nodes in the box only by visiting nodes outside this box. For a demonstration of these definitions see Fig. 14. Burning with the Diameter `B , and the Compact-BoxBurning (CBB) Algorithm The basic idea of the CBB algorithm for the generation of a box is to start from a given box center and then expand the box so that it includes the maximum possible number of nodes, satisfying at the same time the maximum distance between nodes in the box `B . The CBB algorithm is as follows (see Fig. 15): 1. Initially, mark all nodes as uncovered. 2. Construct the set C of all yet uncovered nodes.

Fractal and Transfractal Scale-Free Networks, Figure 14 Our definitions for a box that is a non-compact for `B D 3, i. e. could include more nodes, b compact, c connected, and d disconnected (the nodes in the right box are not connected in the box). e For this box, the values `B D 5 and rB D 2 verify the relation `B D 2rB C 1. f One of the pathological cases where this relation is not valid, since `B D 3 and rB D 2

Fractal and Transfractal Scale-Free Networks, Figure 15 Illustration of the CBB algorithm for `B D 3. a Initially, all nodes are candidates for the box. b A random node is chosen, and nodes at a distance further than `B from this node are no longer candidates. c The node chosen in b becomes part of the box and another candidate node is chosen. The above process is then repeated until the box is complete

3. Choose a random node p from the set of uncovered nodes C and remove it from C. 4. Remove from C all nodes i whose distance from p is `pi  `B , since by definition they will not belong in the same box. 5. Repeat steps (3) and (4) until the candidate set is empty. 6. Repeat from step (2) until all the network has been covered.

Random Box Burning In 2006, J. S. Kim et al. presented a simple algorithm for the calculation of fractal dimension in networks [42,43,44]: 1. Pick a randomly chosen node in the network as a seed of the box. 2. Search using breath-first search algorithm until distance lB from the seed. Assign all newly burned nodes

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to the new box. If no new node is found, discard and start from (1) again. 3. Repeat (1) and (2) until all nodes have a box assigned. This Random Box Burning algorithm has the advantage of being a fast and simple method. However, at the same time there is no inherent optimization employed during the network coverage. Thus, this simple Monte-Carlo method is almost certain that will yield a solution far from the optimal and one needs to implement many different realizations and only retain the smallest number of boxes found out of all these realizations. Burning with the Radius rB , and the MaximumExcluded-Mass-Burning (MEMB) Algorithm A box of size `B includes nodes where the distance between any pair of nodes is less than `B . It is possible, though, to grow a box from a given central node, so that all nodes in the box are within distance less than a given box radius rB (the maximum distance from a central node). This way, one can still recover the same fractal properties of a network. For the original definition of the box, `B corresponds to the box diameter (maximum distance between any two nodes in the box) plus one. Thus, `B and rB are connected through the simple relation `B D 2rB C 1. In general this relation is exact for loopless configurations, but in general there may exist cases where this equation is not exact (Fig. 14). The MEMB algorithm always yields the optimal solution for non scale-free homogeneous networks, since the choice of the central node is not important. However, in inhomogeneous networks with wide-tailed degree distribution, such as scale-free networks, this algorithm fails to achieve an optimal solution because of the presence of hubs. The MEMB, as a difference from the Random Box Burning and the CBB, attempts to locate some optimal central nodes which act as the burning origins for the boxes. It contains as a special case the choice of the hubs as centers of the boxes, but it also allows for low-degree nodes to be burning centers, which sometimes is convenient for finding a solution closer to the optimal. In the following algorithm we use the basic idea of box optimization, in which each box covers the maximum possible number of nodes. For a given burning radius rB , we define the excluded mass of a node as the number of uncovered nodes within a chemical distance less than rB . First, calculate the excluded mass for all the uncovered nodes. Then, seek to cover the network with boxes of maximum excluded mass. The details of this algorithm are as follows (see Fig. 17):

Fractal and Transfractal Scale-Free Networks, Figure 16 Burning with the radius rB from a a hub node or b a non-hub node results in very different network coverage. In a we need just one box of rB D 1 while in b 5 boxes are needed to cover the same network. This is an intrinsic problem when burning with the radius. c Burning with the maximum distance `B (in this case `B D 2rB C 1 D 3) we avoid this situation, since independently of the starting point we would still obtain NB D 1

Fractal and Transfractal Scale-Free Networks, Figure 17 Illustration of the MEMB algorithm for rB D 1. Upper row: Calculation of the box centers a We calculate the excluded mass for each node. b The node with maximum mass becomes a center and the excluded masses are recalculated. c A new center is chosen. Now, the entire network is covered with these two centers. Bottom row: Calculation of the boxes d Each box includes initially only the center. Starting from the centers we calculate the distance of each network node to the closest center. e We assign each node to its nearest box

1. Initially, all nodes are marked as uncovered and noncenters. 2. For all non-center nodes (including the already covered nodes) calculate the excluded mass, and select the node p with the maximum excluded mass as the next center. 3. Mark all the nodes with chemical distance less than rB from p as covered. 4. Repeat steps (2) and (3) until all nodes are either covered or centers. Notice that the excluded mass has to be updated in each step because it is possible that it has been modified during this step. A box center can also be an already covered node, since it may lead to a larger box mass. After the above procedure, the number of selected centers coincides with the

Fractal and Transfractal Scale-Free Networks

number of boxes NB that completely cover the network. However, the non-center nodes have not yet been assigned to a given box. This is performed in the next step: 1. Give a unique box id to every center node. 2. For all nodes calculate the “central distance”, which is the chemical distance to its nearest center. The central distance has to be less than rB , and the center identification algorithm above guarantees that there will always exist such a center. Obviously, all center nodes have a central distance equal to 0. 3. Sort the non-center nodes in a list according to increasing central distance. 4. For each non-center node i, at least one of its neighbors has a central distance less than its own. Assign to i the same id with this neighbor. If there exist several such neighbors, randomly select an id from these neighbors. Remove i from the list. 5. Repeat step (4) according to the sequence from the list in step (3) for all non-center nodes. Comparison Between Algorithms The choice of the algorithm to be used for a problem depends on the details of the problem itself. If connected boxes are a requirement, MEMB is the most appropriate algorithm; but if one is only interested in obtaining the fractal dimension of a network, the greedy-coloring or the random box burning are more suitable since they are the fastest algorithms.

Fractal and Transfractal Scale-Free Networks, Figure 18 Comparison of the distribution of NB for 104 realizations of the four network covering methods presented in this paper. Notice that three of these methods yield very similar results with narrow distributions and comparable minimum values, while the random burning algorithm fails to reach a value close to this minimum (and yields a broad distribution)

As explained previously, any algorithm should intend to find the optimal solution, that is, find the minimum number of boxes that cover the network. Figure 18 shows the performance of each algorithm. The greedy-coloring, the CBB and MEMB algorithms exhibit a narrow distribution of the number of boxes, showing evidence that they cover the network with a number of boxes that is close to the optimal solution. Instead, the Random Box Burning returns a wider distribution and its average is far above the average of the other algorithms. Because of the great ease and speed with which this technique can be implemented, it would be useful to show that the average number of covering boxes is overestimated by a fixed proportionality constant. In that case, despite the error, the predicted number of boxes would still yield the correct scaling and fractal dimension. Bibliography 1. Albert R, Barabási A-L (2002) Rev Mod Phys 74:47; Barabási AL (2003) Linked: how everything is connected to everything else and what it means. Plume, New York; Newman MEJ (2003) SIAM Rev 45:167; Dorogovtsev SN, Mendes JFF (2002) Adv Phys 51:1079; Dorogovtsev SN, Mendes JFF (2003) Evolution of networks: from biological nets to the internet and WWW. Oxford University Press, Oxford; Bornholdt S, Schuster HG (2003) Handbook of graphs and networks. Wiley-VCH, Berlin; PastorSatorras R, Vespignani A (2004) Evolution and structure of the internet. Cambridge University Press, Cambridge; Amaral LAN, Ottino JM (2004) Complex networks – augmenting the framework for the study of complex systems. Eur Phys J B 38: 147–162 2. Albert R, Jeong H, Barabási A-L (1999) Diameter of the world wide web. Nature 401:130–131 3. Albert R, Jeong H, Barabási AL (2000) Nature 406:p378 4. Bagrow JP, Bollt EM (2005) Phys Rev E 72:046108 5. Bagrow JP (2008) Stat Mech P05001 6. Bagrow JP, Bollt EM, Skufca JD (2008) Europhys Lett 81:68004 7. Barabási A-L, Albert R (1999) Sience 286:509 8. ben-Avraham D, Havlin S (2000) Diffusion and reactions in fractals and disordered systems. Cambridge University Press, Cambridge 9. Berker AN, Ostlund S (1979) J Phys C 12:4961 10. Beygelzimer A, Grinstein G, Linsker R, Rish I (2005) Physica A Stat Mech Appl 357:593–612 11. Binney JJ, Dowrick NJ, Fisher AJ, Newman MEJ (1992) The theory of critical phenomena: an introduction to the renormalization group. Oxford University Press, Oxford 12. Bollobás B (1985) Random graphs. Academic Press, London 13. Bollt E, ben-Avraham D (2005) New J Phys 7:26 14. Bunde A, Havlin S (1996) Percolation I and Percolation II. In: Bunde A, Havlin S (eds) Fractals and disordered systems, 2nd edn. Springer, Heidelberg 15. Burch H, Chewick W (1999) Mapping the internet. IEEE Comput 32:97–98 16. Butler D (2006) Nature 444:528 17. Cardy J (1996) Scaling and renormalization in statistical physics. Cambridge University Press, Cambridge

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18. Clauset A, Newman MEJ, Moore C (2004) Phys Rev E 70:066111 19. Cohen R, Erez K, ben-Avraham D, Havlin S (2000) Phys Rev Lett 85:4626 20. Cohen R, Erez K, ben-Avraham D, Havlin S (2001) Phys Rev Lett 86:3682 21. Cohen R, ben-Avraham D, Havlin S (2002) Phys Rev E 66:036113 22. Comellas F Complex networks: deterministic models physics and theoretical computer science. In: Gazeau J-P, Nesetril J, Rovan B (eds) From Numbers and Languages to (Quantum) Cryptography. 7 NATO Security through Science Series: Information and Communication Security. IOS Press, Amsterdam. pp 275–293. 348 pags. ISBN 1-58603-706-4 23. Cormen TH, Leiserson CE, Rivest RL, Stein C (2001) Introduction to algorithms. MIT Press, Cambridge 24. Data from SCAN project. The Mbone. http://www.isi.edu/scan/ scan.html Accessed 2000 25. Database of Interacting Proteins (DIP) http://dip.doe-mbi.ucla. edu Accessed 2008 26. Dorogovtsev SN, Goltsev AV, Mendes JFF (2002) Phys Rev E 65:066122 ˝ P, Rényi A (1960) On the evolution of random graphs. 27. Erdos Publ Math Inst Hung Acad Sci 5:17–61 28. Faloutsos M, Faloutsos P, Faloutsos C (1999) Comput Commun Rev 29:251–262 29. Feder J (1988) Fractals. Plenum Press, New York 30. Gallos LK, Argyrakis P, Bunde A, Cohen R, Havlin S (2004) Physica A 344:504–509 31. Gallos LK, Cohen R, Argyrakis P, Bunde A, Havlin S (2005) Phys Rev Lett 94:188701 32. Gallos LK, Song C, Havlin S, Makse HA (2007) PNAS 104:7746 33. Gallos LK, Song C, Makse HA (2008) Phys Rev Lett 100:248701 34. Garey M, Johnson D (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman, New York 35. Goh K-I, Salvi G, Kahng B, Kim D (2006) Phys Rev Lett 96:018701 36. Han J-DJ et al (2004) Nature 430:88–93 37. Hinczewski M, Berker AN (2006) Phys Rev E 73:066126 38. Jeong H, Tombor B, Albert R, Oltvai ZN, Barabási A-L (2000) Nature 407:651–654 39. Kadanoff LP (2000) Statistical physics: statics, dynamics and renormalization. World Scientific Publishing Company, Singapore

40. Kaufman M, Griffiths RB (1981) Phys Rev B 24:496(R) 41. Kaufman M, Griffiths RB (1984) Phys Rev B 24:244 42. Kim JS, Goh K-I, Salvi G, Oh E, Kahng B, Kim D (2007) Phys Rev E 75:016110 43. Kim JS, Goh K-I, Kahng B, Kim D (2007) Chaos 17:026116 44. Kim JS, Goh K-I, Kahng B, Kim D (2007) New J Phys 9:177 45. Mandelbrot B (1982) The fractal geometry of nature. W.H. Freeman and Company, New York 46. Maslov S, Sneppen K (2002) Science 296:910–913 47. Milgram S (1967) Psychol Today 2:60 48. Motter AE, de Moura APS, Lai Y-C, Dasgupta P (2002) Phys Rev E 65:065102 49. Newman MEJ (2002) Phys Rev Lett 89:208701 50. Newman MEJ (2003) Phys Rev E 67:026126 51. Newman MEJ, Girvan M (2004) Phys Rev E 69:026113 52. Overbeek R et al (2000) Nucl Acid Res 28:123–125 53. Palla G, Barabási A-L, Vicsek T (2007) Nature 446:664–667 54. Pastor-Satorras R, Vázquez A, Vespignani A (2001) Phys Rev Lett 87:258701 55. Peitgen HO, Jurgens H, Saupe D (1993) Chaos and fractals: new frontiers of science. Springer, New York 56. Rozenfeld H, Havlin S, ben-Avraham D (2007) New J Phys 9:175 57. Rozenfeld H, ben-Avraham D (2007) Phys Rev E 75:061102 58. Salmhofer M (1999) Renormalization: an introduction. Springer, Berlin 59. Schwartz N, Cohen R, ben-Avraham D, Barabasi A-L, Havlin S (2002) Phys Rev E 66:015104 60. Serrano MA, Boguna M (2006) Phys Rev Lett 97:088701 61. Serrano MA, Boguna M (2006) Phys Rev E 74:056115 62. Song C, Havlin S, Makse HA (2005) Nature 433:392 63. Song C, Havlin S, Makse HA (2006) Nature Phys 2:275 64. Song C, Gallos LK, Havlin S, Makse HA (2007) J Stat Mech P03006 65. Stanley HE (1971) Introduction to phase transitions and critical phenomena. Oxford University Press, Oxford 66. Vicsek T (1992) Fractal growth phenomena, 2nd edn. World Scientific, Singapore Part IV 67. Watts DJ, Strogatz SH (1998) Collective dynamics of “smallworld” networks. Nature 393:440–442 68. Xenarios I et al (2000) Nucl Acids Res 28:289–291 69. Zhoua W-X, Jianga Z-Q, Sornette D (2007) Physica A 375: 741–752

Hamiltonian Perturbation Theory (and Transition to Chaos)

Hamiltonian Perturbation Theory (and Transition to Chaos) HENK W. BROER1 , HEINZ HANSSMANN2 Instituut voor Wiskunde en Informatica, Rijksuniversiteit Groningen, Groningen, The Netherlands 2 Mathematisch Instituut, Universiteit Utrecht, Utrecht, The Netherlands

1

Article Outline Glossary Definition of the Subject Introduction One Degree of Freedom Perturbations of Periodic Orbits Invariant Curves of Planar Diffeomorphisms KAM Theory: An Overview Splitting of Separatrices Transition to Chaos and Turbulence Future Directions Bibliography Glossary Bifurcation In parametrized dynamical systems a bifurcation occurs when a qualitative change is invoked by a change of parameters. In models such a qualitative change corresponds to transition between dynamical regimes. In the generic theory a finite list of cases is obtained, containing elements like ‘saddle-node’, ‘period doubling’, ‘Hopf bifurcation’ and many others. Cantor set, Cantor dust, Cantor family, Cantor stratification Cantor dust is a separable locally compact space that is perfect, i. e. every point is in the closure of its complement, and totally disconnected. This determines Cantor dust up to homeomorphisms. The term Cantor set (originally reserved for the specific form of Cantor dust obtained by repeatedly deleting the middle third from a closed interval) designates topological spaces that locally have the structure Rn  Cantor dust for some n 2 N. Cantor families are parametrized by such Cantor sets. On the real line R one can define Cantor dust of positive measure by excluding around each rational number p/q an interval of size 2

; q

> 0;  > 2:

Similar Diophantine conditions define Cantor sets in Rn . Since these Cantor sets have positive measure

their Hausdorff dimension is n. Where the unperturbed system is stratified according to the co-dimension of occurring (bifurcating) tori, this leads to a Cantor stratification. Chaos An evolution of a dynamical system is chaotic if its future is badly predictable from its past. Examples of non-chaotic evolutions are periodic or multi-periodic. A system is called chaotic when many of its evolutions are. One criterion for chaoticity is the fact that one of the Lyapunov exponents is positive. Diophantine condition, Diophantine frequency vector A frequency vector ! 2 Rn is called Diophantine if there are constants > 0 and  > n  1 with jhk; !ij 

jkj

for all k 2 Z n nf0g :

The Diophantine frequency vectors satisfying this condition for fixed and  form a Cantor set of half lines. As the Diophantine parameter tends to zero (while  remains fixed), these half lines extend to the origin. The complement in any compact set of frequency vectors satisfying a Diophantine condition with fixed  has a measure of order O( ) as # 0. Integrable system A Hamiltonian system with n degrees of freedom is (Liouville)-integrable if it has n functionally independent commuting integrals of motion. Locally this implies the existence of a torus action, a feature that can be generalized to dissipative systems. In particular a mapping is integrable if it can be interpolated to become the stroboscopic mapping of a flow. KAM theory Kolmogorov–Arnold–Moser theory is the perturbation theory of (Diophantine) quasi-periodic tori for nearly integrable Hamiltonian systems. In the format of quasi-periodic stability, the unperturbed and perturbed system, restricted to a Diophantine Cantor set, are smoothly conjugated in the sense of Whitney. This theory extends to the world of reversible, volumepreserving or general dissipative systems. In the latter KAM theory gives rise to families of quasi-periodic attractors. KAM theory also applies to torus bundles, in which case a global Whitney smooth conjugation can be proven to exist, that keeps track of the geometry. In an appropriate sense invariants like monodromy and Chern classes thus also can be defined in the nearly integrable case. Also compare with  Kolmogorov– Arnold–Moser (KAM) Theory. Nearly integrable system In the setting of perturbation theory, a nearly integrable system is a perturbation of an integrable one. The latter then is an integrable approximation of the former. See an above item.

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Normal form truncation Consider a dynamical system in the neighborhood of an equilibrium point, a fixed or periodic point, or a quasi-periodic torus, reducible to Floquet form. Then Taylor expansions (and their analogues) can be changed gradually into normal forms, that usually reflect the dynamics better. Often these display a (formal) torus symmetry, such that the normal form truncation becomes an integrable approximation, thus yielding a perturbation theory setting. See above items. Also compare with  Normal Forms in Perturbation Theory. Persistent property In the setting of perturbation theory, a property is persistent whenever it is inherited from the unperturbed to the perturbed system. Often the perturbation is taken in an appropriate topology on the space of systems, like the Whitney Ck -topology [72]. Perturbation problem In perturbation theory the unperturbed systems usually are transparent regarding their dynamics. Examples are integrable systems or normal form truncations. In a perturbation problem things are arranged in such a way that the original system is wellapproximated by such an unperturbed one. This arrangement usually involves both changes of variables and scalings. Resonance If the frequencies of an invariant torus with multi- or conditionally periodic flow are rationally dependent, this torus divides into invariant sub-tori. Such resonances hh; !i D 0, h 2 Z k , define hyperplanes in !-space and, by means of the frequency mapping, also in phase space. The smallest number jhj D jh1 jC  Cjh k j is the order of the resonance. Diophantine conditions describe a measure-theoretically large complement of a neighborhood of the (dense!) set of all resonances. Separatrices Consider a hyperbolic equilibrium, fixed or periodic point or invariant torus. If the stable and unstable manifolds of such hyperbolic elements are codimension one immersed manifolds, then they are called separatrices, since they separate domains of phase space, for instance, basins of attraction. Singularity theory A function H : Rn ! R has a critical point z 2 Rn where DH(z) vanishes. In local coordinates we may arrange z D 0 (and similarly that it is mapped to zero as well). Two germs K : (Rn ; 0) ! (R; 0) and N : (Rn ; 0) ! (R; 0) represent the same function H locally around z if and only if there is a diffeomorphism  on Rn satisfying N D K ı: The corresponding equivalence class is called a singularity.

Structurally stable A system is structurally stable if it is topologically equivalent to all nearby systems, where ‘nearby’ is measured in an appropriate topology on the space of systems, like the Whitney Ck -topology [72]. A family is structurally stable if for every nearby family there is a re-parametrization such that all corresponding systems are topologically equivalent. Definition of the Subject The fundamental problem of mechanics is to study Hamiltonian systems that are small perturbations of integrable systems. Also, perturbations that destroy the Hamiltonian character are important, be it to study the effect of a small amount of friction, or to further the theory of dissipative systems themselves which surprisingly often revolves around certain well-chosen Hamiltonian systems. Furthermore there are approaches like KAM theory that historically were first applied to Hamiltonian systems. Typically perturbation theory explains only part of the dynamics, and in the resulting gaps the orderly unperturbed motion is replaced by random or chaotic motion. Introduction We outline perturbation theory from a general point of view, illustrated by a few examples. The Perturbation Problem The aim of perturbation theory is to approximate a given dynamical system by a more familiar one, regarding the former as a perturbation of the latter. The problem then is to deduce certain dynamical properties from the unperturbed to the perturbed case. What is familiar may or may not be a matter of taste, at least it depends a lot on the dynamical properties of one’s interest. Still the most frequently used unperturbed systems are:  Linear systems  Integrable Hamiltonian systems  Normal form truncations, compare with  Normal Forms in Perturbation Theory and references therein  Etc. To some extent the second category can be seen as a special case of the third. To avoid technicalities in this section we assume all systems to be sufficiently smooth, say of class C 1 or real analytic. Moreover in our considerations " will be a real parameter. The unperturbed case always corresponds to " D 0 and the perturbed one to " ¤ 0 or " > 0.

Hamiltonian Perturbation Theory (and Transition to Chaos)

Examples of Perturbation Problems To begin with consider the autonomous differential equation x¨ C "x˙ C

dV (x) D 0 ; dx

modeling an oscillator with small damping. Rewriting this equation of motion as a planar vector field x˙ D

y

y˙ D "y 

dV (x) ; dx

we consider the energy H(x; y) D 12 y 2 C V(x). For " D 0 the system is Hamiltonian with Hamiltonian function H. ˙ Indeed, generally we have H(x; y) D "y 2 , implying that for " > 0 there is dissipation of energy. Evidently for " ¤ 0 the system is no longer Hamiltonian. The reader is invited to compare the phase portraits of the cases " D 0 and " > 0 for V (x) D  cos x (the 1 pendulum) or V (x) D 12 x 2 C 24 bx 4 (Duffing). Another type of example is provided by the non-autonomous equation dV x¨ C (x) D " f (x; x˙ ; t) ; dx which can be regarded as the equation of motion of an oscillator with small external forcing. Again rewriting as a vector field, we obtain ˙t D

1

x˙ D

y

y˙ D 

dV (x) C " f (x; y; t) ; dx

now on the generalized phase space R3 D ft; x; yg. In the case where the t-dependence is periodic, we can take S1  R2 for (generalized) phase space. Remark  A small variation of the above driven system concerns a parametrically forced oscillator like x¨ C (! 2 C " cos t) sin x D 0 ; which happens to be entirely in the world of Hamiltonian systems.  It may be useful to study the Poincaré or period mapping of such time periodic systems, which happens to be a mapping of the plane. We recall that in the Hamiltonian cases this mapping preserves area. For general reference in this direction see, e. g., [6,7,27,66].

There are lots of variations and generalizations. One example is the solar system, where the unperturbed case consists of a number of uncoupled two-body problems concerning the Sun and each of the planets, and where the interaction between the planets is considered as small [6,9, 107,108]. Remark  One variation is a restriction to fewer bodies, for example only three. Examples of this are systems like Sun–Jupiter–Saturn, Earth–Moon–Sun or Earth– Moon–Satellite.  Often Sun, Moon and planets are considered as point masses, in which case the dynamics usually are modeled as a Hamiltonian system. It is also possible to extend this approach taking tidal effects into account, which have a non-conservative nature.  The Solar System is close to resonance, which makes application of KAM theory problematic. There exist, however, other integrable approximations that take resonance into account [3,63]. Quite another perturbation setting is local, e. g., near an equilibrium point. To fix thoughts consider x˙ D Ax C f (x) ;

x 2 Rn

with A 2 gl(n; R), f (0) D 0 and Dx f (0) D 0. By the scaling x D "x¯ we rewrite the system to x˙¯ D A¯x C "g(x¯ ) : So, here we take the linear part as an unperturbed system. Observe that for small " the perturbation is small on a compact neighborhood of x¯ D 0. This setting also has many variations. In fact, any normal form approximation may be treated in this way  Normal Forms in Perturbation Theory. Then the normalized truncation forms the unperturbed part and the higher order terms the perturbation. Remark In the above we took the classical viewpoint which involves a perturbation parameter controlling the size of the perturbation. Often one can generalize this by considering a suitable topology (like the Whitney topologies) on the corresponding class of systems [72]. Also compare with  Normal Forms in Perturbation Theory and  Kolmogorov–Arnold–Moser (KAM) Theory. Questions of Persistence What are the kind of questions perturbation theory asks? A large class of questions concerns the persistence of cer-

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tain dynamical properties as known for the unperturbed case. To fix thoughts we give a few examples. To begin with consider equilibria and periodic orbits. So we put x˙ D f (x; ") ;

x 2 Rn ; " 2 R ;

(1)

for a map f : RnC1 ! Rn . Recall that equilibria are given by the equation f (x; ") D 0. The following theorem that continues equilibria in the unperturbed system for " ¤ 0, is a direct consequence of the implicit function theorem. Theorem 1 (Persistence of equilibria) f (x0 ; 0) D 0 and that

Suppose that

Dx f (x0 ; 0) has maximal rank : Then there exists a local arc " 7! x(") with x(0) D x0 such that f (x("); ")  0 : Periodic orbits can be approximated in a similar way. Indeed, let the system (1) for D 0 have a periodic orbit 0 . Let ˙ be a local transversal section of 0 and P0 : ˙ ! ˙ the corresponding Poincaré map. Then P0 has a fixed point x0 2 ˙ \ 0 . By transversality, for j"j small, a local Poincaré map P" : ˙ ! ˙ is well-defined for (1). Observe that fixed points x" of P" correspond to periodic orbits " of (1). We now have, again as another direct consequence of the implicit function theorem. Theorem 2 (Persistence of periodic orbits) In the above assume that P0 (x0 ) D x0 and Dx P0 (x0 ) has no eigenvalue 1 : Then there exists a local arc " 7! x(") with x(0) D x0 such that P" (x("))  x" : Remark  Often the conditions of Theorem 2 are not easy to verify. Sometimes it is useful here to use Floquet Theory, see [97]. In fact, if T 0 is the period of 0 and ˝ 0 its Floquet matrix, then Dx P0 (x0 ) D exp(T0 ˝0 ).  The format of the Theorems 1 and 2 with the perturbation parameter " directly allows for algorithmic approaches. One way to proceed is by perturbation series, leading to asymptotic formulae that in the real analytic setting have positive radius of convergence. In the latter case the names of Poincaré and Lindstedt are associated with the method, cf. [10]. Also numerical continuation programmes exist based on the Newton method.

 The Theorems 1 and 2 can be seen as special cases of a a general theorem for normally hyperbolic invariant manifolds [73], Theorem 4.1. In all cases a contraction principle on a suitable Banach space of graphs leads to persistence of the invariant dynamical object. This method in particular yields existence and persistence of stable and unstable manifolds [53,54]. Another type of dynamics subject to perturbation theory is quasi-periodic. We emphasize that persistence of (Diophantine) quasi-periodic invariant tori occurs both in the conservative setting and in many others, like in the reversible and the general (dissipative) setting. In the latter case this leads to persistent occurrence of families of quasiperiodic attractors [125]. These results are in the domain of Kolmogorov–Arnold–Moser (KAM) theory. For details we refer to Sect. “KAM Theory: An Overview” below or to [24],  Kolmogorov–Arnold–Moser (KAM) Theory, the former reference containing more than 400 references in this area. Remark  Concerning the Solar System, KAM theory always has aimed at proving that it contains many quasi-periodic motions, in the sense of positive Liouville measure. This would imply that there is positive probability that a given initial condition lies on such a stable quasi-periodic motion [3,63], however, also see [85].  Another type of result in this direction compares the distance of certain individual solutions of the perturbed and the unperturbed system, with coinciding initial conditions over time scales that are long in terms of ". Compare with [24]. Apart from persistence properties related to invariant manifolds or individual solutions, the aim can also be to obtain a more global persistence result. As an example of this we mention the Hartman–Grobman Theorem, e. g., [7,116,123]. Here the setting once more is x˙ D Ax C f (x) ;

x 2 Rn ;

with A 2 gl(n; R), f (0) D 0 and Dx f (0) D 0. Now we assume A to be hyperbolic (i. e., with no purely imaginary eigenvalues). In that case the full system, near the origin, is topologically conjugated to the linear system x˙ D Ax. Therefore all global, qualitative properties of the unperturbed (linear) system are persistent under perturbation to the full system. For details on these notions see the above references, also compare with, e. g., [30].

Hamiltonian Perturbation Theory (and Transition to Chaos)

It is said that the hyperbolic linear system x˙ D Ax is (locally) structurally stable. This kind of thinking was introduced to the dynamical systems area by Thom [133], with a first, successful application to catastrophe theory. For further details, see [7,30,69,116]. General Dynamics We give a few remarks on the general dynamics in a neighborhood of Hamiltonian KAM tori. In particular this concerns so-called superexponential stickiness of the KAM tori and adiabatic stability of the action variables, involving the so-called Nekhoroshev estimate. To begin with, emphasize the following difference between the cases n D 2 and n  3 in the classical KAM theorem of Subsect. “Classical KAM Theory”. For n D 2 the level surfaces of the Hamiltonian are three-dimensional, while the Lagrangian tori have dimension two and hence codimension one in the energy hypersurfaces. This means that for open sets of initial conditions, the evolution curves are forever trapped in between KAM tori, as these tori foliate over nowhere dense sets of positive measure. This implies perpetual adiabatic stability of the action variables. In contrast, for n  3 the Lagrangian tori have codimension n1 > 1 in the energy hypersurfaces and evolution curves may escape. This actually occurs in the case of so-called Arnold diffusion. The literature on this subject is immense, and we here just quote [5,9,93,109], for many more references see [24]. Next we consider the motion in a neighborhood of the KAM tori, in the case where the systems are real analytic or at least Gevrey smooth. For a definition of Gevrey regularity see [136]. First we mention that, measured in terms of the distance to the KAM torus, nearby evolution curves generically stay nearby over a superexponentially long time [102,103]. This property often is referred to as superexponential stickiness of the KAM tori, see [24] for more references. Second, nearly integrable Hamiltonian systems, in terms of the perturbation size, generically exhibit exponentially long adiabatic stability of the action variables, see e. g. [15,88,89,90,93,103,109,110,113,120],  Nekhoroshev Theory and many others, for more references see [24]. This property is referred to as the Nekhoroshev estimate or the Nekhoroshev theorem. For related work on perturbations of so-called superintegrable systems, also see [24] and references therein. Chaos In the previous subsection we discussed persistent and some non-persistent features of dynamical systems un-

der small perturbations. Here we discuss properties related to splitting of separatrices, caused by generic perturbations. A first example was met earlier, when comparing the pendulum with and without (small) damping. The unperturbed system is the undamped one and this is a Hamiltonian system. The perturbation however no longer is Hamiltonian. We see that the equilibria are persistent, as they should be according to Theorem 1, but that none of the periodic orbits survives the perturbation. Such qualitative changes go with perturbing away from the Hamiltonian setting. Similar examples concern the breaking of a certain symmetry by the perturbation. The latter often occurs in the case of normal form approximations. Then the normalized truncation is viewed as the unperturbed system, which is perturbed by the higher order terms. The truncation often displays a reasonable amount of symmetry (e. g., toroidal symmetry), which generically is forbidden for the class of systems under consideration, e. g. see [25]. To fix thoughts we reconsider the conservative example x¨ C (! 2 C " cos t) sin x D 0 of the previous section. The corresponding (time dependent, Hamiltonian [6]) vector field reads ˙t D 1 x˙ D

y

y˙ D (! 2 C " cos t) sin x : Let P!;" : R2 ! R2 be the corresponding (area-preserving) Poincaré map. Let us consider the unperturbed map P!;0 which is just the flow over time 2 of the free pendulum x¨ C! 2 sin x D 0. Such a map is called integrable, since it is the stroboscopic map of a two-dimensional vector field, hence displaying the R-symmetry of a flow. When perturbed to the nearly integrable case " ¤ 0, this symmetry generically is broken. We list a few of the generic properties for such maps [123]:  The homoclinic and heteroclinic points occur at transversal intersections of the corresponding stable and unstable manifolds.  The periodic points of period less than a given bound are isolated. This means generically that the separatrices split and that the resonant invariant circles filled with periodic points with the same (rational) rotation number fall apart. In any concrete example the issue remains whether or not

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Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 1 Chaos in the parametrically forced pendulum. Left: Poincaré map P!;" near the 1 : 2 resonance ! D Right: A dissipative analogue

it satisfies appropriate genericity conditions. One method to check this is due to Melnikov, compare [66,137], for more sophisticated tools see [65]. Often this leads to elliptic (Abelian) integrals. In nearly integrable systems chaos can occur. This fact is at the heart of the celebrated non-integrability of the three-body problem as addressed by Poincaré [12,59, 107,108,118]. A long standing open conjecture is that the clouds of points as visible in Fig. 1, left, densely fill sets of positive area, thereby leading to ergodicity [9]. In the case of dissipation, see Fig. 1, right, we conjecture the occurrence of a Hénon-like strange attractor [14,22,126]. Remark  The persistent occurrence of periodic points of a given rotation number follows from the Poincaré–Birkhoff fixed point theorem [74,96,107], i. e., on topological grounds.  The above arguments are not restricted to the conservative setting, although quite a number of unperturbed systems come from this world. Again see Fig. 1.

1 2

and for " > 0 not too small.

ishing eigenvalues may be unavoidable and it becomes possible that the corresponding dynamics persist under perturbations. Perturbations may also destroy the Hamiltonian character of the flow. This happens especially where the starting point is a dissipative planar system and e. g. a scaling leads for " D 0 to a limiting Hamiltonian flow. The perturbation problem then becomes twofold. Equilibria still persist by Theorem 1 and hyperbolic equilibria moreover persist as such, with the sum of eigenvalues of order O("). Also for elliptic eigenvalues the sum of eigenvalues is of order O(") after the perturbation, but here this number measures the dissipation whence the equilibrium becomes (weakly) attractive for negative values and (weakly) unstable for positive values. The one-parameter families of periodic orbits of a Hamiltonian system do not persist under dissipative perturbations, the very fact that they form families imposes the corresponding fixed point of the Poincaré mapping to have an eigenvalue one and Theorem 2 does not apply. Typically only finitely many periodic orbits survive a dissipative perturbation and it is already a difficult task to determine their number.

One Degree of Freedom Planar Hamiltonian systems are always integrable and the orbits are given by the level sets of the Hamiltonian function. This still leaves room for a perturbation theory. The recurrent dynamics consists of periodic orbits, equilibria and asymptotic trajectories forming the (un)stable manifolds of unstable equilibria. The equilibria organize the phase portrait, and generically all equilibria are elliptic (purely imaginary eigenvalues) or hyperbolic (real eigenvalues), i. e. there is no equilibrium with a vanishing eigenvalue. If the system depends on a parameter such van-

Hamiltonian Perturbations The Duffing oscillator has the Hamiltonian function H(x; y) D

1 2 1 1 y C bx 4 C x 2 2 24 2

(2)

where b is a constant distinguishing the two cases b D ˙1 and  is a parameter. Under variation of the parameter the equations of motion x˙ D y

Hamiltonian Perturbation Theory (and Transition to Chaos)

1 y˙ D  bx 3  x 6

Dissipative Perturbations

display a Hamiltonian pitchfork bifurcation, supercritical for positive b and subcritical in case b is negative. Correspondingly, the linearization at the equilibrium x D 0 of the anharmonic oscillator  D 0 is given by the matrix   0 1 0 0 whence this equilibrium is parabolic. The typical way in which a parabolic equilibrium bifurcates is the center-saddle bifurcation. Here the Hamiltonian reads H(x; y) D

1 2 1 3 ay C bx C cx 2 6

(3)

where a; b; c 2 R are nonzero constants, for instance a D b D c D 1. Note that this is a completely different unfolding of the parabolic equilibrium at the origin. A closer look at the phase portraits and in particular at the Hamiltonian function of the Hamiltonian pitchfork bifurcation reveals the symmetry x 7! x of the Duffing oscillator. This suggests the addition of the non-symmetric term x. The resulting two-parameter family H; (x; y) D

1 2 1 1 y C bx 4 C x 2 C x 2 24 2

of Hamiltonian systems is indeed structurally stable. This implies not only that all equilibria of a Hamiltonian perturbation of the Duffing oscillator have a local flow equivalent to the local flow near a suitable equilibrium in this two-parameter family, but that every one-parameter family of Z2 -symmetric Hamiltonian systems that is a perturbation of (2) has equivalent dynamics. For more details see [36] and references therein. This approach applies mutatis mutandis to every nondegenerate planar singularity, cf. [69,130]. At an equilibrium all partial derivatives of the Hamiltonian vanish and the resulting singularity is called non-degenerate if it has finite multiplicity, which implies that it admits a versal unfolding H  with finitely many parameters. The family of Hamiltonian systems defined by this versal unfolding contains all possible (local) dynamics that the initial equilibrium may be perturbed to. Imposing additional discrete symmetries is immediate, the necessary symmetric versal unfolding is obtained by averaging HG D

1 X H ı g jGj g2G

along the orbits of the symmetry group G.

In a generic dissipative system all equilibria are hyperbolic. Qualitatively, i. e. up to topological equivalence, the local dynamics is completely determined by the number of eigenvalues with positive real part. Those hyperbolic equilibria that can appear in Hamiltonian systems (the eigenvalues forming pairs ˙) do not play an important role. Rather, planar Hamiltonian systems become important as a tool to understand certain bifurcations triggered off by non-hyperbolic equilibria. Again this requires the system to depend on external parameters. The simplest example is the Hopf bifurcation, a co-dimension one bifurcation where an equilibrium loses stability as the pair of eigenvalues crosses the imaginary axis, say at ˙i. At the bifurcation the linearization is a Hamiltonian system with an elliptic equilibrium (the co-dimension one bifurcations where a single eigenvalue crosses the imaginary axis through 0 do not have a Hamiltonian linearization). This limiting Hamiltonian system has a oneparameter family of periodic orbits around the equilibrium, and the non-linear terms determine the fate of these periodic orbits. The normal form of order three reads     x˙ D y 1 C b(x 2 C y 2 ) C x  C a(x 2 C y 2 )     x˙ D x 1 C b(x 2 C y 2 ) C y  C a(x 2 C y 2 ) and is Hamiltonian if and only if (; a) D (0; 0). The sign of the coefficient distinguishes between the supercritical case a > 0, in which there are no periodic orbits coexisting with the attractive equilibria (i. e. when  < 0) and one attracting periodic orbit for each  > 0 (coexisting with the unstable equilibrium), and the subcritical case a < 0, in which the family of periodic orbits is unstable and coexists with the attractive equilibria (with no periodic orbits for parameters  > 0). As  ! 0 the family of periodic orbits shrinks down to the origin, so also this Hamiltonian feature is preserved. Equilibria with a double eigenvalue 0 need two parameters to persistently occur in families of dissipative systems. The generic case is the Takens–Bogdanov bifurcation. Here the linear part is too degenerate to be helpful, but the nonlinear Hamiltonian system defined by (3) with a D 1 D c and b D 3 provides the periodic and heteroclinic orbit(s) that constitute the nontrivial part of the bifurcation diagram. Where discrete symmetries are present, e. g. for equilibria in dissipative systems originating from other generic bifurcations, the limiting Hamiltonian system exhibits that same discrete symmetry. For more details see [54,66,82] and references therein. The continuation of certain periodic orbits from an unperturbed Hamiltonian system under dissipative per-

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Hamiltonian Perturbation Theory (and Transition to Chaos)

turbation can be based on Melnikov-like methods, again see [66,137]. As above, this often leads to Abelian integrals, for instance to count the number of periodic orbits that branch off. Reversible Perturbations A dynamical system that admits a reflection symmetry R mapping trajectories '(t; z0 ) to trajectories '(t; R(z0 )) is called reversible. In the planar case we may restrict to the reversing reflection R:

R2 (x; y)

! 7!

R2 (x; y) :

(4)

All Hamiltonian functions H D 12 y 2 CV (x) which have an interpretation “kinetic C potential energy” are reversible, and in general the class of reversible systems is positioned between the class of Hamiltonian systems and the class of dissipative systems. A guiding example is the perturbed Duffing oscillator (with the roles of x and y exchanged so that (4) remains the reversing symmetry) 1 x˙ D  y 3  y C "x y 6 y˙ D x that combines the Hamiltonian character of the equilibrium at the origin with the dissipative character of the two other equilibria. Note that all orbits outside the homoclinic loop are periodic. There are two ways in which the reversing symmetry (4) imposes a Hamiltonian character on the dynamics. An equilibrium that lies on the symmetry line fy D 0g has a linearization that is itself a reversible system and consequently the eigenvalues are subject to the same constraints as in the Hamiltonian case. (For equilibria z0 that do not lie on the symmetry line the reflection R(z0 ) is also an equilibrium, and it is to the union of their eigenvalues that these constraints still apply.) Furthermore, every orbit that crosses fy D 0g more than once is automatically periodic, and these periodic orbits form one-parameter families. In particular, elliptic equilibria are still surrounded by periodic orbits. The dissipative character of a reversible system is most obvious for orbits that do not cross the symmetry line. Here R merely maps the orbit to a reflected counterpart. The above perturbed Duffing oscillator exemplifies that the character of an orbit crossing fy D 0g exactly once is undetermined. While the homoclinic orbit of the saddle at the origin has a Hamiltonian character, the heteroclinic orbits between the other two equilibria behave like in a dissipative system.

Perturbations of Periodic Orbits The perturbation of a one-degree-of-freedom system by a periodic forcing is a perturbation that changes the phase space. Treating the time variable t as a phase space variable leads to the extended phase space S1 R2 and equilibria of the unperturbed system become periodic orbits, inheriting the normal behavior. Furthermore introducing an action conjugate to the “angle” t yields a Hamiltonian system in two degrees of freedom. While the one-parameter families of periodic orbits merely provide the typical recurrent motion in one degree of freedom, they form special solutions in two or more degrees of freedom. Arcs of elliptic periodic orbits are particularly instructive. Note that these occur generically in both the Hamiltonian and the reversible context. Conservative Perturbations Along the family of elliptic periodic orbits a pair e˙i˝ of Floquet multipliers passes regularly through roots of unity. Generically this happens on a dense set of parameter values, but for fixed denominator q in e˙i˝ D e˙2ip/q the corresponding energy values are isolated. The most important of such resonances are those with small denominators q. For q D 1 generically a periodic center-saddle bifurcation takes place where an elliptic and a hyperbolic periodic orbit meet at a parabolic periodic orbit. No periodic orbit remains under further variation of a suitable parameter. The generic bifurcation for q D 2 is the period-doubling bifurcation where an elliptic periodic orbit turns hyperbolic (or vice versa) when passing through a parabolic periodic orbit with Floquet multipliers  1. Furthermore, a family of periodic orbits with twice the period emerges from the parabolic periodic orbit, inheriting the normal linear behavior from the initial periodic orbit. In case q D 3, and possibly also for q D 4, generically two arcs of hyperbolic periodic orbits emerge, both with three (resp. four) times the period. One of these extends for lower and the other for higher parameter values. The initial elliptic periodic orbit momentarily loses its stability due to these approaching unstable orbits. Denominators q  5 (and also the second possibility for q D 4) lead to a pair of subharmonic periodic orbits of q times the period emerging either for lower or for higher parameter values. This is (especially for large q) comparable to the behavior at Diophantine e˙i˝ where a family of invariant tori emerges, cf. Sect. “Invariant Curves of Planar Diffeomorphisms” below. For a single pair e˙i˝ of Floquet multipliers this behavior is traditionally studied for the (iso-energetic)

Hamiltonian Perturbation Theory (and Transition to Chaos)

Poincaré-mapping, cf. [92] and references therein. However, the above description remains true in higher dimensions, where additionally multiple pairs of Floquet multipliers may interact. An instructive example is the Lagrange top, the sleeping motion of which is gyroscopically stabilized after a periodic Hamiltonian Hopf bifurcation; see [56] for more details. Dissipative Perturbations There exists a large class of local bifurcations in the dissipative setting, that can be arranged in a perturbation theory setting, where the unperturbed system is Hamiltonian. The arrangement consists of changes of variables and rescaling. An early example of this is the Bogdanov–Takens bifurcation [131,132]. For other examples regarding nilpotent singularities, see [23,40] and references therein. To fix thoughts, consider families of planar maps and let the unperturbed Hamiltonian part contain a center (possibly surrounded by a homoclinic loop). The question then is which of these persist when adding the dissipative perturbation. Usually only a definite finite number persists. As in Subsect. “Chaos”, a Melnikov function can be invoked here, possibly again leading to elliptic (Abelian) integrals, Picard Fuchs equations, etc. For details see [61,124] and references therein. Invariant Curves of Planar Diffeomorphisms This section starts with general considerations on circle diffeomorphisms, in particular focusing on persistence properties of quasi-periodic dynamics. Our main references are [2,24,29,31,70,71,139,140]. For a definition of rotation number, see [58]. After this we turn to area preserving maps of an annulus where we discuss Moser’s twist map theorem [104], also see [24,29,31]. The section is concluded by a description of the holomorphic linearization of a fixed point in a planar map [7,101,141,142]. Our main perspective will be perturbative, where we consider circle maps near a rigid rotation. It turns out that generally parameters are needed for persistence of quasiperiodicity under perturbations. In the area preserving setting we consider perturbations of a pure twist map.

view this two-parameter family as a one-parameter family of maps P" : T 1  [0; 1] ! T 1  [0; 1]; (x; ˛) 7! (x C 2 ˛ C "a(x; ˛; "); ˛) of the cylinder. Note that the unperturbed system P0 is a family of rigid circle rotations, viewed as a cylinder map, where the individual map P˛;0 has rotation number ˛. The question now is what will be the fate of this rigid dynamics for 0 ¤ j"j  1. The classical way to address this question is to look for a conjugation ˚" , that makes the following diagram commute T 1  [0; 1] " ˚"

!

P"

T 1  [0; 1] " ˚"

T 1  [0; 1]

!

P0

T 1  [0; 1] ;

i. e., such that P" ı ˚" D ˚" ı P0 : Due to the format of P" we take ˚" as a skew map ˚" (x; ˛) D (x C "U(x; ˛; "); ˛ C " (˛; ")) ; which leads to the nonlinear equation U(x C 2 ˛; ˛; ")  U(x; ˛; ") D 2 (˛; ") C a (x C "U(x; ˛; "); ˛ C " (˛; "); ") in the unknown maps U and  . Expanding in powers of " and comparing at lowest order yields the linear equation U0 (x C 2 ˛; ˛)  U0 (x; ˛) D 20 (˛) C a0 (x; ˛) which can be directly solved by Fourier series. Indeed, writing X a0k (˛)eikx ; a0 (x; ˛) D k2Z

U0 (x; ˛) D

P˛;" : T ! T ; 1

1

x 7! x C 2 ˛ C "a(x; ˛; ")

of circle maps of class C 1 . It turns out to be convenient to

U0k (˛)eikx

k2Z

we find 0 D 1/(2)a00 and

Circle Maps We start with the following general problem. Given a twoparameter family

X

U0k (˛) D

a0k (˛) 2ik˛ e 1

:

It follows that in general a formal solution exists if and only if ˛ 2 R n Q. Still, the accumulation of e2ik˛  1 on 0 leads to the celebrated small divisors [9,108], also see [24,29,31,55].

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The classical solution considers the following Diophantine non-resonance conditions. Fixing  > 2 and

> 0 consider ˛ 2 [0; 1] such that for all rationals p/q ˇ ˇ ˇ ˇ ˇ˛  p ˇ  q : (5) ˇ qˇ This subset of such ˛s is denoted by [0; 1]; and is wellknown to be nowhere dense but of large measure as > 0 gets small [115]. Note that Diophantine numbers are irrational. Theorem 3 (Circle Map Theorem) For sufficiently small and for the perturbation "a sufficiently small in the C 1 -topology, there exists a C 1 transformation ˚" : T 1  [0; 1] ! T 1  [0; 1], conjugating the restriction P0 j[0;1]; to a subsystem of P" . Theorem 3 in the present structural stability formulation (compare with Fig. 2) is a special case of the results in [29,31]. We here speak of quasi-periodic stability. For earlier versions see [2,9]. Remark  Rotation numbers are preserved by the map ˚" and irrational rotation numbers correspond to quasi-periodicity. Theorem 3 thus ensures that typically quasi-periodicity occurs with positive measure in the parameter space. Note that since Cantor sets are perfect, quasi-periodicity typically has a non-isolated occurrence.

Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 2 Skew cylinder map, conjugating (Diophantine) quasi-periodic invariant circles of P0 and P"

 The map ˚" has no dynamical meaning inside the gaps. The gap dynamics in the case of circle maps can be illustrated by the Arnold family of circle maps [2,7,58], given by P˛;" (x) D x C 2 ˛ C " sin x which exhibits a countable union of open resonance tongues where the dynamics is periodic, see Fig. 3. Note that this map only is a diffeomorphism for j"j < 1.  We like to mention that non-perturbative versions of Theorem 3 have been proven in [70,71,139].  For simplicity we formulated Theorem 3 under C 1 -regularity, noting that there exist many ways to generalize this. On the one hand there exist Ck -versions for finite k and on the other hand there exist fine tunings in terms of real-analytic and Gevrey regularity. For details we refer to [24,31] and references therein. This same remark applies to other results in this section and in Sect. “KAM Theory: An Overview” on KAM theory. A possible application of Theorem 3 runs as follows. Consider a system of weakly coupled Van der Pol oscillators y¨1 C c1 y˙1 C a1 y1 C f1 (y1 ; y˙1 ) D "g1 (y1 ; y2 ; y˙1 ; y˙2 ) y¨2 C c2 y˙2 C a2 y2 C f2 (y2 ; y˙2 ) D "g2 (y1 ; y2 ; y˙1 ; y˙2 ) : Writing y˙ j D z j ; j D 1; 2, one obtains a vector field in the four-dimensional phase space R2 R2 D f(y1 ; z1 ); (y2 ; z2 )g. For " D 0 this vector field has an invariant two-torus, which is the product of the periodic motions of the individual Van der Pol oscillations. This two-torus is normally hyperbolic and therefore persistent for j"j  1 [73]. In

Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 3 Arnold resonance tongues; for "  1 the maps are endomorphic

Hamiltonian Perturbation Theory (and Transition to Chaos)

fact the torus is an attractor and we can define a Poincaré return map within this torus attractor. If we include some of the coefficients of the equations as parameters, Theorem 3 is directly applicable. The above statements on quasi-periodic circle maps then directly translate to the case of quasi-periodic invariant two-tori. Concerning the resonant cases, generically a tongue structure like in Fig. 3 occurs; for the dynamics corresponding to parameter values inside such a tongue one speaks of phase lock. Remark  The celebrated synchronization of Huygens’ clocks [77] is related to a 1 : 1 resonance, meaning that the corresponding Poincaré map would have its parameters in the main tongue with rotation number 0. Compare with Fig. 3.  There exist direct generalizations to cases with n-oscillators (n 2 N), leading to families of invariant n-tori carrying quasi-periodic flow, forming a nowhere dense set of positive measure. An alteration with resonance occurs as roughly sketched in Fig. 3. In higher dimension the gap dynamics, apart from periodicity, also can contain strange attractors [112,126]. We shall come back to this subject in a later section. Area-Preserving Maps The above setting historically was preceded by an area preserving analogue [104] that has its origin in the Hamiltonian dynamics of frictionless mechanics. Let   R2 n f(0; 0)g be an annulus, with sympectic polar coordinates ('; I) 2 T 1  K, where K is an interval. Moreover, let  D d' ^ dI be the area form on . We consider a -preserving smooth map P" :  !  of the form P" ('; I) D (' C 2 ˛(I); I) C O(") ; where we assume that the map I 7! ˛(I) is a (local) diffeomorphism. This assumption is known as the twist condition and P" is called a twist map. For the unperturbed case " D 0 we are dealing with a pure twist map and its dynamics are comparable to the unperturbed family of cylinder maps as met in Subsect. “Circle Maps”. Indeed it is again a family of rigid rotations, parametrized by I and where P0 (:; I) has rotation number ˛(I). In this case the question is what will be the fate of this family of invariant circles, as well as with the corresponding rigidly rotational dynamics. Regarding the rotation number we again introduce Diophantine conditions. Indeed, for  > 2 and > 0 the subset [0; 1]; is defined as in (5), i. e., it contains all

˛ 2 [0; 1], such that for all rationals p/q ˇ ˇ ˇ ˇ ˇ˛  p ˇ  q : ˇ qˇ Pulling back [0; 1]; along the map ˛ we obtain a subset ;  . Theorem 4 (Twist Map Theorem [104]) For sufficiently small, and for the perturbation O(") sufficiently small in C 1 -topology, there exists a C 1 transformation ˚" :  ! , conjugating the restriction P0 j; to a subsystem of P" . As in the case of Theorem 3 again we chose the formulation of [29,31]. Largely the remarks following Theorem 3 also apply here. Remark  Compare the format of the Theorems 3 and 4 and observe that in the latter case the role of the parameter ˛ has been taken by the action variable I. Theorem 4 implies that typically quasi-periodicity occurs with positive measure in phase space.  In the gaps typically we have coexistence of periodicity, quasi-periodicity and chaos [6,9,35,107,108,123,137]. The latter follows from transversality of homo- and heteroclinic connections that give rise to positive topological entropy. Open problems are whether the corresponding Lyapunov exponents also are positive, compare with the discussion at the end of the introduction. Similar to the applications of Theorem 3 given at the end of Subsect. “Circle Maps”, here direct applications are possible in the conservative setting. Indeed, consider a system of weakly coupled pendula @U (y1 ; y2 ) @y1 @U y¨2 C ˛22 sin y2 D " (y1 ; y2 ) : @y2 y¨1 C ˛12 sin y1 D "

Writing y˙ j D z j , j D 1; 2 as before, we again get a vector field in the four-dimensional phase space R2  R2 D f(y1 ; y2 ); (z1 ; z2 )g. In this case the energy H" (y1 ; y2 ; z1 ; z2 ) 1 1 D z12 C z22  ˛12 cos y1  ˛22 cos y2 C "U(y1 ; y2 ) 2 2 is a constant of motion. Restricting to a three-dimensional energy surface H"1 D const:, the iso-energetic Poincaré

667

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Hamiltonian Perturbation Theory (and Transition to Chaos)

map P" is a twist map and application of Theorem 4 yields the conclusion of quasi-periodicity (on invariant two-tori) occurring with positive measure in the energy surfaces of H" .

is the nth convergent in the continued fraction expansion of ˛ then the Bruno-condition reads X log(q nC1 ) n

qn

<1:

Remark As in the dissipative case this example directly generalizes to cases with n oscillators (n 2 N), again leading to invariant n-tori with quasi-periodic flow. We shall return to this subject in a later section.

This condition turns out to be necessary and sufficient for ˚ having positive radius of convergence [141,142].

Linearization of Complex Maps

Cremer’s Example in Herman’s Version consider the map

The Subsects. “Circle Maps” and “Area-Preserving Maps” both deal with smooth circle maps that are conjugated to rigid rotations. Presently the concern is with planar holomorphic maps that are conjugated to a rigid rotation on an open subset of the plane. Historically this is the first time that a small divisor problem was solved [7,101,141,142] and  Perturbative Expansions, Convergence of. Complex Linearization Given is a holomorphic germ F : (C; 0) ! (C; 0) of the form F(z) D z C f (z), with f (0) D f 0 (0) D 0. The problem is to find a biholomorphic germ ˚ : (C; 0) ! (C; 0) such that ˚ ıF D˚ : Such a diffeomorphism ˚ is called a linearization of F near 0. We begin with the formal approach. Given the series P P f (z) D j2 f j z j , we look for ˚ (z) D z C j2  j z j . It turns out that a solution always exists whenever  ¤ 0 is not a root of unity. Indeed, direct computation reveals the following set of equations that can be solved recursively: For j D 2 get the equation (1  )2 D f2 For j D 3 get the equation (1  2 )3 D f3 C 2 f2 2 For j D n get the equation (1  n1 ) n D f n Cknown. The question now reduces to whether this formal solution has a positive radius of convergence. The hyperbolic case 0 < jj ¤ 1 was already solved by Poincaré, for a description see [7]. The elliptic case jj D 1 again has small divisors and was solved by Siegel when for some > 0 and  > 2 we have the Diophantine nonresonance condition p

j  e

2i q

j  jqj :

The corresponding set of  constitutes a set of full measure in T 1 D fg. Yoccoz [141] completely solved the elliptic case using the Bruno-condition. If  D e2 i˛

pn and when qn

As an example

F(z) D z C z2 ; where  2 T 1 is not a root of unity. Observe that a point z 2 C is a periodic point of F with period q if and only if F q (z) D z, where obviously q

F q (z) D q z C    C z2 : Writing   q F q (z)  z D z q  1 C    C z2 1 ; the period q periodic points exactly are the roots of the right hand side polynomial. Abbreviating N D 2q  1, it directly follows that, if z1 ; z2 ; : : : ; z N are the nontrivial roots, then for their product we have z1  z2  : : :  z N D  q  1 : It follows that there exists a nontrivial root within radius jq  1j1/N of z D 0. Now consider the set of  T 1 defined as follows:  2  whenever lim inf jq  1j1/N D 0 : q!1

It can be directly shown that  is residual, again compare with [115]. It also follows that for  2  linearization is impossible. Indeed, since the rotation is irrational, the existence of periodic points in any neighborhood of z D 0 implies zero radius of convergence. Remark  Notice that the residual set  is in the complement of the full measure set of all Diophantine numbers, again see [115].

Hamiltonian Perturbation Theory (and Transition to Chaos)

 Considering  2 T 1 as a parameter, we see a certain analogy of these results on complex linearization with the Theorems 3 and 4. Indeed, in this case for a full measure set of s on a neighborhood of z D 0 the map F D F is conjugated to a rigid irrational rotation. Such a domain in the z-plane often is referred to as a Siegel disc. For a more general discussion of these and of Herman rings, see [101]. KAM Theory: An Overview In Sect. “Invariant Curves of Planar Diffeomorphisms” we described the persistent occurrence of quasi-periodicity in the setting of diffeomorphisms of the circle or the plane. The general perturbation theory of quasi-periodic motions is known under the name Kolmogorov–Arnold–Moser (or KAM) theory and discussed extensively elsewhere in this encyclopedia  Kolmogorov–Arnold–Moser (KAM) Theory. Presently we briefly summarize parts of this KAM theory in broad terms, as this fits in our considerations, thereby largely referring to [4,80,81,119,121,143,144], also see [20,24,55]. In general quasi-periodicity is defined by a smooth conjugation. First on the n-torus T n D Rn /(2 Z)n consider the vector field X! D

n X jD1

!j

@ ; @' j

where !1 ; !2 ; : : : ; !n are called frequencies [43,106]. Now, given a smooth (say, of class C 1 ) vector field X on a manifold M, with T  M an invariant n-torus, we say that the restriction Xj T is parallel if there exists ! 2 Rn and a smooth diffeomorphism ˚ : T ! T n , such that ˚ (Xj T ) D X! . We say that Xj T is quasi-periodic if the frequencies !1 ; !2 ; : : : ; !n are independent over Q. A quasi-periodic vector field Xj T leads to an integer affine structure on the torus T. In fact, since each orbit is dense, it follows that the self conjugations of X! exactly are the translations of T n , which completely determine the affine structure of T n . Then, given ˚ : T ! T n with ˚ (Xj T ) D X! , it follows that the self conjugations of Xj T determines a natural affine structure on the torus T. Note that the conjugation ˚ is unique modulo translations in T and T n . Note that the composition of ˚ by a translation of T n does not change the frequency vector !. However, the composition by a linear invertible map S 2 GL(n; Z) yields S X! D XS! . We here speak of an integer affine structure [43].

Remark  The transition maps of an integer affine structure are translations and elements of GL(n; Z).  The current construction is compatible with the integrable affine structure on the Liouville tori of an integrable Hamiltonian system [6]. Note that in that case the structure extends to all parallel tori. Classical KAM Theory The classical KAM theory deals with smooth, nearly integrable Hamiltonian systems of the form '˙ D !(I) C " f (I; '; ") I˙ D "g(I; '; ") ;

(6)

where I varies over an open subset of Rn and ' over the standard torus T n . Note that for " D 0 the phase space as an open subset of Rn  T n is foliated by invariant tori, parametrized by I. Each of the tori is parametrized by ' and the corresponding motion is parallel (or multi- periodic or conditionally periodic) with frequency vector !(I). Perturbation theory asks for persistence of the invariant n-tori and the parallelity of their motion for small values of j"j. The answer that KAM theory gives needs two essential ingredients. The first ingredient is that of Kolmogorov non-degeneracy which states that the map I 2 Rn 7! !(I) 2 Rn is a (local) diffeomorphism. Compare with the twist condition of Sect. “Invariant Curves of Planar Diffeomorphisms”. The second ingredient generalizes the Diophantine conditions (5) of that section as follows: for  > n  1 and > 0 consider the set n D f! 2 Rn j jh!; kij  jkj ; k 2 Z n nf0gg: (7) R; n The following properties are more or less direct. First R; n has a closed half line geometry in the sense that if ! 2 R; n and s  1 then also s! 2 R; . Moreover, the intersection n is a Cantor set of measure S n1 nRn D O( ) S n1 \R; ; as # 0, see Fig. 4. Completely in the spirit of Theorem 4, the classical KAM theorem roughly states that a Kolmogorov non-degenerate nearly integrable system (6)" , for j"j  1 is smoothly conjugated to the unperturbed version (6)0 , provided that the frequency map ! is co-restricted to the Dion . In this formulation smoothness has to phantine set R; be taken in the sense of Whitney [119,136], also compare with [20,24,29,31,55,121]. As a consequence we may say that in Hamiltonian systems of n degrees of freedom typically quasi-periodic invariant (Lagrangian) n-tori occur with positive measure in

669

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Hamiltonian Perturbation Theory (and Transition to Chaos)

Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 4 The Diophantine set Rn; has the closed half line geometry and the intersection Sn1 \ Rn; is a Cantor set of measure Sn1 n Rn; D O() as  # 0

phase space. It should be said that also an iso-energetic version of this classical result exists, implying a similar conclusion restricted to energy hypersurfaces [6,9,21,24]. The Twist Map Theorem 4 is closely related to the iso-energetic KAM Theorem. Remark  We chose the quasi-periodic stability format as in Sect. “Invariant Curves of Planar Diffeomorphisms”. For regularity issues compare with a remark following Theorem 3.  For applications we largely refer to the introduction and to [24,31] and references therein.  Continuing the discussion on affine structures at the beginning of this section, we mention that by means of the symplectic form, the domain of the I-variables in Rn inherits an affine structure [60], also see [91] and references therein. Statistical Mechanics deals with particle systems that are large, often infinitely large. The Ergodic Hypothesis roughly says that in a bounded energy hypersurface, the dynamics are ergodic, meaning that any evolution in the energy level set comes near every point of this set. The taking of limits as the number of particles tends to infinity is a notoriously difficult subject. Here we dis-

cuss a few direct consequences of classical KAM theory for many degrees of freedom. This discussion starts with Kolmogorov’s papers [80,81], which we now present in a slightly rephrased form. First, we recall that for Hamiltonian systems (say, with n degrees of freedom), typically the union of Diophantine quasi-periodic Lagrangian invariant n-tori fills up positive measure in the phase space and also in the energy hypersurfaces. Second, such a collection of KAM tori immediately gives rise to non-ergodicity, since it clearly implies the existence of distinct invariant sets of positive measure. For background on Ergodic Theory, see e. g. [9,27] and [24] for more references. Apparently the KAM tori form an obstruction to ergodicity, and a question is how bad this obstruction is as n ! 1. Results in [5,78] indicate that this KAM theory obstruction is not too bad as the size of the system tends to infinity. In general the role of the Ergodic Hypothesis in Statistical Mechanics has turned out to be much more subtle than was expected, see e. g. [18,64]. Dissipative KAM Theory As already noted by Moser [105,106], KAM theory extends outside the world of Hamiltonian systems, like to volume preserving systems, or to equivariant or reversible systems. This also holds for the class of general smooth systems, often called dissipative. In fact, the KAM theorem allows for a Lie algebra proof, that can be used to cover all these special cases [24,29,31,45]. It turns out that in many cases parameters are needed for persistent occurrence of (Diophantine) quasi-periodic tori. As an example we now consider the dissipative setting, where we discuss a parametrized system with normally hyperbolic invariant n-tori carrying quasi-periodic motion. From [73] it follows that this is a persistent situation and that, up to a smooth (in this case of class Ck for large k) diffeomorphism, we can restrict to the case where T n is the phase space. To fix thoughts we consider the smooth system '˙ D !() C " f ('; ; ") ˙ D0; 

(8)

where  2 Rn is a multi-parameter. The results of the classical KAM theorem regarding (6)" largely carry over to (8);" . Now, for " D 0 the product of phase space and parameter space as an open subset of T n  Rn is completely foliated by invariant n-tori and since the perturbation does ˙ not concern the -equation, this foliation is persistent. The interest is with the dynamics on the resulting invariant tori that remains parallel after the perturbation; compare

Hamiltonian Perturbation Theory (and Transition to Chaos)

with the setting of Theorem 3. As just stated, KAM theory here gives a solution similar to the Hamiltonian case. The analogue of the Kolmogorov non-degeneracy condition here is that the frequency map  7! !() is a (local) diffeomorphism. Then, in the spirit of Theorem 3, we state that the system (8);" is smoothly conjugated to (8);0 , as before, provided that the map ! is co-restricted to the n . Again the smoothness has to be Diophantine set R; taken in the sense of Whitney [29,119,136,143,144], also see [20,24,31,55]. It follows that the occurrence of normally hyperbolic invariant tori carrying (Diophantine) quasi-periodic flow is typical for families of systems with sufficiently many parameters, where this occurrence has positive measure in parameter space. In fact, if the number of parameters equals the dimension of the tori, the geometry as sketched in Fig. 4 carries over in a diffeomorphic way. Remark  Many remarks following Subsect. “Classical KAM Theory” and Theorem 3 also hold here.  In cases where the system is degenerate, for instance because there is a lack of parameters, a path formalism can be invoked, where the parameter path is required to n , see be a generic subfamily of the Diophantine set R; Fig. 4. This amounts to the Rüssmann non-degeneracy, that still gives positive measure of quasi-periodicity in the parameter space, compare with [24,31] and references therein.  In the dissipative case the KAM theorem gives rise to families of quasi-periodic attractors in a typical way. This is of importance in center manifold reductions of infinite dimensional dynamics as, e. g., in fluid mechanics [125,126]. In Sect. “Transition to Chaos and Turbulence” we shall return to this subject. Lower Dimensional Tori We extend the above approach to the case of lower dimensional tori, i. e., where the dynamics transversal to the tori is also taken into account. We largely follow the setup of [29,45] that follows Moser [106]. Also see [24,31] and references therein. Changing notation a little, we now consider the phase space T n  Rm D fx(mod 2); yg, as well a parameter space fg D P Rs . We consider a C 1 -family of vector fields X(x; y; ) as before, having T n  f0g T n  Rm as an invariant n-torus for  D 0 2 P. x˙ D !() C f (y; ) y˙ D ˝() y C g(y; ) ˙ D0; 

(9)

with f (y; 0 ) D O(jyj) and g(y; 0 ) D O(jyj2 ), so we assume the invariant torus to be of Floquet type. The system X D X(x; y; ) is integrable in the sense that it is T n -symmetric, i. e., x-independent [29]. The interest is with the fate of the invariant torus T n  f0g and its parallel dynamics under small perturbation to a system ˜ X˜ D X(x; y; ) that no longer needs to be integrable. Consider the smooth mappings ! : P ! Rn and ˝ : P ! gl(m; R). To begin with we restrict to the case where all eigenvalues of ˝(0 ) are simple and nonzero. In general for such a matrix ˝ 2 gl(m; R), let the eigenvalues be given by ˛1 ˙ iˇ1 ; : : : ; ˛ N 1 ˙ iˇ N 1 and ı1 ; : : : ; ı N 2 , where all ˛ j ; ˇ j and ı j are real and hence m D 2N1 C N2 . Also consider the map spec : gl(m; R) ! R2N 1 CN 2 , given by ˝ 7! (˛; ˇ; ı). Next to the internal frequency vector ! 2 Rn , we also have the vector ˇ 2 R N 1 of normal frequencies. The present analogue of Kolmogorov non-degeneracy is the Broer–Huitema–Takens (BHT) non-degeneracy condition [29,127], which requires that the product map !  (spec) ı ˝ : P ! Rn  gl(m; R) at  D 0 has a surjective derivative and hence is a local submersion [72]. Furthermore, we need Diophantine conditions on both the internal and the normal frequencies, generalizing (7). Given  > n  1 and > 0, it is required for all k 2 Z n n f0g and all ` 2 Z N 1 with j`j  2 that jhk; !i C h`; ˇij  jkj :

(10)

Inside Rn  R N 1 D f!; ˇg this yields a Cantor set as before (compare Fig. 4). This set has to be pulled back along the submersion !  (spec) ı ˝, for examples see Subsects. “(n 1)-Tori” and “Quasi-periodic Bifurcations” below. The KAM theorem for this setting is quasi-periodic stability of the n-tori under consideration, as in Subsect. “Dissipative KAM Theory”, yielding typical examples where quasi-periodicity has positive measure in parameter space. In fact, we get a little more here, since the normal linear behavior of the n-tori is preserved by the Whitney smooth conjugations. This is expressed as normal linear stability, which is of importance for quasi-periodic bifurcations, see Subsect. “Quasi-periodic Bifurcations” below. Remark  A more general set-up of the normal stability theory [45] adapts the above to the case of non-simple (multiple) eigenvalues. Here the BHT non-degeneracy condition is formulated in terms of versal unfolding of the matrix ˝(0 ) [7]. For possible conditions under which vanishing eigenvalues are admissible see [29,42,69] and references therein.

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Hamiltonian Perturbation Theory (and Transition to Chaos)

 This general set-up allows for a structure preserving formulation as mentioned earlier, thereby including the Hamiltonian and volume preserving case, as well as equivariant and reversible cases. This allows us, for example, to deal with quasi-periodic versions of the Hamiltonian and the reversible Hopf bifurcation [38,41,42,44].  The Parameterized KAM Theory discussed here a priori needs many parameters. In many cases the parameters are distinguished in the sense that they are given by action variables, etc. For an example see Subsect. “(n  1)-Tori” on Hamiltonian (n  1)-tori Also see [127] and [24,31] where the case of Rüssmann nondegeneracy is included. This generalizes a remark at the end of Subsect. “Dissipative KAM Theory”.

Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 5 Range of the energy-momentum map of the spherical pendulum

Global KAM Theory We stay in the Hamiltonian setting, considering Lagrangian invariant n-tori as these occur in a Liouville integrable system with n degrees of freedom. The union of these tori forms a smooth T n -bundle f : M ! B (where we leave out all singular fibers). It is known that this bundle can be non-trivial [56,60] as can be measured by monodromy and Chern class. In this case global action angle variables are not defined. This non-triviality, among other things, is of importance for semi-classical versions of the classical system at hand, in particular for certain spectrum defects [57,62,134,135], for more references also see [24]. Restricting to the classical case, the problem is what happens to the (non-trivial) T n -bundle f under small, non-integrable perturbation. From the classical KAM theory, see Subsect. “Classical KAM Theory” we already know that on trivializing charts of f Diophantine quasi-periodic n-tori persist. In fact, at this level, a Whitney smooth conjugation exists between the integrable system and its perturbation, which is even Gevrey regular [136]. It turns out that these local KAM conjugations can be glued together so to obtain a global conjugation at the level of quasi-periodic tori, thereby implying global quasi-periodic stability [43]. Here we need unicity of KAM tori, i. e., independence of the action-angle chart used in the classical KAM theorem [26]. The proof uses the integer affine structure on the quasi-periodic tori, which enables taking convex combinations of the local conjugations subjected to a suitable partition of unity [72,129]. In this way the geometry of the integrable bundle can be carried over to the nearlyintegrable one. The classical example of a Liouville integrable system with non-trivial monodromy [56,60] is the spherical pendulum, which we now briefly revisit. The configuration

space is S2 D fq 2 R3 j hq; qi D 1g and the phase space T  S2 Š f(q; p) 2 R6 j hq; qi D 1 and hq; pi D 0g. The two integrals I D q1 p2  q2 p1 (angular momentum) and E D 12 hp; pi C q3 (energy) lead to an energy momentum map EM : T  S2 ! R2 , given by (q; p) 7! (I; E) D  q1 p2  q2 p1 ; 12 hp; pi C q3 . In Fig. 5 we show the image of the map EM. The shaded area B consists of regular values, the fiber above which is a Lagrangian two-torus; the union of these gives rise to a bundle f : M ! B as described before, where f D EMj M . The motion in the twotori is a superposition of Huygens’ rotations and pendulum-like swinging, and the non-existence of global action angle variables reflects that the three interpretations of ‘rotating oscillation’, ‘oscillating rotation’ and ‘rotating rotation’ cannot be reconciled in a consistent way. The singularities of the fibration include the equilibria (q; p) D ((0; 0; ˙1); (0; 0; 0)) 7! (I; E) D (0; ˙1). The boundary of this image also consists of singular points, where the fiber is a circle that corresponds to Huygens’ horizontal rotations of the pendulum. The fiber above the upper equilibrium point (I; E) D (0; 1) is a pinched torus [56], leading to non-trivial monodromy, in a suitable bases of the period lattices, given by 

1 0

1 1



2 GL(2; R) :

The question here is what remains of the bundle f when the system is perturbed. Here we observe that locally Kolmogorov non-degeneracy is implied by the non-trivial monodromy [114,122]. From [43,122] it follows that the non-trivial monodromy can be extended in the perturbed case.

Hamiltonian Perturbation Theory (and Transition to Chaos)

Remark  The case where this perturbation remains integrable is covered in [95], but presently the interest is with the nearly integrable case, so where the axial symmetry is broken. Also compare [24] and many of its references.  The global conjugations of [43] are Whitney smooth (even Gevrey regular [136]) and near the identity map in the C 1 -topology [72]. Geometrically speaking these diffeomorphisms also are T n -bundle isomorphisms between the unperturbed and the perturbed bundle, the basis of which is a Cantor set of positive measure. Splitting of Separatrices KAM theory does not predict the fate of close-to-resonant tori under perturbations. For fully resonant tori the phenomenon of frequency locking leads to the destruction of the torus under (sufficiently rich) perturbations, and other resonant tori disintegrate as well. In the case of a single resonance between otherwise Diophantine frequencies the perturbation leads to quasi-periodic bifurcations, cf. Sect. “Transition to Chaos and Turbulence”. While KAM theory concerns the fate of most trajectories and for all times, a complementary theorem has been obtained in [93,109,110,113]. It concerns all trajectories and states that they stay close to the unperturbed tori for long times that are exponential in the inverse of the perturbation strength. For trajectories starting close to surviving tori the diffusion is even superexponentially slow, cf. [102,103]. Here a form of smoothness exceeding the mere existence of infinitely many derivatives of the Hamiltonian is a necessary ingredient, for finitely differentiable Hamiltonians one only obtains polynomial times. Solenoids, which cannot be present in integrable systems, are constructed for generic Hamiltonian systems in [16,94,98], yielding the simultaneous existence of representatives of all homeomorphy-classes of solenoids. Hyperbolic tori form the core of a construction proposed in [5] of trajectories that venture off to distant points of the phase space. In the unperturbed system the union of a family of hyperbolic tori, parametrized by the actions conjugate to the toral angles, form a normally hyperbolic manifold. The latter is persistent under perturbations, cf. [73,100], and carries a Hamiltonian flow with fewer degrees of freedom. The main difference between integrable and non-integrable systems already occurs for periodic orbits. Periodic Orbits A sharp difference to dissipative systems is that it is generic for hyperbolic periodic orbits on compact energy shells in

Hamiltonian systems to have homoclinic orbits, cf. [1] and references therein. For integrable systems these form together a pinched torus, but under generic perturbations the stable and unstable manifold of a hyperbolic periodic orbit intersect transversely. It is a nontrivial task to actually check this genericity condition for a given non-integrable perturbation, a first-order condition going back to Poincaré requires the computation of the so-called Mel’nikov integral, see [66,137] for more details. In two degrees of freedom normalization leads to approximations that are integrable to all orders, which implies that the Melnikov integral is a flat function. In the real analytic case the Melnikov criterion is still decisive in many examples [65]. Genericity conditions are traditionally formulated in the universe of smooth vector fields, and this makes the whole class of analytic vector fields appear to be nongeneric. This is an overly pessimistic view as the conditions defining a certain class of generic vector fields may certainly be satisfied by a given analytic system. In this respect it is interesting that the generic properties may also be formulated in the universe of analytic vector fields, see [28] for more details. (n  1)-Tori The (n 1)-parameter families of invariant (n 1)-tori organize the dynamics of an integrable Hamiltonian system in n degrees of freedom, and under small perturbations the parameter space of persisting analytic tori is Cantorized. This still allows for a global understanding of a substantial part of the dynamics, but also leads to additional questions. A hyperbolic invariant torus T n1 has its Floquet exponents off the imaginary axis. Note that T n1 is not a normally hyperbolic manifold. Indeed, the normal linear behavior involves the n  1 zero eigenvalues in the direction of the parametrizing actions as well; similar to (9) the format x˙ D !(y) C O(y) C O(z2 ) y˙ D O(y) C O(z3 ) z˙ D ˝(y)z C O(z2 ) in Floquet coordinates yields an x-independent matrix ˝ that describes the symplectic normal linear behavior, cf. [29]. The union fz D 0g over the family of (n1)-tori is a normally hyperbolic manifold and constitutes the center manifold of T n1 . Separatrices splitting yields the dividing surfaces in the sense of Wiggins et al. [138]. The persistence of elliptic tori under perturbation from an integrable system involves not only the internal

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frequencies of T n1 , but also the normal frequencies. Next to the internal resonances the necessary Diophantine conditions (10) exclude the normal-internal resonances hk; !i D ˛ j

(11)

hk; !i D 2˛ j

(12)

hk; !i D ˛ i C ˛ j

(13)

hk; !i D ˛ i  ˛ j :

(14)

The first three resonances lead to the quasi-periodic center-saddle bifurcation studied in Sect. “Transition to Chaos and Turbulence”, the frequency-halving (or quasiperiodic period doubling) bifurcation and the quasi-periodic Hamiltonian Hopf bifurcation, respectively. The resonance (14) generalizes an equilibrium in 1 : 1 resonance whence T n1 persists and remains elliptic, cf. [78]. When passing through resonances (12) and (13) the lower-dimensional tori lose ellipticity and acquire hyperbolic Floquet exponents. Elliptic (n  1)-tori have a single normal frequency whence (11) and (12) are the only normal-internal resonances. See [35] for a thorough treatment of the ensuing possibilities. The restriction to a single normal-internal resonance is dictated by our present possibilities. Indeed, already the bifurcation of equilibria with a fourfold zero eigenvalue leads to unfoldings that simultaneously contain all possible normal resonances. Thus, a satisfactory study of such tori which already may form one-parameter families in integrable Hamiltonian systems with five degrees of freedom has to await further progress in local bifurcation theory.

tion [66,75,82], where a periodic solution branches off. In a second transition of similar nature a quasi-periodic two-torus branches off, then a quasi-periodic three-torus, etc. The idea is that the motion picks up more and more frequencies and thus obtains an increasingly complicated power spectrum. In the early 1970s this idea was modified in the Ruelle–Takens route to turbulence, based on the observation that, for flows, a three-torus can carry chaotic (or strange) attractors [112,126], giving rise to a broad band power spectrum. By the quasi-periodic bifurcation theory [24,29,31] as sketched below these two approaches are unified in a generic way, keeping track of measure theoretic aspects. For general background in dynamical systems theory we refer to [27,79]. Another transition to chaos was detected in the quadratic family of interval maps f (x) D x(1  x) ; see [58,99,101], also for a holomorphic version. This transition consists of an infinite sequence of period doubling bifurcations ending up in chaos; it has several universal aspects and occurs persistently in families of dynamical systems. In many of these cases also homoclinic bifurcations show up, where sometimes the transition to chaos is immediate when parameters cross a certain boundary, for general theory see [13,14,30,117]. There exist quite a number of case studies where all three of the above scenarios play a role, e. g., see [32,33,46] and many of their references. Quasi-periodic Bifurcations

Transition to Chaos and Turbulence One of the main interests over the second half of the twentieth century has been the transition between orderly and complicated forms of dynamics upon variation of either initial states or of system parameters. By ‘orderly’ we here mean equilibrium and periodic dynamics and by complicated quasi-periodic and chaotic dynamics, although we note that only chaotic dynamics is associated to unpredictability, e. g. see [27]. As already discussed in the introduction systems like a forced nonlinear oscillator or the planar three-body problem exhibit coexistence of periodic, quasi-periodic and chaotic dynamics, also compare with Fig. 1. Similar remarks go for the onset of turbulence in fluid dynamics. Around 1950 this led to the scenario of Hopf– Landau–Lifschitz [75,76,83,84], which roughly amounts to the following. Stationary fluid motion corresponds to an equilibrium point in an 1-dimensional state space of velocity fields. The first transition is a Hopf bifurca-

For the classical bifurcations of equilibria and periodic orbits, the bifurcation sets and diagrams are generally determined by a classical geometry in the product of phase space and parameter space as already established by, e. g., [8,133], often using singularity theory. Quasi-periodic bifurcation theory concerns the extension of these bifurcations to invariant tori in nearly-integrable systems, e. g., when the tori lose their normal hyperbolicity or when certain (strong) resonances occur. In that case the dense set of resonances, also responsible for the small divisors, leads to a Cantorization of the classical geometries obtained from Singularity Theory [29,35,37,38,39,41,44,45, 48,49,67,68,69], also see [24,31,52,55]. Broadly speaking, one could say that in these cases the Preparation Theorem [133] is partly replaced by KAM theory. Since the KAM theory has been developed in several settings with or without preservation of structure, see Sect. “KAM Theory: An Overview”, for the ensuing quasi-periodic bifurcation theory the same holds.

Hamiltonian Perturbation Theory (and Transition to Chaos)

Hamiltonian Cases To fix thoughts we start with an example in the Hamiltonian setting, where a robust model for the quasi-periodic center-saddle bifurcation is given by H!1 ;!2 ;;" (I; '; p; q) D !1 I1 C !2 I2 C 12 p2 C V (q) C " f (I; '; p; q)

(15)

with V (q) D 13 q3  q, compare with [67,69]. The unperturbed (or integrable) case " D 0, by factoring out the T 2 -symmetry, boils down to a standard center-saddle bifurcation, involving the fold catastrophe [133] in the potential function V D V (q). This results in the existence of two invariant two-tori, one elliptic and the other hyperbolic. For 0 ¤ j"j  1 the dense set of resonances complicates this scenario, as sketched in Fig. 6, determined by the Diophantine conditions jhk; !ij  jkj ; jhk; !i C `ˇ(q)j  jkj



;

for q < 0 ;

(16)

for q > 0

for all k p 2 Z n n f0g and for all ` 2 Z with j`j  2. Here ˇ(q) D 2q is the normal frequency of the elliptic torus p given by q D  for  > 0. As before, (cf. Sects. “Invariant Curves of Planar Diffeomorphisms”, “KAM Theory: An Overview”), this gives a Cantor set of positive measure [24,29,31,45,69,105,106]. For 0 < j"j  1 Fig. 6 will be distorted by a nearidentity diffeomorphism; compare with the formulations of the Theorems 3 and 4. On the Diophantine Cantor set the dynamics is quasi-periodic, while in the gaps generically there is coexistence of periodicity and chaos, roughly

comparable with Fig. 1, at left. The gaps at the border furthermore lead to the phenomenon of parabolic resonance, cf. [86]. Similar programs exist for all cuspoid and umbilic catastrophes [37,39,68] as well as for the Hamiltonian Hopf bifurcation [38,44]. For applications of this approach see [35]. For a reversible analogue see [41]. As so often within the gaps generically there is an infinite regress of smaller gaps [11,35]. For theoretical background we refer to [29,45,106], for more references also see [24]. Dissipative Cases In the general dissipative case we basically follow the same strategy. Given the standard bifurcations of equilibria and periodic orbits, we get more complex situations when invariant tori are involved as well. The simplest examples are the quasi-periodic saddle-node and quasi-periodic period doubling [29] also see [24,31]. To illustrate the whole approach let us start from the Hopf bifurcation of an equilibrium point of a vector field [66,75,82,116] where a hyperbolic point attractor loses stability and branches off a periodic solution, cf. Subsect. “Dissipative Perturbations”. A topological normal form is given by 

y˙1 y˙2



 D

˛ ˇ

ˇ ˛



y1 y2



   y12 C y22



y1 y2



(17) where y D (y1 ; y2 ) 2 R2 , ranging near (0; 0). In this representation usually one fixes ˇ D 1 and lets ˛ D  (near 0) serve as a (bifurcation) parameter, classifying modulo topological equivalence. In polar coordinates (17) so gets the form '˙ D 1 ; r˙ D r  r3 : Figure 7 shows an amplitude response diagram (often called the bifurcation diagram). Observe the occurrence of the attracting periodic solution for  > 0 of amplitude p . Let us briefly consider the Hopf bifurcation for fixed points of diffeomorphisms. A simple example has the form

Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 6 Sketch of the Cantorized Fold, as the bifurcation set of the quasiperiodic center-saddle bifurcation for n D 2 [67], where the horizontal axis indicates the frequency ratio !2 : !1 , cf. (15). The lower part of the figure corresponds to hyperbolic tori and the upper part to elliptic ones. See the text for further interpretations

P(y) D e2(˛Ciˇ ) y C O(jyj2 ) ;

(18)

y 2 C Š R2 , near 0. To start with ˇ is considered a constant, such that ˇ is not rational with denominator less than five, see [7,132], and where O(jyj2 ) should contain generic third order terms. As before, we let ˛ D  serve as a bifurcation parameter, varying near 0. On one side of the

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Hamiltonian Perturbation Theory (and Transition to Chaos)

We now discuss the quasi-periodic Hopf bifurcation [17,29], largely following [55]. The unperturbed, integrable family X D X (x; y) on T n  R2 has the form X (x; y) D [!() C f (y; )]@x C [˝()y C g(y; )]@ y ; (19)

Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 7 Bifurcation diagram of the Hopf bifurcation

bifurcation value  D 0, this system has by normal hyperbolicity and [73], an invariant circle. Here, due to the invariance of the rotation numbers of the invariant circles, no topological stability can be obtained [111]. Still this bifurcation can be characterized by many persistent properties. Indeed, in a generic two-parameter family (18), say with both ˛ and ˇ as parameters, the periodicity in the parameter plane is organized in resonance tongues [7,34,82]. (The tongue structure is hardly visible when only one parameter, like ˛, is used.) If the diffeomorphism is the return map of a periodic orbit for flows, this bifurcation produces an invariant two-torus. Usually this counterpart for flows is called Ne˘ımark–Sacker bifurcation. The periodicity as it occurs in the resonance tongues, for the vector field is related to phase lock. The tongues are contained in gaps of a Cantor set of quasi-periodic tori with Diophantine frequencies. Compare the discussion in Subsect. “Circle Maps”, in particular also regarding the Arnold family and Fig. 3. Also see Sect. “KAM Theory: An Overview” and again compare with [115]. Quasi-periodic versions exist for the saddle-node, the period doubling and the Hopf bifurcation. Returning to the setting with T n  Rm as the phase space, we remark that the quasi-periodic saddle-node and period doubling already occur for m D 1, or in an analogous center manifold. The quasi-periodic Hopf bifurcation needs m  2. We shall illustrate our results on the latter of these cases, compare with [19,31]. For earlier results in this direction see [52]. Our phase space is T n  R2 D fx(mod 2); yg, where we are dealing with the parallel invariant torus T n  f0g. In the integrable case, by T n -symmetry we can reduce to R2 D fyg and consider the bifurcations of relative equilibria. The present interest is with small non-integrable perturbations of such integrable models.

were f D O(jyj) and g D O(jyj2 ) as before. Moreover  2 P is a multi-parameter and ! : P ! Rn and ˝ : P ! gl(2; R) are smooth maps. Here we take   ˛() ˇ() ˝() D ; ˇ() ˛() which makes the @ y component of (19) compatible with the planar Hopf family (17). The present form of Kolmogorov non-degeneracy is Broer–Huitema–Takens stability [29,42,45], requiring that there is a subset   P on which the map  2 P 7! (!(); ˝()) 2 Rn  gl(2; R) is a submersion. For simplicity we even assume that  is replaced by (!; (˛; ˇ)) 2 Rn  R2 : Observe that if the non-linearity g satisfies the well-known Hopf non-degeneracy conditions, e. g., compare [66,82], then the relative equilibrium y D 0 undergoes a standard planar Hopf bifurcation as described before. Here ˛ again plays the role of bifurcation parameter and a closed orbit branches off at ˛ D 0. To fix thoughts we assume that y D 0 is attracting for ˛ < 0. and that the closed orbit occurs for ˛ > 0, and is attracting as well. For the integrable family X, qualitatively we have to multiply this planar scenario with T n , by which all equilibria turn into invariant attracting or repelling n-tori and the periodic attractor into an attracting invariant (n C 1)-torus. Presently the question is what happens to both the n- and the (n C 1)-tori, when we apply a small near-integrable perturbation. The story runs much like before. Apart from the BHT non-degeneracy condition we require Diophantine conditions (10), defining the Cantor set (2) ; D f(!; (˛; ˇ)) 2  j jhk; !i C `ˇj  jkj ;

8k 2 Zn n f0g ; 8` 2 Z with j`j  2g ;

(20)

(2) In Fig. 8 we sketch the intersection of ;

Rn R2 with 2 a plane f!g  R for a Diophantine (internal) frequency vector !, cf. (7).

Hamiltonian Perturbation Theory (and Transition to Chaos)

Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 8 (2) Planar section of the Cantor set ;

From [17,29] it now follows that for any family X˜ on  R2  P, sufficiently near X in the C 1 -topology a near-identity C 1 -diffeomorphism ˚ : T n  R2   ! T n  R2   exists, defined near T n  f0g   , that con(2) . jugates X to X˜ when further restricting to T n  f0g  ; So this means that the Diophantine quasi-periodic invariant n-tori are persistent on a diffeomorphic image of the (2) Cantor set ; , compare with the formulations of the Theorems 3 and 4. Similarly we can find invariant (n C 1)-tori. We first have to develop a T nC1 symmetric normal form approximation [17,29] and  Normal Forms in Perturbation Theory. For this purpose we extend the Diophantine conditions (20) by requiring that the inequality holds for all j`j  N for N D 7. We thus find another large Cantor set, again see Fig. 8, where Diophantine quasi-periodic invariant (n C 1)-tori are persistent. Here we have to restrict to ˛ > 0 for our choice of the sign of the normal form coefficient, compare with Fig. 7. In both the cases of n-tori and of (n C 1)-tori, the nowhere dense subset of the parameter space containing the tori can be fattened by normal hyperbolicity to open subsets. Indeed, the quasi-periodic n- and (n C 1)-tori are infinitely normally hyperbolic [73]. Exploiting the normal form theory [17,29] and  Normal Forms in Perturbation Theory to the utmost and using a more or less standard contraction argument [17,53], a fattening of the parameter domain with invariant tori can be obtained that leaves out only small ‘bubbles’ around the resonances, as sketched and explained in Fig. 9 for the n-tori. For earlier results in the same spirit in a case study of the quasi-periodic saddlenode bifurcation see [49,50,51], also compare with [11].

Tn

Hamiltonian Perturbation Theory (and Transition to Chaos), Figure 9 Fattening by normal hyperbolicity of a nowhere dense parameter set with invariant n-tori in the perturbed system. The curve H is the Whitney smooth (even Gevrey regular [136]) image of the ˇ-axis in Fig. 8. H interpolates the Cantor set Hc that contains the non-hyperbolic Diophantine quasi-periodic invari(2) , see (20). To the points 1;2 2 ant n-tori, corresponding to ; Hc discs A1;2 are attached where we find attracting normally hyperbolic n-tori and similarly in the discs R1;2 repelling ones. The contact between the disc boundaries and H is infinitely flat [17,29]

A Scenario for the Onset of Turbulence Generally speaking, in many settings quasi-periodicity constitutes the order in between chaos [31]. In the Hopf– Landau–Lifschitz–Ruelle–Takens scenario [76,83,84,126] we may consider a sequence of typical transitions as given by quasi-periodic Hopf bifurcations, starting with the standard Hopf or Hopf–Ne˘ımark–Sacker bifurcation as described before. In the gaps of the Diophantine Cantor sets generically there will be coexistence of periodicity, quasi-periodicity and chaos in infinite regress. As said earlier, period doubling sequences and homoclinic bifurcations may accompany this. As an example consider a family of maps that undergoes a generic quasi-periodic Hopf bifurcation from circle to two-torus. It turns out that here the Cantorized fold of Fig. 6 is relevant, where now the vertical coordinate is

677

678

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a bifurcation parameter. Moreover compare with Fig. 3, where also variation of " is taken into account. The Cantor set contains the quasi-periodic dynamics, while in the gaps we can have chaos, e. g., in the form of Hénon like strange attractors [46,112]. A fattening process as explained above, also can be carried out here. Future Directions One important general issue is the mathematical characterization of chaos and ergodicity in dynamical systems, in conservative, dissipative and in other settings. This is a tough problem as can already be seen when considering two-dimensional diffeomorphisms. In particular we refer to the still unproven ergodicity conjecture of [9] and to the conjectures around Hénon like attractors and the principle ‘Hénon everywhere’, compare with [22,32]. For a discussion see Subsect. “A Scenario for the Onset of Turbulence”. In higher dimension this problem is even harder to handle, e. g., compare with [46,47] and references therein. In the conservative case a related problem concerns a better understanding of Arnold diffusion. Somewhat related to this is the analysis of dynamical systems without an explicit perturbation setting. Here numerical and symbolic tools are expected to become useful to develop computer assisted proofs in extended perturbation settings, diagrams of Lyapunov exponents, symbolic dynamics, etc. Compare with [128]. Also see [46,47] for applications and further reference. This part of the theory is important for understanding concrete models, that often are not given in perturbation format. Regarding nearly-integrable Hamiltonian systems, several problems have to be considered. Continuing the above line of thought, one interest is the development of Hamiltonian bifurcation theory without integrable normal form and, likewise, of KAM theory without action angle coordinates [87]. One big related issue also is to develop KAM theory outside the perturbation format. The previous section addressed persistence of Diophantine tori involved in a bifurcation. Similar to Cremer’s example in Subsect. “Cremer’s Example in Herman’s Version” the dynamics in the gaps between persistent tori displays new phenomena. A first step has been made in [86] where internally resonant parabolic tori involved in a quasi-periodic Hamiltonian pitchfork bifurcation are considered. The resulting large dynamical instabilities may be further amplified for tangent (or flat) parabolic resonances, which fail to satisfy the iso-energetic non-degeneracy condition. The construction of solenoids in [16,94] uses elliptic periodic orbits as starting points, the simplest exam-

ple being the result of a period-doubling sequence. This construction should carry over to elliptic tori, where normal-internal resonances lead to encircling tori of the same dimension, while internal resonances lead to elliptic tori of smaller dimension and excitation of normal modes increases the torus dimension. In this way one might be able to construct solenoid-type invariant sets that are limits of tori with varying dimension. Concerning the global theory of nearly-integrable torus bundles [43], it is of interest to understand the effects of quasi-periodic bifurcations on the geometry and its invariants. Also it is of interest to extend the results of [134] when passing to semi-classical approximations. In that case two small parameters play a role, namely Planck’s constant as well as the distance away from integrability.

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Hamilton–Jacobi Equations and Weak KAM Theory

Hamilton–Jacobi Equations and Weak KAM Theory ANTONIO SICONOLFI Dip. di Matematica, “La Sapienza” Università di Roma, Roma, Italy Article Outline Glossary Definition of the Subject Introduction Subsolutions Solutions First Regularity Results for Subsolutions Critical Equation and Aubry Set An Intrinsic Metric Dynamical Properties of the Aubry Set Long-Time Behavior of Solutions to the Time-Dependent Equation Main Regularity Result Future Directions Bibliography Glossary Hamilton–Jacobi equations This class of first-order partial differential equations has a central relevance in several branches of mathematics, both from a theoretical and an application point of view. It is of primary importance in classical mechanics, Hamiltonian dynamics, Riemannian and Finsler geometry, and optimal control theory, as well. It furthermore appears in the classical limit of the Schrödinger equation. A connection with Hamilton’s equations, in the case where the Hamiltonian has sufficient regularity, is provided by the classical Hamilton–Jacobi method which shows that the graph of the differential of any regular, say C1 , global solution to the equation is an invariant subset for the corresponding Hamiltonian flow. The drawback of this approach is that such regular solutions do not exist in general, even for very regular Hamiltonians. See the next paragraph for more comments on this issue. Viscosity solutions As already pointed out, Hamilton– Jacobi equations do not have in general global classical solutions, i. e. everywhere differentiable functions satisfying the equation pointwise. The method of characteristics just yields local classical solutions. This explains the need of introducing weak solutions. The idea for defining those of viscosity type is to con-

sider C1 functions whose graph, up to an additive constant, touches that of the candidate solution at a point and then stay locally above (resp. below) it. These are the viscosity test functions, and it is required that the Hamiltonian satisfies suitable inequalities when its first-order argument is set equal to the differential of them at the first coordinate of the point of contact. Similarly it is defined the notion of viscosity sub, supersolution. Clearly a viscosity solution satisfies pointwise the equation at any differentiability points. A peculiarity of the definition is that a viscosity solution can admit no test function at some point, while the nonemptiness of both classes of test functions is equivalent to the solution being differentiable at the point. Nevertheless powerful existence, uniqueness and stability results hold in the framework of viscosity solution theory. The notion of viscosity solutions was introduced by Crandall and Lions at the beginning of the 1980s. We refer to Bardi and Capuzzo Dolcetta [2], Barles [3], Koike [24] for a comprehensive treatment of this topic. Semiconcave and semiconvex functions These are the appropriate regularity notions when working with viscosity solution techniques. The definition is given by requiring some inequalities, involving convex combinations of points, to hold. These functions possess viscosity test functions of one of the two types at any point. When the Hamiltonian enjoys coercivity properties ensuring that any viscosity solution is locally Lipschitz-continuous then a semiconcave or semiconcave function is the solution if and only if it is classical solution almost everywhere, i. e. up to a set of zero Lebesgue measure. Metric approach This method applies to stationary Hamilton–Jacobi equations with the Hamiltonian only depending on the state and momentum variable. This consists of defining a length functional, on the set of Lipschitz-continuous curves, related to the corresponding sublevels of the Hamiltonian. The associated length distance, obtained by performing the infimum of the intrinsic length of curves joining two given points, plays a crucial role in the analysis of the equation and, in particular, enters in representation formulae for any viscosity solution. One important consequence is that only the sublevels of the Hamiltonian matter for determining such solutions. Accordingly the convexity condition on the Hamiltonian can be relaxed, just requiring quasiconvexity, i. e. convexity of sublevels. Note that in this case the metric is of Finsler type and the sublevels are the unit cotangent balls of it.

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Critical equations To any Hamiltonian is associated a one-parameter family of Hamilton–Jacobi equations obtained by fixing a constant level of Hamiltonian. When studying such a family, one comes across a threshold value under which no subsolutions may exist. This is called the critical value and the same name is conferred to the corresponding equation. If the ground space is compact then the critical equation is unique among those of the family for which viscosity solutions do exist. When, in particular, the underlying space is a torus or, in other terms, the Hamiltonian is Z N periodic then such functions play the role of correctors in related homogenization problems. Aubry set The analysis of the critical equation shows that the obstruction for getting subsolutions at subcritical levels is concentrated on a special set of the ground space, in the sense that no critical subsolution can be strict around it. This is precisely the Aubry set. This is somehow compensated by the fact that critical subsolutions enjoy extra regularity properties on the Aubry set. Definition of the Subject The article aims to illustrate some applications of weak KAM theory to the analysis of Hamilton–Jacobi equations. The presentation focuses on two specific problems, namely the existence of C1 classical subsolutions for a class of stationary (i. e. independent of the time) Hamilton–Jacobi equations, and the long-time behavior of viscosity solutions of an evolutive version of it. The Hamiltonian is assumed to satisfy mild regularity conditions, under which the corresponding Hamilton equations cannot be written. Consequently PDE techniques will be solely employed in the analysis, since the powerful tools of the Hamiltonian dynamics are not available. Introduction Given a continuous or more regular Hamiltonian H(x; p) defined on the cotangent bundle of a boundaryless manifold M, where x and p are the state and the momentum variable, respectively, and satisfying suitable convexity and coercivity assumptions, is considered the family of Hamilton–Jacobi equations H(x; Du) D a

x 2 M;

(1)

with a a real parameter, as well as the time-dependent version w t C H(x; Dw) D 0

x 2 M ; t 2 (0; C1) ;

(2)

where Du and ut stand for the derivative with respect to state and time variable, respectively. As a matter of fact, it will be set, for the sake of simplicity, to either M D R N (noncompact case) or M D T N (compact case), where T N indicates the flat torus endowed with the Euclidean metric and with the cotangent bundle identified to T N  RN . The main scope of this article is to study the existence of the C1 classical subsolution to (1), and the long-time behavior of viscosity solutions to (2) by essentially employing tools issued from weak KAM theory. Some of the results that will be outlined are valid with the additional assumption of compactness for the underlying manifold, in particular those concerning the asymptotics of solutions to (2). For both issues it is crucial to perform a qualitative analysis of (1) for a known value of the parameter a, qualified as critical; accordingly Eq. (1) is called critical when a is equal to the critical value. This analysis leads to detection of a special closed subset of the ground space, named after Aubry, where any locally Lipschitz-continuous subsolution to (1) enjoys some additional regularity properties, and behaves in a peculiar way. This will have a central role in the presentation. The requirements on H will be strengthened to obtain some theorems, but we remain in a setting where the corresponding Hamilton equations cannot be written. Consequently PDE techniques will be solely employed in the analysis, since the powerful tools of Hamiltonian dynamics are not available. Actually the notion of critical value was independently introduced by Ricardo Mañé at the beginning of the 1980s, in connection with the analysis of integral curves of the Euler–Lagrange flow enjoying some global minimizing properties, and by P.L. Lions, S.R.S. Varadhan and G. Papanicolaou [25] in 1987 in the framework of viscosity solutions theory, for studying the periodic homogenization of Hamilton–Jacobi equations. The Aubry set was determined and analyzed by Serge Aubry, in a pure dynamical way, as the union of the supports of integral curves of Euler–Lagrange flow possessing suitable minimality properties. John Mather defined (1986), in a more general framework, a set, contained in the Aubry set, starting from special probability measures invariant with respect to the flow. See Contreras and Iturriaga, [9], for an account on this theory. The first author to point out the link between Aubry– Mather theory and weak solutions to the critical Hamilton–Jacobi equation was Albert Fathi, see [16,17], with the so-called weak KAM theory (1996); he thoroughly investigated the PDE counterpart of the dynamical phenom-

Hamilton–Jacobi Equations and Weak KAM Theory

ena occurring at the critical level. However, his investigation is still within the framework of the dynamical systems theory, requires the Hamiltonian to be at least C2 , and requires the existence of associated Hamiltonian flow as well. The work of Fathi and Siconolfi (2005) [19,20], completely bypassed such assumptions and provided a geometrical analysis of the critical equation independent of the flow, which made it possible to deal with nonregular Hamiltonians. The new idea being the introduction of a length functional, intrinsically related to H, for any curve, and of the related distance, as well. The notion of the Aubry set was suitably generalized to this broader setting. Other important contributions in bridging the gap between PDE and the dynamical viewpoint have been made by Evans and Gomes [14,15]. The material herein is organized as follows: the Sects. “Subsolutions”, “Solutions” are of introductory nature and illustrate the notions of viscosity (sub)solution with their basic properties. Some fundamental techniques used in this framework are introduced as well. Section “First Regularity Results for Subsolutions” deals with issues concerning regularity of subsolutions to (1). The key notion of the Aubry set is introduced in Sect. “Critical Equation and Aubry Set” in connection with the investigation of the critical equation, and a qualitative analysis of it, specially devoted to looking into metric and dynamical properties, is performed in Sects. “An Intrinsic Metric”, “Dynamical Properties of the Aubry Set”. Sections “Long-Time Behavior of Solutions to the Time-Dependent Equation”, “Main Regularity Result” present the main results relative to the long-time behavior of solutions to (2) and the existence of C1 subsolutions to (1). Finally, Sect. “Future Directions” gives some ideas of possible developments in the topic.

@fp : H(x; p)  ag D fp : H(x; p) D ag

(6)

where @, in the above formula, indicates the boundary. The a-sublevel of the Hamiltonian appearing in (5) will be denoted by Z a (x). It is a consequence of the coerciveness and convexity assumptions on H that the set-valued map x 7! Z a (x) possesses convex compact values and, in force of (3), is upper semicontinuous; it is in addition continuous at any point x where int Z a (x) ¤ ;. Here (semi)continuous must be understood with respect to the Hausdorff metric. Next will be given four different definitions of weak subsolutions to Eq. (1) and their equivalence will be proved. From this it can be seen that the family of functions so detected is intrinsically related to the equation. As a matter of fact, it will be proved, under more stringent assumptions, that this family is the closure of the classical (i. e. C1 ) subsolutions in the locally uniform topology. Some notations and definitions must be preliminarily introduced. Given two continuous functions u and v, it is said that v is a (strict) supertangent to u at some point x0 if such point is a (strict) local maximizer of u  v. The notion of subtangent is obtained by replacing a maximizer with a minimizer. Since (sub, super)tangents are involved in the definition of viscosity solution, they will be called in the sequel viscosity test functions. It is necessary to check: Proposition 1 Let u be a continuous function possessing both C1 supertangents and subtangents at a point x0 , then u is differentiable at x0 . Recall that by Rademacher Theorem a locally Lipschitz function is differentiable almost everywhere (for short a.e.), with respect to the Lebesgue measure. For such a function w the (Clarke) generalized gradient at any point x is defined by @w(x) D cofp D lim D w(x i ) : x i i

Subsolutions First I will detail the basic conditions postulated throughout the paper for H. Additional properties required for obtaining some particular result, will be introduced when needed. The Hamiltonian is assumed to be continuous in both variables ;

(3)

to satisfy the coercivity assumption f(x; p) : H(y; p)  ag is compact for any a

(4)

and the following quasiconvexity conditions for any x 2 M, a 2 R fp : H(x; p)  ag is convex

(5)

differentiability point of u ; lim x i D xg ; i

where co denotes the convex hull. Remark 2 Record for later use that this set of weak derivatives can be retrieved even if the differentiability points are taken not in the whole ground space, but just outside a set of vanishing Lebesgue measure. The generalized gradient is nonempty at any point; if it reduces to a singleton at some x, then the function w is strictly differentiable at x, i. e. it is differentiable and Du is continuous at x. The set-valued function x 7! @w(x) possesses convex compact values and is upper semicontinu-

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ous. The following variational property holds: 0 2 @w(x) at any local minimizer or maximizer of w ; (7) furthermore, if @(w 

is C1 then

)(x) D @w(x)  D (x) :

(8)

First definition of weak subsolution A function u is said to be an a.e. subsolution to (1) if it is locally Lipschitz-continuous and satisfies

˛ :D inffjx  yj ; x 2 B1 ; y 2 @B2 g > 0 ; and choose an l0 such that

H(x; Du(x))  a for x in a subset of M with full measure : Second definition of weak subsolution A function u is said to be a viscosity subsolution of first type to (1) if it is continuous and H(x0 ; D (x0 ))  a ;

jZ a j1;B 2 < l0

(10)

sup u  inf u  l0 ˛ < 0

(11)

B2

B1

Since u is not Lipschitz-continuous on B1 , a pair of points x0 , x1 in B1 can be found satisfying u(x1 )  u(x0 ) > l0 jx1  x0 j ;

or equivalently D (x0 ) 2 Z a (x0 ) ; for any x0 2 M, any C1 supertangent

to u at x0 .

The previous definition can be equivalently rephrased by taking test functions of class Ck , 1 < k  C1, or simply differentiable, instead of C1 . Third definition of weak subsolution A function u is a viscosity subsolution of second type if it satisfies the previous definition with subtangent in place of supertangent. Fourth definition of weak subsolution A function u is a subsolution in the sense of Clarke if @u(x) Z a (x) for all x 2 M : Note that this last definition is unique based on a condition holding at any point of M, and not just at the differentiability points of u or at points where some test function exists. This fact will be exploited in the forthcoming definition of strict subsolution. Proposition 3 The previous four definitions are equivalent. We first show that the viscosity subsolutions of both types are locally Lipschitz-continuous. It is exploited that Za , being upper semicontinuous, is also locally bounded, namely for any bounded subset B of M there is a positive r with Z a (x) B(0; r)

where B(0; r) is the Euclidean ball centered at 0 with radius r. The minimum r for which (9) holds true will be indicated by jZ a j1;B . The argument is given for a viscosity subsolution of first type, say u; the proof for the others is similar. Assume by contradiction that there is an open bounded domain B1 where u is not Lipschitz-continuous, then consider an open bounded domain B2 containing B1 such that

for x 2 B ;

(9)

(12)

which shows that the function x 7! u(x)  u(x0 )  l0 jx  x0 j), has a positive supremum in B2 . On the other hand such a function is negative on @B2 by (11), and so it attains its maximum in B2 at a point x 2 B2 , or, in other terms, x 7! u(x0 ) C l0 jx  x0 j) is supertangent to u at x ¤ x0 . Consequently l0

x1  x0 2 Z a (x1 ) ; jx1  x0 j

in contradiction with (10). Since at every differentiability point u can be taken as a test function of itself, then any viscosity subsolution of both types is also an a.e. subsolution. The a.e. subsolutions, in turn, satisfy the fourth definition above, thanks to the definition of generalized gradient, Remark 2, and the fact that H is convex in p and continuous in both arguments. Finally, exploit (7) and (8) to see that the differential of any viscosity test function to u at some point x0 is contained in @u(x0 ). This shows that any subsolution in the Clarke sense is also a viscosity subsolution of the first and second type. In view of Proposition 3, from now on any element of this class of functions will be called a subsolution of (1) without any further specification, similarly the notion of a subsolution in an open subset of M can be given. Note that for any subsolution u, any bounded open domain B, the quantity jZ a j1;B is a Lipschitz constant in B for every subsolution, consequently the family of all subsolutions to (1) is locally equiLipschitz-continuous.

Hamilton–Jacobi Equations and Weak KAM Theory

A conjugate Hamiltonian Hˇ can be associated to H, it is defined by ˇ H(x; p) D H(x; p) for any x, p :

(13)

Note that Hˇ satisfies, as H does, assumptions (3)–(5). The two corresponding conjugate Hamilton–Jacobi equations have the same family of subsolutions, up to a change of sign, as is apparent looking at the first definition of subsolution. Next we will have a closer look at the family of subsolutions to (1), denoted from now on by S a ; the properties deduced will be exploited in the next sections. Advantage is taken of this to illustrate a couple of basic arguments coming from viscosity solutions theory. Proposition 4 The family S a is stable with respect to the local uniform convergence. The key point in the proof of this result is to use the same C1 function, at different points, for testing the limit as well as the approximating functions. This is indeed the primary trick to obtaining stability properties in the framework of viscosity solutions theory. Let un be a sequence in S a and u n ! u locally uniformly in M. Let be a supertangent to u at some point x0 , it can be assumed, without loss of generality, that is a strict subtangent, by adding a suitable quadratic term. Therefore, there is a compact neighborhood U of x0 where x0 itself is the unique maximizer of u  . Any sequence xn of maximizers of u n  in U converges to a maximizer of u  , and so x n ! x0 , and consequently lies in the interior of U for n large enough. In other terms is supertangent to un at xn , when n is sufficiently large. Consequently H(x n ; D (x n )  a, which implies, exploiting the continuity of the Hamiltonian and passing at the limit, H(x0 ; D (x0 ))  a, as desired. Take into account the equiLipschitz character of subsolutions to (1), and the fact that the subsolution property is not affected by addition of a constant, to obtain by slightly adjusting the previous argument, and using the Ascoli Theorem: Proposition 5 Let u n 2 S a n , with an converging to some a. Then the sequence un converges to u 2 S a , up to addition of constants and extraction of a subsequence. Before ending the section, a notion which will have some relevance in what follows will be introduced. A subsolution u is said to be strict in some open subset ˝ M if @u(x) int Z a (x) for any x 2 ˝ ;

where int stands for interior. Since the multivalued map x 7! @u(x) is upper semicontinuous, this is equivalent to ess sup˝ 0 H(x; Du(x)) < a for any ˝ 0 compactly contained in ˝ ; where the expression compactly contained means that the closure of ˝ 0 is compact and is contained in ˝. Accordingly, the maximal (possibly empty) open subset W u where u is strict is given by the formula Wu :D fx : @u(x) int Z a (x)g :

(14)

Solutions Unfortunately the relevant stability properties pointed out in the previous section for the family of subsolutions, do not hold for the a.e. solutions, namely the locally Lipschitzcontinuous functions satisfying the equation up to a subset of M with vanishing measure. Take, for instance, the sequence un in T 1 , obtained by linear interpolation of  un  un

k 2n

k 2n

 D 0 for k even, 0  k  2n

 D

1 2n

for k odd, 0  k  2n ;

then it comprises a.e. solutions of (1), with H(x; p) D jpj and a D 1. But its uniform limit is the null function, which is an a.e. subsolution of the same equation, according to Proposition 4, but fails to be an a.e. solution. This lack of stability motivates the search for a stronger notion of weak solution. The idea is to look at the properties of S a with respect to the operations of sup and inf. Proposition 6 Let S˜ S a be a family of locally equibounded functions, then the function defined as the pointwise supremum, or infimum, of the elements of S˜ is a subsolution to (1). Set u(x) D inffv(x) : v 2 S˜g. Let , un be a C1 subtangent to u at a point x0 , and a sequence of functions in S˜ with u n (x0 ) ! u(x0 ), respectively. Since the sequence un is made up of locally equibounded and locally equiLipschitz-continuous functions, it locally uniformly converges, up to a subsequence, by Ascoli Theorem, to a function w which belongs, in force of Proposition 4, to S a . In addition w(x0 ) D u(x0 ), and w is supertangent to u at x0 by the very definition of u. Therefore, is also subtangent to v at x0 and so H(x0 ; D (x0 ))  a, which shows the assertion. The same

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Hamilton–Jacobi Equations and Weak KAM Theory

proof, with obvious adaptations, allows us to handle the case of the pointwise supremum. Encouraged by the previous result, consider the subsolutions of (1) enjoying some extremality properties. A definition is preliminary. A family S˜ of locally equibounded subsolutions to (1) is said to be complete at some point x0 if there exists "x 0 such that if two subsolutions u1 , u2 agree outside some neighborhood of x0 with radius less than "x 0 and u1 2 S˜ then u2 2 S˜. The interesting point in the next proposition is that the subsolutions which are extremal with respect to a complete family, possess an additional property involving the viscosity test functions. Proposition 7 Let u be the pointwise supremum (infimum) of a locally equibounded family S˜ S a complete at a point x0 , and let be a C1 subtangent (supertangent) to u at x0 . Then H(x0 ; D (x0 )) D a. Only the case where u is a pointwise supremum will be discussed. The proof is based on a push up method that will be again used in the sequel. If, in fact, the assertion were not true, there should be a C1 strict subtangent at x0 , with (x0 ) D u(x0 ), such that H(x0 ; D (x0 )) < a. The function , being C1 , is a (classical) subsolution of (1) in a neighborhood U of x0 . Push up a bit the test function to define ( maxf C " ; ug in B(x0 ; ") vD (15) u otherwise with the positive constant " chosen so that B(x0 ; ") U and " < "x 0 , where "x 0 is the quantity appearing in the definition of a complete family of subsolutions at x0 . By the Proposition 6, the function v belongs to S a , and is equal to u 2 S˜ outside B(x0 ; "). Therefore, v 2 S˜, which is in contrast with the maximality of u because v(x0 ) > u(x0 ). Proposition 7 suggests the following two definitions of weak solution in the viscosity sense, or viscosity solutions for Eq. (1). The function u is a viscosity solution of the first type if it is a subsolution and for any x0 , any C1 subtangent to u at x0 one has H(x0 ; D (x0 )) D a. The viscosity solutions of the second type are definite by replacing the subtangent with supertangent. Such functions are clearly a.e. solutions. The same proof of Proposition 4, applied to C1 subtangents as well as supertangents, gives: Proposition 8 The family of viscosity solutions (of both types) to (1) is stable with respect the local uniform convergence. Moreover, the argument of Proposition 6, with obvious adaptations, shows:

Proposition 9 The pointwise infimum (supremum) of a family of locally equibounded viscosity solutions of the first (second) type is a viscosity solution of the first (second) type to (1). Morally, it can be said that the solutions of the first type enjoy some maximality properties, and some minimality properties hold for the others. Using the notion of the strict subsolution, introduced in the previous section, the following partial converse of Proposition 7 can be obtained: Proposition 10 Let ˝, u, ' be a bounded open subset of M, a viscosity solution to (1) of the first (second) type, and a strict subsolution in ˝ coincident with u on @˝, respectively. Then u  ' (u  ') in ˝. The proof rests on a regularization procedure of ' by mollification. Assume that u is a viscosity solution of the first type, the other case can be treated similarly. The argument is by contradiction, then admit that the minimizers of u' in ˝ (the closure of ˝) are in an open subset ˝ 0 compactly contained in ˝. Define for x 2 ˝ 0 , ı > 0, Z 'ı (x) D ı (y  x)'(y) dy ; where ı is a standard C 1 mollifier supported in B(0; ı). By using the convex character of the Hamiltonian and Jensen Lemma, it can be found Z H(x; D'ı (x))  ı (y  x)H(x; D'(y)) dy: Therefore, taking into account the stability of the set of minimizers under the uniform convergence, and that ' is a strict subsolution, ı can be chosen so small that 'ı is a C 1 strict subsolution of (1) in ˝ 0 and, in addition, is subtangent to u at some point of ˝ 0 . This is in contrast with the very definition of viscosity solution of the first type. The above argument will be used again, and explained with some more detail, in the next section. The family of viscosity solutions of first and second type coincide for conjugate Hamilton–Jacobi equations, ˇ up to a change of sign. More with Hamiltonian H and H, precisely: Proposition 11 A function u is a viscosity solution of the first (second) type to ˇ H(x; Du) D a

in M

(16)

if and only if u is a viscosity solution of the second (first) type to (1).

Hamilton–Jacobi Equations and Weak KAM Theory

In fact, if u, are a viscosity solution of the first type to (16) and a C1 supertangent to u at a point x0 , respectively, then u is a subsolution to (1), and  is supertangent to u at x0 so that ˇ 0 ; D (x0 )) D H(x0 ; D (x0 )) ; a D H(x which shows that u is indeed a viscosity solution of the second type to (1). The other implications can be derived analogously. The choice between the two types of viscosity solutions is just a matter of taste, since they give rise to two completely equivalent theories. In this article those of the first type are selected, and they are referred to from now on as (viscosity) solutions, without any further specification. Next a notion of regularity is introduced, called semiconcavity (semiconvexity), which fits the viscosity solutions framework, and that will be used in a crucial way in Sect. “Main Regularity Result”. The starting remark is that even if the notion of viscosity solution of the first (second) type is more stringent than that of the a.e. solution, as proved above, the two notions are nevertheless equivalent for concave (convex) functions. In fact a function of this type, say u, which is locally Lipschitz-continuous, satisfies the inequality u(y)  () u(x0 ) C p (y  x) ; for any x0 , y, p 2 @u(x0 ). It is therefore apparent that it admits (global) linear supertangents (subtangents) at any point x0 . If there were also a C1 subtangent (supertangent), say , at x0 , then u should be differentiable at x0 by Proposition 1, and Du(x0 ) D D (x0 ), so that if u were an a.e. solution then H(x0 ; D (x0 )) D a, as announced. In the above argument the concave (convex) character of u was not exploited to its full extent, but it was just used to show the existence of C1 super(sub)tangents at any point x and that the differentials of such test functions make up @u. Clearly such a property is still valid for a larger class of functions. This is the case for the family of so-called strongly semiconcave (semiconvex) functions, which are concave (convex) up to the subtraction (addition) of a quadratic term. This point will now be outlined for strongly semiconcave functions, a parallel analysis could be performed in the strongly semiconvex case. A function u is said to be strongly semiconcave if u(x)  kjx  x0 j2 is concave for some positive constant k, some x0 2 M. Starting from the inequality u(x1 C (1  )x2 )  kjx1 C (1  )x2  x0 j2  (u(x1 )  kjx1  x0 j2 ) C (1  )(u(x2 )  kjx2  x0 j2 )

which holds for any x1 , x2 ,  2 [0; 1], it is derived through straightforward calculations u(x1 C (1  )x2 )  u(x1 )  (1  )u(x2 )  k(1  )jx1  x2 j2 ;

(17)

which is actually a property equivalent to strong semiconcavity. This shows that for such functions the subtraction of kjx  x0 j2 , for any x0 2 M, yields the concavity property. Therefore, given any x0 , and taking into account (8), one has u(x)  kjx  x0 j2  u(x0 ) C p(x  x0 ) ; for any x, any p 2 @u(x0 ) which proves that the generalized gradient of u at x0 is made up by the differentials of the C1 supertangents to u at x0 . The outcome of the previous discussion is summarized in the following statement. Proposition 12 Let u be a strongly semiconcave (semiconvex) function. For any x, p 2 @u(x) if and only if it is the differential of a C1 supertangent (subtangent) to u at x. Consequently, the notions of a.e. solution to (1) and viscosity solution of the first (second) type coincide for this class of functions. In Sect. “Main Regularity Result” a weaker notion of semiconcavity will be introduced, obtained by requiring a milder version of (17), which will be crucial to proving the existence of C1 subsolutions to (1). Even if the previous analysis, and in particular the part about the equivalent notions of subsolutions, is only valid for (1), one can define a notion of viscosity solution for a wider class of Hamilton–Jacobi equations than (1), and even for some second-order equations. To be more precise, given a Hamilton–Jacobi equation G(x; u; ; Du) D 0, with G nondecreasing with respect to the second argument, a continuous function u is called the viscosity solution of it, if for any x0 , any C1 supertangent (subtangent) to u at x0 the inequality G(x0 ; u(x0 ); D (x0 ))  () 0 holds true. Loosely speaking the existence of comparison principles in this context is related to the strict monotonicity properties of the Hamiltonian with respect to u or the presence in the equation of the time derivative of the unknown. For instance such principles hold for (2), see Sect. “Long-Time Behavior of Solutions to the Time-Dependent Equation”. Obtaining uniqueness properties for viscosity solutions to (1) is a more delicate matter. Such properties are

689

690

Hamilton–Jacobi Equations and Weak KAM Theory

actually related to the existence of strict subsolutions, since this, in turn, allows one to slightly perturb any solution obtaining a strict subsolution. To exemplify this issue, Proposition 10 is exploited to show: Proposition 13 Let ˝, g be an open-bounded subset of M and a continuous function defined on @˝, respectively. Assume that there is a strict subsolution ' to (1) in ˝. Then there is at most one viscosity solution to (1) in ˝ taking the datum g on the boundary. Assume by contradiction the existence of two viscosity solutions u and v with v > u C " at some point of ˝, where " is a positive constant. The function v :D ' C (1  )v is a strict subsolution to (1) for any  2]0; 1], by the convexity assumption on H. Further,  can be taken so small that the points of ˝ for which v > u C "2 make up a nonempty set, say ˝ 0 , are compactly contained in ˝. This goes against Proposition 10, because u and v C "2 agree on @˝ 0 and the strict subsolution v C "2 exceeds u in ˝ 0 . First Regularity Results for Subsolutions A natural question is when does a classical subsolution to (1) exist? The surprising answer is that it happens whenever there is a (locally Lipschitz) subsolution, provided the assumptions on H introduced in the previous section are strengthened a little. Furthermore, any subsolution can be approximated by regular subsolutions in the topology of locally uniform convergence. This theorem is postponed to Sect. “Main Regularity Result”. Some preliminary results of regularity for subsolution, holding under the assumptions (3)–(6), are presented below. Firstly the discussion concerns the existence of subsolutions to (1) that are regular, say C1 , at least on some distinguished subset of M. More precisely an attempt is made to determine when such functions can be obtained by mollification of subsolutions. The essential output is that this smoothing technique works if the subsolution one starts from is strict, so that, loosely speaking, some room is left to perturb it locally, still obtaining a subsolution. A similar argument has already been used in the proof of Proposition 10. In this analysis it is relevant the critical level of the Hamiltonian which is defined as the one for which the corresponding Hamilton–Jacobi equation possesses a subsolution, but none of them are strict on the whole ground space. It is also important subset of M, named after Aubry and indicated by A, made up of points around which no critical subsolution (i. e. subsolution to (1) with a D c) is strict.

According to what was previously outlined, to smooth up a critical subsolution around the Aubry set, seems particulary hard, if not hopeless. This difficulty will be overcome by performing a detailed qualitative study of the behavior of critical subsolutions on A. The simple setting to be first examined is when there is a strict subsolution, say u, to (1) satisfying H(x; Du(x))  a  "

for a.e. x 2 M, and some " > 0 ; (18)

and, in addition, H is uniformly continuous on M  B

T  M, whenever B is a bounded subset of R N . In this case the mollification procedure plainly works to supply a regular subsolution. The argument of Proposition 10 can be adapted to show this. Define, for any x, any ı > 0 Z uı (x) D ı (y  x)u(y) dy ; where ı is a standard C 1 mollifier supported in B(0; ı), and by using the convex character of the Hamiltonian and Jensen Lemma, get Z H(x; Duı (x))  ı (y  x)H(x; Du(y)) dy ; so that if o() is a continuity modulus of H in M  B(0; r), with r denoting a Lipschitz constant for u, a ı can be selected in such a way that o(ı)  "2 , and consequently uı is the desired smooth subsolution, and is, in addition, still strict on M. Even if condition (18) does not hold on the whole underlying space, the previous argument can be applied locally, to provide a smoothing of any subsolution u, at least in the open subset W u where it is strict (see (14) for the definition of this set), by introducing countable open coverings and associated C 1 partition of the unity. The uniform continuity assumption on H as well as the global Lipschitz-continuity of u can be bypassed as well. It can be proved: Proposition 14 Given u 2 S a with W u nonempty, there exists v 2 S a , which is strict and of class C 1 on W u . Note that the function v appearing in the statement is required to be a subsolution on the whole M. In the proof of Proposition 14 an extension principle for subsolutions will be used that will be explained later. Extension principle Let v and C be a subsolution to (1) and a closed subset of M, respectively. Any continuous extension of vjC which is a subsolution on M n C is also a subsolution in the whole M.

Hamilton–Jacobi Equations and Weak KAM Theory

The argument for showing Proposition 14 will also provide the proof, with some adjustments, of the main regularity result, i. e. Theorem 35 in Sect. “Main Regularity Result”. By the very definition of W u an open neighborhood U x0 , compactly contained in Wu , can be found, for all x 2 Wu , in such a way that H(y; Du(y)) < a"x

for a.e. y 2 U x0 and some "x > 0: (19)

Through regularization of u by means of a C 1 mollifier ı supported in B(0; ı), for ı > 0 suitably small, a smooth function can be then constructed still satisfying (19) in a neighborhood of x slightly smaller than U x0 , say U x . The next step is to extract from fU x g, x 2 Wu , a countable locally finite cover of W u , say fU x i g, i 2 N. In the sequel the notations U i , "i are adopted in place of U x i , "x i , respectively. The regularized function is denoted by ui . Note that such functions are not, in general, subsolutions to (1) on M, since their behavior outside U i cannot be controlled. To overcome this difficulty a C 1 partition of the unity ˇ i subordinated to U i is introduced. The crucial point here is that the mollification parameters, denoted by ı i , can be adjusted in such a way that the uniform distance ju  u i j1;U i is as small as desired. This quantity, more precisely, is required to be small with respect to jDˇ1i j1 , 21i and the "j corresponding to indices j such that U j \ U i ¤ ;. In place of 21i one could take the terms of any positive convergent series with sum 1. Define v via the formula (P ˇ i u i in Wu vD (20) u otherwise : Note that a finite number of terms are involved in the sum defining v in W u since the cover fU i g is locally finite. It can P be surprising at first sight that the quantity ˇ i u i , with ui subsolution in U i and ˇ i supported in U i , represents a subsolution to (1), since, by differentiating, one gets X  X X D ˇi ui D ˇ i Du i C Dˇ i u i ; and the latter term does not seem easy to handle. The trick is to express it through the formula X X Dˇ i (u i  u) ; (21) Dˇ i u i D P which holds true because ˇ i  1, by the very definition P of partition of unity, and so Dˇ i  0. From (21) deduce ˇX ˇ X ˇ ˇ Dˇ i u i ˇ  jDˇ i j1 ju  u i j1;U i ; ˇ

and consequently, recalling that ju  u i j1;U i is small with respect to jDˇ1i j1 D

X

 X ˇ i Du i : ˇi ui

Since the Hamiltonian is convex in p, ˇ i is supported in U i and ui is a strict subsolution to (1) in U i , we finally discover that v, defined by (20), is a strict subsolution in W u . Taking into account the extension principle for subsolutions, it is left to show, for proving Proposition 14, that v is continuous. For this, first observe that for any n 2 N the set [ in U i is compact and disjoint from @Wu , and consequently minfi : x 2 U i g ! C1 when x 2 Wu approaches @Wu : This, in turn, implies, since ju u i j1;U i is small compared to 21i , ˇX ˇ X ˇ ˇ ˇ i (x)u i (x)  u(x)ˇ  ˇ i (x)ju  u i j1;U i ˇ fi : x2U i g



X fi : x2U i g

ˇ i (x)

1 ! 0; 2i

whenever x 2 Wu approaches @Wu . This shows the assertion. The next step is to look for subsolutions that are strict in a subset of M as large as possible. In particular a strict subsolution to (1) on the whole M does apparently exist at any level a of the Hamiltonian with a > inffb : H(x; Du) D b has a subsolutiong :

(22)

The infimum on the right-hand side of the previous formula is the critical value of H; it will be denoted from now on by c. Accordingly the values a > c (resp. a < c) will be qualified as supercritical (resp. subcritical). The inf in (22) is actually a minimum in view of Proposition 5. By the coercivity properties of H, the quantity min p H(x; p) is finite for any x, and clearly c  sup min H(x; p) ; x

p

which shows that c > 1, but it can be equal to C1 if M is noncompact. In this case no subsolutions to (1) should exist for any a. In what follows it is assumed that c is finite. Note that the critical value for the conjugate Hamiltonian Hˇ does not change, since, as already noticed in

691

692

Hamilton–Jacobi Equations and Weak KAM Theory

Sect. “Subsolutions”, the family of subsolutions of the two corresponding Hamilton–Jacobi equations are equal up to a change of sign. From Proposition 14 can be derived: Theorem 15 There exists a smooth strict subsolution to (1) for any supercritical value a.

If x belongs to the full measure set where v and all the vi are differentiable, one finds, by exploiting the convex character of H 1 0 X X  i Du i (x)A   i H(x; Du i (x)) H @x; in

in

0

C @1 

Critical Equation and Aubry Set

(23)

A significant progress in the analysis is achieved by showing that there is a critical subsolution v with W v , see (14) for the definition, enjoying a maximality property. More precisely the following statement holds: Proposition 16 There exists v 2 Sc with Wv D W0 :D

[

fWu : u is a critical subsolutiong :

This result, combined with Proposition 14, gives the Proposition 17 There exists a subsolution to (23) that is strict and of class C 1 on W 0 . To construct v appearing in the statement of Proposition 16 a covering technique to W 0 , as in Proposition 14, is applied and then the convex character of H is exploited. Since no regularity issues are involved, there is no need to introduce smoothing procedures and partitions of unity, so the argument is altogether quite simple. Any point y 2 W0 possesses a neighborhood U y where some critical subsolution vy satisfies H(x; Dv y (x)  c" y

1  i A H(x; 0) ;

in

Here the attention is focused on the critical equation H(x; Du) D c :

X

for a.e. x 2 U y , some positive " y :

A locally finite countable subcover fU y i g, i 2 N, can be extracted, the notations U i , vi , "i are used in place of U y i , v y i , " y i . The function v is defined as an infinite convex P 1 v. combination of ui , more precisely v D 2i i To show that v has the properties asserted in the statement, note that the functions vi are locally equiLipschitz-continuous, being critical subsolutions, and can be taken, in addition, locally equibounded, up to addition of P P a constant. The series  i u i ,  i Du i are therefore locally uniformly convergent by the Weierstrass M-test. This shows that the function v is well defined and is Lipschitzcontinuous, in addition X Dv(x) D  i Du i (x) for a.e. x :

for any fixed n. This implies, passing to the limit for n ! C1 ! 1 X  i Du i (x) H(x; Dv(x)) D H x; iD1



1 X

 i H(x; Du i (x)) :

iD1

The function v is thus a critical subsolution, and, in addition, one has X H(x; Dv(x))  H(x; Dv i (x)) C  j H(x; Dv j (x)) i¤ j (24)  c  j"j ; for any j and a.e. x 2 U j . This yields Proposition 16 since fU j g, j 2 N, is a locally finite open cover of W 0 , and so it comes from (24) that the essential sup of v, on any open set compactly contained in W 0 , is strictly less that c. The Aubry set A is defined as M n W0 . According to Propositions 14, 16 it is made up by the bad points around which no function of Sc can be regularized through mollification still remaining a critical subsolution. The points of A are actually characterized by the fact that no critical subsolution is strict around them. Note that a local as well as a global aspect is involved in such a property, for the subsolutions under investigation must be subsolutions on the whole space. Note further that A is also the Aubry set ˇ for the conjugate critical equation with Hamiltonian H. A qualitative analysis of A is the main subject of what follows. Notice that the Aubry set must be nonempty if M is compact, since, otherwise, one could repeat the argument used for the proof of Proposition 16 to get a finite open cover fU i g of M and a finite family ui of critical subsolutions satisfying H(x; Du i (x)  c  " i

for a.e. x 2 U i and some " i > 0; P and to have for a finite convex combination u D i  i u i H(x; Du(x))  c  minf i " i g ; i

Hamilton–Jacobi Equations and Weak KAM Theory

in contrast with the very definition of critical value. If, on the contrary, M is noncompact Hamiltonian such that the corresponding Aubry set is empty, then it can be easily exhibited. One example is given by H(x; p) D jpj  f (x), in the case where the potential f has no minimizers. It is easily seen that the critical level is given by  inf M f , since, for a less than this value, the sublevels Z a () are empty at some x 2 M and consequently the corresponding Hamilton–Jacobi equation does not have any subsolution; on the other side M

which shows that any constant function is a strict critical subsolution on M. This, in turn, implies the emptiness of A. In view of Proposition 14, one has: Proposition 18 Assume that M is noncompact and the Aubry set is empty, then there exists a smooth strict critical subsolution. The points y of A are divided into two categories according to whether the sublevel Z c (y) has an empty or nonempty interior. It is clear that a point y with int Z c (y) D ; must belong to A because for such a point for all p 2 Z c (y) ;

y

S˜a D fu 2 S a : u(y) D 0g ;

(26)

and to define y

w a (x) D sup u(x) ;

(27)

y

S˜a

y

H(x; 0) D  f (x) <  inf f ;

H(y; p) D c

the push-up argument introduced in Sect. “Subsolutions” for proving Proposition 7. By using the previous proposition, the issue of the existence of (viscosity) solutions to (23) or, more generally, to (1) can be tackled. The starting idea is to fix a point y in M, to consider the family

(25)

and, since any critical subsolution u must satisfy @u(y)

Z c (y), it cannot be strict around y. These points are called equilibria, and E indicates the set of all equilibria. The reason for this terminology is that if the regularity assumptions on H are enough to write the Hamilton’s equations on T  M, then (y; p0 ) is an equilibrium of the related flow with H(y; p0 ) D c if and only if y 2 E and Z c (y) D fp0 g. This point of view will not be developed further herein. From now on the subscript c will be omitted to ease notations. Next the behavior of viscosity test functions of any critical subsolution at points belonging to the Aubry set is investigated. The following assertion holds true: Proposition 19 Let u, y, be a critical subsolution, a point of the Aubry set and a viscosity test function to u at y, respectively. Then H(y; D (y)) D c. Note that the content of the proposition is an immediate consequence of (25) if, in addition, y is an equilibrium. In the general case it is not restrictive to prove the statement when is a strict subtangent. Actually, if the inequality H(y; D (y)) < c takes place, a contradiction is reached by constructing a subsolution v strict around y by means of

Since S˜a is complete (this terminology was introduced in y Sect. “Solutions”) at any x ¤ y, the function w a is a subsolution to (1) on M, and a viscosity solution to M n fyg, by Propositions 6, 7. If a D c, and the point y belongs to A then, in view y of Proposition 19, w c is a critical solution on the whole M. On the contrary, the fact that y 62 A i. e. y 2 W0 , prevents this function from being a global solution. In fact in this case, according to Propositions 14, 16, there is a critical subsolution ', which is smooth and strict around y, and it can be also assumed, without any loss of generality, y to vanish at y. Therefore, ' is subtangent to w c at y for the y maximality property of w c and H(y; D'(y)) < c. A characterization of the Aubry set then follows: First characterization of A A point y belongs to the y Aubry set if and only if the function w c , defined in (27) with a D c, is a critical solution on the whole M. If A ¤ ;, which is true when M is compact, then the existence of a critical solution can be derived. Actually in the compact case the critical level is the unique one for which a viscosity solution to (1) does exist. If, in fact a > c, then, by Theorem 15, the equation possesses a smooth strict critical subsolution, say ', which is subtangent to any other function f defined on M at the minimizers of f  u, which do exist since M is assumed to be compact. This rules out the possibility of having a solution of (1) since H(x; D'(x)) < a for any x. Next is discussed the issue that in the noncompact case a solution does exist at the critical as well as at any supercritical level. Let a be supercritical. The idea is to exploit the noncompact setting, and to throw away the points where the property of being a solution fails, by letting them go to infinity. y Let w n :D w a n be a sequence of subsolutions given by (27), with jy n j ! C1. The wn are equiLipschitzcontinuous, being subsolutions to (1), and locally equibounded, up to the addition of a constant. One then gets,

693

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Hamilton–Jacobi Equations and Weak KAM Theory

using Ascoli Theorem and arguing along subsequences, a limit function w. Since the wn are solutions around any fixed point, for n suitably large, then, in view of the stability properties of viscosity solutions, see Proposition 8, w is a solution to (1) around any point of M, which means that w is a viscosity solution on the whole M, as announced. The above outlined properties are summarized in the next statement. Proposition 20

at any point possessing a sublevel with a nonempty interior. The positive homogeneity property implies that the line integral in (28) is invariant under change of parameter preserving the orientation. The intrinsic length ` a is moreover lower semicontinuous for the uniform convergence of a equiLipschitz-continuous sequence of curves, by standard variational argument, see [7]. Let S a denote the length distance associated to ` a , namely S a (y; x) D inff` a () :  connects y to xg :

(i) If M is compact then a solution to (1) does exist if and only if a D c. (ii) If M is noncompact then (1) can be solved in the viscosity sense if and only if a  c.

An Intrinsic Metric Formula (27) gives rise to a nonsymmetric semidistance S a (; ) by simply putting y

S a (y; x) D w a (x) : This metric viewpoint will allow us to attain a deeper insight into the structure of the subsolutions to (1) as well as of the geometric properties of the Aubry set. It is clear that Sa satisfies the triangle inequality and S a (y; y) D 0 for any y. It fails, in general to be symmetric and non-negative. It will be nevertheless called, from now on, distance to ease terminology. An important point to be discussed is that Sa is a length distance, in the sense that a suitable length functional ` a can be introduced in the class of Lipschitz-continuous curves of M in such a way that, for any pairs of points x and y, S a (y; x) is the infimum of the lengths of curves joining them. Such a length, will be qualified from now on as intrinsic to distinguish it from the natural length on the ground space, denoted by `. It only depends on the corresponding sublevels of the Hamiltonian. More precisely one defines for a (Lipschitz-continuous) curve  parametrized in an interval I Z ˙ dt ; (28) ` a () D  a (; ) I

where  a stands for the support function of the a-sublevel of H. More precisely, the function  a is defined, for any (x; q) 2 TM as  a (x; q) D maxfp q : p 2 Z a (x)g ; it is accordingly convex and positively homogeneous in p, upper semicontinuous in x, and, in addition, continuous

The following result holds true: Proposition 21 S a and Sa coincide. Note that by the coercivity of the Hamiltonian ` a ()  r`() for some positive r. Taking into account that the Euclidean segment is an admissible junction between any pair of points, deduce the inequality jS a (y; x)j  rjy  xj for any y, x ; which, combined with the triangle inequality, implies that the function x 7! S a (y; x) is locally Lipschitz-continuous, for any fixed y. Let y0 , x0 be a pair of points in M. Since u :D S a (y0 ; ) is locally Lipschitz-continuous, one has Z d S a (y0 ; x0 ) D u(x0 )  u(y0 ) D u((t) dt dt I for any curve  connecting y0 to x0 , defined in some interval I. It is well known from [8] that d ˙ u((t)) D p(t) (t) dt for a.e. t 2 I, some p(t) 2 @u((t)) ; and, since @u(x) Z a (x) for any x, derive S a (y0 ; x0 )  ` a () for all every curve  joining y0 to x0 ; which, in turn, yields the inequality S a (y0 ; x0 )  S a (x0 ; y0 ). The converse inequality is obtained by showing that the function w :D S a (y0 ; ) is a subsolution to (1), see [20,29]. From now on the subscript from Za , Sa and  a will be omitted in the case where a D c. It is clear that, in general, the intrinsic length of curves can have any sign. However, if the curve is a cycle such a length must be non-negative, according to Proposition 21, otherwise going several times through the same loop the identity S a  1 would be obtained. This remark will have some relevance in what follows. Proposition 21 allows us to determine the intrinsic metric related to the a-sublevel of the conjugate Hamilˇ denoted by Zˇ a (). Since Zˇ a (x) D Z a (x), for tonian H,

Hamilton–Jacobi Equations and Weak KAM Theory

ˇ the correspondany x, because of the very definition of H, ing support function ˇ satisfies ˇ a (x; q) D  a (x; q)

for any x, q :

Therefore, the intrinsic lengths ` a and `ˇa coincide up to a change of orientation. In fact, given , defined in [0; 1], and denoted by (s) D (1  s) the curve with opposite orientation, one has Z 1 ˙ ds ; `ˇa () D  a (; ) 0

and using r D 1  t as a new integration variable one obtains Z 1 Z 1 ˙ ds D  a (; )  a ( ; ˙ ) dr D ` a ( ): 0

(i) if a curve connects two points belonging to C then the corresponding variation of u is estimated from above by the its intrinsic length because of Proposition 23, and since u coincides with a subsolution to (1) on C, (ii) the same estimate holds true for any pair of points if the curve joining them lies outside C in force of the property that u is a subsolution in M n C. Let " be a positive constant, x, y a pair of points and  a curve joining them, whose intrinsic length approximates S a (y; x) up to ". The interval of definition of  can be partitioned in such a way that the portion of the curve corresponding to each subinterval satisfies the setting of one of the previous items (i) and (ii). By exploiting the additivity of the intrinsic length one finds

0

u(x)  u(y)  ` a ()  S a (y; x) C " ;

This yields Sˇa (x; y) D S a (y; x) for any x, y ; where Sˇa stands for the conjugate distance. The function S a (; y) is thus the pointwise supremum of the family fv : v is a subsolution to (16) and v(y) D 0g ; and accordingly S a (; y) the pointwise infimum of fu 2 S a : u(y) D 0g. Summing up:

Proposition 22 Given u 2 S a , y 2 M, the functions S a (y; ), S a (; y) are supertangent and subtangent, respectively, to u at y. The Extension Principle for subsolutions can now be proved. Preliminarily the fifth characterization of the family of subsolutions is given. Proposition 23 A continuous function u is a subsolution to (1) if and only if u(x)  u(y)  S a (y; x) for any x, y :

(29)

It is an immediate consequence that any subsolution satisfies the inequality in the statement. Conversely, let be a C1 subtangent to u at some point y. By the inequality (29) the subsolution x 7! S a (y; x) is supertangent to u at y, for any y. Therefore, is subtangent to x 7! S a (y; x) at the same point, and so one has H(y; D (y))  a, which shows the assertion taking into account the third definition of subsolution given in Sect. “Subsolutions”. To prove the Extension Lemma, one has to show that a function w coincident with some subsolution to (1) on a closed set C, and being a subsolution on M n C, is a subsolution on the whole M. The intrinsic length will play a main role here. Two facts are exploited:

and the conclusion is reached taking into account the characterization of a subsolution given by Proposition 23, and the fact that " is arbitrary. To carry on the analysis, it is in order to discuss an apparent contradiction regarding the Aubry set. Let y0 2 A n E and p0 2 int Z(y), then p0 is also in the interior of the c-sublevels at points suitably close to y, say belonging to a neighborhood U of y, since Z is continuous at y. This implies p (x  y0 )  `c () ;

(30)

for any x 2 U, any curve  joining y to X and lying in U. However, this inequality does not imply by any means that the function x 7! p(x  y) is subtangent to x 7! S(y; x) at y. This, in fact, should go against Proposition 19, since H(y; p0 ) < c. The unique way to overcome the contradiction is to admit that, even for points very close to y, the critical distance from y is realized by the intrinsic lengths of curves going out of U. In this way one could not deduce from the inequality (30) the previously indicated subtangency property. This means that S is not localizable with respect to the natural distance, and the behavior of the Hamiltonian in points far from y in the Euclidean sense can affect it. There thus exist a sequence of points xn converging to y and a sequence of curves joining y to xn with intrinsic length approximating S(y; x n ) up to n1 and going out U. By juxtaposition of  n and the Euclidean segment from xn to y, a sequence of cycles n can be constructed based on y (i. e. passing through y) satisfying `c ( n ) ! 0 ;

inf `( n ) > 0 : n

(31)

695

696

Hamilton–Jacobi Equations and Weak KAM Theory

This is a threshold situation, since the critical length of any cycle must be non-negative. Next it is shown that (31) is indeed a metric characterization of the Aubry set. Metric characterization of the Aubry set A point y belongs to A if and only if there is a sequence n of cycles based on y and satisfying (31). What remains is to prove that the condition (31) holds at any equilibrium point and, conversely, that if it is true at some y, then such a point belongs to A. If y 2 E then this can be directly proved exploiting that int Z(y) is empty and, consequently the sublevel, being convex, is contained in the orthogonal of some element, see [20]. Conversely, let y 2 M n A, according to Proposition 17, there is a critical subsolution u which is of class C 1 and strict in a neighborhood U of y. One can therefore find a positive constant ı such that Du(x) q  (x; q)ı

for any x 2 U, any unit vector q: (32)

Let now  be a cycle based on y and parametrized by the Euclidean arc-length in [0; `()], then (t) 2 U, for t belonging to an interval that can be assumed without loss of generality of the form [0; t1 ] for some t1  dist(y; @U) (where dist indicates the Euclidean distance of a point from a set). This implies, taking into account (32) and that  is a cycle  ˇ   ˇ  `c () D `c  ˇ[0;t1 ] C `c  ˇ[t1 ;T]  (u((t1 )  u((0)) C ıt1 C (u((T)  u((t1 ))  ı dist(y; @U))) : This shows that the condition (31) cannot hold for sequences of cycles passing through y. By slightly adapting the previous argument, a further property of the intrinsic critical length, to be used later in Sect. “Long-Time Behavior of Solutions to the Time-Dependent Equation”, can be deduced. Proposition 24 Let M be compact. Given ı > 0, there are two positive constants ˛, ˇ such that any curve  lying at a distance greater than ı from A satisfies `c ()  ˛ C ˇ`() : An important property of the Aubry set is that it is a uniqueness set for the critical equation, at least when the ground space is compact. This means that two critical solutions coinciding on A must coincide on M. More precisely it holds:

Proposition 25 Let M be compact. Given an admissible trace g on A, i. e. satisfying the compatibility condition g(y2 )  g(y1 )  S(y1 ; y2 ) ; the unique viscosity solution taking the value g on A is given by minfg(y) C S(y; ) : y 2 Ag : The representation formula yields indeed a critical solution thanks to the first characterization of the Aubry set and Proposition 9. The uniqueness can be obtained taking into account that there is a critical subsolution which is strict and C 1 in the complement of A (see Proposition 17), and arguing as in Proposition 13. Some information on the Aubry set in the one-dimensional case can be deduced from both the characterizations of A Proposition 26 Assume M to have dimension 1, then (i) if M is compact then either A D E or A D M, (ii) if M is noncompact then A D E , and, in particular, A D ; if E D ;. In the one-dimensional case the c-sublevels are compact intervals. Set Z(x) D [˛(x); ˇ(x)] with ˛, ˇ continuous, and consider the Hamiltonian H(x; p) D H(x; p  ˛(x)) : It is apparent that u is a critical (sub)solution for H if and only if u C F, where F is any antiderivative of ˛, is a (sub)solution to H D c, and in addition, u is strict as a subsolution in some ˝ M if and only if u C F is strict in the same subset. This proves that c is also the critical value for H. y y Further, u 2 S˜c for some y, with S˜c defined as in (26), if and only if u C F0 , where F 0 is the antiderivative of ˛ vanishing at y, is in the corresponding family of subsolutions to H D c. Bearing in mind the first characterization of A, it comes that the Aubry sets of the two Hamiltonians H and H coincide. The advantage of using H is that the corresponding critical sublevels Z(x) equal [0; ˇ(x)  ˛(x)], for any x, and accordingly the support function, denoted by  , satisfies

q (ˇ(x)  ˛(x)) if q > 0  (x; q) D 0 if q  0 for any x, q. This implies that the intrinsic critical length related to H, say `c , is non-negative for all curves. Now,

Hamilton–Jacobi Equations and Weak KAM Theory

assume M to be noncompact and take y 62 E , the claim is that y 62 A. In fact, let " > 0 be such that m :D inffˇ(x)  ˛(x) : x 2 I" :D]y  "; y C "[ g > 0 ; given a cycle  based on y, there are two possibilities: either  intersects @I" or is entirely contained in I" . In the first case `c ()  m " ;

(33)

in the second case  can be assumed, without loosing generality, to be parametrized by the Euclidean arc-length; since it is a cycle one has Z

`()

˙ ds D 0 ;

0

˙ D 1 for t belonging to a set of one-dimensional so that (t) measure `() 2 . One therefore has `c ()  m

`() : 2

(34)

Inequalities (33), (34) show that `c () cannot be infinitesimal unless `() is infinitesimal. Hence item (ii) of Proposition 26 is proved. The rest of the assertion is obtained by suitably adapting the previous argument.

curve is contained in A. In fact if such a curve is supported on a point, say x0 , then L(x0 ; 0) D c and, consequently, the critical value is the minimum of p 7! H(x; p). This, in turn, implies, in view of (35), that the sublevel Z(x0 ) has an empty interior so that x0 2 E A. If, on the contrary, a critical curve is nonconstant and x1 , x2 are a pair of different points lying in its support, then S(x1 ; x2 ) C S(x2 ; x1 ) D 0 and there exist two sequences of curves,  n and n whose intrinsic length approximates the S(x1 ; x2 ) and S(x2 ; x1 ), respectively. Hence the trajectories obtained through juxtaposition of  n and n are cycles with critical length infinitesimal and natural length estimated from below by a positive constant, since they contain x1 and x2 , with x1 ¤ x2 . This at last implies that such points, and so the whole support of the critical curve, are contained in the Aubry set. Next the feature of the parametrization of a critical curve is investigated, since it apparently matters for a curve to be critical. For this purpose let x0 , q0 ¤ 0, and p0 2 Z(x0 ) with  (x0 ; q0 ) D p0 q0 , it comes from the definition of the Lagrangian  (x0 ; q0 )  c D p0 q0  H(x0 ; p0 )  L(x0 ; q0 ) ; by combining this formula with (37), and recalling the relationship between intrinsic length and distance, one gets L( ; ˙ ) C c D  ( ; ˙ )

Dynamical Properties of the Aubry Set In this section the convexity and the coercivity assumptions on H are strengthened and it is required, in addition to (3), H is convex in p lim

jpj!C1

(35)

H(x; p) D C1 uniformly in x: jpj

(36)

The Lagrangian L can be therefore defined through the formula L(x; q) D maxfp q  H(x; p) : p 2 R N g : A curve , defined in some interval I, is said to be critical provided that Z

t2

L( ; ˙ )C c ds D S( (t2 ); (t1 )) ;

S( (t1 ); (t2 )) D t1

(37) for any t1 , t2 in I. It comes from the metric characterization of A, given in the previous section, that any critical

for a.e. t :

(38)

A parametrization is called Lagrangian if it satisfies the above equality. As a matter of fact it is possible to prove that any curve , which stays far from E , can be endowed with such a parametrization, see [10]. A relevant result to be discussed next is that the Aubry set is fully covered by critical curves. This property allows us to obtain precious information on the behavior of critical subsolution on A, and will be exploited in the next sections in the study of long-time behavior of solutions to (2). More precisely the following result can be shown: Theorem 27 Given y0 2 A, there is a critical curve, defined in R, taking the value y0 at t D 0. If y0 2 E then the constant curve (t)  y0 is critical, as pointed out above. It can therefore be assumed y0 62 E . It is first shown that a critical curve taking the value y0 at 0, and defined in a bounded interval can be constructed. For this purpose start from a sequence of cycles  n , based on y0 , satisfying the properties involved in the metric characterization of A, and parametrized by (Euclidean) arc length in [0; Tn ], with Tn D `( n ). By exploiting Ascoli Theorem, and arguing along subsequences, one obtains a uniform limit curve  of the  n , with (0) D y0 ,

697

698

Hamilton–Jacobi Equations and Weak KAM Theory

in an interval [0; T], where T is strictly less than infn Tn . It is moreover possible to show that  is nonconstant. A new sequence of cycles n can be defined through juxtapositionˇ of , the Euclidean segment joining (T) to  n (T) and n ˇ[T;Tn ] . By the lower semicontinuity of the intrinsic length, the fact that n (T) converges to (T), and consequently the segment between them has infinitesimal critical length, one gets lim `c ( n ) D 0 :

(39)

n

The important thing is that all the n coincide with  in [0; T], so that if t1 < t2 < T, S((t1 ); (t2 )) is estimated from above by  ˇ    `c  ˇ[t ;t ] D `c n j[t1 ;t2 ] ; 1

2

and S((t2 ); (t1 )) by the intrinsic length of the portion of

n joining (t2 ) to (t1 ). Taking into account (39) one gets 0 D lim `c ( n )  S((t1 ); (t2 )) C S((t2 ); (t1 )) ; n

which yields the crucial identity S((t2 ); (t1 )) D S((t1 ); (t2 )) :

(40)

ˇ In addition the previous two formulae imply that  ˇ

[t 1 ;t 2 ]

is a minimal geodesic whose intrinsic length realizes S((t1 ; (t2 )), so that (40) can be completed as follows: Z t2 ˙ ds D S((t1 ); (t2 )) : (41) (; ) S((t2 ); (t1 )) D

Next is presented a further result on the behavior of critical subsolutions on A. From this it appears that, even in the broad setting presently under investigation (the Hamiltonian is supposed to be just continuous), such subsolutions enjoy some extra regularity properties on A. Proposition 29 Let  be a critical curve, there is a negligible set E in R such that, for any critical subsolution u the function u ı  is differentiable whenever in R n E and d ˙ u((t)) D  ((t); (t)) : dt More precisely E is the complement in R of the set of ˙ where, in addition,  is differLebesgue points of  (; ) entiable. E has a vanishing measure thanks to Rademacher and Lebesgue differentiability theorem. See [10] for a complete proof of the proposition. The section ends with the statement of a result, that will be used for proving the forthcoming Theorem 34. The proof can be obtained by performing Lagrangian reparametrizations. Proposition 30 Let  be a curve defined in [0; 1]. Denote by ) the set of curves obtained through reparametrization of  in intervals with right endpoint 0, and for 2 ) indicate by [0; T( )] its interval of definition. One has (Z ) T( )

(L( ; ˙ ) C c) ds : 2 )

` c () D inf

:

0

t1

Finally,  has a Lagrangian parametrization, up to a change of parameter, so that it is obtained in the end Z t2 Z t2 ˙ ds D ˙ C c) ds : (; ) (L(; ) t1

t1

This shows that  is a critical curve. By applying Zorn lemma  can be extended to a critical curve defined in R, which concludes the proof of Theorem 27. As a consequence a first, perhaps surprising, result on the behavior of a critical subsolution on the Aubry set is obtained. Proposition 28 All critical subsolutions coincide on any critical curve, up to an additive constant. If u is such a subsolution and  any curve, one has S((t1 ); (t2 ))  u((t2 ))  u((t1 ))  S((t2 ); (t1 )) ; by Proposition 22. If in addition  is critical then the previous formula holds with equality, which proves Proposition 28.

Long-Time Behavior of Solutions to the Time-Dependent Equation In this section it is assumed, in addition to (3), (36), M is compact

(42)

H is strictly convex in p :

(43)

A solution of the time-dependent Eq. (2) is said to be stationary if it has the variable-separated form u0 (x)  at ;

(44)

for some constant a. Note that if is a supertangent (subtangent) to u0 at some point x0 , then  at is supertangent (subtangent) to u0  at at (x0 ; t) for any t, so that the inequality a C H(x0 ; D (x0 ))  () 0 holds true. Therefore, u0 is a solution to (1) in M. Since such a solution does exist only when a D c, see Proposition 20, it is the case that in (44) u0 is a critical solution

Hamilton–Jacobi Equations and Weak KAM Theory

and a is equal to c. The scope of this section is to show that any solution to the time-dependent equation uniformly converge to a stationary solution, as t goes to C1. In our setting there is a comparison principle for (2), stating that two solutions v, w, issued from initial data v0 , w0 , with v0  w0 , satisfies v  w, from any x 2 M, t > 0. In addition there exists a viscosity solution v, for any continuous initial datum v0 , which is, accordingly, unique, and is given by the Lax–Oleinik representation formula:

Z

t

v(x; t) D inf v0 ((0)) C

for any ". Hence, we exploit the stability properties that have been illustrated in Sects “Subsolutions”, “Solutions”, to prove the claim. The following inequality will be used L(x; q) C c   (x; q)

for any x, q;

which yields, by performing a line integration Z

t

(L( ; ˙ ) C c) ds  `c ( ) ;

0



 is a curve with (t) D x :

(45)

This shows that Eq. (2) enjoys the semigroup property, namely if w and v are two solutions with w(; 0) D v(; t0 ), for some t0 > 0, then w(x; t) D v(x; t0 C t) : It is clear that the solution of (2), taking a critical solution u0 as initial datum, is stationary and is given by (44). For any continuous initial datum v0 it can be found, since the underlying manifold is compact, a critical solution u0 and a pair of constants ˛ > ˇ such that u0 C ˛ > v0 > u0 C ˇ ; and consequently by the comparison principle for (2) u0 C ˛ > v(; t) C ct > u0 C ˇ

(46)

0

˙ ds : L(; )

for any t :

This shows that the family of functions x 7! v(x; t) C ct, for t  0, is equibounded. It can also be proved, see [10], that it is also equicontinuous, so that, by Ascoli Theorem, every sequence v(; t n ) C ct n , for t n ! C1, is uniformly convergent in M, up to extraction of a subsequence. The limits obtained in this way will be called !-limit of v C ct. The first step of the analysis is to show:

for any t > 0, any curve defined in [0; t], moreover, taking into account Lax–Oleinik formula and that M is assumed in this section to be compact v(x; t) C c t  v0 (y) C S(y; x) for some y depending on x ;

(47)

for a solution v of (2) taking a function v0 as initial datum. If, in addition, v0 is a critical subsolution, invoke Proposition 23 to derive from (47) v(x; t)  v0 (x)  c t

for any x, t :

(48)

A crucial point to be exploited, is that, for such a v0 , the evolution induced by (2) on the Aubry set takes place on the critical curves. Given t > 0 and x 2 A, pick a critical curve  with (t) D x0 , whose existence is guaranteed by Theorem 27, and then employ Proposition 29, about the behavior of the subsolution to (23) on critical curves, to obtain Z t ˙ ds  v(x; t) : (49) L(; ) v0 (x)  c t D v0 ((0)) C 0

By combining (49) and (48), one finally has Z

t

v(x; t) D v0 ((0)) C

˙ ds D v0 (x)  c t ; L(; )

0

Proposition 31 Let v be a solution to (2). The pointwise supremum and infimum of the !-limit of v C ct are critical subsolutions. The trick is to introduce a parameter " small and consider the functions v " (x; t) D v(x; t/")

which actually shows the announced optimal character of critical curves with respect to the Lax–Oleinik formula, and, at the same time, the following Proposition 32 Let v be a solution to (2) taking a critical subsolution v0 as initial datum at t D 0. Then

for " > 0 :

Arguing as above, it can be seen that the family v " C ct is equibounded, moreover v " C ct is apparently the solution to "w t C H(x; Dw) D c ;

v(x; t) D v0 (x)  c t

for any x 2 A :

Summing up: stationary solutions are derived by taking as initial datum solutions to (23); more generally, solutions issued from a critical subsolution are stationary at least on

699

700

Hamilton–Jacobi Equations and Weak KAM Theory

the Aubry set. The next step is to examine the long-time behavior of such solutions on the whole M. Proposition 33 Let v be a solution to (2) taking a critical subsolution v0 as initial datum at t D 0. One has lim v(x; t) C c t D u0 (x) uniformly in x;

t!C1

Theorem 34 Let v be a viscosity solution to (2) taking a continuous function v0 as initial datum for t D 0, then lim v(x; t) C c t D u0 (x) uniformly in x;

t!C1

where u0 is the critical solution given by the formula   u0 (x) D inf inf v0 (z) C S(z; y) C S(y; x) : y2A z2M

where u0 is the critical solution with trace v0 on A. The starting remark for getting the assertion is that, for any given x0 , an "-optimal curve for v(x0 ; t0 ), say , must be close to A for some t 2 [0; t0 ], provided " is sufficiently small and t0 large enough. If in fact  stayed far from A for any t 2 [0; t0 ] then L((s); 0) C c could be estimated from below by a positive constant, since E A, and the same should hold true, by continuity, for L((s); q) C c if jqj is small. One should then deduce that `(), and consequently (in view of Proposition 24) `c () were large. On the other side

(51)

The claim is that u0 , as defined in (51), is the critical solution with trace w0 :D inf v0 (z) C S(z; )

(52)

z2M

on the Aubry set. This can indeed be deduced from the representation formula given in Proposition 25, once it is proved that w0 is a critical subsolution. This property, in turn, comes from the characterization of critical subsolutions in terms of critical distance, presented in Proposition 23, the triangle inequality for S, and the inequalities w0 (x1 )  w0 (x2 )

u0 (x0 )  v(x0 ; t0 ) C c t0  v0 ((0)) C `c ()  " ; (50) by (46) and the comparison principle for (2), which shows that the critical length of  is bounded from above, yielding a contradiction. It can be therefore assumed that, up to a slight modification, the curve  intersects A at a time s0 2 [0; t0 ] and satisfies Z t0 ˙ ds  " v(x0 ; t0 )  v0 ((0)) C L(; ) 0 Z t0 ˙ ds  " :  v((s0 ); s0 )) C L(; ) s0

It is known from Proposition 32 that v((s0 ); s0 )) D v0 ((s0 ))  c s0 , so we have from the previous inequality, in view of (46)  ˇ  v(x0 ; t0 )  v0 ((s0 ))  c t0 C `c  ˇ[s 0 ;t0 ]  "  v0 ((s0 ))  c t0 C S((s0 ); x0 )  " : Bearing in mind the representation formula for u0 given in Proposition 25, we obtain in the end v(x0 ; t0 )  u0 (x0 )  c t0  " ; and conclude exploiting u0  v0 and the comparison principle for (2). The previous statement can be suitably generalized by removing the requirement of v0 being a critical subsolution. One more precisely has:

 v0 (z2 )CS(z2 ; x1 )v0 (z2 )S(z2 ; x2 )  S(x2 ; x1 ) ; which hold true if z2 is a point realizing the infimum for w0 (x2 ). If, in particular, v0 itself is a critical subsolution, then it coincides with w0 , so that, as announced, Theorem 34 includes Proposition 33. In the general case, it is apparent that w0  v0 , moreover if z 2 M and w 0 is a critical subsolution with w 0  v0 , one deduces from Proposition 23 w 0 (x)  v0 (z) C S(z; x)

for any z;

which tells that w 0  w0 , therefore w0 is the maximal critical subsolution not exceeding v0 . A complete proof of Theorem 34 is beyond the scope of this presentation. To give an idea, we consider the simplified case where the equilibria set E is a uniqueness set for the critical equation. Given x0 2 E , " > 0, take a z0 realizing the infimum for w0 (x0 ), and a curve , connecting z0 to x0 , whose intrinsic length approximates S(z0 ; x0 ) up to ". By invoking Proposition 30 one deduces that, up to a change of the parameter, such a curve, defined in [0; T], for some T > 0, satisfies Z T ˙ C c) ds < `c () C " : (L(; ) 0

Therefore, taking into account the Lax–Oleinik formula, one discovers Z T ˙ C c) ds  2" w0 (x0 )  v0 (z0 ) C (L(; ) (53) 0  v(x0 ; T) C cT  2" :

Hamilton–Jacobi Equations and Weak KAM Theory

Since L(x0 ; 0) C c D 0, by the very definition of equilibrium, it can be further derived from Lax–Oleinik formula that t 7! v(x0 ; t) is nonincreasing, so that the inequality (53) still holds if T is replaced by every t > T. This, together with the fact that " in (53) is taken arbitrarily, shows in the end, that any !-limit of v C ct satisfies w0 (x0 ) 

(x0 ) for any x0 2 E :

(54)

On the other side, the initial datum v0 is greater than or equal to w0 , and the solution to (2) with initial datum w0 , say w, has as unique !-limit the critical solution u0 with trace w0 on E [recall that E is assumed to be a uniqueness set for (23)]. Consequently, by the comparison principle for (2), one obtains u0 

in M ;

(55)

which, combined with (54), implies that and w0 coincide on E . Further, by Proposition 31, the maximal and minimal are critical subsolutions, and u0 is the maximal critical subsolution taking the value w0 on E . This finally yields D u0 for any , and proves the assertion of Theorem 34. In the general case the property of the set of !-limits of critical curves (i. e. limit points for t ! C1) of being a uniqueness set for the critical equation must be exploited. In this setting the strict convexity of H is essential, in [4,10] there are examples showing that Theorem 34 does not hold for H just convex in p. Main Regularity Result This section is devoted to the discussion of Theorem 35 If the Eq. (1) has a subsolution then it also admits a C1 subsolution. Moreover, the C1 subsolutions are dense in S a with respect to the local uniform convergence. Just to sum up: it is known that a C1 subsolution does exist when a is supercritical, see Theorem 15, and if the Aubry set is empty, see Proposition 18. So the case where a D c and A ¤ ; is left. The starting point is the investigation of the regularity properties of any critical subsolution on A. For this, and for proving Theorem 35 conditions (43), (35) on H are assumed, and (3) is strengthened by requiring H is locally Lipschitz-continuous in both arguments. (56) This regularity condition seems unavoidable to show that the functions S a (y; ) and Sˇa (y; ) D S a (; y) enjoy a weak form of semiconcavity in M n fyg, for all y, namely,

if v is any function of this family, x1 , x2 are different from y and  2 [0; 1], then v(x1 C (1  )x2 )  v(x1 )  (1  )v(x2 ) can be estimated from below by a quantity of the same type as that appearing on the right-hand side of (17), with jx1  x2 j2 replaced by a more general term which is still infinitesimal for jx1  x2 j ! 0. Of course the fundamental property of possessing C1 supertangents at any point different from y and that the set made up by their differentials coincides with the generalized gradient is maintained in this setting. To show the validity of this semiconcavity property, say for S a (y; ), at some point x it is crucial that for any neighborhood U of x suitably small there are curves joining y to x and approximating S a (y; x) which stays in U for a (natural) length greater than a fixed constant depending on U. This is clearly true if x ¤ y and explains the reason why the initial point y has been excluded. However, if a D c and y 2 A, exploiting the metric characterization of the Aubry set, it appears that this restriction on y can be removed, so that the following holds: Proposition 36 Let y 2 A, then the functions S(y; ) and S(; y) are semiconcave (in the sense roughly explained above) on the whole M. This, in particular, implies that both functions possess C1 supertangents at y and their differentials comprise the generalized gradient. From this the main regularity property of critical subsolutions on A are deduced. This reinforces the results given in Propositions 28, 29 under less stringent assumptions. Theorem 37 Every critical subsolution is differentiable on A. All have the same differential, denoted by p(y), at any point y 2 A, and H(y; p(y)) D c : Furthermore, the function y 7! p(y) is continuous on A. It is known from Proposition 22 that S(y; ), S(; y) are supertangent and subtangent, respectively, to every critical subsolution u at any y. If, in particular, y 2 A then S(y; ) and S(; y) admit C1 supertangents and subtangents, respectively, thanks to Proposition 36. This, in turn, implies that u is differentiable at y by Proposition 1 and, in addition, that all the differentials of supertangents to S(y; ) coincide with Du(y), which shows that its generalized gradient reduces to a singleton and so S(y; ) is strictly differentiable at y by Proposition 36. If p(y) denotes the differential of S(y; ) at y, then Du(y) D p(y) for any critical subsolution and, in addition, H(y; p(y)) D c by Proposition 19.

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Finally, the strict differentiability of S(y; ) at y gives that p() is continuous on A. The first application of Theorem 37 is relative to critical curves, and confirm their nature of generalized characteristics. If  is such a curve (contained in A by the results of Sect. “Dynamical Properties of the Aubry Set”) and u a critical subsolution, it is known from Proposition 29 that d ˙ D ((t); (t)) ˙ u((t) D p((t)) (t) dt ˙ D L((t); (t)) Cc; for a.e. t 2 R. Bearing in mind the definition of L, it ˙  is deduced that p((t)) is a maximizer of p 7! p (t) H((t); p), then by invoking (7) one obtains: Proposition 38 Any critical curve  satisfies the differential inclusion ˙ 2 @ p H(; p()) for a.e. t 2 R : In the statement @ p denotes the generalized gradient with respect to the variable p. The proof of Theorem 35 is now attacked. Combining Proposition 17 and Theorem 37, it is shown that there exists a critical subsolution, say w, differentiable at any point 1 of M, ˇ strict and of class C outside the Aubry set, and with ˇ Dw A D p() continuous. The problem is therefore to adjust the proof of Proposition 14 in order to have continuity of the differential on the whole M. The first step is to show a stronger version of the Proposition 16 asserting that it is possible to find a critical subsolution u, which is not only strict on W0 D M n A, but also strictly differentiable on A. Recall that this means that if y 2 A and xn are differentiability points of u with x n ! y, then Du(x n ) ! Du(y) D p(y). This implies that if p() is a continuous extension of p() in M, then p(x) and Du(x) are close at every differentiability point of u close to A. Starting from a subsolution u enjoying the previous property, the idea is then to use for defining the sought C1 subsolution, say v, the same formula (20) given in Proposition 14, i. e. (P ˇ i u i in W0 vD u in A where ˇ i is a C 1 partition of unity subordinated to a countable locally finite open covering U i of W 0 , and the ui are obtained from u through suitable regularization in U i by mollification. Look at the sketch of the proof of

Proposition 16 for the precise properties of these objects. It must be shown X  D ˇ i u i (x n ) ! Du(y) D p(y) ; for any sequence xn of elements of W 0 converging to y 2 A, or equivalently

ˇ X ˇ ˇ ˇ ˇ i u i (x n ) ˇ ! 0 : ˇp(x n )  D One has ˇ X ˇ X ˇ ˇ ˇ i u i (x n ) ˇ  ˇ i (x n )jDu i (x n ) ˇp(x n )  D i

 p(x n )jDˇ i (x n )ju i (x n )  u(x n )j : The estimates given in the proof of Proposition 14 show that the second term of the right-hand side of the formula is small. To estimate the first term calculate Z  jDu i (x n )  p(x n )j  ı i (z  x n ) jDu(z)  p(z)j  C jp(z)  p(x n )j dz ; and observe first that jDu(z)  p(z)j is small, as previously explained, since x n ! y 2 A and z is close to xn , second, that the mollification parameter ı i can be chosen in such a way that jp(z)  p(x n )j is also small. What is left is to discuss the density issue of the C1 subsolutions. This is done still assuming a D c and A nonempty. The proof in the other cases is simpler and goes along the same lines. It is clear from what was previously outlined that the initial subsolution u and v obtained as a result of the regularization procedure are close in the local uniform topology. It is then enough to show that any critical subsolution w can be approximated in the same topology by a subsolution enjoying the same property of u, namely being strict in W 0 and strictly differentiable on the Aubry set. The first property is easy to obtain by simply performing a convex combination of w with a C1 subsolution, strict in W 0 , whose existence has been proved above. It can be, in turn, suitably modified in a neighborhood of A in order to obtain the strict differentiability property, see [20]. Future Directions This line of research seems still capable of relevant developments. In particular to provide exact and approximate correctors for the homogenization of Hamilton–Jacobi equations in a stationary ergodic environment, see Lions and Souganidis [26] and Davini and Siconolfi [11,12], or

Hamilton–Jacobi Equations and Weak KAM Theory

in the direction of extending the results about long-time behavior of solutions of time-dependent problems to the noncompact setting, see Ishii [22,23]. Another promising field of utilization is in mass transportation theory, see Bernard [5], Bernard and Buffoni [6], and Villani [30]. The generalization of the model in the case where the Hamiltonian presents singularities should also make it possible to tackle through these techniques the N-body problem. With regard to applications, the theory outlined in the paper could be useful for dealing with topics such as the analysis of dielectric breakdown as well as other models in fracture mechanics. Bibliography 1. Arnold WI, Kozlov WW, Neishtadt AI (1988) Mathematical aspects of classical and celestial mechanics. In: Encyclopedia of Mathematical Sciences: Dynamical Systems III. Springer, New York 2. Bardi M, Capuzzo Dolcetta I (1997) Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston 3. Barles G (1994) Solutions de viscosité des équations de Hamilton–Jacobi. Springer, Paris 4. Barles G, Souganidis PE (2000) On the large time behavior of solutions of Hamilton–Jacobi equations. SIAM J Math Anal 31:925–939 5. Bernard P (2007) Smooth critical subsolutions of the Hamilton– Jacobi equation. Math Res Lett 14:503–511 6. Bernard P, Buffoni B (2007) Optimal mass transportation and Mather theory. J Eur Math Soc 9:85–121 7. Buttazzo G, Giaquinta M, Hildebrandt S (1998) One-dimensional Variational Problems. In: Oxford Lecture Series in Mathematics and its Applications, 15. Clarendon Press, Oxford 8. Clarke F (1983) Optimization and nonsmooth analysis. Wiley, New York 9. Contreras G, Iturriaga R (1999) Global Minimizers of Autonomous Lagrangians. In: 22nd Brazilian Mathematics Colloquium, IMPA, Rio de Janeiro 10. Davini A, Siconolfi A (2006) A generalized dynamical approach to the large time behavior of solutions of Hamilton–Jacobi equations. SIAM J Math Anal 38:478–502 11. Davini A, Siconolfi A (2007) Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case. to appear in Mathematische Annalen 12. Davini A, Siconolfi A (2007) Hamilton–Jacobi equations in the stationary ergodic setting: existence of correctors and Aubry set. (preprint)

13. Evans LC (2004) A survey of partial differential methods in weak KAM theory. Commun Pure Appl Math 57 14. Evans LC, Gomes D (2001) Effective Hamiltonians and averaging for Hamilton dynamics I. Arch Ration Mech Anal 157: 1–33 15. Evans LC, Gomes D (2002) Effective Hamiltonians and averaging for Hamilton dynamics II. Arch Rattion Mech Anal 161: 271–305 16. Fathi A (1997) Solutions KAM faibles et barrières de Peierls. C R Acad Sci Paris 325:649–652 17. Fathi A (1998) Sur la convergence du semi–groupe de Lax– Oleinik. C R Acad Sci Paris 327:267–270 18. Fathi A () Weak Kam Theorem in Lagrangian Dynamics. Cambridge University Press (to appear) 19. Fathi A, Siconolfi A (2004) Existence of C 1 critical subsolutions of the Hamilton–Jacobi equations. Invent Math 155:363–388 20. Fathi A, Siconolfi A (2005) PDE aspects of Aubry–Mather theory for quasiconvex Hamiltonians. Calc Var 22:185–228 21. Forni G, Mather J (1994) Action minimizing orbits in Hamiltonian systems. In: Graffi S (ed) Transition to Chaos in Classical and Quantum Mechanics. Lecture Notes in Mathematics, vol 1589. Springer, Berlin 22. Ishii H (2006) Asymptotic solutions for large time Hamilton– Jacobi equations. In: International Congress of Mathematicians, vol III. Eur Math Soc, Zürich, pp 213–227 23. Ishii H () Asymptotic solutions of Hamilton–Jacobi equations in Euclidean n space. Anal Non Linéaire, Ann Inst H Poincaré (to appear) 24. Koike S (2004) A beginner’s guide to the theory of viscosity solutions. In: MSJ Memoirs, vol 13. Tokyo 25. Lions PL (1987) Papanicolaou G, Varadhan SRS, Homogenization of Hamilton–Jacobi equations. Unpublished preprint 26. Lions PL, Souganidis T (2003) Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting. Commun Pure Appl Math 56:1501–1524 27. Roquejoffre JM (2001) Convergence to Steady States of Periodic Solutions in a Class of Hamilton–Jacobi Equations. J Math Pures Appl 80:85–104 28. Roquejoffre JM (2006) Propriétés qualitatives des solutions des équations de Hamilton–Jacobi et applications. Séminaire Bourbaki 975, 59ème anné. Société Mathématique de France, Paris 29. Siconolfi A (2006) Variational aspects of Hamilton–Jacobi equations and dynamical systems. In: Encyclopedia of Mathematical Physics. Academic Press, New York 30. Villani C () Optimal transport, old and new. http://www.umpa. ens-lyon.fr/cvillani/. Accessed 28 Aug 2008 31. Weinan E (1999) Aubry–Mather theory and periodic solutions of the forced Burgers equation. Commun Pure and Appl Math 52:811–828

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Hybrid Control Systems ANDREW R. TEEL1 , RICARDO G. SANFELICE2 , RAFAL GOEBEL3 1 Electrical and Computer Engineering Department, University of California, Santa Barbara, USA 2 Department of Aerospace and Mechanical Engineering, University of Arizona, Tucson, USA 3 Department of Mathematics and Statistics, Loyola University, Chicago, USA Article Outline Glossary Notation Definition of the Subject Introduction Well-posed Hybrid Dynamical Systems Modeling Hybrid Control Systems Stability Theory Design Tools Applications Discussion and Final Remarks Future Directions Bibliography Glossary Global asymptotic stability The typical closed-loop objective of a hybrid controller. Often, the hybrid controller achieves global asymptotic stability of a compact set rather than of a point. This is the property that solutions starting near the set remain near the set for all time and all solutions tend toward the set asymptotically. This property is robust, in a practical sense, for well-posed hybrid dynamical systems. (Well-posed) Hybrid dynamical system System that combines behaviors typical of continuous-time and discrete-time dynamical systems, that is, combines both flows and jumps. The system is said to be well-posed if the data used to describe the evolution (consisting of a flow map, flow set, jump map, and jump set) satisfy mild regularity conditions; see conditions (C1)–(C3) in Subsect. “Conditions for Wellposedness”. Hybrid controller Algorithm that takes, as inputs, measurements from a system to be controlled (called the plant) and combines behaviors of continuous-time and discrete-time controllers (i. e. flows and jumps) to produce, as outputs, signals that are to control the plant.

Hybrid closed-loop system The hybrid system resulting from the interconnection of a plant and a controller, at least one of which is a hybrid dynamical system. Invariance principle A tool for studying asymptotic properties of bounded solutions to (hybrid) dynamical systems, applicable when asymptotic stability is absent. It characterizes the sets to which such solutions must converge, by relying in part on invariance properties of such sets. Lyapunov stability theory A tool for establishing global asymptotic stability of a compact set without solving for the solutions to the hybrid dynamical system. A Lyapunov function is one that takes its minimum, which is zero, on the compact set, that grows unbounded as its argument grows unbounded, and that decreases in the direction of the flow map on the flow set and via the jump map on the jump set. Supervisor of hybrid controllers A hybrid controller that coordinates the actions of a family of hybrid controllers in order to achieve a certain stabilization objective. Patchy control Lyapunov functions provide a means of constructing supervisors. Temporal regularization A modification to a hybrid controller to enforce a positive lower bound on the amount of time between jumps triggered by the hybrid control algorithm. Zeno (and discrete) solutions A solution (to a hybrid dynamical system) that has an infinite number of jumps in a finite amount of time. It is discrete if, moreover, the solution never flows, i. e., never changes continuously. Notation  Rn denotes n-dimensional Euclidean space. R denotes the real numbers. R denotes the nonnegative real numbers, i. e., R D [0; 1). Z denotes the integers. N denotes the natural numbers including 0, i. e., N D f0; 1; : : :g.  Given a set S, S denotes its closure.  Given a vector x 2 Rn , jxj denotes its Euclidean vector norm.  B is the closed unit ball in the norm j  j.  Given a set S Rn and a point x 2 Rn , jxj S :D inf y2S jx  yj.  Given sets S1 ; S2 subsets of Rn , S1 C S2 :D fx1 C x2 j x1 2 S1 ; x2 2 S2 g.  A function is said to be positive definite with respect to a given compact set in its domain if it is zero on that compact set and positive elsewhere. When the compact set is the origin, the function will be called positive definite.

Hybrid Control Systems

 Given a function h : Rn ! R, h1 (c) denotes its c-level set, i. e. h1 (c) :D fz 2 Rn jh(z) D c g.  The double-arrow notation e. g., g : D  Rn , indicates a set-valued mapping, in contrast to a single arrow used for functions. Definition of the Subject Control systems are ubiquitous in nature and engineering. They regulate physical systems to desirable conditions. The mathematical theory behind engineering control systems developed over the last century. It started with the elegant stability theory of linear dynamical systems and continued with the more formidable theory of nonlinear dynamical systems, rooted in knowledge of stability theory for attractors in nonlinear differential or difference equations. Most recently, researchers have recognized the limited capabilities of control systems modeled only by differential or difference equations. Thus, they have started to explore the capabilities of hybrid control systems. Hybrid control systems contain dynamical states that sometimes change continuously and other times change discontinuously. These states, which can flow and jump, together with the output of the system being regulated, are used to produce a (hybrid) feedback control signal. Hybrid control systems can be applied to classical systems, where their added flexibility permits solving certain challenging control problems that are not solvable with other methods. Moreover, a firm understanding of hybrid dynamical systems allows applying hybrid control theory to systems that are, themselves, hybrid in nature, that is, having states that can change continuously and also change discontinuously. The development of hybrid control theory is in its infancy, with progress being marked by a transition from ad-hoc methods to systematic design tools. Introduction This article will present a general framework for modeling hybrid control systems and analyzing their dynamical properties. It will put forth basic tools for studying asymptotic stability properties of hybrid systems. Then, particular aspects of hybrid control will be described, as well as approaches to successfully achieving control objectives via hybrid control even if they are not solvable with classical methods. First, some examples of hybrid control systems are given. Hybrid dynamical systems combine behaviors typical of continuous-time dynamical systems (i. e., flows) and behaviors typical of discrete-time dynamical systems (i. e., jumps). Hybrid control systems exploit state variables that may flow as well as jump to achieve control objectives that

are difficult or impossible to achieve with controllers that are not hybrid. Perhaps the simplest example of a hybrid control system is one that uses a relay-type hysteresis element to avoid cycling the system’s actuators between “on” and “off ” too frequently. Consider controlling the temperature of a room by turning a heater on and off. As a good approximation, the room’s temperature T is governed by the differential equation T˙ D T C T0 C T u ;

(1)

where T 0 represents the natural temperature of the room, T  represents the capacity of the heater to raise the temperature in the room by being always on, and the variable u represents the state of the heater, which can be either 1 (“on”) or 0 (“off ”). A typical temperature control task is to keep the temperature between two specified values Tmin and Tmax where T0 < Tmin < Tmax < T0 C T : For purposes of illustration, consider the case when Tmin D 70ı F, Tmax D 80ı F. An algorithm that accomplishes this control task is input T, u if u=1 and T >= 80 then u=0 elseif u = 0 and T <= 70 then u=1 end In words, if the heater is “on” and the temperature is larger than 80ı F, then the heater is turned off, while if the heater is “off ” and the temperature is smaller than 70ı F, then the heater is turned on. By programming the controller with the algorithm above and closing the loop, the temperature of the system will evolve as shown in Fig. 1 (the values T0 D 60ı F and T D 30ı F were used in these simulations). Following the logic in the algorithm above, the controller can be expressed by the following conditional difference equation uC D 0 u

C

D1

if u D 1; T  80 if u D 0; T  70 ;

where uC is the value of u after a jump. Combining this difference equation with the differential equation (1) leads to the closed-loop system ) T˙ D T C T0 C T u u D 0; T  70 or (2) u D 1; T  80 u˙ D 0

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Hybrid Control Systems, Figure 1 Temperature control. a Flow diagram of control algorithm. b Evolution of temperature with control algorithm

TC D T uC D 1  q

)

u D 1; T  80 or u D 0; T  70 :

x12 , respectively, as follows: (3)

x˙11 D x12

(4)

x˙12 D  ; This closed-loop system is a hybrid dynamical system. The state variables are T and u; the continuous dynamics or flows as well as the constraints on the continuous evolution are given by (2); the discrete dynamics or jumps as well as the constraints on the discrete evolution are given by (3). As in the temperature control problem above, a hybrid control system can arise from controlling a classical system with a hybrid controller. In a more general setting, hybrid control systems emerge when controlling hybrid systems with nonlinear and/or hybrid controllers. Consider controlling the vertical motion of a ball through collisions with an actuated robot. Figure 2 depicts such a scenario. The impacts between the ball and the robot generate a jump in their velocity. When the control task is to stabilize the ball to a periodic motion, like the height pattern in Fig. 2b, this system is referred to as the one degree-of-freedom juggler. The dynamics of the ball in between the collisions are given by classical physics laws and can be written in terms of the ball’s height and vertical velocity, denoted by x11 and

where is the gravity constant. We denote the mass of the ball by m1 . We consider a robot with a vertical velocity control input u and denote the robot’s height and velocity by x21 and x22 , respectively. Then, its dynamics are given by x˙21 D x21

(5)

x˙22 D u :

The mass of the actuated robot is denoted by m2 . Following [10,58], impacts between the ball and the robot are assumed to conserve momentum, i. e., C C C m2 x22 D m1 x12 C m2 x22 ; m1 x12

(6)

and, for some restitution coefficient e 2 (0; 1), satisfy C C  x22 D e(x12  x22 ) : x12

(7)

Defining  :D m1 /(m1 C m2 ), Eqs. (6) and (7) can be combined to obtain the update law for velocities

C

  x12 e C (1 C e) (1  )(1 C e) x12 D C x22 (1 C e) 1  (1 C e) x12

 x D:  (; e) 12 : x22 The update law for positions is given by C D x11 ; x11

Hybrid Control Systems, Figure 2 One degree-of-freedom juggler and a juggling task. Positions are denoted by x11 ; x21 and velocities by x12 ; x22 , respectively. The control input is denoted by u. A desired height pattern is denoted by r11 . a Ball and actuated robot. b Desired ball’s height pattern

C x21 D x21 :

Impacts between the ball and the actuated robot occur when their heights are the same, that is, x11 D x21 , and when their velocities indicate that they are not moving away from each other, that is, x12  x22 . The derivation above leads to the following model for the one degree-of-freedom juggler system in Fig. 2:

Hybrid Control Systems

Hybrid Control Systems, Figure 3 Trajectories to the one degree-of-freedom juggler. Their positions are denoted by x11 ; x21 and their velocities by x12 ; x22 , respectively. The desired height pattern is denoted by r11 . a Juggling on ball. b Ball’s position and velocity

Flows:

Well-posed Hybrid Dynamical Systems x˙11 D x12 ;

x˙12 

x˙21 D x22 ;

x˙22 D u

) x11  x21  0 :

Jumps: C D x11 x11

 C x12 D 1 C x21

D x21

 C x22 D 1

9 > >

 > > > x > 0  (; e) 12 > > x22 = x11  x21 D 0 and > > > x12  x22  0 :

 > > > > x ; 0  (; e) 12 > x22

A controller designed to accomplish stabilization of the ball to a periodic pattern will only be able to measure the ball’s state at impacts. During flows, it will be able to control the robot’s velocity through u. Regardless of the nature of the controller, the closed-loop system will be a hybrid system by virtue of the dynamics of the one degree-of-freedom system. Figure 3 shows a trajectory to the closed-loop system with a controller that stabilizes the ball state to the periodic pattern in Fig. 2b (note the discontinuity in the velocity of the ball at impacts); see the control strategy in [55]. Following the modeling techniques illustrated by the examples above, the next section introduces a general modeling framework for hybrid dynamical systems. The framework makes possible the development of a robust stability theory for hybrid dynamical systems and prepares the way for insights into the design of robust hybrid control systems.

For numerous mathematical problems, well-posedness refers to the uniqueness of a solution and its continuous dependence on parameters, for example on initial conditions. Here, well-posedness will refer to some mild regularity properties of the data of a hybrid system that enable the development of a robust stability theory. Hybrid Behavior and Model Hybrid dynamical systems combine continuous and discrete dynamics. Such a combination may emerge when controlling a continuous-time system with a control algorithm that incorporates discrete dynamics, like in the temperature control problem in Sect. “Introduction”, when controlling a system that features hybrid phenomena, like in the juggling problem in Sect. “Introduction”, or as a modeling abstraction of complex dynamical systems. Solutions to hybrid systems (sometimes referred to as trajectories, executions, runs, or motions) can evolve both continuously, i. e. flow, and discontinuously, i. e. jump. Figure 4 depicts a representative behavior of a solution to a hybrid system.

Hybrid Control Systems, Figure 4 Evolution of a hybrid system: continuous motion during flows (solid), discontinuous motion at jumps (dashed)

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For a purely continuous-time system, flows are usually modeled by differential equations, and sometimes by differential inclusions. For a purely discrete-time system, jumps are usually modeled by difference equations, and sometimes by difference inclusions. Set-valued dynamics naturally arise as regularizations of discontinuous difference and differential equations and represent the effect of state perturbations on such equations, in particular, the effect of state measurement errors when the equations represent a system in closed loop with a (discontinuous) feedback controller. For the continuous-time case, see the work by Filippov [22] and Krasovskii [34], as well as [26,27]; for the discrete-time case, see [33]. When working with hybrid control systems, it is appropriate to allow for set-valued discrete dynamics in order to capture decision making capabilities, which are typical in hybrid feedback. Difference inclusions, rather than equations, also arise naturally in modeling of hybrid automata, for example when the discrete dynamics are generated by multiple so-called “guards” and “reset maps”; see, e. g. [8,11,40] or [56] for details on modeling guards and resets in the current framework. Naturally, differential equations and difference inclusions will be featured in the model of a hybrid system. In most hybrid systems, or even in some purely continuoustime systems, the flow modeled by a differential equation is allowed to occur only on a certain subset of the state space Rn . Similarly, the jumps modeled by a difference equation may only be allowed from a certain subset of the state space. Hence, the model of the hybrid system stated above will also feature a flow set, restricting the flows, and a jump set, restricting the jumps. More formally, a hybrid system will be modeled with the following data:    

The flow set C Rn ; The flow map f : C ! Rn ; The jump set D Rn ; The (set-valued) jump map G : D  Rn .

A shorthand notation for a hybrid system with this data will be H D ( f ; C; G; D). Such systems can be written in the suggestive form ( x˙ D f (x) ; x2C n H : x2R (8) C x 2 G(x) ; x 2 D ; where x 2 Rn denotes the state of the system, x˙ denotes its derivative with respect to time, and x C denotes its value after jumps. In several control applications, the state x of the hybrid system can contain logic states that take value in discrete sets (representing, for example, “on” or “off ”

states, like in the temperature control problem in Sect. “Introduction”). Two parameters will be used to specify “time” in solutions to hybrid systems: t, taking values in R0 , and representing the elapsed “real” time; and j, taking values in N, and representing the number of jumps that have occurred. For each solution, the combined parameters (t; j) will be restricted to belong to a hybrid time domain, a particular subset of R0  N. Hybrid time domains corresponding to different solutions may differ. Note that with such a parametrization, both purely continuous-time and purely discrete-time dynamical systems can be captured. Furthermore, for truly hybrid solutions, both flows and jumps are parametrized “symmetrically” (cf. [8,40]). A subset E of R0  N is a hybrid time domain if it is the union of infinitely many intervals of the form [t j ; t jC1 ]  f jg, where 0 D t0  t1  t2  : : : , or of finitely many such intervals, with the last one possibly of the form [t j ; t jC1 ]  f jg, [t j ; t jC1 )  f jg, or [t j ; 1)  f jg. On each hybrid time domain there is a natural ordering of points: we write (t; j) (t 0 j0 ) for (t; j); (t 0 ; j0 ) 2 E if t  t 0 and j  j0 . Solutions to hybrid systems are given by functions, which are called hybrid arcs, defined on hybrid time domains and satisfying the dynamics and the constraints given by the data of the hybrid system. A hybrid arc is a function x : dom x ! Rn , where dom x is a hybrid time domain and t 7! x(t; j) is a locally absolutely continuous function for each fixed j. A hybrid arc x is a solution to H D ( f ; C; G; D) if x(0; 0) 2 C [ D and it satisfies Flow condition: x˙ (t; j) D f (x(t; j)) and

(9)

x(t; j) 2 C

for all j 2 N and almost all t such that (t; j) 2 dom x; Jump condition: x(t; j C 1) 2 G(x(t; j)) x(t; j) 2 D

and

(10)

for all (t; j) 2 dom x such that (t; j C 1) 2 dom x. Figure 5 shows a solution to a hybrid system H D ( f ; C; G; D) flowing (as solutions to continuous-time systems do) while in the flow set C and jumping (as solutions to discrete-time systems do) from points in the jump set D. A hybrid arc x is said to be nontrivial if dom x contains at least one point different from (0; 0) and complete

Hybrid Control Systems

Hybrid Control Systems, Figure 5 Evolution of a solution to a hybrid system. Flows and jumps of the solution x are allowed only on the flow set C and on the jump set D, respectively

if dom x is unbounded (in either the t or j direction, or both). It is said to be Zeno if it has an infinite number of jumps in a finite amount of time, and discrete if it has an infinite number of jumps and never flows. A solution x to a hybrid system is maximal if it cannot be extended, i. e., there is no solution x 0 such that dom x is a proper subset of dom x 0 and x agrees with x 0 on dom x. Obviously, complete solutions are maximal.

between solutions. In contrast to purely continuous-time systems or purely discrete-time systems, the uniform metric is not a suitable indicator of distance: two solutions experiencing jumps at close but not the same times will not be close in the uniform metric, even if (intuitively) they represent very similar behaviors. For example, Fig. 6 shows two solutions to the juggling problem in Sect. “Introduction” starting at nearby initial conditions for which the velocities are not close in the uniform metric. A more appropriate distance notion should take possibly different jump times into account. We use the following notion: given T; J; " > 0, two hybrid arcs x : dom x ! Rn and y : dom y ! Rn are said to be (T; J; ")-close if: (a) for all (t; j) 2 dom x with t  T, j  J there exists s such that (s; j) 2 dom y, jt  sj < ", and jx(t; j)  y(s; j)j < " ; (b) for all (t; j) 2 dom y with t  T, j  J there exists s such that (s; j) 2 dom x, jt  sj < ", and jy(t; j)  x(s; j)j < " :

Conditions for Well-Posedness Many desired results in stability theory for dynamical systems, like invariance principles, converse Lyapunov theorems, or statements about generic robustness of stability, hinge upon some fundamental properties of the space of solutions to the system. These properties may involve continuous dependence of solutions on initial conditions, completeness and sequential compactness of the space of solutions, etc. To begin addressing these or similar properties for hybrid systems, one should establish a concept of distance

An appealing geometric interpretation of (T; J; ")closeness of x and y can be given. The graph of a hybrid arc x : dom x ! Rn is the subset of RnC2 given by ˚ gph x :D (t; j; z) j (t; j) 2 dom x; z D x(t; j) : Hybrid arcs x and y are (T; J; ")-close if the restriction of the graph of x to t  T, j  J, i. e., the set f(t; j; z)j(t; j) 2 dom x; t  T; j  J; z D x(t; j)g, is in the "neighborhood of gph y, and vice versa: the restriction of the graph of y is in the "-neighborhood of gph x. (The

Hybrid Control Systems, Figure 6 Two solutions to the juggling system in Sect. “Introduction” starting at nearby initial conditions. Velocities are not close in the uniform metric near the jump times. a Ball’s heights. b Ball’s velocities

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neighborhoods of gph x and gphy should be understood in the norm for which the unit ball is [1; 1][1; 1]B.) The (T; J; ")-closeness can be used to quantify the concept of graphical convergence of a sequence of hybrid arcs; for details, see [23]. Here, it is only noted that graphical convergence of a sequence of mappings is understood as convergence of the sequence of graphs of these mappings. Such a convergence concept does have solid intuitive motivation when the mappings considered are associated with solutions to hybrid systems. It turns out that when (T; J; ")-closeness and graphical convergence are used to study the properties of the space of solutions to a hybrid system H D ( f ; C; G; D), only mild and easy to verify conditions on the data of H are needed to ensure that H is “well-posed”. These conditions are: (C1) the flow set C and jump set D are closed; (C2) the flow map f : C ! Rn is continuous; (C3) the jump map G : D  Rn is outer semicontinuous and locally bounded. Only (C3) requires further comment: G : D  Rn is outer semicontinuous if for every convergent sequence x i 2 D with x i ! x and every convergent sequence y i 2 G(x i ) with y i ! y, one has y 2 G(x); G is locally bounded if for each compact set K Rn there exists a compact set K 0 Rn such that G(x) K 0 for all x 2 K. Any system H D ( f ; C; G; D) meeting (C1), (C2), and (C3) will be referred to as well-posed. One important consequence of a hybrid system H being well-posed is the following: (?) Every sequence of solutions to H has a subsequence that graphically converges to a solution to H , which holds under very mild boundedness assumptions about the sequence of solutions in question. The assumptions hold, for example, if the sequence is uniformly bounded, i. e., there exists a compact set K Rn such that, for each i, x i (t; j) 2 K for all (t; j) 2 dom x i , where fx i g1 iD1 is a sequence of solutions. Another important consequence of well-posedness is the following outer-semicontinuity (or upper-semicontinuity) property: (??) For every x 0 2 Rn , every desired level of closeness of solutions " > 0, and every (T; J), there exists a level of closeness for initial conditions ı > 0 so that for every solution xı to H with jxı (0; 0)  x 0 j < ı, there exist a solution x to H with x(0; 0) D x 0 such that xı and x are (T; J; ")-close.

This property holds at each x 0 2 Rn from which all maximal solutions to H are either complete or bounded. Properties (?) and (??), while being far weaker than any kind of continuous dependence of solutions on initial conditions, are sufficient to develop basic stability characterizations. Continuous dependence of solutions on initial conditions is rare in hybrid systems, as in its more classical meaning it entails uniqueness of solutions from each initial point. If understood in a set-valued sense, in order for inner-semicontinuity (or lower-semicontinuity) to be present, it still requires many further assumptions on the data. Property (?) is essentially all that is needed to establish invariance principles, which will be presented in Subsect. “Invariance Principles”. Property (??) is useful in describing, for example, uniformity of convergence and of overshoots in an asymptotically stable hybrid system. For the analysis of robustness properties of well-posed hybrid systems, strengthened versions of (?) and (??) are available. They take into account the effect of small perturbations; for example a stronger version of (?) makes the same conclusion not about a sequence of solutions to H , but about a sequence of solutions to H generated with vanishing perturbations. (More information can be found in [23].) It is the stronger versions of the two properties that make converse Lyapunov results possible; see Subsect. “Converse Lyapunov Theorems and Robustness”, where more precise meaning to perturbations is also given. In the rest of this article, the analysis results will assume that (C1)–(C3) hold, i. e., the hybrid system under analysis is well-posed. The control algorithms will be constructed so that the corresponding closed-loop systems are well-posed. Such algorithms will be called well-posed controllers. Modeling Hybrid Control Systems Hybrid Controllers for Classical Systems Given a nonlinear control system of the form ( P:

x˙ D f (x; u) ;

x 2 CP

y D h(x) ;

(11)

where CP is a subset of the state space where the system is allowed to evolve, a general output-feedback hybrid controller K D (; ; C K ; ; D K ) takes the form

K:

8 ˆ < u D (y; ) ˙ D (y; ) ; ˆ : C  2 (y; ) ;

(y; ) 2 C K (y; ) 2 D K ;

Hybrid Control Systems

 During the rest of the sampling period, it will hold the values of u and z constant.

Hybrid Control Systems, Figure 7 Sample-and-hold control of a nonlinear system

where the output of the plant y 2 R p is the input to the controller, the input to the plant u 2 Rm is the output of the controller, and  2 R k is the controller state. When system (11) is controlled by K , their interconnection results in a hybrid closed-loop system given by ) x˙ D f (x; (h(x); )) (x; ) 2 C ˙ D (h(x); ) (12) ) xC D x (x; ) 2 D ; C 2 (h(x); ) where C :D f(x; ) jx 2 C P ; (h(x); ) 2 C K g D :D f(x; ) j(h(x); ) 2 D K g : We now cast some specific situations into this framework. Sample-and-hold Control Perhaps the simplest example of a hybrid system that arises in control system design is when a continuous-time plant is controlled via a digital computer connected to the plant through a sample-and-hold device. This situation is ubiquitous in feedback control applications. A hybrid system emerges by considering the nonlinear control system (11), with C P D Rn , where the measurements y are sampled every T > 0 seconds, producing a sampled signal ys that is processed through a discrete-time algorithm

C z D (z; y s ) us to generate a sequence of input values us each of which is held for T seconds to generate the input signal u. When combined with the continuous-time dynamics, this algorithm will do the following:  At the beginning of the sample period, it will update the value of u and the value of the controller’s internal state z, based on the values of z and y (denoted ys ) at the beginning of the sampling period.

A complete model of this behavior is captured by defining the controller state to be 2 3 z 6 7  :D 4  5 ;  where  keeps track of the input value to hold during a sampling period and  is a timer state that determines when the state variables z and  should be updated. The hybrid controller is specified in the form of the previous section as 9 u D> > > > z˙ D 0 = (y; z; ; ) 2 C K ˙ D 0 > > > > ; ˙ D 1 9

C > z D (z; y) =  (y; z; ; ) 2 D K ; > ; C  D0 where C K :D f(y; z; ; ) j 2 [0; T] g ; D K :D f(y; z; ; ) j D T g : The overall closed-loop hybrid system is given by 9 x˙ D f (x; ) > > > > = z˙ D 0 (x; z; ; ) 2 C > ˙ D 0 > > > ; ˙ D 1 9 xC D x > > > >

C = z (x; z; ; ) 2 D ; D (z; h(x)) >  > > > ; C D 0 where C :D f(x; z; ; ) j 2 [0; T] g ; D :D f(x; z; ; ) j D T g : Notice that if the function  is discontinuous then this may fail to be a well-posed hybrid system. In such a case, it becomes well-posed by replacing the function  by its setvalued regularization \ ¯ y) :D (z; ((z; y) C ıB) : ı>0

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This corresponds to allowing, at points of discontinuity, values that can be obtained with arbitrarily small perturbations of z and y. Allowing these values is reasonable in light of inevitable control systems perturbations, like measurement noise and computer round-off error. Networked Control Systems Certain classes of networked control systems can be viewed as generalizations of systems with a sample-andhold device. The networked control systems generalization allows for multiple sample-and-hold devices operating simultaneously and asynchronously, and with a variable sampling period. Compared to the sample-and-hold closed-loop model in Subsect. “Sample-and-hold Control”, one can think of u as a large vector of inputs to a collection of i plants, collectively modeled by x˙ D f (x; u). The update rule for u may only update a certain part of u at a given jump time. This update rule may depend not only on z and y but perhaps also on u and a logic variable, which we denote by `, that may be cycling through the list of i indices corresponding to connections to different plants. Several common update protocols use algorithms that are discontinuous functions, so this will be modeled explicitly by allowing a set-valued update rule. Finally, due to time variability in the behavior of the network connecting the plants, the updates may occur at any time in an interval [Tmin ; Tmax ] where Tmin > 0 represents the minimum amount of time between transmissions in the network and Tmax > Tmin represents the maximum amount of time between transmissions. The overall control system for the network of plants is given by 9 u D  z˙ D 0 > = ˙ D 0 `˙ D 0 (y; z; ; `;  ) 2 C K > ; ˙ D 1 9

C > z > 2 (z; y; ; `) > > =  (y; z; ; `; ) 2 D K ; `C D (` mod i) C 1 > > > > ; C D 0 where C K :D f(y; z; ; `; ) j 2 [0; Tmax ] g ; D K :D f(y; z; ; `; ) j 2 [Tmin ; Tmax ] g : The closed-loop networked control system has the form 9 > x˙ D f (x; ) > = z˙ D 0 ˙ D 0 (x; z; ; `;  ) 2 C > > ; `˙ D 0 ˙ D 1

9 xC D x > > > >

C > > > z 2 (z; h(x); ; `) =  > > > `C D (` mod i) C 1 > > > > ; C  D0

(x; z; ; `;  ) 2 D ;

where C :D f(x; z; ; `;  ) j 2 [0; Tmax ] g ; D :D f(x; z; ; `;  ) j 2 [Tmin ; Tmax ] g : Reset Control Systems The first documented reset controller was created by Clegg [19]. Consisting of an operational amplifier, resistors, and diodes, Clegg’s controller produced an output that was the integral of its input subject to the constraint that the sign of the output and input agreed. This was achieved by forcing the state of the circuit to jump to zero, a good approximation of the behavior induced by the diodes, when the circuit’s input changed sign with respect to its output. Consider such a circuit in a negative feedback loop with a linear control system x˙ D Ax C Bu ; y D Cx ;

x 2 Rn ; u 2 R

y 2R:

Use  to denote the state of the integrator. Then, the hybrid model of the Clegg controller is given by uD ˙ D y ; C



D0;

(y; ) 2 C K (y; ) 2 D K ;

where C K :D f(y; ) jy  0 g ; D K :D f(y; ) jy  0 g : One problem with this model is that it exhibits discrete solutions, as defined in Subsect. “Hybrid Behavior and Model”. Indeed, notice that the jump map takes points with  D 0, which are in the jump set DK , back to points with  D 0. Thus, there are complete solutions that start with  D 0 and never flow, which corresponds to the definition of a discrete solution. There are several ways to address this issue. When a reset controller like the Clegg integrator is implemented through software, a temporal regularization, as discussed next in Subsect. “Zeno Solutions and Temporal Regularization”, can be used to force a small amount of flow time

Hybrid Control Systems

between jumps. Alternatively, and also for the case of an analog implementation, one may consider a more detailed model of the reset mechanism, as the model proposed above is not very accurate for the case where  and y are small. This modeling issue is analogous to hybrid modeling issues for a ball bouncing on a floor where the simplest model is not very accurate for small velocities. Zeno Solutions and Temporal Regularization Like for reset control systems, the closed-loop system (12) may exhibit Zeno solutions, i. e., solutions with an infinite number of jumps in a finite amount of time. These are relatively easy to detect in systems with bounded solutions, as they exist if and only if there exist discrete solutions, i. e., solutions with an infinite number of jumps and no flowing time. Discrete solutions in a hybrid control system are problematic, especially from an implementation point of view, but they can be removed by means of temporal regularization. The temporal regularization of a hybrid controller K D (; ; C K ; ; D K ) is generated by introducing a timer variable  that resets to zero at jumps and that must pass a threshold defined by a parameter ı 2 (0; 1) before another jump is allowed. The regularization produces a well-posed hybrid controller Kı : D ˜ C K;ı ; ˜ ; D K;ı ) with state ˜ :D (; ) 2 R kC1 , where (˜ ; ; ˜ (˜ ) :D () ˜ (˜ ) :D ()  f1  g ;

in a practical sense. This aspect is discussed in more detail in Subsect. “Zeno Solutions, Temporal Regularization, and Robustness”. Hybrid Controllers for Hybrid Systems Another interesting scenario in hybrid control is when a hybrid controller 8 ˆ < u D (y; ) (y; ) 2 C K K ˙ D (y; ) ; ˆ : C  2 (y; ) ; (y; ) 2 D K is used to control a plant that is also hybrid, perhaps modeled as x˙ D f (x; u) ; x

C

D g(x) ;

x 2 CP x 2 DP

y D h(x) : This is the situation for the juggling example presented in Sect. “Introduction”. In this scenario, the hybrid closedloop system is modeled as ) x˙ D f (x; (h(x); )) (x; ) 2 C ˙ D (h(x); )

C x 2 G(x; ) ; (x; ) 2 D  where

˜ (˜ ) :D

C :D f(x; ) jx 2 C P ; (h(x); ) 2 C K g

C K;ı

D :D f(x; ) jx 2 D P or (h(x); ) 2 D K g

()  f0g ;   :D C K  R0 [ (R k  [0; ı]) ;

D K;ı :D D K  [ı; 1] : This regularization is related to one type of temporal regularization introduced in [32]. The variable  is initialized in the interval [0; 1] and remains there for all time. When ı D 0, the controller accepts flowing only if (y; ) 2 C K , since ˙ D 1   and the flow condition for  when (y; ) … C K is  D 0. Moreover, jumping is possible for ı D 0 if and only if (y; ) 2 D K . Thus, the controller with ı D 0 has the same effect on the closed loop as the original controller K . When ı > 0, the controller forces at least ı seconds between jumps since ˙  1 for all  2 [0; ı]. In particular, Zeno solutions, if there were any, are eliminated. Based on the remarks at the end of Subsect. “Conditions for Well-Posedness”, we expect the effect of the controller for small ı > 0 to be close to the effect of the controller with ı D 0. In particular, we expect that this temporal regularization for small ı > 0 will not destroy the stability properties of the closed-loop hybrid system, at least

and

G P (x; ) :D

G K (x; ) :D

g(x) 



fxg (h(x); )



8 ˆ ˆ G P (x; ) ; x 2 D P ; (h(x); ) … D K ˆ ˆ < G (x; ) ; x … D ; (h(x); ) 2 D K P K G(x; ) :D ˆ G (x; ) [ G (x; ) ; ˆ P K ˆ ˆ : x 2 D P ; (h(x); ) 2 D K : As long as the data of the controller and plant are wellposed, the closed-loop system is a well-posed hybrid system. Also, it can be verified that the only way this model can exhibit discrete solutions is if either the plant exhibits discrete solutions or the controller, with constant y, exhibits discrete solutions. Indeed, if the plant does not exhibit discrete solutions then a discrete solution for the

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closed-loop system would eventually have to have x, and thus y, constant. Then, if there are no discrete solutions to the controller with constant y, there can be no discrete solutions to the closed-loop system. Stability Theory Lyapunov stability theory for dynamical systems typically states that asymptotically stable behaviors can be characterized by the existence of energy-like functions, which are called Lyapunov functions. This theory has served as a powerful tool for stability analysis of nonlinear dynamical systems and has enabled systematic design of robust control systems. In this section, we review some recent advances on Lyapunov-based stability analysis tools for hybrid dynamical systems. Global (Pre-)Asymptotic Stability In a classical setting, say of differential equations with Lipschitz continuous right-hand sides, existence of solutions and completeness of maximal ones can be taken for granted. These properties, together with a Lyapunov inequality ensuring that the Lyapunov function decreases along each solution, lead to a classical concept of asymptotic stability. On its own, the Lyapunov inequality does not say anything about the existence of solutions. It is hence natural to talk about a concept of asymptotic stability that is related only to the Lyapunov inequality. This appears particularly natural for the case of hybrid systems, where existence and completeness of solutions can be problematic. The compact set A Rn is stable for H if for each " > 0 there exists ı > 0 such that any solution x to H with jx(0; 0)jA  ı satisfies jx(t; j)jA  " for all (t; j) 2 dom x; it is globally pre-attractive for H if any solution x to H is bounded and if it is complete then x(t; j) ! A as t C j ! 1; it is globally pre-asymptotically stable if it is both stable and globally pre-attractive. When every maximal solution to H is complete, the prefix “pre” can be dropped and the “classical” notions of stability and asymptotic stability are recovered. Globally Pre-Asymptotically Stable ˝-Limit Sets Suppose all solutions to the hybrid system are bounded, there exists a compact set S such that all solutions eventually reach and remain in S, and there exists a neighborhood of S from which this convergence to S is uniform. (We are not assuming that the set S is forward invariant and thus it may not be stable.) Moreover, assume there is at least one complete solution starting in S. In this case, the hybrid system admits a nonempty compact set A S that is

globally pre-asymptotically stable. Indeed, one such set is the so-called ˝-limit set of S, defined as ˇ 9 8 ˇ y D lim x i (t i ; j i ) ; > ˆ ˇ > ˆ > ˆ i!1 > ˆ ˇ > ˆ = < ˇ n ˇ t i C j i ! 1 ; (t i ; j i ) 2 dom x i : ˝H (S) :D y 2 R ˇ > ˆ ˇ x is a solution to H > ˆ i > ˆ ˇ > ˆ > ˆ ˇ ; : ˇ with x (0; 0) 2 S i

In fact, this ˝-limit set is the smallest compact set in S that is globally pre-asymptotically stable. To illustrate this concept, consider the temperature control system in Sect. “Introduction” with additional heater dynamics given by h˙ D 3h C (2h C h)u ; where h is the heater temperature and h is a constant that determines how hot the heater can get due to being on. That is, when the heater is “on” (u D 1), its temperature rises asymptotically towards h . There is a maximum temperature h¯ < h for which the heater can operate safely, and another temperature h < h¯ corresponding to a temperature far enough below h¯ that it is considered safe to turn the heater back on. For the desired range of temperatures for T given by Tmin D 70ı F, Tmax D 80ı F and T D 30ı F, let the overheating constant be h D 200ı F, the maximum safe temperature be h¯ D 150ı F, and the lower temperature h D 50ı F. Then, to keep the temperature T in the desired range and prevent overheating, the following algorithm is used:  When the heater is “on” (u D 1) and either T  80 or h  150, then turn the heater off (uC D 0).  When the heater is “off ” (u D 0) and T  70 and h  50, then turn the heater on (uC D 1). These rules define the jump map for u, which is given by uC D 1u, and the jump set of the hybrid control system. The resulting hybrid closed-loop system, denoted by H T , is given by

T˙ D T C T0 C T u h˙ D 3h C (2h C h)u u˙ D 0 TC D T hC D h

9 > > =

> > uC D 1  u ;

9 > = > ;

8 u D 1; ˆ ˆ ˆ ˆ ˆ ˆ < (T  80 and h  150) or ˆ ˆ ˆ ˆ u D 0; ˆ ˆ : (T  70 or h  50)

8 ˆ < u D 1; (T  80 or h  150) or ˆ : u D 0; (T  70 and h  50) :

Hybrid Control Systems

For this system, the set S :D [70; 80]  [0; 150]  f0; 1g

(13)

is not forward invariant. Indeed, consider the initial condition (T; h; u) D (70; 150; 0) which is not in the jump set, so there will be some time where the heater remains off, cooling the room to a value below 70ı F. Nevertheless, all trajectories converge to the set S and a neighborhood of initial conditions around S produce solutions that reach the set S in a uniform amount of time. Thus, the set ˝H T (S) S is a compact globally pre-asymptotically stable set for the system H T . Converse Lyapunov Theorems and Robustness For purely continuous-time and discrete-time systems satisfying some regularity conditions, global asymptotic stability of a compact set implies the existence of a smooth Lyapunov function. Such results, known as converse Lyapunov theorems, establish a necessary condition for global asymptotic stability. For hybrid systems H D ( f ; C; G; D), the conditions for well-posedness also lead to a converse Lyapunov theorem. If a compact set A Rn is globally pre-asymptotically stable for the hybrid system H D ( f ; C; G; D), then there exists a smooth Lyapunov function; that is, there exists a smooth function V : Rn ! R0 that is positive definite with respect to A, radially unbounded, and satisfies hrV(x); f (x)i  V(x) 8x 2 C ; V(x) max V(g)  2 g2G(x)

8x 2 D :

Converse Lyapunov theorems are not of mere theoretical interest as they can be used to characterize robustness of asymptotic stability. Suppose that H D ( f ; C; G; D) is well-posed and that V : Rn ! R0 is a smooth Lyapunov function for some compact set A. The smoothness of V and the regularity of the data of H imply that V decreases along solutions when the data is perturbed. More precisely: Global pre-asymptotic stability of the compact set A Rn for H is equivalent to semiglobal practical pre-asymptotic stability of A in the size of perturbations to H , i. e., to the following: there exists a con-

tinuous, nondecreasing in the first argument, nonincreasing in second argument function ˇ : R0  R0 ! R0 with the property that lims&0 ˇ(s; t) D

lim t!1 ˇ(s; t) D 0 and, for each " > 0 and each compact set K Rn , there exists  > 0, such that for each perturbation level 2 (0;  ] each solution x, x(0; 0) 2 K, to the -perturbed hybrid system ( H :

x2R

n

x˙ 2 F (x) ; x

C

2 G (x) ;

x 2 C

x 2 D ;

where, for each x 2 Rn , F (x) :D co f ((x C B) \ C) C B ; G (x) :D fv 2 Rn jv 2 z C B; z 2 G((x C B) \ D) g ; and C :D fz 2 Rn j(z C B) \ C ¤ ; g ; D :D fz 2 Rn j(z C B) \ D ¤ ; g ; satisfies jx(t; j)jA  maxfˇ(jx(0; 0)jA ; tC j); "g 8(t; j) 2 dom x: The above result can be readily used to derive robustness of (pre-)asymptotic stability to various types of perturbations, such as slowly-varying and weakly-jumping parameters, “average dwell-time” perturbations (see [16] for details), and temporal regularizations, as introduced in Subsect. “Zeno Solutions and Temporal Regularization”. We clarify the latter robustness now. Zeno Solutions, Temporal Regularization, and Robustness Some of the control systems we will design later will have discrete solutions that evolve in the set we are trying to asymptotically stabilize. So, these solutions do not affect asymptotic stability adversely, but they are somewhat problematic from an implementation point of view. We indicated in Subsect. “Zeno Solutions and Temporal Regularization” how these solutions arising in hybrid control systems can be removed via temporal regularization. Here we indicate how doing so does not destroy the asymptotic stability achieved, at least in a semi-global practical sense. The assumption is that stabilization via hybrid control is achieved as a preliminary step. In particular, assume that there is a well-posed closed-loop hybrid system H :D ( f ; G; C; D) with state  2 Rn , and suppose that the compact set A Rn is globally pre-asymptotically stable. Following the prescription for a temporal regularization of a hybrid controller in Subsect. “Zeno Solutions and Temporal Regularization”, we consider the hybrid system ˜ Dı ) with the state x : D (; ) 2 RnC1 , Hı :D ( f˜; Cı ; G;

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Hybrid Control Systems

where ı 2 (0; 1) and f˜(x) :D f ()  f1  g ; ˜ G(x) :D G()  f0g ;   Cı :D C  R0 [ (Rn  [0; ı]) ; Dı :D D  [ı; 1] : ˜ D0 ) has As observed before, the system H0 D ( f˜; C0 ; G; ˜ the compact set A :D A  [0; 1] globally pre-asymptotically stable. When ı > 0, in each hybrid time domain of each solution, each time interval is at least ı seconds long, since ˙  1 for all  2 [0; ı]. In particular, Zeno solutions, if there were any, have been eliminated. Regarding pre-asymptotic stability, note that ˇ ˚ Cı z 2 RnC1 ˇ (z C ıB) \ C0 6D ; (with B RnC1 ), while Dı D0 . Hence, following the ˜ discussion above, one can conclude that for Hı , the set A is semi-globally practically asymptotically stable in the size of the temporal regularization parameter ı. Broadly speaking, temporal regularization does not destroy (practical) pre-asymptotic stability of A. Lyapunov Stability Theorem A Lyapunov function is not only necessary for asymptotic stability but also sufficient. It is a convenient tool for establishing asymptotic stability because it eliminates the need to solve explicitly for solutions to the system. In its sufficiency form, the requirements of a Lyapunov function can be relaxed somewhat compared to the conditions of the previous subsection. For a hybrid system H D ( f ; C; G; D) the compact set A Rn is globally pre-asymptotically stable if there exists a continuously differentiable function V : Rn ! R that is positive definite with respect to A, radially unbounded, and, with the definitions, ( u c (x) :D ud (x) :D

hrV(x); f (x)i

x2C

otherwise ( 1 max g2G(x) V(g)  V(x) x 2 D 1

otherwise ;

(14) (15)

satisfies ud (x)  0 8x 2 Rn u c (x) < 0 ;

ud (x) < 0

8x 2 Rn n A :

(16) (17)

In light of the converse theorem of the previous subsection, this sufficient condition for global asymptotic stability is reasonable. Nevertheless, finding a Lyapunov function is often difficult to do. Thus, there is motivation for stability analysis tools that relax the Lyapunov conditions. There are several directions in which to go. One is in the direction of invariance principles, which are presented next. Invariance Principles An important tool to study the convergence of solutions to dynamical systems is LaSalle’s invariance principle. LaSalle’s invariance principle [35,36] states that bounded and complete solutions converge to the largest invariant subset of the set where the derivative or the difference (depending whether the system is continuous-time or discrete-time, respectively) of a suitable energy function is zero. In situations where the condition (17) holds with nonstrict inequalities, the invariance principle provides a tool to extract information about convergence of solutions. By relying on the sequential compactness property of solutions in Subsect. “Conditions for Well-Posedness”, several versions of LaSalle-like invariance principles can be stated for hybrid systems. Like for continuous-time and discrete-time systems, to make statements about the convergence of a solution one typically assumes that the solution is bounded, that its hybrid time domain is unbounded (i. e., the solution is complete), and that a Lyapunov function does not increase along it. To obtain information about the set to which the solution converges, an invariant set is to be computed. For hybrid systems, since solutions may not be unique, the standard concept of invariance needs to be adjusted appropriately. Following [35], but in the setting of hybrid systems, we will insist that a (weakly) invariant set be both weakly forward invariant and weakly backward invariant. The word “weakly” indicates that only one solution, rather than all, needs to meet some invariance conditions. By requiring both forward and backward invariance we refine the sets to which solutions converge. For a given set M and a hybrid system H , these notions were defined, in [54], as follows:  Forward invariance: if for each point x 0 2 M there exists at least one complete solution x to H that starts at x0 and stays in the set M for all (t; j) 2 dom x.  Backward invariance: if for each point q 2 M and every positive number N there exists a point x0 from which there exists at least one solution x to H and (t  ; j ) 2 dom x such that x(t  ; j ) D q and x(t; j) 2 M for all (t; j) 2 dom x ; (t; j) (t  ; j ).

Hybrid Control Systems

Then, the following invariance principle can be stated: Let V : Rn ! R be continuously differentiable and suppose that U Rn is nonempty. Let x be a bounded and complete solution to a hybrid system H :D ( f ; C; G; D). If x satisfies x(t; j) 2 U for each (t; j) 2 dom x and u c (z)  0;

ud (z)  0

for all z 2 U ;

then, for some constant r 2 V(U), the solution x approaches the largest weakly invariant set contained in  1   1  u c (0) [ u1 \ V 1 (r) \ U : d (0) \ G u d (0) (18) Note that the statement and the conclusion of this invariance principle resemble the ones by LaSalle for differential/difference equations. In particular, the definition of the set (18) involves both the zero-level set of uc and ud as the continuous and discrete-time counterparts of the principle. For more details and other invariance principles for hybrid systems, see [54]. The invariance principle above leads to the following corollary on global pre-asymptotic stability: For a hybrid system H D ( f ; C; G; D) the compact set A Rn is globally pre-asymptotically stable if there exists a continuously differentiable function V : Rn ! R that is positive definite with respect to A, radially unbounded, such that, with the definitions (14)–(15), u c (x)  0;

ud (x)  0 8x 2 Rn ;

and, for every r > 0, the largest weakly invariant subset in (18) is empty. This corollary will be used to establish global asymptotic stability in the control application in Subsect. “Source Localization”.

and suppose that we have constructed a finite family of well-posed hybrid controllers Kq , that work well individually, on a particular region of the state space. We will make this more precise below. For simplicity, the controllers will share the same state. This can be accomplished by embedding the states of the individual controllers into a common state space. Each controller Kq , with q 2 Q and Q being a finite index set not containing 0, is given by 8 ˆ < u D q (x; ) ˙ D  q (x; ) ; Kq ˆ : C  2 q (x; ) ;

(20)

(x; ) 2 D q :

The sets Cq and Dq are such that (x; ) 2 C q [ D q implies x 2 C0 . The union of the regions over which these controllers operate is the region over which we want to obtain robust, asymptotic   stability. We define this set as  :D [q2Q C q [ D q . To achieve our goal, we will construct a hybrid supervisor that makes decisions about which of the hybrid controllers should be used based on the state’s location relative to a collection of closed sets q C q [D q that cover . The hybrid supervisor will have its own state q 2 Q. The composite controller, denoted K with flow set C and jump set D, should be such that 1) C [ D D   Q, 2) all maximal solutions of the interconnection of K with (19), denoted H , starting in   Q are complete, 3) and the compact set A  Q, a subset of   Q, is globally asymptotically stable for the system H . We now clarify what we mean by the family of hybrid controllers working well individually. Let H q denote the closed-loop interconnection of the system (19) with the hybrid controller (20). For each q 2 Q, the solutions to the system H q satisfy: I. The set A is globally pre-asymptotically stable. II. Each maximal solution is either complete or ends in 0 @

Design Tools

(x; ) 2 C q

[

1

   i A [ n C q [ D q :

i2Q;i>q

Supervisors of Hybrid Controllers In this section, we discuss how to construct a single, globally asymptotically stabilizing, well-posed hybrid controller from several individual well-posed hybrid controllers that behave well on particular regions of the statespace but that are not defined globally. Suppose we are trying to control a nonlinear system x˙ D f (x; u) ;

x 2 C0

(19)

III. No maximal solution starting in  q reaches 2

0

 n 4C q [ D q [ @

[

13  i A5 n A :

i2Q;i>q

Item III holds for free for the minimum index qmin since q min C q min [ D q min and [ i2Q  i D . The combina-

717

718

Hybrid Control Systems

tion of the three items for the maximum index qmax implies that the solutions to H q max that start in q max converge to A. Intuitively, the hybrid supervisor will attempt to reach its goal by guaranteeing completeness of solutions and making the evolution of q eventually monotonic while (x; ) does not belong to the set A. In this way, the (x; ) component of the solutions eventually converges to A since q is eventually constant and because of the first assumption above. The hybrid supervisor can be content with sticking with controller q as long as (x; ) 2 C q [ D q , it can increment q if (x; ) 2 [ i2Q;i>q  i , and it can do anything it wants if (x; ) 2 A. These are the only three situations that should come up when starting from  q . Otherwise, the hybrid controller would be forced to decrease the value of q, taking away any guarantee of convergence. This provides the motivation for item III above. Due to a disturbance or unfortunate initialization of q, it may be that the state reaches a point that would not otherwise be reached from  q . From such conditions, the important thing is that the solution is either complete (and thus converges to A) or else reaches a point where either q can be incremented or where q is allowed to be decremented. This is the motivation for item II above. The individual hybrid controllers are combined into a single, well-posed hybrid controller K as follows: Define ˚q :D [ i2Q;i>q  i and then 8 u D q (x; ) ˆ ˆ ˆ ˆ < ˙ D  (x; ) ; (x; ) 2 C˜ q q K

 C ˆ  ˆ ˆ ˜q ; ˆ 2 G q (x; ) ; (x; ) 2 D : q

(21)

where ˜ q : D D q [ ˚q [ n(C q [ D q ) ; D C˜ q is closed and satisfies C q n˚q C˜ q C q ; and the set-valued mapping Gq is constructed via the following definitions: D q;a :D ˚q D q n˚q D q;b D q   D q;c :D n C q [ D q [ ˚q

and

 f g fi 2 Q ji > q ; (x; ) 2  i g

 q (x; ) G q;b (x; ) :D fqg

 f g G q;c (x; ) :D fi 2 Q j(x; ) 2  i g [ G q; j (x; ) : G q (x; ) :D f j2fa;b;cg ; (x;)2D q; j g

G q;a (x; ) :D

(22)

This hybrid controller is well-posed and induces complete solutions from   Q and global asymptotic stability of the compact set A  Q. Uniting Local and Global Controllers As a simple illustration, consider a nonlinear control system x˙ D f (x; u), x 2 Rn , and the task of globally asymptotically stabilizing the origin using state feedback while insisting on using a particular state feedback  2 in a neighborhood of the origin. In order to solve this problem, one can find a state feedback  1 that globally asymptotically stabilizes the origin and then combine it with  2 using a hybrid supervisor. Suppose that the feedback  2 is defined on a closed neighborhood of the origin, denoted C2 , and that if the state x starts in the closed neighborhood 2 C2 of the origin then the closed-loop solutions when using  2 do not reach the boundary of C2 . Then, using the notation of this section, we can take 1 D C1 D Rn and D1 D D2 D ;. With these definitions, the assumptions above are satisfied and a hybrid controller can be constructed to solve the posed problem. The controller need not use the additional variable . Its data is defined as G q (x) :D 3  q, ˜ 1 :D 2 D ˚1 , D ˜ 2 :D Rn nC2 , C˜1 :D C1 n2 and D ˜ C2 :D C2 . Additional examples of supervisors will appear in the applications section later. Patchy Control Lyapunov Functions A key feature of (smooth) control Lyapunov functions (CLFs) is that their decrease along solutions to a given control system can be guaranteed by an appropriate choice of the control value, for each state value. It is known that, under mild assumptions on the control system, the existence of a CLF yields the existence of a robust (non hybrid) stabilizing feedback. It is also known that many nonlinear control systems do not admit a CLF. This can be illustrated by considering the question of robust stabilization of a single point on a circle, which faces a similar obstacle as the question of robust stabilization of the set A D f0; 1g for

Hybrid Control Systems

the control system on R given by x˙ D f (x; u) :D u. Any differentiable function on R that is positive definite with respect to A must have a maximum in the interval (0; 1). At such a maximum, say x¯ , one has rV(x¯ ) D 0 and no choice of u can lead to hrV(x¯ ); f (x¯ ; u)i < 0. (Smooth) patchy control Lyapunov functions (PCLFs) are, broadly speaking, objects consisting of several local CLFs the domains of which cover Rn and have certain weak invariance properties. PCLFs turn out to exist for far broader classes of nonlinear systems than CLFs, especially if an infinite number of patches (i. e., of local CLFs) is allowed. They also lead to robust hybrid stabilizing feedbacks. This will be outlined below. A brief illustration of the concept, for the control system on R mentioned above, would be to consider functions V1 (x) D x 2 on (1; 2/3) and V2 (x) D (x  1)2 on (1/3; 1). These functions are local CLFs for the points, respectively, 0 and 1; their domains cover R; and for each function, an appropriate choice of control will not only lead to the function’s decrease, but will also ensure that solutions starting in the function’s domain will remain there. While the example just mentioned outlines a general idea of a PCLF, the definition is slightly more technical. For the purposes of this article, a smooth patchy control Lyapunov function for a nonlinear system x˙ D f (x; u)

x 2 Rn ; u 2 U Rm

(23)

with respect to the compact set A consists of a finite set Q Z and a collection of functions V q and sets ˝ q , ˝q0 for each q 2 Q, such that: (i)

f˝q gq2Q and f˝q0 gq2Q are families of nonempty open subsets of Rn such that [ [ ˝q D ˝q0 ; Rn D q2Q

q2Q

and for all q 2 Q, the unit (outward) normal vector S to ˝ q is continuous on @˝q n i>q ˝ i0 , and ˝q0 ˝q ; (ii) for each q, V q is a smooth function defined on S a neighborhood of ˝q n i>q ˝ i0 ; and the following conditions are met: There exist a continuous, positive definite function ˛ : R0 ! R0 , and positive definite, radially unbounded functions , ¯ such that S (iii) for all q 2 Q, all x 2 ˝q n i>q ˝ i0 ,

(jxjA )  Vq (x)  ¯ (jxjA ) ;

(iv) for all q 2 Q, all x 2 ˝q n u q;x 2 U such that

S

i>q

˝ i0 , there exists

hrVq (x); f (x; u q;x )i  ˛(jxjA ) ; S (v) for all q 2 Q, all x 2 @˝q n i>q ˝ i0 , the u q;x of (iii) can be chosen such that hn q (x); f (x; u q;x )i  ˛(jxjA ); where n q (x) is the unit (outward) normal vector to ˝ q at x. Suppose that, for each x; v 2 Rn and c 2 R, the set fu 2 U j hv; f (x; u)i  cg is convex, as always holds if f (x; u) is affine in u and U is convex. For each q 2 Q let [ ˝ i0 C q D ˝q n i>q

and

[

q D ˝q0 n

˝ i0 :

i2Q;i>q

It can be shown, in part via arguments similar to those one would use when constructing a feedback from a CLF, that for each q 2 Q there exists a continuous mapping kq : Cq ! U such that, for all x 2 C q , hrVq (x); f (x; k q (x))i  

˛(jxjA ) ; 2

all maximal solutions to x˙ D f (x; k q (x)) are either complete or end in 0 1 [ @  i A [ Rn n C q ; i>q;i2Q

and no maximal solution starting in  q reaches 1 0 [ Rn n @C q [ i A : i2Q;i>q

The feedbacks kq can now be combined in a hybrid feedback, by taking D q D ; for each q 2 Q, and following the construction of Subsect. “Supervisors of Hybrid Controllers”. Indeed, the properties of maximal solutions just mentioned ensure conditions I, II and III of that section; the choice of Cq and  q also ensures that q C q and the union of  q ’s covers Rn . Among other things, this construction illustrates that the idea of hybrid supervision of hybrid controllers applies also to combining standard, non hybrid controllers.

719

720

Hybrid Control Systems

Applications In this section, we make use of the following:  For vectors in R2 , we will use the following multiplication rule, conjugate rule, and identity element:

z ˝ x :D







x1 z1 x 1  z2 x 2 ; x c :D ; z2 x 1 C z1 x 2 x2

 1 1 :D : 0

The multiplication rule is commutative, associative, and distributive. Note that x D 1˝ x D x ˝ 1 and note that x c ˝ x D x ˝ x c D jxj2 1. Also, (z ˝ x)c D x c ˝ z c .  For vectors in R4 , we will use the following multiplication rule, conjugate rule, and identity element (vectors are partitioned as x D [x1 x2T ]T where x2 2 R3 ):

 z1 x1  z2T x2 ; z2 x 1 C z1 x 2 C z2  x 2

 x1 ; xc D x2

 1 1D : 0

z˝x D

The multiplication rule is associative and distributive but not necessarily commutative. Note that x D 1 ˝ x D x˝1 and x c ˝x D x˝x c D jxj2 1. Also, (z˝x)c D x c ˝ zc .

Overcoming Stabilization Obstructions Global Stabilization and Tracking on the Unit Circle In this section, we consider stabilization and tracking control of the constrained system ˙ D  ˝ v(!) ;

v(!) :D

 0 !

 2 S1 ;

(24)

where S1 denotes the unit circle and ! 2 R is the control variable. Notice that S1 is invariant regardless of the choice of ! since h;  ˝ v(!)i D 0 for all  2 S 1 and all ! 2 R. This model describes the evolution of orientation angle of a rigid body in the plane as a function of the angular velocity !, which is the control variable. We discuss robust, global asymptotic stabilization and tracking problems which cannot be solved with classical feedback control, even when discontinuous feedback laws are allowed, but can be solved with hybrid feedback control. Stabilization First, we consider the problem of stabilizing the point  D 1. We note that the (classical) feedback control ! D 2 would almost solve this problem. We

would have ˙1 D 22 D 1  12 and the derivative of the energy function V() :D 1  1 would satisfy   hrV();  ˝ v(!)i D  1  12 : We note that the energy will remain constant if  starts at ˙1. Thus, since the goal was (robust) global asymptotic stability, this feedback does not achieve the desired goal. One could also consider the discontinuous feedback ! D sgn(2 ) where the function “sgn” is defined arbitrarily in the set f1; 1g when its argument is zero. This feedback is not robust to arbitrarily small measurement noise which can keep the trajectories of the system arbitrarily close to the point  D 1 for all time. To visualize this, note that from points on the circle with 2 < 0 and close to  D 1, this control law steers the trajectories towards  D 1 counterclockwise, while from points on the circle with 2 > 0 and close to  D 1, it steers the trajectories towards 1 clockwise. Then, from points on the circle arbitrarily close to  D 1, one can generate an arbitrarily small measurement noise signal e that changes sign appropriately so that sgn(2 C e) is always pushing trajectories towards 1. In order to achieve a robust, global asymptotic stability result, we consider a hybrid controller that uses the controller ! D 2 when the state is not near 1 and uses a controller that drives the system away from 1 when it is near that point. One way to accomplish the second task is to build an almost global asymptotic stabilizer for a point different from 1 such that the basin of attraction contains all points in a neighborhood of 1. For example, consider the feedback controller ! D 1 D: 1 () which would almost globally asymptotically stabilize the point  :D (0; 1) with the only point not in the basin of attraction being the point  c . Strangely enough, each of the two controllers can be thought of as globally asymptotically stabilizing the point 1 if their domains are limited. In particular, let the domain of applicability for the controller ! D 1 be C1 :D S 1 \ f j1  1/3 g ; and let the domain of applicability for the controller ! D 2 be C2 : D S 1 \ f j1  2/3 g : Notice that C1 [C2 D S 1 D: . Thus, we are in a situation where a hybrid supervisor, as discussed in Subsect. “Supervisors of Hybrid Controllers”, may be able to give us a hybrid, global asymptotic stabilizer. (There is no state  in the controllers we are working with here.) Let us take 1 :D C1 ;

2 :D S 1 nC1 :

Hybrid Control Systems

We have 1 [ 2 D S 1 . Next, we check the assumptions of Subsect. “Supervisors of Hybrid Controllers”. For each q 2 f1; 2g, the solutions of H q (the system we get by using ! D q () and restricting the flow to Cq ), are such that the point 1 is globally pre-asymptotically stable. For q D 1, this is because there are no complete solutions and 1 does not belong to C1 . For q D 2, this is because C2 is a subset of the basin of attraction for 1. We note that every maximal solution to H1 ends in  2 . Every maximal solution to H2 is complete and every maximal solution to H2 starting in  2 does not reach S 1 nC2 . Thus, the assumptions for a hybrid supervisor are in place. We follow the construction in Subsect. “Supervisors of Hybrid Controllers” to define the hybrid supervisor that combines the feedback laws  1 and  2 . We take ! :D q () C˜ q :D C q

  ˜ 1 :D 2 [ S 1 nC1 D S 1 nC1 D ˜ 2 :D S 1 nC2 : D For this particular problem, jumps toggle the mode q in the set f1; 2g. Thus, the jump map Gq can be simplified to G q :D 3  q. Tracking Let  : R0 ! S 1 be continuously differentiable. Suppose we want to find a hybrid feedback controller so that the state of (24) tracks the signal . This problem can be reduced to the stabilization problem of the previous section. Indeed, first note that  c ˝  D 1 and thus the following properties hold: ˙ c ˝  D  c ˝ ˙

 0 c ˙  ˝ D : 1 ˙2  2 ˙1

(25)

Then, with the coordinate transformation  D z ˝ , we have: I.

By multiplying the coordinate transformation on the right by  c we get z D  ˝ c and z ˝ z c D  ˝ c D 1, so that z 2 S 1 . II.  D  () z D 1. III. The derivative of z satisfies z˙ D ˙ ˝  c C  ˝ ˙ c D  ˝ v(!) ˝  c C  ˝ ˙ c ˝  ˝  c h i D  ˝ v(!) C ˙ c ˝  ˝  c h i D z ˝  ˝ v(!)   c ˝ ˙ ˝  c :

Our desire is to pick ! so that we have z˙ D z ˝ v(˝) and then to choose ˝ to globally asymptotically stabilize the point z D 1. Due to (25) and the properties of multiplication, the vectors  c ˝ ˙ and  c ˝v(˝)˝ are in the range of v(!). So, we can pick v(!) D  c ˝ ˙ C  c ˝ v(˝) ˝  to achieve the robust, global tracking goal. In fact, since the multiplication operation is commutative in R2 , this is equivalent to the feedback v(!) D  c ˝ ˙ C v(˝). Global Stabilization and Tracking for Unit Quaternions In this section, we consider stabilization and tracking control of the constrained system

 0 (26) ˙ D  ˝ v(!) ; v(!) :D  2 S3 ; ! where S3 denotes the hypersphere in R4 and ! 2 R3 is the control variable. Notice that S3 is invariant regardless of the choice of ! since h;  ˝ v(!)i D 0 for all  2 S 3 and all ! 2 R3 . This model describes the evolution of orientation angle of a rigid body in space as a function of angular velocities !, which are the control variables. The state  corresponds to a unit quaternion that can be used to characterize orientation. We discuss robust, global asymptotic stabilization and tracking problems which cannot be solved with classical feedback control, even when discontinuous feedback laws are allowed, but can be solved with hybrid feedback control. Stabilization First, we consider the problem of stabilizing the point  D 1. We note that the (classical) feedback control  ! :D 0 I v(2 ) D: 2 () (i. e., ! D 2 where  2 refers to the last three components of the vector ) would almost solve this problem. We would have ˙1 D 2T 2 D 1  12 and the derivative of the energy function V() :D 1  1 would satisfy   hrV();  ˝ v(!)i D  1  12 : We note that the energy will remain constant if  starts at ˙1. Thus, since the goal is (robust) global asymptotic stabilization, this feedback does not achieve the desired goal. One could also consider the discontinuous feedback ! D sgn(2 ) where the function “sgn” is the componentwise sign and each component is defined arbitrarily in the set f1; 1g when its argument is zero. This feedback is not robust to arbitrarily small measurement noise which can

721

722

Hybrid Control Systems

keep the trajectories of the system arbitrarily close to the point 1. In order to achieve a robust, global asymptotic stability result, we consider a hybrid controller that uses the controller above when the state is not near 1 and uses a controller that drives the system away from 1 when it is near that point. One way to accomplish the second task is to build an almost global asymptotic stabilizer for a point different from 1 and so that the basin of attraction contains all points in a neighborhood of 1. For example, consider stabilizing the point  : D (0; 1; 0; 0)T using the feedback controller (for more details see the next subsection)    z D  ˝  c ; ! D 0 I  c ˝ v(z2 ) ˝  D: 1 () (i. e. ! D (1 ; 4 ; 3 )T where now the subscripts refer to the individual components of ). This feedback would almost globally asymptotically stabilize the point  with the only point not in the basin of attraction being the point  c . The two feedback laws, ! D 2 () and ! D 1 (), are combined into a single hybrid feedback law via the hybrid supervisor approach given in Subsect. “Supervisors of Hybrid Controllers”. In fact, the construction is just like the construction for the case of stabilization on a circle with the only difference being that S1 is replaced everywhere by S3 . Tracking Let  : R0 ! S 3 be continuously differentiable. Suppose we want to find a hybrid feedback controller so that the state of (26) tracks the signal . This problem can be reduced to the stabilization problem of the previous section. Indeed, first note that  c ˝  D 1 and thus the following properties hold: ˙ c ˝  D  c ˝ ˙

 0  c ˝ ˙ D : 1 ˙2  ˙1 2  2  ˙2

(27)

Then, with the coordinate transformation  D z ˝ , we have: I.

By multiplying the coordinate transformation on the right by  c we get z D  ˝  c and z c ˝ z D  c ˝  D 1 so that z 2 S 3 . II.  D  () z D 1. III. The derivative of z satisfies z˙ D ˙ ˝  c C  ˝ ˙ c D ˝b ! ˝  c C  ˝ ˙ c ˝  ˝  c h i D˝ b ! C ˙ c ˝  ˝  c h i D z˝ ˝ b !   c ˝ ˙ ˝  c :

Our desire is to pick ! so that we have z˙ D z ˝ v(˝) and then to choose ˝ to globally asymptotically stabilize the point z D 1. Due to (27) and the properties of multiplication, the vectors  c ˝ ˙ and  c ˝v(˝)˝ are in the range of v(!). So, we can pick v(!) D  c ˝ ˙ C  c ˝ v(˝) ˝  to achieve the robust, global tracking goal. Stabilization of a Mobile Robot Consider the global stabilization problem for a model of a unicycle or mobile robot, given as x˙ D # ˙ D  ˝ v(!)

(28)

where x 2 R2 denotes planar position from a reference point (in meters),  2 S 1 denotes orientation, # 2 V :D [3; 30] denotes velocity (in meters per second), and ! 2 [4; 4] denotes angular velocity (in radians per second). Both # and ! are control inputs. Due to the specification of the set V , the vehicle is able to move more rapidly in the forward direction than in the backward direction. We define A0 to be the point (x; ) D (0; 1). The controllers below will all use a discrete state p 2 P :D f1; 1g. We take A :D A0  P and  : D R2  S 1  P . This system also can be modeled as

x˙ D

 cos( ) # sin( )

˙ D ! where D 0 corresponds to  D 1 and > 0 is in the counterclockwise direction. The set A0 in these coordinates is given by the set f0g  f j D 2k ; k 2 Zg . Even for the point (x; ) D (0; 0), this control system fails Brockett’s well-known condition for robust local asymptotic stabilization by classical (even discontinuous) timeinvariant feedback [9,26,52]. Nevertheless, for the control system (28), the point (0; 1) can be robustly, globally asymptotically stabilized by hybrid feedback. This is done by building three separate hybrid controllers and combining them with a supervisor. The three controllers are the following:  The first hybrid controller, K1 , uses # D ProjV (k1  T x), where k1 < 0 and ProjV denotes the projection onto V , while the feedback for ! is given by the hybrid controller in Subsect. “Global Stabilization and Tracking on the Unit Circle” for tracking on the

Hybrid Control Systems

unit circle with reference signal for  given by x/jxj. The two different values for q in that controller should be associated with the two values in P . The particular association does not matter. Note that the action of the tracking controller causes the vehicle eventually to use positive velocity to move x toward zero. The controller’s flow and jump sets are such that ˚ C1 [ D1 D x 2 R2 jjxj  "11  S 1  P where "11 > 0, and C1 ; D1 are constructed from the hybrid controller in Subsect. “Global Stabilization and Tracking on the Unit Circle” for tracking on the unit circle.  The second hybrid controller, K2 , uses # D ProjV (k2  T x), k2  0, while the feedback for ! is given as in Subsect. “Global Stabilization and Tracking on the Unit Circle” for stabilization of the point 1 on the unit circle. Again, the q values of that controller should be associated with the values in P and the particular association does not matter. The controller’s flow and jump sets are such that C2 [ D 2 D

 x 2 R2 jjxj  "21  S1 ˇ  ˚ \ (x; ) ˇ1  1  "22 jxj2  P ;

˚

where "21 > "11 , "22 > 0, and C2 ; D2 are constructed from the hybrid controller in Subsect. “Global Stabilization and Tracking on the Unit Circle” for stabilization of the point 1 on the unit circle.  The third hybrid controller, K3 , uses # D ProjV (k3  T x), k3 < 0, while the feedback for ! is hybrid as defined below. The controller’s flow and jump sets are designed so that   C3 [ D3 D fx jjxj  "31 g  S 1 ˇ ˚ \ (x; ) ˇ1  1  "32 jxj2  P D: 3 ; where "31 > "21 and "32 > "22 . The control law for ! is given by ! D pk, where k > 0 and the discrete state p has dynamics given by p˙ D 0;

pC D p :

The flow and jump sets are given by 

C3 :D3 \ f(p)2  0g  ˚ [ (p)2  0; 1  1  "22 jxj2 D3 :D3 nC3 :

This design accomplishes the following: controller K1 makes  track x/jxj as long as jxj is not too small, and thus the vehicle is driven towards x D 0 eventually using only positive velocity; controller K2 drives  towards 1 to get the orientation of the vehicle correct; and controller K3 stabilizes  to 1 in a persistently exciting manner so that # can be used to drive the vehicle to the origin. This control strategy is coordinated through a supervisor by defining 1 :DC1 [ D1   2 :D n1 \ (C2 [ D2 )   3 :D fx jjxj  "21 g  S 1 ˇ  ˚ \ (x; ) ˇ1  1  "22 jxj2  P : It can be verified that [q2Q q D  and that the conditions in Subsect. “Supervisors of Hybrid Controllers” for a successful supervisor are satisfied. Figure 8 depicts simulation results of the mobile robot with the hybrid controller proposed above for global asymptotic stabilization of A  Q. From the initial condition x(0; 0) D (10; 10) (in meters), (0; 0) corresponding to an angle of 4 radians, the mobile robot backs up using controller K1 until its orientation  corresponds to about 3 4 radians, at which x is approximately (10; 9:5). The green ? denotes a jump of the hybrid controller K1 . From this configuration, the mobile robot is steered towards a neighborhood of the origin with orientation given by x/jxj. About a fifth of a meter away from it, a jump of the hybrid supervisor connects controller K3 to the vehicle input (the location at which the jump occurs is denoted by the red ?). Note that the trajectory is such that controller K2 is bypassed since at the jump of the hybrid supervisor, while jxj is small the orientation  is such that the system state does not belong to  2 but to  3 ; that is, the orientation is already close enough to 1 at that jump. Figure 8a shows a zoomed version of the x trajectory in Fig. 8b. During this phase, controller K3 is in closed-loop with the mobile robot. The vehicle is steered to the origin with orientation close to 1 by a sequence of “parking” maneuvers. Note that after about seven of those maneuvers, the vehicle position is close to (0; 0:01) with almost the desired orientation. Source Localization Core of the Algorithm Consider the problem of programming an autonomous vehicle to find the location of a maximum for a continuously differentiable function by noting how the function values change as the vehicle

723

724

Hybrid Control Systems

Hybrid Control Systems, Figure 8 Global stabilization of a mobile robot to the origin with orientation (1; 0). Vehicle starts at x(0; 0) D (10; 10) (in meters) and (0; 0) corresponding to an angle of 4 radians. a The vehicle is initially steered to a neighborhood of the origin with orientation x/jxj. At about 1/5 meters away from it, controller K3 is enabled to accomplish the stabilisation task. b Zoomed version of trajectory in a around the origin. Controller K3 steers the vehicle to x D (0; 0) and  D 1 by a sequence of “parking” maneuvers

moves. Like before, the vehicle dynamics are given by x˙ D # ˙ D  ˝ v(!) ;

 2 S1 :

The vehicle is to search for the maximum of the function ' : R2 ! R. The function is assumed to be such that its maximum is unique, denoted x  , that r'(x) D 0 if and only if x D x  , and the union of its level sets over any interval of the form [c; 1), c 2 R, yields a compact set. For simplicity, we will assume that the sign of the derivative of the function ' in the direction x˙ D # is available as a measurement. We will discuss later how to approximate this quantity by considering the changes in the value of the function ' along solutions. We also assume that the vehicle’s angle, , can make jumps, according to C D  ˝  ;

(; ) 2 S 1  S 1 :

We will discuss later how to solve the problem when the angle  cannot change discontinuously. We propose the dynamic controller 9 # D #¯ > > = " # (x; ; z) 2 C z˙ D (z) > > ; ! ) " # zC 2 (z) (x; ; z) 2 D  where #¯ is a positive constant, ˇ ˚ ¯  0 ;  2 S1 ; z 2  C :D (x; ; z) ˇhr'(x);  #i ˇ ˚ ¯  0 ;  2 S1 ; z 2  ; D :D (x; ; z) ˇhr'(x);  #i

(note that C and D use information about the sign of the derivative of ' in the direction of the flow of x) and the required properties for the set  , the function , and the set-valued mapping  are as follows: I. (a) The set  is compact. (b) The maximal solutions of the continuous-time system

 z˙ D (z) z 2  ! and the maximal solutions to the discrete-time system

C z 2 (z) z 2   are complete. II. There are no non-trivial solutions to the system 9 > x˙ D  #¯ > > > ˙ D  ˝ v(!) = (x; ; z) 2 C0

 > > z˙ > > D (z) ; ! where

ˇ ˚ ¯ D 0 ;  2 S1 ; z 2  : C0 :D (x; ; z) ˇhr'(x);  #i

III. The only complete solutions to the system 9 > xC D x > > > = C  D˝ (x; ; z) 2 D

C > > z > 2 (z) > ;  start from xı D x  .

Hybrid Control Systems

The first assumption on (; ; ) above guarantees that the control algorithm can generate commands by either flowing exclusively or jumping exclusively. This permits arbitrary combinations of flows and jumps. Thus, since C [ D D R2  S 1   , all solutions to the closed-loop system are complete. Moreover, the assumption that  is compact guarantees that the only way solutions can grow unbounded is if x grows unbounded. The second assumption on (; ; ) guarantees that closed-loop flows lead to an increase in the function '. One situation where the assumption is easy to check is when ! D 0 for all z 2  and the maxima of ' along each search direction are isolated. In other words, the maxima of the function ';x : R ! R given by  7! '(x C ) are isolated for each (; x) 2 S 1  R2 . In this case it is not possible to flow while keeping ' constant. The last assumption on (; ; ) guarantees that the discrete update algorithm is rich enough to be able to find eventually a direction of decrease for ' for every point x. (Clearly the only way this assumption can be satisfied is if r'(x) D 0 only if x D x  , which is what we are assuming.) The assumption prevents the existence of discrete solutions at points where x ¤ x  . One example of data (; ; ) satisfying the three conditions above is

 D f0g ;

 0 ; (z) D 0

3 z

 (z) D 4 0 5 : 1 2

For this system, the state z does not change, the generated angular velocity ! is always zero, and the commanded rotation at each jump is /2 radians. Other more complicated orientation-generating algorithms that make use of the dynamic state z are also possible. For example, the algorithm in [42] uses the state variable z to generate conjugate directions at the update times. With the assumptions in place, the invariance principle of Subsect. “Invariance Principles” can be applied with the function ' to conclude that the closed-loop system has the compact set A :D fx  gS 1  globally asymptotically stable. Moreover, because of the robustness of global asymptotic stability to small perturbations, the results that we obtain are robust, in a practical sense, to slow variations in the characterization of the function ', including the point where it obtains its maximum. Practical Modifications The assumptions of the previous section that we would like to relax are that  can change discontinuously and that the derivative of ' is available as a measurement.

The first issue can be addressed by inserting a mode after every jump where the forward velocity is set to zero and a constant angular velocity is applied for the correct amount of time to drive  to the value  ˝ . If it is not possible to set the velocity to zero then some other openloop maneuver can be executed so that, after some time, the orientation has changed by the correct amount while the position has not changed. The second issue can be addressed by making sure that, after the direction is updated, values of the function ' along the solution are stored and compared to the current value of ' to determine the sign of the derivative. The comparison should not take place until after a sufficient amount of time has elapsed, to enhance robustness to measurement noise. The robustness of the nominal algorithm to temporal regularization and other perturbations permits such a practical implementation. Discussion and Final Remarks We have presented one viewpoint on hybrid control systems, but the field is still developing and other authors will give a different emphasis. For starters, we have stressed a dynamical systems view of hybrid systems, but authors with a computer science background typically will emphasize a hybrid automaton point of view that separates discrete-valued variables from continuous-valued variables. This decomposition can be found in the early work [61] and [59], and in the more recent work [2,8,58]. An introduction to this modeling approach is given in [7]. Impulsive systems, as described in [4], are closely related to hybrid systems but rely on ordinary time domains and usually do not consider the case where the flow set and jump set overlap. The work in [18] on “left-continuous systems” is closely linked to such systems. Passing to hybrid time domains (under different names) can be seen in [20,24,39,40]; other generalized concepts of time domains can be found in [43] and in the literature on dynamical systems on time scales, see [41] for an introduction. For simplicity, we have taken the flow map to be a function rather than a set-valued mapping. A motivation for set-valued mappings satisfying basic conditions is given in [56] where the notion of generalized solutions is developed and shown to be equivalent to solutions in the presence of vanishing perturbations or noise. Set-valued dynamics, with some consideration of the regularity of the mappings defining them, can be found in [1,2]. Implications of data regularity on the basic structural properties of the set of solutions to a system were outlined concurrently in [20,24]. The work in [24] emphasized the impli-

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cations for robustness of stability theory. The latter work preceded the rigorous derivations in [23] where the proofs of statements in the Subsect. “Conditions for Well-Posedness” can be found. To derive stronger results on continuous dependence of solutions to initial conditions, extra assumptions must be added. An early result in this direction appeared in [59]; see also [11,20,40]. The work in [11] exhibited a continuous selection of solutions, continuous dependence in the set-valued sense was addressed in [17]. Sufficient Lyapunov stability conditions, for hybrid or switching systems, and relying on various concepts of a solution, appeared in [6,21,40,62]. The results in Subsect. “Lyapunov Stability Theorem” are contained in [54] which develops a general invariance principle for hybrid systems. A part of the latter result is quoted in Subsect. “Invariance Principles”. Other invariance results for hybrid or switching systems have appeared in [3,18,28,30,40]. Some early converse results for hybrid systems, relying on nonsmooth and possibly discontinuous Lyapunov functions, were given in [62]. The results quoted in Subsect. “Converse Lyapunov Theorems and Robustness” come from [15]. The development of hybrid control theory is still in its formative stages. This article has focused on the development of supervisors of hybrid controllers and related topics, as well as to applications where hybrid control gives solutions that overcome obstacles faced by classical control. For hybrid supervisors used in the context of adaptive control, see [63] and the references therein. Other results related to supervisors include [47], which considers the problem discussed at the end of Subsect. “Supervisors of Hybrid Controllers”, and [57]. The field of hybrid control systems is moving in the direction of systematic design tools, but the capabilities of hybrid control have been recognized for some time. Many of the early observations were in the context of nonholonomic systems, like the result for mobile robots we have presented as an application of supervisors. For example, see [29,31,37,44,49]. More recently, in [48] and [50] it has been established that every asymptotically controllable nonlinear system can be robustly asymptotically stabilized using logic-based hybrid feedback. Other recent results include the work in [25] and its predecessor [12], and the related work on linear reset control systems as considered in [5,45] and the references therein. This article did not have much to do with the control of hybrid systems, other than the discussion of the juggling problem in the introduction and the structure of hybrid controllers for hybrid systems in Subsect. “Hybrid Controllers for Hybrid Systems”. A significant amount of work on the control of hybrid systems has been done, although

typically not in the framework proposed here. Notable references include [10,46,51] and the references therein. Other interesting topics and open questions in the area of hybrid control systems are developed in [38]. Future Directions What does the future hold for hybrid control systems? With a framework in place that mimics the framework of ordinary differential and difference equations, it appears that many new results will become available in directions that parallel results available for nonlinear control systems. These include certain types of separation principles, control algorithms based on zero dynamics (results in this direction can be found in [60] and [13]), and results based on interconnections and time-scale separation. Surely there will be unexpected results that are enabled by the fundamentally different nature of hybrid control as well. The theory behind the control of systems with impacts will continue to develop and lead to interesting applications. It is also reasonable to anticipate further developments related to the construction of robust, embedded hybrid control systems and robust networked control systems. Likely the research community will also be inspired by hybrid control systems discovered in nature. The future is bright for hybrid control systems design and it will be exciting to see the progress that is made over the next decade and beyond. Bibliography Primary Literature 1. Aubin JP, Haddad G (2001) Cadenced runs of impulse and hybrid control systems. Internat J Robust Nonlinear Control 11(5):401–415 2. Aubin JP, Lygeros J, Quincampoix M, Sastry SS, Seube N (2002) Impulse differential inclusions: a viability approach to hybrid systems. IEEE Trans Automat Control 47(1):2–20 3. Bacciotti A, Mazzi L (2005) An invariance principle for nonlinear switched systems. Syst Control Lett 54:1109–1119 4. Bainov DD, Simeonov P (1989) Systems with impulse effect: stability, theory, and applications. Ellis Horwood, Chichester; Halsted Press, New York 5. Beker O, Hollot C, Chait Y, Han H (2004) Fundamental properties of reset control systems. Automatica 40(6):905–915 6. Branicky M (1998) Multiple Lyapunov functions and other analysis tools for switched hybrid systems. IEEE Trans Automat Control 43(4):475–482 7. Branicky M (2005) Introduction to hybrid systems. In: Levine WS, Hristu-Varsakelis D (eds) Handbook of networked and embedded control systems. Birkhäuser, Boston, pp 91–116 8. Branicky M, Borkar VS, Mitter SK (1998) A unified framework for hybrid control: Model and optimal control theory. IEEE Trans Automat Control 43(1):31–45

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9. Brockett RW (1983.) Asymptotic stability and feedback stabilization. In: Brockett RW, Millman RS, Sussmann HJ (eds) Differential Geometric Control Theory. Birkhauser, Boston, MA, pp 181–191 10. Brogliato B (1996) Nonsmooth mechanics models, dynamics and control. Springer, London 11. Broucke M, Arapostathis A (2002) Continuous selections of trajectories of hybrid systems. Syst Control Lett 47:149–157 12. Bupp RT, Bernstein DS, Chellaboina VS, Haddad WM (2000) Resetting virtual absorbers for vibration control. J Vib Control 6:61 13. Cai C, Goebel R, Sanfelice R, Teel AR (2008) Hybrid systems: limit sets and zero dynamics with a view toward output regulation. In: Astolfi A, Marconi L (eds) Analysis and design of nonlinear control systems – In Honor of Alberto Isidori. Springer, pp 241–261 http://www.springer.com/west/home/ generic/search/results?SGWID=4-40109-22-173754110-0 14. Cai C, Teel AR, Goebel R (2007) Results on existence of smooth Lyapunov functions for asymptotically stable hybrid systems with nonopen basin of attraction. In: Proc. 26th American Control Conference, pp 3456–3461, http://www.ccec.ece.ucsb. edu/~cai/ 15. Cai C, Teel AR, Goebel R (2007) Smooth Lyapunov functions for hybrid systems - Part I: Existence is equivalent to robustness. IEEE Trans Automat Control 52(7):1264–1277 16. Cai C, Teel AR, Goebel R (2008) Smooth Lyapunov functions for hybrid systems - Part II: (Pre-)asymptotically stable compact sets. IEEE Trans Automat Control 53(3):734–748. See also [14] 17. Cai C, Goebel R, Teel A (2008) Relaxation results for hybrid inclusions. Set-Valued Analysis (To appear) 18. Chellaboina V, Bhat S, Haddad W (2003) An invariance principle for nonlinear hybrid and impulsive dynamical systems. Nonlin Anal 53:527–550 19. Clegg JC (1958) A nonlinear integrator for servomechanisms. Transactions AIEE 77(Part II):41–42 20. Collins P (2004) A trajectory-space approach to hybrid systems. In: 16th International Symposium on Mathematical Theory of Networks and Systems, CD-ROM 21. DeCarlo R, Branicky M, Pettersson S, Lennartson B (2000) Perspectives and results on the stability and stabilizability of hybrid systems. Proc of IEEE 88(7):1069–1082 22. Filippov A (1988) Differential equations with discontinuous right-hand sides. Kluwer, Dordrecht 23. Goebel R, Teel A (2006) Solutions to hybrid inclusions via set and graphical convergence with stability theory applications. Automatica 42(4):573–587 24. Goebel R, Hespanha J, Teel A, Cai C, Sanfelice R (2004) Hybrid systems: Generalized solutions and robust stability. In: Proc. 6th IFAC Symposium in Nonlinear Control Systems, pp 1–12, http://www-ccec.ece.ucsb.edu/7Ersanfelice/ Preprints/final_nolcos.pdf 25. Haddad WM, Chellaboina V, Hui Q, Nersesov SG (2007) Energy- and entropy-based stabilization for lossless dynamical systems via hybrid controllers. IEEE Trans Automat Control 52(9):1604–1614, http://ieeexplore.ieee.org/iel5/ 9/4303218/04303228.pdf 26. Hájek O (1979) Discontinuous differential equations, I. J Diff Eq 32:149–170 27. Hermes H (1967) Discontinuous vector fields and feedback control. In: Differential Equations and Dynamical Systems, Academic Press, New York, pp 155–165

28. Hespanha J (2004) Uniform stability of switched linear systems: Extensions of LaSalle’s invariance principle. IEEE Trans Automat Control 49(4):470–482 29. Hespanha J, Morse A (1999) Stabilization of nonholonomic integrators via logic-based switching. Automatica 35(3): 385–393 30. Hespanha J, Liberzon D, Angeli D, Sontag E (2005) Nonlinear norm-observability notions and stability of switched systems. IEEE Trans Automat Control 50(2):154–168 31. Hespanha JP, Liberzon D, Morse AS (1999) Logic-based switching control of a nonholonomic system with parametric modeling uncertainty. Syst Control Lett 38:167–177 32. Johansson K, Egerstedt M, Lygeros J, Sastry S (1999) On the regularization of zeno hybrid automata. Syst Control Lett 38(3):141–150 33. Kellet CM, Teel AR (2004) Smooth Lyapunov functions and robustness of stability for differential inclusions. Syst Control Lett 52:395–405 34. Krasovskii N (1970) Game-Theoretic Problems of capture. Nauka, Moscow 35. LaSalle JP (1967) An invariance principle in the theory of stability. In: Hale JK, LaSalle JP (eds) Differential equations and dynamical systems. Academic Press, New York 36. LaSalle J (1976) The stability of dynamical systems. SIAM’s Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia 37. Lucibello P, Oriolo G (1995) Stabilization via iterative state steering with application to chained-form systems. In: Proc. 35th IEEE Conference on Decision and Control, pp 2614–2619 38. Lygeros J (2005) An overview of hybrid systems control. In: Levine WS, Hristu-Varsakelis D (eds) Handbook of Networked and Embedded Control Systems. Birkhäuser, Boston, pp 519–538 39. Lygeros J, Johansson K, Sastry S, Egerstedt M (1999) On the existence of executions of hybrid automata. In: Proc. 38th IEEE Conference on Decision and Control, pp 2249–2254 40. Lygeros J, Johansson K, Simi´c S, Zhang J, Sastry SS (2003) Dynamical properties of hybrid automata. IEEE Trans Automat Control 48(1):2–17 41. M Bohner M, Peterson A (2001) Dynamic equations on time scales. An introduction with applications. Birkhäuser, Boston 42. Mayhew CG, Sanfelice RG, Teel AR (2007) Robust source seeking hybrid controllers for autonomous vehicles. In: Proc. 26th American Control Conference, pp 1185–1190 43. Michel A (1999) Recent trends in the stability analysis of hybrid dynamical systems. IEEE Trans Circuits Syst – I Fund Theory Appl 45(1):120–134 44. Morin P, Samson C (2000) Robust stabilization of driftless systems with hybrid open-loop/feedback control. In: Proc. 19th American Control Conference, pp 3929–3933 45. Nesic D, Zaccarian L, Teel A (2008) Stability properties of reset systems. Automatica 44(8):2019–2026 46. Plestan F, Grizzle J, Westervelt E, Abba G (2003) Stable walking of a 7-dof biped robot. IEEE Trans Robot Automat 19(4): 653–668 47. Prieur C (2001) Uniting local and global controllers with robustness to vanishing noise. Math Contr, Sig Syst 14(2): 143–172 48. Prieur C (2005) Asymptotic controllability and robust asymptotic stabilizability. SIAM J Control Opt 43:1888–1912

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49. Prieur C, Astolfi A (2003) Robust stabilization of chained systems via hybrid control. IEEE Trans Automat Control 48(10):1768–1772 50. Prieur C, Goebel R, Teel A (2007) Hybrid feedback control and robust stabilization of nonlinear systems. IEEE Trans Automat Control 52(11):2103–2117 51. Ronsse R, Lefèvre P, Sepulchre R (2007) Rhythmic feedback control of a blind planar juggler. IEEE Transactions on Robotics 23(4):790–802, http://www.montefiore.ulg.ac.be/ services/stochastic/pubs/2007/RLS07 52. Ryan E (1994) On Brockett’s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J Control Optim 32(6):1597–1604 53. Sanfelice R, Goebel R, Teel A (2006) A feedback control motivation for generalized solutions to hybrid systems. In: Hespanha JP, Tiwari A (eds) Hybrid Systems: Computation and Control: 9th International Workshop, vol LNCS, vol 3927. Springer, Berlin, pp 522–536 54. Sanfelice R, Goebel R, Teel A (2007) Invariance principles for hybrid systems with connections to detectability and asymptotic stability. IEEE Trans Automat Control 52(12):2282– 2297 55. Sanfelice R, Teel AR, Sepulchre R (2007) A hybrid systems approach to trajectory tracking control for juggling systems. In: Proc. 46th IEEE Conference on Decision and Control, pp 5282–5287 56. Sanfelice R, Goebel R, Teel A (2008) Generalized solutions to hybrid dynamical systems. ESAIM: Control, Optimisation and Calculus of Variations 14(4):699–724 57. Sanfelice RG, Teel AR (2007) A “throw-and-catch” hybrid control strategy for robust global stabilization of nonlinear systems. In: Proc. 26th American Control Conference, pp 3470– 3475

58. van der Schaft A, Schumacher H (2000) An introduction to hybrid dynamical systems. Lecture notes in control and information sciences. Springer, London 59. Tavernini L (1987) Differential automata and their discrete simulators. Nonlinear Anal 11(6):665–683 60. Westervelt E, Grizzle J, Koditschek D (2003) Hybrid zero dynamics of planar biped walkers. IEEE Trans Automat Control 48(1):42–56 61. Witsenhausen HS (1966) A class of hybridstate continuoustime dynamic systems. IEEE Trans Automat Control 11(2):161– 167 62. Ye H, Mitchel A, Hou L (1998) Stability theory for hybrid dynamical systems. IEEE Trans Automat Control 43(4):461–474 63. Yoon TW, Kim JS, Morse A (2007) Supervisory control using a new control-relevant switching. Automatica 43(10):1791– 1798

Books and Reviews Aubin JP, Cellina A (1984) Differential inclusions. Springer, Berlin Haddad WM, Chellaboina V, Nersesov SG (2006) Impulsive and hybrid dynamical systems: stability, dissipativity, and control. Princeton University Press, Princeton Levine WS, Hristu-Varsakelis D (2005) Handbook of networked and embedded control systems. Birkhäuser, Boston Liberzon D (2003) Switching in systems and control. Systems and control: Foundations and applications. Birkhäuser, Boston Matveev AS, Savkin AV (2000) Qualitative theory of hybrid dynamical systems. Birkhäuser, Boston Michel AN, Wang L, Hu B (2001) Qualitative theory of dynamical systems. Dekker Rockafellar RT, Wets RJ-B (1998) Variational analysis. Springer, Berlin

Hyperbolic Conservation Laws

Hyperbolic Conservation Laws ALBERTO BRESSAN Department of Mathematics, Penn State University, University Park, USA Article Outline Glossary Definition of the Subject Introduction Examples of Conservation Laws Shocks and Weak Solutions Hyperbolic Systems in One Space Dimension Entropy Admissibility Conditions The Riemann Problem Global Solutions Hyperbolic Systems in Several Space Dimensions Numerical Methods Future Directions Bibliography Glossary Conservation law Several physical laws state that certain basic quantities such as mass, energy, or electric charge, are globally conserved. A conservation law is a mathematical equation describing how the density of a conserved quantity varies in time. It is formulated as a partial differential equation having divergence form. Flux function The flux of a conserved quantity is a vector field, describing how much of the given quantity moves across any surface, at a given time. Shock Solutions to conservation laws often develop shocks, i. e. surfaces across which the basic physical fields are discontinuous. Knowing the two limiting values of a field on opposite sides of a shock, one can determine the speed of propagation of a shock in terms of the Rankine–Hugoniot equations. Entropy An entropy is an additional quantity which is globally conserved for every smooth solution to a system of conservation laws. In general, however, entropies are not conserved by solutions containing shocks. Imposing that certain entropies increase (or decrease) in correspondence to a shock, one can determine a unique physically admissible solution to the mathematical equations. Definition of the Subject According to some fundamental laws of continuum physics, certain basic quantities such as mass, momen-

tum, energy, electric charge. . . , are globally conserved. As time progresses, the evolution of these quantities can be described by a particular type of mathematical equations, called conservation laws. Gas dynamics, magneto-hydrodynamics, electromagnetism, motion of elastic materials, car traffic on a highway, flow in oil reservoirs, can all be modeled in terms of conservation laws. Understanding, predicting and controlling these various phenomena is the eventual goal of the mathematical theory of hyperbolic conservation laws. Introduction Let u D u(x; t) denote the density of a physical quantity, say, the density of mass. Here t denotes time, while x D (x1 ; x2 ; x3 ) 2 R3 is a three-dimensional space variable. A conservation law is a partial differential equation of the form @ u C div f D 0 : @t

(1)

which describes how the density u changes in time. The vector field f D ( f1 ; f2 ; f3 ) is called the flux of the conserved quantity. We recall that the divergence of f is div f D

@ f1 @ f2 @ f3 C C : @x1 @x2 @x3

To appreciate the meaning of the above Eq. (1), consider a fixed region ˝ R3 of the space. The total amount of mass contained inside ˝ at time t is computed as Z u(x; t)dx : ˝

This integral may well change in time. Using the conservation law (1) and then the divergence theorem, one obtains d dt

Z

Z ˝

u(x; t)dx D

˝

Z @ div fdx u(x; t)dx D  @t Z ˝ D f  nd˙ : ˙

Here ˙ denotes the boundary of ˝, while the integrand f  n denotes the inner product of the vector f with the unit outer normal n to the surface ˙ . According to the above identities, no mass is created or destroyed. The total amount of mass contained inside the region ˝ changes in time only because some of the mass flows in or out across the boundary ˙ . Assuming that the flux f can be expressed as a function of the density u alone, one obtains a closed equation. If the initial density u¯ at time t D 0 is known, then the values of

729

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the function u D u(x; t) at all future times t > 0 can be found by solving the initial-value problem u t C div f(u) D 0

¯ u(0; x) D u(x) :

More generally, a system of balance laws is a set of partial differential equations of the form 8@ ˆ < @t u1 C div f1 (u1 ; : : : ; u n ) D 1 ; (2)  ˆ :@ u C div f (u ; : : : ; u ) D  : n 1 n n @t n Here u1 ; : : : ; u n are the conserved quantities, f1 ; : : : ; fn are the corresponding fluxes, while the functions  i D  i (t; x; u1 ; : : : ; u n ) represent the source terms. In the case where all  i vanish identically, we refer to (2) as a system of conservation laws. Systems of this type express the fundamental balance equations of continuum physics, when small dissipation effects are neglected. A basic example is provided by the equations of non-viscous gases, accounting for the conservation of mass, momentum and energy. This subject is thus very classical, having a long tradition which can be traced back to Euler [21] and includes contributions by Stokes, Riemann, Weyl and von Neumann, among several others. In spite of continuing efforts, the mathematical theory of conservation laws is still largely incomplete. Most of the literature has been concerned with two main cases: (i) a single conservation law in several space dimensions, and (ii) systems of conservation laws in one space dimension. For systems of conservation laws in several space dimensions, not even the global-in-time existence of solutions is presently known, in any significant degree of generality. Several mathematical studies are focused on particular solutions, such as traveling waves, multi-dimensional Riemann problems, shock reflection past a wedge, etc. . . Toward a rigorous mathematical analysis of solutions, the main difficulty that one encounters is the lack of regularity. Due to the strong nonlinearity of the equations and the absence of dissipation terms with regularizing effect, solutions which are initially smooth may become discontinuous within finite time. In the presence of discontinuities, most of the classical tools of differential calculus do not apply. Moreover, the Eqs. (2) must be suitably reinterpreted, since a discontinuous function does not admit derivatives in a classical sense. Topics which have been more extensively investigated in the mathematical literature are the following:  Existence and uniqueness of solutions to the initialvalue problem. Continuous dependence of the solutions on the initial data [8,10,24,29,36,49,57].

 Admissibility conditions for solutions with shocks, characterizing the physically relevant ones [23,38,39, 46].  Stability of special solutions, such as traveling waves, w.r.t. small perturbations [34,47,49,50,62].  Relations between the solutions of a hyperbolic system of conservation laws and the solutions of various approximating systems, modeling more complex physical phenomena. In particular: vanishing viscosity approximations [6,20,28], relaxations [5,33,48], kinetic models [44,53].  Numerical algorithms for the efficient computation of solutions [32,41,41,56,58]. Some of these aspects of conservation laws will be outlined in the following sections. Examples of Conservation Laws We review here some of the most common examples of conservation laws. Throughout the sequel, subscripts such as u t ; f x will denote partial derivatives. Example 1 (Traffic flow) Let u(x, t) be the density of cars on a highway, at the point x at time t. This can be measured as the number of cars per kilometer (see Fig. 1). In first approximation, following [43] we shall assume that u is continuous and that the speed s of the cars depends only on their density, say s D s(u). Given any two points a, b on the highway, the number of cars between a and b therefore varies according to Z

b

Z d b u(x; t)dx dt a D [inflow at x D a]  [outflow at x D b]

u t (x; t)dx D a

D s(u(a; t))  u(a; t)  s(u(b; t))  u(b; t) Z b D [s(u)u]x dx : a

Since the above equalities hold for all a, b, one obtains the conservation law in one space dimension u t C [s(u)u]x D 0 ;

(3)

where u is the conserved quantity and f (u) D s(u)u is the flux function. Based on experimental data, an appropriate flux function has the form  a  2 u (0 < u  a2 ) ; f (u) D a1 ln u for suitable constants a1 ; ~a2 .

Hyperbolic Conservation Laws

A typical choice here is p(v) D kv  , with 2 [1; 3]. In particular  1:4 for air. In general, p is a decreasing function of v. Near a constant state v0 , one can approximate p by a linear function, say p(v)  p(v0 )  c 2 (v  v0 ). Here c 2 D p0 (v0 ). In this case the Eq. (4) reduces to the familiar wave equation Hyperbolic Conservation Laws, Figure 1 Modelling the density of cars by a conservation law

Example 2 (Gas dynamics) The a compressible, non-viscous gas in take the form 8 @ ˆ

C div( v) ˆ @t ˆ ˆ ˆ ˆ (conservation of mass) ; ˆ ˆ ˆ < @ ( v ) C div( v v) C @ p @t

i

i

@x i

v t t  c2 vx x D 0 :

Euler equations for Eulerian coordinates

i D 1; 2; 3;

ˆ (conservation of momentum) ˆ ˆ ˆ ˆ @ ˆ E C div((E C p)v) D0 ˆ @t ˆ ˆ : (conservation of energy) :

jvj2 C e ; 2

where the first term accounts for the kinetic energy while e is the internal energy density (related to the temperature). The system is closed by an additional equation p D p( ; e), called the equation of state, depending on the particular gas under consideration [18]. Notice that here we are neglecting small viscous forces, as well as heat conductivity. Calling fe1 ; e2 ; e3 g the standard basis of unit vectors in R3 , one has @p/@x i D div(pe i ). Hence all of the above equations can be written in the standard divergence form (1).

(5)

Here u is the conserved quantity while f is the flux. As long as the function u is continuously differentiable, using the chain rule the equation can be rewritten as u t C f 0 (u)u x D 0 ;

Here is the mass density, v D (v1 ; v2 ; v3 ) is the velocity vector, E is the energy density, and p the pressure. In turn, the energy can be represented as a sum ED

A single conservation law in one space dimension is a first order partial differential equation of the form u t C f (u)x D 0 :

D0 D0

Shocks and Weak Solutions

(6)

According to (6), in the x–t-plane, the directional derivative of the function u in the direction of the vector v D ( f 0 (u); 1) vanishes. ¯ At time t D 0, let an initial condition u(x; 0) D u(x) be given. As long as the solution u remains smooth, it can be uniquely determined by the so-called method of characteristics. For every point x0 , consider the straight line (see Fig. 2)   ¯ 0) t : x D x0 C f 0 u(x On this line, by (6) the value of u is constant. Hence     ¯ 0 ) t ; t D u(x ¯ 0) u x0 C f 0 u(x

Example 3 (Isentropic gas dynamics in Lagrangian variables) Consider a gas in a tube. Particles of the gas will be labeled by a one-dimensional variable y determined by their position in a reference configuration with constant unit density. Using this Lagrangian coordinate, we denote by u(y, t) the velocity of the particle y at time t, and by v(y; t) D 1 (y; t) its specific volume. The so-called p-system of isentropic gas dynamics [57] consists of the two conservation laws vt  ux D 0 ;

ut C px D 0 :

(4)

The system is closed by an equation of state p D p(v) expressing the pressure as a function of the specific volume.

Hyperbolic Conservation Laws, Figure 2 Solving a conservation law by the method of characteristics. The function u D u(x; t) is constant along each characteristic line

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Hyperbolic Conservation Laws, Figure 3 Shock formation

for all t  0. This allows to construct a solution up to the first time T where two or more characteristic lines meet. Beyond this time, the solution becomes discontinuous. Figure 3 shows the graph of a typical solution at three different times. Points on the graph of u move horizontally with speed f 0 (u). If this speed is not constant, the shape of the graph will change in time. In particular, there will be an instant T at which one of the tangent lines becomes vertical. For t > T, the solution u(; t) contains a shock. The position y(t) of this shock can be determined by imposing that the total area of the region below the graph of u remains constant in time. In order to give a meaning to the conservation law (6) when u D u(x; t) is discontinuous, one can multiply both sides of the equation by a test function ' and integrate by parts. Assuming that ' is continuously differentiable and vanishes outside a bounded set, one formally obtains “ fu' t C f (u)'x gdxdt D 0 : (7)

u D (u1 ; : : : ; u n ) is called a weak solution to the system of conservation laws (8) if the integral identity (7) holds true, for every continuously differentiable test function ' vanishing outside a bounded set. Consider the n  n Jacobian matrix of partial derivatives f at the point u: 1 0 @ f1 /@u1    @ f1 /@u n : A:  A(u) D D f (u) D @ @ f n /@u1    @ f n /@u n Using the chain rule, the system (8) can be written in the quasilinear form u t C A(u)u x D 0 :

(9)

We say that the above system is strictly hyperbolic if every matrix A(u) has n real, distinct eigenvalues, say 1 (u) <    < n (u). In this case, one can find a basis of right eigenvectors of A(u), denoted by r1 (u); : : : ; r n (u), such that, for i D 1; : : : ; n, A(u)r i (u) D  i (u)r i (u) ;

jr i (u)j D 1 :

Example 4 Assume that the flux function is linear: f (u) D Au for some constant matrix A 2 Rnn . If 1 < 2 <    < n are the eigenvalues of A and r1 ; : : : ; r n are the corresponding eigenvectors, then any vector function of the form u(x; t) D

n X

g i (x   i t)r i

iD1

Since the left hand side of the above equation does not involve partial derivatives of u, it remains meaningful for a discontinuous function u. A locally integrable function u is defined to be a weak solution of the conservation law (6) if the integral identity (7) holds true for every test function ', continuously differentiable and vanishing outside a bounded set. Hyperbolic Systems in One Space Dimension A system of n conservation laws in one space dimension can be written as 8@ @ ˆ < @t u1 C @x [ f1 (u1 ; : : : ; u n )] D 0 ; (8)  ˆ :@ @ u C [ f (u ; : : : ; u )] D 0 : n 1 n @t n @x For convenience, this can still be written in the form (5), but keeping in mind that now u D (u1 ; : : : ; u n ) 2 Rn is a vector and that f D ( f1 ; : : : ; f n ) is a vector-valued function. As in the case of a single equation, a vector function

provides a weak solution to the system (8). Here it is enough to assume that the functions g i are locally integrable, not necessarily continuous. Example 3 (continued) For the system (4) describing isentropic gas dynamics, the Jacobian matrix of partial derivatives of the flux is     0 1 @u/@v @u/@u D : Df D p0 (v) 0 @p/@v @p/@u Assuming that p0 (v) < p 0, this matrix has the two real distinct eigenvalues ˙ p0 (v). Therefore, the system is strictly hyperbolic. Entropy Admissibility Conditions Given two states u ; uC 2 Rn and a speed , consider the piecewise constant function defined as ( u if x < t ; u(x; t) D uC if x > t :

Hyperbolic Conservation Laws

Then one can show that the discontinuous function u is a weak solution of the hyperbolic system (8) if and only if it satisfies the Rankine–Hugoniot equations (uC  u ) D f (uC )  f (u ) :

(10)

More generally, consider a function u D u(x; t) which is piecewise smooth in the x–t-plane. Assume that these discontinuities are located along finitely many curves x D

˛ (t), ˛ D 1; : : : ; N, and consider the left and right limits u˛ (t) D u˛C (t) D

lim

u(x; t) ;

lim

u(x; t) :

x!˛ (t) x!˛ (t)C

Then u is a weak solution of the system of conservation laws if and only if it satisfies the quasilinear system (9) together with the Rankine–Hugoniot equations      

˛0 (t) u˛C (t)  u˛ (t) D f u˛C (t)  f u˛ (t) along each shock curve. Here ˛0 D d ˛ /dt. ¯ Given an initial condition u(x; 0) D u(x) containing jumps, however, it is well known that the Rankine– Hugoniot conditions do not determine a unique weak solution. Several “admissibility conditions” have thus been proposed in the literature, in order to single out a unique physically relevant solution. A basic criterion relies on the concept of entropy: A continuously differentiable function  : Rn 7! R is called an entropy for the system of conservation laws (8), with entropy flux q : Rn 7! R if D(u)  D f (u) D Dq(u) : If u D (u1 ; : : : ; u n ) is a smooth solution of (8), not only the quantities u1 ; : : : ; u n are conserved, but the additional conservation law (u) t C q(u)x D 0 holds as well. Indeed, D(u)u t C Dq(u)u x D D(u)[D f (u)u x ] C Dq(u)u x D 0 :

On the other hand, if the solution u is not smooth but contains shocks, the quantity  D (u) may no longer be conserved. The admissibility of a shock can now be characterized by requiring that certain entropies be increasing (or decreasing) in time. More precisely, a weak solution u of (8) is said to be entropy-admissible if the inequality (u) t C q(u)x  0 holds in the sense of distributions, for every pair (; q), where  is a convex entropy and q is the corresponding flux. Calling u ; uC the states to the left and right of the shock, and  its speed, the above condition implies [(uC )  (u )]  q(uC )  q(u ) : Various alternative conditions have been studied in the literature, in order to characterize the physically admissible shocks. For these we refer to Lax [39], or Liu [46]. The Riemann Problem Toward the construction of general solutions for the system of conservation laws (8), a basic building block is the so-called Riemann problem [55]. This amounts to choosing a piecewise constant initial data with a single jump at the origin: ( u if x < 0 ; u(x; 0) D (11) uC if x > 0 : In the special case where the system is linear, i. e. f (u) D Au, the solution is piecewise constant in the x– t-plane. It contains n C 1 constant states u D !0 ; !1 ; : : : ; !n D uC (see Fig. 4, left). Each jump ! i  ! i1 is an eigenvector of the matrix A, and is located along the line x D  i t, whose speed equals the corresponding eigenvalue  i . For nonlinear hyperbolic systems of n conservation laws, assuming that the amplitude juC u j of the jump is

Hyperbolic Conservation Laws, Figure 4 Solutions of a Riemann problem. Left: the linear case. Right: a nonlinear example

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Hyperbolic Conservation Laws

sufficiently small, the general solution was constructed in a classical paper of Lax [38], under the additional hypothesis (H) For each i D 1; : : : ; n, the ith field is either genuinely nonlinear, so that D i (u)  r i (u) > 0 for all u, or linearly degenerate, with D i (u)  r i (u) D 0 for all u. The solution is self-similar: u(x; t) D U(x/t). It still consists of n C 1 constant states !0 D u , !1 ; : : : ; !n D uC (see Fig. 4, right). Each couple of adjacent states ! i1 , ! i is separated either by a shock satisfying the Rankine Hugoniot equations, or else by a centered rarefaction. In this second case, the solution u varies continuously between ! i1 and ! i in a sector of the t–x-plane where the gradient ux coincides with an i-eigenvector of the matrix A(u). Further extensions, removing the technical assumption (H), were obtained by T. P. Liu [46] and by S. Bianchini [3]. Global Solutions Approximate solutions to a more general Cauchy problem can be constructed by patching together several solutions of Riemann problems. In the Glimm scheme [24], one works with a fixed grid in the x–t plane, with mesh sizes x, t. At time t D 0 the initial data is approximated by a piecewise constant function, with jumps at grid points (see Fig. 5, left). Solving the corresponding Riemann problems, a solution is constructed up to a time t sufficiently small so that waves generated by different Riemann problems do not interact. By a random sampling procedure, the solution u(t; ) is then approximated by a piecewise constant function having jumps only at grid points. Solving the new Riemann problems at every one of these points, one can prolong the solution to the next time interval [t; 2t], etc. . . An alternative technique for constructing approximate solutions is by wave-front tracking (Fig. 5, right). This method was introduced by Dafermos [17] in the scalar case and later developed by various authors [7,19,29]. It now

Hyperbolic Conservation Laws, Figure 5 Left: the Glimm scheme. Right: a front tracking approximation

provides an efficient tool in the study of general n  n systems of conservation laws, both for theoretical and numerical purposes. The initial data is here approximated with a piecewise constant function, and each Riemann problem is solved approximately, within the class of piecewise constant functions. In particular, if the exact solution contains a centered rarefaction, this must be approximated by a rarefaction fan, containing several small jumps. At the first time t1 where two fronts interact, the new Riemann problem is again approximately solved by a piecewise constant function. The solution is then prolonged up to the second interaction time t2 , where the new Riemann problem is solved, etc. . . The main difference is that with the Glimm scheme one specifies a priori the nodal points where the the Riemann problems are to be solved. On the other hand, in a solution constructed by wave-front tracking the locations of the jumps and of the interaction points depend on the solution itself. Moreover, no restarting procedure is needed. In the end, both algorithms produce a sequence of approximate solutions, whose total variation remains uniformly bounded. We recall here that the total variation of a function u : R 7! Rn is defined as N

X : Tot.Var. fug D sup ju(x i )  u(x i1 )j ; iD1

where the supremum is taken over all N  1 and all N-tuples of points x0 < x1 <    < x N . For functions of several variables, a more general definition can be found in [22]. Relying on a compactness argument, one can then show that these approximations converge to a weak solution to the system of conservation laws. Namely, one has: Theorem 1 Let the system of conservation laws (8) be strictly hyperbolic. Then, for every initial data u¯ with sufficiently small total variation, the initial value problem u t C f (u)x D 0

¯ u(x; 0) D u(x)

(12)

Hyperbolic Conservation Laws

has a unique entropy-admissible weak solution, defined for all times t  0. The existence part was first proved in the famous paper of Glimm [24], under the additional hypothesis (H), later removed by Liu [46]. The uniqueness of the solution was proved more recently, in a series of papers by the present author and collaborators, assuming that all shocks satisfy suitable admissibility conditions [8,10]. All proofs are based on careful analysis of solutions of the Riemann problem and on the use of a quadratic interaction functional to control the formation of new waves. These techniques also provided the basis for further investigations of Glimm and Lax [25] and Liu [45] on the asymptotic behavior of weak solutions as t ! 1. It is also interesting to compare solutions with different initial data. In this direction, we observe that a function of two variables u(x, t) can be regarded as a map t 7! u(; t) from a time interval [0; T] into a space L1 (R) of integrable functions. Always assuming that the total variation remains small, the distance between two solutions u; v at any time t > 0 can be estimated as ku(t)  v(t)kL1  L ku(0)  v(0)kL1 ; where L is a constant independent of time. Estimates on the rate of convergence of Glimm approximations to the unique exact solutions are available. For every fixed time T  0, letting the grid size x; x tend to zero keeping the ratio t/x constant, one has the error estimate [11]   Glimm u (T; )  uexact (T; )L1 D0: p lim x!0 x  j ln xj An alternative approximation procedure involves the addition of a small viscosity. For " > 0 small, one considers the viscous initial value problem u"t C f (u" )x D " u"x x ;

¯ u" (x; 0) D u(x) :

(13)

For initial data u¯ with small total variation, the analysis in [6] has shown that the solutions u" have small total variation for all times t > 0, and converge to the unique weak solution of (12) as " ! 0. Hyperbolic Systems in Several Space Dimensions In several space dimensions there is still no comprehensive theory for systems of conservation laws. Much of the literature has been concerned with three main topics: (i) Global solutions to a single conservation law. (ii) Smooth solutions to a hyperbolic system, locally in time. (iii) Particular solutions to initial or initial-boundary value problems.

Scalar Conservation Laws The single conservation law on Rm u t C div f(u) D 0 has been extensively studied. The fundamental works of Volpert [59] and Kruzhkov [36] have established the global existence of a unique, entropy-admissible solution to the initial value problem, for any initial data u(x; 0) D ¯ u(x) measurable and globally bounded. This solution can be obtained as the unique limit of vanishing viscosity approximations, solving u"t C div f(u" ) D "u" ; u" (x; 0) D u¯ : As in the one-dimensional case, solutions which are initially smooth may develop shocks and become discontinuous in finite time. Given any two solutions u; v, the following key properties remain valid also in the presence of shocks: (i) If at the initial time t D 0 one has u(x; 0)  v(x; 0) for all x 2 Rm , then u(x; t)  v(x; t) for all x and all t  0. (ii) The L1 distance between any two solutions does not increase in time. Namely, for any 0  s  t one has ku(t)  v(t)kL1 (Rm )  ku(s)  v(s)kL1 (Rm ) : Alternative approaches to the analysis of scalar conservation laws were developed by Crandall [16] using nonlinear semigroup theory, and by Lions, Perthame and Tadmor [44] using a kinetic formulation. Regularity results can be found in [30]. Smooth Solutions to Hyperbolic Systems Using the chain rule, one can rewrite the system of conservation laws (2) in the quasi-linear form ut C

m X

A˛ (u)u x ˛ D 0 :

(14)

˛D1

Various definitions of hyperbolicity can be found in the literature. Motivated by several examples from mathematical physics, the system (14) is said to be symmetrizable hyperbolic if there exists a positive definite symmetric matrix S D S(u) such that all matrices S ˛ (u) D SA˛ are symmetric. In particular, this condition implies that each n  n matrix A˛ (u) has real eigenvalues and admits a basis of linearly independent eigenvectors. As shown in [23], if a system of conservation laws admits a strictly convex entropy (u), such that the Hessian matrix of second deriva-

735

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Hyperbolic Conservation Laws

tives D2u (u) is positive definite at every point u, then the system is symmetrizable. A classical theorem states that, for a symmetrizable hyperbolic system with smooth initial data, the initial value problem has a unique smooth solution, locally in time. This solution can be prolonged in time up to the first time where the spatial gradient becomes unbounded at one or more points. In this general setting, however, it is not known whether the solution can be extended beyond this time of shock formation. Special Solutions In two space dimensions, one can study special solutions which are independent of time, so that u(x1 ; x2 ; t) D U(x1 ; x2 ). In certain cases, one can regard one of the variables, say x1 as a new time and derive a one-dimensional hyperbolic system of equations for U involving the remaining one-dimensional space variable x2 . Another important class of solutions relates to two-dimensional Riemann problems. Here the initial data, assigned on the x1 -x2 plane, is assumed to be constant along rays through the origin. Taking advantage of this self-similarity, the solution can be written in the form u(x1 ; x2 ; t) D U(x1 /t; x2 /t). This again reduces the problem to an equation in two independent variables [35]. Even for the equation of gas dynamics, a complete solution to the Riemann problem is not available. Several particular cases are analyzed in [42,61]. Several other examples, in specific geometries have been analyzed. A famous problem is the reflection of a shock hitting a wedge-shaped rigid obstacle [15,51]. Numerical Methods Generally speaking, there are three major classes of numerical methods suitable for partial differential equations: finite difference methods (FDM), finite volume methods (FVM) and finite element methods (FEM). For conservation laws, one also has semi-discrete methods, such as the method of lines, and conservative front tracking methods. The presence of shocks and the rich structure of shock interactions cause the main difficulties in numerical computations. To illustrate the main idea, consider a uniform grid in the x–t-plane, with step sizes x and t. Consider the times t n D nt and let I i D [x i1/2 ; x iC1/2 ] be a cell. We wish to compute an approximate value for the cell averages u¯ i over I i . Integrating the conservation law over the rectangle I i  [t n ; t nC1 ] and dividing by x, one obtains D u¯ ni C u¯ nC1 i

t [F i1/2  F iC1/2] ; x

where F iC1/2 is the average flux Z t nC1 1 f (u(x iC1/2 ))dt : F iC1/2 D t t n FVM methods seek a suitable approximation to this average flux FiC1/2 . First order methods, based on piecewise constant approximations, are usually stable, but contain large numerical diffusion which smears out the shock profile. High order methods are achieved by using polynomials of higher degree, but this produces numerical oscillations around the shock. The basic problem is how to accurately capture the approximated solution near shocks, and at the same time retain stability of the numerical scheme. A common technique is to use a high order scheme on regions where the solution is smooth, and switch to a lower order method near a discontinuity. Well-known methods of this type include the Godunov methods and the MUSCL schemes, wave propagation methods [40,41], the central difference schemes [52,58] and the ENO/WENO schemes [56]. The conservative front tracking methods combine the FDM/FVM with the standard front tracking [26,27]. Based on a high order FDM/FVM, the methods in addition track the location and the strength of the discontinuities, and treat them as moving boundaries. The complexity increases with the number of fronts. In the FEM setting, the discontinuous Galerkin’s methods are widely used [13,14]. The method uses finite element discretization in space, with piecewise polynomials approximation, but allows the approximation to be discontinuous at cell boundaries. Some numerical methods can be directly extended to the multi-dimensional case, but others need to use a dimensional splitting technique, which introduces additional diffusion. The performance of numerical algorithms is usually tested with some benchmark problem, and little is known theoretically, apart from the case of a scalar conservation law. Moreover, it remains a challenging problem to construct efficient high order numerical methods for systems of conservation laws, both in one and in several space dimensions. Future Directions In spite of extensive research efforts, the mathematical theory of hyperbolic conservation laws is still largely incomplete. For hyperbolic systems in one space dimension, a major challenge is to study the existence and uniqueness of solutions to problems with large initial data. In this direction, some counterexamples show that, for particular sys-

Hyperbolic Conservation Laws

tems, solutions can become unbounded in finite time [31]. However, it is conjectured that for many physical systems, endowed with a strictly convex entropy, such pathological behavior should not occur. In particular, the so-called “p-system” describing isentropic gas dynamics (4) should have global solutions with bounded variation, for arbitrarily large initial data [60]. It is worth mentioning that, for large initial data, the global existence of solutions is known mainly in the scalar case [36,59]. For hyperbolic systems of two conservation laws, global existence can still be proved, relying on a compensated compactness argument [20]. This approach, however, does not provide information on the uniqueness of solutions, or on their continuous dependence on the initial data. Another major open problem is to theoretically analyze the convergence of numerical approximations. Error bounds on discrete approximations are presently available only in the scalar case [37]. For solutions to hyperbolic systems of n conservation laws, proofs of the convergence of viscous approximations [6], semidiscrete schemes [4], or relaxation schemes [5] have always relied on a priori bounds on the total variation. On the other hand, the counterexample in [2] shows that in general one cannot have any a priori bounds on the total variation of approximate solutions constructed by fully discrete numerical schemes. Understanding the convergence of these discrete approximations will likely require a new approach. At present, the most outstanding theoretical open problem is to develop a fundamental existence and uniqueness theory for hyperbolic systems in several space dimensions. In order to achieve an existence proof, a key step is to identify the appropriate functional space where to construct solutions. In the one-dimensional case, solutions are found in the space BV of functions with bounded variation. In several space dimensions, however, it is known that the total variation of an arbitrary small solution can become unbounded almost immediately [54]. Hence the space BV does not provide a suitable framework to study the problem. For a special class of systems, a positive result and a counterexample, concerning global existence and continuous dependence on initial data can be found in [1] and in [9], respectively. Bibliography Primary Literature 1. Ambrosio L, Bouchut F, De Lellis C (2004) Well-posedness for a class of hyperbolic systems of conservation laws in several space dimensions. Comm Part Diff Equat 29:1635–1651

2. Baiti P, Bressan A, Jenssen HK (2006) An instability of the Godunov scheme. Comm Pure Appl Math 59:1604–1638 3. Bianchini S (2003) On the Riemann problem for non-conservative hyperbolic systems. Arch Rat Mach Anal 166:1–26 4. Bianchini S (2003) BV solutions of the semidiscrete upwind scheme. Arch Ration Mech Anal 167:1–81 5. Bianchini S (2006) Hyperbolic limit of the Jin-Xin relaxation model. Comm Pure Appl Math 59:688–753 6. Bianchini S, Bressan A (2005) Vanishing viscosity solutions to nonlinear hyperbolic systems. Ann Math 161:223–342 7. Bressan A (1992) Global solutions to systems of conservation laws by wave-front tracking. J Math Anal Appl 170:414–432 8. Bressan A (2000) Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem. Oxford University Press, Oxford 9. Bressan A (2003) An ill posed Cauchy problem for a hyperbolic system in two space dimensions. Rend Sem Mat Univ Padova 110:103–117 10. Bressan A, Liu TP, Yang T (1999) L1 stability estimates for n  n conservation laws. Arch Ration Mech Anal 149:1–22 11. Bressan A, Marson A (1998) Error bounds for a deterministic version of the Glimm scheme. Arch Rat Mech Anal 142: 155–176 12. Chen GQ, Zhang Y, Zhu D (2006) Existence and stability of supersonic Euler flows past Lipschitz wedges. Arch Ration Mech Anal 181:261–310 13. Cockburn B, Shu CW (1998) The local discontinuous Galerkin finite element method for convection diffusion systems. SIAM J Numer Anal 35:2440–2463 14. Cockburn B, Hou S, Shu C-W (1990) The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. Math Comput 54:545–581 15. Courant R, Friedrichs KO (1948) Supersonic Flow and Shock Waves. Wiley Interscience, New York 16. Crandall MG (1972) The semigroup approach to first-order quasilinear equations in several space variables. Israel J Math 12:108–132 17. Dafermos C (1972) Polygonal approximations of solutions of the initial value problem for a conservation law. J Math Anal Appl 38:33–41 18. Dafermos C (2005) Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer, Berlin, pp 325–626 19. DiPerna RJ (1976) Global existence of solutions to nonlinear hyperbolic systems of conservation laws. J Differ Equ 20: 187–212 20. DiPerna R (1983) Convergence of approximate solutions to conservation laws. Arch Ration Mech Anal 82:27–70 21. Euler L (1755) Principes généraux du mouvement des fluides. Mém Acad Sci Berlin 11:274–315 22. Evans LC, Gariepy RF (1992) Measure Theory and Fine Properties of Functions. C.R.C. Press, Boca Raton, pp viii–268 23. Friedrichs KO, Lax P (1971) Systems of conservation laws with a convex extension. Proc Nat Acad Sci USA 68:1686–1688 24. Glimm J (1965) Solutions in the large for nonlinear hyperbolic systems of equations. Comm Pure Appl Math 18:697–715 25. Glimm J, Lax P (1970) Decay of solutions of systems of nonlinear hyperbolic conservation laws. Am Math Soc Memoir 101:xvii–112

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26. Glimm J, Li X, Liu Y (2001) Conservative Front Tracking in One Space Dimension. In: Fluid flow and transport in porous media: mathematical and numerical treatment, (South Hadley, 2001), pp 253–264. Contemp Math 295, Amer Math Soc, Providence 27. Glimm J, Grove JW, Li XL, Shyue KM, Zeng Y, Zhang Q (1998) Three dimensional front tracking. SIAM J Sci Comput 19: 703–727 28. Goodman J, Xin Z (1992) Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch Ration Mech Anal 121:235–265 29. Holden H, Risebro NH (2002) Front tracking for hyperbolic conservation laws. Springer, New York 30. Jabin P, Perthame B (2002) Regularity in kinetic formulations via averaging lemmas. ESAIM Control Optim Calc Var 8: 761–774 31. Jenssen HK (2000) Blowup for systems of conservation laws. SIAM J Math Anal 31:894–908 32. Jiang G-S, Levy D, Lin C-T, Osher S, Tadmor E (1998) High-resolution non-oscillatory central schemes with non-staggered grids for hyperbolic conservation laws. SIAM J Numer Anal 35:2147–2168 33. Jin S, Xin ZP (1995) The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm Pure Appl Math 48:235–276 34. Kawashima S, Matsumura A (1994) Stability of shock profiles in viscoelasticity with non-convex constitutive relations. Comm Pure Appl Math 47:1547–1569 35. Keyfitz B (2004) Self-similar solutions of two-dimensional conservation laws. J Hyperbolic Differ Equ 1:445–492 36. Kruzhkov S (1970) First-order quasilinear equations with several space variables. Math USSR Sb 10:217–273 37. Kuznetsov NN (1976) Accuracy of some approximate methods for computing the weak solution of a first order quasilinear equation. USSR Comp Math Math Phys 16:105–119 38. Lax P (1957) Hyperbolic systems of conservation laws II. Comm Pure Appl Math 10:537–566 39. Lax P (1971) Shock waves and entropy. In: Zarantonello E (ed) Contributions to Nonlinear Functional Analysis. Academic Press, New York, pp 603–634 40. Leveque RJ (1990) Numerical methods for conservation laws. Lectures in Mathematics. Birkhäuser, Basel 41. Leveque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge 42. Li J, Zhang T, Yang S (1998) The two dimensional problem in gas dynamics. Pitman, Longman Essex 43. Lighthill MJ, Whitham GB (1955) On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc Roy Soc Lond A 229:317–345 44. Lions PL, Perthame E, Tadmor E (1994) A kinetic formulation of multidimensional scalar conservation laws and related equations. JAMS 7:169–191 45. Liu TP (1977) Linear and nonlinear large-time behavior of solutions of general systems of hyperbolic conservation laws. Comm Pure Appl Math 30:767–796 46. Liu TP (1981) Admissible solutions of hyperbolic conservation laws. Mem Am Math Soc 30(240):iv–78 47. Liu TP (1985) Nonlinear stability of shock waves for viscous conservation laws. Mem Am Math Soc 56(328):v–108 48. Liu TP (1987) Hyperbolic conservation laws with relaxation. Comm Math Pys 108:153–175

49. Majda A (1984) Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York 50. Metivier G (2001) Stability of multidimensional shocks. Advances in the theory of shock waves. Birkhäuser, Boston, pp 25–103 51. Morawetz CS (1994) Potential theory for regular and Mach reflection of a shock at a wedge. Comm Pure Appl Math 47: 593–624 52. Nessyahu H, Tadmor E (1990) Non-oscillatory central differencing for hyperbolic conservation laws J Comp Phys 87:408–463 53. Perthame B (2002) Kinetic Formulation of Conservation Laws. Oxford Univ. Press, Oxford 54. Rauch J (1986) BV estimates fail for most quasilinear hyperbolic systems in dimensions greater than one. Comm Math Phys 106:481–484 55. Riemann B (1860) Über die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite. Gött Abh Math Cl 8:43–65 56. Shu CW (1998) Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Advanced numerical approximation of nonlinear hyperbolic equations. (Cetraro, 1997), pp 325–432. Lecture Notes in Mathematics, vol 1697. Springer, Berlin 57. Smoller J (1994) Shock Waves and Reaction-Diffusion Equations, 2nd edn. Springer, New York 58. Tadmor E (1998) Approximate solutions of nonlinear conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol 1697. (1997 C.I.M.E. course in Cetraro, Italy) Springer, Berlin, pp 1–149 59. Volpert AI (1967) The spaces BV and quasilinear equations. Math USSR Sb 2:225–267 60. Young R (2003) Isentropic gas dynamics with large data. In: Hyperbolic problems: theory, numerics, applications. Springer, Berlin, pp 929–939 61. Zheng Y (2001) Systems of Conservation Laws. Two-dimensional Riemann problems. Birkhäuser, Boston 62. Zumbrun K (2004) Stability of large-amplitude shock waves of compressible Navier–Stokes equations. With an appendix by Helge Kristian Jenssen and Gregory Lyng. Handbook of Mathematical Fluid Dynamics, vol III. North-Holland, Amsterdam, pp 311–533

Books and Reviews Benzoni-Gavage S, Serre D (2007) Multidimensional hyperbolic partial differential equations. First-order systems and applications. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford Boillat G (1996) Nonlinear hyperbolic fields and waves. In: Recent Mathematical Methods in Nonlinear Wave Propagation (Montecatini Terme, 1994), pp 1–47. Lecture Notes in Math, vol 1640. Springer, Berlin Chen GQ, Wang D (2002) The Cauchy problem for the Euler equations for compressible fluids. In: Handbook of Mathematical Fluid Dynamics, vol I. North-Holland, Amsterdam, pp 421–543 Courant R, Hilbert D (1962) Methods of Mathematical Physics, vol II. John Wiley & Sons - Interscience, New York Garavello M, Piccoli B (2006) Traffic Flow on Networks. American Institute of Mathematical Sciences, Springfield Godlewski E, Raviart PA (1996) Numerical approximation of hyperbolic systems of conservation laws. Springer, New York

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Gurtin ME (1981) An Introduction to Continuum Mechanics. Mathematics in Science and Engineering, 158. Academic Press, New York, pp xi–265 Hörmander L (1997) Lectures on Nonlinear Hyperbolic Differential Equations. Springer, Berlin Jeffrey A (1976) Quasilinear Hyperbolic Systems and Waves. Research Notes in Mathmatics, vol 5. Pitman Publishing, London, pp vii–230 Kreiss HO, Lorenz J (1989) Initial-boundary value problems and the Navier–Stokes equations. Academic Press, San Diego Kröner D (1997) Numerical schemes for conservation laws. In: Wiley-Teubner Series Advances in Numerical Mathematics. John Wiley, Chichester Landau LD, Lifshitz EM (1959) Fluid Mechanics. translated form the Russian by Sykes JB, Reid WH, Course of Theoretical Physics, vol 6. Pergamon press, London, Addison-Wesley, Reading, pp xii–536 Li T-T, Yu W-C (1985) Boundary value problems for quasilinear hy-

perbolic systems. Duke University Math. Series, vol 5. Durham, pp vii–325 Li T-T (1994) Global classical solutions for quasilinear hyperbolic systems. Wiley, Chichester Lu Y (2003) Hyperbolic conservation laws and the compensated compactness method. Chapman & Hall/CRC, Boca Raton Morawetz CS (1981) Lecture Notes on Nonlinear Waves and Shocks. Tata Institute of Fundamental Research, Bombay Rozhdestvenski BL, Yanenko NN (1978) Systems of quasilinear equations and their applications to gas dynamics. Nauka, Moscow. English translation: American Mathematical Society, Providence Serre D (2000) Systems of Conservation Laws, I Geometric structures, oscillations, and initial-boundry value problem, II Hyperbolicity, entropies, shock waves. Cambridge University Press, pp xii–263 Whitham GB (1999) Linear and Nonlinear Waves. Wiley-Interscience, New York

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Hyperbolic Dynamical Systems VITOR ARAÚJO1,2 , MARCELO VIANA 3 1 CMUP, Porto, Portugal 2 IM-UFRJ, Rio de Janeiro, Brazil 3 IMPA, Rio de Janeiro, Brazil Article Outline Glossary Definition Introduction Linear Systems Local Theory Hyperbolic Behavior: Examples Hyperbolic Sets Uniformly Hyperbolic Systems Attractors and Physical Measures Obstructions to Hyperbolicity Partial Hyperbolicity Non-Uniform Hyperbolicity – Linear Theory Non-Uniformly Hyperbolic Systems Future Directions Bibliography Glossary Homeomorphism, diffeomorphism A homeomorphism is a continuous map f : M ! N which is one-toone and onto, and whose inverse f 1 : N ! M is also continuous. It may be seen as a global continuous change of coordinates. We call f a diffeomorphism if, in addition, both it and its inverse are smooth. When M D N, the iterated n-fold composition f ı : n: : ı f is denoted by f n . By convention, f 0 is the identity map, and f n D ( f n )1 D ( f 1 )n for n  0. Smooth flow A flow f t : M ! M is a family of diffeomorphisms depending in a smooth fashion on a parameter t 2 R and satisfying f sCt D f s ı f t for all s, t 2 R. This property implies that f 0 is the identity map. Flows usually arise as solutions of autonomous differential equations: let t 7!  t (v) denote the solution of X˙ D F(X) ; X(0) D v ;

(1)

and assume solutions are defined for all times; then the family  t thus defined is a flow (at least as smooth as the vector field F itself). The vector field may be recovered from the flow, through the relation F(X) D @ t  t (X) j tD0 .

Ck topology Two maps admitting continuous derivatives are said to be C1 -close if they are uniformly close, and so are their derivatives. More generally, given any k  1, we say that two maps are Ck -close if they admit continuous derivatives up to order k, and their derivatives of order i are uniformly close, for every i D 0; 1; : : : ; k. This defines a topology in the space of maps of class Ck . Foliation A foliation is a partition of a subset of the ambient space into smooth submanifolds, that one calls leaves of the foliation, all with the same dimension and varying continuously from one point to the other. For instance, the trajectories of a vector field F, that is, the solutions of Eq. (1), form a 1-dimensional foliation (the leaves are curves) of the complement of the set of zeros of F. The main examples of foliations in the context of this work are the families of stable and unstable manifolds of hyperbolic sets. Attractor A subset  of the ambient space M is invariant under a transformation f if f 1 () D , that is, a point is in  if and only if its image is.  is invariant under a flow if it is invariant under f t for all t 2 R. An attractor is a compact invariant subset  such that the trajectories of all points in a neighborhood U converge to  as times goes to infinity, and  is dynamically indecomposable (or transitive): there is some trajectory dense in . Sometimes one asks convergence only for points in some “large” subset of a neighborhood U of , and dynamical indecomposability can also be defined in somewhat different ways. However, the formulations we just gave are fine in the uniformly hyperbolic context. Limit sets The !-limit set of a trajectory f n (x); n 2 Z is the set !(x) of all accumulation points of the trajectory as time n goes to C1. The ˛-limit set is defined analogously, with n ! 1. The corresponding notions for continuous time systems (flows) are defined analogously. The limit set L(f ) (or L( f t ), in the flow case) is the closure of the union of all 0 !-limit and all ˛-limit sets. The non-wandering set ˝( f ) (or ˝( f t ), in the flow case) is that set of points such that every neighborhood U contains some point whose orbit returns to U in future time (then some point returns to U in past time as well). When the ambient space is compact all these sets are non-empty. Moreover, the limit set is contained in the non-wandering set. Invariant measure A probability measure  in the ambient space M is invariant under a transformation f if ( f 1 (A)) D (A) for all measurable subsets A. This means that the “events” x 2 A and f (x) 2 A have equally probable. We say  is invariant under a flow if it is invariant under f t for all t. An invariant proba-

Hyperbolic Dynamical Systems

bility measure  is ergodic if every invariant set A has either zero or full measure. An equivalently condition is that  can not be decomposed as a convex combination of invariant probability measures, that is, one can not have  D a1 C (1  a)2 with 0 < a < 1 and 1 ; 2 invariant. Definition In general terms, a smooth dynamical system is called hyperbolic if the tangent space over the asymptotic part of the phase space splits into two complementary directions, one which is contracted and the other which is expanded under the action of the system. In the classical, socalled uniformly hyperbolic case, the asymptotic part of the phase space is embodied by the limit set and, most crucially, one requires the expansion and contraction rates to be uniform. Uniformly hyperbolic systems are now fairly well understood. They may exhibit very complex behavior which, nevertheless, admits a very precise description. Moreover, uniform hyperbolicity is the main ingredient for characterizing structural stability of a dynamical system. Over the years the notion of hyperbolicity was broadened (non-uniform hyperbolicity) and relaxed (partial hyperbolicity, dominated splitting) to encompass a much larger class of systems, and has become a paradigm for complex dynamical evolution. Introduction The theory of uniformly hyperbolic dynamical systems was initiated in the 1960s (though its roots stretch far back into the 19th century) by S. Smale, his students and collaborators, in the west, and D. Anosov, Ya. Sinai, V. Arnold, in the former Soviet Union. It came to encompass a detailed description of a large class of systems, often with very complex evolution. Moreover, it provided a very precise characterization of structurally stable dynamics, which was one of its original main goals. The early developments were motivated by the problem of characterizing structural stability of dynamical systems, a notion that had been introduced in the 1930s by A. Andronov and L. Pontryagin. Inspired by the pioneering work of M. Peixoto on circle maps and surface flows, Smale introduced a class of gradient-like systems, having a finite number of periodic orbits, which should be structurally stable and, moreover, should constitute the majority (an open and dense subset) of all dynamical systems. Stability and openness were eventually established, in the thesis of J. Palis. However, contemporary results of M. Levinson, based on previous work by M. Cartwright and J. Littlewood, provided examples of open subsets of dy-

namical systems all of which have an infinite number of periodic orbits. In order to try and understand such phenomenon, Smale introduced a simple geometric model, the now famous “horseshoe map”, for which infinitely many periodic orbits exist in a robust way. Another important example of structurally stable system which is not gradient like was R. Thom’s so-called “cat map”. The crucial common feature of these models is hyperbolicity: the tangent space at each point splits into two complementary directions such that the derivative contracts one of these directions and expands the other, at uniform rates. In global terms, a dynamical system is called uniformly hyperbolic, or Axiom A, if its limit set has this hyperbolicity property we have just described. The mathematical theory of such systems, which is the main topic of this paper, is now well developed and constitutes a main paradigm for the behavior of “chaotic” systems. In our presentation we go from local aspects (linear systems, local behavior, specific examples) to the global theory (hyperbolic sets, stability, ergodic theory). In the final sections we discuss several important extensions (strange attractors, partial hyperbolicity, non-uniform hyperbolicity) that have much broadened the scope of the theory. Linear Systems Let us start by introducing the phenomenon of hyperbolicity in the simplest possible setting, that of linear transformations and linear flows. Most of what we are going to say applies to both discrete time and continuous time systems in a fairly analogous way, and so at each point we refer to either one setting or the other. In depth presentations can be found in e. g. [6,8]. The general solution of a system of linear ordinary differential equations X˙ D AX ; X(0) D v ; where A is a constant n  n real matrix and v 2 Rn is fixed, is given by X(t) D e tA  v ; t 2R; P n where e tA D 1 nD0 (tA) /n!. The linear flow is called hyperbolic if A has no eigenvalues on the imaginary axis. Then the exponential matrix e A has no eigenvalues with norm 1. This property is very important for a number of reasons.

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Stable and Unstable Spaces

Hyperbolic Linear Systems

For one thing it implies that all solutions have well-defined asymptotic behavior: they either converge to zero or diverge to infinity as time t goes to ˙1. More precisely, let

There is a corresponding notion of hyperbolicity for discrete time linear systems

 Es (stable subspace) be the subspace of Rn spanned by the generalized eigenvector associated to eigenvalues of A with negative real part.  Eu (unstable subspace) be the subspace of Rn spanned by the generalized eigenvector associated to eigenvalues of A with positive real part Then these subspaces are complementary, meaning that Rn D E s ˚ E u , and every solution e tA  v with v 62 E s [ E u diverges to infinity both in the future and in the past. The solutions with v 2 E s converge to zero as t ! C1 and go to infinity as t ! 1, and analogously when v 2 E u , reversing the direction of time.

X nC1 D CX n ; X0 D v ; with C a n  n real matrix. Namely, we say the system is hyperbolic if C has no eigenvalue in the unit circle. Thus a matrix A is hyperbolic in the sense of continuous time systems if and only if its exponential C D eA is hyperbolic in the sense of discrete time systems. The previous observations (well-defined behavior, robustness, denseness and stability) remain true in discrete time. Two hyperbolic matrices are conjugate by a homeomorphism if and only if they have the same index, that is, the same number of eigenvalues with norm less than 1, and they both either preserve or reverse orientation.

Robustness and Density Another crucial feature of hyperbolicity is robustness: any matrix that is close to a hyperbolic one, in the sense that corresponding coefficients are close, is also hyperbolic. The stable and unstable subspaces need not coincide, of course, but the dimensions remain the same. In addition, hyperbolicity if dense: any matrix is close to a hyperbolic one. That is because, up to arbitrarily small modifications of the coefficients, one may force all eigenvalues to move out of the imaginary axis. Stability, Index of a Fixed Point In addition to robustness, hyperbolicity also implies stability: if B is close to a hyperbolic matrix A, in the sense we have just described, then the solutions of X˙ D BX have essentially the same behavior as the solutions of X˙ D AX. What we mean by “essentially the same behavior” is that there exists a global continuous change of coordinates, that is, a homeomorphism h : Rn ! Rn , that maps solutions of one system to solutions of the other, preserving the time parametrization:   h e tA  v D e tB  h(v) for all t 2 R :

Local Theory Now we move on to discuss the behavior of non-linear systems close to fixed or, more generally, periodic trajectories. By non-linear system we understand the iteration of a diffeomorphism f , or the evolution of a smooth flow f t , on some manifold M. The general philosophy is that the behavior of the system close to a hyperbolic fixed point very much resembles the dynamics of its linear part. A fixed point p 2 M of a diffeomorphism f : M ! M is called hyperbolic if the linear part D f p : Tp M ! Tp M is a hyperbolic linear map, that is, if Df p has no eigenvalue with norm 1. Similarly, an equilibrium point p of a smooth vector field F is hyperbolic if the derivative DF(p) has no pure imaginary eigenvalues. Hartman–Grobman Theorem This theorem asserts that if p is a hyperbolic fixed point of f : M ! M then there are neighborhoods U of p in M and V of 0 in the tangent space Tp M such that we can find a homeomorphism h : U ! V such that h ı f D D fp ı h ;

More generally, two hyperbolic linear flows are conjugated by a homeomorphism h if and only if they have the same index, that is, the same number of eigenvalues with negative real part. In general, h can not be taken to be a diffeomorphism: this is possible if and only if the two matrices A and B are obtained from one another via a change of basis. Notice that in this case they must have the same eigenvalues, with the same multiplicities.

whenever the composition is defined. This property means that h maps orbits of Df (p) close to zero to orbits of f close to p. We say that h is a (local) conjugacy between the nonlinear system f and its linear part Df p . There is a corresponding similar theorem for flows near a hyperbolic equilibrium. In either case, in general h can not be taken to be a diffeomorphism.

Hyperbolic Dynamical Systems

Stable Sets

Hyperbolic Behavior: Examples

The stable set of the hyperbolic fixed point p is defined by ˚ W s (p) D x 2 M : d( f n (x); f n (p)) ! 0 :

Now let us review some key examples of (semi)global hyperbolic dynamics. Thorough descriptions are available in e. g. [6,8,9].

Given ˇ > 0 we also consider the local stable set of size ˇ > 0, defined by

A Linear Torus Automorphism

n!C1

˚

Wˇs (p) D x 2 M : d( f n (x); f n (p))  ˇ

for all n  0 :

The image of Wˇs under the conjugacy h is a neighborhood of the origin inside Es . It follows that the local stable set is an embedded topological disk, with the same dimension as Es . Moreover, the orbits of the points in Wˇs (p) actually converges to the fixed point as time goes to infinity. Therefore, z 2 W s (p) , f n (z) 2 Wˇs (p) for some n  0 : Stable Manifold Theorem The stable manifold theorem asserts that Wˇs (p) is actually a smooth embedded disk, with the same order of differentiability as f itself, and it is tangent to Es at the point p. It follows that W s (p) is a smooth submanifold, injectively immersed in M. In general, W s (p) is not embedded in M: in many cases it has self-accumulation points. For these reasons one also refers to W s (p) and Wˇs (p) as stable manifolds of p. Unstable manifolds are defined analogously, replacing the transformation by its inverse. Local Stability We call index of a diffeomorphism f at a hyperbolic fixed point p the index of the linear part, that is, the number of eigenvalues of Df p with negative real part. By the Hartman–Grobman theorem and previous comments on linear systems, two diffeomorphisms are locally conjugate near hyperbolic fixed points if and only if the stable indices and they both preserve/reverse orientation. In other words, the index together with the sign of the Jacobian determinant form a complete set of invariants for local topological conjugacy. Let g be any diffeomorphism C1 -close to f . Then g has a unique fixed point pg close to p, and this fixed point is still hyperbolic. Moreover, the stable indices and the orientations of the two diffeomorphisms at the corresponding fixed points coincide, and so they are locally conjugate. This is called local stability near of diffeomorphisms hyperbolic fixed points. The same kind of result holds for flows near hyperbolic equilibria.

Consider the linear transformation A: R2 ! R2 given by the following matrix, relative to the canonical base of the plane:   2 1 : 1 1 The 2-dimensional torus T 2 is the quotient R2 /Z2 of the plane by the equivalence relation (X1 ; y1 ) (x2 ; y2 )

,

(x1  x2 ; y1  y2 ) 2 Z2 :

Since A preserves the lattice Z2 of integer vectors, that is, since A(Z2 ) D Z2 , the linear transformation defines an invertible map f A : T 2 ! T 2 in the quotient space, which is an example of linear automorphism of T 2 . We call affine line in T 2 the projection under the quotient map of any affine line in the plane. The linear transformation A is hyperbolic, with eigenvalues 0 < 1 < 1 < 2 , and the corresponding eigenspaces E1 and E2 have irrational slope. For each point z 2 T 2 , let W i (z) denote the affine line through z and having the direction of Ei , for i D 1; 2:  distances along W1 (z) are multiplied by 1 < 1 under forward iteration of f A  distances along W2 (z) are multiplied by 1/2 < 1 under backward iteration of f A . Thus we call W1 (z) stable manifold and W2 (z) unstable manifold of z (notice we are not assuming z to be periodic). Since the slopes are irrational, stable and unstable manifolds are dense in the whole torus. From this fact one can deduce that the periodic points of f A form a dense subset of the torus, and that there exist points whose trajectories are dense in T 2 . The latter property is called transitivity. An important feature of this systems is that its behavior is (globally) stable under small perturbations: given any diffeomorphism g : T 2 ! T 2 sufficiently C1 -close to f A , there exists a homeomorphism h : T 2 ! T 2 such that h ı g D f A ı h. In particular, g is also transitive and its periodic points form a dense subset of T 2 . The Smale Horseshoe Consider a stadium shaped region D in the plane divided into three subregions, as depicted in Fig. 1: two half disks,

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Hyperbolic Dynamical Systems, Figure 2 The solenoid attractor

Hyperbolic Dynamical Systems, Figure 1 Horseshoe map

A and C, and a square, B. Next, consider a map f : D ! D mapping D back inside itself as described in Fig. 1: the intersection between B and f (B) consists of two rectangles, R0 and R1 , and f is affine on the pre-image of these rectangles, contracting the horizontal direction and expanding the vertical direction. The set  D \n2Z f n (B), formed by all the points whose orbits never leave the square B is totally disconnected, in fact, it is the product of two Cantor sets. A description of the dynamics on  may be obtained through the following coding of orbits. For each point z 2  and every time n 2 Z the iterate f n (z) must belong to either R0 or R1 . We call itinerary of z the sequence fs n gn2Z with values in the set f0; 1g defined by f n (z) 2 Rs n for all n 2 Z. The itinerary map  ! f0; 1gZ ; z 7! fs n gn2Z is a homeomorphism, and conjugates f restricted to  to the so-called shift map defined on the space of sequences by f0; 1gZ ! f0; 1gZ ; fs n gn2Z 7! fs nC1 gn2Z : Since the shift map is transitive, and its periodic points form a dense subset of the domain, it follows that the same is true for the horseshoe map on . From the definition of f we get that distances along horizontal line segments through points of  are contracted at a uniform rate under forward iteration and, dually, distances along vertical line segments through points of  are contracted at a uniform rate under backward iteration. Thus, horizontal line segments are local stable sets and vertical line segments are local unstable sets for the points of .

A striking feature of this system is the stability of its dynamics: given any diffeomorphism g sufficiently C1 -close to f , its restriction to the set  g D \n2Z g n (B) is conjugate to the restriction of f to the set  D  f (and, consequently, is conjugate to the shift map). In addition, each point of  g has local stable and unstable sets which are smooth curve segments, respectively, approximately horizontal and approximately vertical. The Solenoid Attractor The solid torus is the product space S1  D, where S1 D R/Z is the circle and D D fz 2 C : jzj < 1g is the unit disk in the complex plane. Consider the map f : S1 D ! S1  D given by ( ; z) 7! (2 ; ˛z C ˇe i /2 ) ; 2 R/Z and ˛; ˇ 2 R with ˛ C ˇ < 1. The latter condition ensures that the image f (S1  D) is strictly contained in S1  D. Geometrically, the image is a long thin domain going around the solid torus twice, as described in Fig. 2. Then, for any n  1, the corresponding iterate f n (S1  D) is an increasingly thinner and longer domain that winds 2k times around S1  D. The maximal invariant set  D \n0 f n (S1  D) is called solenoid attractor. Notice that the forward orbit under f of every point in S1  D accumulates on . One can also check that the restriction of f to the attractor is transitive, and the set of periodic points of f is dense in . In addition  has a dense subset of periodic orbits and also a dense orbit. Moreover every point in a neighborhood of  converges to  and this is why this set is called an attractor. Hyperbolic Sets The notion we are now going to introduce distillates the crucial feature common to the examples presented pre-

Hyperbolic Dynamical Systems

viously. A detailed presentation is given in e. g. [8,10]. Let f : M ! M be a diffeomorphism on a manifold M. A compact invariant set  M is a hyperbolic set for f if the tangent bundle over  admits a decomposition T M D E u ˚ E s ; invariant under the derivative and such that kD f 1 j E u k <  and kD f j E s k <  for some constant  < 1 and some choice of a Riemannian metric on the manifold. When it exists, such a decomposition is necessarily unique and continuous. We call Es the stable bundle and Eu the unstable bundle of f on the set . The definition of hyperbolicity for an invariant set of a smooth flow containing no equilibria is similar, except that one asks for an invariant decomposition T M D E u ˚ E 0 ˚ E s , where Eu and Es are as before and E0 is a line bundle tangent to the flow lines. An invariant set that contains equilibria is hyperbolic if and only it consists of a finite number of points, all of them hyperbolic equilibria. Cone Fields The definition of hyperbolic set is difficult to use in concrete situations, because, in most cases, one does not know the stable and unstable bundles explicitly. Fortunately, to prove that an invariant set is hyperbolic it suffices to have some approximate knowledge of these invariant subbundles. That is the contents of the invariant cone field criterion: a compact invariant set is hyperbolic if and only if there exists some continuous (not necessarily invariant) decomposition T M D E 1 ˚ E 2 of the tangent bundle, some constant  < 1, and some cone field around E1

ously to some small neighborhood U of , and then so do the cone fields. By continuity, conditions (a) and (b) above remain valid on U, possibly for a slightly larger constant . Most important, they also remain valid when f is replaced by any other diffeomorphism g which is sufficiently C1 close to it. Thus, using the cone field criterion once more, every compact set K U which is invariant under g is a hyperbolic set for g. Stable Manifold Theorem Let  be a hyperbolic set for a diffeomorphism f : M ! M. Assume f is of class Ck . Then there exist "0 > 0 and 0 <  < 1 and, for each 0 < "  "0 and x 2 , the local stable manifold of size " W"s (x) D fy 2 M : dist( f n (y); f n (x))  " for all n  0g; and the local unstable manifold of size " W"u (x) D fy 2 M : dist( f n (y); f n (x))  " for all n  0g are Ck embedded disks, tangent at x to E sx and E ux , respectively, and satisfying  f (W"s (x)) W"s ( f (x)) and f 1 (W"u (x))

W"u ( f 1 (x));  dist( f (x); f (y))   dist(x; y) for all y 2 W"s (x)  dist( f 1 (x); f 1 (y))   dist(x; y) for all y 2 W"u (x)  W"s (x) and W"u (x) vary continuously with the point x, in the Ck topology. Then, the global stable and unstable manifolds of x, W s (x) D

C 1a (x)

D fv D v1 C v2 2

E 1x

˚

E 2x

: kv2 k  akv1 kg ; x 2 ;

which is

[

  f n W"s ( f n (x))

and

n0

W u (x) D

[

  f n W"u ( f n (x)) ;

n0

D f x (C 1a (x))

(a) forward invariant: (b) expanded by forward for every v 2 C 1a (x)

1 ( f (x)) and

Ca iteration: kD f x (v)k  1 kvk

and there exists a cone field Cb2 (x) around E2 which is backward invariant and expanded by backward iteration.

are smoothly immersed submanifolds of M, and they are characterized by W s (x) D fy 2 M : dist( f n (y); f n (x)) ! 0 as n ! 1g W u (x) D fy 2 M : dist( f n (y); f n (x)) ! 0 as n ! 1g :

Robustness An easy, yet very important consequence is that hyperbolic sets are robust under small modifications of the dynamics. Indeed, suppose  is a hyperbolic set for f : M ! M, and let C 1a (x) and Cb2 (x) be invariant cone fields as above. The (non-invariant) decomposition E 1 ˚ E 2 extends continu-

Shadowing Property This crucial property of hyperbolic sets means that possible small “errors” in the iteration of the map close to the set are, in some sense, unimportant: to the resulting “wrong” trajectory, there corresponds a nearby genuine orbit of the

745

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Hyperbolic Dynamical Systems

map. Let us give the formal statement. Recall that a hyperbolic set is compact, by definition. Given ı > 0, a ı-pseudo-orbit of f : M ! M is a sequence fx n gn2Z such that dist(x nC1 ; f (x n ))  ı

for all n 2 Z :

Given " > 0, one says that a pseudo-orbit is "-shadowed by the orbit of a point z 2 M if dist( f n (z); x n )  " for all n 2 Z. The shadowing lemma says that for any " > 0 one can find ı > 0 and a neighborhood U of the hyperbolic set  such that every ı-pseudo-orbit in U is "-shadowed by some orbit in U. Assuming " is sufficiently small, the shadowing orbit is actually unique. Local Product Structure In general, these shadowing orbits need not be inside th hyperbolic set . However, that is indeed the case if  is a maximal invariant set, that is, if it admits some neighborhood U such that  coincides with the set of points whose orbits never leave U: \ D f n (U) : n2Z

A hyperbolic set is a maximal invariant set if and only if it has the local product structure property stated in the next paragraph. Let  be a hyperbolic set and " be small. If x and y are nearby points in  then the local stable manifold of x intersects the local unstable manifold of y at a unique point, denoted [x; y], and this intersection is transverse. This is because the local stable manifold and the local unstable manifold of every point are transverse, and these local invariant manifolds vary continuously with the point. We say that  has local product structure if there exists ı > 0 such that [x; y] belongs to  for every x; y 2  with dist(x; y) < ı. Stability The shadowing property may also be used to prove that hyperbolic sets are stable under small perturbations of the dynamics: if  is a hyperbolic set for f then for any C1 close diffeomorphism g there exists a hyperbolic set  g close to  and carrying the same dynamical behavior. The key observation is that every orbit f n (x) of f inside  is a ı-pseudo-orbits for g in a neighborhood U, where ı is small if g is close to f and, hence, it is shadowed by some orbit g n (z) of g. The correspondence h(x) D z thus defined is injective and continuous. For any diffeomorphism g close enough to f , the orbits of x in the maximal g-invariant set  g (U) inside U

are pseudo-orbits for f . Therefore the shadowing property above enables one to bijectively associate g-orbits of  g (U) to f -orbits in . This provides a homeomorphism h :  g (U) !  which conjugates g and f on the respective hyperbolic sets: f ı h D h ı g. Thus hyperbolic maximal sets are structurally stable: the persistent dynamics in a neighborhood of these sets is the same for all nearby maps. If  is a hyperbolic maximal invariant set for f then its hyperbolic continuation for any nearby diffeomorphism g is also a maximal invariant set for g. Symbolic Dynamics The dynamics of hyperbolic sets can be described through a symbolic coding obtained from a convenient discretization of the phase space. In a few words, one partitions the set into a finite number of subsets and assigns to a generic point in the hyperbolic set its itinerary with respect to this partition. Dynamical properties can then be read out from a shift map in the space of (admissible) itineraries. The precise notion involved is that of Markov partition. A set R  is a rectangle if [x; y] 2 R for each x; y 2 R. A rectangle is proper if it is the closure of its interior relative to . A Markov partition of a hyperbolic set  is a cover R D fR1 ; : : : ; R m g of  by proper rectangles with pairwise disjoint interiors, relative to , and such W u ( f (x)) \ R j f (W u (x) \ R i ) and f (W s (x) \ R i ) W s ( f (x)) \ R j for every x 2 int (R i ) with f (x) 2 int (R j ). The key fact is that any maximal hyperbolic set  admits Markov partitions with arbitrarily small diameter. Given a Markov partition R with sufficiently small diameter, and a sequence j D ( j n )n2Z in f1; : : : ; mg, there exists at most one point x D h(j) such that f n (x) 2 R j n

for each n 2 Z :

We say that j is admissible if such a point x does exist and, in this case, we say x admits j as an itinerary. It is clear that f ı h D h ı  , where  is the shift (left-translation) in the space of admissible itineraries. The map h is continuous and surjective, and it is injective on the residual set of points whose orbits never hit the boundaries (relative to ) of the Markov rectangles. Uniformly Hyperbolic Systems A diffeomorphism f : M ! M is uniformly hyperbolic, or satisfies the Axiom A, if the non-wandering set ˝( f )

Hyperbolic Dynamical Systems

is a hyperbolic set for f and the set Per( f ) of periodic points is dense in ˝( f ). There is an analogous definition for smooth flows f t : M ! M; t 2 R. The reader can find the technical details in e. g. [6,8,10]. Dynamical Decomposition The so-called “spectral” decomposition theorem of Smale allows for the global dynamics of a hyperbolic diffeomorphism to be decomposed into elementary building blocks. It asserts that the non-wandering set splits into a finite number of pairwise disjoint basic pieces that are compact, invariant, and dynamically indecomposable. More precisely, the non-wandering set ˝( f ) of a uniformly hyperbolic diffeomorphism f is a finite pairwise disjoint union ˝( f ) D 1 [    [  N of f -invariant, transitive sets  i , that are compact and maximal invariant sets. Moreover, the ˛-limit set of every orbit is contained in some  i and so is the !-limit set. Geodesic Flows on Surfaces with Negative Curvature Historically, the first important example of uniform hyperbolicity was the geodesic flow Gt on Riemannian manifolds of negative curvature M. This is defined as follows. Let M be a compact Riemannian manifold. Given any tangent vector v, let v : R ! TM be the geodesic with initial condition v D v (0). We denote by ˙v (t) the velocity vector at time t. Since k ˙v (t)k D kvk for all t, it is no restriction to consider only unit vectors. There is an important volume form on the unit tangent bundle, given by the product of the volume element on the manifold by the volume element induced on each fiber by the Riemannian metric. By integration of this form, one obtains the Liouville measure on the unit tangent bundle, which is a finite measure if the manifold itself has finite volume (including the compact case). The geodesic flow is the flow G t : T 1 M ! T 1 M on the unit tangent bundle T 1 M of the manifold, defined by G t (v) D ˙v (t) : An important feature is that this flow leaves invariant the Liouville measure. By Poincaré recurrence, this implies that ˝(G) D T 1 M. A major classical result in Dynamics, due to Anosov, states that if M has negative sectional curvature then this measure is ergodic for the flow. That is, any invariant set has zero or full Liouville measure. The special case when M is a surface, had been dealt before by Hedlund and Hopf.

The key ingredient to this theorem is to prove that the geodesic flow is uniformly hyperbolic, in the sense we have just described, when the sectional curvature is negative. In the surface case, the stable and unstable invariant subbundles are differentiable, which is no longer the case in general in higher dimensions. This formidable obstacle was overcome by Anosov through showing that the corresponding invariant foliations retain, nevertheless, a weaker form of regularity property, that suffices for the proof. Let us explain this. Absolute Continuity of Foliations The invariant spaces E sx and E ux of a hyperbolic system depend continuously, and even Hölder continuously, on the base point x. However, in general this dependence is not differentiable, and this fact is at the origin of several important difficulties. Related to this, the families of stable and unstable manifolds are, usually, not differentiable foliations: although the leaves themselves are as smooth as the dynamical system itself, the holonomy maps often fail to be differentiable. By holonomy maps we mean the projections along the leaves between two given cross-sections to the foliation. However, Anosov and Sinai observed that if the system is at least twice differentiable then these foliations are absolutely continuous: their holonomy maps send zero Lebesgue measure sets of one cross-section to zero Lebesgue measure sets of the other cross-section. This property is crucial for proving that any smooth measure which is invariant under a twice differentiable hyperbolic system is ergodic. For dynamical systems that are only once differentiable the invariant foliations may fail to be absolutely continuous. Ergodicity still is an open problem. Structural Stability A dynamical system is structurally stable if it is equivalent to any other system in a C1 neighborhood, meaning that there exists a global homeomorphism sending orbits of one to orbits of the other and preserving the direction of time. More generally, replacing C1 by Cr neighborhoods, any r  1, one obtains the notion of Cr structural stability. Notice that, in principle, this property gets weaker as r increases. The Stability Conjecture of Palis–Smale proposed a complete geometric characterization of this notion: for any r  1; C r structurally stable systems should coincide with the hyperbolic systems having the property of strong transversality, that is, such that the stable and unstable manifolds of any points in the non-wandering set are transversal. In particular, this would imply that the prop-

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erty of Cr structural stability does not really depend on the value of r. That hyperbolicity and strong transversality suffice for structural stability was proved in the 1970s by Robbin, de Melo, Robinson. It is comparatively easy to prove that strong transversality is also necessary. Thus, the heart of the conjecture is to prove that structurally stable systems must be hyperbolic. This was achieved by Mañé in the 1980s, for C1 diffeomorphisms, and extended about ten years later by Hayashi for C1 flows. Thus a C1 diffeomorphism, or flow, on a compact manifold is structurally stable if and only if it is uniformly hyperbolic and satisfies the strong transversality condition. ˝-stability A weaker property, called ˝-stability is defined requiring equivalence only restricted to the non-wandering set. The ˝-Stability Conjecture of Palis–Smale claims that, for any r  1; ˝-stable systems should coincide with the hyperbolic systems with no cycles, that is, such that no basic pieces in the spectral decomposition are cyclically related by intersections of the corresponding stable and unstable sets. The ˝-stability theorem of Smale states that these properties are sufficient for C r ˝stability. Palis showed that the no-cycles condition is also necessary. Much later, based on Mañé’s aforementioned result, he also proved that for C1 diffeomorphisms hyperbolicity is necessary for ˝-stability. This was extended to C1 flows by Hayashi in the 1990s. Attractors and Physical Measures A hyperbolic basic piece  i is a hyperbolic attractor if the stable set W s ( i ) D fx 2 M : !(x)  i g contains a neighborhood of  i . In this case we call W s ( i ) the basin of the attractor  i , and denote it B( i ). When the uniformly hyperbolic system is of class C2 , a basic piece is an attractor if and only if its stable set has positive Lebesgue measure. Thus, the union of the basins of all attractors is a full Lebesgue measure subset of M. This remains true for a residual (dense Gı ) subset of C1 uniformly hyperbolic diffeomorphisms and flows. The following fundamental result, due to Sinai, Ruelle, Bowen shows that, no matter how complicated it may be, the behavior of typical orbits in the basin of a hyperbolic attractor is well-defined at the statistical level: any hyperbolic attractor  of a C2 diffeomorphism (or flow) supports

a unique invariant probability measure  such that Z n1 1X j lim '( f (z)) D ' d n!1 n

(2)

jD0

for every continuous function ' and Lebesgue almost every point x 2 B(). The standard reference here is [3]. Property (2) also means that the Sinai–Ruelle–Bowen measure  may be “observed”: the weights of subsets may be found with any degree of precision, as the sojourn-time of any orbit picked “at random” in the basin of attraction: (V ) D fraction of time the orbit of z spends in V for typical subsets V of M (the boundary of V should have zero -measure), and for Lebesgue almost any point z 2 B(). For this reason  is called a physical measure. It also follows from the construction of these physical measures on hyperbolic attractors that they depend continuously on the diffeomorphism (or the flow). This statistical stability is another sense in which the asymptotic behavior is stable under perturbations of the system, distinct from structural stability. There is another sense in which this measure is “physical” and that is that  is the zero-noise limit of the stationary measures associated to the stochastic processes obtained by adding small random noise to the system. The idea is to replace genuine trajectories by “random orbits” (z n )n , where each z n C 1 is chosen "-close to f (zn ). We speak of stochastic stability if, for any continuous function ', the random time average lim

n!1

n1 1X '(z j ) n jD0

R

is close to ' d for almost all choices of the random orbit. One way to construct such random orbits is through randomly perturbed iterations, as follows. Consider a family of probability measures " in the space of diffeomorphisms, such that each " is supported in the "-neighborhood of f . Then, for each initial state z0 define z nC1 D f nC1 (z n ), where the diffeomorphisms f n are independent random variables with distribution law " . A probability measure " on the basin B() is stationary if it satisfies Z " (E) D " (g 1 (E)) d" (g) : Stationary measures always exist, and they are often unique for each small " > 0. Then stochastic stability corresponds to having " converging weakly to  when the noise level " goes to zero.

Hyperbolic Dynamical Systems

Hyperbolic Dynamical Systems, Figure 3 A planar flow with divergent time averages

The notion of stochastic stability goes back to Kolmogorov and Sinai. The first results, showing that uniformly hyperbolic systems are stochastically stable, on the basin of each attractor, were proved in the 1980s by Kifer and Young. Let us point out that physical measures need not exist for general systems. A simple counter-example, attributed to Bowen, is described in Fig. 3: time averages diverge over any of the spiraling orbits in the region bounded by the saddle connections. Notice that the saddle connections are easily broken by arbitrarily small perturbations of the flow. Indeed, no robust examples are known of systems whose time-averages diverge on positive volume sets. Obstructions to Hyperbolicity Although uniform hyperbolicity was originally intended to encompass a residual or, at least, dense subset of all dynamical systems, it was soon realized that this is not the case: many important examples fall outside its realm. There are two main mechanisms that yield robustly nonhyperbolic behavior, that is, whole open sets of non-hyperbolic systems. Heterodimensional Cycles Historically, the first such mechanism was the coexistence of periodic points with different Morse indices (dimensions of the unstable manifolds) inside the same transitive set. See Fig. 4. This is how the first examples of C1 -open subsets of non-hyperbolic diffeomorphisms were obtained by Abraham, Smale on manifolds of dimension d  3. It was also the key in the constructions by Shub and Mañé of non-hyperbolic, yet robustly transitive diffeomorphisms, that is, such that every diffeomorphism in a C1 neighborhood has dense orbits. For flows, this mechanism may assume a novel form, because of the interplay between regular orbits and singularities (equilibrium points). That is, robust non-hyperbolicity may stem from the coexistence of regular and singular orbits in the same transitive set. The first, and

Hyperbolic Dynamical Systems, Figure 4 A heterodimensional cycle

very striking example was the geometric Lorenz attractor proposed by Afraimovich, Bykov, Shil’nikov and Guckenheimer, Williams to model the behavior of the Lorenz equations, that we shall discuss later. Homoclinic Tangencies Of course, heterodimensional cycles may exist only in dimension 3 or higher. The first robust examples of nonhyperbolic diffeomorphisms on surfaces were constructed by Newhouse, exploiting the second of these two mechanisms: homoclinic tangencies, or non-transverse intersections between the stable and the unstable manifold of the same periodic point. See Fig. 5. It is important to observe that individual homoclinic tangencies are easily destroyed by small perturbations of the invariant manifolds. To construct open examples of surface diffeomorphisms with some tangency, Newhouse started from systems where the tangency is associated to a periodic point inside an invariant hyperbolic set with rich geometric structure. This is illustrated on the right hand side of Fig. 5. His argument requires a very delicate control of distortion, as well as of the dependence of the fractal dimension on the dynamics. Actually, for this reason, his construction is restricted to the Cr topology for

Hyperbolic Dynamical Systems, Figure 5 Homoclinic tangencies

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Hyperbolic Dynamical Systems

r  2. A very striking consequence of this construction is that these open sets exhibit coexistence of infinitely many periodic attractors, for each diffeomorphism on a residual subset. A detailed presentation of his result and consequences is given in [9]. Newhouse’s conclusions have been extended in two ways. First, by Palis, Viana, for diffeomorphisms in any dimension, still in the Cr topology with r  2. Then, by Bonatti, Díaz, for C1 diffeomorphisms in any dimension larger or equal than 3. The case of C1 diffeomorphisms on surfaces remains open. As a matter of fact, in this setting it is still unknown whether uniform hyperbolicity is dense in the space of all diffeomorphisms. Partial Hyperbolicity Several extensions of the theory of uniform hyperbolicity have been proposed, allowing for more flexibility, while keeping the core idea: splitting of the tangent bundle into invariant subbundles. We are going to discuss more closely two such extensions. On the one hand, one may allow for one or more invariant subbundles along which the derivative exhibits mixed contracting/neutral/expanding behavior. This is generically referred to as partial hyperbolicity, and a standard reference is the book [5]. On the other hand, while requiring all invariant subbundles to be either expanding or contraction, one may relax the requirement of uniform rates of expansion and contraction. This is usually called non-uniform hyperbolicity. A detailed presentation of the fundamental results about this notion is available e. g. in [6]. In this section we discuss the first type of condition. The second one will be dealt with later. Dominated Splittings Let f : M ! M be a diffeomorphism on a closed manifold M and K be any f -invariant set. A continuous splitting Tx M D E1 (x) ˚    ˚ E k (x); x 2 K of the tangent bundle over K is dominated if it is invariant under the derivative Df and there exists ` 2 N such that for every i < j, every x 2 K, and every pair of unit vectors u 2 E i (x) and v 2 E j (x), one has 1 kD f x`  uk < ; 2 kD f x`  vk

one has kD f xn  uk < Cn kD f xn  vk

for all n  1 :

Let f be a diffeomorphism and K be an f -invariant set having a dominated splitting TK M D E1 ˚    ˚ E k . We say that the splitting and the set K are  partially hyperbolic the derivative either contracts uniformly E1 or expands uniformly Ek : there exists ` 2 N such that either kD f ` j E1 k <

1 2

or k(D f ` j E k )1 k <

1 : 2

 volume hyperbolic if the derivative either contracts volume uniformly along E1 or expands volume uniformly along Ek : there exists ` 2 N such that either j det(D f ` j E1 )j <

1 2

or j det(D f ` j E k )j > 2 :

The diffeomorphism f is partially hyperbolic/volume hyperbolic if the ambient space M is a partially hyperbolic/volume hyperbolic set for f . Invariant Foliations An crucial geometric feature of partially hyperbolic systems is the existence of invariant foliations tangent to uniformly expanding or uniformly contracting invariant subbundles: assuming the derivative contracts E1 uniformly, there exists a unique family F s D fF s (x) : x 2 Kg of injectively Cr immersed submanifolds tangent to E1 at every point of K, satisfying f (F s (x)) D F s ( f (x)) for all x 2 K, and which are uniformly contracted by forward iterates of f . This is called strong-stable foliation of the diffeomorphism on K. Strong-unstable foliations are defined in the same way, tangent to the invariant subbundle Ek , when it is uniformly expanding. As in the purely hyperbolic setting, a crucial ingredient in the ergodic theory of partially hyperbolic systems is the fact that strong-stable and strong-unstable foliations are absolutely continuous, if the system is at least twice differentiable.

(3)

and the dimension of Ei (x) is independent of x 2 K for every i 2 f1; : : : ; kg. This definition may be formulated, equivalently, as follows: there exist C > 0 and  < 1 such that for every pair of unit vectors u 2 E i (x) and v 2 E j (x),

Robustness and Partial Hyperbolicity Partially hyperbolic systems have been studied since the 1970s, most notably by Brin, Pesin and Hirsch, Pugh, Shub. Over the last decade they attracted much attention as the key to characterizing robustness of the dynamics.

Hyperbolic Dynamical Systems

More precisely, let  be a maximal invariant set of some diffeomorphism f : \ f n (U) for some neighborhood U of  : D n2Z

The set  is robust, or robustly transitive, if its continuation  g D \n2Z g n (U) is transitive for all g in a neighborhood of f . There is a corresponding notion for flows. As we have already seen, hyperbolic basic pieces are robust. In the 1970s, Mañé observed that the converse is also true when M is a surface, but not anymore if the dimension of M is at least 3. Counter-examples in dimension 4 had been given before by Shub. A series of results of Bonatti, Díaz, Pujals, Ures in the 1990s clarified the situation in all dimensions: robust sets always admit some dominated splitting which is volume hyperbolic; in general, this splitting needs not be partially hyperbolic, except when the ambient manifold has dimension 3. Lorenz-like Strange Attractors Parallel results hold for flows on 3-dimensional manifolds. The main motivation are the so-called Lorenz-like strange attractors, inspired by the famous differential equations x˙ D  x C  y

 D 10

y˙ D x  y  xz

D 28

z˙ D x y  ˇz

ˇ D 8/3

(4)

like attractors are the only ones which are robust. Indeed, they prove that any robust invariant set of a flow in dimension 3 is singular hyperbolic. Moreover, if the robust set contains equilibrium points then it must be either an attractor or a repeller. A detailed presentation of this and related results is given in [1]. An invariant set  of a flow in dimension 3 is singular hyperbolic if it is a partially hyperbolic set with splitting E 1 ˚ E 2 such that the derivative is volume contracting along E1 and volume expanding along E2 . Notice that one of the subbundles E1 or E2 must be one-dimensional, and then the derivative is, actually, either norm contracting or norm expanding along this subbundle. Singular hyperbolic sets without equilibria are uniformly hyperbolic: the 2-dimensional invariant subbundle splits as the sum of the flow direction with a uniformly expanding or contracting one-dimensional invariant subbundle. Non-Uniform Hyperbolicity – Linear Theory In its linear form, the theory of non-uniform hyperbolicity goes back to Lyapunov, and is founded on the multiplicative ergodic theorem of Oseledets. Let us introduce the main ideas, whose thorough development can be found in e. g. [4,6,7]. The Lyapunov exponents of a sequence fAn ; n  1g of square matrices of dimension d  1, are the values of (v) D lim sup n!1

introduced by E. N. Lorenz in the early 1960s. Numerical analysis of these equations led Lorenz to realize that sensitive dependence of trajectories on the initial conditions is ubiquitous among dynamical systems, even those with simple evolution laws. The dynamical behavior of (4) was first interpreted by means of certain geometric models, proposed by Guckenheimer, Williams and Afraimovich, Bykov, Shil’nikov in the 1970s, where the presence of strange attractors, both sensitive and fractal, could be proved rigorously. It was much harder to prove that the original Eqs. (4) themselves have such an attractor. This was achieved just a few years ago, by Tucker, by means of a computer assisted rigorous argument. An important point is that Lorenz-like attractors cannot be hyperbolic, because they contain an equilibrium point accumulated by regular orbits inside the attractor. Yet, these strange attractors are robust, in the sense we defined above. A mathematical theory of robustness for flows in 3-dimensional spaces was recently developed by Morales, Pacifico, and Pujals. In particular, this theory shows that uniformly hyperbolic attractors and Lorenz-

1 log kAn  vk n

(5)

over all non-zero vectors v 2 Rd . For completeness, set (0) D 1. It is easy to see that (cv) D (v) and (v C v 0 )  maxf(v); (v 0 )g for any non-zero scalar c and any vectors v; v 0 . It follows that, given any constant a, the set of vectors satisfying (v)  a is a vector subspace. Consequently, there are at most d Lyapunov exponents, henceforth denoted by 1 <    <  k1 <  k , and there exists a filtration F0 F1    F k1 F k D Rd into vector subspaces, such that (v) D  i

for all v 2 Fi n F i1 ;

and every i D 1; : : : ; k (write F0 D f0g). In particular, the largest exponent is given by  k D lim sup n!1

1 log kAn k : n

(6)

One calls dim F i  dim F i1 the multiplicity of each Lyapunov exponent  i . There are corresponding notions for continuous families of matrices At ; t 2 (0; 1), taking the limit as t goes to infinity in the relations (5) and (6).

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Hyperbolic Dynamical Systems

Lyapunov Stability Consider the linear differential equation v˙(t) D B(t)  v(t) ;

(7)

where B(t) is a bounded function with values in the space of d  d matrices, defined for all t 2 R. The theory of differential equations ensures that there exists a fundamental matrix At ; t 2 R such that v(t) D At  v0 is the unique solution of (7) with initial condition v(0) D v0 . If the Lyapunov exponents of the family At ; t > 0 are all negative then the trivial solution v(t)  0 is asymptotically stable, and even exponentially stable. The stability theorem of A. M. Lyapunov asserts that, under an additional regularity condition, stability is still valid for nonlinear perturbations w(t) D B(t)  w C F(t; w) ; > 0. That is, the trivwith kF(t; w)k  ial solution w(t)  0 is still exponentially asymptotically stable. The regularity condition means, essentially, that the limit in (5) does exist, even if one replaces vectors v by elements v1 ^    ^ v l of any lth exterior power of Rd ; 1  l  d. By definition, the norm of an l-vector v1 ^    ^ v l is the volume of the parallelepiped determined by the vectors v1 ; : : : ; v k . This condition is usually tricky to check in specific situations. However, the multiplicative ergodic theorem of V. I. Oseledets asserts that, for very general matrix-valued stationary random processes, regularity is an almost sure property. Multiplicative Ergodic Theorem Let f : M ! M be a measurable transformation, preserving some measure , and let A: M ! GL(d; R) be any measurable function such that log kA(x)k is -integrable. The Oseledets theorem states that Lyapunov exponents exist for the sequence An (x) D A( f n1 (x)) : : : A( f (x))A(x) for -almost every x 2 M. More precisely, for -almost every x 2 M there exists k D k(x) 2 f1; : : : ; dg, a filtration

Fx1



Fxk1



Fxk

DR ; d

and numbers 1 (x) <    <  k (x) such that lim

n!1

1 log kAn (x)  vk D  i (x) ; n

A(x)  Fxi D F fi (x) : It is in the nature of things that, usually, these objects are not defined everywhere and they depend discontinuously on the base point x. When the transformation f is invertible one obtains a stronger conclusion, by applying the previous result also to the inverse automorphism: assuming that log kA(x)1 k is also in L1 (), one gets that there exists a decomposition Vx D E 1x ˚    ˚ E xk ;

const kwk1Cc ; c

Fx0

for all v 2 Fxi n Fxi1 and i 2 f1; : : : ; kg. More generally, this conclusion holds for any vector bundle automorphism V ! V over the transformation f , with A x : Vx ! V f (x) denoting the action of the automorphism on the fiber of x. The Lyapunov exponents  i (x), and their number k(x), are measurable functions of x and they are constant on orbits of the transformation f . In particular, if the measure  is ergodic then k and the  i are constant on a full -measure set of points. The subspaces F ix also depend measurably on the point x and are invariant under the automorphism:

defined at almost every point and such that A(x)  E xi D E if (x) and lim

n!˙1

1 log kAn (x)  vk D  i (x) ; n

for all v 2 E xi different from zero and all i 2 f1; : : : ; kg. These Oseledets subspaces Eix are related to the subspaces F ix through j

j

Fx D ˚ iD1 E xi : Hence, dim E xi D dim Fxi  dim Fxi1 is the multiplicity of the Lyapunov exponent  i (x). The angles between any two Oseledets subspaces decay sub-exponentially along orbits of f : ! M j M 1 i E f n (x) ; E f n (x) D 0 ; log angle lim n!˙1 n i2I

j…I

for any I f1; : : : ; kg and almost every point. These facts imply the regularity condition mentioned previously and, in particular, k

lim

n!˙1

X 1  i (x) dim E xi : log j det An (x)j D n iD1

Consequently, if det A(x) D 1 at every point then the sum of all Lyapunov exponents, counted with multiplicity, is identically zero.

Hyperbolic Dynamical Systems

Non-Uniformly Hyperbolic Systems The Oseledets theorem applies, in particular, when f : M ! M is a C1 diffeomorphism on some compact manifold and A(x) D D f x . Notice that the integrability conditions are automatically satisfied, for any f -invariant probability measure , since the derivative of f and its inverse are bounded in norm. Lyapunov exponents yield deep geometric information on the dynamics of the diffeomorphism, especially when they do not vanish. We call  a hyperbolic measure if all Lyapunov exponents are non-zero at -almost every point. By non-uniformly hyperbolic system we shall mean a diffeomorphism f : M ! M together with some invariant hyperbolic measure. A theory initiated by Pesin provides fundamental geometric information on this class of systems, especially existence of stable and unstable manifolds at almost every point which form absolutely continuous invariant laminations. For most results, one needs the derivative Df to be Hölder continuous: there exists c > 0 such that

surably on x. Another key difference with respect to the uniformly hyperbolic setting is that the numbers Cx and x can not be taken independent of the point, in general. Likewise, one defines local and global unstable manifolds, tangent to E ux at almost every point. Most important for the applications, both foliations, stable and unstable, are absolutely continuous. In the remaining sections we briefly present three major results in the theory of non-uniform hyperbolicity: the entropy formula, abundance of periodic orbits, and exact dimensionality of hyperbolic measures. The Entropy Formula The entropy of a partition P of M is defined by h ( f ; P ) D lim

n!1

where P n is the partition into sets of the form P D P0 \ f 1 (P1 ) \    \ f n (Pn ) with Pj 2 P and X H (P n ) D (P) log (P) :

kD f x  D f y k  const d(x; y)c : These notions extend to the context of flows essentially without change, except that one disregards the invariant line bundle given by the flow direction (whose Lyapunov exponent is always zero). A detailed presentation can be found in e. g. [6]. Stable Manifolds An essential tool is the existence of invariant families of local stable sets and local unstable sets, defined at -almost every point. Assume  is a hyperbolic measure. Let Eux and Esx be the sums of all Oseledets subspaces corresponding to positive, respectively negative, Lyapunov exponents, and let x > 0 be a lower bound for the norm of every Lyapunov exponent at x. Pesin’s stable manifold theorem states that, for -als (x) most every x 2 M, there exists a C1 embedded disk Wloc s tangent to E x at x and there exists C x > 0 such that dist( f n (y); f n (x))  C x e nx  dist(y; x) s for all y 2 Wloc (x) : s (x)g is invariant, in the sense Moreover, the family fWloc s s that f (Wloc (x)) Wloc ( f (x)) for -almost every x. Thus, one may define global stable manifolds

W s (x) D

1 [

  s f n Wloc (x)

for -almost every x :

nD0

In general, the local stable disks W s (x) depend only mea-

1 H (P n ) ; n

P2P n

The Kolmogorov–Sinai entropy h ( f ) of the system is the supremum of h ( f ; P ) over all partitions P with finite entropy. The Ruelle–Margulis inequality says that h ( f ) is bounded above by the averaged sum of the positive Lyapunov exponents. A major result of the theorem, Pesin’s entropy formula, asserts that if the invariant measure  is smooth (for instance, a volume element) then the entropy actually coincides with the averaged sum of the positive Lyapunov exponents ! Z X k h ( f ) D maxf0;  j g d : jD1

A complete characterization of the invariant measures for which the entropy formula is true was given by F. Ledrappier and L. S. Young. Periodic Orbits and Entropy It was proved by A. Katok that periodic motions are always dense in the support of any hyperbolic measure. More than that, assuming the measure is non-atomic, there exist Smale horseshoes H n with topological entropy arbitrarily close to the entropy h ( f ) of the system. In this context, the topological entropy h( f ; H n ) may be defined as the exponential rate of growth lim

k!1

1 log #fx 2 H n : f k (x) D xg : k

of the number of periodic points on H n .

753

754

Hyperbolic Dynamical Systems

Dimension of Hyperbolic Measures Another remarkable feature of hyperbolic measures is that they are exact dimensional: the pointwise dimension

The reader is referred to the bibliography, especially the book [2] for a review of much recent progress. Bibliography

log (Br (x)) r!0 log r

d(x) D lim

exists at almost every point, where Br (x) is the neighborhood of radius r around x. This fact was proved by L. Barreira, Ya. Pesin, and J. Schmeling. Note that this means that the measure (Br (x)) of neighborhoods scales as r d(x) when the radius r is small.

Future Directions The theory of uniform hyperbolicity showed that dynamical systems with very complex behavior may be amenable to a very precise description of their evolution, especially in probabilistic terms. It was most successful in characterizing structural stability, and also established a paradigm of how general “chaotic” systems might be approached. A vast research program has been going on in the last couple of decades or so, to try and build such a global theory of complex dynamical evolution, where notions such as partial and non-uniform hyperbolicity play a central part.

1. Araujo V, Pacifico MJ (2007) Three Dimensional Flows. In: XXV Brazillian Mathematical Colloquium. IMPA, Rio de Janeiro 2. Bonatti C, Díaz LJ, Viana M (2005) Dynamics beyond uniform hyperbolicity. In: Encyclopaedia of Mathematical Sciences, vol 102. Springer, Berlin 3. Bowen R (1975) Equilibrium states and the ergodic theory of Anosov diffeomorphisms. In: Lecture Notes in Mathematics, vol 470. Springer, Berlin 4. Cornfeld IP, Fomin SV, Sina˘ı YG (1982) Ergodic theory. In: Grundlehren der Mathematischen Wissenschaften Fundamental Principles of Mathematical Sciences, vol 245. Springer, New York 5. Hirsch M, Pugh C, Shub M (1977) Invariant manifolds. In: Lecture Notes in Mathematics, vol 583. Springer, Berlin 6. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. Cambridge University Press, Cambridge 7. Mañé R (1987) Ergodic theory and differentiable dynamics. Springer, Berlin 8. Palis J, de Melo W (1982) Geometric theory of dynamical systems. An introduction. Springer, New York 9. Palis J, Takens F (1993) Hyperbolicity and sensitive-chaotic dynamics at homoclinic bifurcations. Cambridge University Press, Cambridge 10. Shub M (1987) Global stability of dynamical systems. Springer, New York

Infinite Dimensional Controllability

Infinite Dimensional Controllability OLIVIER GLASS Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, France

Article Outline Glossary Definition of the Subject Introduction First Definitions and Examples Linear Systems Nonlinear Systems Some Other Problems Future Directions Bibliography Glossary Infinite dimensional control system A infinite dimensional control system is a dynamical system whose state lies in an infinite dimensional vector space—typically a Partial Differential Equation (PDE)—and depending on some parameter to be chosen, called the control. Exact controllability The exact controllability property is the possibility to steer the state of the system from any initial data to any target by choosing the control as a function of time in an appropriate way. Approximate controllability The approximate controllability property is the possibility to steer the state of the system from any initial data to a state arbitrarily close to a target by choosing a suitable control. Controllability to trajectories The controllability to trajectories is the possibility to make the state of the system join some prescribed trajectory by choosing a suitable control. Definition of the Subject Controllability is a mathematical problem, which consists of determining the targets to which one can drive the state of some dynamical system, by means of a control parameter present in the equation. Many physical systems such as quantum systems, fluid mechanical systems, wave propagation, diffusion phenomena, etc., are represented by an infinite number of degrees of freedom, and their evolution follows some partial differential equation (PDE). Finding active controls in order to properly influence the dynamics

of these systems generate highly involved problems. The control theory for PDEs, and among this theory, controllability problems, is a mathematical description of such situations. Any dynamical system represented by a PDE, and on which an external influence can be described, can be the object of a study from this point of view. Introduction The problem of controllability is a mathematical description of the general following situation. We are given an evolution system (typically a physical one), on which we can exert a certain influence. Is it possible to use this influence to make the system reach a certain state? More precisely, the system takes generally the following form: y˙ D F(t; y; u) ;

(1)

where y is a description of the state of the system, y˙ denotes its derivative with respect to the time t, and u is the control, that is, a parameter which can be chosen in a suitable range. The standard problem of controllability is the following. Given a time T > 0, an initial state y0 and a target y1 , is it possible to find a control function u (depending on the time), such that the solution of the system, starting from y0 and provided with this function u reaches the state y1 at time T? If the state of the system can be described by a finite number of degrees of freedom (typically, if it belongs to an Euclidean space or to a manifold), we call the problem finite dimensional. The present article deals with the case where y belongs to an infinite-dimensional space, typically a Banach space or a Hilbert space. Hence the systems described here have an infinite number of degrees of freedom. The potential range of the applications of the theory is extremely wide: the models from fluid dynamics (see for instance [52]) to quantum systems (see [6,54]), networks of structures (see [23,44]), wave propagation, etc., are countless. In the infinite dimensional setting, the Eq. (1) is typically a partial differential equation, where F acts as a differential operator on the function y, and the influence of u can take multiple different forms: typically, u can be an additional (force) term in the right-hand side of the equation, localized in a part of the domain; it can also appear in the boundary conditions; but other situations can clearly be envisaged (we will describe some of them). Of course the possibilities to introduce a control problem for partial differential equations are virtually infinite. The number of results since the beginning of the theory in the 1960s has been constantly growing and has reached

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Infinite Dimensional Controllability

huge dimensions: it is, in our opinion, hopeless to give a fair general view of all the activity in the theory. This paper will only try to present some basic techniques connected to the problem of infinite-dimensional controllability. As in the finite dimensional setting, one can distinguish between the linear systems, where the partial differential equation under view is linear (as well as the action of the control), and the nonlinear ones. Structure of the Paper In Sect. “First Definitions and Examples”, we will introduce the problems that are under view and give some examples. The main parts of this paper are Sects. “Linear Systems” and “Nonlinear Systems”, where we consider linear and nonlinear systems, respectively. First Definitions and Examples General Framework We define an infinite-dimensional control system as the following data: 1. an evolution system (typically a PDE)

amples below concern the wave and the heat equations with Dirichlet boundary conditions: these classical equations are reversible and irreversible, respectively, which, as we will see, is of high importance when considering controllability problems. Example 1 Distributed control for the wave/heat equation with Dirichlet boundary conditions. We consider:  ˝ a regular domain in Rn , which is in general required to be bounded,  ! a nonempty open subdomain in ˝,  the wave/heat equation is posed in [0; T]˝, with a localized source term in !:

wave equation v :D  v D 1! u ; vj@˝ D 0 ; @2t t v

heat equation @ t v  v D 1! u ; vj@˝ D 0 :

 In the first case, the state y of the system is given by the couple (v(t; ); @ t v(t; )), for instance considered in the space H01 (˝)  L2 (˝) or in L2 (˝)  H 1 (˝).  In the second case, the state y of the system is given by the function v(t; ), for instance in the space L2 (˝).  In both cases, the control is the function u, for instance considered in L2 ([0; T]; L2 (!)).

y˙ D F(t; y; u) ; 2. the unknown y is the state of the system, which is a function depending on time: t 2 [0; T] 7! y(t) 2 Y , where the set Y is a functional space (for instance a Banach or a Hilbert space), or a part of a functional space, 3. a parameter u called the control, which is a time-dependent function t 2 [0; T] 7! u(t) 2 U, where the set U of admissible controls is again some part of a functional space. As a general rule, one expects that for any initial data yjtD0 and any appropriate control function u there exists a unique solution of the system (at least locally in time). In some particular cases, one can find problems “of controllability” type for stationary problems (such as elliptic equations), see for instance [56].

Example 2 Boundary control of the wave/heat equation with Dirichlet boundary conditions. We consider:  ˝ a regular domain in Rn , typically a bounded one,  ˙ an open nonempty subset of the boundary @˝,  the heat/wave equation in [0; T]  ˝, with nonhomogeneous boundary conditions inside ˙ : wave equation

v :D @2t t v  v D 0 ; vj@˝ D 1˙ u ;

heat equation @ t v  v D 0; vj@˝ D 1˙ u :

The states are the same as in the previous example, but here the control u is imposed on a part of the boundary. One can for instance consider the set of controls as L2 ([0; T]; L2 (˙ )) in the first case, as C01 ((0; T)  ˙ ) in the second case.

Examples Let us give two classical examples of the situation. These are two types of acting control frequently considered in the literature: in one case, the control acts as a localized source term in the equation, while in the second one, the control acts on a part of the boundary conditions. The ex-

Needless to say, one can consider other boundary conditions than Dirichlet’s. Let us emphasize that, while these two types of control are very frequent, these are not by far the only ones: for instance, one can consider the following “affine control”: the heat equation with a right hand side

Infinite Dimensional Controllability

u(t)g(x) where the control u depends only on the time, and g is a fixed function:

@ t v  v D u(t)g(x) ; vj@˝ D 0 ; see an example of this below. Also, one could for instance consider the “bilinear control,” which takes the form:

@ t v  v D g(x)u(t)v ; vj@˝ D 0 : Main Problems Now let us give some definitions of the typical controllability problems, associated to a control system. Definition 1 A control system is said to be exactly controllable in time T > 0 if and only if, for all y0 and y1 in Y , there is some control function u : [0; T] ! U such that the unique solution of the system

y˙ D F(t; y; u) (2) yjtD0 D y0 ; satisfies yjtDT D y1 : Definition 2 We suppose that the space Y is endowed with a metric d. The control system is said to be approximately controllable in time T > 0 if and only if, for all y0 and y1 in Y , for any " > 0, there exists a control function u : [0; T] ! U such that the unique solution of the system (2) satisfies d(yjtDT ; y1 ) < " : Definition 3 We consider a particular element 0 of Y . A control system is said to be zero-controllable in time T > 0 if and only if, for all y0 in Y , there exists a control function u : [0; T] ! U such that the unique solution of the system (2) satisfies

Definition 5 All the above properties are said to be fulfilled locally, if they are proved for y0 sufficiently close to the target y1 or to the starting point of the trajectory y(0); they are said to be fulfilled globally if the property is established without such limitations. Remarks We can already make some remarks concerning the problems that we described above. 1. The different problems of controllability should be distinguished from the problems of optimal control, which give another viewpoint on control theory. In general, problems of optimal control look for a control u minimizing some functional J(u; y(u)) ; where y(u) is the trajectory associated to the control u. 2. It is important to notice that the above properties of controllability depend in a crucial way on the choice of the functional spaces Y ; U. The approximate controllability in some space may be the exact controllability in another space. In the same way, we did not specify the regularity in time of the control functions in the above definitions: it should be specified for each problem. 3. A very important fact for controllability problems is that when a problem of controllability has a solution, it is almost never unique. For instance, if a time-invariant system is controllable regardless of the time T, it is clear that one can choose u arbitrarily in some interval [0; T/2], and then choose an appropriate control (for instance driving the system to 0) during the interval [T/2; T]. In such a way, one has constructed a new control which fulfills the required task. The number of controls that one can construct in this way is clearly infinite. This is of course already true for finite dimensional systems. 4. Formally, the problem of interior control when the control zone ! is the whole domain ˝ is not very difficult, since it suffices to consider the trajectory

yjtDT D 0 : v(t) :D v0 C Definition 4 A control system is said to be controllable to trajectories in time T > 0 if and only if, for all y0 in Y and any trajectory y of the system (typically but not necessarily satisfying (2) with u D 0), there exists a control function u : [0; T] ! U such that the unique solution of the system (2) satisfies yjtDT D y(T) :

t (v1  v0 ) ; T

to compute the left-hand side of the equation with this trajectory, and to choose it as the control. However, by doing so, one obtains in general a control with very low regularity. Note also that, on the contrary, as long as the boundary control problem is concerned, the case when ˙ is equal to the whole boundary @˝ is not that simple.

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Infinite Dimensional Controllability

5. Let us also point out a “principle” which shows that interior and boundary control problems are not very different. We give the main ideas. Suppose for instance that controllability holds for any domain and subdomain ˝ and !. When considering the controllability problem on ˝ via boundary control localized on ˙ , one may introduce an extension ˝˜ of ˝, obtained by gluing along ˙ an “additional” open set ˝ 2 , so that ˝˜ D ˝ [ ˝2 ; ˝ \ ˝2 ˙ and ˝˜ is regular : Consider now ! ˝2 , and obtain a controllability result on ˝˜ via interior control located in ! (one has, of ˜ course, to extend initial and final states from ˝ to ˝). Consider y a solution of this problem, driving the system from y0 to y1 , in the case of exact controllability, for instance. Then one gets a solution of the boundary controllability problem on ˝, by taking the restriction of y on ˝, and by fixing the trace of y on ˙ as the corresponding control (in the case of Dirichlet boundary conditions), the normal derivative in the case of Neumann boundary conditions, etc. Conversely, when one has some boundary controllability result, one can obtain an interior control result in the following way. Consider the problem in ˝ with interior control distributed in !. Solve the boundary control problem in ˝ n ! via boundary controls in @!. Consider y the solution of this problem. Extend properly the solution y to ˝, and as previously, compute the left hand side for the extension, and consider it as a control (it is automatically distributed in !). Of course, in both situations, the regularity of the control that we obtain has to be checked, and this might need a further treatment. 6. Let us also remark that for linear systems, there is no difference between controllability to zero and controllability to trajectories. In that case it is indeed equivalent to bring y0 to y(T) or to bring y0  y(0) to zero. Note that even for linear systems, on the contrary, approximate controllability and exact controllability differ: an affine subspace of an infinite dimensional space can be dense without filling all the space.

Linear Systems In this section, we will briefly describe the theory of controllability for linear systems. The main tool here is the duality between controllability and observability of the adjoint system, see in particular the works by Lions, Russell, and Dolecki and Russell [24,50,51,63,65]. This duality

is also of primary importance for finite-dimensional systems. Let us first describe informally the method of duality for partial differential equations in two cases given in the examples above. Two Examples The two examples that we wish to discuss are the boundary controllability of the wave equation and the interior controllability of the heat equation. The complete answers to these problems have been given by Bardos, Lebeau and Rauch [9] for the wave equation (see also Burq and Gérard [14] and Burq [13] for another proof and a generalization), and independently by Lebeau and Robbiano [48] and Fursikov and Imanuvilov, see [35] for the heat equation. The complete proofs of these deep results are clearly out of the reach of this short presentation, but one can explain rather easily with these examples how the corresponding controllability problems can be transformed into some observability problems. These observability problems consist of proving a certain inequality. We refer for instance to Lions [51] or Zuazua [76] for a more complete introduction to these problems. First, we notice an important difference between the two equations, which will clearly have consequences concerning the controllability problems. It is indeed wellknown that, while the wave equation is a reversible equation, the heat equation is on the contrary irreversible and has a strong regularizing effect. From the latter property, one sees that it is not possible to expect an exact controllability result for the heat equation: outside the control zone !, the state u(T) will be smooth, and in particular one cannot attain an arbitrary state. As a consequence, while it is natural to seek the exact controllability for the wave equation, it will be natural to look either for approximate controllability or controllability to zero as long as the heat equation is concerned. For both systems we will introduce the adjoint system (typically obtained via integration by parts): it is central in the resolution of the control problems of linear equations. In both cases, the adjoint system is written backward in time form. We consider our two examples separately. Wave Equation We first consider the case of the wave equation with boundary control on ˙ and Dirichlet boundary conditions on the rest of the boundary: 8 2 < @ t t v  v D 0 ; v D 1˙ u ; : j@˝ (v; v t )jtD0 D (v0 ; v00 ) :

Infinite Dimensional Controllability

The problem considered is the exact controllability in L2 (˝)  H 1 (˝) (recall that the state of the system is (v; v t )), by means of boundary controls in L2 ((0; T)  ˙ ). For this system, the adjoint system reads: 8 2 ˆ < @ t t j@˝

ˆ :

( ;

˝

Z

(T; x)v 0T dx

˝

 D0; t )jtDT

Z 

D0; (3) D(

T;

0 T)

:

Notice that this adjoint equation is well-posed: here it is trivial since the equation is reversible. The key argument which connects the controllability problem of the equation with the study of the properties of the adjoint system is the following duality formula. It is formally easily obtained by multiplying the equation with the adjoint state and in integrating by parts. One obtains

Z

From (4), we see that reaching (v T ; v 0T ) from (0; 0) will be achieved if and only if the relation

Z (; x)v t (; x)dx  ˝ “ D

T t (; x)v(; x)dx

(0;T)˙

0

@ 1˙ udt d : @n

(4)

In other words, this central formula describes in a simple manner the jump in the evolution of the state of the system in terms of the control, when measured against the dual state. To make the above computation more rigorous, one can consider for dual state the “classical” solutions in C 0 ([0; T]; H01 (˝)) \ C 1 ([0; T]; L2 (˝)) (these solutions are typically obtained by using a diagonalizing basis for the Dirichlet laplacian, or by using evolution semi-group theory), while the solutions of the direct problem for (v0 ; v1 ; u) 2 L2 (˝)  H 1 (˝)  L2 (˙ ) are defined in C 0 ([0; T]; L2 (˝)) \ C 1 ([0; T]; H 1 (˝)) via the transposition method; for more details we refer to the book by Lions [51]. Now, due to the linearity of the system and because we consider the problem of exact controllability (hence things will be different for what concerns the heat equation), it is not difficult to see that it is not restrictive to consider the problem of controllability starting from 0 (that is the problem of reaching any y1 when starting from y0 :D (v0 ; v00 ) D 0). Denote indeed R(T; y0 ) the affine subspace made of states that can be reached from y0 at time T for some control. Then calling y(T) the final state of the system for yjtD0 and u D 0, then one has R(T; y0 ) D y(T) C R(T; 0). Hence R(T; y0 ) D Y , R(T; 0) D Y.

0

(T; x)v T dx “ @ D 1˙ udt d (0;T)˙ @n ˝

(5)

is satisfied for all choices of ( T ; T0 ). On the left-hand side, we have a linear form on ( T ; 0 ) in H 1 (˝)  L2 (˝), while on the right hand side, we 0 T have a bilinear form on (( T ; T0 ); u). Suppose that we make the choice of looking for a control in the form uD

@ 1˙ ; @n

(6)

for some solution of (3). Then one sees, using Riesz’ theorem, that to solve this problem for (v T ; v 0T ) 2 L2 (˝)  H 1 (˝), it is sufficient to prove that the map ( T ; T0 ) 7! k@ /(@n) 1˙ kL 2 ((0;T)˙ ) is a norm equivalent to the H01 (˝)  L2 (˝) one: for some C > 0, k(

T;

0 T )k H 10 (˝)L 2 (˝)

 Ck

@ 1˙ kL 2 ((0;T)˙ ) : (7) @n

This is the observability inequality to be proved to prove the controllability of the wave equation. Let us mention that the inequality in the other sense, that is, the fact that the linear map ( T ; T0 ) 7! @ /(@n) 1˙ is well-defined and continuous from H01 (˝)  L2 (˝) to L2 (@˝) is true but not trivial: it is a “hidden” regularity result, see [66]. When this inequality is proved, a constructive way to 0 select the control is to determine a minimum ( T ; T ) of the functional (

T;

0 T)

“ ˇ ˇ ˇ @ ˇ2 ˇ dt d ˇ ˙ @n ˝ ˛ C h T ; v1 iH 1 H 1  T0 ; v0 L 2 L 2 ;

7! J(

T;

0 T)

:D 0

0

1 2

(8)

then to associate to ( T ; T ) the solution  of (5), and finally to set u as in (6). The way described above to determine a particular control—it is clear that not all controls are in the form (6)—is referred as Lions’s HUM method (see [51]). This particular control can be proved to be optimal in the L2 norm, that is, any other control answering to the controllability problem has a larger norm in L2 ((0; T)  ˙ ). As a matter of fact, looking for the L2 optimal control among those which answer to the problem is a way to justify the choice (6), see [51].

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Infinite Dimensional Controllability

Heat Equation Now let us consider the heat equation with Dirichlet boundary conditions and localized distributed control: 8 < @ t v  v D 1! u ; D0; v : j@˝ vjtD0 D v0 : In this case, we consider in the same way the dual problem: 8 ˆ < @ t    D 0 ; j@˝ D 0 ; (9) ˆ : (T) D  T : Notice that this adjoint equation is well-posed. Here it is very important that the problem is formulated in a backward way: the backward in time setting compensates the opposite sign before the time derivative. It is clear that the above equation is ill-posed when considering initial data at t D 0. In the same way as for the wave equation, multiplying the equation by the adjoint state and integrating by parts yields, at least formally: “ Z Z  T vjtDT dx (0)v0 dx D udtdx: (10) ˝

˝

results; however it requires the analyticity of the coefficients of the operator, and in many situations one cannot use it directly. Let us also mention that as for the exact controllability of the wave equation above, and the zero-controllability for the heat equation below, one can single out a control for the approximate controllability by using a variational approach consisting of minimizing some functional as in (8): see [26,53]. Controllability to zero. Now considering the problem of controllability to zero, we see that, in order that the control u brings the system to 0, it is necessary and sufficient that for all choice of T 2 L2 (˝), we have “

Z 

˝

(0)v0 dx D

Here one would like to reason as for the wave equation, that is, make the choice to look for u in the form u :D  1! ; for some solution  of the adjoint system. But here the application

(0;T)!

Note that standard methods yield regular solutions for both direct and adjoint equations when v0 and  T belong to L2 (˝), and u belongs to L2 ((0; T)  !). Now let us discuss the approximate controllability and the controllability to zero problems separately. Approximate controllability. Because of linearity, the approximate controllability is equivalent to the approximate controllability starting from 0. Now the set R(0; T) of all final states v(T) which can be reached from 0, via controls u 2 L2 ((0; T)  !), is a vector subspace of L2 (˝). The density of R(0; T) in L2 (˝) amounts to the existence of a nontrivial element in (R(0; T))? . Considering  T such an element, and introducing it in (10) together with v0 D 0, we see that this involves the existence of a nontrivial solution of (9), satisfying j(0;T)! D 0 : Hence to prove the approximate controllability, we have to prove that there is no such nontrivial solution, that is, we have to establish a unique continuation result. In this case, this can be proved by using Holmgren’s unique continuation principle (see for instance [40]), which establishes the result. Note in passing that Holmgren’s theorem is a very general and important tool to prove unique continuation

udtdx : (0;T)!

2

!1

“

2

2

N :  T 2 L (˝) 7!

 dtdx

:

(0;T)!

determines a norm (as seen from the above unique continuation result), but this norm is no longer equivalent to the usual L2 (˝) norm (if it was, one could establish an exact controllability result!). A way to overcome the problem is to introduce the Hilbert space X obtained by completing L2 (˝) for the norm N. In this way, we see that to solve the zero-controllability problem, it is sufficient to prove that the linear mapping  T 7! (0) is continuous with respect to the norm N: for some C > 0, k(0)kL 2 (˝)  Ck 1! kL 2 ((0;T)!) :

(11)

This is precisely the observability inequality which one has to prove to establish the zero controllability of the heat equation. It is weaker than the observability inequality when the left-hand side is k(T)kL 2 (˝) (which as we noticed is false). When this inequality is proven, a constructive way to determine a suitable control is to determine a minimum  T in X of the functional  T 7!

1 2

“ (0;T)!

jj2 dtdx C

Z ˝

(0)v0 dx ;

Infinite Dimensional Controllability

then to associate to  T the solution  of (3), and finally to set u :D  1! : (That the mapping  T 7!  1! can be extended to a mapping from X to L2 (!) comes from the definition of X.) So in both situations of exact controllability and controllability to zero, one has to establish a certain inequality in order to get the result; and to prove approximate controllability, one has to establish a unique continuation result. This turns out to be very general, as described in Paragraph Subsect. “Abstract Approach”. Remarks Let us mention that concerning the heat equation, the zero-controllability holds for any time T > 0 and for any nontrivial control zone !, as shown by means of a spectral method—see Lebeau and Robbiano [48], or by using a global Carleman estimate in order to prove the observability property—see Fursikov and Imanuvilov [35]. That the zero-controllability property does not require the time T or the control zone ! to be large enough is natural since parabolic equations have an infinite speed of propagation; hence one should not require much time for the information to propagate from the control zone to the whole domain. The zero controllability of the heat equation can be extended to a very wide class of parabolic equations; for such results global Carleman estimates play a central role, see for instance the reference book by Fursikov and Imanuvilov [35] and the review article by Fernández-Cara and Guerrero [28]. Note also that Carleman estimates are not used only in the context of parabolic equations: see for instance [35] and Zhang [70]. Let us also mention two other approaches for the controllability of the one-dimensional heat equation: the method of moments of Fattorini and Russell (see [27] and a brief description below), and the method of Laroche, Martin and Rouchon [45] to get an explicit approximate control, based on the idea of “flatness” (as introduced by Fliess, Lévine, Rouchon and Martin [33]). On the other hand, the controllability for the wave equation does not hold for any T or any ˙ . Roughly speaking, the result of Bardos, Lebeau and Rauch [9] states that the controllability property holds if and only if every ray of the geometric optics in the domain (reflecting on the boundary) meets the control zone during the time interval [0; T]. In this case, it is also natural that the time should be large enough, because of the finite speed of propagation of the equation: one has to wait for the information coming from the control zone to influence the whole domain. Let us emphasize that in some cases, the geometry

of the domain and the control zone is such that the controllability property does not hold, no matter how long the time T is. An example of this is for instance a circle in which some antipodal regions both belong to the uncontrolled part of the boundary. This is due to the existence of Gaussian beams, that is, solutions that are concentrated along some ray of the geometrical optics (and decay exponentially away from it); when this ray does not meet the control zone, this contradicts in particular (7). The result of [9] relies on microlocal analysis. Note that another important tool used for proving observability inequalities is a multiplier method (see in particular [42,51,60]): however in the case of the wave equation, this can be done only in particular geometric situations. Abstract Approach The duality between the controllability of a system and the observability of its adjoint system turns out to be very general, and can be described in an abstract form due to Dolecki and Russell [24]. For more recent references see [46,69]. Here we will only describe a particular simplified form of the general setting of [24]. Consider the following system y˙ D Ay C Bu ;

(12)

where the state y belongs to a certain Hilbert space H, on which A, which is densely defined and closed, generates a strongly continuous semi-group of bounded operators. The operator B belongs to L(U; H), where U is also a Hilbert space. The solutions of (12) are given by the method of variation of constants so that Z T e(T)A Bu()d : y(T) D e tA y(0) C 0

We naturally associate with the control equation (12), the following observation system:

z˙ D A z ; z(T) D z T ; (13) c :D B z : The dual operator A generates a strongly continuous semi-group for negative times, and c in L2 (0; T; U) is obtained by 

c(t) D B e(Tt)A z T : In the above system, the dynamics of the (adjoint) state z is free, and c is called the observed quantity of the system. The core of the method is to connect the controllability properties of (12) with observability properties of (13) such as described below.

761

762

Infinite Dimensional Controllability

Definition 6 The system (13) is said to satisfy the unique continuation property if and only if the following implication holds true: c D 0 in [0; T] H) z D 0 in [0; T] : Definition 7 The system (13) is said to be observable if and only if there exists C > 0 such that the following inequality is valid for all solutions z of (13): kz(T)kH  CkckL 2 (0;T;U ) :

(14)

Hence the equivalences 1 and 3 are derived from a classical result from functional analysis (see for instance the book of Brezis [12]): the range of S is dense if and only if S  is one-to-one; S is surjective if and only if S  satisfies for some C > 0: kz T k  CkS  z T k ; that is, when (14) is valid. 2. In this case, still due to linearity, the zero-controllability for system (12) is equivalent to the following inclusion: Range (e TA ) Range (S) :

Definition 8 The system (13) is said to be observable at time 0 if and only if there exists C > 0 such that the following inequality is valid for all solutions z of (13): kz(0)kH  CkckL 2 (0;T;U ) :

(15)

k(e TA ) hk  CkS  hk ;

The main property is the following. Duality property 1. The exact controllability of (12) is equivalent to the observability of (13). 2. The zero controllability of (12) is equivalent to the observability at time 0 of (13). 3. The approximate controllability of (12) in H is equivalent to the unique continuation property for (13). Brief explanation of the duality property. The main fact is the following duality formula hy(T); z(T)i H  hy(0); z(0)i H Z TD E  u(); B e(T)A z(T) d ; D U

0

(16)

which in fact can be used to define the solutions of (12) by transposition. 1 and 3. Now it is rather clear that by linearity, we can reduce the problems of exact and approximate controllability to the ones when y(0) D 0. Now the property of exact controllability for system (12) is equivalent to the surjectivity of the operator S : u 2 L2 (0; T; U) 7!

Z

T

e(T)A Bu() d 2 H ;

0

and the approximate controllability is equivalent to the density of its range. By (16) its adjoint operator is S  : z T 2 H 7!

Z 0

T

In this case there is also a functional analysis result (generalizing the one cited above) which asserts the equivalence of this property with the existence of C > 0 such that



B e(T)A z T d 2 L2 (0; T; U) :

see Dolecki and Russell [24] and Douglas [25] for more general results. It follows that in many situations, the exact controllability of a system is proved by establishing an observability inequality on the adjoint system. But this general method is not the final point of the theory: not only is this only valid for linear systems, but importantly as well, it turns out that these types of inequalities are in general very difficult to establish. In general when a result of exact controllability is established by using this duality, the largest part of the proof is devoted to establishing the observability inequality. Another information given by the observability is an estimate of the size of the control. Indeed, following the lines of the proof of the correspondence between observability and controllability, one can see that one can find a control u which satisfies:  Case of exact controllability: kukL 2 (0;T;U)  Cobs ky1  e TA y0 kH ;  Case of zero controllability: kukL 2 (0;T;U)  Cobs ky0 kH ; where in both cases Cobs determines a constant for which the corresponding observability inequality is true. (Obviously, not all the controls answering to the question satisfy this estimate). This gives an upper bound on the size of a possible control. This can give some precise estimates on the cost of the control in terms of the parameters of the problem, see for instance the works of Fernández-Cara and Zuazua [30] and Miller [58] and Seidman [66] concerning the heat equation.

Infinite Dimensional Controllability

Some Different Methods Herein we will discuss two methods that do not rely on controllability/observability duality. This does not pretend to give a general vision of all the techniques that can be used in problems of controllability of linear partial differential equations. We just mention them in order to show that in some situations, duality may not be the only tool available. Characteristics First, let us mention that in certain situations, one may use a characteristics method (see e. g. [22]). A very simple example is the first-dimensional wave equation, 8 < v t t  vx x D 0 ; v D0; : xjxD0 v xjxD1 D u(t) ;

v t C A(x)v x C B(x)v D 0 ;

L2 (0; T).

where the control is u(t) 2 This is taken from Russell [62]. The problem is transformed into @w D @t



0 1

1 0



@w @x

w :D

with 

w1 w2



w2 (t; 0) D 0

 :D

vt vx

 ;

and w2 (t; 0) D u(t) :

Of course, this means that w1 w2 and w1 Cw2 are constant along characteristics, which are straight lines of slope 1 and 1, respectively (this is d’Alembert’s decomposition). Now one can deduce an explicit appropriate control for the controllability problem in [0; 1] for T > 2, by constructing the solution w of the problem directly (and one takes the values of w2 at x D 1 as the “resulting” control u). The function w is completely determined from the initial and final values in the domains of determinacy D1 and D2 : D1 :D f(t; x) 2 [0; T]  [0; 1] / x C t  1g

and

D2 :D f(t; x) 2 [0; T]  [0; 1] / t  x  T  1g : That T > 2 means that these two domains do not intersect. Now it suffices to complete the solution w in D3 :D [0; T]  [0; 1] n (D1 [ D2 ). For that, one chooses w1 (0; t) arbitrarily in the part ` of the axis f0g  [0; 1] located between D1 and D2 , that is for xD0

the symmetric role of x and t, one considers x as the time. Then the initial condition is prescribed on `, as well as the boundary conditions on the two characteristic lines x C t D 1 and x  t D T  1. One can solve elementarily this problem by using the characteristics, and this finishes the construction of w, and hence of u. Note that as a matter of fact, the observability inequality in this (one-dimensional) case is also elementary to establish, by relying on Fourier series or on d’Alembert’s decomposition, see for instance [23]. The method of characteristics described above can be generalized in broader situations, including for instance the problem of boundary controllability of one-dimensional linear hyperbolic systems

and 1  t  T  1 :

Once this choice is made, it is not difficult to solve the Goursat problem consisting of extending w in D3 : using

where A is a real symmetric matrix with eigenvalues bounded away from zero, and A and B are smooth; see for instance Russell [65]. As a matter of fact, in some cases this method can be used to establish the observability inequality from the controllability result (while in most cases the other implication in the equivalence is used). Of course, the method of characteristics may be very useful for transport equations @f C v:r f D g @t

or

@f C div (v f ) D g : @t

An example of this is the controllability of the Vlasov– Poisson equation, see [36]. Let us finally mention that this method is found also to tackle directly several nonlinear problems, as we will see later. Moments Another method which we would like to briefly discuss is the method of moments (see for instance Avdonin and Ivanov [4] for a general reference, see also Russell [65]), which can appear in many situations; in particular this method was used by Fattorini and Russell to prove the controllability of the heat equation in one space dimension, see [27]. Consider for instance the problem of controllability of the one-dimensional heat equation

v t  v x x D g(x)u(t) ; vj[0;T]f0;1g D 0 :

Actually in [27], much more general situations are considered: in particular it concerns more general parabolic equations and boundary conditions, and boundary controls can also be included in the discussion. It is elementary to develop the solution in the L2 (0; 1) orthonormal basis (sin(k x)) k2N  . One obtains that the

763

764

Infinite Dimensional Controllability

state zero is reached at time T > 0 if and only if X

2

ek T v k sin(k x)

k>0

D

XZ

T

ek

2 (Tt)

g k u(t) sin(k x)dt ;

k>0 0

where vk and g k are the coordinates of v0 and g in the basis. Clearly, this means that we have to find u such that for all k 2 N, Z

T

ek

2 (Tt)

2

g k u(t)dt D ek T v k :

0

The classical Muntz–Szász theorem states that for an increasing family (n )n2N  of positive numbers, the family fen t ; n 2 N  g is dense in L2 (0; T) if and only if X 1 D C1 ;  n0 n and in the opposite case, the family is independent and spans a proper closed subspace of L2 (0; T). Here the exponential family which we consider is n D n2 and we are in the second situation. The same method applies for other problems in which n cannot be completely computed but is a perturbation of n2 ; this allows us to treat a wider class of problems. Now in this situation (see e. g. [65]), one can construct in L2 (0; T) a biorthogonal family to fen t ; n 2 N  g, that is a family (p n (t))n2N  2 (L2 (0; T))N satisfying Z

T

2

p n (t)ek t dt D ı kn :

0

Once such a family is obtained, one has formally the following solution, under the natural assumption that jg k j  ck ˛ : u(t) D

X k2N 



ek T v k p k (t) : gk

which allows us to conclude.

The most frequent method for dealing with control problems of nonlinear systems is the natural one (as for the Cauchy problem): one has to linearize (at least in some sense) the equation, try to prove some controllability result on the linear equation (for instance by using the duality principle), and then try to pass to the nonlinear system via typical methods such as inverse mapping theorem, fixed point theory, iterative schemes. . . As for the usual inverse mapping theorem, it is natural to hope for a local result from the controllability of the linear problem. Here we find an important difference between linear and nonlinear systems: while for linear systems, no distinction has to be made between local and global results, concerning the nonlinear systems, the two problems are really of different nature. One should probably not expect a very general result indicating that the controllability of the linearized equation involves the local controllability of the nonlinear system. The linearization principle is a general approach which one has to adapt to the different situations that one can meet. We give below some examples that present to the reader some existing approaches. Some Linearization Situations Let us first discuss some typical situations where the linearized equation has good controllability properties, and one can hope to get a result (in general local) from this information. In some situations where the nonlinearity is not too strong, one can hope to get global results. As previously, the situation where the underlying linear equation is reversible and the situation when it is not have to be distinguished. We briefly describe this in two different examples. Semilinear Wave Equation Let us discuss first an example with the wave equation. This is borrowed from the work of Zuazua [71,72], where the equation considered is

To actually get a control in L2 (0; T), one has to estimate kp k kL 2 , in order that the above sum is well-defined in L2 . In [27] it is proven that one can construct pk in such a way that kp k kL 2 (0;T)  K0 exp(K1 ! k ) ;

Nonlinear Systems

! k :D

p k ;

v t t  v x x C f (v) D u 1! in [0; 1] ; vjxD0 D 0; vjxD1 D 0 ;

where ! :D (l1 ; l2 ) is the control zone and the nonlinearity f 2 C 1 (R; R) is at most linear at infinity (see however the remark below) in the sense that for some C > 0, j f (x)j  C(1 C jxj) on R :

(17)

The global exact controllability in H01  L2 (recall that the state of the system is (v; v t )) by means of a control in

Infinite Dimensional Controllability

L2 ((0; T)  !) is proved by using the following linearization technique. In [72], it is proven that the following linearized equation:

v t t  v x x C a(x)v D u 1! in [0; 1] ; vjxD0 D 0; vjxD1 D 0 :

is controllable in H01 (0; 1)  L2 (0; 1) through u(t) 2 L2 ((0; T)  !), for times T > 2 max(l1 ; 1  l2 ), for any a 2 L1 ((0; T)  (0; 1)). The corresponding observability inequality is

Using the above information, one can deduce estimates for vˆ in C 0 ([0; T]; L2 (0; 1)) \ L2 (0; T; H01 ) independently of v, and then show by Schauder’s fixed point theorem that the above process has a fixed point, which shows a global controllability result for this semilinear wave equation. Remark 1 As a matter of fact, [72] proves the global exact controllability for f satisfying the weaker assumption lim

jxj!C1

k(

T;

0 T )k L 2 (˝)H 1 (˝)

 Cobs k 1! kL 2 ((0;T)!) :

Moreover, the observability constant that one can obtain can be bounded in the following way:

lim inf

f (x) D f (0) C x g(x) ;

vˆt t  vˆx x C vˆ g(v) D  f (0) C u 1! in [0; 1] ; vˆjxD0 D 0; vˆjxD1 D 0 :

(Note that the “drift” term f (0) on the right hand side can be integrated in the final state—just consider the solution of the above system with u D 0 and vˆjtD0 D 0 and withdraw it—or integrated in the formulation (5)). Here an issue is that, as we recalled earlier, a controllability problem has almost never a unique solution. The method in [72] consists of selecting a particular control, which is the one of smallest L2 -norm. Taking the fact that g is bounded and (18) into account, this particular control satisfies kukL 2 (0;T)  C(kv0 kH 1 C kv00 kL 2 C kv1 kH 1 C kv10 kL 2 C j f (0)j); for some C > 0 independent of v.

for some p > 2 (and ! 6D (0; 1)) then the system is not globally controllable due to blow-up phenomena. To obtain this result one has to use Leray–Schauder’s degree theory instead of Schauder’s fixed point theorem. For analogous conditions on the semilinear heat equation, see Fernández-Cara and Zuazua [31]. Burgers Equation Now let us discuss a parabolic example, namely the local controllability to trajectories of the viscous Burgers equation:

where g is continuous and bounded. The idea is to associate to any v 2 L1 ((0; T)  (0; 1)) a solution of the linear control problem

0

j f (x)j >0 (1 C jxj) log p (jxj)

(18)

where ˛ is nondecreasing in the second variable. One would like to describe a fixed-point scheme as follows. Given v, one considers the linearized problem around v, solves this problem and deduces a solution vˆ of the problem of controllability from (v0 ; v00 ) to (v1 ; v10 ) in time T. More precisely, the scheme is constructed in the following way. We write

D0:

This is optimal since it is also proven in [72] that if

jxj!C1

Cobs  ˛(T; kak1 ) ;

j f (x)j (1 C jxj) log2 (jxj)

0

v t C (v 2 )x  v x x D 0 in (0; T)  (0; 1) ; vjxD0 D u0 (t) and vjxD1 D u1 (t) ;

(19)

controlled on the boundary via u0 and u1 . This is taken from Fursikov and Imanuvilov [34]. Now the linear result concerns the zero-controllability via boundary controls of the system

v t C (zv)x  v x x D 0 in (0; T)  (0; 1) ; vjxD0 D u0 (t) and vjxD1 D u1 (t) :

(20)

Consider X the Hilbert space composed of functions z in L2 (0; T; H 2 (0; 1)) such that z t 2 L2 (0; 1; L2 (0; 1)). In [34] it is proved that given v0 2 H 1 (0; 1) and T > 0, one can construct a map which to any z in X, associates v 2 X such that v is a solution of (20) which drives v0 to 0 during time interval [0; T], and moreover this map is compact from X to X. As in the previous situation, the particular control (u0 ; u1 ) has to be singled out in order for the above mapping to be single-valued (and compact). But here, the criterion is not quite the optimality in L2 -norm of the control. The idea is the following: first, one transforms the controllability problem for (20) from y0 to 0 into

765

766

Infinite Dimensional Controllability

a problem of “driving” 0 to 0 for a problem with righthand side:

w t C (zw)x  w x x D f0 in (0; T)  (0; 1) ; (21) wjxD0 D u0 (t) and wjxD1 D u1 (t) ; for some f 0 supported in (T/3; 2T/3)  (0; 1). For this, one introduces  2 C 1 ([0; T]; R) such that  D 1 during [0; T/3] and  D 0 during [2T/3; T], and vˆ the solution of (20) starting from v0 with u0 D u1 D 0, and considers w :D v  ˆv . Now the operator mentioned above is the one which to z associates the solution of the controllability problem which minimizes the L2 -norm of w among all the solutions of this controllability problem. The optimality criterion yields a certain form for the solution. That the corresponding control exists (and is unique) relies on a Carleman estimate, see [34] (moreover this allows estimates on the size of w). As a matter of fact, to obtain the compactness of the operator, one extends the domain, solves the above problem in this extended domain, and then uses an interior parabolic regularity result to have bounds in smaller spaces, we refer to [34] for more details. Once the operator is obtained, the local controllability to trajectories is obtained as follows. One considers v a trajectory of the system (19), belonging to X. Withdrawing (19) for v to (19) for the unknown, we see that the problem to solve through boundary controls is

y t  y x x C [(2v C y)y]x D f0 in (0; T)  (0; 1) ; yjxD0 D v0  v(0) and yjtDT D 0 :

Now consider v0 2 H 1 (0; 1) such that kv0  v(0)kH 1 (0;1) < r ; for r > 0 to be chosen. To any y 2 X, one associates the solution of the controllability problem (21) constructed above, driving v0  vˆ(0) to 0, for z :D (2ˆv C y). The estimates on the solution of the control problem allow one to establish that the unit ball of X is stable by this process provided r is small enough. The compactness of the process is already proved, so Schauder’s theorem allows one to conclude. Note that it is also proved in [34] that the global approximate controllability does not hold. Some Other Examples Let us also mention two other approaches that may be useful in this type of situation. The first is the use of Kakutani–Tikhonov fixed-point theorem for multivalued maps (cf. for instance [68]), see in particular [38,39] and [26]. Such a technique is particularly

useful, because it avoids the selection process of a particular control. One associates to v the set T(v) of all vˆ solving the controllability problem for the equation linearized around v, with all possible controls (in a suitable class). Then under appropriate conditions, one can find a fixed point in the sense that v 2 T(v). Another approach that is very promising is the use of a Nash–Moser process, see in particular the work [10] by Beauchard. In this paper the controllability of a Schrödinger equation via a bilinear control is considered. In that case, one can solve some particular linearized equation (as a matter of fact, the return method described in the next paragraph is used), but with a loss of derivative; as a consequence the approaches described above fail, but the use of Nash–Moser’s theorem allows one to get a result. Note finally that in certain other functional settings, the controllability of this system fails, as shown by using a general result on bilinear control by Ball, Marsden and Slemrod [5]. The Return Method It occurs that in some situations, the linearized equation is not systematically controllable, and one cannot hope by applying directly the above process to get even local exact controllability. The return method was introduced by Coron to deal with such situations (see in particular [18]). As a matter of fact, this method can be useful even when the linearized equation is controllable. The principle of the method is the following: find a particular trajectory y of the nonlinear system, starting at some base point (typically 0) and returning to it, such that the linearized equation around this is controllable. In that case, one can hope to find a solution of the nonlinear local controllability problem close to y. A typical situation of this is the two-dimensional Euler equation for incompressible inviscid fluids (see Coron [16]), which reads 8 < @ t y C (y:r)y D r p in ˝ ; divy D 0 in ˝ ; (22) : y:n D 0 on @˝ n ˙ ; where the unknown is the velocity field y : ˝ ! R2 (the pressure p can be eliminated from the equation), ˝ is a regular bounded domain (simply connected to simplify), n is the unit outward normal on the boundary, and ˙ @˝ is the control zone. On ˙ , the natural control which can be assigned is the normal velocity y:n and the vorticity ! :D curly :D @1 y 2  @2 y 1 at “entering” points, that is points where y:n < 0. Let us discuss this example in an informal way. As noticed by J.-L. Lions, the linearized equation around the null

Infinite Dimensional Controllability

state 8 < @ t y D r p in ˝ ; divy D 0 in ˝; : y:n D 0 on @˝ n ˙ ; is trivially not controllable (even approximately). Now the main goal is to find the trajectory y such that the linearized equation near y is controllable. It will be easier to work with the vorticity formulation

@ t ! C (y:r)! D 0 in ˝ ; curl y D !; divy D 0 :

(23)

In fact, one can show that assigning y(T) is equivalent to assigning both !(T) in ˝ and y(T):n on ˙ , and since this latter is a part of the control, it is sufficient to know how to assign the vorticity of the final state. We can linearize (23) in the following way: to y one associates yˆ through

@ t ! C (y:r)! D 0 in ˝ ; curl yˆ D !; div yˆ D 0 :

(24)

Considering Eq. (24), we see that if the flow of y is such that any point in ˝ at time T “comes from” the control zone ˙ , then one can assign easily ! through a method of characteristics. Hence one has to find a solution y of the system, starting and ending at 0, and such that in its flow, all points in ˝ at time T, come from ˙ at some stage between times 0 and T. Then a simple Gronwall-type argument shows that this property holds for y in a neighborhood of y, hence Eq. (24) is controllable in a neighborhood of y. Then a fixed-point scheme allows one to prove a controllability result locally around 0. But the Euler equation has some time-scale invariance: y(t; x) is a solution on [0; T] ) y  (t; x) :D 1 y(1 t; x) is a solution on [0; T] : Hence given y0 and y1 , one can solve the problem of driving y0 to y1 for  small enough. Changing the variables, one see that it is possible to drive y0 to y1 in time T, that is, in smaller times. Hence one deduces a global controllability result from the above local result. As a consequence, the central part of the proof is to find the function y. This is done by considering a special type of solution of the Euler equation, namely the potential solution: any y :D r (t; x) with regular satisfying x (t; x) D 0 in ˝;

8t 2 [0; T];

@n D 0 on @˝;

8t 2 [0; T] ;

satisfies (22). In [16] it is proven that there exists some satisfying the above equation and whose flow makes all points at time T come from ˙ . This concludes the argument. This method has been used in various situations; see [18] and references therein. Let us underline that this method can be of great interest even in the cases where the linearized equation is actually controllable. An important example obtained by Coron concerns the Navier–Stokes equation and is given in [17] (see also [19]): here the return method is used to prove some global approximate result, while the linearized equation is actually controllable but yields in general a local controllability result (see in particular [29,35,41]). Some Other Methods Let us finally briefly mention that linearizing the equation (whether using the standard approach or the return method) is not systematically the only approach to the controllability of a nonlinear system. Sometimes, one can “work at the nonlinear level.” An important example is the control of one-dimensional hyperbolic systems: v t C A(v)v x D F(v); v : [0; T]  [0; 1] ! Rn ;

(25)

via the boundary controls. In (25), A satisfies the hyperbolicity property that it possesses at every point n real eigenvalues; these are moreover supposed to be strictly separated from 0. In the case of regular C1 solutions, this was approached by a method of characteristics to give general local results, see in particular the works by Cirinà [15] and Li and Rao [49]. Interestingly enough, the linear tool of duality between observability and controllability has some counterpart in this particular nonlinear setting, see Li [55]. In the context of weak entropy solutions, some results for particular systems have been obtained via the ad hoc method of front-tracking, see in particular Ancona, Bressan and Coclite [2], the author [37] and references therein. Other nonlinear tools can be found in Coron’s book [18]. Let us mention two of them. The first one is power series expansion. It consists of considering, instead of the linearization of the system, the development to higher order of the nonlinearity. In such a way, one can hope to attain the directions which are unreachable for the linearized system. This has for instance applications for the Korteweg–de-Vries equation, see Coron and Crépeau [20] and the earlier work by Rosier [61]. The other one is quasistatic deformations. The general idea is to find an (explicit) “almost trajectory” (y("t); u("t)) during [0; T/"] of

767

768

Infinite Dimensional Controllability

the control system y˙ D f (y; u), in the sense that d y("t)  f (y("t); u("t)) dt is of order " (due to the “slow” motion). Typically, the trajectory (y; u) is composed of equilibrium states (that is f (y(); u()) D 0). In such a way, one can hope to connect y(0) to a state close to y(T), and then to exactly y(T) via a local result. This was used for instance by Coron and Trélat in the context of a semilinear heat equation, see [21]. Finally, let us cite a recent approach by Agrachev and Sarychev [1] (see also [67]), which uses a generalization of the “Lie bracket” approach (a standard approach from finite-dimensional nonlinear systems), to obtain some global approximate results on the Navier–Stokes equation with a finite-dimensional (low modes) affine control.

Future Directions There are many future challenging problems for control theory. Controllability is one of the possible approaches to try to construct strategies for managing complex systems. On the road towards systems with increasing complexity, many additional difficulties have to be considered in the design of the control law: one should expect the control to take into account the possible errors of modelization, of measurement of the state, of the control device, etc. All of these should be included in the model so some robustness of the control can be expected, and to make it closer to the real world. Of course, numerics should play an important role in the direction of applications. Moreover, one should expect more and more complex systems, such as environmental or biological systems (how are regulation mechanisms designed in a natural organism?), to be approached from this point of view. We refer for instance to [32,59] for some discussions on some perspectives of the theory.

Some Other Problems As we mentioned earlier, many aspects of the theory have not been referred to herein. Let us cite some of them. Concerning the connection between the problem of controllability and the problem of stabilization, we refer to Lions [50], Russell [63,64,65] and Lasiecka and Triggiani [47]. A very important problem which has recently attracted great interest is the problem of numerics and discretization of distributed systems (see in particular [74,75] and references therein). The main difficulty here comes from the fact that the operations of discretizing the equation and controlling it do not commute. Other important questions considered in particular in the second volume of Lions’s book [51] are the problems of singular perturbations, homogenization, thin domains in the context controllability problems. Here the questions are the following: considering a “perturbed” system, for which we have some controllability result, how does the solution of the controllability problem (for instance associated to the L2 -optimal control) behave as the system converges to its limit? This kind of question is the source of numerous new problems. Another subject is the controllability of equations with some stochastic terms (see for instance [8]). One can also consider systems with memory (see again [51] or [7]). Finally, let us mention that many partial differential equations widely studied from the point of view of Cauchy theory, still have not been studied from the point of view of controllability. The reader looking for more discussion on the subject can consider the references below.

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33. Fliess M, Lévine J, Martin P, Rouchon P (1995) Flatness and defect of non-linear systems: introductory theory and examples. Int J Control 61(6):1327–1361 34. Fursikov A, Imanuvilov OY (1995) On controllability of certain systems simulating a fluid flow, Flow control, Minneapolis, 1992. IMA Math Appl, vol 68. Springer, New York, pp 149–184 35. Fursikov A, Imanuvilov OY (1996) Controllability of evolution equations. Lecture Notes Series, vol 34. Seoul National University Research Institute of Mathematics Global Analysis Research Center, Seoul 36. Glass O (2003) On the controllability of the Vlasov-Poisson system. J Differ Equ 195(2):332–379 37. Glass O (2007) On the controllability of the 1-D isentropic Euler equation. J Eur Math Soc 9:427–486 38. Henry J (1978) Controllability of some non linear parabolic equations. In: Ruberti A (ed) Distributed Parameter Systems: Modelling and Identification. Proceedings of the IFIP working conference, Rome, 21–24 June 1976. Lecture Notes in Control and Information Sciences, vol 1. Springer, Berlin 39. Henry J (1977) Étude de la contrôlabilité de certaines équations paraboliques non linéaires. Thèse, Paris VI 40. Hörmander L (1983) The analysis of linear partial differential operators, vol I. Grundlehren der mathematischen Wissenschaften, vol 256. Springer, Berlin 41. Imanuvilov OY (2001) Remarks on exact controllability for the Navier–Stokes equations. ESAIM Control Optim Calc Var 6:39– 72 42. Komornik V (1994) Exact controllability and stabilization, The multiplier method. RAM: Research in Applied Mathematics. Masson, Paris 43. Komornik V, Loreti P (2005) Fourier series in control theory. Springer Monographs in Mathematics. Springer, New York 44. Lagnese JE, Leugering G, Schmidt EJPG (1994) Modeling, analysis and control of dynamic elastic multi-link structures. Systems and Control: Foundations and Applications. Birkhäuser, Boston 45. Laroche B, Martin P, Rouchon P (2000) Motion planning for the heat equation. Int J Robust Nonlinear Control 10(8):629–643 46. Lasiecka I, Triggiani R (1983) Regularity of hyperbolic equations under L2 (0; T; L2 ( ))-Dirichlet boundary terms. Appl Math Optim 10(3):275–286 47. Lasiecka I, Triggiani R (2000) Control theory for partial differential equations: continuous and approximation theories, vol I. Abstract parabolic systems and II, Abstract hyperbolic-like systems over a finite time horizon. Encyclopedia of Mathematics and its Applications, vols 74, 75. Cambridge University Press, Cambridge 48. Lebeau G, Robbiano L (1995) Contrôle exact de l’équation de la chaleur. Comm PDE 20:335–356 49. Li TT, Rao BP (2003) Exact boundary controllability for quasilinear hyperbolic systems. SIAM J Control Optim 41(6):1748– 1755 50. Lions JL (1988) Exact controllability, stabilizability and perturbations for distributed systems. SIAM Rev 30:1–68 51. Lions JL (1988) Contrôlabilité exacte, stabilisation et perturbations de systèmes distribués, Tomes 1, 2. RMA 8, 9. Masson, Paris 52. Lions JL (1990) Are there connections between turbulence and controllability? In: Bensoussan A, Lions JL (eds) Analysis and Optimization of Systems. Lecture Notes Control and Inform Sci, vol 144. Springer, Berlin

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Inverse Scattering Transform and the Theory of Solitons

Inverse Scattering Transform and the Theory of Solitons TUNCAY AKTOSUN Department of Mathematics, University of Texas at Arlington, Arlington, USA Article Outline Glossary Definition of the Subject Introduction Inverse Scattering Transform The Lax Method The AKNS Method Direct Scattering Problem Time Evolution of the Scattering Data Inverse Scattering Problem Solitons Future Directions Bibliography Glossary AKNS method A method introduced by Ablowitz, Kaup, Newell, and Segur in 1973 that identifies the nonlinear partial differential equation (NPDE) associated with a given first-order system of linear ordinary differential equations (LODEs) so that the initial value problem (IVP) for that NPDE can be solved by the inverse scattering transform (IST) method. Direct scattering problem The problem of determining the scattering data corresponding to a given potential in a differential equation. Integrability A NPDE is said to be integrable if its IVP can be solved via an IST. Inverse scattering problem The problem of determining the potential that corresponds to a given set of scattering data in a differential equation. Inverse scattering transform A method introduced in 1967 by Gardner, Greene, Kruskal, and Miura that yields a solution to the IVP for a NPDE with the help of the solutions to the direct and inverse scattering problems for an associated LODE. Lax method A method introduced by Lax in 1968 that determines the integrable NPDE associated with a given LODE so that the IVP for that NPDE can be solved with the help of an IST. Scattering data The scattering data associated with a LODE usually consists of a reflection coefficient which is a function of the spectral parameter ; a finite number of constants  j that correspond to the poles

of the transmission coefficient in the upper half complex plane, and the bound-state norming constants whose number for each bound-state pole  j is the same as the order of that pole. It is desirable that the potential in the LODE is uniquely determined by the corresponding scattering data and vice versa. Soliton The part of a solution to an integrable NPDE due to a pole of the transmission coefficient in the upper half complex plane. The term soliton was introduced by Zabusky and Kruskal in 1965 to denote a solitary wave pulse with a particle-like behavior in the solution to the Korteweg-de Vries (KdV) equation. Time evolution of the scattering data The evolvement of the scattering data from its initial value S(; 0) at t D 0 to its value S(; t) at a later time t: Definition of the Subject A general theory to solve NPDEs does not seem to exist. However, there are certain NPDEs, usually first order in time, for which the corresponding IVPs can be solved by the IST method. Such NPDEs are sometimes referred to as integrable evolution equations. Some exact solutions to such equations may be available in terms of elementary functions, and such solutions are important to understand nonlinearity better and they may also be useful in testing accuracy of numerical methods to solve such NPDEs. Certain special solutions to some of such NPDEs exhibit particle-like behaviors. A single-soliton solution is usually a localized disturbance that retains its shape but only changes its location in time. A multi-soliton solution consists of several solitons that interact nonlinearly when they are close to each other but come out of such interactions unchanged in shape except for a phase shift. Integrable NPDEs have important physical applications. For example, the KdV equation is used to describe [14,23] surface water waves in long, narrow, shallow canals; it also arises [23] in the description of hydromagnetic waves in a cold plasma, and ion-acoustic waves in anharmonic crystals. The nonlinear Schrödinger (NLS) equation arises in modeling [24] electromagnetic waves in optical fibers as well as surface waves in deep waters. The sine-Gordon equation is helpful [1] in analyzing the magnetic field in a Josephson junction (gap between two superconductors). Introduction The first observation of a soliton was made in 1834 by the Scottish engineer John Scott Russell at the Union Canal between Edinburgh and Glasgow. Russell reported [21] his observation to the British Association of the Advance-

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ment of Science in September 1844, but he did not seem to be successful in convincing the scientific community. For example, his contemporary George Airy, the influential mathematician of the time, did not believe in the existence of solitary water waves [1]. The Dutch mathematician Korteweg and his doctoral student de Vries published [14] a paper in 1895 based on de Vries’ Ph.D. dissertation, in which surface waves in shallow, narrow canals were modeled by what is now known as the KdV equation. The importance of this paper was not understood until 1965 even though it contained as a special solution what is now known as the one-soliton solution. Enrico Fermi in his summer visits to the Los Alamos National Laboratory, together with J. Pasta and S. Ulam, used the computer named Maniac I to computationally analyze a one-dimensional dynamical system of 64 particles in which adjacent particles were joined by springs where the forces also included some nonlinear terms. Their main goal was to determine the rate of approach to the equipartition of energy among different modes of the system. Contrary to their expectations there was little tendency towards the equipartition of energy but instead the almost ongoing recurrence to the initial state, which was puzzling. After Fermi died in November 1954, Pasta and Ulam completed their last few computational examples and finished writing a preprint [11], which was never published as a journal article. This preprint appears in Fermi’s Collected Papers [10] and is also available on the internet [25]. In 1965 Zabusky and Kruskal explained [23] the Fermi-Pasta-Ulam puzzle in terms of solitary wave solutions to the KdV equation. In their numerical analysis they observed “solitary-wave pulses”, named such pulses “solitons” because of their particle-like behavior, and noted that such pulses interact with each other nonlinearly but come out of interactions unaffected in size or shape except for some phase shifts. Such unusual interactions among solitons generated a lot of excitement, but at that time no one knew how to solve the IVP for the KdV equation, except numerically. In 1967 Gardner, Greene, Kruskal, and Miura presented [12] a method, now known as the IST, to solve that IVP, assuming that the initial profile u(x; 0) decays to zero sufficiently rapidly as x ! ˙1: They showed that the integrable NPDE, i. e. the KdV equation, u t  6uu x C u x x x D 0 ;

(1)

is associated with a LODE, i. e. the 1-D Schrödinger equation 

d2 C u(x; t) dx 2

D k2

;

(2)

and that the solution u(x, t) to (1) can be recovered from the initial profile u(x; 0) as explained in the diagram given in Sect. “Inverse Scattering Transform”. They also explained that soliton solutions to the KdV equation correspond to a zero reflection coefficient in the associated scattering data. Note that the subscripts x and t in (1) and throughout denote the partial derivatives with respect to those variables. In 1972 Zakharov and Shabat showed [24] that the IST method is applicable also to the IVP for the NLS equation iu t C u x x C 2ujuj2 D 0 ;

(3)

p where i denotes the imaginary number 1. They proved that the associated LODE is the first-order linear system 8 d ˆ ˆ D i C u(x; t)  ; < dx ˆ ˆ : d D i  u(x; t)  ; dx

(4)

where  is the spectral parameter and an overline denotes complex conjugation. The system in (4) is now known as the Zakharov–Shabat system. Soon afterwards, again in 1972 Wadati showed in a one-page publication [22] that the IVP for the modified Korteweg-de Vries (mKdV) equation u t C 6u2 u x C u x x x D 0 ;

(5)

can be solved with the help of the inverse scattering problem for the linear system 8 d ˆ ˆ D i C u(x; t)  ; < dx ˆ ˆ : d D i  u(x; t)  : dx

(6)

Next, in 1973 Ablowitz, Kaup, Newell, and Segur showed [2,3] that the IVP for the sine-Gordon equation u x t D sin u ; can be solved in the same way by exploiting the inverse scattering problem associated with the linear system 8 d 1 ˆ ˆ D i  u x (x; t)  ; < dx 2 ˆ d 1 ˆ : D i C u x (x; t)  : dx 2 Since then, many other NPDEs have been discovered to be solvable by the IST method.

Inverse Scattering Transform and the Theory of Solitons

direct scattering for LODE at tD0

u(x; 0) ! S(; 0) ? ? ? ?time evolution of scattering data solution to NPDEy y u(x; t)

 S(; t)

inverse scattering for LODE at time t

Inverse Scattering Transform and the Theory of Solitons, Diagram 1 The method of inverse scattering transform

Our review is organized as follows. In the next section we explain the idea behind the IST. Given a LODE known to be associated with an integrable NPDE, there are two primary methods enabling us to determine the corresponding NPDE. We review those two methods, the Lax method and the AKNS method, in Sect. “The Lax Method” and in Sect. “The AKNS Method”, respectively. In Sect. “Direct Scattering” we introduce the scattering data associated with a LODE containing a spectral parameter and a potential, and we illustrate it for the Schrödinger equation and for the Zakharov–Shabat system. In Sect. “Time Evolution of the Scattering Data” we explain the time evolution of the scattering data and indicate how the scattering data sets evolve for those two LODEs. In Sect. “Inverse Scatering Problem” we summarize the Marchenko method to solve the inverse scattering problem for the Schrödinger equation and that for the Zakharov–Shabat system, and we outline how the solutions to the IVPs for the KdV equation and the NLS equation are obtained with the help of the IST. In Sect. “Solitons” we present soliton solutions to the KdV and NLS equations. A brief conclusion is provided in Sect. “Future Directions”.

the direct scattering problem for the LODE. On the other hand, the problem of determining u(x, t) from S(; t) is known as the inverse scattering problem for that LODE. The IST method for an integrable NPDE can be explained with the help of Diagram 1. In order to solve the IVP for the NPDE, i. e. in order to determine u(x, t) from u(x; 0); one needs to perform the following three steps: (i)

Solve the corresponding direct scattering problem for the associated LODE at t D 0; i. e. determine the initial scattering data S(; 0) from the initial potential u(x; 0): (ii) Time evolve the scattering data from its initial value S(; 0) to its value S(; t) at time t: Such an evolution is usually a simple one and is particular to each integrable NPDE. (iii) Solve the corresponding inverse scattering problem for the associated LODE at fixed t; i. e. determine the potential u(x, t) from the scattering data S(; t): It is amazing that the resulting u(x, t) satisfies the integrable NPDE and that the limiting value of u(x, t) as t ! 0 agrees with the initial profile u(x; 0):

Inverse Scattering Transform Certain NPDEs are classified as integrable in the sense that their corresponding IVPs can be solved with the help of an IST. The idea behind the IST method is as follows: Each integrable NPDE is associated with a LODE (or a system of LODEs) containing a parameter  (usually known as the spectral parameter), and the solution u(x, t) to the NPDE appears as a coefficient (usually known as the potential) in the corresponding LODE. In the NPDE the quantities x and t appear as independent variables (usually known as the spatial and temporal coordinates, respectively), and in the LODE x is an independent variable and  and t appear as parameters. It is usually the case that u(x, t) vanishes at each fixed t as x becomes infinite so that a scattering scenario can be created for the related LODE, in which the potential u(x, t) can uniquely be associated with some scattering data S(; t): The problem of determining S(; t) for all  values from u(x, t) given for all x values is known as

The Lax Method In 1968 Peter Lax introduced [15] a method yielding an integrable NPDE corresponding to a given LODE. The basic idea behind the Lax method is the following. Given a linear differential operator L appearing in the spectral problem L D  ; find an operator A (the operators A and L are said to form a Lax pair) such that: The spectral parameter  does not change in time, i. e.  t D 0: (ii) The quantity t  A remains a solution to the same linear problem L D  : (iii) The quantity L t C LA  AL is a multiplication operator, i. e. it is not a differential operator.

(i)

From condition (ii) we get L( t  A ) D ( t  A ) ;

(7)

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Inverse Scattering Transform and the Theory of Solitons

and with the help of L obtain L t  LA

D

D

t

and  t D 0; from (7) we

 A ( ) D @ t ( )  AL

D @ t (L )  AL

D Lt

CL

t

 AL

;

A D 4@3x C 6u@x C 3u x :

(8)

where @ t denotes the partial differential operator with respect to t: After canceling the term L t on the left and right hand sides of (8), we get (L t C LA  AL)

D0;

which, because of (iii), yields

L :D @2x C u(x; t) ;

(10)

where the notation :D is used to indicate a definition so that the quantity on the left should be understood as the quantity on the right hand side. Given the linear differential operator L defined as in (10), let us try to determine the associated operator A by assuming that it has the form A D ˛3 @3x C ˛2 @2x C ˛1 @x C ˛0 ;

(11)

where the coefficients ˛ j with j D 0; 1; 2; 3 may depend on x and t; but not on the spectral parameter : Note that L t D u t : Using (10) and (11) in (9), we obtain (

C(

C(

)@3x

C(

)@2x

C ( )@x C ( ) D 0 ;

(12)

where, because of (iii), each coefficient denoted by () must vanish. The coefficient of @5x vanishes automatically. Setj ting the coefficients of @x to zero for j D 4; 3; 2; 1; we obtain ˛3 D c1 ;

˛2 D c2 ;

For the Zakharov–Shabat system in (4), we proceed in a similar way. Let us write it as L D  ; where the linear differential operator L is defined via # " # " 0 u(x; t) 1 0 : L :D i @x  i 0 1 u(x; t) 0

"

(9)

Note that (9) is an evolution equation containing a firstorder time derivative, and it is the desired integrable NPDE. The equation in (9) is often called a compatibility condition. Having outlined the Lax method, let us now illustrate it to derive the KdV equation in (1) from the Schrödinger equation in (2). For this purpose, we write the Schrödinger equation as L D  with  :D k 2 and

)@4x

(13)

Then, the operator A is obtained as

L t C LA  AL D 0 :

)@5x

in (1). Moreover, by letting c2 D c4 D 0; we obtain the operator A as

3 ˛1 D c3  c1 u; 2

3 ˛0 D c4  c1 u x  c2 u ; 4 with c1 ; c2 ; c3 ; and c4 denoting arbitrary constants. Choosing c1 D 4 and c3 D 0 in the last coefficient in (12) and setting that coefficient to zero, we get the KdV equation

A D 2i

1

0

0

1

#

" @2x 2i

0

u

u

0

#

"

juj2

ux

ux

juj2

@x i

# ;

(14) and the compatibility condition (9) gives us the NLS equation in (3). For the first-order system (6), by writing it as L D  ; where the linear operator L is defined by " " # # 1 0 0 u(x; t) L :D i @x  i ; 0 1 u(x; t) 0 we obtain the corresponding operator A as " " " # # A D 4

1

0

0

1

@3x 6

u2

u x

ux

u2

@x 

6uu x

3u x x

3u x x

6uu x

# ;

and the compatibility condition (9) yields the mKdV equation in (5). The AKNS Method In 1973 Ablowitz, Kaup, Newell, and Segur introduced [2,3] another method to determine an integrable NPDE corresponding to a LODE. This method is now known as the AKNS method, and the basic idea behind it is the following. Given a linear operator X associated with the first-order system v x D X v; we are interested in finding an operator T (the operators T and X are said to form an AKNS pair) such that: The spectral parameter  does not change in time, i. e.  t D 0: (ii) The quantity v t  T v is also a solution to v x D X v; i. e. we have (v t  T v)x D X (v t  T v): (iii) The quantity X t  Tx C XT  T X is a (matrix) multiplication operator, i. e. it is not a differential operator. (i)

Inverse Scattering Transform and the Theory of Solitons

Then from the (1; 2)-entry in (17) we obtain

From condition (ii) we get

u t C 12 x x x  u x  2 x (u  ) D 0 :

v tx  Tx v  T v x D X v t  XT v D (X v) t  X t v  XT v

Assuming a linear dependence of on the spectral parameter and hence letting D  C  in (19), we get

D (v x ) t  X t v  XT v D v x t  X t v  XT v :

(15)

Using v tx D v x t and replacing T v x by T X v on the left side and equating the left and right hand sides in (15), we obtain (X t  Tx C XT  T X )v D 0 ;

2x 2 C

(16)

We can view (16) as an integrable NPDE solvable with the help of the solutions to the direct and inverse scattering problems for the linear system v x D X v: Like (9), the compatibility condition (16) yields a nonlinear evolution equation containing a first-order time derivative. Note that X contains the spectral parameter ; and hence T also depends on  as well. This is in contrast with the Lax method in the sense that the operator A does not contain : Let us illustrate the AKNS method by deriving the KdV equation in (1) from the Schrödinger equation in (2). For this purpose we write the Schrödinger equation, by replacing the spectral parameter k2 with ; as a first-order linear system v x D X v; where we have defined " # " # 0 u(x; t)   x v :D ; X :D : 1 0

1

  2x u C 2x  u x     C u t C 12 x x x  2x u  u x  D 0 :

2 x x x

Equating the coefficients of each power of  to zero, we have  D c1 ;

which in turn, because of (iii), implies X t  T x C XT  T X D 0 :

(19)

 D 12 c1 u C c2 ; u t  32 c1 uu x  c2 u x C 14 c1 u x x x D 0 ;

(20)

with c1 and c2 denoting arbitrary constants. Choosing c1 D 4 and c2 D 0; from (20) we obtain the KdV equation given in (1). Moreover, with the help of (18) we get ˛ D u x C c3 ; ˇ D 42 C 2u C 2u2  u x x ;

D 4 C 2u;  D c3  u x ; where c3 is an arbitrary constant. Choosing c3 D 0; we find " # 42 C 2u C 2u2  u x x ux T D : 4 C 2u u x As for the Zakharov–Shabat system in (4), writing it as v x D X v; where we have defined # " i u(x; t) ; X :D i u(x; t)

Let us look for T in the form " # ˛ ˇ T D ;



we obtain the matrix operator T as " # 2i2 C ijuj2 2u C iu x T D ; 2u C iu x 2i2  ijuj2

where the entries ˛; ˇ; ; and  may depend on x; t; and : The compatibility condition (16) yields

and the compatibility condition (16) yields the NLS equation in (3). As for the first-order linear system (6), by writing it as v x D X v; where " # i u(x; t) X :D ; u(x; t) i

 ˛x  ˇ C (u  ) u t  ˇx C (u  )  ˛(u  ) x C ˇ  (u  )  x C ˛   " # 0 0 D : 0 0



(17) The (1; 1); (2; 1); and (2; 2)-entries in the matrix equation in (17) imply ˇ D ˛x C (u  ) ;

 D ˛  x ;

x D ˛x : (18)

we obtain the matrix operator T as T D



4i3 C 2iu 2 42 u C 2iu x  u x x  2u 3 42 u C 2iu x C u x x C 2u 3 4i3  2iu 2



775

776

Inverse Scattering Transform and the Theory of Solitons

and the compatibility condition (16) yields the mKdV equation in (5). As for the first-order system v x D X v; where 2 3 1 i  (x; t) u x 6 7 2 7; X :D 6 41 5 i u x (x; t) 2 we obtain the matrix operator T as " # sin u i cos u T D : 4 sin u  cos u Then, the compatibility condition (16) gives us the sineGordon equation u x t D sin u : Direct Scattering Problem The direct scattering problem consists of determining the scattering data when the potential is known. This problem is usually solved by obtaining certain specific solutions, known as the Jost solutions, to the relevant LODE. The appropriate scattering data can be constructed with the help of spatial asymptotics of the Jost solutions at infinity or from certain Wronskian relations among the Jost solutions. In this section we review the scattering data corresponding to the Schrödinger equation in (2) and to the Zakharov–Shabat system in (4). The scattering data sets for other LODEs can similarly be obtained. Consider (2) at fixed t by assuming that the potential u(x, t) belongs R 1to the Faddeev class, i. e. u(x, t) is real valued and 1 dx (1 C jxj) ju(x; t)j is finite. The Schrödinger equation has two types of solutions; namely, scattering solutions and bound-state solutions. The scattering solutions are those that consist of linear combinations of eikx and eikx as x ! ˙1; and they occur for k 2 R n f0g; i. e. for real nonzero values of k: Two linearly independent scattering solutions fl and fr ; known as the Jost solution from the left and from the right, respectively, are those solutions to (2) satisfying the respective asymptotic conditions fl (k; x; t) D e i kx C o(1) ; fl0 (k; x; t) D ikei kx C o(1) ;

x ! C1 ;

(21)

fr (k; x; t) D ei kx C o(1) ; fr0 (k; x; t) D ikei kx C o(1) ;

x ! 1 ;

where the notation o(1) indicates the quantities that vanish. Writing their remaining spatial asymptotics in the

form fl (k; x; t) D

fr (k; x; t) D

e i kx L(k; t) ei kx C C o(1) ; T(k; t) T(k; t) x ! 1 ;

(22)

ei kx R(k; t) e i kx C C o(1) ; T(k; t) T(k; t) x ! C1 ;

we obtain the scattering coefficients; namely, the transmission coefficient T and the reflection coefficients L and R; from the left and right, respectively. Let CC denote the upper half complex plane. A boundstate solution to (2) is a solution that belongs to L2 (R) in the x variable. Note that L2 (R) denotes the set of complexvalued functions whose absolute squares are integrable on the real line R: When u(x, t) is in the Faddeev class, it is known [5,7,8,9,16,17,18,19] that the number of bound states is finite, the multiplicity of each bound state is one, and the bound-state solutions can occur only at certain k-values on the imaginary axis in CC : Let us use N to denote the number of bound states, and suppose that the bound states occur at k D i j with the ordering 0 < 1 <    <  N : Each bound state corresponds to a pole of T in CC : Any bound-state solution at k D i j is a constant multiple of fl (i j ; x; t): The left and right boundstate norming constants cl j (t) and cr j (t); respectively, can be defined as

Z 1 1/2 dx fl (i j ; x; t)2 ; cl j (t) :D

Z cr j (t) :D

1 1 1

dx fr (i j ; x; t)2

1/2 ;

and they are related to each other through the residues of T via Res (T; i j ) D i cl j (t)2 j (t) D i

cr j (t)2 ;

j (t)

(23)

where the j (t) are the dependency constants defined as

j (t) :D

fl (i j ; x; t) : fr (i j ; x; t)

(24)

The sign of j (t) is the same as that of (1) N j ; and hence cr j (t) D (1) N j j (t) cl j (t) : The scattering matrix associated with (2) consists of the transmission coefficient T and the two reflection coefficients R and L; and it can be constructed from f j g NjD1 and one of the reflection coefficients. For example, if we start with the right reflection coefficient R(k, t) for k 2 R; we get

Inverse Scattering Transform and the Theory of Solitons

0

1 N Y k C i j A T(k; t) D @ k  i j jD1   Z 1 1 log(1  jR(s; t)j2 )  exp ds ; 2 i 1 s  k  i0C k 2 CC [ R ; where the quantity i0+ indicates that the value for k 2 R must be obtained as a limit from CC : Then, the left reflection coefficient L(k, t) can be constructed via L(k; t) D 

R(k; t) T(k; t) T(k; t)

;

k 2R:

We will see in the next section that T(k; t) D T(k; 0); jR(k; t)j D jR(k; 0)j, and jL(k; t)j D jL(k; 0)j. For a detailed study of the direct scattering problem for the 1-D Schrödinger equation, we refer the reader to [5,7, 8,9,16,17,18,19]. It is important to remember that u(x, t) for x 2 R at each fixed t is uniquely determined [5,7,8,9, 16,17,18] by the scattering data fR; f j g; fcl j (t)gg or one of its equivalents. Letting c j (t) :D cl j (t)2 ; we will work with one such data set, namely fR; f j g; fc j (t)gg, in Sect. “Time Evolution of the Scattering Data” and Sect. “Inverse Scattering Problem”. Having described the scattering data associated with the Schrödinger equation, let us briefly describe the scattering data associated with the Zakharov–Shabat system in (4). Assuming that u(x, t) for each t is integrable in x on R; the two Jost solutions (; x; t) and (; x; t); from the left and from the right, respectively, are those unique solutions to (4) satisfying the respective asymptotic conditions " # 0 (; x; t) D ix C o(1); x ! C1 ; e (25) 

ix e C o(1) ; x ! 1 : (; x; t) D 0 The transmission coefficient T; the left reflection coefficient L; and the right reflection coefficient R are obtained via the asymptotics 2 3 L(; t) eix 6 T(; t) 7 6 7 (; x; t) D 6 7 C o(1) ; x ! 1 ; ix 4 5 e 2

T(; t) eix T(; t)

3

7 6 7 6 (; x; t) D 6 7 C o(1) ; 4 R(; t) e ix 5

x ! C1 :

T(; t) (26)

The bound-state solutions to (4) occur at those  values corresponding to the poles of T in CC : Let us use f j g NjD1 to denote the set of such poles. It should be noted that such poles are not necessarily located on the positive imaginary axis. Furthermore, unlike the Schrödinger equation, the multiplicities of such poles may be greater than one. Let us assume that the pole  j has multiplicity n j : Corresponding to the pole  j ; one associates [4,20] nj boundstate norming constants c js (t) for s D 0; 1; : : : ; n j  1: We assume that, for each fixed t; the potential u(x, t) in the Zakharov–Shabat system is uniquely determined by the scattering data fR; f j g; fc js (t)gg and vice versa. Time Evolution of the Scattering Data As the initial profile u(x; 0) evolves to u(x, t) while satisfying the NPDE, the corresponding initial scattering data S(; 0) evolves to S(; t): Since the scattering data can be obtained from the Jost solutions to the associated LODE, in order to determine the time evolution of the scattering data, we can analyze the time evolution of the Jost solutions with the help of the Lax method or the AKNS method. Let us illustrate how to determine the time evolution of the scattering data in the Schrödinger equation with the help of the Lax method. As indicated in Sect. “The Lax Method”, the spectral parameter k and hence also the values  j related to the bound states remain unchanged in time. Let us obtain the time evolution of fl (k; x; t); the Jost solution from the left. From condition (ii) in Sect. “The Lax Method”, we see that the quantity @ t fl  A fl remains a solution to (2) and hence we can write it as a linear combination of the two linearly independent Jost solutions fl and fr as   @ t fl (k; x; t) 4@3x C 6u@x C 3u x fl (k; x; t) D p(k; t) fl (k; x; t) C q(k; t) fr (k; x; t) ; (27) where the coefficients p(k, t) and q(k, t) are yet to be determined and A is the operator in (13). For each fixed t; assuming u(x; t) D o(1) and u x (x; t) D o(1) as x ! C1 and using (21) and (22) in (27) as x ! C1; we get @ t e i kx C 4@3x e i kx D p(k; t) e i kx

 1 R(k; t) i kx i kx C q(k; t) C e e C o(1) : T(k; t) T(k; t) (28) Comparing the coefficients of eikx and eikx on the two sides of (28), we obtain q(k; t) D 0 ;

p(k; t) D 4ik 3 :

777

778

Inverse Scattering Transform and the Theory of Solitons

Thus, fl (k; x; t) evolves in time by obeying the linear thirdorder PDE @ t fl  A fl D 4ik 3 fl :

(29)

Proceeding in a similar manner, we find that fr (k; x; t) evolves in time according to @ t fr  A fr D 4ik 3 fr :

(30)

Notice that the time evolution of each Jost solution is fairly complicated. We will see, however, that the time evolution of the scattering data is very simple. Letting x ! 1 in (29), using (22) and u(x; t) D o(1) and u x (x; t) D o(1) as x ! 1; and comparing the coefficients of eikx and eikx on both sides, we obtain @ t T(k; t) D 0 ;

The norming constants cj (t) appearing in the Marchenko kernel (38) are related to cl j (t) as c j (t) :D cl j (t)2 ; and hence their time evolution is described as

@ t L(k; t) D 8ik 3 L(k; t);

c j (t) D c j (0) e

8 3j t

:

(35)

As for the NLS equation and other integrable NPDEs, the time evolution of the related scattering data sets can be obtained in a similar way. For the former, in terms of the operator A in (14), the Jost solutions (; x; t) and (; x; t) appearing in (25) evolve according to the respective linear PDEs t

A

D 2i2

;

 t  A D 2i2  :

The scattering coefficients appearing in (26) evolve according to T(; t) D T(; 0) ;

yielding

2

R(; t) D R(; 0) e4i t ;

T(k; t) D T(k; 0) ;

L(k; t) D L(k; 0) e

8i k 3 t

:

L(; t) D L(; 0) e

In a similar way, from (30) as x ! C1; we get 3

R(k; t) D R(k; 0) e8i k t :

(31)

Thus, the transmission coefficient remains unchanged and only the phases of the reflection coefficients change as time progresses. Let us also evaluate the time evolution of the dependency constants j (t) defined in (24). Evaluating (29) at k D i j and replacing fl (i j ; x; t) by j (t) fr (i j ; x; t); we get fr (i j ; x; t) @ t j (t) C j (t) @ t fr (i j ; x; t)  j (t)A fr (i j ; x; t) D 4 3j j (t) fr (i j ; x; t) :

(32)

On the other hand, evaluating (30) at k D i j ; we obtain

j (t) @ t fr (i j ; x; t)  j (t) A fr (i j ; x; t) D 4 3j j (t) fr (i j ; x; t) :

(33)

Comparing (32) and (33) we see that @ t j (t) D 8 3j j (t); or equivalently

j (t) D j (0) e

8 3j t

:

(34)

Then, with the help of (23) and (34), we determine the time evolutions of the norming constants as cl j (t) D cl j (0) e

4 3j t

;

cr j (t) D cr j (0) e

4 3j t

:

4i2 t

(36) :

Associated with the bound-state pole  j of T; we have the bound-state norming constants c js (t) appearing in the Marchenko kernel ˝(y; t) given in (41). Their time evolution is governed [4] by 

c j(n j 1) (t) c j(n j 2) (t) : : : c j0 (t) 4i A2 t  j ; D c j(n j 1) (0) c j(n j 2) (0) : : : c j0 (0) e (37)

where the n j  n j matrix Aj appearing in the exponent is defined as 2 3 1 0 ::: 0 0 i j 6 0 1 : : : 0 0 7 i j 6 7 6 0 7 0 i : : : 0 0 j 6 7 A j :D 6 : 7: : : : : : :: 7 :: :: :: :: 6 :: 6 7 4 0 0 0 : : : i j 1 5 0 0 0 ::: 0 i j Inverse Scattering Problem In Sect. “Direct Scattering Problem” we have seen how the initial scattering data S(; 0) can be constructed from the initial profile u(x; 0) of the potential by solving the direct scattering problem for the relevant LODE. Then, in Sect. “Time Evolution of the Scattering Data” we have seen how to obtain the time-evolved scattering data S(; t) from the initial scattering data S(; 0): As the final step in the IST, in this section we outline how to obtain

Inverse Scattering Transform and the Theory of Solitons

u(x, t) from S(; t) by solving the relevant inverse scattering problem. Such an inverse scattering problem may be solved by the Marchenko method [5,7,8,9,16,17,18, 19]. Unfortunately, in the literature many researchers refer to this method as the Gel’fand–Levitan method or the Gel’fand–Levitan–Marchenko method, both of which are misnomers. The Gel’fand–Levitan method [5,7,16,17, 19] is a different method to solve the inverse scattering problem, and the corresponding Gel’fand–Levitan integral equation involves an integration on the finite interval (0; x) and its kernel is related to the Fourier transform of the spectral measure associated with the LODE. On the other hand, the Marchenko integral equation involves an integration on the semi-infinite interval (x; C1); and its kernel is related to the Fourier transform of the scattering data. In this section we first outline the recovery of the solution u(x, t) to the KdV equation from the corresponding time-evolved scattering data fR; f j g; fc j (t)gg appearing in (31) and (35). Later, we will also outline the recovery of the solution u(x, t) to the NLS equation from the corresponding time-evolved scattering data fR; f j g; fc js (t)gg appearing in (36) and (37). The solution u(x, t) to the KdV equation in (1) can be obtained from the time-evolved scattering data by using the Marchenko method as follows: (a) From the scattering data fR(k; t); f j g; fc j (t)gg appearing in (31) and (35), form the Marchenko kernel ˝ defined via Z 1 N X 1 ˝(y; t) :D dk R(k; t) e i k y C c j (t) e j y : 2 1 jD1

(38) (b) Solve the corresponding Marchenko integral equation K(x; y; t) C ˝(x C y; t) Z 1 dz K(x; z; t) ˝(z C y; t) D 0 ; C x

x < y < C1 ;

(39)

and obtain its solution K(x; y; t): (c) Recover u(x, t) by using u(x; t) D 2

@K(x; x; t) : @x

(40)

The solution u(x, t) to the NLS equation in (3) can be obtained from the time-evolved scattering data by using the Marchenko method as follows: (i)

From the scattering data fR(; t); f j g; fc js (t)gg appearing in (36) and (37), form the Marchenko kernel

˝ as 1 ˝(y; t) :D 2

Z

1

d R(; t) e iy

1

C

j 1 N nX X

jD1 sD0

c js (t)

y s i j y : e s!

(41)

(ii) Solve the Marchenko integral equation Z 1 K(x; y; t)  ˝(x C y; t) C dz x Z 1  ds K(x; s; t) ˝(s C z; t) ˝(z C y; t) D 0 ; x

x < y < C1 ; and obtain its solution K(x; y; t): (iii) Recover u(x, t) from the solution K(x; y; t) to the Marchenko equation via u(x; t) D 2K(x; x; t) : (iv) Having determined K(x; y; t); one can alternatively get ju(x; t)j2 from ju(x; t)j2 D 2

@G(x; x; t) ; @x

where we have defined Z 1 dz K(x; z; t) ˝(z C y; t) : G(x; y; t) :D  x

Solitons A soliton solution to an integrable NPDE is a solution u(x, t) for which the reflection coefficient in the corresponding scattering data is zero. In other words, a soliton solution u(x, t) to an integrable NPDE is nothing but a reflectionless potential in the associated LODE. When the reflection coefficient is zero, the kernel of the relevant Marchenko integral equation becomes separable. An integral equation with a separable kernel can be solved explicitly by transforming that linear equation into a system of linear algebraic equations. In that case, we get exact solutions to the integrable NPDE, which are known as soliton solutions. For the KdV equation the N-soliton solution is obtained by using R(k; t) D 0 in (38). In that case, letting  X(x) :D e1 x e2 x : : : eN x ; 2 3 c1 (t) e1 y 6 c2 (t) e2 y 7 6 7 Y(y; t) :D 6 7; :: 4 5 :  y N c N (t) e

779

780

Inverse Scattering Transform and the Theory of Solitons

we get ˝(x C y; t) D X(x) Y(y; t). As a result of this separability the Marchenko integral equation can be solved algebraically and the solution has the form K(x; y; t) D H(x; t) Y(y; t), where H(x, t) is a row vector with N entries that are functions of x and t. A substitution in (39) yields K(x; y; t) D X(x)  (x; t)1 Y(y; t);

(42)

where the N  N matrix  (x; t) is given by Z 1  (x; t) :D I C dz Y(z; t) X(z);

(43)

x

with I denoting the or N  N identity matrix. Equivalently, the (j, l)-entry of  is given by 2 xC8 3j t

c j (0) e j  jl D ı jl C  j C l

;

with ı jl denoting the Kronecker delta. Using (42) in (40) we obtain

x

As for the NLS equation, the well-known N-soliton solution (with simple bound-state poles) is obtained by choosing R(; t) D 0 and n j D 1 in (41). Proceeding as in the KdV case, we obtain the N-soliton solution in terms of the triplet A; B; C with A :D diagfi1 ; i2 ; : : : ; i N g;

(46)

where the complex constants  j are the distinct poles of the transmission coefficient in CC ; B and C are as in (45) except for the fact that the constants c j (0) are now allowed to be nonzero complex numbers. In terms of the matrices P(x; t); M; and Q defined as P(x; t) :D 2

2

2

diagfe2i1 xC4i1 t ; e2i2 xC4i2 t ; : : : ; e2iN xC4iN t g;

@  X(x)  (x; t)1 Y(x; t) u(x; t) D 2 @x @  Y(x; t) X(x)  (x; t)1 ; D 2 tr @x

M jl :D

where tr denotes the matrix trace (the sum of diagonal entries in a square matrix). From (43) we see that Y(x; t) X(x) is equal to the x-derivative of  (x; t) and hence the N-soliton solution can also be written as

 @ @ (x; t) u(x; t) D 2 tr  (x; t)1 @x @x " @ # det  (x; t) @ @x D 2 ; (44) @x det  (x; t) where det denotes the matrix determinant. When N D 1; we can express the one-soliton solution u(x, t) to the KdV equation in the equivalent form   u(x; t) D 2 12 sech2 1 x  413 t C ; p with :D log 21 /c1 (0) : Let us mention that, using matrix exponentials, we can express [6] the N-soliton solution appearing in (44) in various other equivalent forms such as 3

u(x; t) D 4CeAxC8A t  (x; t)1 A (x; t)1 eAx B ; where A :D diagf1 ; 2 ; : : : ;  N g ;  B :D 1 1 : : : 1 ;  C :D c1 (0) c2 (0) : : : c N (0) :

Note that a dagger is used for the matrix adjoint (transpose and complex conjugate), and B has N entries. In this notation we can express (43) as Z 1 3  (x; t) D I C dz ezA BCezA e8tA :

(45)

i  j  l

;

Q jl :D

ic j c l  j  l

:

we construct the N-soliton solution u(x, t) to the NLS equation as u(x; t) D 2B [I C P(x; t) QP(x; t) M]1 P(x; t) C  ; (47) or equivalently as

xC4i(A )2 t

u(x; t) D 2B eA x  (x; t)1 eA

C  ; (48)

where we have defined

Z 1  As4i A2 t  As4i A2 t  (x; t) :D I C ds (Ce ) (Ce ) x

Z 1   dz (eAz B)(eAz B) : (49) x

Using (45) and (46) in (49), we get the (j, l)-entry of  (x; t) as 2

 jl D ı jl 

2

N X c j c l e i(2m  j  l )xC4i(m  j )t mD1

(m   j )(m   l )

Note that the absolute square of u(x, t) is given by

  @ @ (x; t)  (x; t)1 ju(x; t)j2 D tr @x @x " @ # @ @x det  (x; t) D : @x det  (x; t)

:

Inverse Scattering Transform and the Theory of Solitons

For the NLS equation, when N D 1; from (47) or (48) we obtain the single-soliton solution 2t

u(x; t) D

8c 1 (Im[1 ])2 e2i1 x4i(1 )

2

4(Im[1 ])2 C jc1 j2 e4x(Im[1 ])8t(Im[1 ])

;

where Im denotes the imaginary part. Future Directions There are many issues related to the IST and solitons that cannot be discussed in such a short review. We will briefly mention only a few. Can we characterize integrable NPDEs? In other words, can we find a set of necessary and sufficient conditions that guarantee that an IVP for a NPDE is solvable via an IST? Integrable NPDEs seem to have some common characteristic features [1] such as possessing Lax pairs, AKNS pairs, soliton solutions, infinite number of conserved quantities, a Hamiltonian formalism, the Painlevé property, and the Bäcklund transformation. Yet, there does not seem to be a satisfactory solution to their characterization problem. Another interesting question is the determination of the LODE associated with an IST. In other words, given an integrable NPDE, can we determine the corresponding LODE? There does not yet seem to be a completely satisfactory answer to this question. When the initial scattering coefficients are rational functions of the spectral parameter, representing the timeevolved scattering data in terms of matrix exponentials results in the separability of the kernel of the Marchenko integral equation. In that case, one obtains explicit formulas [4,6] for exact solutions to some integrable NPDEs and such solutions are constructed in terms of a triplet of constant matrices A; B; C whose sizes are pp; p1; and 1p; respectively, for any positive integer p: Some special cases of such solutions have been mentioned in Sect. “Solitons”, and it would be interesting to determine if such exact solutions can be constructed also when p becomes infinite. Bibliography Primary Literature 1. Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge 2. Ablowitz MJ, Kaup DJ, Newell AC, Segur H (1973) Method for solving the sine–Gordon equation. Phys Rev Lett 30:1262– 1264 3. Ablowitz MJ, Kaup DJ, Newell AC, Segur H (1974) The inverse scattering transform-Fourier analysis for nonlinear problems. Stud Appl Math 53:249–315

4. Aktosun T, Demontis F, van der Mee C (2007) Exact solutions to the focusing nonlinear Schrödinger equation. Inverse Problems 23:2171–2195 5. Aktosun T, Klaus M (2001) Chapter 2.2.4, Inverse theory: problem on the line, In: Pike ER, Sabatier PC (eds) Scattering. Academic Press, London, pp 770–785 6. Aktosun T, van der Mee C (2006) Explicit solutions to the Korteweg-de Vries equation on the half-line. Inverse Problems 22:2165–2174 7. Chadan K, Sabatier PC (1989) Inverse problems in quantum scattering theory. 2nd edn. Springer, New York 8. Deift P, Trubowitz E (1979) Inverse scattering on the line. Commun Pure Appl Math 32:121–251 9. Faddeev LD (1967) Properties of the S–matrix of the one– dimensional Schrödinger equation. Amer Math Soc Transl (Ser. 2) 65:139–166 10. Fermi E (1965) Collected papers, vol II: United States, 1939– 1954. University of Chicago Press, Chicago 11. Fermi E, Pasta J, Ulam S (1955) Studies of non linear problems, I. Document LA-1940, Los Alamos National Laboratory 12. Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg–de Vries equation. Phys Rev Lett 19:1095–1097 13. Gel’fand IM, Levitan BM (1955) On the determination of a differential equation from its spectral function. Amer Math Soc Transl (Ser. 2) 1:253–304 14. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary waves. Phil Mag 39:422–443 15. Lax PD (1968) Integrals of nonlinear equations of evolution and solitary waves. Commun Pure Appl Math 21:467–490 16. Levitan BM (1987) Inverse Sturm–Liouville problems. Science VNU Press, Utrecht 17. Marchenko VA (1986) Sturm–Liouville operators and applications. Birkhäuser, Basel 18. Melin A (1985) Operator methods for inverse scattering on the real line. Commun Partial Differ Equ 10:677–766 19. Newton RG (1983) The Marchenko and Gel’fand–Levitan methods in the inverse scattering problem in one and three dimensions. In: Bednar JB, Redner R, Robinson E, Weglein A (eds) Conference on inverse scattering: theory and application. SIAM, Philadelphia, pp 1–74 20. Olmedilla E (1987) Multiple pole solutions of the nonlinear Schrödinger equation. Phys D 25:330–346 21. Russell JS (1845) Report on waves, Report of the 14th meeting of the British Association for the Advancement of Science. John Murray, London, pp 311–390 22. Wadati M (1972) The exact solution of the modified Korteweg– de Vries equation. J Phys Soc Jpn 32:1681 23. Zabusky NJ, Kruskal MD (1965) Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243 24. Zakharov VE, Shabat AB (1972) Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Phys JETP 34:62–69 25. http://www.osti.gov/accomplishments/pdf/A80037041/ A80037041.pdf

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Books and Reviews Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, Philadelphia Aktosun T (2004) Inverse scattering transform, KdV, and solitons, In: Ball JA, Helton JW, Klaus M, Rodman L (ed), Current trends in operator theory and its applications. Birkhäuser, Basel, pp 1–22 Aktosun T (2005) Solitons and inverse scattering transform, In: Clemence DP, Tang G (eds) Mathematical studies in nonlinear wave propagation, Contemporary Mathematics, vol 379. Amer Math Soc, Providence, pp 47–62

Dodd RK, Eilbeck JC, Gibbon JD, Morris HC (1982) Solitons and nonlinear wave equations. Academic Press, London Drazin PG, Johnson RS (1988) Solitons: an introduction. Cambridge University Press, Cambridge Lamb GL Jr (1980) Elements of soliton theory. Wiley, New York Novikov S, Manakov SV, Pitaevskii LP, Zakharov VE (1984) Theory of solitons. Consultants Bureau, New York Scott AC, Chu FYF, McLaughlin D (1973) The soliton: a new concept in applied science. Proc IEEE 61:1443–1483

Isomorphism Theory in Ergodic Theory

Isomorphism Theory in Ergodic Theory CHRISTOPHER HOFFMAN Department of Mathematics, University of Washington, Seattle, USA Article Outline Glossary Definition of the Subject Introduction Basic Transformations Basic Isomorphism Invariants Basic Tools Isomorphism of Bernoulli Shifts Transformations Isomorphic to Bernoulli Shifts Transformations not Isomorphic to Bernoulli Shifts Classifying the Invariant Measures of Algebraic Actions Finitary Isomorphisms Flows Other Equivalence Relations Non-invertible Transformations Factors of a Transformation Actions of Amenable Groups Future Directions Bibliography Glossary Almost everywhere A property is said to hold almost everywhere (a.e.) if the set on which the property does not hold has measure 0. Bernoulli shift A Bernoulli shift is a stochastic process such that all outputs of the process are independent. Conditional measure For any measure space (X; B; ) is a C and -algebra C B the conditional measure R measurable function g such that (C) D C g d for all C 2 C . Coupling of two measure spaces A coupling of two measure spaces (X; ; B) and (Y; ; C ) is a measure on X  Y such that (B  Y) D (B) for all B 2 B and

(X  C) D (C) for all C 2 C . Ergodic measure preserving transformation A measure preserving transformation is ergodic if the only invariant sets ((A4T 1 (A)) D 0) have measure 0 or 1. Ergodic theorem The pointwise ergodic theorem says that for any measure preserving transformation (X; B; ) and T and any L1 function f the time average n 1 X lim f (T i (x)) n!1 n iD1

converges a.e. If the transformation is ergodic then the R limit is the space average, f d a.e. Geodesic A geodesic on a Riemannian manifold is a distance minimizing path between points. Horocycle A horocycle is a circle in the hyperbolic disk which intersects the boundary of the disk in exactly one point. Invariant measure Likewise a measure  is said to be invariant with respect to (X; T) provided that (T 1 (A)) D (A) for all measurable A 2 B. Joining of two measure preserving transformations A joining of two measure preserving transformations (X; T) and (Y; S) is a coupling of X and Y which is invariant under T  S. Markov shift A Markov shift is a stochastic process such that the conditional distribution of the future outputs (fx n gn>0 ) of the process conditioned on the last output (x0 ) is the same as the distribution conditioned on all of the past outputs of the process (fx n gn0 ). Measure preserving transformation A measure preserving transformation consists of a probability space (X; T) and a measurable function T : X ! X such that (T 1 (A)) D (A) for all A 2 B. Measure theoretic entropy A numerical invariant of measure preserving transformations that measures the growth in complexity of measurable partitions refined under the iteration of the transformation. Probability space A probability space X D (X; ; B) is a measure space such that (B) D 1. Rational function, rational map A rational function f (z) D g(z)/h(z) is the quotient of two polynomials. The degree of f (z) is the maximum of the degrees of g(z) and h(z). The corresponding rational maps T f : z ! f (z) on the Riemann sphere C are a main object of study in complex dynamics. Stochastic process A stochastic process is a sequence of measurable functions fx n gn2Z (or outputs) defined on the same measure space, X. We refer to the value of the functions as outputs. Definition of the Subject Our main goal in this article is to consider when two measure preserving transformations are in some sense different presentations of the same underlying object. To make this precise we say two measure preserving maps (X; T) and (Y; S) are isomorphic if there exists a measurable map  : X ! Y such that (1)  is measure preserving, (2)  is invertible almost everywhere and (3) (T(x)) D S((x)) for almost every x.

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The main goal of the subject is to construct a collection of invariants of a transformation such that a necessary condition for two transformations to be isomorphic is that the invariant be the same for both transformations. Another goal of the subject is to solve the much more difficult problem of constructing invariants such that the invariants being the same for two transformations is a sufficient condition for the transformation to be isomorphic. Finally we apply these invariants to many natural classes of transformation to see which of them are (or are not) isomorphic. Introduction In this article we look at the problem of determining of which measure preserving transformations are isomorphic. We look at a number of isomorphism invariants, the most important of which is the Kolmogorov–Sinai entropy. The central theorem in this field is Ornstein’s proof that any two Bernoulli shifts of the same entropy are isomorphic. We also discuss some of the consequences of this theorem, which transformations are isomorphic to Bernoulli shifts as well as generalizations of Ornstein’s theory. Basic Transformations In this section we list some of the basic classes of measure preserving transformations that we study in this article. Bernoulli shifts Some of the most fundamental transformations are the Bernoulli shifts. A probability vecP tor is a vector fp i gniD1 such that niD1 p i D 1 and p i  0 for all i. Let p D fp i gniD1 be a probability vector. The Bernoulli shift corresponding to p has state space f1; 2; : : : ; ngZ , the shift operator T(x) i D x iC1 . To specify the measure we only need to specify it on cylinder sets A D fx 2 X : x i D a i 8i 2 fm; : : : ; kgg for some m  k 2 Z and a sequence a m ; : : : ; a k 2 f1; : : : ; ng. The measure on cylinder sets is defined by ˚  x 2 X : xi D ai

k Y for all i such that m  i  k D pai : iDm

For any d 2 N if p D (1/d; : : : ; 1/d) we refer to Bernoullip as the Bernoulli d shift. Markov shifts A Markov shift on state n symbols is defined by an n  n matrix, M, such that fM(i; j)gnjD1 is a probability vector for each i. The Markov shift is

a measure preserving transformation with state space f1; 2; : : : ; ngZ , transformation T(x) i D x iC1 fx1 D a1 j x0 D a0 g D fx1 D a1 j x0 D a0 ; x1 D a1 ; x2 D a2 ; : : : g for all choices of a i , i  1. Let m D fm(i)gniD1 be a vector such that Mm D m. Then an invariant measure is defined by setting the measure on cylinder sets to be A D fx : x0 D a0 ; x1 D a1 ; : : : ; x n D a n g is Q given by (A) D m(a0 ) niD1 M(a i1; a i ). Shift maps More generally the shift map  is the map  : N Z ! N Z where  (x) i D x iC1 for all x 2 N Z and i 2 Z. We also let  designate the shift map on N N . For each measure that is invariant under the shift map there is a corresponding measure defined on N N that is invariant under the shift map. Let  be an invariant measure under the shift map. For any measurable set of A N N we define A˜ on N Z by A˜ D f: : : ; x1 ; x0 ; x1 ; : : : : x0 ; x1 ; : : : 2 Ag : ˜ D ˜ defined by ( ˜ A) Then it is easy to check that  (A) is invariant. If the original transformation was a Markov or Bernoulli shift then refer to the resulting transformations as a one sided Markov shifts or one sided Bernoulli shift respectively. Rational maps of the Riemann sphere We say that f (z) D g(z)/h(z) is a rational function of degree d  2 if both g(z) and h(z) are polynomials with max(deg(g(z)); deg(h(z))) D d. Then f induces a natural action on the Riemann sphere T f : z ! f (z) which is a d to one map (counting with multiplicity). In Subsect. “Rational Maps” we shall see that for every rational function f there is a canonical measure f such that T f is a measure preserving transformation. Horocycle flows The horocycle flow acts on SL(2; R)/ where  is a discrete subgroup of SL(2; R) such that SL(2; R)/ has finite Haar measure. For any g 2 SL(2; R) and t 2 R we define the horocycle flow by   1 0 : h t ( g) D  g t 1 Matrix actions Another natural class of actions is given by the action of matrices on tori. Let M be an invertible n  n integer valued matrix. We define TM : [0; 1)n ! [0; 1)n by TM (x) i D M(x) i mod 1 for all i, 1  i  n. It is easy to check that if M is surjective then Lebesgue measure is invariant under T M . T M is a jdet(M)j to one map. If n D 1 then M is an integer and we refer to the map as times M.

Isomorphism Theory in Ergodic Theory

The [T; T 1 ] transformations Let (X; T) be any invertible measure preserving transformation. Let  be the shift operator on Y D f1; 1gZ . The space Y comes equipped with the Bernoulli (1/2; 1/2) product measure . The [T; T 1 ] transformation is a map on Y  X which preserves   . It is defined by T; T 1 (y; x) D (S(y); T y0 (x)) : Induced transformations Let (X; T) be a measure preserving transformation and let A X with 0 < (A) < 1. The transformation induced by A, (A; TA ; A ), is defined as follows. For any x 2 A TA (x) D T

n(x)

(x)

where n(x) D inffm > 0 : T m (x) 2 Ag. For any B A we have that A (B) D (B)/(A). Basic Isomorphism Invariants The main purpose of isomorphism theory is to classify which pairs of measure preserving transformation are isomorphic and which are not isomorphic. One of the main ways that we can show that two measure preserving transformation are not isomorphic is using isomorphism invariants. An isomorphism invariant is a function f defined on measure preserving transformations such that if (X; T) is isomorphic to (Y; S) then f ((X; T)) D f ((Y; S)). A measure preserving action is said to be ergodic if (A) D 0 or 1 for every A with    A4T 1 (A) D 0 : A measure preserving action is said to be weak mixing if for every measurable A and B 1 ˇ  1 X ˇˇ   A \ T n (B)  (A)(B)ˇ D 0 : n nD1

A measure preserving action is said to be mixing if for every measurable A and B   lim  A \ T n (B) D (A)(B) : n!1

It is easy to show that all three of these properties are isomorphism invariants and that (X; T) is mixing ) (X; T) is weak mixing ) (X; T) is ergodic. We include one more definition before introducing an even stronger isomorphism invariant. The action

of a group (or a semigroup) G on a probability space (X; ; B) is a family of measure preserving transformations f f g g g2G such that for any g; h 2 G we have that f g ( f h (x)) D f gCh (x) for almost every x 2 X. Thus any invertible measure preserving transformation (X; T) induces a Z action by f n (x) D T n (x). We say that a group action (X; Tg ) is mixing of all orders if for every n 2 N and every collection of set A1 ; : : : ; A n 2 B then   A1 \ Tg 2 (A2 ) \ Tg 2 Cg 3 (A3 ) : : : n Y  \ Tg 2 Cg 3 C:::g n (A n ) ! (A i ) iD1

as the g i go to infinity. Basic Tools Birkhoff’s ergodic theorem states that for every measure preserving action the limits fˆ(x) D lim

n!1

n1 1X  k  f T x n kD0

exists for almost every x and if (X; T) is ergodic then the R limit is f¯ D f d [2]. A partition P of X is a measurable function defined on X. (For simplicity we often assume that a partition is a function to Z or some subset of Z.) We write Pi for P1 (i). For any partition P of (X; T) define the partition T i P by P ı T i . Thus for invertible T this is given by W i T i P(x) D P(T i (x)). Then define (P)T D i2Z T P. i Thus (P)T is the smallest  -algebra which contains T (Pj ) for all i 2 Z and j 2 N. We say that (P)T is the  -algebra generated by P. A partition P is a generator of (X; T) if (P)T D B. Many measure preserving transformations come equipped with a natural partition. Rokhlin’s theorem For any measure preserving transformation (X; T), any > 0 and any n 2 N there exists A X such that T i (A) \ T  j (A) for all 0  i < j  n and ([niD0 T i (A) > 1  . Moreover for any finite partition P of X we can choose A such that (Pi \ A) D (A)(Pi ) for all i 2 N [63]. Shannon–McMillan–Breiman theorem [3,57] For any measure preserving system (X; T) and any > 0 there exists n 2 N and a set G with (G) > 1  with the following property. For any sequence g1 ; : : : g n 2 N T let g D niD1 P(T i (x)) D g i . Then if (g \ G) > 0 then   (g) 2 2 h((X;T)) ; 2 h((X;T))C :

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Krieger generator theorem If H((X; T)) < 1 then there exists a finite partition P such that (P)T D B. Thus every measure preserving transformation with finite entropy is isomorphic to a shift map on finitely many symbols [33]. Measure-theoretic entropy Entropy was introduced in physics by Rudolph Clausius in 1854. In 1948 Claude Shannon introduced the concept to information theory. Consider a process that generates a string of data of length n. The entropy of the process is the smallest number h such that you can condense the data to a string of zeroes and one of length hn and with high probability you can reconstruct the original data from the string of zeroes and ones. Thus the entropy of a process is the average amount of information transmitted per symbol of the process. Kolmogorov and Sinai introduced the concept of entropy to ergodic theory in the following way [31,32]. They defined the entropy of a partition Q is defined to be H(Q) D 

k X

(Q i ) log (Q i ) :

(Y; S) is trivial if Y consists of only one point. We say that a transformation (X; T) has completely positive entropy if every non-trivial factor of (X; T) has positive entropy. Isomorphism of Bernoulli Shifts Kolmogorov–Sinai A long standing open question was for which p and q are Bernoullip and Bernoulliq isomorphic. In particular are the Bernoulli 2 shift and the Bernoulli 3 shift isomorphic. Both of these transformations have completely positive entropy and all other isomorphism invariants which were known at the time are the same for the two transformations. The first application of the Kolmogorov–Sinai entropy was to show that the answer to this question is no. Fix a probability vector p. The transformation Bernoullip has Qp : x ! x0 as a generating partition. By Sinai’s theorem H(Bernoulli p ) D H(Qp ) D

Finally, the measure-theoretic entropy of a dynamical system (X; T) is defined as h((X; T)) D sup h(X; T; Q) Q

where the supremum is taken over all finite measurable partitions. A theorem of Sinai showed that if Q is a generator of (X; T) then h(T) D h(T; Q) [73]. This shows that for every measure preserving function (X; T) there is an associated entropy h(T) 2 [0; 1]. It is easy to show from the definition that entropy is an isomorphism invariant. We say that (Y; S) is a factor of (X; T) if there exists a map  : X ! Y such that (1)  is measure preserving and (2) (T(x)) D S((x)) for almost every x. Each factor (Y; S) can be associated with  1 (C ), which is an invariant sub -algebra of B. We say that

p i log2 (p i ) :

iD1

iD1

The measure-theoretic entropy of a dynamical system (X; T) with respect to a partition Q : X ! f1; : : : ; kg is then defined as ! N _ 1 n T Q : H h(X; T; Q) D lim N!1 N nD1

n X

Thus the Bernoulli 2 shift (with entropy 1) is not isomorphic to the Bernoulli 3 shift (with entropy log2 (3)). Sinai also made significant progress toward showing that Bernoulli shifts with the same entropy are isomorphic by proving the following theorem. Theorem 1 [72] If (X; T) is a measure preserving system of entropy h and (Y; S) is a Bernoulli shift of entropy h0  h then (Y; S) is a factor of (X; T). This theorem implies that if p and q are probability vectors and H(p) D H(q) then Bernoullip is a factor in Bernoulliq and Bernoulliq is a factor in Bernoullip. Thus we say that Bernoullip and Bernoulliq are weakly isomorphic. Explicit Isomorphisms The other early progress on proving that Bernoulli shifts with the same entropy are isomorphic came from Meshalkin. He considered pairs of probability vectors p and q with H(p) D H(q) and all of the pi and qi are related by some algebraic relations. For many such pairs he was able to prove that the two Bernoulli shifts are isomorphic. In particular he proved the following theorem. Theorem 2 [40] Let p D (1/4; 1/4; 1/4; 1/4) and q D (1/2; 1/8; 1/8; 1/8; 1/8). Then Bernoullip and Bernoulliq are isomorphic.

Isomorphism Theory in Ergodic Theory

Ornstein The central theorem in the study of isomorphisms of measure preserving transformations is Ornstein’s isomorphism theorem. Theorem 3 [46] If p and q are probability vectors and H(p) D H(q) then the Bernoulli shifts Bernoullip and Bernoulliq are isomorphic. To see how central this is to the field most of the rest of this article is a summary of: (1) (2) (3) (4)

The proof of Ornstein’s theorem, The consequences of Ornstein’s theorem, The generalizations of Ornstein’s theorem, and How the properties that Ornstein’s theorem implies that Bernoulli shifts must have differ from the properties of every other class of transformations.

The key to Ornstein’s proof was the introduction of the finitely determined property. To explain the finitely determined property we first define the Hamming distance of length n between sequences x; y 2 N Z by jfk 2 f1; : : : ; ng : x k D y k gj d¯n (x; y) D 1  : n Let (X; T) and (Y; S) be a measure preserving transformation and let P and Q be finite partitions of X and Y respectively. We say that (X; T) and P and (Y; S) and Q are within ı in n distributions if ˇ n  o  X ˇ ˇ x : P T i (x) D a i 8i D 1; : : : ; n ˇ (a 1 ;:::;a n )2Z n

 o ˇˇ n    y : Q T i (y) D a i 8i D 1; : : : ; n ˇˇ < ı :

A process (X; T) and P are finitely determined if for every " there exist n and ı such that if (Y; S) and Q are such that (1) (X; T) and P and Yand Q are within ı in n distributions and (2) jH(X; P)  H(Y; Q)j < ı then there exists a joining of X and Y such that for all m Z d¯m (x; y) d (x; y) < :

the Rokhlin lemma and the Shannon–McMillan–Breiman theorem to prove a more robust version of Theorem 1. To describe Ornstein’s proof we use the a description due to Rothstein. We say that for a joining of (X; ) and (Y; ) that P ; C if there exists a partition P˜ of C such that X  

Pi 4P˜i < : i

If P ; C for all > 0 then it is possible to show that there exists a partition P˜ of C such that X  

Pi 4P˜i D 0 i

and we write P  C . If P  C then (X; T) is a factor of (Y; S) by the map  that sends y ! x where P(T i x) D ˜ i y) for all i. P(S In this language Ornstein proved that if (X; T) is finitely determined, P is a generating partition of X and h((Y; S))  h((X; T)) then for every > 0 the set of joinings such that P ; C is an open and dense set. Thus by the Baire category theorem there exists such that P  C . This reproves Theorem 1. Moreover if (X; T) and (Y; S) are finitely determined, h((X; T)) D h((Y; S)) and P and Q are generating partition of X and Y then by the Baire category theorem there exists such that P  C and Q  B. Then the map  ˜ i y) for all i is an that sends y ! x where P(T i x) D P(S isomorphism. Properties of Bernoullis Now we define the very weak Bernoulli property which is the most effective property for showing that a measure preserving transformation is isomorphic to a Bernoulli shift. Given X and a partition P define the past of x by n o   Ppast (x) D x 0 : T i P x 0 D T i P(x) 8i 2 N and denote the measure  conditioned on Ppast (x) by j Ppast (x) . Define d¯n;x; D inf

Z



    d¯n x; x 0 d x; x 0 ;

x;y

A transformation (X; T) is finitely determined if it is finitely determined for every finite partition P. It is fairly straightforward to show that Bernoulli shifts are finitely determined. Ornstein used this fact along with

where the inf is taken over all which are couplings of j Ppast (x) and . Also define d¯n;Ppast ;(X;T) D

Z

d¯n;x; d :

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We say that (X; T) and P are very weak Bernoulli if for every > 0 there exists n such that d¯n;Ppast ;(X;T) < . We say that (X; T) is very weakly Bernoulli if there exists a generating partition P such that (X; T) and P are very weak Bernoulli. Ornstein and Weiss were able to show that the very weak Bernoulli property is both necessary and sufficient to be isomorphic to a Bernoulli shift. Theorem 4 [45,53] For transformations (X; T) the following conditions are equivalent: (1) (X; T) is finitely determined, (2) (X; T) is very weak Bernoulli and (3) (X; T) is isomorphic to a Bernoulli shift.

to a Bernoulli shift then the very weak Bernoulli property is more useful. There have been many classes of transformations that have been proven to be isomorphic to a Bernoulli shift. Here we mention two. The first class are the Markov chains. Friedman and Ornstein proved that if a Markov chain is mixing then it is isomorphic to a Bernoulli shift [9]. The second are automorphisms of [0; 1)n . Let M be any n  n matrix with integer coefficients and jdet(M)j D 1. If none of the eigenvalues f i gniD1 of M are roots of unity then Katznelson proved that T M is isomorphic to the Bernoulli shift with P entropy niD1 max(0; log( i )) [29].

Transformations not Isomorphic to Bernoulli Shifts Using the fact that a transformation is finitely determined or very weak Bernoulli is equivalent to it being isomorphic to a Bernoulli shift we can prove the following theorem. Theorem 5 [44] (1) If (X; T n ) is isomorphic to a Bernoulli shift then (X; T) is isomorphic to a Bernoulli shift. (2) If (X; T) is a factor of a Bernoulli shift then (X; T) is isomorphic to a Bernoulli shift. (3) If (X; T) is isomorphic to a Bernoulli shift then there exists a measure preserving transformation (Y; S) such that (Y; S n ) is isomorphic to (X; T).

Rudolph Structure Theorem An important application of the very weak Bernoulli condition is the following theorem of Rudolph. Theorem 6 [67] Let (X; T) be isomorphic to a Bernoulli shift, G be a compact Abelian group with Haar measure G and  : X ! G be a measurable map. Then let S : X G ! X  G be defined by S(x; g) D (T(x); g C (x)) : Then (X  G; S;   G ) is isomorphic to a Bernoulli shift. Transformations Isomorphic to Bernoulli Shifts One of the most important features of Ornstein’s isomorphism theory is that it can be used to check whether specific transformations (or families of transformations) are isomorphic to Bernoulli shifts. The finitely determined property is the key to the proof of Ornstein’s theorem and the proof of many of the consequences listed in Subsect. “Properties of Bernoullis”. However if one wants to show a particular transformation is isomorphic

Recall that a measure preserving transformation (X; T) has completely positive entropy if for every nontrivial (Y; S) which is a factor of (X; T) we have that H((Y; S)) > 0. It is easy to check that Bernoulli shifts have completely positive entropy. It is natural to ask if the converse true? We shall see that the answer is an emphatic no. While the isomorphism class of Bernoulli shifts is given by just one number, the situation for transformations of completely positive entropy is infinitely more complicated. Ornstein constructed the first example of a transformation with completely positive entropy which is not isomorphic to a Bernoulli shift [47]. Ornstein and Shields built upon this construction to prove the following theorem. Theorem 7 [50] For every h > 0 there is an uncountable family of completely positive entropy transformations which all have entropy h but no two distinct members of the family are isomorphic. Now that we see there are many isomorphic transformations that have completely positive entropy it is natural to ask if (X; T) is not isomorphic to a Bernoulli shift then is there any reasonable condition we can put on (Y; S) that implies the two transformations are isomorphic. For example if (X; T 2 ) and (Y; S 2 ) are completely positive entropy transformations which are isomorphic does that necessarily imply that (X; T) and (Y; S) are isomorphic? The answer turns out to be no [66]. We could also ask if (X; T) and (Y; S) are completely positive entropy transformations which are weakly isomorphic does that imply that (X; T) and (Y; S) are isomorphic? Again the answer is no [17]. The key insight to answering questions like this is due to Rudolph who showed that such questions about the isomorphism of transformations can be reduced to questions about conjugacy of permutations.

Isomorphism Theory in Ergodic Theory

Rudolph’s Counterexample Machine

Theorem 10 [17]

Given any transformation (X; T) and any permutation  in Sn (or SN ) we can define the transformation (X n ; T ; n ; B) by

(1) There exist measure preserving transformations (X; T) and (Y; S) which both have completely positive entropy and are weakly isomorphic but not isomorphic. (2) There exist measure preserving transformations (X; T) and (Y; S) which both have completely positive entropy and are not isomorphic but (X; T k ; ) is isomorphic to (Y; S k ; ) for every k > 1.

T (x1 ; x2 ; : : : ; x n )        D T x(1) ; T x(2) ; : : : ; T x(n) where n is the direct product of n copies of . Rudolph introduced the concept of a transformation having minimal self joinings. If a transformation has minimal self joinings then for every  it is possible to list all of the measures on X n which are invariant under T . If there exists an isomorphism  between T and (Y; S) then there is a corresponding measure on X  Y which is supported on points of the form (x; (x)) and has marginals  and . Thus if we know all of the measures on X  Y which are invariant under T  S then we know all of the isomorphisms between (X; T) and (Y; S). Using this we get the following theorem. Theorem 8 [65] There exists a nontrivial transformation with minimal self joinings. For any transformation (X; T) with minimal self joinings the corresponding transformation T1 is isomorphic to T2 if and only if the permutations 1 and 2 are conjugate. There are two permutations on two elements, the flip 1 D (12) and the identity 2 D (1)(2). For both permutations, the square of the permutation is the identity. Thus there are two distinct permutations whose square is the same. Rudolph showed that this fact can be used to generate two transformations which are mixing that are not isomorphic but their squares are isomorphic. The following theorem gives more examples of the power of this technique. Theorem 9 [65] (1) There exists measure preserving transformations (X; T) and (Y; S) which are weakly isomorphic but not isomorphic. (2) There exists measure preserving transformations (X; T) and (Y; S) which are not isomorphic but (X; T k ; ) is isomorphic to (Y; S k ; ) for every k > 1, and (3) There exists a mixing transformation with no non trivial factors. If (X; T) has minimal self joinings then it has zero entropy. However Hoffman constructed a transformation with completely positive entropy that shares many of the properties of transformations with minimal self joinings listed above.

T, T Inverse All of the transformations that have completely positive entropy but are not isomorphic to a Bernoulli shift described above are constructed by a process called cutting and stacking. These transformations have little inherent interest outside of their ergodic theory properties. This led many people to search for a “natural” example of such a transformation. The most natural examples are the [T; T 1 ] transformation and many other transformations derived from it. It is easy to show that the [T; T 1 ] transformation has completely positive entropy [39]. Kalikow proved that for many T the corresponding [T; T 1 ] transformation is not isomorphic to a Bernoulli shift. Theorem 11 [24] If h(T) > 0 then the [T; T 1 ] transformation is not isomorphic to a Bernoulli shift. The basic idea of Kalikow’s proof has been used by many others. Katok and Rudolph used the proof to construct smooth measure preserving transformations on infinite differentiable manifolds which have completely positive entropy but are not isomorphic to Bernoulli shifts [27,68]. Den Hollander and Steif did a thorough study of the ergodic theory properties of [T; T 1 ] transformations where T is simple random walk on a wide family of graphs [4]. Classifying the Invariant Measures of Algebraic Actions This problem of classifying all of the invariant measures like Rudolph did with his transformation with minimal self joinings comes up in a number of other settings. Ratner characterized the invariant measures for the horocycle flow and thus characterized the possible isomorphisms between a large family of transformations generated by the horocycle flow [59,60,61]. This work has powerful applications to number theory. There has also been much interest in classifying the measures on [0; 1) that are invariant under both times 2 and times 3. Furstenberg proved that the only closed infinite set on the circle which is invariant under times 2 and

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times 3 is [0; 1) itself and made the following measure theoretic conjecture.

Schmidt proved that this happens only in the trivial case that p and q are rearrangements of each other.

Conjecture 1 [10] The only nonatomic measure on [0; 1) which is invariant under times 2 and times 3 is Lebesgue measure.

Theorem 14 [70] If p and q are probability vectors and the Bernoulli shifts Bernoullip and Bernoulliq are isomorphic and there exists an isomorphism  such that  and  1 are both finitary and have finite expected coding time then p is a rearrangement of q.

Rudolph improved on the work of Lyons [36] to provide the following partial answer to this conjecture. Theorem 12 [69] The only measure on [0; 1) which is invariant under multiplication by 2 and by 3 and has positive entropy under multiplication by 2 is Lebesgue measure. Johnson then proved that for all relatively prime p and q that p and q can be substituted for 2 and 3 in the theorem above. This problem can be generalized to higher dimensions by studying the actions of commuting integer matrices of determinant greater than one on tori. Katok and Spatzier [28] and Einsiedler and Lindenstrauss [5] obtained results similar to Rudolph’s for actions of commuting matrices. Finitary Isomorphisms By Ornstein’s theorem we know that there exists an isomorphism between any two Bernoulli shifts (or mixing Markov shifts) of the same entropy. There has been much interest in studying how “nice” the isomorphism can be. By this we mean can  be chosen so that the map x ! ((x))0 is continuous and if so what is its best possible modulus of continuity? We say that a map  from N Z to N Z is finitary if in order to determine ((x))0 we only need to know finitely many coordinates of x. More precisely if for almost every x there exists m(x) such that ˚  x 0 : x 0i D x i for all jij  m(x)     and  x 0 0 ¤ (x)0 D 0 : R We say that m(x) has finite tth moment if m(x) t d< 1 and that  has finite expected coding length if the first moment of m(x) is finite. Keane and Smorodinsky proved the following strengthening of Ornstein’s isomorphism theorem. Theorem 13 [30] If p and q are probability vectors with H(p) D H(q) then the Bernoulli shifts Bernoullip and Bernoulliq are isomorphic and there exists an isomorphism  such that  and  1 are both finitary. The nicest that we could hope  to be is if both  and  1 are finitary and have finite expected coding length.

The best known result about this problem is the following theorem of Harvey and Peres. Theorem 15 [14] If p and q are probability vectors with P P H(p) D H(q) then i (p i )2 log(p i ) D i (q i )2 log(q i ) if and only if the Bernoulli shifts Bernoullip and Bernoulliq are isomorphic and there exists an isomorphism  such that  and  1 are both finitary and have finite one half moment. Flows A flow is a measure preserving action of the group R on a measure space (X; T). A cross section is any measurable set in C X such that for almost every x 0 < inf ft : Tt (x) 2 Cg < 1 : For any flow (X; T) and fTt g t2R and any cross section C we define the return time map for C R : C ! C as follows. For any x 2 C define t(x) D inf ft : Tt (x) 2 Cg then set R(x) D Tt(x) (x). There is a standard method to project the probability measure  on X to an invariant probability measure C on C as well as the -algebra B on X to a  -algebra BC on C such that (C; C ; BC ) and R is a measure preserving transformation. First we show that there is a natural analog of Bernoulli shifts for flows. Theorem 16 [45] There exists a flow (X; T) and fTt g t2R such that for every t > 0 the map (X; T) and T t is isomorphic to a Bernoulli shift. Moreover for any h 2 (0; 1] there exists (X; T) and fTt g t2R such that h(T1 ) D h. We say that such a flow (X; f f t g t2R ; ; B) is a Bernoulli flow. This next version of Ornstein’s isomorphism theorem shows that up to isomorphism and a change in time (considering the flow X and fTc t g instead of X and fTt g) there are only two Bernoulli flows, one with positive but finite entropy and one with infinite entropy. Theorem 17 [45] If (X; T) and fTt g t2R and (Y; S) fS t g t2R are Bernoulli flows and h(T1 ) D h(S1 ) then they are isomorphic.

Isomorphism Theory in Ergodic Theory

As in the case of actions of Z there are many natural examples of flows that are isomorphic to the Bernoulli flow. The first is for geodesic flows. In the 1930s Hopf proved that geodesic flows on compact surfaces of constant negative curvature are ergodic [22]. Ornstein and Weiss extended Hopf’s proof to show that the geodesic flow is also Bernoulli [52]. The second class of flows comes from billiards on a square table with one circular bumper. The state space X consists of all positions and velocities for a fixed speed. The flow T t is frictionless movement for time t with elastic collisions. This flow is also isomorphic to the Bernoulli flow [11]. Other Equivalence Relations In this section we will discuss a number of equivalence relations between transformations that are weaker than isomorphism. All of these equivalence relations have a theory that is parallel to Ornstein’s theory. Kakutani Equivalence We say that two transformations (X; T) and (Y; S) are Kakutani equivalent if there exist subsets A X and B Y such that (TA ; A; A ) and (S B ; B; B ) are isomorphic. This is equivalent to the existence of a flow (X; T) and fTt g t2R with cross sections C and C 0 such that the return time maps of C and C 0 are isomorphic to (X; T) and (Y; S) respectively. Using the properties of entropy of the induced map we have that if (X; T) and (Y; S) are Kakutani equivalent then either h((X; T)) D h((Y; S)) D 0, 0 < h((X; T)); h((Y; S)) < 1 or h((X; T)) D h((Y; S)) D 1. In general the answer to the question of which pairs of measure preserving transformations are isomorphic is quite poorly understood. But if one of the transformations is a Bernoulli shift then Ornstein’s theory gives a fairly complete answer to the question. A similar situation exists for Kakutani equivalence. In general the answer to the question of which pairs of measure preserving transformation are Kakutani equivalent is also quite poorly understood. But the more specialized question of which transformations are isomorphic to a Bernoulli shift has a more satisfactory answer. Feldman constructed a transformation (X; T) which has completely positive entropy but (X; T) is not Kakutani equivalent to a Bernoulli shift. Ornstein, Rudolph and Weiss extended Feldman’s work to construct a complete theory of the transformations that are Kakutani equivalent to a Bernoulli shift [49] for positive entropy transformations and a theory of the transformations that are

Kakutani equivalent to an irrational rotation [49] for zero entropy transformations. (The zero entropy version of this theorem had been developed independently (and earlier) by Katok [26].) They defined two class of transformations called loosely Bernoulli and finitely fixed. The definitions of these properties are the same as the definitions of very weak Bernoulli and finitely determined except that the d¯ metric is replaced by the f¯ metric. For x; y 2 N Z we define k f¯n (x; y) D 1  n where k is the largest number such that sequences 1  i1 < i2 <    < i k  n and 1  j1 < j2 <    < j k  n such that x i l D y j l for all j, 1  j  k. (In computer science this metric is commonly referred to as the edit distance.) Note that d¯n (x; y)  f¯n (x; y). They proved the following analog of Theorem 5. Theorem 18 For transformations (X; T) with h((X; T)) > 0 the following conditions are equivalent: (1) (2) (3) (4)

(X; T) is finitely fixed, (X; T) is loosely Bernoulli, (X; T) is Kakutani equivalent to a Bernoulli shift and There exists a Bernoulli flow Y and fF t g t2R and a cross section C such that the return time map for C is isomorphic to (X; T).

Restricted Orbit Equivalence Using the d¯ metric we got a theory of which transformations are isomorphic to a Bernoulli shift. Using the f¯ metric we got a strikingly similar theory of which transformations are Kakutani equivalent to a Bernoulli shift. Rudolph showed that it is possible to replace the d¯ metric (or the f¯ metric) with a wide number of other metrics and produce parallel theories for other equivalence relations. For instance, for each of these theories we get a version of Theorem 5. This collection of theories is called restricted orbit equivalence [64]. Non-invertible Transformations The question of which noninvertible measure preserving transformations are isomorphic turns out to be quite different from the same question for invertible transformations. In one sense it is easier because of an additional isomorphism invariant. For any measure preserving transformation (X; T) the probability measure j T 1 (x) on T 1 (x) is defined for almost every x 2 X. (If (X; T) is invertible then this measure

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is trivial as jfT 1 (x)gj D 1 and j T 1 (x) (T 1 (x)) D 1 for almost every x.) It is easy to check that if  is an isomorphism from (X; T) to (Y; S) then for almost every x and x 0 2 T 1 (x) we have ˇ ˇ       ˇ T 1 (x) x 0 D  ˇS 1 ((x))  x 0 : From this we can easily see that if p D fp i gniD1 and q D fq i gm iD1 are probability vectors then the corresponding one sided Bernoulli shifts are isomorphic only if m D n and there is a permutation  2 S n such that p(i) D q i for all i. (In this case we say p is a rearrangement of q.) If p is a rearrangement of q then it is easy to construct an isomorphism between the corresponding Bernoulli shifts. Thus the analog of Ornstein’s theorem for Bernoulli endomorphisms is trivial. However we will see that there still is an analogous theory classifying the class of endomorphism that are isomorphic to Bernoulli endomorphisms. We say that an endomorphism is uniformly d to 1 if for almost every x we have that jfT 1 (x)gj D d and j T 1 (x) (y) D 1/d for all y 2 T 1 (x). Hoffman and Rudolph defined two classes of noninvertible transformations called tree very week Bernoulli and tree finitely determined and proved the following theorem. Theorem 19 The following three conditions are equivalent for uniformly d to 1 endomorphisms. (1) (X; T) is tree very weak Bernoulli, (2) (X; T) is tree finitely determined, and (3) (X; T) is isomorphic to the one sided Bernoulli d shift. Jong extended this theorem to say that if there exists a probability vector p such that for almost every x the distribution of j T  1 (x) is the same as the distribution of p then (X; T) is isomorphic to the one sided Bernoulli d shift if and only if it is tree finitely determined and if and only if it is tree very weak Bernoulli [23]. Markov Shifts We saw that mixing Markov chains are isomorphic if they have the same entropy. As we have seen there are additional isomorphism invariants for noninvertible transformations. Ashley, Marcus and Tuncel managed to classify all one sided mixing Markov chains up to isomorphism [1]. Rational Maps Rational maps are the main object of study in complex dynamics. For every rational function f (z) D p(z)/q(z) there is a nonempty compact set J f which is called the Julia set.

Roughly speaking this is the set of points for which every neighborhood acts “chaotically” under repeated iterations of f . In order to consider rational maps as measure preserving transformations we need to specify an invariant measure. The following theorem of Gromov shows that for every rational map there is one canonical measure to consider. Theorem 20 [12] For every f rational function of degree d there exists a unique invariant measure f of maximal entropy. We have that h( f ) D log2 d and  f (J f ) D 1. The properties of this measure were studied by Freire, Lopes and Mañé [8]. Mañé that analysis to prove the following theorem. Theorem 21 [38] For every rational function f of degree d there exists n such that (C; f n ;  f ) (where f n (z) D f ( f ( f : : : (z))) is composition) is isomorphic to the one sided Bernoulli dn shift. Heicklen and Hoffman used the tree very weak Bernoulli condition to show that we can always take n to be one. Theorem 22 [16] For every rational function f of degree d  2 the corresponding map ((C;  f ; B); T f ) is isomorphic to the one sided Bernoulli d shift. Differences with Ornstein’s Theory Unlike Kakutani equivalence and restricted orbit equivalence which are very close parallels to Ornstein’s theory, the theory of which endomorphisms are isomorphic to a Bernoulli endomorphism contains some significant differences. One of the principal results of Ornstein’s isomorphism theory is that if (X; T) is an invertible transformation and (X; T 2 ) is isomorphic to a Bernoulli shift then (X; T) is also isomorphic to a Bernoulli shift. There is no corresponding result for noninvertible transformations. Theorem 23 [19] There is a uniformly two to one endomorphism (X; T) which is not isomorphic to the one sided Bernoulli 2 shift but (X; T 2 ) is isomorphic to the one sided Bernoulli 4 shift. Factors of a Transformation In this section we study the relationship between a transformation and its factors. There is a natural way to associate a factor of (X; T) with a sub  -algebra of B. Let (Y; S) be a factor of (X; T) with factor map  : X ! Y. Then the  -algebra associated with (Y; S) is BY D  1 (C ). Thus the study of factors of a transformation is the study of its sub  -algebras.

Isomorphism Theory in Ergodic Theory

Almost every property that we have discussed above has an analog in the study of factors of a transformation. We give three such examples. We say that two factors C and D of (X; T) are relatively isomorphic if there exists an isomorphism : X ! X of (X; T) with itself such that (C ) D D. We say that (X; T) has relatively completely positive entropy with respect to C if every factor D which contains D has h(D) > h(C ). We say that C is relatively Bernoulli if there exists a second factor D which is independent of D and B D C _ D. Thouvenot defined properties of factors called relatively very weak Bernoulli and relatively finitely determined. Then he proved an analog of Theorem 3. This says that a factor being relatively Bernoulli is equivalent to it being relatively finitely determined (and also equivalent to it being relatively very weak Bernoulli). The Pinsker algebra is the maximal -algebra P such that h(P ) D 0. The Pinsker conjecture was that for every measure preserving transformation (X; T) there exists a factor C such that (1) C is independent of the Pinsker algebra P (2) B D C _ P and (3) (X; C ) has completely positive entropy. Ornstein found a counterexample to the Pinsker conjecture [48]. After Thouvenot developed the relative isomorphism theory he came up with the following question which is referred to as the weak Pinsker conjecture. Conjecture 2 For every measure preserving transformation (X; T) and every > 0 there exist invariant -algebras C ; D B such that (1) (2) (3) (4)

C is independent of D BDC_D (X; T; ; D) is isomorphic to a Bernoulli shift h((X; T; ; C )) < :

There is a wide class of transformations which have been proven to satisfy the weak Pinsker conjecture. This class includes almost all measure preserving transformations which have been extensively studied. Actions of Amenable Groups All of the discussion above has been about the action of a single invertible measure preserving transformation (actions of N and Z) or flows (actions of R). We now consider more general group actions. If we have two actions S and T on a measure space (X; ) which commute (S(T(x)) D T(S(x)) for almost every x) then there is an action of Z2 on (X; ) given by f(n;m) (x) D S n (T m (x)).

A natural question to ask is do there exist a version of entropy theory and Ornstein’s isomorphism theory for actions of two commuting automorphisms. More generally for each of the results discussed above we can ask what is the largest class of groups such that an analogous result is true. It turns out that for most of the properties described above the right class of groups is discrete amenable groups. A Følner sequence F n in a group G is a sequence of subsets F n of G such that for all g 2 G we have that lim n!1 jg(Fn )j/jFn j. A countable group is amenable if and only if it has a Følner sequence. For nonamenable groups it is much more difficult to generalize Birkhoff’s ergodic theorem [41,42]. Lindenstrauss proved that for every discrete amenable group there is an analog of the ergodic theorem [35]. For every amenable group G and every probability vector p we can define a Bernoulli action of G. There are also analogs of Rokhlin’s lemma and the Shannon–McMillan– Breiman theorem for actions of all discrete amenable groups [33,51,54]. Thus we have all of the ingredients to prove a version of Ornstein’s isomorphism theorem. Theorem 24 If p and q are probability vectors and H(p) D H(q) then the Bernoulli actions of G corresponding to p and q are isomorphic. Also all of the aspects of Rudolph’s theory of restricted orbit equivalence can be carried out for actions of amenable groups [25]. Differences Between Actions of Z and Actions of Other Groups Although generalizations of Ornstein theory and restricted orbit equivalence carry over well to the actions of discrete amenable groups there do turn out to be some significant differences between the possible actions of Z and those of other discrete amenable groups. Many of these have to do with the generalization of Markov shifts. For actions of Z2 these are called Markov random fields. By the result of Friedman and Ornstein if a Markov chain is mixing then it has completely positive entropy and it is isomorphic to a Bernoulli shift. Mixing Markov random fields can have very different properties. Ledrappier constructed a Z2 action which is a Markov random field and is mixing but has zero entropy [34]. Even more surprising even though it is mixing it is not mixing of all orders. The existence of a Z action which is mixing but not mixing of all orders is one of the longest standing open questions in ergodic theory [13]. Even if we try to strengthen the hypothesis of Friedman and Ornstein’s theorem to assume that the Markov

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random field has completely positive entropy we will not succeed as there exists a Markov random field which has completely positive entropy but is not isomorphic to a Bernoulli shift [18]. Future Directions In the future we can expect to see progress of isomorphism theory in a variety of different directions. One possible direction for future research is better understand the properties of finitary isomorphisms between various transformations and Bernoulli shifts described in Sect. “Finitary Isomorphisms”. Another possible direction would be to find a theory of equivalence relations for Bernoulli endomorphisms analogous to the one for invertible Bernoulli transformations described in Sect. “Other Equivalence Relations”. As the subject matures the focus of research in isomorphism theory will likely shift to connections to other fields. Already there are deep connections between isomorphism theory and both number theory and statistical physics. Finally one hopes to see progress made on the two dominant outstanding conjectures in the field: Thouvenot weak Pinsker conjecture (Conjecture 2) and Furstenberg’s conjecture (Conjecture 1) about measures on the circle invariant under both the times 2 and times 3 maps. Progress on either of these conjectures would invariably lead the field in exciting new directions. Bibliography 1. Ashley J, Marcus B, Tuncel S (1997) The classification of onesided Markov chains. Ergod Theory Dynam Syst 17(2):269–295 2. Birkhoff GD (1931) Proof of the ergodic theorem. Proc Natl Acad Sci USA 17:656–660 3. Breiman L (1957) The individual ergodic theorem of information theory. Ann Math Statist 28:809–811 4. Den Hollander F, Steif J (1997) Mixing E properties of the generalized T; T 1 -process. J Anal Math 72:165–202 5. Einsiedler M, Lindenstrauss E (2003) Rigidity properties of Zd actions on tori and solenoids. Electron Res Announc Amer Math Soc 9:99–110 6. Einsiedler M, Katok A, Lindenstrauss E (2006) Invariant measures and the set of exceptions to Littlewood’s conjecture. Ann Math (2) 164(2):513–560 7. Feldman J (1976) New K-automorphisms and a problem of Kakutani. Isr J Math 24(1):16–38 8. Freire A, Lopes A, Mañé R (1983) An invariant measure for rational maps. Bol Soc Brasil Mat 14(1):45–62 9. Friedman NA, Ornstein DS (1970) On isomorphism of weak Bernoulli transformations. Adv Math 5:365–394 10. Furstenberg H (1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math Syst Theory 1:1–49 11. Gallavotti G, Ornstein DS (1974) Billiards and Bernoulli schemes. Comm Math Phys 38:83–101

12. Gromov M (2003) On the entropy of holomorphic maps. Enseign Math (2) 49(3–4):217–235 13. Halmos PR (1950) Measure theory. D Van Nostrand, New York 14. Harvey N, Peres Y. An invariant of finitary codes with finite expected square root coding length. Ergod Theory Dynam Syst, to appear 15. Heicklen D (1998) Bernoullis are standard when entropy is not an obstruction. Isr J Math 107:141–155 16. Heicklen D, Hoffman C (2002) Rational maps are d-adic Bernoulli. Ann Math (2) 156(1):103–114 17. Hoffman C (1999) A K counterexample machine. Trans Amer Math Soc 351(10):4263–4280 18. Hoffman C (1999) A Markov random field which is K but not Bernoulli. Isr J Math 112:249–269 19. Hoffman C (2004) An endomorphism whose square is Bernoulli. Ergod Theory Dynam Syst 24(2):477–494 20. Hoffman C (2003) The scenery factor of the [T; T 1 ] transformation is not loosely Bernoulli. Proc Amer Math Soc 131(12):3731–3735 21. Hoffman C, Rudolph D (2002) Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann Math (2) 156(1):79–101 22. Hopf E (1971) Ergodic theory and the geodesic flow on surfaces of constant negative curvature. Bull Amer Math Soc 77: 863–877 23. Jong P (2003) On the isomorphism problem of p-endomorphisms. Ph D thesis, University of Toronto 24. Kalikow SA (1982) T; T 1 transformation is not loosely Bernoulli. Ann Math (2) 115(2):393–409 25. Kammeyer JW, Rudolph DJ (2002) Restricted orbit equivalence for actions of discrete amenable groups. Cambridge tracts in mathematics, vol 146. Cambridge University Press, Cambridge 26. Katok AB (1975) Time change, monotone equivalence, and standard dynamical systems. Dokl Akad Nauk SSSR 223(4):789–792; in Russian 27. Katok A (1980) Smooth non-Bernoulli K-automorphisms. Invent Math 61(3):291–299 28. Katok A, Spatzier RJ (1996) Invariant measures for higherrank hyperbolic abelian actions. Ergod Theory Dynam Syst 16(4):751–778 29. Katznelson Y (1971) Ergodic automorphisms of T n are Bernoulli shifts. Isr J Math 10:186–195 30. Keane M, Smorodinsky M (1979) Bernoulli schemes of the same entropy are finitarily isomorphic. Ann Math (2) 109(2):397–406 31. Kolmogorov AN (1958) A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl Akad Nauk SSSR (NS) 119:861–864; in Russian 32. Kolmogorov AN (1959) Entropy per unit time as a metric invariant of automorphisms. Dokl Akad Nauk SSSR 124:754–755; in Russian 33. Krieger W (1970) On entropy and generators of measure-preserving transformations. Trans Amer Math Soc 149:453–464 34. Ledrappier F (1978) Un champ markovien peut être d’entropie nulle et mélangeant. CR Acad Sci Paris Sér A–B 287(7):A561– A563; in French 35. Lindenstrauss E (2001) Pointwise theorems for amenable groups. Invent Math 146(2):259–295 36. Lyons R (1988) On measures simultaneously 2- and 3-invariant. Isr J Math 61(2):219–224 37. Mañé R (1983) On the uniqueness of the maximizing measure for rational maps. Bol Soc Brasil Mat 14(1):27–43

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38. Mañé R (1985) On the Bernoulli property for rational maps. Ergod Theory Dynam Syst 5(1):71–88 39. Meilijson I (1974) Mixing properties of a class of skew-products. Isr J Math 19:266–270 40. Meshalkin LD (1959) A case of isomorphism of Bernoulli schemes. Dokl Akad Nauk SSSR 128:41–44; in Russian 41. Nevo A (1994) Pointwise ergodic theorems for radial averages on simple Lie groups. I. Duke Math J 76(1):113–140 42. Nevo A, Stein EM (1994) A generalization of Birkhoff’s pointwise ergodic theorem. Acta Math 173(1):135–154 43. Ornstein DS (1973) A K automorphism with no square root and Pinsker’s conjecture. Adv Math 10:89–102 44. Ornstein D (1970) Factors of Bernoulli shifts are Bernoulli shifts. Adv Math 5:349–364 45. Ornstein D (1970) Two Bernoulli shifts with infinite entropy are isomorphic. Adv Math 5:339–348 46. Ornstein D (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 4:337–352 47. Ornstein DS (1973) An example of a Kolmogorov automorphism that is not a Bernoulli shift. Adv Math 10:49–62 48. Ornstein DS (1973) A mixing transformation for which Pinsker’s conjecture fails. Adv Math 10:103–123 49. Ornstein DS, Rudolph DJ, Weiss B (1982) Equivalence of measure preserving transformations. Mem Amer Math Soc 37(262). American Mathematical Society 50. Ornstein DS, Shields PC (1973) An uncountable family of K-automorphisms. Adv Math 10:63–88 51. Ornstein D, Weiss B (1983) The Shannon–McMillan–Breiman theorem for a class of amenable groups. Isr J Math 44(1): 53–60 52. Ornstein DS, Weiss B (1973) Geodesic flows are Bernoullian. Isr J Math 14:184–198 53. Ornstein DS, Weiss B (1974) Finitely determined implies very weak Bernoulli. Isr J Math 17:94–104 54. Ornstein DS, Weiss B (1987) Entropy and isomorphism theorems for actions of amenable groups. J Anal Math 48:1–141 55. Parry W (1981) Topics in ergodic theory. Cambridge tracts in mathematics, vol 75. Cambridge University Press, Cambridge 56. Parry W (1969) Entropy and generators in ergodic theory. WA Benjamin, New York 57. Petersen K (1989) Ergodic theory. Cambridge studies in advanced mathematics, vol 2. Cambridge University Press, Cambridge 58. Pinsker MS (1960) Dynamical systems with completely pos-

59. 60. 61. 62. 63. 64. 65.

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itive or zero entropy. Dokl Akad Nauk SSSR 133:1025–1026; in Russian Ratner M (1978) Horocycle flows are loosely Bernoulli. Isr J Math 31(2):122–132 Ratner M (1982) Rigidity of horocycle flows. Ann Math (2) 115(3):597–614 Ratner M (1983) Horocycle flows, joinings and rigidity of products. Ann Math (2) 118(2):277–313 Ratner M (1991) On Raghunathan’s measure conjecture. Ann Math (2) 134(3):545–607 Halmos PR (1960) Lectures on ergodic theory. Chelsea Publishing, New York Rudolph DJ (1985) Restricted orbit equivalence. Mem Amer Math Soc 54(323). American Mathematical Society Rudolph DJ (1979) An example of a measure preserving map with minimal self-joinings, and applications. J Anal Math 35:97–122 Rudolph DJ (1976) Two nonisomorphic K-automorphisms with isomorphic squares. Isr J Math 23(3–4):274–287 Rudolph DJ (1983) An isomorphism theory for Bernoulli free Zskew-compact group actions. Adv Math 47(3):241–257 Rudolph DJ (1988) Asymptotically Brownian skew products give non-loosely Bernoulli K-automorphisms. Invent Math 91(1):105–128 Rudolph DJ (1990) 2 and 3 invariant measures and entropy. Ergod Theory Dynam Syst 10(2):395–406 Schmidt K (1984) Invariants for finitary isomorphisms with finite expected code lengths. Invent Math 76(1):33–40 Shields P (1973) The theory of Bernoulli shifts. Chicago lectures in mathematics. University of Chicago Press, Chicago Sina˘ı JG (1962) A weak isomorphism of transformations with invariant measure. Dokl Akad Nauk SSSR 147:797–800; in Russian Sina˘ı J (1959) On the concept of entropy for a dynamic system. Dokl Akad Nauk SSSR 124:768–771; in Russian Thouvenot J-P (1975) Quelques propriétés des systèmes dynamiques qui se décomposent en un produit de deux systèmes dont l’un est un schéma de Bernoulli. Conference on ergodic theory and topological dynamics, Kibbutz, Lavi, 1974. Isr J Math 21(2–3):177–207; in French Thouvenot J-P (1975) Une classe de systèmes pour lesquels la conjecture de Pinsker est vraie. Conference on ergodic theory and topological dynamics, Kibbutz Lavi, 1974. Isr J Math 21(2–3):208–214; in French

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Minimal self-joinings Let k  2 be an integer. The ergodic measure-preserving dynamical system T has k-fold minimal self-joinings if, for any ergodic joining  of k copies of T, we can partition the set f1; : : : ; kg of coordinates into subsets J1 ; : : : ; J` such that

Joinings in Ergodic Theory THIERRY DE LA RUE Laboratoire de Mathématiques Raphaël Salem, CNRS – Université de Rouen, Saint Étienne du Rouvray, France

1. For j1 and j2 belonging to the same J i , the marginal of  on the coordinates j1 and j2 is supported on the graph of T n for some integer n (depending on j1 and j2 ); 2. For j1 2 J1 ; : : : ; j` 2 J` , the coordinates j1 ; : : : ; j` are independent.

Article Outline Glossary Definition of the Subject Introduction Joinings of Two or More Dynamical Systems Self-Joinings Some Applications and Future Directions Bibliography Glossary Disjoint measure-preserving systems The two measurepreserving dynamical systems (X; A; ; T) and (Y; B; ; S) are said to be disjoint if their only joining is the product measure  ˝ . Joining Let I be a finite or countable set, and for each i 2 I, let (X i ; A i ;  i ; Ti ) be a measure-preserving dynamical system. A joining of these systems is a probability Q measure on the Cartesian product i2I X i , which has the i ’s as marginals, and which is invariant under the N product transformation i2I Ti . Marginal of a probability measure on a product space Let  be a probability measure on the Cartesian product of aQ finite or countable  collection of measurable N spaces X ; A i , and let J D f j1 ; : : : ; j k g i2I i i2I be a finite subset of I. The k-fold marginal of  on X j 1 ; : : : ; X j k is the probability measure  defined by: 8A1 2 A j 1 ; : : : ; A k 2 A j k ; 0 (A1      A k ) :D  @ A1      A k 

Y

1 Xi A :

i2InJ

Markov intertwining Let (X; A; ; T) and (Y; B; ; S) be two measure-preserving dynamical systems. We call Markov intertwining of T and S any operator P : L2 (X; ) ! L2 (Y; ) enjoying the following properties:  PU T D U S P, where U T and U S are the unitary operators on L2 (X; ) and L2 (Y; ) associated respectively to T and S (i. e. U T f (x) D f (Tx), and U S g(y) D g(Sy)).  P1 X D 1Y ,  f  0 implies P f  0, and g  0 implies P g  0, where P is the adjoint operator of P.

We say that T has minimal self-joinings if T has k-fold minimal self-joinings for every k  2. Off-diagonal self-joinings Let (X; A; ; T) be a measure-preserving dynamical system, and S be an invertible measure-preserving transformation of (X; A; ) commuting with T. Then the probability measure S defined on X  X by S (A  B) :D (A \ S 1 B)

(1)

is a 2-fold self-joining of T supported on the graph of S. We call it an off-diagonal self-joining of T. Process in a measure-preserving dynamical systems Let (X; A; ; T) be a measure-preserving dynamical system, and let (E; B(E)) be a measurable space (which may be a finite or countable set, or Rd , or C d . . . ). For any E-valued random variable  defined on the probability space (X; A; ), we can consider the stochastic process ( i ) i2Z defined by  i :D  ı T i : Since T preserves the probability measure , ( i ) i2Z is a stationary process: For any ` and n, the distribution of (0 ; : : : ; ` ) is the same as the probability distribution of ( n ; : : : ;  nC` ). Self-joining Let T be a measure-preserving dynamical system. A self-joining of T is a joining of a family (X i ; A i ;  i ; Ti ) i2I of systems where each T i is a copy of T. If I is finite and has cardinal k, we speak of a k-fold self-joining of T. Simplicity For k  2, we say that the ergodic measurepreserving dynamical system T is k-fold simple if, for any ergodic joining  of k copies of T, we can partition the set f1; : : : ; kg of coordinates into subsets J1 ; : : : ; J` such that 1. for j1 and j2 belonging to the same J i , the marginal of  on the coordinates j1 and j2 is supported on the graph of some S 2 C(T) (depending on j1 and j2 );

Joinings in Ergodic Theory

2. for j1 2 J1 ; : : : ; j` 2 J` , the coordinates j1 ; : : : ; j` are independent. We say that T is simple if T is k-fold simple for every k  2. Definition of the Subject The word joining can be considered as the counterpart in ergodic theory of the notion of coupling in probability theory (see e. g. [48]): Given two or more processes defined on different spaces, what are the possibilities of embedding them together in the same space? There always exists the solution of making them independent of each other, but interesting cases arise when we can do this in other ways. The notion of joining originates in ergodic theory from pioneering works of H. Furstenberg [14], who introduced the fundamental notion of disjointness, and D.J. Rudolph, who laid the basis of joining theory in his article on minimal self-joinings [41]. It has today become an essential tool in the classification of measure-preserving dynamical systems and in the study of their intrinsic properties. Introduction A central question in ergodic theory is to tell when two measure-preserving dynamical systems are essentially the same, i. e. when they are isomorphic. When this is not the case, a finer analysis consists of asking what these two systems could share in common: For example, do there exist stationary processes which can be observed in both systems? This latter question can also be asked in the following equivalent way: Do these two systems have a common factor? The arithmetical flavor of this question is not fortuitous: There are deep analogies between the arithmetic of integers and the classification of measure-preserving dynamical systems, and these analogies were at the starting point of the study of joinings in ergodic theory. In the seminal paper [14] which introduced the concept of joinings in ergodic theory, H. Furstenberg observed that two operations can be done with dynamical systems: We can consider the product of two dynamical systems, and we can also take a factor of a given system. Like the multiplication of integers, the product of dynamical systems is commutative, associative, it possesses a neutral element (the trivial single-point system), and the systems S and T are both factors of their product S  T. It was then natural to introduce the property for two measure-preserving systems to be relatively prime. As far as integers are concerned, there are two equivalent ways of characterizing the relative primeness: First, the integers a and b are relatively prime if their unique positive common factor is 1.

Second, a and b are relatively prime if, each time both a and b are factors of an integer c, their product ab is also a factor of c. It is a well-known theorem in number theory that these two properties are equivalent, but this was not clear for their analog in ergodic theory. Furstenberg reckoned that the second way of defining relative primeness was the most interesting property in ergodic theory, and called it disjointness of measure-preserving systems (we will discuss precisely in Subsect. “From Disjointness to Isomorphy” what the correct analog is in the setting of ergodic theory). He also asked whether the non-existence of a non-trivial common factor between two systems was equivalent to their disjointness. He was able to prove that disjointness implies the impossibility of a non-trivial common factor, but not the converse. And in fact, the converse turns out to be false: In 1979, D.J. Rudolph exhibited a counterexample in his paper introducing the important notion of minimal self-joinings. The relationships between disjointness and the lack of common factor will be presented in details in Sect. “Joinings and Factors”. Given two measure-preserving dynamical systems S and T, the study of their disjointness naturally leads one to consider all the possible ways these two systems can be both seen as factors of a third system. As we shall see, this is precisely the study of their joinings. The concept of joining turns out to be related to many important questions in ergodic theory, and a large number of deep results can be stated and proved inside the theory of joinings. For example, the fact that the dynamical systems S and T are isomorphic is equivalent to the existence of a special joining between S and T, and this can be used to give a joining proof of Krieger’s finite generator theorem, as well as Ornstein’s isomorphism theorem (see Sect. “Joinings Proofs of Ornstein’s and Krieger’s Theorems”). As it already appears in Furstenberg’s article, joinings provide a powerful tool in the classification of measure-preserving dynamical systems: Many classes of systems can be characterized in terms of their disjointness with other systems. Joinings are also strongly connected with difficult questions arising in the study of the convergence almost everywhere of nonconventional averages (see Sect. “Joinings and Pointwise Convergence”). Amazingly, a situation in which the study of joinings leads to the most interesting results consists of considering two or more identical systems. We then speak of the self-joinings of the dynamical system T. Again, the study of self-joinings is closely related to many ergodic properties of the system: Its mixing properties, the structure of its factors, the transformations which commute with T, and so on. . . We already mentioned minimal self-joinings, and we will see in Sect. “Minimal Self-Joinings” how this

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property may be used to get many interesting examples, such as a transformation with no root, or a process with no non-trivial factor. In the same section we will also discuss a very interesting generalization of minimal self-joinings: the property of being simple. The range of applications of joinings in ergodic theory is very large; only some of them will be given in Sect. “Some Applications and Future Directions”: The use of joinings in proving Krieger’s and Ornstein’s theorems, the links between joinings and some questions of pointwise convergence, and the strong connections between the study of self-joinings and Rohlin’s famous question on multifold mixing, which was first posed in 1949 [39]. Joinings of Two or More Dynamical Systems In the following, we are given a finite or countable family (X i ; A i ;  i ; Ti ) i2I of measure-preserving dynamical systems: T i is an invertible measure-preserving transformation of the standard Borel probability space (X i ; A i ;  i ). When it is not ambiguous, we shall often use the symbol T i to denote both the transformation and the system. A joining  of the T i ’s (see the definition in the Glossary) defines a new measure-preserving dynamical system: The product transformation O

Ti : (x i ) i2I 7! (Ti x i ) i2I

i2I

Q acting on the Cartesian product i2I X i , and preserving the N probability  measure . We will denote this big system by T . Since all marginals of  are given by the i i2I  original probabilities i , observing only the coordinate i in the big system is the same as observing N only the system T i . Thus, each system T i is a factor of i2I Ti  , via the homomorphism  i which maps any point in the Cartesian product to its ith coordinate. Conversely, if we are given a measure-preserving dynamical system (Z; C ; ; R) admitting each T i as a factor via some homomorphism ' i : Z ! X i , then we can conQ struct the map ' : Z ! i2I X i sending z to (' i (z)) i2I . We can easily check that the image of the probability measure is then a joining of the T i ’s. Therefore, studying the joinings of a family of measure-preserving dynamical system amounts to studying all the possible ways these systems can be seen together as factors in another big system. The Set of Joinings The set of all joinings of the T i ’s will be denoted by J(Ti ; i 2 I). Before anything else, we have to observe

that this set is never empty. Indeed, whatever the systems N are, the product measure i2I  i always belongs to this set. Note also that any convex combination of joinings is a joining: J(Ti ; i 2 I) is a convex set. The set of joinings is turned into a compact metrizable space, equipped with the topology defined by the following notion of convergence: n !  if and only if, for all n!1 Q families of measurable subsets (A i ) i2I 2 i2I A i , finitely many of them being different from X i , we have ! ! Y Y n A i !  Ai : (2) i2I

n!1

i2I

We can easily construct a distance defining this topology by observing that it is enough to check (2) when each of the Ai ’s is chosen in some countable algebra C i generating the  -algebra A i . We can also point out that, when the X i ’s are themselves compact metric spaces, this topology on the set of joinings is nothing but the restriction to J(Ti ; i 2 I) of the usual weak* topology. It is particularly interesting to study ergodic joinings of the T i ’s, whose set will be denoted by Je (Ti ; i 2 I). Since any factor of an ergodic system is itself ergodic, a necessary condition for Je (Ti ; i 2 I) not to be empty is that all the T i ’s be themselves ergodic. Conversely, if all the T i ’s are ergodic, we can prove by considering the ergodic deN composition of the product measure i2I  i that ergodic joinings do exist: Any ergodic measure appearing in the ergodic decomposition of some joining has to be itself a joining. This result can also be stated in the following way: Proposition 1 If all the T i ’s are ergodic, the set of their ergodic joinings is the set of extremal points in the compact convex set J(Ti ; i 2 I). From Disjointness to Isomorphy In this section, as in many others in this article, we are focusing on the case where our family of dynamical systems is reduced to two systems. We will then rather call them S and T, standing for (Y; B; ; S) and (X; A; ; T). We are interested here in two extremal cases for the set of joinings J(T; S). The first one occurs when the two systems are as far as possible from each other: They have nothing to share in common, and therefore their set of joinings is reduced to the singleton f ˝ g: This is called the disjointness of S and T. The second one arises when the two systems are isomorphic, and we will see how this property shows through J(T; S). Disjointness Many situations where disjointness arises were already given by Furstenberg in [14]. Particularly in-

Joinings in Ergodic Theory

teresting is the fact that classes of dynamical systems can be characterized through disjointness properties. We list here some of the main examples of disjoint classes of measure-preserving systems. Theorem 2 1. T is ergodic if and only if it is disjoint from every identity map. 2. T is weakly mixing if and only if it is disjoint from any rotation on the circle. 3. T has zero entropy if and only if it is disjoint from any Bernoulli shift. 4. T is a K-system if and only if it is disjoint from any zeroentropy system. The first result is the easiest, but is quite important, in particular when it is stated in the following form: If  is a joining of T and S, with T ergodic, and if  is invariant by T  Id, then  D  ˝ . The second, third and fourth results were originally proved by Furstenberg. They can also be seen as corollaries of the theorems presented in Sect. “Joinings and Factors”, linking the non-disjointness property with the existence of a particular factor. Both the first and the second results can be derived from the next theorem, giving a general spectral condition in which disjointness arises. The proof of this theorem can be found in [49]. It is a direct consequence of the fact that, if f and g are square-integrable functions in a given dynamical system, and if their spectral measures are mutually singular, then f and g are orthogonal in L2 . Theorem 3 If the reduced maximum spectral types of T and S are mutually singular, then T and S are disjoint. As we already said in the introduction, disjointness was recognized by Furstenberg as the most pertinent way to define the analog of the arithmetic property “a and b are relatively prime” in the context of measure-preserving dynamical systems. We must however point out that the statement (i) S and T are disjoint is, in general, strictly stronger than the straightforward translation of the arithmetic property: (ii) Each time both S and T appear as factors in a third dynamical system, then their product S  T also appears as a factor in this system. Indeed, contrary to the situation in ordinary arithmetic, there exist non-trivial dynamical systems T which are isomorphic to T  T: For example, this is the case when T is the product of countably many copies of a single non-triv-

ial system. Now, if T is such a system and if we take S D T, then S and T do not satisfy statement (i): A non-trivial system is never disjoint from itself, as we will see in the next Section. However they obviously satisfy the statement (ii). A correct translation of the arithmetic property is the following: S and T are disjoint if and only if, each time T and S appear as factors in some dynamical system through the respective homomorphisms  T and  S , T  S also appears as a factor through a homomorphism TS such that  X ı TS D T and Y ı TS D S , where  X and  Y are the projections on the coordinates in the Cartesian product X  Y (see the diagram below).

Joinings and Isomorphism We first introduce some notations: For any probability measure  on a measurable 

space, let ‘A D B’ stand for ‘(AB) D 0’. Similarly, if C 

and D are  -algebras of measurable sets, we write ‘C D’ if, for any C 2 C , we can find some D 2 D such that 



C D D, and by ‘C D D’ we naturally mean that both 



C D and D C hold.

Let us assume now that our two systems S and T are isomorphic: This means that we can find some measurable one-to-one map ' : X ! Y, with T() D , and ' ı T D S ı '. With such a ', we construct the measurable map : X ! X  Y by setting   (x) :D (x; '(x) : Let ' be the image measure of  by . This measure is supported on the graph of ', and is also characterized by 8A 2 A ; 8B 2 B ; ' (A  B) D (A \ ' 1 B) : (3) We can easily check that, ' being an isomorphism of T and S, ' is a joining of T and S. And this joining satisfies very special properties: '

 For any measurable A X, A  Y D X  '(A); '

 Conversely, for any measurable B Y, X  B D ' 1 (B)  Y.

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satisfies the weaker one: 

fX; ;g ˝ B A ˝ fY; ;g :

(5)

The existence of a joining satisfying (5) is a criterion for S being a factor of T. For more details on the results stated in this section, we refer the reader to [6]. Joinings and Factors

Joinings in Ergodic Theory, Figure 1 The joining ' identifies the sets A  Y and X  '(A)

Thus, in the case where S and T are isomorphic, we can find a special joining of S and T, which is supported on the graph of an isomorphism, and which identifies the two -algebras generated by the two coordinates. What is remarkable is that the converse is true: The existence of an isomorphism between S and T is characterized by the existence of such a joining, and we have the following theorem: Theorem 4 The measure-preserving dynamical systems S and T are isomorphic if and only if there exists a joining  of S and T such that 

fX; ;g ˝ B D A ˝ fY; ;g :

(4)

When  is such a joining, it is supported on the graph of an isomorphism of T and S, and both systems are isomorphic to the joint system (T ˝ S) . This theorem finds nice applications in the proof of classical isomorphism results. For example, it can be used to prove that two discrete-spectrum systems which are spectrally isomorphic are isomorphic (see [49] or [6]). We will also see in Sect.“ Joinings Proofs of Ornstein’s and Krieger’s Theorems” how it can be applied in the proofs of Krieger’s and Ornstein’s deep theorems. Consider now the case were T and S are no longer isomorphic, but where S is only a factor of T. Then we have a factor map  : X ! Y which has the same properties as an isomorphism ', except that it is not one-to-one ( is only onto). The measure  , constructed in the same way as ' , is still a joining supported on the graph of , but it does not identify the two -algebras generated by the two coordinates anymore: Instead of condition (4),  only

The purpose of this section is to investigate the relationships between the disjointness of two systems S and T, and the lack of a common factor. The crucial fact which was pointed out by Furstenberg is that the existence of a common factor enables one to construct a very special joining of S and T: The relatively independent joining over this factor. Let us assume that our systems S and T share a common factor (Z; C ; ; R), which means that we have measurable onto maps  X : X ! Z and Y : Y ! Z, respectively sending  and  to , and satisfying  X ı T D R ı  X and Y ı S D R ı Y . We can then consider the joinings supported on their graphs  X 2 J(T; R) and Y 2 J(S; R), as defined in the preceding section. Next, we construct a joining  of the three systems S, T and R. Heuristically,  is the probability distribution of the triple (x; y; z) when we first pick z according to the probability distribution , then x and y according to their conditional distribution knowing z in the respective joinings  X and Y , but independently of each other. More precisely,  is defined by setting, for all A 2 A, B 2 B and C2C (A  B  C) :D Z E X [1x2A jz]EY [1 y2B jz] d (z) :

(6)

C

Observe that the two-fold marginals of  on X  Z and Y  Z are respectively  X and Y , which means that we have z D  X (x) D Y (y) -almost surely. In other words, we have identified in the two systems T and S the projections on their common factor R. The two-fold marginal of  on X  Y is itself a joining of T and S, which we call the relatively independent joining over the common factor R. This joining will be denoted by ˝ R . (Be careful: The projections  X and  Y are hidden in this notation, but we have to know them to define this joining.) From (6), we immediately get the formula defining ˝R : 8A 2 A ; 8B 2 B ; ˝ R (A  B) :D Z E X [1x2A jz]EY [1 y2B jz] d (z) : Z

(7)

Joinings in Ergodic Theory

rived from the proof of Theorem 6: For any joining  of S and T, for any bounded measurable function f on X, the factor  -algebra of S generated by the function E [ f (x)jy] is a T-factor  -algebra of S. With the notion of T-factor, Theorem 6 has been extended in [31] in the following way, showing the existence of a special T-factor  -algebra of S concentrating anything in S which could lead to a non-trivial joining between T and S. Theorem 7 Given two measure-preserving dynamical systems (X; A; ; T) and (Y; B; ; S), there always exists a maximum T-factor  -algebra of S, denoted by FT . Under any joining  of T and S, the  -algebras A ˝ f;; Yg and f;; Xg ˝ B are independent conditionally to the  -algebra f;; Xg ˝ FT .

Joinings in Ergodic Theory, Figure 2 The relatively independent joining ˝R  and its disintegration over Z

This definition of the relatively independent joining over a common factor can easily be extended to a finite or countable family of systems sharing the same common factor. Note that ˝ R  coincides with the product measure ˝ if and only if the common factor is the trivial onepoint system. We therefore get the following result: Theorem 5 If S and T have a non-trivial common factor, then these systems are not disjoint. As we already said in the introduction, Rudolph exhibited in [41] a counterexample showing that the converse is not true. There exists, however, an important result, which was published in [20,29] allowing us to derive some information on factors from the non-disjointness of two systems. Theorem 6 If T and S are not disjoint, then S has a nontrivial common factor with some joining of a countable family of copies of T. This result leads to the introduction of a special class of factors when some dynamical system T is given: For any other dynamical system S, call T-factor of S any common factor of S with a joining of countably many copies of T. If (Z; C ; ; R) is a T-factor of S and  : Y ! Z is a factor map, we say that the -algebra  1 (C ) is a T-factor -algebra of S. Another way to state Theorem 6 is then the following: If S and T are not disjoint, then S has a non-trivial T-factor. In fact, an even more precise result can be de-

Theorem 6 gives a powerful tool to prove some important disjointness results, such as those stated in Theorem 2. These results involve properties of dynamical systems which are stable under the operations of taking joinings and factors. We will call these properties stable properties. This is, for example, the case of the zero-entropy property: We know that any factor of a zero-entropy system still has zero entropy, and that any joining of zero-entropy systems also has zero entropy. In other words, T has zero entropy implies that any T-factor has zero entropy. But the property of S being a K-system is precisely characterized by the fact that any non-trivial factor of S has positive entropy. Hence a K-system S cannot have a non-trivial T-factor if T has zero entropy, and is therefore disjoint from T. The converse is a consequence of Theorem 5: If S is not a K-system, then it possesses a non-trivial zero-entropy factor, and therefore there exists some zero-entropy system from which it is not disjoint. The same argument also applies to the disjointness of discrete-spectrum systems with weakly mixing systems, since a discrete spectrum is a stable property, and weakly mixing systems are characterized by the fact that they do not have any discrete-spectrum factor. Markov Intertwinings and Composition of Joinings There is another way of defining joinings of two measurepreserving dynamical systems involving operators on L2 spaces, mainly put to light by Ryzhikov (see [47]): Observe that for any joining  2 J(T; S), we can consider the operator P : L2 (X; ) ! L2 (Y; ) defined by  P ( f ) :D E f (x)jy : It is easily checked that P is a Markov intertwining of T and S. Conversely, given any Markov intertwining P of T

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and S, it can be shown that the measure P defined on X  Y by P (A  B) :D hP1A ; 1B iL 2 (Y;) is a joining of T and S. This reformulation of the notion of joining is useful when joinings are studied in connection with spectral properties of the transformations (see e. g. [22]). It also provides us with a convenient setting to introduce the composition of joinings: If we are given three dynamical systems (X; A; ; T), (Y; B; ; S) and (Z; C ; ; R), a joining  2 J(T; S) and a joining 0 2 J(S; R), the composition of the Markov intertwinings P and P0 is easily seen to give a third Markov intertwining, which itself corresponds to a joining of T and R denoted by  ı 0 . When R D S D T, i. e. when we are speaking of 2-fold self-joinings of a single system T (cf. next section), this operation turns J(T; T) D J 2 (T) into a semigroup. Ahn and Lema´nczyk [1] have shown that the subset Je2 (T) of ergodic two-fold self-joinings is a sub-semigroup if and only if T is semisimple (see Sect. “Simple Systems”). Self-Joinings We now turn to the case where the measure-preserving dynamical systems we want to join together are all copies of a single system T. For k  2, any joining of k copies of T is called a k-fold self-joining of T. We denote by J k (T) the set of all k-fold self-joinings of T, and by Jek (T) the subset of ergodic k-fold self-joinings. Self-Joinings and Commuting Transformations As soon as T is not the trivial single-point system, T is never disjoint from itself: Since T is obviously isomorphic to itself, we can always find a two-fold self-joining of T which is not the product measure by considering self-joinings supported on graphs of isomorphisms (see Sect. “Joinings and Isomorphism”). The simplest of them is obtained by taking the identity map as an isomorphism, and we get that J 2 (T) always contains the diagonal measure 0 :D Id . In general, an isomorphism of T with itself is an invertible measure-preserving transformation S of (X; A; ) which commutes with T. We call commutant of T the set of all such transformations (it is a subgroup of the group of automorphisms of (X; A; )), and denote it by C(T). It always contains, at least, all the powers T n , n 2 Z. Each element S of C(T) gives rise to a two-fold selfjoining S supported on the graph of S. Such self-joinings are called off-diagonal self-joinings. They also belong to Jek (T) if T is ergodic.

It follows that properties of the commutant of an ergodic T can be seen in its ergodic joinings. As an example of application, we can cite Ryzhikov’s proof of King’s weak closure theorem for rank-one transformations1 . Rank-one measure-preserving transformations form a very important class of zero-entropy, ergodic measure-preserving transformations. They have many remarkable properties, among which the fact that their commutant is reduced to the weak limits of powers of T. In other words, if T is rank one, for any S 2 C(T) there exists a subsequence of integers (n k ) such that,   8A 2 A ;  T n k AS 1 A ! 0 : k!1

(8)

King proved this result in 1986 [26], using a very intricate coding argument. Observing that (8) was equivalent to the convergence, in J 2 (T), of T n k to S , Ryzhikov showed in [45] that King’s theorem could be seen as a consequence of the following general result concerning two-fold selfjoinings of rank-one systems: Theorem 8 Let T be a rank-one measure-preserving transformation, and  2 Je2 (T). Then there exist t  1/2, a subsequence of integers (n k ) and another two-fold self-joining 0 of T such that T n k ! t C (1  t)0 : k!1

Minimal Self-Joinings For any measure-preserving dynamical system T, the set of two-fold self-joinings of T contains at least the product measure ˝, the off-diagonal joinings T n for each n 2 Z, and any convex combination of these. Rudolph [41] discovered in 1979 that we can find systems for which there are no other two-fold self-joinings than these obvious ones. When this is the case, we say that T has 2-foldminimal self-joinings, or for short: T 2 MSJ(2). It can be shown (see e. g. [42]) that, as soon as the underlying probability space is not atomic (which we henceforth assume), two-fold minimal self-joinings implies that T is weakly mixing, and therefore that  ˝  and T n , n 2 Z, are ergodic two-fold self-joinings of T. That is why two-fold minimal self-joinings are often defined by the following: T 2 MSJ(2) () Je2 (T) D f ˝ g [ fT n ; n 2 Zg :

(9)

1 An introduction to finite-rank transformations can be found e. g. in [32]; we also refer the reader to the quite complete survey [12].

Joinings in Ergodic Theory

Systems with two-fold minimal self-joinings have very interesting properties. First, since for any S in C(T), S belongs to Je2 (T), we immediately see that the commutant of T is reduced to the powers of T. In particular, it is impossible to find a square root of T, i. e. a measure-preserving S such that S ı S D T. Second, the existence of a non-trivial factor -algebra of T would lead, via the relatively independent self-joining over this factor, to some ergodic two-fold self-joining of T which is not in the list prescribed by (9). Therefore, any factor -algebra of a system with two-fold minimal self-joinings must be either the trivial -algebra f;; Xg or the whole -algebra A. This has the remarkable consequence that if  is any random variable on the underlying probability space which is not almost-surely constant, then the process ( ı T n )n2Z always generates the whole -algebra A. This also implies that T has zero entropy, since positive-entropy systems have many non-trivial factors. The notion of two-fold minimal self-joinings extends for any integer k  2 to k-fold minimal self-joinings, which roughly means that there are no other k-fold ergodic selfjoinings than the “obvious” ones: Those for which the k coordinates are either independent or just translated by some power of T (see the Glossary for a more precise definition). We denote in this case: T 2 MSJ(k). If T has k-fold minimal self-joinings for all k  2, we simply say that T has minimal self joinings. Rudolph’s construction of a system with two-fold minimal self-joinings [41] was inspired by a famous work of Ornstein [35], giving the first example of a transformation with no roots. It turned out that Ornstein’s example is a mixing rank-one system, and all mixing rank-one systems were later proved by J. King [27] to have 2-fold minimal self-joinings. This can also be viewed as a consequence of Ryzhikov’s Theorem 8. Indeed, in the language of joinings, the mixing property of T translates as follows: T is mixing () T n !  ˝  : jnj!1

(10)

Therefore, if in Theorem 8 we further assume that T is mixing, then either the sequence (n k ) we get in the conclusion is bounded, and then  is some T n , or it is unbounded and then  D  ˝ . T 2 MSJ(k) obviously implies T 2 MSJ(k 0 ) for any 2  k 0  k, but the converse is not known. The question whether two-fold minimal self-joinings implies k-fold minimal self-joinings for all k is related to the important open problem of pairwise independent joinings (see Sect. “Pairwise-Independent Joinings”). But the latter problem is solved for some special classes of systems,

in particular in the category of mixing rank-one transformations. It follows that, if T is mixing and rank one, then T has minimal self-joinings. In 1980, Del Junco, Rahe and Swanson proved that Chacon’s transformation also has minimal self-joinings [9]. This well-known transformation is also a rankone system, but it is not mixing (it had been introduced by R.V. Chacon in 1969 [4] as the first explicit example of a weakly mixing transformation which is not mixing). For another example of a transformation with two-fold minimal self-joinings, constructed as an exchange map on three intervals, we refer to [7]. The existence of a transformation with minimal selfjoinings has been used by Rudolph as a wonderful tool to construct a large variety of striking counterexamples, such as  A transformation T which has no roots, while T 2 has roots of any order,  A transformation with a cubic root but no square root,  Two measure-preserving dynamical systems which are weakly isomorphic (each one is a factor of the other) but not isomorphic. . . Let us now sketch the argument showing that we can find two systems with no common factor but which are not disjoint: We start with a system T with minimal self-joinings. Consider the direct product of T with an independent copy T 0 of itself, and take the symmetric factor S of T ˝ T 0 , that is to say the factor we get if we only look at the non-ordered pair of coordinates fx; x 0 g in the Cartesian product. Then S is surely not disjoint from T, since the pair fx; x 0 g is not independent of x. However, if S and T had a non-trivial common factor, then this factor should be isomorphic to T itself (because T has minimal self-joinings). Therefore we could find in the direct product T ˝ T 0 a third copy T˜ of T, which is measurable with respect to the symmetric factor. In particular, T˜ is invariant by the flip map (x; x 0 ) 7! (x 0 ; x), and this prevents T˜ from being measurable with respect to only one coordinate. Then, since T 2 MSJ(3), the systems T, T 0 and T˜ have no choice but being independent. But this contradicts the fact that T˜ is measurable with respect to the  -algebra generated by x and x 0 . Hence, T and S have no non-trivial common factor. We can also cite the example given by Glasner and Weiss [21] of a pair of horocycle transformations which have no nontrivial common factor, yet are not disjoint. Their construction relies on the deep work by Ratner [38], which describes the structure of joinings of horocycle flows.

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Simple Systems An important generalization of two-fold minimal selfjoinings has been proposed by William A. Veech in 1982 [51]. We say that the measure-preserving dynamical system T is two-fold simple if it has no other ergodic two-fold self-joinings than the product measure f ˝ g and joinings supported on the graph of a transformation S 2 C(T). (The difference with MSJ(2) lies in the fact that C(T) may contain other transformations than the powers of T.) It turns out that simple systems may have non-trivial factors, but the structure of these factors can be explicitly described: They are always associated with some compact subgroup of C(T). More precisely, if K is a compact subgroup of C(T), we can consider the factor -algebra F K :D fA 2 A : 8S 2 K ; A D S(A)g ;

and the corresponding factor transformation TjFK (called a group factor). Then Veech proved the following theorem concerning the structure of factors of a two-fold simple system. Theorem 9 If the dynamical system T is two-fold simple, and if F A is a non-trivial factor -algebra of T, then there exists a compact subgroup K of the group C(T) such that F D F K . There is a natural generalization of Veech’s property to the case of k-fold self-joinings, which has been introduced by A. Del Junco and D.J. Rudolph in 1987 [10] (see the precise definition of simple systems in the Glossary). In their work, important results concerning the structure of factors and joinings of simple systems are proved. In particular, they are able to completely describe the structure of the ergodic joinings between a given simple system and any ergodic system (see also [18,49]). Recall that, for any r  1, the symmetric factor of T ˝r is the system we get if we observe the r coordinates of the point in X r and forget their order. This is a special case of group factor, associated with the subgroup of C(T ˝r ) consisting of all permutations of the coordinates. We denote this symmetric factor by T hri . Theorem 10 Let T be a simple system and S an ergodic system. Assume that  is an ergodic joining of T and S which is different from the product measure. Then there exists a compact subgroup K of C(T) and an integer r  1 such that  (TjFK )hri is a factor of S,   is the projection on X Y of the relatively independent joining of T ˝r and S over their common factor (TjFK )hri .

If we further assume that the second system is also simple, then in the conclusion we can take r D 1. In other words, ergodic joinings of simple systems S and T are either the product measure or relatively independent joinings over a common group factor. This leads to the following corollary: Theorem 11 Simple systems without non-trivial common factor are disjoint. As for minimal self-joining, it is not known in general whether two-fold simplicity implies k-fold simplicity for all k. This question is studied in [19], where sufficient spectral conditions are given for this implication to hold. It is also proved that any three-fold simple weakly mixing transformation is simple of all order. Relative Properties with Respect to a Factor In fact, Veech also introduced a weaker, “relativized”, version of the two-fold simplicity. If F A is a non-trivial factor  -algebra of T, let us denote by J 2 (T; F ) the twofold self-joinings of T which are “constructed over F ”, which means that their restriction to the product  -algebra F ˝ F coincides with the diagonal measure. (The relatively-independent joining over F is the canonical example of such a joining.) For the conclusion of Theorem 9 to hold, it is enough to assume only that the ergodic elements of J 2 (T; F ) be supported on the graph of a transformation S 2 C(T). This is an important situation where the study of J 2 (T; F ) gives strong information on the way F is embedded in the whole system T, or, in other words, on the relative properties of T with respect to the factor TjF . A simple example of such a relative property is the relative weak mixing with respect to F , which is characterized by the ergodicity of the relatively-independent joining over F (recall that weak-mixing is itself characterized by the ergodicity of the direct product T ˝ T). For more details on this subject, we refer the reader to [30]. We also wish to mention the generalization of simplicity called semisimplicity proposed by Del Junco, Lema´nczyk and Mentzen in [8], which is precisely characterized by the fact that, for any  2 Je2 (T), the system (T ˝ T) is a relatively weakly mixing extension of T. Some Applications and Future Directions Joinings Proofs of Ornstein’s and Krieger’s Theorems We have already seen that joinings could be used to prove isomorphisms between systems. This fact found a nice application in the proofs of two major theorems in ergodic theory: Ornstein’s isomorphism theorem [34], stating that

Joinings in Ergodic Theory

two Bernoulli shifts with the same entropy are isomorphic, and Krierger’s finite generator theorem [28], which says that any dynamical system with finite entropy is isomorphic to the shift transformation on a finite-valued stationary process. The idea of this joining approach to the proofs of Krieger’s and Ornstein’s theorems was originally due to Burton and Rothstein, who circulated a preliminary report on the subject which was never published [3]. The first published and fully detailed exposition of these proofs can be found in Rudolph’s book [42] (see also in Glasner’s book [18]). In fact, Ornstein’s theorem goes far more beyond the isomorphism of two given Bernoulli shifts: It also gives a powerful tool for showing that a specific dynamical system is isomorphic to a Bernoulli shift. In particular, Ornstein introduced the property for an ergodic stationary process to be finitely determined. We shall not give here the precise definition of this property (for a complete exposition of Ornstein’s theory, we refer the reader to [36]), but simply point out that Bernoulli shifts and mixing Markov chains are examples of finitely determined processes. Rudolph’s argument to show Ornstein’s theorem via joinings makes use of Theorem 4, and of the topology of J(T; S). Theorem 12 (Ornstein’s Isomorphism Theorem) Let T and S be two ergodic dynamical systems with the same entropy, and both generated by finitely determined stationary processes. Then the set of joinings of T and S which are supported on graphs of isomorphisms forms a dense Gı in Je (T; S). Krieger’s theorem is not as easily stated in terms of joinings, because it does not refer to the isomorphism of two specific systems, but rather to the isomorphism of one given system with some other system which has to be found. We have therefore to introduce a larger set of joinings: Given an integer n, we denote by Y n the set of doublesided sequences taking values in f1; : : : ; ng. We consider on Y n the shift transformation S, but we do not determine yet the invariant measure. Now, for a specific measurepreserving dynamical system T, consider the set J(n; T) of all possible joinings of T with some system (Yn ; ; S), when  ranges over all possible shift-invariant probability measures on Y n . J(n; T) can also be equipped with a topology which turns it into a compact convex metric space, and as soon as T is ergodic, the set Je (n; T) of ergodic elements of J(n; T) is not empty. In this setting, Krieger’s theorem can be stated as follows: Theorem 13 (Krieger’s Finite Generator Theorem) Let T be an ergodic dynamical system with entropy h(T) < log2 n. Then the set of  2 J(n; T) which are supported

on graphs of isomorphisms between T and some system (Yn ; ; S) forms a dense Gı in Je (n; T). Since any system of the form (Yn ; ; S) obviously has an n-valued generating process, we obtain as a corollary that T itself is generated by an n-valued process. Joinings and Pointwise Convergence The study of joinings is also involved in questions concerning pointwise convergence of (non-conventional) ergodic averages. As an example, we present here the relationships between disjointness and the following wellknown open problem: Given two commuting measurepreserving transformations S and T acting on the same probability space (X; A; ), is it true that for any f and g in L2 (), the sequence ! n1 X 1 k k f (T x)g(S x) (11) n kD0

n>0

converges -almost surely? It turns out that disjointness of T and S is a sufficient condition for this almost-sure convergence to hold. Indeed, let us first consider the case where T and S are defined on a priori different spaces (X; A; ) and (Y; B; ) respectively, and consider the ergodic average in the product 1 n

n1 X

f (T k x)g(S k y) ;

(12)

kD0

which can be viewed as the integral of the function f ˝ g with respect to the empirical distribution ın (x; y) :D

1 n

n1 X

ı(T k x;S k y) :

kD0

We can always assume that T and S are continuous transformations of compact metric spaces (indeed, any measure-preserving dynamical system is isomorphic to such a transformation on a compact metric space; see e. g. [15]). Then the set of probability measures on X  Y equipped with the topology of weak convergence is metric compact. Now, here is the crucial point where joinings appear: If T and S are ergodic, we can easily find subsets X 0 X and Y0 Y with (X0 ) D (Y0 ) D 1, such that for all (x; y) 2 X 0  Y0 , any cluster point of the sequence (ın (x; y))n>0 is automatically a joining of T and S. (We just have to pick x and y among the “good points” for the ergodic theorem in their respective spaces.) When T and S are disjoint, there is therefore only one possible cluster

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point to the sequence ın (x; y) which is ˝. This ensures that, for continuous f and g, (12) converges to the product of the integrals of f and g as soon as (x; y) is picked in X 0 Y0 . The subspace of continuous functions being dense in L2 , the classical ergodic maximal inequality (see [17]) ensures that, for any f and g in L2 (), (12) converges for any (x; y) in a rectangle of full measure X 0  Y0 . Coming back to the original question where the spaces on which T and S act are identified, we observe that with probability one, x belongs both to X 0 and Y 0 , and therefore the sequence (11) converges. The existence of a rectangle of full measure in which the sequence of empirical distributions (ın (x; y))n>0 always converges to some joining has been studied in [31] as a natural generalization of the notion of disjointness. This property was called weak disjointness of S and T, and it is indeed strictly weaker than disjointness, since there are examples of transformations which are weakly disjoint from themselves. There are other situations in which joinings can be used in the study of everywhere convergence, among which we can cite Rudolph’s joinings proof of Bourgain’s return time theorem [43]. Joinings and Rohlin’s Multifold Mixing Question We have already seen that the property of T being mixing could be expressed in terms of two-fold self-joinings of T (see (10)). Rohlin proposed in 1949 [39] a generalization of this property, called multifold mixing: The measure-preserving transformation T is said to be k-fold mixing if 8A1 ; A2 ; : : : ; A k 2 A, lim

n 2 ;n 3 ;:::;n k !1

(A1 \T n 2 A2 \    \T

(n 2 CCn k )

Ak ) D

k Y

(A i ) :

iD1

Again, this definition can easily be translated into the language of joinings: T is k-fold mixing when the sequence (T n2 ;:::;T n2 CCn k ) converges in J k (T) to ˝k as n2 ; : : : ; n k go to infinity, where (T n2 ;:::;T n2 CCn k ) is the obvious generalization of T n to the case of k-fold selfjoinings. The classical notion of mixing corresponds in this setting to two-fold mixing. (We must point out that Rohlin’s original definition of k-fold mixing involved k C1 sets, thus the classical mixing property was called 1-fold mixing. However it seems that the convention we adopt here is now used by most authors, and we find it more coherent when translated in the language of multifold selfjoinings.)

Obviously, 3-fold mixing is stronger than 2-fold mixing, and Rohlin asked in his article whether the converse is true. This question is still open today, even though many important works have dealt with it and supplied partial answers. Most of these works directly involve self-joinings via the argument exposed in the following section. Pairwise-Independent Joinings Let T be a two-fold mixing dynamical system. If T is not three-fold mixing, (T n ;T nCm ) does not converge to the product measure as n and m go to 1. By compactness of J 3 (T), we can find subsequences (n k ) and (m k ) such that (T n k ;T n k Cm k ) converges to a cluster point  ¤  ˝  ˝ . However, by two-fold mixing, the three coordinates must be pairwise independent under . We therefore get a three-fold selfjoining  with the unusual property that  has pairwise independent coordinates, but  is not the product measure. In fact, systems with this kind of pairwise-independent but non-independent three-fold self-joining are easy to find (see e. g. [6]), but the examples we know so far are either periodic transformations (which cannot be counterexamples to Rohlin’s question since they are not mixing!), or transformations with positive entropy. But using an argument provided by Thouvenot, we can prove that, if there exists a two-fold mixing T which is not threefold mixing, then we can find such a T in the category of zero-entropy dynamical systems (see e. g. [5]). Therefore, a negative answer to the following question would solve Rohlin’s multifold mixing problem: Question 14 Does there exist a zero-entropy, weakly mixing dynamical system T with a self-joining  2 J 3 (T) for which the coordinates are pairwise independent but which is different from  ˝  ˝ ? (non) existence of such pairwise-independent joinings is also related to the question of whether MSJ(2) implies MSJ(3), or whether two-fold simplicity implies three-fold simplicity. Indeed, any counter-example to one of these implication would necessarily be of zero entropy, and would possess a pairwise-independent three-fold self-joining which is not the product measure. Question 14 has been answered by B. Host and V. Ryzhikov for some special classes of zero-entropy dynamical systems. Host’s and Ryzhikov’s Theorems The following theorem, proved in 1991 by B. Host [23] (see also [18,33]), establishes a spectacular connection between the spectral properties of a finite family of dynamical systems and the non-existence of a pairwise-independent, non-independent joining:

Joinings in Ergodic Theory

Theorem 15 (Host’s Theorem on Singular Spectrum) Let (X i ; A i ;  i ; Ti )1ir be a finite family of measure-preserving dynamical systems with purely singular spectrum. Then any pairwise-independent joining  2 J(T1 ; : : : ; Tr ) is the product measure 1 ˝    ˝ r . Corollary 16 If a dynamical system with singular spectrum is two-fold mixing, then it is k-fold mixing for any k  2. The multifold-mixing problem for rank-one measure-preserving systems was solved in 1984 by S. Kalikow [25], using arguments which do not involve the theory of joinings. In 1993, V. Ryzhikov [46] extended Kalikow’s result to finite-rank systems, by giving a negative answer to Question 14 in the category of finite-rank mixing systems: Theorem 17 (Ryzhikov’s Theorem for Finite Rank Systems) Let T be a finite-rank mixing transformation, and k  2. Then the only pairwise-independent k-fold self-joining of T is the product measure. Corollary 18 If a finite-rank transformation is two-fold mixing, then it is k-fold mixing for any k  2. Future Directions A lot of important open questions in ergodic theory involve joinings, and we already have cited several of them: Joinings are a natural tool when we want to deal with some problems of pointwise convergence involving several transformations (see Sect. “Joinings and Pointwise Convergence”). Their use is also fundamental in the study of Rohlin’s question on multifold mixing. As far as this latter problem is concerned, we may mention a recent approach to Question 14: Start with a transformation for which some special pairwise-independent self-joining exists, and see what this assumption entails. In particular, we can ask under which conditions there exists a pairwise-independent three-fold self-joining of T under which the third coordinate is a function of the two others. It has already been proven in [24] that if this function is sufficiently regular (continuous for some topology), then T is periodic or has positive entropy. And there is strong evidence leading to the conjecture that, when T is weakly mixing, such a situation can only arise when T is a Bernoulli shift of entropy log n for some integer n  2. A question in the same spirit was raised by Ryzhikov, who asked in [44] under which conditions we can find a factor of the direct product T  T which is independent of both coordinates. There is also a lot of work to do with joinings in order to understand the structure of factors of some dynamical systems, and how different classes of systems are related. An example of such a work is given in the class of

Gaussian dynamical systems, i. e. dynamical systems constructed from the shift on a stationary Gaussian process: For some of them (which are called GAG, from the French Gaussien à Autocouplages Gaussiens), it can be proven that any ergodic self-joining is itself a Gaussian system (see [29,50]), and this gives a complete description of the structure of their factors. This kind of analysis is expected to be applicable to other classes of dynamical systems. In particular, Gaussian joinings find a nice generalization in the notion of infinitely divisible joinings, studied by Roy in [40]). These ID joinings concern a wider class of dynamical systems of probabilistic origin, among which we can also find Poisson suspensions. The counterpart of Gaussian joinings in this latter class are Poisson joinings, which have been introduced by Derriennic, Fraczek, ˛ Lema´nczyk and Parreau in [11]. As far as Poisson suspensions are concerned, the analog of the GAG property in the Gaussian class can also be considered, and a family of Poisson suspension for which the only ergodic self-joinings are Poisson joinings has been given recently by Parreau and Roy [37]. In [11], a general joining property is described: T satisfies the ELF property (from the French: Ergodicité des Limites Faibles) if any joining which can be obtained as a limit of off-diagonal joinings T n k is automatically ergodic. It turns out that this property is satisfied by any system arising from an infinitely divisible stationary process (see [11,40]). It is proven in [11] that ELF property implies disjointness with any system which is two-fold simple and weakly mixing but not mixing. ELF property is expected to give a useful tool to prove disjointness between dynamical systems of probabilistic origin and other classes of systems (see e. g. [13] in the case of R-action for disjointness between ELF systems and a class of special flows over irrational rotations). Many other questions involving joinings have not been mentioned here. We should at least cite filtering problems, which were one of the motivations presented by Furstenberg for the introduction of the disjointness property in [14]. Suppose we are given two real-valued stationary processes (X n ) and (Yn ), with their joint distribution also stationary. We can interpret (X n ) as a signal, perturbed by a noise (Yn ), and the question posed by Furstenberg is: Under which condition can we recover the original signal (X n ) from the observation of (X n C Yn )? Furstenberg proved that it is always possible if the two processes (X n ) and (Yn ) are integrable, and if the two measure-preserving dynamical systems constructed as the shift of the two processes are disjoint. Furstenberg also observed that the integrability assumption can be removed if a stronger disjointness property is satisfied: A perfect filtering exists if the system T generated by (X n ) is dou-

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bly disjoint from the system S generated by (Yn ), in the sense that T is disjoint from any ergodic self-joining of S. Several generalizations have been studied (see [2,16]), but the question of whether the integrability assumption of the processes can be removed is still open.

Bibliography ´ 1. Ahn Y-H, Lemanczyk M (2003) An algebraic property of joinings. Proc Amer Math Soc 131(6):1711–1716 (electronic) ´ 2. Bułatek W, Lemanczyk M, Lesigne E (2005) On the filtering problem for stationary random Z2 -fields. IEEE Trans Inform Theory 51(10):3586–3593 3. Burton R, Rothstein A (1977) Isomorphism theorems in ergodic theory. Technical report, Oregon State University 4. Chacon RV (1969) Weakly mixing transformations which are not strongly mixing. Proc Amer Math Soc 22:559–562 5. de la Rue T (2006) 2-fold and 3-fold mixing: why 3-dot-type counterexamples are impossible in one dimension. Bull Braz Math Soc (NS) 37(4):503–521 6. de la Rue T (2006) An introduction to joinings in ergodic theory. Discret Contin Dyn Syst 15(1):121–142 7. del Junco A (1983) A family of counterexamples in ergodic theory. Isr J Math 44(2):160–188 ´ 8. del Junco A, Lemanczyk M, Mentzen MK (1995) Semisimplicity, joinings and group extensions. Studia Math 112(2):141–164 9. del Junco A, Rahe M, Swanson L (1980) Chacon’s automorphism has minimal self-joinings. J Analyse Math 37:276–284 10. del Junco A, Rudolph DJ (1987) On ergodic actions whose selfjoinings are graphs. Ergod Theory Dynam Syst 7(4):531–557 ´ 11. Derriennic Y, Fraczek ˛ K, Lemanczyk M, Parreau F (2008) Ergodic automorphisms whose weak closure of off-diagonal measures consists of ergodic self-joinings. Colloq Math 110:81–115 12. Ferenczi S (1997) Systems of finite rank. Colloq Math 73(1): 35–65 ´ 13. Fraczek ˛ K, Lemanczyk M (2004) A class of special flows over irrational rotations which is disjoint from mixing flows. Ergod Theory Dynam Syst 24(4):1083–1095 14. Furstenberg H (1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math Syst Theory 1:1–49 15. Furstenberg H (1981) Recurrence in ergodic theory and combinatorial number theory. In: M.B. Porter Lectures. Princeton University Press, Princeton 16. Furstenberg H, Peres Y, Weiss B (1995) Perfect filtering and double disjointness. Ann Inst H Poincaré Probab Stat 31(3):453–465 17. Garsia AM (1970) Topics in almost everywhere convergence. In: Lectures in Advanced Mathematics, vol 4. Markham Publishing Co, Chicago, IL 18. Glasner E (2003) Ergodic theory via joinings. In: Mathematical Surveys and Monographs, vol 101. American Mathematical Society, Providence 19. Glasner E, Host B, Rudolph DJ (1992) Simple systems and their higher order self-joinings. Isr J Math 78(1):131–142 20. Glasner E, Thouvenot J-P, Weiss B (2000) Entropy theory without a past. Ergod Theory Dynam Syst 20(5):1355–1370 21. Glasner S, Weiss B (1983) Minimal transformations with no common factor need not be disjoint. Isr J Math 45(1):1–8

22. Goodson GR (2000) Joining properties of ergodic dynamical systems having simple spectrum. Sankhy¯a Ser A 62(3):307– 317, Ergodic theory and harmonic analysis (Mumbai, 1999) 23. Host B (1991) Mixing of all orders and pairwise independent joinings of systems with singular spectrum. Isr J Math 76(3):289–298 24. Janvresse É, de la Rue T (2007) On a class of pairwise-independent joinings. Ergod Theory Dynam Syst (to appear) 25. Kalikow SA (1984) Twofold mixing implies threefold mixing for rank one transformations. Ergod Theory Dynam Syst 4(2): 237–259 26. King J (1986) The commutant is the weak closure of the powers, for rank-1 transformations. Ergod Theory Dynam Syst 6(3):363–384 27. King J (1988) Joining-rank and the structure of finite rank mixing transformations. J Anal Math 51:182–227 28. Krieger W (1970) On entropy and generators of measure-preserving transformations. Trans Amer Math Soc 149:453–464 ´ 29. Lemanczyk M, Parreau F, Thouvenot J-P (2000) Gaussian automorphisms whose ergodic self-joinings are Gaussian. Fund Math 164(3):253–293 ´ 30. Lemanczyk M, Thouvenot J-P, Weiss B (2002) Relative discrete spectrum and joinings. Monatsh Math 137(1):57–75 31. Lesigne E, Rittaud B, and de la Rue T (2003) Weak disjointness of measure-preserving dynamical systems. Ergod Theory Dynam Syst 23(4):1173–1198 32. Nadkarni MG (1998) Basic ergodic theory. In: Birkhäuser Advanced Texts: Basler Lehrbücher, 2nd edn. Birkhäuser, Basel 33. Nadkarni MG (1998) Spectral theory of dynamical systems. In: Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Basel 34. Ornstein DS (1970) Bernoulli shifts with the same entropy are isomorphic. Adv Math 4:337–352 35. Ornstein DS (1972) On the root problem in ergodic theory. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Univ. California, Berkeley, 1970/1971, vol II: Probability theory. Univ California Press, Berkeley, pp 347–356 36. Ornstein DS (1974) Ergodic theory, randomness, and dynamical systems. In: James K Whittemore Lectures in Mathematics given at Yale University, Yale Mathematical Monographs, vol 5. Yale University Press, New Haven 37. Parreau F, Roy E (2007) Poisson joinings of Poisson suspension. Preprint 38. Ratner M (1983) Horocycle flows, joinings and rigidity of products. Ann of Math 118(2):277–313 39. Rohlin VA (1949) On endomorphisms of compact commutative groups. Izvestiya Akad Nauk SSSR Ser Mat 13:329–340 40. Roy E (2007) Poisson suspensions and infinite ergodic theory. Ergod Theory Dynam Syst(to appear) 41. Rudolph DJ (1979) An example of a measure preserving map with minimal self-joinings, and applications. J Anal Math 35:97–122 42. Rudolph DJ (1990) The title is Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces. Oxford University Press, New York 43. Rudolph DJ (1994) A joinings proof of Bourgain’s return time theorem. Ergod Theory Dynam Syst 14(1):197–203 44. Ryzhikov VV (1992) Stochastic wreath products and joinings of dynamical systems. Mat Zametki 52(3):130–140, 160

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45. Ryzhikov VV (1992) Mixing, rank and minimal self-joining of actions with invariant measure. Mat Sb 183(3):133–160 46. Ryzhikov VV (1993) Joinings and multiple mixing of the actions of finite rank. Funktsional Anal Prilozhen 27(2):63–78, 96 47. Ryzhikov VV (1993) Joinings, wreath products, factors and mixing properties of dynamical systems. Izv Ross Akad Nauk Ser Mat 57(1):102–128 48. Thorisson H (2000) Coupling, stationarity, and regeneration. In: Probability and its Applications (New York). Springer, New York 49. Thouvenot J-P (1995) Some properties and applications of

joinings in ergodic theory. In: Ergodic theory and its connections with harmonic analysis, Alexandria, 1993. London Math Soc Lecture Note Ser, vol 205. Cambridge Univ Press, Cambridge, pp 207–235 50. Thouvenot J-P (1987) The metrical structure of some Gaussian processes. In: Proceedings of the conference on ergodic theory and related topics, II, Georgenthal, 1986. Teubner-Texte Math, vol 94. Teubner, Leipzig, pp 195–198 51. Veech WA (1982) A criterion for a process to be prime. Monatsh Math 94(4):335–341

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Kolmogorov–Arnold–Moser (KAM) Theory LUIGI CHIERCHIA Dipartimento di Matematica, Università “Roma Tre”, Roma, Italy Article Outline Glossary Definition of the Subject Introduction Kolmogorov Theorem Arnold’s Scheme The Differentiable Case: Moser’s Theorem Future Directions A The Classical Implicit Function Theorem B Complementary Notes Bibliography Glossary Action-angle variables A particular set of variables (y; x) D ((y1 ; : : : ; y d ); (x1 ; : : : ; x d )), xi (“angles”) defined modulus 2, particularly suited to describe the general behavior of an integrable system. Fast convergent (Newton) method Super-exponential algorithms, mimicking Newton’s method of tangents, used to solve differential problems involving small divisors. Hamiltonian dynamics The dynamics generated by a Hamiltonian system on a symplectic manifold, i. e., on an even-dimensional manifold endowed with a symplectic structure. Hamiltonian system A time reversible, conservative (without dissipation or expansion) dynamical system, which generalizes classical mechanical systems (solutions of Newton’s equation m i x¨ i D f i (x), with 1  i  d and f D ( f1 ; : : : ; f d ) a conservative force field); they are described by the flow of differential equations (i. e., the time t map associating to an initial condition, the solution of the initial value problem at time t) on a symplectic manifold and, locally, look like the flow associated with the system of differential equation p˙ D H q (p; q), q˙ D H p (p; q) where p D (p1 ; : : : ; p d ), q D (q1 ; : : : ; qd ). Integrable Hamiltonian systems A very special class of Hamiltonian systems, whose orbits are described by linear flows on the standard d-torus: (y; x) ! (y; x C ! t) where (y; x) are action-angle variables and t is time; the ! i ’s are called the “frequencies” of the orbit.

Invariant tori Manifolds diffeomorphic to tori invariant for the flow of a differential equation (especially of Hamiltonian differential equations); establishing the existence of tori invariant for Hamiltonian flows is the main object of KAM theory. KAM Acronym from the names of Kolmogorov (Andrey Nikolaevich Kolmogorov, 1903–1987), Arnold (Vladimir Igorevich Arnold, 1937) and Moser (Jürgen K. Moser, 1928–1999), whose results, in the 1950’s and 1960’s, in Hamiltonian dynamics, gave rise to the theory presented in this article. Nearly-integrable Hamiltonian systems Hamiltonian systems which are small perturbations of an integrable system and which, in general, exhibit a much richer dynamics than the integrable limit. Nevertheless, KAM theory asserts that, under suitable assumptions, the majority (in the measurable sense) of the initial data of a nearly-integrable system behaves as in the integrable limit. Quasi-periodic motions Trajectories (solutions of a system of differential equations), which are conjugate to linear flow on tori. Small divisors/denominators Arbitrary small combinaP tions of the form !  k :D djD1 ! i k i with ! D (!1 ; : : : ; !d ) 2 Rd a real vector and k 2 Zd an integer vector different from zero; these combinations arise in the denominators of certain expansions appearing in the perturbation theory of Hamiltonian systems, making (when d > 1) convergent arguments very delicate. Physically, small divisors are related to “resonances”, which are a typical feature of conservative systems. Stability The property of orbits of having certain properties similar to a reference limit; more specifically, in the context of KAM theory, stability is normally referred to as the property of action variables of staying close to their initial values. Symplectic structure A mathematical structure (a differentiable, non-degenerate, closed 2-form) apt to describe, in an abstract setting, the main geometrical features of conservative differential equations arising in mechanics. Definition of the Subject KAM theory is a mathematical, quantitative theory which has as its primary object the persistence, under small (Hamiltonian) perturbations, of typical trajectories of integrable Hamiltonian systems. In integrable systems with bounded motions, the typical trajectory is quasi-periodic, i. e., may be described through the linear flow x 2 T d ! x C ! t 2 T d where T d denotes the standard d-dimen-

Kolmogorov–Arnold–Moser (KAM) Theory

sional torus (see Sect. “Introduction” below), t is time, and ! D (!1 ; : : : ; !d ) 2 Rd is the set of frequencies of the trajectory (if d D 1, 2/! is the period of the motion). The main motivation for KAM theory is related to stability questions arising in celestial mechanics which were addressed by astronomers and mathematicians such as Kepler, Newton, Lagrange, Liouville, Delaunay, Weierstrass, and, from a more modern point of view, Poincaré, Birkhoff, Siegel, . . . The major breakthrough in this context, was due to Kolmogorov in 1954, followed by the fundamental work of Arnold and Moser in the early 1960s, who were able to overcome the formidable technical problem related to the appearance, in perturbative formulae, of arbitrarily small divisors1 . Small divisors make the use of classical analytical tools (such as the standard Implicit Function Theorem, fixed point theorems, etc.) impossible and could be controlled only through a “fast convergent method” of Newton-type2 , which allowed, in view of the super-exponential rate of convergence, counterbalancing the divergences introduced by small divisors. Actually, the main bulk of KAM theory is a set of techniques based, as mentioned, on fast convergent methods, and solving various questions in Hamiltonian (or generalizations of Hamiltonian) dynamics. By now, there are excellent reviews of KAM theory – especially Sect. 6.3 of [6] and [60] – which should complement the reading of this article, whose main objective is not to review but rather to explain the main fundamental ideas of KAM theory. To do this, we re-examine, in modern language, the main ideas introduced, respectively, by the founders of KAM theory, namely Kolmogorov (in Sect. “Kolmogorov Theorem”), Arnold (in Sect. “Arnold’s Scheme”) and Moser (Sect. “The Differentiable Case: Moser’s Theorem”). In Sect. “Future Directions” we briefly and informally describe a few developments and applications of KAM theory: this section is by no means exhaustive and is meant to give a non technical, short introduction to some of the most important (in our opinion) extensions of the original contributions; for more detailed and complete reviews we recommend the above mentioned articles Sect. 6.3 of [6] and [60]. Appendix A contains a quantitative version of the classical Implicit Function Theorem. A set of technical notes (such as notes 17, 18, 19, 21, 24, 26, 29, 30, 31, 34, 39), which the reader not particularly interested in technical mathematical arguments may skip, are collected in Appendix B and complete the mathematical expositions. Appendix B also includes several other complementary notes, which contain either standard material or further references or side comments.

Introduction In this article we will be concerned with Hamiltonian flows on a symplectic manifold (M; dy ^ dx); for general information, see, e. g., [5] or Sect. 1.3 of[6]. Notation, main definitions and a few important properties are listed in the following items. (a) As symplectic manifold (“phase space”) we shall consider M :D B  T d with d  2 (the case d D 1 is trivial for the questions addressed in this article) where: B is an open, connected, bounded set in Rd ; T d :D Rd /(2Zd ) is the standard flat d-dimensional torus with periods3 2 P (b) dy ^ dx :D diD1 dy i ^ dx i , (y 2 B, x 2 T d ) is the standard symplectic form4 (c) Given a real-analytic (or smooth) function H : M ! R, the Hamiltonian flow governed by H is the oneparameter family of diffeomorphisms  Ht : M ! M, which to z 2 M associates the solution at time t of the differential equation5 z˙ D J2d rH(z) ;

z(0) D z ;

(1)

where z˙ D dz/dt, J2d is the standard symplectic (2d  2d)-matrix  J2d D

0 1d

1d 0

 ;

1d denotes the unit (d  d)-matrix, 0 denotes a (d  d) block of zeros, and r denotes gradient; in the symplectic coordinates (y; x) 2 B  T d , equations (1) reads

y˙ D H x (y; x) x˙ D H y (y; x)

;

y(0) D y x(0) D x

(2)

Clearly, the flow  Ht is defined until y(t) eventually reaches the border of B. Equations (1) and (2) are called the Hamilton’s equations with Hamiltonian H; usually, the symplectic (or “conjugate”) variables (y; x) are called action-angle variables6; the number d (= half of the dimension of the phase space) is also referred to as “the number of degrees of freedom7 ”. The Hamiltonian H is constant over trajectories  Ht (z), as it follows immediately by differentiating t ! H( Ht (z)). The constant value E D H( Ht (z)) is called the energy of the trajectory  Ht (z). Hamilton equations are left invariant by symplectic (or “canonical”) change of variables, i. e., by diffeomorphisms on M which preserve the 2-form dy ^ dx; i. e.,

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if  : (y; x) 2 M ! (; ) D (y; x) 2 M is a diffeomorphism such that d ^ d D dy ^ dx, then t  ı  Ht ı  1 D  Hı 1 :

(3)

An equivalent condition for a map  to be symplectic is that its Jacobian  0 is a symplectic matrix, i. e., 0T

 J2d  0 D J2d

(4)

where J2d is the standard symplectic matrix introduced above and the superscript T denotes matrix transposition. By a (generalization of a) theorem of Liouville, the Hamiltonian flow is symplectic, i. e., the map (y; x) ! (; ) D  Ht (y; x) is symplectic for any H and any t; see Corollary 1.8, [6]. A classical way of producing symplectic transformations is by means of generating functions. For example, if g(; x) is a smooth function of 2d variables with det

@2 g ¤0; @@x

then, by the Implicit Function Theorem (IFT; see [36] or Sect. “A The Classical Implicit Function Theorem” below), the map  : (y; x) ! (; ) defined implicitly by the relations yD

@g ; @x

D

@g ; @

 : 2 T d ! ( ) :D (v( ); U( )) 2 M If U is a smooth diffeomorphism of T d (so that, in particular9 det U ¤ 0) then  is an embedding of T d into M and the set T! D T!d :D (T d )

d X iD1

!i ni D 0 H) n D 0 ;

(5)

and if there exist smooth (periodic) functions v; u : T d ! Rd such that8 ( y(t) D v(! t) (6) x(t) D ! t C u(! t) :

(7)

is an embedded d-torus invariant for  Ht and on which the motion is conjugated to the linear (Kronecker) flow ! C ! t, i. e.,  1 ı  Ht ı ( ) D C ! t ;

8 2 T d :

(8)

Furthermore, the invariant torus T! is a graph over T d and is Lagrangian, i. e., (T! has dimension d and) the restriction of the symplectic form dy ^ dx on T! vanishes10 . (f) In KAM theory a major role is played by the numerical properties of the frequencies !. A typical assumption is that ! is a (homogeneously) Diophantine vector: ! 2 Rd is called Diophantine or (;  )-Diophantine if, for some constants 0 <   min i j! i j and   d  1, it verifies the following inequalities: j!  nj 

yields a local symplectic diffeomorphism; in such a case, g is called the generating function of the transformation . For example, the function   x is the generating function of the identity map. For general information about symplectic changes of coordinates, generating functions and, in general, about symplectic structures we refer the reader to [5] or [6]. (d) A solution z(t) D (y(t); x(t)) of (2) is a maximal quasi-periodic solution with frequency vector ! D (!1 ; : : : ; !d ) 2 Rd if ! is a rationally-independent vector, i. e., 9n 2 Zd s:t: !  n :D

(e) Let !, u and v be as in the preceding item and let U and  denote, respectively, the maps ( U : 2 T d ! U( ) :D C u( ) 2 T d

 ; jnj

8n 2 Zd n f0g ;

(9)

(normally, for integer vectors n, jnj denotes jn1 jC  C jnd j, but other norms may well be used). We shall refer to  and  as the Diophantine constants of !. The set of Diophantine numbers in Rd with constants  and  d ; the union over all  > 0 of will be denoted by D; d D; will be denoted by Dd and the union over all   d  1 of Dd will be denoted by Dd . Basic facts about these sets are11 : if  < d  1 then Dd D ;; if  > d  1 then the Lebesgue measure of Rd n Dd is zero; if  D d  1, the Lebesgue measure of Dd is zero but its intersection with any open set has the cardinality of R. (g) The tori T! defined in (e) with ! 2 Dd will be called maximal KAM tori for H. (h) A Hamiltonian function (; ) 2 M ! H(; ) having a maximal KAM torus (or, more generally, a maximal invariant torus as in (e) with ! rationally independent) T! , can be put into the form12 K(y; x) :D E C !  y C Q(y; x) ; with @˛y Q(0; x) D 0; 8˛ 2 N d ; j˛j  1 ;

(10)

Kolmogorov–Arnold–Moser (KAM) Theory

compare, e. g., Sect. 1 of [59]; in the variables (y; x), the torus T! is simply given by fy D 0g  T d and E is its (constant) energy. A Hamiltonian in the form (10) is said to be in Kolmogorov normal form. If deth@2y Q(0; )i ¤ 0 ;

(11)

(where the brackets denote an average over T d and @2y the Hessian with respect to the y-variables) we shall say that the Kolmogorov normal form K in (10) is nondegenerate; similarly, we shall say that the KAM torus T! for H is non-degenerate if H can be put in a nondegenerate Kolmogorov normal form.

Thus, if the “frequency map” y 2 B ! !(y) is a diffeomorphism (which is guaranteed if det K y y (y0 ) ¤ 0, for some y0 2 B and B is small enough), in view of (f), for almost all initial data, the trajectories (13) belong to maximal KAM tori fygT d with !(y) 2 Dd . The main content of (classical) KAM theory, in our language, is that, if the frequency map ! D K y of a (real-analytic) integrable Hamiltonian K(y) is a diffeomorphism, KAM tori persist under small (smooth enough) perturbations of K; compare Remark 7–(iv) below. The study of the dynamics generated by the flow of a one-parameter family of Hamiltonians of the form K(y) C "P(y; x; ") ;

Remark 1 (i)

A classical theorem by H. Weyl says that the flow 2 Td ! C !t 2 Td ;

t2R

is dense (ergodic) in T d if and only if ! 2 Rd is rationally independent (compare [6], Theorem 5.4 or Sect. 1.4 of [33]). Thus, trajectories on KAM tori fill them densely (i. e., pass in any neighborhood of any point). (ii) In view of the preceding remark, it is easy to see that if ! is rationally independent, (y(t); x(t)) in (6) is a solution of (2) if and only if the functions v and u satisfy the following quasi-linear system of PDEs on Td: ( D! v D H x (v( ); C u( )) (12) ! C D! u D H y (v( ); C u( )) where D! denotes the directional derivative !  @ D Pd @ iD1 ! i @ i . (iii) Probably, the main motivation for studying quasi-periodic solutions of Hamiltonian systems on Rd  T d comes from perturbation theory of nearly-integrable Hamiltonian systems: a completely integrable system may be described by a Hamiltonian system on M :D B(y0 ; r)  T d Rd  T d with Hamiltonian H D K(y) (compare Theorem 5.8, [6]); here B(y0 ; r) denotes the open ball fy 2 Rd : jy  y0 j < rg centered at y0 2 Rd . In such a case the Hamiltonian flow is simply 



 Kt (y; x) D y; x C !(y)t ; !(y) :D K y (y) :D

@K (y) : @y

(13)

0<" 1;

(14)

was called by H. Poincaré le problème général de la dynamique, to which he dedicated a large part of his monumental Méthodes Nouvelles de la Mécanique Céleste [49]. (iv) A big chapter in KAM theory, strongly motivated by applications to PDEs with Hamiltonian structure (such as nonlinear wave equation, Schrödinger equation, KdV, etc.), is concerned with quasi-periodic solutions with 1  n < d frequencies, i. e., solutions of (2) of the form ( y(t) D v(! t) (15) x(t) D U(! t) ; where v : T n ! Rd , U : T n ! T d are smooth functions, ! 2 Rn is a rationally independent n-vector. Also in this case, if the map U is a diffeomorphism onto its image, the set T!n :D f(y; x) 2 M :

y D v( ) ; x D U( ) ; 2 T n g

(16)

defines an invariant n-torus on which the flow  Ht acts by the linear translation ! C ! t. Such tori are normally referred to as lower dimensional tori. Even though this article will be mainly focused on “classical KAM theory” and on maximal KAM tori, we will briefly discuss lower dimensional tori in Sect. “Future Directions”. Kolmogorov Theorem In the 1954 International Congress of Mathematicians, in Amsterdam, A.N. Kolmogorov announced the following fundamental (for the terminology, see (f), (g) and (h) above).

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Theorem 1 (Kolmogorov [35]) Consider a one-parameter family of real-analytic Hamiltonian functions on M :D B(0; r)  T d given by H :D K C "P

(" 2 R) ;

(17)

where: (i) K is a non-degenerate Kolmogorov normal form (10)–(11); (ii) ! 2 Dd is Diophantine. Then, there exists "0 > 0 and for any j"j  "0 a real-analytic symplectic transformation  : M :D B(0; r )  T d ! M, for some 0 < r < r, putting H in non-degenerate Kolmogorov normal form, H ı  D K , with K :D E C !  y 0 C Q (y 0 ; x 0 ). Furthermore13 , k  idkC 1 (M ) , jE  Ej, and kQ  QkC 1 (M ) are small with ". Remark 2 (i)

From Theorem 1 it follows that the torus T!;" :D  (0; T d )

is a maximal non-degenerate KAM torus for H and the H-flow on T!;" is analytically conjugated (by  ) to the translation x 0 ! x 0 C ! t with the same frequency vector of T!;0 :D f0g  T d , while the energy of T!;" , namely E , is in general different from the energy E of T!;0 . The idea of keeping the frequency fixed is a key idea introduced by Kolmogorov and its importance will be made clear in the analysis of the proof. (ii) In fact, the dependence upon " is analytic and therefore the torus T!;" is an analytic deformation of the unperturbed torus T!;0 (which is invariant for K); see Remark 7–(iii) below. (iii) Actually, Kolmogorov not only stated the above result but gave also a precise outline of its proof, which is based on a fast convergent “Newton” scheme, as we shall see below; compare also [17]. The map  is obtained as  D lim 1 ı    ı  j ; j!1

where the  j ’s are ("-dependent) symplectic transformations of M closer and closer to the identity. It is enough to describe the construction of  1 ;  2 is then obtained by replacing H0 :D H with H1 D H ı 1 and so on. We proceed to analyze the scheme of Kolmogorov’s proof, which will be divided into three main steps. Step 1: Kolmogorov Transformation The map  1 is close to the identity and is generated by   g(y 0 ; x) :D y 0  x C " b  x C s(x) C y 0  a(x)

where s and a are (respectively, scalar and vector-valued) real-analytic functions on T d with zero average and b 2 Rd : setting ˇ0 D ˇ0 (x) :D b C s x ; A D A(x) :D a x

and

0

(18) 0

ˇ D ˇ(y ; x) :D ˇ0 C Ay ; (s x D @x s D (s x 1 ; : : : ; s x d ) and ax denotes the matrix i (a x ) i j :D @a )  1 is implicitly defined by @x j (

y D y 0 C "ˇ(y 0 ; x) :D y 0 C "(ˇ0 (x) C A(x)y 0 ) x 0 D x C "a(x) :

(19)

Thus, for " small, x 2 T d ! x C "a(x) 2 T d defines a diffeomorphism of T d with inverse x D '(x 0 ) :D x 0 C "˛(x 0 ; ") ;

(20)

for a suitable real-analytic function ˛, and  1 is explicitly given by ( 0

0

1 : (y ; x ) !

  y D y 0 C "ˇ y 0 ; '(x 0 ) x D '(x 0 ) :

(21)

Remark 3 (i) Kolmogorov transformation  1 is actually the composition of two “elementary” symplectic transformations: 1 D 1(1) ı 1(2) where 1(2) : (y 0 ; x 0 ) ! (; ) is the symplectic lift of the T d -diffeomorphism given by x 0 D  C "a() (i. e., 1(2) is the symplectic map generated by y 0   C "y 0  a()), while 1(1) : (; ) ! (y; x) is the angle-dependent action translation generated by   x C "(b  x C s(x)); 1(2) acts in the “angle direction” and will be needed to straighten out the flow up to order O("2 ), while 1(1) acts in the “action direction” and will be needed to keep the frequency of the torus fixed. (ii) The inverse of  1 has the form ( (y; x) !

y 0 D M(x)y C c(x) x 0 D (x)

(22)

with M a (d  d)-invertible matrix and  a diffeomorphism of T d (in the present case M D (1d C "A(x))1 D 1d C O(") and  D id C "a) and it is easy to see that the symplectic diffeomorphisms of the form (22) form a subgroup of the symplectic diffeomorphisms, which we shall call the group of Kolmogorov transformations.

Kolmogorov–Arnold–Moser (KAM) Theory

Determination of Kolmogorov transformation Following Kolmogorov, we now try to determine b, s and a so that the “new Hamiltonian” (better: “the Hamiltonian in the new symplectic variables”) takes the form H1 :D H ı 1 D K1 C "2 P1 ;

(23)

with K 1 in the Kolmogorov normal form K1 D E1 C !  y 0 C Q1 (y 0 ; x 0 ) ;

Q1 D O(jy 0 j2 ) : (24)

To proceed we insert y D y 0 C "ˇ(y 0 ; x) into H and, after some elementary algebra and using Taylor formula, we find14

where, defining 8 (1) Q :D Q y (y 0 ; x)  (a x y 0 ) ˆ ˆ ˆ ˆ ˆ ˆ ˆ Q (2) :D [Q y (y 0 ; x)  Q y y (0; x)y 0 ]  ˇ0 ˆ ˆ ˆ Z 1 ˆ ˆ ˆ ˆ D (1  t)Q y y y (ty 0 ; x)y 0  y 0  ˇ0 dt ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ ˆ Q (3) :D P(y 0 ; x)  P(0; x)  Py (0; x)y 0 ˆ ˆ ˆ Z 1 ˆ ˆ ˆ ˆ D (1  t)Py y (ty 0 ; x)y 0  y 0 dt ˆ ˆ ˆ 0 < 1 P(1) :D 2 [Q(y 0 C "ˇ; x)  Q(y 0 ; x) ˆ ˆ ˆ " ˆ ˆ ˆ ˆ  "Q y (y 0 ; x)  ˇ] ˆ ˆ ˆ Z ˆ 1 ˆ ˆ ˆ ˆ (1  t)Q y y (y 0 C t"ˇ; x)ˇ  ˇdt D ˆ ˆ ˆ 0 ˆ ˆ ˆ ˆ 1 ˆ ˆ P(2) :D [P(y 0 C "ˇ; x)  P(y 0 ; x)] ˆ ˆ " ˆ ˆ Z 1 ˆ ˆ ˆ ˆ : Py (y 0 C t"ˇ; x)  ˇdt ; D

(25)

(26)

!j

jD1

Q0

F0

F 0 (y 0 ; x)

(i) F 0 is a first degree polynomial in y 0 so that (28) is equivalent to ( !  b C D! s C P(0; x) D const ; (29) D! a C Q y y (0; x)ˇ0 C Py (0; x) D 0 :

D! u D f ;

(30)

for some given function f real-analytic on the average over T d shows that h f i D 0, and we see that (30) can be solved only if f has vanishing mean value h f i D f0 D 0 ;

@ @x j Q 0 (y 0 ; x),

(28)

T d . Taking

(recall that Q y (0; x) D 0) and denoting the !-directional derivative d X

F 0 (y 0 ; x) D const :

Indeed, the second equation is necessary to keep the torus frequency fixed and equal to ! (which, as we shall see in more detail later, is a key ingredient introduced by Kolmogorov). (ii) In solving (28) or (29), we shall encounter differential equations of the form

0

D! :D

recall, also, that Q D O(jyj2 ) so that Q y D O(y) and Q 0 D O(jy 0 j2 ). Notice that, as an intermediate step, we are considering H as a function of mixed variables y 0 and x (and this causes no problem, as it will be clear along the proof). Thus, recalling that x is related to x 0 by the (y 0 -independent) diffeomorphism x D x 0 C "˛(x 0 ; ") in (21), we see that in order to achieve relations (23)–(24), we have to determine b, s and a so that

Remark 4

H(y 0 C "ˇ; x) D E C !  y 0 C Q(y 0 ; x) C "Q 0 (y 0 ; x) C "F 0 (y 0 ; x) C "2 P0 (y 0 ; x)

where D! a is the vector function with kth entry Pd Pd @a k @a k 0 0 0 jD1 ! j @x j ; D! a  y D !  (a x y ) D j;kD1 ! j @x j y k ;

P0

one sees that D D and D P0 (y 0 ; x) are given by, respectively 8 ˆ Q 0 (y 0 ; x) :D Q (1) C Q (2) C Q (3) D O(jy 0 j2 ) ˆ ˆ ˆ ˆ < 0 0 F (y ; x) :D !  b C D! s C P(0; x) (27) ˚ ˆ ˆ C D! a C Q y y (0; x)ˇ0 C Py (0; x)  y 0 ˆ ˆ ˆ : 0 P :D P(1) C P(2) ;

in such a case, expanding in Fourier series15 , one sees that (30) is equivalent to X X i!  nu n e i nx D f n e i nx ; (31) n2Z d n¤0

n2Z d n¤0

so that the solutions of (30) are given by X fn e i nx ; u D u0 C i!  n d

(32)

n2Z n¤0

for an arbitrary u0 . Recall that for a continuous functionf over T d to be analytic it is necessary and sufficient that its Fourier coefficients f n decay exponentially fast in n, i. e., that there exist positive constants M and  such that j f n j  Mejnj ;

8n :

(33)

815

816

Kolmogorov–Arnold–Moser (KAM) Theory

d one has that (for n ¤ 0) Now, since ! 2 D;

jnj 1  j!  nj 

(34)

and one sees that if f is analytic so is u in (32) (although the decay constants of u will be different from those of f ; see below) Summarizing, if f is real-analytic on T d and has vanishing mean value f 0 , then there exists a unique realanalytic solution of (30) with vanishing mean value, which is given by 1 D! f :D

X n2Z d n¤0

f n i nx ; e i!  n

(35)

all other solutions of (30) are obtained by adding an 1 f as in (32) with u arbitrary. arbitrary constant to D! 0 Taking the average of the first relation in (29), we may the determine the value of the constant denoted const, namely, const D !  b C P0 (0) :D !  b C hP(0; )i :

(36)

Thus, by (ii) of Remark 4, we see see that 1 s D D! (P(0; x)  P0 (0)) D 

X Pn (0) e i nx ; (37) i!  n d

n2Z n¤0

Finally, if '(x 0 ) D x 0 C "˛(x 0 ; ") is the inverse diffeomorphism of x ! x C "a(x) (compare (20)), then, by Taylor’s formula, Z 1 Q x (y 0 ; x 0 C "˛t)  ˛dt : Q(y 0 ; '(x 0 )) D Q(y 0 ; x 0 ) C " 0

In conclusion, we have Proposition 1 If  1 is defined in (19)–(18) with s, b and a given in (37), (39) and (40) respectively, then (23) holds with 8 E1 :D E C "e E ˆ ˆ ˆ ˆ ˆ e ˆ E :D !  b C P0 (0) ˆ ˆ ˆ < Q (y 0 ; x 0 ) :D Q(y 0 ; x 0 ) C "e Q(y 0 ; x 0 ) 1 Z 1 ˆ ˆ ˆ e ˆ Q x (y 0 ; x 0 C t"˛)  ˛dt C Q 0 (y 0 ; '(x 0 )) Q :D ˆ ˆ ˆ 0 ˆ ˆ : P1 (y 0 ; x 0 ) :D P0 (y 0 ; '(x 0 )) (41) with Q 0 and P0 defined in (26), (27) and ' in (20). Remark 5 The main technical problem is now transparent: because of the appearance of the small divisors !  n 1 f (which may become arbitrarily small), the solution D! is less regular than f so that the approximation scheme cannot work on a fixed function space. To overcome this fundamental problem – which even Poincaré was unable to solve notwithstanding his enormous efforts (see, e. g., [49]) – three ingredients are necessary: (i)

where Pn (0) denote the Fourier coefficients of x ! P(0; x); indeed s is determined only up to a constant by the relation in (29) but we select the zero-average solution. Thus, s has been completely determined. To solve the second (vector) equation in (29) we first have to require that the left hand side (l.h.s.) has vanishing mean value, i. e., recalling that ˇ0 D b C s x (see (18)), we must have hQ y y (0; )ib C hQ y y (0; )s x i C hPy (0; )i D 0 :

(38)

In view of (11) this relation is equivalent to   b D hQ y y (0; )i1 hQ y y (0; )s x i C hPy (0; )i ; (39) which uniquely determines b. Thus ˇ 0 is completely determined and the l.h.s. of the second equation in (29) has average zero; thus its unique zero-average solution (again zero-average of a is required as a normalization condition) is given by   1 Q y y (0; x)ˇ0 C Py (0; x) : a D D!

(40)

To set up a Newton scheme: this step has just been performed and it has been summarized in the above Proposition 1; such schemes have the following features: they are “quadratic” and, furthermore, after one step one has reproduced the initial situation (i. e., the form of H 1 in (23) has the same properties of H 0 ). It is important to notice that the new perturbation "2 P1 is proportional to the square "; thus, if one could iterate j times, at the jth step, would find j

H j D H j1 ı  j D K j C "2 Pj :

(42)

The appearance of the exponential of the exponential of " justifies the term “super-convergence” used, sometimes, in connection with Newton schemes. (ii) One needs to introduce a scale of Banach function spaces fB :  > 0g with the property that B 0 B when  <  0 : the generating functions  j will belong to B j for a suitable decreasing sequence  j ; (iii) One needs to control the small divisors at each step and this is granted by Kolmogorov’s idea of keeping the frequency fixed in the normal form so that one can systematically use the Diophantine estimate (9).

Kolmogorov–Arnold–Moser (KAM) Theory

Kolmogorov in his paper very neatly explained steps (i) and (iii) but did not provide the details for step (ii); in this regard he added: “Only the use of condition (9) for proving the convergence of the recursions,  j , to the analytic limit for the recursion  is somewhat more subtle”. In the next paragraph we shall introduce classical Banach spaces and discuss the needed straightforward estimates.

(P5) Assume that x ! f (x) 2 B has a zero average (all above definitions may be easily adapted to functions d (recall depending only on x); assume that ! 2 D; Sect. “Introduction”, point (f)), and let p 2 N. Then, there exist constants B¯ p D B¯ p (d; )  1 and k p D k p (d; )  1 such that, for any multi-index ˇ 2 N d with jˇj  p and for any 0   0 <  one has 1 f k 0  B¯ p k@ˇx D!

Step 2: Estimates For   1, we denote by B the space of function f : B(0; )  T d ! R analytic on W :D D(0; )  Td ;

(43)

where D(0; ) :D fy 2 C d : jyj < g and Td :D fx 2 C d : jImx j j < g/(2Zd )

sup

(i)

A proof of (48) is easily obtained observing that by (35) and (46), calling ı :D    0 , one has 1 k@ˇx D! f k 0 

X jnjjˇ j j f n j 0 e jnj j!  nj d

n2Z n¤0

(44)

 k f k jfj ;

D(0;)T d

D

(in other words, Td denotes the complex points x with real parts Rex j defined modulus 2 and imaginary part Imx j with absolute value less than ). The following properties are elementary:

j f n (y)j  k f k ejnj ;

8n 2 Zd ; 8y 2 D(0; ) : (46)

Another elementary property, which together with (P3) may found in any book of complex variables (e. g., [1]), is the following “Cauchy estimate” (which is based on Cauchy’s integral formula): (P4) let f 2 B and let p 2 N then there exists a constant B p D B p (d)  1 such that, for any multi-index (˛; ˇ) 2 N d  N d with j˛j C jˇj  p (as above P for integer vectors ˛, j˛j D j j˛ j j) and for any 0   0 <  one has k@˛y @ˇx f k 0  B p k f k (   0 )(j˛jCjˇ j) :

(47)

1 f , i. e., on solutions Finally, we shall need estimates on D! of (30):

X jnjjˇ jC eıjnj  d

n2Z n¤0

(45)

(P1) B equipped with the k  k norm is a Banach space; (P2) B 0 B when  <  0 and k f k  k f k 0 for any f 2 B 0 ; (P3) if f 2 B , and f n (y) denotes the n-Fourier coefficient of the periodic function x ! f (y; x), then

(48)

Remark 6

with finite sup-norm k f k :D

k f k (   0 )k p : 

k f k (jˇ jCCd) ı  X  [ıjnj]jˇ jC eıjnj ı d n2Z d n¤0

 const

k f k (   0 )(jˇ jCCd) ; 

where the last estimate comes from approximating the sum with the Riemann integral Z jyjjˇ jC ejyj dy : Rd

More surprising (and much more subtle) is that (48) holds with k p D jˇj C ; such an estimate has been obtained by Rüssmann [54,55]. For other explicit estimates see, e. g., [11] or [12]. (ii) If jˇj > 0 it is not necessary to assume that h f i D 0. (iii) Other norms may be used (and, sometimes, are more useful); for example, rather popular are Fourier norms X j f n jejnj ; (49) k f k0 :D n2Z d

see, e. g., [13] and references therein. By the hypotheses of Theorem 1 it follows that there exist 0 <   1,  > 0 and   d  1 such that H 2 B and d . Denote ! 2 D; T :D hQ y y (0; )i1 ;

M :D kPk :

(50)

817

818

Kolmogorov–Arnold–Moser (KAM) Theory

and let C > 1 be a constant such that16 jEj; j!j; kQk ; kTk < C

(51)

(i. e., each term on the l.h.s. is bounded by the r.h.s.); finally, fix 0<ı<

2 and define ¯ :D   ı ;  0 :D   ı : (52) 3

The parameter  0 will be the size of the domain of analyticity of the new symplectic variables (y 0 ; x 0 ), domain on which we shall bound the Hamiltonian H1 D H ı1 , while ¯ is an intermediate domain where we shall bound various functions of y 0 and x. By (P4) and (P5), it follows that there exist constants ¯ 2 ZC and ¯ D ¯ (d; ) > 1 such c¯ D c¯(d; ; ) > 1,  that17 8 ks x k¯ ; jbj; je Ej; kak¯ ; ka x k¯ ; kˇ0 k¯ ; kˇk¯ ; ˆ ˆ < kQ 0 k¯ ; k@2y 0 Q 0 (0; )k0  c¯C ¯ ı ¯ M D: L¯ ; (53) ˆ ˆ : 0 ¯ : kP k¯  c¯C ¯ ı ¯ M 2 D LM The estimate in the first line of (53) allows us to construct, for " small enough, the symplectic transformation  1 , whose main properties are collected in the following Lemma 1 If j"j  "0 and "0 satisfies "0 L¯ 

ı ; 3

(54)

then the map " (x) :D x C "a(x) has an analytic inverse '(x 0 ) D x 0 C "˛(x 0 ; ") such that, for all j"j < "0 , k˛k 0  L¯

and

' D id C "˛ : Td0 ! T¯d :

(55)

Furthermore, for any (y 0 ; x) 2 W¯ , jy 0 C "ˇ(y 0 ; x)j < , so that   1 D y 0 C "ˇ(y 0 ; '(x 0 )); '(x 0 ) : W 0 ! W ; and k1  idk 0  j"j L¯ ; (56) finally, the matrix 1d C "a x is, for any x 2 T¯d , invertible with inverse 1d C "S(x; ") satisfying kSk¯ 

ka x k¯ 1  j"jka x k¯

<

3¯ L; 2

(57)

so that  1 defines a symplectic diffeomorphism. The simple proof18 of this statement is based upon standard tools in mathematical analysis such as the contraction

mapping theorem or the inversion of close-to-identity matrices by Neumann series (see, e. g., [36]). From the Lemma and the definition of P1 in (41), it follows immediately that kP1 k 0  L¯ :

(58)

Next, by the same technique used to derive (53), one can easily check that ke Qk 0 ; 2C 2 k@2y 0 e Q(0; )k0  cC  ı  M D L ;

(59)

¯  , ¯   (the factor 2C 2 for suitable constants c  c¯,  has been introduced for later convenience; notice also that ¯ But, then, if L  L). "0 L :D "0 cC  ı  M 

ı ; 3

(60)

Tk  L; this bound, together there follows that19 ke with (53), (59), (56), and (58), shows that ( ˜ ke Tk; k1  idk 0  L j Ej; Qk 0 ; ke (61) kP1 k 0  LM ; provided (60) holds (notice that (60) implies (54)). One step of the iteration has been concluded and the needed estimates obtained. The idea is to iterate the construction infinitely many times, as we proceed to describe. Step 3: Iteration and Convergence In order to iterate Kolmogorov’s construction, analyzed in Step 2, so as to construct a sequence of symplectic transformations  j : W jC1 ! W j ;

(62)

closer and closer to the identity, and such that (42) hold, the first thing to do is to choose the sequence  j : this sequence must be convergent, so that ı j D  j   jC1 has to go to zero rather quickly. Inverse powers of ı j (which, at the jth step will play the role of ı in the previous paragraph) appear in the smallness conditions (see, e. g., (54)): this “divergence” will, however, be beaten by the super-fast j decay of "2 . Fix 0 <  <  ( will be the domain of analyticity of  and K in Theorem 1 and, for j  0, let 8 8 ı0 ˆ ˆ ˆ  :D  < ı j :D j < 0 2 (63) ˆ ı0 ˆ : ı :D    ˆ :  jC1 :D  j  ı j D  C j 0 2 2

Kolmogorov–Arnold–Moser (KAM) Theory

and observe that  j #  . With this choice20 , Kolmogorov’s algorithm can be iterated infinitely many times, provided "0 is small enough. To be more precise, let c,  and  be as in (59), and define ˚ C :D 2 max jEj; j!j; kQk ; kTk; 1 : (64) Smallness Assumption: Assume that j"j  "0 and that "0 satisfies "0 DBkPk  1 where D :D 3cı0(C1) C  ; B :D 2C1 ;

(65)

notice that the constant C in (64) satisfies (51) and that (65) implies (54). Then the following claim holds. Claim C: Under condition (65) one can iteratively construct a sequence of Kolmogorov symplectic maps  j as in (62) so j that (42) holds in such a way that "2 Pj , ˚ j :D 1 ı 2 ı    ı  j , Ej , K j , Qj converge uniformly on W to, respectively, 0,  , E , K , Q , which are real-analytic on W and H ı  D K D E C !  y C Q with Q D O(jyj2 ). Furthermore, the following estimates hold for any j"j  "0 and for any i  0:

(The correspondence with the above constants being:  D , ı0 D (1  )/2,  D  C 1, D D 3c(2/(1  ))C1 C  , c D 3c(4/(1  ))C1 ). (ii) From Cauchy estimates and (67), it follows that k  idkC p and kQ  Q kC p are small for any p (small in j"j but not uniformly in21 p). (iii) All estimates are uniform in ", therefore, from Weierstrass theorem (compare note 18) it follows that  and K are analytic in " in the complex ball of radius "0 . Power series expansions in " were very popular in the nineteenth and twentieth centuries22 , however convergence of the formal "-power series of quasi-periodic solutions was proved for the first time only in the 1960s thanks to KAM theory [45]. Some of this matter is briefly discussed in Sect. “Future Directions” below. (iv) The Nearly-Integrable Case In [35] it is pointed out that Kolmogorov’s Theorem easily yields the existence of many KAM tori for nearly-integrable systems (14) for j"j small enough, provided K is nondegenerate in the sense that

i

i

i

j"j2 M i :D j"j2 kPi k i 

(j"jDBM)2 ; DB iC1

(66)

k  idk ; jE  E j; kQ  Q k ; kT  T k  j"jDBM ; where T :D generate.

h@2y Q (0; )i1 ,

(67)

showing that K is non-de-

Remark 7 (i)

From Claim C Kolmogorov Theorem 1 follows at once. In fact we have proven the following quantid tative statement: Let ! 2 D; with   d  1 and 0 <  < 1; let Q and P be real-analytic on W D D d (0; )  Td for some 0 <   1 and let 0 < < 1; let T and C be as in, respectively, (50) and (64). There exist c D c (d; ; ; ) > 1 and positive integers  D (d; ),  such that if j"j  " :D

 c kPk C 

(68)

then one can construct a near-to-identity Kolmogorov transformation (Remark 3–(ii))  : W  ! W such that the thesis of Theorem 1 holds together with the estimates k  idk  ; jE  E j; kQ  Q k  ; kT  T k 

j"j D j"jc kPk C    : "

(69)

det K y y (y0 ) ¤ 0 :

(70)

In fact, without loss of generality we may assume that ! :D H00 is a diffeomorphism on B(y0 ; 2r) and det K y y (y) ¤ 0 for all y 2 B(y0 ; 2r). Furthermore, letting B D B(y0 ; r), fixing  > d 1 and denoting by `d the Lebesgue measure on Rd , from the remark in note 11 and from the fact that ! is a diffeomorphism, there follows that there exists a constant c# depending only on d,  and r such that d d `d (!(B)nD; ); `d (fy 2 B : !(y) … D; g) < c#  :

(71) d g (which, Now, let B; :D fy 2 B : !(y) 2 D; by (71) has Lebesgue measure `d (B; )  `d (B)  c# ), then for any y¯ 2 B; we can make the trivial symplectic change of variables y ! y¯ C y, x ! x so that K can be written as in (10) with

E :D K( y¯) ;

! :D K y ( y¯ ) ;

Q(y; x) D Q(y) :D K(y)  K( y¯)  K y ( y¯)  y ; (where, for ease of notation, we did not change names to the new symplectic variables) and P( y¯ C y; x) replacing (with a slight abuse of notation) P(y; x). By Taylor’s formula, Q D O(jyj2 ) and, furthermore (since Q(y; x) D Q(y), h@2y Q(0; x)i D Q y y (0) D

819

820

Kolmogorov–Arnold–Moser (KAM) Theory

K y y ( y¯), which is invertible according to out hypotheses. Thus K is Kolmogorov non-degenerate and Theorem 1 can be applied yielding, for j"j < "0 , a KAM torus T!;" , with ! D K y ( y¯), for each y¯ 2 B; . Notice that the measure of initial phase points, which, perturbed, give rise to KAM tori, has a small complementary bounded by c#  (see (71)). (v) In the nearly-integrable setting described in the preceding point, the union of KAM tori is usually called the Kolmogorov set. It is not difficult to check that the dependence upon y¯ of the Kolmogorov transformation  is Lipschitz23 , implying that the measure of the complementary of Kolmogorov set itself is also bounded by cˆ#  with a constant cˆ# depending only on d,  and r. Indeed, the estimate on the measure of the Kolmogorov set can be made more quantitative (i. e., one can see how such an estimate depends upon " as " ! 0). In fact, revisiting the estimates discussed in Step 2 above one sees easily that the constant c defined in (53) has the form24 c D cˆ 4 :

(72)

where cˆ D cˆ(d; ) depends only on d and  (here the Diophantine constant  is assumed, without loss of generality, to be smaller than one). Thus the small¯  1 with some ness condition (65) reads "0  4 D ¯ independent of : such condition is satconstant D ¯ 0 )1/4 and since cˆ#  was an isfied by choosing  D ( D" upper bound on the complementary of Kolmogorov set, we see that the set of phase points which do not lie on KAM tori may be bounded by a constant times p 4 " . Actually, it turns that this bound is not optimal, 0 as we shall see in the next section: see Remark 10. (vi) The proof of claim C follows easily by induction on the number j of the iterative steps25 . Arnold’s Scheme The first detailed proof of Kolmogorov Theorem, in the context of nearly-integrable Hamiltonian systems (compare Remark 1–(iii)), was given by V.I. Arnold in 1963. Theorem 2 (Arnold [2]) Consider a one-parameter family of nearly-integrable Hamiltonians H(y; x; ") :D K(y) C "P(y; x) (" 2 R)

(73)

with K and P real-analytic on M :D B(y0 ; r)  T d (endowed with the standard symplectic form dy^dx) satisfying K y (y0 ) D ! 2

d D;

;

det K y y (y0 ) ¤ 0 :

(74)

Then, if " is small enough, there exists a real-analytic embedding  : 2 Td ! M

(75)

close to the trivial embedding (y0 ; id), such that the d-torus   T!;" :D  T d (76) is invariant for H and  Ht ı ( ) D ( C ! t) ;

(77)

showing that such a torus is a non-degenerate KAM torus for H. Remark 8 (i)

The above Theorem is a corollary of Kolmogorov Theorem 1 as discussed in Remark 7–(iv). (ii) Arnold’s proof of the above Theorem is not based upon Kolmogorov’s scheme and is rather different in spirit – although still based on a Newton method – and introduces several interesting technical ideas. (iii) Indeed, the iteration scheme of Arnold is more classical and, from the algebraic point of view, easier than Kolmogorov’s, but the estimates involved are somewhat more delicate and introduce a logarithmic correction, so that, in fact, the smallness parameter will be :D j"j(log j"j1 )

(78)

(for some constant D (d; )  1) rather than j"j as in Kolmogorov’s scheme; see, also, Remark 9–(iii) and (iv) below. Arnold’s Scheme Without loss of generality, one may assume that K and P have analytic and bounded extension to Wr; (y0 ) :D D(y0 ; r)  Td for some  > 0, where, as above, D(y0 ; r) denotes the complex ball of center y0 and radius r. We remark that, in what follows, the analyticity domains of actions and angles play a different role The Hamiltonian H in (73) admits, for " D 0 the (KAM) invariant torus T!;0 D fy0 g  T d on which the K-flow is given by x ! x C ! t. Arnold’s basic idea is to find a symplectic transformation 1 : W1 :D D(y1 ; r1 )Td1 ! W0 :D D(y0 ; r)Td ; (79) so that W1 W0 and ( H1 :D H ı 1 D K1 C "2 P1 ; @ y K1 (y1 ) D ! ;

det @2y K1 (y1 )

K1 D K1 (y) ; ¤0

(80)

Kolmogorov–Arnold–Moser (KAM) Theory

(with abuse of notation we denote here the new symplectic variables with the same name of the original variables; as above, dependence on " will, often, not be explicitly indicated). In this way the initial set up is reconstructed and, for " small enough, one can iterate the scheme so as to build a sequence of symplectic transformations  j : Wj :D D(y j ; r j )  Tdj ! Wj1 so that ( j H j :D H j1 ı  j D K j C "2 Pj ; @ y K j (y j ) D ! ;

det @2y K j (y j )

determined by a generating function of the form ( y D y 0 C "g x (y 0 ; x) 0 0 y  x C "g(y ; x) ; x 0 D x C "g y 0 (y 0 ; x) : Inserting y D y 0 C "g x (y 0 ; x) into H, one finds

(81)

K j D K j (y) ;

¤0:

(82)

H(y 0 C "g x ; x) D K(y 0 )    C " K y (y 0 )  g x C P(y 0 ; x) C "2 P(1) C P(2)

1 [K(y 0 C "g x )  K(y 0 )  "K y (y 0 )  g x ] "2 Z 1 (1  t)K y y (y 0 C t"g x ; x) g x  g x dt D

P(1) :D

0

( ) :D lim ˚ j (y j ; ) ;

P

(83)

˚ j :D 1 ı    ı  j : Wj ! W0 ; defines a real-analytic embedding of T d into the phase space B(y0 ; r)  T d , which is close to the trivial embedding (y0 ; id); furthermore, the torus T!;" :D (T d ) D lim ˚ j (y j ; T d )

(87)

with (compare (26))

Arnold’s transformations, as in Kolmogorov’s case, are closer and closer to the identity, and the limit j!1

(86)

(2)

1 :D [P(y 0 C "g x ; x)  P(y 0 ; x)] " Z 1 Py (y 0 C t"g x ; x)  g x dt : D

(88)

0

Remark 9 (i)

The (naive) idea is to try determine g so that

(84)

K y (y 0 )  g x C P(y 0 ; x) D function of y 0 only; (89)

is invariant for H and (77) holds as announced in Theorem 2. Relation (77) follows from the following argument. The radius rj will turn out to tend to 0 but in a much slower j way than "2 Pj . This fact, together with the rapid convergence of the symplectic transformation ˚ j in (83) implies

however, such a relation is impossible to achieve. First of all, by taking the x-average of both sides of (89) one sees that the “function of y 0 only” has to be the mean of P(y 0 ; ), i. e., the zero-Fourier coefficient P0 (y 0 ), so that the formal solution of (89), is (by Fourier expansion) 8 X Pn (y 0 ) ˆ ˆ
j!1

   Ht ı ( ) D lim  Ht ˚ j (y j ; ) j!1

D lim ˚ j ı  Ht j (y j ; ) j!1

y

D lim ˚ j (y j ; C ! t)

x

0

j!1

D ( C ! t)

(85)

where: the first equality is just smooth dependence upon initial data of the flow  Ht together with (83); the second equality is (3); the third equality is due to the fact j that (see (82))  Ht j (y j ; ) D  Kt j (y j ; ) C O("2 kPj k) D j

j

(y j ; C! t)CO("2 kPj k) and O("2 kPj k) goes very rapidly to zero; the fourth equality is again (83). Arnold’s Transformation Let us look for a near-to-the-identity transformation  1 so that the first line of (80) holds; this transformation will be

But, (at difference with Kolmogorov’s scheme) the frequency K y (y 0 ) is a function of the action y 0 and since, by the Inverse Function Theorem (Appendix “A The Classical Implicit Function Theorem”), y ! K y (y) is a local diffeomorphism, it follows that, in any neighborhood of y0 , there are points y such that K y (y)  n D 0 for some26 n 2 Zd . Thus, in any neighborhood of y0 , some divisors in (90) will actually vanish and, therefore, an analytic solution g cannot exist27 . (ii) On the other hand, since K y (y0 ) is rationally independent, it is clearly possible (simply by continuity) to control a finite number of divisors in a suitable

821

822

Kolmogorov–Arnold–Moser (KAM) Theory

neighborhood of y0 , more precisely, for any N 2 N one can find r¯ > 0 such that K y (y)  n ¤ 0 ; (91)

the important quantitative aspects will be shortly discussed below. (iii) Relation (89) is also one of the main “identity” in Averaging Theory and is related to the so-called Hamilton–Jacobi equation. Arnold’s proof makes such a theory rigorous and shows how a Newton method can be built upon it in order to establish the existence of invariant tori. In a sense, Arnold’s approach is more classical than Kolmogorov’s. (iv) When (for a given y and n) it occurs that K y (y)n D 0 one speaks of an (exact) resonance. As mentioned at the end of point (i), in the general case, resonances are dense. This represents the main problem in Hamiltonian perturbation theory and is a typical feature of conservative systems. For generalities on Averaging Theory, Hamilton–Jacobi equation, resonances etc. see, e. g., [5] or Sects. 6.1 and 6.2 of [6]. The key (simple!) idea of Arnold is to split the perturbation into two terms X 8 ˆ Pˆ :D Pn (y)e i nx ˆ ˆ < jnjN (92) P D Pˆ C Pˇ where X ˆ ˆ Pˇ :D Pn (y)e i nx ˆ : jnj>N

choosing N so that Pˇ D O(")

(93)

(this is possible because of the fast decay of the Fourier coefficients of P; compare (33)). Then, for " ¤ 0, (87) can be rewritten as follows ˆ 0 ; x) H(y C "g x ; x) D K(y ) C " K y (y )  g x C P(y   C "2 P(1) C P(2) C P(3) (94) 0



0

1ˇ 0 P(y ; x) : "

X 0<jnjN

(97)

P0 (y 0 ; x) :D P(1) C P(2) C P(3) :

Now, by the IFT (Appendix “A The Classical Implicit Function Theorem”), for " small enough, the map x ! x C g y 0 (y 0 ; x) can be inverted with a real-analytic map of the form '(y 0 ; x 0 ; ") :D x 0 C "˛(y 0 ; x 0 ; ")

Pn e i nx ; iK y (y 0 )  n

(99)

so that Arnold’s symplectic transformation is given by   0 8 0 0 0 ˆ < y D y C "g x y ; '(y ; x ; ") (100) 1 : (y 0 ; x 0 ) ! x D '(y 0 ; x 0 ; ") ˆ : D x 0 C "˛(y 0 ; x 0 ; ; ") (compare (21)). To finish the construction, observe that, from the IFT (see Appendix “A The Classical Implicit Function Theorem” and the quantitative discussion below) it follows that there exists a (unique) point y1 2 B(y0 ; r¯) so that the second line of (80) holds, provided " is small enough. In conclusion, the analogue of Proposition 1 holds, describing Arnold’s scheme: Proposition 2 If  1 is defined in (100) with g given in (96) (with N so that (93) holds) and ' given in (99), then (80) holds with K 1 as in (98) and P1 (y 0 ; x 0 ) :D P0 (y 0 ; '(y 0 ; x 0 )) with P 0 defined in (98), (95) and (88). Estimates and Convergence If f is a real-analytic function with analytic extension to Wr; , we denote, for any r0  r and  0  , k f kr 0 ; 0 :D

sup

j f (y; x)j ;

(101)

M :D kPkr; ;

(102)

Wr0 ; 0 (y 0 )

furthermore, we define

and assume (without loss of generality)  < 1; r < 1;   1; maxf1; kK y kr ; kK y y kr ; kTkg < C ;

(y 0 )

(98)

(95)

Thus, letting28 gD

K1 (y 0 ) :D K(y 0 ) C "P0 (y 0 ) ;

T :D K y y (y0 )1 ;

with P (1) and P(2) as in (88) and P(3) (y 0 ; x) :D

H(y 0 C "g x ; x) D K1 (y 0 ) C "2 P0 (y 0 ; x) where

8y 2 D(y0 ; r¯) ; 80 < jnj  N ;

0

one gets

(96)

(103)

for a suitable constant C (which, as above, will not change during the iteration).

Kolmogorov–Arnold–Moser (KAM) Theory

We begin by discussing how N and r¯ depend upon ". From the exponential decay of the Fourier coefficients (33), it follows that, choosing N :D 5ı 1  ;

where  :D log j"j1 ;

(104)

then ˇ kPk  j"jM r; ı

(105)

2

provided j"j  const ı

(106)

for a suitable29 const D const(d). The second key inequality concerns the control of the small divisors K y (y 0 )  n appearing in the definition of g (see (96)), in a neighborhood D(y0 ; r¯) of y0 : this will determine the size of r¯. d , by Taylor’s forRecalling that K y (y0 ) D ! 2 D; mula and (9), one finds, for any 0 < jnj  N and any y 0 2 D(y0 ; r¯), ˇ   ˇ jK y (y 0 )  nj D ˇ!  n C K y (y 0 )  K y (y0 )  nˇ   kK y y kr  j!  nj 1  jnj¯r j!  nj    C C1 ¯  1  r jnj jnj     C C1 ¯  r N 1  jnj  1   ; (107) 2 jnj provided r¯  r satisfies also (108)

Equation (107) allows us to easily control Arnold’s generating function g. For example:

2



sup D(y 0 ;¯r )T d ı

 2

0<jnjN



X n2Z d

 const

ˇ ˇ ˇ X ˇ 0) ˇ ˇ nP (y n i nx ˇ ˇ e ˇ ˇ 0 ˇ0<jnjN K y (y )  n ˇ

sup D(y0 ;r) jPn (y 0 )j

X

M

jK y (y 0 )  nj

r¯ 

r 2

(110)

(allowing the use of Cauchy estimates for y-derivatives of K or P in D(y0 ; r¯)), it is not difficult to see that the quantitative IFT of Appendix “A The Classical Implicit Function Theorem” implies that there exists a unique y1 2 D(y0 ; r¯) such that (80) holds and, furthermore jy1  y0 j  4CMr1 j"j ;

ı

jnje( 2 )jnj

2jnjC1  ı jnj e 2 

M (C1Cd) ; ı  (109)

(111)

and @2y K1 (y1 ) :D K y y (y1 ) C "@2y P0 (y1 ) D: T 1 (1d C A)

(112)

with a matrix A satisfying kAk  10C 3 Mj"j 

1 2

(113)

provided30 8 < 8C 2 r¯  1 ; r : 8CM r¯2 j"j  1

(114)

Equation (113) shows that @2y K(y1 ) is invertible (Neumann series) and that31 @2y K1 (y1 )1 D T C "e T;

  (104) r¯  D : C1 C1 2CN 25 C(ı 1 )C1

kg x kr¯; ı D

where “const” denotes a constant depending on d and  only; compare also Remark 6–(i). Let us now discuss, from a quantitative point of view, how to choose the new “center” of the action variables y1 , which is determined by the requirements in (80). Assuming that

ke Tk  20C 3 M :

(115)

Finally, notice that the second conditions in (114) and (111) imply that jy1  y0 j < r¯/2 so that D(y1 ; r¯/2) D(y0 ; r¯) :

(116)

Now, all the estimating tools are set up and, writing K1 :D K C "e K D K C "P0 (y 0 ) ; y1 :D y0 C " y˜ ;

(117)

one can easily prove (along the lines that led to (53)) the following estimates, where as in Sect. “Kolmogorov Theorem”, ¯ :D   23 ı and r¯ is as above: 8 kg x kr¯;¯ ˆ ˆ K y kr¯ ; ke K y y k; j y˜j; ke Tk ; kg y kr¯;¯ ; ke ˆ < r (118) 2  

 c C ı  M D: L ; ˆ ˆ ˆ : 0 kP kr¯;¯  c 2 C  ı   M 2 D LM ;

823

824

Kolmogorov–Arnold–Moser (KAM) Theory

where c D c(d; ) > 1,  2 ZC ,  and are positive integers depending on d and . Now, by32 Lemma 1 and (118), one has that map x ! x C "g y (y 0 ; x) has, for any y 0 2 Dr¯ (y0 ), an analytic inverse ' D x 0 C "˛(x 0 ; y 0 ; ") D: '(y 0 ; x 0 ) on T ¯d ı provided (54), with L¯ replaced by L  3

in (118), holds, in which case (55) holds (for any j"j  "0 and any y 0 2 Dr (y0 )). Furthermore, under the above hypothesis, it follows that33  0  8 0 0 0 0 0 ˆ < 1 :D y C "g x (y ; '(y ; x )); '(y ; x ) : Wr¯/2;ı (y1 ) ! Wr; (y0 ) ˆ : k1  idkr¯/2;ı  j"jL :

(119)

Finally, letting P1 (y 0 ; x 0 ) :D P0 (y 0 ; '(y 0 ; x 0 )) one sees that P1 is real-analytic on Wr¯/2;ı ¯ (y 1 ) and bounded on that domain by kP1 kr¯/2;ı  LM :

(120)

In order to iterate the above construction, we fix 0 <  <  and set C :D 2 maxf1; kK y kr ; kK y y kr ; kTkg ;

:D 3C ; ı0 :D

(121)

(  1)(   ) ;

 j and ı j as in (63) but with ı 0 as in (121); we also define, for any j  0, j

;  j :D 2 j  D log "2 0  ; r j :D C1 4  5C1 C(ı 1 j j)

(122)

(this part is adapted from Step 3 in Sect. “Kolmogorov Theorem”; see, in particular, (103)). With such choices it is not difficult to check that the iterative construction may be carried out infinitely many times yielding, as a byproduct, Theorem 2 with  real-analytic on Td , provided j"j  "0 with "0 satisfying34 8 ˆ ˇ ˆ ˆ < "0  e "0 DBkPk  1 ˆ ˆ ˆ :

with with

1 ı0     C1 5 Cr D :D 3c 2 ı0(C1) C  ;

ˇ :D

B :D C1 (log "1 0 ) :

(123) Remark 10 Notice that the power of  1 (the inverse of the Diophantine constant) in the second smallness condition in (123) is two, which implies (compare Remark 7–

(v)) that the measure of the complement of the Kolp mogorov set may be bounded by a constant times "0 . This bound is optimal as the trivial example (y12 C y22 )/2 C " cos(x1 ) shows: the Hamiltonian is integrable and the phase portrait shows that the separatrices ofpthe pendulum y12 /2 C " cos x1 bound a region of area j"j with no KAM tori (as the librational curves within such region are not graphs over the angles).

The Differentiable Case: Moser’s Theorem J.K. Moser, in 1962, proved a perturbation (KAM) Theorem, in the framework of area-preserving twist mappings of an annulus35 [0; 1]  S1 , for integrable analytic systems perturbed by a Ck perturbation, [42] and [43]. Moser’s original setup corresponded to the Hamiltonian case with d D 2 and the required smoothness was Ck with k D 333. Later, this number was brought down to 5 by H. Rüssmann, [53]. Moser’s original approach, similarly to the approach that led J. Nash to prove its theorem on the smooth embedding problem of compact Riemannian manifolds [48], is based on a smoothing technique (via convolutions), which re-introduces at each step of the Newton iteration a certain number of derivatives which one loses in the inversion of the small divisor operator. The technique, which we shall describe here, is again due to Moser [46] but is rather different from the original one and is based on a quantitative analytic KAM Theorem (in the style of statement in Remark 7–(i) above) in conjunction with a characterization of differentiable functions in terms of functions, which are real-analytic on smaller and smaller complex strips; see [44] and, for an abstract functional approach, [65], [66]. By the way, this approach, suitably refined, leads to optimal differentiability assumptions (i. e., the Hamiltonian may be assumed to be C ` with ` > 2d); see, [50] and the beautiful exposition [59], which inspires the presentation reported here. Let us consider a Hamiltonian H D K C"P (as in (17)) with K a real-analytic Kolmogorov normal form as in (10) d with ! 2 D; and Q real-analytic; P is assumed to be ` d d a C (R ; T ) function with ` D `(d; ) to be specified later36 . Remark 11 The analytic KAM theorem, we shall refer to is the quantitative Kolmogorov Theorem as stated in Remark 7–(i) above, with (69) strengthened by including in the left hand side of (69) also37 k@(  id)k  and k@(Q  Q )k  (where “@” denotes, here, “Jacobian” with regard to (y; x) for (  id) and “gradient” for (Q  Q )).

Kolmogorov–Arnold–Moser (KAM) Theory

The analytic characterization of differentiable functions, suitable for our purposes, is explained in the following two lemmata38 Lemma 2 (Jackson, Moser, Zehnder) Let f 2 C l (Rd ) with l > 0. Then, for any  > 0 there exists a real-analytic function f : Xd :D fx 2 C d : jImx j j < g ! C such that 8 sup j f j  ck f kC 0 ; ˆ ˆ ˆ Xd  <

(124) ˆ sup j f  f 0 j  ck f kC l  l ; 8 0 <  0 <  ; ˆ ˆ : d X 0

where c D c(d; l) is a suitable constant; if f is periodic in some variable xj , so is f . Lemma 3 (Bernstein, Moser) Let l 2 RC nZ and  j :D 1/2 j . Let f0 D 0 and let, for any j  1, f j be real analytic functions on X dj :D fx 2 C d : jImx j j < 2 j g such that sup j f j  f j1 j  A2 jl

(125)

X dj

Rd

to for some constant A. Then, f j tends uniformly on a function f 2 C l (Rd ) such that, for a suitable constant C D C(d; l) > 0, k f kC l (Rd )  CA :

(126)

Finally, if the f i ’s are periodic in some variable xj then so is f . Now, denote by X D Xd  T d C 2d and define (compare Lemma 2) P j :D P j ;

 j :D

1 : 2j

(127)

Claim M: If j"j is small enough and if ` >  C 1, then there exists a sequence of Kolmogorov symplectic transformations f˚ j g j0 , j"j-close to the identity, and a sequence of Kolmogorov normal forms K j such that H j ı ˚ j D K jC1 on W jC1

(128)

where H j :D K C "P j ˚0 D 0

and ˚ j :D ˚ j1 ı  j ; ( j  1)

 j : W jC1 ! W˛ j ; ˚ j1 : W˛ j ! X j ; 1 j  1 and ˛ :D p ; 2 sup x2T d

j˚ j (0; x)  ˚ j1 (0; x)j  constj"j2(`) j :

jC1

(129)

The proof of Claim M follows easily by induction39 from Kolmogorov’s Theorem (compare Remark 11) and Lemma 2. From Claim M and Lemma 3 (applied to f (x) D ˚ j (0; x)  ˚0 (0; x) and l D `   , which may be assumed not integer) it then follows that ˚ j (0; x) converges in the C1 norm to a C1 function  : T d ! Rd  T d , which is "-close to the identity, and, because of (128), (x C ! t) D lim ˚ j (0; x C ! t) D lim  Ht ı ˚ j (0; x) D  Ht ı (x)

(130)

showing that (T d ) is a C1 KAM torus for H (note that the map  is close to the trivial embedding x ! (y; x)). Future Directions In this section we review in a schematic and informal way some of the most important developments, applications and possible future directions of KAM theory. For exhaustive surveys we refer to [9], Sect. 6.3 of [6] or [60]. 1. Structure of the Kolmogorov set and Whitney smoothness The Kolmogorov set (i. e., the union of KAM tori), in nearly-integrable systems, tends to fill up (in measure) the whole phase space as the strength of the perturbation goes to zero (compare Remark 7–(v) and Remark 10). A natural question is: what is the global geometry of KAM tori? It turns out that KAM tori smoothly interpolate in the following sense. For " small enough, there exists a C 1 symplectic diffeomorphism  of the phase space M D B  T d of the nearly-integrable, non-degenerate Hamiltonians H D K(y) C "P(y; x) and a Cantor set C B such that, for each y 0 2 C , the set 1 (fy 0 g  T d ) is a KAM torus for H}; in other words, the Kolmogorov set is a smooth, symplectic deformation of the fiber bundle C  T d . Still another way of describing this result is that there exists a smooth function K : B ! R such that (K C "P) ı  and K agree, together with their derivatives, on C  T d : we may, thus, say that, in general, nearly-integrable Hamiltonian systems are integrable on Cantor sets of relative big measure. Functions defined on closed sets which admit Ck extensions are called Whitney smooth; compare [64], where H. Whitney gives a sufficient condition, based on Taylor uniform approximations, for a function to be Whitney Ck . The proof of the above result – given, independently, in [50] and [19] in, respectively, the differentiable and the analytic case – follows easily from the following lemma40 :

825

826

Kolmogorov–Arnold–Moser (KAM) Theory

Lemma 4 Let C Rd a closed set and let f f j g, f0 D 0, be a sequence of functions analytic on Wj :D [ y2C D(y; r j ). P < 1. Then, f j Assume that j1 supW j j f j  f j1 jrk j converges uniformly to a function f , which is Ck in the sense of Whitney on C . Actually, the dependence upon the angles x 0 of  is analytic and it is only the dependence upon y 0 2 C which is Whitney smooth (“anisotropic differentiability”, compare Sect. 2 in [50]). For more information and a systematic use of Whitney differentiability, see [9]. 2. Power series expansions KAM tori T!;" D " (T d ) of nearly-integrable Hamiltonians correspond to quasi-periodic trajectories z(t; ; ") D " ( C! t) D  Ht (z(0; ; ")); compare items (d) and (e) of Sect. “Introduction” and Remark 2–(i) above. While the actual existence of such quasi-periodic motions was proven, for the first time, only thanks to KAM theory, the formal existence, in terms of formal "-power series41 was well known in the nineteenth century to mathematicians and astronomers (such as Newcombe, Lindstedt and, especially, Poincaré; compare [49], vol. II). Indeed, formal power solutions of nearly-integrable Hamiltonian equations are not difficult to construct (see, e. g., Sect. 7.1 of [12]) but direct proofs of the convergence of the series, i. e., proofs not based on Moser’s “indirect” argument recalled in Remark 7–(iii) but, rather, based upon direct estimates on the kth "-expansion coefficient, are quite difficult and were carried out only in the late eighties by H. Eliasson [27]. The difficulty is due to the fact that, in order to prove the convergence of the Taylor–Fourier expansion of such series, one has to recognize compensations among huge terms with different signs42 . After Eliasson’s breakthrough based upon a semi-direct method (compare the “Postscript 1996” at p. 33 of [27]), fully direct proofs were published in 1994 in [30] and [18]. 3. Non-degeneracy assumptions Kolmogorov’s non-degeneracy assumption (70) can be generalized in various ways. First of all, Arnold pointed out in [2] that the condition   Kyy Ky ¤0; (131) det Ky 0 (this is a (d C 1)  (d C 1) symmetric matrix where last column and last row are given by the (d C 1)-vector (K y ; 0)) which is independent from condition (70), is also sufficient to construct KAM tori. Indeed, (131) may be used to construct iso-energetic KAM tori, i. e., tori on a fixed energy level43 E.

More recently, Rüssmann [57] (see, also, [58]), using results of Diophantine approximations on manifolds due to Pyartly [52], formulated the following condition (the “Rüssmann non-degeneracy condition”), which is essentially necessary and sufficient for the existence of a positive measure set of KAM tori in nearly-integrable Hamiltonian systems: the image !(B) Rd of the unperturbed frequency map y ! !(y) :D K y (y) does not lie in any hyperplane passing through the origin. We simply add that one of the prices that one has to pay to obtain these beautiful general results is that one cannot fix the frequency ahead of time. For a thorough discussion of this topic, see Sect. 2 of [60]. 4. Some physical applications We now mention a short (and non-exhaustive) list of important physical application of KAM theory. For more information, see Sect. 6.3.9 of [6] and references therein. 4.1. Perturbation of classical integrable systems As mentioned above (Remark 1–(iii)), one of the main original motivations of KAM theory is the perturbation theory for nearly-integrable Hamiltonian systems. Among the most famous classical integrable systems we recall: one-degree-of freedom systems; Keplerian two-body problem, geodesic motion on ellipsoids; rotations of a heavy rigid body with a fixed point (for special values of the parameters: Euler’s, Lagrange’s, Kovalevskaya’s and Goryachev–Chaplygin’s cases); Calogero–Moser’s system of particles; see, Sect. 5 of [6] and [47]. A first step, in order to apply KAM theory to such classical systems, is to explicitly construct actionangle variables and to determine their analyticity properties, which is in itself a technically non-trivial problem. A second problem which arises, especially in Celestial Mechanics, is that the integrable (transformed) Hamiltonian governing the system may be highly degenerate (proper degeneracies – see Sect. 6.3.3, B of [6]), as is the important case of the planetary n-body problem. Indeed, the first complete proof of the existence of a positive measure set of invariant tori44 for the planetary (n C 1) problem (one body with mass 1 and n bodies with masses smaller than ") has been published only in 2004 [29]. For recent reviews on this topic, see [16]. 4.2. Topological trapping in low dimensions The general 2-degree-of-freedom nearly-integrable Hamiltonian exhibits a kind of particularly strong stability: the phase space is 4-dimensional and the energy levels are 3-dimensional; thus KAM tori

Kolmogorov–Arnold–Moser (KAM) Theory

(which are two-dimensional and which are guaranteed, under condition (131), by the iso-energetic KAM theorem) separate the energy levels and orbits lying between two KAM tori will remain forever trapped in the invariant region. In particular the evolution of the action variables stays forever close to the initial position (“total stability”). This observation is originally due to Arnold [2]; for recent applications to the stability of threebody problems in celestial mechanics see [13] and item 4.4 below. In higher dimension this topological trapping is no longer available, and in principle nearby any point in phase space it may pass an orbit whose action variables undergo a displacement of order one (“Arnold’s diffusion”). A rigorous complete proof of this conjecture is still missing45 . 4.3. Spectral Theory of Schrödinger operators KAM methods have been applied also very successfully to the spectral analysis of the one-dimensional Schrödinger (or “Sturm–Liouville”) operator on the real line R L :D 

d2 C v(t) ; dt 2

t 2R:

(132)

If the “potential” v is bounded then there exists a unique self-adjoint operator on the real Hilbert space L2 (R) (the space of Lebesgue square-integrable functions on R) which extends L above on C02 (the space of twice differentiable functions with compact support). The problem is then to study the spectrum (L) of L; for generalities, see [23]. If v is periodic, then (L) is a continuous band spectrum, as it follows immediately from Floquet theory [23]. Much more complicated is the situation for quasi-periodic potentials v(t) :D V (! t) D V (!1 t; : : : ; !n t), where V is a (say) real-analytic function on T n , since small-divisor problems appear and the spectrum can be nowhere dense. For a beautiful classical exposition, see [47], where, in particular, interesting connections with mechanics are discussed46 ; for deep developments of generalization of Floquet theory to quasi-periodic Schrödinger operators (“reducibility”), see [26] and [7]. 4.4. Physical stability estimates and break-down thresholds KAM Theory is perturbative and works if the parameter " measuring the strength of the perturbation is small enough. It is therefore a fundamental question: how small " has to be in order for KAM

results to hold. The first concrete applications were extremely discouraging: in 1966, the French astronomer M. Hénon [32] pointed out that Moser’s theorem applied to the restricted three-body problem (i. e., the motion of an asteroid under the gravitational influence of two unperturbed primary bodies revolving on a given Keplerian ellipse) yields existence of invariant tori if the mass ratio of the primaries is less than47 1052 . Since then, much progress has been made and very recently, in [13], it has been shown via a computer-assisted proof48 , that, for a restricted-three body model of a subsystem of the Solar system (namely, Sun, Jupiter and Asteroid Victoria), KAM tori exist for the “actual” physical values (in that model the Jupiter/Sun mass ratio is about 103 ) and, in this mathematical model – thanks to the trapping mechanism described in item 4.2 above – they trap the actual motion of the subsystem. From a more theoretical point of view, we notice that, (compare Remark 2–(ii)) KAM tori (with a fixed Diophantine frequency) are analytic in "; on the other hand, it is known, at least in lower dimensional settings (such as twist maps), that above a certain critical value KAM tori (curves) cannot exist ([39]). Therefore, there must exist a critical value "c (!) (“breakdown threshold”) such that, for 0  " < "c !, the KAM torus (curve) T!;" exists, while for " > "c (!) does not. The mathematical mechanism for the breakdown of KAM tori is far from being understood; for a brief review and references on this topic, see, e. g., Sect. 1.4 in [13]. 5. Lower dimensional tori In this item we consider (very) briefly, the existence of quasi-periodic solutions with a number of frequencies smaller than the number of degrees of freedom49 . Such solutions span lower dimensional (non Lagrangian) tori. Certainly, this is one of the most important topics in modern KAM theory, not only in view of applications to classical problems, but especially in view of extensions to infinite dimensional systems, namely PDEs (Partial Differential Equations) with a Hamiltonian structure; see, item 6 below. For a recent, exhaustive review on lower dimensional tori (in finite dimensions), we refer the reader to [60]. In 1965 V.K. Melnikov [41] stated a precise result concerning the persistence of stable (or “elliptic”) lower dimensional tori; the hypotheses of such results are, now, commonly referred to as “Melnikov conditions”. However, a proof of Melnikov’s statement was given only later by Moser [45] for the case n D d  1 and, in

827

828

Kolmogorov–Arnold–Moser (KAM) Theory

the general case, by H. Eliasson in [25] and, independently, by S.B. Kuksin [37]. The unstable (“partially hyperbolic”) case (i. e., the case for which the lower dimensional tori are linearly unstable and lie in the intersection of stable and unstable Lagrangian manifolds) is simpler and a complete perturbation theory was already given in [45], [31] and [66] (roughly speaking, the normal frequencies to the torus do not resonate with the inner (or “proper”) frequencies associated with quasi-periodic motion). Since then, Melnikov conditions have been significantly weakened and much technical progress has been made; see [60], Sects. 5, 6 and 7, and references therein. To illustrate a typical situation, let us consider a Hamiltonian system with d D n C m degrees of freedom, governed by a Hamiltonian function of the form H(y; x; v; u; ) D K(y; v; u; ) C "P(y; x; v; u; ) ;

(133)

where (y; x) 2 T n  Rn , (v; u) 2 R2m are pairs of standard symplectic coordinates and  is a real parameter running over a compact set ˘ Rn of positive Lebesgue measure50; K is a Hamiltonian admitting the n-torus T0n () :D fy D 0g  T n  fv D u D 0g ;

 2˘;

as invariant linearly stable invariant torus and is assumed to be in the normal form: m

K D E() C !()  y C

1X ˝ j ()(u2j C v 2j ) : (134) 2 jD1

The  Kt flow decouples in the linear flow x 2 T n ! x C !()t times the motion of m (decoupled) harmonic oscillators with characteristic frequencies ˝ j () (sometimes referred to as normal frequencies). Melnikov’s conditions (in the form proposed in [51]) reads as follows: assume that ! is a Lipschitz homeomorphism; let ˘ k;l denote the “resonant parameter set” f 2 ˘ : !()  k C ˝  () D 0g and assume (

˝ i () > 0 ; ˝ i () ¤ ˝ j () ; 8 2 ˘ ; 8i ¤ j meas ˘ k;l D 0 ; 8k 2 Z n nf0g ; 8l 2 Z m : jlj  2 : (135) Under these assumptions and if j"j is small enough, there exists a (Cantor) subset of parameters ˘ ˘ of positive Lebesgue measure such that, to each  2 ˘ , there

corresponds a n-dimensional, linearly stable H-invariant torus T"n () on which the H flow is analytically conjugated to x ! x C ! ()t where ! is a Lipschitz homeomorphism of ˘ assuming Diophantine values and close to !. This formulation has been borrowed from [51], to which we refer for the proof; for the differentiable analog, see [22]. Remark 12 The small-divisor problems arising in the perturbation theory of the above lower dimensional tori are of the form !kl ˝ ;

jlj  2 ; jkj C jlj ¤ 0 ;

(136)

where one has to regard the normal frequency ˝ as functions of the inner frequencies ! and, at first sight, one has – in J. Moser words – a lack-of-parameter problem. To overcome this intrinsic difficulty, one has to give up full control of the inner frequencies and construct, iteratively, n-dimensional sets (corresponding to smaller and smaller sets of -parameters) on which the small divisors are controlled; for more motivations and informal explanations on lower dimensional small divisor problems, see, Sects. 5, 6 and 7 of [60]. 6. Infinite dimensional systems As mentioned above, the most important recent developments of KAM theory, besides the full applications to classical n-body problems mentioned above, is the successful extension to infinite dimensional settings, so as to deal with certain classes of partial differential equations carrying a Hamiltonian structure. As a typical example, we mention the non-linear wave equation of the form u t t  u x x C V (x)u D f (u) ; f (u) D O(u2 ) ; 0 < x < 1 ; t 2 R :

(137)

These extensions allowed, in the pioneering paper [63], establishing the existence of small-amplitude quasi-periodic solutions for (137), subject to Dirichlet or Neumann boundary conditions (on a finite interval for odd and analytic nonlinearities f ); the technically more difficult periodic boundary condition case was considered later; compare [38] and references therein. A technical discussion of these topics goes far beyond the scope of the present article and, for different equations, techniques and details, we refer the reader to the review article [38].

Kolmogorov–Arnold–Moser (KAM) Theory

A The Classical Implicit Function Theorem Here we discuss the classical Implicit Function Theorem for complex functions from a quantitative point of view. The following Theorem is a simple consequence of the Contraction Lemma, which asserts that a contraction ˚ on a closed, non-empty metric space51 X has a unique fixed point, which is obtained as lim j!1 ˚ j (u0 ) for any52 u0 2 X. As above, D n (y0 ; r) denotes the ball in C n of center y0 and radius r. Theorem 3 (Implicit Function Theorem) Let F : (y; x) 2 D n (y0 ; r)  D m (x0 ; s) C nCm ! F(y; x) 2 C n be continuous with continuous Jacobian matrix F y ; assume that F y (y0 ; x0 ) is invertible and denote by T its inverse; assume also that sup

D(y 0 ;r)D(x 0 ;s)

1 ; 2 r : sup jF(y0 ; x)j  2kTk D(x 0 ;s)

k1n  TF y (y; x)k 

(138)

sup jgj  2kTk sup jF(y0 ; )j : D(x 0 ;s)

Additions: (i)

If F is periodic in x or/and real on reals, then (by uniqueness) so is g; (ii) If F is analytic, then so is g (Weierstrass Theorem, since g is attained as uniform limit of analytic functions); (iii) The factors 1/2 appearing in the right-hand sides of (138) may be replaced by, respectively, ˛ and ˇ for any positive ˛ and ˇ such that ˛ C ˇ D 1. Taking n D m and F(y; x) D f (y)  x for a given C 1 (D(y0 ; r); C n ) function, one obtains the Theorem 4 (Inverse Function Theorem) Let f : y 2 D n (y0 ; r) ! C n be a C1 function with invertible Jacobian f y (y0 ) and assume that sup k1n  T f y k 

Then, all solutions (y; x) 2 D(y0 ; r) D(x0 ; s) of F(y; x) D 0 are given by the graph of a unique continuous function g : D(x0 ; s) ! D(y0 ; r) satisfying, in particular, D(x 0 ;s)

which implies that y1 D g(x1 ) and that all solutions of F D 0 in D(y0 ; r)  D(x0 ; s) coincide with the graph og g. Finally, (139) follows by observing that kg  y0 k D k˚ (g)  y0 k  k˚(g)  ˚(y0 )k C k˚ (y0 )  y0 k  1  2 kg  y 0 k C kTkkF(y 0 ; )k, finishing the proof.

(139)

Proof Let X D C(D m (x0 ; s); D n (y0 ; r)) be the closed ball of continuous function from D m (x0 ; s) to D n (y0 ; r) with respect to the sup-norm k  k (X is a non-empty metric space with distance d(u; v) :D ku  vk) and denote ˚ (y; x) :D y  TF(y; x). Then, u ! ˚(u) :D ˚ (u; ) maps C(D m (x0 ; s)) into C(C m ) and, since @ y ˚ D 1n  TF y (y; x), from the first relation in (138), it follows that u ! ˚(u) is a contraction. Furthermore, for any u 2 C(D m (x0 ; s); D n (y0 ; r)), j˚(u)  y0 j  j˚(u)  ˚(y0 )j C j˚(y0 )  y0 j 1  ku  y0 k C kTkkF(y0 ; x)k 2 1 r  r C kTk Dr; 2 2kTk showing that ˚ : X ! X. Thus, by the Contraction Lemma, there exists a unique g 2 X such that ˚(g) D g, which is equivalent to F(g; x) D 0 8x. If F(y1 ; x1 ) D 0 for some (y1 ; x1 ) 2 D(y0 ; r)  D(x0 ; s), it follows that jy1  g(x1 )j D j˚(y1 ; x1 )  ˚(g(x1 ); x1 )j  ˛jy1  g(x1 )j,

D(y 0 ;r)

1 ; 2

T :D f y (y0 )1 ;

(140)

then there exists a unique C1 function g : D(x0 ; s) ! D(y0 ; r) with x0 :D f (y0 ) and s :D r/(2kTk) such that f ı g(x) D id D g ı f . Additions analogous to the above also hold in this case. B Complementary Notes 1

2

Actually, the first instance of a small divisor problem solved analytically is the linearization of the germs of analytic functions and is due to C.L. Siegel [61]. The well-known Newton’s tangent scheme is an algorithm, which allows us to find roots (zeros) of a smooth function f in a region where the derivative f 0 is bounded away from zero. More precisely, if xn is an “approximate solution” of f (x) D 0, i. e., f (x n ) :D " n is small, then the next approximation provided by Newton’s tangent scheme is x nC1 :D x n  f (x n )/ f 0 (x n ) [which is the intersection with x-axis of the tangent to the graph of f passing through (x n ; f (x n ))] and, in view of the definition of "n and Taylor’s formula, one has that "nC1 :D f (x nC1) D 1 00 0 2 2 2 f ( n )" n /( f n (x n ) (for a suitable  n ) so that " nC1 D 2 2 O(" n ) D O("1 ) and, in the iteration, xn will converge (at a super-exponential rate) to a root x¯ of f . This type of extremely fast convergence will be typical in the analyzes considered in the present article.

829

830

Kolmogorov–Arnold–Moser (KAM) Theory

3

4

5

6

7

8

9 10

11

12

The elements of T d are equivalence classes x D x¯ C 2Zd with x¯ 2 Rd . If x D x¯ C 2Zd and y D y¯ C 2Zd are elements of T d , then their distance d(x; y) is given by minn2Zd j x¯  y¯ C 2 nj where j  j denotes the standard euclidean norm in Rn ; a smooth (analytic) function on T d may be viewed as (“identified with”) a smooth (analytic) function on Rd with period 2 in each variable. The torus T d endowed with the above metric is a real-analytic, compact manifold. For more information, see [62]. A symplectic form on an (even dimensional) manifold is a closed, non-degenerate differential 2-form. The symplectic form ˛ D dy ^ dx is actually exact symP plectic, meaning that ˛ D d( iD1 y i dx i ). For general information see [5]. For general facts about the theory of ODE (such as Picard theorem, smooth dependence upon initial data, existence times, . . . ) see, e. g., [23]. This terminology is due to that fact the the xj are “adimensional” angles, while analyzing the physical dimensions of the quantities appearing in Hamilton’s equations one sees that dim(y)  dim(x) D dim H  dim(t) so that y has the dimension of an energy (the Hamiltonian) times the dimension of time, i. e., by definition, the dimension of an action. This terminology is due to the fact that a classical mechanical system of d particles of masses m i > 0 and subject to a potential V(q) with q 2 A Rd is govP erned by a Hamiltonian of the form djD1 p2j /2m j C V (q) and d may be interpreted as the (minimal) number of coordinates necessary to physically describe the system. To be precise, (6) should be written as y(t) D v(T d (! t)), x(t) D T d (! t C u(T d (! t))) where T d denotes the standard projection of Rd onto T d , however we normally omit such a projection. As standard, U denotes the (d  d) Jacobian matrix with entries (@U i )/(@ j ) D ı i j C (@u i )/(@ j ). For generalities, see [5]; in particular, a Lagrangian manifold L M which is a graph over T d admits a “generating function”, i. e., there exists a smooth function g : T d ! R such that L D f(y; x) : y D g x (x), x 2 T d g. Compare [54] and references therein. We remark that, if B(!0 ; r) denote the ball in Rd of radius r centered at ! 0 and fix  > d  1, then one can prove that the d can be bounded by Lebesgue measure of B(y0 ; r)nD; d1 for a suitable constant cd depending only on d; cd r for a simple proof, see, e.g, [21]. The sentence “can be put into the form” means “there exists a symplectic diffeomorphism  : (y; x) 2 M !

13

14

15

16

17

(; ) 2 M such that H ı  has the form (10)”; for multi-indices ˛, j˛j D ˛1 C    C ˛d and @˛y D ˛ @˛y11    @ y dd ; the vanishing of the derivatives of a function f (y) up to order k in the origin will also be indicated through the expression f D O(jyj kC1 ). Notation: If A is an open set and p 2 N, then the Cp -norm of a function f : x 2 A ! f (x) is defined as k f kC p (A) : supj˛jp supA j@˛x f j. Notation: If f is a scalar function f y is a d-vector; f yy the Hessian matrix ( f y i y j ); f yyy the symmetric 3-tensor of third derivatives acting as follows: P f y y y a  b  c :D di; j;kD1 (@3 f )/(@y i @y j @y k )a i b j c k . Notation: If f is (a regular enough) function over d coefficients are defined as f np :D T R , its Fourier i nx dx/(2)d ; where, as usual, i D f (x)e 1 Td denotes imaginary unit; for general information about Fourier series see, e. g., [34]. The choice of norms on finite dimensional spaces (Rd , C d , space of matrices, tensors, etc.) is not particularly relevant for the analysis in this article (since changing norms will change d-depending constants); however for matrices, tensors (and, in general, linear operators), it is convenient to work with the “operator norm”, i. e., the norm defined as kLk D supu¤0 kLuk/kuk, so that kLuk  kLkkuk, an estimate, which will be constantly be used; for a general discussion on norms, see, e. g., [36]. As an example, let us work out the first two estimates, i. e., the estimates on ks x k¯ and jbj: actually these estimates will be given on a larger intermediate domain, namely, Wı/3 , allowing to give the remaining bounds on the smaller domain W¯ (recall that W s denotes the complex domain D(0; s)  Tsd ). Let f (x) :D P(0; x)hP(0; )i. By definition of kk and M, it follows that k f k  kP(0; x)k CkhP(0; )ik  2M. By (P5) with p D 1 and  0 D   ı/3, one gets 2M k 1 k 1 ks x k ı  B¯ 1 ; 3 ı 3  which is of the form (53), provided c¯  (B¯ 1 2  3 k 1 )/ and ¯  k1 . To estimate b, we need to bound first jQ y y (0; x)j and jPy (0; x)j for real x. To do this we can use Cauchy estimate: by (P4) with p D 2 and, respectively, p D 1, and  0 D 0, we get kQ y y (0; )k0  mB2 C 2  mB2 Cı 2 ; kPy (0; x)k0  mB1 Mı

1

and

;

where m D m(d)  1 is a constant which depend on the choice of the norms, (recall also that ı < ). Putting these bounds together, one gets that jbj can be

Kolmogorov–Arnold–Moser (KAM) Theory

18

19

20

21

bounded by the r.h.s. of (53) provided c¯  m(B2 B¯ 1 2  3 k 1  1 C B1 ),   2 and ¯  k1 C2. The other bounds in (53) follow easily along the same lines. We sketch here the proof of Lemma 1. The defining relation " ı ' D id implies that ˛(x 0 ) D a(x 0 C "˛(x 0 )), where ˛(x 0 ) is short for ˛(x 0 ; ") and that equation is a fixed point equation for the non-linear operator f : u ! f (u) :D a(id C "u). To find a fixed point for this equation one can use a standard contraction Lemma (see [36]). Let Y denote the closed ball (with respect to the sup-norm) of continuous func¯ By (54), tions u : Td0 ! C d such that kuk 0  L. ¯ for any jIm(x 0 C "u(x 0 ))j <  0 C "0 L¯ <  0 C ı/3 D , u 2 Y, and any x 0 2 Td0 ; thus, k f (u)k 0 ;"  kak¯  L¯ by (53), so that f : Y ! Y; notice that, in particular, this means that f sends periodic functions into periodic functions. Moreover, (54) implies also that f is a contraction: if u; v 2 Y, then, by the mean value the¯ orem, j f (u) f (v)j  Lj"jjuvj (with a suitable choice of norms), so that, by taking the sup-norm, one has ¯  vk 0 < 1 ku  vk 0 showk f (u)  f (v)k 0 < "0 Lku 3 ing that f is a contraction. Thus, there exists a unique ˛ 2 Y such that f (˛) D ˛. Furthermore, recalling that the fixed point is achieved as the uniform limit limn!1 f n (0) (0 2 Y) and since f (0) D a is analytic, so is f n (0) for any n and, hence, by Weierstrass Theorem on the uniform limit of analytic function (see [1]), the limit ˛ itself is analytic. In conclusion, ' 2 B 0 and (55) holds. Next, for (y 0 ; x) 2 W¯ , by (53), one has jy 0 C "ˇ(y 0 ; x)j < ¯ C "0 L¯ < ¯ C ı/3 D  so that (56) holds. Furthermore, since k"a x k¯ < "0 L¯ < 1/3 the matrix 1d C "a x is invertible with inverse given by the “Neumann series” (1d C "a x )1 D 1d C P1 k k kD1 (1) ("a x ) D: 1d C"S(x; "), so that (57) holds. The proof is finished. From (59), it follows immediately that h@2y 0 Q1 (0; )i D h@2y Q(0; )iC"h@2y 0 e Q(0; )i D T 1 (1d C"Th@2y 0 e Q(0; )i) 1 D: T (1d C "R) and, in view of (51) and (59), we see that kRk < L/(2C). Therefore, by (60), "0 kRk < 1/6 < 1/2, implying that (1C"R) is invertible P k k k and (1d C "R)1 D 1d C 1 kD1 (1) " R D: 1 C "D with kDk  kRk/(1  j"jkRk) < L/C. In conclusion, T1 D (1 C "R)1 T D T C "DT D: T C "e T, ke Tk  kDkC  (L/C)C D L. Actually, there is quite some freedom in choosing the sequence f j g provided the convergence is not too fast; for general discussion, see, [56], or, also, [10] and [14]. In fact, denoting by B the real d-ball centered at 0 and of radius  for 2 (0; 1), from Cauchy estimate (47) with  D  and  0 D  , one has k 

ˇ

22 23

24

25

idkC p (B T d ) D supB T d supj˛jCjˇ jp j@˛y @x (  ˇ id)j  supj˛jCjˇ jp k@˛y @x (  id)k   B p k  idk 1/(  ) p  const p j"j with const p :D B p DBM1/(  ) p . An identical estimate holds for kQ  QkC p (B T d ) . Also very recently "-power series expansions have been shown to be a very powerful tool; compare [13]. A function f : A Rn ! Rn is Lipschitz on A if there exists a constant (“Lipschitz constant”) L > 0 such that j f (x)  f (y)j  Ljx  yj for all x; y 2 A. For a general discussion on how Lebesgue measure changes under Lipschitz mappings, see, e. g., [28]. In fact, the dependence of  on y¯ is much more regular, compare Remark 11. In fact, notice that inverse powers of  appear through (48) (inversion of the operator D! ), therefore one sees that the terms in the first line of (53) may be replaced by c˜ 2 (in defining a one has to apply the 1 twice) but then in P (1) (see (26)) there operator D! appears kˇk2 , so that the constant c in the second line of (53) has the form (72); since  < 1, one can replace in (53) c with cˆ 4 as claimed. Proof of Claim C Let H0 :D H, E0 :D E, Q0 :D Q, K0 :D K, P0 :D P, 0 :D  and let us assume (inductive hypothesis) that we can iterate the Kolmogorov transformation j times obtaining j symplectic transformations  iC1 : W iC1 ! W i , for 0  i  j  1, and j i

Hamiltonians H iC1 D H i ı  iC1 D K i C "2 Pi realanalytic on W i such that j!j; jE i j; kQ i k i ; kTi k < C ; j"j2 L i :D j"j2 cC  ı0 2 i M i  i

i

ıi ; 3

(*)

80  i  j  1 : By ( ), Kolmogorov iteration (Step 2) can be applied to H i and therefore all the bounds described in paragraph Step 2 hold (having replaced H; E; : : : ; ; ı; H 0 ; E 0 ; : : : ;  0 with, respectively, H i ; E i ; : : : ;  i ; ı i ; H iC1 ; E iC1 ; : : : ;  iC1 ); in particular (see (61)) one has, for 0  i  j  1 (and for any j"j  "0 ), 8 2i ˆ ˆ jE iC1 j  jE i j C j"j L i ; ˆ ˆ ˆ 2i < kQ k iC1  iC1  kQ i k i C j"j L i ; (C.1) ˆ 2i ˆ  idk  j"j L k ˆ iC1 i  iC1 ˆ ˆ : M iC1  M i L i Observe that the definition of D, B and LI , j j 2j i j"j2 L j (3Cı 1 j ) D: DB j"j M j , so that L i < DB M i ,

831

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Kolmogorov–Arnold–Moser (KAM) Theory

thus by the second line in (C:1), for any 0  i  j  1, iC1 i j"j2 M iC1 < DB i (M i j"j2 )2 , which iterated, yields (66) for 0  i  j. Next, we show that, thanks to (65), ( ) holds also for i D j (and this means that Kolmogorov’s step can be iterated an infinite number of times). In fact, by ( ) and the definition of C P P j1 i in (64): jE j j  jEj C iD0 "20 L i  jEj C 13 i0 ı i < P jEjC 16 i1 2i < jEjC1 < C. The bounds for kQ i k and kTi k are proven in an identical manner. Now, j j 2j by (66) iD j and (65), j"j2 L j (3ı 1 j ) D DB j"j M j 

30

j

26

27

28

29

DB j (DB"0 M)2 /(DB jC1 )  1/B < 1, which implies the second inequality in ( ) with i D j; the proof of the induction is finished and one can construct an infinite sequence of Kolmogorov transformations satisfying ( ), (C:1) and (66) for all i  0. To check (67), i i we observe that j"j2 L i D ı0 /(3  2 i )DB i j"j2 M i  i (1/2 iC1 )(j"jDBM)2  (j"jDBM/2) iC1 and therefore P P 2i (j"jDBM/2) i  j"jDBM. Thus, i0 j"j L i  Pi1 2i ˜ kQ  Q k  i0 kQ i k i  j"j L i  j"jDBM; and analogously for jE  E j and kT  T k. To estimate k  idk , observe that k˚ i  idk i  k˚ i1 ı i  i   i k i C k i  idk i  k˚ i1  idk i1 C j"j2 L i , P k which iterated yields k˚ i  idk i  ikD0 j"j2 L k  j"jDBM: taking the limit over i completes the proof of (67) and the proof of Claim C. In fact, observe: (i) given any integer vector 0 ¤ n 2 Zd with d  2, one can find 0 ¤ m 2 Zd such n  m D 0; (ii) the set ftn : t > 0 and n 2 Zd g is dense in Rd ; (iii) if U is a neighborhood of y0 , then K y (U) is a neighborhood of ! D K y (y0 ). Thus, by (ii) and (iii), in K y (U) there are infinitely many points of the form tn with t > 0 and n 2 Zd to which correspond points y(t; n) 2 U such that K y (y(t; n)) D tn and for any of such points one can find, by (i), m 2 Z such that m  n D 0, whence K y (y(t; n))  m D tn  m D 0. This fact was well known to Poincaré, who based on the above argument his non-existence proof of integral of motions in the general situation; compare Sect. 7.1.1, [6]. Compare (90) but observe, that, since Pˆ is a trigonometric polynomial, in view of Remark 9–(ii), g in (96) defines a real-analytic function on D(y0 ; r¯)  Td0 with a suitable r¯ D r¯(") and  0 < . Clearly it is important to see explicitly how the various quantities depend upon "; this is shortly discussed after Proposition 2. P jnjı/2  Me(ı/4)N ˇ r;ı/2  M In fact: kPk jnj>N e P P jnjı/4  Me(ı/4)N jnj>0 ejnjı/4  jnj>N e const Me(ı/4)N ı d  j"jM if (106) holds and N is taken as in (104).

31

32

33

34

35

36

Apply the IFT of Appendix “A The Classical Implicit Function Theorem” to F(y; ) :D K y (y) C @ y P0 (y)  K y (y0 ) defined on D d (y0 ; r¯)  D1 (0; j"j): using the mean value theorem, Cauchy estimates and (114), k1d  TF y k  k1d  TK y y k C j"jk@2y P0 k  kTkkK y y y k¯r C kTkj"jk@2y P0 k  C 2 2¯r/r C Cj"j4/r2 M  14 C 18 < 12 ; also: 2kTk kF(y0 ; k D 2kTkjj@ y P0 (y0 )k < 2Cj"jM2/r  2CM¯r1 j"j < 14 r¯ (where last inequality is due to (114)), showing that conditions (138) are fulfilled. Equation (111) comes from (139) and (113) follows easily by repeating the above estimates. Recall note 18 and notice that (1d C A)1 D 1d C D with kDk  kAk/(1  kAk)  2kAk  20C 3 Mj"j, where last two inequalities are due to (113). Lemma 1 can be immediately extended to the y 0 -dependent case (which appear as a dummy parameter) as far as the estimates are uniform in y 0 (which is the case). By (118) and (54), j"jkg x kr¯;¯  j"jrL  r/2 so that, by (116), if y 0 2 Dr¯/2 (y1 ), then y 0 C"g x (y 0 ; '(y 0 ; x 0 )) 2 Dr (y0 ). The first requirement in (123) is equivalent to require that r0  r, which implies that if r¯ is defined as the r.h.s. of (108), then r¯  r/2 as required in (110). Next, the first requirement in (114) at the ( jC1)th step of the iteration translates into 16C 2 r jC1 /r j  1, which is satisfied, since, by definition, r jC1 /r j D (1/(2 ))C1  (1/(2 ))2 D 1/(36C 2 ) < 1/(16C 2 ). The second condition in (114), which at the ( j C 1)th step, reads 2j 2j 2CM j r2 jC1 j"j is implied by j"j L j  ı j /(3C) (corresponding to (54)), which, in turn, is easily controlled along the lines explained in note 25. An area-preserving twist mappings of an annulus A D [0; 1]  S1 , (S1 D T 1 ), is a symplectic diffeomorphism f D ( f1 ; f2 ) : (y; x) 2 A ! f (y; x) 2 A, leaving invariant the boundary circles of A and satisfying the twist condition @ y f2 > 0 (i. e., f twists clockwise radial segments). The theory of area preserving maps, which was started by Poincaré (who introduced such maps as section of the dynamics of Hamiltonian systems with two degrees of freedom), is, in a sense, the simplest nontrivial Hamiltonian context. After Poincaré the theory of area-preserving maps became, in itself, a very rich and interesting field of Dynamical Systems leading to very deep and important results due to Herman, Yoccoz, Aubry, Mather, etc; for generalities and references, see, e. g., [33]. It is not necessary to assume that K is real-analytic, but it simplify a little bit the exposition. In our case, we shall see that ` is related to the number  in (66). We

Kolmogorov–Arnold–Moser (KAM) Theory

37

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recall the definition of Hölder norms: If ` D `0 C  with `0 2 ZC and  2 (0; 1), then k f kC ` :D k f kC ` C supj˛jD`0 sup0<jxyj<1 j@˛ f (x)  @˛ f (y)j/jx  yj ; C ` (Rd ) denotes the Banach space of functions with finite C ` norm. To p obtain these new estimates, one can, first replace by and then use the remark in the note 21 with p D 1; clearly the constant  has to be increased by one unit with respect to the constant  appearing in (69). For general references and discussions about Lemma 2 and 3, see, [44] and [65]; an elementary detailed proof can be found, also, in [15]. Proof of Claim M The first step of the induction consists in constructing ˚0 D 0 : this follows from Kolmogorov’s Theorem (i. e., Remark 7–(i) and Remark 11) with  D 1 D 1/2 (assume, for simplicity, that Q is analytic on W 1 and note that j"jkP0 k1  j"jkPkC 0 by the first inequality in (124)). Now, assume that (128) and (129) holds together with C i < 4C p and k@(˚ i  id)k˛ iC1 < ( 2  1) for 0  i  j (C0 D C and Ci are as in (64) for, respectively, K0 :D K and K i ). To determine  jC1 , observe that, by (128), one has H jC1 ı˚ jC1 D (K jC1 C"PjC1 )ı jC1 where Pj :D (P jC1  P j ) ı ˚ j , which is real-analytic on W˛ jC1 ; thus we may apply Kolmogorov’s Theorem to K jC1 C "PjC1 with  D ˛ jC1 and D ˛; in fact, by the second inequality in (124), kPjC1 k˛ jC1  kP j1  P j k X jC1  ckPkC `  `jC1 and the smallness condition (66) becomes j"jD ` jC1 (with D :D c ckPkC ` (4C)b 2/2 ), which is clearly satisfied for j"j < D1 . Thus,  jC1 has been determined and (notice that ˛ 2  jC1 D  jC1 /2 D  jC2 ) k jC1 idk jC2 , @(k jC1  id)k jC2  j"jD jC1 . Let us now check the domain constraint ˚ j : W˛ jC1 ! X  jC1 . By the inductive assumptions and the real-analyticity of ˚ j , one has that, for z 2 W˛ jC1 , jIm˚ j (z)j D jIm(˚ j (z)  k@˚ j k˛ jC1 jImzj  ˚ j (Rez))j  j˚ j (z)  ˚ j (Rez)j  p (1 C k@(˚ i  id)k˛ iC1 )˛ jC1 < 2˛ jC1 D  jC1 so that ˚ j : W˛ jC1 ! X jC1 . The remaining inductive assumptions in (129) with j replaced by j C 1 are easily checked by arguments similar to those used in the induction proof of Claim C above. See, e. g., the Proposition at page 58 of [14] with g j D f j  f j1 . In fact, the lemma applies to the Hamiltonians H j and to the symplectic map  j in (82) in Arnold’s scheme with W j in (81) and taking C D C :D fy 0 D d lim j!1 y j (!) : ! 2 B \ K 1 y (D; )g and y j (!) :D y j is as in (82). A formal "-power series quasi-periodic trajectory, with rationally-independent frequency !, for a nearly-integrable Hamiltonian H(y; x; ") :D K(y) C "P(y; x)

42

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44 45

46

is, by definition, a sequence of functions fz k g :D (fv k g; fu k g), real-analytic on T d and such that P j D! z k D J2d  k (rH( k1 jD0 " z j )) where  k () :D 1 k @ ()j ; compare Remark 1–(ii) above. "D0 k! " In fact, Poincaré was not at all convinced of the convergence of such series: see chapter XIII, no 149, entitled “Divergence des séries de M. Lindstedt”, of his book [49]. Equation (70) guarantees that the map from y in the (d  1)-dimensional manifold fK D Eg to the (d  1)-dimensional real projective space f!1 : !2 :    : !d g RP d1 (where ! i D K y i ) is a diffeomorphism. For a detailed proof of the “iso-energetic KAM Theorem”, see, e. g., [24]. Actually, it is not known if such tori are KAM tori in the sense of the definitions given above! The first example of a nearly-integrable system (with two parameters) exhibiting Arnold’s diffusion(in a certain region of phase space) was given by Arnold in [4]; a theory for “a priori unstable systems” (i. e., the case in which the integrable system carries also a partially hyperbolic structure) has been worked out in [20] and in recent years a lot of literature has been devoted to study the “a priori unstable” case and to to try to attack the general problem (see, e. g., Sect. 6.3.4 of [6] for a discussion and further references). We mention that J. Mather has recently announced a complete proof of the conjecture in a general case [40]. Here, we mention briefly a different and very elementary connection with classical mechanics. To study the spectrum  (L) (L as above with a quasi-periodic potential V (!1 t; : : : ; !n t)) one looks at the equation q¨ D (V (! t)  )q, which is the q-flow of the Hamiltonian  Ht H D H(p; q; I; '; ) :D p2 /2 C [  V (')]q2 /2 where (p; q) 2 R2 and (I; ') 2 Rn  T n w.r.t. the standard form dp ^ dq C dI ^ d' and  is regarded as a parameter. Notice that '˙ D ! so that ' D '0 C ! t and that the (p; q) decouples from the I-flow, which is, then, trivially determined one the (p; q) flow is known. Now, the action-angle variables(J; ) for the harmonic p oscillator p2 /2 C q2 /2 are given by JpD r2 /  and (r; ) are polar coordinates in the (p; q)-plane; p in J  such variables, H takes the form H D !  I C p V (')/  sin2 . Now, if, for example V is small, this Hamiltonian is seen to be a perturbation p of (nC1) harmonic oscillator with frequencies (!; ) and it is remarkable that one can provide a KAM scheme, which preserves the linear-in-action structure of this Hamiltonian and selects p the (Cantor) set of values of the frequency ˛ D  for which the KAM scheme can be carried out so as to conjugate H to a Hamiltonian of

833

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48

49

50

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52

the form !  I C ˛J, proving the existence of (generalized) quasi-periodic eigen-functions. For more details along these lines, see [14]. The value 1052 is about the proton-Sun mass ratio: the mass of the Sun is about 1:991  1030 kg, while the mass of a proton is about 1:6721021 kg, so that (mass of a proton)/(mass of the Sun) ' 8:4  1052 . “Computer-assisted proofs” are mathematical proofs, which use the computers to give rigorous upper and lower bounds on chains of long calculations by means of so-called “interval arithmetic”; see, e. g., Appendix C of [13] and references therein. Simple examples of such orbits are equilibria and periodic orbits: in such cases there are no small-divisor problems and existence was already established by Poincaré by means of the standard Implicit Function Theorem; see [49], Volume I, chapter III. Typically,  may indicate an initial datum y0 and y the distance from such point or (equivalently, if the system is non-degenerate in the classical Kolmogorov sense)  ! !() might be simply the identity, which amounts to consider the unperturbed frequencies as parameter; the approach followed here is that in [51], where, most interestingly, m is allowed to be 1. I. e., a map ˚ : X ! X for which 90 < ˛ < 1 such that d(˚(u); ˚ (v))  ˛d(u; v), 8u; v 2 X, d(; ) denoting the metric on X; for generalities on metric spaces, see, e. g., [36]. ˚ j D ˚ ı    ı ˚ j-times. In fact, let u j :D ˚ j (u0 ) and notice that, for each j  1 d(u jC1 ; u j )  ˛d(u j ; u j1 )  ˛ j d(u1 ; u0 ) D: ˛ j ˇ, so that, for each P h1 j; h  1, d(u jCh ; u j )  iD0 d(u jCiC1 ; u jCi )  P h1 jCi j ˇ/(1  ˛), showing that fu g is ˛ ˇ  ˛ j iD0 a Cauchy sequence. Uniqueness is obvious.

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Books and Reviews Arnold VI, Novikov SP (eds) (1985) Dynamical Systems IV, Symplectic Geometry and its Applications. In: Encyclopaedia of Mathematical Sciences, vol 4, Volume package Dynamical Systems. Original Russian edition published by VINITI, Moscow. 2nd expanded and rev. edn., 2001 Berretti A, Celletti A, Chierchia L, Falcolini C (1992) Natural Boundaries for Area-Preserving Twist Maps. J Stat Phys 66:1613– 1630 Bost JB (1984/1985) Tores invariants des systèmes dynamiques hamiltoniens. Séminaire Bourbaki, exp. 639 Bourgain J (1997) On Melnikov’s persistency problem. Math Res Lett 4(4):445–458 Broer HW, Sevryuk MB KAM theory: Quasi-periodicity in dynamical systems. In: Handbook of Dynamical Systems. Elsevier, to appear Celletti A, Chierchia L (2006) KAM tori for N-body problems: a brief history. Celest Mech Dynam Astronom 95(1–4):117–139 Chierchia L, Falcolini C (1996) Compensations in small divisor problems. Comm Math Phys 175:135–160 Gallavotti G (1983) The Elements of Mechanics. Springer, New York Giorgilli A, Locatelli U (1997) On classical series expansion for quasiperiodic motions. MPEJ 3(5):1–25 Herman M (1986) Sur les courbes invariantes pour les difféomorphismes de l’anneau. Astérisque 2:248 Herman M (1987) Recent results and some open questions on Siegel’s linearization theorems of germs of complex analytic diffeomorphisms of Cn near a fixed point. In: VIIIth International Congress on Mathematical Physics, Marseille, 1986. World Sci. Publishing, Singapore, pp 138–184 Herman M (1989) Inégalités a priori pour des tores Lagrangiens invariants par des difféomorphismes symplectiques. Publ Math IHÈS 70:47–101 Hofer H, Zehnder E (1995) Symplectic invariants and Hamiltonian dynamics. The Floer memorial volume. Progr. Math., 133. Birkhäuser, Basel, pp 525–544 Kappeler T, Pöschel J (2003) KdV & KAM. Springer, Berlin, XIII,279 pp Koch H (1999) A renormalization group for Hamiltonians, with applications to KAM tori. Ergod Theor Dyn Syst 19:475–521

Kuksin SB (1993) Nearly integrable infinite dimensional Hamiltonian systems. Lecture Notes in Mathematics, vol 1556. Springer, Berlin Lanford OE III (1984) Computer assisted proofs in analysis. Phys A 124:465–470 Laskar J (1996) Large scale chaos and marginal stability in the solar system. Celest Mech Dyn Astronom 64:115–162 de la Llave R (2001) A tutorial on KAM theory. Smooth ergodic theory and its applications. In: Proc. Sympos. Pure Math., vol 69, Seattle, WA, 1999. Amer Math Soc, Providence, pp 175– 292 de la Llave R, González A, Jorba À, Villanueva J (2005) KAM theory without action-angle variables. Nonlinearity 18(2):855– 895 de la Llave R, Rana D (1990) Accurate strategies for small divisor problems. Bull Amer Math Soc (N.S.) 22:(1)85–90 Lazutkin VF (1993) KAM theory and semiclassical approximations to eigenfunctions. Springer, Berlin MacKay RS (1985) Transition to chaos for area-preserving maps. Lect Notes Phys 247:390–454 Meyer KR, Schmidt D (eds) (1991) Computer Aided Proofs in Analysis. Springer, Berlin Moser J (1973) Stable and random motions in dynamical systems. With special emphasis on celestial mechanics. Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J Annals of Mathematics Studies, No. 77. Princeton University Press, Princeton; University of Tokyo Press, Tokyo Pöschel J (2001) A lecture on the classical KAM theorem. Smooth ergodic theory and its applications. In: Proc Sympos Pure Math, vol 69 Seattle,WA, 1999. Amer Math Soc, Providence, pp 707– 732 Salamon D, Zehnder E (1989) KAM theory in configuration space. Comment Math Helvetici 64:84–132 Sevryuk MB (1995) KAM-stable Hamiltonians. J Dyn Control Syst 1(3):351–366 Sevryuk MB (1999) The lack-of-parameters problem in the KAM theory revisited, Hamiltonian systems with three or more degrees of freedom (S’Agaró, 1995). NATO Adv Sci Inst Ser C Math Phys Sci, vol 533. Kluwer, Dordrecht, pp 568–572 Siegel CL, Moser JK (1971) Lectures on Celestial Mechanics. Springer, Berlin Weinstein A (1979) Lectures on symplectic manifolds, 2nd edn., CBMS Regional Conference Series in Mathematics, vol 29. American Mathematical Society, Providence Xu J, You J, Qiu Q (1997) Invariant tori for nearly integrable Hamiltonian systems with degeneracy. Math. Z 226(3):375–387 Yoccoz J-C (1992) Travaux de Herman sur les tores invariants. Séminaire Bourbaki, 1991/92, Exp. 754. Astérisque 206:311–344 Yoccoz J-C (1995) Théorème de Siegel, pôlynomes quadratiques et nombres de Brjuno. Astérisque 231:3–88

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the YU-JIE REN1,2,3 1 Department of Mathematics, Shantou University, Shantou, People’s Republic of China 2 Department of Mathematics and Physics, Dalian Polytechnic University, Dalian, People’s Republic of China 3 Beijing Institute of Applied Physics and Computational Mathematics, Beijing, People’s Republic of China

Article Outline Glossary Definition of the Subject Introduction The Generalized Hyperbolic Function–Bäcklund Transformation Method and Its Application in the (2 + 1)-Dimensional KdV Equation The Generalized F-expansion Method and Its Application in Another (2 + 1)-Dimensional KdV Equation The Generalized Algebra Method and Its Application in (1 + 1)-Dimensional Generalized Variable – Coefficient KdV Equation A New Exp-N Solitary-Like Method and Its Application in the (1 + 1)-Dimensional Generalized KdV Equation The Exp-Bäcklund Transformation Method and Its Application in (1 + 1)-Dimensional KdV Equation Future Directions Acknowledgment Bibliography

Glossary Analytical method for solving an equation The method for obtaining the exact solutions of an equation. Solitons The special solitary waves which retain their original shapes and speeds after collision and exhibit only a small overall phase shift. " ˙1 t the time (x; y) Cartesian coordinates of a point R @1 an indefinite integrate operator dx. x

Definition of the Subject This paper presents the analytical methods for obtaining the exact solutions of the Korteweg–de Vries Equation (KdV equation). The KdV equation and its exact solutions can describe and explain many physical problems. In addition, it is a typical, relatively simple and classical equation among the many nonlinear equations in physics. Much of the literature of nonlinear equation theory customarily uses solving soliton solutions of the KdV equation as an example to introduce the nonlinear theory, method and character of soliton solutions. During the last five decades, the construction of exact solution for a wide class of nonlinear equations has been an exciting and extremely active area of research. This includes the most famous nonlinear example of the KdV equation. In the family of the KdV equations, a well known KdV equation is expressed in its simplest form as [1,2,3,4]: u t (x; t) C ˛u(x; t)u x (x; t) C ˇu x x x (x; t) D 0 :

(1)

where the coefficient of the nonlinear term ˛ and the coefficient of the dispersive term ˇ are independent of x and t. The KdV equation was derived in 1885 by the two scientists, Korteweg and de Vries [1], to describe long wave propagation on shallow water. Its importance comes from its many applications in weakly nonlinear and weakly dispersive physical systems and its interesting mathematical properties. It can be considered as a paradigm in nonlinear science. The equation is so powerful it has been used to simulate the red spots on Jupiter. Moreover, it describes non-linear waves in rotating fluids [5], the giant ocean waves known as Tsunami, and other aspects of solitary waves in plasma. The giant internal waves in the interior of the ocean arising from temperature differences which may destroy marine vessels can be also described by the powerful KdV equation. It possesses infinitely many generalized symmetries and infinitely many integrals of motion [6], and can be viewed as a completely integrable Hamiltonian system and solved by the inverse scattering transform. It also provides resources for studying integrability of nonlinear differential (and difference) equations, serving as a typical model of integrable equations. More remarkable is that various physically important solutions to the KdV equation can be presented explicitly in a simple way, among which are solitons, rational solutions, positons and negatons. The KdV equation has been widely investigated and developed continually in recent decades. Several generalizations of the KdV equation have found applications in many areas, including quantum field theory, plasma

837

838

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

physics, solid-state physics, liquid-gas bubble mixtures, and anharmonic crystals. For instance, the classical generalized Korteweg–de Vries equation (gKdV) has the following form [7,8,9] u t (x; t) C ˛u p (x; t)u x (x; t) C ˇu x x x (x; t) D 0 :

(2)

where the coefficient of the nonlinear term ˛ and the coefficient of the dispersive term ˇ are independent of x and t. The more generalized form of the KdV equation is defined by parameters (l; p) as follows [10,11,12,13,14] u t (x; t) D u l 2 (x; t)u x (x; t) C ˛[2u p (x; t)u x x x (x; t) C 4pu p1 (x; t)u x (x; t)u x x (x; t) C p(p  1)u p2 (x; t)u3x (x; t)] :

(3)

Propagation of weakly nonlinear long waves in an inhomogeneous waveguide is governed by a variable-coefficient KdV equation of the form [15] u t (x; t) C 6u(x; t)u x (x; t) C B(t)u x x x (x; t) D 0 : (4) where u(x; t) is the wave amplitude, t the propagation coordinate, x the temporal variable and B(t) is the local dispersion coefficient. The applicability of the variable-coefficient KdV equation (4) arises in many areas of physics as, for example, the description of the propagation of gravity-capillary and interfacial-capillary waves, internal waves and Rossby waves [15]. In order to study the propagation of weakly nonlinear, weakly dispersive waves in the inhomogeneous media, Eq. (4) is rewritten as follows [16] u t (x; t)C 6A(t)u(x; t)u x (x; t)C B(t)u x x x (x; t) D 0 : (5) which includes a variable nonlinearity coefficient A(t). In studying long waves, tides, solitary waves, and related phenomena, one is led to an equation of the form [17] u t (x; t) C (m C 1)(m C 2)u m (x; t)u x (x; t) C u x x x (x; t) D f (x; t) :

(6)

where f (x; t) is a given function and m D 1; 2; : : :, with u(x; t); u x (x; t), u x x (x; t) ! 0 as jxj ! C1. This equation is referred to as a generalized Korteweg–de Vries equation in [15]. The extended Korteweg–de Vries (eKdV) equation [18] u t (x; t) C ˛u(x; t)u x (x; t) C ˇu2 (x; t)u x (x; t) C ıu x x x (x; t) D 0 :

(7)

incorporates both quadratic and cubic nonlinearities, and serves as a useful model in the propagation of long waves in physical oceanography. A system of two coupled Korteweg–de Vries equations in [4] is given by the following equation: u i t (x; t) C

X

Ahk i u h (x; t)u kx (x; t)

h;k

C

X

Aki u kx x x (x; t) D 0 ; i; k; h D 1; 2 :

(8)

k

The (2 C 1)-dimensional KdV equation is presented by the following equation [19]: 8 ˆ
Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

the exp-function method [31], the Adomian Pade approximation [32], the Jacobi elliptic function expansion method [33], the F-expansion method [34], the Weierstrass semi-rational expansion method [35], the unified rational expansion method [36], the algebraic method [37, 50,51], the auxiliary equation method [38], and so on. In recent years, based on the ideas of unification methods, algorithm realization and mechanization for solving NLPDEs, we have improved and presented some analytical methods for obtaining exact solutions of NLPDEs, such as the generalized hyperbolic function – the Bäcklund transformation method [39], the method for constructing higher order and higher dimension Bäcklund transformation [40], the generalized hyperbolic function – the Riccati method [40], the generalized F-expansion method [41, 42], the extended generalized algebraic method [43,44], the Exp-Bäcklund method [45,46], the Exp-N soliton-like method [47,48], the extended sine-cosine method [52], and so on. The present article is motivated by the desire to introduce and make use of our works published in [39, 40,41,42,43,44,45,46,47,48] to construct more general exact solutions, which contain not only the results obtained by using the methods [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15, 16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34, 35,36,37,38,49,50,51] and Ablowitz and Clarkson 1991, Dodd et al. 1984, Drazin and Johnson 1989, Fan 2004, Guo 1996, Liu and Liu 2000 Miura 1976, and Miurs 1978, but also a series of new and more general exact solutions. For illustration, we apply our analytical methods above to the different types of the KdV equations, and successfully obtain many new and more general exact solutions. The remainder of this article is organized as follows. In Sect. “The Generalized Hyperbolic Function– Bäcklund Transformation Method and Its Application in the (2 + 1)-Dimensional KdV Equation”, in order to construct the unification solutions of the hyperbolic function solution and exponential function solution of NLPDEs, we first find a new theory of generalized hyperbolic functions which includes two new definitions of generalized hyperbolic functions and generalized hyperbolic function transformations and their properties. Then we present a new higher order and higher dimension Bäcklund transformation method and a new generalized hyperbolic function – Bäcklund transformation method. The validity of the methods are tested by their application in the (2 C 1)-dimensional KdV equation (9). In Sect. “The Generalized F-expansion Method and Its Application in Another (2 + 1)-Dimensional KdV Equation”, we present a new and more general formal transformation and a new generalized F-expansion method and apply them to another (2C1)-dimensional KdV equation (10). In Sect. “The Gen-

eralized Algebra Method and Its Application in (1 + 1)-Dimensional Generalized Variable – Coefficient KdV Equation”, in order to develop the algebraic method [50,51] for constructing the traveling wave solutions of NLPDEs, we first present two new and general transformations, two new theorems and their proofs, by using Maple, and a new mechanization method to find the exact solutions of a first-order nonlinear ordinary differential equations with any degree. The validity of the method is tested by its application in the first-order nonlinear ordinary differential equation with six degrees. Next, we present a new and more general transformation and a new generalized algebra method and apply them to the (1 C 1)-dimensional generalized variable – coefficient KdV equation (5). In Sect. “A New Exp-N Solitary-like Method and Its Application in the (1 + 1)-Dimensional Generalized KdV Equation”, in order to develop the Exp-function method [31], we present two new generic transformations, a new ExpN solitary-like method, and its algorithm. In addition, we apply our methods to construct new exact solutions of the (1 C 1)-dimensional classical generalized KdV (gKdV) equation (2); In Sect. “The the Exp-Bäcklund Transformation Method and Its Application in (1 + 1)-Dimensional KdV Equation”, a new Exp-Bäcklund transformation method and its algorithm is presented to find more exact solutions of NLPDEs. We choose the (1 C 1)-dimensional KdV equation (1) to illustrate the effectiveness and convenience of our algorithm. As a result, we obtain more general exact solutions including non-traveling wave solutions and traveling wave solutions. In addition, the longterm behavior of the non-traveling wave solutions and traveling wave solutions are illustrated by some Figures. In Sect. “Future Directions”, future directions for Solving NLPDEs are given. The Generalized Hyperbolic Function–Bäcklund Transformation Method and Its Application in the (2 + 1)-Dimensional KdV Equation In this section, in order to construct the unification solutions of the hyperbolic function solution and the exponential function solution of NLPDEs, we establish a new theory of generalized hyperbolic functions, which includes two new definitions of generalized hyperbolic functions and generalized hyperbolic function transformations and their properties that we first presented in [39,40]. Then we present a new higher order and higher dimension Bäcklund transformation method and a new generalized hyperbolic function-Bäcklund transformation method that we first presented in [39,40]. With the aid of symbolic computation, we choose the (2 C 1)-dimensional Korteweg–

839

840

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

de Vries equations to illustrate the validity and advantages of the methods. As a result, many new and more general exact non-traveling waves are obtained. The Definition and Properties of Generalized Hyperbolic Functions

Definition 1 Suppose that  is an independent variable, p; q and k are constants. The generalized hyperbolic sine function is sinh pq k () D

 2

qek

(12)

generalized hyperbolic cosine function is cosh pq k () D

pe k C qek 2

(13)

generalized hyperbolic tangent function is tanh pq k () D

pe k  qek pe k C qek

(14)

pe k C qek pe k  qek

tanh pq k () D coth pq k () D tanh pq k () D

1 ; cosh pq k () 1 ; sinh pq k () sinh pq k () ; cosh pq k () cosh pq k () ; sinh pq k () 1 ; coth pq k ()

2 pe k C qek

pe k

2  qek

(22) (23) (24) (25)

cosh pq k () D cosh q pk () ;

(27)

tanh pq k () D  tanh q pk () ;

(28)

coth pq k () D  coth q pk () ;

(29)

sech pq k () D sechq pk () ;

(30)

csch pq k () D  csch q pk () ;

(31)

1 [sinhq pk ()  cosh pq k () pq  cosh pq k ()  sinh q pk ()] ;

(32)

(15) sinh p2 q 2 k ( C ) D sinh q pk ()  cosh pq k () C cosh pq k ()  sinh q pk () ; (16)

C sinh pq k ()  sinh q pk () ; (17)

the above six kinds of functions are said generalized hyperbolic functions.

(33)

cosh p2 q 2 k ( C ) D cosh q pk ()  cosh pq k ()

generalized hyperbolic cosecant function is csch pq k () D

(21)

(26)

generalized hyperbolic secant function is sech pq k () D

(20)

sinh pq k () D  sinhq pk () ;

sinh11k (  ) D

generalized hyperbolic cotangent function is coth pq k () D

sech pq k () D csch pq k () D

In [39], we first defined the following new functions which named generalized hyperbolic functions and studied the properties of these functions for constructing new exact solutions of NLPDEs.

pe k

1  coth2pq k () D pq  csch2pq k () ;

cosh11k (  ) D

(34)

1 [cosh q pk ()  cosh pq k () pq  sinh pq k ()  sinh q pk ()] ;

(35)

sinh q 2 p2 k (2) D 2 sinh pq k ()  cosh pq k () ;

(36)

Thus we can prove the following theory of generalized hyperbolic functions on the basis of Definition 1.

cosh p2 q 2 k (2) D sinh2q pk () C cosh2q pk () :

(37)

Theorem 1 The generalized hyperbolic functions satisfy the following relations:

sinh p2 q 2 k ( C ) tanhq pk () C tanhq pk () D ; 1  tanhq pk () tanhq pk () p  q  cosh11k (  )

cosh2pq k ()  sinh2pq k () D pq ;

(18)

(38)

1  tanh2pq k () D pq  sech2pq k () ;

(19)

tanhq pk ()  tanhq pk () p  q  sinh11k (  ) D : (39) 1 C tanhq pk () tanhq pk () cosh p2 q 2 k ( C )

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

cosh(); tanh(); coth(); sech() and csch(), respectively. In addition, when p D 0 or q D 0 in (12)–(17), sinh pq k (); cosh pq k (); sech pq k (); csch pq k (), tanh pq k () and coth pq k () degenerate as exponential functions 1 1 k k ; 2pek ; ˙2qe k and ˙1, respectively. 2 pe ; ˙ 2 qe

For simplicity, only a sample of these are proved here. Proof By (18) and (21), we obtain 1  tanh2pq k () D 1  D D

sinh2pq k () cosh2pq k ()

cosh2pq k ()  sinh2pq k ()

A New Higher Order and Higher Dimension Bäcklund Transformation Method to Construct an Auto–Bäcklund Transformation of the (2 + 1)-Dimensional KdV Equation

cosh2pq k () pq cosh2pq k ()

D pq  sech2pq k () :

Similarly, we can prove other conclusions in Theorem 1. Theorem 2 The derivative formulae of generalized hyperbolic functions are as follows: d(sinh pq k ()) D k  cosh pq k () ; d d(cosh pq k ()) D k  sinh pq k () ; d d(tanh pq k ()) D kpq  sech2pq k () ; d d(coth pq k ()) D kpq  csch2pq k () ; d d(sech pq k ()) D k  sech pq k ()  tanh pq k () ; d d(csch pq k ()) D k  csch pq k ()  coth pq k () : d

(40)

D

D



sinh pq k () cosh pq k ()

8 ˆ
(41) (42)

(46) (43) (44)

The auto-Bäcklund transformations of Eqs. (46) have the following forms: (

(45)

Proof According to (40) and (41), we can get d(tanh pq k ()) D d

In this section, we obtain an auto-Bäcklund transformation of the following (2C1)-dimensional KdV equation by making use of the method for constructing higher order and higher dimension Bäcklund transformations which we presented in [39,40] via symbolic computation. The (2 C 1)-dimensional KdV equations [19] take the form

0

(sinh pq k ())0 cosh pq k ()  sinh pq k ()(cosh pq k ())0 coshpqk 2 () k  cosh pq k ()  cosh pq k ()  sinh pq k () (k  sinh pq k ()) cosh2pq k ()

D kpq  sech2pq k () :  Similarly, we can prove other differential coefficient formulae in Theorem 2. Remark 1 We see that when p D 1; q D 1; k D 1 in (12)– (17), new generalized hyperbolic functions sinh pq k (); cosh pq k (); tanh pq k (); coth pq k (); sech pq k () and csch pq k () degenerate as hyperbolic functions sinh();

u(x; y; t) D v(x; y; t) D

Pm 1

j D0 u j 1 (x;

y; t) f j 1 m1 (x; y; t) ;

j 2 D0 v j 2 (x;

y; t) f j 2 m2 (x; y; t) ;

Pm12

(47)

where f (x; y; t); u j 1 (x; y; t); j1 D 0; 1; 2; : : : ; m1  1 and v j 2 (x; y; t); j2 D 0; 1; 2; : : : ; m2  1 are all differential functions to be determined later, and u m1 (x; y; t); v m2 (x; y; t) are the trivial seed solutions of Eqs. (46). By balancing the highest order linear term and nonlinear terms in Eqs. (46), we get m1 D 2 and m2 D 2, and (47) has the following formal 8 ˆ u(x; y; t) D u0 (x; y; t) f 2 (x; y; t) ˆ ˆ ˆ < Cu1 (x; y; t) f 1 (x; y; t) C u2 (x; y; t) ; ˆv(x; y; t) D v0 (x; y; t) f 2 (x; y; t) ˆ ˆ ˆ : Cv (x; y; t) f 1 (x; y; t) C v (x; y; t) ; 1

2

(48) We take the trivial seed solutions as u2 D u2 (x; y; t); v2 D v2 (x; y; t) :

(49)

With the aid of Maple symbolic computation software, substituting (48) and (49) into (46), and collecting

841

842

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

all terms with f i (x; y; t); i D 0; 1; 2; : : :, we obtain 



   24u0 f x3 C 12u0 v0 f x f 5 C 12u1 f x3 C 36 f x2 f y f x 2  C 18v1 f x2 f y C 24 f x3 f x y f 4 C    D 0 ; (50)     2u0 f x C 2v0 f y f 3 C u0x  u1 f x C v1 f y  v0y   f 2 C u1x  v1y f 1 C u2x  v2y D 0 ; (51)

Setting the coefficient of f 5 in (50) and f 3 in (51) to be zero, we obtain a differential equation  24u0 f x3 C 12u0 v0 f x D 0 ;

(52)

 2u0 f x C 2v0 f y D 0 ;

(53)

v0 D 2 f x2 :

(54)

Setting the coefficients of f 4 in (50) and f 2 in (51) to be zero, we obtain a differential equation 12u1 f x3 C 36 f x2 f y f x 2 C 18v1 f x2 f y C 24 f x3 f x y D 0 ; (55) u0x  u1 f x C v1 f y  v0y D 0 ;

We first give the definition of a generalized hyperbolic function transformation [40] as follows: Definition 2 If a transformation includes the generalized hyperbolic functions, then the transformation is defined to be a generalized hyperbolic function transformation. In this section, we will use the generalized hyperbolic function–Bäcklund transformation method to seek new exact solutions of Eqs. (46). We take f (x; y; t) in (48), (49), (54), (57) and (59) as being of a new form which is the following generalized hyperbolic function transformation f (x; y; t) D H(t)

which has the solution u0 D 2 f x f y ;

The Generalized Hyperbolic Function–Bäcklund Transformation Method and Its Application in the (2 + 1)-Dimensional KdV Equation

C

n X

K i (t)  F i ( i (x; y; t))  G i ( i (x; y; t)) ;

(60)

iD1

where F i ( i ) and G i ( i ) may take any two generalized hyperbolic functions among sinh pq k (); cosh pq k (); tanh pq k (); coth pq k (); sech pq k ()and csch pq k (). And then (60) has twenty–one types of general forms. For example

(56)

from which we can get the following expression:

f (x; y; t) D H(t) C

n X

K i (t)  sech p 1i q 1i k 1i ( i (x; y; t))

iD1

u1 D 2 f yx ;

v1 D 2 f x x :

(57)

By (48), (49), (54) and (57), we obtain an auto-Bäcklund transformation of Eqs. (46) u D 2@ yx ln( f (x; y; t)) C u2 (x; y; t) ; v D 2@x x ln( f (x; y; t)) C v2 (x; y; t) ;

(58)

where u2 (x; y; t); v2 (x; y; t) are the trivial seed solutions of Eqs. (46), f D f (x; y; t) satisfies the Eqs. (52), (53), (55), (56) and the following equations

 tanh p 2i q 2i k 2i ( i (x; y; t)) ; f (x; y; t) D H(t)C

n X

(61)

K i (t)csch2p i q i k i ( i (x; y; t)); (62)

iD1

We see that when p D 1; q D 1; k D 1 in (12)–(12), new generalized hyperbolic function sinh pq k (); cosh pq k (); tanh pq k (); coth pq k (); sech pq k () and csch pq k () degenerate as hyperbolic functions sinh(); cosh(); tanh(); coth(); sech() and csch() respectively. Therefore (3.49) includes twenty-one types of hyperbolic function forms. For example

 u0 v1x  u1 v0x C 2u1 v1 f x C 2u2 v0 f x  2/3u0 f t  2/3u0 f x 3 C 2u1 f x f x 2  2u0x f x 2

f (x; y; t) D H(t) C

C 2u1x f x2  2u0x 2 f x  v0 u1x  v1 u0x C 2v2 u0 f x D 0 ; u0x 3  u1 f x 3  3u0 v2x  3u1 v1x  3u2 v0x C 3u2 v1 f x  3u1x f x 2  u1 f t  3v0 u2x  3v1 u1x  3v2 u0x C 3v2 u1 f x C u0t  3u1x 2 f x D 0 :

n X

K i (t)  sinh( i (x; y; t))

iD1

(59)

 sech( i (x; y; t)) ;

(63)

and thirty-six types of the omnibus forms of generalized hyperbolic functions and hyperbolic functions. For example

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

f (x; y; t) D H(t) C

n X

Pn

iD1

 csch p 2i q 2i k 2i ( i (x; y; t)) ;

(64)

In addition, when p D 0 or q D 0 in (12)–(17), sinh pq k (); cosh pq k (); sech pq k (); csch pq k (); tanh pq k () and coth pq k () degenerate as exponential functions 1 1 k k ; 2pek ; ˙2qe k and ˙1 respectively. 2 pe ; ˙ 2 qe For example, we take F i ( i (x; y; t)) D tanh p 1i q 1i k 1i ( i ); G i ( i (x; y; t)) D sinh p 2i q 2i k 2i ( i ). Then (60) has the new form n X K i (t) tanh p 1 q 1 k 1 ( i ) f (x; y; t) D H(t) C iD1

 sech p 2 q 2 k 2 ( i ) ;

(65)

where  i D ˛ i (t)x Cˇ i (t)yCr i (t);  i D  i (t)x C i (t)yC l i (t); (i D 1; 2; : : : ; n), and H(t); K i (t); ˛ i (t), ˇ i (t); r i (t);  i (t);  i (t) and l i (t) are the functions of t, p1i ; q1i ; k1i ; P p2i ; q2i ; k2i are the constants, and niD1 K 2i (t) ¤ 0(i D 0; 1; 2; : : : ; n). We take the initial solution of Eqs. (46) as u2 (x; y; t) D u2 (t), and v2 (x; y; t) D v2 (t) in (58) for convenience. So based on the Bäcklund transformation (58), the generalized hyperbolic function transformation (65), and d 2 (tanh pq k ()) D 2k 2 pqsech2pq k ()tanh pq k (); (66) d 2 d 2 (sech pq k ()) D k 2 sech pq k ()(2 tanh2pq k ()1); (67) d 2 we can get the following new multi-soliton-like solution which we call a generalized hyperbolic function solution of Eq. (46): Pn

p1 q1 k1 k2 K i (t) i (t)˛ i (t) tanh p 2 q 2 k 2 ( i ) sech2p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i ) Pn uD H(t) C iD1 K i (t) tanh p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i ) 2

iD1

Pn

k22 K i (t) i (t) i (t) tanh p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i )(1  2 tanh2p 2 q 2 k 2 ( i )) P C H(t) C niD1 K i (t) tanh p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i ) 2

iD1

Pn

p1 q1 k1 k2 K i (t)ˇ i (t) i (t) sech2p 1 q 1 k 1 ( i ) tanh p 2 q 2 k 2 ( i ) sech p 2 q 2 k 2 ( i ) P C H(t) C niD1 K i (t) tanh p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i ) 2

Pn

iD1

K i (t)(p1 q1 k1 ˇ i (t) sech2p 1 q 1 k 1 ( i )  k2  i (t) tanh p 1 q 1 k 1 ( i ) tanh p 2 q 2 k 2 ( i )) sech p 2 q 2 k 2 ( i ) P H(t) C niD1 K i (t) tanh p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i )

2

iD1

K i (t)(p1 q1 k1 ˛ i (t) sech2p 1 q 1 k 1 ( i )  k2  i (t) tanh p 1 q 1 k 1 ( i ) tanh p 2 q 2 k 2 ( i )) sech p 2 q 2 k 2 ( i ) P H(t) C niD1 K i (t) tanh p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i ) iD1

K i (t)  sech( i (x; y; t))

C u2 ;

(68) Pn

2 iD1 4p1 q1 k1 ˛ i (t)K i (t) sech p 1 q 1 k 1 ( i )

sech p 2 q 2 k 2 ( i )(k1 ˛ i (t) tanh p 1 q 1 k 1 ( i ) Ck2  i (t) tanh p 2 q 2 k 2 ( i )) Pn vD H(t) C iD1 K i (t) tanh p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i ) Pn

K i (t)k22 ( i (t))2 tanh p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i )(1  2 tanh2p 2 q 2 k 2 ( i )) P C H(t) C niD1 K i (t) tanh p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i )  Pn 2 iD1 K i (t) sech p 2 q 2 k 2 ( i )(p1 q1 k1 ˛ i (t) sech p 1 q 1 k 1 2 ( i )  k2  i (t) tanh p 1 q 1 k 1 ( i ) tanh p 2 q 2 k 2 ( i ))  2 P H(t) C niD1 K i (t) tanh p 1 q 1 k 1 ( i ) sech p 2 q 2 k 2 ( i ) 2

iD1

C v2 ;

(69)

where  i D ˛ i (t)x Cˇ i (t)yCr i (t);  i D  i (t)x C i (t)yC l i (t). For example, when we take n D 2; H; K i ; p i ; q i ; k i ; ˛ i ; ˇ i and r i ; i D 1; 2 are all constants, then (65) becomes f (x; y; t) D H C K1 tanh p 1 q 1 k 1 (˛1 x C ˇ1 y C r1 )  sech p 1 q 1 k 1 (˛1 x C ˇ1 y C r1 ) C K2 tanh p 2 q 2 k 2  (˛2 x C ˇ2 y C r2 ) sech p 2 q 2 k 2 (˛2 x C ˇ2 y C r2 ) ; (70) With the help of symbolic computation, substituting (70) and u2 (x; y; t) D u2 (t); v2 (x; y; t) D v2 (t) into (52), (53), (55), (56) and (59), we can get H; K i ; p i ; q i ; k i ; ˛ i ; ˇ i and r i ; i D 1; 2 respectively as follows: Case 1 u2 D C1 ; v2 D

C1 (ˇ2 2 ˛1 2 C ˇ1 2 ˛2 2 C ˇ1 ˛2 ˇ2 ˛1 ) ; 3ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )

p 1 D p 1 ; p 2 D p 2 ; q1 D 0 ; q2 D 0 ; p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (2ˇ1 ˛2 C ˇ2 ˛1 ) or k2 D " ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )˛2 p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (2ˇ1 ˛2 C ˇ2 ˛1 ) ; k2 D " ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )˛2 p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (ˇ1 ˛2 C 2ˇ2 ˛1 ) ; k1 D " (ˇ1 ˇ2 2 ˛1 C ˇ1 2 ˇ2 ˛2 )˛1

843

844

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

Case 5

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; K1 D K1 ; K2 D K2 :

(71)

Case 2 p 2 D p 2 ; u2 D C1 ;

p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (2ˇ1 ˛2 C ˇ2 ˛1 ) k2 D " or ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )˛2 p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (2ˇ1 ˛2 C ˇ2 ˛1 ) ; k2 D " ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )˛2 C1 (ˇ2 2 ˛1 2 C ˇ1 2 ˛2 2 C ˇ1 ˛2 ˇ2 ˛1 ) ; 3ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 ) p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (ˇ1 ˛2 C 2ˇ2 ˛1 ) ; k1 D " (ˇ1 ˇ2 2 ˛1 C ˇ1 2 ˇ2 ˛2 )˛1

v2 D

C1 (ˇ2 2 ˛1 2 C ˇ1 2 ˛2 2 C ˇ1 ˛2 ˇ2 ˛1 ) v2 D ; 3ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 ) p 1 D p 1 ; q1 D 0 ; p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (ˇ1 ˛2 C 2ˇ2 ˛1 ) ; k1 D " (ˇ1 ˇ2 2 ˛1 C ˇ1 2 ˇ2 ˛2 )˛1

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

q2 D q2 ;

q2 D 0 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

k2 D 0 ; H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

u2 D C1 ;

K1 D K1 ; K2 D K2 :

p1 D 0 ;

q1 D q1 ;

p2 D p2 ;

K1 D K1 ; K2 D K2 : (72)

Case 3

(75) Case 6

p 2 D 0 ; u2 D C1 ;

p1 D 0 ;

C1 (ˇ2 2 ˛1 2 C ˇ1 2 ˛2 2 C ˇ1 ˛2 ˇ2 ˛1 ) v2 D ; 3ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )

u2 D C1 ;

p 1 D p 1 ; q1 D 0 ; K 1 D K 1 ; K 2 D K 2 ; p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (ˇ1 ˛2 C 2ˇ2 ˛1 ) ; k1 D " (ˇ1 ˇ2 2 ˛1 C ˇ1 2 ˇ2 ˛2 )˛1

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

q2 D q2 ;

p2 D p2 ;

v2 D v2 ;

q1 D 0 ; q2 D q2 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ; K1 D K1 ; K2 D K2 :

k2 D 0 ; H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 : (73) Case 4

p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (2ˇ1 ˛2 C ˇ2 ˛1 ) or k2 D " ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )˛2 p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (2ˇ1 ˛2 C ˇ2 ˛1 ) ; k2 D " ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )˛2 p2 D 0 ;

k2 D k2 ; k1 D k1 ;

u2 D C1 ;

(76)

Case 7 p 2 D 0 ; q2 D 0 ; p 1 D 0 ; q1 D q1 ; v2 D C1 ; p p p 3C1 3C1 3C1 k2 D " or k2 D " ; k1 D " ; ˛2 ˛2 ˛1 u2 D 0 ; H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ; ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; K1 D K1 ; K2 D K2 :

(77)

Case 8 p 3ˇ1 (C2 ˇ1 C C1 ˛1 ) k1 D " or ˇ1 ˛1

C1 (ˇ2 2 ˛1 2 C ˇ1 2 ˛2 2 C ˇ1 ˛2 ˇ2 ˛1 ) v2 D ; 3ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )

v2 D C 2 ; p 2 D 0 ; p 3ˇ1 (C2 ˇ1 C C1 ˛1 ) ; q1 D q1 ; u2 D C1 ; k1 D " ˇ1 ˛1 p 3ˇ2 (C2 ˇ2 C C1 ˛2 ) ; q2 D 0 ; p 1 D 0 ; k2 D " ˇ2 ˛2

p 1 D p 1 ; q1 D 0 ; p ˇ1 ˇ2 (ˇ1 ˛2 C ˇ2 ˛1 )C1 (ˇ1 ˛2 C 2ˇ2 ˛1 ) ; k1 D " (ˇ1 ˇ2 2 ˛1 C ˇ1 2 ˇ2 ˛2 )˛1 q2 D q2 ; H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

K1 D K1 ; K2 D K2 : (74)

K1 D K1 ; K2 D K2 :

(78)

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Case 9 v2 D C 2 ;

Case 14

p 2 D 0 ; p 1 D 0 ; q1 D q1 ; p 3ˇ2 (C2 ˇ2 C C1 ˛2 ) ; k1 D 0 ; k2 D " ˇ2 ˛2

p2 D 0 ;

v2 D "

p1 D 0 ;

q1 D q1 ;

u2 D C1 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

q2 D q2 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

C1 ˛2 ; ˇ2

q1 D q1 ;

(84)

(79) Case 15

p1 D 0 ;

u2 D C1 ;

K1 D K1 ; K2 D K2 :

K1 D K1 ; K2 D K2 :

p2 D 0 ;

k2 D 0 ;

q2 D 0 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

Case 10 v2 D C 2 ;

k1 D k1 ;

p 3ˇ1 (C2 ˇ1 C C1 ˛1 ) k1 D " ; ˇ1 ˛1

v2 D C2 ;

p2 D 0 ;

p 3ˇ2 (C2 ˇ2 C C1 ˛2 ) k2 D " ; ˇ2 ˛2

k2 D 0 ;

k1 D k1 ;

q2 D 0 ;

p1 D 0 ;

u2 D C1 ;

q1 D q1 ;

q2 D q2 ; H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

u2 D C1 ; H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; K1 D K1 ; K2 D K2 :

K1 D K1 ; K2 D K2 :

(80)

Case 11 p 2 D p 2 ; q2 D 0 ; p 1 D 0 ; q1 D q1 ; p p 3C1 3C1 k2 D " ; v2 D C1 or k2 D " ; ˛2 ˛2 p 3C1 v2 D C 1 ; k 1 D " ; u2 D 0 ; ˛1 H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

Case 16 p2 D 0 ; q1 D q1 ;

q2 D 0 ;

p1 D 0 ;

k2 D k2 ;

(86)

Case 17 C1 ˛2 C1 ˛2 v2 D " or v2 D " ; ˇ2 ˇ2 p 3ˇ2 ˇ1 C1 (ˇ1 ˛2  ˇ2 ˛1 ) ; k2 D 0 ; k1 D " ˇ2 ˇ1 ˛1 q1 D 0 ;

p 1 D p 1 ; u2 D C1 ; q2 D q2 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ; ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; K1 D K1 ; K2 D K2 : (82)

p 3ˇ1 (C2 ˇ1 C C1 ˛1 ) k1 D " ; ˇ1 ˛1 q1 D q1 ;

u2 D C1 ;

p2 D 0 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

p2 D 0 ;

k2 D k2 ;

p1 D 0 ;

K1 D K1 ; K2 D K2 :

Case 12 p 2 D p 2 ; q2 D 0 ; p 1 D 0 ; q1 D q1 ; p p 3C1 3C1 k2 D " or k2 D " ; v2 D C1 ; ˛2 ˛2 p 3C1 k1 D " ; ˛1 u2 D 0 ; K 1 D K 1 ; K 2 D K 2 ;

Case 13 v2 D C2 ;

v2 D v2 ;

q2 D 0 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

(81)

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 :

k1 D k1 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; K1 D K1 ; K2 D K2 :

(85)

u2 D C1 ;

Case 18 C1 ˛2 C1 ˛2 v2 D " or v2 D " ; ˇ2 ˇ2 p 3ˇ2 ˇ1 C1 (ˇ1 ˛2  ˇ2 ˛1 ) ; k2 D 0 ; k1 D ˇ2 ˇ1 ˛1 p2 D p2 ;

q1 D 0 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

(87)

p 1 D p 1 ; u2 D C1 ;

q2 D q2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

K1 D K1 ; K2 D K2 :

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; (83)

K1 D K1 ; K2 D K2 :

(88)

845

846

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

Case 19 p2 D 0 ;

q1 D q1 ;

k1 D 0 ;

C1 ˛1 v2 D " or ˇ1

C1 ˛1 v2 D " ; ˇ1 p 3ˇ2 ˇ1 C1 (ˇ1 ˛2  ˇ2 ˛1 ) ; k2 D " ˇ2 ˇ1 ˛2 u2 D C1 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; K1 D K1 ; K2 D K2 : Case 23

p1 D p1 ;

q2 D q2 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

q1 D q1 ;

v2 D C2 ;

k1 D 0 ;

p 3ˇ2 (C2 ˇ2 C C1 ˛2 ) k2 D " ; ˇ2 ˛2 p1 D p1 ;

u2 D C1 ;

q2 D q2 ; H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; K1 D K1 ; K2 D K2 :

p2 D 0 ;

(92)

(89)

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; K1 D K1 ; K2 D K2 : (93)

Case 20 p 3ˇ2 (C2 ˇ2 C C1 ˛2 ) k2 D " ˇ2 ˛2

p2 D 0 ;

v2 D C2 ; p 3ˇ2 (C2 ˇ2 C C1 ˛2 ) or k2 D " ; ˇ2 ˛2 p 3ˇ1 (C2 ˇ1 C C1 ˛1 ) ; q2 D 0 ; q1 D 0 ; k1 D " ˇ1 ˛1

Case 24

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

p 3C1 or p 2 D p 2 ; v2 D C1 ; k1 D " ˛1 p 3C1 k1 D " ; u2 D 0 ; q2 D 0 ; q1 D 0 ; ˛1 p 3C1 p1 D p1 ; k2 D " ; ˛2 H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ;

K1 D K1 ; K2 D K2 :

K1 D K1 ; K2 D K2 :

p1 D p1 ;

u2 D C1 ;

(94)

(90) Case 21 p p 3C1 3C1 v2 D C1 ; k2 D " or k2 D " ; ˛2 ˛2 p 3C1 k1 D " ; u2 D 0 ; p 2 D 0 ; q1 D 0 ; ˛1 p 1 D p 1 ; q2 D q2 ; H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

q1 D q1 ;

p 3ˇ2 (C2 ˇ2 C C1 ˛2 ) q1 D q1 ; k2 D " ˇ2 ˛2 p 3ˇ2 (C2 ˇ2 C C1 ˛2 ) or k2 D " ; ˇ2 ˛2 p p 3 ˇ1 (C2 ˇ1 C C1 ˛1 ) ; v2 D C2 ; k1 D " ˇ1 ˛1 p1 D p1 ;

u2 D C1 ;

p1 D p1 ;

u2 D C1 ;

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; K1 D K1 ; K2 D K2 :

Case 22

q2 D 0 ;

q2 D 0 ;

v2 D v2 ;

H D H ; ˛1 D ˛1 ; ˛2 D ˛2 ;

(91)

p2 D 0 ;

k2 D k2 ; k1 D k1 ;

(95)

We get the following new solution (called generalized hyperbolic function solution) of Eq. (46):

ˇ1 D ˇ1 ; ˇ2 D ˇ2 ; 1 D 1 ; 2 D 2 ; K1 D K1 ; K2 D K2 :

Case 25 p2 D 0 ;

u1 D 2

K1 k1 ˇ1 (t) sech p 1 q 1 k 1 ()(1  2p1 q1 sech2p 1 q 1 k 1 ()) C K2 k2 ˇ2 (t) sech p 2 q 2 k 2 () (1  p2 q2 (1 C p2 q2 ) sech2p 2 q 2 k 2 ()) H C K1 tanh p 1 q 1 k 1 () sech p 1 q 1 k 1 () CK2 tanh p 2 q 2 k 2 () sech p 2 q 2 k 2 ()

K1 k1 ˛1 (t) sech p 1 q 1 k 1 ()(1  2p1 q1 sech2p 1 q 1 k 1 ()) CK2 k2 ˛2 (t) sech p 2 q 2 k 2 ()(1  p2 q2 (1 C p2 q2 ) sech2p 2 q 2 k 2 ()) H C K1 tanh p 1 q 1 k 1 () sech p 1 q 1 k 1 () CK2 tanh p 2 q 2 k 2 () sech p 2 q 2 k 2 ()

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

2

2

k1 2 K1 ˛1 (t)ˇ1 (t) tanh p 1 q 1 k 1 () sech p 1 q 1 k 1 () (1  6p1 q1 sech2p 1 q 1 k 1 ()) H C K1 tanh p 1 q 1 k 1 () sech p 1 q 1 k 1 () CK2 tanh p 2 q 2 k 2 () sech p 2 q 2 k 2 ()

k2 2 K2 ˛2 (t)ˇ2 (t) tanh p 2 q 2 k 2 () sech p 2 q 2 k 2 () (1  4p2 q2 (1 C p2 q2 ) sech2p 2 q 2 k 2 ())

C u2 ;

H C K1 tanh p 1 q 1 k 1 () sech p 1 q 1 k 1 () CK2 tanh p 2 q 2 k 2 () sech p 2 q 2 k 2 ()

(96)

v1 D 2

2

u i (t; x1 ; x2 ; : : : ; x m ) D u i (!) ;

(H C K1 tanh p 1 q 1 k 1 () sech p 1 q 1 k 1 () CK2 tanh p 2 q 2 k 2 () sech p 2 q 2 k 2 ())2

H C K1 tanh p 1 q 1 k 1 () sech p 1 q 1 k 1 () CK2 tanh p 2 q 2 k 2 () sech p 2 q 2 k 2 () (t))2

H C K1 tanh p 1 q 1 k 1 () sech p 1 q 1 k 1 () CK2 tanh p 2 q 2 k 2 () sech p 2 q 2 k 2 ()

(99)

where x D (x1 ; x2 ; x3 ; : : : ; x m ; t) and !(x) is a function to be determined later. For example, when n D 2, we may take !(x) D p(x2 ; t)x1 C q(x2 ; t) for convenience, here p(x2 ; t) and q(x2 ; t) are functions to be determined later.

tanh p 1 q 1 k 1 () sech p 1 q 1 k 1 () k1 K1 (˛1 (1  6p1 q1 sech2p 1 q 1 k 1 ())

tanh p 2 q 2 k 2 () sech p 2 q 2 k 2 () K2 k2 (˛2 (1  2p2 q2 (2 C p2 q2 ) sech2p 2 q 2 k 2 ())

! D !(x) ;

(i D 1; 2; : : : ; n) ;

(t))2

2

2

We first consider the following general formal solutions of the Eqs. (98)

(K1 k1 ˛1 (t) sech p 1 q 1 k 1 ()(1  2p1 q1 sech2p 1 q 1 k 1 ()) C K2 k2 ˛2 (t) sech p 2 q 2 k 2 () (1  p2 q2 (1 C p2 q2 ) sech2p 2 q 2 k 2 ()))2

2

(i D 1; 2; : : : ; n) in m C 1 independent variables t; x1 ; x2 ; : : : ; x m , 8 F (u ; : : : ; u n ; u1t ; : : : ; u nt ; u1x 1 ; : : : ; u nx m ; ˆ ˆ ˆ 1 1 ˆ ˆ u1tx 1 ; : : : ; u ntx m ; : : :) D 0 ; ˆ ˆ ˆ ˆ ˆ (u F ˆ < 2 1 ; : : : ; u n ; u1t ; : : : ; u nt ; u1x 1 ; : : : ; u nx m ; (98) u1tx 1 ; : : : ; u ntx m ; : : :) D 0 ; ˆ ˆ ˆ ˆ : : : : : : : : : : : : ; ˆ ˆ ˆ ˆ ˆ ˆFn (u1 ; : : : ; u n ; u1t ; : : : ; u nt ; u1x 1 ; : : : ; u nx m ; ˆ : u1tx 1 ; : : : ; u ntx m ; : : :) D 0 :

C v2 ;

(97) where  D ˛1 (t)x Cˇ1 (t)y C r1 (t);  D ˛2 (t)x Cˇ2 (t)y C r2 (t), here H; K i ; p i ; q i ; k i ; ˛ i ; ˇ i and r i ; i D 1; 2 satisfy (71)–(95), respectively. The Generalized F-expansion Method and Its Application in Another (2 + 1)-Dimensional KdV Equation In the section, we will introduce the main steps of the generalized F-expansion method for constructing exact solutions of NLPDEs which we first presented in [40,41,42]. Then we will make use of the method to find new exact solutions of the (2 C 1)-dimensional KdV equation. Summary of the Generalized F-expansion Method In the following we would like to outline the main steps of our general method (called the generalized F-expansion method) in [40,41,42]. Step 1 For a given nonlinear partial differential equation system with some physical fields u i (t; x1 ; x2 ; : : : ; x m ),

Step 2 We introduce new and more general formal transformations of the Eq. (98), if available, in the forms which we first presented in [42]: 8  Pk1 i 1 1 a (x) ˆ u (x) D a (x) C ( f (!(x))) ˆ 1 1 i1 0 ˆ i 1 D1 ˆ  ˆ ˆ c i1 (x) d i1 (x) ˆ ˆ f (!(x)) C b (x)g(!(x)) C C i 1 ˆ f (!(x)) g(!(x)) ; ˆ ˆ  ˆ Pk2 ˆ ˆ i 2 1 a (x) ˆ i2 ˆ
847

848

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Step 3 Determine k j ( j D 1; 2; : : : ; n) in (100) by balancing the highest nonlinear terms and the highest-order partial differential terms in the given Eq. (98). If kj is a nonnegative integer, then we first make the transformation kj

u j (x) D v j (x). Step 4 Substitute (100) into the Eq. (98) with (101) and p collecting coefficients of polynomials of f (!); g(!); l1 f 4 (!) C m1 f 2 (!) C n1 , with the aid of Maple, then setting each coefficient to zero to get a set of over-determined partial differential equations with respect to l1 ; l2 ; m1 ; m2 ; n1 ; n2 ; a j 0 (x); a i j (x), b i j (x); c i j (x); d i j (x)( j D 1; 2; : : : ; n; i j D 1; 2; : : : ; k j ) and !(x). Step 5 Solving the over-determined partial differential equations with Maple, we then can determine a j 0 (x); a i j (x); b i j (x); c i j (x); d i j (x)( j D 1; 2; : : : ; n; i j D 1; 2; : : : ; k j ) and !(x). Step 6 Selecting the proper value of parameters l1 ; l2 ; m1 ; m2 ; n1 ; n2 to determine the corresponding the solutions of the Jacobi elliptic functions f (!) and g(!) in (101). The relations between the parameters and their corresponding Jacobi elliptic functions are known and are given in the following Table 1. Remark 2 Relations between values of l1 ; m1 ; n1 ; l2 ; m2 ; n2 and corresponding f (!), g(!) in ODEs (101) are in the following Table 1. Step 7 By using the results obtained in the above step, we can derive a series of generalized solutions, such as different kinds of Jacobi elliptic function solutions in terms

of Remark 2. Finally, substituting a j 0 (x); a i j (x); b i j (x); c i j (x); d i j (x)( j D 1; 2; : : : ; n; i j D 1; 2; : : : ; k j ) and !(x) into the generalized solutions with the corresponding solutions of f (!); g(!), we can get the the new exact solutions of the given Eq. (98). Remark 3 As is known to all, when k ! 1, the Jacobi elliptic functions can degenerate as hyperbolic functions, and when k ! 0, the Jacobi elliptic functions degenerate as trigonometric functions. So by this method we can obtain many other new exact solutions to Eq. (98). Remark 4 The main advantages of the generalized F-expansion method are simpler, more powerful, and more convenient than the Jacobi elliptic function expansion method and the F-expansion method. First of all, with the aid of nonlinear ordinary differential equations (ODEs) (101), one only needs to calculate the functions f (!) and g(!) which are the solutions of the ODEs (101), instead of calculating the Jacobi elliptic functions one by one. Secondly, the values of the coefficients l1 ; m1 ; n1 ; l2 ; m2 ; n2 of the ODEs (101) can be selected so that the corresponding solutions of the coupled functions f (!) and g(!) are Jacobi elliptic functions in the above Table 1. The relations between the coefficients and the corresponding Jacobi elliptic function solutions are known and are given in Remark 2. Thus, in terms of Remark 2, one can simultaneously obtain more periodic wave solutions expressed by various Jacobi elliptic functions. Thirdly, we present a new transformations (100) that are more general than the transformations of the Jacobi elliptic function expansion method and the F-expansion method.

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, Table 1 The ODEs (101) and Jacobi Elliptic Functions Relation between values of li ; mi ; ni (i D 1; 2) and corresponding f(!) and g(!) in ODEs (101) l1 k2 k2 k2 1  k2 1 1 1 1 k2 k2 (1  k2 ) k2 (1  k2 ) 1  k2

m1 1  k2 1  k2 2k2  1 2k2  1 2  k2 1  k2 1  k2 1  k2 1  k2 2k2  1 2k2  1 2  k2

n1 1 1 1  k2 k2 1  k2 k2 k2 k2 1 1 1 1

l2 k2 1 1 1  k2 1 1  k2 1 1 1 C k2 1 C k2 k2 1

m2 2k2  1 2  k2 2  k2 2  k2 2k2  1 2k2  1 2  k2 2k2  1 2  k2 2  k2 1  k2 1  k2

n2 1  k2 1 C k2 k2  1 1 k2 (1  k2 ) k2 1  k2 k2 (1  k2 ) 1 1 1 k2

f (!) sn(!) sn(!) cn(!) nc(!) cs(!) dc(!) ns(!) ns(!) cd(!) sd(!) sd(!) sc(!)

g(!) cn(!) dn(!) dn(!) sc(!) ds(!) nc(!) cs(!) ds(!) nd(!) nd(!) cd(!) dc(!)

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

The Generalized F-expansion Method to Find the Exact Solutions of the (2 + 1)-Dimensional KdV Equation In this section, we will make use of our method [40,41,42] and symbolic computation to find new exact solutions of the (2 C 1)-dimension KdV equation. The (2 C 1)-dimension KdV equation in [20] is written in the following form u t (x; y; t)  4u(x; y; t)u y (x; y; t)  4u x (x; y; t)@1 x u y (x; y; t)  u x x y (x; y; t) D 0 :

(102)

R where @1 x is an indefinite integrate operator dx, possessing some interesting coherent structures. We discuss the similar solution of (102) in its potential form (u D v x ), v x t (x; y; t)  v x x x y (x; y; t)  4v x (x; y; t)v x y (x; y; t)  4v x x (x; y; t)v y (x; y; t) D 0 :

(103)

By balancing the highest nonlinear terms and the highestorder partial derivative terms in (103), we suppose (103) to have the following formal solution:

C  D 0 ; 23328c2 (y; t)(˛(y; t))3 l2 l1 5

C

c4 (y; t) c3 (y; t) C ; g(!) f (!)

(104)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) are all functions to be determined later. And the new variables f (!) and g(!) satisfy the relation (101). Substitute (104) into the Eq. (103) with (101). Collect coefficients of polynomials of f (!); g(!); p l1 f 4 (!) C m1 f 2 (!) C n1 , with the aid of Maple, then set each coefficient to zero to get a set of over-determined partial differential equations with respect to l1 ; l2 ; m1 ; m2 ; n1 ; n2 ; ˛(y; t); p(y; t); c i (y; t); (i D 0; 1; 2; 3; 4) and q(t) as follows: 

 @ 120(˛(y; t)) c4 (y; t)c1 (y; t) ˛(y; t) xm1 4 m2 2 @y   @ 2 C 168(˛(y; t)) c4 (y; t)l2 c3 (y; t) p(y; t) m1 4 m2 @y   @ C 120(˛(y; t))2 c4 (y; t)c1 (y; t) p(y; t) m2 2 m1 4 @y   @ C 48(˛(y; t))2 c4 (y; t)l2 c3 (y; t) ˛(y; t) xm1 2 m2 3 @y 2



 @ p(y; t) m2 @y

d C 1458c2 (y; t)˛(y; t)l2 l1 5 q(t)  37908c2 (y; t) dt   @ 5 3 p(y; t) m1 C 23328c2 (y; t) (˛(y; t)) l2 l1 @y   @ 5 3 (˛(y; t)) l2 l1 ˛(y; t) xm2 C 1458c2 (y; t) @y   @ ˛(y; t)l2 l1 5 ˛(y; t) x @t   @  5832(˛(y; t))2 c0 (y; t) l2 l1 5 c2 (y; t) @y   @ 5 3  37908c2 (y; t)(˛(y; t)) l2 l1 ˛(y; t) xm1 @y C  D 0

 @ ˛(y; t) xm2 3 l1 @y   @  540(˛(y; t))3 c1 (y; t)l2 n2 p(y; t) m2 4 l1 @y   @ 3 C 2160(˛(y; t)) c1 (y; t)l2 n2 ˛(y; t) xm1 m2 3 l1 @y   @  3240(˛(y; t))3 c1 (y; t)l2 n2 ˛(y; t) xm1 2 m2 2 l1 @y   @ C 2160(˛(y; t))3 c1 (y; t)l2 n2 ˛(y; t) xm1 3 m2 l1 @y   @ 3 2  1296c3 (y; t)(˛(y; t)) l2 n2 ˛(y; t) x l1 m2 3 @y   @ C 180c3 (y; t)(˛(y; t))3 l2 l1 m1 3 ˛(y; t) xm2 2 @y  60(˛(y; t))3 c1 (y; t)m1 3

v D c0 (y; t) C c1 (y; t)g(!) C c2 (y; t) f (!)



 @ ˛(y; t) xm2 3 m1 3 @y   @ 2  72(˛(y; t)) c4 (y; t)c1 (y; t) p(y; t) m1 5 m2 @y   @ C 24(˛(y; t))2 c4 (y; t)c1 (y; t) ˛(y; t) xm2 5 m1 @y  80(˛(y; t))2 c4 (y; t)c1 (y; t)



C  D 0 ; :::

(105)

Because there are so many over-determined partial differential equations, only a few of them are shown here for convenience. With the aid of Maple symbolic computation software, we solve the over-determined partial differential equations, and we get the following solutions of ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t):

849

850

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Case 1 c4 (y; t) D

C1 ; F1 (t)

p(y; t) D F2 (t) ;

C2 ; F1 (t) C4 c1 (y; t) D ; F1 (t)

c3 (y; t) D

C3 ; ˛(y; t) D F1 (t) ; c2 (y; t) D F1 (t)   Z d F1 (t) x q(t) D 4F1 (t)F3 (t)  dt d  F2 (t)dt C C5 ; c0 (y; t) D F3 (t)y C F4 (t) ; dt

(106) where Ci , i D 1; 2; 3; 4; 5 are arbitrary constants and F i , i D 1; 2; 3; 4 are all arbitrary functions with respect to variable t. Case 2 48C1 l1 c2 (y; t) D  ; F1 (t)(m1  m2 )   Z d F1 (t) x q(t) D 4F1 (t)F3 (t)  dt d  F2 (t)dt C C4 ; dt C3 ˛(y; t) D F1 (t) ; c1 (y; t) D ; F1 (t) C1 ; p(y; t) D F2 (t) ; c4 (y; t) D F1 (t) C2 ; c0 (y; t) D F3 (t)y C F4 (t) ; (107) c3 (y; t) D F1 (t) where Ci , i D 1; 2; 3; 4 are arbitrary constants and F i , i D 1; 2; 3; 4 are all arbitrary functions with respect to variable t. Case 3 c2 (y; t) D 

48C1 l1 ; F1 (t)(m1  m2 )

c4 (y; t) D

C1 ; F1 (t)

C2 ; ˛(y; t) D F1 (t) ; F1 (t) c0 (y; t) D F3 (t)y C F4 (t) ; c1 (y; t) D 0 ;   Z d q(t) D 4F1 (t)F3 (t)  F1 (t) x dt d  F2 (t)dt C C3 ; dt (108) p(y; t) D F2 (t) ;

c3 (y; t) D

where Ci , i D 1; 2; 3; 4 are arbitrary constants and F i , i D 1; 2; 3; 4 are all arbitrary functions with respect to variable t.

Case 4 c3 (y; t) D c3 (y; t) ; q(t) D q(t) ;

c4 (y; t) D c4 (y; t) ;

c0 (y; t) D c0 (y; t) ;

p(y; t) D p(y; t) ; c2 (y; t) D c2 (y; t) ;

c1 (y; t) D c1 (y; t) ;

(109)

˛(y; t) D 0 ;

where C1 is an arbitrary constant and c3 (y; t); c4 (y; t); q(t); c0 (y; t); p(y; t); c1 (y; t); c2 (y; t) are all arbitrary functions with respect to variable y and t. Select the proper value of parameters l1 ; l2 ; m1 ; m2 ; n1 ; n2 to determine the corresponding Jacobi elliptic functions f (!) and g(!) in (101). The relations between the parameters and the corresponding Jacobi elliptic function are known and given in Table 1, above. By using the results obtained in the above step, we can derive a series of generalized solutions, such as different kinds of Jacobi elliptic functions solutions. Finally, by substituting (106)–(109) into the generalized solutions with the corresponding solutions of f (!) and g(!), respectively, we can get the the new exact solutions of the given Eqs. (103). Type 1 If we select l1 D k 2 ; m1 D (1 C k 2 ); n1 D 1; l2 D k 2 ; m2 D (2k 2  1); n2 D (1  k 2 ), corresponding to (101), we can get f D sn(!); g D cn(!). Thus we get the following Jacobi elliptic function solutions of the equation (103): v1 (x; y; t) D c0 (y; t) C c1 (y; t)cn(!) C c2 (y; t)sn(!) C

c3 (y; t) c4 (y; t) C ; cn(!) sn(!)

(110)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109) respectively. For example, when we select ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) to satisfy (106), namely Case 1, we can easily get the following Jacobi elliptic function solution of Eq. (103): C4 cn(!) C3 sn(!) C F1 (t) F1 (t) C1 C2 C ; (111) C F1 (t)cn(!) F1 (t)sn(!)

v2 (x; y; t) D F3 (t)y C F4 (t) C

where F i ; i D 1; 2; 3; 4 are arbitrary functions of t; C i (i D 1; 2; 3; 4; 5) are arbitrary constants, and Z ! D F1 (t)x C F2 (t) C 4F1 (t)F3 (t)   d d F1 (t) x  F2 (t)dt C C5 :  dt dt

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

When k ! 1; sn(!) ! tanh(!); cn(!) ! sech(!), so we get degenerative soliton-like solutions from the solution (111): C4 sech(!) C3 tanh(!) C F1 (t) F1 (t) C1 C2 C : (112) C F1 (t) sech(!) F1 (t) tanh(!)

where F i (t); i D 1; 2; 3; 4 are arbitrary functions of t; C i (i D 1; 2; 3; 4) are arbitrary constants, and 

Z ! D F1 (t)x C F2 (t) C

4F1 (t)F3 (t) 

v3 (x; y; t) D F3 (t)yCF4 (t)C

When k ! 0; sn(!) ! sin(!); cn(!) ! cos(!), so we get the degenerative trigonometric function solutions from the solution (111):



 d F1 (t) x dt

d F2 (t)dt C C4 : dt

When k ! 1; cn(!) ! sech(!); dn(!) ! sech(!), so we get a degenerative soliton-like solution from the solution (115): C3 sech(!) F1 (t) C1 C C2 C1 l1 sech(!) C ;  48 F1 (t)(m1  m2 ) F1 (t) sech(!)

v7 (x; y; t) D F3 (t)y C F4 (t) C C4 cos(!) F1 (t) C2 C1 C3 sin(!) C C : C F1 (t) F1 (t) cos(!) F1 (t) sin(!)

v4 (x; y; t) D F3 (t)y C F4 (t) C

(113)

So from Case 1, we can obtain Jacobi elliptic function solutions, degenerative soliton-like solutions and trigonometric function solutions of Eq. (103). We can also get many other solutions if we make use of Case 2–Case 4. Therefore, through selecting l1 D k 2 ; m1 D (1 C k 2 ); n1 D 1; l2 D k 2 ; m2 D (2k 2  1); n2 D (1  k 2 ), we can get families of new exact solutions of Eq. (103). Type 2 If we select l1 D k 2 ; m1 D (2k 2  1); n1 D 1k 2 ; l2 D 1; m2 D (2k 2 ); n2 D k 2 1, corresponding to (101), we can get f D cn(!); g D dn(!). So we get the following Jacobi elliptic function solutions of Eq. (103): v5 (x; y; t) D c0 (y; t) C c1 (y; t)dn(!) C c2 (y; t)cn(!) C

c3 (y; t) c4 (y; t) C ; dn(!) cn(!)

 48

When k ! 0; cn(!) ! cos(!); dn(!) ! 1, so we get the degenerative trigonometric function solutions from the solution (115): C2 C C3 F1 (t) C1 C1 l1 cos(!) C :  48 F1 (t)(m1  m2 ) F1 (t) cos(!)

v8 D F3 (t)y C F4 (t) C

(117)

where F i (t); i D 1; 2; 3; 4 are arbitrary functions of t and C i (i D 1; 2; 3; 4) are arbitrary constants. So from Case 2, we we can get the Jacobi elliptic function solutions, degenerative soliton-like solutions and trigonometric function solutions of Eq. (103). We can also get some other solutions if we make use of other cases. Therefore, through selecting l1 D k 2 ; m1 D (2k 2  1); n1 D 1  k 2 ; l2 D 1; m2 D (2  k 2 ); n2 D k 2  1, we can get families of new exact solutions of Eq. (103).

(114)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109), respectively. For example, when we select ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) to satisfy (107), namely Case 2, we can easily get the following Jacobi elliptic function solution of Eq. (103):

v6 (x; y; t) D F3 (t)y C F4 (t) C

(116)

C3 dn(!) F1 (t)

C2 C1 C1 l1 cn(!) C C : F1 (t)(m1  m2 ) F1 (t)dn(!) F1 (t)cn(!) (115)

Type 3 If we select l1 D 1; m1 D (2  k 2 ); n1 D 1  k 2 ; l2 D 1; m2 D (2k 2  1); n2 D k 2 (1  k 2 ), corresponding to (101), we can get f D cs(!); g D ds(!). So we get the following Jacobi elliptic function solutions of Eq. (103): v9 (x; y; t) D c0 (y; t) C c1 (y; t)ds(!) C c2 (y; t)cs(!) C

c4 (y; t) c3 (y; t) C ; ds(!) cs(!)

(118)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109), respectively. For example, when we select ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) to satisfy

851

852

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

(108), namely Case 3, we can easily get the following Jacobi elliptic function solution of Eq. (103): C1 l1 cs(!) F1 (t)(m1  m2 ) C1 C2 C ; (119) C F1 (t)ds(!) F1 (t)cs(!)

v10 (x; y; t) D F3 (t)y C F4 (t)  48

where F i (t); i D 1; 2; 3; 4 are arbitrary functions of t; C i (i D 1; 2; 3; 4) are arbitrary constants, and ! D F1 (t)x C F2 (t)q(t)   Z d d D 4F1 (t)F3 (t)  F1 (t) x  F2 (t)dt C C3 : dt dt When k ! 1; cs(!) ! sech(!) coth(!); ds(!) ! sech(!) coth(!), so we get a degenerative soliton-like solution from the solution (119): v11 (x; y; t) D F3 (t)y C F4 (t) 48

C1 C C2 C1 l1 sech(!) coth(!) C : F1 (t)(m1  m2 ) F1 (t) sech(!) coth(!) (120)

When k ! 0; cs(!) ! cot(!); ds(!) ! csc(!), so we get the triangular function solution from the solution (119): v12 (x; y; t) D F3 (t)y C F4 (t) 48

C2 C1 C1 l1 cot(!) C C : F1 (t)(m1  m2 ) F1 (t) csc(!) F1 (t) cot(!) (121)

So from Case 3, we can get Jacobi elliptic function solutions, degenerative soliton-like solutions and trigonometric function solutions of Eq. (103). We can also get many other solutions if we make use of other cases. Therefore, through selecting l1 D 1; m1 D (2  k 2 ); n1 D 1  k 2 ; l2 D 1; m2 D (2k 2  1); n2 D k 2 (1  k 2 ), we can get families of new exact solutions of Eq. (103). Type 4 If we select l1 D 1  k 2 ; m1 D (2k 2  1); n1 D k 2 ; l2 D (1  k 2 ); m2 D (2  k 2 ); n2 D 1, corresponding to (101), we can get f D nc(!); g D sc(!). So we get the following Jacobi elliptic function solutions of Eq. (103): v13 (x; y; t) D c0 (y; t) C c1 (y; t)sc(!) C c2 (y; t)nc(!) C

c3 (y; t) c4 (y; t) C ; sc(!) nc(!)

(122)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109), respectively.

So, in the same way, we can get many other Jacobi elliptic function solutions of Eq. (103). When k ! 1, or k ! 0, we can also get families of new exact solutions of Eq. (103). Type 5 If we select l1 D k 2 ; m1 D (1 C k 2 ); n1 D 1; l2 D 1; m2 D (2  k 2 ); n2 D (1  k 2 ), corresponding to (101), we can get f D sn(!); g D dn(!). So we get the following Jacobi elliptic function solutions of Eq. (103): v14 (x; y; t) D c0 (y; t) C c1 (y; t)dn(!) C c2 (y; t)sn(!) C

c3 (y; t) c4 (y; t) C ; dn(!) sn(!)

(123)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109), respectively. So, in the same way, we can get many other Jacobi elliptic function solutions of Eq. (103). When k ! 1, or k ! 0, we can also get families of new exact solutions of Eq. (103). Type 6 If we select l1 D 1; m1 D (1 C k 2 ); n1 D k 2 ; l2 D 1  k 2 ; m2 D (2k 2  1); n2 D k 2 , corresponding to (101), we can get f D dc(!); g D nc(!). So we get the following Jacobi elliptic function solutions of Eq. (103): v15 (x; y; t) D c0 (y; t) C c1 (y; t)nc(!) C c2 (y; t)dc(!) C

c3 (y; t) c4 (y; t) C ; nc(!) dc(!)

(124)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109), respectively. So, in the same way, we can get many other Jacobi elliptic function solutions of Eq. (103). When k ! 1, or k ! 0, we can also get families of new exact solutions of Eq. (103). Type 7 If we select l1 D 1; m1 D (1 C k 2 ); n1 D k 2 ; l2 D 1; m2 D 2  k 2 ; n2 D 1  k 2 , corresponding to (101), we can get f D ns(!); g D cs(!). So we get the following Jacobi elliptic function solutions: v16 (x; y; t) D c0 (y; t) C c1 (y; t)cs(!) C c2 (y; t)ns(!) C

c3 (y; t) c4 (y; t) C ; cs(!) ns(!)

(125)

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109), respectively. So, in the same way, we can get many other Jacobi elliptic function solutions of Eq. (103). When k ! 1, or k ! 0, we can also get families of new exact solutions of Eq. (103). Type 8 If we select l1 D 1; m1 D (1 C k 2 ); n1 D k 2 ; l2 D 1; m2 D 2k 2  1; n2 D k 2 (1  k 2 ), corresponding to (101), we can get f D ns(!); g D ds(!). So we get the following Jacobi elliptic function solutions: v17 (x; y; t) D c0 (y; t) C c1 (y; t)ds(!) C c2 (y; t)ns(!) C

c3 (y; t) c4 (y; t) C ; ds(!) ns(!)

(126)

v19 (x; y; t) D c0 (y; t) C c1 (y; t)nd(!) C c2 (y; t)sd(!) C

c3 (y; t) c4 (y; t) C ; nd(!) sd(!)

(128)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y;  t);p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 y; t satisfy (106)–(109), respectively. So, in the same way, we can get many other Jacobi elliptic function solutions of Eq. (103). When k ! 1, or k ! 0, we can also get families of new exact solutions of Eq. (103). Type 11 If we select l1 D k 2 (1  k 2 ); m1 D 2k 2  1; n1 D 1; l2 D k 2 ; m2 D (1 C k 2 ); n2 D 1, corresponding to (101), we can get f D sd(!); g D cd(!). So we get the following Jacobi elliptic function solutions of Eq. (103): v20 (x; y; t) D c0 (y; t) C c1 (y; t)cd(!)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109), respectively. So, in the same way, we can get many other Jacobi elliptic function solutions of Eq. (103). When k ! 1, or k ! 0, we can also get families of new exact solutions of Eq. (103). Type 9 If we select l1 D k 2 ; m1 D (1 C k 2 ); n1 D 1; l2 D (1  k 2 ); m2 D 2  k 2 ; n2 D 1, corresponding to (101), we can get f D cd(!); g D nd(!). So we get the following Jacobi elliptic function solutions of Eq. (103): v18 (x; y; t) D c0 (y; t) C c1 (y; t)nd(!) C c2 (y; t)cd(!) C

c3 (y; t) c4 (y; t) C ; nd(!) cd(!)

(127)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109), respectively. So, in the same way, we can get many other Jacobi elliptic function solutions of Eq. (103). When k ! 1, or k ! 0, we can also get families of new exact solutions of Eq. (103). Type 10 If we select l1 D k 2 (1  k 2 ); m1 D 2k 2  1; n1 D 1; l2 D (1  k 2 ); m2 D 2  k 2 ; n2 D 1, corresponding to (101), we can get f D sd(!); g D nd(!). So we get the following Jacobi elliptic function solutions of Eq. (103):

C c2 (y; t)sd(!) C

c3 (y; t) c4 (y; t) C ; cd(!) sd(!)

(129)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109), respectively. So, in the same way, we can get many other Jacobi elliptic function solutions of Eq. (103). When k ! 1, or k ! 0, we can also get families of new exact solutions of Eqs. (103). Type 12 If we select l1 D 1  k 2 ; m1 D 2  k 2 ; n1 D 1; l2 D 1; m2 D (1 C k 2 ); n2 D k 2 , corresponding to (101), we can get f D sc(!); g D dc(!). So we get the following Jacobi elliptic function solutions: v21 (x; y; t) D c0 (y; t) C c1 (y; t)dc(!) C c2 (y; t)sc(!) C

c3 (y; t) c4 (y; t) C ; dc(!) sc(!)

(130)

where ! D ˛(y; t)x C p(y; t) C q(t); ˛(y; t); p(y; t); q(t); c0 (y; t); c1 (y; t); c2 (y; t); c3 (y; t) and c4 (y; t) satisfy (106)– (109), respectively. So, in the same way, we can get many other Jacobi elliptic function solutions of Eq. (103). When k ! 1, or k ! 0, we can also get families of new exact solutions of Eq. (103). Remark 5 From the details above, it is very easy to see that we have gotten many new exact solutions of the (2 C 1)-dimensional KdV equation (103). The solutions we get include Jacobi elliptic doubly periodic function solutions,

853

854

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

soliton-like solutions and triangular function solutions. These solutions are more general than the solutions which the extended F-expansion method gets, and they are more abundant than the solutions in [34]. Besides, our method is more convenient than the method in [34]. Among the solutions, the arbitrary functions imply that these solutions have rich local structures. In this paper, we have provided some figures to describe the character of the new exact solutions of Eq. (103). The Generalized Algebra Method and Its Application in (1 + 1)-Dimensional Generalized Variable – Coefficient KdV Equation In this section, with the aid of the Maple symbolic computation system, we will develop the algebraic methods [50,51] for constructing the traveling wave solutions of NLPDEs and present the following new methods and theorems: 1. A new and general transformation, a new theorem and its proof by using Maple, are presented in [40]. 2. A new mechanization method to find the exact solutions of a first-order nonlinear ordinary differential equation with any degree. The validity and reliability of the method are tested by its application to the firstorder nonlinear ordinary differential equation with six degrees, eight degrees, ten degrees, and twelve degrees in [40,43,44]. 3. A general transformation, a new generalized algebraic method and their algorithms are suggested based on a nonlinear ordinary differential equation with any degree. The (1 C 1)-Dimensional Generalized Variable – Coefficient KdV Equations are chosen to illustrate our algorithm so that more families of new exact solutions are obtained, which contain both non-traveling and traveling wave solutions in [40,43,44]. Recently, much research work has been concentrated on the various extensions and applications of the algebraic method with computerized symbolic computation [50,51]. The algebraic method can obtain many types of traveling wave solutions based on a nonlinear ordinary differential equation with four degrees and the following theorem:

admits many kinds of fundamental solutions, some of which are listed as follows. Case 1 If c0 D c1 D c3 D 0, Eq. (131) admits: a bell shaped solitary wave solution r p  c2 () D  sech c2  ; c2 > 0 ; c4 < 0 ; (132) c4 a triangular type solution r p  c2 () D  sec c2  ; c4

c2 < 0 ; c4 > 0 ; (133)

a rational polynomial type solution 1 () D  p ; c4 

c2 D 0 ; c4 > 0 :

(134)

Case 2 If c0 D d22 /(4c4 ); c1 D c3 D 0, Eq. (131) admits a kink shaped solitary wave solution r r  c2 c2 tanh   ; c2 < 0 ; c4 > 0 ; () D  2c4 2 (135) a triangular type solution r r  c2 c2 tan  ; () D 2c4 2

c2 > 0 ; c4 > 0 ; (136)

a rational polynomial type solution 1 () D  p ; c4 

c2 D 0 ; c4 > 0 :

(137)

Case 3 If c1 D c3 D 0, Eq. (131) admits three Jacobian elliptic functions type solutions s r  c2 c2 m2 () D  cn  ; c4 (2m2  1) 2m2  1 c0 D s

d22 m2 (m2  1) ; c4 (2m2  1)2

c2 m2  sn c4 (m2 C 1)

() D

c0 D

Theorem 3 The following nonlinear ordinary differential equation with four degrees

r 

c2 > 0 ;

c2  m2 C 1

d22 m2 ; c4 (m2 C 1)2

(138)

 ;

c2 < 0 ;

(139)

and d () d p D c0 C c1 () C c2  2 () C c3  3 () C c4  4 () ; c i D constant ; i D 0; 1; 2; 3; 4

(131)

r () D



 r c2 c2  ; dn c4 (2  m2 ) 2  m2 d 2 (1  m2 ) ; c2 > 0 : c0 D 2 c4 (2  m2 )2

(140)

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Case 4 If c0 D c1 D c4 D 0, Eq. (131) admits two bell shaped solitary wave solutions p  c2 c2 2 (141)  ; c2 > 0 ; () D  sech c3 2 and () D 

c2 sech2 c3

 p c2  ; 2

c2 < 0 ;

(142)

a rational polynomial type solution () D

1 ; c3  2

c2 D 0 ;

(143)

A New Transformation and a New Theorem In Yu-Jie Ren’s Ph.D Dissertation of Dalian University of Technology [40], in order to develop the above algebraic method, we first presented the following new transformation (150) and new Theorem 4. Then we proved the theorem by means of the Maple computer algebraic system. Theorem 4 ([40]) Suppose n and m are any integers, a first-order nonlinear ordinary differential equation with any degree in the form of v u r X u d c i   i () ; () D "tc0 C d iD1

Case 5 If c3 > 0; c4 D 0, Eq. (131) admits a Weiertrass elliptic functions type solution  p c3 4c1 4c0 ; ;  : (144) () D } 2 c3 c3

If we substitute the following new transform into Eq. (149)

Let’s specifically see how the algebraic method works. For a given nonlinear differential equation, say, in two variables x; t

then Eq. (149) can be transformed into an ordinary differential equation

F(u; u t ; u x ; u x x ; u x t ; u t t ; : : :) D 0

(145)

where F is a polynomial function with respect to the indicated variables or some function which can be reduced to a polynomial function by using some transformations. By using the traveling wave transformation u D u();  D x  t ;

(146)

Eq. (145) is reduced to an ordinary differential equation with constant coefficients G(U; U 0 ; U 00 ; U 000 ; : : :) D 0

(147)

A transformation was presented by Fan [5] in the form u(x) D A0 C

n X

A i  i1 () ;

(148)

iD1

with the new variable () satisfying Eq. (131), where A0 ; A i ; d j are constants. Substituting (148) into (147) along with (131), we can determine the parameter n in (148). And then by substituting (148) with the concrete n into (147) and equating the coefficients of these terms ! i ! 0 j (i D 0; 1; 2; : : : ; j D 0; 1), we obtain a system of algebraic equations with respect to other parameters A0 ; A i ; d j ; . By solving the system, if available, we may determine these parameters. Therefore we establish a transformation (147) between (146) and (148). If we know the solutions of (148), then we can obtain the solutions of (147) (or (145)) by using (146).

c i D consts ; i D 1; 2; : : : ; r :

n

() D (g()) m ;



(149)

(150)

2 r X n(i2) d c i (g())2C m : g() D m2 n2 d

(151)

iD0

We give the proof of Theorem 4 [40] by using Maple as follows: Proof Step 1 Importing the following Maple program at the Maple Command Window eq :D diff(phi(x i); x i)2 sum(c[i] phi(x i) i ; i D 0::r); Eq. (149) is displayed at a computer screen (after implementing the above program) as follows:  eq :D

2 X r d c i (()) i : ()  d iD0

Step 2 Importing the following Maple program at the Maple Command Window eq :D subs(phi(x i) D (g(x i))( n/m); eq) ; the following result is displayed at the screen (after running the above program): 2    n 2 d eq :D (g()) m n2 g() m2 (g())2 d r   X n i  c i (g()) m : iD0

855

856

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Step 3 Importing the following Maple program at the Maple Command Window eq :D simplify(eq) ;

eq :D expand(eq) ; eq :D numer(eq) ;

the following result is displayed at the screen (after running the above program):  2   d n 2 g() eq :D (g()) m n2 d r   X n i  c i (g()) m m2 (g())2 : iD0

Step 4 Importing the following Maple program at the Maple Command Window eq :D subs(diff(g(x i); x i) D G(x i); g(x i) D g; eq) ; eq :D eq (g ( n/m))(  2) ;

C i n/m); i D 0::r)) m2 ; eq :D simplify(eq); the following result is displayed at the screen (after running the above program): r X

nm m

eq :D n2 (G())2 

ci g

2nC2mCni m

m2 :

Step 5 Importing the following Maple program at the Maple Command Window eq :D subs(g D g(x i); G(x i) D diff(g(x i); x i); eq) ; the following result is displayed at the screen (after running the above program): eq :D n2

g(x i)( (2 nC2 mCn i)/m); i D 0::r) m2 ; Eq. (151) is displayed at a computer screen (after implementing the above program) as follows 2 X  r 2nC2mCni d 2 m c i (g()) m2 g()  eq :D n d iD0

eq621 :D subs(r D 6; m D 2; n D 1; eq) ;

iD0



Step 1 Import the following Maple program at the Maple Command Window

Step 2 Import the following Maple programs with some values of the degree r and parameter (n; m) in new transform (150) at the Maple Command Window. For example,

 n i c i g m m2

iD0 r X

According to Theorem 4 in Subsect. “The the Exp-Bäcklund Transformation Method and Its Application in (1 + 1)-Dimensional KdV Equation”, in [40] we first presented the following mechanization method to find the exact solutions of a first-order nonlinear ordinary differential equation with any degree. The validity and reliability of our method are tested by its application to a firstorder nonlinear ordinary differential equation with six degrees [40]. Now, we simply describe our mechanization method as follows:

eq :D n2 diff(g(x i); x i)2  sum(c[i]

eq :D simplify(eq) ;

eq :D n2 G(x i)2  (sum(c[i] g ( (2 (n  m)/m)

eq :D n2 (G())2  g 2

A New Mechanization Method to Find the Exact Solutions of a First-Order Nonlinear Ordinary Differential Equation with any Degree Using Maple and Its Application

2 X r 2nC2mCni d m c i (g()) m2 : g()  d

eq621 :D simplify(eq621) ; the following result is displayed at the screen (after running the above program):  eq621 :D

2 d g()  4c0 g() d

 4c1 (g())3/2  4c2 (g())2  4c3 (g())5/2  4c4 (g())3  4c5 (g())7/2  4c6 (g())4 :

iD0

(153)

(152) We can reduce Eq. (152) to (151).  Remark 6 The above transformation (150), Theorem 4, and its proof by means of Maple were first presented by us in Yu-Jie Ren’s Ph.D Dissertation of Dalian University of Technology [40].

Step 3 According to the output results in Step 2, we choose the coefficients of ci, i D 1; 2; : : : ; m(< n) to be zero. Then we import their Maple program. For example, we import the Maple program as follows: eq246 :D subs(c[0] D 0; c[1] D 0; c[3] D 0; c[5] D 0; eq621) ;

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

the following result is displayed at the screen (after running the above program): 

4c6 c2 e4C 1

2 d g() d

eq246 :D

g4 () D 4 p

c 2 Ce4

p

p

c 2 (C 1C) c 2 p c 2  2e2 c 2 (C 1C) c

e2

4 Cc4

2 e4C 1

p

c2

:

(160)

 4c2 (g())2  4c4 (g())3  4c6 (g())4 :

(154)

Step 4 Import the Maple program for solving the output equation in Step 3. For example, we import the Maple program for solving the output Eq. (154) as follows:

Step 5 We discuss the above solutions under different conditions in Step 3. For example, we discuss the solutions under different conditions c42  4c6 c2 < 0 or c42  4c6 c2 > 0 or c42  4c6 c2 D 0 or c2 < 0 or c2 > 0 or c2 D 0. When

dsolve(eq246; g(x i)); the following formal solutions of (154) are displayed at the screen (after running the above program):

g1 () D 1/2

g2 () D 1/2 

c4 

q

c42  4c6 c2

c6 q c4 C c42  4c6 c2

p

c6 2



p

;

(155)

;

2

 p 2  p 2 g4 () D 4 e c 2 c2 eC 1 c 2 0 11  p 4  p 2 e c 2 e c 2 c4 B 2C @4c6 c2   p 4 C 2  p 2  c4 A : C c C c e 1 2 e 1 2 (158) Import the Maple program for reducing the above solutions of (154). For example, we import the Maple program for reducing the (157) and (158) as follows: g3 :D simplify(g3); g4 :D simplify(g4) ; the following results are displayed at a computer screen (after running the above program):

4c6 c2 e4

e2 c2

C e4C 1

p

p

we may import the following Maple program: assume(c[2] > 0; (4 c[2] c[6]  c[4]2 ) < 0);

(156)

(157)

p

(161)

eq246 jie :D dsolve(eq246; g(x i));

g3 () D 4 eC 1 c 2 c2 e c 2 0 11  p 4  p 2 eC 1 c 2 e C 1 c 2 c4 B 2C @4c6 c2 C  p 4  2  p 2 C c4 A ;  c  c e 2 e 2

g3 () D 4

c2 > 0; 4c6 c2  c42 < 0 ;

c 2 (C 1 C) c 2 p c 2  2e2 c 2 (C 1 C) c

4

C c42 e4

p

c2

;

(159)

six solutions of a first-order nonlinear ordinary differential equation with six degrees under condition (161), which includes two new solutions, are displayed at a computer screen (after running the above program). Here we omit them due to the length of our article. Importing the following Maple program for reducing the two new solutions above, eq246 jie1 :D subs(2 x i sqrt(c[2])  2 C 1 sqrt(c[2]) D eta; eq246 jie1); eq246 jie2 :D subs(2 x i sqrt(c[2])  2 C 1 sqrt(c[2]) D eta; eq246 jie2); the following results are displayed at a computer screen (after running the above program): (11 ())2 D  c2  c4 2 C (tanh())2 c4 2  tanh()  (162) p p c4 2  4c6 c2 c4 2 ((tanh())2  1)   2 ; 4c6 (tanh())2 c2  c4 2 c4 (12 ())2 D  c2  c4 2 C (tanh())2 c4 2 C tanh()  (163) p p c4 2  4c6 c2 c4 2 ((tanh())2  1) 2 : (4c6 (tanh())2 c2  c4 2 )c4

857

858

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Importing the following Maple program for reducing (162) and (163): eq246 jie3 :D subs(sqrt(c[4]2 (tanh(eta)2  1)) D I abs(c[4] sech(eta)); c[4] C tanh(eta)2 c[4] D  sech(eta)2 c[4]; eq246 jie3); eq246 jie4 :D subs(sqrt(c[4]2 (tanh(eta)2  1)) D I abs(c[4] sech(eta)); c[4] C tanh(eta)2 c[4] D  sech(eta)2 c[4]; eq246 jie4); phi[1](x i) :D epsilon (eq246 jie3)( 1/2); phi[2](x i) :D epsilon (eq246 jie4)( 1/2); the following results are displayed at a computer screen (after running the above program): v  u u 2c2 c42 sech2 () C ijc4 u  u u sech()j tanh()p c 2  4c c u 4 6 2 1 () D "t ;   c4 4c6 tanh2 ()c2  c4 2 (164) v  u u 2c2 c4 2 sech2 ()  ijc4 u  u u sech()j tanh()p c 2  4c c u 4 6 2 2 () D "t :   c4 4c6 tanh2 ()c2  c4 2 (165) p

where  D 2 c2 (  C1 ). Step 6 We need to further discuss the results found by sometimes adding new conditions so that the results are simpler in form. For example, we can import the following Maple program for getting rid of the absolute value sign in the above results: assume(c[4] sech(eta) < 0); eq246 jie3 :D 2 c[2] (c[4]2 sech(eta)2 C I

tanh(eta) sqrt(c[4]2  4 c[6] c[2])

abs(c[4] sech(eta)))/(4 c[6] tanh(eta)2

c[2]  c[4]2 )/c[4]; eq246 jie4 :D 2 c[2] (c[4]2 sech(eta)2  I

tanh(eta)sqrt(c[4]2  4 c[6] c[2]) abs(c[4] sech(eta)))/(4 c[6] tanh(eta)2

c[2]  c[4]2 )/c[4]; phi[1; 1](x i) :D e psilon

(eq246 jie3)( 1/2); (

phi[1; 2](x i) :D e psilon (eq246 jie4) 1/2);

the following results are displayed at a computer screen (after running the above program): 1;1 () D v   u u 2c2 sech() c4 sech()  i tanh()p c4 2  4c6 c2 t " ; 4c6 (tanh())2 c2  c4 2 (166) 1;2 () D v   p u u 2c2 sech() c4 sech() C i tanh() c4 2  4c6 c2 t " : 4c6 (tanh())2 c2  c4 2 (167) Importing the following Maple program for reducing and discussing above results: assume(c[4] sech(eta) > 0); eq246 jie3 :D 2 c[2] (c[4]2 sech(eta)2 C I

tanh(eta) sqrt(c[4]2  4 c[6] c[2]) abs(c[4] sech(eta)))/(4 c[6] tanh(eta)2

c[2]  c[4]2 )/c[4]; eq246 jie4 :D 2 c[2] (c[4]2 sech(eta)2  I tanh(eta) sqrt(c[4]2  4 c[6] c[2]) abs(c[4] sech(eta)))/(4 c[6] tanh(eta)2

c[2]  c[4]2 )/c[4]; phi[2; 1](x i) :D e psilon (eq246 jie3)( 1/2); phi[2; 2](x i) :D e psilon (eq246 jie4)( 1/2); the following results are shown at a computer screen (after running the above program): 2;1 () D v   p u u 2c2 sech() c4 sech() C i tanh() c4 2  4c6 c2 t " ; 4c6 (tanh())2 c2  c4 2 (168) 2;2 () D v   p u u 2c2 sech() c4 sech()  i tanh() c4 2  4c6 c2 t " : 4c6 (tanh())2 c2  c4 2 (169) By using this method, we obtained some new types of general solution of a first-order nonlinear ordinary dif-

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

ferential equation with six degrees and presented the following theorem in [40,43,44]. Theorem 5 The following nonlinear ordinary differential equation with six degrees d() D" d

s

c0 C c1 () C c2  2 () C c3  3 () ; Cc4  4 () C c5  5 () C c6  6 () (170)

admits many kinds of fundamental solutions which depend on the values and constraints between c i ; i D 0; 1; 2; 3; 4; 5; 6. Some of the solutions are listed in the following sections. Case 1. If c0 D c1 D c3 D c5 D 0, Eq. (170) admits the following constant solutions

1;2 () D " s 3;4 () D "

c4 C

c4 

p

c4 2  4c6 c2 ; 2c6

(171)

v u u 7;8 () D 2"u u t

p

c4 2  4c6 c2 ; 2c6

(172)

e2

p

c 2 (CC 1 ) c 2 p p 4C c 1 2 4c c e C e4 c 2p p6 2 2e2 c 2 (CC 1 ) c4 C c4 2 e4C 1 c 2

e2

;

(173)

c2 > 0 ;

and the following tan-sec triangular type solutions v h u u 2c2 sec()  c4 sec() u i p u u C" c4 2  4c6 c2 tan() t 5;6 () D " ; c2 < 0 ; 4c6 c2 tan2 () C c4 2 (177)

7;8 () D v h i p u u 2c2 sec() c4 sec()  " c4 2  4c6 c2 tan() t " ; 4c6 c2 tan2 () C c4 2 c2 < 0 ;

c 2 (CC 1 ) c 2 p p 4 c 2 C e4C 1 c 2 4c c e 6 2 p p 2e2 c 2 (CC 1 ) c4 C c4 2 e4 c 2

:

(174)

Case 1.2. If 4c2 c6  c4 2 > 0, Eq. (170) admits the following sinh-cosh hyperbolic type solutions v h i p u u 2c2 c4 C " 4c2 c6  c4 2 sinh() t ; 9;10 () D " 4c2 c6 sinh2 ()  c4 2 cosh2 () c2 > 0 ;

4c6 c2 tanh2 ()  c4 2

;

11;12 () D "

c2 > 0 ; (175)

(179)

v h i p u u 2c2 c4  " 4c2 c6  c4 2 sinh() t 4c2 c6 sinh2 ()  c4 2 cosh2 () c2 > 0 ;

Case 1.1. If 4c2 c6  c4 2 < 0, Eq. (170) admits the following tanh-sech hyperbolic type solutions v h u u 2c sech() c4 sech() 2 u i p u 2 u C"i c4  4c6 c2 tanh() t

(178)

p

p

When we take different values and constraints of 4c2 c6  c4 2 , the solutions (5.43) and (5.44) can be written in different formats as follows:

1;2 () D "

;

where  D 2 c2 (  C1 ) and C1 is any constant,  D p 2 c2 (  C1 ) and C1 is any constant.

and the following exponential type solutions v u u 5;6 () D 2"u u t

4c6 c2 tanh2 ()  c4 2

(176)

c i D constant ; i D 0; 1; 2; 3; 4; 5; 6

s

3;4 () D "

v h u u 2c2 sech() c4 sech() u i p u u "i c4 2  4c6 c2 tanh() t

;

(180)

and the following sin-cos triangular type solutions v h i p u u 2c2 c4 C "i 4c2 c6  c4 2 sin() t ; 13;14 () D " 4c6 c2 sin2 () C c4 2 cos2 () c2 < 0

(181)

v h i p u u 2c2 c4  "i 4c6 c2  c4 2 sin() t 15;16 () D " ; 4c6 c2 sin2 () C 4 2 cos2 () c2 < 0 ;

(182)

859

860

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

p p where  D 2 c2 (  C1 ),  D 2 c2 (  C1 ) and C1 is any constant. Case 1.3. If 4c2 c6  c4 2 D 0, Eq. (170) admits the following tanh hyperbolic type solutions s 17;18;19;20() D " s D"

e"2

p

2c2 c 2 (C 1 )

 c4

  p 2c2 1 C " tanh c2 (  C1 ) p  ; c2 > 0 ; 1  c4  "(1 C c4 ) tanh c2 (  C1 ) (183)

and the following tan triangular type solutions

F j (u1 ; : : : ; u n ; u1;t ; : : : ; u n;t ; u1;x 1 ; : : : ; u n;x m ; u1;t t ; : : : ; u n;t t ; u1;tx 1 ; : : : u n;tx m ; : : :) D 0 ; j D 1; 2; : : : ; n ;

(185)

where j D 1; 2; : : : ; n. By using the following more general transformation, which we first present here, u i (t; x1 ; x2 ; : : : ; x m ) D U i () ;  D u i (t; x1 ; x2 ; : : : ; x m ) D U i () ;  D ˛0 (t) C

m1 X

(186)

˛ i (x i ; x iC1 ; : : : ; x m ; t)  ˇ i (x i ) ;

iD1

s 21;22;23;24() D "

in m C 1 independent variables t; x1 ; x2 ; : : : ; x m ,

p

2c2

e"2i c 2 (C 1 )  c4 s   p 2c2 1 C " tan i c2 (  C1 ) p  ; c2 < 0 D" 1  c4  "(1 C c4 ) tan i c2 (  C1 ) (184)

where C1 is any constant. Case 2. If c5 D c6 D 0, Eq. (170) admits the solutions in Theorem 6.1. Remark 7 By using our mechanization method using Maple to find the exact solutions of a first-order nonlinear ordinary differential equation with any degree, we can obtain some new types of general solution of a firstorder nonlinear ordinary differential equation with r(D 7; 8; 9; 10; 11; 12; : : :) degree [40]. We do not list the solutions here in order to avoid unnecessary repetition. Summary of the Generalized Algebra Method In this section, based on a first-order nonlinear ordinary differential equation with any degree (149) and its the exact solutions obtained by using our mechanization method via Maple, we will develop the algebraic methods [50,51] for constructing the traveling wave solutions and present a new generalized algebraic method and its algorithms [40, 43,44]. The KdV equation is chosen to illustrate our algorithm so that more families of new exact solutions are obtained which contain both non-traveling solutions and traveling wave solutions. We outline the main steps of our generalized algebraic method as follows: Step 1. For given nonlinear differential equations, with some physical fields u i (t; x1 ; x2 ; : : : ; x m ), (i D 1; 2; : : : ; n)

where ˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t) and ˇ i (x i ); i D 1; 2; : : : ; m  1 are functions to be determined later. For example, when n D 1, we may take  D ˛0 (t) C ˛1 (x1 t)  ˇ1 (x1 ) ; where ˛0 (t); ˛1 (x1 t) and ˇ1 (x1 ) are undetermined functions. Then Eq. (185) is reduced to nonlinear differential equations G j (U1 ; : : : ; U n ; U10 ; : : : ; U n0 ; U100 ; : : : ; U n00 ; : : :) D 0 ; j D 1; 2; : : : ; n ;

(187)

where G j ( j D 1; 2; : : : ; n) are all polynomials of U i (i D 1; 2; : : : ; n), ˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t), ˇ i (x i ); i D 1; 2; : : : ; m  1 and their derivatives. If Gk of them is not a polynomial of U i (i D 1; 2; : : : ; n), ˛ i (x i ; x iC1 ; : : : ; x m ; t), ˇ i (x i ); i D 1; 2; : : : ; m  1, ˛0 (t) and their derivatives, then we may use new variable v i ()(i D 1; 2; : : : ; n) which makes Gk become a polynomial of v i ()(i D 1; 2; : : : ; n), ˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t) and ˇ i (x i ); i D 1; 2; : : : ; m 1 and their derivatives. Otherwise the following transformation will fail to seek solutions of Eq. (185). Step 2. We introduce a new variable () which is a solution of the following ODE d() D" d

s

c0 C c1 () C c2  2 () C c3  3 () ; Cc4  4 () C    C cr  r () r D 0; 1; 2; 3; : : : :

(188)

Then the derivatives with respect to the variable  become the derivatives with respect to the variable .

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Step 3. By using the new variable , we expand the solution of Eq. (185) in the form:

U i D a i;0 (X) C

ni X

k

(a i;k (X) ((X))

kD1

C b i;k (X) k ((X))) :

(189)

where X D (x1 ; x2 ; : : : ; x m ; t),  D (X), a i;0(X); a i;k (X); b i;k (X)(i D 1; 2; : : : ; n; k D 1; 2; : : : ; n i ) are all differentiable functions of X to be determined later. Step 4. In order to determine n i (i D 1; 2; : : : ; n) and r, we may substitute (188) into (187) and balance the highest derivative term with the nonlinear terms in Eq. (187). By using the derivatives with respect to the variable , we can obtain a relation for ni and r, from which the different possible values of ni and r can be obtained. These values lead to the series expansions of the solutions for Eq. (185). Step 5. Substituting (189) into the given Eq. (185) and k k collecting qPcoefficients of polynomials of  ;  , and r j  i  k jD1 c j  (), with the aid of Maple, then setting each coefficient to zero, we will get a system of over-determined partial differential equations with respect to a i;0 (X); a i;k (X); b i;k (X); i D 1; 2; : : :, n; k D 1; 2; : : : ; n i c j ; j D 0; 1; : : : ; r˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t) and ˇ i (x i ); i D 1; 2; : : : ; m  1. Step 6. Solving the over-determined partial differential equations with Maple,then we can determine a i;0(X); a i;k (X); b i;k (X); i D 1; 2; : : : ; n; k D 1; 2; : : : ; n i )c j ; j D 0; 1; : : : ; r˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t) and ˇ i (x i ); i D 1; 2; : : : ; m  1 . Step 7. From the constants a i;0(X); a i;k (X); b i;k (X)(i D 1; 2; : : : ; n; k D 1; 2; : : : ; n i )c j ( j D 0; 1; : : : ; r), ˛0 (t); ˛ i (x i ; x iC1 ; : : : ; x m ; t) and ˇ i (x i ); i D 1; 2; : : : ; m  1 obtained in Step 6 to Eq. (188), we can then obtain all the possible solutions. Remark 8 When c5 D c6 D 0 and b i;k D 0, Eq. (170) and transformation (189) just become the ones used in our previous method [50,51]. However, if c5 ¤ 0 or c6 ¤ 0, we may obtain solutions that cannot be found by using the methods [50,51]. It should be pointed out that there is no method to find all solutions of nonlinear PDEs. But our method can be used to find more solutions of nonlinear PDEs, and with the exact solutions obtained by using our mechanization method via Maple, we will develop the algebraic methods [50,51] for constructing the travel-

ing wave solutions and present a new generalized algebraic method and its algorithms [40,43,44]. Remark 9 By the above description, we find that our method is more general than the method in [50,51]. We have improved the method [50,51] in five aspects: First, we extend the ODE with four degrees (131) into the ODE with any degree (188) and get its new general solutions by using our mechanization method via Maple [40,43,44]. Second, we change the solution of Eq. (185) into a more general solution (189) and get more types of new rational solutions and irrational solutions. Third, we replace the traveling wave transformation (146) in [50,51] by a more general transformation (186). Fourth, we suppose the coefficients of the transformation (186) and (189) are undetermined functions, but the coefficients of the transformation (146) in [50,51] are all constants. Fifth, we present a more general algebra method than the method given in [50,51], which is called the generalized algebra method, to find more types of exact solutions of nonlinear differential equations based upon the solutions of the ODE (188). This can obtain more general solutions of the NPDEs than the number obtained by the method in [50,51]. The Generalized Algebra Method to Find New Non-traveling Waves Solutions of the (1 + 1)-Dimensional Generalized Variable-Coefficient KdV Equation In this section, we will make use of our generalized algebra method and symbolic computation to find new nontraveling waves solutions and traveling waves solutions of the following (1C1) – dimensional Generalized Variable – Coefficient KdV equation [16]. Propagation of weakly nonlinear long waves in an inhomogeneous waveguide is governed by a variable – coefficient KdV equation of the form [15] u t (x; t) C 6u(x; t)u x (x; t) C B(t)u x x x (x; t) D 0 : (190) where u(x; t) is the wave amplitude, t the propagation coordinate, x the temporal variable and B(t) is the local dispersion coefficient. The applicability of the variable-coefficient KdV equation (190) arises in many areas of physics as, for example, for the description of the propagation of gravity-capillary and interfacial-capillary waves, internal waves and Rossby waves [15]. In order to study the propagation of weakly nonlinear, weakly dispersive waves in inhomogeneous media, Eq. (190) is rewritten as follows [16] u t (x; t)C6A(t)u(x; t)u x (x; t)CB(t)u x x x (x; t) D 0: (191)

861

862

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

which now has a variable nonlinearity coefficient A(t). Here, Eq. (191) is called a (1 C 1)-dimensional generalized variable-coefficient KdV (gvcKdV) equation. In order to find new non-traveling waves solutions and traveling waves solutions of the following (1 C 1)-dimensional gvcKdV equation (191) by using our generalized algebra method and symbolic computation, we first take the following new general transformation, which we first present here u(x; t) D u() ;

 D ˛(x; t)ˇ(t)  r(t) ;

(192)

where ˛(x; t); ˇ(t) and r(t) are functions to be determined later. By using the new variable  D () which is a solution of the following ODE d() D" d

s

c0 C c1 () C c2  2 () C c3  3 () : (193) Cc4  4 () C c5  5 () C c6  6 ()

we expand the solution of Eq. (191) in the form [40,43]:

u D a0 (X)C

n X

(a i (X) i ((X))Cb i (X) i ((X))) (194)

iD1

where X D (x; t), a0 (X); a i (X); b i (X) (i D 1; 2; : : : ; n) are all differentiable functions of X to be determined later. Balancing the highest derivative term with the nonlinear terms in Eq. (191) by using the derivatives with respect to the variable , we can determine the parameter n D 2 in (194). In addition, we take a0 (X) D a0 (t); a i (X) D a i (t); b i (X) D b i (t); i D 1; 2 in (194) for simplicity, then substituting them and (192) into (194) along with n D 2, leads to: u(x; t) D a0 (t) C a1 (t)() C a2 (t)(())2 C

b1 (t) b2 (t) : C () (())2

(195)

where  D ˛(x; t)ˇ(t)  r(t), and ˛(x; t); ˇ(t); r(t); a0 (t); a i (t) and b i (t); i D 1; 2 are all differentiable functions of x or t to be determined later. By substituting (195) into the given Eq. (191) along with (193) and the derivatives of ,q and collecting coeffiP6 k  j cients of polynomials of  and  jD1 c j  (), with the aid of Maple, then setting each coefficient to zero, we will get a system of over-determined partial differential equations with respect to A(t); B(t); ˛(x; t); ˇ(t); r(t);

a0 (t); a i (t) and b i (t), i D 1; 2 as follows:  3 @ 21B(t)(ˇ(t))3  "a1 (t) ˛(x; t) c4 c6 C 18A(t) @x   @ " ˛(x; t) ˇ(t)a1 (t)a2 (t)c6  3B(t)(ˇ(t))3 @x  3 @  "b1 (t) ˛(x; t) c6 2 D 0 ; @x   d 12b2 (t)  " r(t) c2  8B(t)(ˇ(t))3 dt  3 @  "b2 (t) ˛(x; t) c2 2  2b2 (t) @x   @ " ˛(x; t) ˇ(t)c2  2B(t)ˇ(t) @t  3  @ ˛(x; t) c2  2b2 (t)  "b2 (t) @x 3   d  "˛(x; t) ˇ(t) c2  6A(t) dt   @ " ˛(x; t) ˇ(t)(b1 (t))2 c2 @x   @  12A(t)  " ˛(x; t) ˇ(t)a0 (t)b2 (t)c2  12A(t) @x   @ " ˛(x; t) ˇ(t)(b2 (t))2 c4 D 0 ; @x     d d b1 (t)  " r(t) c6  a1 (t)  " r(t) c4 dt dt   @ C 6A(t)  " ˛(x; t) ˇ(t)a2 (t)b1 (t)c4 @x   d C a1 (t)  "˛(x; t) ˇ(t) c4  b1 (t) dt   d  "˛(x; t) ˇ(t) c6 C 6A(t) dt   @ " ˛(x; t) ˇ(t)a0 (t)a1 (t)c4  6A(t) @x   @ " ˛(x; t) ˇ(t)a0 (t)b1 (t)c6 @x  3 @  4B(t)(ˇ(t))3  "b1 (t) ˛(x; t) c2 c6 @x  3  @ C B(t)ˇ(t)  "a1 (t) ˛(x; t) c4 @x 3   @ C a1 (t)  " ˛(x; t) ˇ(t)c4  b1 (t) @t   @ " ˛(x; t) ˇ(t)c6  6A(t) @t   @ ˛(x; t) ˇ(t)a1 (t)b2 (t)c6 C 7B(t)(ˇ(t))3 " @x

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

 3 @  "a1 (t) ˛(x; t) c2 c4 C 18A(t) @x   @ " ˛(x; t) ˇ(t)a1 (t)a2 (t)c2  B(t)ˇ(t) @x  3  @ ˛(x; t) c6 D 0 ;  "b1 (t) @x 3

Case 4 b2 (t) D 0 ;

A(t) D A(t) ;

˛(x; t) D F1 (t) ; (196)

::::::::::::

B(t) D B(t) ;

ˇ(t) D ˇ(t) ;

a2 (t) D C1 ;

a0 (t) D C4 ; a1 (t) D C2 ;   Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C5 ; dt dt b1 (t) D C3 ;

because there are so many over-determined partial differential equations, only a few of them are shown here for convenience. Solving the over-determined partial differential equations with Maple, we have the following solutions.

where A(t), B(t), F1 (t), ˇ(t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4; 5 are arbitrary constants.

Case 1

Case 5

(200)

A(t) D A(t) ;

B(t) D B(t) ;

˛(x; t) D F1 (t) ;

b2 (t) D 0 ;

ˇ(t) D ˇ(t) ;

a2 (t) D C1 ;

a1 (t) D C2 ;

B(t) D B(t) ;

b2 (t) D C3 ; a0 (t) D C5 ;   Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C6 ; dt dt b1 (t) D C4 ; (197) where A(t), B(t), ˇ(t), F1 (t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4; 5 are arbitrary constants.

A(t) D A(t) ;

˛(x; t) D F1 (t) ;

B(t) D B(t) ;

ˇ(t) D ˇ(t) ;

r(t) D C5 ;

a0 (t) D C4 ; b1 (t) D C3 ;

(201)

˛(x; t) D ˛(x; t) ; where ˛(x; t) are arbitrary functions of x and t, A(t), B(t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4; 5 are arbitrary constants. Case 6 C2 c4 ; ˇ(t) D 0 ; b2 (t) D 0 ; c6 A(t) D A(t) ; B(t) D B(t) ; a2 (t) D C1 ; a1 (t) D C2 ;

a2 (t) D C1 ;

a0 (t) D C4 ; a1 (t) D C2 ;   Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C5 ; dt dt b2 (t) D C3 ; (198) where A(t), B(t), ˇ(t), F1 (t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4 are arbitrary constants. Case 3 ˇ(t) D 0 ;

a1 (t) D C2 ;

A(t) D A(t) ;

a2 (t) D C1 ;

b1 (t) D

Case 2 b1 (t) D 0 ;

ˇ(t) D 0 ;

A(t) D A(t) ;

B(t) D B(t) ;

a2 (t) D C1 ;

a1 (t) D C2 ;

b2 (t) D C3 ;

a0 (t) D C5 ;

b1 (t) D C4 ;

r(t) D C6 ;

(199)

˛(x; t) D ˛(x; t) ; where ˛(x; t) are arbitrary functions of x and t, A(t), B(t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4; 5 are arbitrary constants.

r(t) D C4 ;

(202)

a0 (t) D C3 ;

˛(x; t) D ˛(x; t) ; where ˛(x; t) are arbitrary functions of x and t, A(t), B(t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4 are arbitrary constants. Case 7 B(t) D B(t) ; A(t) D 0 ;

b1 (t) D 0 ;

a0 (t) D C3 ;

b2 (t) D C2 ;

a1 (t) D 0 ; a2 (t) D 0 ; C1 ˛(x; t) D F1 (t)x C F2 (t) ; ˇ(t) D ; F1 (t)    d d C1 ; F1 (t) dt F2 (t)  dt F1 (t)  Z F2 (t) C 4c2 B(t)C1 2 (F1 (t))2 r(t) D dt C C4 ; (F1 (t))2 (203) where F1 (t), B(t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4 are arbitrary constants.

863

864

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

So we get new general forms of solutions of equations (191):

Case 8 B(t) D B(t) ;

b1 (t) D 0 ;

˛(x; t) D F1 (t) ;

ˇ(t) D ˇ(t) ;

A(t) D 0 ;

a1 (t) D 0 ;

u(x; t) D a0 (t) C a1 (t)(˛(x; t)ˇ(t)  r(t))

a2 (t) D 0 ;   Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C3 ; dt dt a0 (t) D C2 ;

b2 (t) D C1 ;

C a2 (t)((˛(x; t)ˇ(t)  r(t)))2 C

b1 (t) b2 (t) : C (˛(x; t)ˇ(t)  r(t)) ((˛(x; t)ˇ(t)  r(t)))2 (208)

(204) where A(t), B(t), ˇ(t), F1 (t) are arbitrary functions of t, and Ci , i D 1; 2; 3 are arbitrary constants. Case 9 A(t) D A(t) ;

B(t) D B(t) ;

˛(x; t) D F1 (t) ; a2 (t) D C1 ;

b1 (t) D 0 ;

ˇ(t) D ˇ(t) ;

a0 (t) D C3 ;

b2 (t) D C2 ;

a1 (t) D 0 ;   Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C4 ; dt dt

(205)

where A(t), B(t), ˇ(t), F1 (t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4 are arbitrary constants. Case 10 A(t) D A(t) ;

B(t) D B(t) ;

˛(x; t) D F1 (t) ;

b1 (t) D 0 ;

ˇ(t) D ˇ(t) ;

a2 (t) D C1 ; b2 (t) D 0 ; a1 (t) D 0 ;   Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C3 ; dt dt

where ˛(x; t), ˇ(t), r(t), a0 (t), a i (t), b i (t), i D 1; 2 satisfy (197)–(207) respectively, and the variable (˛(x; t)ˇ(t)  r(t)) takes the solutions of the Eq. (193). For example, we may take the variable (˛(x; t)ˇ(t)r(t)) as follows: Type 1. If 4c2 c6  c4 2 < 0, corresponding to Eq. (193), (˛(x; t)ˇ(t)  r(t)) is taken, we get the following four tanh-sech hyperbolic type solutions 1;2 ((˛(x; t)ˇ(t)  r(t))) D s p 2c2 sech()[c4 sech() C "i c4 2  4c6 c2 tanh()] ; " 4c6 c2 tanh2 ()  c4 2 c2 > 0 ;

(209)

3;4 ((˛(x; t)ˇ(t)  r(t))) D s p 2c2 sech()[c4 sech()  "i c4 2  4c6 c2 tanh()] ; " 4c6 c2 tanh2 ()  c4 2 c2 > 0 ;

(210)

and the following tan-sec triangular type solutions

a0 (t) D C2 ; (206) where A(t), B(t), ˇ(t), F1 (t) are arbitrary functions of t, and Ci , i D 1; 2; 3; 4 are arbitrary constants.

c2 < 0 ;

Case 11 A(t) D A(t) ; a2 (t) D C1 ; ˇ(t) D 0 ;

5;6 ((˛(x; t)ˇ(t)  r(t))) D s p 2c2 sec()[c4 sec() C " c4 2  4c6 c2 tan()] " ; 4c6 c2 tan2 () C c4 2

B(t) D B(t) ; b2 (t) D 0 ;

b1 (t) D 0 ;

a1 (t) D 0 ;

r(t) D C3 ;

˛(x; t) D ˛(x; t) ;

(207)

a0 (t) D C2 ;

7;8 ((˛(x; t)ˇ(t)  r(t))) D s p 2c2 sec()[c4 sec()  " c4 2  4c6 c2 tan()] " ; 4c6 c2 tan2 () C c4 2 c2 < 0 ;

where ˛(x; t) are arbitrary functions of x and t, A(t), B(t) are arbitrary functions of t, and Ci , i D 1; 2 are arbitrary constants. Because there are so many solutions, only a few of them are shown here for convenience.

(211)

(212)

p where  D 2 c2 ((˛(x; t)ˇ(t)  r(t))  C1 ) and C1 is any p constant,  D 2 c2 ((˛(x; t)ˇ(t)  r(t))  C1 ) and C1 is any constant.

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Type 2. If 4c2 c6  c4 2 > 0, corresponding to Eq. (193), (˛(x; t)ˇ(t)  r(t)) is taken, we get the following four sinh-cosh hyperbolic type solutions 9;10 ((˛(x; t)ˇ(t)  r(t))) s p 2c2 [c4 C " 4c2 c6  c4 2 sinh()] D" ; 4c2 c6 sinh2 ()  c4 2 cosh2 ()

c2 > 0 ; (213)

11;12 ((˛(x; t)ˇ(t)  r(t))) s p 2c2 [c4  " 4c2 c6  c4 2 sinh()] ; D" 4c2 c6 sinh2 ()  c4 2 cosh2 ()

Substituting (209)–(218) and (197)–(207) into (208), respectively, we get many new irrational solutions and rational solutions of combined hyperbolic type or triangular type solutions of Eq. (191). For example, when we select A(t); B(t); ˛(x; t); ˇ(t); r(t); a0 (t); a i (t) and b i (t); i D 1; 2 to satisfy Case 1, we can easily get the following irrational solutions of combined hyperbolic type or triangular type solutions of Eq. (191): u(x; t) D C5 C C2 (F1 (t)ˇ(t)  r(t))

c2 > 0 ;

C C1 ((F1 (t)ˇ(t)  r(t)))2 C4 C3 C ; C (F1 (t)ˇ(t)  r(t)) ((F1 (t)ˇ(t)  r(t)))2 (219)

(214) and the following sin-cos triangular type solutions 13;14 ((˛(x; t)ˇ(t)  r(t))) s p 2c2 [c4 C "i 4c2 c6  c4 2 sin()] D" ; 4c6 c2 sin2 () C c4 2 cos2 ()

c2 < 0 (215)

15;16 ((˛(x; t)ˇ(t)  r(t))) s p 2c2 [c4  "i 4c6 c2  c4 2 sin()] D" ; 4c6 c2 sin2 () C 4 2 cos2 ()

c2 < 0; (216)

p where  D 2 c2 ((˛(x; t)ˇ(t)  r(t))  C1 ),  D p 2 c2 ((˛(x; t)ˇ(t)  r(t))  C1 ),and C1 is any constant. Type 3. If 4c2 c6  c4 2 D 0, corresponding to Eq. (193), (˛(x; t)ˇ(t)  r(t)) is taken, we get the following two sech-tanh hyperbolic type solutions 4c2 c6  c4 2 D 0, Eq. (170) admits the following tanh hyperbolic type solutions 17;18;19;20(˛(x; t)ˇ(t)  r(t)) D s p 2c2 f1 C " tanh[ c2 (˛(x; t)ˇ(t)  r(t)  C1 )]g ; " p 1  c4 "(1 C c4 ) tanh[ c2 (˛(x; t)ˇ(t)  r(t)  C1 )] c2 > 0 ;

(217)

and the following tan triangular type solutions 21;22;23;24(˛(x; t)ˇ(t)  r(t)) D v u 2c f1 C " tan[i pc (˛(x; t)ˇ(t)  r(t)  C )]g 2 2 1 u "u ; t 1  c4 p "(1 C c4 ) tan[i c2 (˛(x; t)ˇ(t)  r(t)  C1 )] c2 < 0 ; where C1 is any constant.

where A(t); B(t); ˇ(t); F1 (t) are arbitrary functions of t, Ci , i D 1; 2; 3; 4; 5 are arbitrary constants, and Z r(t) D

F1 (t)

d ˇ(t) C dt



 d F1 (t) ˇ(t)dt C C6 : dt

Substituting (209)–(218) into (219), respectively, we get the following new irrational solutions and rational solutions of combined hyperbolic type or triangular type solutions of Eq. (191). Case 1. If 4c2 c6  c4 2 < 0; c2 > 0, corresponding to (209), (210) and (197), Eq. (191) admits the following four tanh-sech hyperbolic type solutions u1;2 (x; t) D C5 C r "

C4

p

2c 2 sech()[c 4 sech() C"i c 42 4c 6 c 2 tanh()] 4c 6 c 2 tanh2 ()c 4 2

C3 (4c6 c2 tanh2 ()  c4 2 ) p 2c2 sech()[c4 sech() C "i c4 2  4c6 c2 tanh()] v u u 2c2 sech() u u p u [c4 sech() C "i c4 2  4c6 c2 tanh()] t C C2 " 4c6 c2 tanh2 ()  c4 2

C

C

p 2c2 C1 sech()[c4 sech() C "i c4 2  4c6 c2 tanh()] 4c6 c2 tanh2 ()  c4 2

;

(220)

(218) and

865

866

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

u3;4 (x; t) D C5 C3 (4c6 c2 tanh2 ()  c4 2 ) p 2c2 sech()[c4 sech()  "i c4 2  4c6 c2 tanh()] C4 C r p sech()"i c 4 2 4c 6 c 2 tanh()] " 2c 2 sech()[c 44c 2 2 6 c 2 tanh ()c 4 v u u 2c2 sech() u p u [c4 sech()  "i c4 2  4c6 c2 tanh()] t C C2 " 4c6 c2 tanh2 ()  c4 2 p 2c2 C1 sech()[c4 sech()  "i c4 2  4c6 c2 tanh()] C ; 4c6 c2 tanh2 ()  c4 2 (221) C

p 2c2 C1 sec()[c4 sec()  " c4 2  4c6 c2 tan()] C ; 4c6 c2 tan2 () C c4 2 (223) p where  D 2 c2 ((˛(x; t)ˇ(t)  r(t))  C1 ) and the rest of the parameters are the same as with Case 1. Case 3. If 4c2 c6  c4 2 > 0; c2 > 0, corresponding to (213), (214) and (197), Eq. (191) admits the following four sinh-cosh hyperbolic type solutions u9;10 (x; t) D C5 C C2 "

p

where  D 2 c2 ((˛(x; t)ˇ(t)  r(t))  C1 ), A(t); B(t); ˇ(t); F1 (t) are arbitrary functions of t, Ci , i D 1; 2; 3; 4; 5 are arbitrary constants, and   Z d d r(t) D F1 (t) ˇ(t) C F1 (t) ˇ(t)dt C C6 : dt dt Case 2. If 4c2 c6  c4 2 < 0; c2 < 0, corresponding to (211), (212) and (197), Eq. (191) admits the following four tan-sec triangular type solutions u5;6 (x; t) D C5 C r "

C4 p 2c 2 sec()[c 4 sec()C" c 4 2 4c 6 c 2 tan()] 2 4c 6 c 2 tan ()Cc 4 2

C3 (4c6 c2 tan2 () C c4 2 ) p 2c2 sec()[c4 sec() C " c4 2  4c6 c2 tan()] s p 2c2 sec()[c4 sec() C " c4 2  4c6 c2 tan()] C C2 " 4c6 c2 tan2 () C c4 2 p 2c2 C1 sec()[c4 sec() C " c4 2  4c6 c2 tan()] C ; 4c6 c2 tan2 () C c4 2 (222) C

s

C

p 2c2 [c4 C " 4c2 c6  c4 2 sinh()]

4c2 c6 sinh2 ()  c4 2 cosh2 () p 2c2 C1 [c4 C " 4c2 c6  c4 2 sinh()]

4c2 c6 sinh2 ()  c4 2 cosh2 () C4 C r p 4c 2 c 6 c 4 2 sinh()] " 2c4c2 [cc4 C" 2 2 2 2 6 sinh ()c 4 cosh () p 2c2 C3 [c4 C " 4c2 c6  c4 2 sinh()] C ; 4c2 c6 sinh2 ()  c4 2 cosh2 ()

(224)

and u11;12 (x; t) D C5 C C2 " C

s

p 2c2 [c4 C " 4c2 c6  c4 2 sinh()]

4c2 c6 sinh2 ()  c4 2 cosh2 () p 2c2 C1 [c4 C " 4c2 c6  c4 2 sinh()]

4c2 c6 sinh2 ()  c4 2 cosh2 () C4 C r p 2 " 2c 2 [c 4 C" 24c 2 c 6 c24 sinh()] 4c 2 c 6 sinh ()c 4 cosh2 () p 2c2 C3 [c4 C " 4c2 c6  c4 2 sinh()] C ; 4c2 c6 sinh2 ()  c4 2 cosh2 ()

(225)

p where  D 2 c2 ((˛(x; t)ˇ(t)  r(t))  C1 ) and the rest of the parameters are the same as with Case 1.

and u7;8 (x; t) D C5 C r "

C4 p 2c 2 sec()[c 4 sec()" c 4 2 4c 6 c 2 tan()] 2 4c 6 c 2 tan ()Cc 4 2

C3 (4c6 c2 tan2 () C c4 2 ) p 2c2 sec()[c4 sec()  " c4 2  4c6 c2 tan()] s p 2c2 sec()[c4 sec()  " c4 2  4c6 c2 tan()] C C2 " 4c6 c2 tan2 () C c4 2

C

Case 4. If 4c2 c6  c4 2 > 0; c2 < 0, corresponding to (215), (216) and (197), Eq. (191) admits the following four sin-cos triangular type solutions u13;14 (x; t) C4 D C5 C r p 2 4 C"i 4c 2 c 6 c 4 sin()] " 2c 2 [c 4c c sin2 ()Cc 2 cos2 () 6 2

4

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

C3 (4c6 c2 sin2 () C c4 2 cos2 ()) p 2c2 [c4 C "i 4c2 c6  c4 2 sin()] s p 2c2 [c4 C "i 4c2 c6  c4 2 sin()] C C2 " 4c6 c2 sin2 () C c4 2 cos2 () i h p 2c2 C1 c4 C "i 4c2 c6  c4 2 sin() C ; 4c6 c2 sin2 () C c4 2 cos2 ()

tan triangular type solutions

C

u21;22;23;24 (x; t)

v u u u D C 5 C C 2  "t

h  i 2c2 1 C " tan i2   1  c4  "(1 C c4 ) tan i2 h  i 2c2 C1 1 C " tan i2    1  c4  "(1 C c4 ) tan i2

(226)

and u15;16 (x; t) D C5 C r "

C4 i h p 2c 2 c 4 "i 4c 2 c 6 c 4 2 sin()

"

h

p where  D 2 c2 ((˛(x; t)ˇ(t)  r(t))  C1 ) and the rest of the parameters are the same as with Case 1.

u17;18;19;20 (x; t)

Remark 10 We may further generalize (194) as follows: u D a0 (X)

C

m X

P

P

r i1 CCr in Di a r i1 ;:::;r ni (X) r i1 r ni G1i (1i (X)) : : : G ni ( ni (X))

l i1 CCl in Di b l i1 ;:::;l ni (X) l i1 l ni (1i (X)) : : : G ni ( ni (X)) G1i

;

i C c i (X) (230)

P where l i1 CCl in Di b2l i1 ;:::;l ni (X) C c 2i (X) ¤ 0, a0 (X), 1i (X); : : :,  ni (X), a r i1 ;:::;r ni (X), b l i1 ;:::;l ni (X) and c i (X); i D 0; 1; 2; : : : ; m are all differentiable functions to be determined later,  is a constant, G1i (1i (X)); : : :, G ni ( ni (X)) are all ( ki (X)) or  1 ( ki (X)) or some derivatives  ( j) ( ki (X)); k D 1; 2; : : : ; n;i D 0; 1; 2; : : : ; m; j D ˙1; ˙2; : : :. We can get many new explicit solutions of Eq. (191).

s

   2c2 1 C " tanh 2   D C5 C C2  " 1  c4  "(1 C c4 ) tanh 2    2c2 C1 1 C " tanh 2   C 1  c4  "(1 C c4 ) tanh 2 C4 C r  2c 1C" tanh( 2 ) " 1c 2"(1Cc ) tanh ( 2 ) 4 4    C3 1  c4  "(1 C c4 ) tanh 2     ; C 2c2 1 C " tanh 2

(229)

p where  D 2 c2 ((˛(x; t)ˇ(t)  r(t))  C1 ) and the rest of the parameters are the same as with Case 1.

iD0

Case 5. If 4c2 c6  c4 2 D 0; c2 > 0, corresponding to (217) and (197), Eq. (191) admits the following four tanh hyperbolic type solutions

C4 h  i 2c 2 1C" tan i 2   1c 4 "(1Cc 4 ) tan i 2

   C3 1  c4  "(1 C c4 ) tan i2 h  i ;  2c2 1 C " tan i2

4c 6 c 2 sin2 ()Cc 4 2 cos2 () C3 (4c6 c2 sin2 () C c4 2 cos2 ())

i p 2c2 c4  "i 4c2 c6  c4 2 sin() v i h p u u 2c2 c4  "i 4c2 c6  c4 2 sin() t C C2 " 4c6 c2 sin2 () C c4 2 cos2 () h i p 2c2 C1 c4  "i 4c2 c6  c4 2 sin() C ; (227) 4c6 c2 sin2 () C c4 2 cos2 ()

C

s

C

(228)

p where  D 2 c2 ((˛(x; t)ˇ(t)  r(t))  C1 ) and the rest of the parameters are the same as with Case 1. Case 6. If 4c2 c6  c4 2 D 0; c2 < 0, corresponding to (218) and (197), Eq. (191) admits the following two sec-

A New Exp-N Solitary-Like Method and Its Application in the (1 + 1)-Dimensional Generalized KdV Equation In this section, in order to develop the Exp-function method [31], we present two new generic transformations, a new Exp-N solitary-like method, and its algorithm [40, 47]. In addition, we apply our method to construct new exact solutions of the (1 C 1)-dimensional classical generalized KdV(gKdV) equation.

867

868

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Summary of the Exp-N Solitary-Like Method In the following we would like to outline the main steps of our Exp-N solitary-like method [40,47]: Step 1. For a given NLEE system with some physical fields u m (t; x1 ; x2 ; : : : ; x n1 ), (m D 1; 2; : : : ; n) in n independent variables t; x1 ; x2 ; : : : ; x n1 , Fm (u1 ; u2 ; : : : ; u n ; u1;t ; u2;t ; : : : ; u n;t ; u1;x 1 ; u2;x 1 ; : : : ; u n;x 1 ; u1;t t ; u2;t t ; : : : ; u n;t t ; : : :) D 0; (m D 1; 2; : : : ; n) :

(231)

We introduce a new generic transformation 8 ˆ
jDc

a m j (X)ex p( j1 )

kDp

b mk (X)ex p(k2 )

u m (1 ; 2 ) D g m (X) C P q

(m D 1; 2; : : : ; n) ;

; (233)

where c; d; p and q are any positive integers, 1 ; 2 ; g m (X); a m j (X); j D c; : : : ; d, and b mk (X); k D p; : : : ; q; m D 1; 2; : : : ; n are functions to be determined later. Step 3. We take a numerical value of c; d; p and q, substitute (233) into the partial differential equations obtained in Step 1, and then set all coefficients of exp(k); k D 0; 1; 2; : : : to zero to get an over-determined partial differential equation with respect to g m (X); a m j (X); j D c; : : : ; d, b mk (X); k D p; : : : ; q; m D 1; 2; : : : ; n, p i j (t; x1 ; x2 ; : : : ; x i1 ); q i j (x i ); j D 1; 2; i D 1; 2; : : : ; n  1 and p0 j (t).

Step 4. Solving the over-determined partial differential equations in Step 3 by use of Maple, we would end up with explicit expressions for g m (X); a m j (X); j D c; : : : ; d, b mk (X); k D p; : : : ; q; m D 1; 2; : : : ; n, p i j (t; x1 ; x2 ; : : : ; x i1 ); q i j (x i ); j D 1; 2; i D 1; 2; : : : ; n  1 and p0 j (t). Step 5. Substituting the results obtained in Step 4 into (232) and (233), we can then obtain new rational solutions of Exp-functions as follows u m (X) D g m (X)C Pd a (X) exp   jDc m j P p (t; x ; x ; : : : ; x )  q (x ) j p01 (t) C n1 i1 1 2 i1 i1 i iD1 ; Pq b (X) exp mk kDp    P k p02 (t) C n1 iD1 p i2 (t; x 1 ; x2 ; : : : ; x i1 )  q i2 (x i ) (m D 1; 2; : : : ; n)

(234)

Step 6. If we again take a numerical value of c; d; p and q and repeat Step 3, Step 4, and Step 5, then we can obtain other rational solutions of Exp-functions Eq. (231). Now, repeat the process. We can obtain a family of new N solitary-like solutions of Eq. (231). Remark 11 The new transformations (232) and (233) proposed here are more general than the transformation in the Exp-function method [31]. We have obtained many families of non-traveling waves solutions and traveling waves solutions of the (2 C 1)-dimensional generalized KdV–Burgers equation by using the two transformations in [40,47]. Remark 12 Our methods are more powerful, simpler, and more convenient than the Exp-function method [31]. The Exp-function method [31] is used only to obtain a traveling waves solution of lower dimensional NLPDE because it makes use of the homogeneous balance method. In this paper we improve the method to be more powerful such that it can be used to obtain many families of non-traveling waves solutions and traveling waves solutions of higher dimensional NLPDEs because we don’t use the homogeneous balance method and make use of our transformations (232) and (233). The Application of the Exp-N Solitary-Like Method in the (1 + 1)-Dimensional Generalized KdV Equation In this section, we will present a new transformation, and then make use of the transformation and our method via symbolic computation to find the solutions of the (1 C

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

1)-dimensional classical generalized Korteweg–de Vries (gKdV) equation. The (1 C 1)-dimensional gKdV equation has the following form [7,8,9]: u t (x; t) C ˛u p (x; t)u x (x; t) C ˇu x x x (x; t) D 0 ; (235) where the coefficient of the nonlinear term ˛ and the coefficient of the dispersive term ˇ are independent of x and t. According to the above steps, to seek traveling wave solutions of Eq. (235), we present the following new transformation here. When p  2 2

u(x; t) D f p (x; t) :

(236)

By substituting (236) into Eq. (235), we get the following equation: p2 f 2 (x; t) f t (x; t) C ˛ p2 f 4 (x; t) f x (x; t) C 4ˇ f x3(x; t) C 6pˇ f (x; t) f x (x; t) f x x (x; t)  6pˇ f x3 (x; t) C ˇ p2 f x x x (x; t) f 2 (x; t)  3ˇ p2 f (x; t) f x (x; t) f x x (x; t) C 2ˇ p2 f x3 (x; t) D 0 : (237) We introduce a new generic transformation [40,47] Pm 2

iDm 1

k i e i(˛1 xCˇ1 tC1 )

jDn 1

h j e j(˛2 xCˇ2 tC2 )

f (x; t) D g C Pn 2

;

(238)

where m1 ; m2 ; n1 ; n2 are any positive integers, g, ˛1 ; ˇ1 , ˛2 ; ˇ2 ; 1 ; 2 , k i ; i D m1 ; : : : ; m2 , and h j ; j D n1 ; : : : ; n2 are all constants to be determined later. With the aid of Maple, substituting (238) into (237) and setting  D ˛1 x Cˇ1 t C 1 ;  D ˛2 x Cˇ2 t C 2 , yields the following equation (because the equation is so long, just part of the equation is shown here for simplification) P 4 P Pn 2 m2 i 2 i j ˛ p2 ˛1 m iDm 1 k i e iDm 1 k i ie jDn 1 h j e P 4 P n2 m2 j i  p2 jDn 1 h j e iDm 1 k i e ˇ2 Pn 2 j 2 2 jDn 1 h j je ˇ p g P 5 P n2 2 3 i j C ˛1 3 m g2 iDm 1 k i i e jDn 1 h j e P i 3 2  ˇ p2 m iDm 1 k i e ˛2  4 Pn 2 Pn 2 3 j j g2 jDn 1 h j j e jDn 1 h j e P 5 P n2 j 2 i C ˛ p2 g 4 ˛1 m jDn 1 h j e iDm 1 k i ie   4 Pn2 P 2 i j C 6 ˇ p˛1 3 m iDm 1 k i ie jDn 1 h j e Pm 2 2 i iDm 1 k i i e g P 2 P 2 m2 i C6ˇ p k e ˛2 3 njDn h j je j i iDm 1 1

2 n2 j g jDn 1 h j e 2 P 3 n i 2 j  12 ˇ p ˛2 3 iDm 1 k i e jDn 1 h j je P 2 g njDn h j e j P1m2 P 2 C 12 ˇ ˛1 iDm1 k i ie i njDn h j e j 1 P 2 P 2 m2 n2 i j 2 k e ˛ h je i 2 j iDm 1 jDn 1 P 2 P 2 m2 n2 i j 2  12 ˇ ˛1 iDm 1 k i ie jDn 1 h j e Pm2 Pn 2 i j iDm 1 k i e ˛2 jDn 1 h j je P 2 P 3 n2 m2 j i C 4 ˛ p2 g ˛1 jDn 1 h j e iDm 1 k i e Pm2 i iDm 1 k i ie P 3 P 2 n2 m2 j i  4 ˛ p2 g 3 h e k e ˛2 j i jDn 1 iDm 1 Pn2 j jDn 1 h j je P 3 P 2 n2 m2 j i C 6 ˛ p2 g 2 ˛1 jDn 1 h j e iDm 1 k i e Pm2 i iDm 1 k i ie P 3 P 2 n2 m2 j i  2 p2 ˇ2 jDn 1 h j e iDm 1 k i e Pn2 j jDn 1 h j je g P 4 P 2 n2 i j  3 ˇ p2 ˛1 3 m iDm 1 k i ie jDn 1 h j e Pm2 2 i iDm 1 k i i e g P 4 P 2 m2 j i  4 ˛ p2 g njDn h e k e j i iDm 1 1 P 2 ˛2 njDn h j je j 1 P 4 P 2 n2 3 e i j C 2 ˇ p2 ˛1 3 m k i h e i j iDm 1 jDn 1 P 2 i g m k e i iDm 1 P 3 P 2 n2 i 3 j  6 ˇ p2 m iDm 1 k i e ˛2 jDn 1 h j je P 2 n2 j g2 jDn 1 h j e P 3 P 2 m2 i C 3 ˇ p2 ˛2 3 njDn h j je j iDm 1 k i e 1 Pn2 P n2 2 j j jDn 1 h j j e jDn 1 h j e 3 P P 2 m2 i C 6ˇ p k e ˛2 3 njDn h j je j i iDm 1 1 Pn2 P n2 2 j j jDn 1 h j j e jDn 1 h j e  2 Pn 2 P 2 i 2 j C 6 ˇ p2 ˛1 m iDm 1 k i ie ˛2 jDn 1 h j je P 3 n2 j g2 jDn 1 h j e P Pn 2 2 i 2 2 j  3 ˇ p2 ˛1 m iDm 1 k i ie ˛2 jDn 1 h j j e P 4 n2 j g2 jDn 1 h j e P 3 P 2 n2 i j C 6 ˇ p˛1 3 m k ie h e i j iDm 1 jDn 1 Pm2 Pm 2 2 e i i k i k e i i iDm 1 iDm 1 P 3 P 3 m2 n2 i j  6 ˇ p˛1 3 iDm 1 k i ie jDn 1 h j e P 3 P 3 m2 n2 i j  2 ˇ p2 ˛2 3 iDm 1 k i e jDn 1 h j je Pn2

h j j2 e j P m2

jDn 1

P

869

870

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

P

P 3 n2 j ˛2 3 jDn 1 h j je P 3 P 3 m2 n2 i j C 2 ˇ p2 ˛1 3 iDm 1 k i ie jDn 1 h j e P 2 P 3 n2 m2 j i  6 ˛ p2 g 2 jDn 1 h j e iDm 1 k i e P 2 ˛2 njDn h j je j C P 1 2 P 2 m2 n2 i j k ie h e 6 ˇ p˛1 2 i j iDm 1 jDn 1 Pm2 Pn 2 i ˛ j k e h je i 2 j iDm 1 jDn 1 P 2 P 2 m2 i C 9 ˇ p2 ˛2 3 njDn h j je j iDm 1 k i e 1   2 Pn2 Pn2 2 j j g jDn 1 h j j e jDn 1 h j e P 2 P m2 n i 2  6ˇ p ˛2 jDn h j je j iDm 1 k i e 1  2 Pn 2 P 2 2 i j ˛1 2 m iDm 1 k i i e jDn 1 h j e P 2 P 3 m2 n2 i j C 6 ˇ p2 ˛1 2 iDm 1 k i ie jDn 1 h j e P 2 ˛2 njDn h j je j g 1 P 2 P 2 n2 i j k ie h e  6 ˇ p˛1 m i j iDm 1 jDn 1 P 2 Pn2 m2 i 2 j 2 ˛2 iDm 1 k i e jDn 1 h j j e Pm2 P 2 C 6 ˇ p˛1 iDm1 k i ie i njDn h j e j 1 P 2 P 2 m2 n2 i j ˛2 2 iDm 1 k i e jDn 1 h j je P 2 P 2 m2 i  2 ˇ p2 ˛2 3 njDn h j j3 e j iDm 1 k i e 1 P 3 n2 j g jDn 1 h j e P 4 P n2 m2 j i C 4 ˛ p2 g 3 jDn 1 h j e iDm 1 k i e Pm2 ˛1 iDm1 k i ie i P 4 P n2 j 2 i C 2 p2 h e ˇ1 m j jDn 1 iDm 1 k i ie Pm2 g iDm1 k i e i P 3 P 2 n2 i j  6 ˇ p˛1 m iDm 1 k i ie jDn 1 h j e Pm2 Pn 2 i 2 2 j iDm 1 k i e ˛2 jDn 1 h j j e g P 2 P 3 m2 n2 i j 3  6 ˇ p2 k e ˛ h je i 2 j iDm 1 jDn 1 P 2 j g njDn h e j 1 P 3 P 2 n2 i j  3 ˇ p2 ˛1 3 m iDm 1 k i ie jDn 1 h j e Pm2 Pm 2 2 i i iDm 1 k i i e iDm 1 k i e  3 Pn 2 P 2 i j  3 ˇ p2 ˛1 m k ie h e i j iDm 1 jDn 1 Pm2 Pn 2 i ˛ 2 2 j k e h i 2 iDm 1 jDn 1 j j e g  2 Pn2 P 2 i j C 24 ˇ p˛1 m iDm 1 k i ie jDn 1 h j e P 2 Pm2 n2 i 2 j g iDm 1 k i e ˛2 jDn 1 h j je Pn2 Pm2 i j  6 ˇ p iDm1 k i e ˛2 jDn 1 h j je ˛1 2 P 3 Pm2 n2 2 e i j k i h e g i j iDm 1 jDn 1  6ˇ p

m2 iDm 1

k i e i

3

 3 ˇ p2 Pm2 

Pm 2

iDm 1

iDm 1 3 ˇ p2 ˛1 2

P

ki i2e Pm 2

n2 jDn 1

k i e i ˛2 P n2 i

Pn2

h j je j ˛1 2 3 j g jDn 1 h j e Pn2 2 i k i i e ˛2 jDn 1 h j je j jDn 1

iDm 1 4 h j e j g 2

P 2 P 3 m2 n2 i j  12 ˇ p˛1 2 iDm 1 k i ie jDn 1 h j e P 2 ˛2 njDn h j je j g P m21 Pn2 2 i 3 j C 6ˇ p iDm 1 k i e ˛2 jDn 1 h j je   3 Pn2 Pn2 2 j j g2 jDn 1 h j j e jDn 1 h j e P 3 P n2 j 2 i C p2 ˇ1 m jDn 1 h j e iDm 1 k i ie P 2 m2 i iDm 1 k i e P 3 P 2 n2 3 i j C ˇ p2 ˛1 3 m iDm 1 k i i e jDn 1 h j e P 2 m2 i iDm 1 k i e P 3 P 2 m2 i  ˇ p2 ˛2 3 njDn h j j3 e j iDm 1 k i e 1 P 2 n2 j jDn 1 h j e P 3 P 3 m2 n2 i j  4ˇ ˛2 3 iDm 1 k i e jDn 1 h j je P 3 P 3 m2 n2 i j C 4 ˇ ˛1 3 iDm 1 k i ie jDn 1 h j e P 2 P 3 n2 m2 j i  p2 jDn 1 h j e iDm 1 k i e P 2 ˇ2 njDn h j je j P 1 5 P m2 i 2  ˛ p2 k e ˛2 njDn h j je j C p2 i iDm 1 1

0 @

n2 X jDn 1

15 hje 0

 ˛ p2 g 4 @

j A

ˇ1

m2 X

k i ie i g 2

iDm 1 n2 X jDn 1

14

h j e j A

m2 X iDm 1

k i e i ˛2

n2 X

h j je j

jDn 1

D 0:

(239)

where m1 ; m2 ; n1 ; n2 are all any positive integers which we may take arbitrarily. For example, we can take n2 D 2;m2 D 2;n1 D 1; m1 D 1 and substitute them into (239), and then set all coefficients of ex p(k) and ex p( j); j; k D 0; 1; 2; : : : to zero to get an over-determined equation with respect to h1 ; h0 ; h1 ; h2 ; k0 ; k1 ; k2 ; k1 ; ˛1 ; ˇ1 ; ˛2 ; ˇ2 and g. Solving the over-determined equations by use of Maple, we obtain the following solutions of h1 ; h0 ; h1 ; h2 ; k0 ; k1 ; k2 ; k1 ; ˛1 ; ˇ1 ; ˛2 ; ˇ2 and g:

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Case 6.

Case 1. 4ˇ˛2 3 ˇ2 D  2 ; p

p2 ˛k0 2 h1 D ; 8˛2 2 ˇh1 (3p C 2 C p2 )

k1 D 0 ;

g D 0 ; 2 D 2 ;

k0 D k0 ;

˛1 D ˛1 ;

h1 D h1 ;

˛2 D ˛2 ;

ˇ1 D ˇ1 ;

1 D 1 ;

k2 D 0 ;

k1 D 0 ;

h0 D 0 ;

h2 D 0 ;

k0 D k0 ;

˛1 D ˛1 ;

ˇ1 D ˇ1 ;

1 D 1 ;

k2 D k2 ;

k1 D k1 ;

h1 D 0 ;

ˇ2 D ˇ2 ;

˛2 D ˛2 ;

k1 D k1 ;

h0 D 0 ;

(240)

h2 D 0 ;

h1 D 0 ;

ˇ2 D ˇ2 ; h2 D 0 ;

1 D 1 ; h2 D 0 ;

2 D 2 ;

h1 D h1 ; k1 D 0 ;

h1 D 0 ;

˛1 D ˛1 ; h0 D 0 ;

k0 D 0 ;

gDg;

k1 D k1 ;

(245)

h1 D h1 ; ˛2 D ˛2 ;

k2 D 0 ;

k1 D k1 ;

h0 D 0 ;

h1 D 0 ;

k0 D 0 ;

gDg;

(246)

Case 8.

ˇ1 D 0 ;

k0 D k0 ; ˛1 D 0 ;

h1 D 0 ;

1 D 1 ; k2 D k2 ;

ˇ2 D ˇ2 ;

k0 D 0 ;

k2 D k2 ; g D g ;

h0 D 0 ;

where ˛; ˇ; 2 ; 1 ; h1 ; ˇ2 ; ˛2 ; k1 ; ˛1 and g are any constants.

2 D 2 ; ˇ2 D 2ˇ1 ;

k1 D 0 ;

˛1 D ˛2 ;

Case 3. ˛2 D 2˛1 ;

h1 D h1 ;

1 D 1 ;

ˇ1 D ˇ2 ;

(241)

˛1 D ˛1 ;

Case 7.

h2 D 0 ; gDg;

2 D 2 ;

where ˛; ˇ; ˇ1 ; ˛1 ; 2 ; 1 ; h1 ; g and k1 are any constants.

k1 D 0 ;

where ˛; ˇ; 2 ; k0 ; ˛1 ; ˇ1 ; 1 ; k2 ; k1 ; ˇ2 ; ˛2 ; k1 and g are any constants.

h0 D h0 ; k1 D k1 ;

˛2 D ˛2 ;

h2 D 0 ;

k1 D k1 ;

h1 D 0 ;

gDg;

(247)

(242)

where ˛; ˇ; ˛1 ; ˇ1 ; 2 ; ˇ1 ; 1 ; h1 ; k2 and g are any constants.

where ˛; ˇ; 2 ; k0 ; 1 ; h0 ; k2 ; k1 ; ˇ2 ; ˛2 ; k1 and g are any constants. Case 9.

Case 4.

2 D 2 ;

˛2 D ˛2 ;

1 D 1 ;

h1 D h1 ;

k1 D 0 ;

1 D 1 ;

2 D 2 ;

2 D 2 ;

˛2 D ˛1 ;

h1 D 0 ;

Case 2.

k1 D 0 ;

ˇ1 D ˇ1 ; k2 D 0 ;

where ˛; ˇ; ˛2 ; 2 ; ˛2 ; k0 ; ˛1 ; ˇ1 ; 1 and k0 are any constants.

ˇ1 D ˇ1 ;

ˇ2 D ˇ1 ;

˛1 D ˛1 ; k2 D 0 ;

ˇ2 D ˇ2 ;

h1 D h1 ;

h0 D h0 ;

k0 D 0 ;

2 D 2 ;

ˇ1 D ˇ1 ;

˛1 D ˛2 ;

k1 D 0 ;

h1 D 0 ;

h2 D h2 ; gDg;

1 D 1 ;

k2 D 0 ;

ˇ1 D ˇ2 ; ˇ2 D ˇ2 ;

g D0;

˛2 D ˛2 ;

h2 D 0 ;

(243)

k1 D 0 ;

h0 D 0 ;

h1 D h1 ; k1 D k1 ; k0 D 0 ;

(248)

where 2 ; ˛2 ; ˛1 ; ˇ1 ; 1 ; h1 ; ˇ2 ; h1 ; h2 and g are any constants.

where ˛; ˇ; 2 ; 1 ; ˛2 ; ˇ2 ; h1 ; ˇ2 ; ˛2 and k1 are any constants.

Case 5.

Case 10.

2 D 2 ; k0 D k0 ; ˛1 D ˛1 ;

1 D 1 ;

h1 D h1 ;

k1 D 0 ;

h1 D h1 ;

ˇ1 D ˇ1 ;

k2 D 0 ; h2 D h2 ;

˛2 D 0; ˇ2 D 0 ;

2 D 2 ; ˇ1 D ˇ1 ; 1 D 1 ;

k1 D 0 ;

k1 D 0 ;

h0 D h0 ; gDg;

(244)

where ˛; ˇ; 2 ; k0 ; ˛1 ; ˇ1 ; 1 ; h1 ; h1 ; h2 ; h0 and g are any constants.

k2 D 0 ;

˛1 D ˛2 ; h1 D h1 ; ˇ2 D ˇ1 ;

h1 D 0 ; ˛2 D ˛2 ;

k1 D k1 ; h2 D 0 ; h0 D 0; k0 D 0; g D g ;

(249)

where ˛; ˇ; 2 ; ˇ1 ; 1 ; ˛2 ; h1 ; k1 and g are any constants.

871

872

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Case 16.

Case 11. 1 ˇ1 D ˇ2 ; 2 D 2 ; 1 D 1 ; k 1 D 0 ; 2 k1 D 0 ; h1 D h1 ; ˛2 D 0 ; ˛1 D 0 ; k2 D k2 ;

h1 D 0 ;

ˇ2 D ˇ2 ;

h0 D 0 ;

ˇ1 D 2ˇ2 ;

2 D 2 ; ˛1 D 2˛2 ;

h2 D 0 ;

k0 D 0 ;

gDg;

h1 D 0 ;

1 D 1 ;

k2 D 0 ;

h2 D h2 ;

k1 D 0 ;

k1 D k1 ;

˛2 D ˛2 ;

h1 D 0 ;

h0 D 0 ;

(250)

ˇ2 D ˇ2 ;

k0 D 0 ;

gDg;

(255)

where ˛; ˇ; ˇ2 ; 2 ; 1 ; k1 ; ˛2 ; ˇ2 and g are any constants. where ˇ1 D 12 ˇ2 ; ˛; ˇ; 2 ; 1 ; h1 ; k2 ; ˇ2 and g are any constants.

˛1 D 2˛2 ;

Case 12. 1 ˇ2 D 2ˇ1 ; ˛1 D ˛2 ; 2 D 2 ; 1 D 1 ; 2 k1 D 0 ; k1 D 0 ; h1 D h1 ; ˛2 D ˛2 ; ˇ1 D ˇ1 ;

k2 D k2 ;

h1 D 0 ;

h0 D 0 ;

h2 D 0 ;

k0 D 0 ;

gDg;

(251)

where ˛; ˇ; 2 ; 1 ; h1 ; ˛2 ; ˇ1 ; k2 and g are any constants. Case 13. ˇ1 D 2ˇ2 ; g D0;

h1 D 0 ;

h2 D h2 ;

2 D 2 ;

k2 D 0 ;

˛2 D 1/2˛1 ;

k1 D k1 ;

1 D 1 ;

k1 D 0 ;

ˇ2 D ˇ2 ;

h0 D 0 ;

k0 D 0 ;

ˇ1 D 2ˇ2 ;

k2 D 0; h2 D h2 ; k1 D 0 ;

˛2 D ˛2 ;

2 D 2 ;

k1 D k1 ;

(252)

h0 D 0 ;

ˇ2 D ˇ2 ;

k0 D 0 ;

gDg;

(253)

where ˛; ˇ; ˛2 ; ˇ2 ; h2 ; 2 ; 1 ; k1 ; ˇ2 and g are any constants. Case 15. ˇ1 D 2ˇ2 ;

h1 D 0 ;

h2 D h2 ;

k2 D 0 ;

˛2 D 1/2˛1 ;

k1 D k1 ;

1 D 1 ; ˇ2 D ˇ2 ;

k1 D 0 ;

˛1 D ˛1 ; k0 D 0 ;

h2 D h2 ;

k1 D 0 ;

ˇ1 D 2ˇ2 ;

h1 D 0 ;

h0 D 0 ;

k0 D 0 ;

ˇ2 D ˇ2 ; gDg;

(256)

where ˛; ˇ; ˛2 ; h2 ; 2 ; 1 ; ˇ2 ; k1 ; ˇ2 and g are any constants. Case 18. h2 D h2 ;

2 D 2 ;

1 D 1 ;

k1 D 0 ;

˛2 D ˛2 ;

˛1 D ˛2 ;

ˇ1 D ˇ2 ;

k1 D 0 ;

h1 D 0 ; k2 D k2 ;

h1 D 0 ;

ˇ2 D ˇ2 ;

k0 D 0 ;

gDg;

(257)

where ˛; ˇ; h2 ; 2 ; 1 ; ˛2 ; k2 ; ˇ2 and g are any constants. Case 19. h1 D 0 ;

k0 D k0 ;

ˇ1 D 0 ;

˛1 D 0 ;

h0 D h0 ;

2 D 2 ;

˛2 D ˛2 ;

h2 D 0 ;

1 D 1 ;

k2 D k2 ; ˇ2 D ˇ2 ;

k1 D k1 ; gDg;

(258)

where ˛; ˇ; k0 ; h0 ; 2 ; 1 ; k1 ; ˛2 ; k2 ; k1 ; ˇ2 and g are any constants. Substituting (240)–(258) into (236) and (238), respectively, we can obtain the exact solutions of Eq. (235). u(x; t) D v u u k0 C 2 cosh k 1 ;k 1 ;1 (˛1 x C ˇ1 t C 1 ) u u Ck2 e2(˛1 xCˇ1 tC1 ) 2/p ug C ; u h0 C 2 cosh h1 ;h1 ;1 (˛2 x C ˇ2 t C 2 ) t Ch2 e2(˛2 xCˇ2 tC2 ) (259)

h1 D 0 ;

h0 D 0 ;

k2 D 0 ; ˛2 D ˛2 ;

h1 D 0 ;

h1 D 0 ;

1 D 1 ;

h1 D 0 ;

1 D 1 ;

k1 D k1 ;

k1 D k1 ;

˛1 D 2˛2 ;

h1 D 0 ;

h0 D 0 ;

h1 D 0 ;

Case 14.

g D0;

2 D 2 ;

˛1 D ˛1 ;

where ˛; ˇ; ˇ2 ; ˛1 ; h2 ; 2 ; 1 ; ˇ2 and k1 are any constants.

2 D 2 ;

Case 17.

(254)

where ˛; ˇ; ˇ2 ; ˛1 ; h2 ; ˛1 ; 2 ; 1 ; ˇ2 and k1 are any constants.

where cosh k 1 ;k 1 ;1 (˛1 x C ˇ1 t C 1 ) and cosh h1 ;h1 ;1 (˛2 x C ˇ2 t C 2 ) are all the generalized hyperbolic cosine functions, here k i ; h i ; i D 0; ˙1; 2; ˛ j ; ˇ j ; j ; j D 1; 2 and g satisfy (240)–(258), respectively.

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

For example, if we substitute (240) into (236) and (238), we obtain the following solutions of Eq. (235). v u u k0 C 2 cosh0;0;1 (˛1 x C ˇ1 t C 1 ) t u1 (x; t) D 2/p ; 3 2 cosh h1 ;h1 ;1 (˛2 x  4 ˇ ˛p22 t C 2 ) (260)

We obtain an auto-Bäcklund transformation of Eqs. (261) using our method of constructing autoBäcklund transformation (if it exist) u i (t; x1 ; : : : ; x n1 ) D

mi X

u i j (t; x1 ; : : : ; x n1 ) f jm i (t; x1 ; : : : ; x n1 ) ;

jD0

where cosh0;0;1 (˛1 x C ˇ1 t C 1 ) and cosh h1 ;h1 ;1 (˛2 x  4(ˇ˛2 3 t)/p2 C 2 ) are all the generalized hyperbolic cosine function, here h1 D (p2 ˛k0 2 )/(8˛2 2 ˇh1 (3p C 2 C p2 )), and ˛; ˇ; ˛2 ; 2 ; ˛2 ; k0 ; ˛1 ; ˇ1 ; 1 ; k0 are any constants. Remark 13 When m1 ; m2 ; n1 ; n2 D 1; 2; : : :, we can obtain a family of N solitary-like solutions of Eq. (235). In summary, we suggest two new generic transformations, a new Exp–N solitary-like method and its algorithm, to find new non-traveling waves solutions and traveling waves solutions of higher dimensional NLPDEs by our transformations. The suggested method is more powerful than the method proposed by Dr. He [31] to seek more exact solutions of higher dimensional NLPDEs. The (1 C 1)-dimensional generalized KdV equation is chosen to illustrate our algorithm such that new exact solutions are found.

i D 1; 2; : : : ; n :

(262)

where u i j (t; x1 ; : : : ; x n1 ); i D 1; 2; : : : ; n; j D 0; 1; 2; : : : ; m i  1 are differentiable functions which have already been obtained. But f (t; x1 ; : : : ; x n1 ) are any differentiable functions, u i;m i ; i D 1; 2; : : : ; n are the seed solutions of Eq. (261), and m i ; i D 1; 2; : : : ; n are positive integers which have already been obtained. Step 2. Substituting (262) into Eq. (261), and setting the coefficients of these terms 1/( f k (t; x; y)); k D 0; 1; 2; : : : to zero yields a set of over-determined partial differential equations with respect to f (t; x; y), u i;m i ; i D 1; 2; : : : ; n. Step 3.

We consider the variable in the form

 i D p i0 (t) C

n1 X

p i k (t; x1 ; : : : ; x k1 )  x k ;

i D 1; 2

kD1

The Exp-Bäcklund Transformation Method and Its Application in (1 + 1)-Dimensional KdV Equation In this section, a new method and its algorithm, which is called Exp-Bäcklund transformation method, is presented to find more exact solutions of NLPDEs based on the idea of the Exp-function method [31]. We choose the (1 C 1)-dimensional KdV equations to illustrate the effectiveness and convenience of our algorithm. As a result, many new solutions are obtained. Summary of the Exp-Bäcklund Transformation Method In the following we would like to outline the main steps of our method: Step 1. For a given NLEEs, with u i (t; x1 ; x2 ; : : : ; x n1 ) (i D 1; 2; : : : ; n) in n independent variables t; x1 ; x2 ; : : : ; x n1 , F j (u1 ; : : : ; u n ; u1;t ; : : : ; u n;t ; u1;x 1 ; : : : ; u n;x n1 ; u1;t t ; : : : ; u n;t t ; u1;tx 1 ; : : : u n;tx n1 ; : : :) D 0 ; where j D 1; 2; : : : ; n.

(261)

(263) and make the transformation f (t; x1 ; : : : ; x n1 ) D f (1 ; 2 ) :

(264)

Then we express the f (t; x1 ; : : : ; x n1 ) in (262) and (264) to be the following forms Pd a n ex p(n1 ) f (1 ; 2 ) D PqnDc ; b mDp m ex p(m2 )

(265)

P where n1 kD0 p k ¤ 0, c; d; p and q are any positive integers, a n ; n D c; : : : ; d, b m ; m D p; : : : ; q, and p i k ; k D 0; 1; 2; : : : ; n  1are functions or constants to be determined later. Step 4. If we substitute (262)–(265) into the over-determined partial differential equations in Step 2 and yield a set of over-determined partial differential equations with respect to ex p(k i ); i D 1; 2; k D 0; ˙1; ˙2; : : :, a n ; n D c; : : : ; d, b m ; m D p; : : : ; q, u i;m i ;i D 1; 2; : : : ; n, and p i k ; k D 0; 1; 2; : : : ; n  1. Then we collect the coefficients of the polynomials with respect to ex p(k i ); i D 1; 2; k D 0; ˙1; ˙2; : : : and set each coefficient to zero,

873

874

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

we will get a system of over-determined partial differential of a n ; n D c; : : : ; d, b m ; m D p; : : : ; q, and p i k ; k D 0; 1; 2; : : : ; n  1, u i;m i ; i D 1; 2; : : : ; n. Step 5. We solve the over-determined partial differential in Step 4 and obtain a n ; n D c; : : : ; d, b m ; m D p; : : : ; q, u i;m i ; i D 1; 2; : : : ; n, and p i k ;k D 0; 1; 2; : : : ; n  1. Step 6. Substituting the results obtained in Step 5 into (262)–(265), then we can obtain a family of rational solutions of Exp-functions as follows u i (t; x1 ; : : : ; x n1 ) D

mi X

(270)

With the aid of Maple symbolic computation software, substituting (269) and (270) into (267), and collecting all terms with f i (x; t); i D 0; 1; 2; : : :, we obtain

[2˛u02 (x; t) f x (x; t)  24ˇu0 (x; t) f 3 (x; t)]

1 f5

C 6u1 (x; t) f x3 (x; t)]

u i j (t; x1 ; : : : ; x n1 )

C ˛u0 (x; t)[u0x (x; t)  u1 (x; t) f x (x; t)] 1  2˛u1 (x; t)u0 (x; t) f x (x; t)g 4 f

jD0 nDc a n ex p(np10 (t) B Cn P n1 p (t; x ; : : : ; x )  x ) 1 k1 k B kD1 1k B Pq @ mDp b m ex p(mp20 (t) P Cm n1 kD1 p2k (t; x 1 ; : : : ; x k1 )  x k )

1 jm i C C C A

i D 1; 2; : : : ; n ;

;

(266)

C 6u0x (x; t) f x x (x; t)  6u1x (x; t) f x2 (x; t)]  2u0 (x; t) f t (x; t) C ˛u0 (x; t)u1x (x; t)

The Application of the Exp-Bäcklund Transformation Method in (1 + 1)-Dimensional KdV Equation In this section, we will make use of our Exp-Bäcklund transformation method [40,46] and symbolic computation to find the exact solutions of the following (1 C 1)-dimensional KdV equation [1,2,3,4] u t (x; t) C ˛u(x; t)u x (x; t) C ˇu x x x (x; t) D 0 ; (267) where the coefficient of the nonlinear term ˛ and the coefficient of the dispersive term ˇ is independent of x and t. Set the auto-Bäcklund transformation of Eq. (267) to have the following form: u j (x; t) f jm (x; t) ;

C fˇ[6u1 (x; t) f x (x; t) f x x (x; t) C 6u0x x (x; t) f x (x; t) C 2u0 (x; t) f x x x (x; t)

where c; d; p and q are any positive integers.

u(x; t) D

u2 D u2 (x; t) :

C fˇ[18u0 (x; t) f x (x; t) f x x (x; t)  18u0x (x; t) f x2 (x; t)

0 Pd

m X

We take the trivial seed solution as

C ˛u1 (x; t)[u0x (x; t)  u1 (x; t) f x (x; t)] 1  2˛u2 (x; t)u0 (x; t) f x (x; t)g 3 f C fˇ[u0x x x (x; t) C 3u1x (x; t) f x x (x; t) C 3u1x x (x; t) f x (x; t) C u1 (x; t) f x x x (x; t)] C ˛u0 (x; t)u2x (x; t) C ˛u1 (x; t)u1x (x; t) C ˛u2 (x; t)[u0x (x; t)  u1 (x; t) f x (x; t)] 1 C u0t (x; t)  u1 (x; t) f t (x; t)g 2 f C (u1t (x; t) C ˛u1 (x; t)u2x (x; t) C ˛u2 (x; t)u1x (x; t) 1 C ˇu1x x x (x; t)) C u2t (x; t) C ˛u2 (x; t)u2x (x; t) f C ˇu2x x x (x; t) D 0 ; (271)

(268)

j 1 D0

where f (x; t) and u j (x; t); j D 0; 1; 2; : : : ; m  1 are all differential functions to be determined later, and u m (x; t) is the trivial seed solution of Eq. (267). By balancing the highest order linear term and nonlinear terms in Eq. (267), we get m D 2 and (268) has the following formal u(x; t) D u0 (x; t) f

2

(x; t)Cu1 (x; t) f

1

Setting the coefficient of f 5 in (271) to be zero, we obtain a differential equation  2˛u02 (x; t) f x (x; t)  24ˇu0 (x; t) f 3 (x; t) D 0 ; (272) which has solution

(x; t)Cu2 (x; t); (269)

u0 (x; t) D 

12ˇ 2 f (x; t) : ˛ x

(273)

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Setting the coefficient of f 4 in (271) to be zero, we obtain a differential equation  ˇ[18u0 (x; t) f x (x; t) f x x (x; t)  18u0x (x; t) f x2 (x; t) C 6u1 (x; t) f x3 (x; t)] C ˛u0 (x; t)[u0x (x; t)  u1 (x; t) f x (x; t)]  2˛u1 (x; t)u0 (x; t) f x (x; t) D 0 ;

(274)

we can get the following expressions: u1 (x; t) D

12ˇ f x x (x; t) : ˛

(275)

By (273), (275), (270) and (269), we obtain an autoBäcklund transformation of Eq. (267) as follows 12ˇ u(x; t) D  ˛





f x x (x; t) f 2 (x; t)  x2 Cu2 (x; t) (276) f (x; t) f (x; t)

where u2 (x; t) is a seed solution of Eq. (267), f (x; t) is any function that satisfies Eq. (272), Eq. (274), and the following equations  ˇ[6u1 (x; t) f x (x; t) f x x (x; t) C 6u0x x (x; t) f x (x; t) C 2u0 (x; t) f x x x (x; t) C 6u0x (x; t) f x x (x; t)  6u1x (x; t) f x2 (x; t)]  2u0 (x; t) f t (x; t) C ˛u0 (x; t)u1x (x; t) C ˛u1 (x; t)[u0x (x; t)  u1 (x; t) f x (x; t)]  2˛u2 (x; t)u0 (x; t) f x (x; t) D 0 ;  ˇ[u0x x x (x; t) C 3u1x (x; t) f x x (x; t) C 3u1x x (x; t) f x (x; t) C u1 (x; t) f x x x (x; t)]

By substituting (278) into the given Eq. (277) with (273) and (275), and then collecting the coefficients of the polynomials of ei(p 1 (t)xCq 1 (t)) and e j(p 2 (t)xCq 2 (t)) ; i; j D 0; 1; 2; : : :, then setting each coefficient to zero, we will get a system of over-determined partial differential equations with respect to k i (t); i D m1 ; : : : ; m2 ; h j (t); j D n1 ; : : : ; n2 ; p k (t); q k (t); k D 1; 2 and u2 (x; t). We solve the over-determined partial differential equations and obtain many families of solutions of k i (t);i D m1 ; : : : ; m2 ;h j (t); j D n1 ; : : : ; n2 ;p k (t); q k (t);k D 1; 2 and u2 (x; t). Finally, substituting them into (276) with (278), we can obtain the following solutions of Eq. (267) uD

 P Pn 2 2 i1 j2 12ˇ  m iDm 1 k i (t)i p1 (t)e jDn 1 h j (t)e 2 P 2 Pn2 i1 j2 C m k (t)e h (t) jp (t)e i j 2 iDm 1 jDn 1  2 P 2 P n2 m2 j2 i1 ˛ jDn 1 h j (t)e iDm 1 k i (t)e 2 P m2 i1 k (t)ie 12ˇ p1 2 (t) i iDm 1  2 P m2 i1 ˛ iDm 1 k i (t)e P 2 2 i1 12ˇ p1 2 (t) m iDm 1 k i (t)i e C Pm 2 ˛ iDm1 k i (t)e i1 2 P n2 j2 12ˇ p2 2 (t) jDn 1 h j (t) je C P 2 n2 j2 ˛ jDn 1 h j (t)e Pn2 2 2 12ˇ p2 (t) jDn 1 h j (t) j e j2  C u2 (t) ; Pn2 j2 jDn 1 h j (t)e ˛

C ˛u0 (x; t)u2x (x; t) C ˛u1 (x; t)u1x (x; t)

(279)

C ˛u2 (x; t)[u0x (x; t)  u1 (x; t) f x (x; t)] C u0t (x; t)  u1 (x; t) f t (x; t) D 0 ; u1t (x; t) C ˛u1 (x; t)u2x (x; t) C ˛u2 (x; t)u1x (x; t) C ˇu1x x x (x; t) D 0 : (277) We take f (x; t) to be of the following form Pm 2

iDm 1

k i (t)e i(p 1 (t)xCq 1 (t))

jDn 1

h j (t)e j(p 2 (t)xCq 2 (t))

f (x; t) D Pn 2

;

(278)

where n1 ; n2 ; m1 and m2 are any positive integers, p l (t); q l (t); l D 1; 2; k i (t); i D m1 ; : : : ; m2 and h j (t); j D n1 ; : : : ; n2 are the coefficients to be determined later.

where n1 ; n2 ; m1 and m2 are any positive integers, and 1 D p1 (t)x C q1 (t) and 2 D p2 (t)x C q2 (t). For example, when we take n1 D 1; n2 D 1; m1 D 1 and m2 D 1, substituting (278) into the given Eq. (277) with (273) and (275), and then collecting the coefficients of the polynomials of e i(p 1 (t)xCq 1 (t)) and e j(p 2 (t)xCq 2 (t)) ; i; j D 0; 1; 2; : : :, then setting each coefficient to zero, we will get a system of over-determined partial differential equations with respect to p l (t); q l (t); l D 1; 2; k i (t); i D 1; 0; 1 and h j (t), j D 1; 0; 1 as follows  p2 (t)k0 (t)h14 (t)k1t (t)  p2 (t)k1 (t)h14 (t)k0t (t) C 4ˇ p2 (t)k0 (t)h14 (t)p31 (t)k1 (t) C 2p2 (t)k1 (t)h13 (t)k0 (t)h1t (t)

875

876

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

C p2 (t)k0 (t)h14 (t)k1 (t)q1t (t) C

Case 1 12ˇ(k1 (t))2 (p1 (t))2 (ep 1 (t)xq 1 (t) )2 ˛(k1 (t)ep 1 (t)xq 1 (t) C C1 )2 12ˇk1 (t)(p1 (t))2 ep 1 (t)xq 1 (t) C u2 (t) ; (281) C ˛(k1 (t)ep 1 (t)xq 1 (t) C C1 ) R d k (t)(p 1 (t))3 k 1 (t)ˇ p 1 (t)u 2 (t)˛k 1 (t) dt where q1 (t) D dt 1 k 1 (t) CC3 ; k1 (t); p1 (t) are any functions of t, and ˛; ˇ; C i , i D 1; 3 are any constants.

p1 (t)k1 (t)h14 (t)k0 (t)q2t (t)

u1 (x; t) D 

C 2u2 (x; t)˛ p1 (t)k1 (t)h14 (t)p2 (t)k0 (t) C 2u2 (x; t)˛ p22 (t)k1 (t)h14 (t)k0 (t) C 2p2 (t)k1 (t)h14 (t)k0 (t)q2t (t) C p1 (t)k1 (t)h13 (t)k0 (t)h1t (t)  p1 (t)k1 (t)h14 (t)k0t (t) C 4ˇ p1 (t)k1 (t)h14 (t)p2 3 (t)k0 (t)

Case 2

C 2ˇ p42 (t)k1 (t)h14 (t)k0 (t) u2 (x; t) D 

C 6ˇ p22 (t)k0 (t)h14 (t)p21 (t)k1 (t) D 0 ; p1 (t)k1 (t)h14 (t)k0t (t)  4ˇ p2 (t)k0 (t)h14 (t)p31 (t)k1 (t)

C

 4ˇ p1 (t)k1 (t)h14 (t)p32 (t)k0 (t)

12ˇ(C1 p2 (t) C k1 (t)(p2 (t)  p1 (t))e1 CC2 (p2 (t) C p1 (t))e1 )2 ˛(k1 (t)e1 C C1 C C2 e1 )2

12ˇ(C1 (p2 (t))2 C C2 (p2 (t) C p1 (t))2 e1 C(p1 (t)  p2 (t))2 k1 (t)e1 ) ˛(k1 (t)e1 C C1 C C2 e1 )

; (282)

C 2p2 (t)k0 (t)h14 (t)k1 (t)q2t (t) C 2ˇ p42 (t)k1 (t)h14 (t)k0 (t)

where 1 D p1 (t)x C q1 (t); k1 (t); q1 (t); p1 (t) are any functions of t, and ˛; ˇ; C i , i D 1; 2 are any constants.

C 2u2 (x; t)˛ p22 (t)k0 (t)h14 (t)k1 (t)  2u2 (x; t)˛ p1 (t)k1 (t)h14 (t)p2 (t)k0 (t)  p2 (t)k1 (t)h14 (t)k0t (t) C 6ˇ p22 (t)k0 (t)h14 (t)p21 k1 (t) C 2p2 (t)k0 (t)h13 (t)k1 (t)h1t (t)  p1 (t)k1 (t)h13 (t)k0 (t)h1t (t)  p2 (t)k0 (t)h14 (t)k1 (t)q1t (t)

Case 3

u3 (x; t) D 

 p2 (t)k0 (t)h14 (t)k1t (t) 4 (t)p32 (t)k0 (t)  p1 (t)k1 (t)h14 (t)k0 (t)q2t (t)4ˇ p1 (t)k1 (t)h1

C

12ˇ(k1 (t)(p2 (t)  p1 (t))e1 Ck1 (t)(p2 (t) C p1 (t))e1 C C1 p2 (t))2 ˛(k1 (t)e1 C C1 C k1 (t)e1 )2

12ˇ(C1 (p2 (t))2 C k1 (t)(p2 (t) C p1 (t))2 e1 C(p1 (t)  p2 (t))2 k1 (t)e1 ) ˛(k1 (t)e1 C C1 C k1 (t)e1 )

; (283)

4 4 (t)k1t (t) C p2 (t)k0 (t)h1 (t)k1 (t)q1t (t) C p2 (t)k0 (t)h1

where 1 D p1 (t)x C q1 (t); k1 (t); k1 (t); q1 (t); p1 (t) are any functions of t, and ˛; ˇ; C1 are any constants.

4 (t)k1 (t)q2t (t) C 2p2 (t)k0 (t)h1 4 (t)k0 (t)q2t (t) C p1 (t)k1 (t)h1

Case 4

4 (t)p21 (t)k1 (t) C 6ˇ p22 (t)k0 (t)h1

12ˇ(p1 (t))2 (k1 (t))2 (e p 1 (t)xCq 1 (t) )2 ˛(C1 C k1 (t)e p 1 (t)xCq 1 (t) )2 12ˇ(p1 (t))2 k1 (t)e p 1 (t)xCq 1 (t) C C u2 (t) ; (284) ˛(C1 C k1 (t)e p 1 (t)xCq 1 (t) ) R ˇ (p 1 (t))3 k 1 (t)Cu 2 (t)˛ p 1 (t)k 1 (t)C dtd k 1 (t) dtC where q1 (t) D  k 1 (t) C3 ; k1 (t); p1 (t) are the functions of t, and ˛; ˇ; C i , i D 1; 3 are any constants.

4 (t)p2 (t)k0 (t) C 2u2 (x; t)˛ p1 (t)k1 (t)h1

u4 (x; t) D 

3 (t)k0 (t)h1t (t)  p1 (t)k1 (t)h1 4 (t)k1 (t) C 2u2 (x; t)˛ p22 (t)k0 (t)h1 3 (t)k1 (t)h1t (t)  2p2 (t)k0 (t)h1 4 (t)k0 (t) C 2ˇ p42 (t)k1 (t)h1 4 4 (t)p31 k1 (t) C p1 (t)k1 (t)h1 (t)k0t (t) C 4ˇ p2 (t)k0 (t)h1 4 (t)k0t (t) D 0 ; C p2 (t)k1 (t)h1

(280)

Case 5

:::::::::::::::::::::::: We solve the over-determined partial differential equations and obtain the following many families of solutions:

u5 (x; t) D

2C1 h0 (t)(u2 (t)˛  6ˇ(p2 (t))2 )e2 Cu2 (t)˛C12 e22 C u2 (t)˛(h0 (t))2 (C1 e2 C h0 (t))2 ˛

; (285)

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

where 2 D p2 (t)x C q2 (t); h0 (t); p2 (t); q2 (t) are the functions of t, and ˛; ˇ; C1 are any constants.

Case 10 12ˇ(p1 (t))2 (k1 (t))2 (e p 1 (t)xCq 1 (t) )2 ˛(k0 (t) C k1 (t)e p 1 (t)xCq 1 (t) )2 12ˇ(p1 (t))2 k1 (t)e p 1 (t)xCq 1 (t) C C u2 (t) ; (290) ˛(k0 (t) C k1 (t)e p 1 (t)xCq 1 (t) )

u10 (x; t) D  Case 6

u6 (x; t) D 

2C1 h0 (t)(6ˇ(p2 (t))2  u2 (t)˛)e2 C2C1 h1 (t)(24ˇ(p2 (t))2  u2 (t)˛) (C1 e2 C h0 (t) C h1 (t)e2 )2 ˛

R

2h0 (t)h1 (t)(6ˇ(p2 (t))2  u2 (t)˛)e2 u2 (t)˛((h1 (t))2 C C12 )e22  u2 (t)˛(h0 (t))2



;

(C1 e2 C h0 (t) C h1 (t)e2 )2 ˛

(286)

where q1 (t) D dt C C2 ; k0 (t); k1 (t); q2 (t) are the functions of t, and ˛; ˇ; C2 are any constants. Case 11

where 2 D p2 (t)x C q2 (t); h0 (t); h1 (t); q1 (t); q2 (t) are the functions of t, and ˛; ˇ; C1 are any constants.

u11 (x; t) D

Case 7



u7 (x; t) D

(t))2 e22

C

˛(k1 (t)e1 C k0 (t) C k1 (t)e1 )2 12ˇ(k0 (t)(p2 (t))2 C k1 (t)(p2 (t) C p1 (t))2 e1 C(p1 (t)  p2 (t))2 k1 (t)e1 )

C

˛(k1 (t)e1 C k0 (t) C k1 (t)e1 ) C u2 (t) ;

(t))2 e22 )

C (h1 u2 (t)˛((h1 48ˇh1 (t)(p2 (t))2 h1 (t) Cu2 (t)˛(2h1 (t)h1 (t) C (h0 (t))2 )

(h1 (t)e2 C h0 (t) C h1 (t)e2 )2 ˛

;

(287)

where 2 D p2 (t)x C q2 (t); h1 (t); h0 (t); h1 (t); q1 (t); q2 (t) are the functions of t, and ˛; ˇ are any constants.

(291)

where 1 D p1 (t)x C q1 (t); k1 (t); k0 (t); k1 (t); q1 (t); q2 (t) are the functions of t, and ˛; ˇ are any constants. Case 12 12ˇ(k1 (t))2 (p1 (t))2 (ep 1 (t)xq 1 (t) )2 ˛(k1 (t)ep 1 (t)xq 1 (t) C k0 (t))2 12k1 (t)(p1 (t))2 ep 1 (t)xq 1 (t) ˇ C C u2 (t) ; (292) ˛(k1 (t)ep 1 (t)xq 1 (t) C k0 (t))

u12 (x; t) D 

Case 8

u8 (x; t) D

2C1 k0 (t)(6ˇ(p1 (t))2 C u2 (t)˛)e1 Cu2 (t)˛C12 e21 C u2 (t)˛(k0 (t))2 ˛(C1 e1 C k0 (t))2

; (288)

where 1 D p1 (t)x C q1 (t); k0 (t); q1 (t); q2 (t) are the functions of t, and ˛; ˇ; C1 are any constants. Case 9

u9 (x; t) D

C

12ˇ(k1 (t)(p2 (t)  p1 (t))e1 Ck1 (t)(p2 (t) C p1 (t))e1 C k0 (t)p2 (t))2

2h0 (t)h1 (t)(u2 (t)˛  6ˇ(p2 (t))2 )e2 C2h0 (t)h1 (t)(u2 (t)˛  6ˇ(p2 (t))2 )e2 (h1 (t)e2 C h0 (t) C h1 (t)e2 )2 ˛

k 0 (t) dtd k 1 (t)  k 1 (t) dtd k 0 (t) C k 0 (t)k 1 (t) (p 1 (t))3 ˇ C u 2 (t)˛ p 1 (t)k 1 (t)k 0 (t) k 0 (t)k 1 (t)

u2 (t)˛((k0 (t))2 C 2C1 k1 (t)) C48ˇC1 (p1 (t))2 k1 (t) C u2 (t)˛(k1 (t))2 e21 ˛(C1 e1 C k0 (t) C k1 (t)e1 )2

2k0 (t)k1 (t)(6ˇ(p1 (t))2 C u2 (t)˛)e1 C2C1 k0 (t)(6ˇ(p1 (t))2 C u2 (t)˛)e1 Cu2 (t)˛C12 e21 ˛(C1 e1 C k0 (t) C k1 (t)e1 )2

;

(289)

where 1 D p1 (t)x C q1 (t); k0 (t); k1 (t); q1 (t); q2 (t) are the functions of t, and ˛; ˇ; C1 are any constants.

R

k 0 (t) dtd k 1 (t)  u 2 (t)˛ p 1 (t)k 1 (t)k 0 (t) k 0 (t)(p 1 (t))3 k 1 (t)ˇ  k 1 (t) dtd k 0 (t) k 0 (t)k 1 (t)

dt C where q1 (t) D C2 ; k1 (t); k0 (t); q2 (t) are the functions of t, and ˛; ˇ; C2 are any constants. Remark 14 From the above application, it is easily seen that our method is more powerful, more convenient, and more general than the method presented by He [31]. Firstly, our method is convenient to obtain the exact solutions of higher order and higher dimensional NLEEs because we do not use the homogeneous balance method to get the value of c; d; p; q in (265). Secondly, we can obtain more general exact solutions including non-traveling wave solutions and traveling wave solutions by using our method. For example, the solution (282) of Eq. (267) is a more general exact solution including the non-traveling wave solutions with constant coefficients, the non-traveling wave solutions with variable coefficients, the traveling

877

878

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

wave solutions with constant coefficients, and the traveling wave solutions with variable coefficients of Eq. (267). Example 1 The solution (282) of Eq. (267) includes abundant non-traveling wave solutions with variable coefficients. For example, when we take C1 D 1/2; C2 D 1; p2 (t) D cos(t 2 ); u2 (t) D 1/2; p1 (t) D tanh2 (t); q1 (t) D sin(t 4 ); k1 (t) D cosh(t); k0 (t) D cos(t 2 ); ˇ D 2; ˛ D 3, we can obtain a non-traveling wave solution u2;1 (x; t) with variable coefficients as follows:

8 cos2 (t 2 )(2 cosh(t)e1 C 1 C 2e1 ) C32 tanh2 (t) cos(t 2 )(e1  cosh(t)e1 ) C16 tanh4 (t)(cosh(t)e1 C e1 ) 2 cosh(t)e1 C 1 C 2e1

8

(2 cosh(t)e1 C 1 C 2e1 )2

C 1/2 ; (293)

where 1 D tanh2 (t)x C sin(t 4 ). Some figures of the long-term behavior of the u2;1 (x; t) from f(x; t)jjxj < 0:1; jtj < 0:1; x 2 R; t 2 Rg to f(x; t)jjxj < 109 ; jtj < 109 ; x 2 R; t 2 Rg are shown by following Fig. 1 and Fig. 2. From Fig. 1 and Fig. 2, it is easy to find the global existence, asymptotic behaviors as t ! C1 and jxj ! C1 and the scattering properties of the solution u2;1 (x; t) of Eq. (267). It is worth noting that the shape of u2;1 (x; t) degenerates gradually from a set of waves to a set of solitons on a trigonal curve as the jxj and jtj in the u2;1 (x; t) continue to increase. Example 2 The solution (282) of Eq. (267) includes abundant non-traveling wave solutions with constant coefficients. For example, when we take k1 (t) D 1/2; ˇ D 2; ˛ D 3; k0 (t) D 2; C2 D 1; C1 D 1/2; p1 (t) D 1/2; p2 (t) D 1/3; u2 (t) D 1/2, and q1 (t) D sin(t 4 ), we can obtain a non-traveling wave solution u2;2 (x; t) with constant coefficients as follows: 4

u2;2 (x; t) D 

4

2(ex/2sin(t )  10ex/2Csin(t )  2)2 9(ex/2sin(t 4 ) C 1 C 2ex/2Csin(t 4 ) )2 4

C

2(ex/2t/3  10e x/2Ct/3  2)2 9(ex/2t/3 C 1 C 2e x/2Ct/3 )2 8 C 100ex/2Ct/3 C 2ex/2t/3 C x/2t/3 C 1/2 : 9e C 9 C 18ex/2Ct/3

u2;3 (x; t) D 

(295)

Some figures of the long-term behavior of the u2;3 (x; t) from f(x; t)jjxj < 0:1; jtj < 0:1; x 2 R; t 2 Rg to f(x; t)jjxj < 423; jtj < 423; x 2 R; t 2 Rg are shown by following Fig. 5 and Fig. 6. It is worth noting that the shape of the u2;3 (x; t) degenerates gradually from a wave to a row of solitons on a straight line as the jxj and jtj in u2;3 (x; t) continue to increase.

u2;1 (x; t) D

(2 tanh2 (t)(e1  cosh(t)e1 ) C cos(t 2 )(2 cosh(t)e1 C 1 C 2e1 ))2

Example 3 The solution (282) of Eq. (267) includes abundant traveling wave solutions with constant coefficients. For example, when q1 (t) D sin(t 4 ) in (294) is replaced by q1 (t) D t/3, we can obtain a traveling wave solution u2;3 (x; t) with constant coefficients of Eq. (267) as follows:

4

8 C 100ex/2Csin(t ) C 2ex/2sin(t ) C 1/2 : 9ex/2sin(t 4 ) C 9 C 18ex/2Csin(t 4 )

(294)

Some figures of the long-term behavior of the u2;2 (x; t) from f(x; t)jjxj < 0:1; jtj < 0:1; x 2 R; t 2 Rg to f(x; t)jjxj < 109 ; jtj < 109 ; x 2 R; t 2 Rg are shown by following Fig. 3 and Fig. 4.

Example 4 The solution (282) of Eq. (267) includes abundant traveling wave solutions with variable coefficients. For example, when p1 (t) D tanh2 (t); q1 (t) D sin(t 4 ) in (293) is replaced by p1 (t) D 1/2; q1 (t) D t/3, we can obtain a traveling wave solution u2;4 (x; t) with variable coefficients of Eq. (267) as follows: u2;4 (x; t) 1 C 12 cosh(t)ex/2t/3 C 12e x/2Ct/3 C4exC2t/3 C 72 cosh(t) C 4 cosh2 (t)ex2t/3 : D 2(2 cosh(t)ex/2t/3 C 1 C 2e x/2Ct/3 )2 (296) Some figures of the long-term behavior of the u2;4 (x; t) from f(x; t)jjxj < 0:1; jtj < 0:1; x 2 R; t 2 Rg to f(x; t)jjxj < 109 ; jtj < 109 ; x 2 R; t 2 Rg are shown by following Fig. 7 and Fig. 8. From Fig. 7 and Fig. 8, it is easy to find the global existence, asymptotic behaviors as t ! C1 and jxj ! C1 and the scattering properties of the solution u2;4 (x; t) of Eq. (267). It is worth noting that the shape of u2;4 (x; t) degenerates gradually from a trigonal wave to a set of solitons on a trigonal curve as jxj and jtj in u2;4 (x; t) continue to increase. But the plane has become a scraggly surface over f(x; t)jjxj < 109 ; jtj < 109 ; x 2 R; t 2 Rg, namely, the u2;4 (x; t) appears a mild disorder. Remark 15 From the above long-term behavior of the u2;1 (x; t); u2;2 (x; t); u2;3 (x; t); u2;4 (x; t), we find easily that the stabilization of the traveling wave solutions u2;3 (x; t) with constant coefficients is best as t ! C1 and jxj ! C1. In addition, we still discover that the longterm behavior of a solution, which includes the global existence, asymptotic behaviors as t ! C1 and jxj !

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, Figure 1 The evolution of a non-traveling wave solution u2;1 (x; t) with variable coefficients of Eq. (267) from f(x; t)jjxj < 0:1; jtj < 0:1; x 2 R; t 2 Rg to f(x; t)jjxj < 15; jtj < 15; x 2 R; t 2 Rg

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, Figure 2 The evolution of a non-traveling wave solution u2;1 (x; t) with variable coefficients of Eq. (267) from f(x; t)jjxj < 35; jtj < 35; x 2 R; t 2 Rg to f(x; t)jjxj < 109 ; jtj < 109 ; x 2 R; t 2 Rg

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, Figure 3 The evolution of a non-traveling wave solution u2;2 (x; t) with constant coefficients of Eq. (267) from f(x; t)jjxj < 0:1; jtj < 0:1; x 2 R; t 2 Rg to f(x; t)jjxj < 15; jtj < 15; x 2 R; t 2 Rg

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, Figure 4 The evolution of a non-traveling wave solution u2;2 (x; t) with constant coefficients of Eq. (267) from f(x; t)jjxj < 35; jtj < 35; x 2 R; t 2 Rg to f(x; t)jjxj < 109 ; jtj < 109 ; x 2 R; t 2 Rg

879

880

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, Figure 5 The evolution of a traveling wave solution u2;3 (x; t) with constant coefficients of Eq. (267) from f(x; t)jjxj < 0:1; jtj < 0:1; x 2 R; t 2 Rg to f(x; t)jjxj < 15; jtj < 15; x 2 R; t 2 Rg

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, Figure 6 The evolution of a traveling wave solution u2;3 (x; t) with constant coefficients of Eq. (267) from f(x; t)jjxj < 50; jtj < 50; x 2 R; t 2 Rg to f(x; t)jjxj < 423; jtj < 423; x 2 R; t 2 Rg

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, Figure 7 The evolution of a traveling wave solution u2;4 (x; t) with variable coefficients of Eq. (267) from f(x; t)jjxj < 0:1; jtj < 0:1; x 2 R; t 2 Rg to f(x; t)jjxj < 15; jtj < 15; x 2 R; t 2 Rg

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the, Figure 8 The evolution of a traveling wave solution u2;4 (x; t) with variable coefficients of Eq. (267) from f(x; t)jjxj < 35; jtj < 35; x 2 R; t 2 Rg to f(x; t)jjxj < 109 ; jtj < 109 ; x 2 R; t 2 Rg

Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the

C1, and the scattering properties of the solution, depends mainly on the functions that constitute the solution and the selection of all arbitrary constants and the parameters in the solution. We choose other nonlinear evolution equations [39,40,45,47] to test and obtain similar results. Therefore, we devise a new guess as follows. R-guess The long-term behavior of a solution for a given NLEEs (261), which includes the global existence, asymptotic behaviors as t ! C1 and jxj ! C1, and the scattering properties of the solution, depends mainly on the functions that constitute the solution and the selection of all arbitrary constants and parameters in the solution. Future Directions In recent years, directly searching for exact solutions of NLPDEs has become more and more attractive partly due to the availability of computer symbolic systems like Maple or mathematica which allow us to perform complicated and tedious algebraic calculation on a computer, as well as helping us to find new exact solutions of NLPDEs in mathematical physics. Although many powerful analytical methods for solving NLPDEs have been presented, the methods fail to satisfy the developmental needs in physics, mechanics, chemistry, biology, computer science, etc. There is much new work to be done on analytical methods for solving NLPDEs. The main future directions are as follows: 1. Researching new and uniform analytical methods based on the ideas of unification methods, algorithm realization and mechanization for solving NLPDEs (we have done some new work in Sect. “The Generalized Hyperbolic Function–Bäcklund Transformation Method and Its Application in the (2 + 1)-Dimensional KdV Equation” of this article). 2. Researching mechanization methods and developing their computer software for showing the long-playing traveling state of the exact solutions of NLPDEs (we have done some new work in [40]). 3. Developing and improving existing analytical methods, computer algebraic systems, and software so that analytical methods can be actualized with complete mechanization by their computer software at a familiar computer (we have done some new work in [39,40,41,42,43, 44,45,46,47,48]). 4. Researching new analytical and numerical complex methods and their computer software for solving those NLPDEs that can’t make use of analytical methods. The complex methods can be actualized with complete

mechanization by their computer software at a familiar computer (we have done some new work in [40,53]). 5. One can calculate the solutions of NLPDEs as easily as counting or addition by using a small calculator with the development of mathematics and computer science (we have done some new work in [39,40,41,42,43,44, 45,46,47,48,52,53,54] and Ren 2055 and 2007, and Ren und Sun 2004).

Acknowledgment The author would like to express his gratitude to all the people whose efforts contributed to the various editions of this paper. One reviewer who made useful recommendations for this paper is Professor M. Atef Helal at Mathematics Department, Cairo University, Egypt. The author is grateful to Professor M. Atef Helal for his time and valuable suggestions. The author thanks Editor-in-Chief Robert Meyers of the Encyclopedia of Complexity and System Science, Professor M.S. El Naschie at Frankfurt Institute for the advancement of Theoretical Physics, Frankfurt, Germany, Professor Hong-Qing Zhang at Department of Applied Mathematics, Dalian University of Technology, Dalian, PR China, and Professor Abdul-Majid Wazwaz at Department of Mathematics and Computer Science, Saint Xavier University, Chicago, USA, for their help. This work has been supported partially by a National Key Basic Research Project of China (2004 CB 318000). Bibliography Primary Literature 1. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos Mag Ser 5 39:422–443 2. Khaters AH, El-Kalaawy H, Helal MA (1997) Two new classes of exact solutions for the KdV equation via Bäcklund transformations. Chaos Solitons Fractals 8(12):1901–1909 3. Helal MA (2001) A Chebyshev spectral method for solving Korteweg–de Vries equation with hydrodynamical application. Chaos Solitons Fractals 12:943–950 4. Brugarino T, Sciacca M (2008) A direct method to find solutions of some type of coupled Korteweg–de Vries equations using hyperelliptic functions of genus two. Phys Lett A 372(11):1836– 1840 5. Leibovitch S (1970) Weakly nonlinear waves in rotating fluids. J Fluid Mech 42:803–825 6. RM Miura, CS Gardner, MD Kruskal (1968) Korteweg–de Vries equation and generalizations. II Existence of conservation laws and constants of motion. J Math Phys 9:1204–1209 7. MA Helal, MS Mehanna (2007) A comparative study between two different methods for solving the general Korteweg–

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de Vries equation (GKdV). Chaos Solitons Fractals 33:725– 739 8. Hamdia S, Enrighta WH, Schiesserb WE, Gottliebc JJ (2005) Exact solutions and conservation laws for coupled generalized Korteweg–de Vries and quintic regularized long wave equations. Nonlinear Anal 63:1425–1434 9. Sarma J (2008) Solitary wave solution of higher-order Korteweg–de Vries equation. Chaos Solitons Fractals, in press 10. Karpman VI (1998) Evolution of radiating solitons described by the fifth-order Korteweg–de Vries type equations. Phys Lett A 244:394–396 11. Hassan MM (2004) Exact solitary wave solutions for a generalized KdV–Burgers equation. Chaos Solitons Fractals 19:1201– 1206 12. Khare A, Cooper F (1993) One-parameter family of soliton solutions with compact support in a class of generalized Korteweg–de Vries equation. Phys Rev E 48(6):4843–4 13. Dod’an K (2005) An application for the higher order modified KdV equation by decomposition method. Commun Nonlinear Sci Num Simul 10:693–702 14. Shen J, Xu W (2007) Travelling wave solutions in a class of generalized Korteweg–de Vries equation. Chaos Solitons Fractals 34:1299–1306 15. Clarke S, Malomed BA, Grimshaw R (2002) Dispersion management for solitons in a Korteweg–de Vries system. Chaos 12(1):8–15 16. Houria T (2007) Energy transfer in a dispersion-managed Korteweg–de Vries system. Math Comput Simul 76:283–292 17. Kaya D, Aassila M (2002) An application for a generalized KdV equation by the decomposition method. Physic Lett A 299:201–206 18. Shek ECM, Chow KW (2008) The discrete modified Korteweg– de Vries equation with non-vanishing boundary conditions: Interactions of solitons. Chaos Solitons Fractals 36:296–302 19. Zhang S (2007) A generalized auxiliary equation method and its application to the (2 C 1)-dimensional KdV equations. Appl Math Comput 188:1–6 20. Song L, Zhang H (2007) A new Korteweg–de Vries equationbased sub-equation method and its application to the (2 C 1)-dimensional Korteweg–de Vries equation. Appl Math Comput 187:1368–1372 21. Hirota R (1971) Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys Rev Lett 27:1192– 1194 22. Wadati M, Sanuki H, Konno K (1975) Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws. Prog Theor Phys 53:419–436 23. Weiss J, Tabor M, Carnevale G (1983) The Painlevel property for partial differential equations. J Math Phys 24:522–526 24. Fan EG, Zhang HQ (1998) A note on the homogeneous balance method. Phys Lett A 246:403–406 25. Wazwaz AM (2005) The tanh and the sine-cosine methods for the complex modified KdV and the generalized KdV equations. Comput Math Appl 45:1101–1112 26. Wang ML (1996) Exact solution for a compound KdV–Burgers equations. Phys Lett A 213:279–287 27. He JH (2005) Homotopy perturbation method for bifurcation

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47. Ren YJ, He JH, Zhang HQ (2008) A new Exp-N solitary-like method and its application in the (2 C 1)-dimensional generalized KdV–Burgers equation. Chaos Solitons Fractals (accepted) 48. Ren YJ (2007) A new the Exp-N soliton-like method. Ph D Dissertation. Dalian University of Technology, p 172–178 49. Wu YT, Geng XG, Hu XB, Zhu SM (1999) A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations. Phys Lett A 255:259–264 50. Fan EG, Dai HH (2003) A direct approach with computerized symbolic computation for finding a series of traveling waves to nonlinear equations. Computer Physics Communications 153:17–21 51. Yan ZY (2004) An improved algebra method and its applications in nonlinear wave equations. Chaos Solitons Fractals 21:1013–1021 52. Wang DS, Ren YJ, Zhang HQ (2005) Further extended sih-cosh and sin-cos methods and new non-travelng wave solutions of the (2 C 1)-dimensional dispersive long wave equation. Appl Math E-notes 5:157–163 53. Ren YJ, Li YJ, Song FG, Zhang HQ (2006) A new compound operation method of the numerical solutions and analytic solution of partial differential equation. International Conf. on Multimedia, Information Technology and its Applications, Dalian, China, p 328–330 54. Ren YJ, Song FG, Li YJ, Zhang HQ (2006) A mechanization method on the mixed operation of the numerical solutions, analytic solution, and showing their figures simultaneously for differential equation. International Conf. on Multimedia, Information Technology and its Applications, Dalian, China, p 276– 278

Books and Reviews Ablowitz MJ, Clarkson PA (1991) Soliton, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York Dodd RK, Eilbeck JC, Gibbon JD, Morris HC (1984) Solitons and nonlinear wave equations. Academic Press, London Drazin PG, Johnson RS (1989) Solitons: an introduction. Cambridge University Press, Cambridge Fan EG (2004) Integrable System and Computer Algebra. Science Press, Beijing Guo BL (1996) Nonlinear Evolution Equation [M]. Shanghai Science and Technology Press, Shanghai Liu SK, Liu SD (2000) Nonlinear Equations in Physics. Peking University Press, Beijing Miura RM (ed) (1976) Backlaund Transformation C. Springer Lecture Notes in Mathematica, vol 515. Springer, New York Miurs MR (1978) Bäcklund Transformation. Springer, Berlin Ren YJ (2005) Advanced Mathematics – Multivariable Calculus and Experiments (MATLAB edition). China Machine Press, Beijing Ren YJ (2007) Numerical Analysis and its MATLAB Realization. Higher Education Press, Beijing Ren YJ, Sun WH (2004) Advanced Mathematics – Unitary Calculus and Experiments (MATLAB edition). China Machine Press, Beijing Rogerscand Shaduick WR (1982) Backland Transformation and their Applications. Academic Press, New York Whitham GB (1974) Linear and nonlinear waves. Wiley, New York Zhuo LY et al (2000) Nonlinear Physics Threoy and Application [M]. Science Press, Peking

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Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the YI Z ANG Department of Mathematics, Zhejiang Normal University, Jinhua, China Article Outline Glossary Definition of the Subject Introduction Inverse Scattering Transform for the KdV Equation Exact N-soliton Solutions of the KdV Equation Further Properties of the KdV Equation Future Directions Bibliography Glossary Soliton A soliton is a solitary wave which asymptotically preserves its shape and velocity upon nonlinear interaction with other solitary waves, or more generally, with another (arbitrary) localized disturbance. Korteweg–de Vries equation In mathematics, the Korteweg-de Vries (KdV) equation is a mathematical model of waves on shallow water surfaces. It is particularly famous as the prototypical example of an exactly solvable model, that is, a nonlinear partial differential equation whose solutions can be exactly and precisely specified. Inverse scattering transform The identification of the KdV equation as an isospectral flow of the Schrödinger operator enabled Gardner, Greene, Kruskal and Miura (GGKM) to devise a method of solving the KdV equation (with ’rapidly decreasing’ boundary conditions), called the inverse scattering or inverse spectral transform (IST). This is a direct generalization of the Fourier transform used to solve linear equations and can be represented by essentially the same scheme. Hirota bilinear method In 1971, Hirota developed a direct method for finding N-soliton solutions of nonlinear evolution equations, also named Hirota bilinear method. The key step is to transform the equation into a bilinear form, from which we can get soliton solutions successively by means of a kind of perturbational technique. Definition of the Subject Among integrable equations is the celebrated KdV equation, which serves as a model equation governing weakly

nonlinear long waves whose phase speed attains a simple maximum for waves of infinite length. It motivates us to explore beauty hidden in nonlinear differential (and difference) equations. The equation is named for Diederik Korteweg and Gustav de Vries. The remarkable and exceptional discovery of the inverse scattering transform is one of important developments in the field of applied mathematics, which comes from the study of the KdV equation. There are various algebraic and geometric characteristics that the KdV equation possesses, for example, infinitely many symmetries and infinitely many conserved densities, the Lax representation, bi-Hamiltonian structure, loop group, and the Darboux-Bäcklund transformation. More significantly, many physically important solutions to the KdV equation can be presented explicitly through a simple, specific form, called the Hirota bilinear form. Introduction A nice story about the history and the underlying physical properties of KdV equation can be found at an Internet page of the Herriot-Watt University in Edinburgh (Scotland). The following text is taken from that page: Over one hundred and fifty years ago, which conducting experiments to determine the most efficient design for canal boats, a young Scottish engineer named John Scott Russell (1808–1882) made a remarkable scientific discovery. Here his original text as he described it in Russell [1]: I believe I shall best introduce this phenomenon by describing the circumstances of my own first acquaintance with it. I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped-not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation, a name which it now very generally bears.

Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the

He followed this observation with a number of experiments during which he determined the shape of a solitary wave to be that of a sech2 () function. He also determined the relationship of the speed of the wave to its amplitude. At that time there was no equation describing such water waves and having such a solution. Thus John Scott Russell discovered the solution to an as yet unknown equation! Further investigations were undertaken by Airy, Stokes, Boussinesq and Rayleigh in an attempt to understand this phenomenon. Boussinesq and Rayleigh independently obtained approximate descriptions of the solitary wave; Boussinesq derived a one-dimensional nonlinear evolution equation, which now bears his name, in order to obtain his results. These investigations provoked much lively discussion and controversy as to whether the inviscid equations of water waves would possess such solitary wave solutions. The issue was finally resolved by Korteweg and de Vries [2]. After performing a Galilean and a variety of scaling transformations, the KdV equation can be written in simplified form: u t  6uu x  u x x x D 0 ;

(1)

where subscripts denote partial differentiations. In particular, if we now assume a solution in the form of a traveling wave u(x; t) D f (x C ct), then Eq. (1) can be integrated: imposing the boundary conditions at large distances that u(x; t) tends to 0 sufficiently fast as x ! ˙1, we find the exact solution c 1p u(x; t) D sech2 c(x C ct C ı) ; (2) 2 2 where ı is the phase. This clearly represents the solitary wave observed by John Scott Russell and shows that the peak amplitude is exactly half the speed. Thus larger solitary waves have greater speeds. This suggests a numerical experiment: start with two solitary wave solutions, with centers well separated and the larger to the right. Initially, with negligible overlap, they will evolve independently as solitary wave solutions. However, the larger, faster one will start to overtake the smaller and the non linearity will play a significant role. For most dispersive evolution equations these solitary waves would scatter in elastically and lose ’energy’ to radiation. Not so for the KdV equation: after a fully nonlinear interaction, the solitary waves reemerge, retaining their identities (same speed and form), suffering nothing more than a phase shift (modified ı’s, representing a displacement of their centers). It was after a similar numerical experiment that Kruskal and Zabusky [3] coined the name ’soliton’, to reflect the particle-like behavior of the solitary waves under interaction.

It was not until the mid 1960’s when applied scientists began to use modern digital computers to study nonlinear wave propagation that the soundness of Russell’s early ideas began to be appreciated. He viewed the solitary wave as a self-sufficient dynamic entity, a “thing” displaying many properties of a particle. From the modern perspective it is used as a constructive element to formulate the complex dynamical behavior of wave systems throughout science: from hydrodynamics to nonlinear optics, from plasmas to shock waves, from tornado’s to the Great Red Spot of Jupiter, from the elementary particles of matter to the elementary particles of thought. For a more detailed and technical account of the solitary wave, see [4,5]. Inverse Scattering Transform for the KdV Equation We now briefly discuss the inverse scattering transform for the KdV Eq. (1). First consider the Miura map u D v x  v 2 C , which can be viewed as a Riccati equation for v and thus linearized by the substitution v D x / , giving 

L

xx

Cu

D :

(3)

This is the time-independent Schrödinger equation with u(x; t) playing the role of potential and  the energy. It is important to realize that t is not the time of the time-independent Schrödinger equation. We think of x as the spatial variable and t as a parameter. Considered as a Sturm– Liouville eigenvalue problem it is natural to ask how  and change with t as u(x; t) evolves from some initial state according to the KdV equation. GGKM [6,7] discovered the remarkable fact that the (discrete part of the) spectrum necessarily remains constant in ’time’ while the corresponding wave functions evolve according to a very simple linear differential equation. Today (following Lax [8]) we usually take the opposite route. We postulate that evolves through a linear differential equation: t

DP

:

(4)

Eq. (3) and (4) form an overdetermined system, whose integrability conditions can be written: L t D [P; L] D PL  LP :

(5)

A consequence of (5) is that all eigenvalues corresponding to the function u(x; t) remain constant. When P is given by:   P D  4@3 C 6u@ C 3u x ; (where @ is a differential operator)

(6)

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Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the, Figure 1 Inverse Scattering Transform

(5) reduces to the KdV Eq. (1). Since, under these conditions, the spectrum of L remains constant, the KdV equation is referred to as an isospectral flow. Eq. (5) is called the Lax representation of the KdV equation. The identification of the KdV equation as an isospectral flow of the Schrödinger operator enabled GGKM to devise a method of solving the KdV equation (with ’rapidly decreasing’ boundary conditions), called the inverse scattering or inverse spectral transform (IST). This is a direct generalization of the Fourier transform used to solve linear equations and can be represented by essentially the same scheme (Fig. 1). For these boundary conditions, the solutions of (3) are characterized by their asymptotic properties as x ! ˙1. For a given potential function, the continuous and discrete spectrum are treated separately. Corresponding to the continuous spectrum ( D k 2 ) the solutions are asymptotically oscillatory, characterized by two coefficients:

T(k)ei kx x ! 1 (7) ei kx C R(k)e i kx x ! C1 subject to the condition jRj2 C jTj2 D 1. The constants R(k) and T(k) are respectively called the reflection and transmission coefficients, from their quantum mechanical interpretation. Under mild conditions on the potential function, the Schrödinger operator L has only a finite number of discrete eigenvalues fn2 g. The corresponding eigenfunctions are square integrable and are the ’bound states’ of quantum mechanics with asymptotic properties (for n > 0):

e c n en x x ! 1 : (8) n c n en x x ! C1 The direct scattering transform constructs the quantities fT(k); R(k); n ; c n g from a given potential function. The

important inversion formula were derived by Gel’fand and Levitan in 1955 [9]. These enable the potential u to be constructed out of the spectral or scattering data S D fR(k); n ; c n g. This is considerably more complicated than the Inverse Fourier Transform, involving the solution of a nontrivial integral equation, whose kernel is built out of the scattering data (see [10,11,12,13] for descriptions of this). To solve the KdV equation we first construct the scatting data S(0) from the initial condition u(x; 0). As a consequence of (4) (with an additional constant) with the given boundary conditions, the scatting data evolves in a very simple way. Indeed, we can give explicit formula: 3

R(k; t) D R(k; 0)e8i k t ;

3

c n (t) D c n (0)e4n t :

(9)

Using the inverse scattering transform on the scattering data S(t), we obtain the potential u(x; t) and thus the solution to the initial value problem for the KdV equation. This process cannot be carried out explicitly for arbitrary initial data, although, in this case, it gives a great deal of information about the solution u(x; t). However, whenever the reflection coefficient is zero, the kernel of Gel’fand-Levitan integral equation becomes separable and explicit solutions can be found. It is in this way that the N-soliton solution is constructed by IST from the initial condition: u(x; 0) D N(N C 1)sech2 x :

(10)

The general formula for the multi-soliton solution is given by: u(x; t) D 2(ln detM)x x ;

(11)

where M is a matrix built out of the discrete scattering data. Exact N-soliton Solutions of the KdV Equation Besides the IST, there are several analytical methods for obtaining solutions of the KdV equation, such as Hirota bilinear method [14,15,16], Bäcklund transformation [17], Darboux transformation [18], and so on. The existence of such analytical methods reflects a rich algebraic structure of the KdV equation. In Hirota’s method, we transform the equation into a bilinear form, from which we can get soliton solutions successively by means of a kind of perturbational technique. The Bäcklund transformation is also employed to obtain solutions from a known solution of the concerned equation. In what follows, we will mainly discuss the Hirota bilinear method to derive the N-soliton solutions of the KdV equation.

Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the

It is well known that Hirota developed a direct method for finding N-soliton solutions of nonlinear evolution equations. In particular, we shall discuss the KdV bilinear form   D x D t C D3x f  f D 0 ; (12) by the dependent variable transformation u(x; t) D 2(ln f )x x :

u(x; t) D 2(ln f2 )x x ; where f2 D 1 C exp(1 ) C exp(2 ) C exp(1 C 2 C A12 ) : (21)

(13) Frequently the N-soliton solutions are obtained as follows

Here the Hirota bilinear operators are defined by D xm D nt ab

For N D 2, the two-soliton solution for the KdV equation is similarly obtained from

m

n

D (@x @x0 ) (@ t @ t0 ) a(x; t)b(x0; t0)j x0Dx;t0Dt :

fN D

f (x; t) D 1 C f (1) " C f (2) "2 C    C f ( j) " j C    : (15) Substituting (15) into (12) and equating coefficients of powers of " gives the following recursion relations for the f (n) ":

f x(1) xxx

C

f x(1) t

D0;

(16)

1 (2) 4 (1) "2 : f x(2)  f (1) ; x x x C f x t D  (D x D t C D x ) f 2   (3) 4 (1) "3 : f x(3)  f (2) ; x x x C fx t D  Dx D t C Dx f

(17) (18)

and so on. N-soliton solutions of the KdV equation are found by assuming that f (1) has the form f (1) D

N X

exp( j );

 j D k j x  ! j t C x j0 ;

(19)

jD1

and k j ; ! j and x j0 are constants, provided that the series (15) truncates. For N D 1, we take

f

(n)

D 0;

Therefore we have !1 D k13 ;

and u(x; t) D

 1 k12 sech2 k1 x  k13 t C x10 : 2 2

jD1

1  j  l A jl A ;

(22)

1 j
Conservation Laws It was this numerical evidence which prompted Kruskal and his co-workers in Princeton to analytically investigate the KdV equation. Initially, Miura investigated local conservation laws: @ t T C @x F D 0 ;

(23)

where T and F are polynomials in u(x, t) and its x-derivatives and where @ t and @x denote total derivatives. With appropriate boundary conditions this leads to a conserved quantity (constant of the motion). Integrating (23) with respect to x we get: Z

B

T C [F]BA D 0 :

(24)

Under periodic (in x) boundary conditions with A  B an integer multiple of the period or with u(x, t) rapidly decreasing as x ! ˙1 and (A; B) D (1; C1), the square bracket R Bin (24) vanishes and we have the constant of motion @ t A Tdx. The quantities T and F are respectively called conserved density and flux. Each T is, in fact, only determined up to an exact x-derivative, so defines an equivalence class of conserved densities:

for n D 2; 3; : : : :

f1 D 1 C exp(1 );

jj C

N X

Further Properties of the KdV Equation

A

and by solving (16) we find that

exp @

N X

P where the first D0;1 means a summation over all possiPN ble combinations of  j D 0; 1 and 1 j
@t

f (1) D exp(1 );

0

D0;1

(14) We expand f as power series in a parameter "

X

(20)

T T C @x S ;

(25)

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Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the

R since this only adds @ t S to F and leaves the value of Tdx unchanged. For the KdV equation the first three are:

This led Lax to an interesting generalization: L

T0 D u ; 1 T1 D u2 ; 2 1 T2 D u3  u2x ; 2

F0 D u x x  3u2 ; 1 F1 D uu x x C u2x  2u3 ; 2 (26) 1 F2 D u x u x x x  u2x x 2 9  3u2 u x x C 6uu2x  u4 : 2

The first of these is just the equation itself. These three conservation laws have same physical interpretation, so it was no surprise that they exist. However, Miura discovered several more by direct calculation and was led to the conjecture that there should be infinitely many. In order to ascertain whether the KdV equation was the only such equation with so many conservation laws Miura investigated equations of the form: u t D u x x x C 6u n u x ;

(27)

and found that for n D 1 and n D 2 (and only these values) there existed many conservation laws. With a slight change in notation the second of these, called the modified KdV (MKdV) equation, can be written: 

v t D v x x x  6v 2 v x D v x x  2v

 3 x

:

(28)

The first few conservation laws correspond to conserved densities and fluxes: T˜1 D v ; T˜0 D v 2 ;  1 T˜1 D v 2x C v 4 ; 2

The Lax Hierarchy In [8] Lax reformulated GGKM’s discovery [6,7] of the isospectral nature of the KdV equation in algebraic form:

DP

D P(m)

) L t m D [P(m) ; L] :

(31)

The integrability condition L t m D [P(m) ; L] is explicitly written as: u t m D (2m C 1)u x  2b2m1x @2m C    C (P(m) u  Lb0 ) : (32) Equating coefficients of @ i ; i D 0; : : : ; 2m; gives us 2m C 1 equations, from which we deduce the 2m coefficients b0 ; : : : ; b2m1 : 1 (2m C 1)u; 2 1 D (2m C 1)(2m  1)u x ; : : : 4

b2m1 D b2m2

(33)

together with the isospectral flow: u t m D P(m)u  Lb0 :

(34)

Nontrivial flows only exist for odd-order equations, since the adjoint of the integrability condition implies P D P. There exists an infinite hierarchy of such isospectral flows, the first three of which are: u t0 D u x ;

(35)

1 F˜1 D v x x x C 2v 3 ; u t1 D (u x x x C 6uu x ) ; (36) 4 F˜0 D 2vv x x C v 2x C 3v 4 ; 1 u t2 D (u x x x x x C10uu x x x C20u x u x x C30u2 u x ); (37) 1 16 F˜1 D v x v x x x C v 2x x 2  2v 3 v x x C 6v 2 v 2x C 2v 6 : corresponding respectively to operators: (29)

t

tm

9  (@2 C u) D  > > > 2m1 = X i 2mC1 @ C bi @ > > iD0 > ; t D 0

9 L   (@2 C u) D  =   4@3 C 6u@ C 3u x ; t D 0 ) L t D [P; L]  PL  LP :

P(0) D @ ;

(38)

3 P(1)n D @3 C (u@ C @u) ; 4

(39)

5 P(2) D @5 C (u@3 C @3 u) 4 5 C ((3u2  uxx)@ C @(3u2  u x x )) : 16

(40)

Future Directions (30)

In mathematics, the KdV equation is a mathematical model of waves on shallow water surfaces. It is particu-

Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the

larly famous as the prototypical example of an exactly solvable model, that is, a nonlinear partial differential equation whose solutions can be exactly and precisely specified. The solutions in turn are the prototypical examples of solitons; these may be found by means of the inverse scattering transform. The mathematical theory behind the KdV equation is rich and interesting, and, in the broad sense, is a topic of active mathematical research. The KdV equation has several connections to physical problems. In addition to being the governing equation of the string in the Fermi–Pasta–Ulam problem in the continuum limit, it approximately describes the evolution of long, one-dimensional waves in many physical settings, including shallow-water waves with weakly nonlinear restoring forces, long internal waves in a densitystratified ocean, ion-acoustic waves in a plasma, acoustic waves on a crystal lattice, and more. The scientists’ interest for analytical solutions of the KdV equation stems from the fact that in applying numerical methods to nonlinear partial differential equations, the KdV equation is well suited as a test object, since having an analytical solution statements can be made on the quality of the numerical solution in comparing the numerical result to the exact result. More significantly, the existence of several analytical methods reflects a rich algebraic structure of the KdV equation. It is hoped that the study of the KdV equation could further assist in understanding, identifying and classifying nonlinear integrable differential equations and their exact solutions.

5. 6.

7.

8. 9.

10.

11. 12. 13. 14. 15.

16.

17.

18.

Springer Series in Nonlinear Dynamics. Springer, New York, Berlin, Heidelberg etc. pp 150–281 Scott AC, Chu FY, Mclaughlin DW (1973) The Soliton: A New Concept in Applied Science. Proc IEEE 61(10):1443–1483 Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg–de Vries equation. Phys Rev Lett 19:1095–1097 Gardner CS, Greene JM, Kruskal MD, Miura RM (1974) The Korteweg–de Vries equation and generalizations VI. Methods for exact solution. Commun Pure Appl Math 27:97–133 Lax PD (1968) Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math 21:467–490 Gel’fand IM, Levitan BM (1951) On the determination of a differential equation from its spectral function. Izv Akad Nauk SSR Ser Math 15:309–360 Ablowitz MJ, Clarkson PA (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Math Soc, vol 149, CUP, Cambridge Fordy AP (1990) Soliton Theory: A Survey of Results. MUP, Manchester Newell AC (1985) Solitons in Mathematics and Physics. SIAM, Philadelphia Novikov SP, Manakov SV, Pitaevskii LP, Zakharov VE (1984) Theory of Solitons. Plenum, NY Hirota R (1971) Exact soliton of the KdV equation for multiple collisions of solitons. Phys Rev Lett 27:1192–1194 Hirota R, Satsuma J (1977) Nonlinear evolution equation generated from the Bäcklund transformation for the Boussinesq equation. Prog Theor Phys Suppl 57:797–807 Hirota R (1980) Direct methods in soliton theory. In: Bullough RK, Caudrey PJ (ed) Topics of Modern Physics, vol 17. Springer, New York, pp 157–176 Wahlquist HD, Estabrook FB (1973) Bäcklund transformations for solitons of the Korteweg–de Vries equation. Phys Rev Lett 31:1386–1390 Matveev VB, Salle MA (1991) Darboux Transformations and Solitons. Springer, Berlin

Bibliography Books and Reviews Primary Literature 1. Russell JS (1845) Report on Waves. Proc Roy Soc, Edinburgh, pp 421–445 2. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos Mag Serl 39(5):422–443 3. Zabusky NJ, Kruskal MD (1965) Interaction of Solitons in a Collisionless Plasma and the recurrence of initial States. Phys Rev Lett 15:240–243 4. Bullough RK (1988) “The wave par excellence”, the solitary, progressive great wave of equilibrium of the fluid – an early history of the solitary wave. In: Lakshmanan M (ed) solitons.

Ablowitz MJ, Clarkson PA (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York Ablowitz MJ, Segur H (1981) Solitons and the Inverse Scattering Transform. SIAM, USA Drazin PG, Johnson RS (1989) Solitons: an Introduction. Cambridge University Press, New York Konopelchenko BG (1993) Solitons in Multidimensions: Inverse Spectral Transform Method. World Scientific, Singapore Rogers C, Shadwick WR (1982) Bäcklund Transformations and Their Application. Academic Press, New York Whitham GB (1974) Linear and Nonlinear Waves. Wiley, New York

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Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

Korteweg–de Vries Equation (KdV) and Modified Korteweg–de Vries Equations (mKdV), Semi-analytical Methods for Solving the ˘ DO GAN KAYA 1,2 1 Department of Mathematics, Firat University, Elazig, Turkey 2 Dean of Engineering Faculty, Ardahan University, Ardahan, Turkey

Article Outline Glossary Definition of the Subject Introduction An Analysis of the Semi-analytical Methods and Their Applications Future Directions Bibliography Glossary Korteweg–de Vries equation The classical nonlinear equations of interest usually admit for the existence of a special type of the traveling wave solutions, which are either solitary waves or solitons. Modified Korteweg–de Vries This equation is a modified form of the classical KdV equation in the nonlinear term. Soliton This concept can be regarded as solutions of nonlinear partial differential equations. Exact solution A solution to a problem that contains the entire physics and mathematics of a problem, as opposed to one that is approximate, perturbative, closed, etc. Adomian decomposition method, Homotopy analysis method, Homotopy perturbation method and Variational iteration method These are some of the semi-analytic/numerical methods for solving ODE or PDE in literature. Definition of the Subject In this study, some semi-analytical/numerical methods are applied to solve the Korteweg–de Vries (KdV) equation and the modified Korteweg–de Vries (mKdV) equation, which are characterized by the solitary wave solutions of the classical nonlinear equations that lead to solitons. Here, the classical nonlinear equations of interest usually

admit for the existence of a special type of the traveling wave solutions which are either solitary waves or solitons. These approaches are based on the choice of a suitable differential operator which may be ordinary or partial, linear or nonlinear, deterministic or stochastic. It does not require discretization, and consequently massive computation. In this scheme the solution is performed in the form of a convergent power series with easily computable components. This section is particularly concerned with the Adomian decomposition method (ADM) and the results obtained are compared to those obtained by the variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM). Some numerical results of these particular equations are also obtained for the purpose of numerical comparisons of those considered approximate methods. The numerical results demonstrate that the ADM is relatively accurate and easily implemented. Introduction In this part, a certain nonlinear partial differential equation is introduced which is characterized by the solitary wave solutions of the classical nonlinear equations that lead to solitons [8,9,11,64,70]. The classical nonlinear equations of interest usually admit for the existence of a special type of the traveling wave solutions which are either solitary waves or solitons. In this study, a few solutions arising from the semi-analytical work on the Korteweg–de Vries (KdV) equation and modified Korteweg– de Vries (mKdV) equation will be reviewed. In this work, a brief history of the above mentioned nonlinear KdV equation is given and how this type of equation leads to the soliton solutions is then presented as an introduction to the theory of solitons. This theory is an important branch of applied mathematics and mathematical physics. In the last decade this topic has become an active and productive area of research, and applications of the soliton equations in physical cases have been considered. These have important applications in fluid mechanics, nonlinear optics, ion plasma, classical and quantum fields theories, etc. The best introduction for this study may be the Scottish naval engineer J. Scott Russell’s seminal 1844 report to the Royal Society. In the time of the eighteen century, Scott Russell’s report was called “Report on Waves” [57]. Around forty years later, starting from Scott Russell’s experimental observations in 1934, the theoretical scientific work of Lord Rayleigh and Joseph Boussinesq around 1870 [56] independently confirmed Russell’s prediction

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the, Figure 1 A solitary wave

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the, Figure 2 Development of solitary waves: a initial profile at t D 0, b profile at t D  1 , and c wave profile at t D (3:6) 1 [13]

and derived formula (1) from the equation of motion for an inviscid incompressible liquid. They also gave the solitary wave profile z D u(x; t) as

a function of t, and this system is written as

 u(x; t) D a sech2 ˇ(x  Ut)

(1)

˚ where ˇ 2 D 3a  4h2 (h C a) any a > 0. They drew the solitary wave profile as in the following Finally, two Dutch scientists, Korteweg and de Vries, developed a nonlinear partial differential equation to model the propagation of shallow water waves applicable to situation in 1895 [43]. This work was really what Scott Russell fortuitously witnessed. This famous classical equation is known simply as the KdV equation. Korteweg and de Vries published a theory of shallow water waves which reduced Russell’s observations to its essential features. The nonlinear classical dispersive equation was formulated by Korteweg and de Vries in the form @u @u @u @3 u Cc C" 3 C u D0; @t @x @x @x

(2)

where c; "; are physical parameters. This equation now plays a key role in soliton theory. After setting the theory and applications of the KdV solitary waves, scientists should know a numerical model for the such nonlinear equations. This model was constructed and published as a Los Alamos Scientific Laboratory Report in 1955 by Fermi, Pasta, and Ulam (FPU). This was a numerical model of a discrete nonlinear mass– spring system [13]. The natural question arises of how the energy would be distributed among all modes in this nonlinear discrete system. The energy would be distributed uniformly among all modes in accordance with the principle of the equipartition of energy stated by Fermi, Pasta and Ulam. This model follows the form of a coupled nonlinear ordinary differential equation system, where wi is

m d2 w i D (w iC1 C w i1  2w i ) k dt 2  C ˛ (w iC1  w i )2  (w i  w i1 )2

(3)

where wi is the displacement of the ith mass from the equilibrium position, i D 1; 2; : : : ; n, w0 D w n D 0, k is the linear spring constant and ˛ (> 0) measures the strength of nonlinearity. In the late 1960s, Zabusky and Kruskal numerically studied, and then analytically solved, the KdV equation [71] from the result of the FPU experiment inspiration. They came to the result that the stable pulse-like waves could exist in a problem described by the KdV equation from their numerical study. A remarkable aspect of the discovered solitary waves is that they retain their shapes and speeds after a collision. After the observation of Zabusky and Kruskal, they called to these waves “solitons”, because the character of the waves had a particlelike nature. In fact, they considered the initial value problem for the KdV equation in the form v t C vv x C ı vxxx D 0 ;

(4)

(h/`)2 ,

` is a typical horizontal length scale, where ı D with the initial condition v(x; 0) D cos  x;

0x2

(5)

and the periodic boundary conditions with period 2, so that v(x; p t) D v(x C 2; t) for all t. Their numerical study with ı D 0:022 produced a lot of new interesting results, which are shown as follows. In 1967, this numerical development was placed on a settled mathematical basis with C.S. Gardner, J.M. Greene, M.D. Kruskal and R.M. Miura’s discovery of the inverse-scattering-transform method [14,15]. This way of calculation was an ingenious method for finding the

891

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Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

exact solution of the KdV equation. Almost simultaneously, Miura [51] and Miura et al. [52] formulated another ingenious method to derive an infinite set of conservation laws for the KdV equations by introducing the so-called Miura transformation. Subsequently, Hirota [28,29,30] constructed analytical solutions of the KdV equation which provide the description of the interaction among N solitons for any positive integral N. The soliton concept can be regarded with solutions of nonlinear partial differential equations. The soliton solution of a nonlinear equation is usually used as single wave. If there are several soliton solutions, these solutions are called “solitons.” On the other hand, if a soliton separates infinitely from other soliton, this soliton is a single wave. Besides, a single wave solution cannot be a sech2 function for equations different from nonlinear equations, such as the KdV equation. But this solution can be sech or tan1 (e˛x ). At this stage, one could ask what the definition of a soliton solution is. It is not easy to define the soliton concept. Wazwaz [64] describes solitons as solutions of nonlinear differential equations such that 1. A long and shallow water wave should not lose its permanent forms; 2. A long and shallow water wave of the solution is localized, which means that either the solutions decay exponentially to zero such as the solitons admitted by the KdV equation, or approach a constant at infinity such as the solitons provided by the SG equation; 3. A long and shallow water wave of the solution can interact with other solitons and still preserve its character. There is also a more formal definition of this concept, but these definitions require substantial mathematics [12]. On the other hand, the phenomena of the solitons use not quite the same as have stated above three properties in all expression of the solutions. For example, the concept referred to as “light bullets” in nonlinear optics are often called solitons despite losing energy during interaction. This idea can be found at an internet web page of the Simon Fraser University British Columbia, Canada [50]. Nonlinear phenomena play a crucial role in applied mathematics and physics. Calculating exact and numerical solutions, particularly traveling wave solutions, of nonlinear equations in mathematical physics play an important role in soliton theory [8,70]. It has recently become more interesting to obtain exact solutions of nonlinear partial differential equations utilizing symbolical computer programs (such as Maple, Matlab, and Mathematica) which facilitate complex and tedious algebraical computations. It is also important to find exact solutions of nonlinear

partial differential equations. These equations exist because of the mathematical models of complex physical phenomenon that arise in engineering, chemistry, biology, mechanics and physics. Various effective methods have been developed to understand the mechanisms of these physical models, to help physicians and engineers and to ensure knowledge for physical problems and its applications. Many explicit exact methods have been introduced in literature [8,9,11,31,32,33,42,62,64,65,66,67,69,70]. Some of them are: Bäcklund transformation, Cole–Hopf transformation, Generalized Miura Transformation, Inverse Scattering method, Darboux transformation, Painleve method, similarity reduction method, tanh (and its variations) method, homogeneous balance method, Exp-function method, sine–cosine method and so on. There are also many numerical methods implemented for nonlinear equations [16,17,27,34,35,37,38,39,40,54,61]. Some of them are: finite difference methods, finite elements method, Sinc–Galerkin method and some approximate or semi-analytic/numerical methods such as Adomian decomposition method, variational analysis method, homotopy analysis method, homotopy perturbation method and so on. An Analysis of the Semi-analytical Methods and Their Applications The recent years have seen a significant development in the use of various methods to find the numerical and semianalytical/numerical solution of a linear or nonlinear, deterministic or stochastic ODE or PDE. In this work, we will only discuss the nonlinear PDE case of the problem. Before giving the semi-analytical and numerical implementations of the considered methods, may draw some conclusions from Liao’s book [46] and the recent new papers [1,7,10] for comparisons of the ADM, HAM, HPM and VIM. ADM can be applied to solve linear or nonlinear ordinary and partial differential equations, no matter whether they contain small/large parameters, and thus is rather general. In some cases, the Adomian decomposition series converge rapidly. However, this method has some restrictions, e. g. if you are solving nonlinear equation you have to sort out Adomian’s polynomials. In the same case, this needs to take more calculations. Liao [46] pointed out that “in general, convergence regions of power series are small, thus acceleration techniques are often needed to enlarge convergence regions. This is mainly due to the fact that a power series is often not an efficient set of base functions to approximate a nonlinear problem, but unfortunately ADM does not provide us with freedom to use

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

different base functions. Like the artificial small parameter method and the d-expansion method, ADM itself also does not provide us with a convenient way to adjust the convergence region and the rate of approximation solutions”. Many authors [1,7,10] made their own studies in which they gave implementations of equations by using HAM, HPM, and then found numerical solutions. Their results show that: (i) Liao’s HAM can produce much better approximations than the previous solutions for nonlinear differential equations. (ii) They compared the approximations of these two methods and they observe that although HAM is faster than HPM (see Figs. 1–6 in [10]), both of the methods converge to the exact solution quite fast. HAM and HPM are in some cases similar to each other; e. g., the obtained solution function of the equation is shorter if HPM is used instead of HAM. The corresponding results converge more rapidly if HAM is used. (iii) The other advantage of the HAM is that it gives the flexibility to choose an auxiliary parameter h to adjust and control the convergence and its rate for the solutions series and the defined different functions which originate from the nature of the considered problems [1]. (iv) If one compares the approximated numerical solution with the corresponding exact solution by using HAM and HPM, one can see that when the small parameter h is increased the error of the first method is less than the second one. Chowdhury and co worker [7], show that their obtained numerical results by the 5-term HAM are exactly the same as the ADM solutions and HPM solutions for the special case of the auxiliary parameter h D 1 and auxiliary function H(x) D 1. Because of this conclusion, they admitted that the HPM and the ADM is a special case of HAM. Adomian Decomposition Method The aim of the present section is to give an outline and implementation of the Adomian decomposition method (ADM) for nonlinear wave equations, and to obtain analytic and approximate solutions which are obtained in a rapidly convergent series with elegantly computable components by this method. The approach is based on the choice of a suitable differential operator which may be ordinary or partial, linear or nonlinear, deterministic or stochastic [4,5,6,63,64]. It allows one to obtain a decomposition series solution of the equation which is calculated in the form of a convergent power series with easily computable components. The inhomogeneous problem is quickly solved by observing the self-canceling “noise” terms, where the sum of

the components vanishes in the limit. Many tests which model problems from mathematical physics, linear and nonlinear, are discussed to illustrate the effectiveness and the performance of the ADM. Adomian and Rach [5] and Wazwaz [63] have investigated the phenomena of the selfcanceling “noise” terms where the sum of the components vanishes in the limit. An important observation was made that the “noise” terms appear for nonhomogenous cases only. Further, it was formally justified that if terms in u0 are canceled by terms in u1 , even though u1 includes further terms, then the remaining non-canceled terms in u1 constitute the exact solution of the equation. ADM is valid for ordinary and partial differential equations, whether or not they contain small/large parameters, and thus is rather general. Moreover, the Adomian approximation series converge quickly. However, this method has some restrictions. Approximates solutions given by ADM often contain polynomials. In general, convergence regions of power series are small, thus acceleration techniques are often needed to enlarge convergence regions. This is mainly due to the fact that power series is often not an efficient set of base functions to approximate a nonlinear problem, but unfortunately ADM does not provide us with freedom to use different base functions. An outline of the method will be given here in order to obtain analytic and approximate solutions by using the ADM. Considering a generalized KdV equation [41] u t C ˛ u m u x C uxxx D 0 ;

u(x; 0) D g(x) ;

(6)

where ˛ and m > 0 are constants. An operator form of this equation can be written as L t (u) C ˛Nu C Lxxx (u) D 0 ;

(7)

where L t  @/(@t), Nu D u m u x and Lxxx  @3 /(@x 3 ). It is 1  assumed that L1 t is an integral operator given by L t Rt 1 0 (:)dt. Operating with the integral operator L t on both sides of (7) with 1 1 L1 t L t (u) D ˛L t (Nu)  L t Lxxx (u) :

(8)

Therefore, it follows that 1 u(x; t) D u(x; 0)  ˛L1 t (Nu)  L t Lxxx (u) :

We find that the zeroth component is obtained by u0 D u(x; 0) ;

(9)

which is defined by all terms that arise from the initial condition and from integrating the source term and decom-

893

894

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

posing the unknown function u(x; t) as a sum of components defined by the decomposition series u(x; t) D

1 X

u n (x; t) :

(10)

nD0

The nonlinear term u m u x can be decomposed into the infinite series of polynomials given by Nu D u m u x D

1 X

An ;

nD0

where the components u i (x; t) will be determined recurrently, and the An polynomials are the so-called Adomian polynomials [64] of u0 ; u1 ; u2 ;    ; u i defined by !#ˇ " 1 ˇ X 1 dn ˇ k ˚  u ; n  0 : (11) An D ˇ k ˇ n! dn kD1

D0

Substituting (11) and (9) into (8) gives rise to 1 u nC1 D L1 t (A n )  L t Lxxx (u n ) ;

n0

(12)

is the previously given integration operawhere L1 t tor. The solution u(x; t) must satisfy the requirements imposed by the initial conditions. The decomposition method provides a reliable technique that requires less work if compared with the traditional techniques. To give a clear overview of the methodology, the following examples will be discussed. Example 1 Let’s consider the mKdV equation (6) for the value of m D 1, which is called the classical KdV equation, and take this equation with the following initial value condition [36], p !  3 3 2 u (x; 0) D  tanh x ; (13) ˛ ˛ 2 and then basically Eq. (6) is taken in an operator form exactly in the same manner as the form of the Eq. (6) and using (9) to find the zeroth component of u0 D u(x; 0). Taking into consideration (12) with (11) and one can calculate the rest of the terms of the series with the aid of Mathematica and then write these terms to the Eq. (10), yielding u (x; t) D ( p !)   p  1 4 x  2 4 3t  2 C cosh x  sech 4˛ 2 (  p  1 C t 4 7  33  26 cosh x  64˛

!

p !) x  sech 2 ( p ! p ! x  1 5 x  3 11 2 C t  sech  11 sinh 8˛ 2 2 p ! !) 3x  C sinh 2 ( p ! p ! 1 x  7 x  5 72 C t  sech 302 sinh 640˛ 2 2 p ! p ! !) 3x  5x   57 sinh C sinh 2 2 ( p ! p !) 5 1 x  x  2 C 9t 2 sech tanh ˛ 2 2 ( p !) p ! 5 x  x  1  6t 2 sech4 tanh ˛ 2 2 ( p ! p !) 5 x  x  1 2 3  6t 2 sech tanh ˛ 2 2 ( p !!) 1 x  C 3 1  tanh2 C  ˛ 2

 p  C cosh 2x 

6

Example 2 Consider the mKdV equation (6) with the values of m D 2 and ˛ D 6, it has the traveling wave solution which can be obtained subject to the initial condition [41]  p p u(x; 0) D c sech k C cx ; (14) for all c  0, where k is an arbitrary constant. Again, to find the solution of this equation, simply take the equation in a operator form exactly in the same manner as the form of Eq. (6) and use (12) to find the zeroth component of u0 D u(x; 0) and obtain sequential terms by using (12) with (11) to determine the other individual terms of the decomposition series with the aid of Mathematica, to get  p  p u (x; t) D c sech k C cx   p  p  1 7  C c 2 t 2 3 C cosh 2k C 2 cx sech3 k C cx 4  p  1 13 4  c 2 t 115  76 cosh 2k C 2 cx C 192   p  p  C cosh 4k C 4 cx sech5 k C cx (   p  19 1 c 2 t 6  11774 C 10543 cosh 2k C 2 cx C 23040   p  p   722 cosh 4k C 4 cx C cosh 6k C 6 cx

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

7

 sech

) p  k C cx

  p  p  1 C c 5 t 3 sech4 k C cx  23 sinh k C cx 4  p  C sinh 3k C 3 cx (   p  p  1 c 8 t 5 sech6 k C cx 1682 sinh k C cx C 1920 )   p  p   237 sinh 3k C 3 cx C sinh 5k C 5 cx (  p  1 C c 11 t 7 sech8 k C cx 322560    p  p   259723 sinh k C cx C 60657 sinh 3k C 3 cx )   p  p   2179 sinh 5k C 5 cx C sinh 7k C 7 cx   p  p  C c 2 t sech3 k C cx tanh k C cx   p  p  C c 2 t sech k C cx tanh3 k C cx C    Homotopy Analysis Method In this section, the homotopy analysis method (HAM) [44,46] is considered. HAM has been constructed and successfully implemented as an approximate and numerical solution for many types of nonlinear problems [18,19,45,46,47,48,49,58,59,60] and the references cited therein. A very nice explanation of the basic ideas of the HAM, its relationships with other analytic techniques, and some of its applications in science and engineering are given in Liao’s book [46]. There are many groups of methods similar to HAM in the literature which are implemented nonlinear problems. Most of these groups of methods are in principle based on a Taylor series in an embedding parameter. If one could guess the initial function and the auxiliary linear operator well, then one can get very good approximations in a few terms, especially for a small value of the variable of the series. For the purpose of illustration of the HAM [44], the mKdV equation is written in the operator form as L (u) C ˛ u m u x C uxxx D 0 ;

fe n (x) t n ; n  0g ; where e n (x) as a coefficient is a function with respect to x. This provides us with the so-called Rule of Solution Expression. Following Liao’s method [44,46], let u (x; 0) D U0 (x) indicate an initial guess of the exact solution u; h ¤ 0, an auxiliary parameter, H (x; t) ¤ 0 an auxiliary function. A zero-order deformation equation is constructed as 

    1  q L ' x; t; q  u0 (x; t)

   D qhH (x; t) N ' x; t; q ;

(17)

with the initial condition   ' x; 0; q D U0 (x) :

(18)

When q D 0 and 1, the above equation has the solution '(x; t; 0) D u0 (x; t)

(19)

and '(x; t; 1) D u(x; t)

(20)

respectively. Assume the auxiliary function H (x; t) and  the auxil iary parameter h are properly chosen so that ' x; t; q can be expressed by the Taylor series 1 X   u n (x; t) q n ; ' x; t; q D u0 (x; t) C

(21)

nD1

where  ˇ 1 @n ' x; t; q ˇˇ u n (x; t) D ˇ ˇ n! @q n

(22) qD0

(15)

where L is a linear operator: L  @/(@t). Equation (15) can be written in a nonlinear operator form as      @' x; t; q N ' x; t; q D @t       @3 ' x; t; q @' x; t; q ; C ˛' m x; t; q C @x @x 3

  where q 2 [0; 1] is an embedding parameter and ' x; t; q is a function. From u (x; 0) D U0 (x), 1 < x < 1, it is straightforward to express the solution u by a set of base functions

and that the above series is convergent at q D 1. Equations (19) and (20) then yield u (x; t) D u0 (x; t) C

1 X

u n (x; t) :

(23)

nD1

For the sake of simplicity, define the vectors (16)

uE n (x; t) D fu0 (x; t) ; u1 (x; t) ; : : : ; u n (x; t)g

(24)

895

896

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

differentiating the zero-order deformation Eq. (17) n times with respect to the embedding parameter q, then setting q D 0, and finally dividing by n!, the nth-order deformation equation is written as L [u n (x; t)  n u n1 (x; t)] D hH (x; t) R n



uEn1 (x; t) ;

(25)

where  R n uEn1 (x; t) " 1 #) ˇ ( ˇ X 1 @n1 ˇ m N u D t) q (x; ˇ m n1 ˇ @q  1)! (n mD0

qD0

(26) and ( n D

0;

n1

1;

n>1

(27)

with the initial condition u n (x; 0) D 0;

n  1:

(28)

Therefore the nth order approximation of u (x; t) is given by u (x; t)  u0 (x; t) C

N X

u m (x; t) :

(29)

mD1

Example 1 Let’s consider the mKdV equation (6) for the value of m D 1, which is called the classical KdV equation, and take this equation with the following initial value condition p !!  3 2 u (x; 0) D u0 D 1  tanh x : (30) ˛ 2 All related formulae are the same as those given from (25), n1

R n [u n1 ] D

X @u n1i @3 u n1 @u n1 ui ; (31) C˛ C @t @x @x 3 iD0

using (24) and (25) with (31) and the initial function (30), three terms of the series (29) can be calculated as p

!!

 3 1  tanh2 x u (x; t) D ˛ 2  p   p  5 3ht 2 sech2 x 2  tanh x 2   ˛

( p ! 5 1 4 x  2 3ht sech C 4˛ 2 )    p   p   2 C cosh x   2 (1 C h) sinh x  ( p ! 1 5 3 2 x  2 12h (1 C h)t 2 C ht sech 8˛ 2 p !   p  2 x   2 C cosh x  sech 2 p ! x   h2 t 2 3 sech3 2 p ! p !! x  3x   11 sinh C sinh 2 2 p ! ) x  24 (1 C h)2 tanh 2 ( p ! 5 x  1 2 C ht 2 sech 64˛ 2

  p  3 2 2  144h (1 C h) t 2 C cosh x  p ! 2 x   sech 2   p   p  9 C h3 t 3  2 33  26 cosh x  C cosh 2x  p ! 4 x   sech  24h2 (1 C h) t 2 3 2 p ! 3 x   sech 2 p ! p !! x  3x   11 sinh C sinh 2 2 p ! ) x  3  192 (1 C h) tanh C  2 Example 2 In this example, the mKdV equation (6) is considered for the values of m D 2 and ˛ D 6. It has the traveling wave solution which can be obtained subject to the initial condition u(x; 0) D

p

 p c sech k C cx :

(32)

For implementation of the HAM for this particular equation with the initial function (32), all related formulae are the same as those given from (15) and (25) with the fol-

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

lowing R n [u n1 ]: @u n1 @t i n1 X X

with the boundary condition   @u D0; r 2 ; B u; @n

R n [u n1 ] D C˛

iD0 jD0

u j u i j

@u n1i @3 u n1 : C @x @x 3

(33)

 p c sech k C cx   p p  c 2 ht sech k C cx tanh k C cx  p 1 C c 2 ht sech k C cx 4   p  3  2c 2 ht 1  2 sech2 k C cx   p   4 (1 C h) tanh k C cx p

A is a general differential operator, B is a boundary operator, f (r) is a given analytic right hand side function, and  is the boundary of the domain ˝. The operator A can be generally divided into two parts L and N, where L is linear and N is the nonlinear term. So, Eq. (34) can be rewritten as follows: (37)

By the homotopy technique [45], a homotopy  v r; p : ˝  [0; 1] ! < is obtained which satisfies

 p  1 2 c ht sech k C cx 24  !   p  3  12c 2 h (1 C h) t 3 C cosh 2 k C cx   p  p   sech2 k C cx C c 3 h2 t 2 sech3 k C cx      p  p   23 sinh k C cx  sinh 3 k C cx  p   24 (1 C h)2 tanh k C cx C    which are the three terms of the approximate series solution (29). Homotopy Perturbation Method In 1998, Liao’s early idea to construct the one-parameter family of equations was introduced by He’s so-called “homotopy perturbation method” [20,21,23,24], which is only a special case of the homotopy analysis method [45]. But this homotopy perturbation method is almost the same as Liao’s early one-parameter family equation which can be found in [44]. There are some minor differences, and detailed discussions and proofs can be seen in literature [20,21,23,24]. The authors in [20,21,23,24], have proved that these two methods lead to the same solutions of the considered equation for high-order approximations. This is also mainly because of the Taylor series properties. In conclusion of this, nothing is new in He’s idea, except the new name “homotopy perturbation method” [58]. To illustrate HPM, consider the following nonlinear differential equation r2˝

(36)

L (u) C N (u)  f (r) D 0 :

C

A (u)  f (r) D 0 ;

where A (u) is written as follows: A (u) D L (u) C N (u) :

Using (24) and (25) with (33) by with the initial function (32), we have u (x; t) D

(35)

(34)

  H v; p    D 1  p [L (v)  L (u0 )] C p A (v)  f (r) D0;

p 2 [0; 1] ;

r2˝;

(38)

where p 2 [0; 1] is an embedding parameter, and u0 is an initial approximation of Eq. (34) which satisfies the boundary conditions. Obviously, from (38) the result will be H (v; 0) D L (v)  L (u0 )

D0;

H (v; 1) D A (v)  f (r)

D0;

changing   process of p from zero to unity is just that of v r; p from u0 (r) to u (r). In topology, this is called deformation, and L (v)  L (u0 ) and A (v)  f (r) are called homotopy. We consider v as the following: v D v0 C pv1 C p2 v2 C p3 v3 C    :

(39)

According to HPM, the best approximation solution of Eq. (37) can be explained as a series of the power of p, u D lim v D v0 C v1 C v2 C v3 C    : p!1

(40)

The above convergence is given in [21]. Some results have been discussed in [20,23,24]. Example 1 Here, the mKdV equation (6) is again considered for the value of m D 1, which is called classical KdV equation; first we construct a homotopy as follows: i h: i  h : : 1  p Y  u0 C p Y C˛YY 0 C Y 000 D 0 ; (41)

897

898

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

where :

YD

@3 Y

@Y @Y ; Y0 D ; Y 000 D @t @x @x 3

and

p 2 [0; 1] :

With the initial approximation 3 3 Y0 D u0 D  tanh2 ˛ ˛

p !  x ; 2

suppose the solution of Eq. (41) has the form: Y D Y0 C pY1 C p2 Y2 C p3 Y3 C : : : D

1 X

p n Yn (x; t) :

(42)

nD0

Substituting Eq. (42) into Eq. (41) and arranging the coefficients of “p” powers, we have :

:

:

:

p0 : Y0  u0 D 0 ;

(43)

p1 : Y1 C u0 C˛Y0 Y00 C Y0000 D 0 ;

(44)

:

p2 : Y2 C˛Y0 Y10 C ˛Y1 Y00 C Y1000 D 0 ;

(45)

:

p3 : Y3 C˛Y0 Y20 C ˛Y1 Y10 C ˛Y2 Y00 C Y2000 D 0 ;

(46)

:: :

Example 2 Let’s consider the mKdV equation (6) with the values of m D 2 and ˛ D 6. For this method’s implementation, a homotopy is constructed as follows: i h: i  h : : 1  p Y  u0 C p Y C6Y 2 Y 0 C Y 000 D 0 : (51) With the initial approximation  p p Y0 D u0 D c sech k C cx ; suppose the solution of Eq. (41) has the form: Y D Y0 C pY1 C p2 Y2 C p3 Y3 C : : :

and finally using Mathematica, the solutions of the equation can be obtained as follows: p !  3 3 2 Y0 D  tanh x ; (47) ˛ ˛ 2 p ! p ! 5 x  3t 2 2 x  Y1 D sech tanh (48) ˛ 2 2 Y2 D

of the considered KdV equation is obtained as follows: p ! 3 3  2 u (x; t) D  tanh x ˛ ˛ 2 p ! p ! 5 3t 2 x  2 x  C sech tanh ˛ 2 2 p !    p  3t 2 4 4 x  C 2 C cosh x  sech 4˛ 2 p ! 11 3t 3  2 x  C sech5 8˛ 2 p ! p !! x  3x  11 sinh C sinh C 2 2

D

1 X

p n Yn (x; t) :

Substituting Eq. (52) into Eq. (51) and arranging the coefficients of “p” powers, we have :

:

:

:

p0 : Y0  u0 D 0 ;

(53)

p1 : Y1 C u0 C6Y02 Y00 C Y0000 D 0 ; p ! x  ; 2

  p  3t 2 4 2 C cosh x  sech4 4˛ p ! 11 3t 3  2 5 x  sech Y3 D 8˛ 2 p ! p !! x  3x   11 sinh C sinh 2 2 :: :

:

(49)

(52)

nD0

p2 : Y2 C12Y0 Y1 Y00 C 6Y02 Y10 C Y1000 D 0 ;

(54) (55)

:

p3 : Y3 C12Y0 Y2 Y00 C 6Y12 Y00 C 12Y0 Y1 Y10 C Y2000 D 0 (50)

The above terms of the series (42) could be calculated. When the series (42) is considered with the terms (43)– (46) and supposing p D 1, an approximate series solution

(56)

:: : Finally, using Mathematica, the few terms of the series solution of the Eq. (51) can be obtained as follows:  p p  u (x; t) D c sech k C cx     p  1 7 C c 2 t 2 3 C cosh 2 k C cx 4

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

 p   sech3 k C cx  p  1 C c 5 t 3 sech4 k C cx 24      p  p   23 sinh k C cx C sinh 3 k C cx   p  p  C c 2 t sech k C cx tanh k C cx C   

determined. Consequently, the solution is given by u D lim u n :

In what follows, the VIM will apply to two nonlinear KdV equations with similar initial functions to illustrate the strength of the method and to compare the given above methods.

Variational Iteration Method In this section, a kind of semi-analytical technique is considered for a non-linear problem called the variational iteration method (VIM) [20,22,25]. It is proposed by He [20], and the method is based on the use of restricted variations and correction functionals. In this technique, a correction functional is defined by a general Lagrange multiplier using the ingenuity of variational theory. This method is implemented to get approximate solutions for many linear and non-linear problems in physics and mathematics [2,3,26,53,55,68]. When the method is implemented on the differential equation, it does not require the presence of small parameters and the solution of the equation is obtained as a sequence of iterates. The method does not require that the nonlinearities be differentiable with respect to the dependent variable and its derivatives [2,3,26,53,55,68]. To illustrate the VIM, let’s consider the nonlinear KdV equation (6): Lu C Nu D 0 ;

(59)

n!1

(57)

where L and N are linear and nonlinear operators, respectively. In [2,3,26,53,55,68], He proposed the VIM where a correction functional for Eq. (57) can be written as:

Example 1 Let’s consider the mKdV equation (6) for the value of m D 1 with the following initial function p !  3 3 2  tanh x ; (60) u (x; 0) D ˛ ˛ 2 where ; ˛ are arbitrary constant. The correction functional for this equation is u nC1 (x; t) D u n (x; t) C

Z

n   ˛  u n (x; )  C u˜ 2n (x; ) x 2 0   o C u n (x; ) xxx d ; n  0 ; (61) t

where  is the general Lagrange multiplier [20] whose optimal value is found using variational theory. u0 is an initial solution with or without unknown parameters. In the case of no unknown parameters, u0 should satisfy initial-boundary conditions and u˜ 2n is the restricted variation [22], i. e., u˜ 2n D 0. To find the optimal value of , we have ıu nC1 (x; t)

Z

t

D ıu n (x; t) C ı 0

u nC1 (x; t)

Z

t

D u n (x; t) C

˚ ˜ n (x; ) d ;  Lu n (x; ) C Nu

0

n  0;

(58)

where  is a general Lagrange multiplier [20], which can be identified optimally via the variational theory, and u˜ n is a restricted variation, which means ı u˜ n D 0. It is required first to determine the Lagrangian multiplier  that will be identified optimally via integration by parts. The successive approximations u nC1 (x; t), n  0, of the solution u (x; t) will be readily obtained upon using the Lagrangian multiplier obtained and by using any selective function u0 . The initial values u (x; 0) and u t (x; 0) are usually used for the selective zeroth approximation u0 . Having  determined, then several approximations u j (x; t), j  0, can be

n   ˛  u n (x; )  C u˜ 2n (x; ) x 2  o  C u n (x; ) xxx d ; (62)

and this yields the stationary conditions 0 () D 0 ;

1 C  ()jDt D 0 :

This in turn gives  D 1 therefore, Eq. (51) can be written as following u nC1 (x; t) D u n (x; t) 

Z tn  0

  ˛ 2 u˜ (x; ) x u n (x; )  C 2 n  o  C u n (x; ) xxx d : (63)

Substituting the value of m D 1 into the mKdV equation (6) with an initial value (60), and then substituting this into Eq. (63) and using Mathematica, the solutions of

899

900

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

the Eq. (6) with initial value (60) can be obtained as follows: 1   p  u (x; t) D 560˛ 1 C cosh x  ( p !

2 x  4 6  105 32 C t  sech 2 p ! n   2 x  2 3  88 C 4 149 C 6t  sech 2 p !   x  C 1095  124t 2 3 sech4 2 p !   x  C 198 3 C t 2 3 sech6 2 p !o

  5 x  99t 2 3 sech8 4t 2 140 6 C t 2 3 2 p !n x  C 9t 2 3 sech2 140  141t 2 3 2 p !   x  2  35 4 C 3t 2 3 sech 2 p !   4 x  2 3 4 6 C 497t   80t  sech 2 p !   6 x  2 3 2 3 C 20t  21 C 20t  sech 2 p ! x   630t 4 6 sech8 2 p ! o 10 x  4 6 C 315t  sech 2 p !) x   tanh C  2 Example 2 Again the mKdV equation (6) is considered with the values of m D 2 and ˛ D 6 and the following initial function  p p u (x; 0) D c sech k C cx : (64) The correction functional for this equation is u nC1 (x; t) D u n (x; t) 

Z tn 

   u n (x; )  C 2 u˜ 3n (x; ) x 0   o C u n (x; ) xxx d ; n  0 :

(65)

Substituting the value of m D 2 into the mKdV equation (6) with initial value (64), and then substituting this

into Eq. (65) and using Mathematica, the solutions of the considered equation with initial value (64) can be obtained as follows:  p  1 u (x; t) D sech k C cx 4 (   p  7  2c 2 t 2 1  2 sech2 k C cx  p  C 2c 5 t 3 sech5 k C cx      p  p   17 sinh k C cx C 3 sinh 3 k C cx     p  13 C 3c 2 t 4 3 C cosh 2 k C cx   p  p   sech4 k C cx tanh2 k C cx ) p  p  2 C  C 4 c C c t tanh k C cx We have just written the series solution for the mKdV as three terms rather than four terms. Numerical Experiments For numerical comparisons purposes, two KdV equations are considered. The first one is the classical KdV and the second one is the modified KdV equation. The formula of numerical results for ADM, HAM and HPM are given as follows lim  n D u(x; t)

n!1

where

 n (x; t) D

n X

u k (x; t) ;

n  0:

(66)

kD0

The formula of numerical results for VIM is as (59). The recurrence relations of the methods are given as in (12), (25), (43)–(46) and (58), respectively. Moreover, the decomposition series of the KdV equation’s solutions generally converge very rapidly in real physical problems [5,6,64]. Results were obtained about the speed of convergence of this method, providing us methods to solve linear and nonlinear functional equations. The convergence of the HAM and VIM are numerically shown in some works [3,55] and references there in. Here, how all these methods are converged to their corresponding exact solutions has been proved. Numerical approximations show a high degree of accuracy and in most cases of  n , the n-term approximation is accurate for quite low values of n. The solutions are very rapidly convergent by utilizing the ADM, VIM, HPM and HAM. The obtained numerical results justify the advantage of these methodologies, and even with few terms the approximation is accurate. Furthermore, as the all these methods do not require discretization of the variables, i. e.

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the, Table 1 Comparison between the absolute error of the solution of KdV equation (6) (m D 1) at various values of t and x D 0:5 for ˛ D 1;  D 0:01 in ADM, VIM and HAM (h D 1) t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.

xD0 ADM 2.9925  108 1.197  107 2.69323  107 4.78795  107 7.48113  107 1.07727  106 1.46628  106 1.91512  106 2.42379  106 2.9923  106

t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.

x D 10 ADM 0.0000207071 0.0000413971 0.0000620701 0.0000827257 0.000103364 0.000123985 0.000144588 0.000165173 0.000185741 0.00020629

HAM 2.9925  108 1.197  107 2.69323  107 4.78795  107 7.48113  107 1.07727  106 1.46628  106 1.91512  106 2.42379  106 2.9923  106

HAM 0.0000229046 0.000045826 0.000068764 0.0000917187 0.00011469 0.000137677 0.000160681 0.0001837 0.000206736 0.000229787

VIM 3.  108 1.2  107 2.69998  107 4.79995  107 7.49987  107 1.07997  106 1.46995  106 1.91992  106 2.42987  106 2.9998  106

VIM 0.0000229046 0.0000458261 0.0000687642 0.000091719 0.00011469 0.000137678 0.000160682 0.000183702 0.000206738 0.00022979

xD5 ADM 0.0000130999 0.0000261535 0.0000391608 0.0000521216 0.000065036 0.0000779036 0.0000907246 0.000103499 0.000116226 0.000128906

x D 15 ADM 0.0000216022 0.0000432119 0.0000648289 0.0000864532 0.000108085 0.000129723 0.000151369 0.000173022 0.000194682 0.000216348

HAM 0.0000145274 0.0000291008 0.0000437201 0.0000583853 0.0000730962 0.0000878527 0.000102655 0.000117502 0.000132395 0.000147333

HAM 0.0000238682 0.0000477289 0.0000715819 0.0000954272 0.000119265 0.000143094 0.000166916 0.00019073 0.000214535 0.000238333

VIM 0.0000145274 0.000029101 0.0000437206 0.0000583862 0.0000730976 0.0000878548 0.000102658 0.000117506 0.0001324 0.000147339

VIM 0.0000238682 0.0000477289 0.0000715819 0.0000954271 0.000119265 0.000143094 0.000166916 0.000190729 0.000214535 0.000238332

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the, Table 2 Comparison between the absolute error of the solution for Eq. (6) (m D 1) by HAM and HPM at various values of t and x D 0:5 with different values of h for HAM, in fact HPM is the value of h D 1 for HAM t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.

HPM 1.60354  106 3.26676  106 4.98966  106 6.77222  106 8.61444  106 0.0000105163 0.0000124777 0.0000144988 0.0000165795 0.0000187197

HAM (h D 1) 1.60354  106 3.26676  106 4.98966  106 6.77222  106 8.61444  106 0.0000105163 0.0000124777 0.0000144988 0.0000165795 0.0000187197

HAM (h D 1:4) 1.60838  106 3.27654  106 5.00446  106 6.79214  106 8.63955  106 0.0000105467 0.0000125135 0.0000145401 0.0000166263 0.0000187722

time and space, it is not effected by computation round off errors and one is not faced with the necessity of large computer memory and time. In order to verify numerically whether the proposed methodologies lead to higher accuracy, the numerical so-

HAM (h D 0:9) 1.60346  106 3.26662  106 4.98946  106 6.77196  106 8.61411  106 0.0000105159 0.0000124773 0.0000144984 0.000016579 0.0000187192

HAM (h D 0:6) 1.59877  106 3.25727  106 4.97551  106 6.75346  106 8.59111  106 0.0000104885 0.0000124455 0.0000144621 0.0000165384 0.0000186744

lutions can be evaluated using the n-term approximation (66) and (59). The numerical results of the Eq. (6) (for KdV m D 1 and for mKdV m D 2) for the various values of x and t are illustrated in Tables 1 and 3 KdV and mKdV, respectively. These tabulated results show the

901

902

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the, Table 3 Comparison between the absolute error of the solution of Eq. (6) (m D 2) for c D 1; k D 7 juExact  '3 j for ADM tjx 0.1 0.1 8.23236  109 0.2 9.098  109 0.3 1.00546  108 0.4 1.11118  108 0.5 1.228  108

0.2 1.29161  107 1.42742  107 1.57752  107 1.74338  107 1.92667  107

0.3 6.41349  107 7.08789  107 7.83317  107 8.65678  107 9.56694  107

0.4 1.98865  106 2.19776  106 2.42885  106 2.68424  106 2.96645  106

0.5 4.76447  106 5.26547  106 5.81914  106 6.43099  106 7.10715  106

juExact  u3 jfor VIM tjx 0.1 0.1 8.19516  109 0.2 9.04779  109 0.3 9.98686  109 0.4 1.10203  108 0.5 1.21566  108

0.2 1.2858  107 1.41958  107 1.56692  107 1.72908  107 1.90738  107

0.3 6.38476  107 7.0491  107 7.78082  107 8.58611  107 9.47155  107

0.4 1.97978  106 2.18579  106 2.4127  106 2.66243  106 2.93702  106

0.5 4.74334  106 5.23696  106 5.78065  106 6.37904  106 7.03703  106

juExact  '3 j for HPM HAM with value of h D 1 tjx 0.1 0.2 0.3 0.1 8.19516  109 1.2858  107 6.38476  107 0.2 9.04779  109 1.41958  107 7.0491  107 0.3 9.98686  109 1.56692  107 7.78082  107 0.4 1.10203  108 1.72908  107 8.58611  107 0.5 1.21566  108 1.90738  107 9.47155  107

0.4 1.97978  106 2.18579  106 2.4127  106 2.66243  106 2.93702  106

0.5 4.74334  106 5.23696  106 5.78065  106 6.37904  106 7.03703  106

juExact  '3 j for HAM with value of h D 0:9 tjx 0.1 0.2 0.3 0.1 2.1809  109 1.26484  107 1.18355  107 0.2 2.41029  109 1.39787  107 1.30802  107 0.3 2.66382  109 1.54489  107 1.44558  107 0.4 2.94403  109 1.70737  107 1.59761  107 0.5 3.25373  109 1.88694  107 1.76562  107

0.4 1.28924  107 1.42489  107 1.57481  107 1.74053  107 1.9237  107

0.5 5.68147  107 6.27919  107 6.93984  107 7.67005  107 8.47719  107

differences of the absolute errors between the exact solution and the numerical solution for the ADM, VIM and HAM. One can conclude that the values on the interval 0  x  15 and for some small values of t all the considered methods give very close numerical solutions to a corresponding KdV equation. If one closely looks at Tables 1 and 3 then it is evident that the numerical results are almost the same for a few terms of the series solutions, such as the first three terms. The calculation for the later terms of the series has stopped because the scheme VIM does not work very well in the sense of computer time. This problem needs to be approached in a completely different way. Moreover, it has also been numerically proved that HPM is a special case of the HAM which is tabulated in Table 2 for the various values of h for interval 1:4  h  0:6 and similar constant values as in Table 1. It is numerically shown that when we take the value of h D 1 in the for-

mula HAM, then (numerically) HAM  HPM in Table 2. It is also our experience that the choice of the h value depends on the chosen equation and the solution of the equation, too. This can be seen in Tables 2 and 3. Overall, one can see that closeness of the approximate solutions and exact solutions of the all methods give almost equal absolute values of error, except for the approximate solutions from HAM. The numerical solutions with a special value of h D 0:9 for the HAM perform very well, as one can see in Table 3. It is also a valuable opportunity for us to use HAM among the considered methods because it gives the users flexibility to choose different values of h, depending on which value is necessary for the problem. The graphs of the numerical values of the exact and approximate solutions of the classical KdV equation are depicted in Figs. 3–6. In Figs. 3–5, and numerical results

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the, Figure 3 A comparison by the ADM, VIM, HPM and HAM with exact solution of Eq. (6) with (m D 1) at t D 0:5 for ˛ D 1;  D 0:01 in a–c and ˛ D 1;  D 0:01; h D 1 in d, respectively

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the, Figure 4 A comparison by the ADM, VIM, HAM and HPM with exact solution of Eq. (6) with (m D 1) at t D 0:5 for ˛ D 1;  D 0:1 in a–c and ˛ D 1;  D 0:1; h D 1 in d, respectively

are given for the KdV equation and corresponding exact solution by using the considered methods. It is to be noted that only four terms were used in evaluating the approximate solutions with different values of the constant

 D 0:01; 0:1; 1 in order to get how close the approximate solutions were to the exact solution. A very good approximation to the actual solution of the equation was achieved by only using the four terms of the series derived

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Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the, Figure 5 A comparison by the ADM, VIM, HAM and HPM with exact solution of Eq. (6) with (m D 1) at t D 0:5 for ˛ D 1;  D 1 in a–c and ˛ D 1;  D 1; h D 1 in d, respectively

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), ˇ Semi-analytical Methods for Solving the, Figure 6 ˇ These two graphs is a comparison of uEr D ˇuExact  uApprox ˇ for ADM, HAM and VIM of the solutions of Eq. (6) with (m D 1) at t D 0:5 for ˛ D 1;  D 0:01; h D 1 in a and ˛ D 1;  D 1; h D 1 in b, respectively

above (59) and (66) for all considered methods with the value of  D 0:01 which are depicted in Fig. 3. This is because of the nature of the series methods. The closeness of the numerical results for approximate solutions and the exact solution diverges for the other value of  D 0:1 in all considered methods which are depicted in Fig. 4. This divergence is very clear in Fig. 5 for the results of the all the methods. But a sharp divergence can be seen in the numerical results of VIM especially at the value of x D 0:5, in Fig. 5b. In fact, these are illustrated in Tables 1– 2 and Figs. 3–4. It is evident that the overall errors can be made smaller by adding new terms of the series (59) and (66).

One can see from Fig. 6a, the absolute values of the numerical results for approximate solutions and exact solutions are very small at the value of 0  x  1 but other than this values of the x neither positive nor negative values of the x gives small errors for all considered methods with the value of  D 0:01. However, in all considered methods with the value of  D 1 a relatively small absolute error is given at the value of 0  x  1, but all the methods are given biggest absolute error at a value of x around ˙2. The VIM is poorly performed at the value of x around 2, but besides this all the other methods are performed relatively well at the other values of the x, which can be seen in Fig. 4b.

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

Lastly, the clear conclusion can be draw from the numerical results that the ADM and HAM algorithms provide highly accurate numerical solutions without spatial discretizations for nonlinear partial differential equations. It is also worth noting that the advantage of the approximation of the series methodologies displays a fast convergence of the solutions. The illustrations show the dependence of the rapid convergence depends on the character and behavior of the solutions just as in a closed form solutions. Finally, it can be pointed out that, for given equations with initial values u(x; 0), the corresponding analytical and numerical solutions are obtained according to the recurrence relations (12), (25), (43)–(46) and (58) using Mathematica package version of Mathematica 4 in PC computer. Future Directions Nonlinear phenomena play a crucial role in applied mathematics and physics. Furthermore, when an original nonlinear equation is directly calculated, the solution will preserve the actual physical characters of solutions. Explicit solutions to the nonlinear equations are of fundamental importance. Various effective methods have been developed to understand the mechanisms of these physical models, to help physicists and engineers, and to ensure knowledge for physical problems and their applications. In the future, the scientist will be very busy doing many works via applications of the nonlinear evolution equations in order to obtain exact and numerical solutions. One has to find out more efficient and effective methods to solve the nonlinear problems in applied mathematical areas of constructing either an exact solution or a numerical solution. Scientists have long way to go in this nonlinear study. Bibliography Primary Literature 1. Abbasbandy S (2006) The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 360:109–13 2. Abdou MA, Soliman AA (2005) New applications of variational iteration method. Phys D 211:1–8 3. Abdou MA, Soliman AA (2005) Variational iteration method for solving Burger’s and coupled Burger’s equations. J Comput Appl Math 181:245–251 4. Adomian G (1988) A Review of the Decomposition Method in Applied Mathematics. J Math Anal Appl 135:501–544 5. Adomian G, Rach R (1992) Noise Terms in Decomposition Solution Series. Comput Math Appl 24:61–64 6. Adomian G (1994) Solving Frontier Problems of Physics: The decomposition method. Kluwer, Boston

7. Chowdhury MSH, Hashim I, Abdulaziz O (2009) Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems. Commun Nonlinear Sci Numer Simul 14:371–378 8. Debtnath L (1997) Nonlinear Partial Differential Equations for Scientist and Engineers. Birkhauser, Boston 9. Debtnath L (2007) A brief historical introduction to solitons and the inverse scattering transform – a vision of Scott Russell. Int J Math Edu Sci Techn 38:1003–1028 10. Domairry G, Nadim N (2008) Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation. Intern Commun Heat Mass Trans 35:93–102 11. Drazin PG, Johnson RS (1989) Solutions: An Introduction. Cambrige University Press, Cambridge 12. Edmundson DE, Enns RH (1992) Bistable light bullets. Opt Lett 17:586 13. Fermi E, Pasta J, Ulam S (1974) Studies of nonlinear problems. American Math Soci, Providence, pp 143–156 14. Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg–de Vries equation. Phys Rev Lett 19:1095–1097 15. Gardner CS, Greene JM, Kruskal MD, Miura RM (1974) Korteweg–de Vries equation and generalizations, VI, Methods for exact solution. Commun Pure Appl Math 27:97–133 16. Geyikli T, Kaya D (2005) An application for a Modified KdV equation by the decomposition method and finite element method. Appl Math Comp 169:971–981 17. Geyikli T, Kaya D (2005) Comparison of the solutions obtained by B-spline FEM and ADM of KdV equation. Appl Math Comp 169:146–156 18. Hayat T, Abbas Z, Sajid M (2006) Series solution for the upperconvected Maxwell fluid over a porous stretching plate. Phys Lett A 358:396–403 19. Hayat T, Sajid M (2007) On analytic solution of thin film flow of a fourth grade fluid down a vertical cylinder. Phys Lett A 361:316–322 20. He JH (1997) A new approach to nonlinear partial differential equations. Commun Nonlinear Sci Numer Simul 2:203–205 21. He JH (1999) Homotopy perturbation technique. Comput Math Appl Mech Eng 178:257–262 22. He JH (1999) Variation iteration method – a kind of non-linear analytical technique: some examples. Int J Nonlinear Mech 34:699–708 23. He JH (2000) A coupling method of homotopy technique and a perturbation technique for nonlinear problems. Int J Nonlinear Mech 35:37–43 24. He JH (2003) Homotopy perturbation method a new nonlinear analytical technique. Appl Math Comput 135:73–79 25. He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 20:1141–1199 26. He JH, Wu XH (2006) Construction of solitary solution and compacton-like solution by variational iteration method. Chaos Solitons Fractals 29:108–113 27. Helal MA, Mehanna MS (2007) A comparative study between two different methods for solving the general Korteweg–de Vries equation. Chaos Solitons Fractals 33:725–739 28. Hirota R (1971) Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys Rev Lett 27:1192– 1194

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29. Hirota R (1973) Exact N-solutions of the wave equation of long waves in shallow water and in nonlinear lattices. J Math Phys 14:810–814 30. Hirota R (1976) Direct methods of finding exact solutions of nonlinear evolution equations. In: Miura RM (ed) Bäcklund Transformations. Lecture Notes in Mathematics, vol 515. Springer, Berlin, pp 40–86 31. Inan IE, Kaya D (2006) Some Exact Solutions to the Potential Kadomtsev–Petviashvili Equation. Phys Lett A 355:314–318 32. Inan IE, Kaya D (2006) Some Exact Solutions to the Potential Kadomtsev–Petviashvili Equation. Phys Lett A 355:314–318 33. Inan IE, Kaya D (2007) Exact Solutions of the Some Nonlinear Partial Differential Equations. Phys A 381:104–115 34. Kaya D (2003) A Numerical Solution of the Sine–Gordon Equation Using the Modified Decomposition Method. Appl Math Comp 143:309–317 35. Kaya D (2006) The Exact and Numerical Solitary-wave Solutions for Generalized Modified Boussinesq Equation. Phys Lett A 348:244–250 36. Kaya D Some Methods for the Exact and Numerical Solutions of Nonlinear Evolution Equations. Paper presented at the 2nd International Conference on Mathematics: Trends and Developments (ICMTD), Cairo, Egypt, Dec 2007, pp 27–30 (in press) 37. Kaya D, Al-Khaled K (2007) A Numerical Comparison of a Kawahara Equation. Phys Lett A 363:433–439 38. Kaya D, El-Sayed SM (2003) An Application of the Decomposition Method for the Generalized KdV and RLW Equations. Chaos Solitons Fractals 17:869–877 39. Kaya D, El-Sayed SM (2003) Numerical Soliton-like Solutions of the Potential Kadomstev–Petviashvili Equation by the Decomposition Method. Phys Lett A 320:192–199 40. Kaya D, El-Sayed SM (2003) On a Generalized Fifth Order KdV Equations. Phys Lett A 310:44–51 41. Kaya D, Inan IE (2005) A Convergence Analysis of the ADM and an Application. Appl Math Comp 161:1015–1025 42. Khater AH, El-Kalaawy OH, Helal MA (1997) Two new classes of exact solutions for the KdV equation via Bäcklund transformations. Chaos Solitons Fractals 8:1901–1909 43. Korteweg DJ, de Vries H (1895) On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos Mag 39:422–443 44. Liao SJ (1992) The proposed homotopy analysis technique for the solution of nonlinear problems. Ph D Thesis, Shanghai Jiao Tong University 45. Liao SJ (1995) An approximate solution technique not depending on small parameters: A special example. Int J Nonlinear Mech 30:371 46. Liao SJ (2003) Beyond Perturbation: Introduction to Homotopy Analysis Method. Chapman & Hall/CRC Press, Boca Raton 47. Liao SJ (2004) On the homotopy analysis method for nonlinear problems. Appl Math Comp 147:499 48. Liao SJ (2004) On the homotopy analysis method for nonlinear problems. Appl Math Comput 147:499–513 49. Liao SJ (2005) Comparison between the homotopy analysis method and homotopy perturbation method. Appl Math Comp 169:1186–1194 50. Light Bullet Home Page, http://www.sfu.ca/~renns/lbullets. html 51. Miura RM (1968) Korteweg–de Vries equations and generalizations, I; A remarkable explicit nonlinear transformations. J Math Phys 9:1202–1204

52. Miura RM, Gardner CS, Kruskal MD (1968) Korteweg–de Vries equations and generalizations, II; Existence of conservation laws and constants of motion. J Math Phys 9:1204–1209 53. Momani S, Abuasad S (2006) Application of He’s variational iteration method to Helmholtz equation. Chaos Solitons Fractals 27:1119–1123 54. Polat N, Kaya D, Tutalar HI (2006) A Analytic and Numerical Solution to a Modified Kawahara Equation and a Convergence Analysis of the Method. Appl Math Comp 179:466–472 55. Ramos JI (2008) On the variational iteration method and other iterative techniques for nonlinear differential equations. Appl Math Comput 199:39–69 56. Rayleigh L (1876) On Waves. Lond Edinb Dublin Philos Mag 5:257 57. Russell JS (1844) Report on Waves. In: Proc 14th meeting of the British Association for the Advancement of Science, BAAS, London 58. Sajid M, Hayat T (2008) Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations, Nonlinear Analysis. Real World Appl 9:2296–2301 59. Sajid M, Hayat T (2008) The application of homotopy analysis method for thin film flow of a third order fluid. Chaos Solitons Fractal 38:506–515 60. Sajid M, Hayat T, Asghar S (2007) Comparison between HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt. Nonlinear Dyn 50:27–35 61. Shawagfeh N, Kaya D (2004) Series Solution to the Pochhammer–Chree Equation and Comparison with Exact Solutions. Comp Math Appl 47:1915–1920 62. Ugurlu Y, Kaya D Exact and Numerical Solutions for the Cahn– Hilliard Equation. Comput Math Appl 63. Wazwaz AM (1997) Necessary Conditions for the Appearance of Noise Terms in Decomposition Solution Series J Math Anal Appl 5:265–274 64. Wazwaz AM (2002) Partial Differential Equations: Methods and Applications. Balkema, Rotterdam 65. Wazwaz AM (2007) Analytic study for fifth-order KdV-type equations with arbitrary power nonlinearities. Comm Nonlinear Sci Num Sim 12:904–909 66. Wazwaz AM (2007) A variable separated ODE method for solving the triple sine-Gordon and the triple sinh-Gordon equations. Chaos Solitons Fractals 33:703–710 67. Wazwaz AM (2007) The extended tanh method for abundant solitary wave solutions of nonlinear wave equations. App Math Comp 187:1131–1142 68. Wazwaz AM (2007) The variational iteration method for solving linear and nonlinear systems of PDEs. Comput Math Appl 54:895–902 69. Wazwaz AM, Helal MA (2004) Variants of the generalized fifthorder KdV equation with compact and noncompact structures. Chaos Solitons Fractals 21:579–589 70. Whitham GB (1974) Linear and Nonlinear Waves. Wiley, New York 71. Zabusky NJ, Kruskal MD (1965) Interactions of solitons in collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243

Books and Reviews The following, referenced by the end of the paper, is intended to give some useful for further reading.

Korteweg–de Vries Equation (KdV) and Modified KdV (mKdV), Semi-analytical Methods for Solving the

For another obtaining of the KdV equation for water waves, see Kevorkian and Cole (1981); one can see the work of the Johnson (1972) for a different water-wave application with variable depth, for waves on arbitrary shears in the work of Freeman and Johnson (1970) and Johnson (1980) for a review of one and two-dimensional KdV equations. In addition to these; one can see the book of Drazin and Johnson (1989) for some numerical solutions of nonlinear evolution equations. In the work of the Zabusky, Kruskal and Deam (F1965) and Eilbeck (F1981), one can see the motion pictures of soliton interactions. See a comparison of the KdV equation with water wave experiments in Hammack and Segur (1974).

For further reading of the classical exact solutions of the nonlinear equations can be seen in the works: the Lax approach is described in Lax (1968); Calogero and Degasperis (1982, A.20), the Hirota’s bilinear approach is developed in Matsuno (1984), the Bäckland transformations are described in Rogers and Shadwick (1982); Lamb (1980, Chap. 8), the Painleve properties is discussed by Ablowitz and Segur (1981, Sect. 3.8), In the book of Dodd, Eilbeck, Gibbon and Morris (1982, Chap. 10) can found review of the many numerical methods to solve nonlinear evolution equations and shown many of their solutions.

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Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the MUSTAFA INC Department of Mathematics, Fırat University, Elazı˘g, Turkey Article Outline Glossary Definition of the Subject Introduction Some Numerical Methods for Solving the Korteweg–de Vries (KdV) Equation The Adomian Decomposition Method (ADM) The Homotopy Analysis Method (HAM) The Variational Iteration Method (VIM) The Homotopy Perturbation Method (HPM) Numerical Applications and Comparisons Conclusions and Discussions Future Directions Bibliography Glossary Absolute error When a real number x is approximated by another number x , the error is x  x  . The absolute error is jx  x  j. Adomian decomposition method The method was introduced and developed by George Adomian. The method proved to be powerful, effective, and can easily handle a wide class of linear or nonlinear differential equations. Adomian polynomials It is well known now that the ADM suggests that the unknown linear function u may be presented by the decomposition series P1 u D nD0 u n , the nonlinear term F(u), such as u2 ; u3 ; sin u; eu ; u2x , etc. can be expressed by an infinite series of the so-called Adomian polynomials An . Finite difference method The method handles the differential equation by replacing the derivatives in the equation with difference quotients. Homotopy analysis method The HAM was introduced by Shi-Jun Liao in 1992. For details see Sect. “The Variational Iteration Method (VIM)”. Homotopy perturbation method The HPM was introduced by Ji-Huan He in 1999. For details see Sect. “Numerical Applications and Comparisons”.

Korteweg–de Vries equation This is one of the simplest and most useful nonlinear model equations for solitary waves. Lagrange multiplier The multiplier in the functional should be chosen such that its correction solution is superior to its initial approximation (trial function) and is the best within the flexibility of trial function, accordingly we can identify the multiplier by variational theory. Initial conditions The PDEs mostly arise to govern physics phenomena such as heat distribution, wave propagation phenomena. Most of the PDEs depend on the time t. The initial values of the dependent variable u at the starting time t D 0 should be prescribed. Solitons It is interesting to point out that there is no precise definition of a soliton. However, a soliton can be defined as a solution of a nonlinear partial differential equation. Variational iteration method The VIM was first proposed by Ji-Huan He. This method has been employed to solve a large variety of linear and nonlinear problems with approximations converging rapidly to accurate solutions. Definition of the Subject In this work, the Adomian decomposition method (ADM), the homotopy analysis method (HAM), the variational iteration method (VIM), the homotopy perturbation method (HPM) and the explicit finite difference method (EFDM) are implemented to investigate numerical solutions of the Korteweg–de Vries (KdV) equation with initial condition. The five methods are compared and it is shown that the HAM and EFDM are more efficient and effective than the ADM, the VIM and the HPM, and also converges to its exact solution more rapidly. The HAM contains the auxiliary parameter ¯ which provides us with a simple way to adjust and control the convergence region of solution series. Introduction John Scott Russell first experimentally observed the solitary wave, a long water wave without change in shape, on the Edinburgh-Glasgow Canal in 1834. He called it the great wave of translation and, then, reported his observations at the British Association in his 1844 paper “Report on waves“ [1]. After 60 years from this discovery, the two scientists D.J. Korteweg and G. de Vries formulated a mathematical model equation to provide an explanation of the phe-

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

nomenon observed by Russell [2]. They derived the famous equation, the Korteweg–de Vries (KdV) equation, for the propagation of waves in one dimension on the surface of water. This is one of the simplest and most useful nonlinear model equations for solitary waves, and it represents the longtime evolution of wave phenomena in which the steepening effect of the nonlinear term is counterbalanced by dispersion. The KdV equation is a nonlinear partial differential equation of third order. Modern developments in the theory and applications of the KdV equation began with the seminal work published as a Los Alamos Scientific Laboratory Report in 1955 by Fermi, Pasta and Ulam (FPU) on a numerical model of a discrete nonlinear mass-spring system [3]. This curious result of the FPU experiment inspired Zabusky and Kruskal [4] marked the birth of the new concept of the soliton, a name intended to signify particle like quantities. They found that stable pulse like waves could exist in a system described by the KdV equation. A remarkable quality of these solitary waves was that they could collide with each other and yet preserve their shapes and speeds after the collision. This means that a collision between KdV solitary waves is elastic. Subsequently, Zabusky [5] confirmed, numerically, the actual physical interaction of two solitons, and Lax [6] gave a rigorous analytical proof that the identities of two distinct solitons are preserved through the nonlinear interaction governed by the KdV equation. Subsequently, a paper by Gardner et al. [7] demonstrated that it was possible to write many exact solutions to the equation by using ideas from scattering theory. They discovered the first integrable nonlinear partial differential equation. Gardner et al. [8] and Hirota [9,10] constructed analytical solutions of the KdV equation that provide the description of the interaction among N solitons for any positive integer N. Experimental confirmation of solitons and their interactions has been provided successfully by Zabusky and Galvin [11], Hammack and Segur [12], and Weidman and Maxworthy [13]. During the past fifteen years a rather complete mathematical description of solitons has been developed. The nondispersive nature of the soliton solutions to the KdV equation arises not because the effects of dispersion are absent, but because they are balanced by nonlinearities in the system [14,15,16,17]. The KdV equation [2,7,18] arises in several areas of nonlinear physics such as hydrodynamics, plasma physics, etc. During the last five decades or so an exciting and extremely active area of research has been devoted to the construction of exact and numerical solutions for a wide class of nonlinear equations. This includes the most famous nonlinear one of Korteweg and de-Vries.

Exact solutions of the KdV equation have been used for many powerful methods such as the application of the Jacobian elliptic function [19,20,21], the powerful inverse scattering transform [22], the Painleve analysis [23], the Backlund transformations method [24], the Lie group theoretical methods [25], the direct algebraic method [26], the tanh-method [27], the sine-cosine method [28], the homogeneous balance method [29], the Riccati expansion method with constant coefficient [30] and the mapping method [31]. In addition, Sawada and Kotera [32], Rosales [33], Whitham [34], Wadati and Sawada [35,36] have all employed perturbation techniques. Recently, Helal and El-Eissa [37], Khater et al. [38,39], Helal [40] have studied analytically and numerically some physical problems that lead to the KdV equation and the soliton solutions have been obtained [17]. The KdV equation is a generic equation for the study of weakly nonlinear long waves. The KdV type of equation has been an important class of nonlinear evolution equations with numerous applications in physical sciences and engineering fields. For example, in plasma physics, these equations give rise to the ion acoustic solitons [41]; in geophysical fluid dynamics, they describe a long wave in shallow seas and deep oceans [42,43]. Their strong presence is exhibited in cluster physics, super deformed nuclei, fission, thin film, radar and rheology [44,45], optical-fiber communications [46] and superconductors [47]. Thus many analytical solutions of the KdV equation are found, and their existence and uniqueness have been studied for a certain class of initial functions [14]. The numerical solutions of the KdV equation are essential because of solutions which are not analytically available. Many methods have been proposed for numerical treatment of the KdV equation for the various boundary and initial conditions. For appropriate initial conditions, Gradner et al. [7] have shown the existence and uniqueness of solutions of the KdV equation. For the KdV equation, several quite successful numerical methods have been available such as spectral/pseudo spectral methods [48,49], finite-difference methods and Fourier spectral methods developed by many authors from both the theoretical and computational points of view [50,51]. The exponential finite-difference method is used to solve the KdV equation with the initial and boundary conditions by Bahadır [52]. Jain et al. [53] developed a numerical method for solving the KdV equation by using splitting method and quintic spline approximation technique. Soliman [54] worked numerical solutions for the KdV equation based on the collocation method using septic splines as element shape functions were set up. Frauendiener and Klein [55] presented the hyperelliptic theta-functions with spectral methods for

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numerical solutions of the KP and KdV equations. Bhatta and Bhatti [56] studied numerical solution of the KdV equation by using modified Bernstein polynomials. Helal and Mehanna [57] presented a comparative study between the Adomian decomposition method and the finite-difference method for solving the general KdV equation. Kutluay et al. [58] used the heat balance integral method to the KdV equation prescribed by appropriate homogenous boundary conditions and a set of initial conditions to obtain its approximate analytical solutions at small times. An analytical-numerical method was applied to the KdV equation with a variant of boundary and initial conditions to obtain its numerical solutions at small times by Özer and Kutluay [59]. Recently, Dehghan and Shokri [60] proposed a numerical scheme to solve the third-order nonlinear KdV equation using collocation points and approximating the solution using multiquadric radial basis function. Dag and Dereli [61] presented numerical solution of the KdV equation by using the meshless method based on the collocation with radial basis functions. Some Numerical Methods for Solving the Korteweg–de Vries (KdV) Equation The Korteweg–de Vries (KdV) equation has the following form: u t  6uu x C u x x x D 0 ;

(1)

subject to initial condition u (x; 0) D f (x) ;

(2)

where u (x; t) is a differentiable function and f (x) is bounded. We shall assume that the solution u(x; t), along with its derivatives, tends to zero as jxj ! 1. The second and third terms uu x and uxxx represent the nonlinear convection and dispersion effects, respectively. The solitons of the KdV equation arise as a balance between nonlinear convection and dispersion terms. The nonlinear effect causes the steepening of the waveform [61], while the dispersion effect makes the waveform spread. Due to the competition of these two effects, a stationary waveform (solitary wave) exists. The solution is obtained among the nonlinear and dispersion by stability. The soliton solutions of nonlinear wave equations shown by Wadati [62,63] have the following properties:  Some certain waves propagate which does not change its special behavior.  Reginal waves do not lose their properties and also they are stable towards the collisions.

The basic purpose of this work is to approach the KdV Eq. (1) with initial condition (2) differently, by using four semi inverse methods and the EFDM. In this work, we will use the Adomian decomposition method (ADM), the homotopy analysis method (HAM), the variational iteration method (VIM), the homotopy perturbation method (HPM) and the explicit finite difference method (EFDM). The results are compared with those obtained using the five methods. More details for the five methods can be found in next sections. The Adomian Decomposition Method (ADM) The ADM was first introduced by Adomian in the beginning of [64,65,66]. The method is useful for obtaining both a closed form and the explicit solution and numerical approximations of linear or nonlinear differential equations and it is also quite straight forward for writing computer codes. This method has been applied to obtain a formal solution to a wide class of stochastic and deterministic problems in science and engineering involving algebraic, differential, integro-differential, differential delay, integral and partial differential equations [67,68,69,70,71,72,73,74,75,76]. The convergence of ADM for partial differential equations was presented by Cherruault [77]. Application and convergence of this method for nonlinear partial differential equations are found in [78,79,80,81]. In general, it is necessary to construct the solution of the problems in the form of a decomposition series solution. In the simplest case, the solution can be developed as a Taylor series expansion about the function, not the point at which the initial condition and integration of the righthand side function of the problem are determined (the first term u0 of the decomposition series for n  0). The sum of the u0 ; u1 ; u2 ; : : : terms are simply the decomposition series u (x; t) D

1 X

u n (x; t) :

(3)

nD0

Suppose that the differential equation operator, including both linear and nonlinear terms, can be formed as Lu C Ru C Nu D F (x; t) ;

(4)

with initial condition u (x; 0) D g (x) ;

(5)

where L is the higher-order derivative which is assumed to be invertible, R is a linear differential operator of order

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

less than L, N is the nonlinear term and F (x; t) is a source term. We next apply the inverse operator L1 to both sides of Eq. (4) and using the given condition (5) to obtain u (x; t) D g (x) C f (x; t)  L1 (Ru)  L1 (Nu) ; (6) where the function f (x; t) represents the terms arising from integrating the source term F (x; t) and from using the given conditions, all are assumed to be prescribed. The nonlinear term can be written as Nu D

1 X

An ;

(7)

nD0

where An is the Adomian polynomials. These polynomials are defined as " !# 1 X 1 dn k  u k (x; t) ; An D N n! dn nD0 kD0

n D 0; 1; 2; : : :

(8)

for example A0 D N (u0 ) ; A1 D u1 N 0 (u0 ) ; 1 A2 D u2 N 0 (u0 ) C u1 N 00 (u0 ) ; 2

and so on, the other polynomials can be constructed in a similar way [66]. As indicated before, Adomian method defines the solution u by an infinite series of components given by Eq. (4) and the components u0 ; u1 ; u2 ; : : : are usually recurrently determined. Thus, the formal recursive relation is defined by u0 (x; t) D g (x) C f (x; t) ; u nC1 (x; t) D L1 (Ru n )  L1 (Nu n ) ;

n 0; (10)

which are obtained for all components of u. As a result, the terms of the series u0 ; u1 ; u2 ; : : : are identified and the exact solution may be entirely determined by using the approximation u (x; t) D lim 'n (x; t) ; n!1

(11)

where 'n (x; t) D

n1 X kD0

u k (x; t) ;

8 ˆ '0 D u0 ; ˆ ˆ ˆ ˆ ˆ ' D u0 C u1 ; ˆ < 1 '2 D u0 C u1 C u2 ; ˆ ˆ :: ˆ ˆ : ˆ ˆ ˆ : 'n D u0 C u1 C u2 C : : : C u n1 ;

(13) n  0:

The Homotopy Analysis Method (HAM) The HAM was developed in 1992 by Liao in [82,83,84,85]. This method has been successfully applied by many authors [86,87,88,89,90,91]. The HAM contains the auxiliary parameter ¯ which provides us with a simple way to adjust and control the convergence region of solution series for large or small values of x and t. We consider the following differential equation N [u (x; t)] D 0 ;

(14)

where N is a nonlinear differential operator, x and t denote independent variables, u (x; t) is an unknown function. By means of the HAM, one first constructs a zero-order deformation equation         1  p £  x; t; p  u0 (x; t) D p ¯ N  x; t; p ;

(9)

1 A3 D u3 N 0 (u0 ) C u1 u2 N 00 (u0 ) C u13 N 000 (u0 ) ; 6

(

or

(12)

(15) where p 2 [0; 1] is the embedding parameter, ¯ ¤ 0 is a non-zero auxiliary parameter, is an auxiliary linear  op- erator, u0 (x; t) is an initial guess of u (x; t) and  x; t; p is a unknown function. It is important that one has great freedom to choose auxiliary things in the HAM. Obviously, when p D 0 and p D 1, it holds  (x; t; 0) D u0 (x; t) ;

 (x; t; 1) D u (x; t) ; (16)   respectively. The solution  x; t; p varies from the initial guess u0 (x; t) to the solution u (x; t) : Expanding  x; t; p in Taylor series about the embedding parameter C1 X   u m (x; t) p m ;  x; t; p D u0 (x; t) C

(17)

mD1

where u m (x; t) D

 ˇ 1 @m  x; t; p ˇˇ : ˇ m! @p m pD0

(18)

The convergence of the series (17) depends upon the auxiliary parameter ¯. If it is convergent at p D 1, one has u (x; t) D u0 (x; t) C

C1 X mD1

u m (x; t) :

(19)

911

912

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

According to the definition (19), the governing equation can be deduced from the zero-order deformation Eq. (15). Define the vector

exact solution. un s are calculated by a correction functional as follows: u nC1 (x; t)

!

u n D fu0 (x; t) ; u1 (x; t) ; : : : ; u n (x; t)g :

Zt

Differentiating Eq. (15) m-times with respect to the embedding parameter p and then setting p D 0 and finally dividing them by m!, we have the so-called mth-order deformation equation !  £ [u m (x; t)  m u m1 (x; t)] D ¯ <m u m1 ; (20) where !  <m u m1 D

   ˇ @m1 N  x; t; p ˇ 1 ˇ ; (21) ˇ @p m1 (m  1)! pD0

and ( m D

0;

m  1;

1;

m > 1:

(22)

D u n (x; t) C

n o

 Lu n () C N u n ()  g () d ;

0

n  0;

where  is the general Lagrange multiplier, which can be identified optimally by the variational theory, the sub script n denotes the nth approximation and u n considered

as a restricted variation, i. e. ı u n D 0. For linear problems, the exact solution can be obtained by only one iteration step due to the fact that the Lagrange multiplier can be exactly identified. In nonlinear problems, in order to determine the Lagrange multiplier in a simple manner, the nonlinear terms have to be considered as restricted variations. Consequently, the exact solution may be obtained by using u (x; t) D lim u n (x; t) : n!1

It should be emphasized that u m (x; t) for m  1 is governed by the nonlinear Eq. (20) with the linear boundary conditions that come from the original problem, which can be easily solved by symbolic computation software such as Maple and Mathematica. The Variational Iteration Method (VIM) The VIM was first proposed by He [92,93,94,95,96]. This method has been employed to solve a large variety of linear and nonlinear problems with approximations converging rapidly to accurate solutions. The idea of VIM is to construct a correction functional by a general Lagrange multiplier. The multiplier in the functional should be chosen such that its correction solution is superior to its initial approximation (trial function) and is the best within the flexibility of the trial function, accordingly we can identify the multiplier by variational theory. The initial approximation can be freely chosen with possible unknowns, which can be determined by imposing the boundary/initial conditions. We consider the following general differential equation: Lu C Nu D g ;

(23)

where L and N are linear and nonlinear operators respectively, and g(t) is the source inhomogeneous term. According to the VIM, the terms of a sequence fu n g are constructed such that this sequence converges to the

(24)

(25)

This method has been employed to solve a large variety of linear and nonlinear problems with approximations converging to accurate solutions. This technique was used to find the solutions of partial differential equations, ordinary differential equations, boundary-value problems and integral equations by many authors [97,98,99,100, 101,102,103,104,105]. The Homotopy Perturbation Method (HPM) The HPM was developed by He [106,107,108,109]. In recent years, the applications of HPM in nonlinear problems has been devoted to scientists and engineers [110,111, 112,113]. We consider the following general nonlinear differential equation:   A y  f (r) D 0 ; r 2 ˝ ; (26) with boundary conditions   B y; @y/@n D 0 ; r 2  ;

(27)

where A is a general differential operator, B is a boundary operator, f (r) is a known analytic function and  is the boundary of the domain ˝. The operator A can be generally divided into two parts L and N, where L is linear and N is nonlinear. Therefore, Eq. (26) can be written as follows:     L y C N y  f (r) D 0 : (28)

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

  We construct a homotopy of Eq. (26) y r; p : ˝  [0; 1] ! R which satisfies            H y; p D 1  p L y  L y0 C p A y  f (r) D0;

r 2 R; (29)

which is equivalent to 

The ADM for Eq. (14) According to the ADM [64,65,66], Eq. (34) can be written in an operator form ( Lu D 6uu x  u x x x ; p  (35) c u (x; 0) D f (x) D  2c sec h2 2 x ; where the differential operator L is



         H y; p D L y  L y0 C pL y0 C p A y  f (r)

@ ; @t

L

D0; (30) where p 2 [0; 1] is an embedding parameter and y0 is an initial approximation which satisfies the boundary conditions. It follows from Eqs. (29) and (30) that       H y; 0 D L y  L y0 D 0 and     (31) H y; 1 D A y  f (r) D 0 : Thus, the changing process of p from 0 to 1 is just that of y r; p from y0 (r)   this is  to y (r).  In topology called deformation and L y  L y0 and A y  f (r) are called homotopic. Here the embedding parameter is introduced much more naturally, unaffected by artificial factors; further it can be considered as a small parameter for 0  p  1. So it is very natural to assume that the solution of (29) and (30) can be expressed as y (x) D u0 (x) C pu1 (x) C p2 u2 (x) C    :

(32)

According to HPM, the approximate solution of (26) can be expressed as a series of the power of p, i. e.,

(36)

and the inverse operator L1 is an integral operator given by Z t L1 (ı) D (37) (ı) dt : 0

The ADM [64,65,66] assumes a series solution for the unknown function u (x; t) given by Eq. (3) and the nonlinear term Nu D uu x can be decomposed into a infinite series of polynomials given by Eq. (5). Operating with the integral operator (37) on both sides of (35) and using the initial condition, we get u (x; t) D f (x) C L1 (6uu x  u x x x ) :

(38)

Substituting (3) and (5) into the functional Eq. (38) yields 1 X

u n (x; t)

nD0

"

D f (x) C L

1

6

1 X

! An

nD0

y D lim y (x) D u0 (x) C u1 (x) C u2 (x) C    : (33)



1 X

!

Numerical Applications and Comparisons We now consider the initial value problem associated with the KdV equation: ( u t  6uu x C u x x x D0 ;  p (34) c u (x; 0) D  2c sec h2 2 x ; with u (x; t) is a sufficiently smooth function and c is the velocity of the wavefront. In this section, we apply the above described four methods and the explicit finite difference method on the KdV Eq. (34) so that the comparisons are made numerically.

:

un

nD0

xxx

(39)

p!1

The convergence of series (33) has been proved by He in [106].

#

The ADM admits that the zeroth component u0 (x; t) be identified by all terms that arise from the initial conditions and from the source terms if exist, and as a result, the remaining components u n (x; t), n  1 can be determined by using the recurrence relation: p  ( c u0 (x; t) D f (x) D  2c sec h2 2 x ; (40)   u kC1 (x; t) D L1 6A k  (u k ) x x x ; k  0 ; where Ak , k  0 are Adomian polynomials that represent the nonlinear term (uu x ) and given by 8 ˆ ˆ ˆA0 D u0 u0x ; ˆ
1 2x

1x 2

0x 3

0 3x

913

914

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

Thus, some of the symbolically computed components are found as: p  c c u0 (x; t) D  sec h2 x ; 2 2 p  p  c c c 5/2 sec h2 x tanh x t; u1 (x; t) D  2 2 2 p   p c c4  cosh cx  2 sec h4 x t2 ; u2 (x; t) D  8 2 p  c c 11/2 5 sec h x u3 (x; t) D  48 2  p   p  c 3 c x C sinh x t3 ;  11 sinh 2 2 p  c c7 sec h6 x u4 (x; t) D  384 2 h  p i p  33  26 cosh cx C cosh 2 cx t 4 ; and so on. In this manner the other components of the decomposition series can be easily obtained using any symbolic program. The solution in a series for is given by

u(x; t) D

1 X

u n (x; t)

nD0

D u0 (x; t) C u1 (x; t) C u2 (x; t) C u3 (x; t) C    ; p  c c 2 D  sec h x 2 2 p  p   c 5/2 c c 2  sec h x tanh x t 2 2 2 p   p  c c4  4 cosh cx  2 sec h x t2  8 2 p  c c 11/2 5  sec h x 48 2  p   p  3 c c  11 sinh x C sinh x t3 2 2 C ::: ; (42) so that the exact solution  p c c 2 u (x; t) D  sec h (x  ct) ; 2 2

(43)

is readily obtained. The convergence analysis of the ADM applied to the KdV Eq. (35) without the initial condition has been conducted by Helal and Mehanna [55].

The HAM for Eq. (34) We will apply the HAM to the Eq. (34) to illustrate the strength of the method and to establish exact solution for this Eq. (34). We choose the linear operator      @ x; t; p ; (44) £  x; t; p D @t with property  £ k D0;

(45)

where k is a constant. We now define a nonlinear operator      @ x; t; p N  x; t; p D @t       @ x; t; p @3  x; t; p :  6 x; t; p C @x @x 3

(46)

We construct the zeroth-order deformation equation 

       1  p £  x; t; p  u0 (x; t) D p ¯ N  x; t; p : (47)

For p D 0 and p D 1, we can write (  (x; t; 0) D u (x; 0) D u0 (x; t) ;  (x; t; 1) D u (x; t) :

(48)

So we obtain mth-order deformation equation !  £ [u m (x; t)  m u m1 (x; t)] D ¯ <m u m1 ; (49) where   !  @ m1 x; t; p <m u m1 D @t   m1 X   @ m1n x; t; p 6  n x; t; p @x nD0

  @3  m1 x; t; p : C @x 3

(50)

Now the solution of the mth-order deformation Eq. (49) for m  1 become h ! i u m (x; t) D m u m1 (x; t)C¯ £1 <m u m1 : (51) Thus we get the following terms by using (51) for Eq. (34): p  c c 2 x ; u0 (x; t) D  sec h 2 2

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

 p  c x tanh x t; 2 2 p  h p  c c 5/2 ¯t sec h4 x c 3/2 ¯t cosh cx u2 (x; t) D  2 2  p  i 3/2 ;  2 c ¯t C (1 C ¯) sinh cx "  pc   pc  c 5/2 u3 (x; t) D ¯t sec h2 x c 3 ¯2 t 2 sec h3 x 48 2 2  3p c   pc   sinh x C 24¯(1 C ¯) tanh x 2 2  p c  p x  12c 3/2 ¯(1 C ¯)t cosh( cx) C sec h2 2 p C 12(1 C ¯) sinh( cx) #   pc  3/2 3/2 ; x C c ¯t 24 C 24¯  11c ¯t tanh 2 u1 (x; t) D

c 5/2 ¯ sec h2 2

p

c

and so on. In this manner the other components of the decomposition series can be easily obtained using any symbolic program. Liao [82,83,84,85] showed that whatever a solution series converges to it will be one of the solutions of considered problem. Liao [82,83,84,85] showed the approximate solutions obtained by the HAM to be controlled by the auxiliary parameter and the rate of convergence of ¯. It is noted that the approximate solutions converge at 2:1  ¯  1:1 for 5th-order approximation and at 4  ¯  2 for 10th-order approximation for the KdV Eq. (34) when x D 10, t D 0:5, c D 1 and c D 5 (see Figs. 2 and 3). The VIM for Eq. (34) To solve the KdV Eq. (34) by means of the VIM, we construct a correction functional which reads as u nC1 (x; t) D u n (x; t) !

Z t @u n (x; ) @3 u n (x; )

 d ;  N u n (x; ) C C @ @x 3 0 (52) where  is the general Lagrange multiplier [109] whose optimal value is found using variational theory, u0 (x; t) is an initial approximation which must be chosen suitable and

N u n is the restricted variation, i. e., ı u n D 0 [107]. To find the optimal values of  we have Z



t



ıu nC1 (x; t) D ıu n (x; t)Cı 0

@u n (x; ) @

 d ; (53)

that results Z

t

ıu nC1 (x; t) D ıu n (x; t) (1 C ) 

ıu n (x; t) 0 d ;

0

(54) which yields 0 () D 0 ÍDt ; 1 C  () D 0 jDt :

(55)

Thus we have  (t) D 1 ;

(56)

and we obtain the following iteration formula Z t

@u n (x; ) @ 0  3 @u n (x; ) @ u n (x; )  6u n (x; ) C d : @x @x 3

u nC1 (x; t) D u n (x; t) 

(57)

Now using (57) we can find the solution of the KdV Eq. (34) as a convergent sequence. We get the following components: p  c c 2 x ; u0 (x; t) D  sec h 2 2 p 

p  c c c x cosh x u1 (x; t) D  sec h3 2 2 2 p   c x ; C c 3/2 t sinh 2 p 

  c c 7 x  2 4c 3 t 2  5 u2 (x; t) D  sec h 32 2  p  p    3 c c 3 2 x C 5  c t cosh x  cosh 2 2  p   p  5 c 5 c C cosh x C c 3 t 2 cosh x 2 2 p  p  c c C 4c 3/2 t sinh x  60c 9/2 t 3 sinh x 2 2  p   p  3 c 3 c C 6c 3/2 t sinh x C 12c 9/2 t 3 sinh x 2 2  p  5 c C 2c 3/2 t sinh x ; 2 and so on. In the same manner the rest of components of the iteration formulae (57) can be obtained using any symbolic packages. So we obtain  p c c 2 u (x; t) D lim u n (x; t) D  sec h (x  ct) : n!1 2 2 (58)

915

916

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

The HPM for Eq. (34)

The EFDM for Eq. (34)

To investigate the approximate solution of Eq. (34), we first construct a homotopy as follows:  :    : : 1  p Y  u0 C p Y  6YY 0 C Y 000 D 0 ; (59)

The finite difference methods are the most frequently used and universally applicable. These methods are approximate in the sense that the derivatives at a point are approximated by difference quotients over a small interval [114]. In order to obtain a finite difference replacement of the KdV Eq. (34) the region to be examined is divided into equal rectangular meshes with sides x and t parallel to the x- and t-axes respectively. The function u(x; t) is approximated by u ni D u (ix; nt) where i and n are integers and i D n D 0 is the origin. Let us define

where “primes” denote differentiation with respect to x, and “dot” denotes differentiation with respect to t. In order to obtain the unknowns of Yi (x; t), i D 1; 2; 3; 4, we must construct and solve the following system which includes four equations with three unknowns, considering the initial condition of Y (x; 0) D u (x; 0) and having the initial approximation of Eq. (34): 0

:

:

p :

Y 0  u0 D 0 ;

p1 :

Y 1 C u0  6Y0 Y00 C Y0000 D 0 ;

p2 :

Y 2  6Y0 Y10  6Y1 Y00 C Y1000 D 0 ;

p3 :

Y 3  6Y0 Y20  6Y1 Y120  6Y2 Y00 C Y2000 D 0 :

:

:

:

(60)

:

k D 0; 1; 2; 3;    :

p!1

(61)

To calculate the terms of the homotopy series (61) for u (x; t), we substitute the initial condition into the systems (60) and finally using Mathematica, the solutions of the equation can be obtained as follows: p  c c u0 (x; t) D  sec h2 x ; 2 2 p  p  c c c 5/2 sec h2 x tanh x t ; (62) u1 (x; t) D  2 2 2 u2 (x; t) c 5/2 2 t sec h4 D 2

p 2

c

x

h

c 3/2 cosh

U x x x Í(i;n) D

1 2 (x)

3

(64) (65)

 n n n n U iC2  2U iC1 : C 2U i1  U i2 (66)

Thus, we will obtain: u (x; t) D lim Yk (x; t) ;

U inC1  U in1 ; 2t n U n  U i1 U x Í(i;n) D iC1 ; 2x

U t Í(i;n) D

i p  cx C2c 3/2 ;

and so on. In this manner the other components can be easily obtained. Substituting Eq. (62) into Eq. (61): u (x; t) D Y0 (x; t) C Y1 (x; t) C Y2 (x; t) C    p  c c 2 x D  sec h 2 2 p   p  c c c 5/2 2 sec h x tanh x t  2 2 2 p    p  c c4 x cosh cx  2 C    :  t 2 sec h4 2 2 (63)

The explicit scheme computes the value of the numerical solution at the forward time step in terms of known values at the previous time step. An explicit scheme for solving the KdV equation produced originally by Zabusky and Galvin [11] is centered in time and space. Substituting (64)–(66) into the KdV Eq. (34) with U Í(i;n) D

 1 n n U ; C U in C U iC1 3 i1

(67)

leads to U inC1

 n  n  n n D U in1 C 2t U i1 C U in C U iC1 U iC1  U i1  t  n n n n C 2U i1  U i2 U iC2  2U iC1 : (68)  3 (x)

For the initial step, we use a scheme which is forward in time and centered in space  0   0 0 0 U iC1  U i1 C U i0 C U iC1 U i1 D U i0 C t U i1  t  0 0 0 0 C 2U i1  U i2 U iC2  2U iC1 :  (x)3 (69) It is clear that Eq. (68) is a three-level scheme of time, i. e. in order to obtain U i at the time level n C 1, we need the following values of U i2 , U i1 , U iC1 and U iC2 at the previous time level n in addition to the value of U i at the time level n  1. The explicit difference scheme (68) has

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

second order accuracy in t and x as the truncation error is O (t)2 C O (x)2 and is also consistent with KdV equation. A stability analysis of the nonlinear numerical scheme (68) using Fourier mode method is not easy to handle unless it is assumed that U, in the nonlinear term, is locally constant. This is equivalent to replacing the term (67) in Eq. (68) by U  . This linearized scheme for the KdV Eq. (34) has stability condition [115]:

 ˇ ˇ t 4 6 ˇU  ˇ C  1: (70) x (x)2 In Table 7, h is x-direction mesh size and k is t-direction time step. Conclusions and Discussions In this paper, the ADM, HAM, VIM, HPM and EFDM were used for the KdV Eq. (34) with initial conditions. We made comparisons of the numerical results obtained by using ADM, HAM, VIM, HPM and EFDM. The approximate solutions to the equation has been also calculated by using the ADM, HAM, VIM, HPM and EFDM without any need to use transformation techniques and linearization of the equation. The ADM, HAM and HPM avoids the difficulties and massive computational work by determining analytic solutions of the nonlinear equations. The main advantage of HAM, VIM and HPM are to overcome the difficulty arising in calculating Adomian’s polynomials in the ADM. But the main disadvantage of VIM and EFDM, in general, very big terms and the consuming time to compute it is big, so we need a large computer memory and time. In addition, it is difficult and takes a lot of time to calculate the terms after the fifth term in the VIM for the KdV Eq. (14). It is convenient to examine Tables 1, 2, 3, 4, 5, 6, 7 and Figs. 1, 3, 4, 5 for comparing exact solution and numerical solutions which one gets

by using these five methods. Tables and figures are calculated for these methods by using five terms and c D 1, except Tables 2 and 4. The results which are obtained by the EFDM are better than the results which are obtained by the other four (semi-inverse) methods for five terms (see Tables 1, 3, 5, 6, 7. If you increase the numbers of terms in the semi-inverse methods, we can see that the results are better (see Tables 2 and 4. Obtaining serial solutions with the semi-inverse methods are easier and more efficient than pure numerical methods (the finite difference methods, the finite element methods, B-spline methods, etc.). Finding numerical results by the EFDM are difficult and take time. The present methods give nearly the same results for c D 1 and large values of x but the HAM is the more sensitive method than the others. Because it is possible to obtain a rapidly convergent solution for large and small values of x and t by choosing convenient helper the auxiliary parameter ¯. For this it is enough to consider Figs. 1, 3, 4, 5. Finally, we point out that, for given equations with initial value, the corresponding analytical and numerical solutions are obtained according to the recurrence equations by using Mathematica. Future Directions The Korteweg–de Vries equation, which is the most important equation of mathematical physics, has solitons and solitary wave solutions. It was solved by using analytical and numerical methods by many authors. We have mentioned these analytical and numerical methods in the introduction. The theoretical studies on the KdV equation have been nearly completed, but we may find new soliton solutions by producing new analytical methods or expanding the old methods. It is also possible to perform same operations on numerical methods. We may show the interaction of KdV solitons with each other by numerical.

Korteweg–de Vries Equation(KdV),  Some Numerical Methods for Solving the, Figure 1 The comparison of the ADM '5 and the exact solutions for Eq. (34) at c D 1

917

918

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the, Figure 2 The for x D 10, t D 0:5, c D 1 for 5th-order approximation and x D 10, t D 0:5, c D 5 for 10th-order approximation of  ¯-curves  u x; t , respectively

Korteweg–de Vries Equation(KdV), Some Numerical Methods for Solving the, Figure 3  The comparison of the HAM 5th-order approximation and the exact solutions for Eq. (34) at c D 1

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the, Figure 4 The comparison of the VIM (five iterations) and the exact solutions for Eq. (34) at c D 1

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the, Figure 5 The comparison of the HPM (five iterations) and the exact solutions for Eq. (34) at c D 1

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

Korteweg–de Vries Equation  (KdV), Some Numerical Methods for Solving the, Table 1 Error between the ADM '5 and exact solutions for the KdV Eq. (14) at c D 1 t/x 0.1 0.2 0.3 0.4 0.5

10 2:00679  104 2:22178  104 2:45102  104 2:70872  104 2:99336  104

15 1:35233  106 1:49452  106 1:65169  106 1:82535  106 2:01722  106

20 9:11171  109 1:00712  108 1:11292  108 1:22991  108 1:35919  108

25 6:13942  1011 6:78512  1011 7:49865  1011 8:28713  1011 9:15815  1011

30 4:13671  1013 4:57177  1013 5:05255  1013 5:58381  1013 6:17071  1013

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the, Table 2  Error between the ADM '10 and exact solutions for the KdV Eq. (34) at c D 5 t/x 0.1 0.2 0.3 0.4 0.5

10 2:00679  108 2:22178  108 2:45102  107 2:70872  107 2:99336  106

15 1:35233  1013 1:49452  1012 1:65169  1012 1:82535  1012 2:01722  1011

20 9:11171  1018 1:00712  1018 1:11292  1017 1:22991  1017 1:35919  1016

25 6:13942  1023 6:78512  1023 7:49865  1022 8:28713  1022 9:15815  1021

30 4:1367  1028 4:5717  1027 5:0525  1027 5:5838  1026 6:1707  1026

Korteweg–de Vries Equation (KdV), Some Numerical   Methods for Solving the, Table 3 Error between the HAM 5th-order approximation and exact solutions for the KdV Eq. (34) at c D 1 and ¯ D 0:5 t/x 0.1 0.2 0.3 0.4 0.5

10 1:94277  104 2:08022  104 2:22925  104 2:39102  104 2:56683  104

15 1:30915  106 1:40179  106 1:50222  106 1:61125  106 1:72975  106

20 8:82101  109 9:44517  109 1:01219  108 1:08565  108 1:16549  108

25 5:94355  1011 6:36412  1011 6:82009  1011 7:31508  1011 7:85304  1011

30 4:00473  1013 4:28811  1013 4:59534  1013 4:92886  1013 5:29134  1013

Korteweg–de Vries Equation (KdV), Some NumericalMethods for Solving the, Table 4  Error between the HAM 10th-order approximation and exact solutions for the KdV Eq. (34) at c D 5 and ¯ D 0:5

t/x 0.1 0.2 0.3 0.4 0.5

10 8:12684  109 3:62126  108 8:86781  107 1:87306  107 5:02718  107

15 1:13334  1013 5:05121  1013 1:23668  1012 2:61211  1012 7:01076  1012

20 1:58053  1018 7:04272  1018 1:72463  1017 3:64277  1017 9:77698  1017

25 2:20415  1023 9:82156  1023 2:40512  1022 5:08009  1022 1:36347  1021

30 3:0738  1028 1:3696  1027 3:3541  1027 7:0845  1027 1:9014  1026

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the, Table 5 Error between the VIM (5 terms) and exact solutions for the KdV Eq. (34) at c D 1 t/x 0.1 0.2 0.3 0.4 0.5

10 2:33263  104 2:99151  104 3:81873  104 4:84308  104 6:09593  104

15 1:57189  106 2:01592  106 2:57322  106 3:26302  106 4:10597  106

20 1:05913  108 1:35833  108 1:73383  108 2:19864  108 2:76657  108

25 7:13638  1011 9:15216  1011 1:16824  1010 1:48141  1010 1:86417  1010

30 4:24233  1013 6:16666  1013 7:87156  1013 9:98164  1013 1:25602  1012

919

920

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the, Table 6 Error between the HPM (five iterations) and exact solutions for the KdV Eq. (34) at c D 1 t/x 0.1 0.2 0.3 0.4 0.5

10 2:00679  104 2:21782  104 2:45102  104 2:70871  104 2:99337  104

15 1:35232  106 1:49452  106 1:65169  106 1:82535  106 2:01722  106

20 9:11171  109 1:00734  108 1:11296  108 1:22991  108 1:35919  108

25 6:13942  1011 6:78513  1011 7:49865  1011 8:28716  1011 9:15815  1011

30 4:13671  1013 4:57177  1013 5:05255  1013 5:58381  1013 6:17071  1013

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the, Table 7 Error between the EFDM and exact solutions for the KdV Eq. (34) at c D 1; h D 0:05 and k D 0:025 t/x 0.1 0.2 0.3 0.4 0.5

10 6:15156  106 7:54177  106 9:14755  106 1:10086  105 1:35102  105

15 4:14842  108 5:09160  108 6:19126  108 7:46977  108 8:94343  108

20 2:79520  1010 3:43074  1010 4:17173  1010 5:03358  1010 6:03380  1010

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52. Bahadır AR (2005) Exponential finite-difference method applied to the KdV equation for small times. Appl Math Comput 160:675–682 53. Jain PC, Shankar R, Bhardwaj D (1997) Numerical solution of the KdV equation. Chaos Solitons Fractals 8:943–951 54. Soliman AA (2004) Collocation solution of the KdV equation using septic splines. Int J Comput Math 81:325–331 55. Frauendiener J, Klein C (2006) Hyperelliptic theta-functions and spectral methods: KdV and KP solutions. Lett Math Phys 76:249–267 56. Bhatta DD, Bhatti MI (2006) Numerical solution of KdV equation using modified Bernstein polynomials. Appl Math Comput 174:1255–1268 57. Helaln MA, Mehanna MS (2007) A comparative study between two different methods for solving the GKdV equation. Chaos Solitons Fractals 33:729–739 58. Kutluay S, Bahadır AR, Özde¸s A (2000) A small time solutions for the KdV equation. Appl Math Comput 107:203–210 59. Özer S, Kutluay S (2005) An analytical numerical method for solving the KdV equation. Appl Math Comput 164:789–797 60. Dehghan M, Shokri A (2007) A numerical method for KdV equation using collocation and radial basis functions. Nonlinear Dyn 50:111–120 61. Dag˘ I, Dereli Y (2008) Numerical solutions of KdV equation using radial basis functions. Appl Math Model 32:535–546 62. Wadati M (1973) The modified Kortweg–de Vries equation. J Phys Soc Jpn 34:1289–1296 63. Wadati M (2001) Introduction to solitons. Pramana J Phys 57(5–6):841–847 64. Adomian G (1989) Nonlinear Stochastic Systems and Applications to Physics. Kluwer, Boston 65. Adomian G (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Boston 66. Adomian G (1998) A review of the decomposition method in applied mathematics. J Math Anal Appl 135:501–544 67. Lesnic D, Elliott L (1999) The decomposition approach to inverse heat conduction. J Math Anal Appl 232:82–98 68. Wazwaz AM (2002) Partial Differential Equations: Methods and Applications. Balkema Publication, Lisse 69. Wazwaz AM (2003) An analytical study on the third-order dispersive partial differential equation. Appl Math Comput 142:511–520 70. Cherruault Y (1998) Modéles et Méthodes Math ématiques pour les Sciences du Vivant. Presses Universitaires de France, Paris 71. Dehghan M (2004) Application of ADM for two-dimensional parabolic equation subject to nonstandard boundary specification. Appl Math Comput 157:549–560 72. Dehghan M (2004) The solution of a nonclassic problem for one-dimensional hyberbolic equation using the decomposition procedure. Int J Comput Math 81:979–989 73. Babolian E, Biazar J, Vahidi AR (2004) The decomposition method applied to systems of Fredholm integral equations of the second kind. Appl Math Comput 148:443–452 74. El-Sayed MS (2002) The modified decomposition method for solving nonlinear algebraic equations. Appl Math Comput 132:589–597 75. Inc M (2006) On numerical Jacobi elliptic function solutions of (1+1)-dimensional dispersive long wave equation by the decomposition method. Appl Math Comput 173:372–382

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76. Inc M (2007) Exact solutions with solitary patterns for the Zakharov-Kuznetsov equations with fully nonlinear dispersion. Chaos Solitons Fractals 33:1783–1790 77. Cherruault Y (1990) Convergence of Adomian’s decomposition method. Math Comput Modelling 14:83–86 78. Ngarhasta N, Some B, Abboui K, Cherruault Y (2002) New numerical study of Adomian method applied to a diffusion model. Kybernetes 31:61–75 79. Mavoungou T, Cherruault Y (1992) Convergence of Adomian’s method and applications to nonlinear partial differential equations. Kybernetes 21:13–25 80. Inc M (2006) On numerical soliton solution of the KK equation and convergence analysis of the decomposition method. Appl Math Comput 172:72–85 81. Hashim I, Noorani MS, Al-Hadidi MRS (2006) Solving the generalized Burgers-Huxley equation using the Adomian decomposition method. Math Comput Model 43:1404–1411 82. Liao SJ (2002) An analytic approximation of the drag coefficient for the viscous flow past a sphere. Int J Nonlinear Mech 37:1–18 83. Liao SJ (2003) Beyond Perturbation: Introduction to the Homotopy Analysis Method. Champan, Boca Raton 84. Liao SJ (2003) On the analytic solution of magnetohydrodynamic flows non-Newtonian fluids over a stretching sheet. J Fluid Mech 488:189–212 85. Liao SJ (2005) A new branch of solutions of boundary-layer flows over an impermeable stretched plate. Int J Heat Mass Transfer 48:2529–2539 86. Sajid M, Hayat T, Asghar S (2006) On the analytic solution of the steady flow of a fourth grade fluid. Phys Lett 355:18–26 87. Hayat T, Abbas Z, Sajid M (2006) Series solution for the upperconvected Maxwell fluid over a porous stretching plate. Phys Lett 358:396–403 88. Hayat T, Khan M (2005) Homotopy solutions for a generalized second-grade fluid past a porous plate. Nonlinear Dynamics 42:395–405 89. Abbasbandy S (2006) The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett 360:109–113 90. Abbasbandy S (2007) The application of homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation. Phys Lett 361:478–483 91. Inc M (2007) On exact solution of Laplace equation with Dirichlet and Neumann boundary conditions by the homotopy analysis method. Phys Lett 365:412–415 92. He JH (1997) A new approach to nonlinear partial differential equations. Commun Nonlinear Sci Numer Simul 2:230–235 93. He JH (1997) Variational iteration method for delay differential equations. Commun Nonlinear Sci Numer Simul 2:235– 236 94. He JH (1999) Variational iteration method a kind of nonlinear analytical technique: Some examples. Int J Nonlinear Mech 34:699–708 95. He JH (2000) Variational iteration method for autonomous ordinary differential systems. Appl Math Comput 114:115–123 96. He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int J Modern Phys B 20:1141–1199 97. Abdou MA, Soliman AA (2005) Variational iteration method for solving Burger’s and coupled Burger’s equations. J Comput Appl Math 181:245–251

98. Abdou MA, Soliman AA (2005) New applications of variational iteration method. Phys D 211:1–8 99. Bildik N, Konuralp A (2006) The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int J Nonlinear Sci Numer Simul 7:65–70 100. Draganescu G (2006) Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives. J Math Phys 47:082902 101. Tatari M, Dehghan M (2007) He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation. Chaos Solitons Fractals 33:671–677 102. Wazwaz AM (2007) The variational iteration method for exact solutions of Laplace equation. Phys Lett 363:260–262 103. Tatari M, Dehghan M (2007) Solution of problems in calculus of variations via He’s variational iteration method. Phys Lett 362:401–406 104. Inc M (2007) An approximate solitary wave solution with compact support for the modified KdV equation. Appl Math Comput 184:631–637 105. Inc M (2007) Exact and numerical solutions with compact support for nonlinear dispersive K(m,p) equations by the variational iteration method. Physica 375:447–456 106. He JH (1999) Homotopy perturbation technique. Comput Meth Appl Mech Eng 178:257–262 107. He JH (2005) Application of homotopy perturbation method to nonlinear wave equations. Chaos Solitons Fractals 26:695– 700 108. He JH (2005) Homotopy perturbation method for bifurcation of nonlinear problems. Int J Nonlinear Sci Numer Simul 6:207– 208 109. He JH (2006) Non-perturbative Methods for Strongly Nonlinear Problems. Dissertation. de-Verlag im Internet GmbH, Berlin 110. Ganji DD, Rajabi A (2006) Assesment of homotopy perturbation and perturbation methods in heat radiation equations. Int Commun Heat Mass Transf 33:391–400 111. Siddiqui AM, Mahmood R, Ghori QK (2006) Homotopy perturbation method for thin film flow of a fourth grade fluid down a vertical cylinder. Phys Lett 352:404–410 112. Ganji DD, Rafei M (2006) Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method. Phys Lett 356:131–137 113. Chowdhury MSH, Hashim I (2007) Solutions of a class of singular second-order IVPs by homotopy-perturbation method. Phys Lett 356(5–6):439–447 114. Smith GD (1987) Numerical Solution of Partial Differential Equation: Finite Difference Methods. Oxford University Press, New York 115. Vilegenthart AC (1971) On finite difference methods for the Korteweg–de Vries equations. J Eng Math 5(2):137–155

Books and Reviews Abbasbandy S (2007) An approximation solution of a nonlinear equation with Riemann–Liouville’s fractional derivatives by He’s variational iteration method. J Comp Appl Math 207: 53–58 Abbasbandy S (2007) A new application of He’s variational itera-

Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the

tion method for quadratic Riccati differential equation by using Adomian’s polynomials. J Comp Appl Math 207:59–63 Deeba E, Khuri SA, Xie S (2000) An algorithm for solving boundary value problems. J Comp Phys 159:125–138 El-Sayed SM (2002) The modified decomposition method for solving nonlinear algebraic equations. Appl Math Comput 132:589–597 Ganji DD, Sadighi A (2007) Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. J Comp Appl Math 207:24–34 Hayat T, Sajid M (2007) Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet. Int J Heat Mass Transf 50:75–84 Kaya D (2005) An application for the higher order modified KdV equation by decomposition method. Commun Nonlinear Sci Numer Simul 10(6):693–702

Kaya D, Aassila M (2002) An application for a generalized KdV equation by the decomposition method. Phys Lett 299(2):201– 206 Khelifa S, Cherruault Y (2000) New results for the Adomian method. Kybernetes 29(3):332–354 Özi¸s T, Yıldırım A (2007) A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity. Chaos Solitons Fractals 34(3):989–991 Wang C, Wu Y, Wu W (2005) Solving the nonlinear periodic wave problems with the Homotopy Analysis Method. Wave Motion 41(4):329–337 Wazwaz AM (1997) A First Course in Integral Equations. World Scientific, London Wu Y, Wang C, Liao S (2005) Solving the one-loop soliton solution of the Vakhnenko equation by means of the Homotopy analysis method. Chaos Solitons Fractals 23(5):1733–1740

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Learning, System Identification, and Complexity

Learning, System Identification, and Complexity M. VIDYASAGAR Tata Consultancy Services, Software Units Layout, Hyderabad, India Article Outline Glossary Definition of the Subject Introduction Least Squares Curve Fitting – A Precursor System Identification Statistical Learning Theory The Complexity of Learning System Identification as a Learning Problem Future Directions Acknowledgment Bibliography Glossary Accuracy A significant parameter in statistical learning theory that describes the error between the current estimate and the truth. Algorithmic complexity The computational complexity of computing a hypothesis in a learning problem, on the basis of observations. ARMA Auto-Regressive Moving Average. It is a specific kind of model for a dynamical system. Confidence An estimate of the confidence with which we can make a particular statement, that is, a bound on the probability that the statement is false. PAC Probably Approximately Correct. It is a widely used model for statistical learning. Sample complexity The number of samples required to learn to a specific accuracy and confidence. Vapnik–Chervonenkis dimension A combinatorial parameter that captures the “richness” of a set of concepts to be learnt. Definition of the Subject Definitions A B Symmetric difference between the sets A and B. A m Algorithm to be applied to m labeled inputs. C A concept class, that is, a collection of subsets. d P (T; H m ) Generalization error with the target set T and the hypothesis H m . H m Hypothesis generated by the algorithm A m . H m (T; x) Hypothesis generated when the target set is T and the sample is x.

I T () Indicator function of the set T. J( ) Cost function (often, though not always, the sum of squares). M(; C ; d P ) -packing number of the concept class C with the pseudometric d P . N(; C ; d P ) -covering number of the concept class C with the pseudometric d P . r(m; ) Learning rate function; depends on the number of inputs m and the accuracy .

Parameter that characterizes the model used in identification.

ˆt Estimate of at time t. Set of possible parameters . u t Input to the system to be identified at time t. y t Output of the system to be identified at time t. System identification refers to the problem of fitting a model, from within a given set of models, to a series of observations (or measurements). By tradition, system identification is recursive, that is, the data is given one observation at a time, and ideally the next model is obtained as a function of the previously fit model and the current observation. One consequence of having a recursive identification procedure is that, once we have used the latest observation to update the model, the observation can be discarded, because the latest model embodies the effect of all the past observations. Statistical learning is a closely related problem in which it is assumed that the data at hand is generated by a “true but unknown” target concept (or function), and the objective is to fit a model to the data at hand. Unlike in system identification, statistical learning is generally not recursive. In other words, at any given point in time, the entire set of past observations is often used to do the model fitting. Perhaps the major difference between system identification theory and statistical theory is that the latter is heavily focused on so-called “finite time estimates.” In other words, by tradition most theorems in system identification theory are asymptotic in nature, and tell us what happens as the number of observations approaches infinity. In contrast, by its very nature, statistical learning theory is devoted to the derivation of explicit upper bounds on just far the current model is from the unknown target. These bounds are a function of the number of observations, and the desired accuracy and confidence parameters (these are defined precisely later on). Introduction One of the earliest known examples of system identification is “curve fitting,” which is the problem of fitting

Learning, System Identification, and Complexity

a curve from within a given family to set of points in the plane (or some higher dimensional space). The problem of “curve fitting” goes back to antiquity, with some of the major contributions being made two centuries ago by Gauss and Laplace. System identification is a natural extension of curve fitting to the situation where the models are not “static” functions, but rather dynamical systems. The dynamical nature of the models introduces additional complexities into the theory. In a learning problem, the issue of “complexity” is central, whether it is “sample complexity” or “algorithmic complexity.” The former refers to the number of samples needed to achieve a desired level of accuracy and confidence, and the latter refers to the difficulty of computing a model on the basis of the given set of observations. Both types of complexity are studied in this article. The difference between system identification and statistical learning is one of emphasis more than anything else. Thus it is not surprising that it is possible to view system identification as a problem in statistical learning. In this article, this connection is brought out explicitly. We begin the article by reviewing the problem of curve fitting using a least squares optimization criterion, and then show how the basic ideas can be extended to the problem of system identification. Then we introduce the problem of PAC (probably approximately correct) learning. Then we discuss the issues of complexity. Then we show how identification can be viewed as a kind of learning problem. We conclude with some directions for future research. Least Squares Curve Fitting – A Precursor Many of the issues that underlie system identification can be readily understood in the context of the much simpler problem of “curve fitting” using a least squares error criterion. Suppose we are given a series of pairs of vectors (x i ; y i ); i D 1; : : : ; l, where x i 2 Rn ; y i 2 Rm 8i. The objective is to fit the data with a linear relationship of the form y D Ax, where A 2 Rmn , such that the least squares error criterion J(A) D

l X

k y i  Ax i k22

iD1

is minimized, where k  k2 denotes the Euclidean norm k v k2 D (vt v)1/2 : Using the notation Y :D [y1 : : : y l ] 2 Rml ; X :D [x1 : : : x l ] 2 Rnl ;

we can easily establish that the optimal choice of the matrix A is given by A D CD1 ;

(1)

where C D YXt ;

D D XXt :

This formula presupposes that the matrix D is nonsingular and therefore invertible. Since D D XXt D

l X

x i x ti ;

iD1

it is easy to see that D is nonsingular if and only if the set of vectors fx1 ; : : : ; x l g contains n linearly independent vectors. This leads to the first observation, namely: Unless there is sufficient data, the curve-fitting problem does not have a unique solution. Now suppose that the data is in fact generated by a “true but unknown” linear relationship, where the data is corrupted by noise. In other words, suppose that y i D A0 x i C v i ; where A0 is the “true” matrix, and fvi g is a sequence of random vectors that are pairwise independent. For the moment, suppose that the vectors x i are deterministic and not random. Then the measurements are described by Y D AX C V ; where V D [v1 : : : v l ] : Now from this formula and (1), the following facts follow readily. 1. Since the measurement noise vectors are random, the optimal choice of A after l measurements, call it A l , is also random. 2. If the measurement noise vectors vi are all zero mean, then the optimal choice A l is unbiased, that is, the expected value of A l equals the true matrix A0 for each value of l. 3. If the measurement noise vectors are normally distributed around zero, then A l is normally distributed around the true value A0 . 4. Under very broad conditions, for example if the noise vectors are a sequence of i.i.d. (independent, identically distributed) vectors with finite variance, then the estimate A l converges almost surely to the true value A0 as l ! 1. We shall see that many of the same features are present in the system identification problem as well.

925

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Learning, System Identification, and Complexity

data, the vector

System Identification System identification refers to the problem of fitting a dynamical system model to a given sequence of temporal (i. e., time-spaced) measurements. Thus one is given a set of input-output pairs f(u t ; y t ); t  0g, as well as a family of models fh( ); 2 g. The idea is to choose the “best possible” model from within this family to represent the data. System identification is a vast and well-developed subject. In a brief essay such as this one, it is difficult to do more than to scratch the surface of the topic. The reader is referred to the book [9] for a very authoritative and comprehensive treatment. Problem Formulation

k X

A i ( )y ti C B i ( )u ti ;

(2)

iD1

where A i ( ); B i ( ) are matrices of appropriate dimension that depend on the parameter 2 , where  is a subset of some Euclidean space. Note that the input ut is not necessarily “white” noise, but could be a combination of a deterministic input and “colored” noise. The relationship (2) can be rewritten in the form [I  A( )  B( )] t D 0 ;

(3)

where A( ) D [A1 ( ) : : : A k ( )];

D yl 

k X

A i ( )y l i C B i ( )u l i

iD1

gives a goodness of fit of the model to the data, when the parameter is chosen as . Therefore we can define the cost function at time t as t

1X k e t k22 t l Dk " #2 t k  X 1 X   D A i ( )y l i C B i ( )u l i   yl    t

J t ( ) D

l Dk

Suppose one is given a sequence of input-output measurements f(u t ; y t ); t  0g, where ut is the input to the system at time t, and y t is the output of the system at time t. The family of models fh( ); 2 g can consist of linear recursive relationships, or nonlinear mappings. Perhaps the simplest class of models is the so-called ARMA (Auto-Regressive Moving Average) model, in the form yt D

e l ( ) D [I  A( )  B( )] l

iD1

2

as a scalar measure of how good a model parameter is. It is easy to see that the normalizing factor 1/t is inserted to ensure that the quantity J t remains well behaved as t ! 1. Thus it is logical to choose the estimate ˆt such that ˆt D arg min J t ( ) : 2

Note that ˆ is our estimate of the parameter , based on the observations up to time t. The above description applies when the model is a linear ARMA model. However, the approach itself is far more general. Define the vector  t as above, and suppose that the family of models consists of a set of relationships of the form f( ;  t ) D 0 : If f( ; ) is a linear relationship of the form (3) for each choice of the parameter 2 , then we get the familiar parametrized ARMA model. As before, given a sequence of observations, the error at time t is defined as e t ( ) D f( ;  t ) ;

2

yt

6 y 6 t1 6 : 6 :: 6 6 B( ) D [B1 ( ) : : : B k ( )];  t D 6 y tk 6 6 u t1 6 6 :: 4 : u tk

3 7 7 7 7 7 7 7: 7 7 7 7 5

To estimate the parameter t at time t, it is common to choose it so as to minimize the accumulated least-squares error. Since we wish to fit a model of the form (3) to the

while the accumulated least squares error criterion is defined as before, as t

J t ( ) D

1X k e t k22 : t l Dk

In case the models are linear, choosing the least-squares error criterion leads to a very elegant solution of the optimization problem. But in case the models are not linear, there is no particular advantage in choosing the leastsquares error criterion. Instead, one can choose some “loss” function ` : RC ! RC such that `() is monotoni-

Learning, System Identification, and Complexity

cally increasing, and choose t

J t ( ) D

1X `(k e t k2 ) t l Dk

as the cost function at time t. Once again, the estimate ˆt is chosen as the minimizer of J t ( ) as varies over . Suppose now that the data comes from a “true but unknown” system whose output measurements are corrupted by measurement noise. Specifically, suppose it is the case that yt D

k X

A i ( true )y ti C B i ( true )u ti C v t ;

iD1

where the measurement noise v t is an i.i.d. zero-mean sequence. For convenience, let us define y0t D

k X

A i ( true )y ti C B i ( true )u ti

iD1

to be the corresponding output sequence of the same system, but without any measurement noise. Because the current output yt depends not only on the current and past inputs ut1 but also the past outputs y ti , the difference y t  y0t is a function of not just the current measurement noise v t , but also of the past measurement noise vectors v ti . As a result, even if the measurement noise v t consists of independent random variables, the dynamical nature of the system ensures that the difference between the actual measured output y t and the ideal, noiseless measurement y0t is a sequence of dependent random variables. In other words, the dynamical nature of the system ensures that we cannot any longer make an analogy with the “static” least squares curve-fitting problem of the previous section. Behavior of Identification Algorithms We have seen that a widely used approach to system identification is to formulate the error criterion J t ( ), and at each instant of time, choose the model parameter ˆt so as to minimize J t ( ). As time goes on, this approach generates a sequence of estimated parameters f ˆt g. Now we can ask three successively stronger questions, as follows: First, as more and more data is provided to the identification algorithm, does the estimation error between the outputs of the identified model and the actual time series approach the minimum possible estimation error achievable by any model within the given model class? To clarify this question, let us define the quantity J( ) as the limit as t ! 1 of the cost function J t ( ). It is easy to see that

the limit exists under very mild conditions. Now define J  D min J( ) 2

to be the best possible error measure. The question that is being asked is: Does the quantity J( ˆt ) approach this minimum value? In other words, is the performance of the identification algorithm asymptotically optimal? Second, does the identified model converge to “the set of best possible models” within the given model class? In other words, suppose that the minimum possible estimation error is achievable by one or more “best possible models,” and let   the set of all such that J( ) D J  . The question is: does the distance between the estimated vector ˆt and the set   approach zero at t ! 1? Note that the question, as posed, permits the sequence f ˆt g not to have a limit in the conventional sense, but to keep bouncing around nearer and nearer the set   . The difference between the first question and the second question is this. The first question requires only that the scalar J( ˆt ) should converge to the optimal value J  , while the second question requires that the estimated parameter vector ˆt should converge to the set of minimizers   . Third, assuming that the data is generated by a “true” system whose output is corrupted by measurement noise, does the identified model converge to the “true” system? In other words, if both the true system and the family of models are parametrized by a vector of parameters, does the estimated parameter vector converge to the true parameter vector? The difference between the second question and the third question is the following: In the second question, it is not assumed that the data is generated by a “true” system corresponding to a “true” parameter true belonging to the set . Accordingly, in the second question, all that we ask is that the sequence of estimates f ˆt g should converge to the set of minimizers of the function J. In contrast, in the third question, we add the assumption that the data to which the identification algorithm is applied has a specific structure, namely that it is generated by a true but unknown parameter vector true . The requirement in the third question is that the sequence of estimates f ˆt g should converge to true as t ! 1. From a technical standpoint, Questions 2 and 3 are easier to answer than Question 1. Since identification is often carried out recursively, the output of the identification algorithm is a sequence of estimates f t g t1 . Suppose that we are able to show that Question 1 has an affirmative answer, i. e., that J( t ) ! J  , where J  denotes the minimum possible estimation error. In such a case, with some additional assumptions it is possible to answer both Questions 2 and 3 in the affirmative.

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Traditionally a positive answer to Question 2 is assured by assuming that  is a compact set, which in turn ensures that the sequence f t g contains at least one convergent subsequence. For convenience, let us relabel this subsequence again as f t g. If the answer to Question 1 is “yes,” if  is any limit point of the sequence, and if J( ) is continuous (or at worst, lower semi-continuous) with respect to , then it readily follows that J( t ) ! J  . In other words, the model h(  ) is an “optimal” fit to the data among the family fh( ); 2 g. Coming now to Question 3, suppose true is the parameter of the “true” system, and let htrue denote the “true” system. In order for Question 3 to have an affirmative answer, the true system htrue must belong to the model family fh( ); 2 g; otherwise we cannot hope that h( t ) will converge to htrue . The traditional way to ensure that true D  is to assume that the input to the true system is “persistingly exciting” or “sufficiently rich,” so that the only way for h(  ) to match the performance of htrue is to have  D true . Because of the arguments presented above, the main emphasis in system identification theory has been to study conditions to ensure that Question 1 has an affirmative answer, i. e., that the identification algorithm is asymptotically optimal. This question can be readily addressed using the following “universal” approach. We have defined J( ) as the limit as t ! 1 of the quantity J t ( ). As remarked earlier, this limit exists under very mild conditions, for each fixed . Thus it can be said without much fuss that J t ( ) ! J( ) as t ! 1, again for each fixed . However, if a stronger property holds, namely that J t ( ) ! J( ) as t ! 1, uniformly with respect , then Question 1 has an affirmative answer. This is the approach adopted in [8]. In many ways, this approach to answering Question 1 is the common thread that binds system identification theory to learning theory. An alternate approach to Question 1 based on ergodic theory can be found in [5,6]. The fourth question that can be asked is: Suppose that the estimated parameter sequence f ˆt g converges to the true parameter true . As we have already seen, since the data that acts as the input to the identification algorithm is generated at random, the resulting estimates ˆt are also random. Thus we may wish to know, not merely that the random variable ˆt converges to true in some probabilistic sense (say in probability, or almost surely), but also, the asymptotic distribution of the estimate ˆt around the true value true . To make this point clear, let us return to the linear least-squares curve-fitting problem. We know that if the measurement noise consists of i.i.d. gaussian variables, then the optimal estimate also has a gaussian distribution centered around the true value, and that the variance around the true value goes to zero at a rate that can

be quantified fairly precisely. When we address this question in the context of dynamical systems however, the situation is considerably more complex, and one is obliged to use very sophisticated mathematics in order to analyze the asymptotic behavior of the estimated parameter sequence. Nevertheless, it is possible to say something about the asymptotic behavior of the estimates. The reader is invited to consult Chap. 9 of [9] for details. To keep the exposition simple, we have discussed only one possible approach to system identification. Minimizing some kind of generalized least squares error criterion is only one way to choose an estimated parameter ˆt . There are other popular methods, such as the “instrument variables” method. In the setting of linear systems especially, it is possible to evolve quite different approaches based on frequency domain (instead of time domain) methods, based on correlating the input and output variables. Here again, the book [9] is an excellent reference. Another excellent reference is [10]. Statistical Learning Theory Problem Formulation A parallel theme to the problem of system identification is statistical learning. This kind of learning theory comes in many varieties, and the type of learning studied here is known as Probably Approximately Correct (PAC) learning theory. Even within PAC learning theory, there are two aspects, namely: statistical and computational. The present section discusses the statistical aspects of PAC learning, while the next section discusses the computational aspects. Suppose one is trying to identify an unknown set on the basis of observation. The notion of PAC (probably approximately correct) learnability gives mathematical formalization of this intuitive idea. Suppose X is a set, and that C is a collection of subsets of X. Suppose also that P is a probability measure on the set X, which may or may not be known ahead of time to the learner. The learning problem is formulated as follows: There is a fixed but unknown concept T 2 C , called the target concept. The objective is to “learn” the target concept on the basis of observation, consisting of i.i.d. samples x1 ; : : : ; x m 2 X drawn in accordance with P. For each sample x j , an “oracle” tells the learner whether or not x j 2 T; equivalently, the oracle returns the value of I T (x j ), where I T () is the indicator function of T. Thus, after m samples have been drawn, the information available to the learner consists of the “labeled multisample” [(x1 ; I T (x1 )); : : : ; (x m ; I T (x m ))] 2 [X  f0; 1g]m :

Learning, System Identification, and Complexity

The objective is to construct a suitable approximation to the unknown target concept T on the basis of the labeled sample, using an appropriate algorithm. For the purposes of the present discussion, an “algorithm” is merely an indexed family of maps fA m gm1 , where m

A m : [X  f0; 1g] ! C : Thus an algorithm merely maps sets of labeled multisamples into elements of the concept class C . Suppose m i.i.d. samples have been drawn, and define H m (T; x) D A m [(x1 ; I T (x1 )); : : : ; (x m ; I T (x m ))] : The set H m (T; x) is referred to as the hypothesis generated by the algorithm. When there is no danger of confusion, one can abbreviate H m (T; x) by H m . In order to analyze how close the hypothesis H m is to the unknown target set T, we need a quantitative measure of the disparity. Given two sets A; B  X, define their symmetric difference AB by AB D (A \ B c ) [ (Ac \ B) D (A [ B)  (A \ B) : Thus AB consists of those points that belong to precisely one of the sets (and not the other). Now define the “generalization error” d P (T; H m ) D P(TH m ) : Then it is easy to see that d P (T; H m ) is precisely the probability that a randomly chosen test sample is misclassified by the hypothesis H m . Roughly speaking, the algorithm fA m g can be said to “learn” the target concept T if the distance d P (T; H m ) approaches zero as m ! 1. However, since the samples x i are random, the resulting hypothesis H m is also “random” and the generalization error d P (T; H m ) is also random. Define r(m; ) D sup P m fx 2 X m : d P [T; H m (T; x)] > g : (4) T2C

The quantity r(m; ) is called the “learning rate” function. Definition 1 The algorithm fA m g is said to be probably approximately correct (PAC) if r(m; ) ! 0 as m ! 1 for each > 0. The concept class C is said to be PAC learnable if there exists an algorithm that is PAC. Thus the algorithm fA m g is PAC if, for each target concept T, the distance d P (T; H m ) converges to zero in probability as m ! 1, and moreover, the rate of convergence is uniform with respect to the unknown target concept T.

Note that, if an algorithm is PAC, then for each pair of numbers ; ı > 0, there exists a corresponding integer m0 D m0 ( ; ı) such that P m fx 2 X m : d P [T; H m (T; x)] > g  ı ; 8m  m0 ; 8T 2 C :

(5)

In other words, if at least m0 i.i.d. samples are drawn and the algorithm is applied to the resulting labeled multisample, it can be said with confidence at least 1  ı that the hypothesis H m produced by the algorithm is within a distance of the true but unknown target concept T. In the above expression, the quantity is referred to as the “accuracy” of the algorithm, while ı is referred to as the “confidence.” The integer m0 ( ; ı) is called the “sample complexity” of the algorithm. In all but the simplest cases, one can only obtain upper bounds for m0 ( ; ı), but this is sufficient. The above statement provides some motivation for the nomenclature “PAC.” If the algorithm fA m g is PAC, the hypothesis H m produced by the algorithm need not exactly equal T; rather, H m is only an approximately correct version of T, in the sense that d P (H m ; T)  . Even this statement is only probably true, with a confidence of at least 1  ı. The notion of PAC learnability is usually credited to Valiant (see [12]), who showed that a certain collection of Boolean formulas is PAC learnable; however, the term “PAC learnability” seems to be due to [1]. It should be mentioned that Valiant had some additional requirements in his formulation beyond the condition that r(m; ) ! 0 as m ! 1. In particular, he required that the mapping A m be computable in polynomial time. These additional requirements are, in some sense, what distinguish computational (PAC) learning theory from statistical (PAC) learning theory. The references on computational learning theory in the bibliography go into computational issues in greater depth. PAC learning is closely related to an older topic in probability theory known as empirical process theory, which is devoted to a study of the behavior of empirical means as the number of samples becomes large. One of the standard references in the field, namely [13], is actually about empirical process theory. In the book [15] PAC learning theory is treated side by side with the uniform convergence problem, and it is shown that each has something to offer to the other. However, in the present write-up we do not discuss the uniform convergence problem. Finally, the book [16] removes the assumption that the learning samples must be independent, and replaces it with the weaker assumption that the learning samples form a so-called mixing process.

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Conditions for PAC Learnability In this section we summarize the main results that provide necessary and sufficient conditions for a collection of sets to be PAC learnable. A central role is played by a combinatorial parameter known as the Vapnik–Chervonenkis (VC)-dimension. This notion is implicitly contained in the paper [14]. Definition 2 Let A be a collection of subsets of X. A set S D fx1 ; : : : ; x n g  X is said to be shattered by A if, for every subset B  S, there exists a set A 2 A such that S \ A D B. The Vapnik–Chervonenkis dimension of A, denoted by VC-dim(A), equals the largest integer n such that there exists a set of cardinality n that is shattered by A. One can think of the VC-dimension of A as a measure of the “richness” of the collection A. A set S is “shattered” by A if A is rich enough to distinguish between all possible subsets of S. Note that S is shattered by A if (and only if) one can “pick off” every possible subset of S by intersecting S with an appropriately chosen set A 2 A. In this respect, perhaps “completely distinguished” or “completely discriminated” would be a better term than “shattered.” However, the latter term has by now become standard in the literature. Note that, if A has finite VC-dimension, say d, then it is not rich enough to distinguish all subsets of any set containing d C 1 elements or more; but it is rich enough to distinguish all subsets of some set containing d elements (but not necessarily all sets of cardinality d). From the definition it is clear that, in order to shatter a set S of cardinality n, the concept class A must contain at least 2n distinct concepts. Thus an immediate corollary of the definition is that every finite collection of sets has finite VC-dimension, and that VC-dim(A)  lg(jAj) : The importance of the VC-dimension in PAC learning theory arises from the following theorem, which is due to Blumer et al.; see [3]. In the statement of the theorem, we use the notion of a “consistent” algorithm. An algorithm is said to be “consistent” if it perfectly reproduces the training data. In other words, an algorithm is consistent if, for every labeled multisample ((x1 ; I T (x  1); : : : ; (x m ; I T (x m )), we have that I H m (x i ) D I T (x i )8i ; where H m is the hypothesis produced by the algorithm. Theorem 1 Suppose the concept class C has finite VC-dimension; then C is PAC-learnable for every probability measure over X. Let d denote the VC-dimension of C , and let fA m g be any consistent algorithm. Then the algorithm

is PAC; moreover, the quantity r(m; ) defined in (4) is bounded by   2em d m /2 r(m; )  2 2 ; (6) d for every probability measure over X. Any consistent algorithm learns to accuracy and confidence ı provided at least 

8d 8e 4 2 lg ; lg (7) m  max ı samples are used. Thus Theorem 1 shows that the finiteness of the VC-dimension of the concept class C is a sufficient condition for so-called “distribution-free” PAC learnability. Moreover, (7) gives an estimate of the sample complexity, that is, the number of samples that suffice to achieve an accuracy of and a confidence of ı. Note that the sample complexity estimate is O(1/ log(1/ )) for fixed ı, and O(log(1/ı)) for fixed . This is often summarized by the statement “confidence is cheaper than accuracy.” There is a converse to Theorem 1. It can be shown that, in order for a concept class C to be PAC learnable for every probability measure on the set X, and for the learning rate to be uniformly bounded with respect to the underlying probability measure, the concept class C must have finite VC-dimension. Theorem 1 provides not only a condition for PAClearnability but also relates the learning rate to the VC-dimension of the concept class. Thus it is clearly worthwhile to determine upper bounds for the VC-dimension of various concept classes. Over the years, several researchers have studied this problem, using a wide variety of approaches, ranging from simple counting arguments to extremely sophisticated mathematical reasoning based on advanced ideas such as algebraic topology, model theory of real numbers, and so on. Since the list of relevant references would be too long, the reader is referred instead to Chap. 10 of [15,16] which contain a derivation of many of the important known results together with citations of the original sources. The Complexity of Learning In this section, we discuss briefly the complexity of a learning problem. Complexity can arise in two ways: sample complexity, and algorithmic complexity. Sample complexity refers to the rate at which the number m0 ( ; ı) grows with respect to 1/ and 1/ı, where m0 ( ; ı) is the smallest value of m0 such that (5) holds. Algorithmic complexity refers to the complexity of computing the hypothesis H m

Learning, System Identification, and Complexity

as a function of the number of samples of m, Each kind of complexity has its own interesting features.

complexity m0 ( ; ı) 

Sample Complexity Let us begin by discussing the notions of covering numbers, and packing numbers. Given a concept class C consisting of subsets of a given set X and a probability measure P on X, let us observe that the quantity d P defines a pseudometric on the collection of (measurable) subsets of X, as follows: Given two subsets A; B  X, define d P (A; B) D P(AB) ; where AB is the symmetric difference between the sets A and B. Note that d P is only a pseudometric because d P (A; B) D 0 whenever AB is a set of measure zero – it need not be the empty set (meaning that A D B). However, this is a minor technicality, and we can still refer to d P as defining a “distance” between subsets of X. Given the concept class C , the probability measure P (which in turn defines the pseudometric d P ), and a number , we say that a finite collection fC1 ; : : : ; C n g where each C i 2 C is an -cover of C if, for every A 2 C , we have d P (A; C i )  for at least one index i. To put it in other words, the collection fC1 ; : : : ; C n g is an -cover of C under the pseudometric d P if every set A 2 C is within a distance of from at least one of the sets C i . The -covering number N( ; C ; d P ) is defined to be the smallest number n such that there exists an -cover of cardinality n. Note that, in some cases, an -cover may not exist, in which case we take the -covering number to be infinity. The next, and related, notion is the -packing number of a concept class. A finite collection fC1 ; : : : ; C m g where each C i 2 C is said to be -separated if d P (C i ; C j ) > 8i ¤ j. In other words, the finite collection fC1 ; : : : ; C m g is -separated if the concepts are pairwise at a distance of more than from each other. The -packing number M( ; C ; d P ) is defined to be the largest number m such that there exists an -separated collection of cardinality m. It is not difficult to show that M(2 ; C ; d P )  N( ; C ; d P )  M( ; C ; d P ) 8 > 0 : See for instance Lemma 2.2 of [16]. The next two theorems, due to Benedek and Itai [2], are fundamental results on sample complexity. Theorem 2 Suppose C is a given concept class, and let > 0 be specified. Then C is PAC-learnable to accuracy provided the covering number N( /2; C ; d P ) is finite. Moreover, there exists an algorithm that is PAC with sample

32 N( /2; C ; d P ) ln : ı

Thus a concept class with a finite covering number N( /2; C ; d P ) is learnable to accuracy . The next theorem provides a kind of converse. Theorem 3 Suppose C is a given concept class, and let > 0 be specified. Then any algorithm that is PAC to accuracy requires at least lg M(2 ; C ; d P ) samples, where M(2 ; C ; d P ) denotes the 2 -packing number of the concept class C with respect to the pseudometric d P . One consequence of the theorem is that a concept class C is learnable to accuracy only if M(2 ; C ; d P ) is finite. This fact is used to construct examples of “non-learnable” concept classes; see for instance Example 6.10 in [16]. However, the question of interest here is whether there exists a concept class that is learnable, but for which the sample complexity m0 ( ; ı) grows faster than any polynomial in 1/ . In Sect. 3.3 in [7], the author constructs a concept class with the property that M(1/n; C ; d P )  22

cn 1/4

;

where c is some constant. Thus it follows from Theorem 3 that the sample complexity satisfies 1/4

m0 ( /2; ı)  2c(1/ )

;

8 :

So it is after all possible to have a situation where a concept class is learnable, but the sample complexity grows faster than any polynomial in the reciprocal accuracy 1/ . Algorithmic Complexity We have seen from Theorem 1 that if a concept class C has finite VC-dimension, then every consistent algorithm is PAC. In the original paper [12], one studies not a single concept class C , but rather an indexed family of concept classes fCn g, for every integer n. The objective is to determine whether there exists an algorithm that PAC-learns each concept class, such that the computational complexity of learning Cn is polynomial in n. Note that we are not referring here to the sample complexity. In most problems in computer science, each concept class Cn is finite and therefore automatically has finite VC-dimension. Rather, we are referring to the difficulty of constructing a consistent hypothesis, and the computational complexity of this task as a function of n. In the original paper [12], it is shown that the problem of learning 3-CNF (3 conjunctive

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normal forms) is PAC learnable in this more restrictive sense. Note that 3-CNF in n Boolean variables consists of all conjunctions of clauses in which each clause is a disjunction of no more than three Boolean variables or their negations. For instance, if n D 5, then (x1 _ :x2 _ x5 ) ^ (x2 _ x3 _ :x4 ) ^ (:x1 _ :x3 ) ^ (:x2 _ :x3 _ :x5 ) belongs to 3-CNF. In this instance, Valiant shows that it is possible to construct a consistent hypothesis in polynomial time in n. Shortly after the publication of this paper, a very simple example was produced of a learning problem in which it is NP-hard to construct a consistent hypothesis. Specifically, consider the family of all conjunctions of at most k clauses, where each clause is a disjunction of various literals or their negations (without limit). For instance, if n D 5, then (x1 _ :x2 _ :x3 _ x5 ) ^ (:x1 _ x3 _ :x4 _ :x5 ) ^ (:x1 _ x2 _ x4 _ :x5 ) is an example of this kind of formula. In this case, it can be shown that, for each fixed k and n, the VC-dimension of the concept class is polynomial in k and n. Thus it follows from Theorem 1 that the sample complexity of learning is polynomial. However, it is shown in [11] that the computational complexity of finding a consistent hypothesis is intractable unless P D N P. In contrast, if each such Boolean formula is expressed as a 3-DNF, then it is possible to construct a consistent hypothesis in polynomial time with respect to n. System Identification as a Learning Problem The Need for a Quantitative Identification Theory By tradition, identification theory is asymptotic in nature. In other words, the theory analyzes what happens as the number of samples approaches infinity. However, there are situations where it is desirable to have finite time, or nonasymptotic, estimates of the rate at which the output of the identification process converges to the best possible model. For instance, in “indirect” adaptive control, one first carries out an identification of an unknown system, and after a finite amount of time has elapsed, designs a controller based on the current model of the unknown system. If one can quantify just how close the identified model is to the true but unknown system after a finite number of samples, then these estimates can be combined with robust control theory to ensure that indirect adaptive control leads to closed-loop stability.

Among the first papers to state that the derivation of finite-time estimates is a desirable property in itself are Weyer, Williamson and Mareels; see [20,21]. In those papers, it is assumed that the time series to which the system identification algorithm is applied consists of so-called “finitely-dependent” data; in other words, it is assumed that the time series can be divided into blocks that are independent. In a later paper [19], the author replaced the assumption of finite dependence by the assumption that the time series is ˇ-mixing. However, he did not derive conditions under which a time series is ˇ-mixing. In a still later paper [4], the authors study the case where the data is generated by general linear systems (Box–Jenkins models) driven by i.i.d. Gaussian noise, and the model family also belongs to the same class. In this paper, the authors say that they are motivated by the observation that “signals generated by dynamical systems are not ˇ-mixing in general”. This is why their analysis is limited to an extremely restricted class. However, their statement that dynamical systems do not generate ˇ-mixing sequences is simply incorrect. Specifically, the all the systems studied in [4] are themselves ˇ-mixing! In another paper [18], the present author has shown that practically any exponentially stable system driven by i.i.d. noise with bounded variance is ˇ-mixing. Hence, far from being restrictive, the results of [19] have broad applicability, though he did not show it at the time. Problem Formulation System identification viewed as a learning problem can be stated as follows: One is given a time series f(u t ; y t )g, where u t denotes the input to the unknown system at time t, and y t denotes the output at time t. We have that u t 2 U  R l ; y t 2 Y  R k for all t, where for simplicity it is assumed that both U and Y are compact sets. One is also given a family of models fh( ); 2 g parametrized by a parameter vector belonging to a set . Usually  is a subset of a finite-dimensional Euclidean space. At time t, the data available to the modeler consists of all the past measurements until time t  1. Based on these measurements, the modeler chooses a parameter t , with the objective of making the best possible prediction of the next measured output y t . The method of choosing t is called the identification algorithm. To make this formulation a little more precise, let us Q introduce some notation. Define U :D 1 1 U, and define Y analogously. Note that the input sequence fu t g belongs to the set U, while the output sequence belongs to Y . Let U01 denote the one-sided infinite cartesian product Q0 U01 :D 1 U, and for a given two-sided infinite se-

Learning, System Identification, and Complexity

quence u 2 U, define ut :D (u t1 ; u t2 ; u t3 ; : : : ) 2

U01

:

Thus the symbol ut denotes the infinite past of the input signal u at time t. The family of models fh( ); 2 g consists of a collection of maps h( ); 2 , where each h( ) maps U01 into Y. Thus, at time t, the quantity h( )  ut D: yˆ t ( ) is the “predicted” output if the model parameter is chosen as . The quality of this prediction is measured by a “loss function” ` : Y  Y ! [0; 1]. Thus `(y t ; h( )  u t ) is the loss we incur if we use the model h( ) to predict the output at time t, and the actual output is y t . Since we are dealing with a time series, all quantities are random. Hence, to assess the quality of the prediction made using the model h( ), we should take the expected value of the loss function `(y t ; h( )ut ) with respect to the law of the time series f(u t ; y t )g, which we denote by P˜u;y . Thus we define the objective function  J( ) :D E `(y t ; h( )  u t ); P˜u;y : (8) Note that J( ) depends solely on and nothing else. A key observation at this stage is that the probability measure P˜u;y is unknown. This is because, if the statistics of the time series are known ahead of time, then there is nothing to identify! Definition 3 System Identification Problem: Given the time series f(u t ; y t )g with unknown law P˜u;y , construct if possible an iterative algorithm for choosing t as a function of t, in such a way that

from time 1 to time t. Note that the least squares error is in general unbounded, whereas the loss function ` is supposed to map into the bounded interval [0; 1]. To circumvent this technical difficulty, we can choose t  1X  Jˆt ( ) D  k y i  yˆ i ( ) k2 ; t iD1

where () is any kind of saturating nonlinear function, for instance (s) D tanh(s). This corresponds to the choice `(y; z) D tanh(k y  z k2 ). ˆ ) Note that, unlike the quantity J( ), the function J( can be computed on the basis of the available data. Hence, in principle at least, it is possible to choose t so as to minimize the “approximate” (but computable) objective funcˆ ) in the hope that, by doing so, we will somehow tion J( minimize the “true” (but uncomputable) objective function J( ). The next theorem gives some sufficient conditions for this approach to work. Specifically, if the estimates Jˆt ( ) converge uniformly to the correct values J( ) as t ! 1, where the uniformity is with respect to 2 , then the above approach works. The importance of this uniform convergence property was recognized very early on in identification theory. In particular, Caines [5,6] uses ergodic theory to establish this property, whereas Ljung [8] takes a more direct approach. Theorem 4 At time t, choose t so as to minimize Jˆt ( ); that is, t D arg min Jˆt ( ) :

J( t ) ! inf J( ) :

2

2

Let A General Result Since the law P˜u;y of the time series is not known, the objective function J( ) cannot be computed exactly. To circumvent this difficulty, one replaces the “true” objective function J() by an “empirical approximation,” as defined next. For each t  1 and each 2 , define the empirical error t

1X Jˆt ( ) :D `[y i ; h( )  ui ] : t

(9)

2

Define the quantity q(t; ) :D P˜u;y f sup j Jˆt ( )  J( )j > g :

k2 ,

then

t

1X k y i  yˆ i ( ) k2 Jˆt ( ) D t iD1

would be the average cumulative mean-squared error between the actual output y i and the predicted error yˆ i ( ),

(10)

2

Suppose it is the case that q(t; ) ! 0 as t ! 1; 8 > 0. Then P˜u;y fJ( t ) > J  C g ! 0

iD1

If we could choose `(y; z) Dk y  z

J  :D inf J( ) :

as t ! 1; 8 > 0 : (11)

In other words, the quantity J( t ) converges to the optimal value J  in probability, with respect to the measure P˜u;y . Corollary 1 Suppose that q(t; ) ! 0 as t ! 1; 8 > 0. Given ; ı > 0, choose t0 D t0 ( ; ı) such that q(t; ) < ı8t  t0 ( ; ı) :

(12)

933

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Learning, System Identification, and Complexity

Then P˜u;y fJ( t )



> J C g < ı8t  t0 ( /3; ı) :

(13)

Proofs of the theorem and corollary can be found in [17]. A Result on the Uniform Convergence of Empirical Means The property of Theorem 4 does indeed hold in the commonly studied case where y t is the output of a “true” system corrupted by additive noise, and the loss function ` is the squared error, provided the model family and the true system satisfy some fairly reasonable conditions, namely: each system is BIBO (bounded input, bounded output) stable, all systems in the model family have exponentially decaying memory, and finally, an associated collection of input-output maps has finite P-dimension. Note that the P-dimension is a generalization of the VC-dimension to the case of maps assuming values in a bounded interval [0; 1]. By identifying a set with its support function, we can think of the VC-dimension as a combinatorial parameter associated with binary-valued maps. The P-dimension is defined in Chap. 4 of [15,16] Define the collection of functions H mapping U into R as follows: g( ) :D u 7!k ( f  h( ))  u0 k2 : U ! R ; G :D fg( ) : 2 g :

Now the various assumptions are listed.

Now we can state the main theorem. Theorem 5 Define the quantity q(t; ) as in (10) and suppose Assumptions A1 through A3 are satisfied. Given an > 0, choose k( ) large enough that  k  /4 for all k  k( ). Then for all t  k( ) we have 

32e 32e ln q(t; )  8k( )

d(k( ))

 exp(bt/k( )c 2 /512M 2 ) ;

(15)

where bt/k( )c denotes the largest integer part of t/k( ). Theorem 6 Let all notation be as in Theorem 5. Then, in order to ensure that the current estimate t satisfies the inequality J( t )  J  C (i. e., is -optimal) with confidence 1  ı, it is enough to choose the number of samples t large enough that "     24k( ) 32e 512M 2 bt/k( )c  ln C d(k( )) ln 2 ı  # 32e C d(k( )) ln ln : (16) Bounds on the P-Dimension

A1. There exists a constant M such that jg( )  u0 j  M; 8 2 ; u 2 U : This assumption can be satisfied, for example, by assuming that the true system and each system in the family fh( ); 2 g is BIBO stable (with an upper bound on the gain, independent of ), and that the set U is bounded (so that fu t g is a bounded stochastic process). A2. For each integer k  1, define g k ( )  ut :D g( )  (u t1 ; u t2 ; : : : ; u tk ; 0; 0; : : : ) :

sentially means that each of the systems in the model family has decaying memory (in the sense that the effect of the values of the input at the distant past on the current output becomes negligibly small). A3. Consider the collection of maps G k D fg k ( ) : 2 g, viewed as maps from U k into R. For each k, this family G k has finite P-dimension, denoted by d(k).

In order for the estimate in Theorem 5 to be useful, it is necessary for us to derive an estimate for the P-dimension of the family of functions defined by G k :D fg k ( ) : 2 g ;

(17)

where g k ( ) : U k ! R is defined by g k ( )(u) :Dk ( f  h( ))  u k k2 ; where u k :D (: : : ; 0; u k ; u k1 ; : : : ; u1 ; 0; 0; : : : ) :

(14)

With this notation, define  k :D sup sup j(g( )  g k ( ))  u0 j : u2U 2

Then the assumption is that  k is finite for each k and approaches zero as k ! 1. This assumption es-

Note that, in the interests of convenience, we have denoted the infinite sequence with only k nonzero elements as u k ; : : : ; u1 rather than u0 ; : : : ; u1k as done earlier. Clearly this makes no difference. In this section, we state and prove such an estimate for the commonly occurring case where each system model h( ) is an ARMA model where the parameter enters linearly. Specifically, it is

Learning, System Identification, and Complexity

supposed that the model h( ) is described by x tC1 D

l X

i  i (x t ; u t ); y t D x t ;

(18)

3.

iD1

where D ( 1 ; : : : ; l ) 2   R l , and each  i (; ) is a polynomial of degree no larger than r in the components of x t ; u t .

4. 5.

Theorem 7 With the above assumptions, we have that P-dim(G k )  9l C 2l lg[2(r kC1  1)/(r  1)]  9l C 2l k lg(2r)

if

r > 1:

6.

(19)

In case r D 1 so that each system is linear, the above bound can be simplified to P-dim(G k )  9l C 2l lg(2k) :

(20)

It is interesting to note that the above estimate is linear in both the number of parameters l and the duration k of the input sequence u, but is only logarithmic in the degree of the polynomials  i . In the practically important case of linear ARMA models, even k appears inside the logarithm.

7. 8. 9. 10. 11. 12.

Future Directions

13.

At present, system identification theory is a mature, widely used subject. Statistical learning theory is newer and still needs to find its feet. In particular, the various estimates for sample complexity given here are deemed to be “too conservative” to be of practical use. Thus considerable research needs to be carried out on the issue of sample complexity. The interplay between system identification theory and statistical learning theory perhaps offers the greatest scope for interesting new work. At present, statistical learning theory allows us to derive finite time estimates of the disparity between the current estimates of a system, and the true but unknown system. However, these estimates do not always translate readily into a metric “distance” between the true but unknown system and the current estimate. In order to be useful in the context of robust controller design, it is necessary to derive estimates for some kind of metric distance. This is an important topic for future research.

14.

Acknowledgment The author thanks Erik Weyer for his careful reading and comments on an earlier draft of this article. Bibliography Primary Literature 1. Angluin D (1987) Queries and concept learning. Mach Learn 2:319–342 2. Benedek GM, Itai A (1988) Learnability by fixed distributions.

15. 16.

17.

18.

19.

20.

21.

First Workshop on Computational Learning Theory. MorganKaufmann, San Mateo, pp 80–90 Blumer A, Ehrenfeucht A, Haussler D, Warmuth M (1989) Learnability and the Vapnik–Chervonenkis dimension. J ACM 36(4):929–965 Campi MC, Weyer E (2002) Finite sample properties of system identification methods. IEEE Trans Auto Control 47:1329–1334 Caines PE (1976) Prediction error identification methods for stationary stochastic processes. IEEE Trans Auto Control AC21:500–505 Caines PE (1978) Stationary linear and nonlinear system identification and predictor set completeness. IEEE Trans Auto Control AC-23:583–594 Hammer B (1999) Learning recursive data, Math of Control. Signals Syst 12(1):62–79 Ljung L (1978) Convergence analysis of parametric identification methods. IEEE Transa Auto Control AC-23:77–783 Ljung L (1999) System Identification: Theory for the User. Prentice-Hall, Englewood Cliffs Pintelon R, Schoukens J (2001) System Identification: A frequency domain approach. IEEE press, Piscataway Pitt L, Valiant L (1988) Computational limits on learning from examples. J ACM 35(4):965–984 Valiant LG (1984) A theory of the learnable. Comm ACM 27(11):1134–1142 Vapnik VN (1982) Estimation of Dependences Based on Empirical Data. Springer, Heidelberg Vapnik VN, Chervonenkis AY (1971) On the uniform convergence of relative frequencies to their probabilities. Theory Prob Appl 16(2):264–280 Vidyasagar M (1997) A Theory of Learning and Generalization. Springer, London Vidyasagar M (2003) Learning and Generalization: With Applications to Neural Networks and Control Systems. Springer, London Vidyasagar M, Karandikar RL (2001) System identification: A learning theory approach. Proc IEEE Conf on Decision and Control, Orlando, pp 2001–2006 Vidyasagar M, Karandikar RL (2001) A learning theory approach to system identification and stochastic adaptive control. In: Calafiore G, Dabbene F (eds) Probabilistic and Randomized Algorithms for Design Under Uncertainty. Springer, Heidelberg, pp 265–302 Weyer E (2000) Finite sample properties of system identification of ARX models under mixing conditions. Automatica 36(9):1291–1299 Weyer E, Williamson RC, Mareels I (1996) Sample complexity of least squares identification of FIR models. Proc 13th World Congress of IFAC, San Francisco, pp 239–244 Weyer E, Williamson RC, Mareels I (1999) Finite sample properties of linear model identification. IEEE Trans Auto Control 44(7):1370–1383

Books and Reviews Anthony M, Biggs N (1992) Computational Learning Theory. Cambridge University Press, Cambridge Devroye L, Györfi L, Lugosi G (1996) A probabilistic theory of pattern recognition. Springer, New York Haussler D (1992) Decision theoretic generalizations of the PAC model for neural net and other learning applications. Info Comput 100:78–150

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Kearns M, Vazirani U (1994) Introduction to Computational Learning Theory. MIT Press, Cambridge Kulkarni SR, Vidyasagar M (2001) Learning decision rules under a family of probability measures. IEEE Trans Info Thy Natarajan BK (1991) Machine Learning: A Theoretical Approach. Morgan-Kaufmann, San Mateo van der Vaart AW, Wallner JA (1996) Weak convergence and Empirical Processes. Springer, New York

Vapnik AN, Chervonenkis AY (1981) Necessary and sufficient conditions for the uniform convergence of means to their expectations. Theory Prob Appl 26(3):532–553 Vapnik VN (1995) The Nature of Statistical Learning Theory. Springer, New York Vapnik VN (1998) Statistical Learning Theory. Wiley, New York Weyer E, Campi MC (2002) Non-asymptotic confidence ellipsoids for the least squares error estimate. Automatica 38:1529–1547

Lyapunov–Schmidt Method for Dynamical Systems

Lyapunov–Schmidt Method for Dynamical Systems ANDRÉ VANDERBAUWHEDE Department of Pure Mathematics and Computer Algebra, Ghent University, Gent, Belgium Article Outline Glossary Definition of the Subject Introduction Lyapunov–Schmidt Method for Equilibria Lyapunov–Schmidt Method in Discrete Systems Lyapunov–Schmidt Method for Periodic Solutions Lyapunov–Schmidt Method in Infinite Dimensions Outlook Bibliography Glossary Bifurcation A change in the local or global qualitative behavior of a system under the change of one or more parameters; the critical parameter values at which the change takes place are called bifurcation values. Saddle-node bifurcation A bifurcation where two equilibria, usually one stable and the other unstable, merge and then disappear. A similar bifurcation can also happen with periodic orbits. Pitchfork bifurcation A bifurcation where an equilibrium loses stability and at the same time two new stable equilibria emerge; this type of bifurcation frequently appears in systems with symmetry. Period-doubling bifurcation A bifurcation where a fixed point of a mapping (discrete system) loses stability and a stable period-two point emerges. In a continuous system this corresponds to a periodic orbits losing stability while a new periodic orbit, with approximately twice the period of the original one, bifurcates. Equivariant system A system with symmetries; these symmetries form a group of (usually linear) transformations commuting with the vectorfield or the mapping generating the system. Reversible system A symmetric system where some of the symmetries involve a reversal of the time. Definition of the Subject The mathematical description of the state and the evolution of systems used to model all kind of applications requires in many cases a large or even an infinite number of

variables. In contrast, a number of basic phenomena observed in such systems (in particular bifurcation phenomena) are known to depend only on a very small number of variables. A simple example is the Hopf bifurcation, where the system changes from an equilibrium state to a periodic regime when the parameters cross a certain critical boundary, a transition which in a typical system essentially only involves two of the state variables. There are several ways in which the original system can be reduced to a much smaller system which captures the phenomenon one wants to analyze. There is a dynamical approach, usually in the form of some kind of center manifold reduction, where one tries to find a subsystem (invariant under the full system flow) which contains the core of the bifurcation under consideration. The Lyapunov–Schmidt method takes a rather different point of view, by concentrating purely on the objects under study, such as equilibria or periodic solutions, setting up an equation for these objects in an appropriate space, and then performing a reduction on this equation without any further reference to the original dynamics. This is done by splitting both the unknown and the equation in two parts, solving one of the resulting partial equations and substituting the solutions in the second equation. The outcome is typically a low-dimensional set of algebraic equations – usually called bifurcation equations – the solutions of which correspond in a one-to-one way with the chosen solutions of the original system. The analysis of these bifurcation equations (using appropriate techniques) allows to prove existence and multiplicity results and to describe bifurcation scenarios. The Liapunov–Schmidt reduction method has a long history, going back to the early 20th century. It has been used in many different forms, and sometimes under different names, to a large variety of problems both within and outside the strict domain of dynamical systems and bifurcation theory: continuous and discrete systems, ordinary and partial differential equations, functional differential equations, integral equations, variational problems, and so on. Within the theory of dynamical systems the method has been widely used (and remains to be used) for the study of bifurcations of equilibria and periodic orbits. Introduction The Lyapunov–Schmidt method is, as its name suggests, a method, a technique or procedure which can be applied to a wide class of problems and equations, ranging from local bifurcation problems for equilibria and periodic solutions in finite-dimensional systems to global existence and multiplicity results for solutions of infinite-dimensional problems. The common theme in each of these applica-

937

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Lyapunov–Schmidt Method for Dynamical Systems

tions is that of a reduction: the original equation is partly solved, and substitution of the partial solution in the remaining equations gives a reduced equation of a lower dimension than the original one. In many cases – but there are notorious exceptions – the reduced problem is finitedimensional, with low dimension (one, in the best case). Lyapunov–Schmidt Method – An Overview The basic idea behind the Lyapunov–Schmidt reduction method is a rather simple splitting algorithm which can in very general terms be described as follows. The starting point is an equation F (x) D 0

(1)

for an unknown x belonging to some space X, and depending on some parameter . The mapping F takes the space X into a space Y; in simple cases Y D X, and typically X is a subspace of Y (although this is not necessary for the method to work). For example, X D Y may be the phase space, and (1) the condition for x 2 X to be an equilibrium of a given dynamical system on X. Or X and Y may be spaces of periodic mappings, while (1) represents the condition for such periodic mapping to be a solution of a given system. The main ingredients for the splitting are two projections: a projection P of the space X onto a subspace U, and a projection Q of the space Y onto a subspace W; in most applications the subspaces U and W are finite-dimensional. The projection P is used to split the variable x: x D Px C (I  P)x D u C v ; with u D Px 2 U and v D (I  P)x 2 V D (I  P)(X). The projection Q allows to split the Eq. (1) into a set of two equations, (I  Q)F (x) D 0 and QF (x) D 0. Bringing the two splittings together shows that (1) is equivalent to the system (I  Q)F (u C v) D 0 ;

QF (u C v) D 0 :

(2)

Of course these equations heavily depend on the choices for the projections P and Q. The main hypothesis which allows to perform a reduction is that the projections P and Q should be such that the first equation in (2) has for each (u; ) in a certain domain a unique solution v D v (u). Depending on the particular problem this solution v (u) will be obtained by different techniques, but mostly some form of the contraction principle or the implicit function theorem is used. Substituting the solution of the first equation of (2) into the second equation results in the following

reduced equation, usually called the bifurcation or determining equation:   G (u) D QF u C v (u) D 0 :

(3)

Under the condition that both U and W are finite-dimensional this represents a finite set of equations for a finite number of unknowns. Every solution (u; ) of (3) corresponds to a solution (x; ) D (u C v (u); ) of (1), and conversely, every solution (x; ) of (1) (in the appropriate domain) can be written as x D uCv (u) for some solution (u; ) of (3). The main difficulty in analyzing the Eq. (3) consists in the fact that typically the solution v (u) of the first equation of (2) is not explicitly known, and therefore also the mapping G (which takes U into W) is not explicitly known. It is therefore important that the set-up (i. e. the projections P and Q used in the reduction) should be such that at least some qualitative properties of v and G , such as continuity and differentiability, can be obtained; when the method is used to study local bifurcation problems also a good approximation of these mappings, such as for example a few terms in their Taylor expansion, is required to obtain relevant conclusions. When the method is used to study (local) bifurcation phenomena one generally assumes that a particular solution (x0 ; 0 ) of (1) is known, and one wants to obtain the full solution set of (1) close to this given solution. In such case a convenient choice for the projections P and Q is related to the linearization L0 D DF0 (x0 ) of the mapping F : the projection P should be such that U D N(L0 ) D fx 2 X j L0 x D 0g, while Q should be a projection of Y onto a subspace W which is complementary to the range R(L0 ) D fA0 x j x 2 Xg of L0 , i. e. (I  Q)(Y) D R(L0 ). With such choice, and under appropriate smoothness conditions for the mapping F , the existence of a unique solution v D v (u) of the first equation in (2) is guaranteed by the implicit function theorem for all (u; ) near (u0 ; 0 ) D (Px0 ; 0 ). Also the approximation of v (u) and G (u) by Taylor expansions works quite well in such case. Lyapunov–Schmidt Method – An Example As a simple but illustrative example on how the method works in practice consider the second order scalar ordinary differential equation y¨ C ˛ y˙ C y C g(y; y˙) D 0 ;

(4)

with ˛ 2 R a parameter and g(y; y˙) a smooth function of two variables such that g(y; y˙) D O((jyj C j y˙j)2 ) as (y; y˙) ! (0; 0). This equation has the equilibrium solution

Lyapunov–Schmidt Method for Dynamical Systems

y(t)  0 for all values of the parameter; the linearization at this equilibrium, y¨ C ˛ y˙ C y D 0 ;

For each ( ; v; ) 2 R  V (2)  R the term g( cos t C v(t); (1 C )( sin t C v˙(t))) appearing in (6) is decomposed as

(5) A( ; v; ) cos t C B( ; v; ) sin t C g˜( ; v; )(t) ;

has for ˛ D 0 a two-dimensional space of periodic solutions given by fy(t) D a cos t C b sin t j (a; b) 2 R2 g. One may then ask which of these periodic solutions survive when the linear Eq. (5) is replaced by the non-linear Eq. (4). Since (5) is an approximation of (4) only for small (y; y˙), and since (5) has periodic solutions only for ˛ D 0, the question can be stated more precisely as follows: determine, for all small values of ˛, all small periodic solutions of (4). An immediate problem which arises is that different periodic solutions may have different periods; indeed, the determination of the period is part of the problem. In the Lyapunov–Schmidt approach this is handled by setting y(t) D x((1 C )t), resulting in the new equation (1 C )2 x¨ C ˛(1 C )x˙ C x C g(x; (1 C )x˙ ) D 0 ; (6) 2-periodic solutions of (6) correspond to 2 (1C )1 -periodic solutions of (4). The new problem is then to determine, for all small values of (; ˛), all small 2-periodic solutions of (6). Observe that this problem is symmetric with respect to phase shifts: if x(t) is a 2-periodic solution, then so is x(t C ), for each . To solve this problem a possible 2-periodic solution x(t) is written in the form x(t) D a cos t C b sin t C v(t) D cos(t C ) C v(t) ; with v(t) a twice continuously differentiable function such that Z 2 Z 2 v(t) cos tdt D v(t) sin tdt D 0 : (7) 0

0

Taking the symmetry with respect to phase shifts into account one can set  D 0, such that x(t) takes the form x(t) D cos t C v(t) ;

with v 2 V (2) ;

(8)

here V (2) denotes the space of 2-periodic C2 -functions satisfying (7), while V (0) will denote the space of continuous such functions. It is an easy exercise to verify that for each v˜ 2 V (0) the non-homogeneous linear equation v¨ C v D v˜(t) has a unique solution v 2 Z

(9) V (2) ,

t

sin(t  s)˜v (s)ds C

v(t) D 0

given by 1 2

Z

2

sin(t  s)s v˜(s)ds : 0

with 1 A( ; v; ) D 

Z

2

g( cos t C v(t); 0

(1 C )( sin t C v˙(t))) cos tdt ; B( ; v; ) D

1 

Z

2

g( cos t C v(t); 0

(1 C )( sin t C v˙(t))) sin tdt ; and g˜( ; v; ) 2 V (0) . The Eq. (6) then splits into two equations, namely [(2 C  2 ) C A( ; v; )] cos t  C ˛(1 C ) C B( ; v; ) sin t D 0

(10)

and (1 C )2 v¨ C ˛(1 C )˙v C v C g˜( ; v; ) D 0 :

(11)

Both equations are satisfied for ( ; v) D (0; 0). Using the solvability property of (9) mentioned above in combination with the implicit function theorem proves that the Eq. (11) has for each sufficiently small ( ; ; ˛) a unique small solution v D v  ( ; ; ˛) 2 V (2) , with v  (0; ; ˛) D 0. Bringing this solution in (10) gives two scalar equations (2 C  2 ) C A( ; v  ( ; ; ˛); ) D 0 and ˛(1 C ) C B( ; v  ( ; ; ˛); ) D 0 :

(12)

It is easy to verify that A( ; v  ( ; ; ˛); ) and B( ; v  ( ; ; ˛); ) are of the order O( 2 ) as ! 0, such that ˜ ; ˛) A( ; v  ( ; ; ˛); ) D A( ;  ˜ and B( ; v ( ; ; ˛); ) D B( ; ; ˛) ; ˜ ; ˛) D B(0; ˜ ; ˛) D 0, and (12) reduces for with A(0; non-trivial solutions (i. e. solutions with ¤ 0) to ˜ ; ˛) D 0 (2 C  2 ) C A( ; ˜ and ˛(1 C ) C B( ; ; ˛) D 0 :

(13)

For small ( ; ; ˛) these equations can be solved by the implicit function theorem for (; ˛) D (  ( ); ˛  ( )), with (  (0); ˛  (0)) D (0; 0). This means that for each small one can find a parameter value ˛ D ˛  ( ) such that (4) has a periodic solution with “amplitude” equal to j j and

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Lyapunov–Schmidt Method for Dynamical Systems

given by x  ( )(t) D cos((1 C   ( ))t) C v  ( ;   ( ); ˛  ( ))((1 C   ( ))t) ; the minimal period of this solution is 2(1 C   ( ))1 . In fact, the periodic orbits corresponding to and  coincide, as can be seen from the following. The phase shift symmetry together with  cos t D cos(t C ) and the uniqueness of the solution v  ( ; ; ˛) of (11) implies that v  ( ; ; ˛)(t) D v  ( ; ; ˛)(t C ), from which one can easily deduce that A( ; v  ( ; ; ˛); ) D A( ; v  ( ; ; ˛); ), B( ; v  ( ; ; ˛); ) D B( ; v  ( ; ; ˛); ) and (  ( ); ˛  ( )) D (  ( ); ˛  ( )). Hence x  ( )(t) D x  ( )(t C (1 C   ( ))1 ) : A further consequence is that   ( ) and ˛  ( ) are in fact functions of 2 , i. e.   ( ) D ˆ ( 2 ) D c 2 C h.o.t. and ˆ 2 ) D d 2 C h.o.t. ˛  ( ) D ˛( It is interesting to notice the difference in treatment between the auxiliary equation (11) and the bifurcation equations (12). The auxiliary equation (11) is solved by a direct application of the implicit function theorem, while the same implicit function theorem can only be used to solve the bifurcation equations after some manipulation, which in this example simply consists in dividing by to “eliminate” the trivial solution branch. A further remark is that when the nonlinearity g in (4) does not depend on y˙ (i. e. g D g(y)) then ˛  ( )  0 and we have what is usually called a vertical bifurcation. For the example this means that the equation y¨ C ˛ y˙ C y C g(y) D 0 can have periodic solutions only for ˛ D 0, and that all small solutions of y¨ C y C g(y) D 0 are indeed periodic. To see how this well known fact fits into the Lyapunov–Schmidt reduction observe that the equation y¨ C y C g(y) D 0 is reversible, in the sense that if y(t) is a solution then so is y(t). Using this reversibility one can show that v  ( ; ; 0)(t) D v  ( ; ; 0)(t); therefore g( cos t C v  ( ; ; 0)(t)) is an even function of t, and B( ; v  ( ; ; 0); ) D 0 for all ( ; ). This immediately implies that ˛  ( )  0, as claimed. The foregoing example forms a very simple case of a Hopf bifurcation; this bifurcation will be treated in a more general context in Subsect. “Hopf Bifurcation”. Lyapunov–Schmidt Method – Historical Background The Lyapunov–Schmidt method has been initiated in the early 20th century by A.M. Lyapunov [48] and E. Schmidt [55] in their (independent) studies of nonlinear integral equations. Later the method was used in diverse

forms in the work of (among many others) Cacciopoli [8], Cesari [9,10,11], Shimizu [56], Cronin [19], Bartle [5], Hale [26], Lewis [46], Vainberg and Trenogin [59], Antosiewicz [2], Rabinowitz [51], Bancroft, Hale and Sweet [4], Mawhin [49], Hall [33,34], Crandall and Rabinowitz [17], etc. Expositions on the method were given by Friedrichs [21], Nirenberg [50], Krasnosel’skii [43,44], Hale [27,28,30], Vainberg and Trenogin [60], and others. In particular the work of Cesari and Hale, who used the name alternative method, has put the method on a strong functional analytic basis. Apparently the name Lyapunov– Schmidt method was first introduced in the paper [59] of Vainberg and Trenogin; it has been used systematically for this type of approach since the seventies. In the seventies and the eighties the method was used extensively in a wide variety of bifurcation studies, not only in finitedimensional systems but also in delay equations [29], reaction-diffusion equations [18,20], boundary value problems [17,52], and so on. In particular it forms together with group representation theory and singularity theory the main cornerstone for equivariant bifurcation theory, the study of bifurcation problems in the presence of symmetry (see e. g. the monographs by Sattinger [54], Vanderbauwhede [61], Golubitsky, Stewart and Schaeffer [23,24], and Chossat and Lauterbach [13], and the more recent theory of bifurcations in cell networks initiated by Golubitsky, Pivato and Stewart [25]). Later the method was extended and adapted to study the bifurcation of periodic orbits in Hamiltonian systems [63], reversible systems [41,42] and mappings [15,62], and there have been several numerical implementations [3,6,38]. Lin’s method to study bifurcations near homoclinic and heteroclinic connections, based on the work of Lin [47] and developed by Sandstede [53] and Knobloch [40], is maybe from a technical point of view not a pure example of a Lyapunov–Schmidt reduction, but it certainly has all the flavors from it. In a different direction the method was used in combination with topological methods, in particular degree theory, to prove all kind of existence and multiplicity results and to obtain more global bifurcation results; a few references where one can read more about this are [22,37] and [39]. A recent account of the use of the Lyapunov–Schmidt method by mainly Russian authors is given in [57]. In recent years the Lyapunov–Schmidt method has been used to study (among many other topics) the existence of transition layers in singularly perturbed reaction-diffusion equations [31,32,58], and, in combination with a Newton scheme to solve small divisor problems, the existence of periodic and quasi-periodic solutions of nonlinear Schrödinger equations in one or more space dimensions [7,16].

Lyapunov–Schmidt Method for Dynamical Systems

The subsequent sections give a more detailed account of the method as it is used in the local bifurcation theory for equilibria and periodic orbits of finite-dimensional systems. Lyapunov–Schmidt Method for Equilibria The easiest example to illustrate the Lyapunov–Schmidt method is to consider the equilibria of a finite-dimensional parameter-dependent system of the form x˙ D f (x; ) ; Rn

(14) Rn

Rk

Rn

with x 2 and f :  ! a smooth mapping. These equilibria are given by the solutions of the equation f (x; ) D 0 :

(15)

Let (x0 ; 0 ) be any solution of this equation; by a translation one can without loss of generality assume that (x0 ; 0 ) D (0; 0), and hence f (0; 0) D 0. The problem is to describe all solutions of (15) which are close to the given solution (0; 0). Let A0 be the Jacobian (n  n)-matrix D x f (0; 0). If A0 is invertible, that is, if the nullspace N(A0 ) is trivial, then by the implicit function theorem the Eq. (15) has for each small parameter value  a unique small solution x D x  () 2 Rn . This means that the equilibrium x D 0 for  D 0 can be continued for all nearby parameter values . Therefore one is mainly interested in the case where A0 is not invertible and N(A0 ) nontrivial, and that is when the Lyapunov–Schmidt method can be used to reduce the Eq. (15), as follows. Let P be the orthogonal projection in Rn onto N(A0 ). This projection allows to split a vector x 2 Rn as x D Px C (I  P)x D u C v, with u D Px 2 N(A0 ) and v D (I  P)x 2 V D N(A0 )? (the orthogonal complement of N(A0 )). It is known from elementary algebra that R(A0 ) D N(AT0 )? , where AT0 is the transpose of A0 , and that dim N(A0 ) D dim N(AT0 ). Hence, if Q is the orthogonal projection of Rn onto W D N(AT0 ), then (I  Q)(Rn ) D R(A0 ) and Rn D W ˚ R(A0 ). Bringing the splitting x D u C v into (15) and using Q to split (15) into Q f (x; ) D 0 and (I  Q) f (x; ) results in the following system of two equations, equivalent to (15):

and

f˜(u; v; ) D Q f (u C v; ) D 0 fˆ(u; v; ) D (I  Q) f (u C v; ) D 0 :

(16)

Clearly f˜(0; 0; 0) D 0 and fˆ(0; 0; 0) D 0; also, Dv fˆ(0; 0; 0) is equal to the restriction of (I  Q)A0 to V, which is an invertible linear operator from V onto R(A0 ). From the implicit function theorem one can then conclude that

the auxiliary equation fˆ(u; v; ) D 0 has, for each sufficiently small (u; ) 2 U  R k , a unique small solution v D v  (u; ) 2 V, with v  : U  R k ! V a smooth mapping such that v  (0; 0) D 0. Moreover, differentiating the identity fˆ(u; v  (u; ); ) D 0 and using the fact that A0 u D 0 for all u 2 U gives (I  Q)A0 Du v  (0; 0) D 0, and hence Du v  (0; 0) D 0. Substitution of the solution v D v  (u; ) in the first equation of (16) gives the bifurcation equation g(u; ) D f˜(u; v  (u; ); ) D Q f (u C v  (u; ); ) D 0 :

(17)

The bifurcation mapping g : U  R k ! W is smooth, and direct calculation shows that g(0; 0) D 0 and Du g(0; 0) D 0 (just use the fact that QA0 x D 0 for all x 2 Rn ). In order to come to conclusions about the equilibria of (14) one has now to study the bifurcation Eq. (17). Two remarks about this equation are in order here. First, since the solution v D v  (u; ) is only implicitly defined, the mapping v  (u; ) and hence also g(u; ) are usually not explicitly known; however, it is in principle possible to obtain the Taylor expansions of these mappings around the point (u; ) D (0; 0). Second, since Du g(0; 0) D 0 it is not possible to use the implicit function theorem to solve (17) for u (or even part of u) as a function of the parameter . Lyapunov–Schmidt Reduction at a Simple Zero Eigenvalue In the case of just one single parameter (i. e. k D 1 and  2 R) one can argue as follows. If at some parameter value one has an equilibrium at which the linearization is invertible, then, as explained before, this equilibrium can be continued for nearby parameter values. This leads to one-parameter branches of equilibria of the form f(x  (); ) j  2 Ig, with I R some interval. This continuation stops at parameter values  D 0 where the linearization A D D x f (x  (); ) becomes singular, i. e. when zero is an eigenvalue of A0 . For one-parameter families of matrices, such as fA j  2 Ig, this typically happens in such a way that zero will be a simple eigenvalue of A0 , which means that dim N(A0 ) D 1 and Rn D N(A0 ) ˚ R(A0 ). Shifting (x  (0 ); 0 ) to the origin and using an appropriate linear transformation this motivates the following hypothesis: f (0; 0) D 0

and

A0 x D (0; Aˆ 0 v);

8x D (u; v) 2 R  Rn1 D Rn ;

(18)

where (as before) A0 D D x f (0; 0) and where Aˆ 0 is an invertible (n  1)  (n  1)-matrix. With this set-up the

941

942

Lyapunov–Schmidt Method for Dynamical Systems

Lyapunov–Schmidt reduction is straightforward: both P and Q are the projections in Rn onto the first component, the decomposition x D (u; v) used in (18) is precisely the one needed to apply the reduction, and the resulting bifurcation equation is a single scalar equation g(u; ) D f1 (u; v  (u; ); ) D 0 :

(19)

As in the general case g(0; 0) D 0 and @g/@u(0; 0) D 0. Depending on some further hypotheses this bifurcation equation leads to different bifurcation scenario’s which are discussed in the next subsections. It should be noted that under appropriate conditions the bifurcation function g(u; ) also contains some stability information for the equilibria of (14) corresponding to solutions (u; ) of (19) (see e. g. [23]): such equilibrium is stable if @g/@u(u; ) < 0, and unstable if @g/@u(u; ) > 0. Saddle-Node Bifurcation

can write the bifurcation function g(u; ) in the form g(u; ) D u g˜(u; ) ; with g˜(u; ) D g˜(u; ) and g˜(0; 0) D 0 ;

hence the non-trivial solutions of (19) (these are solutions with u ¤ 0) are obtained by solving the equation g˜(u; ) D 0 :

g˜(u; ) D u2 C ı C h.o.t. ;

D

1 @3 f1 (0; 0; 0) ; 6 @u3

1 @2 f1 (0; 0; 0) ; 2 @u2

ˇD

@ f1 (0; 0; 0) : @

(20)

Assuming that ˇ ¤ 0 one can solve (19) for  as a function of u to obtain  D  (u) D ˛ˇ 1 u2 C h.o.t.

ıD

@2 f1 (0; 0; 0) : @u@

(24)

Under the condition ı ¤ 0 it is possible to solve (23) for  as a function of u, giving

Just keeping the lowest order terms in u and  the bifurcation function g(u; ) can be written as

˛D

(23)

Further calculations show that

 D  (u) ;

g(u; ) D ˛u2 C ˇ C h.o.t. ;

(22)

with  (0) D 0;  (u) D  (u) and  (u) D  ı 1 u2 C h.o.t.

(25)

When also ¤ 0 this results in the following bifurcation scenario when ı > 0 and < 0: the zero equilibrium is stable for  < 0 and becomes unstable for  > 0; at  D 0 two branches of symmetrically related non-trivial equilibria bifurcate from the trivial branch of equilibria; these nontrivial equilibria exist for  > 0 and are stable. For other signs of and ı similar scenario’s hold. This type of bifurcation is known as a pitchfork bifurcation.

(21)

If also ˛ ¤ 0 then one has, in case ˛ˇ > 0, for  < 0 two equilibria, one stable and one unstable, which merge for  D 0 and then disappear for  > 0; in case ˛ˇ < 0 one has the opposite scenario. This is called a saddle-node or limit point bifurcation. Pitchfork Bifurcation Suppose that the vectorfield f (x; ) appearing in (14) has some symmetry properties, for example f (x; ) D  f (x; ). Then clearly one has for each parameter value  2 R the trivial equilibrium x  () D 0; setting (as before) A D D x f (0; ) one will typically find isolated parameter values where zero is a simple eigenvalue of A . Assuming that  D 0 is such a critical parameter value, and working through the Lyapunov–Schmidt reduction, it is not hard to see that v  (u; ) D v  (u; ), g(u; ) D g(u; ) and g(0; ) D 0. Then clearly both coefficients ˛ and ˇ appearing in (20) will be zero, and the conditions for a saddle-node bifurcation are not satisfied. Instead, one

Transcritical Bifurcation The arguments used to obtain the pitchfork bifurcation remain partly valid in the following situation which is not necessarily related to symmetry. Suppose that the system (14) has a given equilibrium x  () for each  2 R; such equilibrium may be a consequence of symmetry properties (see the case of a pitchfork bifurcation) or of some other property of the system. For example, in population models one always has the zero equilibrium; and in oscillation systems there is usually some basic equilibrium which persists for all parameter values. By a parameter dependent translation one can always arrange that x  () D 0 and hence f (0; ) D 0 for all . Assuming as before that zero is a simple eigenvalue of A0 D D x f (0; 0) and applying the Lyapunov–Schmidt method one finds that v  (0; ) D 0;

g(0; ) D 0 ; 8 2 R ) g(u; ) D u g˜(u; ) ;

(26)

with g˜(u; ) a smooth function such that g˜(0; 0) D 0.

Lyapunov–Schmidt Method for Dynamical Systems

So again non-trivial equilibria are obtained by solving the Eq. (23), in which now the function g˜(u; ) can be written as

1 @2 f1 (0; 0; 0) ; 2 @u2

The discrete analogue of the continuous system (14) has the form x jC1 D f (x j ; ) ;

g˜(u; ) D u C ı C h.o.t. ; D

Lyapunov–Schmidt Method in Discrete Systems

ıD

@2 f1 (0; 0; 0) : @u@

(27)

Assuming that ı ¤ 0 one can solve (23) for  as a function of u, resulting in

j D 0; 1; 2; : : : ;

again with x j 2 Rn ( j 2 N) and f : Rn  R k ! Rn smooth. For a given value  of the parameter the equilibria of (29) are given by the fixed points of f D f (; ), i. e. by the solutions of the equation f (x; ) D x :



 D  (u) D ı

1

u C h.o.t.

(28)

When also  ¤ 0 this represents a transcritical bifurcation: for each  ¤ 0 there exists next to the trivial equilibrium at zero also a non-trivial equilibrium which approaches zero as  tends to zero. For the same parameter value  ¤ 0 the trivial and the nontrivial equilibria have opposite stability properties, and there is an exchange of stability at  D 0. For example, if ı > 0 then the trivial equilibrium is stable for  < 0 and unstable for  > 0, while the non-trivial equilibrium is unstable for  < 0 and stable for  > 0. A remark about the condition ı ¤ 0 used to obtain the pitchfork and the transcritical bifurcation is in order here. In both cases it was assumed that zero is a simple eigenvalue of A0 D AD0 , where A D D x f (0; ). This simple eigenvalue can be continued for small values of , i. e. the matrix A has, for sufficiently small values of , a simple real eigenvalue () which depends smoothly on  and is such that (0) D 0. It is then not very hard to prove that the condition ı ¤ 0 is equivalent to the transversality condition 0 (0) ¤ 0 which states that the simple eigenvalue () crosses zero with non-zero speed when  crosses zero. Similar transversality conditions are needed to obtain most of the elementary bifurcations which appear in continuous systems such as (14) or in discrete systems such as considered in the next section. A further remark has to do with the condition that zero should be a simple eigenvalue of A0 D D x f (0; 0). In certain cases this condition can be satisfied by restricting a larger system to an appropriate invariant subspace. This happens in particular when one studies bifurcations in equivariant systems; in many such systems the simple zero eigenvalue can be obtained by restricting the full system to invariant subsystems consisting of solutions which are invariant under certain isotropy subgroups. Usually the symmetry also forces the origin to be an equilibrium. Pitchfork or transcritical bifurcations observed for such invariant subsystems form then special cases of the Equivariant Branching Lemma [12,61]; see [61] and [24] for more details.

(29)

(30)

The continuation and bifurcation of such fixed points under a change of the parameter can be studied by the same techniques as the ones used in Sect. “Lyapunov–Schmidt Method for Equilibria”: it is sufficient to replace f (x; ) by F(x; ) D f (x; )  x. At a given solution (x0 ; 0 ) 2 Rn  R k of (29) the linearization L0 D D x F(x0 ; 0 ) of F0 D F(; 0 ) is given by L0 D A0  I n , where A0 D D x f (x0 ; 0 ) and I n is the identity matrix on Rn . Therefore, if 1 is not an eigenvalue of A0 then the fixed point x0 can be continued for all  sufficiently close to 0 , and if 1 is a simple eigenvalue of A0 (and k D 1) then, depending on some further conditions, one may have at  D 0 a saddlenode, a pitchfork bifurcation or a transcritical bifurcation of fixed points. Discrete systems such as (29) arise for example frequently in mathematical biology and mathematical economics. A special situation arises when the mapping f in (29) is a Poincaré map constructed near a given periodic orbit  0 of a continuous parameter-dependent autonomous (n C 1)-dimensional system (see e. g. [45]); in that case a fixed point of f corresponds to a periodic orbit of the original continuous system, usually with the origin corresponding to  0 . The periodic orbit  0 has always 1 as a Floquet multiplier; if this multiplier is simple, then 1 is not an eigenvalue of the linearization at the origin of the Poincaré map, and the periodic orbit  0 can be continued for nearby parameter values. If k D 1 (a single parameter) and the multiplier 1 of  0 has multiplicity two, then 1 will be a simple eigenvalue of the linearization of the Poincaré map, and one will typically obtain a saddlenode of fixed points of the Poincaré map, corresponding to a saddle-node of periodic orbits in the original continuous system. In particular cases such as the presence of symmetry one may also see a transcritical or pitchfork bifurcation of periodic orbits. Periodic Points in Discrete Systems Next to the bifurcation of fixed points for (29) one can also study the bifurcation of periodic points from fixed points.

943

944

Lyapunov–Schmidt Method for Dynamical Systems

For any  2 R k and any integer q  1 a q-periodic point of f D f (; ) is a point x 2 Rn such that the orbit fx j j j 2 Ng of (29) starting at x0 D x has the property that x jCq D x j for all j 2 N. In case f is a Poincaré map associated to a periodic solution with minimal period T0 > 0 of a continuous system then such q-periodic point corresponds to a periodic orbit of the continuous system with minimal period near qT 0 . Such periodic solution is called a subharmonic solution (when q  2); the bifurcation of periodic points from fixed points for discrete systems such as (29) is therefore strongly related to subharmonic bifurcation in continuous systems. The periodicity condition x jCq D x j (for all j 2 N) is equivalent to the condition that x q D x0 D x, or to q

f (x) D x;

q

f D f ı f ı    ı f

q

(32) This orbit space is finite-dimensional, with dimension qn. Any map g : Rn ! Rn can be lifted to a map gˆ : Oq ! Oq by setting 8z D (x j ) j2Z 2 Oq :

The shift operator is the linear mapping  : Oq ! Oq defined by (  z) j D x jC1 ;

8j 2 Z ;

fˆ (z) D   z :

(33)

The advantage of this equation over the equivalent Eq. (31) is that (33) is Z q -equivariant: both sides of the equation commute with  , and if (z; ) 2 Oq  R k is a solution, then so is ( j  z; ) for any j 2 Z. This equivariance plays an important role in the bifurcation analysis for q-periodic points of (29) since it puts strong restrictions on the possible form of the bifurcation equation. The Z q -equivariance of (33) also fits nicely together with the Lyapunov– Schmidt reduction, as will be shown next.

(q times) : (31)

Hence a q-periodic point of f is a fixed point of f (the q-th iterate of f ), and therefore one could in principle study the bifurcation of q-periodic points of f by q studying the bifurcation of fixed points of f . Observe that q a fixed point of f is automatically also a fixed point of f , for any q  1. However, there is a better approach which takes explicitly into account an important property of (31), namely that if x is a solution then so are all the other points of the j orbit fx j D f (x) j j 2 Ng generated by x. The basic idea is the following: instead of considering all orbits of (29) and then finding all those which are q-periodic, one considers all q-periodic sequences with elements in Rn and then tries to determine which ones are actually orbits of the system (29). To work this out one has to introduce (for the chosen value of q) the space ˚ Oq D z D (x j ) j2Z j x j 2 Rn ; x jCq D x j ; 8 j 2 Z :

gˆ(z) D (g(x j )) j2Z ;

A q-periodic sequence z D (x j ) j2Z 2 Oq forms an orbit under the iteration of f if and only if f (x j ) D x jC1 for all j 2 Z, that is, if and only if

8z D (x j ) j2Z 2 Oq :

Observe that  q D IOq , and hence  generates a Z q -action on Oq (Z q is the cyclic group of order q, i. e. Z q D Z/qZ). Also gˆ ı  D  ı gˆ for any g : Rn ! Rn . And finally, identifying Oq with (Rn )q and using the standard scalar product it is easily seen that h  z1 ;   z2 i D hz1 ; z2 i for all z1 ; z2 2 Oq , i. e.  is orthogonal and  T D  1 .

Lyapunov–Schmidt Reduction for Periodic Points Assume that (30) has a solution (x0 ; 0 ) 2 Rn  R k ; as before one can without loss of generality translate this solution to the origin and assume that f (0; 0) D 0. Consider then, for a given integer q  1, the following problem: (Pq ) Determine, for all small values of the parameter , all small q-periodic points of f . The bifurcation equation for this problem can be obtained by a Lyapunov–Schmidt reduction near the solution (z; ) D (0; 0) of (33), as follows. Linearization of (33) at (0; 0) gives the equation Aˆ 0  z D   z ;

A0 D D x f (0; 0) ;

(34)

and the corresponding linear operator L0 D Aˆ 0   : Oq ! Oq . The nullspace N(L0 ) consists of the q-periodic orbits of the linear discrete system x jC1 D A0 x j and can be described as follows. Let  q  U D N A0  I n ;

ˇ S D A0 ˇU : U ! U

and  : U ! Oq ; u 7! (u) D (S j u) j2Z :

(35)

Then N(L0 ) D (U), and  is an isomorphism between U and N(L0 ). Observe that the subspace U of Rn is the sum of the eigenspaces of A0 corresponding to all eigenvalues  2 C of A0 for which q D 1, i. e. all eigenvalues  which are q-th roots of unity. A further observation is that the linear operator S generates a Z q -action on U (since S q D I U by definition of U and S), and that (Su) D   (u) ;

8u 2 U :

(36)

The description of R(L0 ) D N(L0T )? and the necessary projections can be somewhat simplified by assuming that all eigenvalues of A0 which are q-th roots of unity are semi-

Lyapunov–Schmidt Method for Dynamical Systems

q

simple, an assumption which implies that Rn D N(A0  q I n ) ˚ R(A0  I n ) D U ˚ Y; writing x 2 Rn as x D (u; y), with u 2 U and y 2 Y, and constructing an appropriate basis of Rn one can then put A0 in the following (diagonal) form: A0 x D A0 (u; y) D (Su; A˜0 y) ; 8x D (u; y) 2 U  Y D Rn ;

(37)

where S is orthogonal, i. e. S T S D I U , and where A˜ 0 : Y ! q Y is such that A˜ 0  I Y is invertible. Under these assumptions (see [62] for the general case) it is easily seen that N(L0T ) D N(b A0T   T ) is equal to N(L0 ) (use the fact that T 1  D  and S T D S 1 ). Therefore R(L0 ) is the orthogonal complement in Oq D (Rn )q of N(L0 ) D (U), and every z 2 Oq has a unique decomposition of the form z D (u)C v ;

with u 2 U and v 2 V D R(L0 ): (38)

In particular, going back to (33) and replacing x by (u) Cv, one can write fˆ ((u) C v) D (g(u; v; )) C h(u; v; ) ; with g : U  V  R k ! U and h : U  V  R k ! V smooth mappings such that g(0; 0; 0) D 0, h(0; 0; 0) D 0, Du g(0; 0; 0) D ˇS, Dv g(0; 0; 0) D 0, Du h(0; 0; 0) D 0, Dv h(0; 0; 0) D Aˆ 0 ˇV , and g(Su;   v; ) D Sg(u; v; ) ; h(Su;   v; ) D   h(u; v; ) :

(39)

This last property is a consequence of (36), the Zq -equivariance of fˆ , and the fact that  leaves the decomposition Oq D (U) ˚ V invariant since  commutes with L0 . It follows from the foregoing that the Eq. (33) can be split in the two equations g(u; v; ) D Su

h(u; v; ) D   v : (40) ˇ Since h(0; 0; 0) D 0, Dv h(0; 0; 0) D Aˆ 0 ˇV and L0 D Aˆ 0  is invertible on V D R(L0 ), the second equation in (40) can, locally near (u; v; ) D (0; 0; 0), be solved by the implicit function theorem for v D v  (u; ); the mapping v  : U R k is smooth, with v  (0; 0) D 0 and Du v  (0; 0) D 0. Also, (39) and the uniqueness of solutions given by the implicit function theorem imply that and

v  (Su; ) D   v  (u; ) :

g  (Su; ) D Sg  (u; ) ;

8(u; ) 2 U  R k :

(43)

This last property ensures that the bifurcation equation (42) is Z q -equivariant, i. e. commutes with the Z q -action generated by S on U, and if (u; ) 2 U  R k is a so˜ ) for any u˜ belonging to the Zq -orbit lution then so is (u; fS j u j j 2 Zg generated by u. Period-Doubling Bifurcation Suppose that f (0; 0) D 0 and that A0 D D x f (0; 0) has 1 as a simple eigenvalue, while +1 is not an eigenvalue. Then, taking q D 2 in the foregoing, dim U D 1, Su D u for all u 2 U, and the bifurcation equation (42) is a scalar equation: g  (u; ) D u ;

@g  (0; 0) D 1 @u and g  (u; ) D g  (u; ) : with

(44)

Due to the symmetry the discussion of this bifurcation equations runs completely analogous to the one in Subsect. “Pitchfork Bifurcation” and leads, for k D 1 and under appropriate genericity conditions, to a pitchfork bifurcation. However, the interpretation of the bifurcation diagram is different in this case: the trivial solution u D 0 of (42) corresponds to a fixed point x0 () of f which changes stability as  crosses zero, while the symmetric ˆ pairs ˙u() of nontrivial solutions which exist either for  < 0 or for  > 0 correspond to the two points xˆ () and f (xˆ ()) of a period-two orbit of f . Such bifurcation is known as a period-doubling bifurcation or a flip bifurcation. In case f represents a Poincaré map near a given (nontrivial) periodic orbit with minimal period T 0 of a continuous system such period-doubling bifurcation means the following: the given periodic orbit persists for nearby parameter values, with minimal period T near T 0 , but when the parameter crosses the critical value there bifurcates a branch of periodic solutions with minimal period near 2T0 ; so also in the continuous system one sees a period-doubling.

(41)

Substituting the solution v D v  (u; ) into the first equation of (40) gives the bifurcation equation g  (u; ) D Su ; with g  (u; ) D g(u; v  (u; ); ) :

This is an equation on the subspace U of Rn ; the mapping g  : U  R k ! U is smooth, with g  (0; 0) D 0; Du g  (0; 0) D S and

(42)

Bifurcation of q-Periodic Points for q  3 Next let q  3 and assume that f (0; 0) D 0 and that A0 has a pair of simple eigenvalues exp(˙i 0 ) ;

with 0 D

2 p ; q

gcd(p; q) D 1 : (45)

945

946

Lyapunov–Schmidt Method for Dynamical Systems

Assume also that A0 has no other eigenvalues  with q D 1; in particular +1 and 1 are no eigenvalues. Then dim U D 2 and the linear operator S in (37) has the matrix form   cos 0  sin 0 SD : sin 0 cos 0 Identifying u D (u1 ; u2 ) 2 U D R2 with z D u1 Ciz2 2 C the expression (37) for A0 takes the more explicit form   A0 x D A0 (z; y) D ei 0 z; A˜0 y ; 8x D (z; y) 2 C  Y D Rn :

(46)

The Lyapunov–Schmidt reduction for q-periodic points as explained in Subsect. “Lyapunov–Schmidt Reduction for Periodic Points” leads to a complex bifurcation equation g  (z; ) D ei 0 z ;

with g  (z; 0) D ei 0 z C O(jzj2 )   and g  ei 0 z;  D ei 0 g  (z; ) : (47)

It can be shown (see e. g. [24] or [62]) that the Z q -equivariance of g  implies that   g  (z; ) D e i 0 C 1 (jzj2 ; ) z C 2 (jzj2 ; )¯z q1 C O(jzj qC1 );

(48)

with  1 and  2 complex valued functions and 1 (0; 0) D 0. The bifurcation equation (47) has for each  2 R k the trivial solution z D 0, corresponding to the continuation of the fixed point x D 0 at  D 0 (remember that it was assumed that +1 is not an eigenvalue of A0 ). To obtain nontrivial solutions one can multiply (47) by z¯, set z D e i and divide by 2 ; the resulting equation has the form g˜( ; ; ) D 1 ( 2 ; ) C 2 ( 2 ; ) q2 ei q C O( q ) D 0 ;

(49)

with g˜( ; C 0 ; ) D g˜( ; ; ) and g˜( ; C ; ) D g˜( ; ; ). The standard way to handle (49) is to assume k D 2 (i. e.  D (1 ; 2 ) 2 R2 ) together with the transversality condition @(<1 ; =1 ) (0; 0) ¤ 0 : @(1 ; 2 )

(50)

Then (49) can be solved for  D q ( ; ), with q (0; ; ) D 0, q ( ; C 0 ; ) D q ( ; ; ) and q ( ; C ; ) D q ( ; ; ). Both the transversality condition and the solution set f( ; ; q ( ; )) j ( ; ) 2 R2 g require some further explanation. Since exp(˙ 0 ) are simple eigenvalues of A0 it follows that these eigenvalues can be smoothly continued for nearby parameter values, i. e. A has for small  a pair

¯  ()g near exp(˙ 0 ). The of simple eigenvalues f ();  transversality condition (50) means that the mapping  7!  () is a local diffeomorphism near  D 0. Under this condition the solution  D q ( ; ) of (49) implies that for each  in the range ˚

Aq D q ( ; ) j ( ; ) 2 R2



the mapping f has at least one q-periodic orbit near x D 0. For q  5 the set Aq typically forms a horn-like domain in the -plane, with tip at the origin; higher values of q correspond to sharper tips. Indeed, it follows from (48) that q ( ; ) D 2 ˛( 2 ) C q2 e q ˇ( 2 ) C O( q ) ; for some complex valued functions ˛( 2 ) and ˇ( 2 ); if ˛(0) ¤ 0 this shows that the “thickness” of Aq at a distance d from the origin is of the order d (q2)/2 (for q D 3 and q D 4 the situation is different, see e. g. [62] or [36]). The horn-like regions Aq are known as Arnol’d tongues. For parameter values  in the interior of Aq and close to  D 0 the mapping f has two q-periodic orbits which annihilate each other in a saddle-node bifurcation when the parameter crosses the boundary of Aq . Moreover, the hypotheses imply that f undergoes a Neimark–Sacker bifurcation: when  crosses the curve defined by j ()j D 1 the stability of the fixed point changes and an invariant curve bifurcates from the fixed point. The Arnol’d tongue Aq is contained in the region where an invariant curve exists, and the two periodic orbits are contained in this invariant curve; on the curve, one periodic orbit is stable, the other unstable. When f represents a Poincaré map the bifurcation picture as described in the foregoing can be lifted to the original continuous system, leading to the bifurcation of invariant tori and subharmonics. For more details see e. g. [36] or [45]. For a discussion of the case where the mappings f are reversible see [15]. Lyapunov–Schmidt Method for Periodic Solutions The ideas described in Sect. “Lyapunov–Schmidt Method in Discrete Systems and in particular in Subsect. “Lyapunov–Schmidt Reduction for Periodic Points” can be adapted for the treatment of the bifurcation of periodic orbits from equilibria in continuous systems (14); however, there are a few complications which arise for this continuous case. First, the orbit space consists of continuous periodic functions defined on the real line, hence this orbit space is infinite-dimensional; this necessitates a more careful handling of splittings and projections. Second, when looking for periodic solutions of (14) the (minimal) period

Lyapunov–Schmidt Method for Dynamical Systems

is one of the unknowns of the problem. As will be seen in the treatment which follows the finite symmetry group Z q which plays an important role in the bifurcation analysis of periodic orbits in discrete systems will be replaced by the continuous circle group S1 in the case of continuous systems. In order to get a more precise description of the problem, assume that f (0; 0) D 0 and consider the linearized equation x˙ D A0 x ;

(51)

(with A0 D D x f (0; 0) as before). The solutions have the form x(t) D e A 0 t x0 with x0 2 Rn ; such solution is T 0 -periodic (for some T0 > 0) if x0 2 N(eA 0 T0  I n ). In order to exclude a possible bifurcation of equilibria one should require that zero is not an eigenvalue of A0 , which entails that without further loss of generality one can also assume that f (0; ) D 0 for all . By a time rescale one can put T0 D 2. The nullspace N(e2 A 0  I n ) is then given by the direct sum of the (real) eigenspaces corresponding to all pairs of eigenvalues of the form ˙ik (k D 1; 2; : : :). A further simplification in the formulation of the Lyapunov–Schmidt reduction occurs if one assumes that all such eigenvalues are semisimple, leading to the following hypotheses about (14) which will be assumed further on: (H) f (0; ) D 0 for all , A0 D D x f (0; 0) is invertible, and 1 is a semisimple eigenvalue of e2 A 0 , i. e. Rn D N(e2 A 0  I n ) ˚ R(e2 A 0  I n ) D U ˚ Y, where U D N(e2 A 0  I n ) and Y D R(e2 A 0  I n ). By an appropriate choice of basis one can then more explicitly assume that A0 x D A0 (u; y) D (Su; A˜0 y) ; 8x D (u; y) 2 U  Y D Rn ;

(52)

with S : U ! U and A˜0 : Y ! Y linear and invertible, S anti-symmetric (S T D S), e2 S D I U and ˜ N(e2 A 0  I Y ) D f0g. The space of 2-periodic solutions of (51) is then given by fe S t u j u 2 Ug. The question one wants to answer is then which of these periodic solutions survive when we turn on the nonlinearities and change parameters. More precisely, the problem can be formulated as follows: (P) Determine, for all (T; ) 2 R  R k near (2; 0), all small T-periodic solutions of (14). Lyapunov–Schmidt Reduction for the Problem (P) Analogically to the treatment for discrete systems in Sect. “Lyapunov–Schmidt Method in Discrete Systems” the first step should be to reformulate (P) in an appropriate orbit space; to do so one has to fix the period, which is done as

follows. Let xˆ (t) be a T-periodic solution of (14), define  2 R by (1 C )T D 2, and set x˜ (t) D xˆ ((1 C )1 t). ˜ is a 2-periodic solution of the equation Then x(t) (1 C )x˙ D f (x; ) :

(53)

Conversely, if x˜ (t) is a 2-periodic solution of (53) then xˆ (t) D x˜ ((1 C )t) is a T-periodic solution of (14), with T D 2(1 C )1 . Therefore the problem (P) is equivalent to (P* ) Determine, for all small (; ) 2 R  R k , all small 2-periodic solutions of (53). 0 of all conThe orbit space for this problem is the space C2 n tinuous 2-periodic mappings x : R ! R ; this is a Banach space when one uses the norm kxk02 D supfkx(t)k j 0 one can define a natural S1 -action (where t 2 Rg. On C2 1 S D R/2R is the circle group) by the phase shift operator 0 ˙ : S 1  C2 ! C20 ; (; x) 7! ˙ (x) ;

with ˙ (x)(t) D x(t C ) :

(54)

S1 -action

The infinitesimal generator of this is defined 1 of continuously differentiable on the dense subspace C2 2-periodic mappings x : R ! Rn and is given by ˇ d ˇˇ 1 0 ! C2 ; x 7! D t x D ˙ (x) ; D t : C2 d ˇD0 1 (D t x)(t) D x˙ (t); 8(x; t) 2 C2 R:

(55)

1 is The subspace C2 1 the norm kxk2 D

itself a Banach space when one uses 1 kxk02 C kD t xk02 ; the space C2 0 is continuously imbedded in C2 . Also, any continuous 0 mapping g : Rn ! Rn can be lifted to a mapping on C2 by defining 0 0 gˆ : C2 ! C2 ; x 7! gˆ(x) ;

with gˆ(x)(t) D g(x(t)) ;

0 R: 8(x; t) 2 C2

Moreover, such a lifting commutes with the shift operator: gˆ ı ˙ D ˙ ı gˆ for all  2 S 1 . Using the foregoing definitions the problem (P ) 1  translates into finding all small solutions (x; ; ) 2 C2 k R  R of the equation b f  (x) D (1 C )D t x :

(56)

(Observe the analogy and the differences with (33)). This equation is S1 -equivariant: both sides commute with ˙ for all  2 S 1 ; therefore, if (x; ; ) is a solution, then so is (˙ (x); ; ) for all  2 S 1 . 1  R  R k ! C 0 by F(x; ; ) D Next define F : C2 2 b f   (1 C )D t ; then F(0; ; ) D 0, and the linearization 1 ! C 0 is given by L D b L0 D D x F(0; 0; 0) : C2 A0  D t . 0 2 1 is the space of all 2-periodic The nullspace N(L0 ) C2

947

948

Lyapunov–Schmidt Method for Dynamical Systems

solutions of (51) which was described before; hence there exists an isomorphism  : U ! N(L0 ) given by (u)(t) D eS t u ;

8u 2 U; 8t 2 R :

(57) ˇ It follows from the definition of U and S D A0 ˇU that S is the infinitesimal generator of an S1 -action on U given by (; u) 2 S 1  U 7! e S u 2 U : Also, (e S u) D ˙ (u) and (Su) D D t (u) for all (; u) 2 S 1  U. In order to describe R(L0 ) one has to find those z 2 0 for which the non-homogeneous linear 2-periodic C2 differential equation x˙ D A0 x C z(t)

(58)

has a 2-periodic solution. The Fredholm alternative for this problem (see e. g. [28] or [61]) states that this will be the case if and only if Z 2 1 hz(t); x(t)idt D 0 ; 8x 2 C2 :  x˙ D AT0 x : 0

0 Stated differently: if we introduce on C2 a scalar product h; iC 0 by 2

Z hx; xˆ iC 0 D

2

ˆ hx(t); x(t)idt ;

2

0

0 8x ; xˆ 2 C2 ;

1 ! C0 and if we define the adjoint operator L0 : C2 2  T  ? b by L0 D A0 C D t , then R(L0 ) D N(L0 ) . It follows easily from (52) that N(L0 ) D N(L0 ), and therefore 0 D N(L0 ) ˚ R(L0 ). Moreover, R(L0 ) D N(L0 )? and C2

N(L0 ) is finite-dimensional and R(L0 ) is a closed sub0 ; these properties imply that L : C 1 ! C 0 space of C2 0 2 2 is a Fredholm operator of index zero (see the next section for the definition). 0 One can associate the orthogonal splitting C2 D N(L0 ) ˚ R(L0 ) D (U) ˚ V (with V D R(L0 )) to a con0 ! C 0 such that R(P) D (U) tinuous projection P : C2 2 ˜ with P˜ : C 0 ! U and N(P) D V; explicitly P D  ı P, 2 given by 1 ˜ P(x) D 2

Z

2

eSs u(s)ds ;

0 0 : 8x D x() D (u(); y()) 2 C2

a similar way b f  ((u) C v) D (g(u; v; )) C h(u; v; ), with g(u; v; ) D P˜ fˆ ((u) C v) 2 U and h(u; v; ) D (IP) fˆ ((u)Cv) 2 V; the mappings g : U V R k ! U and h : U  V  R k ! V are smooth, with g(0; 0; ) D 0, h(0; 0; ) D 0, and satisfy the equivariance relations   g e S u; ˙ v;  D eS g(u; v; ) and   h e S u; ˙ v;  D ˙ (h(u; v; )) ; 8 2 S 1 : Using the foregoing decompositions the Eq. (56) splits into two equations g(u; v; ) D (1 C )Su and

(60)

h(u; v; ) D (1 C )D t v ;

1 )RR k . to be solved for small (u; v; ; ) 2 U (V \C2 Since h(0; 0; ) D 0 and the restriction of L0 to 1 is an isomorphism of V \ C 1 onto V the secV \ C2 2 ond equation in (60) can be solved for v D v  (u; ; ), 1 a smooth mapping, with v  : U  R  R k ! V \ C2   v (0; ; ) D 0, Du v (0; 0; 0) D 0 and v  (e S u; ; ) D ˙ (v  (u; ; )) for all  2 S 1 . Bringing this solution into the first equation of (60) gives the bifurcation equation

g  (u; ; ) D (1 C )Su ; with g  (u; ; ) D g(u; v  (u; ; ); ) :

(61)

The mapping g  : U  R  R k ! U is smooth, with g  (0; ; ) D 0, Du g  (0; ; 0) D S and g  (e S u; ; ) D eS g  (u; ; ) ;

8 2 S 1 :

(62)

This S1 -equivariance of g  implies that for each solution (u; ; ) 2 U RR k of (62) also (e S u; ; ) is a solution, for each  2 S 1 . Solving (61) near the origin and working back through the reduction allows to solve the problems (P ) and (P). The method outlined in this subsection can be adapted for special types of systems, such as Hamiltonian or reversible systems, and can also be generalized to the case where the linearization A0 has some nilpotencies (nonsemisimple eigenvalues); see e. g. [41,63] and [42] for more details. In these papers there is also established a link between the Lyapunov–Schmidt reduction and normal form theory.

(59)

Observe that P˜ ı ˙ D e S ı P˜ and P ı ˙ D ˙ ı P 0 D (U) ˚ for every  2 S 1 , and that the splitting C2 1 V is invariant under the S -action; as a consequence the 1 infinitesimal generator Dt of the S1 -action maps V \ C2 1 into V. Every x 2 C2 can then be written as x D (u)Cv, 1 ; in ˜ with u D P(x) 2 U and v D (I  P)x 2 V \ C2

Hopf Bifurcation The simplest possible case in which the reduction of Subsect. “Lyapunov–Schmidt Reduction for the Problem (P)” can be applied is when A0 has ˙i as simple eigenvalues, while there are no other eigenvalues of the form ˙ki, with k 2 N (this is called a non-resonance condition). Then

Lyapunov–Schmidt Method for Dynamical Systems

dim U D 2 and S can be brought into the matrix form   0 1 SD : 1 0 Identifying u D (u1 ; u2 ) 2 U D R2 with z D u1 C iu2 2 C the linear operator A0 gets the form (compare with (52)) A0 x D A0 (z; y) D (iz; A˜0 y) ; 8x D (z; y) 2 C  Rn2 D Rn :

(63)

The bifurcation equation is then a single complex equation g  (z; ; ) D (1 C )iz ;

(64)

with g  : C  R  R k ! C such that g  (0; ; ) D 0, g  (z; ; 0) D iz C O(jzj2 ) and g  (ei z; ; ) D ei g  (z; ; ) ;

8 2 S 1 :

(65)

This equivariance property of g  implies (see e. g. [24]) that one can write g  (z; ; ) D g˜(jzj2 ; ; )z ;

(67)

The solutions of this equation come in circles f(ei ; ; ) j  2 S 1 g; periodic solutions of (14) corresponding to different points on such circle are related to each other by phase shift, and therefore each circle of solutions of (67) corresponds to a unique periodic orbit of (14). The standard way to handle (67) is to assume k D 1 (i. e.  2 R) and R

@ g˜ (0; 0; 0) ¤ 0 : @

˛ 0 (0) ¤ 0 ;

(69)

this transversality condition means that as  crosses zero the pair of eigenvalues ˛() ˙ iˇ() of the linearization A at the equilibrium x D 0 crosses the imaginary axis with non-zero transversal speed. Similarly as for equilibria the requirement for Hopf bifurcation to have a pair of simple eigenvalues crossing the imaginary axis can sometimes be achieved by restricting (56) to periodic loops with some particular space-time symmetries. Working out this idea leads to the Equivariant Hopf Bifurcation Theorem. There exists a massive literature developing and applying this theorem; for more details see the basic reference [24]. Lyapunov–Schmidt Method in Infinite Dimensions

with g˜(0; ; 0) D i : (66)

The Eq. (64) has for each (; ) the trivial solution z D 0, corresponding to the equilibrium x D 0 of (14) Using (66) the nontrivial solutions of (64) are given by the solutions of the equation g˜(jzj2 ; ; ) D (1 C )i :

Since the eigenvalues ˙i of A0 were assumed to be simple they can be continued to eigenvalues ˛() ˙ iˇ() of A D D x f (0), for small values of . It is not hard to prove that the condition (68) is equivalent to

(68)

Then (67) can be split in real and imaginary parts and solved by the implicit function theorem for (; ) D (  (jzj2 );  (jzj2 )), with (  (0);  (0)) D (0; 0). For each sufficiently > 0 there is a parameter value  D  ( 2 ) such that the vectorfield f has a small periodic orbit of “amplitude” and with period 2(1 C   ( 2 ))1 . Such bifurcation of periodic orbits from an equilibrium is known as a Hopf bifurcation, or Andronov–Hopf bifurcation, or even Poincaré–Andronov–Hopf bifurcation. The papers [35] and [1] contain some of the original work on this topic; an extensive discussion on the history of Hopf bifurcation together with many more references can be found in Chap. 9 of [14].

The Hopf bifurcation problem as explained in the preceding section leads to an Eq. (56) in the infinite-dimensional 0 orbit space C2 . Also bifurcation problems associated to boundary value problems for partial differential equations typically lead to equations in infinite-dimensional spaces (see e. g. [39] for some examples). The aim of this section is to briefly sketch a basic abstract setting which, under the appropriate conditions, allows to reduce these infinite-dimensional problems to finite-dimensional ones. Consider an equation F(x; ) D 0 ;

(70)

with F : X  R k ! Y a smooth parameter-dependent mapping between two Banach spaces X and Y. Assuming that x D 0 is a solution for  D 0, one can ask the question to what extend this solution persists for nearby values of , i. e. one wants to find all solutions (x; ) of (70) near (0; 0). The first step in solving this local problem consists in considering the linearized equation A0 x D 0 ;

with A0 D D x F(0; 0) ;

(71)

and to determine the nullspace and range of A0 . The following hypothesis about A0 allows to proceed with the Lyapunov–Schmidt reduction: (F) The continuous linear operator A0 : X ! Y is a Fredholm operator: (i) the nullspace N(A0 ) is finite-dimensional; (ii) the range R(A0 ) is closed in Y; (iii) the range R(A0 ) has finite codimension in Y.

949

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Lyapunov–Schmidt Method for Dynamical Systems

The index of such Fredholm operator is given by the number dim N(A0 )  codimR(A0 ); in many applications this index is zero, i. e. codimR(A0 ) D dim N(A0 ). Under the condition (F) there exist continuous linear projections P : X ! X and Q : Y ! Y such that R(P) D N(A0 ) and N(Q) D R(A0 ) ;

(72)

resulting in the topological decompositions X D N(A0 ) ˚ N(P) D U ˚V and Y D R(Q)˚R(A0 ) D W˚R(A0 ), with U D N(A0 ), V D N(P) and W D R(Q). The restriction of A0 to V is an isomorphism from V onto R(A0 ). Replacing x 2 X by x D Px C (I  P)x D u C v in (70), and splitting the equation itself using Q results in the following system equivalent to (70): QF(u C v; ) D 0 ;

(I  Q)F(u C v; ) D 0 :

(73)

The second of these equations is sometimes called the auxiliary equation; the corresponding mapping Faux : U  V  R k ! R(A0 ) defined by Faux (u; v; ) D (I Q)F(u Cv; ) is such that Faux (0; 0; 0) D ˇ0, Du Faux (0; 0; 0) D 0 and Dv Faux (0; 0; 0) D (I  Q)A0 ˇV . Since this last operator is an isomorphism from V onto R(A0 ) one can apply the implicit function theorem to solve the auxiliary equation in the neighborhood of (u; v; ) D (0; 0; 0) for v D v  (u; ). The mapping v  : U R k ! V is smooth, with v  (0; 0) D 0 and Du v  (0; 0) D 0. Substituting v D v  (u; ) into the first equation of (73) we obtain the bifurcation equation 

G(u; ) D QF(u C v (u; ); ) D 0 :

(74)

Rk

! W is smooth, The bifurcation mapping G : U  with G(0; 0) D 0, Du G(0; 0) D 0 and D G(0; 0) D QD F(0; 0). The further analysis of the bifurcation Eq. (74) will of course strongly depend on the particularities of the problem, such as the dimensions of U and W, possible symmetries, etc. A well known and widely applied particular case is the by now classical theorem of Crandall and Rabinowitz [17] on bifurcation from simple eigenvalues. The hypotheses for this result are the following: (CR) F : X  R ! Y is smooth, with F(0; ) D 0 for all  2 R, and: (i) the nullspace of D x F(0; 0) is one-dimensional; (ii) the range of D x F(0; 0) is closed and has codimension one in Y; (iii) D D x F(0; 0)  u0 62 R(D x F(0; 0)), where u0 ¤ 0 belongs to N(D x F(0; 0)). Under these hypotheses dim U D dim W D 1 in the foregoing, and therefore U D f u0 j 2 Rg and W D fw0 j  2 Rg for some non-zero elements u0 2 N(D x F(0; 0)) and w0 62 R(D x F(0; 0)). Also v  (0; ) D 0 and G(0; ) D 0

for all  2 R; writing G( u0 ; ) D g( ; )w0 it follows that also g(0; ) D 0 for all . Hence Z g( ; ) D g˜( ; ) ;

1

with g˜( ; ) D 0

@g (s ; )ds : @

@g Clearly g˜(0; 0) D 0, since g˜(0; 0)w0 D @ (0; 0)w0 D Du G(0; 0)  u0 and Du G(0; 0) D 0. Moreover,

@ g˜ @2 g (0; 0)w0 D (0; 0)w0 D D Du G(0; 0)  u0 @ @@ D QD D x F(0; 0)  u0 ¤ 0 ; @ g˜

by the hypothesis (CR)-(iii), and so @ (0; 0) ¤ 0. Nonzero solutions of the bifurcation equation g( ; ) D 0 are given by the solutions of g˜( ; ) D 0, and using the implicit function theorem this last equation can be solved for  D  ( ), with j j sufficiently small and  (0) D 0. This gives, under the hypotheses (CR), a unique branch f(x  ( );  ( )) Dj 0 < j j < 0 g ; with x  ( ) D u0 C v  ( u0 ;  ( )) ; of nontrivial solutions of (70) bifurcation from the trivial branch f(0; ) j  2 Rg at  D 0. Outlook The Liapunov–Schmidt method is a well established reduction technique which can and has been used on a wide variety of problems. The preceding sections just give a small glimpse of its use in local bifurcation theory; the references given in the introduction will point the reader to a whole plethora of other applications. The basics of the method are rather simple, but depending on the mappings and the spaces one is using a number of technical conditions (sometimes easy, sometimes more difficult) need to be verified before the reduction can be carried out. In view of the steady increase in applications which the method has seen over the last half century one can safely say that the Lyapunov–Schmidt method will remain a valuable tool for future research in bifurcation theory, dynamical systems theory and nonlinear analysis. Bibliography Primary Literature 1. Andronov A, Leontovich FA (1938) Sur la théorie de la variation de la structure qualitative de la division du plan en trajectoires. Dokl Akad Nauk 21:427–430 2. Antosiewicz HA (1966) Boundary value problems for nonlinear ordinary differential equations. Pacific J Math 17:191–197

Lyapunov–Schmidt Method for Dynamical Systems

3. Ashwin P, Böhmer K, Mei Z (1993) A numerical Lyapunov– Schmidt method for finitely determined systems. In: Allgower E, Georg K, Miranda R (eds) Exploiting Symmetry in Applied and Numerical Analysis. Lectures in Appl Math, vol 29. AMS, Providence, pp 49–69 4. Bancroft S, Hale JK, Sweet D (1968) Alternative problems for nonlinear functional equations. J Differ Equ 4:40–56 5. Bartle R (1953) Singular points of functional equations. Trans Amer Math Soc 75:366–384 6. Böhmer K (1993) On a numerical Lyapunov–Schmidt method for operator equations. Computing 53:237–269 7. Bourgain J (1998) Quasi-periodic Solutions of Hamiltonian Perturbations of 2D Linear Schrödinger Equations. Ann Math 148(2):363–439 8. Cacciopoli R (1936) Sulle correspondenze funzionali inverse diramata: teoria generale e applicazioni ad alcune equazioni funzionali non lineari e al problema di Plateau. Atti Accad Naz Lincei Rend Cl Sci Fis Mat Natur Sez I 24:258–263, 416–421 9. Cesari L (1940) Sulla stabilità delle soluzioni dei sistemi di equazioni differenziali lineari a coefficienti periodici. Atti Accad Ital Mem Cl Fis Mat Nat 6(11):633–692 10. Cesari L (1963) Periodic solutions of hyperbolic partial differential equations. In: Intern. Symposium on Nonlinear Differential Equations and Nonlinear Mechanics. Academic Press, New York, pp 33–57 11. Cesari L (1964) Functional analysis and Galerkin’s method. Michigan Math J 11:385–414 12. Cicogna C (1981) Symmetry breakdown from bifurcations. Lett Nuovo Cimento 31:600–602 13. Chossat P, Lauterbach R (2000) Methods in Equivariant Bifurcations and Dynamical Systems. Advanced Series in Nonlinear Dynamics, vol 15. World Scientific, Singapore 14. Chow S-N, Hale JK (1982) Methods of Bifurcation Theory. Grundlehren der Mathematischen Wissenschaften, vol 251. Springer, New York 15. Ciocco M-C, Vanderbauwhede A (2004) Bifurcation of Periodic Points in Reversible Diffeomorphisms. In: Elaydi S, Ladas G, Aulbach B (eds) New Progress in Difference Equations – Proceedings ICDEA2001, Augsburg. Chapman & Hall, CRC Press, pp 75–93 16. Craig W, Wayne CE (1993) Newton’s method and periodic solutions of nonlinear wave equations. Comm Pure Appl Math 46:1409–1498 17. Crandall MG, Rabinowitz PH (1971) Bifurcation from simple eigenvalues. J Funct Anal 8:321–340 18. Crandall MG, Rabinowitz PH (1978) The Hopf bifurcation theorem in infinite dimensions. Arch Rat Mech Anal 67:53–72 19. Cronin J (1950) Branch points of solutions of equations in Banach spaces. Trans Amer Math Soc 69:208–231; (1954) 76:207– 222 20. Da Prato G, Lunardi A (1986) Hopf bifurcation for fully nonlinear equations in Banach space. Ann Inst Henri Poincaré 3:315– 329 21. Friedrichs KO (1953–54) Special Topics in Analysis. Lecture Notes. New York University, New York 22. Gaines R, Mawhin J (1977) Coincidence Degree, and Nonlinear Differential Equations. Lect Notes in Math, vol 568. Springer, Berlin 23. Golubitsky M, Schaeffer DG (1985) Singularities and Groups in Bifurcation Theory, vol 1. Appl Math Sci, vol 51. Springer, New York

24. Golubitsky M, Stewart IN, Schaeffer DG (1988) Singularities and Groups in Bifurcation Theory, vol 2. Appl Math Sci, vol 69. Springer, New York 25. Golubitsky M, Pivato M, Stewart I (2004) Interior symmetry and local bifurcation in coupled cell networks. Dyn Syst 19:389–407 26. Hale JK (1954) Periodic solutions of non-linear systems of differential equations. Riv Mat Univ Parma 5:281–311 27. Hale JK (1963) Oscillations in nonlinear systems. McGraw-Hill, New York 28. Hale JK (1969) Ordinary Differential Equations. Wiley, New York 29. Hale JK (1977) Theory of Functional Differential Equations. Appl Math Sciences, vol 3. Springer, New York 30. Hale JK (1984) Introduction to dynamic bifurcation. In: Salvadori L (ed) Bifurcation Theory and Applications. Lecture Notes in Math, vol 1057. Springer, Berlin, pp 106–151 31. Hale JK, Sakamoto K (1988) Existence and stability of transition layers. Japan J Appl Math 5:367–405 32. Hale JK, Sakamoto K (2005) A Lyapunov–Schmidt method for transition layers in reaction-diffusion systems. Hiroshima Math J 35:205–249 33. Hall WS (1970) Periodic solutions of a class of weakly nonlinear evolution equations. Arch Rat Mech Anal 39:294–332 34. Hall WS (1970) On the existence of periodic solutions of the 2p equation Dtt u C (1)p Dx u D "f (t; x; u). J Differ Equ 7:509– 526 35. Hopf E (1942) Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber Math Phys Kl Sachs Acad Wiss Leipzig 94:1–22. English translation in: Marsden J, McCracken M (1976) Hopf Bifurcation and its Applications. Springer, New York 36. Iooss G, Joseph D (1981) Elementary Stability and Bifurcation Theory. Springer, New York 37. Ize J (1976) Bifurcation theory for Fredholm operators. Mem AMS 174 38. Jepson A, Spence A (1989) On a reduction process for nonlinear equations. SIAM J Math Anal 20:39–56 39. Kielhöfer H (2004) Bifurcation Theory, An Introduction with Applications to PDEs. Appl Math Sciences, vol 156. Springer, New York 40. Knobloch J (2000) Lin’s method for discrete dynamical systems. J Differ Eq Appl 6:577–623 41. Knobloch J, Vanderbauwhede A (1996) A general reduction method for periodic solutions in conservative and reversible systems. J Dyn Diff Eqn 8:71–102 42. Knobloch J, Vanderbauwhede A (1996) Hopf bifurcation at k-fold resonances in conservative systems. In: Broer HW, van Gils SA, Hoveijn I, Takens F (eds) Nonlinear Dynamical Systems and Chaos. Progress in Nonlinear Differential Equations and Their Applications, vol 19. Birkhäuser, Basel, pp 155–170 43. Krasnosel’skii MA (1963) Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon Press, Oxford 44. Krasnosel’skii MA, Vainiko GM, Zabreiko PP, Rutitskii YB, Stesenko VY (1972) Approximate solutions of operator equations. Wolters-Noordhoff, Groningen 45. Kuznetsov YA (1998) Elements of Applied Bifurcation Theory, 2nd edn. Appl Math Sci, vol 112. Springer, New York 46. Lewis DC (1956) On the role of first integrals in the perturbations of periodic solutions. Ann Math 63:535–548 47. Lin XB (1990) Using Melnikov’s method to solve Silnikov’s problems. Proc Roy Soc Edinb 116A:295–325

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48. Lyapunov AM (1906) Sur les figures d’équilibre peu différentes des ellipsoï, des d’une masse liquide homogène dotées d’un mouvement de rotation. Zap Akad Nauk St Petersb, pp 1–225; (1908), pp 1–175; (1912), pp 1–228; (1914), pp 1–112 49. Mawhin J (1969) Degré topologique et solutions périodiques des systèmes différentiels non linéaires. Bull Soc Roy Sci Liège 38:308–398 50. Nirenberg L (1960–61) Functional Analysis. Lecture Notes. New York University, New York 51. Rabinowitz PH (1967) Periodic solutions of nonlinear hyperbolic partial differential equations. Comm Pure Appl Math 20:145–206 52. Rabinowitz PH (1976) A Survey of Bifurcation Theory. In: Cesari L, Hale JK, LaSalle J (eds) Dynamical Systems, An International Symposium, vol 1. Academic Press, New York, pp 83–96 53. Sandstede B (1993) Verzweigungstheorie homokliner Verdopplungen. Ph D Thesis, University of Stuttgart, IAAS Report No. 7 54. Sattinger DH (1979) Group Theoretic Methods in Bifurcation Theory. Lecture Notes in Math, vol 762. Springer, Berlin 55. Schmidt E (1908) Zur Theorie des linearen und nichtlinearen Integralgleichungen, 3 Teil. Über die Auflösung der nichtlinearen Integralgleichungen und die Verzweigung ihrer Lösungen. Math Ann 65:370–399 56. Shimizu T (1948) Analytic operations and analytic operational equations. Math Japon 1:36–40 57. Sidorov N, Loginov B, Sinitsyn A, Falaleev M (2002) Lyapunov–Schmidt Methods in Nonlinear Analysis and Applications. Mathematics and Its Applications, vol 550. Kluwer, Dordrecht 58. Taniguchi M (1995) A Remark on Singular Perturbation Meth-

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ods via the Lyapunov–Schmidt Reduction. Publ RIMS, Kyoto Univ 31:1001–1010 Vainberg MM, Trenogin VA (1962) The method of Lyapunov and Schmidt in the theory of nonlinear differential equations and their further development. Russ Math Surv 17:1–60 Vainberg MM, Trenogin VA (1974) Theory of Branching of Solutions of Non-Linear Equations. Noordhoff, Leyden Vanderbauwhede A (1982) Local bifurcation and symmetry. Research Notes in Mathematics, vol 75. Pitman, London Vanderbauwhede A (2000) Subharmonic bifurcation at multiple resonances. In: Elaydi S, Allen F, Elkhader A, Mughrabi T, Saleh M (eds) Proceedings of the Mathematics Conference, Birzeit, August 1998. World Scientific, Singapore, pp 254–276 Vanderbauwhede A, Van der Meer JC (1995) A general reduction method for periodic solutions near equilibria in Hamiltonian systems. In: Langford WF, Nagata W (eds) Normal Forms and Homoclinic Chaos. Fields Institute Comm. AMS, Providence, pp 273–294

Books and Reviews Alligood KT, Sauer T, Yorke JA (1997) Chaos, An Introduction to Dynamical Systems. Springer, New York Chicone C (2006) Ordinary Differential Equations with Applications, 2nd edn. Texts in Applied Math, vol 34. Springer, New York Govaerts W (2000) Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM Publications, Philadelphia Hale JK, Koçak H (1991) Dynamics and Bifurcations. Texts in Applied Mathematics, vol 3. Springer, New York Ma T, Wang S (2005) Bifurcation Theory and Applications. World Scientific Series on Nonlinear Science, vol A53. World Scientific, Singapore

Maximum Principle in Optimal Control

Maximum Principle in Optimal Control VELIMIR JURDJEVIC University of Toronto, Toronto, Canada Article Outline Glossary Definition of the Subject Introduction The Calculus of Variations and the Maximum Principle Variational Problems with Constraints Maximum Principle on Manifolds Abnormal Extrema and Singular Problems Future Directions Bibliography Glossary Manifolds Manifolds M are topological spaces that are covered by a set of compatible local charts. A local chart is a pair (U; ) where U is an open set on M and  is a homeomorphism from U onto an open set in Rn . If  is written as '(x) D (x1 ; : : : ; x n ) then x1 ; : : : ; x n are called coordinates of a point x in U. Charts are said to be compatible if for any two charts (U; ) and (V; ) the mapping ' 1 : (U \ V) ! (U \ V ) is smooth, i. e. admits derivatives of all orders. Such manifolds are also called smooth. Manifolds on which the above mappings ' 1 are analytic are called analytic. Tangent and cotangent spaces The vector space of tangent vectors at a point x in a manifold M will be denoted by Tx M. The tangent bundle TM of M is the union [fTx M : x 2 Mg. The cotangent space at x, consisting of all linear functions on Tx M will be denoted by Tx M. The cotangent bundle of M, denoted by T  M is equal to the union [fTx M : x 2 Mg. Lie groups Lie groups G are analytic manifolds on which the group operations are compatible with the manifold structure, in the sense that both (g; h) ! gh from G  G onto G and g ! g 1 from G onto G are analytic. The set of all n  n non-singular matrices is an n2 dimensional Lie group under matrix multiplication, and so is any closed subgroup of it. Absolutely continuous curves A parametrized curve x : [0; T] ! Rn is said to be absolutely continuous if each derivative R T (dx i )/(dt)(t) exists almost everywhere in [0;RT] 0 j(dx i )/(dt)(t)jdt < 1, and t x(t2 )  x(t1 ) D t12 (dx)/(dt)(t)dt for almost all points

t1 and t2 in [0; T]. This notion extends to curves on manifolds by requiring that the above condition holds in any system of coordinates. Control systems and reachable sets A control system is any system of differential equations in Rn or a manifold M of the form dx dt (t) D F(x(t); u(t)); where u(t) D (u1 (t); : : : ; u m (t)) is a function, called control function. Control functions are assumed to be in some specified class of functions U, called the class of admissible controls. For purposes of optimization U is usually assumed to consist of functions measurable and bounded on compact intervals [t1 ; t2 ] that take values in some a priori specified set U in Rm . Under the usual existence and uniqueness assumptions on the vector fields F, each control u(t) : R ! Rm and each initial condition x0 determine a unique solution x(x0 ; u; t) that passes through x0 at t D 0. These solutions are also called trajectories. A point y is said to be reachable from x0 at time T if there exists a control u(t) defined on the interval [0; T] such that x(x0 ; u; T) D y. The set of points reachable from x0 at time T is called the reachable set at time T. The set of points reachable from x0 at any terminal time T is called the reachable set from x0 . Riemannian manifolds A manifold M together with a positive definite quadratic form h ; i : Tx M  Tx M ! R that varies smoothly with base point x is called Riemannian. If x(t), t 2 [0; T] is a curve on q M then its Riemannian length is given by RT RT 0 0 h(dx)/(dt); (dx)/(dt)i dt. Riemannian metric is a generalization of the Euclidean metric in Rn given P by hx; yi D niD1 x i y i . Definition of the Subject The Maximum Principle provides necessary conditions that the terminal point of a trajectory of a control system belongs to the boundary of its reachable set or to the boundary of its reachable set at a fixed time T. In practice, however, its utility lies in problems of optimization Rt in which a given cost functional t01 f (x(t); u(t))dt is to be minimized over the solutions (x(t); u(t)) of a control system dx dt D F(x(t); u(t)) that conform to some specified boundary conditions at the end points of the inoptimal trajectory terval [t0 ; t1 ]. For then, the extended Rt ¯ ¯ ¯ (x¯0 (t); x(t); u(t)) with x¯0 (t) D t0 f (x¯ (t); u(t))dt must be on the boundary of the reachable set associated with extended control system dx0 D f (x(t); u(t)) ; dt

dx D F(x(t); u(t)) : dt

(1)

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In the original publication [24] the Maximum Principle is stated for control systems in Rn in which the controls u(t) D (u1 (t); : : : ; u m (t)) take values in an arbitrary subset U of Rm and are measurable and bounded on each interval [0; T], under the assumptions that the cost functional f and each coordinate F i of the vector field F in i Eq. (1) together with the derivatives @F are continuous @x j ¯ where U¯ denotes the topological closure of U. on Rn  U, ¯ and the corresponding trajecA control function u(t) ¯ tory x¯ (t) of dx D F(x(t); u(t)) each defined on an interval dt [0; T] are said to be optimal relative to the given bound¯ ary submanifolds S0 and S1 of Rn if x(0) 2 S0 ; x¯ (T) 2 S1 , and Z T Z S ¯ f (x¯ (t); u(t))dt  f (x(t); u(t))dt (2) 0

0

for any other solution x(t) of dx dt D F(x(t); u(t)) that satisfies x(0) 2 S0 and x(S) 2 S1 . Then Proposition 1 (The Maximum Principle (MP)) Suppose ¯ ¯ that (u(t); x(t)) is an optimal pair on the interval [0; T]. Then there exist an absolutely continuous curve p(t) D (p1 (t); : : : ; p n (t)) on the interval [0; T] and a multiplier p0  0 with the following properties: P 1. p20 C niD1 p2i (t) > 0, for all t 2 [0; T], 2.

dp i dt (t)

@H

¯ D  @x pi 0 (x¯ (t); p(t); u(t)), i D 1; : : : ; n, a. e. P in [0; T] where H p 0 (x; p; u) D p0 f (x; u) C niD1 p i F i (x; u).

¯ D M(t) a. e. in [0; T] where M(t) 3. H p 0 (x¯ (t); p(t); u(t)) denotes the maximum value of H p 0 (x¯ (t); p(t); u) relative to the controls u 2 U. Furthermore, 4. M(t) D 0 for all t 2 [0; T]. 5. hp(0); vi D 0, v 2 Tx(0) ¯ S0 and hp(T); vi D 0, v 2 Tx(T) S . These conditions, known as the transversality ¯ 1 conditions, become void when the manifolds S0 and S1 reduce to single points x0 and x1 . The maximum principle is also valid for optimal problems in which the length of the interval [0; T] is fixed, ¯ satisfies RinT the sense that theR Toptimal pair (x¯ (t); u(t)) ¯ ¯ f ( x (t); u(t))dt  f (x(t); u(t))dt for any other tra0 0 jectory (x(t); u(t)) with x(0) 2 S0 and x(T) 2 S1 . In this context the maximum principle asserts the existence of a curve p(t) and the multiplier p0 subject to the same conditions as stated above except that the maximal function M(t) must be constant in the interval [0; T] and need not be necessarily equal to zero. These two versions of the (MP) are equivalent in the sense that each implies the other [1].

Pairs (x(t); p(t)) of curves that satisfy the conditions of the maximum principle are called extremal. The extremal curves that correspond to the multiplier p0 ¤ 0 are called normal while the ones that correspond to p0 D 0 are called abnormal. In the normal case it is customary to reduce the multiplier p0 to p0 D 1. Since the original publication, however, the maximum principle has been adapted to control problems on arbitrary manifolds [12] and has also been extended to more general situations in which the vector fields that define the control system are locally Lipschitz rather than continuously differentiable [9,27]. On this level of generality the Maximum Principle stands out as a fundamental principle in differential topology that not only merges classical calculus of variations with mechanics, differential geometry and optimal control, but also reorients the classical knowledge in two major ways: 1. It shows that there is a natural “energy” Hamiltonian for arbitrary variational problems and not just for problems of mathematical physics. The passage to the appropriate Hamiltonians is direct and bypasses the Euler–Lagrange equation. The merits of this observation are not only limited to problems with inequality constraints for which the Euler-equation is not applicable; they also extend to the integration procedure of the extremal equations obtained through the integrals of motion. 2. The Hamiltonian formalism associated with (MP) further enriched with geometric control theory makes direct contact with the theory of Hamiltonian systems and symplectic geometry. In this larger context, the maximum principle brings fresh insights to these classical fields and also makes their theory available for problems of optimal control. Introduction The Maximum Principle of Pontryagin and his collaborators [1,12,24] is a generalization of C. Weierstrass’ necessary conditions for strong minima [29] and is based on the topological fact that an optimal solution must terminate on the boundary of the extended reachable set formed by the competing curves and their integral costs. An important novelty of Pontryagin’s approach to the calculus of variations consists of liberating the variations along an optimal trajectory of the constricting condition that they must terminate at the given boundary data. Control theoretic context induces a natural class of variations that generates a cone of directions locally tangent to the reachable set at the terminal point defined by the optimal trajectory. As a consequence of optimality, the direction of

Maximum Principle in Optimal Control

the decreasing cost cannot be in the interior of this cone. This observation leads to the separation theorem, a generalization of the classic Legendre transform in the calculus of variations, that ultimately produces the appropriate Hamiltonian function. The maximum principle asserts that the Hamiltonian that corresponds to the optimal trajectory must be maximal relative to the completing directions, and it also asserts that each optimal trajectory is the projection of an integral curve of the corresponding Hamiltonian field. The methodology used in the original publication extends the maximum principle to optimal control problems on arbitrary manifolds where, combined with Lie theoretic criteria for reachable sets, it stands out as a most important tool of optimal control available for problems of mathematical physics and differential geometry. Much of this article, particularly the selection of the illustrating examples is devoted to justifying this claim. The exposition begins with comparisons between (MP) and the classical theory of the calculus of variations in the absence of constraints. Then it proceeds to optimal control problems in Rn with constraints amenable by the (MP) stated in Proposition 1. This section, illustrated by two famous problems of classical theory, the geodesic problem on the ellipsoid of C.J.G. Jacobi and the mechanical problem of C. Newman–J. Moser is deliberately treated by control theoretic means, partly to illustrate the effectiveness of (MP), but mostly to motivate extensions to arbitrary manifolds and to signal important connections to the theory of integrable systems. The exposition then shifts to the geometric version of the maximum principle for control problems on arbitrary manifolds, with a brief discussion of the symplectic structure of the cotangent bundle. The maximum principle is first stated for extremal trajectories (that terminate on the boundary of the reachable sets) and then specialized to optimal control problems. The passage from the first to the second clarifies the role of the multiplier. There is brief discussion of canonical coordinates as a bridge that connects geometric version to the original formulation in Rn and also leads to left invariant adaptations of the maximum principle for problems on Lie groups. Left invariant variational control problems on Lie groups make contact with completely integrable Hamiltonian systems, Lax pairs and the existence of spectral parameters. For that reason there is a section on the Poisson manifolds and the symplectic structure of the coadjoint orbits of a Lie group G which is an essential ingredient of the theory of integrable systems. The exposition ends with a brief discussion of the abnormal and singular extremals.

The Calculus of Variations and the Maximum Principle The simplest problems in the calculus of variations can be formulated as optimal control problems of the form U D Rn , dx dt (t) D u(t), S0 D fx0 g and S1 D fx1 g, with one subtle qualification connected with the basic terminology. In the literature on the calculus of variations one usu¯ defined on an inally assumes that there is a curve x(t) terval [0; T] that provides a “local minimum” for the inteRT gral 0 f (x(t); dx (t))dt in the sense that there is a “neighdt borhood” N in the space of curves on [0; T] such that RT RT d x¯ dx ¯ 0 f (x (t); dt (t))dt  0 f (x(t); dt (t))dt for any other curve x(t) 2 N that satisfies the same boundary conditions as x¯ (t). There are two distinctive topologies, strong and weak on the space of admissible curves relative to which optimality is defined. In strong topology admissible curves consist of absolutely continuous curves with bounded derivatives on [0; T] in which an neighborhood N consists of all admissible curves x(t) such that kx¯ (t)  x(t)k < for all t 2 [0; T]. In this setting local minima are called strong minima. For weak minima admissible curves consist of continuously differentiable curves on [0; T] with ¯ defined by an neighborhood of x(t)    dx¯ dx   (t) < for all t 2 [0; T] : kx¯ (t)  x(t)k C  (t)  dt dt  (3) Evidently, any strong local minimum that is continuously differentiable is also a weak local minimum. The converse, however, may not hold (see p. 341 in [12]). The Maximum Principle is a necessary condition for local strong minima x¯ (t) under suitable restriction of the state space. It is a consequence of conditions (1) and (3) in Proposition 1 that the multiplier p0 can not be equal to 0. Then it may be normalized to p0 D 1 and (MP) can be rephrased in terms of the excess function of Weierstrass [7,29] as 

 d x¯ f (x¯ (t); u)  f x¯ (t); (t) dt   n X d x¯ @f x¯ (t); (t) (u¯ i u i ); for all u 2 Rn ;  @u i dt iD1

(4) because the critical points of H (x¯ (t); p(t); u) D  f (x¯ (t); P u) C niD1 p i (t)u i relative to u 2 Rn are given by p i (t) D @f ¯ yields the maximum of (x¯ (t); u). Since ddtx¯ (t) D u(t) @u i

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Maximum Principle in Optimal Control

H (x¯ (t); p(t); u) it follows that

p i (t) D

@f ¯ (x¯ (t); u(t)) : @u i

(5)

Combining Eq. (5) with condition 2 of the maximum principle yields the Euler–Lagrange equation in integrated R t @f @f ¯ ¯  0 dx i (x¯ (); u()) d D c, with c form @u i (x¯ (t); u(t)) a constant, which under further differentiability assumptions can be stated in its usual form   @f d @f ¯ ¯ (x¯ (t); u(t))  (x¯ (); u()) d D 0 : (6) dt @u i dx i As a way of illustration consider Example 2 (The harmonic oscillator) The problem of minRT imizing 0 21 (mu2 kx 2 )dt over the trajectories of dx dt D u leads to the family of Hamiltonians Hu D  12 (mu2  kx 2 ) C pu. According to the Maximum Principle every optimal trajectory x(t) is the projection of a curve p(t) that dp Hu D kx(t), subject to the maximality satisfies dt D  @@x condition that 1  (mu(t)2  kx(t)2 ) C p(t)u(t) 2 1   (mv 2  kx(t)2 ) C p(t)v 2

(7)

for any choice of v. That implies that the optimal control that generates x(t) is of the form u(t) D

1 p(t) m

(8)

which then further implies that optimal solutions are the integral curves of a single Hamiltonian function H D 1 2 1 2 2m p C 2 kx . This Hamiltonian is equal to the total en2

ergy of the oscillator. The Euler–Lagrange equation ddtx2 D du k dt D  m x(t) is easily obtained by differentiating, but there is no need for it since all the information is already contained in the Hamiltonian equations. It follows from the above that the projections of the q qextremal curves are given by x(t) D A cos t mk CB sin t mk for arbitrary constants A and B. It can be shown, by a separate argument q[12], that the preceding curves are optimal

if and only if t mk  . This example also illustrates that the Principle of Least Action in mechanics may be valid only on small time intervals [t0 ; t1 ] (in the sense that it yields the least action). Variational Problems with Constraints The early applications of (MP) are best illustrated through time optimal problems for linear control systems

dx dt (t) D Ax(t) C Bu(t) with control functions u(t) D (u1 (t); : : : ; u r (t)) taking values in a compact neighborhood U of the origin in Rr . Here A and B are matrices of appropriate dimensions such that the “controllability” matrix [B; AB; A2 B; : : : ; An1 B] is of rank n. In this situation if (x(t); u(t)) is a time optimal pair then the corresponding Hamiltonian H p 0 (x; p; u) D p0 C hp(t); Ax(t) C Bu(t)i defined by the Maximum Principle is equal to 0 almost everywhere and is also maximal almost everywhere relative to the controls u 2 U. The rank condition together with the fact that p(t) is the solution of a linear differendp tial equation dt D AT p(t) easily implies that the control u(t) cannot take value in the interior of U for any convergent sequence of times ft n g otherwise, lim p(t n ) D 0, and therefore p0 D 0, which in turn contradicts condition 1 of Proposition 1. It then follows that each optimal control u(t) must take values on the boundary of U for all but possibly finitely many times. This fact is known as the Bang-Bang Principle, since when U is a polyhedral set then optimal control “bangs” from one face of U to another. In general, however, optimal controls may take values both in the interior and on the boundary of U, and the extremal curves could be concatenations of pieces generated by controls with values in the interior of U and the pieces generated by controls with values on the boundary. Such concatenations may exhibit dramatic oscillations at the juncture points, as in the following example.

RT Example 3 (Fuller’s problem) Minimize 12 0 x12 (t)dt over dx 2 1 the trajectories of dx dt D x2 ; dt D u(t) subject to the constraint that ju(t)j  1. Here T may be taken fixed and sufficiently large that admits an optimal trajectory x(t) D (x1 (t); x2 (t)) that transfers the initial point a to a given terminal point b in T units of time. Evidently, the zero trajectory is generated by zero control and is optimal relative to a D b D 0. The (MP) reveals that any other optimal trajectory is a concatenation of this singular trajectory and a bang-bang trajectory. At the point of juncture, whether leaving or entering the origin, optimal control oscillates infinitely often between the boundary values ˙1 (in fact, the oscillations occur in a geometric sequence [12,19]). This behavior is known as Fuller’s phenomenon. Since the original discovery Fuller’s phenomena have been detected in many situations [18,30]. The Maximum Principle is the only tool available in the literature for dealing with variational problems exhibiting such chattering behavior. However, even for problems of geometry and mechanics which are amenable by the classical methods the (MP) offers certain advantages as the following examples demonstrate.

Maximum Principle in Optimal Control

Example 4 (Ellipsoidal Geodesics) This problem, initiated and solved by C.G. Jacobi in 1839 [23] consists of finding the curves of minimal length on a general ellipsoid hx; A1 xi D

x12 x2 x2 C 22 C    C 2n D 1 : 2 an a1 a2

(9)

Recall that the length of a curve x(t) on an interRT dx val [0; T] is given by 0 k dx dt (t)kdt, where k dt (t)k D p ((dx1 )/(dt))2 C    C ((dx n )/(dt))2 . This problem can be recast as an optimal control problem either as a time optimal problem when the curves are parametrized by arc length, or as the problem of minimizing the energy funcRT 2 tional 12 0 k dx dt (t)k dt over arbitrary curves [15]. In the latter formulation the associated optimal control probRT lems consists of minimizing the integral 12 0 ku(t)k2 dt over the trajectories of dx dt (t) D u(t) that satisfy hx(t); A1 x(t)i D 1. Since there are no abnormal extremals in this case, it follows that the adjoint curve p(t) associated with an optimal trajectory x(t) must maximize Hu D  12 kuk2 C hp(t); ui on the cotangent bundle of the manifold x : hx(t); A1 x(t)i  1 D 0 . The latter is naturally identified with the constrains G1 D G2 D 0, where G1 D hx(t); A1 x(t)i  1 and G2 D hp; A1 xi. According to the Lagrange multiplier rule the correct maximum of Hu subject to these constrains is obtained by maximizing the function Gu D  12 jjujj2 Chp; uiC1 G1 C2 G2 relative to u. The the maximal Hamiltonian is given by H D 12 jjpjj2 C 1 G1 C 2 G2 obtained by substituting u D p. The correct multipliers 1 and 2 are determined by requiring that the integral curves !  of the associated Hamiltonian vector field H respect the hA1 p;pi constraints G1 D G2 D 0. It follows that 1 D 2jjA1 xjj and 2 D 0. Hence the solutions are the projections of the integral curves of hA1 p;

pi 1 H D kpk2 C G1 restricted to G1 D G2 D 0 : 2 2kA1 xk2 (10) The corresponding equations are dx @H D D p; dt @p

and

dp @H hA1 p; pi : D D dt @x kA1 xk2

(11)

The projections of these equations on the ellipsoid that reside on the energy level H D 12 are called geodesics. It is

well known in differential geometry that geodesics are only locally optimal (up to the first conjugate point). The relatively simple case, when the ellipsoid degenerates to the sphere occurs when A D I. Then the above hp;pi Hamiltonian reduces to H D 12 (kpk2 ) C 2kxk2 (kxk2  1) and the corresponding equations are given by dx D p; dt

dp D kpk2 x : dt

(12)

It follows by an easy calculation that the projections x(t) of Eqs. (12) evolve along the great circles because x(t) ^ dx dx dt (t) D x(0) ^ dt (0). The solutions in the general case can be obtained either by the method of separation of variables inspired by the work of C.G.J. Jacobi (see also [2]), or by the isospectral deformation methods discovered by J. Moser [22], which are somewhat mysteriously linked to the following problem. Example 5 (Newmann–Moser problem) This problem concerns the motion of a point mass on the unit sphere hx; xi D 1 that moves in a force field with quadratic potential V D 12 hx; Axi with A an arbitrary symmetric matrix. Then the Principle of Least Action applied to the 1 2 Lagrangian L D 12 k dx defines an optimal dt k  2 hx; Axi RT control problem by maximizing 0 L(x(t); u(t))dt over the trajectories of dx dt D u(t), subject to the constraint G1 D jjxjj2  1 D 0, whose extremal equations etc. In fact, H D 12 (kpk2 C hx; Axi) C 1 G1 C 2 G2 , where hp;xi

G2 D hx; pi; 1 D  kxk2 and 2 D

kpk2 2kxk2

 hAx; xi.

dx dp D p; D Ax C (hAx; x  kpk2 ix dt dt

(13)

Equations (13) can be recast in matrix form dP (t) D [K(t); P(t)] ; dt

dK (t) D [P(t); A] dt

(14)

with P(t) D x(t) ˝ x(t)  I and K(t) D x(t) ^ p(t) D x(t) ˝ p(t)  p(t) ˝ x(t), where [M; N] denotes the matrix commutator N M  M N. Equations (14) admit a Lax pair representation in terms of scalar parameter . 

1 dL D P(t); L (t) ; dt  where L (t) D P(t)  K(t)  2 A ;

(15)

that provides a basis for Moser’s method of proving integrability of Eqs. (13). This method exploits the fact that

957

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Maximum Principle in Optimal Control

the spectrum of L (t) is constant, and hence the functions  k; D Trace(Lk ) are constants of motion for each  and k > 0. Moreover, these functions are in involution with each other (to be explained in the next section). Remarkably, Eqs. (11) can be transformed to Eqs. (13) from which then can be inferred that the geodesic ellipsoidal problem is also integrable [22]. It will be shown later that this example is a particular case of a more general situation in which the same integration methods are available.

1. x(t) is the projection of (t) in [0; T] and d dt (t) D E u(t) ((t)), a. e. in [0; T], where Hu(t) () D (F(x; H ¯ u(t))),  2 Tx M. 2. (t) ¤ 0 for all t 2 [0; T]. 3. Hu(t) ((t)) D (t)(F(x(t); u(t)))  (t)(F(x(t); v)) for all v 2 U a. e. in [0; T]. 4. If the extremal curve is extremal relative to the fixed terminal time then Hu(t) ((t)) is constant a. e. in [0; T], otherwise, Hu(t) ((t)) D 0 a. e. in [0; T].

Maximum Principle on Manifolds

An absolutely continuous curve (t) that satisfies the conditions of the Maximum Principle is called an extremal. R T Problems of optimization in which a cost functional 0 f (x(t); u(t))dt is to be minimized over the trajectories of a control system in M subject to the prescribed boundary conditions with terminal time either fixed or variable are reduced to boundary problems defined above in the same manner as described in the introductory part. Then each optimal trajectory x(t) is equal to the projection of an ˜ D R  M relaextremal for the extended system (1) on M ˜ tive to the extended initial conditions x˜0 D (0; x0 ). If T  M is identified with R  T  M and its points ˜ are written as (0 ; ) the Hamiltonian lifts of the extended control system are of the form

The formulation of the maximum principle for control systems on arbitrary manifolds requires additional geometric concepts and terminology [1,12]. Let M denote an n-dimensional smooth manifold with Tx M and Tx M denoting the tangent and the cotangent space at a point x 2 M. The tangent bundle TM is equal to the union [fTx M : x 2 Mg, and similarly the cotangent bundle T  M is equal to fTx M : x 2 Mg. In each of these cases there is a natural bundle projection  on the base manifold. In particular, x 2 M is the projection of a point  2 T  M if and only if  2 Tx M. The cotangent bundle T  M has a canonical symplectic form ! that turns T  M into a symplectic manifold. This implies that for each function H on T  M there is a vector E V ) for all vecE on T  M defined by V (H) D !(H; field H  tor fields V on T M. In the symplectic context H is called E is called the Hamiltonian vector field Hamiltonian and H corresponding to H. Each vector field X on M defines a function H X () D (X(x)) on T  M. The corresponding Hamiltonian vector E X is called the Hamiltonian lift of X. In particular, field H control systems dx dt (t) D F(x(t); u(t)) lift to Hamiltonians Hu parametrized by controls with Hu () D (X u (x)) D (F(x; u)),  2 Tx M. With these notations in place consider a control system dx dt (t) D F(x(t); u(t)) on M with control functions u(t) taking values in an arbitrary set U in Rm . Suppose that the system satisfies the same assumptions as in Proposition 1. Let Ax 0 (T) denote the reachable set from x0 at time T and let Ax 0 D [fAx 0 (T) : T  0g. The control u that generates trajectory x(t) from x0 to the boundary of either Ax 0 (T) or Ax 0 is called extremal. For extremal trajectories the following version of the Maximum Principle is available. Proposition 6 (Geometric maximum principle (GMP)) Suppose that a trajectory x(t) corresponds to an extremal control u(t) on an interval [0; T]. Then there exists an absolutely continuous curve (t) in T  M in the interval [0; T] that satisfies the following conditions:

¯ Hu (0 ; ) D 0 f (x; u) C (F(x; u(t))); 0 2 R ;  2 Tx M :

(16)

For each extremal curve (0 (t); (t)) associated with the extended system 0 (t) is constant because the Hamiltonian Hu (0 ; ) is constant along the first factor of the extended state space. In particular, 0  0 along the extremals that correspond to optimal trajectories. As before, such extremals are classified as normal or abnormal depending whether  0 is equal to zero or not, and in the normal case the variable  0 is traditionally reduced to 1. The Maximum principle is then stated in terms of the reduced Hamiltonian on T  M in which  0 appear as parameters. In this context Proposition 6 can be rephrased as follows: Proposition 7 Let x(t) denote an optimal trajectory on an interval [0; T] generated by a control u(t). Then there exist a number 0 2 f0; 1g and an absolutely continuous curve (t) in T  M defined on the interval [0; T] that projects onto x(t) and satisfies: 1. (t) ¤ 0 whenever 0 D 0. E 2. d dt (t) D Hu(t) (0 ; (t)) a. e. on [0; T]. 3. Hu(t) (0 ; (t))  Hv (0 ; (t)) for any v 2 U, a. e. in [0; T]. When the initial and the terminal points are replaced

Maximum Principle in Optimal Control

by the submanifolds S0 and S1 there are transversality conditions: 4. (0)(v) D 0, v 2 Tx(0) S0 and (T)(v) D 0, v 2 Tx(T) S1 . The above version of (MP) coincides with the Euclidean version when the variables are expressed in terms of the canonical coordinates. Canonical coordinates are defined as follows. Any choice of coordinates (x1 ; x2 ; : : : ; x n ) on M induces coordinates (v1 ; : : : ; v n ) of vectors in Tx M relative to the basis @x@ ; : : : ; @x@ and it also induces co1 n ordinates (p1 ; p2 ; : : : ; p n ) of covectors in Tx M relative to the dual basis dx1 ; dx2 ; : : : ; dx n . Then (x1 ; x2 ; : : : ; x n ; p1 ; p2 ; : : : ; p n ) serves as a system of coordinates for points  in T  M. These coordinates in turn define coordinates for tangent vectors in T (T  M) relative to the basis @x@ 1 ; : : : ; @x@n ; @p@ 1 ; : : : ; @p@ n . The symplectic form ! can then be expressed in terms of vector fields X D Pn Pn @ @ @ @ iD1 Vi @x i C Pi @p i and Y D iD1 Wi @x i C Q i @p i as !(x;p) (X; Y) D

n X

Q i Vi  Pi Wi :

(17)

iD1

The correspondence between functions H and their E is given by Hamiltonian fields H E H(x; p) D

n X @H @ @H @  ; @p i @x i @x i @p i

(18)

iD1

E are given by the and the integral curves (x(t); p(t)) of H usual differential equations dx i @H ; D dt @p i

dp i @H ; D dt @x i

i D 1; : : : ; n :

(19)

Any choice of coordinates on T  M that preserves Eq. (17) is called canonical. Canonical coordinates could be defined alternatively as the coordinates that preserve the Hamiltonian equations (19). In terms of the canonical coordinates the Maximum Principle of Proposition 7 coincides with the original version in Proposition 1. Optimal Control Problems on Lie Groups Canonical coordinates are not suitable for all variational problems. For instance, variational problems in geometry and mechanics often have symmetries that govern the solutions; to take advantage of these symmetries it may be necessary to use coordinates that are compatible with the symmetries and which may not necessarily be canonical. That is particularly true for optimal problems on Lie groups that are either right or left invariant.

To elaborate further, assume that G denotes a Lie group (matrix group for simplicity) and that g denotes its Lie algebra. A vector field X on G is called left-invariant if for every g 2 G, X(g) D gA for some matrix A in g, i. e., X is determined by its tangent vector at the group identity. Similarly, right-invariant vector fields are defined as the right translations of matrices in g. Both the left and the right invariant vector fields form a frame on G, that is, Tg G D fgA: A 2 gg D fAg : A 2 gg for all g 2 G. Therefore, the tangent bundle TG can be realized as the product G  g either via the left translations (g; A) ! gA, or via the right translations (g; A) ! Ag. Similarly, the cotangent bundle T  G can be realized in two ways as G  g with g equal to the dual of g. In the left-invariant realization  2 Tg G is identified with (g; l) 2 G  g via the formula l(A) D (gA) for all A 2 g. For optimal control problems which are left-invariant it is natural to identify T  G as G  g via the left translations, and likewise identify T  G as G  g via the right translations for right-invariant problems. In both cases the realization T  G D G  g rules out canonical coordinates (assuming that G is non-abelian) and hence the Hamiltonian equations (19) take on a different form. For concreteness sake assume that T  G D G  g is realized via the left translations. Then it is natural to realize (T  G) as (G  g )  (g  g ) where ((g; l); (A; f )) 2 (G  g )  (g  g ) denotes tangent vector (A; f ) at the point (g; l). In this representation of T(T  G) the symplectic form ! is given by the following expression: !(g;l ) ((A1 ; f1 ); (A2 ; f2 )) D f2 (A1 )  f1 (A2 )  l([A1 ; A2 ]) :

(20)

Functions on Gg that are constant over the first factor, i. e., functions on g , are called left-invariant Hamiltonians. If H is left invariant then integral curves (g(t); l(t)) E are given of the corresponding Hamiltonian vector field H by dg (t) D g(t) dH(l(t)) ; dt

(21)

dl (t) D ad  (dH(l(t))(l(t))) dt where dH denotes the differential of H, and where ad  (A) : g ! g is given by (ad  (A)(l))(X) D l([A; X]), l 2 g , X 2 g, [12,13]. On semi-simple Lie groups linear functions l in g can be identified with matrices L in g via an invariant

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quadratic form h ; i so that Eqs. (21) become dg (t) D g(t) dH(l(t)) ; dt dL (t) D [dH(l(t); L(t))] : dt 1 2

(22)

For instance, the problem of minimizing the integral RT 2 0 ku(t)k dt over the trajectories of ! m X dg u i (t)A i ; u 2 Rm (23) (t) D g(t) A0 C dt iD1

with A0 ; A1 ; : : : ; A m matrices in g leads to the following Hamiltonians: P H 2i C H0 . 1. (Normal extrema) H D 12 m iD1 P 2. (Abnormal extrema) H D H0 C m iD1 u i (t)H i , subject to H i D 0, i D 1; : : : ; m, with each H i equal to the Hamiltonian lift of the left invariant vector field g ! gA i . In the left invariant representation of T  G each H i is a linear function on g , i. e., H i (l) D l(A i ) and consequently both Hamiltonians above are left-invariant. In the abnormal case dH D P A0 C m iD1 u i (t)A i , and ! m X dg u i (t)A i ; (t) D g(t) A0 C dt iD1 # (24) " m X dL u i (t)A i ; L(t) ; (t) D A0 C dt iD1

H1 (t) D H2 (t) D    D H m (t) D 0 ;

(25)

are the corresponding extremal equations. In the normal case the extremal controls are given by u i D H i , i D 1; : : : ; m, and the corresponding Hamiltonian system is given by Eqs. (22) with dH D A0 C Pm iD1 H i (t)A i . Left invariant Hamiltonian systems [Eqs. (21) and (22)] always admit certain functions, called integrals of motion, which are constant along their solutions. Hamiltonians which admit a “maximal” number of functionally independent integrals of motion are called integrable. For left invariant Hamiltonians on Lie groups there is a deep and beautiful theory directed to characterizing integrable systems [10,13,23]. This topic is discussed in more detail below. Poisson Bracket, Involution and Integrability Integrals of motion are most conveniently discussed in terms of the Poisson bracket. For that reason it becomes

necessary to introduce the notion of a Poisson manifold. A manifold M that admits a bilinear and skew symmetric form f ; g : C 1 (M)  C 1 (M) ! C 1 (M) that satisfies the Jacobi identity f f ; fg; hgg C fh; f f ; ggg C fg; fh; f gg D 0 and is a derivation f f g; hg D f fg; hg C gf f ; hg is called Poisson. It is known that every symplectic manifold is Poisson, and it is also known that every Poisson manifold admits a foliation in which each leaf is symplectic. In particular, the cotangent bundle T  M is a Poisson manifold E with f f ; hg() D ! ( fE(); h()) for all functions f and h. It is easy to show that F is an integral of motion for H if and only if fF; Hg D 0 from which it follows that F is an integral of motion for H if and only if H is an integral of motion for F. Functions F and H for which fF; Hg D 0 are also said to be in involution. A function H on a 2n dimensional symplectic manifold S is said to be integrable or completely integrable if there exist n functions '1 ; : : : ; 'n with '1 D H which are functionally independent and further satisfy f' i ; ' j g D 0 for each i and j. It is known that such a system of functions is dimensionally maximal. On Lie groups the dual g of the Lie algebra g inherits a Poisson structure from the symplectic form ! [Eq. (17)], with f f ; hg(l) D l([d f ; dh]) for any functions f and h on g . In the literature on Hamiltonian systems this structure is often called Lie–Poisson. The symplectic leaves induced by the Poisson–Lie structure coincide with the coadjoint orbits of G and the solutions of the equation dl  dt (t) D ad (dH(l(t))(l(t))) associated with Eqs. (21) evolve on coadjoint orbits of G. Most of the literature on integrable systems is devoted to integrability properties of the above equation considered as a Hamiltonian equation on a coadjoint orbit, or to its semi-simple counterpart dL dt (t) D [dH(L(t)); L(t)]. In this setting integrability is relative to the Poisson–Lie structure on each orbit, which may be of different dimensions. However, integrability can also be defined relative to the entire cotangent structure in which case the system is integrable whenever the number of independent integrals in involution is equal to the dimension of G. Leaving these subtleties aside, left-invariant Hamiltonian systems on Lie groups (22) always admit certain integrals of motion. They fall into two classes: 1. Hamiltonian lifts of right-invariant vector fields on G Poisson commute with any left-invariant Hamiltonian because right-invariant vector fields commute with leftinvariant vector fields. If X(g) D Ag denote a right invariant vector field then its Hamiltonian lift F A is equal to FA (L; g) D hL; g 1 Agi. In view of the formula fFA ; FB g D F[A;B] , the maximal number of functionally independent Hamiltonian lifts of right-invariant vector fields is equal to the rank of g. The rank of a semi-sim-

Maximum Principle in Optimal Control

ple Lie algebra g is equal to the dimension of a maximal abelian subalgebra of g. Such maximal abelian subalgebras are called Cartan subalgebras. 2. Each eigenvalue of L(t) is a constant of motion for dL dt (t) D [dH(L(t); L(t))]. If (L) and (L) denote eigenvalues of L then f; g D 0. Equivalently, the spectral functions ' k (L) D Trace(L k ) are in involution and Poisson commute with H. For three-dimensional Lie groups the above integrals are sufficient for complete integrability. For instance, every left-invariant Hamiltonian is completely integrable on SO3 [12]. In general, it is difficult to determine when the above integrals of motion can be extended to a completely integrable system of functions for a given Hamiltonian H [10,13,25]. Affirmative answers are known only in the exceptional cases in which there are additional symmetries. For instance, integrable system (15) is a particular case of the following more general situation. Example 8 Suppose that a semi-simple Lie group G admits an involutive automorphism  ¤ I that splits the Lie algebra g of G into a direct sum g D p C k with k D fA:  (A) D Ag and p D fA:  (A) D Ag. Such a decomposition is known as a Cartan decomposition and the following Lie algebraic conditions hold [p; p] D k;

[p; k] D p;

[k; k]  k :

(26)

Then L 2 g can be written as L D K C P with P 2 p and K 2 k. Assume that H(L) D 12 hK; Ki C hA; Pi, for some A 2 p where h ; i denotes a scalar multiple of the Cartan– Killing form that is positive definite on k. This Hamiltonian describes normal R Textrema for the problem of minimizing the integral 12 0 kU(t)k2 dt over the trajectories of dg (t) D g(t)(A C U(t)); dt

U(t) 2 k :

(27)

With the aid of the above decomposition the HamiltoE are given by nian equations (22) associated with H dg D g(A C K) ; dt dK D [A; P] ; dt dP D [A; K] C [K; P] : dt

(28)

dP Equations dK dt D [A; P]; dt D [A; K] C [K; P] admit two distinct types of integrals of motion. The first type is a con-

sequence of the spectral parameter representation dM D [N ; M ]; with M D P  K C (2  1)A dt 1 and N D (P  A) ; (29)  from which it follows that ;k D Trace(Mk ) are constants of motion for each  2 R and k 2 Z C . The second type follows from the observation that [A; P] is orthogonal (relative to the Cartan–Killing form) to the subalgebra k0 D fX 2 k : [A; X] D 0g. Hence the projection of K(t) on k0 is constant. In many situations these two types of integrals of motion are sufficient for complete integrability [23]. Abnormal Extrema and Singular Problems For simplicity of exposition the discussion will be confined to control affine systems written explicitly as m

X dx u i (t)X i (x) ; D X0 (x) C dt iD1

u D (u1 ; : : : ; u m ) 2 U ;

(30)

with X 0 ; X1 ; : : : ; X m smooth vector fields on a smooth manifold M. Recall that abnormal extrema are absolutely continuous curves (t) ¤ 0 in T  M satisfying Pm E E 1. d iD1 u i (t)H i ((t)) a. e. for some dt (t) D H0 ((t)) C E m denote the E 0; : : : ; H admissible control u(t) where H Hamiltonian vector fields associated with Hamiltonian lifts H i () D (X i (x)),  2 Tx M, i D 0; : : : ; m, and P Pm 2. H0 ((t)) C m iD1 u i (t)H i ((t))  H0 ((t)) C iD1 v i H i ((t)) a. e. for any v D (v1 ; : : : ; v m ) 2 U. Abnormal extremals satisfy the conditions of the Maximum Principle independently of any cost functional and can be studied in their own right. However, in practice, they are usually linked to some fixed optimization problem, such as for instance the problem of minimizing R 1 T 2 2 0 ku(t)k dt. In such a case there are several situations that may arise: 1. An optimal trajectory is only the projection of a normal extremal curve. 2. An optimal trajectory is the projection of both a normal and an abnormal extremal curve. 3. An optimal trajectory is only the projection of an abnormal curve (strictly abnormal case). When U is an open subset of R the maximality condition (2) implies that H i ((t)) D 0, i D 1; : : : ; m.

961

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Maximum Principle in Optimal Control

Then extremal curves which project onto optimal trajectories satisfy another set of constraints fH i ; H j g((t)) D (t)([X i ; X j ](x(t))) D 0, 1  i; j  m, known as the Goh condition in the literature on control theory [1]. Case (1) occurs when X1 (x(t)); : : : ; X m (x(t)); [X i ; X j ](x(t)), 1  i; j  m span Tx(t) M. The remaining cases occur in the situations where higher order Lie brackets among X0 ; : : : ; X m are required to generate the entire tangent space Tx(t) M along an optimal trajectory x(t). In the second case abnormal extrema can be ignored since every optimal trajectory is the projection of a normal extremal curve. However, that is no longer true in Case (3) as the following example shows. Example 9 (Montgomery [21]) In this example M D R3 with its points parametrized by cylindrical coordinates r; ; z.R The optimal control problem consists of minimizT ing 12 0 (u12 C u22 )dt over the trajectories of dx (t) D u1 (t)X 1 (x(t)) C u2 (t)X 2 (x(t)); dt   @ 1 @ @ ; X2 D  A(r) ; X1 D @r r @ @z 1 and A(r) D r2  2

where (31)

1 4 r ; 4

(32)

or more explicitly over the solutions of the following system of equations: dr D u1 ; dt

d u2 D ; dt r

dz u2 D  A(r) : dt r

(33)

Then normal extremal curves are integral curves of the Hamiltonian vector field associated to H D 12 (H12 C H22 ), with H1 D p r ; H2 D 1r (p  A(r)p z ), where p r ; p ; p z denote dual coordinates of co-vectors p defined by p D p r dr C p d C p z dz. @ An easy calculation shows that [X1 ; X2 ] D  dA dr @z 2

@ and [X1 ; [X 1 ; X2 ]] D  ddrA2 @z . Hence, X1 (x); X 2 (x); [X 1 ; X2 ](x) spans R3 except at x D (r; ; z) where dA dr D 0, that is, on the cylinder r D 1. Since [X 1 ; [X 1 ; X2 ]] ¤ 0 on r D 1, it follows that X1 ; X2 ; [X 1 ; X2 ]; [X 1 ; [X 1 ; X2 ]] span R3 at all points x 2 R3 . The helix r D 1; z( ) D A(1) ; 2 R, generated by u1 D 0; u2 D 1, is a locally optimal trajectory (shown in [21]). It is the projection of an abnormal extremal curve and not the projection of a normal extremal curve.

Trajectories of a control system that are the projections of a constrained Hamiltonian system are called singular [3]. For instance, the helix in the above example is singular. The terminology derives from the singularity

theory of mappings, and in the control theoretic context it is associated to the end point mapping E : u[0;T] ! x(x0 ; u; T), where x(x0 ; u; t) denotes the trajectory of dx dt D F(x(t); u(t)), x(x 0 ; u; 0) D x0 , with the controls u(t) in the class of locally bounded measurable with values in Rm . It is known that the singular trajectories are the projections of the integral curves (t) of the constrained Hamiltonian system, obtained by the Maximum Principle: d E D H((t); u(t)) ; dt

dH ((t); u(t)) D 0 ; du

(34)

where H(; u) D (F(x; u)),  2 Tx M. For an extensive theory of singular trajectories see [3]. Future Directions Since the original publications there has been a considerable effort to obtain the maximum principle under more general conditions and under different technical assumptions. This effort seems to be motivated by two distinct objectives: the first motivation is a quest for a high order maximum principle [4,16,17], while the second motivation is an extension of the maximum principle to differential inclusions and non-smooth problems [9,20,28]. Although there is some indication that the corresponding theoretical approaches do not lead to common theory [5], there still remains an open question how to incorporate these diverse points of view into a universal maximum principle. Bibliography Primary Literature 1. Agrachev AA, Sachkov YL (2005) Control theory from the geometric viewpoint. Encycl Math Sci 87. Springer, Heidelberg 2. Arnold VI (1989) Mathematical methods of classical mechanics. Graduate texts in Mathematics, vol 60. Springer, Heidelberg 3. Bonnard B, Chyba M (2003) Singular trajectories and their role in control. Springer, Heidelberg 4. Bianchini RM, Stefani G (1993) Controllability along a trajectory; a variational approach. SIAM J Control Optim 31:900–927 5. Bressan A (2007) On the intersection of a Clarke cone with a Boltyanski cone (to appear) 6. Berkovitz LD (1974) Optimal control theory. Springer, New York 7. Caratheodory C (1935) Calculus of variations. Teubner, Berlin (reprinted 1982, Chelsea, New York) 8. Clarke FH (1983) Optimization and nonsmooth analysis. Wiley Interscience, New York 9. Clarke FH (2005) Necessary conditions in dynamic optimization. Mem Amer Math Soc 816(173) 10. Fomenko AT, Trofimov VV (1988) Integrable systems on Lie algebras and symmetric spaces. Gordon and Breach 11. Gamkrelidze RV (1978) Principles of optimal control theory. Plenum Press, New York

Maximum Principle in Optimal Control

12. Jurdjevic V (1997) Geometric control theory. Cambridge Studies in Advanced Mathematics vol 51. Cambridge University Press, Cambridge 13. Jurdjevic V (2005) Hamiltonian systems on complex Lie groups and their homogeneous spaces. Mem Amer Math Soc 178(838) 14. Lee EB, Markus L (1967) Foundations of optimal control theory. Wiley, New York 15. Liu WS, Sussmann HJ (1995) Shortest paths for SR metrics of rank 2 distributions. Mem Amer Math Soc 118(564) 16. Knobloch H (1975) High order necessary conditions in optimal control. Springer, Berlin 17. Krener AJ (1977) The high order maximum principle and its application to singular extremals. SIAM J Control Optim 17:256– 293 18. Kupka IK (1990) The ubiquity of Fuller’s phenomenon. In: Sussmann HJ (ed) Non-linear controllability and Optimal control. Marcel Dekker, New York, pp 313–350 19. Marchal C (1973) Chattering arcs and chattering controls. J Optim Theory App 11:441–468 20. Morduchovich B (2006) Variational analysis and generalized differentiation: I. Basic analysis, II. Applications. Grundlehren Series (Fundamental Principle of Mathematical Sciences). Springer, Berlin 21. Montgomery R (1994) Abnormal minimizers. SIAM J Control Optim 32(6):1605–1620 22. Moser J (1980) Geometry of quadrics and spectral theory. In: The Chern Symposium 1979. Proceedings of the International symposium on Differential Geometry held in honor of S.S. Chern, Berkeley, California. Springer, pp 147–188 23. Perelomov AM (1990) Integrable systems of classical mechanics and Lie algebras. Birkhauser, Basel

24. Pontryagin LS, Boltyanski VG, Gamkrelidze RV, Mischenko EF (1962) The mathematical theory of optimal processes. Wiley, New York 25. Reiman AG, Semenov Tian-Shansky MA (1994) Group-theoretic methods in the theory of finite dimensional integrable systems. In: Arnold VI, Novikov SP (eds) Encyclopedia of Mathematical Sciences. Springer, Heidelberg 26. Sussmann HJ, Willems J (2002) The brachistochrone problem and modern control theory. In: Anzaldo-Meneses A, Bonnard B, Gauthier J-P, Monroy-Perez F (eds) Contemporary trends in non-linear geometric control theory and its applications. Proceedings of the conference on Geometric Control Theory and Applications held in Mexico City on September 4–6, 2000, to celebrate the 60th anniversary of Velimir Jurdjevic. World Scientific Publishers, Singapore, pp 113–165 27. Sussmann HJ () Set separation, approximating multicones and the Lipschitz maximum principle. J Diff Equations (to appear) 28. Vinter RB (2000) Optimal control. Birkhauser, Boston 29. Young LC (1969) Lectures in the calculus of variations and optimal control. Saunders, Philadelphia 30. Zelikin MI, Borisov VF (1994) Theory of chattering control with applications to aeronautics, Robotics, Economics, and Engineering. Birkhauser, Basel

Books and Reviews Bressan A (1985) A high-order test for optimality of bang-bang controls. SIAM J Control Optim 23:38–48 Griffiths P (1983) Exterior differential systems and the calculus of variations. Birkhauser, Boston

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Measure Preserving Systems

KARL PETERSEN Department of Mathematics, University of North Carolina, Chapel Hill, USA

tems theory. These systems arise from science and technology as well as from mathematics itself, so applications are found in a wide range of areas, such as statistical physics, information theory, celestial mechanics, number theory, population dynamics, economics, and biology.

Article Outline

Introduction: The Dynamical Viewpoint

Glossary Definition of the Subject Introduction: The Dynamical Viewpoint Where do Measure-Preserving Systems Come from? Construction of Measures Invariant Measures on Topological Dynamical Systems Finding Finite Invariant Measures Equivalent to a Quasi-Invariant Measure Finding -finite Invariant Measures Equivalent to a Quasi-Invariant Measure Some Mathematical Background Future Directions Bibliography

Sometimes introducing a dynamical viewpoint into an apparently static situation can help to make progress on apparently difficult problems. For example, equations can be solved and functions optimized by reformulating a given situation as a fixed point problem, which is then addressed by iterating an appropriate mapping. Besides practical applications, this strategy also appears in theoretical settings, for example modern proofs of the Implicit Function Theorem. Moreover, the introduction of the ideas of change and motion leads to new concepts, new methods, and even new kinds of questions. One looks at actions and orbits and instead of always seeking exact solutions begins perhaps to ask questions of a qualitative or probabilistic nature: what is the general behavior of the system, what happens for most initial conditions, what properties of systems are typical within a given class of systems, and so on. Much of the credit for introducing this viewpoint should go to Henri Poincaré [29].

Measure Preserving Systems

Glossary Dynamical system A set acted upon by an algebraic object. Elements of the set represent all possible states or configurations, and the action represents all possible changes. Ergodic A measure-preserving system is ergodic if it is essentially indecomposable, in the sense that given any invariant measurable set, either the set or its complement has measure 0. Lebesgue space A measure space that is isomorphic with the usual Lebesgue measure space of a subinterval of the set of real numbers, possibly together with countably or finitely many point masses. Measure An assignment of sizes to sets. A measure that takes values only between 0 and 1 assigns probabilities to events. Stochastic process A family of random variables (measurable functions). Such an object represents a family of measurements whose outcomes may be subject to chance. Subshift, shift space A closed shift-invariant subset of the set of infinite sequences with entries from a finite alphabet. Definition of the Subject Measure-preserving systems model processes in equilibrium by transformations on probability spaces or, more generally, measure spaces. They are the basic objects of study in ergodic theory, a central part of dynamical sys-

Two Examples Consider two particular examples, one simple and the other not so simple. Decimal or base 2 expansions of numbers in the unit interval raise many natural questions about frequencies of digits and blocks. Instead of regarding the base 2 expansion x D :x0 x1 : : : of a fixed x 2 [0; 1] as being given, we can regard it as arising from a dynamical process. Define T : [0; 1] ! [0; 1] by Tx D 2x mod 1 (the fractional part of 2x) and let P D fP0 D [0; 1/2); P1 D [1/2; 1]g be a partition of [0; 1] into two subintervals. We code the orbit of any point x 2 [0; 1] by 0’s and 1’s by letting x k D i if T k x 2 Pi ; k D 0; 1; 2; : : : . Then reading the expansion of x amounts to applying to the coding the shift transformation and projection onto the first coordinate. This is equivalent to following the orbit of x under T and noting which element of the partition P is entered at each time. Reappearances of blocks amount to recurrence to cylinder sets as x is moved by T, frequencies of blocks correspond to ergodic averages, and Borel’s theorem on normal numbers is seen as a special case of the Ergodic Theorem. Another example concerns Szemerédi’s Theorem [34], which states that every subset A N of the natural numbers which has positive upper density contains arbi-

Measure Preserving Systems

trarily long arithmetic progressions: given L 2 N there are s; m 2 N such that s; s C m; : : : ; s C (L  1)m 2 A. Szemerédi’s proof was ingenious, direct, and long. Furstenberg [15] saw how to obtain this result as a corollary of a strengthening of Poincaré’s Recurrence Theorem in ergodic theory, which he then proved. Again we have an apparently static situation: a set A N of positive density in which we seek arbitrarily long regularly spaced subsets. Furstenberg proposed to consider the characteristic function 1 A of A as a point in the space f0; 1gN of 0’s and 1’s and to form the orbit closure X of this point under the shift transformation . Because A has positive density, it is possible to find a shift-invariant measure  on X which gives positive measure to the cylinder set B D [1] D fx 2 X : x1 D 1g. Furstenberg’s Multiple Recurrence Theorem says that given L 2 N there is m 2 N such that (B \ T m B \    \ T (L1)m B) > 0. If y is a point in this intersection, then y contains a block of L 1’s, each at distance m from the next. And since y is in the orbit closure of 1 A , this block also appears in the sequence 1 A 2 f0; 1gN , yielding the result. Aspects of the dynamical argument remain in new combinatorial and harmonic-analytic proofs of the Szemerédi Theorem by T. Gowers [16,17] and T. Tao [35], as well as the extension to the (density zero) set of prime numbers by B. Green and T. Tao [18,36]. A Range of Actions Here is a sample of dynamical systems of various kinds: 1. A semigroup or group G acts on a set X. There is given a map G  X ! X, (g; x) ! gx, and it is assumed that g1 (g2 x) D (g1 g2 )x for all g1 ; g2 2 G; x 2 X (1) ex D x

2. 3.

4.

5.

for all x 2 X; if G has an identity element e. (2) A continuous linear operator T acts on a Banach or Hilbert space V. B is a Boolean -algebra (a set together with a zero element 0 and operations _; ^;0 which satisfy the same rules as ;; [; \;c (complementation) do for -algebras of sets); N is a -ideal in B (N 2 N ; B 2 B; B ^ N D B implies B 2 N ; and N1 ; N2 ;    2 N implies _1 nD1 N n 2 N ); and S : B ! B preserves the Boolean  -algebra operations and S N N . B is a Boolean -algebra,  is a countably additive positive (nonzero except on the zero element of B) function on B, and S : B ! B is as above. Then (B; ) is a measure algebra and S is a measure algebra endomorphism. (X; B; ) is a measure space (X is a set, B is a -algebra of subsets of X, and  : B ! [0; 1] is countably additive: If B1 ; B2 ;    2 B are pairwise disjoint, then

P1 ([1 nD1 (B n )); T : X ! X is measurnD1 B n ) D able (T 1 B B) and nonsingular ((B) D 0 implies (T 1 B) D 0 – or, more stringently,  and T 1 are equivalent in the sense of absolute continuity). 6. (X; B; ) is a measure space, T : X ! X is a one-toone onto map such that T and T 1 are both measurable (so that T 1 B D B D T B), and (T 1 B) D (B) for all B 2 B. (In practice often T is not one-to-one, or onto, or even well-defined on all of X, but only after a set of measure zero is deleted.) This is the case of most interest for us, and then we call (X; B; ; T) a measurepreserving system. We also allow for the possibility that T is not invertible, or that some other group (such as R or Zd ) or semigroup acts on X, but the case of Z actions will be the main focus of this article. 7. X is a compact metric space and T : X ! X is a homeomorphism. Then (X; T) is a topological dynamical system. 8. M is a compact manifold (C k for some k 2 [1; 1]) and T : M ! M is a diffeomorphism (one-to-one and onto, with T and T 1 both C k ). Then (M; T) is a smooth dynamical system. Such examples can arise from solutions of an autonomous differential equation given by a vector field on M. Recall that in Rn , an ordinary differential equation initial-value problem x 0 D f (x); x(0) D x0 has a unique solution x(t) as long as f satisfies appropriate smoothness conditions. The existence and uniqueness theorem for differential equations then produces a flow according to Tt x0 D x(t), satisfying TsCt x0 D Ts (Tt x0 ). Restricting to a compact invariant set (if there is one) and taking T D T1 (the time 1 map) gives us a smooth system (M; f ). Naturally there are relations and inclusions among these examples of actions. Often problems can be clarified by forgetting about some of the structure that is present or by adding desirable structure (such as topology) if it is not. There remain open problems about representation and realization; for example, taking into account necessary restrictions, which measure-preserving systems can be realized as smooth systems preserving a smooth measure? Sometimes interesting aspects of the dynamics of a smooth system can be due to the presence of a highly nonsmooth subsystem, for example a compact lowerdimensional invariant set. Thus one should be ready to deal with many kinds of dynamical systems. Where do Measure-Preserving Systems Come from? Systems in Equilibrium Besides physical systems, abstract dynamical systems can also represent aspects of biological, economic, or other

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Measure Preserving Systems

real-world systems. Equilibrium does not mean stasis, but rather that the changes in the system are governed by laws which are not themselves changing. The presence of an invariant measure means that the probabilities of observable events do not change with time. (But of course what happens at time 2 can still depend on what happens at time 1, or, for that matter, at time 3.) We consider first the example of the wide and important class of Hamiltonian systems. Many systems that model physical situations, for example a large number of ideal charged particles in a container, can be studied by means of Hamilton’s equations. The state of the entire system at any time is supposed to be specified by a vector (q; p) 2 R2n , the phase space, with q listing the coordinates of the positions of all of the particles, and p listing the coordinates of their momenta. We assume that there is a time-independent Hamiltonian function H(q; p) such that the time development of the system satisfies Hamilton’s equations: dq i @H ; D dt @p i

dp i @H ; D dt @q i

i D 1; : : : ; n :

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Often the Hamiltonian function is the sum of kinetic and potential energy: H(q; p) D K(p) C U(q) :

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The potential energy U(q) may depend on interactions among the particles or with an external field, while the kinetic energy K(p) depends on the velocities and masses of the particles. As discussed above, solving these equations with initial state (q; p) for the system produces a flow (q; p) ! Tt (q; p) in phase space. According to Liouville’s Theorem, this flow preserves Lebesgue measure on R2n . Calculating dH/dt by means of the Chain Rule and using Hamilton’s equations shows that H is constant on orbits of the flow, and thus each set of constant energy X(H0 ) D f(q; p) : H(q; p) D H0 g is an invariant set. Thus one should consider the flow restricted to the appropriate invariant set. It turns out that there are also natural invariant measures on the sets X(H0 ), namely the ones given by rescaling the volume element dS on X(H0 ) by the factor 1/jjOHjj. For details, see [25]. Systems in equilibrium can also be hiding inside systems not in equilibrium, for example if there is an attractor supporting an SRB measure (for Sinai, Ruelle, and Bowen) (for definitions of the terms used here and more explanations, see the article in this collection by A. Wilkinson). Suppose that T : M ! M is a diffeomorphism on a compact manifold as above, and that m is a version of Lebesgue

measure on M, say given by a smooth volume form. We consider m to be a “physical measure”, corresponding to laboratory measurements of observable quantities, whose values can be determined to lie in certain intervals in R. Quite possibly m is not itself invariant under T, and an experimenter might observe strange or chaotic behavior whenever the state of the system gets close to some compact invariant set X. The dynamics of T restricted to X can in fact be quite complicated – maybe a full shift, which represents completely undeterministic behavior (for example if there is a horseshoe present), or a shift of finite type, or some other complicated topological dynamical system. Possibly m(X) D 0, so that X is effectively invisible to the observer except through its effects. It can happen that there is a T-invariant measure  supported on X such that 1 n

n1 X

mT k !  weak ;

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kD0

and then the long-term equilibrium dynamics of the system is described by (X; T; ). For a recent survey on SRB measures, see [39]. Stationary Stochastic Processes A stationary process is a family f f t : t 2 Tg of random variables (measurable functions) on a probability space (˝; F ; P). Usually T is Z; N, or R. For the remainder of this section let us fix T D Z (although the following definition could make sense for T any semigroup). We say that the process f f n : n 2 Zg is stationary if its finitedimensional distributions are translation invariant, in the sense that for each r D 1; 2; : : : , each n1 ; : : : ; nr 2 Z, each choice of Borel sets B1 ; : : : ; Br R, and each s 2 Z, we have Pf! : f n 1 (!) 2 B1 ; : : : ; f n r (!) 2 Br g D Pf! : f n 1 Cs (!) 2 B1 ; : : : ; f n r Cs (!) 2 B r g :

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The f n represent measurements made at times n of some random phenomenon, and the probability that a particular finite set of measurements yield values in certain ranges is supposed to be independent of time. Each stationary process f f n : n 2 Zg on (˝; F ; P) corresponds to a shift-invariant probability measure  on the set RZ (with its Borel  -algebra) and a single observable, namely the projection 0 onto the 0’th coordinate, as follows. Define  : ˝ ! RZ

by (!) D ( f n (!))1 1 ;

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and for each Borel set E RZ , define (E) D P( 1 E). Then examining the values of  on cylinder sets – for Borel

Measure Preserving Systems

B1 ; : : : ; Br R, fx 2 RZ : x n i 2 B i ; i D 1; : : : ; rg D Pf! 2 ˝ : f n i (!) 2 B i ; i D 1; : : : ; rg

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– and using stationarity of ( f n ) shows that  is invariant under . Moreover, the processes ( f n ) on ˝ and 0 ı  n on RZ have the same finite-dimensional distributions, so they are equivalent for the purposes of probability theory. Construction of Measures We review briefly (following [33]) the construction of measures largely due to C. Carathéodory [8], with input from M. Fréchet [13], H. Hahn [19], A. N. Kolmogorov [26], and others, then discuss the application to construction of measures on shift spaces and of stochastic processes in general. The Carathéodory Construction A semialgebra is a family S of subsets of a set X which is closed under finite intersections and such that the complement of any member of S is a finite disjoint union of members of S. Key examples are 1. the family H of half-open subintervals [a; b) of [0; 1); 2. in the space X D AZ of doubly infinite sequences on a finite alphabet A, the family C of cylinder sets (determined by fixing finitely many entries) fx 2 AZ : x n 1 D a1 ; : : : ; x n r D ar g ;

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3. the family C1 of anchored cylinder sets fx 2 AN : x1 D a1 ; : : : ; x r D ar g

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in the space X D AN of one-sided infinite sequences on a finite alphabet A. An algebra is a family of subsets of a set X which is closed under finite unions, finite intersections, and complements. A  -algebra is a family of subsets of a set X which is closed under countable unions, countable intersections, and complements. If S is a semialgebra of subsets of X, the algebra A(S) generated by S is the smallest algebra of subsets of X which contains S. A(S) is the intersection of all the subalgebras of the set 2 X of all subsets of X and consists exactly of all finite disjoint unions of elements of S. Given an algebra A, the -algebra B(A) generated by A is the smallest -algebra of subsets of X which contains A. A nonnegative set function on S is a function  : S ! [0; 1] such that (;) D 0 if ; 2 S. We say that such a  is

 finitely additive if whenever S1 ; : : : ; S n 2 S are pairwise disjoint and S D [niD1 S i 2 S, we have (S) D Pn iD1 (S i );  countably additive if whenever S1 ; S2    2 S are pairwise disjoint and S D [1 iD1 S i 2 S, we have (S) D P1 (S ); and i iD1  countably subadditive if whenever S1 ; S2    2 S and P1 S D [1 iD1 (S i ). iD1 S i 2 S, we have (S)  A measure is a countably additive nonnegative set function defined on a  -algebra. Proposition 1 Let S be a semialgebra and  a nonnegative set function on S. In order that  have an extension to a finitely additive set function on the algebra A(S) generated by S, it is necessary and sufficient that  be finitely additive on S. Proof 1 The stated condition is obviously necessary. Conversely, given  which is finitely additive on S, it is natural to define (

n [

Si ) D

iD1

n X

(S i )

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iD1

whenever A D [niD1 S i (with the S i pairwise disjoint) is in the algebra A(S) generated by S. It is necessary to verify that  is then well defined on A(S), since each element of A(S) may have more than one representation as a finite disjoint union of members of S. But, given two such representations of a single set A, forming the common refinement and applying finite additivity on S shows that  so defined assigns the same value to A both times. Then finite additivity on A(S) of the extended  is clear.  Proposition 2 Let S be a semialgebra and  a nonnegative set function on S. In order that  have an extension to a countably additive set function on the algebra A(S) generated by S, it is necessary and sufficient that  be (i) finitely additive and (ii) countably subadditive on S. Proof 2 Conditions (i) and (ii) are clearly necessary. If  is finitely additive on S, then by Proposition 1  has an extension to a finitely additive nonnegative set function, which we will still denote by , on A(S). Let us see that this extension  is countably subadditive on A(S). Suppose that A1 ; A2 ;    2 A(S) are pairwise disjoint and their union A 2 A(S). Then A is a finite disjoint union of sets in S, as is each A i : AD

AD

1 [ iD1 m [ jD1

A i ; each A i D

ni [

Si k ;

kD1

R j ; each A i 2 A(S) ; each S i k ; R j 2 S :

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Measure Preserving Systems

Since each R j 2 S, by countable subadditivity of  on S, and using R j D R j \ A, (R j ) D (

ni 1 [ [

S i k \R j ) 

iD1 kD1

ni 1 X X

(S i k \R j ); (13)

iD1 kD1

and hence, by finite additivity of  on A(S), (A) D

m X

(R j ) 

jD1

D

ni X 1 X m X

(S i k \ R j )

iD1 kD1 jD1

ni 1 X X

(S i k ) D

iD1 kD1

1 X

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(A i ) :

Theorem 1 In order that a nonnegative set function  on an algebra A of subsets of a set X have an extension to a (countably additive) measure on the -algebra B(A) generated by A, it is necessary and sufficient that  be countably additive on A. Here is a sketch of how the extension can be constructed. Given a countably additive nonnegative set function  on an algebra A of subsets of a set X, one defines the outer measure  that it determines on the family 2 X of all subsets of X by 1 X

In this way, beginning with the semialgebra H of halfopen subintervals of [0; 1) and [a; b) D ba, one arrives at Lebesgue measure on the  -algebra M of Lebesgue measurable sets and on its sub- -algebra B(H ) of Borel sets.

iD1

Now finite additivity of  on an algebra A implies that  is monotonic on the algebra: if A; B 2 A and A B, then (A)  (B). Thus if A1 ; A2 ;    2 A(S) are pairwise disjoint and their union A 2 A(S), then for each n P we have niD1 (A i ) D ([niD1 A i )  (A), and hence P1  iD1 (A i )  (A).

 (E) D inf f

of all subsets of X as above. Then the family M of  measurable subsets of X is a  -algebra containing A (and hence B(A)) and all subsets of X which have  measure 0. The restriction  jM is a (countably additive) measure which agrees on A with . If  is  -finite on A (so that there are X1 ; X2 ;    2 A with (X i ) < 1 for all i and X D [1 iD1 X i ), then  on B(A) is the only extension of  on A to B(A).

(A i ) : A i 2 A; E [1 iD1 A i g : (15)

Measures on Shift Spaces The measures that determine stochastic processes are also frequently constructed by specifying data on a semialgebra of cylinder sets. Given a finite alphabet A, denote by ˝(A) D AZ and ˝ C (A) D AN the sets of two and onesided sequences, respectively, with entries from A. These are compact metric spaces, with d(x; y) D 2n when n D inffjkj : x k ¤ y k g. In both cases, the shift transformation  defined by ( x)n D x nC1 for all n is a homeomorphism. Suppose (cf. [3]) that for every k D 1; 2; : : : we are given a function g k : Ak ! [0; 1], and that these functions satisfy, for all k, 1. g k (B)  0 for all B 2 Ak ; P g (Bi) D g k (B) for all B 2 Ak ; 2. P i2A kC1 3. i2A g 1 (i) D 1. Then Theorems 1 and 2 imply that there is a unique measure  on the Borel subsets of ˝ C (A) such that for all k D 1; 2; : : : and B 2 Ak

iD1

fx 2 ˝ C (A) : x1 : : : x k D Bg D g k (B) :



is a nonnegative, countably subadditive, monoThen tonic set function on 2 X . Define a set E to be  -measurable if for all T X,  (T) D  (T \ E) C  (T \ E c ) :

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This ingenious definition can be partly motivated by noting that if  is to be finitely additive on the family M of  -measurable sets, which should contain X, then at least this condition must hold when T D X. It is amazing that then this definition readily, with just a little set theory and a few "’s, yields the following theorem. Theorem 2 Let  be a countably additive nonnegative set function on an algebra A of subsets of a set X, and let  be the outer measure that it determines on the family 2 X

4.

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If in addition the g k also satisfy P i2A g kC1 (iB) D g k (B) for all k D 1; 2; : : : and all B 2 Ak , then there is a unique shift-invariant measure  on the Borel subsets of ˝ C (A) (also ˝(A)) such that for all n, all k D 1; 2; : : : and B 2 Ak fx 2 ˝ C (A) : x n : : : x nCk1 D Bg D g k (B) : (18) This follows from the Carathéodory theorem by beginning with the semialgebra C1 of anchored cylinder sets or the semialgebra C of cylinder sets determined by finitely many consecutive coordinates, respectively.

There are two particularly important examples of this construction. First, let our finite alphabet be A D f0; : : : ;

Measure Preserving Systems

d  1g, and let p D (p0 ; : : : ; p d1 ) be a probability vecP tor: all p i  0 and d1 iD0 p i D 1. For any block B D b1 : : : b k 2 Ak , define g k (B) D p b 1 : : : p b k :

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The resulting measure  p is the product measure on ˝(A) D AZ of infinitely many copies of the probability measure determined by p on the finite sample space A. The measure-preserving system (˝; B; ; ) (with B the  -algebra of Borel subsets of ˝(A), or its completion), is denoted by B(p) and is called the Bernoulli system determined by p. This system models an infinite number of independent repetitions of an experiment with finitely many outcomes, the ith of which has probability p i on each trial. This construction can be generalized to model stochastic processes which have some memory. Again let A D f0; : : : ; d  1g, and let p D (p0 ; : : : ; p d1 ) be a probability vector. Let P be a d  d stochastic matrix with rows and columns indexed by A. This means that all entries of P are nonnegative, and the sum of the entries in each row is 1. We regard P as giving the transition probabilities between pairs of elements of A. Now we define for any block B D b 1 : : : b k 2 Ak

frequently T D Z; N; R; Zd , or Rd ). We give a brief description, following [4]. Let T be an arbitrary index set. We aim to produce a R-valued stochastic process indexed by T, that is to say, a Borel probability measure P on ˝ D RT , which has prespecified finite-dimensional distributions. Suppose that for every ordered k-tuple t1 ; : : : ; t k of distinct elements of T we are given a Borel probability measure  t1 :::t k on R k . Denoting f 2 RT also by ( f t : t 2 T), we want it to be the case that, for each k, each choice of distinct t1 ; : : : t k 2 T, and each Borel set B R k , Pf( f t : t 2 T) : ( f t1 ; : : : ; f t k ) 2 Bg D  t1 :::t k (B) : (22) For consistency, we will need, for example, that  t1 t2 (B1  B2 ) D  t2 t1 (B2  B1 ) ; since

Pf( f t1 ; f t2 ) 2 A1  A2 g D Pf( f t2 ; f t1 ) 2 A2  A1 g : (24) Thus we assume: 1. For any k D 1; 2; : : : and permutation  of 1; : : : ; k, if  : R k ! R k is defined by  (x1 ; : : : ; x k ) D (x1 ; : : : ; x k ) ;

g k (B) D p b 1 Pb 1 b 2 Pb 2 b 3 : : : Pb k1 b k :

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Using the g k to define a nonnegative set function  p;P on the semialgebra C1 of anchored cylinder subsets of ˝ C (A), one can verify that  p;P is (vacuously) finitely additive and countably subadditive on C1 and therefore extends to a measure on the Borel -algebra of ˝ C (A), and its completion. The resulting stochastic process is a (one-step, finite-state) Markov process. If p and P also satisfy pP D p ;

(21)

then condition 4. above is satisfied, and the Markov process is stationary. In this case we call the (one or two-sided) measure-preserving system the Markov shift determined by p and P. Points in the space are conveniently pictured as infinite paths in a directed graph with vertices A and edges corresponding to the nonzero entries of P. A process with a longer memory, say of length m, can be produced by repeating the foregoing construction after recoding with a sliding block code to the new alphabet Am : for each ! 2 ˝(A), let ((!))n D !n !nC1 : : : !nCm1 2 Am . The Kolmogorov Consistency Theorem There is a generalization of this method to the construction of stochastic processes indexed by any set T. (Most

(23)

(25)

then for all k and all Borel B R k  t1 :::t k (B) D  t1 :::t k (1 B) :

(26)

Further, since leaving the value of one of the f t j free does not change the probability in (22), we also should have 2. For any k D 1; 2; : : : , distinct t1 ; : : : ; t k ; t kC1 2 T, and Borel set B R k ,  t1 :::t k (B) D  t1 :::t k t kC1 (B  R) :

(27)

Theorem 3 (Kolmogorov Consistency Theorem [26]) Given a system of probability measures  t1 :::t k as above indexed by finite ordered subsets of a set T, in order that there exist a probability measure P on RT satisfying (22) it is necessary and sufficient that the system satisfy 1. and 2. above. When T D Z; R, or N, as in the example with the g k above, the problem of consistency with regard to permutations of indices does not arise, since we tacitly use the order in T in specifying the finite-dimensional distributions. In case T is a semigroup, by adding conditions on the given data  t1 :::t k it is possible to extend this construction also to produce stationary processes indexed by T, in parallel with the above constructions for T D Z or N.

969

970

Measure Preserving Systems

Invariant Measures on Topological Dynamical Systems

Ergodicity and Unique Ergodicity

Existence of Invariant Measures Let X be a compact metric space and T : X ! X a homeomorphism (although usually it is enough just that T be a continuous map). Denote by C (X) the Banach space of continuous real-valued functions on X with the supremum norm and by M(X) the set of Borel probability measures on X. Given the weak topology, according to which Z Z f n d ! f d n !  if and only if X

(28)

M(X) is a convex subset of the dual space C (X) of all continuous linear functionals from C (X) to R. With the

weak topology it is metrizable and (by Alaoglu’s Theorem) compact. Denote by MT (X) the set of T-invariant Borel probability measures on X. A Borel probability measure  on X is in M(X) if and only if (T 1 B) D (B) for all Borel sets B X ;

X

Proposition 3 For every compact topological dynamical system (X; T) (with X not empty) there is always at least one T-invariant Borel probability measure on X. Proof 3 Let m be any Borel probability measure on X. For example, we could pick a point x0 2 X and let m be the point mass ıx 0 at x0 defined by ıx 0 ( f ) D f (x0 ) for all f 2 C (X) :

(31)

Form the averages 1 n

n1 X

mT i ;

(32)

iD0

which are also in M(X). By compactness, fA n mg has a weak cluster point , so that there is a subsequence A n k m !  weak :

(33)

Then  2 M(X); and  is T-invariant, because for each f 2 C (X) 1 j( f k!1 n k

j( f T)  ( f )j D lim

Equivalently (using the Ergodic Theorem), (X; B; ; T) is ergodic if and only if for each f 2 L1 (X; B; ) 1 n

n1 X kD1

f (T k x) !

Z f d almost everywhere : (36) X

It can be shown that the ergodic measures on (X; T) are exactly the extreme points of the compact convex set MT (X), namely those  2 MT (X) for which there do not exist 1 ; 2 2 MT (x) with 1 ¤ 2 and s 2 (0; 1) such that  D s1 C (1  s) 2 :

(37)

(29)

equivalently, Z Z f ıT d D f d for all f 2 C (X): (30) ( f T) D

An m D

B 2 B ; (T 1 B4B) D 0 implies (B) D 0 or 1 : (35)

X

for all f 2 C (X) ;

X

Among the T-invariant measures on X are the ergodic ones, those for which (X; B; ; T) (with B the  -algebra of Borel subsets of X) forms an ergodic measure-preserving system. This means that there are no proper T-invariant measurable sets:

T n k )  ( f )j D 0 ; (34)

both terms inside the absolute value signs being bounded. 

The Krein-Milman Theorem states that in a locally convex topological vector space such as C (X) every compact convex set is the closed convex hull of its extreme points. Thus every nonempty such set has extreme points, and so there always exist ergodic measures for (X; T). A topological dynamical system (X; T) is called uniquely ergodic if there is only one T-invariant Borel probability measure on X, in which case, by the foregoing discussion, that measure must be ergodic. There are many examples of topological dynamical systems which are uniquely ergodic and of others which are not. For now, we just remark that translation by a generator on a compact monothetic group is always uniquely ergodic, while group endomorphisms and automorphisms tend to be not uniquely ergodic. Bernoulli and (nonatomic) Markov shifts are not uniquely ergodic, because they have many periodic orbits, each of which supports an ergodic measure. Finding Finite Invariant Measures Equivalent to a Quasi-Invariant Measure Let (X; B; m) be a  -finite measure space, and suppose that T : X ! X is an invertible nonsingular transformation. Thus we assume that T is one-to-one and onto (maybe after a set of measure 0 has been deleted), that T and T 1 are both measurable, so that T B D B D T 1 B ;

(38)

Measure Preserving Systems

and that T and T 1 preserve the -ideal of sets of measure 0: m(B) D 0 if and only if m(T

1

B) D 0

if and only if m(TB) D 0 :

(B) D (39)

In this situation we say that m is quasi-invariant for T. A nonsingular system (X; B; m; T) as above may model a nonequilibrium situation in which events that are impossible (measure 0) at any time are also impossible at any other time. When dealing with such a system, it can be useful to know whether there is a T-invariant measure  that is equivalent to m (in the sense of absolute continuity – they have the same sets of measure 0–in which case we write  m), for then one would have available machinery of the measure-preserving situation, such as the Ergodic Theorem and entropy in their simplest forms. Also, it is most useful if the measures are -finite, so that tools such as the Radon-Nikodym and Tonelli-Fubini theorems will be available. We may assume that m(X) D 1. For if X D [1 iD1 X i with each X i 2 B and m(X i ) < 1, disjointifying (replace X i by X i n X i1 for i  2) and deleting any X i that have measure 0, we may replace m by 1 X iD1

mj X i : 2 i m(X i )

(40)

Definition 1 Let (X; B; m) be a probability space and T : X ! X a nonsingular transformation. We say that A; B 2 B are T-equivalent, and write A T B, if there are two sequences of pairwise disjoint sets, A1 ; A2 ; : : : and B1 ; B2 ; : : : and integers n1 ; n2 ; : : : such that

AD

1 [ iD1

Ai ; B D

1 [

1 X D [1 iD1 X i ; B D [ iD1 B i , then

ni

B i ; and T A i D B i for all i :

iD1

(41)

1 X

(B i ) D

iD1

D

1 X

(T n i X i )

iD1

1 X

(42)

(X i ) D (X) ;

iD1

so that (X n B) D 0 and hence m(X n B) D 0. For the converse, one tries to show that if X is T-nonshrinkable, then for each A 2 B the following limit exists: 1 n!1 n

lim

n1 X

m(T k A) :

(43)

kD0

The condition of T-nonshrinkability not being easy to check, subsequent authors gave various necessary and sufficient conditions for the existence of a finite equivalent invariant measure: 1. Dowker [11]. Whenever A 2 B and m(A) > 0, lim inf n!1 m(T n A) > 0. 2. Calderón [6]. Whenever A 2 B and m(A) > 0; P k lim in f n!1 n1 1 kD0 m(T A) > 0. 3. Dowker [12]. Whenever A 2 B and m(A) > 0, P k lim supn!1 n1 1 kD0 m(T A) > 0. Hajian and Kakutani [20] showed that the condition m(A) > 0 implies lim sup m(T n A) > 0

(44)

n!1

is not sufficient for existence of a finite equivalent invariant measure. They also gave another necessary and sufficient condition. Definition 3 A measurable set W X is called wandering if the sets T i W; i 2 Z, are pairwise disjoint. W is called weakly wandering if there are infinitely many integers n i such that T n i W and T n j W are disjoint whenever ni ¤ n j .

Definition 2 Let (X; B; m; T) be as above. A measurable set A X is called T-nonshrinkable if A is not T-equivalent to any proper subset: whenever B A and B T A we have m(A n B) D 0.

Theorem 5 (Hajian-Kakutani [20]) Let (X; B; m) be a probability space and T : X ! X a nonsingular transformation. There exists a finite invariant measure  m if and only if there are no weakly wandering sets of positive measure.

Theorem 4 (Hopf [23]) Let (X; B; m) be a probability space and T : X ! X a nonsingular transformation. There exists a finite invariant measure  m if and only if X is T-nonshrinkable.

Finding  -finite Invariant Measures Equivalent to a Quasi-Invariant Measure

Proof 4 We present just the easy half. If  m is T-invariant and X T B, with corresponding decompositions

While being able to replace a quasi-invariant measure by an equivalent finite invariant measure would be great, it

First Necessary and Sufficient Conditions

971

972

Measure Preserving Systems

may be impossible, and then finding a -finite equivalent measure would still be pretty good. Hopf’s nonshrinkability condition was extended to the -finite case by Halmos: Theorem 6 (Halmos [21]) Let (X; B; m) be a probability space and T : X ! X a nonsingular transformation. There exists a -finite invariant measure  m if and only if X is a countable union of T-nonshrinkable sets. Another necessary and sufficient condition is given easily in terms of solvability of a cohomological functional equation involving the Radon-Nikodym derivative w of mT with respect to m, defined by Z

w dm for all B 2 B :

m(TB) D

(45)

B

Proposition 4 ([21]) Let (X; B; m) be a probability space and T : X ! X a nonsingular transformation. There exists a  -finite invariant measure  m if and only if there is a measurable function f : X ! (0; 1) such that f (Tx) D w(x) f (x) a.e.

(46)

Proof 5 If  m is -finite and T-invariant, let f D dm/d be the Radon-Nikodym derivative of m with respect to , so that Z

f d for all B 2 B :

m(B) D

(47)

B

Then for all B 2 B, since T D , f d D f T d ; while also ZB ZT B w dm D w f dm ; m(TB) D m(TB) D

B

Definition 4 A nonsingular system (X; B; m; T) (with m(X) D 1) is called conservative if there are no wandering sets of positive measure. It is called completely dissipative if there is a wandering set W such that ! 1 [ i T W D m(X) : (51) m iD1

Note that if (X; B; m; T) is completely dissipative, it is easy to construct a  -finite equivalent invariant measure. With W as above, define  D m on W and push  along the orbit of W, letting  D mT n on each T n W. We want to claim that this allows us to restrict attention to the conservative case, which follows once we know that the system splits into a conservative and a completely dissipative part. Theorem 7 (Hopf Decomposition [24]) Given a nonsingular map T on a probability space (X; B; m), there are disjoint measurable sets C and D such that 1. X D C [ D; 2. C and D are invariant: TC D C D T 1 C, T D D D D T 1 D; 3. TjC is conservative; 4. If D ¤ ;, then Tj D is completely dissipative. Proof 6 Assume that the family W of wandering sets with positive measure is nonempty, since otherwise we can take C D X and D D ;. Partially order W by W1  W2 if m(W1 n W2 ) D 0 :

Z

Z

Conservativity and Recurrence

(48)

B

so that f T D w f a.e. Conversely, given such an f , let

We want to apply Zorn’s Lemma to find a maximal element in W . Let fW :  2 g be a chain (linearly ordered subset) in W . Just forming [2 W may result in a nonmeasurable set, so we have to use the measure to form a measure-theoretic essential supremum of the chain. So let s D supfm(W ) :  2 g ;

Z (B) D B

1 f

dm for all B 2 B :

Then for all B 2 B Z Z 1 1 (TB) D dm D f f T dmT TB B Z Z 1 1 D w dm D fT f dm D (B) : B

(49)

(54)

and let

B



(53)

so that s 2 (0; 1]. If there is a  such that m(W ) D s, let W be that W . Otherwise, for each k choose  k 2  so that s k D m(Wk ) " s ;

(50)

(52)

WD

1 [ kD1

Wk :

(55)

Measure Preserving Systems

We claim that in either case W is an upper bound for the chain fW :  2 g. In both cases we have m(W) D s. Note that if ;  2  are such that m(W )  m(W ), then W  W . For if W  W , then m(W n W ) D 0, and thus m(W ) D m(W \ W ) C m(W n W ) D m(W \ W ) D m(W )  m(W ) ; (56) so that m(W nW ) D 0; W  W , and hence W D W . Thus in the first case W 2 W is an upper bound for the chain. In the second case, by discarding the measure 0 set 1 [

(Wk n WkC1 ) ;

B D

1 [

T i B

(58)

iD1

is wandering. In fact much more is true.

 m(W \ W ) C m(W n W )

ZD

nonsingular system is recurrent: almost every point of each set of positive measure returns at some future time to that set. This is easy to see, because for each B 2 B, the set

(57)

kD1

we may assume that W is the increasing union of the Wk . Then W  Wk for all k, and W is wandering: if some T n W \ W ¤ ;, then there must be a k such that T n Wk \ Wk ¤ ;. Moreover, W  W for all  2 . For let  2  be given. Choose k with s k D m(Wk ) > m(W ). By the above, we have Wk  W . Since W is the increasing union of the Wk , we have W  Wk for all k. Therefore W  W , and W is an upper bound in W for the given chain. By Zorn’s Lemma, there is a maximal element W  i  in W . Then D D [1 iD1 T W is T-invariant, Tj D is completely dissipative, and C D X n D cannot contain any wandering set of positive measure, by maximality of W  , so TjC is conservative.  Because of this decomposition, when looking for a -finite equivalent invariant measure we may assume that the nonsingular system (X; B; m; T) is conservative, for if not we can always construct one on the dissipative part. Remark 1 If (X; B; m) is nonatomic and T : X ! X is nonsingular, invertible, and ergodic, in the sense that if A 2 B satisfies T 1 A D A D TA then either m(A) D 0 or m(Ac ) D 0, then T is conservative. For if W is a wandering set of positive measure, taking any A W with i 0 < m(A) < m(W) and forming [1 iD1 T A will produce an invariant set of positive measure whose complement also has positive measure. We want to reduce the problem of existence of a -finite equivalent invariant measure to that of a finite one by using first-return maps to sets of finite measure. For this purpose it will be necessary to know that every conservative

Theorem 8 ([21]) For any nonsingular system (X; B; m; T) the following properties are equivalent: 1. The system is incompressible: for each B 2 B such that T 1 B B, we have m(B n T 1 B) D 0. 2. The system is recurrent; for each B 2 B, with B defined as above, m(B n B ) D 0. 3. The system is conservative: there are no wandering sets of positive measure. 4. The system is infinitely recurrent: for each B 2 B, almost every point of B returns to B infinitely many times, equivalently, m Bn

1 [ 1 \

! T

i

B Dm Bn

nD0 iDn

1 \

! T

n 

B

D 0:

nD0

(59) There is a very slick proof by F. B. Wright [38] of this result in the even more general situation of a Boolean  -algebra homomorphism (reproduced in [28]). Using First-Return Maps, and Counterexamples to Existence Now given a nonsingular conservative system (X; B; m; T) and a set B 2 B, for each x 2 B there is a smallest n B (x)  1 such that T n B (x) 2 B :

(60)

We define the first-return map TB : B ! B by TB (x) D T n B (x) (x) for all x 2 B :

(61)

Using derivative maps, it is easy to reduce the problem of existence of a  -finite equivalent invariant measure to that of existence of finite equivalent invariant measures, in a way. Theorem 9 (see [14]) Let T be a conservative nonsingular transformation on a probability space (X; B; m). Then there is a  -finite T-invariant measure  m if and only if there is an increasing sequence of sets B n 2 B with [1 nD1 B n D X such that for each n the first-return map TB n

973

974

Measure Preserving Systems

has a finite invariant measure equivalent to m restricted to B n . Proof 7 Given a -finite equivalent invariant measure , let the B n be sets of finite -measure that increase to X. Conversely, given such a sequence B n with finite invariant measures n for the first-return maps TB n , extend 1 in the obvious way to an (at least -finite) invariant measure i on the full orbit A1 D [1 iD1 T B1 . Then replace B2 by  B2 n A1 , and continue. There are many more checkable conditions for existence of a -finite equivalent invariant measure in the literature. There are also examples of invertible ergodic nonsingular systems for which there does not exist any -finite equivalent invariant measure due to Ornstein [27] and subsequently Chacon [9], Brunel [5], L. Arnold [2], and others. Invariant Measures for Maps of the Interval or Circle Finally we mention sample theorems from a huge array of such results about existence of finite invariant measures for maps of an interval or of the circle. Theorem 10 (“Folklore Theorem” [1]) Let X D (0; 1) and denote by m Lebesgue measure on X. Let T : X ! X be a map for which there is a finite or countable partition ˛ D fA i g of X into half-open intervals [a i ; b i ) satisfying the following conditions. Denote by A0i the interior of each interval A i . Suppose that 1. for each i, T : A0i ! X is one-to-one and onto; 2. T is C 2 on each A0i ; 3. there is an n such that inf inf j(T n )0 (x)j > 1 ; i x2A0 i

(62)

4. for each i, sup j x;y;z2A0i

T 00 (x) j<1: T 0 (y)T 0 (z)

(63)

Then for each measurable set B, limn!1 m(T n B) D (B) exists and defines the unique T-invariant ergodic probability measure on X that is equivalent to m. Moreover, the partition ˛ is weakly Bernoulli for T, so that the natural extension of T is isomorphic to a Bernoulli system. A key example to which the theorem applies is that of the Gauss map Tx D 1/x mod 1 with the partition for which A i D [1/(i C 1); 1/i) for each i D 1; 2; : : : . Coding orbits to N N by letting a(x) D a i if x 2 A i carries T to

the shift on the continued fraction expansion [a1 ; a2 ; : : : ] of x. It was essentially known already to Gauss that T preserves the measure whose density with respect to Lebesgue measure is 1/((1 C x) log 2). Theorem 11 (see [22]) Let X D S 1 , the unit circle, and let T : X ! X be a (noninvertible) C 2 map which is expanding, in the sense that jT 0 (x)j > 1 everywhere. Then there is a unique finite invariant measure  equivalent to Lebesgue measure m, and in fact  is ergodic and the Radon-Nikodym derivative d/dm has a continuous version. Some Mathematical Background Lebesgue Spaces Definition 5 Two measure spaces (X; B; ) and (Y; C ; ) are isomorphic (sometimes also called isomorphic mod 0) if there are subsets X 0 X and Y0 Y such that (X0 ) D 0 D (Y0 ) and a one-to-one onto map  : X n X0 ! Y n Y0 such that  and  1 are measurable and ( 1 C) D (C) for all measurable C Y n Y0 . Definition 6 A Lebesgue space is a finite measure space that is isomorphic to a measure space consisting of a (possibly empty) finite subinterval of R with the  -algebra of Lebesgue measurable sets and Lebesgue measure, possibly together with countably many atoms (point masses). The measure algebra of a measure space (X; B; ) conˆ with Bˆ the Boolean  -algebra (see sists of the pair (Bˆ ; ), Sect. “A Range of Actions”, 3.) of B modulo the  -ideal of sets of measure 0, together with the operations induced by ˆ is induced on Bˆ by  on B. Evset operations in B, and  ˆ is a metric space with the metery measure algebra (Bˆ ; ) ˆ ric d(A; B) D (A4B) for all A; B 2 Bˆ . It is nonatomic if whenever A; B 2 Bˆ and A < B (which means A ^ B D A), either A D 0 or A D B. A homomorphism of measure alˆ is a Boolean  -algebra homogebras : (Cˆ; ˆ ) ! (Bˆ ; ) ˆ D ˆ (C) for all Cˆ 2 Cˆ. The inˆ C) morphism such that ( verse of any factor map  : X ! Y from a measure space (X; B; ) to a measure space (Y; C ; ) induces a homoˆ We say morphism of measure algebras (Cˆ; ˆ ) ! (Bˆ ; ). that a measure algebra is normalized if the measure of the ˆ 0 ) D 1. maximal element is 1: (0 We work within the class of Lebesgue spaces because (1) they are the ones commonly encountered in the wide range of naturally arising examples; (2) they allow us to assume if we wish that we are dealing with a familiar space such as [0; 1] or f0; 1gN ; and (3) they have the following useful properties.

Measure Preserving Systems

 (Carathéodory [7]) Every normalized and nonatomic measure algebra whose associated metric space is separable (has a countable dense set) is measure-algebra isomorphic with the measure algebra of the unit interval with Lebesgue measure.  (von Neumann [37]) Every complete separable metric space with a Borel probability measure on the completion of the Borel sets is a Lebesgue space.  (von Neumann [37]) Every homomorphism : (Cˆ; ˆ of the measure algebras of two Lebesgue ˆ ) ! (Bˆ ; ) spaces (Y; C ; ) and (X; B; ) comes from a factor map: there are a set X0 X with (X0 ) D 0 and a measurable map  : X n X 0 ! Y such that coincides with the map induced by  1 from Cˆ to Bˆ .

Every countable or finite measurable partition of a complete measure space is an R-partition. The orbit partition of a measure-preserving transformation is often not an R-partition. (For example, if the transformation is ergodic, the corresponding factor space will be trivial, consisting of just one cell, rather than corresponding to the partition into orbits as required.) For any set B X, let B0 D B and B1 D B c D X n B. Definition 9 A basis C D fC1 ; C2 ; : : : g for a complete measure space (X; B; ) is called complete, and the space is called complete with respect to the basis, if for every 0,1-sequence e 2 f0; 1gN , 1 \

C ie i ¤ ; :

(66)

iD1

Rokhlin Theory V. A. Rokhlin [31] provided an axiomatic, intrinsic characterization of Lebesgue spaces. The key ideas are the concept of a basis and the correspondence of factors with complete sub--algebras and (not necessarily finite or countable) measurable partitions of a special kind. Definition 7 A basis for a complete measure space (X; B; ) is a countable family C D fC1 ; C2 ; : : : g of measurable sets which generates B: For each B 2 B there is C 2 B(C ) (the smallest -algebra of subsets of X that contains C ) such that B C and (C n B) D 0; and separates the points of X: For each x; y 2 X with x ¤ y, there is C i 2 C such that either x 2 C i ; y … C i or else y 2 Ci ; x … Ci . Coarse sub--algebras of B may not separate points of X and thus may lead to equivalence relations, partitions, and factor maps. Partitions of the following kind deserve careful attention. Definition 8 Let (X; B; ) be a complete measure space and  a partition of X, meaning that up to a set of measure 0, X is the union of the elements of , which are pairwise disjoint up to sets of measure 0. We call  an Rpartition if there is a countable family D D fD1 ; D2 ; : : : g of -saturated sets (that is, each D i is a union of elements of ) such that for all distinct E; F 2  ; there is D i such that either E  D i ; F ª D i (so F D ci ) or F  D i ; E ª D i (so E

D ci )

(64)

:

Any such family D is called a basis for . Note that each element of an R-partition is necessarily measurable: if C 2  with basis fD i g, then \ (65) CD fD i : C  D i g :

C is called complete mod 0 (and (X; B; ) is called complete mod 0 with respect to C , if there is a complete measure space (X 0 ; B0 ; 0 ) with a complete basis C 0 such that X is a full-measure subset of X 0 , and C i D C 0i \ X for all i D 1; 2; : : : .

From the definition of basis, each intersection in (66) contains at most one point. The space f0; 1gN with Bernoulli 1/2; 1/2 measure on the completion of the Borel sets has the complete basis C i D f! : ! i D 0g. Proposition 5 If a measure space is complete mod 0 with respect to one basis, then it is complete mod 0 with respect to every basis. Theorem 12 ([31]) A measure space is a Lebesgue space (that is, isomorphic mod 0 with the usual Lebesgue measure space of a possibly empty subinterval of R possibly together with countably many atoms) if and only if it has a complete basis. In a Lebesgue space (X; B; ) there is a one-to-one onto correspondence between complete sub--algebras of B (that is, those for which the restriction of the measure yields a complete measure space) and R-partitions of X: Given an R-partition , let B() denote the  -algebra generated by , which consists of all sets in B that are -saturated – unions of members of  – and let B() denote the completion of B() with respect to . Conversely, given a complete sub- -algebra C B, define an equivalence relation on X by x y if for all A 2 C , either x; y 2 A or else x; y 2 Ac . The measure ˆ has a countable dense set Cˆ0 (take a countalgebra (Cˆ; ) ˆ and, for each i; j for which able dense set fBˆ i g for (Bˆ ; ) it is possible, choose Cˆ i j within distance 1/2 j of Bˆ i ). Then representatives C i 2 C of the Cˆi j will be a basis for the partition  corresponding to the equivalence relation .

975

976

Measure Preserving Systems

Given any family fB g, of complete sub--algebras of B, their join is the intersection of all the sub--algebras that contain their union: [ _ B D B( B ) ; (67) 



and their infimum is just their intersection: \ ^ B D B( B ) : 

(68)



These -algebra operations correspond to the supremum and infimum of the corresponding families of R-partitions. We say that a partition 1 is finer than a partition 2 , and write 1  2 , if every element of 2 is a union of elements of 1 . Given any family f g of R-partitions, there W is a coarsest R-partition   which refines all of them, V and a finest R-partition   which is coarser than all of them. We have _ ^ ^ _ B( ) D B(  ) ; B( ) D B(  ) : (69) 







Now we discuss the relationship among factor maps  : X ! Y from a Lebesgue space (X; B; ) to a complete measure space (Y; C ; ), complete sub--algebras of B, and R-partitions of X. Given such a factor map , BY D  1 C is a complete sub--algebra of B, and the equivalence relation x1 x2 if (x1 ) D (x2 ) determines an R-partition Y . (A basis for  can be formed from a countable dense set in Bˆ Y as above.) Conversely, given a complete sub--algebra C

B, the identity map (X; B; ) ! (X; C ; ) is a factor map. Alternatively, given an R-partition of X, we can form a measure space (X/; B();  ) and a factor map  : X ! X/ as follows. The space X/ is just  itself; that is, the points of X/ are the members (cells, or atoms) of the partition . B() consists of the -saturated sets in B considered as subsets of , and  is the restriction of  to B(). Completeness of (X; B; ) forces completeness of (X/; B();  ). The map  : X ! X/ is defined by letting (x) D (x) D the element of  to which x belongs. Thus for a Lebesgue space (X; B; ), there is a perfect correspondence among images under factor maps, complete sub--algebras of B, and R-partitions of X. Theorem 13 If (X; B; ) is a Lebesgue space and (Y; C ; ) is a complete measure space that is the image of (X; B; ) under a factor map, then (Y; C ; ) is also a Lebesgue space. Theorem 14 Let (X; B; ) be a Lebesgue space, (Y; C ; ) a separable measure space (that is, one with a countable basis as above, equivalently one with a countable dense set in

its measure algebra), and  : X ! Y a measurable map ( 1 C B). Then ' is also forward measurable: if A X is measurable, then (A) Y is measurable. Theorem 15 Let (X; B; ) be a Lebesgue space. 1. Every measurable subset of X, with the restriction of B and , is a Lebesgue space. Conversely, if a subset A of X with the restrictions of B and  is a Lebesgue space, then A is measurable (A 2 B). 2. The product of countably many Lebesgue spaces is a Lebesgue space. ˆ (defined 3. Every measure algebra isomorphism of (Bˆ ; ) as above) is induced by a point isomorphism mod 0. Disintegration of Measures Every R-partition  of a Lebesgue space (X; B; ) has associated with it a canonical system of conditional measures: Using the notation of the preceding section, for  -almost every C 2 X/, there are a  -algebra BC of subsets of C and a measure mC on BC such that: 1. (C; BC ; mC ) is a Lebesgue space; 2. for every A 2 B, A \ C 2 BC for  -almost every C 2 ; 3. for every A 2 B, the map C ! mC (A \ C) is B()measurable on X/; 4. for every A 2 B, Z (A) D mC (A \ C) d (C) : (70) X/

It follows that for f 2 L1 (X), (a version of) its conditional expectation (see the next section) with respect to the factor algebra corresponding to  is given by Z E( f jB()) D f dmC on  - a.e. C 2  ; (71) C

since the right-hand side is B()-measurable and for each A 2 B(), its integral over any B 2 B() is, as required, (A \ B) (use the formula on B/(jB)). It can be shown that a canonical system of conditional measures for an R-partition of a Lebesgue space is essentially unique, in the sense that any two measures mC and m0C will be equal for  -almost all C 2 . Also, any partition of a Lebesgue space that has a canonical system of conditional measures must be an R-partition. These conditional systems of measures can be used to prove the ergodic decomposition theorem and to show that every factor situation is essentially projection of a skew product onto the base (see [32]).

Measure Preserving Systems

Theorem 16 Let (X; B; ) be a Lebesgue space. If  is an R-partition of X, f(C; BC ; mC )g is a canonical system of conditional measures for , and A 2 B, define (AjC) D mC (A \ C). Then: 1. for every A 2 B, (Aj(x)) is a measurable function of x 2 X; 2. if (n ) is an increasing sequence of R-partitions of X, then for each A 2 B _ (Ajn (x)) ! (Aj  n (X)) a.e. d ; (72) n

3. if ( n ) is a decreasing sequence of R-partitions of X, then for each A 2 B ^ (Ajn (x)) ! (Aj  n (X)) a.e. d : (73) n

This is a consequence of the Martingale and Reverse Martingale Convergence Theorems. The statements hold just as well for f 2 L1 (X) as for f D 1 A for some A 2 B.

each F 2 F we know whether or not x 2 F. When F is the  -algebra generated by a finite measurable partition ˛ of X and f is the characteristic function of a set A 2 B, the conditional expectation gives the conditional probabilities of A with respect to all the sets in ˛: E(11 A jF )(x) D (Aj˛(x))

(79) D (A \ F)/(F) if x 2 F 2 ˛ : R We write E( f ) D E( f jf;; Xg) D X f d for the expectation of any integrable function f . A measurable function f on X is independent of a sub- -algebra F B if for each (a; b) R and F 2 F we have ( f 1 (a; b)) \ F) D ( f 1 (a; b)) (F) :

(80)

A function  : R ! R is convex if whenever t1 ; : : : ; t n  0 P and niD1 t i D 1, n n X X ti xi )  t i (x i ) for all x1 ; : : : ; x n 2 R : (81) ( iD1

iD1

Theorem 17 Let (X; B; ) be a probability space and F

B a sub- -algebra.

Conditional Expectation Let (X; B; ) be a -finite measure space, f 2 L1 (X), and F B a sub--algebra of B. Then Z (F) D f d (74) F

defines a finite signed measure on F which is absolutely continuous with respect to  restricted to F . So by the Radon-Nikodym Theorem there is a function g 2 L1 (X; F ; ) such that Z (F) D g d for all F 2 F : (75)

1. E(jF ) is a positive contraction on L p (X) for each p  1. 2. If f 2 L1 (X) is F -measurable, then E( f jF ) D f a.e. If f 2 L1 (X) is F -measurable, then E( f gjF ) D f E(gjF ) for all g 2 L1 (X). 3. If F1 F2 are sub- -algebras of B, then E(E ( f jF2)jF1 ) D E( f jF1 ) a.e. for each f 2 L1 (X). 4. If f 2 L1 (X) is independent of the sub- -algebra F

B, then E( f jF ) D E( f ) a.e. 5. If  : R ! R is convex, f and ı f 2 L1 (X), and F B is a sub- -algebra, then (E( f jF ))  E( ı f jF ) a.e.

F

Any such function g, which is unique as an element of L1 (X; F ; ) (and determined only up to sets of -measure 0) is called a version of the conditional expectation of f with respect to F , and denoted by g D E( f jF ) :

(76)

As an element of L1 (X; B; ), E( f jF ) is characterized by the following two properties: E( f jF ) is F -measurable ; Z Z E( f jF ) d D f d for all F 2 F : F

(77) (78)

F

We think of E( f jF )(x) as our expected value for f if we are given the information in F , in the sense that for

The Spectral Theorem A separable Hilbert space is one with a countable dense set, equivalently a countable orthonormal basis. A normal operator is a continuous linear operator S on a Hilbert space H such that SS  D S  S, S  being the adjoint operator defined by (S f ; g) D ( f ; S  g) for all f ; g 2 H . A continuous linear operator S is unitary if it is invertible and S  D S 1 . Two operators S1 and S2 on Hilbert spaces H1 and H2 , respectively, are called unitarily equivalent if there is a unitary operator U : H1 ! H2 which carries S1 to S2 : S2 U D U S1 . The following brief account follows [10,30]. Theorem 18 Let S : H ! H be a normal operator on a separable Hilbert space H . Then there are mutually singular Borel probability measures 1 ; 1 ; 2 ; : : : such that

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978

Measure Preserving Systems

S is unitarily equivalent to the operator M on the direct sum Hilbert space 2

2

L (C; 1 ) ˚ L (C; 1 ) ˚

2 M kD1

˚ ˚

m M

! 2

L (C; 2 ) !

L2 (C; m ) ˚ : : :

(82)

kD1

defined by

The various L2 spaces and multiplication operators involved in the above theorem can be amalgamated into a coherent whole, resulting in the following convenient form of the Spectral Theorem for normal operators Theorem 19 Let S : H ! H be a normal operator on a separable Hilbert space H . There are a finite measure space (X; B; ) and a bounded measurable function h : X ! C such that S is unitarily equivalent to the operator of multiplication by h on L2 (X; B; ). The form of the Spectral Theorem given in Theorem 18 is useful for discussing absolute continuity and multiplicity properties of the spectrum of a normal operator. Another form, involving spectral measures, has useful consequences such as the functional calculus.

M(( f1;1 (z1;1 ); f1;2 (z1;2 ); : : : ); ( f1;1 (z1;1 )); ( f2;1 (z2;1 ); f2;2 (z2;2 )); : : : ) D (z1;1 f1;1 (z1;1 ); z1;2 f1;2 (z1;2 ); : : : ); (z1;1 f1;1 (z1;1 )); (z2;1 f2;1 (z2;1 ); z2;2 f2;2 (z2;2 )); : : : ) : (83) The measures  i are supported on the spectrum (S) of S, the (compact) set of all  2 C such that S  I does not have a continuous inverse. Some of the  i may be 0. They are uniquely determined up to absolute continuity equivalence. The smallest absolute continuity class with respect to which all the  i are absolutely continuous is called the maximum spectral type of S. A measure representing this P type is i  i /2 i . We have in mind the example for which H D L2 (X; B; ) and S f D f ı T (the “Koopman operator”) for a measure-preserving system (X; B; ; T) on a Lebesgue space (X; B; ), which is unitary: it is linear, continuous, invertible, preserves scalar products, and has spectrum equal to the unit circle. The proof of Theorem 18 can be accomplished by first decomposing H (in a careful way) into the direct sum of pairwise orthogonal cyclic subspaces H n : each H n is the closed linear span of fS i (S  ) j f n : i; j  0g for some f n 2 H . This means that for each n the set fp(S; S  ) f n : p is a polynomial in two variablesg is dense in H n . Similarly, by the Stone-Weierstrass Theorem the set Pn of all polynomials p(z; z) is dense in the set C ((SjH n )) of continuous complex-valued functions on  (SjH n ). We define a bounded linear functional  n on Pn by (p) D (p(S; S  ) f n ; f n )

(84)

and extend it by continuity to a bounded linear functional on C ((SjH n )). It can be proved that this functional is positive, and therefore, by the Riesz Representation Theorem, it corresponds to a positive Borel measure on (SjH n ).

Theorem 20 Let S : H ! H be a normal operator on a separable Hilbert space H . There is a unique projectionvalued measure E defined on the Borel subsets of the spectrum  (S) of S such that E( (S)) D I (D the identity on H ); 1 [

E

!! f D

Ai

iD1

1 X (E(A i )) f

(85)

iD1

whenever A1 ; A2 ; : : : are pairwise disjoint Borel subsets of  (S) and f 2 H , with the series converging in norm; and Z SD

(S)

 dE() :

(86)

Spectral integrals such as the one in (86) can be defined by reducing to complex measures  f ;g (A) D (E(A) f ; g), for f ; g 2 H and A  (S) a Borel set. Given a bounded Borel measurable function  on  (S), the operator Z V D (S) D

(S)

() dE()

(87)

is determined by specifying that Z (V f ; g) D

(S)

() d f ;g for all f ; g 2 H :

(88)

Then Sk D

Z (S)

 k dE() for all k D 0; 1; : : : :

(89)

Measure Preserving Systems

These spectral integrals sometimes behave a bit strangely: Z If V1 D 1 () dE() (S) Z and V2 D 2 () dE() ; (90) (S) Z then V1 V2 D 1 () 2 () dE() : (S)

Finally, if f 2 H and  is a finite positive Borel measure that is absolutely continuous with respect to  f ; f , then there is g in the closed linear span of fS i (S  ) j f : i; j  0g such that  D  g;g . Theorem 20 can be proved by applying Theorem 19, which allows us to assume that H D L2 (X; B; ) and S is multiplication by h 2 L1 (X; B; ). For any Borel set A (S), let E(A) be the projection operator given by multiplication by the 0; 1-valued function 1 A ı h. Future Directions The mathematical study of dynamical systems arose in the late nineteenth and early twentieth century, along with measure theory and probability theory, so it is a young field with many interesting open problems. New questions arise continually from applications and from interactions with other parts of mathematics. Basic aspects of the problems of classification, topological or smooth realization, and systematic construction of measure-preserving systems remain open. There is much work to be done to understand the relations among systems and different types of systems (factors and relative properties, joinings and disjointness, various notions of equivalence with associated invariants). There is a continual need to determine properties of classes of systems and of particular systems arising from applications or other parts of mathematics such as probability, number theory, geometry, algebra, and harmonic analysis. Some of these questions are mentioned in more detail in the other articles in this collection. Bibliography Primary Literature 1. Adler RL (1973) F-expansions revisited. Lecture Notes in Math, vol 318. Springer, Berlin 2. Arnold LK (1968) On  -finite invariant measures. Z Wahrscheinlichkeitstheorie Verw Geb 9:85–97 3. Billingsley P (1978) Ergodic theory and information. Robert E Krieger Publishing Co, Huntington NY, pp xiii,194; Reprint of the 1965 original 4. Billingsley P (1995) Probability and measure, 3rd edn. Wiley, New York, pp xiv,593

5. Brunel A (1966) Sur les mesures invariantes. Z Wahrscheinlichkeitstheorie Verw Geb 5:300–303 6. Calderón AP (1955) Sur les mesures invariantes. C R Acad Sci Paris 240:1960–1962 7. Carathéodory C (1939) Die Homomorphieen von Somen und die Multiplikation von Inhaltsfunktionen. Annali della R Scuola Normale Superiore di Pisa 8(2):105–130 8. Carathéodory C (1968) Vorlesungen über Reelle Funktionen, 3rd edn. Chelsea Publishing Co, New York, pp x,718 9. Chacon RV (1964) A class of linear transformations. Proc Amer Math Soc 15:560–564 10. Conway JB (1990) A Course in Functional Analysis, vol 96, 2nd edn. Springer, New York, pp xvi,399 11. Dowker YN (1955) On measurable transformations in finite measure spaces. Ann Math 62(2):504–516 12. Dowker YN (1956) Sur les applications mesurables. C R Acad Sci Paris 242:329–331 13. Frechet M (1924) Des familles et fonctions additives d’ensembles abstraits. Fund Math 5:206–251 14. Friedman NA (1970) Introduction to Ergodic Theory. Van Nostrand Reinhold Co., New York, pp v,143 15. Furstenberg H (1977) Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J Anal Math 31:204–256 16. Gowers WT (2001) A new proof of Szemerédi’s theorem. Geom Funct Anal 11(3):465–588 17. Gowers WT (2001) Erratum: A new proof of Szemerédi’s theorem. Geom Funct Anal 11(4):869 18. Green B, Tao T (2007) The primes contain arbitrarily long arithmetic progressions. arXiv:math.NT/0404188 19. Hahn H (1933) Über die Multiplikation total-additiver Mengenfunktionen. Annali Scuola Norm Sup Pisa 2:429–452 20. Hajian AB, Kakutani S (1964) Weakly wandering sets and invariant measures. Trans Amer Math Soc 110:136–151 21. Halmos PR (1947) Invariant measures. Ann Math 48(2):735–754 22. Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, pp xviii,802 23. Hopf E (1932) Theory of measure and invariant integrals. Trans Amer Math Soc 34:373–393 24. Hopf E (1937) Ergodentheorie. Ergebnisse der Mathematik und ihrer Grenzgebiete, 1st edn. Springer, Berlin, pp iv,83 25. Khinchin AI (1949) Mathematical Foundations of Statistical Mechanics. Dover Publications Inc, New York, pp viii,179 26. Kolmogorov AN (1956) Foundations of the Theory of Probability. Chelsea Publishing Co, New York, pp viii,84 27. Ornstein DS (1960) On invariant measures. Bull Amer Math Soc 66:297–300 28. Petersen K (1989) Ergodic Theory. Cambridge Studies in Advanced Mathematics, vol 2. Cambridge University Press, Cambridge, pp xii,329 29. Poincaré H (1987) Les Méthodes Nouvelles de la Mécanique Céleste. Tomes I, II,III. Les Grands Classiques Gauthier-Villars. Librairie Scientifique et Technique Albert Blanchard, Paris 30. Radjavi H, Rosenthal P (1973) Invariant subspaces, 2nd edn. Springer, Mineola, pp xii,248 31. Rohlin VA (1952) On the fundamental ideas of measure theory. Amer Math Soc Transl 1952:55 32. Rohlin VA (1960) New progress in the theory of transformations with invariant measure. Russ Math Surv 15:1–22

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33. Royden HL (1988) Real Analysis, 3rd edn. Macmillan Publishing Company, New York, pp xx,444 34. Szemerédi E (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith 27:199–245 35. Tao T (2006) Szemerédi’s regularity lemma revisited. Contrib Discret Math 1:8–28 36. Tao T (2006) Arithmetic progressions and the primes. Collect Math Extra:37–88 37. von Neumann J (1932) Einige Sätze über messbare Abbildungen. Ann Math 33:574–586 38. Wright FB (1961) The recurrence theorem. Amer Math Mon 68:247–248 39. Young L-S (2002) What are SRB measures, and which dynamical systems have them? J Stat Phys 108:733–754

Books and Reviews Billingsley P (1978) Ergodic Theory and Information. Robert E. Krieger Publishing Co, Huntington, pp xiii,194 Cornfeld IP, Fomin SV, Sina˘ı YG (1982) Ergodic Theory. Fundamental Principles of Mathematical Sciences, vol 245. Springer, New York, pp x,486 Denker M, Grillenberger C, Sigmund K (1976) Ergodic Theory on Compact Spaces. Lecture Notes in Mathematics, vol 527. Springer, Berlin, pp iv,360

Friedman NA (1970) Introduction to Ergodic Theory. Van Nostrand Reinhold Co, New York, pp v,143 Glasner E (2003) Ergodic Theory via Joinings. Mathematical Surveys and Monographs, vol 101. American Mathematical Society, Providence, pp xii,384 Halmos PR (1960) Lectures on Ergodic Theory. Chelsea Publishing Co, New York, pp vii,101 Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge, pp xviii,802 Hopf E (1937) Ergodentheorie. Ergebnisse der Mathematik und ihrer Grenzgebiete, 1st edn. Springer, Berlin, pp iv,83 Jacobs K (1960) Neue Methoden und Ereignisse der Ergodentheorie. Jber Dtsch Math Ver 67:143–182 Petersen K (1989) Ergodic Theory. Cambridge Studies in Advanced Mathematics, vol 2. Cambridge University Press, Cambridge, pp xii,329 Royden HL (1988) Real Analysis, 3rd edn. Macmillan Publishing Company, New York, pp xx,444 Rudolph DJ (1990) Fundamentals of Measurable Dynamics. Oxford Science Publications. The Clarendon Press Oxford University Press, New York, pp x,168 Walters P (1982) An Introduction to Ergodic Theory. Graduate Texts in Mathematics, vol 79. Springer, New York, pp ix,250

Mechanical Systems: Symmetries and Reduction

Mechanical Systems: Symmetries and Reduction JERROLD E. MARSDEN1 , TUDOR S. RATIU2 1 Control and Dynamical Systems, California Institute of Technology, Pasadena, USA 2 Section de Mathématiques and Bernoulli Center, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland Article Outline Glossary Definition of the Subject Introduction Symplectic Reduction Symplectic Reduction – Further Discussion Reduction Theory: Historical Overview Cotangent Bundle Reduction Future Directions Acknowledgments Appendix: Principal Connections Bibliography Glossary Lie group action A process by which a Lie group, acting as a symmetry, moves points in a space. When points in the space that are related by a group element are identified, one obtains the quotient space. Free action An action that moves every point under any nontrivial group element. Proper action An action that obeys a compactness condition. Momentum mapping A dynamically conserved quantity that is associated with the symmetry of a mechanical system. An example is angular momentum, which is associated with rotational symmetry. Symplectic reduction A process of reducing the dimension of the phase space of a mechanical system by restricting to the level set of a momentum map and also identifying phase space points that are related by a symmetry. Poisson reduction A process of reducing the dimension of the phase space of a mechanical system by identifying phase space points that are related by a symmetry. Equivariance Equivariance of a momentum map is a property that reflects the consistency of the mapping with a group action on its domain and range. Momentum cocycle A measure of the lack of equivariance of a momentum mapping.

Singular reduction A reduction process that leads to non-smooth reduced spaces. Often associated with non-free group actions. Coadjoint orbit The orbit of an element of the dual of the Lie algebra under the natural action of the group. KKS (Kostant-Kirillov-Souriau) form The natural symplectic form on coadjoint orbits. Cotangent bundle A mechanical phase space that has a structure that distinguishes configurations and momenta. The momenta lie in the dual to the space of velocity vectors of configurations. Shape space The space obtained by taking the quotient of the configuration space of a mechanical system by the symmetry group. Principal connection A mathematical object that describes the geometry of how a configuration space is related to its shape space. Related to geometric phases through the subject of holonomy. In turn related to locomotion in mechanical systems. Mechanical connection A special (principal) connection that is built out of the kinetic energy and momentum map of a mechanical system with symmetry. Magnetic terms These are expressions that are built out of the curvature of a connection. They are so named because terms of this form occur in the equations of a particle moving in a magnetic field. Definition of the Subject Reduction theory is concerned with mechanical systems with symmetries. It constructs a lower dimensional reduced space in which associated conservation laws are enforced and symmetries are “factored out” and studies the relation between the dynamics of the given system with the dynamics on the reduced space. This subject is important in many areas, such as stability of relative equilibria, geometric phases and integrable systems. Introduction Geometric mechanics has developed in the last 30 years or so into a mature subject in its own right, and its applications to problems in Engineering, Physics and other physical sciences has been impressive. One of the important aspects of this subject has to do with symmetry; even things as apparently simple as the symmetry of a system such as the n-body problem under the group of translations and rotations in space (the Euclidean group) or a wheeled vehicle under the planar Euclidean group turns out to have profound consequences. Symmetry often gives conservation laws through Noether’s theorem and these conservation laws can be used to reduce the dimension of a system.

981

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In fact, reduction theory is an old and time-honored subject, going back to the early roots of mechanics through the works of Euler, Lagrange, Poisson, Liouville, Jacobi, Hamilton, Riemann, Routh, Noether, Poincaré, and others. These founding masters regarded reduction theory as a useful tool for simplifying and studying concrete mechanical systems, such as the use of Jacobi’s elimination of the node in the study of the n-body problem to deal with the overall rotational symmetry of the problem. Likewise, Liouville and Routh used the elimination of cyclic variables (what we would call today an Abelian symmetry group) to simplify problems and it was in this setting that the Routh stability method was developed. The modern form of symplectic reduction theory begins with the works of Arnold [7], Smale [168], Meyer [117], and Marsden and Weinstein [110]. A more detailed survey of the history of reduction theory can be found in the first sections of this article. As was the case with Routh, this theory has close connections with the stability theory of relative equilibria, as in Arnold [8] and Simo, Lewis, and Marsden [166]. The symplectic reduction method is, in fact, by now so well known that it is used as a standard tool, often without much mention. It has also entered many textbooks on geometric mechanics and symplectic geometry, such as Abraham and Marsden [1], Arnold [9], Guillemin and Sternberg [57], Libermann and Marle [89] and Mcduff and Salamon [116]. Despite its relatively old age, research in reduction theory continues vigorously today. It will be assumed that the reader is familiar with the basic concepts in [104]. For the statements of the bulk of the theorems, it is assumed that the manifolds involved are finite dimensional and are smooth unless otherwise stated. While many interesting examples are infinite-dimensional, the general theory in the infinite dimensional case is still not in ideal shape; see, for example, Chernoff and Marsden [39], Marsden and Hughes [96] and Mielke [118] and examples and discussion in [92]. Notation To keep things reasonably systematic, we have adopted the following universal conventions for some common maps and other objects: Configuration space of a mechanical system: Q Phase space: P Cotangent bundle projection: Q : T  Q ! Q Tangent bundle projection: Q : TQ ! Q Quotient projection: P;G : P ! P/G Tangent map: T' : TM ! T N for the tangent of a map ': M ! N

Thus, for example, the symbol T  Q;G denotes the quotient projection from T  Q to (T  Q)/G.  The Lie algebra of a Lie group G is denoted g.  Actions of G on a space is denoted by concatenation. For example, the action of a group element g on a point q 2 Q is written as gq or g  q  The infinitesimal generator of a Lie algebra element  2 g for an action of G on P is denoted P , a vector field on P  Momentum maps are denoted J : P ! g .  Pairings between vector spaces and their duals are denoted by simple angular brackets: for example, the pairing between g and g is denoted h; i for  2 g and 2g  Inner products are denoted with double angular brackets: hhu; vii. Symplectic Reduction Roughly speaking, here is how symplectic reduction goes: given the symplectic action of a Lie group on a symplectic manifold that has a momentum map, one divides a level set of the momentum map by the action of a suitable subgroup to form a new symplectic manifold. Before the division step, one has a manifold (that can be singular if the points in the level set have symmetries) carrying a degenerate closed 2-form. Removing such a degeneracy by passing to a quotient space is a differential-geometric operation that was promoted by Cartan [26]. The “suitable subgroup” related to a momentum mapping was identified by Smale [168] in the special context of cotangent bundles. It was Smale’s work that inspired the general symplectic construction by Meyer [117] and the version we shall use, which makes explicit use of the properties of momentum maps, by Marsden and Weinstein [110]. Momentum Maps Let G be a Lie group, g its Lie algebra, and g be its dual. Suppose that G acts symplectically on a symplectic manifold P with symplectic form denoted by ˝. We shall denote the infinitesimal generator associated with the Lie algebra element  by  P and we shall let the Hamiltonian vector field associated to a function f : P ! R be denoted X f . A momentum map is a map J : P ! g , which is defined by the condition  P D XhJ;i

(1)

for all  2 g, and where hJ; i : P ! R is defined by the natural pointwise pairing. We call such a momentum

Mechanical Systems: Symmetries and Reduction

map equivariant when it is equivariant with respect to the given action on P and the coadjoint action of G on g . That is, J(g  z) D Adg 1 J(z)

(2)

for every g 2 G, z 2 P, where g  z denotes the action of g on the point z, Ad denotes the adjoint action, and Ad the coadjoint action. Note that when we write Adg 1 , we literally mean the adjoint of the linear map Ad g 1 : g ! g. The inverse of g is necessary for this to be a left action on g . Some authors let that inverse be understood in the notation. However, such a convention would be a notational disaster since we need to deal with both left and right actions, a distinction that is essential in mechanics. A quadruple (P; ˝; G; J), where (P; ˝) is a given symplectic manifold and J : P ! g is an equivariant momentum map for the symplectic action of a Lie group G, is sometimes called a Hamiltonian G-space. Taking the derivative of the equivariance identity (2) with respect to g at the identity yields the condition of infinitesimal equivariance: Tz J( P (z)) D ad J(z)

(3)

for any  2 g and z 2 P. Here, ad : g ! g;  7! [; ] is the adjoint map and ad : g ! g is its dual. A computation shows that (3) is equivalent to hJ; [; ]i D fhJ; i; hJ; ig

(4)

for any ;  2 g, that is, hJ; i : g ! F (P) is a Lie algebra homomorphism, where F (P) denotes the Poisson algebra of smooth functions on P. The converse is also true if the Lie group is connected, that is, if G is connected then an infinitesimally equivariant action is equivariant (see §12.3 in [104]). The idea that an action of a Lie group G with Lie algebra g on a symplectic manifold P should be accompanied by such an equivariant momentum map J : P ! g and the fact that the orbits of this action are themselves symplectic manifolds both occur already in Lie [90]; the links with mechanics also rely on the work of Lagrange, Poisson, Jacobi and Noether. In modern form, the momentum map and its equivariance were rediscovered by Kostant [78] and Souriau [169,170] in the general symplectic case and by Smale [168] for the case of the lifted action from a manifold Q to its cotangent bundle P D T  Q. Recall that the equivariant momentum map in this case is given explicitly by ˝ ˛ ˝ ˛ J(˛q );  D ˛q ;  Q (q) ; (5) where ˛q 2 Tq Q;  2 g, and where the angular brackets denote the natural pairing on the appropriate spaces.

Smale referred to J as the “angular momentum” by generalization from the special case G D SO(3), while Souriau used the French word “moment”. Marsden and Weinstein [110], followed usage emerging at that time and used the word “moment” for J, but they were soon corrected by Richard Cushman and Hans Duistermaat, who suggested that the proper English translation of Souriau’s French word was “momentum,” which fit better with Smale’s designation as well as standard usage in mechanics. Since 1976 or so, most people who have contact with mechanics use the term momentum map (or mapping). On the other hand, Guillemin and Sternberg popularized the continuing use of “moment” in English, and both words coexist today. It is a curious twist, as comes out in work on collective nuclear motion Guillemin and Sternberg [56] and plasma physics (Marsden and Weinstein [111] and Marsden, Weinstein, Ratiu, Schmid, and Spencer [114]), that moments of inertia and moments of probability distributions can actually be the values of momentum maps! Mikami and Weinstein [119] attempted a linguistic reconciliation between the usage of “moment” and “momentum” in the context of groupoids. See [104] for more information on the history of the momentum map and Sect. “Reduction Theory: Historical Overview” for a more systematic review of general reduction theory. Momentum Cocycles and Nonequivariant Momentum Maps Consider a momentum map J : P ! g that need not be equivariant, where P is a symplectic manifold on which a Lie group G acts symplectically. The map  : G ! g that is defined by  (g) :D J(g  z)  Adg 1 J(z) ;

(6)

where g 2 G and z 2 P is called a nonequivariance or momentum one-cocycle. Clearly,  is a measure of the lack of equivariance of the momentum map. We shall now prove a number of facts about  . The first claim is that  does not depend on the point z 2 P provided that the symplectic manifold P is connected (otherwise it is constant on connected components). To prove this, we first recall the following equivariance identity for infinitesimal generators:   Tq ˚ g  P (q) D (Ad g )P (g  q) ;   (7) i. e.; ˚ g  P D Ad g 1  P : This is an easy Lie group identity that is proved, for example, in [104]; see Lemma 9.3.7. One shows that  (g) is constant by showing that its Hamiltonian vector field vanishes. Using the fact that  (g)

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is independent of z along with the basic identity Ad g h D Ad g Ad h and its consequence Ad(g h)1 D Adg 1 Adh 1 , shows that  satisfies the cocycle identity (gh) D (g) C Adg 1 (h)

(8)

for any g; h 2 G. This identity shows that  produces a new action  : G  g ! g defined by (g; ) :D Adg 1  C (g)

(9)

with respect to which the momentum map J is obviously equivariant. This action  is not linear anymore – it is an affine action. For  2 g, let  (g) D h(g); i. Differentiating the definition of , namely ˝ ˛  (g) D hJ(g  z); i  J(z); Ad g 1  with respect to g at the identity in the direction  2 g shows that Te  () D ˙ (; ) ;

(10)

where ˙ (; ), which is called the infinitesimal nonequivariance two-cocycle, is defined by ˙ (; ) D hJ; [; ]i  fhJ; i; hJ; ig :

(11)

Since  does not depend on the point z 2 P, neither does ˙ . Also, it is clear from this definition that ˙ measures the lack of infinitesimal equivariance of J. Another way to look at this is to notice that from the derivation of Eq. (10), for z 2 P and  2 g, we have Tz J( P (z)) D ad J(z) C ˙ (; ) :

(12)

Comparison of this relation with Eq. (3) also shows the relation between ˙ and the infinitesimal equivariance of J. The map ˙ : g  g ! R is bilinear, skew-symmetric, and, as can be readily verified, satisfies the two-cocycle identity ˙ ([; ]; ) C ˙ ([; ]; ) C ˙ ([; ]; ) D 0 ;

(13)

for all ; ;  2 g. The Symplectic Reduction Theorem There are many precursors of symplectic reduction theory. When G is Abelian, the components of the momentum map form a system of functions in involution (i. e. the Poisson bracket of any two is zero). The use of k such functions to reduce a phase space to one having 2k fewer dimensions may be found already in the work of Lagrange,

Poisson, Jacobi, and Routh; it is well described in, for example, Whittaker [179]. In the nonabelian case, Smale [168] noted that Jacobi’s elimination of the node in SO(3) symmetric problems can be understood as division of a nonzero angular momentum level by the SO(2) subgroup which fixes the momentum value. In his setting of cotangent bundles, Smale clearly stated that the coadjoint isotropy group G of  2 g (defined to be the group of those g 2 G such that g   D , where the dot indicates the coadjoint action), leaves the level set J1 () invariant (Smale [168], Corollary 4.5). However, he only divided by G after fixing the total energy as well, in order to obtain the “minimal” manifold on which to analyze the reduced dynamics. The goal of his “topology and mechanics” program was to use topology, and specifically Morse theory, to study relative equilibria, which he did with great effectiveness. Marsden and Weinstein [110] combined Souriau’s momentum map for general symplectic actions, Smale’s idea of dividing the momentum level by the coadjoint isotropy group, and Cartan’s idea of removing the degeneracy of a 2-form by passing to the leaf space of the form’s null foliation. The key observation was that the leaves of the null foliation are precisely the (connected components of the) orbits of the coadjoint isotropy group (a fact we shall prove in the next section as the reduction lemma). An analogous observation was made in Meyer [117], except that Meyer worked in terms of a basis for the Lie algebra g and identified the subgroup G as the group which left the momentum level set J1 () invariant. In this way, he did not need to deal with the equivariance properties of the coadjoint representation. In the more general setting of symplectic manifolds with an equivariant momentum map for a symplectic group action, the fact that G acts on J1 () follows directly from equivariance of J. Thus, it makes sense to form the symplectic reduced space which is defined to be the quotient space P D J1 ()/G :

(14)

Roughly speaking, the symplectic reduction theorem states that, under suitable hypotheses, P is itself a symplectic manifold. To state this precisely, we need a short excursion on level sets of the momentum map and some facts about quotients. Free and Proper Actions The action of a Lie group G on a manifold M is called a free action if g  m D m for some g 2 G and m 2 M implies that g D e, the identity element.

Mechanical Systems: Symmetries and Reduction

An action of G on M is called proper when the map G  M ! M  M; (g; m) 7! (g  m; m) is a proper map – that is, inverse images of compact sets are compact. This is equivalent to the statement that if mk is a convergent sequence in M and if g k  m k converges in M, then g k has a convergent subsequence in G. As is shown in, for example, [2] and Duistermaat and Kolk [48], freeness, together with properness implies that the quotient space M/G is a smooth manifold and that the projection map  : M ! M/G is a smooth surjective submersion. Locally Free Actions An action of G on M is called infinitesimally free at a point m 2 M if  M (m) D 0 implies that  D 0. An action of G on M is called locally free at a point m 2 M if there is a neighborhood U of the identity in G such that g 2 U and g  m D m implies g D e. Proposition 1 An action of a Lie group G on a manifold M is locally free at m 2 M if and only if it is infinitesimally free at m. A free action is obviously locally free. The converse is not true because the action of any discrete group is locally free, but need not be globally free. When one has an action that is locally free but not globally free, one is lead to the theory of orbifolds, as in Satake [164]. In fact, quotients of manifolds by locally free and proper group actions are orbifolds, which follows by the use of the Palais slice theorem (see Palais [144]). Orbifolds come up in a variety of interesting examples involving, for example, resonances; see, for instance, Cushman and Bates [44] and Alber, Luther, Marsden, and Robbins [3] for some specific examples. Symmetry and Singularities If  is a regular value of J then we claim that the action is automatically locally free at the elements of the corresponding level set J1 (). In this context it is convenient to introduce the notion of the symmetry algebra at z 2 P defined by gz D f 2 g j P (z) D 0g : The symmetry algebra gz is the Lie algebra of the isotropy subgroup Gz of z 2 P defined by Gz D fg 2 G j g  z D zg : The following result (due to Smale [168] in the special case of cotangent bundles and in general to Arms, Marsden,

and Moncrief [5]), is important for the recognition of regular as well as singular points in the reduction process. Proposition 2 An element  2 g is a regular value of J if and only if gz D 0 for all z 2 J1 (). In other words, points are regular points precisely when they have trivial symmetry algebra. In examples, this gives an easy way to recognize regular points. For example, for the double spherical pendulum (see, for example, Marsden and Scheurle [108] or [95]), one can say right away that the only singular points are those with both pendula pointing vertically (either straight down or straight up). This result holds whether or not J is equivariant. This result, connecting the symmetry of z with the regularity of , suggests that points with symmetry are bifurcation points of J. This observation turns out to have many important consequences, including some related key convexity theorems. Now we are ready to state the symplectic reduction theorem. We will be making two sets of hypotheses; other variants are discussed in the next section. The following notation will be convenient in the statement of the results. SR (P; ˝) is a symplectic manifold, G is a Lie group that acts symplectically on P and has an equivariant momentum map J : P ! g . SRFree G acts freely and properly on P. SRRegular Assume that  2 g is a regular value of J and that the action of G on J1 () is free and proper From the previous discussion, note that condition SRFree implies condition SRRegular. The real difference is that SRRegular assumes local freeness of the action of G (which is equivalent to  being a regular value, as we have seen), while SRFree assumes global freeness (on all of P). Theorem 3 (Symplectic reduction theorem) Assume that condition SR and that either the condition SRFree or the condition SRRegular holds. Then P is a symplectic manifold, and is equipped with the reduced symplectic form ˝ that is uniquely characterized by the condition  ˝;  ˝ D i

(15)

where  : J1 () ! P is the projection to the quotient space and where i : J1 () ! P is the inclusion. The above procedure is often called point reduction because one is fixing the value of the momentum map at a point  2 g . An equivalent reduction method called orbit reduction will be discussed shortly.

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Coadjoint Orbits

Mathematical Physics Links

A standard example (due to Marsden and Weinstein [110]) that we shall derive in detail in the next section, is the construction of the coadjoint orbits in g of a group G by reduction of the cotangent bundle T  G with its canonical symplectic structure and with G acting on T  G by the cotangent lift of left (resp. right) group multiplication. In this case, one finds that (T  G) D O , the coadjoint orbit through  2 g . The reduced symplectic form is given by the Kostant, Kirillov, Souriau coadjoint form, also referred to as the KKS form:

Another example in Marsden and Weinstein [110] came from general relativity, namely the reduction of the cotangent bundle of the space of Riemannian metrics on a manifold M by the action of the group of diffeomorphisms of M. In this case, restriction to the zero momentum level is the divergence constraint of general relativity, and so one is led to a construction of a symplectic structure on a space closely related to the space of solutions of the Einstein equations, a question revisited in Fischer, Marsden, and Moncrief [51] and Arms, Marsden, and Moncrief [6]. Here one sees a precursor of an idea of Atiyahf and Bott [11], which has led to some of the most spectacular applications of reduction in mathematical physics and related areas of pure mathematics, especially low-dimensional topology.

!O ()(ad ; ad ) D h; [; ]i ;

(16)

where ;  2 g;  2 O , ad : g ! g is the adjoint operator defined by ad  :D [; ] and ad : g ! g is its dual. In this formula, one uses the minus sign for the left action and the plus sign for the right action. We recall that coadjoint orbits, like any group orbit is always an immersed manifold. Thus, one arrives at the following result (see also Theorem 7): Corollary 4 Given a Lie group G with Lie algebra g and any point  2 g , the reduced space (T  G) is the coadjoint orbit O through the point ; it is a symplectic manifold with symplectic form given by (16). This example, which “explains” Kostant, Kirillov and Souriau’s formula for this structure, is typical of many of the ensuing applications, in which the reduction procedure is applied to a “trivial” symplectic manifold to produce something interesting. Orbit Reduction An important variant of the symplectic reduction theorem is called orbit reduction and, roughly speaking, it constructs J1 (O)/G, where O is a coadjoint orbit in g . In the next section – see Theorem 8 – we show that orbit reduction is equivalent to the point reduction considered above. Cotangent Bundle Reduction The theory of cotangent bundle reduction is a very important special case of general reduction theory. Notice that the reduction of T  G above to give a coadjoint orbit is a special case of the more general procedure in which G is replaced by a configuration manifold Q. The theory of cotangent bundle reduction will be outlined in the historical overview in this chapter, and then treated in some detail in the following chapter.

Singular Reduction In the preceding discussion, we have been making hypotheses that ensure the momentum levels and their quotients are smooth manifolds. Of course, this is not always the case, as was already noted in Smale [168] and analyzed (even in the infinite-dimensional case) in Arms, Marsden, and Moncrief [5]. We give a review of some of the current literature and history on this singular case in Sect. “Reduction Theory: Historical Overview”. For an outline of this subject, see [142] and for a complete account of the technical details, see [138]. Reduction of Dynamics Along with the geometry of reduction, there is also a theory of reduction of dynamics. The main idea is that a G-invariant Hamiltonian H on P induces a Hamiltonian H on each of the reduced spaces, and the corresponding Hamiltonian vector fields X H and X H are  -related. The reverse of reduction is reconstruction and this leads one to the theory of classical geometric phases (Hannay–Berry phases); see Marsden, Montgomery, and Ratiu [98]. Reduction theory has many interesting connections with the theory of integrable systems; we just mention some selected references Kazhdan, Kostant, and Sternberg [72]; Ratiu [154,155,156]; Bobenko, Reyman, and Semenov-Tian-Shansky [22]; Pedroni [148]; Marsden and Ratiu [103]; Vanhaecke [174]; Bloch, Crouch, Marsden, and Ratiu [19], which the reader can consult for further information. Symplectic Reduction – Further Discussion The symplectic reduction theorem leans on a few key lemmas that we just state. The first refers to the reflexivity of

Mechanical Systems: Symmetries and Reduction

the operation of taking the symplectic orthogonal complement. Lemma 5 Let (V; ˝) be a finite dimensional symplectic vector space and W V be a subspace. Define the symplectic orthogonal to W by W ˝ D fv 2 V j ˝(v; w) D 0 for all w 2 Wg : Then  ˝ W˝ DW:

(17)

In what follows, we denote by G  z and G  z the G and G -orbits through the point z 2 P; note that if z 2 J1 () then G  z J1 (). The key lemma that is central for the symplectic reduction theorem is the following. Lemma 6 (Reduction lemma) Let P be a Poisson manifold and let J : P ! g be an equivariant momentum map of a Lie group action by Poisson maps of G on P. Let G   denote the coadjoint orbit through a regular value  2 g of J. Then J1 (G  ) D G  J1 () D fg  z j g 2 G and J(z) D g; (ii) G  z D (G  z) \ J1 (); (iii) J1 () and G  z intersect cleanly, i. e., (i)

Tz (G  z) D Tz (G  z) \ Tz (J1 ()); (iv) if (P; ˝) is symplectic, then Tz (J1 ()) D (Tz (G  z))˝ ; i. e., the sets Tz (J1 ()) and Tz (G  z) are ˝-orthogonal complements of each other. Refer to Fig. 1 for one way of visualizing the geometry associated with the reduction lemma. As it suggests, the two manifolds J1 () and G  z intersect in the orbit of the isotropy group G  z and their tangent spaces Tz J1 () and Tz (G z) are orthogonal and intersect in  symplectically  the space Tz G  z . Notice from the statement (iv) that Tz (J1 ())˝ Tz (J1 ()) provided that G  z D G  z. Thus, J1 () is coisotropic if G D G; for example, this happens if  D 0 or if G is Abelian. Remarks on the Reduction Theorem 1. Even if ˝ is exact; say ˝ D d and the action of G leaves  invariant, ˝ need not be exact. Perhaps the simplest example is a nontrivial coadjoint orbit of SO(3), which is a sphere with symplectic form given by the area form (by Stokes’ theorem, it cannot be ex-

Mechanical Systems: Symmetries and Reduction, Figure 1 The geometry of the reduction lemma

act). That this is a symplectic reduced space of T  SO(3) (with the canonical symplectic structure, so is exact) is shown in Theorem 7 below. 2. Continuing with the previous remark, assume that ˝ D d and that the G principal bundle J1 () ! P :D J1 ()/G is trivializable; that is, it admits  2 a global section s : P ! J1 (). Let  :D s  i 1 ˝ (P ). Then the reduced symplectic form ˝ D d is exact. This statement does not imply that the one-form  descends to the reduced space, only that the reduced symplectic form is exact and one if its primitives is  . In fact, if one changes the global section, another primitive of ˝ is found which differs from  by a closed one-form on P . 3. The assumption that  is a regular value of J can be relaxed. The only hypothesis needed is that  be a clean value of J, i. e., J1 () is a manifold and Tz (J1 ()) D ker Tz J. This generalization applies, for instance, for zero angular momentum in the three dimensional two body problem, as was noted by Marsden and Weinstein [110] and Kazhdan, Kostant, and Sternberg [72]; see also Guillemin and Sternberg [57]. Here are the general definitions: If f : M ! N is a smooth map, a point y 2 N is called a clean value if f 1 (y) is a submanifold and for each x 2 f 1 (y), Tx f 1 (y) D ker Tx f . We say that f intersects a submanifold L N cleanly if f 1 (L) is a submanifold of M and Tx ( f 1 (L)) D (Tx f )1 (T f (x) L). Note that regular values of f are clean values and that if f intersects the submanifold L transversally, then it intersects it cleanly. Also note that the definition of clean intersection of two manifolds is equivalent to the statement that the inclusion map of either one of them intersects the other cleanly. The reduction lemma is an example of this situation.

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4. The freeness and properness of the G action on J1 () are used only to guarantee that P is a manifold; these hypotheses can thus be replaced by the requirement that P is a manifold and  : J1 () ! P a submersion; the proof of the symplectic reduction theorem remains unchanged. 5. Even if  is a regular value (in the sense of a regular value of the mapping J), it need not be a regular point (also called a generic point) in g ; that is, a point whose coadjoint orbit is of maximal dimension. The reduction theorem does not require that  be a regular point. For example, if G acts on itself on the left by group multiplication and if we lift this to an action on T  G by the cotangent lift, then the action is free and so all  are regular values, but such values (for instance, the zero element in so(3) ) need not be regular. On the other hand, in many important stability considerations, a regularity assumption on the point  is required; see, for instance, Patrick [145], Ortega and Ratiu [136] and Patrick, Roberts, and Wulff [146].

Nonequivariant Reduction We now describe how one can carry out reduction for a nonequivariant momentum map. If J : P ! g is a nonequivariant momentum map on the connected symplectic manifold P with nonequivariance group one-cocycle  consider the affine action (9) and let e G  be the isotropy subgroup of  2 g relative to this action. Then, under the same regularity assumptions (for example, assume that G acts freely and properly on P, or that  is a regular value of J and that e G acts freely and properly on J1 ()), the quotient manifold P :D J1 ()/e G  is a symplectic manifold whose symplectic form is uniquely determined by the relation  ˝ D   ˝ . The proof of this statement is identical to i   the one given above with the obvious changes in the meaning of the symbols. When using nonequivariant reduction, one has to remember that G acts on g in an affine and not a linear manner. For example, while the coadjoint isotropy subgroup at the origin is equal to G; that is, G0 D G, this is no longer the case for the affine action, where e G 0 in general does not equal G. Coadjoint Orbits as Symplectic Reduced Spaces We now examine Corollary 4 – that is, that coadjoint orbits may be realized as reduced spaces – a little more closely. Realizing them as reduced spaces shows that they are symplectic manifolds See, Chap. 14 in [104] for a “di-

rect” or “bare hands” argument. Historically, a direct argument was found first, by Kirillov, Kostant and Souriau in the early 1960’s and the (minus) coadjoint symplectic structure was found to be ! (ad ; ad ) D  h; [; ]i

(18)

Interestingly, this is the symplectic structure on the symplectic leaves of the Lie–Poisson bracket, as is shown in, for example, [104]. (See the historical overview in Sect. “Reduction Theory: Historical Overview” below and specifically, see Eq. (21) for a quick review of the Lie– Poisson bracket). The strategy of the reduction proof, as mentioned in the discussion in the last section, is to show that the coadjoint symplectic form on a coadjoint orbit O of the point , at a point  2 O, may be obtained by symplectically reducing T  G at the value . The following theorem (due to Marsden and Weinstein [110]), and which is an elaboration on the result in Corollary 4, formulates the result for left actions; of course there is a similar one for right actions, with the minus sign replaced by a plus sign. Theorem 7 (Reduction to coadjoint orbits) Let G be a Lie group and let G act on G (and hence on T  G by cotangent lift) by left multiplication. Let  2 g and let JL : T  G ! g be the momentum map for the left action. Then  is a regular value of JL , the action of G is free and proper, the symplectic reduced space J1 L ()/G is identified via left translation with O , the coadjoint orbit through , and the reduced symplectic form coincides with !  given in Eq. (18). Remarks 1. Notice that, as in the general Symplectic Reduction Theorem 3, this result does not require  to be a regular (or generic) point in g ; that is, arbitrarily nearby coadjoint orbits may have a different dimension. 2. The form !  on the orbit need not be exact even though ˝ is. An example that shows this is SO(3), whose coadjoint orbits are spheres and whose symplectic structure is, as shown in [104], a multiple of the area element, which is not exact by Stokes’ Theorem. Orbit Reduction So far, we have presented what is usually called point reduction. There is another point of view that is called orbit reduction, which we now summarize. We assume the same set up as in the symplectic reduction theorem, with P connected, G acting symplectically, freely, and properly on P with an equivariant momentum map J : P ! g .

Mechanical Systems: Symmetries and Reduction

The connected components of the point reduced spaces P can be regarded as the symplectic leaves of the Poisson manifold  (P/G; f; gP/G ) in the following way. Form a map i : P ! P/G defined by selecting an equivalence class [z]G for z 2 J1 () and sending it to the class [z]G . This map is checked to be well-defined and smooth. We then have the commutative diagram

 One then checks that i is a Poisson injective immersion. Moreover, the i -images in P/G ofthe con- nected components of the symplectic manifolds P ; ˝ are its symplectic leaves (see [138] and references therein for details). As sets,      i P D J1 O /G ; where O g is the coadjoint orbit through  2 g . The set   PO :D J1 O /G is called the orbit reduced space associated to the orbit O . The smooth manifold structure (and hence  the topology) on PO is the one that makes the map i : P ! PO into a diffeomorphism. For the next theorem, which characterizes the symplectic form and the Hamiltonian dynamics on PO , recall the coadjoint orbit symplectic structure of Kirillov, Kostant and Souriau that was established in the preceding Theorem 7: !O ()(g (); g ()) D h; [; ]i ;

(19)

for ;  2 g and  2 O . We also recall that an injectively immersed submanifold of S of Q is called an initial submanifold of Q when for any smooth manifold P, a map g : P ! S is smooth if and only if  ı g : P ! Q is smooth, where  : S ,! Q is the inclusion. Theorem 8 (Symplectic orbit reduction theorem) In the setup explained above: (i)

The momentum map J is transverse to the coadjoint orbit O and hence J1 (O ) is an initial submanifold

  of P. Moreover, the projection O : J1 O ! PO is a surjective submersion. (ii) PO is a symplectic manifold with the symplectic form ˝O uniquely characterized by the relation  O ˝O D JO !O C iO ˝;

(20)

  where JO is the restriction of J to J1 O and iO : J1 O ,! P is the inclusion. (iii) The map i : P ! PO is a symplectic diffeomorphism. (iv) (Dynamics). Let H be a G-invariant function on P and e : P/G ! R by H D e define H H ı . Then the Hamiltonian vector field X H is also G-invariant and hence induces a vector field on P/G, which coincides with the Hamiltonian vector field Xe H . Moreover, the flow of Xe H leaves the symplectic leaves PO of P/G invariant. This flow restricted to the symplectic leaves is again Hamiltonian relative to the symplectic form ˝O and the e O given by Hamiltonian function H e H O ı O D H ı i O : Note that if O is an embedded submanifold of g then J is transverse to O and hence J1 (O ) is automatically an embedded submanifold of P. The proof of this theorem when O is an embedded submanifold of g can be found in Marle [91], Kazhdan, Kostant, and Sternberg [72], with useful additions given in Marsden [94] and Blaom [17]. For nonfree actions and when O is not an embedded submanifold of g see [138]. Further comments on the historical context of this result are given in the next section. Remarks 1. A similar result holds for right actions. 2. Freeness and properness of the G -action on J1 () are only needed indirectly. In fact these conditions are sufficient but not necessary for P to be a manifold. All that is needed is for P to be a manifold and  to be a submersion and the above result remains unchanged. 3. Note that the description of the symplectic structure on J1 (O)/G is not as simple as it was for J1 ()/G, while the Poisson bracket description is simpler on J1 (O)/G. Of course, the symplectic structure depends only on the orbit O and not on the choice of a point  on it. Cotangent Bundle Reduction Perhaps the most important and basic reduction theorem in addition to those already presented is the cotangent bundle reduction theorem. We shall give an exposition of

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the key aspects of this theory in Sect. “Cotangent Bundle Reduction” and give a historical account of its development, along with references in the next section. At this point, to orient the reader, we note that one of the special cases is cotangent bundle reduction at zero (see Theorem 10). This result says that if one has, again for simplicity, a free and proper action of G on Q (which is then lifted to T  Q by the cotangent lift), then the reduced space at zero of T  Q is given by T  (Q/G), with its canonical symplectic structure. On the other hand, reduction at a nonzero value is a bit more complicated and gives rise to modifications of the standard symplectic structure; namely, one adds to the canonical structure, the pull-back of a closed two form on Q to T  Q. Because of their physical interpretation (discussed, for example, in [104]), such extra terms are called magnetic terms. In Sect. “Cotangent Bundle Reduction”, we state the basic cotangent bundle reduction theorems along with providing some of the other important notions, such as the mechanical connection and the locked inertia tensor. Other notions that are important in mechanics, such as the amended potential, can be found in [95]. Reduction Theory: Historical Overview We have already given bits an pieces of the history of symplectic reduction and momentum maps. In this section we take a broader view of the subject to put things in historical and topical context. History before 1960 In the preceding sections, reduction theory has been presented as a mathematical construction. Of course, these ideas are rooted in classical work on mechanical systems with symmetry by such masters as Euler, Lagrange, Hamilton, Jacobi, Routh, Riemann, Liouville, Lie, and Poincaré. The aim of their work was, to a large extent, to eliminate variables associated with symmetries in order to simplify calculations in concrete examples. Much of this work was done using coordinates, although the deep connection between mechanics and geometry was already evident. Whittaker [179] gives a good picture of the theory as it existed up to about 1910. A highlight of this early theory was the work of Routh [161,163] who studied reduction of systems with cyclic variables and introduced the amended potential for the reduced system for the purpose of studying, for instance, the stability of a uniformly rotating state – what we would call today a relative equilibrium, terminology introduced later by Poincaré. Smale [168] eventually put the amended potential into a nice geometric setting. Routh’s

work was closely related to the reduction of systems with integrals in involution studied by Jacobi and Liouville around 1870; the Routh method corresponds to the modern theory of Lagrangian reduction for the action of Abelian groups. The rigid body, whose equations were discovered by Euler around 1740, was a key example of reduction – what we would call today either reduction to coadjoint orbits or Lie–Poisson reduction on the Hamiltonian side, or Euler– Poincaré reduction on the Lagrangian side, depending on one’s point of view. Lagrange [81] already understood reduction of the rigid body equations by a method not so far from what one would do today with the symmetry group SO(3). Many later authors, unfortunately, relied so much on coordinates (especially Euler angles) that there is little mention of SO(3) in classical mechanics books written before 1990, which by today’s standards, seems rather surprising! In addition, there seemed to be little appreciation until recently for the role of topological notions; for example, the fact that one cannot globally split off cyclic variables for the S1 action on the configuration space of the heavy top. The Hopf fibration was patiently waiting to be discovered in the reduction theory for the classical rigid body, but it was only explicitly found later on by H. Hopf [64]. Hopf was, apparently, unaware that this example is of great mechanical interest – the gap between workers in mechanics and geometers seems to have been particularly wide at that time. Another noteworthy instance of reduction is Jacobi’s elimination of the node for reducing the gravitational (or electrostatic) n-body problem by means of the group SE(3) of Euclidean motions, around 1860 or so. This example has, of course, been a mainstay of celestial mechanics. It is related to the work done by Riemann, Jacobi, Poincaré and others on rotating fluid masses held together by gravitational forces, such as stars. Hidden in these examples is much of the beauty of modern reduction, stability and bifurcation theory for mechanical systems with symmetry. While both symplectic and Poisson geometry have their roots in the work of Lagrange and Jacobi, it matured considerably with the work of Lie [90], who discovered many remarkably modern concepts such as the Lie– Poisson bracket on the dual of a Lie algebra. See Weinstein [176] and Marsden and Ratiu [104] for more details on the history. How Lie could have viewed his wonderful discoveries so divorced from their roots in mechanics remains a mystery. We can only guess that he was inspired by Jacobi, Lagrange and Riemann and then, as mathematicians often do, he quickly abstracted the ideas, losing valuable scientific and historical connections along the way.

Mechanical Systems: Symmetries and Reduction

As we have already hinted, it was the famous paper Poincaré [153] where we find what we call today the Euler– Poincaré equations – a generalization of the Euler equations for both fluids and the rigid body to general Lie algebras. (The Euler–Poincaré equations are treated in detail in [104]). It is curious that Poincaré did not stress either the symplectic ideas of Lie, nor the variational principles of mechanics of Lagrange and Hamilton – in fact, it is not clear to what extent he understood what we would call today Euler–Poincaré reduction. It was only with the development and physical application of the notion of a manifold, pioneered by Lie, Poincaré, Weyl, Cartan, Reeb, Synge and many others, that a more general and intrinsic view of mechanics was possible. By the late 1950’s, the stage was set for an explosion in the field. 1960–1972 Beginning in the 1960’s, the subject of geometric mechanics indeed did explode with the basic contributions of people such as (alphabetically and nonexhaustively) Abraham, Arnold, Kirillov, Kostant, Mackey, MacLane, Segal, Sternberg, Smale, and Souriau. Kirillov and Kostant found deep connections between mechanics and pure mathematics in their work on the orbit method in group representations, while Arnold, Smale, and Souriau were in closer touch with mechanics. The modern vision of geometric mechanics combines strong links to important questions in mathematics with the traditional classical mechanics of particles, rigid bodies, fields, fluids, plasmas, and elastic solids, as well as quantum and relativistic theories. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to the “hidden” symmetries underlying integrable systems. As we have already mentioned, reduction theory concerns the removal of variables using symmetries and their associated conservation laws. Variational principles, in addition to symplectic and Poisson geometry, provide fundamental tools for this endeavor. In fact, conservation of the momentum map associated with a symmetry group action is a geometric expression of the classical Noether theorem (discovered by variational, not symplectic methods). Arnold and Smale The modern era of reduction theory began with the fundamental papers of Arnold [7] and Smale [168]. Arnold focused on systems whose configuration manifold is a Lie group, while Smale focused on bifurcations of relative equilibria. Both Arnold and Smale linked their theory strongly with examples. For Arnold, they were the same

examples as for Poincaré, namely the rigid body and fluids, for which he went on to develop powerful stability methods, as in Arnold [8]. With hindsight, we can say that Arnold [7] was picking up on the basic work of Poincaré for both rigid body motion and fluids. In the case of fluids, G is the group of (volume preserving) diffeomorphisms of a compact manifold (possibly with boundary). In this setting, one obtains the Euler equations for (incompressible) fluids by reduction from the Lagrangian formulation of the equations of motion, an idea exploited by Arnold [7] and Ebin and Marsden [49]. This sort of description of a fluid goes back to Poincaré (using the Euler–Poincaré equations) and to the thesis of Ehrenfest (as geodesics on the diffeomorphism group), written under the direction of Boltzmann. For Smale, the motivating example was celestial mechanics, especially the study of the number and stability of relative equilibria by a topological study of the energymomentum mapping. He gave an intrinsic geometric account of the amended potential and in doing so, discovered what later became known as the mechanical connection. (Smale appears to not to have recognized that the interesting object he called ˛ is, in fact, a principal connection; this was first observed by Kummer [79]). One of Smale’s key ideas in studying relative equilibria was to link mechanics with topology via the fact that relative equilibria are critical points of the amended potential. Besides giving a beautiful exposition of the momentum map, Smale also emphasized the connection between singularities and symmetry, observing that the symmetry group of a phase space point has positive dimension if and only if that point is not a regular point of the momentum map restricted to a fiber of the cotangent bundle (Smale [168], Proposition 6.2) – a result we have proved in Proposition 2. He went on from here to develop his topology and mechanics program and to apply it to the planar n-body problem. The topology and mechanics program definitely involved reduction ideas, as in Smale’s construction of the quotients of integral manifolds, as in I c;p /S 1 (Smale [168], page 320). He also understood Jacobi’s elimination of the node in this context, although he did not attempt to give any general theory of reduction along these lines. Smale thereby set the stage for symplectic reduction: he realized the importance of the momentum map and of quotient constructions, and he worked out explicit examples like the planar n-body problem with its S1 symmetry group. (Interestingly, he pointed out that one should really use the nonabelian group SE(2); his feeling of unease with fixing the center of mass of an n-body system is remarkably perceptive).

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Synthesis The problem of synthesizing the Lie algebra reduction methods of Arnold [7] with the techniques of Smale [168] on the reduction of cotangent bundles by Abelian groups, led to the development of reduction theory in the general context of symplectic manifolds and equivariant momentum maps in Marsden and Weinstein [110] and Meyer [117], as we described in the last section. Both of these papers were completed by 1972. Poisson Manifolds Meanwhile, things were also gestating from the viewpoint of Poisson brackets and the idea of a Poisson manifold was being initiated and developed, with much duplication and rediscovery (see, Section 10.1 in [104] for additional information). A basic example of a noncanonical Poisson bracket is the Lie–Poisson bracket on g , the dual of a Lie algebra g. This bracket (which comes with a plus or minus sign) is given on two smooth functions on g by 

 ı f ıg ; ; (21) f f ; gg˙ () D ˙ ; ı ı where ı f /ı is the derivative of f , but thought of as an element of g. These Poisson structures, including the coadjoint orbits as their symplectic leaves, were known to Lie [90], although, as we mentioned previously, Lie does not seem to have recognized their importance in mechanics. It is also not clear whether or not Lie realized that the Lie Poisson bracket is the Poisson reduction of the canonical Poisson bracket on T  G by the action of G. (See, Chap. 13 in [104] for an account of this theory). The first place we know of that has this clearly stated (but with no references, and no discussion of the context) is Bourbaki [24], Chapter III, Section 4, Exercise 6. Remarkably, this exercise also contains an interesting proof of the Duflo–Vergne theorem (with no reference to the original paper, which appeared in 1969). Again, any hint of links with mechanics is missing. This takes us up to about 1972. Post 1972 An important contribution was made by Marle [91], who divides the inverse image of an orbit by its characteristic foliation to obtain the product of an orbit and a reduced manifold. In particular, as we saw in Theorem 8, P is symplectically diffeomorphic to an “orbit-reduced” space P Š J 1 (O )/G, where O is a coadjoint orbit of G. From this it follows that the P are symplectic leaves in the

Poisson space P/G. The related paper of Kazhdan, Kostant, and Sternberg [72] was one of the first to notice deep links between reduction and integrable systems. In particular, they found that the Calogero–Moser systems could be obtained by reducing a system that was trivially integrable; in this way, reduction provided a method of producing an interesting integrable system from a simple one. This point of view was used again by, for example, Bobenko, Reyman, and Semenov–Tian–Shansky [22] in their spectacular group theoretic explanation of the integrability of the Kowalewski top. Noncanonical Poisson Brackets The Hamiltonian description of many physical systems, such as rigid bodies and fluids in Eulerian variables, requires noncanonical Poisson brackets and constrained variational principles of the sort studied by Lie and Poincaré. As discussed above, a basic example of a noncanonical Poisson bracket is the Lie–Poisson bracket on the dual of a Lie algebra. From the mechanics perspective, the remarkably modern book (but which was, unfortunately, rather out of touch with the corresponding mathematical developments) by Sudarshan and Mukunda [172] showed via explicit examples how systems such as the rigid body could be written in terms of noncanonical brackets, an idea going back to Pauli [147], Martin [115] and Nambu [129]. Others in the physics community, such as Morrison and Greene [128] also discovered noncanonical bracket formalisms for fluid and magnetohydrodynamic systems. In the 1980’s, many fluid and plasma systems were shown to have a noncanonical Poisson formulation. It was Marsden and Weinstein [111,112] who first applied reduction techniques to these systems. The reduction philosophy concerning noncanonical brackets can be summarized by saying Any mechanical system has its roots somewhere as a cotangent bundle and one can recover noncanonical brackets by the simple process of Poisson reduction. For example, in fluid mechanics, this reduction is implemented by the Lagrange-to-Euler map. This view ran contrary to the point of view, taken by some researchers, that one should proceed by analogy or guesswork to find Poisson structures and then to try to limit the guesses by the constraint of Jacobi’s identity. In the simplest version of the Poisson reduction process, one starts with a Poisson manifold P on which a group G acts by Poisson maps and then forms the quotient space P/G, which, if not singular, inherits a natural Poisson structure itself. Of course, the Lie–Poisson struc-

Mechanical Systems: Symmetries and Reduction

ture on g is inherited in exactly this way from the canonical symplectic structure on T  G. One of the attractions of this Poisson bracket formalism was its use in stability theory. This literature is now very large, but Holm, Marsden, Ratiu, and Weinstein [63] is representative. The way in which the Poisson structure on P is related to that on P/G was clarified in a generalization of Poisson reduction due to Marsden and Ratiu [103], a technique that has also proven useful in integrable systems (see, e. g., Pedroni [148] and Vanhaecke [174]). Reduction theory for mechanical systems with symmetry has proven to be a powerful tool that has enabled key advances in stability theory (from the Arnold method to the energy-momentum method for relative equilibria) as well as in bifurcation theory of mechanical systems, geometric phases via reconstruction – the inverse of reduction – as well as uses in control theory from stabilization results to a deeper understanding of locomotion. For a general introduction to some of these ideas and for further references, see Marsden, Montgomery, and Ratiu [98]; Simo, Lewis, and Marsden [166]; Marsden and Ostrowski [99]; Marsden and Ratiu [104]; Montgomery [122,123,124,125,126]; Blaom [16,17] and Kanso, Marsden, Rowley, and Melli-Huber [71]. Tangent and Cotangent Bundle Reduction The simplest case of cotangent bundle reduction is the case of reduction of P D T  Q at  D 0; the answer is simply P0 D T  (Q/G) with the canonical symplectic form. Another basic case is when G is Abelian. Here, (T  Q) Š T  (Q/G), but the latter has a symplectic structure modified by magnetic terms, that is, by the curvature of the mechanical connection. An Abelian version of cotangent bundle reduction was developed by Smale [168]. Then Satzer [165] studied the relatively simple, but important case of cotangent bundle reduction at the zero value of the momentum map. The full generalization of cotangent bundle reduction for nonabelian groups at arbitrary values of the momentum map appears for the first time in Abraham and Marsden [1]. It was Kummer [79] who first interpreted this result in terms of a connection, now called the mechanical connection. The geometry of this situation was used to great effect in, for example, Guichardet [54,66], Iwai [67], and Montgomery [120,123,124]. We give an account of cotangent bundle reduction theory in the following section. The Gauge Theory Viewpoint Tangent and cotangent bundle reduction evolved into what we now term as the “bundle picture” or the “gauge

theory of mechanics”. This picture was first developed by Montgomery, Marsden, and Ratiu [127] and Montgomery [120,121]. That work was motivated and influenced by the work of Sternberg [171] and Weinstein [175] on a “Yang–Mills construction” which is, in turn, motivated by Wong’s equations, i. e., the equations for a particle moving in a Yang–Mills field. The main result of the bundle picture gives a structure to the quotient spaces (T  Q)/G and (TQ)/G when G acts by the cotangent and tangent lifted actions. The symplectic leaves in this pic´ ture were analyzed by Zaalani [182], Cushman and Sniatycki [47] and Marsden and Perlmutter [102]. The work of Perlmutter and Ratiu [149] gives a unified study of the Poisson bracket on (T  Q)/G in both the Sternberg and Weinstein realizations of the quotient. As mentioned earlier, we shall review some of the basics of cotangent bundle reduction theory in Sect. “Cotangent Bundle Reduction”. Further information on this theory may be found in [1,95], and [92], as well as a number of the other references mentioned above. Lagrangian Reduction A key ingredient in Lagrangian reduction is the classical work of Poincaré [153] in which the Euler–Poincaré equations were introduced. Poincaré realized that the equations of fluids, free rigid bodies, and heavy tops could all be described in Lie algebraic terms in a beautiful way. The importance of these equations was realized by Hamel [58,59] and Chetayev [40], but to a large extent, the work of Poincaré lay dormant until it was revived in the Russian literature in the 1980’s. The more recent developments of Lagrangian reduction were motivated by attempts to understand the relation between reduction, variational principles and Clebsch variables in Cendra and Marsden [34] and Cendra, Ibort, and Marsden [33]. In Marsden and Scheurle [109] it was shown that, for matrix groups, one could view the Euler–Poincaré equations via the reduction of Hamilton’s variational principle from TG to g. The work of Bloch, Krishnaprasad, Marsden and Ratiu [21] established the Euler–Poincaré variational structure for general Lie groups. The paper of Marsden and Scheurle [109] also considered the case of more general configuration spaces Q on which a group G acts, which was motivated by both the Euler–Poincaré case as well as the work of Cendra and Marsden [34] and Cendra, Ibort, and Marsden [33]. The Euler– Poincaré equations correspond to the case Q D G. Related ideas stressing the groupoid point of view were given in Weinstein [177]. The resulting reduced equations were

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called the reduced Euler–Lagrange equations. This work is the Lagrangian analogue of Poisson reduction, in the sense that no momentum map constraint is imposed. Lagrangian reduction proceeds in a way that is very much in the spirit of the gauge theoretic point of view of mechanical systems with symmetry. It starts with Hamilton’s variational principle for a Lagrangian system on a configuration manifold Q and with a symmetry group G acting on Q. The idea is to drop this variational principle to the quotient Q/G to derive a reduced variational principle. This theory has its origins in specific examples such as fluid mechanics (see, for example, Arnold [7] and Bretherton [25]), while the systematic theory of Lagrangian reduction was begun in Marsden and Scheurle [109] and further developed in Cendra, Marsden, and Ratiu [35]. The latter reference also introduced a connection to realize the space (TQ)/G as the fiber product T(Q/G)  g˜ of T(Q/G) with the associated bundle formed using the adjoint action of G on g. The reduced equations associated to this construction are called the Lagrange–Poincaré equations and their geometry has been fairly well developed. Note that a G-invariant Lagrangian L on TQ induces a Lagrangian l on (TQ)/G. Until recently, the Lagrangian side of the reduction story had lacked a general category that is the Lagrangian analogue of Poisson manifolds in which reduction can be repeated. One candidate is the category of Lie algebroids, as explained in Weinstein [177]. Another is that of Lagrange–Poincaré bundles, developed in Cendra, Marsden, and Ratiu [35]. Both have tangent bundles and Lie algebras as basic examples. The latter work also develops the Lagrangian analogue of reduction for central extensions and, as in the case of symplectic reduction by stages, cocycles and curvatures enter in a natural way. This bundle picture and Lagrangian reduction has proven very useful in control and optimal control problems. For example, it was used in Chang, Bloch, Leonard, Marsden and Woolsey [38] to develop a Lagrangian and Hamiltonian reduction theory for controlled mechanical systems and in Koon and Marsden [76] to extend the falling cat theorem of Montgomery [123] to the case of nonholonomic systems as well as to nonzero values of the momentum map. Finally we mention that the paper Cendra, Marsden, Pekarsky, and Ratiu [37] develops the reduction theory for Hamilton’s phase space principle and the equations on the reduced space, along with a reduced variational principle, are developed and called the Hamilton–Poincaré equations. Even in the case Q D G, this collapses to an interesting variational principle for the Lie–Poisson equations on g .

Legendre Transformation Of course the Lagrangian and Hamiltonian sides of the reduction story are linked by the Legendre transformation. This mapping descends at the appropriate points to give relations between the Lagrangian and the Hamiltonian sides of the theory. However, even in standard cases such as the heavy top, one must be careful with this approach, as is already explained in, for example, Holm, Marsden, and Ratiu [61]. For field theories, such as the Maxwell–Vlasov equations, this issues is also important, as explained in Cendra, Holm, Hoyle and Marsden [31] (see also Tulczyjew and Urba´nski [173]). Nonabelian Routh Reduction Routh reduction for Lagrangian systems, which goes back Routh [161,162,163] is classically associated with systems having cyclic variables (this is almost synonymous with having an Abelian symmetry group). Modern expositions of this classical theory can be found in Arnold, Koslov, and Neishtadt [10] and in [104], §8.9. Routh Reduction may be thought of as the Lagrangian analog of symplectic reduction in that a momentum map is set equal to a constant. A key feature of Routh reduction is that when one drops the Euler–Lagrange equations to the quotient space associated with the symmetry, and when the momentum map is constrained to a specified value (i. e., when the cyclic variables and their velocities are eliminated using the given value of the momentum), then the resulting equations are in Euler–Lagrange form not with respect to the Lagrangian itself, but with respect to a modified function called the Routhian. Routh [162] applied his method to stability theory; this was a precursor to the energy-momentum method for stability that synthesizes Arnold’s and Routh’s methods (see Simo, Lewis and Marsden [166]). Routh’s stability method is still widely used in mechanics. The initial work on generalizing Routh reduction to the nonabelian case was that of Marsden and Scheurle [108]. This subject was further developed in Jalnapurkar and Marsden [69] and Marsden, Ratiu and Scheurle [105]. The latter reference used this theory to give some nice formulas for geometric phases from the Lagrangian point of view. Semidirect Product Reduction In the simplest case of a semidirect product, one has a Lie group G that acts on a vector space V (and hence on its dual V  ) and then one forms the semidirect product S D GsV , generalizing the semidirect product structure of the Euclidean group SE(3) D SO(3) s R3 .

Mechanical Systems: Symmetries and Reduction

Consider the isotropy group G a 0 for some a0 2 V  . The semidirect product reduction theorem states that each of the symplectic reduced spaces for the action of G a 0 on T  G is symplectically diffeomorphic to a coadjoint orbit in (g s V ) , the dual of the Lie algebra of the semidirect product. This semidirect product theory was developed by Guillemin and Sternberg [55,56], Ratiu [154,157,158], and Marsden, Ratiu, and Weinstein [106,107]. The Lagrangian reduction analog of semidirect product theory was developed by Holm, Marsden and Ratiu [61,62]. This construction is used in applications where one has advected quantities (such as the direction of gravity in the heavy top, density in compressible fluids and the magnetic field in MHD) as well as to geophysical flows. Cendra, Holm, Hoyle and Marsden [31] applied this idea to the Maxwell–Vlasov equations of plasma physics. Cendra, Holm, Marsden, and Ratiu [32] showed how Lagrangian semidirect product theory fits into the general framework of Lagrangian reduction. The semidirect product reduction theorem has been proved in Landsman [82], Landsman [83], Chap. 4 as an application of a stages theorem for his special symplectic reduction method. Even though special symplectic reduction generalizes Marsden–Weinstein reduction, the special reduction by stages theorem in Landsman [82] studies a setup that, in general, is different to the ones in the reduction by stages theorems of [92]. Singular Reduction Singular reduction starts with the observation of Smale [168] that we have already mentioned: z 2 P is a regular point of a momentum map J if and only if z has no continuous isotropy. Motivated by this, Arms, Marsden, and Moncrief [5,6] showed that (under hypotheses which include the ellipticity of certain operators and which can be interpreted more or less, as playing the role of a properness assumption on the group action in the finite dimensional case) the level sets J1 (0) of an equivariant momentum map J have quadratic singularities at points with continuous symmetry. While such a result is easy to prove for compact group actions on finite dimensional manifolds (using the equivariant Darboux theorem), the main examples of Arms, Marsden, and Moncrief [5] were, in fact, infinite dimensional – both the phase space and the group. Singular points in the level sets of the momentum map are related to convexity properties of the momentum map in that the singular points in phase space map to corresponding singular points in the the image polytope. The paper of Otto [143] showed that if G is a Lie group acting properly on an almost Kähler manifold then the or-

bit space J1 ()/G decomposes into symplectic smooth manifolds constructed out of the orbit types of the G-action on P. In some related work, Huebschmann [65] has made a careful study of the singularities of moduli spaces of flat connections. The detailed structure of J1 (0)/G for compact Lie groups acting on finite dimensional manifolds was determined by Sjamaar and Lerman [167]; their work was extended to proper Lie group actions and to J1 (O )/G by Bates and Lerman [12], with the assumption that O be locally closed in g . Ortega [130] and [138] redid the entire singular reduction theory for proper Lie group actions starting with the point reduced spaces J1 ()/G and also connected it to the more algebraic approach of Arms, Cushman, and Gotay [4]. Specific examples of singular reduction, with further references, may be found in Lerman, Montgomery, and Sjamaar [84] and [44]. One of these, the “canoe” is given in detail in [92]. In fact, this is an example of singular reduction in the case of cotangent bundles, and much more can be said in this case; see Olmos and Dias [150,151]. Another approach to singular reduction based on the technique of blowing up singularities, and which was also designed for the case of singular cotangent bundle reduction, was started in Hernandez and Marsden [60] and Birtea, Puta, Ratiu, and Tudoran [14], a technique which requires further development. Singular reduction has been extensively used in the study of the persistence, bifurcation, and stability of relative dynamical elements; see [41,42,53,85,86,132,134,135, 136,139,146,159,160,180,181]. Symplectic Reduction Without Momentum Maps The reduction theory presented so far needs the existence of a momentum map. However, more primitive versions of this procedure based on foliation theory (see Cartan [26] and Meyer [117]) do not require the existence of this object. Working in this direction, but with a mathematical program that goes beyond the reduction problem, Condevaux, Dazord, and Molino [43] introduced a concept that generalizes the momentum map. This object is defined via a connection that associates an additive holonomy group to each canonical action on a symplectic manifold. The existence of the momentum map is equivalent to the vanishing of this group. Symplectic reduction has been carried out using this generalized momentum map in Ortega and Ratiu [140,141]. Another approach to symplectic reduction that is able to avoid the possible non-existence of the momentum map is based on the optimal momentum map introduced and studied in Ortega and Ratiu [137], Ortega [131], and [92].

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This distribution theoretical approach can also deal with reduction of Poisson manifolds, where the standard momentum map does not exist generically.

Ratiu [36] and Bloch [18] for a more detailed historical review. Multisymplectic Reduction

Reduction of Other Geometric Structures Besides symplectic reduction, there are many other geometric structures on which one can perform similar constructions. For example, one can reduce Kähler, hyperKähler, Poisson, contact, Jacobi, etc. manifolds and this can be done either in the regular or singular cases. We refer to [138] for a survey of the literature for these topics. The Method of Invariants This method seeks to parametrize quotient spaces by group invariant functions. It has a rich history going back to Hilbert’s invariant theory. It has been of great use in bifurcation with symmetry (see Golubitsky, Stewart, and Schaeffer [52] for instance). In mechanics, the method was developed by Kummer, Cushman, Rod and coworkers in the 1980’s; see, for example, Cushman and Rod [45]. We will not attempt to give a literature survey here, other than to refer to Kummer [80], Kirk, Marsden, and Silber [73], Alber, Luther, Marsden, and Robbins [3] and the book of Cushman and Bates [44] for more details and references. Nonholonomic Systems Nonholonomic mechanical systems (such as systems with rolling constraints) provide a very interesting class of systems where the reduction procedure has to be modified. In fact this provides a class of systems that gives rise to an almost Poisson structure, i. e. a bracket which does not necessarily satisfy the Jacobi identity. Reduction theory for nonholonomic systems has made a lot of progress, but many interesting questions still remain. In these types of systems, there is a natural notion of a momentum map, but in general it is not conserved, but rather obeys a momentum equation as was discovered by Bloch, Krishnaprasad, Marsden, and Murray [20]. This means, in particular, that point reduction in such a situation may not be appropriate. Nevertheless, Poisson reduction in the almost Poisson and almost symplectic setting is interesting and from the mathematical point of view, point reduction is also interesting, although, as remarked, one has to be cautious with how it is applied to, for example, nonholonomic systems. A few references are Koiller [75], Bates and Sniatycki [13], Bloch, Krishnaprasad, Marsden, and Murray [20], Koon and Marsden [77], Blankenstein and Van Der Schaft [15], ´ Cushman, Sniatycki [46], Planas-Bielsa [152], and Ortega and Planas-Bielsa [133]. We refer to Cendra, Marsden, and

Reduction theory is by no means completed. For example, for PDE’s, the multisymplectic (as opposed to symplectic) framework seems appropriate, both for relativistic and nonrelativistic systems. In fact, this approach has experienced somewhat of a revival since it has been realized that it is rather useful for numerical computation (see Marsden, Patrick, and Shkoller [100]). Only a few instances and examples of multisymplectic and multi-Poisson reduction are really well understood (see Marsden, Montgomery, Morrison, and Thompson [97]; CastrillónLópez, Ratiu and Shkoller [30], Castrillón-López, Garcia Pérez and Ratiu [27], Castrillón-López and Ratiu [28], Castrillón-López and Marsden [29]), so one can expect to see more activity in this area as well. Discrete Mechanical Systems Another emerging area, also motivated by numerical analysis, is that of discrete mechanics. Here the idea is to replace the velocity phase space TQ by Q  Q, with the role of a velocity vector played by a pair of nearby points. This has been a powerful tool for numerical analysis, reproducing standard symplectic integration algorithms and much more. See, for example, Wendlandt and Marsden [178], Kane, Marsden, Ortiz and West [70], Marsden and West [113], Lew, Marsden, Ortiz, and West [87] and references therein. This subject, too, has its own reduction theory. See Marsden, Pekarsky, and Shkoller [101], Bobenko and Suris [23] and Jalnapurkar, Leok, Marsden and West [68]. Discrete mechanics also has some intriguing links with quantization, since Feynman himself first defined path integrals through a limiting process using the sort of discretization used in the discrete action principle (see Feynman and Hibbs [50]). Cotangent Bundle Reduction As mentioned earlier, the cotangent bundle reduction theorems are amongst the most basic and useful of the symplectic reduction theorems. Here we only present the regular versions of the theorems. Cotangent bundle reduction theorems come in two forms – the embedding cotangent bundle reduction theorem and the bundle cotangent bundle reduction theorem. We start with a smooth, free, and proper, left action ˚: GQ ! Q

Mechanical Systems: Symmetries and Reduction

of the Lie group G on the configuration manifold Q and lift it to an action on T  Q. This lifted action is symplectic with respect to the canonical symplectic form on T  Q, which we denote ˝can , and has an equivariant momentum map J : T  Q ! g given by ˝

˛ hJ(˛q ); i D ˛q ;  Q (q) ; where  2 g. Letting  2 g , the aim of this section is to determine the structure of the symplectic reduced space ((T  Q) ; ˝ ), which, by Theorem 3, is a symplectic manifold. We are interested in particular in the question of to what extent ((T  Q) ; ˝ ) is a synthesis of a cotangent bundles and a coadjoint orbit.

For  2 g and q 2 Q, notice that, under the condition CBR1, (iQ ˛ )(q) D hJ(˛ (q)); i D h0 ; i ; and so iQ ˛ is a constant function on Q. Therefore, for  2 g , iQ d˛ D £Q ˛  diQ ˛ D 0 ;

since the Lie derivative is zero by G -invariance of ˛ . It follows that There is a unique two-form ˇ on Q such that  ˇ D d˛ : Q;G 

Since Q;G is a submersion, ˇ is closed (it need not be exact). Let

Cotangent Bundle Reduction: Embedding Version In this version of the theorem, we first form the quotient manifold Q :D Q/G ; which we call the -shape space. Since the action of G on Q is smooth, free, and proper, so is the action of the isotropy subgroup G and therefore, Q is a smooth manifold and the canonical projection Q;G : Q ! Q is a surjective submersion. Consider the G -action on Q and its lift to T  Q. This lifted action is of course also symplectic with respect to the canonical symplectic form ˝can and has an equivariant momentum map J : T  Q ! g obtained by restricting J; that is, for ˛q 2 Tq Q, J (˛q ) D J(˛q )jg : Let 0 :D jg 2 g be the restriction of  to g . Notice that there is a natural inclusion of submanifolds J1 () (J )1 (0 ) :

(23)

(22)

Since the actions are free and proper,  and 0 are regular values, so these sets are indeed smooth manifolds. Note that, by construction, 0 is G -invariant. There will be two key assumptions relevant to the embedding version of cotangent bundle reduction. Namely, CBR1 In the above setting, assume there is a G -invariant one-form ˛ on Q with values in (J )1 (0 ); and the condition (which by (22), is a stronger condition) CBR2 Assume that ˛ in CBR1 takes values in J1 ().

B D Q ˇ ; where Q : T  Q ! Q is (following our general conventions for maps) the cotangent bundle projection. Also, to avoid confusion with the canonical symplectic form ˝can on T  Q, we shall denote the canonical symplectic form on T  Q , the cotangent bundle of -shape space, by !can . Theorem 9 (Cotangent bundle reduction – embedding version) (i)

If condition CBR1 holds, then there is a symplectic embedding ' : ((T  Q) ; ˝ ) ! (T  Q ; !can  B ) ;

onto a submanifold of T  Q covering the base Q/G . (ii) The map ' in (i) gives a symplectic diffeomorphism of ((T  Q) ; ˝ ) onto (T  Q ; !can  B ) if and only if g D g . (iii) If CBR2 holds, then the image of ' equals the vector subbundle [TQ;G (V )]ı of T  Q , where V TQ is the vector subbundle consisting of vectors tangent to the G-orbits, that is, its fiber at q 2 Q equals Vq D f Q (q) j  2 gg, and ı denotes the annihilator relative to the natural duality pairing between TQ and T  Q . Remarks 1. A history of this result can be found in Sect. “Reduction Theory: Historical Overview”. 2. As shown in the appendix on Principal Connections (see Proposition A2) the required one form ˛ may be constructed satisfying condition CBR1 from a connec-

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tion on the -shape space bundle Q;G : Q ! Q/G and an ˛ satisfying CBR2 can be constructed using a connection on the shape space bundle Q;G : Q ! Q/G. 3. Note that in the case of Abelian reduction, or, more generally, the case in which G D G , the reduced space is symplectically diffeomorphic to T  (Q/G) with the symplectic structure given by ˝can  B . In particular, if  D 0, then the symplectic form on T  (Q/G) is the canonical one, since in this case one can choose ˛ D 0 which yields B D 0. 4. The term B on T  Q is usually called a magnetic term, a gyroscopic term, or a Coriolis term. The terminology “magnetic” comes from the Hamiltonian description of a particle of charge e moving according to the Lorentz force law in R3 under the influence of a magnetic field B. This motion takes place in T  R3 but with the nonstandard symplectic structure dq i ^ dp i  ec B; i D 1; 2; 3, where c is the speed of light and B is regarded as a closed two-form: B D B x dy ^ dz  B y dx ^ dz C Bz dx ^ dy (see §6.7 in [104] for details) The strategy for proving this theorem is to first deal with the case of reduction at zero and then to treat the general case using a momentum shift. Reduction at Zero The reduced space at  D 0 is, as a set, (T  Q)0 D J1 (0)/G since, for  D 0; G D G. Notice that in this case, there is no distinction between orbit reduction and symplectic reduction. Theorem 10 (Reduction at zero) Assume that the action of G on Q is free and proper, so that the quotient Q/G is a smooth manifold. Then 0 is a regular value of J and there is a symplectic diffeomorphism between (T  Q)0 and T  (Q/G) with its canonical symplectic structure. The Case G D G If one is reducing at zero, then clearly G D G . However, this is an important special case of the general cotangent bundle reduction theorem that, for example, includes the case of Abelian reduction. The key assumption here is that G D G , which indeed is always the case if G is Abelian. Theorem 11 Assume that the action of G on Q is free and proper, so that the quotient Q/G is a smooth manifold. Let  2 g , assume that G D G , and assume that CBR2 holds. Then  is a regular value of J and there is a symplectic

diffeomorphism between (T  Q) and T  (Q/G), the latter with the symplectic form !can B ; here, !can is the canoni ˇ , where cal symplectic form on T  (Q/G) and B D Q/G  the two form ˇ on Q/G is defined by  ˇ D d˛ : Q;G

Example Consider the reduction of a general cotangent bundle T  Q by G D SO(3). Here G Š S 1 , if  ¤ 0, and so the reduced space is embedded into the cotangent bundle T  (Q/S 1 ). A specific example is the case of Q D SO(3). 2 , the sphere Then the reduced space (T  SO(3)) is Skk of radius kk which is a coadjoint orbit in so(3) . In this 2 and the embedding of case, Q/G D SO(3)/S 1 Š Skk 2  2 Skk into T Skk is the zero section. Magnetic Terms and Curvature Using the results of the preceding section, we will now show how one can interpret the magnetic term B as the curvature of a connection on a principal bundle. We saw in the preamble to the Cotangent Bundle Reduction Theorem 9 that iQ d˛ D 0 for any  2 g , which was used to drop d˛ to the quotient. In the language of principal bundles, this may be rephrased by saying that d˛ is horizontal and thus, once a connection is introduced, the covariant exterior derivative of ˛ coincides with d˛ . There are two methods to construct a form ˛ with the properties in Theorem 9. We continue to work under the general assumption that G acts on Q freely and properly. First Method Construction of ˛ from a connection A 2 ˝ 1 (Q; g ) on the principal bundle Q;G : Q ! Q/G . To carry this out, one shows that the choice ˛ :D h0 ; A i 2 ˝ 1 (Q) satisfies the condition CBR1 in Theorem 9, where, as above, 0 D jg . The two-form d˛ may be interpreted in terms of curvature. In fact, one shows that d˛ is the 0 -component of the curvature two-form. We summarize these results in the following statement. Proposition 12 If the principal bundle Q;G : Q ! Q/G with structure group G has a connection A , then ˛ (q) can be taken to equal A (q) 0 and B is induced on T  Q by d˛ (a two-form on Q), which equals the 0 -component of the curvature B of A .

Mechanical Systems: Symmetries and Reduction

Second Method Construction of ˛ from a connection A 2 ˝ 1 (Q; g) on the principal bundle Q;G : Q ! Q/G. One can show that the choice (A1), that is, ˛ :D h; Ai 2 ˝ 1 (Q) satisfies the condition CBR2 in Theorem 9. As with the first method, there is an interpretation of the two-form d˛ in terms of curvature as follows. Proposition 13 If the principal bundle Q;G : Q ! Q/G with structure group G has a connection A, then ˛ (q) can be taken to equal A(q)  and B is the pull back to T  Q of d˛ 2 ˝ 2 (Q), which equals the -component of the two form B C [A; A] 2 ˝ 2 (Q; g), where B is the curvature of A.

(T  Q) embeds as a vector subbundle of T  (Q/G ). The bundle version of this theorem says, roughly speaking, that (T  Q) is a coadjoint orbit bundle over T  (Q/G) with fiber the coadjoint orbit O through . Again we utilize a choice of connection A on the shape space bundle Q;G : Q ! Q/G. A key step in the argument is to utilize orbit reduction and the identification (T  Q) Š (T  Q)O . Theorem 15 (Cotangent bundle reduction – bundle version) The reduced space (T  Q) is a locally trivial fiber bundle over T  (Q/G) with typical fiber O. This point of view is explored further and the exact nature of the coadjoint orbit bundle is identified and its symplectic structure is elaborated in [92]. Poisson Version

Coadjoint Orbits We now apply the Cotangent Bundle Reduction Theorem 9 to the case Q D G and with the G-action given by left translation. The right Maurer–Cartan form R is a flat connection associated to this action (see Theorem A13) and hence ˝ ˛ d˛ (g)(u g ; v g ) D ; [ R ; R ](g)(u g ; v g ) ˛ ˝ D ; [Tg R g 1 u g ; Tg R g 1 v g ] : Recall from Theorem 7 that the reduced space (T  G) is the coadjoint orbit O endowed with the negative orbit  and, according to the Cotangent Bunsymplectic form ! dle Reduction Theorem, it symplectically embeds as the zero section into (T  O ; !can B ), where B D O ˇ , O : T  O ! O is the cotangent bundle projection,  G;G ˇ D d˛ , and G;G : G ! O is given by  G;G (g) D Adg . The derivative of G;G is given by ˇ d ˇˇ Ad  D ad Adg  Tg G;G (Te L g ) D dt ˇtD0 g exp(t) . for any  2 g. Then a computation shows that ˇ D ! Thus, the embedding version of the cotangent bundle reduction theorem produces the following statement which, of course, can be easily checked directly.  ) symplectically Corollary 14 The coadjoint orbit (O ; ! embeds as the zero section into the symplectic manifold  ). (T  O ; !can C O !

Cotangent Bundle Reduction: Bundle Version The embedding version of the cotangent bundle reduction theorem presented in the preceding section states that

This same type of argument as above shows the following, which we state slightly informally. Theorem The Poisson reduced space (T  Q)/G is diffeomorphic to the coadjoint bundle of Q;G : Q ! Q/G. This diffeomorphism is implemented by a connection A 2 ˝ 1 (Q; g). Thus the fiber of (T  Q)/G ! T  (Q/G) is isomorphic to the Lie–Poisson space g . There is an interesting formula for the Poisson structure on (T  Q)/G that was originally computed in Montgomery, Marsden, and Ratiu [127], Montgomery [121]. Further developments in Cendra, Marsden, Pekarsky, and Ratiu [37] and Perlmutter and Ratiu [149] gives a unified study of the Poisson bracket on (T  Q)/G in both the Sternberg and Weinstein realizations of the quotient. Finally, we refer to, for instance, Lewis, Marsden, Montgomery and Ratiu [88] for an application of this result; in this case, the dynamics of fluid systems with free boundaries is studied. Coadjoint Orbit Bundles The details of the nature of the bundle and its associated symplectic structure that was sketched in Theorem 15 is due to Marsden and Perlmutter [102]; see also Za´ alani [182] Cushman and Sniatycki [47], and [149]. An exposition may be found in [92]. Future Directions One of the goals of reduction theory and geometric mechanics is to take the analysis of mechanical systems with symmetries to a deeper level of understanding. But much more needs to be done. As has already been explained,

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there is still a need to put many classical concepts, such as quasivelocities, into this context, with a resultant strengthening of the theory and its applications. In addition, links with Dirac structures, groupoids and algebroids is under development and should lead to further advances. Finally we mention that while much of this type of work has been applied to field theories (such as electromagnetism and gravity), greater insight is needed for many topics, stressenergy-momentum tensors being one example. Acknowledgments This work summarizes the contributions of many people. We are especially grateful to Alan Weinstein, Victor Guillemin and Shlomo Sternberg for their incredible insights and work over the last few decades. We also thank Hernán Cendra and Darryl Holm, our collaborators on the Lagrangian context and Juan-Pablo Ortega, a longtime collaborator on Hamiltonian reduction and other projects; he along with Gerard Misiolek and Matt Perlmutter were our collaborators on [92], a key recent project that helped us pull many things together. We also thank many other colleagues for their input and invaluable support over the years; this includes Larry Bates, Tony Bloch, Marco Castrillón-López, Richard Cushman, Laszlo Fehér, Mark Gotay, John Harnad, Eva Kanso, Thomas Kappeler, P.S. Krishnaprasad, Naomi Leonard, Debra Lewis, James Montaldi, George Patrick, Mark Roberts, Miguel Rodríguez´ Olmos, Steve Shkoller, Je¸drzej Sniatycki, Leon Takhtajan, Karen Vogtmann, and Claudia Wulff. Appendix: Principal Connections

One can alternatively use right actions, which is common in the principal bundle literature, but we shall stick with the case of left actions for the main exposition. Vectors that are infinitesimal generators, namely those of the form Q (q) are called vertical since they are sent to zero by the tangent of the projection map Q;G . Definition A1 A connection, also called a principal connection on the bundle Q;G : Q ! Q/G is a Lie algebra valued 1-form A : TQ ! g

where g denotes the Lie algebra of G, with the following properties: (i) the identity A( Q (q)) D  holds for all  2 g; that is, A takes infinitesimal generators of a given Lie algebra element to that same element, and (ii) we have equivariance: A(Tq ˚ g (v)) D Ad g (A(v)) for all v 2 Tq Q, where ˚ g : Q ! Q denotes the given action for g 2 G and where Ad g denotes the adjoint action of G on g. A remark is noteworthy at this point. The equivariance identity for infinitesimal generators noted previously (see (7)), namely,   Tq ˚ g  Q (q) D (Ad g )Q (g  q) ; shows that if the first condition for a connection holds, then the second condition holds automatically on vertical vectors. If the G-action on Q is a right action, the equivariance condition (ii) in Definition A1 needs to be changed to A(Tq ˚ g (v)) D Ad g 1 (A(v)) for all g 2 G and v 2 Tq Q.

In preparation for the next section which gives a brief exposition of the cotangent bundle reduction theorem, we now give a review and summary of facts that we shall need about principal connections. An important thing to keep in mind is that the magnetic terms in the cotangent bundle reduction theorem will appear as the curvature of a connection.

Since A is a Lie algebra valued 1-form, for each q 2 Q, we get a linear map A(q) : Tq Q ! g and so we can form its dual A(q) : g ! Tq Q. Evaluating this on  produces an ordinary 1-form:

Principal Connections Defined

This 1-form satisfies two important properties given in the next Proposition.

We consider the following basic set up. Let Q be a manifold and let G be a Lie group acting freely and properly on the left on Q. Let

Proposition A2 For any connection A and  2 g , the corresponding 1-form ˛ defined by (A1) takes values in J1 () and satisfies the following G-equivariance property:

Q;G : Q ! Q/G denote the bundle projection from the configuration manifold Q to shape space S D Q/G. We refer to Q;G : Q ! Q/G as a principal bundle.

Associated One-Forms

˛ (q) D A(q) () :

(A1)

˚ g ˛ D ˛Adg  : Notice in particular, if the group is Abelian or if  is G-invariant, (for example, if  D 0), then ˛ is an invariant 1-form.

Mechanical Systems: Symmetries and Reduction

Horizontal and Vertical Spaces Associated with any connection are vertical and horizontal spaces defined as follows. Definition A3 Given the connection A, its horizontal space at q 2 Q is defined by H q D fv q 2 Tq Q j A(v q ) D 0g and the vertical space at q 2 Q is, as above, Vq D fQ (q) j  2 gg : The map v q 7! verq (v q ) :D [A(q)(v q )]Q (q) is called the vertical projection, while the map v q 7! horq (v q ) :D v q  verq (v q ) is called the horizontal projection. Because connections map infinitesimal generators of a Lie algebra elements to that same Lie algebra element, the vertical projection is indeed a projection for each fixed q onto the vertical space and likewise with the horizontal projection. By construction, we have v q D verq (v q ) C hor q (v q ) and so Tq Q D H q ˚ Vq and the maps hor q and verq are projections onto these subspaces. It is sometimes convenient to define a connection by the specification of a space H q declared to be the horizontal space that is complementary to V q at each point, varies smoothly with q and respects the group action in the sense that H gq D Tq ˚ g (H q ). Clearly this alternative definition of a principal connection is equivalent to the definition given above. Given a point q 2 Q, the tangent of the projection map Q;G restricted to the horizontal space H q gives an isomorphism between H q and T[q] (Q/G). Its inverse 1  Tq Q;G j H q : TQ;G (q) (Q/G) ! H q is called the horizontal lift to q 2 Q.

manifold Q is a Riemannian manifold. Of course, the Riemannian structure will often be that defined by the kinetic energy of a given mechanical system. Thus, assume that Q is a Riemannian manifold, with metric denoted hh; ii and that G acts freely and properly on Q by isometries, so Q;G : Q ! Q/G is a principal G-bundle. In this context we may define the horizontal space at a point simply to be the metric orthogonal to the vertical space. This therefore defines a connection called the mechanical connection. Recall from the historical survey in the introduction that this connection was first introduced by Kummer [79] following motivation from Smale [168] and [1]. See also Guichardet [54], who applied these ideas in an interesting way to molecular dynamics. The number of references since then making use of the mechanical connection is too large to survey here. In Proposition A5 we develop an explicit formula for the associated Lie algebra valued 1-form in terms of an inertia tensor and the momentum map. As a prelude to this formula, we show the following basic link with mechanics. In this context we write the momentum map on TQ simply as J : TQ ! g . Proposition A4 The horizontal space of the mechanical connection at a point q 2 Q consists of the set of vectors v q 2 Tq Q such that J(v q ) D 0. For each q 2 Q, define the locked inertia tensor I(q) to be the linear map I(q) : g ! g defined by hI(q); i D hhQ (q); Q (q)ii

(A2)

for any ;  2 g. Since the action is free, I(q) is nondegenerate, so (A2) defines an inner product. The terminology “locked inertia tensor” comes from the fact that for coupled rigid or elastic systems, I(q) is the classical moment of inertia tensor of the rigid body obtained by locking all the joints of the system. In coordinates, j

I ab D g i j K ai K b ;

(A3)

where [Q (q)] i D K ai (q) a define the action functions K ai . Define the map A : TQ ! g which assigns to each v q 2 Tq Q the corresponding angular velocity of the locked system: A(q)(v q ) D I(q)1 (J(v q )) ;

(A4)

where L is the kinetic energy Lagrangian. In coordinates, The Mechanical Connection As an example of defining a connection by the specification of a horizontal space, suppose that the configuration

A a D I ab g i j K bi v j

since J a (q; p) D p i K ai (q).

(A5)

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We defined the mechanical connection by declaring its horizontal space to be the metric orthogonal to the vertical space. The next proposition shows that A is the associated connection one-form. Proposition A5 The g-valued one-form defined by (A4) is the mechanical connection on the principal G-bundle Q;G : Q ! Q/G. Given a general connection A and an element  2 g , we can define the -component of A to be the ordinary oneform ˛ given by ˛ (q) D A(q)  2 Tq Q; ˛ ˝ D ; A(q)(v q )

i. e.;

˛ ˝ ˛ (q); v q

Curvature The curvature B of a connection A is defined as follows.

for all v q 2 Tq Q. Note that ˛ is a G -invariant one-form. It takes values in J1 () since for any  2 g, we have ˝ ˛ ˝ ˛ J(˛ (q));  D ˛ (q); Q D h; A(q)(Q (q))i D h; i : In the Riemannian context, Smale [168] constructed ˛ by a minimization process. Let ˛q] 2 Tq Q be the tangent vector that corresponds to ˛q 2 Tq Q via the metric hh; ii on Q. A(q) 

Tq Q

Proposition A6 The 1-form ˛ (q) D 2 associated with the mechanical connection A given by (A4) is characterized by K(˛ (q)) D inffK(ˇq ) j ˇq 2 J1 () \ Tq Qg ; (A6) where K(ˇ q ) D T  Q. See Fig. 2.

] 2 1 2 kˇ q k

proof in Smale [168] or of Proposition 4.4.5 in [1]. Thus, one of the merits of the previous proposition is to show easily that this variational definition of ˛ does indeed yield a smooth one-form on Q with the desired properties. Note also that ˛ (q) lies in the orthogonal space to Tq Q \ J1 () in the fiber Tq Q relative to the bundle metric on T  Q defined by the Riemannian metric on Q. It also follows that ˛ (q) is the unique critical point of the kinetic energy of the bundle metric on T  Q restricted to the fiber Tq Q \ J1 ().

is the kinetic energy function on

The proof is a direct verification. We do not give here it since this proposition will not be used later in this book. The original approach of Smale [168] was to take (A6) as the definition of ˛ . To prove from here that ˛ is a smooth one-form is a nontrivial fact; see the

Mechanical Systems: Symmetries and Reduction, Figure 2 The extremal characterization of the mechanical connection

Definition A7 The curvature of a connection A is the Lie algebra valued two-form on Q defined by B(q)(u q ; v q ) D dA(horq (u q ); hor q (v q )) ;

(A7)

where d is the exterior derivative. When one replaces vectors in the exterior derivative with their horizontal projections, then the result is called the exterior covariant derivative and one writes the preceding formula for B as B D dA A :

For a general Lie algebra valued k-form ˛ on Q, the exterior covariant derivative is the k C 1-form dA ˛ defined on tangent vectors v0 ; v1 ; : : : ; v k 2 Tq Q by dA ˛(v0 ; v1 ; : : : ; v k )   D d˛ horq (v0 ); hor q (v1 ); : : : ; hor q (v k ) :

(A8)

Here, the symbol dA reminds us that it is like the exterior derivative but that it depends on the connection A. Curvature measures the lack of integrability of the horizontal distribution in the following sense. Proposition A8 On two vector fields u, v on Q one has   B(u; v) D A [hor(u); hor(v)] : Given a general distribution D TQ on a manifold Q one can also define its curvature in an analogous way directly in terms of its lack of integrability. Define vertical vectors at q 2 Q to be the quotient space Tq Q/D q and define the curvature acting on two horizontal vector fields u, v (that is, two vector fields that take their values in the distribution) to be the projection onto the quotient of their Jacobi–Lie bracket. One can check that this operation depends only on the point values of the vector fields, so indeed defines a two-form on horizontal vectors.

Mechanical Systems: Symmetries and Reduction

Cartan Structure Equations We now derive an important formula for the curvature of a principal connection. Theorem A9 (Cartan structure equations) For any vector fields u, v on Q we have B(u; v) D dA(u; v)  [A(u); A(v)] ;

(A9)

where the bracket on the right hand side is the Lie bracket in g. We write this equation for short as B D dA  [A; A] :

If the G-action on Q is a right action, then the Cartan Structure Equations read B D dA C [A; A]. The following Corollary shows how the Cartan Structure Equations yield a fundamental equivariance property of the curvature. Corollary A10 For all g 2 G we have ˚ g B D Ad g ı B. If the G-action on Q is on the right, equivariance means ˚ g B D Ad g 1 ı B. Bianchi Identity The Bianchi Identity, which states that the exterior covariant derivative of the curvature is zero, is another important consequence of the Cartan Structure Equations. Corollary A11 If B D dA A 2 ˝ 2 (Q; g) is the curvature two-form of the connection A, then the Bianchi Identity holds: dA B D 0 : This form of the Bianchi identity is implied by another version, namely dB D [B; A]^ ; where the bracket on the right hand side is that of Lie algebra valued differential forms, a notion that we do not develop here; see the brief discussion at the end of §9.1 in [104]. The proof of the above form of the Bianchi identity can be found in, for example, Kobayashi and Nomizu [74]. Curvature as a Two-Form on the Base We now show how the curvature two-form drops to a twoform on the base with values in the adjoint bundle. The associated bundle to the given left principal bundle Q;G : Q ! Q/G via the adjoint action is called the

adjoint bundle. It is defined in the following way. Consider the free proper action (g; (q; )) 2 G  (Q  g) 7! (gq; Ad g ) 2 Qg and form the quotient g˜ :D QG g :D (Q  g)/G which is easily verified to be a vector bundle g˜ : g˜ ! Q/G, where g˜ (g; ) :D Q;G (q). This vector bundle has an additional structure: it is a Lie algebra bundle; that is, a vector bundle whose fibers are Lie algebras. In this case the bracket is defined pointwise: [g˜ (g; ); g˜ (g; )] :D g˜ (g; [; ]) for all g 2 G and ;  2 g. It is easy to check that this defines a Lie bracket on every fiber and that this operation is smooth as a function of Q;G (q). The curvature two-form B 2 ˝ 2 (Q; g) (the vector space of g-valued two-forms on Q) naturally induces a two-form B on the base Q/G with values in g˜ by   B(Q;G (q)) Tq Q;G (u); Tq Q;G (v)   :D g˜ q; B(u; v) (A10) for all q 2 Q and u; v 2 Tq Q. One can check that B is well defined.  B D Since (A10) can be equivalently written as Q;G   g˜ ı id Q  B and Q;G is a surjective submersion, it follows that B is indeed a smooth two-form on Q/G with values in g˜ . Associated Two-Forms Since B is a g-valued two-form, in analogy with (A1), for every  2 g we can define the -component of B, an ordinary two-form B 2 ˝ 2 (Q) on Q, by ˝ ˛ B (q)(u q ; v q ) :D ; B(q)(u q ; v q ) (A11) for all q 2 Q and u q ; v q 2 Tq Q. The adjoint bundle valued curvature two-form B induces an ordinary two-form on the base Q/G. To obtain it, we consider the dual g˜  of the adjoint bundle. This is a vector bundle over Q/G which is the associated bundle relative to the coadjoint action of the structure group G of the principal (left) bundle Q;G : Q ! Q/G on g . This vector bundle has additional structure: each of its fibers is a Lie–Poisson space and the associated Poisson tensors on each fiber depend smoothly on the base, that is, g˜  : g˜  ! Q/G is a Lie–Poisson bundle over Q/G. Given  2 g , define the ordinary two-form B on Q/G by    B Q;G (q) Tq Q;G (u q ); Tq Q;G (v q )  ˛ ˝ :D g˜  (q; ); B (Q;G (q)) Tq Q;G (u q ); Tq Q;G (v q ) ˛ ˝ D ; B(q)(u q ; v q ) D B (q)(u q ; v q ) ; (A12)

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where q 2 Q, u q ; v q 2 Tq Q, and in the second equality h; i : g˜   g˜ ! R is the duality pairing between the coadjoint and adjoint bundles. Since B is well defined and smooth, so is B . Proposition A12 Let A 2 ˝ 1 (Q; g) be a connection oneform on the (left) principal bundle Q;G : Q ! Q/G and B 2 ˝ 2 (Q; g) its curvature two-form on Q. If  2 g , the corresponding two-forms B 2 ˝ 2 (Q) and B 2 ˝ 2 (Q/G) defined by (A11) and (A12), respectively, are re B D B . In addition, B satisfies the follated by Q;G    lowing G-equivariance property: ˚ g B D BAdg  :  B , where Thus, if G D G then d˛ D B D Q;G   ˛ (q) D A(q) ().

Further relations between ˛ and the -component of the curvature will be studied in the next section when discussing the magnetic terms appearing in cotangent bundle reduction. The Maurer–Cartan Equations A consequence of the structure equations relates curvature to the process of left and right trivialization and hence to momentum maps. Theorem A13 (Maurer–cartan equations) Let G be a Lie group and let R : TG ! g be the map (called the right Maurer–Cartan form) that right translates vectors to the identity: R (v g ) D Tg R g 1 (v g ) : Then d R  [ R ; R ] D 0 : There is a similar result for the left trivialization L , namely the identity d L C [ L ; L ] D 0 : Of course there is much more to this subject, such as the link with classical connection theory, Riemannian geometry, etc. We refer to [92] for further basic information and references and to Bloch [18] for applications to nonholonomic systems, and to Cendra, Marsden, and Ratiu [35] for applications to Lagrangian reduction.

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Navier–Stokes Equations: A Mathematical Analysis

Navier–Stokes Equations: A Mathematical Analysis GIOVANNI P. GALDI University of Pittsburgh, Pittsburgh, USA Article Outline Glossary Definition of the Subject Introduction Derivation of the Navier–Stokes Equations and Preliminary Considerations Mathematical Analysis of the Boundary Value Problem Mathematical Analysis of the Initial-Boundary Value Problem Future Directions Acknowledgment Bibliography Glossary Steady-State flow Flow where both velocity and pressure fields are time-independent. Three-dimensional (or 3D) flow Flow where velocity and pressure fields depend on all three spatial variables. Two-dimensional (or planar, or 2D) flow Flow where velocity and pressure fields depend only on two spatial variables belonging to a portion of a plane, and the component of the velocity orthogonal to that plane is identically zero. Local solution Solution where velocity and pressure fields are known to exist only for a finite interval of time. Global solution Solution where velocity and pressure fields exist for all positive times. Regular solution Solution where velocity and pressure fields satisfy the Navier–Stokes equations and the corresponding initial and boundary conditions in the ordinary sense of differentiation and continuity. At times, we may interchangeably use the words “flow” and “solution”. Basic notation N is the set of positive integers. R is the field of real numbers and RN , N 2 N, is the set of all N-tuple x D (x1 ; : : : ; x N ). The canonical base in R N is denoted by fe 1 ; e 2 ; e 3 ; : : : ; e N g  fe i g. For a; b 2 R, b > a, we set (a; b) D fx 2 R : a < x < bg, [a; b] D fx 2 R : a  x  bg, [a; b) D fx 2 R : a  x < bg and (a; b] D fx 2 R : a < x  bg. ¯ we indicate the closure of the subset A of R N . By A

A domain is an open connected subset of R N . Given a second-order tensor A and a vector a, of components fA i j g and fa i g, respectively, in the basis fe i g, by a  A [respectively, A  a] we mean the vector with components A i j a i [respectively, A i j a j ]. (We use the Einstein summation convention over repeated indices, namely, if an index occurs twice in the same expression, the expression is implicitly summed over all possible p values for that index.) Moreover, we set jAj D A i j A i j . If h(z)  fh i (z)g is a vector field, by r h we denote the second-order tensor field whose components fr hg i j in the given basis are given by f@h j /@z i g. Function spaces notation If A  R N and k 2 N [ f0g, ¯ )] we denote the class of by C k (A) [respectively, C k (A functions which are continuous in A up to their kth derivatives included [respectively, are bounded and uniformly continuous in A up to their kth derivatives included]. The subset of C k (A) of functions vanishing outside a compact subset of A is indicated by C0k (A). If u 2 C k (A) for all k 2 N [ f0g, we shall write ¯) u 2 C 1 (A). In an analogous way we define C 1 (A and C01 (A). The symbols L q (A), W m;q (A), m  0, 1  q  1, denote the usual Lebesgue and Sobolev spaces, respectively (W 0;q (A) D L q (A)). Norms in L q (A) and W m;q (A) are denoted by k  kq;A , k  km;q;A . The trace space on the boundary, @A, of A for functions from W m;q (A) will be denoted by W m1/q;q (@A) and its norm by k  km1/q;q;@A . By D k;q (A), k  1, 1 < q < 1, we indicate the homogeneous Sobolev space of order (m; q) on A, that is, the class of functions u that are (Lebesgue) locally integrable in A and with Dˇ u 2 L q (A), jˇj D k, where ˇ ˇ ˇ Dˇ D @jˇ j /@x1 1 @x2 2 : : : @x NN , jˇj D ˇ1 C ˇ2 C    C ˇ N . For u 2 D k;q (A), we put 0 juj k;q;A D @

X Z jˇ jDk

A

11/q j Dˇ u jq A

:

Notice that, whenever confusion does not arise, in the integrals we omit the infinitesimal volume or surface elements. Let D(A) D f' 2 C01 (A) : div ' D 0g : q

By L (A) we denote the completion of D(A) in the norm k  kq . If A is any domain in R N we have L2 (A) D L2 (A) ˚ G(A), where G(A) D fh 2 L2 (A) : h D r p ; for some p 2 D1;2 (A)g; (see Sect. III.1 in [31]). We denote by P the orthogonal projection operator from L2 (A) onto L2 (A). By D1;2 0 (A) we mean the completion of D(A) in the norm j 

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j1;2;A . D1;2 0 (A)R is a Hilbert space with scalar product [v 1 ; v 2 ] :D A (@v 1 /@x i )  (@v 2 /@x i ). Furthermore, 1;2 1;2 D0 (A) is the dual space of D0 (A) and h; iA is the associated duality pairing. If g  fg i g and h  fh i g are vector fields on A, we set Z (g; h)A D g i hi ; A

whenever the integrals make sense. In all the above notation, if confusion will not arise, we shall omit the subscript A. Given a Banach space X, and an open interval (a; b), we denote by L q (a; b; X) the linear space of (equivalence classes of) functions f : (a; b) ! X whose X-norm is in L q (a; b). Likewise, for r a nonnegative integer and I a real interval, we denote by C r (I; X) the class of continuous functions from I to X, which are differentiable in I up to the order r included. If X denotes any space of real functions, we shall use, as a rule, the same symbol X to denote the corresponding space of vector and tensor-valued functions. Definition of the Subject The Navier–Stokes equations are a mathematical model aimed at describing the motion of an incompressible viscous fluid, like many common ones as, for instance, water, glycerin, oil and, under certain circumstances, also air. They were introduced in 1822 by the French engineer Claude Louis Marie Henri Navier and successively re-obtained, by different arguments, by a number of authors including Augustin-Louis Cauchy in 1823, Siméon Denis Poisson in 1829, Adhémar Jean Claude Barré de Saint-Venant in 1837, and, finally, George Gabriel Stokes in 1845. We refer the reader to the beautiful paper by Olivier Darrigol [17], for a detailed and thorough analysis of the history of the Navier–Stokes equations. Even though, for quite some time, their significance in the applications was not fully recognized, the Navier– Stokes equations are, nowadays, at the foundations of many branches of applied sciences, including Meteorology, Oceanography, Geology, Oil Industry, Airplane, Ship and Car Industries, Biology, and Medicine. In each of the above areas, these equations have collected many undisputed successes, which definitely place them among the most accurate, simple and beautiful models of mathematical physics. Notwithstanding these successes, up to the present time, a number of unresolved basic mathematical questions remain open – mostly, but not only, for the physically relevant case of three-dimensional flow.

Undoubtedly, the most celebrated is that of proving or disproving the existence of global 3D regular flow for data of arbitrary “size”, no matter how smooth (global regularity problem). Since the beginning of the 20th century, this notorious question has challenged several generations of mathematicians who have not been able to furnish a definite answer. In fact, to date, 3D regular flows are known to exist either for all times but for data of “small size”, or for data of “arbitrary size” but for a finite interval of time only. The problem of global regularity has become so intriguing and compelling that, in the year 2000, it was decided to put a generous bounty on it. In fact, properly formulated, it is listed as one of the seven $1M Millennium Prize Problems of the Clay Mathematical Institute. However, the Navier–Stokes equations present also other fundamental open questions. For example, it is not known whether, in the 3D case, the associated initialboundary value problem is (in an appropriate function space) well-posed in the sense of Hadamard. Stated differently, in 3D it is open the question of whether solutions to this problem exist for all times, are unique and depend continuously upon the data, without being necessarily “regular”. Another famous, unsettled challenge is whether or not the Navier–Stokes equations are able to provide a rigorous model of turbulent phenomena. These phenomena occur when the magnitude of the driving mechanism of the fluid motion becomes sufficiently “large”, and, roughly speaking, it consists of flow regimes characterized by chaotic and random property changes for velocity and pressure fields throughout the fluid. They are observed in 3D as well as in two-dimensional (2D) motions (e. g., in flowing soap films). We recall that a 2D motion occurs when the relevant region of flow is contained in a portion of a plane, $ , and the component of the velocity field orthogonal to $ is negligible. It is worth emphasizing that, in principle, the answers to the above questions may be unrelated. Actually, in the 2D case, the first two problems have long been solved in the affirmative, while the third one remains still open. Nevertheless, there is hope that proving or disproving the first two problems in 3D will require completely fresh and profound ideas that will open new avenues to the understanding of turbulence. The list of main open problems can not be exhausted without mentioning another outstanding question pertaining to the boundary value problem describing steadystate flow. The latter is characterized by time-independent velocity and pressure fields. In such a case, if the flow region, R, is multiply connected, it is not known (neither in 2D nor in 3D) if there exists a solution under a given ve-

Navier–Stokes Equations: A Mathematical Analysis

locity distribution at the boundary of R that merely satisfy the physical requirement of conservation of mass. Introduction Fluid mechanics is a very intricate and intriguing discipline of the applied sciences. It is, therefore, not surprising that the mathematics involved in the study of its properties can be, often, extremely complex and difficult. Complexities and difficulties, of course, may be more or less challenging depending on the mathematical model chosen to describe the physical situation. Among the many mathematical models introduced in the study of fluid mechanics, the Navier–Stokes equations can be considered, without a doubt, the most popular one. However, this does not mean that they can correctly model any fluid under any circumstance. In fact, their range of applicability is restricted to the class of so-called Newtonian fluids; see Remark 3 in Sect. “Derivation of the Navier–Stokes Equations and Preliminary Considerations”. This class includes several common liquids like water, most salt solutions in water, aqueous solutions, some motor oils, most mineral oils, gasoline, and kerosene. As mentioned in the previous section, these equations were proposed in 1822 by the French engineer Claude Navier upon the basis of a suitable molecular model. It is interesting to observe, however, that the law of interaction between the molecules postulated by Navier were shortly recognized to be totally inconsistent from the physical point of view for several materials and, in particular, for liquids. It was only more than two decades later, in 1845, that the same equations were re derived by the 26-year-old George Stokes in a quite general way, by means of the theory of continua. The role of mathematics in the investigation of the fluid properties and, in particular, of the Navier–Stokes equations, is, as in most branches of the applied sciences, twofold and aims at the accomplishment of the following objectives. The first one, of a more fundamental nature, is the validation of the mathematical model, and consists in securing conditions under which the governing equations possess the essential requirements of well-posedness, that is, existence and uniqueness of corresponding solutions and their continuous dependence upon the data. The second one, of a more advanced character, is to prove that the model gives an adequate interpretation of the observed phenomena. This paper is, essentially, directed toward the first objective. In fact, its goals consist of (1) formulating the primary problems, (2) describing the related known results, and (3) pointing out the remaining basic open questions,

for both boundary and initial-boundary value problems. Of course, it seemed unrealistic to give a detailed presentation of such a rich and diversified subject in a few number of pages. Consequently, we had to make a choice which, we hope, will still give a fairly good idea of this fascinating and intriguing subject of applied mathematics. The plan of this work is the following. In Sect. “Derivation of the Navier–Stokes Equations and Preliminary Considerations”, we shall first give a brief derivation of the Navier–Stokes equations from continuum theory, then formulate the basic problems and, further on, discuss some basic properties. Section “Mathematical Analysis of the Boundary Value Problem” is dedicated to the boundary value problem in both bounded and exterior domains. Besides existence and uniqueness, the main topics include the study of the structure of solution set at large Reynolds number, as well as a condensed treatment of bifurcation issues. Section “Mathematical Analysis of the InitialBoundary Value Problem” deals with the initial-boundary value problem in a bounded domain and with the related questions of well-posedness and regularity. Finally, in Sect. “Future Directions” we shall outline some future directions of research. Companion to this last section, is the list of a number of significant open questions that are mentioned throughout the paper. Derivation of the Navier–Stokes Equations and Preliminary Considerations In the continuum mechanics modeling of a fluid, F , one assumes that, in the given time interval, I  [0; T], T > 0, of its motion, F continuously fills a region, ˝, of the three-dimensional space, R3 . We call points, surfaces, and volumes of F , material points (or particles), material surfaces, and material volumes, respectively. In most relevant applications, the region ˝ does not depend on time. This happens, in particular, whenever the fluid is bounded by rigid walls like, for instance, in the case of a flow past a rigid obstacle, or a flow in a bounded container with fixed walls. However, there are also some significant situations where ˝ depends on time as, for example, in the motion of a fluid in a pipe with elastic walls. Throughout this work, we shall consider flow of F where ˝ is time-independent. Balance Laws In order to describe the motion of F it is convenient to represent the relevant physical quantities in the Eulerian form. Precisely, if x D (x1 ; x2 ; x3 ) is a point in ˝ and t 2 [0; T], we let D (x; t), v D v(x; t) D (v1 (x; t); v2 (x; t); v3 (x; t)) and a D a(x; t) D (a1 (x; t); a2 (x; t); a3 (x; t)) be the density, velocity, and acceleration, re-

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spectively, of that particle of F that, at the time t, passes through the point x. Furthermore, we denote by f D f (x; t) D ( f1 (x; t); f2 (x; t); f3 (x; t)) the external force per unit volume (body force) acting on F . In the continuum theory of non-polar fluids, one postulates that in every motion the following equations must hold 9 @ @ > > ( (x; t)v i (x; t)) D 0 ; (x; t) C > > @t @x i > > > > for all =

(x; t) a i (x; t) (x; t) 2 ˝ @T ji > > C (x; t) f i (x; t) ; i D 1; 2; 3 ;> D >  (0; T) : > @x j > > > ; Ti j (x; t) D T ji (x; t) ; i; j D 1; 2; 3 (1) These equations represent the local form of the balance laws of F in the Eulerian description. Specifically, (1)1 expresses the conservation of mass, (1)2 furnishes the balance of linear momentum, while (1)3 is equivalent to the balance of angular momentum. The function T D T(x; t) D fT ji (x; t)g is a second-order, symmetric tensor field, the Cauchy stress tensor, that takes into account the internal forces exerted by the fluid. More precisely, let S denote a (sufficiently smooth) fixed, closed surface in R3 bounding the region RS , let x be any point on S, and let n D n(x) be the outward unit normal to S at x. Furthermore, let R(e) S be the region exterior to RS , with R(e) \ ˝ ¤ ;. Then the S vector t D t(x; t) defined as t(x; t) :D n(x)  T(x; t) ;

(2)

represents the force per unit area exerted by the portion of the fluid in R(e) S on S at the point x and at time t; see Fig. 1. Constitutive Equation An important kinematical quantity associated with the motion of F is the stretching tensor field, D D D(x; t), whose components, Dij , are defined as follows:   1 @v j @v i ; i; j D 1; 2; 3 : (3) C Di j D 2 @x i @x j

The stretching tensor is, of course, symmetric and, roughly speaking, it describes the rate of deformation of parts of F . In fact, it can be shown that a necessary and sufficient condition for a motion of F to be rigid (namely, the mutual distance between two arbitrary particles of F does not change in time), is that D(x; t) D 0 at each (x; t) 2 ˝  I. @v i Moreover, div v(x; t)  trace D(x; t)  @x (x; t) D 0 for i all (x; t) 2 ˝  I, if and only if the motion is isochoric, namely, every material volume does not change with time. A noteworthy class of fluids whose generic motion is isochoric is that of fluids having constant density. In fact, if is a positive constant, from (1)1 we find div v(x; t) D 0, for all (x; t) 2 ˝I. Fluids with constant density are called incompressible. Herein, incompressible fluids will be referred to as liquids. During the generic motion, the internal forces will, in general, produce a deformation of parts of F . The relation between internal forces and deformation, namely, the functional relation between T and D, is called constitutive equation and characterizes the physical properties of the fluid. A liquid is said to be Newtonian if and only if the relation between T and D is linear, that is, there exist a scalar function p D p(x; t) (the pressure) and a constant  (the shear viscosity) such that T D pI C 2D ;

(4)

where I denotes the identity matrix. In a viscous Newtonian liquid, one assumes that the shear viscosity satisfies the restriction  > 0:

(5)

We will discuss further the meaning of this assumption in Remark 2. Navier–Stokes Equations In view of the condition div v D 0, it easily follows from (4) that @Tji @p D C v i @x j @x i where  :D @x@@x is the Laplace operator. Therefore, by l l (1) and (4) we deduce that the equations governing the motion of a Newtonian viscous liquid are furnished by 2

9 @p > C v i C f i ; i D 1; 2; 3> = @x i in ˝(0; T) : > @v i > ; D0 @x i (6)

a i D 

Navier–Stokes Equations: A Mathematical Analysis, Figure 1 Stress vector at the point x of the surface S

Navier–Stokes Equations: A Mathematical Analysis

It is interesting to observe that both equations in (6) are linear in all kinematical variables. However, in the Eulerian description, the acceleration is a nonlinear functional of the velocity, and we have ai D

@v i @v i ; C vl @t @x l

or, in a vector form, aD

@v C v  rv : @t

Replacing this latter expression in (6) we obtain the Navier–Stokes equations:   9 @v > >

C v  rv > = @t in ˝  (0; T) : (7) D r p C v C f ;> > > ; divv D 0 In these equations, the (constant) density , the shear viscosity  [satisfying (5)], and the force f are given quantities, while the unknowns are velocity v D v(x; t) and pressure p D p(x; t) fields. Some preliminary comments about the above equations are in order. Actually, we should notice that the unknowns v; p do not appear in a “symmetric” way. In other words, the equation of conservation of mass (7)2 does not involve the pressure field. This is due to the fact that, from the mechanical point of view, the pressure plays the role of reaction force (Lagrange multiplier) associated with the isochoricity constraint div v D 0. In other words, whenever a portion of the liquid “tries to change its volume” the liquid “reacts” with a suitable distribution of pressure to keep that volume constant. Thus, the pressure field must be generally deduced in terms of the velocity field, once this latter has been determined; see Remark 1. Initial-Boundary Value Problem In order to find solutions to the problem (7), we have to append suitable initial, at time t D 0, and boundary conditions. As a matter of fact, these conditions may depend on the specific physical problem we want to model. We shall assume that the region of flow, ˝, is bounded by rigid walls, @˝, and that the liquid does not “slip” at @˝. The appropriate initial and boundary conditions then become, respectively, the following ones v(x; 0) D v 0 (x) ; v(x; t) D v 1 (x; t) ;

x2˝ (x; t) 2 @˝  (0; T) ;

where v 0 and v 1 are prescribed vector fields.

(8)

Steady-State Flow and Boundary-Value Problem An important, special class of solutions to (7), called steadystate solutions, is that where velocity and pressure fields are independent of time. Of course, a necessary requirement for such solutions to exist is that f does not depend on time as well. From (7) we thus obtain that a generic steady-state solution, (v D v(x); p D p(x)), must satisfy the following equations )

v  rv D r p C v C f ; in ˝ : (9) divv D 0 Under the given assumptions on the region of flow, from (8)2 it follows that the appropriate boundary conditions are v(x) D v  (x) ;

x 2 @˝ ;

(10)

where v  is a prescribed vector field. Two-Dimensional Flow In several mathematical questions related to the unique solvability of problems (7)–(8) and (9)–(10), separate attention deserve two-dimensional solutions describing the planar motions of F . For these solutions the fields v and p depend only on x1 , x2 (say) [and t in the case (7)–(8)], and, moreover, v3  0. Consequently, the relevant (spatial) region of motion, ˝, becomes a subset of R2 . Remark 1 By formally operating with “div” on both sides of (7)1 and by taking into account (8)2 , we find that, at each time t 2 (0; T) the pressure field p D p(x; t) must satisfy the following Neumann problem p D (v  rv  f ) in ˝ @p D [v  (v 1  rv  f )]  n @n

at @˝

(11)

where n denotes the outward unit normal to @˝. It follows that the prescription of the pressure at the bounding walls or at the initial time independently of v, could be incompatible with (8) and, therefore, could render the problem ill-posed. Remark 2 We can give a simple qualitative explanation of the assumption (5). To this end, let us consider a steadystate flow of a viscous liquid between two parallel, rigid walls ˘ 1 , ˘ 2 , set at a distance d apart and parallel to the plane x2 D 0 see Fig. 2. The flow is induced by a force per unit area F D F e1 , F > 0, applied to the wall ˘ 1 , that moves ˘ 1 with a constant velocity V D V e 1 , V > 0, while ˘ 2 is kept fixed; see Fig. 2. No external force is acting on the liquid. It is then easily checked that the velocity

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presentation into two parts, depending on the “geometry” of the region of flow ˝. Specifically, we shall treat the cases when ˝ is either a bounded domain or an exterior domain (flow past an obstacle). For each case we shall describe methods, main results, and fundamental open questions. Navier–Stokes Equations: A Mathematical Analysis, Figure 2 Pure shear flow between parallel plates induced by a force F

and pressure fields v D V (x2 /d)e 1 ; p D p0 D const. (pure shear flow) satisfy (9) with f  0, along with the appropriate boundary conditions v(0) D 0, v(d) D V e 1 . From (2) and (4), we obtain that the force t per unit area exerted by the fluid on ˘ 1 is given by t(x1 ; d; x3 ) D e 2  T(x1 ; d; x3 ) D p0 e 2  

V e1 ; d

that is, t has a “purely pressure” component t p D p0 e 2 , and a “purely shear viscous” component t v D (V /d)e 1 . As expected in a viscous liquid, t v is directed parallel to F and, of course, if it is not zero, it should also act against F, that is, t v  e 1 < 0. However, (V /d) > 0, and so we must have  > 0. Since the physical properties of the fluid are independent of the particular flow, this simple reasoning justifies the assumption made in (5). Remark 3 As mentioned in the Introduction, the constitutive Eq. (4) and, as a consequence, the Navier–Stokes equations, provide a satisfactory model only for a certain class of liquids, while, for others, their predictions are at odds with experimental data. These latter include, for example, biological fluids, like blood or mucus, and aqueous polymeric solutions, like common paints, or even ordinary shampoo. In fact, these liquids, which, in contrast to those modeled by (4), are called non-Newtonian, exhibit a number of features that the linear relation (4) is not able to predict. Most notably, in liquids such as blood, the shear viscosity is no longer a constant and, in fact, it p decreases as the magnitude of shear (proportional to D i j D i j ) increases. Furthermore, liquids like paints or shampoos show, under a given shear rate, a distribution of stress (other than that due to the pressure) in the direction orthogonal to the direction of shear (the so-called normal stress effect). Modeling and corresponding mathematical analysis of a variety of non-Newtonian liquids can be found, for example, in [42].

Flow in Bounded Domains In this section we shall analyze problem (9)–(10) where ˝ is a bounded domain of R3 . A similar (and simpler) analysis can be performed in the case of planar flow, with exactly the same results. Variational Formulation and Weak Solutions To show existence of solutions, one may use, basically, two types of methods that we shall describe in some detail. The starting point of both methods is the so-called variational formulation of (9)–(10). Let ' 2 D(˝). Since div ' D div v D 0 in ˝ and 'j@˝ D 0, we have (by a formal integration by parts) Z p'  n D 0 ; ('; r p) D @˝ Z n  rv  ' D [v; '] (v; ') D [v; '] C @˝ Z (v  rv; ') D v  nv  '  (v  r'; v) D (v  r'; v) @˝

(12) where [; ] is the scalar product in D1;2 0 (˝). Thus, if we dot-multiply both sides of (9)1 by ' 2 D(˝) and integrate by parts over ˝, we (formally) obtain with  :D /  [v; '](v r'; v) D h f ; 'i ;

for all ' 2 D(˝) ; (13)

where we assume that f belongs to D1;2 (˝). Equa0 tion. (13) is the variational (or weak) form of (9)1 . We observe that (13) does not contain the pressure. Moreover, 1;2 (˝). every term in (13) is well-defined provided v 2 Wloc Definition 1 A function v 2 W 1;2 (˝) is called a weak solution to (9)–(10) if and only if: (i) divv D 0 in ˝; (ii) vj@˝ D v  (in the trace sense); (iii) v satisfies (13). If, in particular, v   0, we replace conditions (i), (ii) with the single requirement: (i)0 v 2 D1;2 0 (˝) . Throughout this work we shall often use the following result, consequence of the Hölder inequality and of a simple approximating procedure. Lemma 1 The trilinear form

Mathematical Analysis of the Boundary Value Problem We begin to analyze the properties of solutions to the boundary-value problem (9)–(10). We shall divide our

(a; b; c) 2 L q (˝)  W 1;2 (˝)  Lr (˝) 7! (a  rb; c) 2 R ;

1 1 1 C D ; q r 2

Navier–Stokes Equations: A Mathematical Analysis

is continuous. Moreover, (a  rb; c) D (a  rc; b), for any b; c 2 D01;2 (˝), and for any a 2 L2 (˝). Thus, in particular, (a  rb; b) D 0. Regularity of Weak Solutions If f is sufficiently regular, the corresponding weak solution is regular as well and, moreover, there exists a scalar function, p, such that (9) is satisfied in the ordinary sense. Also, if @˝ and v are smooth enough, the solution (v; p) is smooth up to the boundary and (10) is satisfied in the classical sense. A key tool in the proof of these properties is the following lemma, which is a special case of a more general result due to Cattabriga [12]; (see also Lemma IV.6.2 and Theorem IV.6.1 in [31]). Lemma 2 Let ˝ be a bounded domain of R3 , of m;q (˝), u  2 class C mC2 , m  0, and let g 2 W R mC21/q;q (@˝), 1 < q < 1, with @˝ u  n D 0. W Moreover, let u 2 W 1;2 (˝) satisfy the following conditions (a)  [u; '] D (g; ') for all ' 2 D(˝) ; (b) div u D 0 ; (c) u D u at @˝ in the trace sense. Then, u 2 W mC2;q R (˝) and there exists a unique  2 W mC1;q (˝), with ˝  D 0, such that the pair (u; ) satisfies the following Stokes equations ) u D r C g in ˝ : div u D 0 Furthermore, there exists a constant C D C(˝; m; q) > 0 such that kukmC2;q C kkmC1;q    C kgkm;q C ku  kmC21/q;q;@˝ : To give an idea of how to prove regularity of weak solutions by means of Lemma 2, we consider the case f 2 ¯ ˝ of class C 1 and v  2 C 1 (@˝). (For a general C 1 (˝), regularity theory of weak solutions, see Theorem VIII.52 in [32].) Thus, in particular, by the embedding Theorem II.2.4 in [31] W 1;2 (˝) L q (˝) ;

for all q 2 [1; 6] ;

(14)

and by the Hölder inequality we have that g :D f vrv 2 L3/2 (˝). From (13) and Lemma 2, we then deduce that v 2 W 2;3/2 (˝) and that there exists a scalar field p 2 W 1;3/2 such that (v; p) satisfy (9) a.e. in ˝. Therefore, because of the embedding W 2;3/2 (˝) W 1;3 (˝) Lr (˝), arbitrary r 2 [1; 1), Theorem II.2.4 in [31], we obtain the improved

regularity property g 2 W 1;s (˝), for all s 2 [1; 3/2). Using again Lemma 2, we then deduce v 2 W 3;s (˝) and p 2 W 2;s (˝) which, in particular, gives further regularity for ¯ g. By induction, we then prove v; p 2 C 1 (˝). Existence Results. Homogeneous Boundary Conditions As we mentioned previously, there are, fundamentally, two kinds of approaches to show existence of weak solutions to (9)–(10), namely, the finite-dimensional method and the function-analytic method. We will first describe these methods in the case of homogeneous boundary conditions, v   0, deferring the study of the non-homogeneous problem to Subsect. “Existence Results. NonHomogeneous Boundary Conditions”. In what follows, we shall refer to (9)–(10) with v  D 0 as (9)–(10)hom . A. The Finite-Dimensional Method This popular approach, usually called Galerkin method, was introduced by Fujita [28] and, independently, by Vorovich and Yudovich [96]. It consists in projecting (13) on a suitable finite dimensional space, V N , of D1;2 0 (˝) and then in finding a solution, v N 2 VN , of the “projected” equation. One then passes to the limit N ! 1 to show, with the help of an appropriate uniform estimate, that fv N g contains at least one subsequence converging to some v 2 D1;2 0 (˝) that satisfies condition (13). Precisely, let f k g D(˝) be an orthonormal basis of D1;2 0 (˝), and PN set v N D c , where the coefficients ciN are reiD1 i N i quested to be solutions of the following nonlinear algebraic system [v N ;

k] 

(v N  r

k; vN )

D hf;

ki ;

k D 1; : : : ; N :

(15)

By means of the Brouwer fixed point theorem, it can be shown (see Lemma VIII.3.2 in [32]) that a solution (c1N ; : : : ; c N N ) to (15) exists, provided the following estimate holds jv N j1;2  M

(16)

where M is a finite, positive quantity independent of N. Let us show that (16) indeed occurs. Multiplying through both sides of (15) by ckN , summing over k from 1 to N, and observing that, by Lemma 1, (v N  rv N ; v N ) D 0, we obtain jv N j21;2 D h f ; v N i  j f j1;2 jv N j1;2 ; which proves the desired estimate (16) with M :D j f j1;2 . Notice that the validity of this latter estimate can be obtained by formally replacing in (13) ' with v 2 D1;2 0 (˝) and by using Lemma 1. From classical properties of

1015

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Navier–Stokes Equations: A Mathematical Analysis

Hilbert spaces and from (16), we can select a subsequence fv N 0 g and find v 2 D1;2 0 (˝) such that lim [v N 0 ; '] D [v; '] ;

N 0 !1

for all ' 2 D1;2 0 (˝) : (17)

By Definition 1, to prove that v is a weak solution to (9)– (10)hom it remains to show that v satisfies (13). In view of the Poincaré inequality (Theorem II.4.1 in [31]): '2

k'k2  c P j'j1;2

1;2 D0 (˝) ;

c P D c P (˝) > 0 ;

by (16) it follows that fv N g is bounded in W01;2 (˝) and so, by Rellich compactness theorem (see Theorem II.4.2 in [31]) we can assume that fv N 0 g converges to v in L4 (˝): lim kv N 0  vk4 D 0 :

(19)

N 0 !1

We now consider (15) with N D N 0 and pass to the limit N 0 ! 1. Clearly, by (17), we have for each fixed k lim [v N 0 ;

k]

D [v;

k] :

(20)

Moreover, by Lemma 1 and by (19), lim (v N 0  r

N 0 !1

k ; vN0 )

D (v  r

k ; v) ;

and so, from this latter relation, from (20), and from (15) we conclude that   v;

k



 (v  r

k ; v)

D hf;

1;2 D0 (˝) L4 (˝) ;

(21)

and so, from Lemma 1 and from the Riesz representation theorem, we have that, for each fixed v 2 D1;2 0 (˝), there exists N (v) 2 D1;2 (˝) such that 0 (v  r'; v) D [N (v); '] ;

(18)

N 0 !1

the advantage of furnishing, as a byproduct, significant information on the solutions set. By (14) and (18) we find

ki ;

for all k 2 N ;

Since f k g is a basis in D1;2 0 (˝), this latter relation, along Lemma 1, immediately implies that v satisfies (13), which completes the existence proof. Remark 4 The Galerkin method provides existence of a weak solution corresponding to any given f 2 1;2 D0 (˝). Moreover, it is constructive, in the sense that the solution can be obtained as the limit of a sequence of “approximate solutions” each of which can be, in principle, evaluated by solving the system of nonlinear algebraic Eqs. (15). B. The Function-Analytic Method This approach, that goes back to the work of Leray [61,63] and of Ladyzhenskaya [58], consists, first, in re-writing (13) as a nonlinear equation in the Hilbert space D1;2 0 (˝), and then using an appropriate topological degree theory to prove existence of weak solutions. Though more complicated than, and not constructive like the Galerkin method, this approach has

for all ' 2 D1;2 0 (˝) : (22)

Likewise, we have h f ; 'i D [F; '], for some F 2 D1;2 0 (˝) and for all ' 2 D1;2 0 (˝). Thus, Eq. (13) can be equivalently re-written as N(; v) D F ;

in D1;2 0 (˝) ;

(23)

where the map N is defined as follows N : (; v) 2 (0; 1)  D1;2 0 (˝) 7! v C N (v) 2 D1;2 0 (˝) :

(24)

In order to show the above-mentioned property of solutions, we shall use some basic results related to Fredholm maps of index 0 [84]. This approach is preferable to that originally used by Leray – which applies to maps that are compact perturbations of homeomorphism – because it covers a larger class of problems, including flow in exterior domains [36]. Definition 2 A map M : X 7! Y, X, Y Banach spaces, is Fredholm, if and only if: (i) M is of class C1 (in the sense of Fréchet differentiability) and denoted by D x M(x) its derivative at x 2 X (iii) the integers. The integers ˛ :D dim fz 2 X : [D x M(x)](z) D 0g and ˇ :D codimfy 2 Y : [D x M(x)](z) D yg for some z 2 X are both finite. The integer m :D ˛  ˇ is independent of the particular x 2 X (Sect. 5.15 in [100]), and is called the index of M. Definition 3 A map M : X 7! Y, is said to be proper if K1 :D fx 2 X : M(x) D y ; y 2 Kg is compact in X, whenever K is compact in Y. By using the properties of proper Fredholm maps of index 0 [84], and of the associated Caccioppoli–Smale degree [10,84], one can prove the following result (see Theorem I.2.2 in [37]) Lemma 3 Let M be a proper Fredholm map of index 0 and of class C2 , satisfying the following. (i) There exists y¯ 2 Y such that the equation M(x) D y¯ has one and only one solution x¯ ; (ii) [D x M(x¯ )](z) D 0 ) z D 0.

Navier–Stokes Equations: A Mathematical Analysis

Then: (a) M is surjective ; (b) There exists an open, dense set Y0 Y such that for any y 2 Y0 the solution set fx 2 X : M(x) D yg is finite and constituted by an odd number,  D (y), of points ; (c) The integer  is constant on every connected component of Y 0 . The next result provides the required functional properties of the map N. Proposition 1 The map N defined in (24) is of class C 1 . 1;2 Moreover, for any  > 0, N(; ) : D1;2 0 (˝) 7! D0 (˝) is proper and Fredholm of index 0. Proof It is a simple exercise to prove that N is of class C 1 (see Example I.1.6 in [37]). By the compactness of the embedding W01;2 (˝) L4 (˝) Theorem II.4.2 in [31], from Lemma 1 and from the definition of the map N [see (22)], one can show that N is compact, that is, it maps bounded sequences of D1;2 0 (˝) into relatively compact sequences. Therefore, N(; ) is a compact perturbation of a multiple of the identity operator. Moreover, by (22) and by Lemma 1 we show that [N (v); v] D 0, which implies, [N(; v); v] D jvj21;2 :

(25)

Using the Schwartz inequality on the left-hand side of (25), we infer that jN(; v)j1;2  jvj1;2 and so, for each fixed  > 0 we have jN(; v)j1;2 ! 1 as jvj1;2 ! 1, that is, N(; ) is (weakly) coercive. Consequently, N(; ) is proper (see Theorem 2.7.2 in [7]). Finally, since the derivative of a compact map is compact (see Theorem 2.4.6 in [7]), the derivative map Dv N(; v)  I C Dv N(v); I identity operator, is, for each fixed  > 0, a compact perturbation of a multiple of the identity, which implies that N(; ) is Fredholm of index 0 (see Theorem 5.5.C in [100]).  It is easy now to show that, for any fixed  > 0, N(; ) satisfies all the assumptions of Lemma 3. Actually, in view of Proposition 1, we have only to prove the validity of (i) and (ii). We take y¯  0, and so, from (25) we obtain that the equation N(; v) D 0 has only the solution x¯  v D 0, so that (i) is satisfied. Furthermore, from (22) it follows that the equation Dv N(; 0)(w) D 0 is equivalent to  w D 0, and so also condition (ii) is satisfied for  > 0. Thus, we have proved the following result (see also [22]). Theorem 1 For any  > 0 and F 2 D1;2 0 (˝), Eq. (23) has at least one solution v 2 D1;2 (˝). Moreover, for each fixed 0  > 0, there exists open and dense O D O() D1;2 0 (˝)

Navier–Stokes Equations: A Mathematical Analysis, Figure 3 Sketch of the manifold S(F)

with the following properties: (i) For any F 2 O the number of solutions to (23), n D n(F; ) is finite and odd ; (ii) the integer n is constant on each connected component of O. Next, for a given F 2 D1;2 0 (˝), consider the solution manifold n o 1;2 S(F) D (; v) 2 (0; 1)  D0 (˝) : N(; v) D F : By arguments similar to those used in the proof of Theorem 1, one can show the following “generic” characterization of S(F) see Example I.2.4 in [37] and also Chap. 10.3 in [93]. Theorem 2 There exists dense P D1;2 0 (˝) such that, for every F 2 P , the set S(F) is a C 1 1-dimensional manifold. Moreover, there exists an open and dense subset of (0; 1),  D (F), such that for each  2 , Eq. (23) has a finite number m D m(F; ) > 0 of solutions. Finally, the integer m is constant on every open interval contained in . In other words, Theorem 2 expresses the property that, for every F 2 P , the set S(F) is the union of smooth and non-intersecting curves. Furthermore, “almost all” lines  D 0 D const. intersect these curves at a finite number of points, m(0 ; F), each of which is a solution to (23) corresponding to  0 and to F. Finally, m(0 ; F) D m(1 ; F) whenever  0 and  1 belong to an open interval of a suitable dense set of (0; 1). A sketch of the manifold S(F) is provided in Fig. 3. Existence Results. Non-Homogeneous Boundary Conditions Existence of solutions to (9)–(10) when v  6 0 leads to one of the most challenging open questions in the mathematical theory of the Navier–Stokes equations, in the case when the boundary @˝ is constituted by more than one connected component. In order to explain the problem, let Si , i D i; : : : ; K, K  1, denote these components. Conservation of mass (9)2 along with Gauss theorem imply the following compatibility condition on the

1017

1018

Navier–Stokes Equations: A Mathematical Analysis

data v  K Z X iD1

v  n i  Si

K X

˚i D 0 ;

(26)

iD1

where n i denotes the outward unit normal to Si . From the physical point of view, the quantity ˚ i represents the mass flow-rate of the liquid through the surface Si . Now, assuming v  and ˝ sufficiently smooth (for example, v  2 W 1/2;2 (@˝) and ˝ locally Lipschitzian Sect. VIII.4 in [32]), we look for a weak solution to (9)–(10) in the form v D u C V, where V 2 W 1;2 (˝) is an extension of v  with div V D 0 in ˝. Thus, if we use, for example, the Galerkin method of Subsect. IV.1.1(A), from (15) we obPN tain that the “approximate solution” u N D iD1 c i N i must satisfy the following equations (k D 1; : : : ; N) [u N ;

k]

 (V  r

 (u N  r k; uN )

k ; uN )

 (V  r

 (u N  r

k; V)

k; V)

C [V ; D hf;

k] k i:

(27)

Therefore, existence of a weak solution will be secured provided we show that the sequence fv N :D u N C V g satisfies the bound (16). In turn, this latter is equivalent to showing ju N j1;2  M1 ;

(28)

where M 1 is a finite, positive quantity independent of N. Multiplying through both sides of (27) by ckN , summing over k from 1 to N, and observing that, by Lemma 1, (u N  ru N ; u N ) D (V  ru N ; u N ) D 0, we find ju N j21;2 D (u N  ru N ; V ) C (V  ru N ; V )  [V ; u N ] C h f ; u N i : By using (14), the Hölder inequality and the Cauchy– Schwartz inequality ab  "a2 C (1/4")b2 ;

a; b; " > 0

(29)

on the last three terms on the right-hand side of this latter equation, we easily find, for a suitable choice of ",  ju N j21;2  (u N  ru N ; V ) C C ; 2

(30)

where C D C(V ; f ; ˝) > 0. From (30) it follows that, in order to obtain the bound (28) without restrictions on the magnitude of , it suffices that V meets the following requirement: Given ";

there is V D V("; x) 2 (0; /4)

such that ('  r'; V )  "j'j21;2

for all ' 2 D(˝) : (31)

As indicated by Leray pp. 28–30 in [61] and clarified by Hopf [52], if ˝ is smooth enough and if K D 1, that is, if @˝ is constituted by only one connected component, S1 , it is possible to construct a family of extensions satisfying (31). Notice that, in such a case, condition (26) reduces to the single condition ˚1 D 0. If K > 1, the same construction is still possible but with the limitation that ˚ i D 0, for all i D 1; : : : ; K. It should be emphasized that this condition is quite restrictive from the physical point of view, in that it does not allow for the presence, in the region of flow, of isolated “sources” and “sinks” of liquid. Nevertheless, one may wonder if, by using a different construction, it is still possible to satisfy (31). Unfortunately, as shown by Takeshita [90] by means of explicit examples, in general, the existence of extensions satisfying (31) implies ˚ i D 0, for all i D 1; : : : ; K; see also § VIII.4 in [32]. We wish to emphasize that the same type of conclusion holds if, instead of the Galerkin method, we use the function-analytic approach; see [61], notes to Chap. VIII in [32]. Finally, it should be remarked that, in the special case of two-dimensional domains possessing suitable symmetry and of symmetric boundary data, Amick [1] and Fujita [29] have shown existence of corresponding symmetric solutions under the general assumption (26). However, we have the following. Open Question Let ˝ be a smooth domain in Rn , n D 2; 3, with @˝ constituted by K > 1 connected components, Si , i D 1; : : : ; K, and let v  be any smooth field satisfying (26). It is not known if the corresponding problem (9)–(10) has at least one solution. Uniqueness and Steady Bifurcation It is a well-established experimental fact that a steady flow of a viscous incompressible liquid is “observed”, namely, it is unique and stable, if the magnitude of the driving force, usually measured through a dimensionless number  2 (0; 1), say, is below a certain threshold, c . However, if  > c , this flow becomes unstable and another, different flow is instead observed. This latter may be steady or unsteady (typically, time-periodic). In the former case, we say that a steady bifurcation phenomenon has occurred. From the physical point of view, bifurcation happens because the liquid finds a more “convenient” motion (than the original one) to dissipate the increasing energy pumped in by the driving force. From the mathematical point of view, bifurcation occurs when, roughly speaking, two solutioncurves (parametrized with the appropriate dimensionless number) intersect. (As we know from Theorem 2, the intersection of these curves is not “generic”.) It can happen

Navier–Stokes Equations: A Mathematical Analysis

that one curve exists for all values of , while the other only exists for  > c (supercritical bifurcation). The point of intersection of the two curves is called the bifurcation point. Thus, in any neighborhood of the bifurcation point, we must have (at least) two distinct solutions and so a necessary condition for bifurcation is the occurrence of nonuniqueness. This section is dedicated to the above issues.

Theorem 4 (Non-Uniqueness) Let ˝ be a bounded smooth body of revolution around an axis r, that does not include points of r. For example, ˝ is a torus of arbitrary bounded smooth section. Then there are smooth fields f and v  and a value of  > 0 such that problem (9)–(10) corresponding to these data admits at least two distinct and smooth solutions.

Uniqueness Results It is simple to show that, if jvj1;2 is not “too large”, then v is the only weak solution to (9)–(10) corresponding to the given data.

Some Bifurcation Results By using the functional setting introduced in Sect. “B. The Function-Analytic Method”, it is not difficult to show that steady bifurcation can be reduced to the study of a suitable nonlinear eigenvalue problem in the space D1;2 0 (˝). To this end, we recall certain basic definitions. Let U be an open interval of R and let

Theorem 3 (Uniqueness) Let ˝ be locally Lipschitzian and let jvj1;2 < / ;

(32)

with  D (˝) > 0. Then, there is only one weak solution to (9)–(10). Proof Let v, v 1 D v C u be two different solutions. From (13) we deduce [u; '] D (u r'; v)C(v 1 r'; u) ;

for all ' 2 D(˝) : (33)

If ˝ is locally Lipschitzian, then u 2 D1;2 0 (˝) (see Theorem II.3.2 and § II.3.5 in [31]), and since D(˝) is dense in 1;2 D0 (˝), with the help of Lemma 1 we can replace ' with u in (33) to get juj21;2 D (u  ru; v) : We now use (14) and Lemma 1 on the right-hand side of this equation to find (   jvj1;2 )juj21;2  0 ; with  D (˝) > 0, which proves u D 0 in D1;2 0 (˝), namely, uniqueness, if v satisfies (33). 

M : (x; ) 2 X  U 7! Y :

(34)

Definition 4 The point (x0 ; 0 ) is called a bifurcation point of the equation M(x; ) D 0

(35)

if and only if (a) M(x0 ; 0 ) D 0, and (b) there are (at least)  ;  )g, to two sequences of solutions, f(x m ; m )g and f(x m m  (35), with x m ¤ x m , for all m 2 N, such that (x m ; m ) !  ;  ) ! (x ;  ) as m ! 1. (x0 ; 0 ) and (x m m 0 0 If M is suitably smooth around (x0 ; 0 ), a necessary condition for (x0 ; 0 ) to be a bifurcation point is that D x M(x0 ; 0 ) is not a bijection. In fact, we have the following result which is an immediate corollary of the implicit function theorem (see, e. g., Lemma III.1.1 in [37]). Lemma 4 Suppose that D x M exists in a neighborhood of (x0 ; 0 ), and that both M and D x M are continuous at (x0 ; 0 ). Then, if (x0 ; 0 ) is a bifurcation point of (34), D x M(x0 ; 0 ) is not a bijection. If, in particular, D x M(x0 ; 0 ) is a Fredholm operator of index 0 (see Definition 2), then the equation D x M(x0 ; 0 )x D 0 has at least one nonzero solutions.

Remark 5 As we have seen previously, if v   0, weak solutions satisfy jvj1;2  j f j1;2 . Thus, (32) holds if j f j1;2   2 /. If v  ¤ 0, then one can show that (32) holds if j f j1;2 C (1 C )kv  k1/2;2;@˝ C kv  k21/2;2;@˝   2 /1 , where  1 has the same properties as ; (see Theorem VIII.4.2 in [32]). Notice that these conditions are satisfied if a suitable non-dimensional parameter    j f j1;2 C kv  k1/2;2;@˝ is “sufficiently small”.

Let v 0 D v 0 (),  2 (0; 1) be a family of weak solution to (9)–(10) corresponding to given data f and v  . We assume that f and v  are fixed. Denoting by v any other solution corresponding to the same data, by an argument completely similar to that leading to (23), we find that u :D v  v 0 , satisfies the following equation in D1;2 0 (˝)

Remarkably enough, one can give explicit examples of nonuniqueness, if condition (32) is violated. More specifically, we have the following result (see Theorem VIII.2.2 in [32]).

with B(v 0 ) :D Dv N (v 0 ) and N defined in (22). Obviously, (v 0 (0 ); 0 ) is a bifurcation point for the original equation if and only if (0; 0 ) is such for (36). Thus, in view of Lemma 4, a necessary condition for (0; 0 ) to be

u C B(v 0 )(u) C N (u) D 0 ;

(36)

1019

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Navier–Stokes Equations: A Mathematical Analysis

a bifurcation point for (36) is that the equation 0 v C B(v 0 (0 ))(v) D 0

(37)

has at least one nonzero solution v 2 D1;2 0 (˝). In several significant situations it happens that, after a suitable non-dimensionalization of (36), the family of solutions v 0 (),  2 (0; 1) is independent of the parameter  which, this time, has to be interpreted as the inverse of an appropriate dimensionless number (Reynolds number), like, for instance, in the Taylor–Couette problem; see the following subsection. Now, from Proposition 1, we know that B(u) is compact at each u 2 D1;2 0 (˝), so that I C B(u) is Fredholm of index 0, at each u 2 D1;2 0 (˝), for all  > 0 (see Theorem 5.5.C in [100]). Therefore, whenever v 0 does not depend on , in a neighborhood of  0 , from Lemma 4 we find that a necessary condition for (v 0 ; 0 ) to be a bifurcation point for (35) is that  0 is an eigenvalue of the (compact) linear operator B(v 0 ). The stated condition becomes also sufficient, provided we make the additional assumptions that  0 is a simple eigenvalue of B(v 0 ). This is a consequence of the following theorem (see, e. g., Lemma III.1.2 in [37]). Theorem 5 Let X Y and let the operator M in (34) be of the form M D  I C T, where I is the identity in X and T is of class C1 . Furthermore, set L :D D x T(0). Suppose that 0 I C L is Fredholm of index 0 (Definition 2), for some 0 2 U, and that 0 is a simple eigenvalue for L, namely, the equation 0 x C L(x) D 0 has one and only one (nonzero) independent solution, x1 , while the equation 0 x C L(x) D x1 has no solutions. Then, (0; 0 ) is a bifurcation point for the equation M(x; ) D 0.

Navier–Stokes Equations: A Mathematical Analysis, Figure 4 Sketch of the streamlines of Taylor vortices, at the generic section =const

solve (9), with f D 0, for all values of the Reynolds number  :D ! r12 /. Moreover, u0 satisfies the boundary conditions u0 (r) D 0

at r D 1 ;

u 0 (r) D e

at r D R :

Experiments show that, if  exceeds a critical value, c , a flow with entirely different features than the flow (38) is observed. In fact, this new flow is dominated by large toroidal vortices, stacked one on top of the other, called Taylor vortices. They are periodic in the z-direction and axisymmetric (independent of ); see Fig. 4. Therefore, we look for bifurcating solutions of the form u 0 (r) C w(r; z), p0 (r) C p(r; z), satisfying (39), and where w and p are periodic in the z-direction with period P, and w satisfies the following parity conditions: wr (r; z) D wr (r; z) ;

w (r; z) D w (r; z) ; wz (r; z) D wz (r; z) :

Bifurcation of Taylor–Couette Flow A notable application of Theorem 5 is the bifurcation of Taylor–Couette flow. In this case the liquid fills the space between two coaxial, infinite cylinders, C1 and C2 , of radii R1 and R2 > R1 , respectively. C1 rotates around the common axis, a, with constant angular velocity !, while C2 is at rest. Denote by (r; ; z) a system of cylindrical coordinates with z along a and oriented as ! and let (e r ; e ; e z ) the associated canonical base. The components of a vector w in such a base are denoted by wr ; w and wz , respectively. If we introduce the non-dimensional quantities   u D v/(!r1 ) ; x¯ D x/r1 ; p D p/ ! 2 r12 ; R D R2 /R1 ; with ! D j!j, we see at once that the following velocity and pressure fields   1  u 0 D 1  R2 r  R2 /r e ;

p0 D ju0 j2 ln rCconst ; (38)

(39)

(40)

Moreover, the relevant region of flow becomes the periodicity cell ˝ :D (1; R)  (0; P). If we now introduce the stream function @ D wr ; @z

@(r ) D r wz ; @r (r; z) D  (r; z) ;

(41)

and the vector u :D ( ; w ), it can be shown (see § 72.7 in [99]) that u satisfies an equation of the type (36), namely,   ¯ 0 )(u) C N ¯ (u) D 0 in H (˝) u C  B(v (42) ¯ 0 ) and N ¯ obey the same funcwhere the operators B(v tional properties as B(v 0 ) and N , and where H (˝) is the Banach space of functions u :D ( ; w ) 2 C 4;˛ (˝)  C 2;˛ (˝); ˛ 2 (0; 1), such that: (i) u(r; 0) D

Navier–Stokes Equations: A Mathematical Analysis

u(r; P) for all r 2 (1; R), (ii) u satisfies the parity conditions in (40), (41), and (iii) D @ /@r D w D 0 at r D 1; R. Thus, in view of Theorem 5 and of the properties of the operators involved in (42), we will obtain that (0; 0 ) is a bifurcation point for (41) if the following two conditions are met ¯ 0 )(u) D 0 (a) u C 0 B(v has one and only one independent solution, u1 2 H (˝) ; (b)

¯ 0 )(u) D u1 the equation u C 0 B(v has no solution in H (˝) :

(43)

It is in Lemma 72.14 in [99] that there exists a period P for which both conditions in (43) are satisfied. In addition, for ¯ 0 )(u) D 0 has only all  2 (0; 0 ) the equation u C B(v the trivial solution in H (˝), which, by Lemma 4, implies that no bifurcation occurs for  2 (0; 0 ), which, in turn, means that the bifurcation is supercritical.

Extension of the Boundary Data If ˝ isRsmooth enough (locally Lipschitzian, for example), since @˝ e 1  n D 0, by what we observed in Sect. “Existence Results. NonHomogeneous Boundary Conditions”, for any  > 0 we find V D V (; x), with div V D 0 in ˝, and satisfying (31) with " D 1/(2), say. Actually, as shown in Proposition 3.1 in [36], the extension V can be chosen, in particular, to be of class C 1 ((0; 1)  ˝) and such that the support of V(; ) is contained in a bounded set, independent of . The proof given in [36] is for ˝ R3 , but it can be easily extended to the case ˝ R2 . Variational Formulation and Weak Solutions Setting u D w  V in (44)1 , dot-multiplying through both sides of this equation by ' 2 D(˝), integrating by parts and taking into account 12, we find   @u ; '   (u  r'; u)  f(u  r'; V) [u; ']   @x1 C (V  r'; u)g C (H; ') D h f ; 'i ;

for all ' 2 D(˝) :

Flow in Exterior Domains

(45)

One of the most significant questions in fluid mechanics is to determine the properties of the steady flow of a liquid past a body B of simple symmetric shape (such as a sphere or cylinder), over the entire range of the Reynolds number  :D U L/ 2 (0; 1); see, e. g., [§ 4.9] in [4]. Here L is a length scale representing the linear dimension of B, while v 1 D U e 1 , U D const > 0, is the uniform velocity field of the liquid at large distances from B. In the mathematical formulation of this problem, one assumes that the liquid fills the whole space, ˝, outside the closure of the domain B Rn , n D 2; 3. Thus, by scaling v by U and x by L, from (9)–(10) we obtain that the steady flow past a body consists in solving the following non-dimensional exterior boundary-value problem 9 @w w   C w  rw D r p C f = @x1 in ˝ ; divw D 0 w(x) D e 1 ;

x 2 @˝ ;

lim w(x) D 0 ; (44)

jxj!1

where w :D v C e 1 and where we assume, for simplicity, that v   0. (However, all the stated results continue to hold, more generally, if v  belongs to a suitable trace space.) Whereas in the three-dimensional case (e. g., flow past a sphere) the investigation of (44) is, to an extent, complete, in the two-dimensional case (e. g., flow past a circular cylinder) there are still fundamental unresolved issues. We shall, therefore, treat the two cases separately. We need some preliminary considerations.

where H D H(V) :D V C 

@V  V  rV : @x1

Definition 5 A function w 2 D1;2 (˝) is called a weak solution to (44) if and only if w D u C V where u 2 D1;2 0 (˝) and u satisfies (45). Regularity of Weak Solutions (A) Local Regularity. The proof of differentiability properties of weak solutions can be carried out in a way similar to the case of a bounded domain. In particular, if f and ˝ are of class C 1 , one can show that w 2 ¯ and there exists a scalar field p 2 C 1 (˝) ¯ C 1 (˝) such that (44)1;2;3 holds in the ordinary sense. For this and other intermediate regularity results, we refer to Theorem IX.1.1 in [32]. (B) Regularity at Infinity. The study of the validity of (44)4 and, more particularly, of the asymptotic structure of weak solutions, needs a much more involved treatment, and the results depend on whether the flow is three- or two-dimensional. In the three-dimensional case, if f satisfies suitable summability assumptions outside a ball of large radius, by using a local representation of a weak solution, (namely, at all points of a ball of radius 1 and centered at x 2 ˝, dist (x; @˝) > 1) one can prove that w and all its derivatives tend to zero at infinity uniformly pointwise and that a similar property holds for p (see Theorem X.6.1 in [32]). The starting point of this analysis

1021

1022

Navier–Stokes Equations: A Mathematical Analysis

to (45) of the form u N D

is the crucial fact that if ˝ R ; 3

1;2 D0 (˝)

is continuously embedded in L6 (˝) ;

(46)

which implies that w tends to zero at infinity in a suitable sense. However, the existence of the “wake” behind the body along with the sharp order of decay of w and p requires a more complicated analysis based on the global representation of a weak solution (namely, at all points outside a ball of sufficiently large radius) by means of the Oseen fundamental tensor, along with maximal regularity estimates for the solution of the linearized Oseen problem, this latter being obtained by suppressing the nonlinear term w  rw in (44) [§§ IX.6, IX.7 and IX.8] in [32]. In particular, one can prove the following result concerning the behavior of w; (see Theorems IX.7.1, IX.8.1 and Remark IX.8.1 in [32]). Theorem 6 Let ˝ be a three-dimensional exterior domain of class C2 , and let w be a weak solution to (44) corresponding to a given f 2 L q (˝), for all q 2 (1; q0 ], q0 > 3. Then w 2 Lr (˝) ;

if and only if r > 2 :

(47)

If, in addition, f is of bounded support, then, denoting by the angle made by a ray starting from the origin of coordi¯ with the nates (taken, without loss of generality, in R3  ˝) positively directed x1 -axis, we have jw(x)j 

M  ; jxj 1 C jxj(1 C cos )

x 2˝;

(48)

where M D M(; ˝) > 0. Remark 6 Two significant consequences of Theorem 6 are: (i) the total kinetic energy of the flow past an obstacle ( 12 kwk22 ) is infinite , see (47) ; and (ii) the asymptotic decay of w is faster outside any semi-infinite cone with its axis coinciding with the negative x1 -axis (existence of the “wake”); see (48). The study of the asymptotic properties of solutions in the two-dimensional case is deferred till Sect. “Two-Dimensional Flow. The Problem of Existence”. Three-Dimensional Flow. Existence of Solutions and Related Properties As in the case of a bounded domain, we may use two different approaches to the study of existence of weak solutions. A. Finite-Dimensional Method Assume f 2 D1;2 (˝). 0 With the same notation as in Subsect. “A. The Finite-Dimensional Method”, we look for “approximate solutions”

 uN ;

k



 

 f(u N  r

PN

@u ; @N x 1

k; V)

iD1 c i N

 k

   uN  r

C (V  r

D hf;

i , where

ki ;

k ; u N )g C

k; uN



(H;

k D 1; : : : ; N :

k)

(49)

As in the case of a bounded domain, existence to the algebraic system (49), in the unknowns ciN , will be achieved if we show the uniform estimate (28). By dot-multiplying through both sides of (49) by ckN , by summing over k between 1 and N and by using Lemma 1 we find ju N j21;2 D  (u N  ru N ; V )  (H; u N ) C h f ; u N i : (50) From the properties of the extension of the boundary data, we have that V satisfies (31) with " D 1/(2) and that, moreover (H; u N )  Cju N j1;2 , for some C D C(˝; ) > 0. Thus, from this and from (50) we deduce the uniform bound (28). This latter implies the existence of a subsequence fu N 0 g converging to some u 2 D1;2 0 (˝) weakly. Moreover, by Rellich theorem, u N 0 ! u in L4 (K), where K :D ˝ \ fjxj > g, all sufficiently large and finite

> 0 (see Theorem II.4.2 in [31]). Consequently, recalling that k is of compact support in ˝, we proceed as in the bounded domain case and take the limit N  N 0 ! 1 in (49) to show that u satisfies (45) with '  k . Successively, by taking into account that every ' 2 D(˝) can be approximated in L3 (˝) by linear combinations of k (see Lemma VII.2.1 in [31]) by (46) and by Lemma 1 we conclude that u satisfies (45). Notice that, as in the case of flow in bounded domains, this method furnishes existence for (˝). any  > 0 and any f 2 D1;2 0 B. The Function-Analytic Method As in the case of flow in a bounded domain, Subsect. “The Function-Analytic Method”ion IV.1.(B), our objective is to rewrite (45) as a nonlinear operator equation in an appropriate Banach space, where the relevant operator satisfies the assumptions of Lemma 3. In this way, we may draw the same conclusions of Theorem 1 also in the case of a flow past an obstacle. However, unlike the case of flow in a bounded domain, the map ' 2 D(˝) 7! (u  r'; u) 2 R can not be extended to a linear, bounded functional in 1;2 1;2 D0 (˝), if u merely belongs to D0 (˝). Analogous conclusion holds for the map ' 2 D(˝) 7! (@u/@x1 ; ') 2 R. The reason is because, in an exterior domain, the Poincaré inequality (18) and, consequently, the embedding 21 are, in general, not true. It is thus necessary to consider the

Navier–Stokes Equations: A Mathematical Analysis

above functionals for u in a space strictly contained in 1;2 D0 (˝). Set ˇ ˇ ˇ ˇ @u ˇ @x 1 ; ' ˇ [juj] :D sup j'j1;2 '2D(˝) and let n X D X(˝) :D u 2 D1;2 0 (˝) :

o [juj] < 1 :

Clearly, X(˝) endowed with the norm j  j1;2 C [j  j] is a Banach space. Moreover, X(˝) L4 (˝), continuously; see Proposition 1.1 in [36]. We may thus conclude, by Riesz theorem, by Hölder inequality and by the properties of the extension V that, for any u 2 X(˝), there exist L(u), M(u), V (u), and H () in D1;2 0 (˝) such that, for all ' 2 D(˝),   @u  ; ' D [L(u); '] ; @x1 (u  r'; u) D [M(u); '] ; (u  r'; V )  (V  r'; u) D [V (u); '] ; (H; ') D [H (); '] : Consequently, we obtain that (45) is equivalent to the following equation M(; u) D F

in D1;2 0 (˝) ;

(51)

where M : (; u) 2 (0; 1)  X(˝) 7! uC (L(u) C V (u) C M(u))CH () 2 D1;2 0 (˝) : A detailed study of the properties of the operator M is done in § 5 in [36], where, in particular, the following result is proved. Lemma 5 The operator M is of class C 1 . Moreover, for each  > 0, M(; ) : X(˝) 7! D1;2 0 (˝) is proper and Fredholm of index 0, and the two equations M(; u) D H () and D u M(; 0)(w) D 0 only have the solutions u D w D 0. From this lemma and with the help of Lemma 3, we obtain the following result analogous to Theorem 1. Theorem 7 For any  > 0 and F 2 D1;2 0 (˝) the Eq. (51) has at least one solution u 2 X(˝). Moreover, for each fixed  > 0, there exists open and dense Q D Q()

1;2 D0 (˝) with the following properties: (i) For any F 2 Q the number of solutions to (51), n D n(F; ) is finite and odd ; (ii) the integer n is finite on each connected component of Q.

Navier–Stokes Equations: A Mathematical Analysis, Figure 5 Visualization of a flow past a sphere at increasing Reynolds numbers ; after [91]

Finally, concerning the geometric structure of the set of pairs (; u) 2 (0; 1)  X(˝) satisfying (51) for a fixed F, a result entirely similar to Theorem 2 continues to hold; see Theorem 6.2 in [36]. Three-Dimensional Flow. Uniqueness and Steady Bifurcation There is both experimental [91,97] and numerical [68,95] evidence that a steady flow past a sphere is unique (and stable) if the Reynolds number  is sufficiently small. Moreover, experiments report that a closed recirculation zone first appears at  around 20–25, and the flow stays steady and axisymmetric up to at least  '130. This implies that the first bifurcation occurs through a steady motion. For higher values of , the wake behind the sphere becomes time-periodic, thus suggesting the occurrence of unsteady (Hopf) bifurcation; see Fig. 5. In this section we shall collect the relevant results available for uniqueness and steady bifurcation of a flow past a three-dimensional obstacle. Uniqueness Results Unlike the case of a flow in a bounded domain where uniqueness in the class of weak solutions is simply established (see Theorem 3), in the situation of a flow past an obstacle, the uniqueness proof requires the detailed study of the asymptotic behavior of a weak solution mentioned in Sect. “Flow in Exterior Domains”; see also Theorem 6. Precisely, we have the following result, for whose proof we refer to Theorem IX.5.3 in [32]. ¯ Theorem 8 Suppose f 2 L6/5 (˝) \ L3/2 (˝),  2 (0; ], ¯ > 0, and let w be the corresponding weak for some  solution.(Notice that, under the given assumptions, f 2 1;2 ¯ > 0 such that, if D0 (˝).) There exists C D C(˝; ) k f k6/5 C  < C ; then w is the only weak solution corresponding to f .

1023

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Navier–Stokes Equations: A Mathematical Analysis

Some Bifurcation Results The rigorous study of steady bifurcation of a flow past a body is a very challenging mathematical problem. However, the function-analytic framework developed in the previous section allows us to formulate sufficient conditions for steady bifurcation, that are formally analogous to those discussed in Sect. “Uniqueness and Steady Bifurcation” for the case of a flow in a bounded domain; see § VII in [36]. To this end, fix f 2 D1;2 (˝), once and for all, and let u0 D u 0 (), 0  in some open interval I  (0; 1), be a given curve in X(˝), constituted by solutions to (51) corresponding to the prescribed f . If u C u0 , is another solution, from (51) we easily obtain that u satisfies the following equation in 1;2 D0 (˝) u C  (L(u) C B(u 0 ())(u) C M(u)) D 0 ; u 2 X(˝) ;

(52)

where B(u 0 ) :D Du M(u 0 ). In this setting, the branch u 0 () becomes the solution u  0 and the bifurcation problem thus reduces to find a nonzero branch of solutions u D u() to (52) in every neighborhood of some bifurcation point (0; 0 ); see Definition 4. Define the map F : (; u) 2 (0; 1)  X(˝) 7! u C (L(u) C B(u 0 ())(u) C M(u)) 2 D1;2 0 (˝) : (53) In § VII in [36] the following result is shown. Lemma 6 The map F is of class C 1 . Moreover, the derivative Du F(; 0)(w) D w C  (L(w) C B(u0 ())(w)) ; is Fredholm of index 0. Therefore, from this lemma and from Lemma 4 we obtain that a necessary condition for (0; 0 ) to be a bifurcation point to (52) is that the linear problem w 1 C 0 (L(w 1 ) C B(u 0 (0 ))(w 1 )) D 0 ; w 1 2 X(˝) ;

(54)

has a non-zero solution w 1 . Once this necessary condition is satisfied, one can formulate several sufficient conditions for the point (0; 0 ) to be a bifurcation point. For a review of different criteria for global and local bifurcation for Fredholm maps of index 0, we refer to Sect. 6 in [33]. Here we wish to use the criterion of Theorem 5 to present a very simple (in principle) and familiar sufficient condition in the particular case when the given curve u0 can be made (locally, in a neighborhood of 0 ) independent of . This may depend on the particular non-dimensionalization of

the Navier–Stokes equations and on the special form of the family of solutions u 0 . In fact, there are several interesting problems formulated in exterior domains where this circumstance takes place, like, for example, the problem of steady bifurcation considered in the previous section and the one studied in Sect. 6 in [39]. Now, if u0 does not depend on , from Theorem 5 and from (54) we immediately find that a sufficient condition in order that (0; 0 ) be a bifurcation point is that the following problem w C 0 L (w) D w 1 ;

w 2 X(˝) ;

(55)

with L :D L C B(u 0 ) and w 1 solving (54), has no solution. In different words, a sufficient condition for (0; 0 ) to be a bifurcation point is that 1/0 is a simple eigenvalue of the operator L . It is interesting to observe that this condition is formally the same as the one arising in steady bifurcation problems for flow in a bounded domain; see Sect.“Uniqueness and Steady Bifurcation”. However, while in this latter case L ( B(v 0 )) is compact and defined on the whole of D1;2 0 (˝), in the present situation L , with domain D :D X(˝) D1;2 0 (˝), is an unbounded operator. As such, we can not even be sure that L has real simple eigenvalues. Nevertheless, since, by Lemma 6, I C L is Fredholm of index 0 for all  2 (0; 1), and since one can show (Lemma 7.1 in [36]) that L is graph-closed, if u 0 2 L3 (˝) (this latter condition is satisfied under suitable hypotheses on f ; see Theorem 6) from well-known results of spectral theory (see Theorem XVII.2.1 in [46]), it follows that the set  of real eigenvalues of L satisfies the following properties: (a)  is at most countable; (b)  is constituted by isolated points of finite algebraic and geometric multiplicities, and (c) points in  can only cluster at 0. Consequently, the bifurcation condition requiring the simplicity of 1/0 (namely, algebraic multiplicity 1) is perfectly meaningful. Two-Dimensional Flow. The Problem of Existence The planar motion of a viscous liquid past a cylinder is among the oldest problems to have received a systematic mathematical treatment. Actually, in 1851, it was addressed by Sir George Stokes in his work on the motion of a pendulum in a viscous liquid [89]. In the wake of his successful study of the flow past a sphere in the limit of vanishing  (Stokes approximation), Stokes looked for solutions to (44), with f  0 and  D 0, in the case when ˝ is the exterior of a circle. However, to his surprise, he found that this linearized problem has no solution, and he concluded with the following (wrong) statement [89], p. 63, “It appears that the supposition of steady motion is inadmissible”.

Navier–Stokes Equations: A Mathematical Analysis

Such an observation constitutes what we currently call Stokes Paradox. This is definitely a very intriguing starting point for the resolution of the boundary-value problem (44), in that it suggests that, if the problem has a solution, the nonlinear terms have to play a major role. In this regard, by using, for instance, the Galerkin method and proceeding exactly as in Subsection IV.2.1(A), we can prove the existence of a weak solution to (44), for any  > 0 and f 2 D1;2 (˝). 0 In addition, this solution is as smooth as allowed by the regularity of ˝ and f (see Sect. “Flow in Exterior Domains”) and, in such a case, it satisfies (44)1;2;3 in the ordinary sense. However, unlike the three-dimensional case [see Eq. (46)], the space D1;2 0 (˝) is not embedded in any Lq -space and, therefore, we can not be sure that, even in an appropriate generalized sense, this solution vanishes at infinity, as requested by (44)4 . Actually, if ˝ R2 there are functions in D1;2 0 (˝) becoming unbounded at infinity. Take, for example, w D (ln jxj)˛ e , ˛ 2 (0; 1), and ˝ the exterior of the unit circle. In this sense, we call these solutions weak, and not because of lack of local regularity (they are as smooth as allowed by the smoothness of ˝ and f ). This problem was first pointed out by Leray (pp. 54–55 in [61]). The above partial results leave open the worrisome possibility that a Stokes paradox could also hold for the fully nonlinear problem (45). If this chance turned out to be indeed true, it would cast serious doubts on the Navier– Stokes equations as a reliable fluid model, in that they would not be able to catch the physics of a very elementary phenomenon, easily reproduced experimentally. The possibility of a nonlinear Stokes paradox was ruled out by Finn and Smith in a deep paper published in 1967 [20], where it is shown that if f  0 and if ˝ is sufficiently regular, then (45) has a solution, at least for “small” (but nonzero!) . The method used by these authors is based on the representation of solutions and careful estimates of the Green tensor of the Oseen problem. Another approach to existence for small , relying upon the Lq theory of the Oseen problem, was successively given by Galdi [30] where one can find the proof of the following result. Theorem 9 Let ˝ be of class C2 and let f 2 L q (˝), for some q 2 (1; 6/5). Then there exists 1 > 0 and C D C(˝; q; 1 ) > 0 such that if, for some  2 (0; 1 ], j log j1 C 2(1/q1) k f kq < C ; problem (45) has at least one weak solution that, in addition, satisfies (45)4 uniformly pointwise.

It can be further shown that the above solutions meet all the basic physical requirements. In particular, as in the three-dimensional case (see Sect. “Flow in Exterior Domains”), they exhibit a “wake” in the region x1 < 0. Moreover, the solutions are unique in a ball of a suitable Banach space, centered at the origin and of “small” radius. For the proof of these two statements, we refer to §§ X.4 and X.5 in [32]. Though significant, these results leave open several fundamental questions. The most important is, of course, that of whether problem (44) is solvable for all  > 0 and all f in a suitable space. As we already noticed, this solvability would be secured if we can show that the weak solution does satisfy (44)4 , even in a generalized sense. It should be emphasized that, since, as shown previously, there are functions in D1;2 0 (˝) that become unbounded at infinity, the proof of this asymptotic property must be restricted to functions satisfying (44)1;2;3 . This question has been taken up in a series of remarkable papers by Gilbarg and Weinberger [44], [45] and by Amick [2] in the case when f  0. Some of their results lead to the following one due to Galdi; see Theorem 3.4 in [35]. Theorem 10 Let w be a weak solution to (44) with f  0. Then, there exists 2 R2 such that lim w(x) D

jxj!1

uniformly :

Open Question It is not known if D 0, and so it is not known if w satisfies (44)4 . Thus, the question of the solvability of (44) for arbitrary  > 0, even when f  0, remains open. When ˝ is symmetric around the direction of e 1 , in [33] Galdi has suggested a different approach to the solvability of (44) (with f  0) for arbitrary large  > 0. Let C be the class of vector fields, v D (v1 ; v2 ), and scalar fields  such that (i) v1 and  are even in x2 and v2 is odd in x2 , and (ii) jvj1;2 < 1. The following result holds. Theorem 11 Let ˝ be symmetric around the x1 -axis. Assume that the homogeneous problem: ) u D u  ru C r in ˝ divu D 0 (56) lim u(x) D 0 uniformly ; uj@˝ D 0 ; jxj!1

has only the zero solution, u  0 ; p D const: , in the class C . Then, there is a set M with the following properties: (i) M (0; 1); (ii) M  (0; c) for some c D c(˝) > 0; (iii) M is unbounded;

1025

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Navier–Stokes Equations: A Mathematical Analysis

(iv) For any  2 M, problem (44) has at least one solution in the class C . Open Question The difficulty with the above theorem relies in establishing the validity of its hypothesis, namely, whether or not (56) has only the zero solution in the class C . Moreover, supposing that the hypothesis holds true, the other fundamental problem is the study of the properties of the set M. For a detailed discussion of these issues, we refer to § 4.3 in [35]. Mathematical Analysis of the Initial-Boundary Value Problem Objective of this section is to present the main results and open questions regarding the unique solvability of the initial-boundary value problem (7)–(8), and significant related properties. Preliminary Considerations In order to present the basic problems for (7)–(8) and the related results, we shall, for the sake of simplicity, restrict ourselves to the case when f  v 1  0. In what follows, we shall denote by (7)–(8)hom this homogeneous problem. The first, fundamental problem that should be naturally set for (7)–(8)hom is the classical one of (global) well-posedness in the sense of Hadamard. Problem 1 Find a Banach space, X, such that for any initial data v 0 in X there is a corresponding solution (v; p) to (7)(8)hom satisfying the following conditions: (i) it exists for all T > 0, (ii) it is unique and (iii) it depends continuously on v 0 . In different words, the resolution of Problem 1 will ensure that the Navier–Stokes equations furnish, at all times, a deterministic description of the dynamics of the liquid, provided the initial data are given in a “sufficiently rich” class. It is immediately seen that the class X should meet some necessary requirements for Problem 1 to be solvable. For instance, if we take v 0 only bounded, we find that problem (7)–(8), with ˝ D Rn and f D 0, admits the following two distinct solutions v 1 (x; t) D 0 ;

p1 (x; t) D 0 ;

v 2 (x; t) D sin te 1 ;

p2 (x; t) D x1 cos t ;

corresponding to the same initial data v 0 D 0. Furthermore, we observe that the resolution of Problem 1, does not exclude the possibility of the formation of a “singularity”, that is, the existence of points in the spacetime region where the solution may become unboundedly

large in certain norms. This possibility depends, of course, on the regularity of the functional class where well-posedness is established. One is thus led to considering the next fundamental (and most popular) problem. Problem 2 Given an initial distribution of velocity v 0 , no matter how smooth, with Z jv 0 (x)j2 < 1 ; (57) ˝

determine a corresponding regular solution v(x; t); p(x; t) to (7)–(8)hom for all times t 2 (0; T) and all T > 0. By “regular” here we mean that v and p are both of class C 1 in the open cylinder ˝  (0; T), for all T > 0. When ˝  R3 , Problem 1 is, basically, the third Millennium Prize Problem posted by the Clay Mathematical Institute in May 2000. The requirement (57) on the initial data is meaningful from the physical point of view, in that it ensures that the kinetic energy of the liquid is initially finite. Moreover, it is also necessary from a mathematical viewpoint because, if we relax (57) to the requirement that the initial distribution of velocity is, for example, only bounded, then Problem 2 has a simple negative answer. In fact, the following pair v(x; t) D

1 e1  t

p(x; t) D 

x1 ; (  t)2 t 2 [0; ) ;

 >0

is a solution to (7)–(8)hom with f  0 and ˝ D R3 , that becomes singular at t D , for any given positive . An alternative way of formulating Problem 2 in “more physical” terms is as follows. Problem 20 Can a spontaneous singularity arise in a finite time in a viscous liquid that is initially in an arbitrarily smooth state? Though, perhaps, the gut answer to this question could be in the negative, one can bring very simple examples of dissipative nonlinear evolution equations where spontaneous singularities do occur, if the initial data are sufficiently large. For instance, the initial-value problem v 0 C  v D v 2 , v(0) D v0 ,  > 0, has the explicit solution  v0 v(t) D v0  e  t (v0   ) which shows that, if v0   , then v is smooth for all t > 0, while if v0 > , then v becomes unbounded in a finite time:   1 1 v0 v(t)  ; t 2 [0; ) ;  :D log :  t  v0  

Navier–Stokes Equations: A Mathematical Analysis

If the occurrence of singularities for problem (7)– (8)hom can not be at all excluded, one can still theorize that singularities are unstable and, therefore, “undetectable”. Another plausible explanation could be that singularity may appear in the Navier–Stokes equations due to the possible break-down of the continuum model at very small scales. It turns out that, in the case of two-dimensional (2D) flow, both problems 1 and 2 are completely resolved, while they are both open in the three-dimensional (3D) case. The following section will be dedicated to these issues.

v 2 L1 (0; T; L2 (˝)) \ L2 (0; T; D1;2 0 (˝)) ;

On the Solvability of Problems 1 and 2 As in the steady-state case, a basic tool for the resolution of both Problems 1 and 2 is the accomplishment of “good” a priori estimates. By “good”, we mean that (i) they have to be global, namely, they should hold for all positive times, and (ii) they have to be valid in a sufficiently regular function class. These estimates can then be used along suitable “approximate solutions” which eventually will converge, by an appropriate limit procedure, to a solution to (7)–(8)hom . To date, “good” estimates for 3D flow are not known. Unless explicitly stated, throughout this section we assume that ˝ is a bounded, smooth (of class C2 , for example) domain of Rn , n D 2; 3. Derivation of Some Fundamental A Priori Estimates We recall that, for simplicity, we are assuming that the boundary data, v 1 , in (8) is vanishing. Thus, if we formally dot-multiply through both sides of (7)1 (with f  0) by v, integrate by parts over ˝ and take into account 12 and Lemma 1, we obtain the following equation 1 d kv(t)k22 C  jv(t)j21;2 D 0 : 2 dt

(58)

The physical interpretation of (58) is straightforward. Actually, if we dot-multiply both sides of the identity div D(v) D v by v, where D D D(v) is the stretching tensor (see 3), and integrate by parts over ˝, we find that jvj1;2 D kD(v)k2 . Since, as we observed in Sect. “Derivation of the Navier–Stokes Equations and Preliminary Considerations”, D takes into account the deformation of the parts of the liquid, Eq. (58) simply relates the rate of decreasing of the kinetic energy to the dissipation inside the liquid, due to the combined effect of viscosity and deformation. If we integrate (58) from s  0 to t  s, we obtain the so-called energy equality Z t jv( )j21;2 d D kv(s)k22 0  s  t : (59) kv(t)k22 C2 s

Notice that the nonlinear term v  rv does not give any contribution to Eq. (58) [and, consequently, to Eq. (59)], in virtue of the fact that (v  rv; v) D 0; see Lemma 1. By taking s D 0 in (59), we find a bound on the kinetic energy and on the total dissipation for all times t  0, in terms of the initial data only, provided the latter satisfy (57). In concise words, the energy equality (59) is a global a priori estimate. It should be emphasized that the energy equality is, basically, the only known global a priori estimate for 3D flow. From (59) it follows, in particular, all T > 0 : (60) A second estimate can be obtained by dot-multiplying through both sides of (7)1 by Pv and by integrating by parts over ˝. Taking into account that 

@v ; Pv @t



 D

@v ; v @t

 D

1 d 2 jvj ; 2 dt 1;2 (r p; Pv) D 0 ;

we deduce 1 d 2 jvj C kPvk22 D (v  rv; Pv) : 2 dt 1;2

(61)

Since the right-hand side of this equation need not be zero, we get that, unlike (58), the nonlinear term does contribute to (61). In addition, since the sign of this contribution is basically unknown, in order to obtain useful estimates we have to increase it appropriately. To this end, we recall the validity of the following inequalities 8 1 1 ˆ c1 kuk22 kPuk22 ˆ ˆ ˆ ˆ ˆ ˆ if n D 2 ; for all u 2 L2 (˝) ; ˆ ˆ ˆ ˆ < with Pu 2 L2 (˝) and uj@˝ D 0 ; kuk1  (62) 1 1 ˆ 2 2 ˆ juj kPuk c ˆ 2 1;2 2 ˆ ˆ ˆ ˆ ˆ if n D 3 ; for all u 2 D1;2 ˆ 0 (˝) ; ˆ ˆ : 2 with Pu 2 L (˝) ; where c i D c i (˝) > 0, i D 1; 2. These relations follow from the Sobolev embedding theorems along with Lemma 2. We shall sketch a proof in the case n D 3. By the property of the projection operator P, u satisfies the assumptions of Lemma 2 with g :D Pu, and, consequently, we have, on the one hand, that u 2 W 2;2 (˝), and, on the other hand, kuk2;2  c kPuk2 ;

(63)

1027

1028

Navier–Stokes Equations: A Mathematical Analysis

with c D c(˝) > 0. We now recall that there exists an extension operator E : u 2 W 2;2 (˝) 7! E(u) 2 W 2;2 (R3 ) such that kE(u)k k;2  C k kuk k;2 ;

k D 0; 1; 2 ;

(64)

see Chap. VI, Theorem 5 in [88]. Next, take ' 2 From the identity (j'j2 ) D 2('  ' C jr'j2 ) we have the representation C01 (R3 ).

j'(x)j2 D 

1 2

Z R3

'(y)  '(y) C jr'(y)j2 dy : (65) jx  yj

namely, we do not know if  D 1. Integrating both sides of (66) from 0 to t < , and taking into account (67) we find that Z

t 0

kPv(s)k22 ds  M1 (˝; ; t; kv 0 k1;2 ) ; for all t 2 [0; ) ;

with M 1 continuous in t. From (68), (67), (60), and 63 one can then show that v 2 L1 (0; t; W01;2 (˝)) \ L2 (0; t; W 2;2 (˝)) ;

Using Schwarz inequality on the right-hand side of (65) along with the classical Hardy inequality § II.5 in [31]: Z R3

j'(y)j2 d y  4j'j21;2 ; jx  yj2 1

1

2 2 j'j2;2 . Since C01 (R3 ) is we recover k'k1  (2/)1/2 j'j1;2 2;2 3 dense in W (R ), from this latter inequality we deduce, in particular, 1

1

2 2 jE(u)j2;2 ; kuk1  (2/)1/2 jE(u)j1;2

which, in turn, by (64) and Poincaré’s inequality (18) implies that 1 2

1 2

kuk1  c5 juj1;2 kuk2;2 ; with c5 D c5 (˝) > 0. Equation (62)2 then follows from this latter inequality and from (63). (For the proof of (62) in more general domains, as well as in domains with less regularity, we refer to [9,66,98]. I am not aware of the validity of (62) in an arbitrary (smooth) domain.) We now employ (62) and 29 into (61) to obtain ( c3 kvk22 jvj41;2 if n D 2 ; 1 d 2  2 jvj1;2 C kPvk2  (66) 2 dt 2 if n D 3 ; c4 jvj61;2 where c i D c i (˝; ) > 0, i D 3; 4. Thus, observing that, from (59), kv(t)k2  kv 0 k2 , if we assume, further, that v 0 2 D1;2 0 (˝), from the previous differential inequality, we obtain the following uniform bound jv(t)j1;2  M(˝; ; t; kv 0 k1;2 ) ; for all t 2 [0; ) ;

and some   1/(K jv 0 j˛1;2 ) ;

(67)

where M is a continuous function in t and K D 8c3 , ˛ D 2 if n D 2, while K D 4c4 , ˛ D 4 if n D 3. Equation (67) provides the second a priori estimate. Notice that, unlike (59), we do not know if in (67) we can take t arbitrary large,

(68)

t 2 [0; ) :

(69)

A third a priori estimate, on the time derivative of v, can be formally obtained by dot-multiplying both sides of (7)1 and by integrating by parts over ˝. By using arguments similar to those leading to (61) we find  2    @v   d 2  D  v  rv; @v ; jvj1;2 C   @t  2 dt @t 2

(70)

and so, employing Hölder inequality on the right-hand side of (62) along with 29, (67) and (68) we show the following estimate Z t  2  @v    ds  M2 (˝; ; t; kv 0 k1;2 ) ;  @s  0 2 for all t 2 [0; ) ;

(71)

with M 2 continuous in t. This latter inequality implies that @v 2 L2 (0; t; L2 (˝)) ; @t

t 2 [0; ) :

(72)

Existence, Uniqueness, Continuous Dependence, and Regularity Results We shall now use estimates (59), (67), (68), (71) along a suitable approximate solution constructed by the finite-dimensional (Galerkin) method to show existence to (7)–(8)hom in an appropriate function class. We shall briefly sketch the argument. Similarly to the steady-state case, we look for an approximate solution to P (7)–(8)hom of the form v N (x; t) D N iD0 c i N (t) i , where 2 f i g is a base of L (˝) constituted by the eigenvectors of the operator P, namely,  

i

D i

i

C r˚ i ;

div

i

D 0 in ˝ ;

i j@˝ D 0 ;

i 2N;

(73)

where i are the corresponding eigenvalues. The coefficients c i N (t) are requested to satisfy the following system

Navier–Stokes Equations: A Mathematical Analysis

it depends continuously on v 0 in the norm of L2 (˝), in the time interval [0; ).

of ordinary differential equations 

@v N ; @t

 k

C (v N  rv N ;

k)

D (v N ;

k) ;

k D 1; : : : ; N ;

(74)

with initial conditions c i N (0) D (v 0 ; i ), i D 1; : : : ; N. Multiplying both sides of (74), in the order, by ckN , by  k c k N and by dc k N /dt and summing over k from 1 to N, we at once obtain, with the help of (73) and of Lemma 1, that v N satisfies (59), (66), and (70). Consequently, v N satisfies the uniform (in N) bounds (58) (evaluated at s D 0) (67), (68), and (71). Employing these bounds together with, more or less, standard limiting procedures, we can show the existence of a field v in the classes defined by (69) and (72) satisfying the relation 



@v C v  rv  v; ' D 0 ; @t for all ' 2 D(˝) and a.a. t 2 [0; ) :

(75)

Proof Let (v; p) and (v C u; p C p1 ) be two solutions corresponding to data v 0 and v 0 C u0 , respectively. From (7)– (8)hom we then find @u C u  ru C v  ru C u  rv D r(p1 / ) C u ; @t div u D 0 ; a.e. in ˝  (0; ) : (77) Employing the properties of the function v and u, it is not hard to show the following equation 1 d kuk22 C juj21;2 D (u  rv; u) ; 2 dt

(78)

that is formally obtained by dot-multiplying both sides of (77)1 by u, and by using (77)2 and Lemma 1 along with the fact that u has zero trace at @˝. By Hölder inequality and inequalities (14), (18), and (29), we find j(u  rv; u)j  kuk2 kuk4 jvj1;4  c1 kuk2 juj1;2 kvk2;2   c2 kvk22;2 kuk22 C juj21;2 ; 2

Because of (69) and (72), the function involving v in (75) belongs to L2 (˝), a.e. in [0; ), and therefore, in view of the orthogonal decomposition L2 (˝) D L2 (˝) ˚ G(˝) and of the density of D(˝) in L2 (˝), we find p 2 L2 (0; t; W 1;2 (˝)), t 2 [0; ), such that (v; p) satisfies (7)1 for a.a. (x; t) 2 ˝  [0; ). We thus find the following result, basically due to G. Prodi, to whose paper [72] we refer for all missing details; see also Chapter V.4 in [86].

where c1 D c1 (˝) > 0 and c2 D c2 (˝; ) > 0. If we replace this inequality back into (78) we deduce

Theorem 12 (Existence) For every v 0 2 D1;2 0 (˝), there exist v D v(x; t) and p D p(x; t) such that

and R t so, 2by Gronwall’s lemma and by the fact that 0 kv(s)k2;2 ds < 1, t 2 [0; ) (see Theorem 12), we prove the desired result. 

v 2 L1 (0; ; L2 (˝)) \ L2 (0; ; D1;2 0 (˝)) ; 9 1;2 v 2 C([0; t]; D0 (˝)) > > > > > \ L2 (0; t; W 2;2 (˝)) ; > = for all t 2 [0; ) ; @v > 2 L2 (0; t; L2 (˝)) ; > > > @t > > ; p 2 L2 (0; t; W 1;2 (˝)) ;

(76)

with  given in (67), satisfying (7) for a.a. (x; t) 2 ˝  [0; ), and (8)2 (with v 1  0) for a.a. (x; t) 2 @˝  [0; ). Moreover, the initial condition (8)1 is attained in the following sense:

d kuk22  2c2 kvk22;2 kuk22 ; dt

Finally, we have the following result concerning the regularity of solutions determined in Theorem 12, for whose proof we refer to [38] and Theorem 5.2 in [34]. Theorem 14 (Regularity) Let ˝ be a bounded domain of class C 1 . Then, the solution (v; p) constructed in Theorem 12 is of class C 1 (˝¯  (0; )). Remark 7 The results of Theorems 12–14 can be extended to arbitrary domains of Rn , provided their boundary is sufficiently smooth; see [34,50]. Moreover, the continuous dependence result in Theorem 13 can be proved in the stronger norm of D1;2 0 (˝).

lim kv(t)  v 0 k1;2 D 0 :

t!0C

We also have. Theorem 13 (Uniqueness and Continuous Dependence on the Initial Data) Let v 0 be as in Theorem 12. Then the corresponding solution is unique in the class (76). Moreover,

Times of Irregularity and Resolution of Problems 1 and 2 in 2D Theorems 12–14 furnish a complete and positive answer to both Problems 1 and 2, provided we show that  D 1. Our next task is to give necessary and sufficient conditions for this latter situation to occur. To this end, we give the following.

1029

1030

Navier–Stokes Equations: A Mathematical Analysis

Conversely, if  < 1 and (81) holds, then  is a time of irregularity. Moreover, if  is a time of irregularity, for all t 2 (0; ) the following growth estimates hold 8 C ˆ ˆ if n D 2 ; <  t 2 jv(t)j1;2  (82) C ˆ ˆ if n D 3 ; :p  t

Definition 6 Let (v; p) be a solution to (7)–(8)hom in the class (76). We say that  is a time of irregularity if and only if (i)  < 1, and (ii) (v; p) can not be continued, in the class (76), to an interval [0; 1 ) with 1 > . If  is a time of irregularity, we expect that some norms of the solution may become infinite at t D , while being bounded for all t 2 [0; ). In order to show this rigorously, we premise a simple but useful result. Lemma 7 Assume that v is a solution to (7)–(8)hom in the class (76) for some  > 0. Then, jv(t)j1;2 < 1, for all t 2 [0; ). Furthermore, for all q 2 (n; 1], and all t 2 [0; ) Z t 2 n kv(s)krq ds < 1 ; C D 1: r q 0 Proof The proof of the first statement is obvious. Moreover, by the Sobolev embedding theorem (see Theorem at p. 125 in [70]) we find, for n D 2, 2/q

12/q

kvkq  c1 kvk2 jvj1;2

;

q 2 (2; 1)

(79)

and kvk1  c2 kvk2;2 ; with c1 D c1 (q) > 0, c2 D c2 (˝) > 0 while, if n D 3, (6q)/2q

kvkq  c3 ; kvk2 2q/(q3)

kvkq

3(q2)/2q

jvj1;2

(q6)/(q3)

 c4 kvk2;2

;

;

(80)

if q 2 (6; 1] ;

Z 0

t2[0;]

t

kv(s)k22;2 ds < 1 ; t 2 [0; ) ;

the lemma follows by noting that (q  6)/(q  3) < 2 for q > 3.  We shall now furnish some characterization of the possible times of irregularity in terms of the behavior, around them, of the norms of the solution considered in Lemma 7. Lemma 8 (Criteria for the Existence of a Time of Irregularity) Let (v; p) be a solution to (7)–(8)hom in the class (76) for some  2 (0; 1]. Then, the following properties hold: (i)

If  is a time of irregularity, then lim jv(t)j1;2 D 1 :

t! 

Proof (i)

Clearly, if  < 1 and (81) holds, then  is a time of irregularity. Conversely, suppose  is a time of irregularity and assume, by contradiction, that there exists a sequence ft k g in [0; ) and M > 0, independent of k, such that jv(t k )j1;2  M :

Since v(t k ) 2 D1;2 0 (˝), by Theorem 12 we may construct a solution (¯v ; p¯) with initial data v(t k ), in a time interval [t k ; t k C   ) where see (67)    A/jv(t k )j˛1;2  AM ˛   ;

where c i D c i (˝; q) > 0, i D 3; 4. Since, by (76), sup kv(t)k2 C jv(t)j1;2 C

Conversely, if  < 1 and (83) holds for some q D q¯ 2 (n; 1], then,  is a time of irregularity. (iii) If n D 3, there exists K D K(˝; ) > 0 such that, if kv 0 k2 jv0 j1;2 < K, then  D 1.

tk !  ;

if q 2 [2; 6] ;

(6Cq)/(q3)

jvj1;2

with C D C(˝; ) > 0. (ii) If  is a time of irregularity, then, for all q 2 (n; 1], Z  2 n kv(s)krq ds D 1 ; C D 1: (83) r q 0

and A depends only on ˝ and . By Theorem 12, v¯ belongs to the class (76) in the time interval [t k ; t k C  ], with  independent of k and, by Theorem 13, v¯ must coincide with v in the time interval [t k ; ). We may now choose t k  0 such that 0 C  > , contradicting the assumption that  is time of irregularity. We next show (82) when n D 3, the proof for n D 2 being completely analogous. Integrating (66) (with n D 3) we find 1 1   c4 (s  t) ; 4 jv(t)j1;2 jv(s)j41;2

0 < t < s < :

Letting s !  and recalling (81), we prove (82). (ii) Assume that  is a time of irregularity. Then, (82)2 holds. Now, by the Sobolev embedding theorems, one can show that (see [proof of Lemma 5.4] in [34]) 2q/(qn)

(v  rv; Pv)  C kvkq (81)

˛ D 2(n  1) ;

 kPvk22 ; 2 for all q 2 (n; 1] ;

jvj21;2 C

Navier–Stokes Equations: A Mathematical Analysis

where C D C(˝; ) > 0. If we replace this relation into (61) and integrate the resulting differential inequality from 0 to t < , we find jv(t)j21;2



jv 0 j21;2 expf2C

Z

t 0

kv(s)krq dsg ;

for all t 2 [0; ) :

(84)

If condition (83) is not true for some q 2 (n; 1], then (84) evaluated for that particular q, would contradict (82). Conversely, assume (83) holds for some q D q¯ 2 (n; 1], but that, by contradiction, the solution of Theorem 12 can be extended to [0; 1 ) with 1 > . Then, by Lemma 7 we would get the invalid¯ and the proof of (ii) ity of condition (83) with q D q, is completed. (iii) By integrating the differential inequality (66)2 , we find jv(t)j21;2 

jv 0 j21;2 ; Rt 1  2c4 jv 0 j21;2 0 jv(s)j21;2 ds

jv 0 j21;2 1  c5 jv0 j21;2 kv 0 k22

;

t 2 [0; ) :

t 2 [0; ) ;

with c5 D c5 (˝; ) > 0, which shows that (82)2 can not occur if the initial data satisfy the imposed “smallness” restriction.  A fundamental consequence of Lemma 8 is that, in the case n D 2, a time of irregularity can not occur. In fact, for example, (82) is incompatible with the fact that v 2 L2 (0; ; D1;2 0 (˝)) (see (77)1 ). We thus have the following theorem, which answers positively both Problems 1 and 2 in the 2D case. Theorem 15 (Resolution of Problems 1 and 2 in 2D) Let ˝ R2 . Then, in Theorems 12–14 we can take  D 1. The conclusion of Theorem 15 also follows from Lemma 8 (ii). In fact, in the case n D 2, by (79) we at once find that L2q/(q2) (0; T : L q (˝)) L1 (0; T; L2 (˝)) \ L2 (0; T; D1;2 0 (˝)) ; for all T > 0 and all q 2 (2; 1) : (85) so that, from (76)1 , we deduce Z  2q/(2q) kv(t)kq dt < 1 ; 0

Z

 0

for all q 2 (2; 1) :

kv(t)krq dt < 1 ; 2 3 C D 1; r q

some q 2 (3; 1] :

(86)

However, from (80)1 and from (76), it is immediately verified that, in the case n D 3, the solutions constructed in Theorem 12 satisfy the following condition Z

 0

Thus, by (59) with s D 0 in this latter inequality we find jv(t)j21;2 

which, by Lemma 8 (ii), excludes the occurrence of time of irregularity. Unfortunately, from all we know, in the case n D 3, we can not draw the same conclusion. Actually, in such a case, (82)2 and (77)1 are no longer incompatible. Moreover, from Lemma 8 (ii) it follows that a sufficient condition for  not to be a time of irregularity is that

kv(t)krq dt < 1 ; 2 3 1 C D1C ; r q 2

all q 2 [2; 6] :

(87)

Therefore, in view of Lemma 8 (iii), the best conclusion we can draw is that for 3D flow Problems 1 and 2 can be positively answered if the size of the initial data is suitably restricted. Remark 8 In the case n D 3, besides (86), one may furnish other sufficient conditions for the absence of a time of irregularity. We refer, among others, to the papers [6,8,15,18,56,57,69,80]. In particular, we would like to direct attention to the work [18,80], where the difficult borderline case q D n D 3 in condition (86) is worked out by completely different methods than those used here. Open Question In the case n = 3, it is not known whether or not condition (86) (or any of the other conditions referred to in Remark 8) holds along solutions of Theorem 12. Less Regular Solutions and Partial Regularity Results in 3D As shown in the previous section, we do not know if, for 3D flow, the solutions of Theorem 12 exist in an arbitrarily large time interval, without restricting the magnitude of the initial data: they are local solutions. However, following the line of thought introduced by J. Leray [62], we may extend them to solutions defined for all times, for initial data of arbitrary magnitude, namely, to global solutions, but belonging to a functional class, C , a priori less regular than that given in (76) (weak solutions). Thus, if, besides existence, we could prove in C also uniqueness and

1031

1032

Navier–Stokes Equations: A Mathematical Analysis

continuous dependence, Problem 1 would receive a positive answer. Unfortunately, to date, the class where global solutions are proved to exist is, in principle, too large to secure the validity of these latter two properties and some extra assumptions are needed. Alternatively, in relation to Problem 2, one may investigate the “size” of the space-time regions where these generalized solutions may (possibly) become irregular. As a matter of fact, singularities, if they at all occur, have to be concentrated within “small” sets of space-time. Our objective in this section is to discuss the above issues and to present the main results. For future purposes, we shall present some of these results also in space dimension n D 2, even though, as shown in Theorem 15, in this case, both Problems 1 and 2 are answered in the affirmative. Weak Solutions and Related Properties We begin to introduce the corresponding definition of weak solutions in the sense of Leray–Hopf [54,62]. By formally dot-multiplying through both sides of (7) by ' 2 D(˝) and by integrating by parts over ˝, with the help of 12 we find d (v(t); ') C (rv(t); r') C (v(t)  rv(t); ') D 0 ; dt for all ' 2 D(˝) : (88) Definition 7 Let ˝ Rn , n D 2; 3. A field v : ˝  (0; 1) 7! Rn is a weak solution to (7)–(8)hom if and only if: (i) v 2 L1 (0; T; L2 (˝)) \ L2 (0; T; D1;2 0 (˝)) , for all T > 0; (ii) v satisfies (88) for a.a. t  0 , and (iii) lim (v(t)  v 0 ; ') D 0, for all ' 2 D(˝) . t!0C

A. Existence The proof of existence of weak solutions is easily carried out, for example, by the finite-dimensional (Galerkin) method indicated in Subsection V.2.2. This time, however, along the “approximate” solutions v N to (74), we only use the estimate corresponding to the energy equality (59). We thus obtain kv N (t)k22 C 2

Z

t s

jv N ( )j21;2 d D kv N (s)k22 ; 0  s  t:

(89)

As we already emphasized, the important feature of this estimate is that it holds for all t  0 and all data v0 2 L2 (˝). From (89) it follows, in particular, that the sequence fv N g is uniformly bounded in the class of functions specified in (i) of Definition 7. Using this fact together with classical weak and strong compactness arguments in (74), we can show the existence of at least one subsequence converging, in suitable topologies, to a weak solution v. We then

have the following result for whose complete proof we refer, e. g., to Theorem 3.1 in [34]. Theorem 16 For any v 0 2 L2 (˝) there exists at least one weak solution to (7)–(8)hom . This solution verifies, in addition, the following properties. (i) The energy inequality: kv(t)k22

Z

t

C 2 s

jv( )j21;2 d  kv(s)k22 ;

for a.a. s 2 [0; 1); 0 included, and all t  s : (90) (ii) lim t!0C kv(t)  v 0 k2 D 0 . B. On the Energy Equality In the case n D 3, weak solutions in Theorem 16 only satisfy the energy inequality (90) instead of the energy equality [see (59)]. (For the case n D 2, see Remark 9.) This is an undesired feature that is questionable from the physical viewpoint. As a matter of fact, for fixed s  0, in time intervals [s; t] where (90) were to hold as a strict inequality, the kinetic energy would decrease by an amount which is not only due to the dissipation. From a strictly technical viewpoint, this happens because the convergence of (a subsequence of) the sequence fv N g to the weak solution v can be proved only in the weak topology of L2 (0; T; D1;2 0 (˝)) and this only ensures that, as N ! 1, the second term on the left-hand side of (89) tends to a quantity not less than the one given by the second term on the left-hand side of (90). One may think that this circumstance is due to the special method used for constructing the weak solutions. Actually, this is not the case because, in fact, we have the following. Open Question If n D 3, it is not known if there are solutions satisfying (90) with the equality sign and corresponding to initial data v 0 2 L2 (˝) of unrestricted magnitude. Remark 9 A sufficient condition for a weak solution, v, to satisfy the energy equality (59) is that v 2 L4 (0; t; L4 (˝)), for all t > 0 (see Theorem 4.1 in [34]). Consequently, from (85) it follows that, if n D 2, weak solutions satisfy (59) for all t > 0. Moreover, from (80)1 , we find that the solutions of Theorem 12, for n D 3, satisfy the energy equality (59), at least for all t 2 [0; ). Remark 10 For future purposes, we observe that the definition of weak solution and the results of Theorem 16 can be extended easily to the case when f 6 0. In fact, it is enough to change Definition 7 by requiring that

Navier–Stokes Equations: A Mathematical Analysis

v satisfies the modification of (88) obtained by adding to its right-hand side the term ( f ; '). Then, if f 2 L2 (0; T; D1;2 (˝)), for all T > 0, one can show the ex0 istence of a weak solution satisfying condition (ii) of TheoremR16 and the variant of (90) obtained by adding the t term 0 ( f ; v) ds on its right-hand side. C. Uniqueness and Continuous Dependence The following result, due to Serrin (Theorem 6 in [83]) and Sather (Theorem 5.1 in [75]), is based on ideas of Leray [§ 32] in [62] and Prodi [71]. A detailed proof is given in Theorem 4.2 in [34]. Theorem 17 Let v, u be two weak solutions corresponding to data v 0 and u 0 . Assume that u satisfies the energy inequality (90) with s D 0, and that v 2 Lr (0; T; L q (˝)) ;

for some q 2 (n; 1] n 2 such that C D 1 : r q

(91)

Then, kv(t)  u(t)k22  Ckv 0  u 0 k22 expf

Z

t 0

kv( )krq d g ;

for all t 2 [0,T]g where C D C(˝; ) > 0. Thus, in particular, if v 0 D u0 , then v D u a.e. in ˝  [0; T]. Remark 11 If n D 2, from (85) and Remark 9 we find that every weak solution satisfies the assumptions of Theorem 17. Therefore, in such a case, every weak solution is unique in the class of weak solutions and depends continuously upon the data. Furthermore, if n D 3, the uniqueness result continues to hold if in condition (91), we take q D n D 3; see [55,87]. Open Question While the existence of weak solutions u satisfying the hypothesis Theorem 17 is secured by Theorem 16, in the case n D 3 it is not known if there exist weak solutions having the property stated for v. [In principle, as a consequence of Definition 7(i) and (80)1 , v only satisfies (87).] Consequently in the case n D 3 uniqueness and continuous dependence in the class of weak solutions remains open, and so does the resolution of Problem 1. Remark 12 As a matter of fact, weak solutions possess more regularity than that implied by their very definition. Actually, if n D 2, they are indeed smooth (see Remark 13). If n D 3, by means of sharp estimates for solutions to the linear problem obtained from (7)–(8) by neglecting the nonlinear term v  rv (Stokes problem) one

can show that every corresponding weak solution satisfies the following additional properties (see Theorem 3.1 in [43], Theorem 3.4 in [87]) @v 2 L l (ı; T; Ls (˝)) ; v 2 L l (ı; T; W 2;s (˝)) ; @t for all T > 0 and all ı 2 (0; T) ; where the numbers l; s obey the following conditions 2 3 C  4; l s

l 2 [1; 2] ;

 s 2 1; 32 :

Moreover, there exists p 2 L l (ı; T; W 1;s (˝)) \ L l (ı; T; L3s/(3s) (˝)) such that the pair (v; p) satisfies (7) for a.a. (x; t) 2 ˝  (0; 1). If, in addition, v 0 lies in a sufficiently regular subspace of L2 (˝), we can take ı D 0. However, the above properties are still not enough to ensure the validity of condition (91). Weak solutions enjoying further regularity properties are constructed in [14,67] and Theorem 3.1 and Corollary 3.2 in [24]. D. Partial Regularity and “Suitable” Weak Solutions A problem of fundamental importance is to investigate the set of space-time where weak solutions may possibly become irregular, and to give an estimate of how “big” this set can be. To this end, we recall that, for a given S RdC1 , d 2 N [ f0g, and  2 (0; 1), the -dimensional (spherical) Hausdorff measure H  of S is defined as H  (S) D lim Hı (S) ; ı!0

P where Hı (S) D inf i ri , the infimum being taken over all at most countable coverings fB i g of S of closed balls B i Rd of radius ri with r i < ı; see, e. g., [19]. If d 2 N, the -dimensional parabolic Hausdorff measure P  of S is defined as above, by replacing the ball Bi with a parabolic cylinder of radius ri : Qr i (x; t) D f(y; s) 2 Rd R : jy  xj < r i ; js  tj < r2i g : (92) In general, it is H  (S)  C P  (S), C > 0; see § 2.10.1 in [19] for details. The following lemma is a direct consequence of the preceding definition. Lemma 9 For any S RdC1 , d 2 N [ f0g (respectively, d 2 N), we have H  (S) D 0 (respectively, P  (S) D 0) if and only if, for each ı > 0, S can be covered by closed balls fB i g (respectively, parabolic cylinders fQ i g) of radii ri , P  i 2 N, such that 1 iD1 r i < ı .

1033

1034

Navier–Stokes Equations: A Mathematical Analysis

We begin to consider the collection of times where weak solutions are (possibly) not smooth and show that they constitute a very “small” region of (0; 1). Specifically, we have the following result, basically, due to Leray pp. 244– 245 in [62] and completed by Scheffer [76]. Theorem 18 Let ˝ be a bounded domain of class C 1 . Assume v is a weak solution determined in Theorem 16. Then, there exists a union of disjoint and, at most, countable open time intervals T D T (v) (0; 1) such that: (i) (ii) (iii) (iv)

v is of class C 1 in ˝¯  T , There exists T  2 (0; 1) such that T  (T  ; 1); If v 0 2 D1;2 0 (˝) then T  (0; T1 ) for some T1 > 0 ; Let (s; ) be a generic bounded interval in T (v) and suppose v 62 C 1 (˝¯  (s; 1 )), 1 > . Then, both following conditions must hold jv(t)j21;2 Z lim

t!

t

C  ; (  t)1/2 2q/(q3)

kv(s)kq

ds D 1 ;

t 2 (s; ) and for all q > 3 ;

s

(93) where C D C(˝; ) > 0 ; (v) The 12 -dimensional Hausdorff measure of I (v) :D (0; 1)  T is zero; Proof By (90) we may select T  > 0 with the following properties: (a) kv(T  )k2 jv(T  )j1;2 < K, and (b) the energy inequality (90) holds with s D T  , where K is the constant introduced in Lemma 8 (iii). Let us denote by v˜ the solution of Theorem 12 corresponding to the data v(T  ). By Lemma 8 (iii), v˜ exists for all times t  T  and, by Theorem 14, it is of class C 1 in ˝  (T  ; 1). By Lemma 7 and by Theorem 17 we must have v  v˜ in ˝  (T  ; 1), and part (ii) is proved. Next, denote by I the set of those t 2 [0; T  ) such that (a) kv(t)k1;2 < 1, and (b) the energy inequality (90) holds with s 2 I. Clearly, [0; T  ]  I is of zero Lebesgue measure. Moreover, for every t0 2 I we can construct in (t0 ; t0 C T(t0 )) a solution v˜ assuming at t0 the initial data v(t0 ) (2 D1;2 0 (˝)); see Theorem 12. From Theorem 14, Lemma 7, and Theorem 17, we know that v˜ is of class C 1 in ˝  (t0 ; t0 C T(t0 )) and that it coincides with v, since this latter satisfies the energy inequality with s D t0 . Furthermore, if v 0 2 D1;2 0 (˝), then 0 2 I. Properties (ii)–(iv) thus follow with T  S  i2I (s i ;  i ) [ (T ; 1), where (s i ;  i ) are the connected components in I. Notice that (s i ;  i ) [0; T  ] ;

for all i 2 I ;

(s i ;  i ) \ (s j ;  j ) D ; ;

i ¤ j;

(94)

and that, moreover, the (1-dimensional) Lebesgue measure of I :D T  (0; 1) is 0. Finally, property (iv) is an immediate consequence of Lemma 8 and Theorem 14. It remains to show (v). From (iv) and (90) we find X

( i  s i )1/2  1/(2C)

i2I

XZ

si i

i2I

krv()k22 dt  kv 0 k22 /(4C) :

Thus, for every ı > 0 we can find a finite part Iı of I such that X

X

( i  s i ) < ı;

i…Iı

( i  s i )1/2 < ı :

(95)

i…Iı

By (94)1 , [ i2I (s i ;  i ) [0; T  ] and so the set [0; T  ]  [ i2Iı (s i ;  i ) consists of a finite number of disjoint closed intervals Bj , j D 1; : : : ; N. Clearly, N [

B j  I (v):

(96)

jD1

By (94)2 , each interval (s i ;  i ), i … Iı , is included in one and only one Bj . Denote by I j the set of all indices i satisfying B j  (s i ;  i ). We thus have 0 I D Iı [ @ 0 Bj D @

N [

1 IjA ;

jD1

[

1

(97) 

 (s i ;  i )A [ B j \ I (v) :

i2I j

Since I has zero Lebesgue measure, from(97)2 we have P diam B j D i2I j ( i  s i ). Thus, by (95) and (97)1 , diam B j 

X

( i  s i ) < ı

(98)

i…Iı

and, again by (95) and (97)1 , N X

(diam B j )1/2 

jD1

N X

0 @

jD1



X

X

11/2 ( i  s i )A

i2I j

( i  s i )1/2 < ı :

(99)

i…Iı

Therefore, property (v) follows from (96), (98), (99), and Lemma 9. 

Navier–Stokes Equations: A Mathematical Analysis

Remark 13 If ˝ is of class C 1 , from 85, Remark 11 and from Theorem 18 (v) we at once obtain that, for n D 2, every weak solution is of class C 1 (˝¯ (0; t)), for all t > 0. We shall next analyze, in more detail, the set of points where weak solutions may possibly lose regularity. In view of Remark 13, we shall restrict ourselves to consider the case n=3 only. Let (s; ) be a bounded interval in T (v) and assume that, at t D , the weak solution v becomes irregular. We may wish to estimate the spatial set ˙ D ˙ ()  ˝ where v() becomes irregular. By defining ˙ as the set of x 2 ˝ where v(x; ) is not continuous, in the case ˝ D R3 , Scheffer has shown that H 1 (˙ ) < 1 [77]. More generally, one may wish to estimate the “size” of the region of space-time where points of irregularity (appropriately defined, see Definition 9 below) may occur. This study, initiated by Scheffer [77,78] and continued and deepened by Caffarelli, Kohn and Nirenberg [11], can be performed in a class of solutions called suitable weak solutions which, in principle, due to the lack of an adequate uniqueness theory, is more restricted than that of weak solutions. Definition 8 A pair (v; p) is called a suitable weak solution to (7)–(8)hom if and only if: (i) v satisfies Definition 7(i) and p 2 L3/2 ((0; T); L3/2 (˝)), for all T > 0 ; (ii) (v; p) satisfies d (v(t); ) C (rv(t); r ) C (v(t)  rv(t); ) dt (p(t); div ) D 0 ; for all 2 C01 (˝) ; and (iii) (v; p) obeys the following localized energy inequality Z TZ 2 jrvj2  dxdt 0

Z 

0

T

Z

˝

˝

fjvj2 (

@ C ) C (jvj2 C 2p)v  rg dxdt ; @t

for all non-negative  2 C01 (˝  (0; T)). Remark 14 By taking l D 3/2, s D 9/8 in Remark 12, it follows that every weak solution, corresponding to sufficiently regular initial data, matches requirements (i) and (ii) of Definition 8 (recall that ˝ is bounded). However, it is not known, to date, if it satisfies also condition (iii). Moreover, it is not clear if the finite-dimensional (Galerkin) method used for Theorem 16 is appropriate to construct solutions obeying such a condition (see, [47] for a partial answer). Nevertheless, by using different methods, one can show the existence of at least one weak solution satisfying the properties stated in Theorem 16, and

which, in addition, is a suitable weak solution; (see [5], Theorem A.1 in [11], and Theorem 2.2 in [64]). Definition 9 A point P :D (x; t) 2 ˝T :D ˝  (0; T) is called regular for a suitable weak solution (v; p), if and only if there exists a neighborhood, I, of P such that v is in L1 (I). A point which is not regular will be called irregular. Remark 15 The above definition of a regular point is reinforced by a result of Serrin [82], from which we deduce that, in the neighborhood of every regular point, a suitable weak solution is, in fact, of class C 1 in the space variables. The next result is crucial in assessing the “size” of the set of possible irregular points in the space-time domain. For its proof, we refer to [64], Theorem 2.2 in [60], Proposition 2 in [11]. We recall that Qr (x; t) is defined in (92). Lemma 10 Let (v; p) be a suitable weak solution and let (x; t) 2 ˝T . There exists K > 0 such that, if Z jrv(y; s)j2 dyds < K ; (100) lim sup r1 r!0

Q r (x;t)

then (x; t) is regular. Now, let S D S(v; p)  ˝ (0; T) be the set of possible irregular points for a suitable weak solution (v; p), and let V be a neighborhood of S. By Lemma 10 we then have that, for each ı > 0, there is Qr (x; t) V with r < ı, such that Z 1 jrv(y; s)j2 d y ds > r : (101) K Q r (x;t)

Let Q D fQr (x; t)g be the collection of all Q satisfying this property. Since ˝  (0; T) is bounded, from Lemma 6.1 in [11] we can find an at most countable, disjoint subfamily of Q, fQr i (x i ; t i )g, such that S [ i2I Q5r i (x i ; t i ). From (101) it follows, in particular, that X i2I

ri  K

1

XZ i2I

Q r i (x i ;t i )

jrv(y; s)j2 dy ds  K 1

Z

jrvj2 :

(102)

V

Since ı is arbitrary, 102 implies, on the one hand, that S is of zero Lebesgue measure and, on the other hand, that Z 5 P 1 (S)  jrvj2 ; K V for every neighborhood V of S. Thus, by the absolute continuity of the Lebesgue integral, from this latter inequality and from Lemma 9 we have the following Theorem B in [11].

1035

1036

Navier–Stokes Equations: A Mathematical Analysis

Theorem 19 Let (v; p) be a suitable weak solution and let S D S(v; p)  ˝  (0; T) be the corresponding set of possible irregular points. Then P 1 (S) D 0. Remark 16 Let Z D Z(v; p)

ting u :D v  v¯ , P :D p  p¯, from (7)–(10) we find 9 @u > > C u  ru C u  r v¯ > = @t ¯ C v  ru D rP C u ;> > > ; div u D 0

u(x; 0) D v 0 (x)  v¯ (x) ; x 2 ˝ ; u D 0 at @˝  (0; T) :

:D ft 2 (0; T) : (x; t) 2 S(v; p) ; for some x 2 ˝g : Clearly, t 2 Z if and only if v becomes essentially unbounded around (x; t), for some x 2 ˝. Namely, for any M > 0 there is a neighborhood of (x; t), I M , such that jv(y; s)j > M for a.a. (y; s) 2 I M . From Theorem 18(ii) we deduce at once that Z(v; p) I (v), where I (v) is the set of all possible times of irregularity; see Definition 6. Thus, from Theorem 18(i) we find H 1/2 (Z) D 0. Remark 17 There is a number of papers dedicated to the formulation of sufficient conditions for the absence of irregular points, (x; t), for a suitable weak solution, when x is either an interior or a boundary point. In this latter case, the definition of the regular point as well as that of the suitable weak solution must be, of course, appropriately modified. Among others, we refer to [11,48,49,64,81].

Long-Time Behavior and Existence of the Global Attractor Suppose a viscous liquid moves in a fixed spatial bounded domain, ˝, under the action of a given time-independent driving mechanism, m, and denote by  > 0 a non-dimensional parameter measuring the “magnitude” of m. To fix the ideas, we shall assume that the velocity field of the liquid, v 1 , vanishes at @˝, for each time t  0. This assumption is made for the sake of simplicity. All main results can be extended to the case v 1 6 0, provided v 1 (x; t)  nj@˝ D 0, for all t  0, where n is the unit normal on @˝. We then take m to be a (non-conservative) time-independent body force, f 2 L2 (˝) with  k f k2 . We shall denote by (7)–(8)Hom the initial-boundary value problem (7)–(8) with v 1  0. As we know from Theorem 3 and Remark 5, if  is “sufficiently” small, less than c , say, there exists one and only one (steady-state) solution, (¯v ; p¯) to the boundary-value problem (9)–(10) (with v   0). Actually, it is easy to show that, with the above restriction on , every solution to the initial-boundary value problem (8)–(8)Hom , belonging to a sufficiently regular function class and corresponding to the given f and to arbitrary v 0 2 L2 (˝), decays exponentially fast in time to (¯v ; p¯). In fact, by set-

in ˝  (0; T) ;

(103) If we formally dot-multiply through both sides of (103)1 by u, integrate by parts over ˝ and take into account 12 and Lemma 1, we obtain 1 d kuk22 C juj21;2 D (u  r v¯ ; u) : 2 dt

(104)

From Lemma 1, (21), and (18) we find j(u  r v¯ ; u)j  c1 juj21;2 j¯v j1;2 , with c1 D c1 (˝) > 0. Thus, by Remark 5, it follows that j(u  r v¯ ; u)j 

c2 k f k2 juj21;2 ; 

with c2 D c2 (˝) > 0, which, in turn, once replaced in (104), furnishes   c2 1 d kuk22 C   k f k2 juj21;2  0 : 2 dt  Therefore, if :D  (c2 /)k f k2 > 0, from (18) and from the latter displayed equation we deduce ku(t)k22  ku(0)k22 e 2( /C P ) t ;

(105)

which gives the desired result. Estimate (105) can be read, in “physical terms” in the following way: after a certain amount of time (depending on how close is to 0, namely, on how close  is to c ), the transient motion will die out exponentially fast and the “true” dynamics of the fluid will be described by the unique steady-state flow corresponding to the given force f . From a mathematical point of view, the steady-state (¯v ; p¯ ) is, in a suitable function space, a one-point set which is invariant under the flow. Now, let us increase  higher and higher beyond c . Following Eberhard Hopf [53], we expect that, after a while, the transient motion will yet die out, and that the generic flow will approach a certain manifold, M D M() which need not reduce to a single point. (There are, however, explicit examples where M() remains a single point for any  > 0; see [65].) Actually, in principle, the structure of M can be very involved. Nevertheless, we envisage that M is still invariant under the flow, and that it is M where, eventually, the “true” dynamics of the liquid will take place. For obvious reasons, the manifold M is called the global attractor.

Navier–Stokes Equations: A Mathematical Analysis

The existence of a global attractor and the study of the dynamics of the liquid on it, could be of the utmost importance in the effort of formulating a mathematical theory of turbulence. Actually, as is well known, if the magnitude of the driving force becomes sufficiently large, the corresponding flow becomes chaotic and the velocity and pressure of the liquid exhibit large and completely random variation in space and time. According to the ideas proposed by Smale [85] and by Ruelle and Takens [74], this chaotic behavior could be explained by the existence of a very complicated global attractor, where, as mentioned before, the ultimate dynamics of the liquid occurs. Existence of the Global Attractor for Two-Dimensional Flow, and Related Properties Throughout this section we shall consider two-dimensional flow, so that, in particular, ˝ R2 . Let f 2 L2 (˝) and  > 0 be given. Consider the oneparameter family of operators S t : a 2 L2 (˝) 7! S t (a) :D v(t) 2 L2 (˝) ;

Theorem 20 For any f 2 L2 (˝) and  > 0, the corresponding semi-flow fL2 (˝); S t g admits a global attractor M D M( f ; ) which is also connected. Proof In view of Theorem 1.1 in [94] and of Rellich compactness theorem (Theorem II.4.2 in [31]), it suffices to show the following two properties: (a) existence of a bounded absorbing set, and (b) given M > 0, there is t0 D t0 (M; f ; ) > 0 such that kak2  M implies jS t (a)j1;2  C, for all t  t0 and for some C > 0 independent of t. The starting point is the analog of (58) and (61) which, this time, take the form 1 d kv(t)k22 C jv(t)j21;2 D ( f ; v) ; 2 dt 1 d jv(t)j21;2 C kPv(t)k22 D (v  rv; Pv)  ( f ; Pv) 2 dt (106)

t 2 [0; 1)

where v(t) is, at each t, the weak solution to (7)–(8)Hom corresponding to the initial data a; see Remark 10. From Remark 9 and Remark 11 we deduce that the family fS t g t0 defines a (strongly) continuous semigroup in L2 (˝), namely (i) S t1 S t2 (a) D S t1 Ct2 (a) for all a 2 L2 (˝) and all t1 ; t2 2 [0; 1); (ii) S0 (a) D a , and (iii) the map t 7! S t (a) is continuous for all a 2 L2 (˝). L2 (˝), the

correspondDefinition 10 For any given f 2 ing pair fL2 (˝); S t g is called semi-flow associated to (7)– (8)Hom Our objective is to study the asymptotic properties (as t ! 1) of the semi-flow fL2 (˝); S t g. To this end, we need to recall some basic facts. Given A1 ; A2 L2 (˝), we set d(A1 ; A2 ) D sup

the semiflow admits a bounded absorbing set on which the semiflow becomes, eventually, relatively compact; (see Theorem 1.1 in [94]). The following result holds.

inf ku 1  u2 k2 :

u 1 2 A1 u 2 2 A2

¯ 2 . Moreover, we Notice that d(A1 ; A2 ) D 0 ) A1  A denote by A the class of all bounded subset of L2 (˝). Definition 11 B L2 (˝) is called: (i) absorbing iff for any A 2 A there is t0 D t0 (A)  0 such that S t (A)  B, for all t  t0 ; (ii) attracting iff lim d(S t (A); B) D 0, for t!1

all A 2 A ; (iii) invariant iff S t (B) D B, for all t  0 ; (iv) maximal invariant iff it is invariant and contains every invariant set in A ; (v) global attractor iff it is compact, attracting and maximal invariant. Clearly, if a global attractor exists, it is unique. Furthermore, roughly speaking, its existence is secured whenever

By using in these equations the Schwarz inequality, and inequalities (18), 29, (62)1 and (63), we deduce d kv(t)k22 C (/C P )kv(t)k22  F ; dt d jv(t)j21;2  g(t)jv(t)j21;2  F ; dt

(107)

where F :D k f k22 /, g(t) :D c1 kv(t)k22 jv(t)j21;2 , and c1 D c1 (˝) > 0. By integrating (107)1 , we find kS t (a)k22 :D kv(t)k22  kak22 e t/C P C(1e  t/C P )F/C P : (108) Thus, setting B :D f' 2 L2 (˝) : k'k2  (2F/C P )1/2  g ;

(109)

from (107) we deduce that, whenever kak2 < M, there exists t1 D t1 (M; F; ) > 0 such that S t (a) 2 B, which shows that B is absorbing. Next, again from Schwarz inequality and from (98)1 , we obtain Z

tC1

1 k f k2   4 2 1/2 Z tC1

M CP kv(s)k2 ds   1 2 2 t t

jv(s)j21;2 ds 

for all t  t1 ; (110)

which, in particular, implies jv( ¯t )j21;2  12 ;

for some ¯t 2 (t; t C 1) ;

all t  t1 : (111)

1037

1038

Navier–Stokes Equations: A Mathematical Analysis

We next integrate (107)2 from ¯t to t C 1 and use (110), to get R tC1

jv(t C 1)j21;2  ( 12 C F)e

t

 ( 12 C F)e

 ( 12 C F)e

g() d

R 2 tC1 t

2 2 1

;

which proves also property (b).

jv ()j21;2 d

for all t  t1 ; 

Remark 18 In the proof of the previous theorem the assumption of the boundedness of ˝ is crucial, in order to ensure the validity of Rellich’s compactness theorem. However, a different approach, due to Rosa (see Theorem 3.2 in [73]) allows us to draw the same conclusion of Theorem 20 under the more general assumption that in ˝ the Poincaré inequality (18) holds. This happens whenever ˝ is contained in a strip of finite width (like in a flow in an infinite channel). We shall now list some further properties of the global attractor M, for whose proofs we refer to the monographs [26,59,94] A. Smoothness The restriction of the semigroup St to M can be extended to a group, S˜t , defined for all t 2 (1; 1). Therefore, the pair fM; S˜t g constitutes a flow (dynamical system). This flow is as smooth as allowed by f and ˝. In particular, if f and ˝ are of class C 1 , then the solutions to (7)–(8)Hom belonging to M are of class C 1 in space and time as well. Further significant regularity properties can be found in [23]. B. Finite Dimensionality Let X be a bounded set of a metric space and let N(X; ") be the smallest number of balls of radius " necessary to cover X. The non-negative (possibly infinite) number d f (X) D lim sup "!0C

ln N(X; ") ; ln(1/")

is called the fractal dimension of X. If X is closed with d f (X) < 1, then there exists a Lipschitz-continuous function, g : X 7! Rm , m > 2d f (X), possessing a Höldercontinuous inverse on g(X) (see Theorem 1.2 in [21]). The fundamental result states that d f (M) is finite and that, moreover, d f (M)  ck f k2 /( 2 C P ) :D cG ;

(112)

where c is a positive constant depending only on the “shape” of ˝; see [92]. The quantity G is non-dimensional (often called Grashof number). Consequently, M

is (in particular) homeomorphic to a compact set of Rm , with m D 2c G C 1. Notice that, in agreement with what conjectured by E. Hopf, (112) gives a rigorous estimate of how the dimension of M is expected to increase with the magnitude of the driving force. (Recall, however, that, as remarked previously, there are examples where d f (M(G)) D 0, for all G > 0.) Open Question Since M can be parametrized by a finite number of parameters, or, equivalently, it can be “smoothly” embedded in a finite-dimensional space, it is a natural question to ask whether or not one can construct a finite-dimensional dynamical system having a global attractor on which the dynamics is “equivalent” to the Navier–Stokes dynamics on M. This question, which is still unresolved, has led to the introduction of the idea of inertial manifold [27] and of the associated approximate inertial manifold [25], for whose definitions and detailed properties we refer to Chap. VIII in [94]; see, however, also [51]. Further Questions Related to the Existence of the Global Attractor In this section we shall address two further important aspects of the theory of attractors for the Navier–Stokes equations, namely, the three-dimensional case (in a bounded domain) and the case of a flow past an obstacle. A. Three-Dimensional Flow in a Bounded Domain If we go through the first part of the proof of Theorem 20, we see that the assumption of planar flow has not been used. In fact, by the same token, we can still prove for three-dimensional flow the existence of an “absorbing set”, in the sense that every solution departing from any bounded set of L2 , A, will end up in the set B defined in (109), after a time t0 , dependent on A and F. However, the difficulty in extending the results of Theorem 20 to threedimensional flow, resides, fundamentally, in the lack of well-posedness of (7)–(8)Hom in the space L2 (˝); see Subsect. “Uniqueness and continous Dependence”. In order to overcome this situation, several strategies of attack have been proposed. One way is to make an unproved assumption on all possible solutions, which guarantees the existence of a semiflow on L2 (˝); see [Chapter I] in [16]. In such a case, all the fundamental results proven for the 2D flow, continue to hold in 3D as well [16]. Another way is to weaken the definition of global attractor, by requiring the attractivity property in the weak topology of L2 ; see [Chapter III.3] in [26], and a third way is to generalize the definition of semiflow in such a way that the uniqueness property is no longer required; see [3,13,79].

Navier–Stokes Equations: A Mathematical Analysis

B. Flow Past an Obstacle In this case, the relevant initial-boundary value to be investigated is (7)–(8) with f  v 1  0, endowed with the condition at infinity limjxj!1 v(x; t) D U, where U D U e 1 is a given, non-zero constant vector. For the reader’s convenience, we shall rewrite here this set of equations and put them in a suitable non-dimensional form: 9 @v C v  rv D v  rp= @t in˝  (0; 1) ; div v D 0 (113) v(x; t)j@˝ D 0 ; lim v(x; t) D e 1 ; t > 0 ; jxj!1

(115)

where v is a solution to (113). Then, we can find an un¯ 2 L2 (˝) (posbounded sequence, ft m g, and an element w sibly depending on the sequence) such that ¯ ') ; lim (v(t m )  e 1 ; ') D (w;

for all ' 2 D(˝) : (116)

In (113),  :D jUjd/, with d a length scale, is the appropriate Reynolds number which furnishes the magnitude of the driving mechanism. As we observed in Remark 16, the least requirement on the spatial domain for the existence of a global attractor in a two-dimensional flow, is that there holds Poincaré’s inequality (18). Since this inequality is no longer valid in an exterior domain, the problem of the existence of an attractor for a flow past an obstacle is, basically, unresolved. Actually, the situation is even more complicated than what we just described. In fact, from Theorem 9 and from the considerations developed after it, we know that there is c > 0 such that if  < c , the corresponding boundary-value problem has one and only one solution, (¯v ; p¯), in a suitable function class. Now, it is not known if this solution is attracting. More precisely, in the two-dimensional flow, it is not known if, for sufficiently small , solutions to (113), defined in an appropriate class, tend, as t ! 1, to the only corresponding steady solution. In the three-dimensional case, the situation is slightly better, but still, the question of the existence of an attractor is completely open. We would like to go into more detail about this point. We begin to observe that there is 0 > 0 such that if  < 0 , for any given v 0 with (v 0  e 1 ) 2 L3 (˝), problem (113) has one and only one (smooth) solution that tends to the (uniquely determined) corresponding steady-state solution, (¯v ; p¯). In particular lim kv(t)  v¯ k3 D 0 ;

kv(t)  e 1 k2  K ;

m!1

v(x; 0) D v 0 (x) :

t!1

general, can not be bounded in L2 (˝), uniformly in time, even when  < 0 . This means that the kinetic energy associated to the motion described by (113) has to grow unbounded for large times. To see this, assume  < 0 and that there exists K > 0, independent of t, such that

(114)

see [41]. The fundamental question that stays open is then that of investigating the behavior of solutions to (113) for large t, when  > 0 . As a matter of fact, it is not known whether there exists a norm with respect to which solutions to (113), in a suitable class, remain bounded uniformly in time, for all  > 0. In this respect, it is readily seen that, unlike the bounded domain situation, solutions to (113), in

¯ D v¯  e 1 , which By (114) and (116) we thus must have w in turn implies (¯v  e 1 ) 2 L2 (˝), which, from Theorem 6, we know to be impossible. Consequently, (115) can not be true. Thus, the basic open question is whether or not there exists a function space, Y, where the solution v(t)  e 1 remains uniformly bounded in t 2 (0; 1), for all  > 0. (The bound, of course, may depend on .) The above considerations along with Theorem 6 suggest then that a plauq sible candidate for Y is L (˝), for some q > 2. However, the proof of this property for q  3 appears to be overwhelmingly challenging because, in view of Theorem 18 and Remark 8, it would be closely related to the existence of a global, regular solution. Nevertheless, one could investigate the validity of the following weaker property kv(t)  e 1 kq  K1 ;

for some q 2 (2; 3) ;

(117)

where K 1 is independent of t 2 (0; 1). Of course, the requirement is that (117) holds for all  > 0 and for all corresponding solutions. It is worth emphasizing that the proof of (117) would be of “no harm” to the outstanding global regularity Problem 2, since, according to the available regularity criteria for weak solutions that we discussed in Sect. “Less Regular Solutions and Partial Regularity Results in 3D”, the corresponding solutions, while global in time, will still be weak, even though more regular than those described in Theorem 16. However, notwithstanding its plausibility and “harmlessness”, the property (117) appears to be very difficult to establish. Future Directions The fundamental open questions that we have pointed out throughout this work constitute as many topics for future investigation. Actually, it is commonly believed that the answer to most of these questions (in the affirmative or in the negative) will probably shed an entirely new light

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not only on the mathematical theory of the Navier–Stokes equations but also on other disciplines of applied mathematics. Acknowledgment This work was partially supported by the National Science Foundation, Grants DMS-0404834 and DMS-0707281. Bibliography 1. Amick CJ (1984) Existence of Solutions to the Nonhomogeneous Steady Navier–Stokes Equations. Indiana Univ Math J 33:817–830 2. Amick CJ (1988) On Leray’s Problem of Steady Navier–Stokes Flow Past a Body in the Plane. Acta Math 161:71–130 3. Ball JM (1997) Continuity Properties and Global Attractors of Generalized Semiflows and the Navier–Stokes Equations. J Nonlinear Sci 7:475–502 4. Batchelor GK (1981) An Introduction to Fluid Mechanics. Cambridge University Press, Cambridge 5. Beirão da Veiga H (1985) On the Construction of Suitable Weak Solutions to the Navier–Stokes Equations via a General Approximation Theorem. J Math Pures Appl 64:321–334 6. Beirão da Veiga H (1995) A New Regularity Class for the Navier–Stokes Equations in Rn . Chinese Ann Math Ser B 16:407–412 7. Berger MS (1977) Nonlinearity and Functional Analysis. Lectures on Nonlinear Problems in Mathematical Analysis. Academic Press, New York 8. Berselli LC, Galdi GP (2002) Regularity Criteria Involving the Pressure for the Weak Solutions to the Navier–Stokes Equations. Proc Amer Math Soc 130:3585–3595 9. Brown RM, Shen Z (1995) Estimates for the Stokes Operator in Lipschitz Domains. Indiana Univ Math J 44:1183–1206 10. Caccioppoli R (1936) Sulle Corrispondenze Funzionali Inverse Diramate: Teoria Generale ed Applicazioni al Problema di Plateau. Rend Accad Lincei 24:258–263; 416–421 11. Caffarelli L, Kohn R, Nirenberg L (1982) Partial Regularity of Suitable Weak Solutions of the Navier–Stokes Equations. Comm Pure Appl Math 35:771–831 12. Cattabriga L (1961) Su un Problema al Contorno Relativo al Sistema di Equazioni di Stokes. Rend Sem Mat Padova 31:308–340 13. Cheskidov A, Foia¸s C (2006) On Global Attractors of the 3D Navier–Stokes Equations. J Diff Eq 231:714–754 14. Constantin P (1990) Navier–Stokes Equations and Area of Interfaces. Comm Math Phys 129:241–266 15. Constantin P, Fefferman C (1993) Direction of Vorticity and the Problem of Global Regularity for the Navier–Stokes Equations. Indiana Univ Math J 42:775–789 16. Constantin P, Foia¸s C, Temam R (1985) Attractors Representing Turbulent Flows. Mem Amer Math Soc 53 no. 314 vii, pp 67 17. Darrigol O (2002) Between Hydrodynamics and Elasticity Theory: The First Five Births uof the Navier–Stokes Equation. Arch Hist Exact Sci 56:95–150 18. Escauriaza L, Seregin G, Sverák V (2003) Backward Uniqueness for Parabolic Equations. Arch Ration Mech Anal 169:147–157

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57. Kozono H, Yatsu N (2004) Extension Criterion via TwoComponents of Vorticity on Strong Solutions to the 3D Navier–Stokes Equations. Math Z 246:55–68 58. Ladyzhenskaya OA (1963) The mathematical theory of viscous incompressible flow. Revised English edition. Gordon and Breach Science, New York-London 59. Ladyzhenskaya OA (1991) Attractors for Semigroups and Evolution Equations. Lezioni Lincee. Cambridge University Press, Cambridge 60. Ladyzhenskaya OA, Seregin GA (1999) On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier– Stokes Equations. J Math Fluid Mech 1:356–387 61. Leray J (1933) Etude de Diverses Òquations Intégrales non Linéaires et de Quelques Problèmes que Pose l’ Hydrodynamique. J Math Pures Appl 12:1–82 62. Leray J (1934) Sur le Mouvement d’un Liquide Visqueux Emplissant l’Espace. Acta Math 63:193–248 63. Leray J (1936) Les Problémes non Linéaires. Enseignement Math 35:139–151 64. Lin F (1998) A New Proof of the Caffarelli–Kohn-Nirenberg Theorem. Comm. Pure Appl Math 51:241–257 65. Marchioro C (1986) An Example of Absence of Turbulence for any Reynolds Number. Comm Math Phys 105:99–106 66. Maremonti P (1998) Some Interpolation Inequalities Involving Stokes Operator and First Order Derivatives. Ann Mat Pura Appl 175:59–91 67. Málek J, Padula M, Ruzicka M (1995) A Note on Derivative Estimates for a Hopf Solution to the Navier–Stokes System in a Three-Dimensional Cube. In: Sequeira A (ed) Navier– Stokes Equations and Related Nonlinear Problems. Plenum, New York 141–146 68. Natarajan R, Acrivos A (1993) The Instability of the Steady Flow Past Spheres and Disks. J Fluid Mech 254:323–342 69. Neustupa J, Novotný A, Penel P (2002) An Interior Regularity of a Weak Solution to the Navier–Stokes Equations in Dependence on one Component of Velocity. In: Galdi GP, Rannacher R (eds) Topics in mathematical fluid mechanics. Quad Mat Aracne, Rome 10:163–183 70. Nirenberg L (1959) On Elliptic Partial Differential Equations. Ann Scuola Norm Sup Pisa 13:115–162 71. Prodi G (1959) Un Teorema di Unicità per le Equazioni di Navier–Stokes. Ann Mat Pura Appl 48:173–182 72. Prodi G (1962) Teoremi di Tipo Locale per il Sistema di Navier– Stokes e Stabilità delle Soluzioni Stazionarie. Rend Sem Mat Univ Padova 32:374–397 73. Rosa R (1998) The Global Attractor for the 2D Navier– Stokes Flow on Some Unbounded Domains. Nonlinear Anal 32:71–85 74. Ruelle D, Takens F (1971) On the Nature of Turbulence. Comm Math Phys 20:167–192 75. Sather J (1963) The Initial Boundary Value Problem for the Navier–Stokes Equations in Regions with Moving Boundaries. Ph.D. thesis, University of Minnesota 76. Scheffer V (1976) Partial Regularity of Solutions to the Navier– Stokes Equations. Pacific J Math 66:535–552 77. Scheffer V (1977) Hausdorff Measure and the Navier–Stokes Equations. Comm Math Phys 55:97–112 78. Scheffer V (1980) The Navier–Stokes Equations on a Bounded Domain. Comm Math Phys 73:1–42 79. Sell GR (1996) Global Attractors for the Three-Dimensional Navier-Stokes Equations. J Dynam Diff Eq 8:1–33

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n-Body Problem and Choreographies

n-Body Problem and Choreographies SUSANNA TERRACINI Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, Milano, Italia Article Outline Glossary Definition of the Subject Introduction Simple Choreographies and Relative Equilibria Symmetry Groups and Equivariant Orbits The 3-Body Problem Minimizing Properties of Simple Choreographies Generalized Orbits and Singularities Asymptotic Estimates at Collisions Absence of Collision for Locally Minimal Paths Future Directions Bibliography Glossary Central configurations Are the critical points of the potential constrained on the unitary moment of inertia ellipsoid. Central configurations are associated to particular solutions to the n-body problem: the relative equilibrium and the homographic motions defined in Definition 1. Choreographical solution A choreographical solution of the n-body problem is a solution such that the particles move on the same curve, exchanging their positions after a fixed time. This property can be regarded as a symmetry of the trajectory. This notion finds a natural generalization in that of G-equivariant trajectory defined in Definition 2 for a given group of symmetries-G. The G-equivariant minimization technique consists in seeking action minimizing trajectories among all G-equivariant paths. Collision and singularities When a trajectory can not be extended beyond a certain time b we say that a singularity occurs. Singularities can be collisions if the solution admits a limit configuration as t ! b. In such a case we term b a collision instant. n-Body problem The n-body problem is the system of differential equations (1) associated with suitable initial or boundary value data. A solution or trajectory is a doubly differentiable path q(t) D (q1 (t); : : : ; q n (t)) satisfying (1) for all t. The weaker notion of generalized solution is defined in Definition 5 applies to trajectories found by variational methods.

Variational approach The variational approach to the n-body problem consists in looking at trajectories as critical points of the action functional defined in (4). Such critical points can be (local) minimizers, or constrained minimizers or mountain pass, or other type. Definition of the Subject The motion of n-point particles of positions x i (t) 2 R3 and masses m i > 0, interacting in accordance with Newton’s law of gravitation, satisfies the system of differential equations: n X xi  x j  m i x¨ i (t) D G mi m j ; jx i  x j j3 j¤i; jD1

i D 1; : : : ; n; ;

t 2R:

(1)

The n-body problem consists in solving Eq. (1) associated with initial or boundary conditions. A simple choreography is a periodic solution to the n-body problem Eq. (1) where the bodies lie on the same curve and exchange their mutual positions after a fixed time, namely, there exists a function x : R ! R such that x i (t) D x(t C (i  1)) ; i D 1; : : : ; n ;

t 2 R;

(2)

where  D 2/n. Introduction The two-body problem can be reduced, by the conservation of the linear momentum, to the one center Kepler problem and can be completely solved either by exploiting the conservation laws (angular momentum, energy and the Lenz vector), or by performing the Levi–Civita change of coordinates reducing the problem to that of an harmonic oscillator [58]. The three-body problem is much more complicated than the two-body and can not be solved in a simple way. A major study of the Earth–Moon–Sun system was undertaken by Delaunay in his La Théorie du mouvement de la lune. In the restricted three-body problem, the mass of one of the bodies is negligible; the circular restricted threebody problem is the special case in which two of the bodies are in circular orbits and was worked on extensively by many famous mathematicians and physicists, notably Lagrange in the 18th century, Poincaré at the end of the 19th century and Moser in the 20th century. Poincaré’s work on the restricted three-body problem was the foundation of deterministic chaos theory.

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A very basic – still fundamental – question concerns the real number of degrees of freedom of the n-body problem. As the motion of each point particle is represented by 3-dimensional vectors, the n-body problem has 3n degrees of freedom and hence it is 6n-dimensional. First integrals are: the center of mass, the linear momentum, the angular momentum, and the energy. Hence there are 10 independent algebraic integrals. This allows the reduction of variables to 6n  10. It was proved in 1887 by Bruns that these are the only linearly independent integrals of the n-body problem, which are algebraic with respect to phase and time variables. This theorem was later generalized by Poincaré. A second very natural problem is whether there exists a power series expressing – if not every – at least a large and relevant class of trajectories for the n-body problem. In 1912, after pioneering works of Mittag–Leffler and Levi– Civita, Sundman proved the existence of a series solution in powers of t 1/3 for the 3-body problem. This series is convergent for all real t, except for those initial data which correspond to vanishing angular momentum. These initial data have Lebesgue measure zero and therefore are not generic. An important issue in proving this result is the fact that the radius of convergence for this series is determined by the distance to the nearest singularity. Since then, the study of singularities became the main point of interest in the study of the n-body problem. Sundman’s result was later generalised to the case of n > 3 bodies by Q. Wang in [88]. However, the rate and domain of convergence of this series are so limited to make it hardly applicable to practical and theoretical purposes. Finally, the n-body problem can be faced from the point of view of the theory of perturbations and represents both its starting point and its most relevant application. Delaunay and Poincaré described the spatial three-body problem as a four-dimensional Hamiltonian system and already encountered some trajectories featuring a chaotic behavior. When one of the bodies is much heavier than the other two (a system of one “star” and two “planets”), one can neglect the interaction between the small planets; hence the system can be seen as a perturbation of two decoupled two-body problems whose motions are known to be all periodic from Kepler’s laws. In the planetary three-body problem, hence, two harmonic oscillators interact nonlinearly through the perturbation. Resonances between the two oscillators can be held responsible of high sensitivity with respect to initial data and other chaotic features. A natural question regards the coexistence of the irregular trajectories with regular (periodic or quasi-periodic) ones. A modern approach to the problem of stability of solutions to nearly integrable systems goes through the

application of Kolmogorov–Arnold–Moser (KAM) Theorem [8,56], whose main object indeed is indeed the persistence, under perturbations, of invariant tori. The n-body problem is paradigmatic of any complex system of many interacting objects, and it can neither be solved nor it can be simplified in an efficient way. A possible starting point for its analysis is to seek selected trajectories whose motion is particularly simple, in the sense that it repeats after a fixed period: the periodic solutions. Following Poincaré in his Méthodes Nouvelles de la Mécanique Céleste, tome I, 1892, “. . . D’ailleurs, ce qui nous rends ces solutions périodiques si précieuses, c’est qu’elles sont, pour ainsi dire, la seule brèche par où nous puissons essayer de pénetrer dans une place jusqu’ici réputée inabordable.” Indeed, just before, Poincaré conjectured that periodic trajectories are dense in the phase space: “. . . Voici un fait que je n’ai pas pu démontrer rigourousement, mais qui me parait pourtant très vraisemblable. Étant donneées des équations de la forme définie dans le n.131 et une solution quelconque de ces équations, on peut toujours trouver une solution périodique (dont la période peut, il est vrai, être trés longue), telle que la différence entre les deux solutions soit aussi petite qu’on le veut, pendant un temps aussi long qu’on le veut.”

Singular Hamiltonian Systems From an abstract point of view, the n-body problem is a Hamiltonian System of the form m i x¨ i D

@U (t; x) ; @x i

i D 1; : : : ; n ;

(3)

@U are undefined on a singular set , where the forces @x i the set of collisions between two or more particles in the nbody problem. Such singularities play a fundamental role in the phase portrait (see, e. g. [43]) and strongly influence the global orbit structure, as they can be held responsible, among others, of the presence of chaotic motions (see, e. g. [39]) and of motions becoming unbounded in a finite time [64,90]. Two are the major steps in the analysis of the impact of the singularities in the n-body problem: the first consists in 1 Formula N. 13 quoted by Poincaré is Hamilton equation and covers our class od Dynamical Systems Eq. (3)

n-Body Problem and Choreographies

performing the asymptotic analysis along a single collision (total or partial) trajectory and goes back, in the classical case, to the works by Sundman ([84]), Wintner ([89]) and, in more recent years by Sperling, Pollard, Saari and other authors (see for instance [40,47,75,76,79,83]). The second step consists in blowing-up the singularity by a suitable change of coordinates introduced by McGehee in [65] and replacing it by an invariant boundary – the collision manifold – where the flow can be extended in a smooth manner. It turns out that, in many interesting applications, the flow on the collision manifold has a simple structure: it is a gradient-like, Morse–Smale flow featuring a few stationary points and heteroclinic connections (see, for instance, the surveys [39,68]). The analysis of the extended flow allows us to obtain a full picture of the behavior of solutions near the singularity, despite the flow fails to be fully regularizable (except for binary collisions).

Simple Choreographies and Relative Equilibria A possible starting point for the study of the n-body problem is to find selected trajectories which are particularly simple, when regarded from some point of view. Examples of such particular solutions are the collinear periodic orbits (found in 1767 by Euler), in which three bodies of any masses move such that they oscillate along a rotation line, and the Lagrange triangular solutions, where the bodies lie at the vertices of a rotating equilateral triangle that shrinks and expands periodically, discovered in 1772. Both these trajectories are stationary in a rotating frame. A second remarkable class of trajectories – the choreographies – can be found when the masses are all equal, by exploiting the fact that particles can be interchanged without changing the structure of the system. Among all periodic solutions of the planar 3-body problem, the relative equilibrium motions – the equilateral Lagrange and the collinear Euler–Moulton solutions – are definitely the simplest and most known. In general such simple periodic motions exist for any number of bodies. Definition 1 A relative equilibrium trajectory is a solution of (1) whose configuration remains constant in a rotating frame. A homographic trajectory is a solution of (1) whose configuration remains constant up to homotheties. The normalized configurations of such trajectories are named central and are the critical points of the potential X mi m j U(x) D ; jx i  x j j i< j

constrained to the ellipsoid of unitary momentum of inertia: X m i jx i j2 D 1 : I(x) D i

Relative equilibria feature an evident symmetry (SO(2) and O(2) respectively), that is, they are equivariant with respect to the symmetry group of dimension 1 acting as SO(2) (resp. O(2)) on the time circle and on the plane, and trivially on the set of indexes f1; 2; 3g. In fact, they are minimizers of the Lagrangian action functional in the space of all loops having their same symmetry group. Therefore, generally speaking, G-equivariant minimizers for the action functional (given a symmetry group G) can be thought as the natural generalization of relative equilibrium motions. This perspective has known a wide popularity in the recent literature and has produced a new boost in the study of periodic trajectories in the n-body problem; the recent proof of the existence of the Chenciner–Montgomery eight-shaped orbit is emblematic of this renewed interest (see [7,20,22,25,29,52] and the major part of our bibliographical references). In all these papers, periodic and quasi-periodic solutions of the n-body problem can be found as critical points of the Lagrangian action functional restricted to suitable spaces of symmetric paths. Basic Definitions and Notations Let us consider n point particles with masses m1 ; m2 ; : : : ; m n and positions x1 ; x2 ; : : : ; x n 2 Rd , with d  2. We denote by X the space of configurations with center of mass in 0, and by Xˆ D X n the set of collision-free configurations (collision means x i D x j for some i ¤ j). On the configuration space we define the homogeneous (Newton) potential of degree ˛ < 0: X mi m j U(x) D U i; j (jx i x j j); U i; j (jx i x j j) ' : jx i  x j j˛ i< j

In many cases one can simply require the U i; j to be asymptotically homogeneous at the singularity. Furthermore, a major part of our analysis can be extended to logarithmic potentials. Relative equilibria correspond to those configurations (termed central) which are critical for the restriction of the potential U to the ellipsoid I D  where I denotes the momentum of inertia: X I(x) D m i jx i j2 : i

On collisions the potential U D C1. We are interested in (relative) periodic (such that 8t : x(t C T) D x(t))

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solutions to the system of differential equations: m i x¨ i D

@U : @x i

We associate with the equation the Lagrangian integrand L(x; x˙ ) D L D KCU D

X1 i

2

2

m i j x˙ i j C

X

U i; j (jx i x j j)

i< j

A(x) D

T

L(x(t); x˙ (t))dt :

Then G acts on time (translation and reversal) T via  and it acts on the configuration space X via and  in the following way 8i D 1 : : : n : (gx) i D (g)x(g)1 (i) :

and the action functional: Z

 an orthogonal representation of dimension 2  : G ! O(2) (on cyclic time T Š S 1 ),  an orthogonal representation (on the euclidean space Rd ) : G ! O(d),  and an homomorphism on the symmetric group on n elements (n D f1; 2; : : : ; ng)  : G ! ˙n .

(4)

0

Sometimes it will be preferable to consider the problem in a frame rotating uniformly about the vertical axis, with an angular speed !; the corresponding action A! then contains a gyroscopic term, associated with Coriolis force. We shall seek periodic solutions as critical points of the action functional on the Sobolev space of T-periodic trajectories:  D H 1 (T ; X ) or, to be more precise, of the action constrained on suitable linear subspaces 0 . Two are the major difficulties to be faced in following the variational approach; the first is due to the lack of coercivity (or of Palais–Smale) due to the vanishing at infinity of the force fields: indeed sequences of almost-critical points (such as minimizing sequences) may very well diverge. Furthermore, as the potential U is singular on collisions, minimizers or other critical points can a priori be collision trajectories. Compactness can be successfully recovered by the symmetry of the problem, as we are going to explain here below. We shall expose in the last part of this article some of the strategies which has been developed in order to overcome the problem of collisions. Symmetry Groups and Equivariant Orbits We can generalize the concept of relative equilibria as follows: we impose the permutation of the positions of the particles, after a given time, up to isometries of the space. This gives rise to a class of symmetric trajectories (the generalized choreoghapies). It is worthwhile noticing that these trajectories arise as a common feature of any system of interacting objects, regardless whether they come from models in Celestial mechanics or not. It applies to atoms, or molecules, galaxies or whatever system, provided the objects can be freely interchanged. Let us start by introducing some basics concepts and definitions from [52]. Let G be a finite group endowed with:

As a consequence we have an action on the space of trajectories: Definition 2 A continuous function x(t) D (x1 (t); : : : ; x n (t)) is G-equivariant if 8g 2 G : x(g t) D (gx)(t) : The linear subspace 0 D G  denotes the set of periodic trajectories in  which are equivariant with respect to the G-action: The Palais Principle of symmetric criticality [73] ensures that critical points of an invariant functional restricted to the space of equivariant trajectories are indeed free critical points. We stress that by the particular form of the interaction potentials, in our setting invariance is simply implied by equality of those masses which are interchanged by the action of G on the set of the indices. When the action of  is transitive, all the masses must be equal and the associated G–equavariant trajectories give rise to generalized choreographies. Cyclic and Dihedral Actions Consider the normal subgroup ker G G and the quotient G¯ D G/ker. Since G¯ acts effectively on T , it is either a cyclic group or a dihedral group.  If the group G¯ acts trivially on the orientation of T , then G¯ is cyclic and we say that the action of G on  is of cyclic type.

 If the group G¯ consists of a single reflection on T , then we say that action of G on  is of brake type.

n-Body Problem and Choreographies

 Otherwise, we say that the action of G on  is of dihedral type.

With this first classification, we can easily associate to our minimization problem some proper boundary conditions.

Proposition 1 Let I be the fundamental domain (for a dihedral type), or any interval having as length the minimal angle of time-rotation in G¯ (for a cyclic type). Then the G-equivariant minimization problem is equivalent to problem of minimizing the action over all paths x : I ! X ker  subject to the boundary conditions x(0) 2 X H 0 and x(1) 2 X H 1 , where H 0 and H 1 are the maximal isotropy subgroups of the boundary of I.

Of course, beside seeking minimizers, one can look for other type of critical points, such as mountain pass or others. As the potential U is singular on collisions, minimizers (or other critical points) can a priori be collision trajectories. Many strategies were proposed in the literature in order to overcome these obstacles. The development of a suitable Critical Point theory taking into account of the contribution of fake periodic solutions (the critical points at infinity) was proposed by some authors [9,60,77] and returned a good estimate of the number of periodic trajectories satisfying an appropriate bound on the length, with the main disadvantage of requiring a strong order of infinity at the collisions (the strong force condition). Another strategy to recover coercivity of action functional consists in imposing a symmetry constraint on the loop space. Surprisingly enough, once coercivity is recovered, also the problem of collisions becomes much less dramatic. This fact was remarked for the first time in [34] and widely exploited in the literature, also thanks to the neat idea, due to C. Marchal, of averaging over all possible variations (generalized and exposed in Sect. “Generalized Orbits and Singularities”) to avoid the occurrence of collisions for extremals of the action (Marchal idea was first exposed in [25]). This argument can be used in most of the known cases to prove that absence of collisions for minimizing trajectories and will be outlined in the last section. The Eight Shaped Three-Body Solution

The Variational Approach Let us consider the A restricted to the space of symmetric loops G . We recall that the action functional is said to be coercive if limjxj!C1 A(x) D C1. Coercivity implies the validity of the direct method of Calculus of Variations and, consequently, the existence of a minimizer. Proposition 2 The action functional A is coercive in G if and only if X G D 0. Consequently, if X G D 0 then a minimizer of AG in G exists. Given and , we can compute dimX G 1 X dimX G D Tr( (g))#Fix( (g))  d : jGj g2G

In the frame rotating with constant angular speed ! the action A! is generally coercive, except for a (possible) discrete set values of !.

In their paper [29], Chenciner and Montomery exploited a variational argument in order to prove the existence of a periodic trajectory for the three body problem, where the three particles move one a single eight-shaped curve, interchanging their positions after a fixed time. C. Moore [71] was the first to find numerically the eight, lead by topological reasons which turned out to be insufficient to insure its existence. One of the the simplest symmetries giving rise to the eight shaped trajectory is the following. Denote x(t) D (x1 (t); x2 (t); x3 (t)) 2 R6 : Let G D D6 be the dihedral group generated by two following reflections: the first x1 (t) D x1 (t) ; x3 (t) D x2 (t) ;

x2 (t) D x3 (t) ;

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n-Body Problem and Choreographies

with g1 D (1 ; 1 ; 1 ) is 1 (t) D t, 1 (x) D x and 1 (1; 2; 3) D (1; 3; 2). And the second x1 (1  t) D x2 (t) ;

x2 (1  t) D x1 (t) ;

x3 (1  t) D x3 (t) ; with g2 D (2 ; 2 ; 2 ) is 2 (t) D 1  t, 2 (x) D x and 2 (1; 2; 3) D (2; 1; 3). There are three possible groups yielding an eightshaped trajectory. First, we consider the group of cyclic action type C6 (the cyclic eight having order 6), which acts cyclically on T (i. e. by a rotation of angle /3), by a reflection in the plane E, and by the cyclic permutation (1; 2; 3) in the index set. The second group, which we denote by D12 , is the group of order 12 obtained by extending C6 with the element h defined as follows: (h) is a reflection in T , (h) is the antipodal map in E (thus, the rotation of angle ), and (h) is the permutation (1; 2). This is the symmetry group used by Chenciner and Montgomery. The third group is the subgroup of D12 generated by h and the subgroup C3 of order 3 of C6 D12 . We denote this group D6 (since it is a dihedral group of order 6). The symmetry groups D12 and D6 are of dihedral type. The choreography group C3 is a subgroup of all the three groups, thus the action is coercive on G-equivariant loops.

Definition 3 A group G acts with the rotating circle property if for every T -isotropy subgroup G t G and for at least n  1 indexes i 2 n there exists in Rd a rotating circle S under Gt for i. In most of the known examples the property is fulfilled. In [52] the following results were proved. Theorem 1 Consider a finite group K acting on  with the rotating circle property. Then a minimizer of the K-equivariant fixed-ends (Bolza) problem is free of collisions. Corollary 1 For every ˛ > 0, minimizers of the fixed-ends (Bolza) problem are free of interior collisions. Corollary 2 If the action of G on  is of cyclic type and ker has the rotating circle property then any local minimizer of AG in G is collisionless. Corollary 3 If the action of G on  is of cyclic type and ker D 1 is trivial then any local minimizer of AG in G is collisionless. Theorem 2 Consider a finite group G acting on  so that every maximal T -isotropy subgroup of G either has the rotating circle property or acts trivially on the index set n. Then any local minimizer of AG yields a collision-free periodic solution of the Newton equations for the n-body problem in Rd .

The Rotating Circle Property (RCP) As a second step of the variational approach to the n-body problem, one has to prove that the output of the minimization (or some other variational method) procedure is free of collisions. This point involves a deep analysis of the structure of the possible singularities and will be outlined in the last part of this article. After performing the analysis of collisions, we find that if the action of G on T and X fulfills some conditions (computable) then (local) minimizers of the action functional A in G  do not have collisions. For a group H acting orthogonally on Rd , a circle S

d R (with center in 0) is termed rotating under H if S is invariant under H (that is, for every g 2 H gS D S) and for every g 2 H the restriction gjS : S ! S is a rotation (the identity is meant as a~rotation of angle 0). Let i 2 n be an index and H G a subgroup. A circle S Rd D V (with center in 0) is termed rotating for i under H if S is rotating under H and S V H i V D Rd ; where H i H denotes the isotropy subgroup of the index i in H relative to the action of H on the index set n induced by restriction (that is, the isotropy H i D fg 2 Hjg i D ig).

Examples In this section, for the sake of illustrating the power and the limitations of the approach through the G-equivariant minimization, we include a few examples fitting in our theoretical framework and we give some hints of which symmetry groups satisfy or not assumptions of Theorem 2. Well-known examples are the celebrated Chenciner–Montgomery “eight” [29], Chenciner– Venturelli “Hip–Hop” solutions [30], Chenciner “generalized Hip–Hops” solutions [26], Chen’s orbit [20,22] and Terracini–Venturelli generalized Hip–Hops [85]. One word about the pictures of planar orbits: the configurations at the boundary points of the fundamental domain I are denoted with an empty circle (starting point x i (0)) and a black disc (ending point x i (t), with t appropriate), with a label on the starting point describing the index of the particle. The trajectories of the particles with the times in I are painted as thicker lines (thus it is possible to recover the direction of the movement from x i (0) to x i (t)). Unfortunately this feature was not possible with the threedimensional images. Also, in all the following examples but 4 and 5 existence of the orbits follows directly from the results of The-

n-Body Problem and Choreographies

orem 2. The existence of the orbits described in Examples 4 and 5, which goes beyond the scope of this article, has been recently proved by Chen in [22]. Thousands of other suitable actions and the corresponding orbits have been found by a special-purpose computer program based on GAP [53]. Example 1 (Choreographies) Consider the cyclic group G D Z n of order n acting trivially on V, with a cyclic permutation of order n on the index set n D f1; : : : ; ng and with a rotation of angle 2/n on the time circle T . Since X G D 0, by Proposition 2 the action functional AG is coercive. Moreover, since the action of G on T is of cyclic type and ker D 1, by Corollary 2 the minimum exists and it has no collisions. For several numerical results and a description of choreographies we refer the reader to [32]. An insight of the variational properties of choreographies is provided in Sect. “Minimizing Properties of Simple Choreographies”. Example 2 Let n be odd. Consider the dihedral group G D D2n of order 2n, with the presentation ˛ ˝ G D g1 ; g2 jg12 D g2n D (g1 g2 )2 D 1 : Let  be the homomorphism defined by 

0 1 ; and (g1 ) D 1 0

 cos 2  sin 2 n n (g2 ) D : sin 2 cos 2 n n Furthermore, let the homomorphism be defined by

 1 0 ; and

(g1 ) D 0 1 

1 0 :

(g2 ) D 0 1 Finally, let G act on n by the homomorphism  defined as (g1 ) D (1; n  1)(2; n  2) : : : ((n  1)/2; (n C 1)/2), (g2 ) D (1; 2; : : : ; n), where (i1 ; i2 ; : : : ; i k ) means the standard cycle-decomposition notation for permutation groups. By the action of g 2 it is easy to show that all the loops in G are choreographies, and thus that, since X G D 0, the action functional is coercive. The maximal T -isotropy subgroups are the subgroups of order 2 generated by the elements g1 g2i with i D 0 : : : n  1. Since they are all conjugated, it is enough to show that one of them acts with the rotating circle property. Thus consider H D hg1 i G. For every index i 2 f1; 2; : : : ; n  1g the isotropy H i H relative to the action of H on n is trivial, and g 1 acts by rotation on V D R2 . Therefore for every i 2 f1; 2; : : : ; n1g it is possible to choose a circle rotating

n-Body Problem and Choreographies, Figure 1 The (D6 -symmetric) eight for n D 3

n-Body Problem and Choreographies, Figure 2 The (D10 -symmetric) eight with n D 5

under H for i, since, being H i trivial, V H i D V. The resulting orbits are not homographic (since all the particles pass through the origin 0 at some time of the trajectory and the configurations are centered). For n D 3 this is the eight with less symmetry of [25]. Possible trajectories are shown in Figs. 1 and 2. Example 3 As in the previous example, let n  3 be an odd integer. Let G D C2n Š Z2 C Z n be the cyclic group of order 2n, presented as G D hg1 ; g2 jg12 D g2n D g1 g2 g11 g21 D 1i. The action of G on T is given by (g1 g2 ) D 2n , where 2n denotes the rotation of angle /n (hence the action will be of cyclic type). Now, G can act on the plane V D R2 by the homomorphism defined by 

1 0 ; and

(g1 ) D 0 1

 1 0

(g2 ) D : 0 1 Finally, the action of G on n D f1; 2; : : : ; ng is given by the homomorphism  : G ! ˙n defined by  (g1 ) D (),  (g2 ) D (1; 2; : : : ; n). The cyclic subgroup H2 D hg2 i

G gives the symmetry constraints of the choreographies, hence loops in G are choreographies and the functional is coercive. Furthermore, since the action is of cyclic type, by Corollary 3 the minimum of the action functional is collisionless. It is possible that such minima coincide with the minima of the previous example: this would imply that the symmetry group of the minimum contains the two groups above.

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n-Body Problem and Choreographies, Figure 3 Another symmetry constraint for an eight-shaped orbit (n D 5)

Example 4 Consider four particles with equal masses and an odd integer q  3. Let G D D4q  C2 be the direct product of the dihedral group of order 4q with the group C2 of order 2. Let D4q be presented by D4q D hg1 ; g2 jg12 D 2q g2 D (g1 g2 )2 D 1i, and let c 2 C2 be the non-trivial element of C2 . Now define the homomorphisms ,  and  as follows:

 1 0 ;

(g1 ) D (g1 ) D 0 1 " # cos 2  sin 2 2q 2q

(g2 ) D (g2 ) D ; sin 2 cos 2 2q 2q  



1 0 1 0 ; ; (c) D

(c) D 0 1 0 1 (g1 ) D (1; 2)(3; 4) ;

(g2 ) D (1; 3)(2; 4) ;

(c) D (1; 2)(3; 4) : It is not difficult to show that X G D 0, and thus the action is coercive. Moreover, ker D C2 , which acts on R2 with the rotation of order 2, hence ker acts with the rotating circle property. Thus, by Proposition 2 and Theorem 1 the minimizer exists and does not have interior collisions. To exclude boundary collisions we cannot invoke Theorem 2, since the maximal T -isotropy subgroups do not act with the rotating circle property. A possible graph for such a minimum can be found in Fig. 4, for q D 3 (one needs to prove that the minimum is not the homographic solution – with a level estimate – and that there are no boundary collisions – with an argument similar to [20]). See also [22] for an updated and much generalized treatment of such orbits. Example 5 Consider four particles with equal masses and an even integer q  4. Let G D D q C2 be the direct product of the dihedral group of order 2q with the group C2 of q order 2. Let D4q be presented by D4q D hg1 ; g2 jg12 D g2 D (g1 g2 )2 D 1i, and let c 2 C2 be the non-trivial element of C2 . As in Example 4, define the homomorphisms ,  and  as follows.

 1 0 ;

(g1 ) D (g1 ) D 0 1

n-Body Problem and Choreographies, Figure 4 The orbit of Example 4 with q D 3

"

# cos 2  sin 2 q q

(g2 ) D (g2 ) D ; sin 2 cos 2 q q



 1 0 1 0

(c) D ; (c) D ; 0 1 0 1  (g1 ) D (1; 2)(3; 4) ;

(g2 ) D (1; 3)(2; 4) ;

 (c) D (1; 2)(3; 4) : Again, one can show that a minimizer without interior collisions exists since ker D C2 acts with the rotating circle property (a possible minimizer is shown in Fig. 5). This generalizes Chen’s orbit [22]. See also [20]. Example 6 (Hip–Hops) If G D Z2 is the group of order 2 acting trivially on n, acting with the antipodal map on V D R3 and on the time circle T , then again X G D 0, so that Proposition 2 holds. Furthermore, since the action is of cyclic type Corollary 2 ensures that minimizers have no collisions. Such minimizers were called generalized Hip–

n-Body Problem and Choreographies, Figure 5 A possible minimizer for Example 5

n-Body Problem and Choreographies

n-Body Problem and Choreographies, Figure 6 The Chenciner–Venturelli Hip–Hop n-Body Problem and Choreographies, Figure 7 The planar equivariant minimizer of Example 7

Hops in [25]. See also [26]. A subclass of symmetric trajectories leads to a generalization of such a Hip–Hop. Let n  4 an even integer. Consider n particles with equal masses, and the group G D C n  C2 direct product of the cyclic group of order n (with generator g 1 ) and the group C2 of order 2 (with generator g 2 ). Let the homomorphisms ,  and  be defined by 2

 sin 2 cos 2 n n 2 4

(g1 ) D sin n cos 2 n 0 0 2 3 1 0 0

(g2 ) D 4 0 1 0 5 ; 0 0 1

 1 0 (g2 ) D ; 0 1  (g1 ) D (1; 2; 3; 4) ;

3 0 05 ; 1

(g1 ) D

1 0

 0 ; 1

(g2 ) D () :

Example 7 Consider the direct product G D D6  C3 of the dihedral group D6 (with generators g 1 and g 2 of order 3 and 2 respectively) of order 6 and the cyclic group C3 of order 3 generated by c 2 C3 . Let us consider the planar n-body problem with n D 6 with the symmetry constraints given by the following G-action. 

1 0 1 ; (g2 ) D 0 1 0

2 2  cos 3  sin 3

(c) D ; sin 2 cos 2 3 3

 (g1 ) D (1; 3; 2)(4; 5; 6) ;

 0 ; 1

(g2 ) D (1; 4)(2; 5)(3; 6) ;

 (c) D (1; 2; 3)(4; 5; 6) :

It is easy to see that X G D 0, and thus a minimizer exists. Since the action is of cyclic type, it suffices to exclude interior collisions. But this follows from the fact that ker D C n has the rotating circle property. This example is the natural generalization of the Hip–Hop solution of [30] to n  4 bodies. We can see the trajectories in Fig. 6.

(g1 ) D



cos 2  sin 2 3 3 ; sin 2 cos 2 3 3



0 1 1 ; (c) D (g2 ) D 1 0 0

(g1 ) D

 0 ; 1

By Proposition 2 one can prove that a minimizer exists, and since G acts with the rotating circle property (actually, the elements of the image of are rotations) on T -maximal isotropy subgroups, the conclusion of Theorem 2 holds. It is not difficult to see that configurations in X ker are given by two centered equilateral triangles. Now, to guarantee that the minimizer is not a homographic solution, of course it suffices to show that there are no homographic solutions in G (like in the case of Example 2). This follows from the easy observation that at some times t 2 T with maximal isotropy it happens that x1 D x4 , x2 D x5 and x3 D x6 , while at some other times it happens that x1 D x5 , x2 D x6 and x3 D x4 or that x1 D x6 , x2 D x4 and x3 D x5 and this implies that there are no homographic loops in G . With no difficulties the same action can be defined for n D 2k, where k is any odd integer. We can see a possible trajectory in Fig. 7. Also, it is not difficult to consider a similar example in dimension 3. With n D 6 and the notation of D6 and C3 as above, consider the group G D D6  C3  C2 . Let g 1 , g 2 , c be as above, and let c2 be the generator of C2 . The homomorphisms ,  and  are defined in a similar way by 2 3 2 3 1 0 0 1 0 0

(g1 ) D 40 1 05 ; (g2 ) D 4 0 1 05 ; 0 0 1 0 0 1 2 3 2 2 cos 3  sin 3 0

(c) D 4 sin 2 cos 2 05 ; 3 3 0 0 1

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n-Body Problem and Choreographies, Figure 8 The three-dimensional equivariant minimizer of Example 7

2

3

1 0 0

(c2 ) D 40 1 0 5 ; 0 0 1



 cos 2 0  sin 2 3 3 (g1 ) D ; (g2 ) D 2 sin 2 cos 1 3 3  



1 0 1 0 ; ; (c2 ) D (c) D 0 1 0 1 (g1 ) D (1; 3; 2)(4; 5; 6) ; (c) D (1; 2; 3)(4; 5; 6) ;

 1 ; 0

(g2 ) D (1; 4)(2; 5)(3; 6) ; (c2 ) D () :

In the resulting collisionless minimizer (again, it follows by Proposition 2 and Theorem 2) two equilateral triangles rotate in opposite directions and have a “brake” motion on the third axis. The likely shape of the trajectories can be found in Fig. 8. Example 8 (Marchal’s P12 –symmetry revisited) Let k  2 be an integer, and consider the cyclic group G D C6k of order 6k generated by the element c 2 G. Now consider orbits for n D 3 bodies in the space of dimension d D 3. With a minimal effort and suitable changes the example can be generalized for every n  3. We leave the details to the reader. The homomorphisms ,  and  are defined by 3 2 0 cos k  sin k cos k 05;

(c) D 4 sin k 0 0 1

2 2  cos 6k  sin 6k ; (c) D sin 2 cos 2 6k 6k (c) D (1; 2; 3) : Straightforward calculations show that X G D 0 and hence Proposition 2 can be applied. Furthermore, the action is of cyclic type with ker D 1, and hence by Corollary 2 the minimizer does not have collisions. It is left to show that this minimum is not a homographic motion. The

n-Body Problem and Choreographies, Figure 9 The non-planar choreography of Example 8 for k D 4

only homographic motion in G is a Lagrange triangle y(t) D (y1 ; y2 ; y3 )(t), rotating with angular velocity 32k (assume that the period is 2, i. e. that T D jT j D 2) in the plane u3 D 0 (let u1 ; u2 ; u3 denote the coordinates in R3 ). To be a minimum it needs to be inscribed in the horizontal circle of radius ((˛3˛/2 )/(2(32k)2 ))1/(2C˛) . Now, for every function (t) defined on T such that (c 3 t) D (t), the loop given by v1 (t) D (0; 0; (t)), v2 (t) D (0; 0; (c 2 t)) and v3 (t) D (0; 0; (c 2 t)) is G-equivariant, and thus belongs to G . If one computes the value of Hessian of the Lagrangian action A in y and in the direction of the loop v one finds that ˇ Dv2 Aˇ y D 3

Z

2

˙ 2 (t)dt  2(3  2k)2

0

Z

2



((t) C (ct))2 dt :

0

In particular, if we set the function (t) D sin(kt), which has the desired property, elementary integration yields Dv2 Aj y D 3(k 2  2(3  2k)2 ) ; which does not depend on ˛ and is negative for every k  3. Thus for every k  3 the minimizer is not homographic. We see a possible trajectory in Fig. 9. Remark 1 In the previous example, if k 6 0 mod 3, the cyclic group G can be written as the sum C3 C C2k . The generator of C3 acts trivially on V, acts with a rotation of order 3 on T and with the cyclic permutation (1; 2; 3) on f1; 2; 3g. This means that for all k 6 0 mod 3 the orbits of Example 8 are non-planar choreographies. Furthermore, it is possible to define a cyclic action of the same kind by setting  and  as above and 2 3 p p cos 3k   sin 3k  0 6 7 p

(c) D 4 sin p  cos 3k  05; 3k 0 0 1

n-Body Problem and Choreographies



n-Body Problem and Choreographies, Figure 10 A non-planar symmetric orbit of Remark 1, with k D 3 and p D 1

where p is a non-zero integer. If p D 3 one obtains the same action as in Example 8. One can perform similar computations and obtain that the Lagrange orbit (with angular velocity p  2k, this time) is not a minimizer for all (p; k) such that 0 < p < 3k and k 2  2(p  2k)2 < 0.

1 Y

 (1; 1) 1

is isomorphic to the direct product I  Y of order 120, and acts on the euclidean space R3 as the full icosahedron group. The action on T is cyclic and given by the fact that ker D 1  Y. The isotropy is generated by the central inversion  1, and hence the set of bodies is G/I Š Y. Thus at any time t the 60 point particles are constrained to be a Y-orbit in R3 (which does not mean they are vertices of a icosahedron, simply that the configuration is Y-equivariant). After half period every body is in the antipodal position: x i (t C T/2) D x i (in other words, the group contains the anti-symmetry), also known as Italian symmetry – see [25,26]. Of course, the group Y is just an example: one can choose also the tetrahedral group T or the octahedral O and obtain anti-symmetric orbits for 12 (tetrahedral) or 24 (octahedral) bodies, as depicted in Fig. 11. The action is by its definition transitive and coercive; local minimizers are collisionless since the maximal T -isotropy group acts as a subgroup of SO(3) (i. e. orientation–preserving).

More Examples with non Trivial Core

Example 10 Let G be the group with Krh data   1 (1; 1) ; Dk 1

Now consider the core of G, ker D K G. With an abuse of notation, using the factorization of the action, we can identify K with a subgroup of O(3). A very remarkable class of symmetry groups with non trivial core has been defined by Ferrario in [50], through its Krh decomposition:

where Dk is the rotation dihedral group of order 2k. As in the previous example, the action is such that the action functional is coercive and its local minima collisionless. At every time instant the bodies are Dk -equivariant in R k and the anti-symmetry holds. Approximations of minima can be seen in Fig. 12.

Definition 4 Let G be a symmetry group. Then define

Example 11 To illustrate the case of non-transitive symmetry group, consider the following (cyclic) Krh data:   1 (3; 1) ; 1 1

1. K D ker, 2. [r] 2 WO(3) K as the image in the Weyl group of the generator mod K of ker det  G/K (corresponding to the time-shift with minimal angle). If ker det  D K, then [r] D 1. 3. [h] 2 WO(3) K as the image in the Weyl group of one of the time-reflections mod K in G/K, in the cases such an element exists. Otherwise it is not defined. In short, the triple (K; [r]; [h])

which yields a group of order 6 acting cyclically on 3 bodies, and with the antipodal map on R3 . Since ker is trivial and the group is of cyclic type, local minima are collisionless. Now, by adding k copies of such group one obtains a symmetry group having k copies of it as its transitive components, where still local minimizers are collisionless and the restricted functional is coercive. Some possible minima can be found in Fig. 13, for k D 3; 4.

is said the Krh data of G. Example 9 Consider the icosahedral group Y of order 60. The group G with Krh data

The 3-Body Problem The major achievement of [14] is to give the complete description of the outcome of the equivariant minimization

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n-Body Problem and Choreographies, Figure 11 60-icosahedral Y, 12-tetrahedral T and 24-octahedral O periodic minimizers (chiral)

n-Body Problem and Choreographies, Figure 12 4-dihedral D2 and 6-dihedral D6 symmetric periodic minimizers

n-Body Problem and Choreographies, Figure 13 9 and 12 bodies in anti-choreographic constraints grouped by 3

n-Body Problem and Choreographies

procedure for the planar three-body problem. First we can ensure that minimizers are always collisionless. Theorem 3 Let G a symmetry group of the Lagrangian in the 3-body problem (in a rotating frame or not). If G is not bound to collision (i. e. every equivariant loop has collisions), then any (possible) local minimizer is collisionless. A symmetry group G of the Lagrangian functional A is termed  bound to collisions if all G-equivariant loops actually have collisions,  fully uncoercive if for every possible rotation vecG in the frame rotating tor ! the action functional A! around ! with angular speed j!j is not coercive in the space of G-equivariant loops (that is, its global minimum escapes to infinity);  homographic if all G-equivariant loops are constant up to orthogonal motions and rescaling.  The core of the group G is the subgroup of all the elements which do not move the time t 2 T . If, for every angular velocity, G is a symmetry group for the Lagrangian functional in the rotating frame, then we will say that G is of type R. This is a fundamental property for symmetry groups. In fact, if G is not of type R, it turns out that the angular momentum of all G-equivariant trajectories vanishes. The Classification of Planar Symmetry Groups for 3-body Theorem 4 Let G be a symmetry group of the Lagrangian action functional in the planar 3-body problem. Then, up to a change of rotating frame, G is either bound to collisions, fully uncoercive, homographic, or conjugated to one of the symmetry groups listed in Table 1 (RCS stands for Rotating

Circle Property and HGM for Homographic Global Minimizer). Planar Symmetry Groups  The trivial symmetry Let G be the trivial subgroup of order 1. It is clear that it is of type R, it has the rotating circle property. It yields a coercive functional on G D  only when ! is not an integer. If ! D 12 mod 1 then the minimizers are minimizers for the anti-symmetric symmetry group (also known as Italian symmetry) x(at) D ax(t), where a is the antipodal map on T and E. The masses can be different. Proposition 3 For every ! … Z and every choice of masses the minimum for the trivial symmetry occurs in the relative equilibrium motion associated to the Lagrange central configuration.  The line symmetry Another case of symmetry group that can be extended to rotating frames with arbitrary masses is the line symmetry: the group is a group of order 2 acting by a reflection on the time circle T , by a reflection on the plane E, and trivially on the set of indexes. That means, at time 0 and  the masses are collinear, on a fixed line l E. It is coercive only when ! 62 Z. In this case the Lagrangian solution cannot be a minimum, while it can the relative equilibrium associated with the Euler configuration.  The 2  1-choreography symmetry Consider the group of order 2 acting as follows: (g) D 1, (g) D 1 (that is, the translation of half-period) and  (g) D (1; 2) (that is,  (g)(1) D 2,  (g)(2) D 1,

n-Body Problem and Choreographies, Table 1 Planar symmetry groups with trivial core Name jGj Type R Trivial 1 yes Line 2 yes 2  1-choreography 2 yes Isosceles 2 yes Hill 4 yes 3-choreography 3 yes Lagrange 6 yes C6 6 no D6 6 no D12 12 no

Action type trans. dec. 1C1C1 brake 1C1C1 cyclic 2C1 brake 2C1 dihedral 2C1 cyclic 3 dihedral 3 cyclic 3 dihedral 3 dihedral 3

RCP yes (no) yes no no yes no yes yes no

HGM yes no no yes no yes yes no no no

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n-Body Problem and Choreographies, Figure 14 The poset of symmetry groups for the planar 3-body problem

n-Body Problem and Choreographies, Figure 15 Action levels for the line symmetry

and (g)(3) D 3). That is, it is a half-period choreography for the bodies 1 and 2. It can be extended to rotating frames and coercive for a suitable choice of ! ¤ 0; 1 mod 2. The Euler’s orbit with k D 1 and the Hill’s orbits with k D ˙1 are equivariant for the 2  1-choreography symmetry, while the Euler’s orbit with k D 0 is not equivariant for this symmetry. In Figures 15 and 16 Euler 1 represents the action levels on the Euler’s orbit with k D 1, Hill 1–2 the ones on the Hill’s with k D ˙1.  The isosceles symmetry The isosceles symmetry can be obtained as follows: the group is of order 2, generated by h; (h) is a reflection

n-Body Problem and Choreographies, Figure 16 Action levels for the 2  1-choreography symmetry

in the time circle T , (h) is a reflection along a line l in E, and  (h) D (1; 2) as above. The constraint is therefore that at time 0 and  the 3-body configuration is an isosceles triangle with one vertex on 1 (the third). Proposition 4 For every ! … Z and every choice of masses the minimum for the isosceles symmetry occurs in the relative equilibrium motion associated to the Lagrange configuration.  The Euler–Hill symmetry Now consider the symmetry group with a cyclic generator r of order 2 (i. e. (r) D 1) and a time reflection h (i. e. (h) is a reflection of T ) given by (r) D 1,

n-Body Problem and Choreographies

(r) D (1; 2), (h) is a reflection and (h) D (). It contains the 2  1-choreography (as the subgroup ker det()), the isosceles symmetry (as the isotropy of /2 2 T ) and the line symmetry (as the isotropy of 0 2 T ) as subgroups. Proposition 5 The minimum of the Euler–Hill symmetry is not homographic, provided that the angular velocity ! is close to 0.5 and the values of the masses are close to 1.  The choreography symmetry The choreography symmetry is given by the group C3 of order 3 acting trivially on the plane E, by a rotation of order 3 in the time circle T and by the cyclic permutation (1; 2; 3) of the indices. Proposition 6 For every ! the minimal choreography of the 3 body problem is a rotating Lagrange configuration.  The Lagrange symmetry The Lagrange symmetry group is the extension of the choreography symmetry group by the isosceles symmetry group. Thus, it is a dihedral group of order 6, the action is of type R. Hence, the relative equilibrium motions associated to the Lagrange configuration are admissible motions for this symmetry and, again, the minimizer occurs in the relative equilibrium motion associated to the Lagrange configuration.  The Chenciner–Montgomery symmetry group and the eights There are three symmetry groups (up to change of coordinates) that yield the Chenciner–Montgomery figure eight orbit: they are the only symmetry groups which do not extend to the rotating frame and we have already described them in Subsect. “The Eight Shaped Three-Body Solution”. One can prove that all G-equivariant trajectories have vanishing angular momentum, whenever the group is not of type R. Moreover, we were able to partially answer to the open question (posed by Chenciner) whether their minimizers coincide or not: for two of them (D6 and D12 ) the minimizer is necessarily the same. Space Three-body Problem Based on the classification of planar groups, by introducing a natural notion of space extension of a planar group, Ferrario gave in [51] a complete answer to the classification problem for the three-body problem in the space and at the same time to determine the resulting minimizers and describe its more relevant properties.

n-Body Problem and Choreographies, Table 2 Space extensions of planar symmetry groups with trivial core Name Trivial Line Isosceles Hill 3-choreography Lagrange D6 D12

Extensions C1 LC; , L;C 2 2 C; H2 , H2;C H4C; , H4;C C3C , C3 LC;C , LC; , L;C 6 6 6 C; ;C D6 , D6 D;C 12

Theorem 5 Symmetry groups not bound to collisions, not fully uncoercive and not homographic are, up to a change of rotating frame, either the three-dimensional extensions of planar groups (if trivial core) listed in table below or the vertical isosceles triangle (if non-trivial core). The next theorem is the answer to the natural questions about collisions and description of some main features of minimizers. Theorem 6 Let G be a symmetry group not bound to collisions and not fully uncoercive. Then 1. Local minima of A! do not have collisions. 2. In the following cases minimizers are planar trajectories: ;C a) If G is not of type R: D6C; , D6;C and D12 (and then G-equivariant minimizers are Chenciner– Montgomery eights). b) If there is a G-equivariant minimal Lagrange rotating solution: C1 , H2C; , C3C , LC;C , LC; (and then 6 6 the Lagrange solution is of course the minimizer). c) If the core is non-trivial and it is not the vertical isosceles (and then minimizers are homographic). 3. In the following cases minimizers are always non-planar: a) The groups L;C and C3 for all ! 2 (1; 1) C 6Z, 6 ! ¤ 0 (the minimizers for L;C are the elements of 6 Marchal family P12 , and minimizers of C3 are a less0 2 ). symmetric family P12 b) The extensions of line and Hill–Euler type groups, for on open subset of mass distributions and angular speeds !: LC; , L;C , H4C; and H4;C (for L;C 2 2 2 this happens also with equal masses). c) The vertical isosceles for suitable choices of masses and !. 2 Highly likely they are not distinct families: this is the recurring phenomenon of “more symmetries than expected” in n-body problems.

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n-Body Problem and Choreographies, Figure 17 and LC; , plotted in the fixed frame Non planar minimizers for the groups L;C 6 2

n-Body Problem and Choreographies, Figure 18 Examples for Theorem 10 close to an integer that divides n

Minimizing Properties of Simple Choreographies In the space of symmetric (choregraphical) loops, the action takes the form A(x) D

n1 Z 1X  1 j x˙ (t C h)j2 dt C 2 2 0 hD0



n1 Z X h;l D0

 0

dt : jx(t C l)  x(t C h)j˛

h¤l

A natural question concerns the nature of the minimizers under the sole choregraphical constraint. Unfortunately, the bare minimization among choreographical loops gives returns only trivial motions:

Theorem 7 For every ˛ 2 RC and d  2, the absolute minimum of A on  is attained on a relative equilibrium motion associated to the regular n-gon. This theorem extends some related result for the italian symmetry by Chenciner and Desolneux [27], and the results in Sect. “The 3-Body Problem”. The proof is based on a (quite involved) convexity argument together with the analysis of some spectral properties related to the choreographical constraint. Now, in order to find nontrivial minimizers, we look at the same problem in a rotating frame. In order to take into account of the Coriolis force, the new action functional has to contain a gyroscopic term: A(y) D

1 2

Z

2 0

j y˙(t) C J! y(t)j2dt

n-Body Problem and Choreographies

C

n1 Z dt 1 X 2 : 2 jy(t)  y(t C h)j˛ 0 hD1

Consider the function h : RC ! N, h(!) D minn2N  (!  n)2 /n2 and let !  D 43 . The same technique used for the inertial system extends to rotating systems having small angular velocity; this gives the following result. Theorem 8 If ! 2 (0; !  )nf1g, then the action attains its minimum on a circle with minimal period 2 and radius depending on n, ˛ and !.

undergoes some transitions (for example it has to pass from relative equilibrium having different winding numbers). This scenario suggests the presence of other critical points, such as local minimizers or mountain pass. This was indeed discovered numerically in [12] and then proved by a computer assisted proof in [7]. To begin with, let us look at the following figure where the values of the action functional A! on the branches of circular orbits L!k are plotted:

When ! is Close to an Integer The situation changes dramatically when ! is close to some integer. To understand this phenomenon, let us first check the result of the minimization procedure when ! is an integer: Proposition 7 (1) If ! D n, then the action has a continuum of minimizers. (2) If ! D k, coprime with n, then the action does not achieve its infimum (escape of minimizing sequences). (3) If ! D k and k divides n, then the action does not achieve its infimum (clustering of minimizing sequences). As a consequence, we have the following result: Theorem 9 Suppose that n and k are coprime. Then there exist D (˛; n; k) such that if ! 2 (k  ; k C ) the minimum of the action is attained on a circle with minimal period 2/k that lies in the rotating plane with radius depending on n, ˛ and !. An interesting situation appears when the integer closest to the angular velocity is not coprime with the number of bodies. In this case we prove that the minimal orbit it is not circle anymore, as the following theorem states. Theorem 10 Take k 2 N and g:c:d:(k; n) D k˜ > 1, k˜ ¤ n. Then there exists D (˛; n; k) > 0 such that if ! 2 (k  ; k C )nfkg the minimum of the action is attained on a planar 2-periodic orbit with winding number k which is not a relative equilibrium motion. Also, it has be noticed that, for large number of bodies and angular velocities close to the half on an integer, the minimizer apparently is not anymore planar. Mountain Pass Solutions for the Choreographical 3-Body Problem The discussion carried in the previous section shows that, as the angular velocity varies, the minimizer’s shape must

The analysis of this picture suggests the presence of critical points different from the Lagrange motions. Indeed, let us take the angular velocity ! D 1:5: in this case there are two distinct global minimizers, the uniform circular motions with minimal period 2 and , lying in the plane orthogonal to the rotation direction. This is a well known structure in Critical Point Theory, referred as the Mountain Pass geometry and gives the existence of a third critical point, provided the Palais–Smale condition is fulfilled, with an additional information on the Morse index. Next theorem follows from the application of the Mountain Pass Theorem to the action functional A3/2 : Theorem 11 There exists a (possibly collision) critical point for the action functional A3/2 with Morse index smaller then 1 and distinct from any Lagrange motion. Once the existence of a Mountain Pass critical point was theoretically established, we studied its main properties in order to understand whether it belonged to some known families of periodic trajectories. To this aim we applied the bisection algorithm proposed in [13] to approximate the maximal of a locally optimal path joining the two strict global minimizers, finding in this a good numerical candidate. Of course, there could be a gap between the mountain pass solution whose existence is ensured by Theorem 11

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n-Body Problem and Choreographies, Figure 19 Non planar minimizers of the action with angular velocities close to the half on an integer

and the numerical candidate found by applying the bisection algorithm. In order to fill this gap proved the existence of an actual solution very close to the numerical output of the Mountain Pass algorithm. The argument was based upon a fixed point principle and involved a rigorous computer assisted proof. As a consequence, we obtained the existence of a new branch of solution for the spatial 3-body problem (see Fig. 8). Here are some relevant features of the new solution: the orbit is not planar, its winding number with respect, for instance, to the line x D 0:2, y D 0 is 2 and it does not intersect itself. A natural question is whether this solution can be continued as a function of the parameter !. We were able to extend the numerical–rigorous argument to cover a full interval of values of the angular velocity, providing the existence of a full branch of solution. Theorem 12 There exists a smooth map B(!) giving the (locally unique, up to symmetries) branch of solution of the choreographical 3-body problem for all ! 2 [1; 2], starting at the Mountain Pass solutions for ! D 1:5. A natural question is whether this mountain pass branch meets one of the known branches of choreographical periodic orbits: either one of the Lagrange or Marchal’s P12 (described by Marchal in [62]) families. Surprisingly enough, it turns out from further numerical computation that this branch does not emanates from any of Lagrange motions L1 or L2 ; apparently, it emanates from the branch of P12 solutions. The details of the bifurcation diagram are depicted in Fig. 8. Generalized Orbits and Singularities The existence of a G-equivariant minimizer of the action is a simple consequence of the direct method of the Calculus

of Variations. These trajectories, however, solve the associated differential equations only where they are far from the singular set. Indeed, generally speaking, G-equivariant minimizers may present singularities, even though the set of collision instants must clearly be of vanishing Lebesgue measure. A first natural question concerns the number of possible collision instants. Here below we follow [15]. Definition 5 A path x : (a; b) ! X is called a locally minimizing solution of the N-body problem if, for every t0 2 (a; b), there exists ı > 0 such that the restriction of x to [t0  ı; t0 C ı] is a local minimizer for the action with the same boundary conditions. A path which is the uniform limit of a sequence of locally minimal solutions will be called a generalized solution. We remark that:  The definition can be modified in a obvious manner to include symmetries.  Equivariant minimizers are generalized solutions.  If the potential is of class C 2 outside collisions then every non collision solution is a generalized solution.  The generalized solutions possess an index: the minimal number of intervals I j needed to cover (a; b) such that the restriction of x to I j is a local minimizer for the action.  There is a natural notion of maximal existence interval for generalized solutions, even without the unique extension property. Singularities and Collisions Generally speaking we are dealing with systems of the form M x¨ D rU(t; x) ;

t 2 (a; b) ;

M i j D mi ıi j ;

n-Body Problem and Choreographies

n-Body Problem and Choreographies, Figure 20 Mountain pass solution with angular velocities close to the half on an integer 12

7.8 mountain pass L1 L2

10

mountain pass P12 L2

7.75 7.7

8

7.65 6

7.6

4

7.55

2 0

a

7.5 0

0.5

1

1.5

2

2.5

3

7.45 0.93 0.94 0.95 0.96 0.97 0.98 0.99

b

1

1.01

n-Body Problem and Choreographies, Figure 21 Action levels for the Lagrange and the mountain pass solution in the 3-body problem. On the x-axes the angular velocity varies in the interval [0; 3). The left picture focuses on the bifurcation from the P12 family

where the potential U possesses a singular set (a; b)  , in the sense that [U0]

lim U(t; x) D C1 ;

x!

uniformly in t :

The set  is clearly the collision set. We assume  to be a cone in Rnd : x 2  H) x 2  : Definition 6 We say that a generalized solution x on the interval (a; b), has a singularity at t  < C1 if

The Theorems of Painlevé and Von Zeipel In the usual mathematical language a classical solution x on the interval (a; b), has a singularity at t  < C1 if it is not possible to extend x as a (classical) solution to a larger interval (a; t  C ı). A classical results relates singularities of solutions with those of the potential. Theorem 13 (Painlevé’s Theorem) Let x¯ be a classical solution for the n-body dynamical system on the interval [0; t  ). If x¯ has a singularity at t  < C1, then lim U(x¯ (t)) D C1 :

t!t 

lim sup U(t; x(t)) D C1 : t!t 

When t  2 (a; b) we will say that x has an interior singularity at t D t  , while when t  D a or t  D b (when finite) we will talk about a boundary singularity.

Painlevé’s Theorem does not necessarily imply that a collision (i. e. that is a singularity such the configuration has a definite limit) occurs when there is a singularity at a finite time (on this subject we refer to [66,75,76]). The next

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result has been stated by Von Zeipel in 1908 and definitely proved by Sperling in 1970.

With these assumptions, we can extend then Von Zeipel’s Theorem to generalized solutions:

Theorem 14 (Von Zeipel’s Theorem) If x¯ is a classical solution for the n-body dynamical system on the interval (a; t  ) with a singularity at t  < C1 and

Theorem 15 Let x¯ be a generalized solution on the bounded interval (a; b). If x¯ is bounded on the whole interval (a; b) then the singularities of x¯ are collisions.

lim kx¯ (t)k < C1 ;

t!t 

Asymptotic Estimates at Collisions

then x¯ (t) has a definite limit configuration x  a s t tends to t  .

From now on, we require our potential U to satisfy some one-side homogeneity conditions:

Von Zeipel’s Theorem and the Structure of the Collision Set To the aim of extending the Von Zeipel’s Theorem to our notion of generalized solutions, we need to introduce some assumptions on the potential U and its singular set . [ D V ; (5) 2M

[U1] there exists C1  0 such that for every (t; x) 2 ((a; b)  Rnd n) ˇ ˇ ˇ ˇ @U ˇ  C1 U(t; x) : ˇ (t; x) ˇ ˇ @t [U2] there exist ˛ 2 (0; 2), > 0 and C2  0 such that

Rk

and M where the V ’s are distinct linear subspaces of is a finite set; observe that the set  is a cone, as required before. We endow the family of the V ’s with the inclusion partial ordering and we assume the family to be closed with respect to intersection (thus we are assuming that M is a semilattice of linear subspaces of R k : it is the intersection semilattice generated by the arrangement of maximal subspaces V ’s). With each  2  we associate \ V : () D minf :  2 V g i. e. ; V() D 2V

Fixed  2 M we define the set of collision configurations satisfying  D f 2  : () D g and we observe that this is an open subset of V and its closure  is V . We also notice that the map  ! dim(V() ) is lower semicontinuous. We denote by p the orthogonal projection onto V and we write x D p (x) C w (x) ; where, of course, w D I  p . We assume that, near the collision set, the potential depends, roughly, only on the projection orthogonal to the collision set: more precisely we assume [U5] For every  2 ; there is " > 0 such that U(t; x)  U(t; w() (x)) D W(t; x) 2 C 1 ((a; b)  B" ()) ; where B" () D fx : jx  j < "g :

One Side Conditions on the Potential and Its Radial Derivative

rU(t; x)  x C ˛U(t; x)  C2 jxj U(t; x) ; whenever jxj is small : We observe that when U is homogeneous functions of degree ˛, the equality in condition (U2) is attained with C2 D 0; this assumption is satisfied also when we take into account potentials of the form U(t; x) D U˛ (x) C Uˇ (x) where U˛ is homogeneous of degree ˛, Uˇ is homogeneous of degree ˇ, 0 < ˇ < ˛, and U˛ is positive. Isolatedness of Collisions Instants In order to study the behavior of the solution at a collision, we firs perform a translation in such a way that the collision holds at the origin; next we introduce polar coordinates: p p x r D Mx  x D I ; s D ; r where s 2 E D fx : I 2 (x) D Mx  x D 1g belongs the ellipsoid of all the configurations having unitary moment of inertia. [U3h ] there exists a function U˜ defined on (a; b)  (E n), such that (on compact subsets of ((a; b)  E n)): ˜ s) ; lim r˛ U(t; rs) D U(t;

r!0

[U3l ] there exists a function U˜ defined on (a; b)  (E n), such that (on compact subsets of ((a; b)  E n)):  ˜ s) ; lim U(t; x) C M(t) log jxj D U(t; jxj!0

n-Body Problem and Choreographies

The following regularity theorem is proved in [15] based on a suitable variant of Sundman’s inequality and the asymptotic analysis of possible collisions outlined in the sequel. Theorem 16 Let x : (a; b) ! X be a generalized solution of the N-body problem, with a potential U satisfying [U0– U3]. Then collision instants are isolated in (a; b). Furthermore, if (a; b) is the finite maximal extension interval of x and no escape in finite time occurs then the number of collision instants is finite. Some remarks are in order:  A generalized solution does not solve the Euler–Lagrange equation in a distributional sense (the force field can not be locally integrable). Moreover its action needs not to be finite: one needs to prove it.  a priori our solution can have a huge set of collision instants (more than countable, but null measure);

the ˛ is bounded variation and d 2  ˛ 1C˛ 2 r˙j˙sj  C1 r˛ U(t; rs) ˛ (r; s)   r dt 2 r˙ C C2 r˛C U(t; rs) : r Generalized Sundman–Sperling Estimates The asymptotic analysis along a single collision (total or partial) trajectory goes back, in the classical case, to the works by Sundman ([84]), Wintner ([89]) and, in more recent years by Sperling, Pollard, Saari, Diacu and other authors (see for instance [40,47,75,76,79,83]). Now we are in a position to extend such estimate also to generalized solutions. Theorem 18

 there is no a priori bound on the total action on the whole interval (a; b), nor on the energy;

 The following asymptotic estimates hold: p 2 r ( t) 2C˛ (r jtj  log(jtj)) 2˛ 2 1 ¨  2 ( t) 2C˛ I K U 4  2˛ (2 C ˛)2 (  log jtj) :

 we have very weak assumptions on the potential and only a one side inequality on the radial component

 Let s be the normalized configuration of the colliding cluster s D x/r. Then

 there may be accumulation of partial collisions at a collision having more bodies;

 the theorem extends to the logarithmic potentials.

lim r2 j˙sj2 D 0

t!0

lim U(s(t)) D b < C1

t!0

Conservation Laws Even though they can not satisfy the differential equations in a distributional sense, generalized solutions satisfy a number of conservation laws. Theorem 17 Let x be a generalized solutions on (a; b). Then  The action A(x; [a; b]) on (a; b) is finite  The energy h is bounded and belongs to the Sobolev space W 1;1 ((a; b); R).  Lagrange–Jacobi inequality holds in the sense of measures: 1¨ ¯ I(x¯ (t))  2h(t)(x¯ (t)) C (2  ˛)U(t; x(t)) 2  ¯ ; 8t 2 (a; b) :  C2 j x¯ j U(t; x)  A monotonicity formula holds (extending Sundman’s inequality): let us consider the angular energy ˛ (r; s) :D r˛



 1 2 2 r j˙sj  U(t; rs) 2

 Moreover there exists the angular blow-up, that is angular scaled family (s(t)) is precompact for the topology of uniform convergence on compact sets of Rnf0g. As a further consequence we have the vanishing of the total angular momentum. In the same setting of the Theorem, assume that the potential U verifies the further assumption: ˜ s). [U4] limr!0 r˛C1 rT U(t; x) D rT U(t; Then we have lim dist(C b ; s(t)) D lim inf js(t)  s¯j D 0 ;

t!t 0

t!t 0 s¯2C b

where C b is the set of central configurations for U˜ at level b. Dissipation and McGehee Coordinates The main tool in proving the asymptotic estimates follow is the monotonicity formula, which is somehow equivalent to Sundman’s inequality. To fix our mind, let us think to a homogeneous potential U˛ . A possible way to to see this

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dissipation is to perform the change of variables (reminiscent of the McGehee coordinates):

Dr

2˛ 4

;

0 D

2˛  2C˛ 4 r0 4 r

to obtain the action functional depending on ( ; s); Z   4 2 0 2 1 A( ; s) D ( ) 2 2˛ 0   C 2 12 js 0 j2 C U˛ (s)   ˇ d ; where ˇ :D 2(2 C ˛)/(2  ˛) > 2. Here we have reparametrized the time as dt D r

2C˛ 2

d ;

vDr

t 2R;

where  D ( i ) i2k is a central configuration with k bodies. Proposition 8 The sequences x n and (dx n )/(dt) converge to the blow-up x¯ and its derivative x˙¯ respectively, in the H 1–topology. Moreover x¯ is a minimizing trajectory in the sense of Morse. Z T ¯  0]dt : [L(x¯ C ')  L(q)

Logarithmic Type Potentials

This is the coupling between a Duffing equation and the N-body angular system. When is increasing, it acts as a viscosity coefficient on the angular Lagrangian. A classical framework for the study of collisions is given by the McGehee coordinates [65] (here and below we assume, for simplicity of notations, all the masses be equal to one): ˛/2

i 2k;

for any compactly supported variation '.

0

r D jxj D I

x¯ i (t) D jtj2/(2C˛)  i ;

0

one proves that Z 1 2C˛  r 2 dt D C1 :  D

1/2

 The blow-up x¯ is parabolic: where a parabolic collision trajectory for the cluster k is the path

;

(y  s) ;

sD

x r

uDr

lim j x¯ n (t)j D lim

 n !0

 n !0

(y  y  ss) :

r(n t) p n t 2M(0) log(n t)

p  t 2M(0) log(n t) D C1

; ˛/2

The equation of motions become (here 0 denotes differentiation with respect to the new time variable ): ˛ r0 D rv ; v 0 D v 2 C juj2  ˛U˛ (s) ; s 0 D u ; ˛  2 0  1 vu  juj2 s C ˛U˛ (s)s C rU˛ (s) : u D 2 It is worthwhile noticing that the three last equations do not depen on r and hence are defined also for r D 0. In this constext, the monotonicity formula means that v is a Lyapunov function for the system.

for every t > 0. On the other, hand, looking at the differential equation, the (right) blow-up should be: ¯ :D t¯s ; q(t)

i 2k;

where s¯ is a central configuration for the system limit of a sequence s(n ) where (n )n is such that n ! 0. This limiting function is the pointwise limit of the normalized sequence x¯ n (t) :D

1 x¯ (n t) : p n 2M(0) log n

Unfortunately this path is not locally minimal for the limiting problem, indeed since, the sequence (x¨¯ n )n converges to 0 as n tends to C1, and hence this blow-up minimizes only the kinetic part of the action functional.

Blow-ups For every  > 0 let x  (t) D 2/(2C˛) x(t) : If fn gn is a sequence of positive real numbers such that s(n ) converges to a normalized configuration s¯, then 8t 2 (0; 1) : lim s(n t) D lim s(n ) D s¯ : n!1

It is not possible to define a blow-up suitable for logarithmic type potentials. Indeed, the natural scaling should be ¯ (n t), which x¯ n (t) :D 1 n x pdoes not converge, since, by the asymptotics jx(t)j ' jtj  log jtj, we have:

n!1

Hence the rescaled sequence will converge uniformly to the blow-up of x(t) relative to the colliding cluster k n (in t D 0).

Absence of Collision for Locally Minimal Paths As a matter of fact, solutions to the Newtonian n-body problem which are minimals for the action are, very likely, free of any collision. This fact was observed by the construction of suitable local variation arguments for the 2 and 3-body cases by Serra and Terracini [81,82]. The 4body case was treated afterward by Dell’Antonio ([35],

n-Body Problem and Choreographies

with a non completely rigorous argument) and then by A. Venturelli [87]. In general, the proof goes by the sake of the contradiction and involves the construction of a suitable variation that lowers the action in presence of a collision. A recent breakthrough in this direction is due of the neat idea, due to C. Marchal, [63], of averaging over a family of variations parametrized on a sphere. The method of averaged variations for Newtonian potentials has been outlined and exposed by Chenciner [25], and then fully proved and extended to ˛-homogeneous potentials and various constrained minimization problems by Ferrario and Terracini in [52]. This technique can be used in many of the known cases to prove that minimizing trajectories are collisionless. Of course, in some specific situations, other arguments can be useful, such as level estimates, on the infimum of the action on colliding paths [17,20,21,25]. However, these argument require global conditions on the potentials and can not be applied in the present setting, where we work under local assumptions about the singularities. We are in a position to prove the absence of collisions for locally minimal solutions when the potentials have quasi-homogeneous or logarithmic singularities. The first case is simpler, because one can take advantage the blow-up technique already exploited in [52]. On the other hand, when dealing with logarithmic potentials, the blowup technique is no longer available and we conclude proving directly some averaging estimates that can be used to show the nonminimality of large classes of colliding motions. The following results are exposed in [15]. Quasi-Homogeneous Potentials Let U˜ be the C 1 function defined on (a; b)(R k n) in the following way: ˜ rx/jxj) : ˜ x) D jxj˛ lim r˛ U(t; U(t; r!0

˜ With a slight abuse of notation, we denote U(x) D ˜  ; x). U˜ is homogeneous of degree ˛. U(t ? , [U6] there is a 2-dimensional linear subspace of V() say W, where U˜ is rotationally invariant:

˜ i w) D U(w) ˜ U(e ;

8w 2 W ;

Theorem 19 In addition to [U0], [U1], [U2 h ], [U3 h ], [U4 h ], [U5], assume that, for all  2  [U6] and [U7] hold. Then generalized solutions do not have collisions at the time t  . Remark 2 As our potential U˜ is homogeneous of degree ˛ the function '(x) D U˜ 1/˛ (x) is a non negative, homogeneous of degree one function, having now  as zero set. In most of our applications ' will be indeed a quadratic form. Assume that ' 2 splits in the following way: ' 2 (x) D KjW (x)j2 C ' 2 (W ? (x)) for some positive constant K. Then [U6] and [U7] are satisfied. Indeed, denoting w D W (x) and z D x  w we have, for every ı 2 W, ' 2 (x C ı) D Kjw C ıj2 C ' 2 (z) ˇ ˇ ˇ '(x) ' 2 (z) '(w) ˇˇ2 ˇ D Kˇ wC ı ˇ C K 2 jıj2 '(w) '(x) ' (x) ˇ ˇ2 ˇ '(x) ˇ '(w)  K ˇˇ wC ıˇ '(w) '(x) ˇ   '(w) 2 '(x) D' wC ı ; '(w) '(x) which is obviously equivalent to [U7]. ˜ Proposition 9 Assume U(x) D Q˛/2 (x) for some non negative quadratic form Q(x) D hAx; xi. Then assumptions [U6] and [U7] are satisfied whenever W is included in an eigenspace of A associated with a multiple eigenvalue. Given two potentials satisfying [U6] and [U7] for a common subspace W, their sum enjoys the same properties. On the other hand, the class of potentials satisfying [U6] and [U7] is not stable with respect to the sum of potentials. In order to deal with a class of potentials which is closed with respect to the sum, we introduce the following variant of the last Theorem. Theorem 20 In addition to [U0], [U1], [U2h], [U3h], [U4h], [U5], assume that U˜ has the form

8 2 [0; 2] ;

Rk

˜ U(x) D

N X

K ; (dist(x; V ))˛ D1

and ı 2 W there holds: [U7] for every x 2 1/˛  ˜ U(W (x)) ˜ ˜ W (x) U(x C ı)  U ˜ U(x) 1/˛   ˜ U(x) C ı ; ˜ W (x)) U(

where K are positive constants and V is a family of linear subspaces, with codim(V )  2, for every  D 1; : : : ; N. Then locally minimizing trajectories do not have collisions at the time t  .

where  W denotes the orthogonal projection onto W.

These two theorems extend to logarithmic potentials.

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Neumann Boundary Conditions and G-equivariant Minimizers Our analysis allows to prove that minimizers to the fixedends (Bolza) problems are free of collisions: indeed all the variations of our class have compact support. However, other type of boundary conditions (generalized Neumann) can be treated in the same way. Indeed, consider a trajectory which is a (local) minimizer of the action among all paths satisfying the boundary conditions x(0) 2 X 0 ;

x(T) 2 X 1 ;

where X 0 and X 1 are two given linear subspaces of the configuration space. Consider a (locally) minimizing path x¯ : of course it has not interior collisions. In order to exclude boundary collisions we have ensure that the class of variations preserve the boundary conditions. This can be achieved by imposing assumptions [U6] and [U7] to be fulfilled also by the restriction of the potential to the boundary subspaces X i . The analysis of boundary conditions was a key point in the paper [52], where symmetric periodic trajectories where constructed by reflections about given subspaces. Our results can be used to prove the absence of collisions also for G-equivariant (local) minimizers, provided the group G satisfies the Rotating Circle Property defined in Definition 3. Hence, existence of G-equivariant collisionless periodic solutions can be proved for the wide class of symmetry groups described in [14,52], for a much larger class of interacting potentials, including quasi-homogeneous and logarithmic ones. On the other hand, our results can be applied to prove that G-equivariant minimals are collisionless for many relevant symmetry groups violating the rotating circle property, such as the groups of rotations recently introduced in [50,51]. The Standard Variation In order to prove that local minimizers are free of collisions we are going to make use of the following class of variations: Definition 7 The standard variation associated to ı 2 ) and T is defined as 8 if 0  jtj  T  jıj ˆ <ı ı ı v (t) D (T  t) jıj if T  jıj  jtj  T : ˆ : 0 if jtj  T Our next goal is to find a standard variation v ı such that the trajectory q C v ı does not have a collision at t D 0 and Z C1 A :D [Lk (q C v ı )  Lk (q)]dt < 0 : 1

Let us introduce the function  Z C1  1 1 dt ; S(; ı) D  j t 2/(2C˛)  ıj˛ j t 2/(2C˛) j˛ 0 where ; ı 2 R2 . We first estimate the variation of the action as ı ! 0: Theorem 21 Let q D fqg i D ft 2/(2C˛)  i g, i D 1; : : : ; k be a parabolic collision trajectory and v ı a standard variation. Then, as ı ! 0   X ıi  ı j A D 2jıj1˛/2 mi m j S i   j ; CO(jıj): jıj i< j i; j2k

We observe that S(; ı) D jj1˛/2 jj1˛/2 S(; ı) and hence the sign of S depends on the angle between  and ı. Let Z C1 1 ˚ (#) D ˛/2  4 2 0 t ˛C2  2 cos # t ˛C2 C 1 

1 2˛

dt ;

˛ 2 (0; 2) :

t ˛C2 ˚ ( ) represents the potential differential needed for displacing the colliding particle from zero to ei . Expanding, we can express ˚ ( ) as a hypergeometric function:

  1 ˛C2 ˛C2 ˛(˛ C 2) ˚(#) D ˇ ; 2 ˛2 4 4 ! C1 X ˛/2 1 (1) k 2 k1 (cos #) k ˇ C k ˛ kD1   ˛ 1 k ˛ 1 k  C ; C C :  4 2 2 4 2 2 Some Properties of ˚ The value of ˚( ) ranges from C1 to some negative value, depending on ˛. However, thanks to some harmonic analysis one can prove that suitable averages are always negative: the first inequality is particularity useful for dealing with reflected triple collisions from the Lagrange central configuration:     2 2 ˚ C C˚  < 0 ; 8 2 [0; /2] : 3 3 A key remark was made by Christian Marchal: being the Newton potential a harmonic map averaging it on a sphere

n-Body Problem and Choreographies

results in a truncation in the interior. In fact, is not so much a matter of harmonicity. A crucial estimate was proved in [FT] about the averages of ˚ on circles: Theorem 22 For every ˛ > 0,  2 R3 nf0g and for every circle S Rd with center in 0, Z ˜ S) D 1 S(; ı)dı S(; jSj S Z 2 1 ˚( )d < 0 : D jj1˛/2 jıj1˛/2 2 0 Consider  D x i  x j and ı ranging in a circle. Then the above inequality implies the following principle, a generalization of Marcha’s statement in [25]: “it is more convenient (from the point of view of the integral of the potential on the time line) to replace one of the point particles with a homogeneous circle of same mass and fixed radius which is moving keeping its center in the position of the original particle.” Future Directions This article mainly focuses on the variational approach to the search of selected trajectories to the n-body problem. In our examples, the masses can very well be equal: hence the problem can not be regarded as a small perturbation of a simpler one and a full picture of the dynamics is out of reach. In contrast, the planetary n-body problem deals with systems where one of the bodies (a “star”) is much heavier than the others (the “planets”) and can be seen as a perturbation of a decoupled systems of one center problems. A small parameter " represents the order of the ratio between the mass of the planets and that of the star. The system is then called nearly integrable as it can be associated with a Hamiltonian of the form: H(I; ') D h(I) C " f (I; ') ; where I 2 R N and ' 2 T N (N is the number of degree of freedom) and " is a small parameter. In the three body problem we have N D 4, but the integrable limit possesses motions lying on T 2 (the product of two Keplerian orbits) so the integrable limit depends on less action variables than the number of degrees of freedom. For this reason the system is properly degenerate. The integrable limit possesses only quasi-periodic motions: in [56], the question whether these motions survive for positive values of " is settled. KAM Theory deals with the problem of persistence of such invariant tori but can not be directly applied to the planetary n-body problem, for its degeneracy. Nevertheless, the

existence of invariant tori has been recently achieved in the planar and spatial three body problem [8,16,18,57,78] and in the planetary many body problem [19,48]. The future researches will extend these results to some non perturbative settings, finding both regular motions, such as periodic, quasi–periodic or quasi–periodic trajectories and irregular, chaotic ones, through the application of suitable variational methods taking into account of collision trajectories.

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Books and Reviews Arnold VI, Kozlov V, Neishtadt A (2006) Mathematical aspects of classical and celestial mechanics [Dynamical systems. III]. 3rd edn. In: Encyclopaedia of Mathematical Sciences, vol 3. Springer, Berlin, xiv and p 518, translated from the Russian original by Khukhro E Diacu F, Holmes P (1996) Celestial encounters. The origins of chaos and stability. Princeton University Press, Princeton Meyer, Kenneth R (1999) Periodic solutions of the N-body problem, Lecture Notes in Mathematics, 1719. Springer, Berlin Moser J (1973) Stable and random motions in dynamical systems, With special emphasis on celestial mechanics, Hermann Weyl Lectures, the Institute for Advanced Study, Princeton NJ, Annals of Mathematics Studies, No. 77. Princeton University Press, Princeton Pollard H (1976) Celestial mechanics, Carus Mathematical Monographs, 18. Mathematical Association of America, Washington Saari D (2005) Collisions, rings, and other Newtonian N-body problems, CBMS Regional Conference Series in Mathematics, 104. American Mathematical Society, Providence RI, Washington, Published for the Conference Board of the Mathematical Sciences Siegel CL, Moser JK (1995) Lectures on celestial mechanics. Classics in Mathematics. Springer, Berlin, Translated from the German by Kalme CI, Reprint of the 1971 translation Stiefel EL, Scheifele G (1971) Linear and regular celestial mechanics. Perturbed two-body motion, numerical methods, canonical theory. Die Grundlehren der mathematischen Wissenschaften, Band 174. Springer, New York, Heidelberg

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Nekhoroshev Theory LAURENT N IEDERMAN1,2 1 Topologie et Dynamique – UMR 8628 du CNRS, Université Paris, Paris, France 2 Astronomie et Systèmes Dynamiques – UMR 8028 du CNRS, IMCCE, Paris, France Article Outline Glossary Definition of the Subject Introduction Exponential Stability of Constant Frequency Systems Nekhoroshev Theory (Global Stability) Applications Future Directions Appendix: An Example of Divergence Without Small Denominators Bibliography Glossary Quasi integrable Hamiltonian A Hamiltonian is quasi integrable if it is close to another Hamiltonian whose associated system is integrable by quadrature. Here, we will consider real analytic Hamiltonians on a domain D which admit a holomorphic extension on a complex strip DC around D and the closeness to the integrable Hamiltonian is measured with the supremum norm jj:jj1 over DC . Exponentially stable Hamiltonian An integrable Hamiltonian governs a system which admits a collection of first integral. We say that an integrable Hamiltonian h is exponentially stable if for any small enough Hamiltonian perturbation of h, the solutions of the perturbed system are at least defined over timescales which are exponentially long with respect to the inverse of the size of the perturbation. Moreover, the first integrals of the integrable system should remain nearly constant along the solutions of the perturbed system over the same amount of time. Nekhoroshev theorem (1977) There exists a generic set of real analytic integrable Hamiltonians which are exponential stable. Definition of the Subject We only know explicitly the solutions of the Hamiltonian systems which are integrable by quadrature. Unfortunately these integrable Hamiltonians are exceptional but many physical problems can be studied by a Hamiltonian system

which differs from an integrable one by a small perturbation. One of the most famous is the motions of the planets around the Sun which can be seen as a perturbation of the integrable system associated to the motion of noninteracting points around a fixed attracting center. Poincaré [70] considered the study of these quasi integrable Hamiltonian systems as the Problème général de la dynamique. This question was tackled with the Hamiltonian perturbation theories which were introduced at the end of the nineteenth century precisely to study the planetary motions (see  Hamiltonian Perturbation Theory (and Transition to Chaos)). Significant theorems concerning the mathematical justification of these methods were not put forward until the 1950s and later. A cornerstone of these results is the Kolmogorov–Arnold–Moser (KAM) theory which states that under suitable nondegeneracy and smoothness assumptions, for a small enough perturbation most of the solutions of a nearly integrable Hamiltonian system are quasi periodic, hence they are stable over infinite times. More accurately, KAM theory provides a set of large measure of invariant tori which support stable and global solutions of the perturbed system. But this set of stable quasi periodic solutions has a complicated topology: it is a Cantor set nowhere dense and without interior points. Numerically, it is extremely difficult to determine if a given solution is quasi periodic or not. Moreover, for a n degrees of freedom Hamiltonian system (hence a 2n-dimensional system) KAM theory provides n-dimensional tori. If n D 2, these two-dimensional invariant tori divide the three-dimensional energy level, therefore the solutions of the perturbed system are global and bounded over infinite times but an arbitrary large drift of the orbits is still possible for n  3. Actually, Arnold [1] proposed examples of quasi integrable Hamiltonian systems where an arbitrary large instability occurs for an arbitrary small perturbation. This is known as “Arnold diffusion”. Thus, results of stability for quasi integrable Hamiltonian systems which are valid for an open set of initial condition can only be proved over finite times. The first theorems of effective stability over finite but very long times were proved by Moser [58] and Littlewood [45] around an elliptic equilibrium point in an Hamiltonian system. Extending this kind of result in a general framework, Nekhoroshev [62,63] proved in 1977 a fundamental theorem of global stability which completes KAM theory and states that, for a generic set of integrable Hamitonian, Arnold diffusion can only occur over exponentially long times with respect to the inverse of the size of the perturbation.

Nekhoroshev Theory

These theories have been applied extensively in celestial mechanics but also to study molecular dynamics, beam dynamics, billiards, geometric numerical integrators, stability of certain solutions of nonlinear PDE (Schrodinger, wave, . . . ). Introduction We first recall that the canonical equations with a given scalar function H (the Hamiltonian) which is differentiable on an open set D R2n equipped with the symplecP tic form of Liouville ! D i dp i ^ dq i for (q1 ; : : : ; q n ) 2 Rn and their conjugate coordinates (p1 ; : : : ; p n ) 2 Rn can be written: p˙ D @q H (p; q) ;

q˙ D @ p H (p; q) :

Actually, this kind of system can be defined over any symplectic manifold. Moreover, a diffeomophism ˚ is symplectic or canonical if it preserves Liouville form: ˚  ! D !. Especially, for a Hamiltonian X, the flow ˚ Xt linked to the canonical system governed by X is a symplectic transformation over its domain of definition [54]. Such a diffeomorphism preserves the Hamilton equations: in the new variables (p; q) D ˚(P; Q) the transformed system is still canonical with the initial Hamiltonian K(P; Q) D H ı ˚(P; Q). Integrable and Quasi-integrable Hamiltonian Systems According to the theorem of Liouville–Arnold, under general topological conditions, a Hamiltonian system integrable by quadrature can be reduced to a system defined over ˝  T n for an open set ˝ Rn and the n-dimensional torus T n with a Hamiltonian which does not depend on the n angles. The new variables (I; ) 2 ˝  T n are called actions-angle variables and the system is linked to a Hamiltonian K(I), hence the equations take the trivial form I˙ D 0; ˙ D rK(I) where rK is the gradient of K and we obtain quasi-periodic motions of frequencies rK(I). If a small Hamiltonian perturbation is added to an integrable system, then in action-angle variables the considered system is governed by H ("; I; ) D h(I) C " f (I; )

where H 2 C ! (˝  T n ; R)

(1)

and the equations become: I˙ D "@ f (I; ) ;

˙ D r h(I) C "@I f (I; ) :

(2)

Hence, the variables are separated into two groups: the fast variables and the other variables which evolve slowly (the actions but also the angles with zero frequencies), over times of order 1/" their evolution may be of order 1.

Averaging Principle The averaging principle consists of the replacement of the initial system by its time average along the unperturbed flow ˚ ht linked to h which means that we consider the averaged Hamiltonian: hH i(I; ) D h(I) C "h f i(I; )   Z t 1 f (I; C sr h(I))ds : withh f i(I; ) D lim t!1 t 0 Actually, this average depends on the commensurability relations which are satisfied by the components of vector r h(I). More accurately, to a submodule M Z n , we associate its resonant zone: ZM D fI 2 Rn such that k :r h(I) D 0

if and only if k 2 Mg :

(3)

Let r 2 f0; : : : ; n  1g be the rank of M, there exists a unimodular matrix R 2 SL(n; Z) such that ! 2 M? if and only if the first r lines of R! are zeros. Hence, the symplectic transformation  D R and I D t R1 J deh(J) C fined over ˝  T n yields the new Hamiltonian e "e f (J; ) where J D (J1 ; J2 ) 2 Rr  Rnr and  D h(J) D (0; !(J)) when (1 ; 2 ) 2 T r  T nr such that re t R1 J 2 Z . Moreover, the second component !(J) 2 M Rnr does not satisfy any commensurability relation. Equivalently, on the set ZM , we can consider the torus T n as the product T r  T nr where the unperturbed flow is constant over T r and ergodic over T nr . Hence, at infinity, the time average he f i(J; ) is equal to the space average: he f i(J1 ; J2 ; 1 ) D



1 2

nr “ T nr

e f (J1 ; J2 ; 1 ; 2 ) d2 ; (4)

and the averaged system involves only variables which evolve slowly. For instance, to determine the solutions numerically, we only need to take a step of integration of order 1/" [41]. Actually, in the expression (4), we have only made a partial averaging by removing n  r angles. Hence, the considered system is simpler but, for a generic perturbation, it is only for the zero module M D f0g that we obtain an integrable Hamiltonian. It should be noticed that, for an arbitrary module M, the solutions of the average system can be unbounded. For instance, the Hamiltonian H (I1 ; I2 ; 1 ; 2 ) D I1 I2 C

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" sin( 2 ) is averaged with respect to the module M D ZEi with Ei D (1; 0) since 1 does not appear in the perturbation and it admits the unbounded solution (I1 (t); I2 (t); 1 (t); 2 (t)) D (0; "t; "t 2 /2; 0) which starts from the origin (0; 0; 0; 0) at t D 0. One can also notice that we have a maximal speed of drift of the action I 2 with respect to the size of the perturbation. On the other hand, a key observation for the proof of Nekhoroshev theorem is the fact that for an arbitrary module M, the canonical equations ensure that the variations of the actions under the averaged vector field with respect to M are located in the subspace spanned by M. This point will be specified in Sect. “The Initial Statement”, it will be the only place in Nekhoroshev’s proof where the canonical form of the equations is used. Hamiltonian Perturbation Theory The averaging principle is based on the idea that the oscillating terms discarded in averaging cause only small oscillations which are superimposed to the solutions of the averaged system. In order to prove the validity of the averaging principle, one should check that any solutions of the perturbed system remain close to the solution of the averaged system with the same initial condition. Especially, this will be the case if one finds a canonical transformation "-close to identity which transforms the perturbed Hamiltonian to its average. Hence we are reduced to a problem of normal form where one looks for a convenient system of coordinates which gives the simplest possible form to the considered system. Here, we consider the transformation ˚ X1 given by the time 1 flow of the Hamiltonian system governed by X(I; ) D "X1 (I; ). With Taylor formula, the transformed Hamiltonian H ı ˚ X1 admits the following expansion with respect to ": H ı ˚ X1 D h C "( f C r h(I):@ X 1 (I; )) C O("2 ) ;

and in order to obtain H ı ˚ X1 D h(I) C "h f i one must solve: r h(I):@ X1 (I; ) D  f C h f i ;

(5)

which is the homological equation, this is the central equation of perturbation theory. Since we are in the analytic setting, the function f admits the expansion X f k (I) exp(ik ) : k2Z n

Hence, for an averaging with respect to a resonant module M around the resonant zone ZM linked to M, the

homological equation admits the formal solution: X 1 (I; ) D

X k…M

f k (I) exp(ik ) ; i (k:r h(I))

(6)

thus one obtains a transformation which normalizes the Hamiltonian at first order in ". In the same way, one can formally eliminate the fast angles at all orders by looking at a transformation generP n ated by a Hamiltonian X(I; ) D n1 " X n (I; ). Indeed, the same type of homological equation appears at all order to determine X n for n > 1. This is the Linstedt method. With the previous construction, it is obvious that the normalizing transformation admits an expansion which is divergent if the denominators k:r h(I) for k … M are too close to zero on the considered domain in the action space. This is the well-known problem of the small denominators which was emphasized by Poincaré [70] in his celebrated theorem about nonexistence of analytic first integrals for a generic quasi integrable Hamiltonian. It is less known that even without small denominators, the classical perturbation theory can yield divergent expansions. This is the problem of the great multipliers according to Poincaré terminology which come from the successive differentiations. This problem is presented in the next subsection. Exponential Stability of Constant Frequency Systems The Case of a Single Frequency System The normalization of an analytic quasi-integrable Hamiltonian system with only one fast phase is one of the main problems in perturbation theory. This question appears naturally to compute the time of approximate conservation of the adiabatic invariants [17,49]. It has been accurately studied in [60] and [73] and we will focus our attention on this case where the phenomenon of divergence without small denominators appears in its simplest setting. Indeed, as in the Sect. “Hamiltonian Perturbation Theory”, one can build formally the Hamiltonian X(I; ) D P n n1 " X n (I; ) which generates a normalizing symplectic transformation and eliminates the fast angle in the perturbation. But, for a generic analytic quasi-integrable Hamiltonian, it can be shown [60,73] that perturbation theory yields a Gevrey-2 normalizing transformation over U  T n (i. e.: T 2 C 1 (U  T n ) with jj@ k T jj1  CM 2jkj k!2 where C; M are positive constants and k D (k1 ; : : : ; k2n ) 2 N 2n ; jkj D jk1 j C : : : C jk2n j; k! D k1 ! : : : k2n !) such that the initial perturbed Hamiltonian

Nekhoroshev Theory

H D h C " f is transformed into an integrable Hamiltonian h" (e I) with a one parameter family h" of scalar functions Gevrey-2 over U. Hence, the normalizing transformations are usually divergent. On the other hand, by general properties of Gevrey functions (see Sect. 3.3 in [53], and the references therein) if one considers the transformation generated by the truncated expansion N X

"n X n (I; )

n1

obtained after N steps of perturbation theory, then the transformed Hamiltonian is normalized up to a remainder of size " NC1 N!. In the appendix, we will study an example of a quasiintegrable Hamiltonian where this latter estimate cannot be improved and where the source of divergence of the normalizing transformation which comes from the successive differentiations in the construction can be emphasized. We see that the remainder of size " NC1 N! decreases rapidly before increasing to infinity, following Poincaré [70] this is a “convergent expansion according to astronomers” and a “divergent expansion according to geometers”. Now, we can use the process of “summation at the smallest term”: for a fixed " > 0, one obtains an optimal normalization with a truncation at order N such that jj@ 2 X N1 jj1 ' "jj@ 2 X N jj1 which yields N D E(1/") : Finally,pthe Stirling formula yields the size of the remainder: 2" exp(1/") which is exponentially small with respect to the inverse of the size of the perturbation. More generally, Marco and Sauzin [53] have proved in the same setting that starting from a Gevrey-˛ Hamiltonian (jj@ k H (I; )jj1  CM ˛jkj (k!)˛ ), one can build a normalizing transformation which is Gevrey-˛+1 and these estimates cannot be improved usually but the previous construction is still possible. Indeed, one can still make a summation at the smallest term and obtain a canonical change of coordinates which normalizes the Hamiltonian up to an exponentially small remainder with respect to the size of the perturbation. In the case where the averaged Hamiltonian is integrable, according to the mean value theorem, the speed of drift of the normalized action variables is at most exponentially slow. Since the size of the normalizing transformation is of order ", the initial actions admit at most a drift of size " over an exponentially long time.

The Case of a Strongly Nonresonant Constant Frequency System Systems with constant frequencies, hence h(I) D !:I for some constant vector ! 2 Rn , appear when we consider small nonlinear interactions of linear oscillatory systems or the action of quasi-periodic perturbations on linear oscillatory systems. In any case, the considered Hamiltonian can be written: H D !:I C f (I; ) with a small function f 2 C ! (˝  T n ; R) where ˝ is an open set in Rn . Moreover, we assume here that the frequency ! is a ( ; )-Diophantine vector for some positive constants

and , hence ! 2 ˝; :

˝; D ! 2 Rn such that jk:!j 

jjkjj1

 for all k 2 Z n nf(0; : : : ; 0g :

(7)

We recall that the measure of the complementary set of ˝; is of order O( ) for  > n  1. Under these assumptions, one can prove [13,31,45,71] that for a small enough analytic perturbation, the action variables of the unperturbed problem become quasi integrals of the perturbed system over exponentially long times, more specifically: Theorem 1 ([13,31,45,71]) Consider a Hamiltonian !:I C f (I; ) real analytic over a domain U  T n

Rn  T n which admit a holomorphic extension on a complex strip of width > 0 around U  T n in C 2n . The supremum norm for a holomorphic function on this complex strip is denoted jj:jj . There exists positive constants C1 ; C2 ; C3 ; C4 which depend only on , ; ; n such that if " D jj f jj < C1 , an arbitrary solution (I(t); '(t)) of the perturbed system associated to !:I C f (I; ) with an initial action I(t0 ) 2 U is defined at least over an exponentially long time and satisfies: 1

jjI(t)  I(0)jj  C2 " if jtj  C3 exp(C4 " 1C ) : (8)

The proof is based on the existence of a normalizing transformation up to an exponentially small error. This is possible since we have lower bounds on the small denominators and the growth of the coefficients in the normalizing expansion is reduced to a combinatorial problem. Finally, since the averaged Hamiltonian is integrable, the speed of drift of the action variables is at most exponentially slow.

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Nekhoroshev Theory (Global Stability) The Initial Statement Thirty years ago, Nekhoroshev [62,63] stated a global result of stability which is valid for a generic set of integrable Hamiltonian. Especially, we don’t have anymore a control on the small denominators as in the previous section but we have to handle the resonant zones. Nekhoroshev’s reasonings allow one to prove a global result of stability independent of the arithmetical properties of the unperturbed frequencies by taking into account the geometry of the integrable system. This is really a change of perspective with respect to the previous results. The key ingredient is to find a suitable property of the integrable Hamiltonian, namely the property of steepness introduced in the sequel, which ensures that a drift of the actions in the averaged system with respect to a module M Z n leads to an escape of the resonant zone ZM . More specifically, Nekhoroshev proved global results of stability over open sets of the following type: Definition 2 (exponential stability) Consider an open set ˝ Rn , an analytic integrable Hamiltonian h : ˝ ! R and action-angle variables (I; ') 2 ˝  T n where T D R/Z. For an arbitrary > 0, let O be the space of analytic functions over a complex neighborhood ˝ C 2n of size around ˝  T n equipped with the supremum norm jj:jj over ˝ . We say that the Hamiltonian h is exponentially stae ˝ if there exists positive conble over an open set ˝ stants ; C1 ; C2 ; a; b and "0 which depend only on h and e such that: ˝ i) h 2 O . ii) For any function H (I; ') 2 O such that jjH hjj D " < "0 , an arbitrary solution (I(t); '(t)) of the Hamiltonian system associated to H with an initial action e is defined over a time exp(C2 /" a ) and satisI(t0 ) in ˝ fies: jjI(t)I(t0 )jj  C1 "b

for jtt0 j  exp(C2 /" a ); (9)

a and b are called stability exponents. Remark Along the same lines, the previous definition can be extended to an integrable Hamiltonian in the Gevrey class (see [53]). Hence, for a small enough perturbation, the action variables of the unperturbed problem become quasi integrals of the perturbed system over exponentially long times.

In order to introduce the problem, we begin by a typical example of non-exponentially stable integrable Hamiltonian: h(I1 ; I2 ) D I1 I2 . Indeed, the perturbed system governed by h(I1 ; I2 ) C " sin( 2 ) admits the unbounded solution (I1 (t); I2 (t); 1 (t); 2 (t)) D (0; "t; "t 2 /2; 0) which starts from the origin (0; 0; 0; 0) at t D 0, hence a drift of the actions (I1 (t); I2 (t)) on a segment of length 1 occurs over a timespan of order 1/". The important feature in this example which has to be avoided in order to ensure exponential stability is the fact that the gradient r h(I1 ; 0) remains orthogonal to the first axis. Equivalently, the gradient of the restriction of h on this first axis is identically zero. Nekhoroshev [61,62,63] introduced the class of steep functions where this problem is avoided. The property of steepness is a quantitative condition of transversality for a real valued function differentiable over an open set ˝ Rn which involves all the affine subspaces which intersect ˝. Actually, steepness can be characterized by the following simple geometric criterion proved thanks to theorems of real subanalytic geometry [67]: Theorem 3 ([67]) A real analytic scalar function without critical points is steep if and only if its restriction to any proper affine subspace admits only isolated critical points. This is an extension of a previous similar result in the holomorphic case [42]. In this setting, Nekhoroshev proved the following: Theorem 4 ([62,63]) If the integrable Hamiltonian h is real analytic, does not admit critical points, is nondegenerate (jr 2 h(I)j 6D 0 for any I 2 ˝) and steep then h is exponentially stable. The set of steep functions is generic among sufficiently smooth functions. For instance, we have seen that the function xy is not steep but it can easily be shown that x yC x 3 is steep. Actually, a given function can be transformed into a steep function by adding higher order terms [61,62]. It can be noticed that the (quasi-)convex functions are the steepest functions since their restrictions to any affine subspaces admit at most one critical point which is also nondegenerate. The original proof of Nekhoroshev is global. It is based on a covering of the action space in open sets with controlled resonance properties where one can build resonant normal forms (i. e.: where only resonant harmonics are retained) up to an exponentially small remainder. The averaged Hamiltonian is not necessarily integrable but, thanks to the steepness of the integrable Hamiltonian, if a drift of the normalized actions occurs then it can only lead to a zone associated to resonances of lower multiplicity than

Nekhoroshev Theory

the initial one (i. e.: the resonant module admits a lower dimension). Eventually, after a short distance the orbits reach a resonance-free area (i. e.: the Fourier expansion of the normalized perturbation admits only nonresonant harmonics up to an exponentially small remainder). Then, the local normal form is integrable and yields the confinement of the action variables over the desired amount of time. Improved Versions of Nekhoroshev Theorem The articles of Nekhoroshev remained largely unnoticed in the western countries until Benettin, Galgani and Giorgilli [12] rewrote and clarified the initial proof in the convex case. Benettin and Gallavotti [13] proved that under an assumption of (quasi) convexity of the unperturbed Hamiltonian, the proofs of these theorems can be simplified. Indeed, after an averaging as in the steep case, the quasi convexity and the energy conservation ensure that the normalized Hamiltonian is an approximate Liapunov function over exponentially long time intervals. This allows one to confine the actions in the initial set where the considered orbit was located (we do not have to consider a drift over resonant areas of different multiplicity as in the original proof). Hence, the construction of a single normal form is enough to confine the actions in the convex case. Following this idea, Lochak [46,47] has significantly simplified the proof of Nekhoroshev estimates for the convex quasi integrable Hamiltonians. His reasonings are based on normalization around the periodic orbits of the integrable Hamiltonian which represent the worst type of resonances. Using convexity, Lochak obtains open sets around the periodic orbits which hold exponential stability. Then, Dirichlet theorem about simultaneous Diophantine approximation ensures that these open sets recover the whole action space and yield the global result. A remarkable feature of this proof is the fact that improved estimates can be obtained in the vicinity of resonances thanks to the relative abundance of periodic orbits in these areas. More specifically, periodicity corresponds to n  1 commensurability relations and we have already several commensurability relations at the resonances hence Dirichlet theorem can be applied on a lower dimensional space with better rates of approximation [46,47]. These improvements are important to extend Nekhoroshev estimates for large systems or infinite dimensional systems, they also fit with the speed of drift of the action variables in examples of unstable quasi integrable Hamiltonian (these points will be discussed in the sequel).

It can also be noticed that averaging along the periodic orbits of the integrable system is exactly a one phase averaging without small denominators. Toward sharp estimates, Lochak–Neishtadt [50] and Pöschel [71] have independently obtained the following: Theorem 5 ([50,71]) If the integrable Hamiltonian h is real analytic, does not admit critical points and convex over a domain ˝ Rn then h is exponentially stable over ˝ with the global exponents a D b D 1/2n. Moreover, around the resonant zones linked to a module of rank m < n, the integrable Hamiltonian h is exponentially stable with the improved exponents a D b D 1/2(n  m). The proof in [50], explicitly derived in [51], relies on Lochak periodic orbits method together with a refined procedure of averaging due to Neishtadt [60]. In [71], the original scheme of Nekhoroshev is combined with a refined study of the geometry of resonances which gives an accurate partition of the action space in open sets where the action variables are confined and also Neishtadt’s averaging procedure is used. Pöschel’s study of the geometry of resonances should also be important in the study of Arnold diffusion. This value of the time exponent (a D 1/2n) is expected to be optimal in the convex case according to heuristic reasonings of Chirikov [22], see also [46] on the speed of drift of Arnold diffusion. Actually, Marco–Sauzin [53] in the Gevrey cases and Marco–Lochak [52] in the analytic case have essentially proved the optimality of the improved exponent a D 1/2(n  2) in the doubly resonant zones starting from an example of unstable quasi integrable Hamiltonian given by Michel Herman. On the other hand, the previous studies except the original one of Nekhoroshev do not cover the cases of a time-dependent perturbation or a perturbed steep integrable Hamiltonian, despite their importance in physics. For instance, a time periodic perturbation of a convex Hamiltonian can be reduced to the time-independent perturbation of a quasi convex Hamiltonian in the extended phase space, but this is not the case for a general timedependent perturbation, where energy conservation cannot be used. This problem has been studied in the light of Nekhoroshev theory by Giorgilli and Zehnder [34] in connection with the dynamics of a particle in a time-dependent potential with a high kinetic energy. A new general study of the stability of steep integrable Hamiltonians has been carried out in [66]. This proof of stability relies on the mechanism of Nekhoroshev since we analyze the dynamics around resonances of any multiplicity and use local resonant normal forms but the original

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global construction is substituted with a local construction along each trajectory of the perturbed system. This construction is based on the approximation of the frequencies r h(I(t)) at certain times by rational vectors thanks to Dirichlet theorem of simultaneous Diophantine approximation as in Lochak’s proof of exponential stability in the convex case. This allows significant simplifications with respect to Nekhoroshev’s original proof. Moreover, the results of Lochak and Pöschel are generalized for the steep case since the exponents of stability derived in [66] give back the exponents a D b D 1/2n in the particular case of convex integrable Hamiltonians. In [68], Nekhoroshev estimates are proved for perturbations of integrable Hamiltonians which satisfy a strictly weaker condition of nondegeneracy than the initial condition of steepness. Indeed, the lack of steepness allows a drift of the action variables around the resonant zones. But, due to the exponential decay of the Fourier coefficients in the expansion of the Hamiltonian vector field, such a drift would be extremely slow if one consider resonant zones linked to a module spanned by integer vectors of large length. Thanks to this property, Morbidelli and Guzzo [57] have observed that the Hamiltonian h(I1 ; I2 ) D I12  ıI22 where ı is the square of a Diophantine number is non steep but nevertheless h is exponentially stable since p its isotropic directions are the lines spanned by (1; ˙ ı) which are “far” from the lines with rational slopes. This phenomenon has been studied numerically for the quadratic integrable Hamiltonians in [39]. Starting from this observation, a general weak condition of steepness which involves only the affine subspaces spanned by integer vectors has been stated in [68] with a complete proof of exponential stability in this setting. The point in this refinement lies in the fact that it allows one to exhibit a generic class of real analytic integrable Hamiltonians which are exponentially stable with fixed exponents of stability a and b while Nekhoroshev original theory provides a generic set of exponentially stable integrable Hamiltonians but with exponents of stability which can be arbitrarily small [68]. More specifically, we consider genericity in a measure theoretical sense since: Theorem 6 ([68]) Consider an arbitrary real analytic integrable Hamiltonian h defined on a neighborhood of the (n) closed ball BR of radius R centered at the origin in Rn . For almost any ˝ 2 Rn , the integrable Hamiltonian h˝ (x) D h(I)  ˝:I is exponentially stable with the exponents: aD

b 2 C n2

and

bD

1 : 2(1 C 2n n)

Finally, all these results can be generalized for quasi integrable symplectic mappings thanks to a theorem of inclusion of an analytic symplectic diffeomorphism into the flow linked to a real analytic Hamiltonian up to an exponentially small accuracy ([43,75] or [37] for a direct proof). KAM Stability, Exponential Stability, Nekhoroshev Stability A last point which should be emphasized is the link between KAM stability, exponential stability and Nekhoroshev stability which are cornerstones in the study of stability of analytic quasi integrable Hamiltonian systems. A first problem is the stability of the solutions in a neighborhood of a Lagrangian invariant torus over which an analytic Hamiltonian vectorfield induces a linear flow of frequency !. Then the considered Hamiltonian can be written as H (I; ) D !:I C F (I; ) in action-angle variables where the perturbation F (I; ) is analytic in a neighborhood of the origin and starts at order two in actions. In the case of a KAM torus, the frequency is strongly nonresonant (Diophantine). With a suitable rescaling, the expansion of the Hamiltonian takes the form considered in Theorem 1 and the exponential estimates of stability (9) are valid. In the general case (where the frequency can satisfy resonances of low order), the previous procedure cannot be applied. On the other hand, if one assumes that there are no resonances of order lower or equal to four: 8k 2 Z n nf0g such that jkj D jk1 j C : : : C jk n j  4

then jk:!j 6D 0 ;

it is possible to perform a Birkhoff’s normalization which reduces the studied Hamiltonian to:       f e e e I; e D !:e ICe Q e I; e ; (10) H I CF e (e where e Q is a quadratic form (the torsion) and F I; e ) D O3 (e I). At this point, one introduces the steepness condition required for the application of Nekhoroshev theory by imposing that the quadratic form e Q is sign definite (a weaker condition is considered in [24]). In a neighborhood of the considered torus, the problem is now reduced to the study of perturbations of a nonlinear convex integrable Hamiltonian and an exponential estimate of stability can be derived. These reasonings were stated in ([62], 2.2) and more specifically studied in [46,48]. A remarkable result of Morbidelli and Giorgilli [56] clearly shows that the two previous results which come

Nekhoroshev Theory

respectively from Hamiltonian perturbation theory and Nekhoroshev’s theorem are independent and can be superimposed. Indeed, in the case of a strongly nonresonant invariant torus (especially for a KAM torus) which admits moreover a sign definite torsion, one can state results of stability over superexponentially long times. The proof starts with a Birkhoff’s normal form like (10) but with an exponentially small remainder. Then, Nekhoroshev’s theorem is applied with a perturbation which is already exponentially small, hence we obtain a superexponential time of stability. More specifically, provided that the Birkhoff normal form is quasi-convex, results of stability over times of the order of exp(exp(cR1/ )) can be ensured for the solutions with an initial condition in a ball of radius R small enough around the invariant torus. Actually, the previous results were extended [46,48] for an elliptic equilibrium point or a lower dimensional torus in a Hamiltonian system except for an annoying problem of singularity in the action-angle transformation. This problem does not allow one to prove a stability theorem for a complete neighborhood of an elliptic equilibrium point. The corresponding theorems for all initial data were obtained in [28,65,72] where Nekhoroshev’s estimates were established without action-angle variables. Finally, the relationship between KAM and Nekhoroshev theory was considered in [32] and [23] where the existence of a sequence of nested domains in phase space which converge to the KAM set of invariant tori for the perturbed system and over which stability estimates are valid and growth occurs in a superexponential way was proven. Especially, on the initial domain we recover the usual statement of Nekhoroshev.

The same question arises in connection with the problem of energy equipartition in large systems of Fermi– Pasta–Ulam type and the conjecture of Boltzmann and Jeans [8,30]. The problem of existence of an approximate first integral over a long time but this time for a quasi integrable symplectic mapping appears to study the effective stability of the billiard flow near the boundary of a strictly convex domain in Rn [35]. For several degrees of freedom, results of stability over very long times for a quasi integrable Hamiltonian system were first obtained by Littlewood [45] about triangular Lagrangian equilibria in the three bodies problem. One can reduce this question to the study of a perturbed strongly nonresonant constant frequency system [31]. For effective computations, it is much more efficient to make a numerical summation at the smallest term instead of plugging the data into an abstract theorem which usually gives poor estimates. Following this scheme, Giorgilli and different coauthors have obtained effective stability results in celestial mechanics with realistic physical parameters (see [8,27,30,33,74]). The same situation of a perturbed strongly nonresonant constant frequency system appears in the study of stability of symplectic numerical integrators [14,41,55]. In the realm of PDE’s, results of stability around finite dimensional nonresonant tori can be proved ([4,11,18,44] and  Perturbation Theory for PDEs for surveys). The remarkable paper of Bambusi and Grebert [6] gives an extension of the previous results for infinite dimensional nonresonant tori.

Applications

Application of the Global Nekhoroshev Theory

With the remaining space, we only give glimpses of application of the previous theorems in physics and astronomy. We mainly quote surveys in the sequel and these references do not form at all a complete list.

Here, we look for global results of stability for perturbed nonlinear integrable Hamiltonian systems. This situation appears in celestial mechanics where the unperturbed system is often properly degenerate, namely the number of constants of motion exceeds the number of degrees of freedom. This is the case for the Kepler problem which yields the integrable part of the planetary n-bodies problem (i. e.: the approximation of the n-bodies problem corresponding to the motion of the planets in the solar system). A study of this system in the light of Nekhoroshev’s theory was given in [64], suggesting a modification of the original statement of Nekhoroshev by considering the proper degeneracies of this system. The question of stability in celestial mechanics was also considered for the asteroid belt [57] where additional degeneracies and resonances appear (see also [40]).

The Case of a Constant Frequency Integrable System The question of stability of a perturbed single frequency system corresponds to the problem of preservation of the adiabatic invariants which appears in numerous physical problems [2,59]. Especially, for the applications in plasma physics, we can mention the beautiful survey of Northrop [69]. The problem of charge trapping by strong nonuniform magnetic fields (Van Allen belts, magnetic bottles) can also be tackled by means of the adiabatic invariant theory [15].

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We have seen that Nekhoroshev’s theory also allows one to study stability around an elliptic equilibrium point in a Hamiltonian system with a sign definite torsion. But the most famous example of such an equilibrium in astronomy, namely the stability of an asteroid located at the top of an equilateral triangle with the Sun and Jupiter (the Lagrangian points L4, L5) cannot be tackled with the previous theorems in the convex case. It can be shown that with the actual masses of the Sun and Jupiter, the problem of stability of the Lagrangian points could be reduced to a study of a small perturbation of an integrable Hamiltonian of three degrees of freedom whose 3-jet satisfies a condition which implies steepness [10]. It allows one to prove a confinement over very long time intervals for asteroids located close enough to the Lagrangian points L4 or L5. Other applications of Nekhoroshev’s stability at an elliptic equilibrium point are the stability of the Riemann Ellipsoid [29], the fast rotation of a rigid body [9]. A Nekhoroshev-like theory has also been developed for beam dynamics [25,26]. From a numerical point of view, a new spectral formulation of the Nekhoroshev theorem has been introduced [38]. This allows one to recognize whether or not the motion in a quasi-integrable Hamiltonian system is in a Nekhoroshev regime (i. e. the action coordinates are eventually subject to an exponentially slow drift) by looking at the Fourier spectrum of the solutions. In a geometric setting, an extension of Nekhoroshev’s results for perturbations of convex, noncommutatively integrable Hamiltonian systems has been given in [16]. Finally, no general results like the Nekhoroshev theorem are known yet for large Hamiltonian systems or for Hamiltonian PDE’s seen as infinite dimensional Hamiltonian systems. But a number of quasi-Nekhoroshev theorems for special systems of this type have been proved mostly by Bourgain and Bambusi (see [5] for large systems and [3,7,18] for PDE’s).

nents are at the basis of Nekhoroshev-type results which are obtained in large systems [5] or in PDE [3,7,18] hence their generalization would be important. The global stability results considered so far are obviously valid only if one takes into account the most pessimistic estimations in the whole phase space. On the other hand, we have just seen that these results can be improved locally. It would be relevant to make a study of the average exponent of stability. It could be a space average of these exponents by considering all initial conditions or, in certain examples, the time average of these exponents by taking into account the variations of the speed of drift of the actions under Arnold’s diffusion. The applications of Nekhoroshev’s theory in Astronomy is an active field either analytically or in a numerical way [36]. The relevance of Nekhoroshev’s estimates for statistical mechanics and thermodynamics is an important question which is tackled with the problem of energy equipartition in the Fermi-Pasta-Ulam (FPU) model (see [19,20] and the references therein). Finally, a partial generalization of KAM theory to PDEs has been carried out during the last twenty years by many high level mathematicians, but the theory developed up to now only allows one to show that finite dimensional invariant tori persist under perturbation. Thus, most of the initial data are outside invariant tori. It is therefore clear that it would be very important to understand the behavior of solutions starting outside the tori. This is also related to the problem of estimating the time of existence of the solutions of hyperbolic PDEs in compact domains, a problem that is one of the most important open questions in relation to hyperbolic PDEs. Up to now Nekhoroshev’s theorem has been generalized to PDE’s only in order to deal with small amplitude solutions but the obtention of really global results of stability would be a challenge (Kuksin has announced these kinds of theorems for the KdV equation, see  Perturbation Theory for PDEs).

Future Directions The analyticity of the studied systems is only needed for the construction of the normal forms up to an exponentially high remainder. On the other hand, the steepness condition is generic for Hamiltonians of finite but sufficiently high smoothness [68]. It would be natural to prove the analogous stability theorems in the case of smooth functions which would give stability over polynomially long times. Another question is the extension of Lochak’s mechanism of stabilization around resonances for non quasiconvex integrable Hamiltonians. These improved expo-

Appendix: An Example of Divergence Without Small Denominators We would like to develop the following example of Neishtadt [60] where the phenomenons of divergence without small denominators and summation up to an exponentially small remainder appear in its simplest setting (see also [76] for another example). Consider the quasi integrable system governed by the Hamiltonian H (I1 ; I2 ; 1 ; 2 ) D I1  "[I2  cos( 1 ) f ( 2 )]

defined over R2  T 2 ;

(11)

Nekhoroshev Theory

Especially, with g(z) D 1/exp(z)  1, we have:

with X ˛m cos(m ) ; f ( ) D m m1 where ˛m D e˛m for 0 < ˛  1, this last choice corresponds to the exponential decay of the Fourier coefficients for a holomorphic function. Indeed, f 0 ( ) D Im(g(˛ C i )) where g(z) D (exp(z)  1)1 and the complex pole of f which is closest to the real axis is located at i˛. Conversely, all real function which admits a holomorphic extension over a complex strip of width ˛ (i. e.: jIm(z)j  ˛) has Fourier coefficients bounded by (C˛m )m2N  for some constant C > 0. As in the Sect. “Hamiltonian Perturbation Theory”, it is possible to eliminate formally completely the fast angle 1 in the perturbation without the occurrence of any small denominators. Indeed, one can consider a normalizing transformaP tion generated by X( 1 ; 2 ) D n1 "n X n ( 1 ; 2 ) where the functions X n satisfy the homological equations: @ 1 X1 ( 1 ; 2 ) D cos( 1 ) f ( 2 ) and @ 1 X n ( 1 ; 2 ) D @ 2 X n1 ( 1 ; 2 ) : The solutions can be written X n ( 1 ; 2 ) D cos( 1  n 2 ) f (n1) ( 2 ), hence: X ( 1 ; 2 ) D

X ˛m mn m2   n2N m2N      cos 1  n sin m 2 C n 2 2 X

"n

and, for instance, X

 X X e˛m ("m)2kC1 ;0 D 2 2 m 



m2N k2N

which is divergent for all " > 0 since "  1/m for m large enough. We see that divergence comes from coefficients arising with successive differentiations. On the other hand, if one considers the transformaPN n tion generated by the truncated Hamiltonian n1 " X n ( 1 ; 2 ) then the transformed Hamiltonian becomes H (I1 ; I2 ; 1 ; 2 ) D I1  "I2 C " NC1 @ 2 X N ( 1 ; 2 ) ;

  where @ 2 X N ( 1 ; 2 ) D cos 1  N 2 f (N) ( 2 ).

  @ 2 X2N ( 1 ; 2 ) D  cos( 1 )Re g (2N1) (˛ C i 2 ) 

@ 2 X2NC1 ( 1 ; 2 ) D sin( 1 )Im g

(2N)

for N > 0 ;  (˛ C i 2 ) for N  0 :

Now, around the real axis, the main term in the asymptotic expansion of g (n) (z) as n goes to infinity is given by the derivative of the polar term 1/z in the Laurent expansion of g at 0. Indeed, we consider the function h(z) D g(z)  1/z which is real analytic and admits an analyticity width of size 2 around the real axis. Hence, Cauchy estimates ensure that for n large enough and z in the strip of width  around the real axis, the derivative h(n) (z) becomes negligible with respect to the derivative of 1/z. Consequently, g (n) (z) is equivalent to (1)n n!/z nC1 as n goes to infinity for z in the strip of width  around the real axis. Hence, the remainder @ 2 X N ( 1 ; 2 ) has a size of order (N  1)!/˛ N for N large. This latter estimate cannot be improved, since @ 2 X2N (; 0) and @ 2 X2NC1 (/2; ˛ tan(/4N C 2)) admit the same size of order (N  1)!/˛ N for N large. Now, we can make a “summation at the smallest term” as in Sect. “The Case of a Single Frequency System” to obtain an optimal normalization with an exponentially small remainder. Bibliography Primary Literature 1. Arnold VI (1964) Instability of dynamical systems with several degrees of freedom. Sov Math Dokl 5:581–585 2. Arnold VI, Kozlov VV, Neishtadt AI (2006) Mathematical aspects of classical and celestial mechanics, 3rd revised edn. In: Encyclopaedia of Mathematical Sciences 3. Dynamical Systems 3. Springer, New York 3. Bambusi D (1999) Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations. Math Z 230(2):345–387 4. Bambusi D (1999) On long time stability in Hamiltonian perturbations of non-resonant linear PDEs. Nonlinearity 12(4):823– 850 5. Bambusi D, Giorgilli A (1993) Exponential stability of states close to resonance in infinite-dimensional Hamiltonian systems. J Stat Phys 71(3–4):569–606 6. Bambusi D, Grébert B (2006) Birkhoff normal form for partial differential equations with tame modulus. Duke Math J 135(3):507–567

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7. Bambusi D, Nekhoroshev NN (2002) Long time stability in perturbations of completely resonant PDE’s. Acta Appl Math 70(1– 3):1–22 8. Benettin G (2005) Physical applications of Nekhoroshev theorem and exponential estimates. In: Giorgilli A (ed) Cetraro (2000) Hamiltonian Dynamics, Theory and Applications. Springer, New York 9. Benettin G, Fasso F (1996) Fast rotations of the rigid body: A study by Hamiltonian perturbation theory, I. Nonlinearity 9(1):137–186 10. Benettin G, Fasso F, Guzzo M (1998) Nekhorochev stability of L4 and L5 in the spatial restricted three body problem. Regul Chaotic Dyn 3(3):56–72 11. Benettin G, Fröhlich J, Giorgilli A (1988) A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom. Commun Math Phys 119(1):95–108 12. Benettin G, Galgani L, Giorgilli A (1985) A proof of Nekhorochev’s theorem for the stability times in nearlyintegrable Hamiltonian systems. Celest Mech 37:1–25 13. Benettin G, Gallavotti G (1986) Stability of motions near resonances in quasi-integrable Hamiltonian systems. J Stat Phys 44(3–4):293–338 14. Benettin G, Giorgilli A (1994) On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms. J Stat Phys 74(5–6):1117– 1143 15. Benettin G, Sempio P (1994) Adiabatic invariants and trapping of a point charge in a strong non-uniform magnetic field. Nonlinearity 7(1):281–303 16. Blaom AD (2001) A geometric setting for Hamiltonian perturbation theory. Mem Am Math Soc 727(xviii):112 17. Bogolyubov NN, Mitropol’skij YA (1958) Asymptotic Methods in the Theory of Nonlinear Oscillations, 2nd edn. Nauka, Moscow. Engl. Transl. (1961) Gordon and Breach, New York 18. Bourgain J (2004) Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations. Ergod Theory Dyn Syst 24(5):1331–1357 19. Carati A, Galgani L, Giorgilli A, Ponno A (2002) The Fermi– Pasta–Ulam Problem. Nuovo Cimento B 117:1017–1026 20. Carati A, Galgani L, Giorgilli A (2006) Dynamical Systems and Thermodynamics. In: Françoise JP, Naber GL, Tsun TS (eds) Encyclopedia of mathematical physics. Elsevier, Amsterdam 21. Celletti A, Giorgilli A (1991) On the stability of the Lagrangian points in the spatial restricted problem of three bodies. Celest Mech Dyn Astron 50(1):31–58 22. Chirikov BV (1979) A universal instability in many dimensional oscillator systems. Phys Rep 52:263–279 23. Delshams A, Gutierrez P (1996) Effective Stability and KAM Theory. J Diff Eq 128(2):415–490 24. Dullin H, Fassò F (2004) An algorithm for detecting directional quasi-convexity. BIT 44(3):571–584 25. Dumas HS (1993) A Nekhoroshev-like theory of classical particle channeling in perfect crystals. In: Jones CKRT et al (ed) Dynamics reported. Expositions in dynamical systems, new series, vol 2. Springer, Berlin 26. Dumas HS (2005) Mathematical theories of classical particle channeling in perfect crystals. Nucl Inst Meth Phys Res Sect B (Beam Interactions with Materials and Atoms) 234(1–2):3–13 27. Efthymiopoulos C, Giorgilli A, Contopoulos G (2004) Nonconvergence of formal integrals. II: Improved estimates for the op-

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timal order of truncation. J Phys A (Math Gen) 37(45):10831– 10858 Fassò F, Guzzo M, Benettin G (1998) Nekhoroshev-stability of elliptic equilibria of Hamiltonian systems. Commun Math Phys 197(2):347–360 Fassò F, Lewis D (2001) Stability properties of the Riemann ellipsoids. Arch Ration Mech Anal 158(4):259–292 Giorgilli A (1998) On the problem of stability for near to integrable Hamiltonian systems. In: Proceedings of the International Congress of Mathematicians Berlin 1998. Documenta Mathematica III:143–152 Giorgilli A, Delshams A, Fontich E, Galgani L, Simó C (1989) Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three bodies problem. J Diff Equa 77:167–198 Giorgilli A, Morbidelli A (1997) Invariant KAM tori and global stability for Hamiltonian systems. Z Angew Math Phys 48(1):102–134 Giorgilli A, Skokos C (1997) On the stability of the Trojan asteroids. Astron Astroph 317:254–261 Giorgilli A, Zehnder E (1992) Exponential stability for time dependent potential. ZAMP 43:827–855 Gramchev T, Popov G (1995) Nekhoroshev type estimates for billiard ball maps. Ann Inst Fourier 45(3):859–895 Guzzo M (2003) Nekhoroshev stability of asteroids. Triennal report 2000–2003 of Commission 7-Celestial Mechanics and dynamical Astronomy of the IAU; Reports on Astronomy, 1999– 2002. Transactions of the International Astronomical Union, vol XXVA, Astronomical Society of the Pacific Guzzo M (2004) A direct proof of the Nekhoroshev theorem for nearly integrable symplectic maps. Ann Henri Poincaré 5(6):1013–1039 Guzzo M, Benettin G (2001) A spectral formulation of the Nekhoroshev theorem and its relevance for numerical and experimental data analysis. Discret Contin Dyn Syst Ser B 1(1):1– 28 Guzzo M, Lega E, Froeschlé C (2006) Diffusion and stability in perturbed non-convex integrable systems. Nonlinearity 19(5):1049–1067 Guzzo M, Morbidelli A (1997) Construction of a Nekhoroshevlike result for the asteroid belt dynamical system Celest. Mech Dyn Astron 66:255–292 Hairer E, Lubich C, Wanner G (2006) Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, 2nd edn. Springer Series in Computational Mathematics, vol 31. Springer, New York Ilyashenko IS (1986) A steepness test for analytic functions. Russ Math Surv 41:229–230 Kuksin S, Pöschel J (1994) On the inclusion of analytic symplectic maps in analytic Hamiltonian flows and its applications. In: Kuksin S, Lazutkin VF, Pöschel J (eds) Proceedings of the 1991 Euler Institute Conference on Dynamical Systems. Prog NonLin Diff Equ App (12) Birkhäuser, Basel Kuksin SB (2006) Hamiltonian PDEs (with an appendix by Dario Bambusi). In: Hasselblatt B (ed) Handbook of dynamical systems, vol 1B. Elsevier, Amsterdam Littlewood JE (1959) On the equilateral configuration in the restricted problem of the three bodies. Proc London Math Soc 9(3):343–372 Lochak P (1992) Canonical perturbation theory via simultaneous approximation. Russ Math Surv 47:57–133

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47. Lochak P (1993) Hamiltonian perturbation theory: Periodic orbits, resonances and intermittency. Nonlinearity 6(6):885– 904 48. Lochak P (1995) Stability of Hamiltonian systems over exponentially long times: the near-linear case. In: Dumas HS, Meyer K, Schmidt D (eds) Hamiltonian dynamical systems – History, theory and applications. IMA conference proceedings series, vol 63. Springer, New York, pp 221–229 49. Lochak P, Meunier C (1988) Multiphase averaging methods for Hamiltonian systems. Appl Math Sci Series, vol 72. Springer, New York 50. Lochak P, Neishtadt AI (1992) Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian. Chaos 2(4):495–499 51. Lochak P, Neishtadt AI, Niederman L (1994) Stability of nearly integrable convex Hamiltonian systems over exponentially long times. In: Kuksin S, Lazutkin VF, Pöschel J (eds) Proceedings of the 1991 Euler Institute Conference on Dynamical Systems. Prog NonLin Diff Equ App (12). Birkhäuser, Basel 52. Marco JP, Lochak P (2005) Diffusion times and stability exponents for nearly integrable analytic systems. Central Eur J Math 3(3):342–397 53. Marco JP, Sauzin D (2003) Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian Systems. Publ Math Inst Hautes Etudes Sci 96:199–275 54. Meyer KR, Hall GR (1992) Introduction to Hamiltonian dynamical systems and the N-Body problem. Applied Mathematical Sciences, vol 90. Springer, New York 55. Moan PC (2004) On the KAM and Nekhoroshev theorems for symplectic integrators and implications for error growth. Nonlinearity 17(1):67–83 56. Morbidelli A, Giorgilli A (1995) Superexponential stability of KAM tori. J Stat Phys 78:1607–1617 57. Morbidelli A, Guzzo M (1997) The Nekhoroshev theorem and the asteroid belt dynamical system. Celest Mech Dyn Astron 65(1–2):107–136 58. Moser J (1955) Stabilitätsverhalten Kanonisher Differentialgleichungssysteme. Nachr Akad Wiss Göttingen, Math Phys Kl IIa 6:87–120 59. Neishtadt AI (1981) On the accuracy of conservation of the adiabatic invariant. Prikl Mat Mekh 45:80–87. Translated in J Appl Math Mech 45:58–63 60. Neishtadt AI (1984) The separation of motions in systems with rapidly rotating phase. J Appl Math Mech 48:133–139 61. Nekhorochev NN (1973) Stable lower estimates for smooth mappings and for gradients of smooth functions. Math. USSR Sb 19(3):425–467 62. Nekhorochev NN (1977) An exponential estimate of the time of stability of nearly integrable Hamiltonian systems. Russ Math Surv 32:1–65 63. Nekhorochev NN (1979) An exponential estimate of the time of stability of nearly integrable Hamiltonian systems 2. Trudy Sem Petrovs 5:5–50. Translated In: Oleinik OA (ed) Topics in Modern Mathematics. Petrovskii Semin, vol 5. Consultant Bureau, New York

64. Niederman L (1996) Stability over exponentially long times in the planetary problem. Nonlinearity 9(6):1703–1751 65. Niederman L (1998) Nonlinear stability around an elliptic equilibrium point in an Hamiltonian system. Nonlinearity 11:1465– 1479 66. Niederman L (2004) Exponential stability for small perturbations of steep integrable Hamiltonian systems. Erg Theor Dyn Syst 24(2):593–608 67. Niederman L (2006) Hamiltonian stability and subanalytic geometry. Ann Inst Fourier 56(3):795–813 68. Niederman L (2007) Prevalence of exponential stability among nearly integrable Hamiltonian systems. Erg Theor Dyn Syst 27(3):905–928 69. Northrop TG (1963) The adiabatic motion of charged particles. Interscience Publishers, New York 70. Poincaré H (1892) Méthodes Nouvelles de la Mécanique Céleste, vol 4. Blanchard, Paris 71. Pöschel J (1993) Nekhorochev estimates for quasi-convex Hamiltonian systems. Math Z 213:187–217 72. Pöschel J (1999) On Nekhoroshev’s estimate at an elliptic equilibrium. Int Math Res Not 1999(4):203–215 73. Ramis JP, Schäfke R (1996) Gevrey separation of fast and slow variables. Nonlinearity 9(2):353–384 74. Steichen D, Giorgilli A (1998) Long time stability for the main problem of artificial satellites. Cel Mech 69:317–330 75. Treschev DV (1994) Continuous averaging in Hamiltonian systems. In: Kuksin S, Lazutkin VF, Pöschel J (eds) Proceedings of the 1991 Euler Institute Conference on Dynamical Systems. Prog NonLin Diff Equ App (12). Birkhäuser, Basel 76. Valdinoci E (2000) Estimates for non-resonant normal forms in Hamiltonian perturbation theory. J Stat Phys 101(3–4):905– 919

Books and Reviews Arnold VI (1983) Geometrical methods in the theory of ordinary differential equations. Transl. from the Russian by Joseph Szuecs, Mark Levi (ed) Grundlehren der Mathematischen Wissenschaften 250. Springer, New York Arnold VI, Kozlov VV, Neishtadt AI (2006) Mathematical aspects of classical and celestial mechanics, 3rd revised edn. Encyclopaedia of Mathematical Sciences 3. Dynamical Systems 3. Springer, New York Benettin G (2005) Physical applications of Nekhoroshev theorem and exponential estimates. In: Giorgilli A (ed) Cetraro (2000) Hamiltonian Dynamics, Theory and Applications. Springer, New York Giorgilli A (2003) Exponential stability of Hamiltonian systems. Dynamical systems, Part I. Hamiltonian systems and celestial mechanics. Selected papers from the Research Trimester held in Pisa, Italy, February 4–April 26, 2002. Pisa: Scuola Normale Superiore. Pubblicazioni del Centro di Ricerca Matematica Ennio de Giorgi. Proceedings, 87–198 Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems, 2nd edn. Applied Mathematical Sciences 59. Springer, New York

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Non-linear Dynamics, Symmetry and Perturbation Theory in GIUSEPPE GAETA Dipartimento di Matematica, Università di Milano, Milan, Italy

Article Outline Glossary Definition of the Subject Introduction Symmetry of Dynamical Systems Perturbation Theory: Normal Forms Perturbative Determination of Symmetries Symmetry Characterization of Normal Forms Symmetries and Transformation to Normal Form Generalizations Symmetry for Systems in Normal Form Linearization of a Dynamical System Further Normalization and Symmetry Symmetry Reduction of Symmetric Normal Forms Conclusions Future Developments Additional Notes Bibliography Glossary Perturbation theory A theory aiming at studying solutions of a differential equation (or system thereof), possibly depending on external parameters, near a known solution and/or for values of external parameters near to those for which solutions are known. Dynamical system A system of first order differential equations dx i /dt D f i (x; t), where x 2 M, t 2 R. The space M is the phase space for the dynamical system, e D M  R is the extended phase space. When f and M is smooth we say the dynamical system is smooth, and for f independent of t, we speak of an autonomous dynamical system. e mapping Symmetry An invertible transformation of M solutions into solutions. If the dynamical system is smooth, smoothness will also be required on symmetry transformations; if it is autonomous, it will be natural e to consider transformations of M rather than of M. Symmetry reduction A method to reduce the equations under study to simpler ones (e. g. with less dependent variables, or of lower degree) by exploiting their symmetry properties.

Normal form A convenient form to which the system of differential equations under study can be brought by means of a sequence of change of coordinates. The latter are in general well defined only in a subset of M, possibly near a known solution for the differential equations. Further normalization A procedure to further simplify the normal form for a dynamical system, in general making use of certain degeneracies in the equations to be solved in the course of the normalization procedure. Definition of the Subject Given a differential equation or system of differential equations  with independent variables  a 2 )  Rq and dependent variables x i 2 M  R p , a symmetry of  is an invertible transformation of the extended phase space e D )  M into itself which maps solutions of  into M (generally, different) solutions of . The presence of symmetries is a non-generic feature; correspondingly, equations with symmetry have some special features. These can be used to obtain information about the equation and its solutions, and sometimes allow one to obtain explicit solutions. The same applies when we consider a perturbative approach to the equations: taking into account the presence of symmetries guarantees the perturbative expansion has certain specific features (e. g. some terms are not allowed) and hence allows one to deal with simplified expansions and equations; thus this approach can be of great help in providing explicit solutions. As mentioned above, symmetry is a non-generic feature: if we take a “generic” equation or system, it will not have any symmetry property. What makes the symmetry approach useful and widely applicable is a remarkable fact: many of the equations encountered in applications, and especially in physical and related ones (mechanical, electronic, etc.) are symmetric; this in turn descends from the fact that the fundamental equations of physics have a high degree of symmetry. Thus, symmetry-based methods are at the same time “non-generic” in a mathematical sense, and “general” in a physical, or more generally real-world, sense. Introduction Symmetry has been a major ingredient in the development of quantum perturbation theory, and is a fundamental ingredient of the theory of integrable (Hamiltonian and nonHamiltonian) systems; yet, the use of symmetry in the context of general perturbation theory is rather recent.

Non-linear Dynamics, Symmetry and Perturbation Theory in

From the point of view of nonlinear dynamics, the use of symmetry has become widespread only through equivariant bifurcation theory; even in this case, attention has been mostly confined to linear symmetries. Also, in recent years the theory and practice of symmetry methods for differential equations became increasingly popular and has been applied to a variety of problems (to a large extent, following the appearance of the book by Olver [151]). This theory is deeply geometrical and deals with symmetries of general nature (provided that they are described by smooth vector fields), i. e. in this context there is no reason to limit attention to linear symmetries. In this article we look at the basic tools of perturbation theory, i. e. normal forms (first introduced by Poincaré more than a century ago for general dynamical systems; the Hamiltonian case being studied in its special features by Birkhoff several decades ago) and study their interaction with symmetries, with no limitation to linear ones. See the articles  Normal Forms in Perturbation Theory,  Hamiltonian Perturbation Theory (and Transition to Chaos) for an introduction to Normal Forms. We focus on the most basic setting, i. e. systems having a fixed point (at the origin) and perturbative expansions around this; thus our theory is entirely local. We also limit to the discussion of general vector fields, i. e. we will not discuss the formulation one would obtain for the special case of Hamiltonian vector fields  Hamiltonian Perturbation Theory (and Transition to Chaos), [111] (in which case one can deal with the Hamiltonian function rather than with the vector field it generates), referring the reader to [51] for this as well as for other extensions and for several proofs. We start by recalling basic notions about the symmetry of differential equations, and in particular of dynamical systems; we will then discuss normal forms in the presence of symmetries, and the problem of taking into normal form the dynamical vector field and the symmetry vector field(s) at the same time. The presence of symmetry causes several peculiar phenomena in the dynamics, and hence also in perturbative expansions. This has been explained in very effective terms by Ian Stewart [175]: Symmetries abound in nature, in technology, and – especially – in the simplified mathematical models we study so assiduously. Symmetries complicate things and simplify them. They complicate them by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They simplify them

by introducing exceptional types of behavior, increasing the number of variables involved, and making vanish things that usually do not vanish. They violate all the hypotheses of our favorite theorems, yet lead to natural generalizations of those theorems. It is now standard to study the “generic” behavior of dynamical systems. Symmetry is not generic. The answer is to work within the world of symmetric systems and to examine a suitably restricted idea of genericity. Here we deal with dynamical systems, and more specially autonomous ones, i. e. systems of equations of the form dx i /dt D f i (x). Now we have a single independent variable, the time t 2 R, and in view of its distinguished role we will mainly focus attention on transformations leaving it unchanged. It is appropriate to point out here connections to several topics which we will not illustrate in this article. First of all, we stress that we will work at the formal level, i. e. without considering the problem of convergence of the power series entering in the theory. This convergence is studied in the articles  Perturbation Theory,  Perturbative Expansions, Convergence of, to which the interested reader is referred in the first instance. As hinted above, perturbation theory for symmetric systems has many points of contact with the topic of Equivariant Bifurcation Theory, which we will not touch upon here. The interested reader is referred to [104,107, 118,160,179] for Bifurcation Theory in general, and then for the equivariant setting to the books [42,105,118]. More compact introductions are provided by the review papers [55,79]. Many facets of the interplay of symmetry and perturbation theory are also discussed in the SPT conference proceedings volumes [1,14,17,57,96,98]. Our discussion is based on the treatment in [51], with integrations and updates where appropriate. Some considerations and remarks are given in additional notes collected in the last section; these are called for by marks(x x) with xx consecutive numbers. Symmetry of Dynamical Systems Symmetry of differential equations – and its use to solve or reduce the differential equations themselves – is a classical and venerable subject, being the very motivation to Sophus Lie when he created what is nowadays known as the theory of Lie groups [121]. The subject is now dealt with in a number of textbooks (see e. g. [3,19,26,27,37,80,115, 125,151,152,174]) and review papers (see e. g. [116,184, 185,191,192]); we will thus refer the reader to these for the

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general theory, and briefly recall here the special formulation one obtains when dealing with symmetries of smooth dynamical systems in Rn . Consider a (possibly non-autonomous) system ˙i

i

x D f (x; t)

i D 1; : : : ; n ;

(1)

we assume x 2 M D Rn ; M is also called the phase space, e D M  R (the second factor representing of course and M time t) is the extended phase space.(1) e these can be writWe consider now vector fields in M; ten in coordinates as n

S D (x; t)

X @ @ ' i (x; t) i : C @t @x

(2)

iD1

Note that (1) is identified with the vector field X f :D

n X

f i (x; t)

iD1

@ : @x i

(3)

A (vector) function x : R ! M is naturally identified e (corresponding to its graph) dewith the subset  x of M fined by e: x D f(y; t) 2 M  R : y i D x i (t)g M

(4)

e by mapping The vector field S acts infinitesimally in M points (y; t) to points (b y;b t) given by b t D t C "(y; t) ;

b y i D y i C "' i (y; t) ;

(5)

as " is small these relations can be inverted, yielding at first order in " t Db t  "(b y;b t) ;

yi D b y i  "' i (b y;b t) :

(6)

Using these relations, it is easy to check that the subset  D x is mapped by S to a (generally) different subset b , corresponding to y D b x(t), with h i b x i (t) D x i (t) C " ' i (x(t); t)  x˙ i (t) (x(t); t) : (7) We say that S is a symmetry for the dynamical system (1) if it maps solutions into (generally, different) solutions. The condition for this to happen turns out to be [51] @' i @f i @' i @ i  fj : (8)  f D 'j j @t @t @x @x j This can be more compactly expressed by introducing the Lie–Poisson bracket f f ; gg :D ( f  r) g  (g  r) f

(9)

e Then (8) reads between vector functions on M. (@'/@t)  (@/@t) f D f'; f g :

(10)

In the following we will consider autonomous dynamical systems; in this case it is rather natural to consider only transformations which leave t invariant, i. e. with  D 0. In this case (10) reduces to (@'/@t) C f f ; 'g D 0 :

(11)

A further reduction is obtained if we only consider transformations for which the action on M is also independent of time, so that @'/@t D 0 and the symmetry condition is f f ; 'g D 0 ;

(12)

in this case one speaks of Lie-point time-independent (LPTI) symmetries. The Eqs. (8) (or its reductions) will be referred to as the determining equations for the symmetries of the dynamical system (1).(2) It should be stressed that (8) are linear in ' and ; it is thus obvious that the solutions will span a linear space. It is also easy to check (the proof of this fact follows from the bilinearity of (9) and the Jacobi identity) that if S1 and S2 are solutions to (8), so is their Lie–Poisson bracket fS1 ; S2 g. The set G X f of vector fields X' with ' solutions to (8) is thus a Lie algebra; it is the symmetry algebra for the dynamical system (1). The symmetry algebra of a dynamical system is infinite dimensional, but has moreover an additional structure. That is, it is a module over the algebra I X f of first integrals for f (that is, scalar functions ˛ : M ! R such that X f (˛)  ( f  r)˛ D 0). Albeit G X f is infinite dimensional as a Lie algebra, it is not so as a Lie module. We have, indeed [186]: Theorem 1 (Walcher) The set G X f is a finitely generated module over I X f . Perturbation Theory: Normal Forms In this section we recall some basic facts about perturbation theory for general dynamical systems, referring to  Normal Forms in Perturbation Theory,  Hamiltonian Perturbation Theory (and Transition to Chaos),  Perturbation Theory for details. For the sake of simplicity, we discuss perturbations around an equilibrium point; see e. g. [6,7,8,111,160,164,165] for more general settings. As is well known – and discussed, e. g. in  Normal Forms in Perturbation Theory,  Hamiltonian Perturbation Theory (and Transition to Chaos) – a central objective of perturbation theory is to set (1) in normal form, i. e.

Non-linear Dynamics, Symmetry and Perturbation Theory in

to eliminate as many nonlinear terms as possible, so that the difference with respect to the linearized equation is as small as possible (see again  Normal Forms in Perturbation Theory,  Hamiltonian Perturbation Theory (and Transition to Chaos), or  Perturbation Theory, for a precise meaning to this statement; a lengthier discussion is given e. g. in [5,7,99,104,179]).(3) We briefly recall how this goes, also in order to fix notation. We consider a C 1 dynamical system x˙ D f (x) in Rn , admitting x D 0 as an equilibrium point – that is with f (0) D 0. By Taylor-expanding f (x) around x D 0 we will write this in the form x˙ D f (x) D

1 X

F k (x)

(13)

kD0

where F k (x) is homogeneous of degree (k C 1) in x (this seemingly odd notation will come out handy in the following). We denote the linear space of vector function f : Rn ! Rn homogeneous of degree (k C 1) by V k . Poincaré–Dulac Normal Forms Let us consider a change of coordinates of the form  i D x i C h ki (x) ;

(14)

where h k 2 V k (k  1); we write  ji D (@h i /@x j ). A change of coordinates of the form (14) is called a Poincaré transformation (or P transformation for short), and the function hk is also called the generator of the P transformation. In the following we will freely drop the subscript k when this cannot generate any confusion. The transformation (14) is, for small x, a near-identity transformation; thus it is surely invertible in a small enough neighborhood of the origin. We apply  :D (I C  )1 on (13), and get the P transformed dynamical system in the form x˙ D e f (x)  

1 X

Fm (x C h k (x)) D

mD0

1 X

e F m (x) : (15)

mD0

In order to identify the e F m we should consider power series expansions for  and for Fm (x C h(x)). With standard computations (we refer again to  Normal Forms in Perturbation Theory, or to [7,51], for details), we obtain that the e F m are given (with [q] the integer part of q) by " p # [m/k] X X ps s s e (1)  ˚ Fmk p : (16) F m D Fm C hk

pD1

sD0

The ˚ hr appearing in (16) are defined as follows. With a multi-index notation, write J D ( j1 ; : : : ; j n ), jJj D

P

j1 jn i j i ; set then @ J :D @1 : : : @ n , and similarly (h1k ) j 1 : : : (h nk ) j n . The operators ˚ hr (representing

h kJ :D all the partial derivatives of order jJj) are defined as ˚ hr D P (1/r!) jJjDr (h J  @ J ). Some special cases following from this general formula should be noted. As well known, the terms of degree smaller than k are not changed at all, i. e. e F m D Fm for m < k, and the term of degree k is changed (writing h  h k ) according to e F k D F k C [˚ h   ] F0 :

(17)

(Similarly, for 0 <  < k, the term of degree k C  is changed into e F kC D F kC C [˚ h   ]F .) Define now, recalling (9), the operators L k D fF k ; :g ;

(18)

note L k : Vm ! VmCk . The operator A D L0 , associated with the linear part A D (D f )(0) of f (that is, F0 (x) D Ax (4) ) is called the homological operator; it leaves the spaces V k invariant, and L1 (m) , hence it admits the decomposition A D mD0 A (m) where A is just the restriction of A to Vm . In the following, we will need to consider the adjoint AC of the operator A. For this we need to introduce S a scalar product in the space V D V k . Actually, we can introduce a scalar product in each of the spaces V k into which V decomposes. A convenient scalar product was introduced in [60] (following [18]); we will only use this one (the reader should be warned that different definitions are also considered in the literature [5,51]). We denote by ;i the vector function whose components are all zero but the ith   one, given by x  :D x1 1 : : : x n n ; with this notation, we define(5) 

n  X ;i ; ; j D h;i ; ;i i ;

(19)

iD1 n 1 where hx  ; x  i  @ x  , and @ D @ 1 : : : @ n . (When in the following we consider adjoint operators, these will be understood in terms of this.) With this scalar product, one has the following lemma (a proof is given, e. g. in [118]).

Lemma 1 If A is the homological operator associated with the matrix A, A D fAx; :g, then its adjoint AC is the homological operator associated with the adjoint matrix AC , i. e., AC D fAC x; :g . We will also consider the projection onto the range of A(k) , denoted by  k . The general homological equations

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(the one for k D 0 corresponds to the standard homological equation) are then A(k) (h k ) D  k F k :

(20)

These are equations for h k 2 V k , and always admit a solution (thanks to the presence of the projection operator  k ). The Eq. (20) maps into a set of algebraic equations once we introduce a basis in V k . The h k 2 V k solving to (20) will be of the form h k D hk C ` k , where hk D A ( k F k ) 2 Ran[(A(k) ) ] (here A is the pseudo-inverse to A; note Ker(A ) D Ker(AC )) is unique, and ` k is any function in Ker[A(k) ]. Remark 1 It should be stressed that, while adding a nonzero ` k to hk does not change the resulting e F k , it could – and in general will – affect the terms of higher order. One can then normalize X f in the standard recursive way, based on solving homological equations; this is described, e. g. in  Normal Forms in Perturbation Theory. In this way, we are reduced to considering only systems with

 h  i? C  : D Ker A(k) F k 2 Ran A(k)

(21)

resonant.(6)

Such terms are also called The presence of resonant terms is related to the existence of resonance relations among eigenvalues i of the matrix A describing the linear part of the system; these are P relations of the form (m  )  i m i  i D s where the P mi are non-negative integers, with jmj D m i > 1 (the restriction jmj > 1 is to avoid trivial cases); the integer jmj is also called the order of the resonance ( Normal Forms in Perturbation Theory, [7,8]). If the system x˙ D f (x) has only resonant nonlinear terms, we say that it is in Poincaré–Dulac normal form ( Normal Forms in Perturbation Theory, [7,8,35, 44]). If all the nonlinear terms of order up to q are resonant, we say that the system is in normal form up to order q. Theorem 2 (Poincaré–Dulac) Any analytic vector field X f with f (0) D 0 can be formally taken to normal form by a sequence of Poincaré transformations. Remark 2 If we do not have an exact resonance, but (m  )  s ' 0, we have a small denominator, and correspondingly a very large coefficient in hk , where k D jmj. Such small denominators are responsible for divergencies in the normalizing series [7,8,9,32,33,35,172].

Lie Transforms In discussing Poincaré normal forms, we have considered near-identity diffeomorphisms of M; these can be expressed as time-one maps for the flow under some vector fields X h . In a number of cases, it is more convenient to deal directly with such vector fields. We are going to briefly discuss this approach, and its relation with the one discussed above; for further detail, see e. g. ( Normal Forms in Perturbation Theory, [25,28,29,30,51,61,84,101,145]). Let the vector field H  X h be given, in the x coordinates, by H D h i (x)(@/@x i ). We denote by  (s; x) the local flow under H starting at x, so that (d/ds) (s; x) D H( (s; x)). We also use exponential notation:  (s; x) D esH x. We will denote  (1; ) as x; the direct and inverse changes of coordinates will be defined as  x D  (1; ) D esH  sD1 ;   D  (1; x) D esH x sD1 :

(22)

Now, consider another vector field X on M, describing the dynamical system we are interested in. If we study the dynamical system ˙ i D f i (), we consider the vector field given in the x coordinates by X D X f D f i (x)(@/@x i ) :

(23)

This also generates a (local) flow, i. e. for any 0 2 M we have a one-parameter family (t)   (t; 0 ) 2 M such that (d(t)/dt) D X(). By means of (22), this also defines a one-parameter family x(t) 2 M, which will satisfy (dx(t)/dt) D e X(x) for some vector field e X on M; this will be the transformed vector field under (22), and is given by(7)  e X D esH XesH (sD1) :

(24)

We call this transformation the Lie-Poincaré transformation generated by h. Notice that this yields, up to order one in s, (and therefore if h 2 V k , up to terms in V k ), just the same result as the Poincaré transformation with the same generator h. The e X can be given (for arbitrary s), in terms of the Baker–Campbell–Haussdorf formula, as e XD

1 X (1) k s k (k) X ; k!

(25)

kD0

where X (0) D X, and X (kC1) are defined recursively by X (kC1) D [X (k) ; H].

Non-linear Dynamics, Symmetry and Perturbation Theory in

Perturbative Determination of Symmetries Let us now consider the problem of determining the symmetries of a given dynamical system. Writing the latter in the form x˙ D f (x) D

1 X

F k (x) (F k 2 V k ) ;

(26)

vector fields, is a Lie algebra. It is also easy to see that, as f (0) D 0, Y 2 G X implies s(0) D 0. We can now expand Y, i. e. s(x), in a perturbative series around x0 D 0 in the same way as we did for X. We write s(x) D

it is quite natural to also look for symmetries in terms of a (possibly only formal) power series; this will be our approach here. Consider a generic smooth vector field (by smooth we mean, here and elsewhere, C 1 ; however, we will soon go on to actually consider vector fields represented by power series) in M  R, n X

' i (x; t)@ i :

(27)

Here and below, @ i  (@/@x i ). As remarked in Sect. “Symmetry of Dynamical Systems”, (27) is too general for our purposes; we are primarily interested in vector fields acting on M alone, and mainly [48,49,51,80] in time-independent vector fields on M (note in this way the dynamical and the symmetry vector fields are on the same footing). Thus we will just consider (we will consistently use X for the vector field X f defined by (26), and Y for the symmetry vector field X s , in order to simplify the notation) n X

s i (x) @ i :

(28)

iD1

Note that at this stage we are not assuming the dynamical system described by X has been taken into normal form; we will see later on the specific features of this case. Determining Equations Now, let us consider the dynamical system identified by X, and look for the determining equation identifying its symmetries Y as in (28). Condition (8) yields f j (x)

@s i (x) @ f i (x)  s j (x)  ( f  r)s  (s  r) f D 0 : j @x @x j (29)

As discussed above, using (9) this is also written as f f ; sg D 0, which just means (as had to be expected) [X; Y] D 0 :

(S k 2 V k ) :

(31)

Plugging this into the determining Eqs. (30) we get, after rearranging the terms, 1 X k X

(30)

We now denote the set of Y satisfying (30) by G X . It is obvious that G X , equipped with the usual commutator of

fFm ; S km g D 0 :

(32)

kD0 mD0

For this to hold, the different homogeneous terms of degree k must vanish separately. Thus, we have a hierarchy (in a sense to be explained in a moment) of equations k X

iD1

YD

S k (x)

kD0

kD0

Y D (x; t) @ t C

1 X

fFm ; S km g D 0

k D 0; 1; 2; 3; : : :

(33)

mD0

It is convenient to isolate the terms containing linear factors, i. e. to rewrite (33) – for k  1 – in the form fF0 ; S k g  fS0 ; F k g D 

k1 X

fFm ; S km g   k ;

(34)

mD1

where we have used the antisymmetry of f:; :g, and 0 D 1 D 0. Recursive Solution of the Determining Equations Let us now consider the problem of concretely solving the determining Eq. (32). As the perturbative series expansion suggests, we can proceed order by order, i. e. start with consideration of the equation for k D 0, then tackle k D 1, and so on. Proceeding in this way we are always reduced to consider equations of the form ! ! i @F0i j @S k j j F0 (35) D Sk C  k (x) @x j @x j j

with the  k known functions of x (as they depend on the known F k and on the Sj with j < k, determined at previous stages). Notice also that F 0 is just the (known) linear part of X. If we write it in matrix form as F0i (x) D A i j x j (we will write similarly S0i (x) D B i j x j for S0 ), then (35) reads simply ! h i @S i j j k Ai j x j (36) D A i j S k C  k (x) : @x j

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Solving the determining equations in such a recursive way only requires one to solve at each stage a system of (inhomogeneous) linear PDEs for the Sk . For further reference, we introduce the notation X M for the linear vector field associated with the matrix M, i. e. X M D (M i j x j )(@/@x i ). We also write LM for the homological operator associated with a matrix M, L M (:) D fMx; :g (see also note 4). In this notation, for any two matrices A; B we have [X A ; X B ] D X[A;B] ; similarly we have, as a consequence of the Jacobi identity: Lemma 2 For any matrices A; B, the commutator of the associated homological operators is given by [L A ; L B ] D L[B;A] D L[A;B] . Let us follow explicitly the iterative procedure for solving (32), (33) for the first steps. For k D 0, we require that fF0 ; S0 g D 0. With our matrix notation, fF0 ; S0 g  fAx; Bxg D  [A; B] i j x j

(37)

and therefore at this stage we only have to determine the matrices B commuting with a given matrix A. For k D 1, we just get fF0 ; S1 g C fF1 ; S0 g D 0; in the matrix notation this just reads fAx; S1 g D fBx; F1 g or, using the homological operators notation, A(S1 ) D B(F1 ) :D 1 (x) :

Note that it could happen where we are only able to determine a commuting vector field for X up to some finite order k (either for a full symmetry does not exist or for our limited capacities, computational or otherwise). If in this case we consider a neighborhood of the origin of small size, say a ball B" of radius "  1, in this we have [X; Y] D O(" k ); thus, Y represents an approximate symmetry for X. Approximate symmetries are interesting and useful in a number of contexts. In particular, in some cases – notably, for Hamiltonian vector fields – there is a connection between symmetries of dynamical systems and conserved quantities (i. e. constants of motion) for it; in this case, approximate symmetries will correspond to approximate constants of motion, i. e. to quantities which are not exactly conserved, but are approximately so. More precisely, an approximate symmetry will correspond to a quantity J whose evolution under the dynamics described by X is slow of order k, i. e. dJ/dt  " k for some finite k. It is rather clear that these can be quite useful in applications, where we are often concerned with study of the dynamics over finite times; see [51] and especially, in the Hamiltonian case, [100].

(38)

Symmetry Characterization of Normal Forms

(39)

Let us now consider the case where the dynamical system (13) has already been taken into Poincaré–Dulac normal form. We start by recalling some notions of linear algebra of use here.

For k D 2 we get in the same way A(S2 ) D B(F2 )  fF1 ; S1 g :D 2 (x) ;

Approximate Symmetries

and so on for any k. Linear Algebra Remark 3 The fact that we can proceed recursively (and only deal with linear PDEs) does not mean that we are guaranteed to find solutions at any stage k. At k D 0 we always have at least the solutions given by B D I and by B D Aq (q D 1; 2; : : :). For k  1 we do not, in general, have solutions to the determining equations apart from S j D cF j for all q D 0; : : : ; k (i. e., as k is generic, Y D cX). This corresponds to the fact that symmetry is not generic. Remark 4 As the relevant equations are not homogeneous, S k D 0 is not, in general, an acceptable solution at stage k. This is quite natural if one thinks that the choice B D I is always acceptable at k D 0. Choosing S k D 0 at all the following stages would leave us with the dilation vector field Y D x i @ i , which is a symmetry only for X linear [80,151,174].

A real matrix T is semisimple if its complexification T C can be diagonalized, and is normal if it commutes with its adjoint, [T; T C ] D 0. A diagonal matrix is normal. For a normal matrix, T : Ker(T C ) ! Ker(T C ). If T is normal we actually have Ker(T) D Ker(T C ). Any semisimple matrix can be transformed into a normal one by a linear transformation. If two semisimple matrices A; B commute, then they can be simultaneously diagonalized (by a linear, in general non-orthogonal, transformation), and so taken simultaneously to be normal. Thus, when considering such a pair of matrices, we can with no loss of generality assume them to be diagonal or, a fortiori, normal. If we want to transform T into a real normal matrix, we just have to consider the transformation of T into a block diagonal matrix, the blocks corresponding to (complex

Non-linear Dynamics, Symmetry and Perturbation Theory in

conjugate) eigenvalues. It is easy to see that in this way we still get a (real) normal matrix.(8) In the following, we will at several points restrict, for ease of discussion, to normal matrices; our statements for normal matrices will be easily extended to semisimple ones up to the appropriate linear transformation.

Normal Forms We now note that F 2 Ker(A) means that the vector field associated with F (which we denote by X F ) commutes with the linear vector field X A associated with A. That is, F 2 Ker(A) , fAx; F(x)g D 0 , [X A ; X F ] D 0 : (40) Thus we have the following characterization for vector fields in normal form (note this uses Lemma 1 and hence the scalar product defined in Sect. “Perturbation Theory: Normal Forms”). Lemma 3 A vector field X D (A i j x j C F i (x))@ i , where the F are nonlinear functions, is in normal form if and only if its nonlinear part X F D F i (x)@ i commutes with the vecj tor field X AC D [AC i j x ]@ i associated with the adjoint of its linear part, i. e. if and only if F 2 Ker(AC ). For the sake of simplicity, we will only consider the case where the matrix A, corresponding to the linear part of X (in both the original and the normal form coordinates), commutes with its adjoint, i. e. we make the following(9) assumption: The matrix A is normal: [A; AC ] D 0. If A is normal, then it follows [L A ; L AC ] D 0 due to Lemma 2; this implies in particular that: Lemma 4 If A is normal, then Ker(A) D Ker(AC ). It is important to recall that with the standard scalar product, we have (L A )C D L AC . It is also important, although trivial, to note that if A is normal, then A is a normal operator (under the standard scalar product), and Ker(A) \ Ran(A) D f0g, but f 2 Ker(A2 ) ) (A( f )) 2 Ker(A) and therefore A( f ) D 0. Hence Lemma 5 If A is normal, Ker(L2A ) D Ker(L A ). This discussion leads to a natural characterization of Poincaré–Dulac normal forms in terms of symmetry properties.(10) Lemma 6 If A is a normal matrix, then X D (Ax C F) i (@/@x i ) is in normal form if and only if X A is a symmetry of X, i. e. F 2 Ker(A).

The General Case In the general case, i. e. when A is not normal, the resonant terms will be those in Ker(AC ); similarly, in this case we could characterize systems in normal form by fF(x); AC xg D 0. However, for a symmetry characterization it is better to proceed in a slightly different way. That is, we recall that any matrix A can be uniquely decomposed as A D As C An where As is semisimple and An is nilpotent, with [As ; An ] D 0. Resonance properties involve eigenvalues of As , and resonant terms will satisfy fF(x); AC s xg D 0. This is a more convenient characterization, in that it shows that the full vector field (in normal form) X will commute with the linear vector field S D (As ) ij x j @ i corresponding to the semisimple part of its linear part. Symmetries and Transformation to Normal Form We want to consider the case where the dynamical system (1), or equivalently the vector field X, admits a symmetry Y (the case of an n-dimensional symmetry algebra will be considered later on); we want to discuss how the presence of the symmetry affects the normalization procedure. Moreover, as the dynamical and symmetry vector fields are on equal footing, it will be natural to investigate if they can be both put into normal form; or even if some kind of joint normal form is possible (as is the case). We will use the notation introduced above for the expression of X; Y in the x coordinates, and denote by y the normal form coordinates. Correspondingly, the bracket f:; :g (which is defined in coordinates) will be denoted as f:; :g(x) or f:; :g(y) when confusion is possible. We have, therefore, X D f i (x)(@/@x i ) D g i (y)(@/@y i ) ; Y D s i (x)(@/@x i ) D r i (y)(@/@y i ) ;

(41)

and similarly for the power series expansions of f ; g; s; r in terms homogeneous of degree (k C 1) D 1; 2; : : : . We will denote the matrices associated with the linear parts of X; Y by, respectively, A and B : (D f )(0) D (Dg)(0) D A, (Ds)(0) D (Dr)(0) D B. The corresponding homological operators will be denoted by A D L A and by B D L B . We assume that both A and B are normal matrices. Nonlinear Symmetries (The General Case) The key, albeit trivial, observation is that the geometric relation [X; Y] D 0 does not depend on the coordinate system we are using. Therefore, if f f ; sg(x) D 0, we must also have fg; rg(y) D 0.

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Another important, and again quite trivial, observation is that when we consider a P-transformation x D y C h k (y), the term F k of order k in X changes according to A(h k ), but the term Sk of order k in Y changes according to B(h k ). The same applies when we consider a LiePoincaré transformation. Thus, although when we choose h k C ` k [with ` k 2 Ker(A)] to generate the Poincaré transformation, we get the same transformation as that generated by hk on the F k (see Remark 1), the transformation on Sk can be different. This means that the freedom left by the Poincaré prescription for construction of the normalizing transformation could, in principle, be used to take the symmetry vector field Y into some convenient form. This is indeed the case, as will be shown below. Two vector fields X; Y on M, as in (35), with A and B semisimple, are in Joint Normal Form if both Gk and Rk are in Ker(A) \ Ker(B) for all k  1. Theorem 3 Let the vector fields X D f i (x)@ i and Y D s i (x)@ i have a fixed point in x0 D 0. Let them commute, [X; Y] D 0, and have normal semisimple linear parts A D (D f )(0) and B D (Ds)(0). Then, by means of a sequence of Poincaré transformations, they can be brought to Joint Normal Form. In this theorem (a proof of this is given in [49,51]; see also [106] for a different approach to a related problem) we did not really use the interpretation of one of the vector fields as describing the dynamics of the system and the other describing a symmetry, but only their commutation relation. From this point of view, it is natural to also consider arbitrary (possibly non-Abelian) algebras of vector fields. Linear Symmetries A special case of symmetries is given by linear symmetries, i. e. by the case where   Y D X B D B i j x j (@/@x i ) :

(42)

In this case, if X; Y are in Joint Normal Form we have in particular that f 2 Ker(B). We have the following corollaries to Theorem 3: Corollary 1 If the linear vector field Y D X B is a symmetry for X D f i (x)@ i , it is possible to normalize f by passing to y coordinates so that X D g i (y)(@/@y i ), Y D (By) i (@/@y i ), and g 2 Ker(A) \ Ker(B). Corollary 2 Let s(x) D Bx C S(x), with S the nonlinear part of s, and let Y D s i (x)@ i be a symmetry of

X D f i (x)@ i . Then when X; Y are put in Joint Normal Form, X B is a linear symmetry of X. When we perform the Poincaré transformations needed to transform X into its normal form, there seems to be no reason, a priori, why the Y should keep its linear form. It is actually possible to prove the following result (the proof of this relies on a similar result by Ruelle [159,160] dealing with the center manifold mapping, and is given e. g. in [60,118]; see also [20,21,22,23]) that we quote from [118]: Theorem 4 If X commutes with a linear vector field Y D X B D (B i j x j )@ i , then it is possible to find a normalizing series of Poincaré transformations with generators h k 2 Ker(B), so that in the new coordinates y, X is taken into normal form and Y is left unchanged, i. e. Y D (B i j y j )@ i . Note that, for resonant B, Theorem 4 is not a special case of Theorem 3 and the above Corollaries.(11) Remark 5 One should avoid confusions between linear symmetries of the dynamical system and symmetries of its linearization, which do not extend in general to symmetries of the full system. Generalizations In this section we are going to discuss some generalizations of the results illustrated in the previous one: we will deal with the problem of transformation into normal form of a (possibly non Abelian) Lie algebra with more than two generators, not necessarily commuting.(12) Abelian Lie Algebra It is actually convenient to drop the distinction between the vector field defining the dynamical system and the symmetry vector fields. Thus, we simply consider an algebra G of vector fields X i :(13) First of all, we consider the case of an Abelian Lie algebra of vector fields. In this case we have the following result (see [51] for a proof). Theorem 5 Let fX1 ; : : : ; X r g commute, and assume that the matrices A(i) identifying the linear parts of the X j are normal. Then fX1 ; : : : ; X r g can be put in Joint Normal Form by a sequence of Poincaré or Lie-Poincaré transformations. Note that if A i are the homological operators correspondTk ing to the A(i) , and we write K D iD0 Ker(A i ), the Theorem states there are coordinates yi such that X j D (e f ( j) (y)) i (@/@y i ) with e f ( j) 2 K for all j.

Non-linear Dynamics, Symmetry and Perturbation Theory in

Nilpotent Lie Algebra For generic Lie algebras one cannot expect results as general as in the Abelian case [4,48,51]. A significant exception to this is met in the case of nilpotent algebras(14) (see also [54] for a group-theoretical approach), in which we can recover essentially the same results obtained in the Abelian case (see [112] for the case of semisimple Lie algebras). Actually, an extension of Theorem 5 to the nilpotent case should be considered with some care, as the only nilpotent algebras of nontrivial semisimple matrices are Abelian. On the other hand, we could have a non-Abelian nilpotent algebra of vector fields with linear parts given by semisimple matrices, provided that some of these vanish. Indeed, although [X i ; X j ] D c ikj X k necessarily implies that [A i ; A j ] D c ikj A k , it could happen that all the Ak for which there exists a nonzero c ikj do vanish, so that the algebra of vector fields is “Abelian at the linear level”. (As a concrete example, consider the algebra spanned by X D x 2 d/dx and Y D xd/dx.) Note that in this case Ker(A k ) would be just the whole space; needless to say, we should consider the full vector fields X k , which will produce (by assumption) a closed Lie algebra. With this remark (and in view of the fact that the proof of Theorem 5 is based on properties of the algebra of the Ai ’s and of the corresponding homological operators, see [51]) it is to be expected that the result for the nilpotent case will not substantially differ from the one holding for the Abelian case (as usual, the key to this extension will be to proceed to normalization of vector fields in an order which respects the structure of the Lie algebra). This is indeed what happens, and one has the following result [48,51]: Theorem 6 Let the vector fields fX1 ; : : : ; X r g form a nilpotent Lie algebra G under [:; :]; assume that the matrices A(i) identifying the linear parts of the X j are normal. Then fX1 ; : : : ; X r g can be put in Joint Normal Form by a sequence of Poincaré or Lie-Poincaré transformations. Corollary 3 If the general Lie algebra G of vector fields fX1 ; : : : ; X n g contains a nilpotent subalgebra G  , then the set of vector fields X i spanning this G  can be put in Joint Normal Form. General Lie Algebra Some “partial” Joint Normal Form can be obtained, even for non-nilpotent algebras, under some special assumptions. We will just quote a result in this direction, referring as usual to [51] for details. A description of normal

forms for systems with symmetry corresponding to simple compact Lie groups is given in [86] Theorem 7 Let G be a d-dimensional algebra spanned by X F a with F a D A a x C F a (a D 1; : : : ; d), and let G admit a non-trivial center C(G ). Let the center of G be spanned by Xw b with w b D Cb x C Wb (b D 1; : : : ; dC , where dC  d), and assume that the semisimple parts Cb;s are normal matrices. Denote by Cb;s the associated homological operators, C and write Ks D \dbD1 Ker(Cb;s ). Then, by means of a sequence of Poincaré transformations, all the F a can be taken into b F a 2 Ks . The same holds G D G ˚ N , with N a nilpotent subalgebra. for b Symmetry for Systems in Normal Form No definite relation exists, in general, between symmetries G X of a vector field X and symmetries GA of its linear part, or between constants of motion I X for X and constants of motion I A for its linear part, but if X is in normal form, one has some interesting results [48,186]: Lemma 7 If X is in normal form, any constant of motion of X must also be a constant of motion of its linearization A, i. e. I X  I A . In general I X ¤ I A , even if X is in normal form. Also, if [B; A] D 0 in general (x)Bx does not belong to G X , even for  2 I X (unless X B 2 L X as well, where L X are the linear symmetries of X), nor to GA (unless  2 I A ). Lemma 8 If X is in normal form, then G X  GA . This result allows for the restricting our search for Y 2 G X to GA rather than considering the full set of vector fields on M. Similarly, Lemmas 7 and 8 can be useful in the determination of the sets L X ; I X , in that we can first solve the easier problem of determining LA ; I A , and then look for L X , I X in the class of vector fields LA and of the functions in I A , rather than in the full set of linear vector fields on M and, respectively, in the full set of scalar functions on M. Moreover, GA can be determined in a relatively simple way, by solving the system of quasi-linear non-homogeneous first order PDEs fAx; gg D 0, which are written explicitly as 

 A i j x j (@g k /@x i ) D Ak j g j :

(43)

By considering this, and introducing the set I A of the meromorphic (i. e., quotients of formal power series) constants of motion of the linear problem x˙ D Ax, one can obtain [48,60,186] the following result:

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Lemma 9 GA is the set of all formal power series in I A ˝ LA . In a more explicit form, as GA  Ker(A), the resonant terms F(x) 2 Ker(A) are power series of the form F(x) D K( (x))  x, where K is a matrix commuting with A when written in terms of its real entries K ij , and where K( (x)) is the same matrix in which the entries K ij are replaced by functions of the constants of motion D (x) 2 I A . The set of the vector fields in GA is of course a Lie algebra. We summarize our discussion for dynamical systems in normal form in the following proposition: Theorem 8 Let X be a vector field in normal form, and let A be the normal matrix corresponding to its linear part. Then, L X  G X  GA ; and L X  LA  GA . Remark 6 It should be mentioned that Kodama considered the problem of determining GA from a more algebraic standpoint [122]. In the same work, Kodama also observed that GA , considered as an algebra, is not only infinite-dimensional, but has the natural structure of a graded Virasoro algebra. Remark 7 We emphasize once again that the above results were given for X in normal form. They can obviously be no longer true if X is not in normal form.(15) Linearization of a Dynamical System An interesting application of the Joint Normal Form deals with the case of linearizable dynamical systems. Clearly, if Ker(A) D f0g, the dynamical system is linearizable by means of a formal Poincaré transformation. But, whatever the matrix A, the linear vector field X A D (Ax  r) comP mutes with the vector field S D i x i (@/@x i ) D ((Ix)  r), which generates the dilation’s in Rn . It is easy to see that, conversely, the only vector fields commuting with S are the linear ones. It is also clear that the identity does not admit resonances. Thus [16,92]: Lemma 10 A vector field X f (or a dynamical system x˙ D f (x)) can be linearized if and only if it admits a (possibly formal) symmetry X g such that B D (Dg)(0) D I. Proceeding in a similar way – but using Joint Normal Forms – we have, more generally: Theorem 9 The vector field X f with A D (D f )(0) can be linearized if and only if it admits a (possibly formal) symmetry X g with B D (Dg)(0) such that A and B do not admit common resonances, i. e. Ker(A) \ Ker(B) D f0g.

This result can be easily extended not only to the case of more than one symmetry, as an obvious consequence of Theorem 4, but also to the non-semisimple case [48]. Another interesting result related to linear dynamical systems, is the following [48,51]: Theorem 10 If a dynamical system in Rn can be linearized, then it admits n independent commuting symmetries, that can be simultaneously linearized. Further Normalization and Symmetry As repeatedly noted above (see in particular Remark 1), when the linear part of the dynamical vector field is resonant the resulting degeneracy in the solution to the homological equation is not a real degeneracy for what concerns effects on higher order terms in the normal form. These higher order terms could – and in general will – generate resonant terms, which cannot be eliminated by the standard algorithm. On the other hand, it is clear that this could be seen as a bonus rather than as a problem: in fact, it is conceivable that by carefully choosing the component of hk lying in Ker(A), one could generate resonant terms which exactly cancel those already present in the vector field. Several algorithms have been designed to take advantage, in one way or the other, of this possibility; some review of different approaches is provided in the paper [39]. Here we are concerned with those based on symmetry properties, and discuss two different approaches, developed respectively by the present author [81,83] and by Palacian and Yanguas [155]. It should be stressed that once the presence of additional symmetries – and in the Hamiltonian context, additional constants of motion – has been determined for the normal form truncated at some finite order N, one should investigate if the set of symmetries (or constants of motion) persist under small perturbations; in particular, when considering terms of higher order as well. A general tool to investigate this kind of question is provided by the Nekhoroshev generalization of the Poincaré-Lyapounov theorem [148,149,150]; see also the discussion in [15,87,88,91]. Further Normalization and Resonant Further Symmetry We will assume again, for ease of discussion, that the matrix A associated with the linear part of the dynamical vector field X is normal. In this case, as discussed above, the normal form is written as X D g i (x)@ i where g(x) D P1 kD0 G k (x), with G k 2 V k and all Gk being resonant.

Non-linear Dynamics, Symmetry and Perturbation Theory in

We can correspondingly write XD

1 X

X k ; X k D G ki @ i :

(44)

kD0

(note H (q)  H (p) for q > p), and denote by M p the restriction of L p to H (p) ; thus Ker(M p ) D H (pC1) . We define spaces F (p) (with F (0) D V ) as C F (p) :D Ker(MC 0 ) \ : : : \ Ker(M p1 ) ;

(47)

As X is in normal forms, we are guaranteed to have [X0 ; X k ] D 0

8k D 0; 1; 2 : : : ;

(45)

this corresponds to the characterization of normal forms in terms of symmetry as discussed in Sect. “Symmetry Characterization of Normal Forms”. In other words, G k 2 Ker(A) and hence, defining G0 as the Lie algebra of vector fields commuting with X 0 , X k 2 G0 . On the other hand, in general it will be [X j ; X k ] 6D 0 for j 6D k and both j; k greater than zero; thus G0 is not, in general, an Abelian Lie algebra: we can only state that X 0 belongs to the center of G0 , X 0 2 Z(G0 ). Suppose we want to operate further Lie-Poincaré transformations generated by functions which are symmetric under X 0 (i. e. are in the kernel of A). It follows from the formulas obtained in Sect. “Perturbation Theory: Normal Forms” that these will map G0 into itself, i. e. G0 is globally invariant under this restricted set of Lie-Poincaré transformations. As for the individual vector fields, it follows from the general formula (24) that each of them is invariant under such a transformation at first order in hk , but not at higher orders. That is, making use again of Remark 1, we can still in this way generate new resonant terms in the normal form, including maybe terms which cancel some of those present in (44). By looking at the explicit formulas (25), it is rather easy to analyze in detail the higher order terms generated in the concerned Lie-Poincaré transformation. Note that we did not take into account problems connected with the convergence of the further normalizing transformations; it has to be expected that each step will reduce the radius of convergence, so that the further normalized forms will be actually (and not just formally) conjugated to the original dynamic in smaller and smaller neighborhoods of the fixed point; we refer to [90] for an illustration of this point by explicit examples and numerical computations; and to  Perturbative Expansions, Convergence of for a general discussion on the convergence of normalizing transformations. In order to state exactly the result obtained by this construction, we need to introduce some function spaces, which require abstract definitions. With(16) L k (:) :D [X k ; :], we set H (p) :D Ker(L0 ) \ : : : \ Ker(L p1 )

(46)

the adjoint should be meant in the sense of the scalar product introduced in Sect. “Perturbation Theory: Normal (p) Forms”. We also write F k D F (p) \ V k . We will say that X is in Poincaré renormalized form up to order N if G k 2 F k(k) for all k  N. Theorem 11 The vector field X can be formally taken into Poincaré renormalized form up to (any finite) order N by means of a (finite) sequence of Lie-Poincaré transformations. For a proof of this theorem, and a detailed description of the renormalizing procedure, see [51,81,83,85]; see [51,81] for the case where additional symmetries are present; an improved procedure, taking full advantage of the Lie algebraic structure of G0 , is described in [84]. Further Normalization and External Symmetry A different approach to further reduction (simplification) of vector fields in normal form has been developed by Palacian and Yanguas [155] (and applied mainly in the context of Hamiltonian dynamics [154,156,157]), making use of a result by Meyer [144]. As discussed in the previous subsection we have [X 0 ; X k ] D 0 for all k. Suppose now there is a linear (in the coordinates used for the decomposition (44)) vector field Y such that [X0 ; Y] D 0. Then the Jacobi identity guarantees that [X 0 ; [X k ; Y]] D 0 8k :

(48)

We assume that Y also corresponds to a normal matrix B, so that the homological operator B associated to it is also normal. We can then proceed to further normalization as above, being guaranteed that – provided we choose h k 2 Ker(L0 ) – the resulting vector fields will not only still be in G0 , but also still satisfy (48). One can use freedom in the choice of the generator h k 2 Ker(L0 ) for the further normalization in a different way than discussed above: that is, we can choose it so that [Y; X k ] D 0; in other words, we will get b X 2 Ker(A) \ Ker(B). Note the advantage of this: we do not have to worry about complicated matters related to relevant homological operators acting between different spaces, as we only make use of the homological operator B associated with

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the “external” symmetry linear vector field and thus mapping each V k into the same V k . The result one can obtain in this way is the following (which we quote in a simplified setting for the sake of brevity; in particular the normality assumption can be relaxed) [155].

Let us recall that vector monomial v;˛ :D x  e˛ is resonant with A if (  ) :D

Applications of this theorem, and more generally of this approach, are discussed e. g. in [154,155,156,157]. We stress that albeit the assumptions of this theorem are rather strong(17) , it points out to the fact that there are cases in which a symmetry – and in the Hamiltonian framework, an integral of motion – of the linear part can be extended to a symmetry of the full normal form.

 i  i D ˛

iD1

jj :D Theorem 12 Let X be in normal form; assume moreover X0 D Ax and there is a normal matrix B such that [A; B] D 0; denote by Y the associated vector field, Y D (Bx) i @ i . Assume moreover for each resonant vector field Rk there is Qk satisfying [Y; Q k ] D 0 and such that b R k D R k C [X 0 ; Q k ] commutes with Y. Then X can be taken to a (different) normal form b X such that [Y; b X] D 0.

n X

n X

with i  0 ; (49)

i  1 ;

iD1

here i are the eigenvalues of A, which we suppose to be semisimple, for the sake of simplicity (in the general case one would consider As rather than A). As mentioned in Sect. “Perturbation Theory: Normal Forms”, the relation (  ) D ˛ is said to be a resonance relation related to the eigenvalue ˛ , and the integer jj is said to be the order of the resonance. In the present context it is useful to include order one resonances in the definition (albeit the trivial order one resonances given by ˛ D ˛ are obviously of little interest). Let us consider again the resonance Eq. (49). It is clear that if there are non-negative integers  i (some of them nonzero) such that n X

i i D 0 ;

(50)

iD1

Symmetry Reduction of Symmetric Normal Forms Symmetry reduction is a general – and powerful – approach to the study of nonlinear dynamical systems. (In the Hamiltonian case, this is also known as (Marsden– Weinstein) moment map [6,134,135].) A general theory based on the geometry of group action has been developed by Louis Michel; this was originally motivated by the study of spontaneous symmetry breaking in high-energy physics [140,141] (see [136,137, 138,139] for the simpler case where only stationary solutions are considered, and [142] for the full theory and applications; see also [2,82,85,89,93,166,167]). A description of this would lead us too far away from the scope of this paper, but as this theory also applies to vector fields in normal form, we will briefly describe the results that can be obtained in this way; we will mainly follow [94] (see [95,97] for further detail, extensions, and a more abstract mathematical formulation). As mentioned in Sect. “Symmetry of Dynamical Systems”, the Lie algebra of vector fields in normal form is infinite dimensional, but also has the structure of a Lie module over the algebra of constants of motion for the linear part X 0 of the vector field (which remains the same under all the considered transformations).

then we always have infinitely many resonances. In this case the monomial ' D x  will be called a resonant scalar monomial. It is an invariant of X 0 , and any multi-index  with  i D k i C ı i˛ provides a resonance relation (  ) D ˛ related to the eigenvalue ˛ ; in other words, any monomial x k x ˛ D ' k x ˛ is resonant, and so is any vector v kCe ˛ ;˛ . Therefore, we say that (50) identifies a invariance relation. The presence of invariance relations is the only way to have infinitely many resonances in a finite dimensional system (see [186]). Any nontrivial resonance (49) which does not originate in an invariance relation, is said to be a sporadic resonance. Sporadic resonances are always in finite number (if any) in a finite dimensional system [186]. Any invariance relation (50) such that there is no  with  i   i (and of course  6D  ) providing another invariance relation, is said to be an elementary invariance relation. Every invariance relation is a linear combination (with nonnegative integer coefficients) of elementary ones. Elementary invariance relations are always in finite number (if any) in a finite dimensional system [186]. If there are m independent elementary invariance relations, each of them of the form (50), we associate to these monomials Q ˇ j D x  D niD1 x i i ( j D 1; : : : ; m).

Non-linear Dynamics, Symmetry and Perturbation Theory in

Similarly, if there are r sporadic resonances (49), we associate resonant monomials ˛ j (x) D x  D Qn ( j) i iD1 x i ( j D 1; : : : ; r) and resonant vectors v;˛ to sporadic resonances. We then introduce two set of new coordinates: these will be the coordinates w 1 ; : : : ; w r in correspondence with sporadic resonances, and other new coordinates ' 1 ; : : : ; ' m in correspondence with elementary invariance relations. The evolution equations for the xi can be written in simplified form using these (note that some ambiguity is present here, in that we can write these in different ways in terms of the x; w; '), but we should also assign evolution equations for them; these will be given in agreement with the dynamics itself. That is, we set dw j @w j dx i D :D h j (x; w; ') ; dt @x i dt

(51)

@' a dx i d' a D :D z a (x; w; ') : dt @x i dt

(52)

We are thus led to consider the enlarged space W D RnCrCm of the (x; w; '), and in this the vector field Y D f i (x; w; ') (@/@x i ) C h j (x; w; ') (@/@w j ) C z a (x; w; ') (@/@' a ) :

(53)

The vector field Y is uniquely defined on the manifold identified by j :D w j  ˛ j (x) D 0, ' a  ˇ a (x) D 0. It is obvious (by construction) that the (n C m)-dimensional manifold M W identified by i :D w i  ˛ (i) D 0 is invariant under the flow of Y, see (51). It is also easy to show that the functions za defined in (52) can be written in terms of the ' variables alone, i. e. @z a /@x i D @z a /@w j D 0. This implies(18) Lemma 11 The evolution of the ' variables is described by a (generally, nonlinear) equation involving the ' variables alone. Note that the equations for x and w depend on ' and are therefore non-autonomous. We have the following result (we refer to [94,95] for a proof; see [97] for extensions). Theorem 13 The analytic functions f i and hj defined above can be written as linear in the x and w variables, the coefficients being functions of the ' variables. Hence the evolution of the x and w variables is described by nonautonomous linear equations, obtained by inserting the solution ' D '(t) of the equations for ' in the equations x˙ D f (x; w; '), w˙ D h(x; w; '). Note that if no invariance relations are present, hence no ' variables are introduced, then the system describing the time evolution of the x; w variables is linear; in this case we can interpret normal forms as projections of a linear

system to an invariant manifold (without symmetry reduction). If there are no sporadic resonances of order greater than one, then upon solving the reduced equation for the ' variables one obtains a non-autonomous linear system. Moreover, if all eigenvalues are distinct then we have a product system of one-dimensional equations. If '(t) converges to some constant ' 0 , the asymptotic evolution of the system is governed by a linear autonomous equation for x and w. Similarly, if there is a pe¯ and '(t) converges to '(t) ¯ for large t, riodic solution '(t) the asymptotic evolution of the system is governed by a linear equation with t-periodic coefficients for x and w. Conclusions We have reviewed the basic notions concerning symmetry of dynamical systems and its determination, in particular in a perturbative setting. We have subsequently considered various situations where the interplay of perturbation theory and symmetry properties produce nontrivial results, either in that the perturbative expansion turns out to be simplified (with respect to the general case) due to symmetry properties, or in that computations of symmetry is simplified by dealing with a system in normal form; we then considered the problem of jointly normalizing an algebra of vector fields (with possibly but not necessarily one of these defining a dynamical system, the other being symmetry vector fields). We also discussed how normal forms can be characterized in terms of symmetry, and how this is extended to a characterization of “renormalized forms”. Finally we considered symmetry reduction applied to systems in normal form. The discussion conducted here illustrates some of the powerful conceptual and computational simplifications arising for systems with symmetry, also in the realm of perturbation theory. As remarked in the Introduction, symmetry is a nongeneric property; on the other hand, it is often the case that equations arising from physical (mechanical, electronic, etc.) systems enjoy some degree of symmetry, as a consequence of the symmetry of the fundamental equations of physics. Disregarding the symmetry properties in these cases would mean renouncing the use of what is often the only handle to grab the behavior of non-linear systems; and correspondingly a symmetry analysis can often on the one hand lead to identifying several relevant properties of the system even without a complete solution, and on the other hand being instrumental in obtaining exact (or approximate, as we are here dealing with perturbation theory) solutions.

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Here we discussed some of the consequences of symmetry for the specific case of dynamical systems, such as those met in analyzing the behavior of nonlinear systems near a known (e. g. trivial) solution. For a more general discussion, as well as for concrete applications, the reader is referred on the one hand to texts discussing symmetry for differential equations [3,19,26,27, 37,80,115,125,151,152,174], on the other to texts and papers specifically dealing with the interplay of symmetry and perturbation theory, quoted in the main text and listed in the ample Bibliography below.

help to anybody having to deal with perturbation of concrete symmetric systems. Finally, another field of future developments can be called for: here we discussed the interplay of perturbation theory and “standard” symmetries. Or, the notion of symmetry of differential equations has been generalized in various directions, producing in some cases a significant advantage in application to concrete systems. It should thus be expected that the interplay between these generalized notions of symmetry and perturbation theory will be investigated in the near future, and most probably will produce interesting and readily applicable results.

Future Developments

Additional Notes

First and foremost, future developments should be expected to concern further application of the general theory in concrete cases, both in nonlinear theoretical mechanics and in specific subfields, ranging from the more applied (e. g., ship dynamics and stability [13,176]; or handling of complex electrical networks [133,183]) to the more theoretical (e. g., galactic dynamics [24,59]). On the other hand, the theory developed so far is in many cases purely formal, in that consideration of convergence properties – and estimation of the convergence region in phase and/or parameter space – of the resulting (perturbative) series is often left aside, with the understanding that in any concrete application one will have explicit series and be ready to analyze their convergence. The story of general (i. e. non-symmetric) perturbation theory shows however that the theoretical analysis of convergence properties can be precious – not only for the conceptual understanding but also in view of concrete applications – and it should thus be expected that future developments will deal with convergence properties in the symmetric case (see e. g.  Perturbative Expansions, Convergence of, [52] for a review of existing results). The same issue of convergence, and estimation of the convergence region, arises in connection with further normalization (under any approach), and has so far been given little consideration. A different kind of generalization is called for when dealing with symmetry reduction of normal forms: in fact, on the one hand it is natural to try applying the same approach (based on quite general geometrical properties) to more general systems than initially considered; on the other hand the method discussed in Sect. “Symmetry Reduction of Symmetric Normal Forms” is algorithmic and could be implemented by symbolic manipulations packages – such as those already existing for computations of symmetry of differential equations – which would be of

We collect here the additional notes called throughout the manuscript. e is naturally a fiber bundle [40,119,146, (1) Note that M 147] over t: that is, it can be decomposed as the union of copies of M, each one in correspondence to a value of t. A section of this bundle is simply the graph of a function  : t ! M. The set  x considered in a moment is a section of this fiber bundle. (2) For general differential equations, one would go along the same lines. A relevant difference is however present: for (systems of) first order equations there is no algorithmic way to find the general solutions to determining equations, as opposed to any other case [151,174]. (3) In the case of Hamiltonian systems, one can work directly on the Hamiltonian H(p; q) rather than on the Hamilton equations of motion  Hamiltonian Perturbation Theory (and Transition to Chaos), [111]. We will however, for the sake of brevity, not discuss the specific features of the Hamiltonian case. (4) Later on, in particular when we deal with different homological operators, it will be convenient to also denote this A as LA , with reference to the matrix A appearing in F0 D Ax. (5) This is equivalent to defining (;i ; ; j ) D ı; ı i; j (!), where for the multi-index  we have defined ! D (1 !) : : : (n !). (6) The name “resonant” is due to the relation existing between eigenvectors of A and resonance relations among eigenvalues of the matrix A D (D f )(0) describing the linear part F0 (x) D Ax of the system. (7) In order to determine e X De f i (x)@/@x i , we write (using the exponential notation) (t C ıt) D eı t X (t). Therefore, x(t C ıt) D [esH e(ı t)X (t)]sD1 . Using (t) D eH x(t), we have [x(t C ıt)  x(t)] D [esH (e(ı t)X  I)esH x(t)](sD1) , and therefore (24).

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(8) Note, however, that when the diagonalizing matrix M is not unitary, this transformation changes implicitly the scalar product. (9) The general case is discussed e. g. in [51]; see also, for a more general discussion on normal forms with non-normal and non-semisimple normal form, the article  Perturbation of Systems with Nilpotent Real Part. (10) Note that the terms in Ker(A) D Ker(AC ) are just the resonant ones. Thus, the present characterization of normal forms is equivalent to the one given earlier on. (11) We also note that in the framework of Lie–Poincaré transformations, i. e. when considering the time one action of a vector field ˚ [25,28,145] (see above), Theorem 4 shows that ˚ can be chosen to admit Y as a symmetry; see [51]. (12) Generalizations can also be obtained in the direction of relaxing the normality assumption for the matrices identifying the linear part of vector fields. We will not discuss this case, referring the reader instead to  Perturbation of Systems with Nilpotent Real Part, [51]. (13) If this is the symmetry algebra of one of the vector fields, say X 0 , we have that [X 0 ; X i ] D 0 for all the i, i. e. X 0 belongs to the center of the algebra G . (14) We stress that we refer here to the case of a nilpotent Lie algebra, not to the case where the relevant matrices are nilpotent! (15) This is readily shown by the following example. Consider the two-dimensional system, which is not in normal form, x˙1 D x1 x2 , x˙2 D x2 . It is easily seen that  D x1 exp(x2 ) is a constant of motion, but not of the linearized problem, i. e.  2 I X , but  … I A . Similarly, we have that Y D x12 exp(x2 )(@/@x1 ) 2 G X but Y 62 GA . (16) It should be noted that X k , and hence L k , change under further normalization transformations; however, they stabilize after a finite number of steps, and in particular will not change anymore after the further normalization reaches their order. (17) Note that the b X k in Theorem 12 satisfies B(b X k ) :D b [Y; X k ] D [Y; X k ] C [Y; [X 0 ; Q k ]]; using the Jacobi identity, and assuming [Y; Q k ] D 0, this reads B(X k )  A[B(Q k )]. On the other hand, [A; B] D 0 Xk) D guarantees [A; B] D 0, hence we get B(b B[X k  A(Q k )]. Thus the assumption of Theorem 12 could be rephrased in terms of kernels of the operators AC and B as follows: for each X k 2 Ker(AC ) \ V k , there exists Q k 2 Ker(B) such that [X k  A(Q k )] 2 Ker(B).

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Non-linear Internal Waves

Non-linear Internal Waves MOUSTAFA S. ABOU-DINA , MOHAMED A. HELAL Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt Article Outline Glossary Definition of the Subject Introduction Problem and Frame of Reference Notation Equations of Motion The Shallow Water Theory Free Surface and Interface Elevations of Different Modes Secular Term Multiple Scale Transformation of Variables Derivation of the KdV Equation Conclusions Future Directions Bibliography Glossary Nonlinear waves Nonlinear waves are such waves which arise as solutions of the nonlinear mathematical models simulating physical phenomena in fluids. Shallow water Shallow water means water waves for which the ratio between the amplitude and the wave length is relatively small. The linear theory of motion is inadequate for the description of shallow water waves. Internal waves Internal waves are gravity waves that oscillate within a fluid medium. They arise from perturbations to hydrostatic equilibrium, where balance is maintained between the force of gravity and the buoyant restoring force. A simple example is a wave propagating on the interface between two fluids of different densities, such as oil and water. Internal waves typically have much lower frequencies and higher amplitudes than surface gravity waves because the density differences (and therefore the restoring forces) within a fluid are usually much smaller than the density of the fluid itself. Pycnocline Pycnocline is a rapid change in water density with depth. In freshwater environments, such as lakes, this density change is primarily caused by water temperature, while in seawater environments such as oceans the density change may be caused by changes in water temperature and/or salinity. Solitary waves Solitary waves are localized traveling waves, which asymptotically tend to zero at large distances.

Solitons Solitons are waves which appear as a result of a balance between a weakly nonlinear convection and a linear dispersion. The solitons are localized highly stable waves that retain their identity (shape and speed) upon interaction, and resemble particle like behavior. In the case of a collision, solitons undergo a phase shift. Baroclinic fluid Baroclinic fluid is such a fluid for which the density depends on both the temperature and the pressure. In atmospheric terms, the baroclinic areas are generally found in the mid-latitude/polar regions. Barotropic fluid Barotropic fluid is such a fluid for which the density depends only on the pressure. In atmospheric terms, the barotropic zones of the Earth are generally found in the central latitudes, or tropics. Definition of the Subject The objective of the present work is to study the generation and propagation of nonlinear internal waves in the frame of the shallow water theory. These waves are generated inside a stratified fluid occupying a semi infinite channel of finite and constant depth by a wave maker situated in motion at the finite extremity of the channel. A distortion process is carried out to the variables and the nonlinear equations of the problem using a certain small parameter characterizing the motion of the wave maker and double series representations for the unknown functions is introduced. This procedure leads to a solution of the problem including a secular term, vanishing at the position of the wave maker. This inconvenient result is remedied using a multiple scale transformation of variables and it is shown that the free surface and the interface elevations satisfy the well known KdV equation. The initial conditions necessary for the solution of the KdV equations are obtained from the results of the first procedure. Introduction The work on internal gravity waves has been started in the middle of the twentieth century by Keulegang [21] followed by Long [25], where they applied Boussinesq’s and Rayleigh’s techniques, respectively, on free surface waves to the problem of internal waves. In physical oceanography, fluid density varies with the depth due to the salinity and the temperature of the fluid [23]. Several works, dealing with variable density, have been worked out (e. g. [2,4,15,19,20,31,32], . . . ). The geophysical problem of the propagation waves in stratified fluid is very important in physical oceanography.

Non-linear Internal Waves

The physical problem is simulated by a suitable model of a free-surface and interface fluid flow over a horizontal bottom or over a certain topography. The fluid in the ocean may be considered as a stratified one according to the variation of the salinity and temperature, due to the weather, with respect to the vertical coordinate normal to the surface. The mathematical model of a two-fluid system (stratified fluid) is a good description to simulate wave motions in physical applications. Based on results of many field observations, it is found that the fluid’s density changes rapidly within the regions of pycnocline. Nonlinear theoretical models for waves in a two-fluid system have been established with various restrictions on length scales. Among these, Benjamin [6] derived the KdV equation for thin layers in comparison with the wave length. Later, Benjamin [7], Davis and Acrivos [14] and Ono [30] constructed the BO equation under the assumptions of a thin upper layer and an infinitely deep lower layer. Kubota et al. [22] and Choi and Camassa [11,12] carried out a series of investigations to derive model equations for weakly and strongly nonlinear wave propagation in stratified fluids. The upper boundary is allowed to be either free or rigid. The assumptions they stated are as follows: 1) the wavelength of the interfacial wave is long. 2) the upper layer is thin. 3) no depth restriction is made on the lower layer. The effect of a submerged obstacle on an incident wave inside an ideal two-laver stratified shallow water has been studied in two dimensions by Abou-Dina and Helal [3] by applying the shallow water approximation theory. In 1995, Pinettes, Renouard and Germain presented a detailed analytical and experimental study of the oblique passing of a solitary wave over a shelf in a two-layer fluid. Later, Barthelemy, Kabbaj and Germain [5] applied the WKBJ technique to investigate theoretically the scattering of a surface long wave by a step bottom in a two-layer fluid. This method enabled them to overcome the main inconsistency of the shallow-water theory in the presence of obstacles. Matsuno [28] attempted to unify the KdV, BO and ILW models using the assumption of small wave steepness. Lynett and Liu [26] assumed a small density difference and derived a set of model equations. Recently, Craig et al. [13] presented a Hamiltonian perturbation theory for the long-wave limit, and provided a uniform treatment of various long-wave models (KdV, BO and ILW models). Their formulation is shown to be very effective for perturbation calculations and represents

a basis for numerical simulations. For a single-fluid system, in the last two decades, the traditional Boussinesq equations have been improved to extend their applicability from shallow water to deep water (see for example: Nwogu [29], Chen and Liu [10], Wei et al. [36], Madsen and Schaffer [27], Gobbi et al. [18]). Most recently, Liu et al. [24] published an excellent study on the essential properties of Boussinesq equations for internal and surface waves in a two-fluid system. Most studies applied the technique of the Padé approximate to allow a much higher-order accuracy without increasing the order of the derivatives. Although the theoretical work for surface waves are excessive, studies of internal waves based on Boussinesq equations are still lack. In the present work, and for a better description of the physical problem, we shall investigate, in the frame of the shallow water theory, the effect of the bounded motion of a vertical plane wave maker on the generation and propagation of waves inside a stratified fluid. The wave maker is situated at the finite end of a semi-infinite channel of constant depth occupied by two non immiscible fluid layers of different constant densities. The problem considered here, is supposed to be a two dimensional one with an irrotational motion, the fluid layers are taken ideal and the surface tension is neglected. Double series representation for the unknown functions, in terms of a certain small parameter characterizing the motion of the wave maker is used. The first few orders of approximations obtained following this technique indicates the presence of a secular term increasing indefinitely in going away from the wave maker and vanishing at the position of this later. To overcome this physically unacceptable result, a multiple scale transformation of variables is carried out and a single series representation, for the unknown functions, is proposed. This technique leads to solutions up to the fifth order of approximations free from secular terms and shows that the generation and propagation processes of internal waves are governed by the well known KdV equation. The initial conditions needed for the solution of this nonlinear partial differential equation are obtained from the results of the preceding method at the position of the wave maker where the solution is free from secular terms. On the light of this result it is expected to obtain an internal solitonic waves. Problem and Frame of Reference A vertical plane wave-maker is situated at the finite extremity of a semi-infinite channel with a constant depth. The channel is occupied by two layers of immiscible and

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Non-linear Internal Waves

Non-linear Internal Waves, Figure 1 Problem and frame of reference

inviscid liquids of constant densities. The wave-maker is set in motion generating waves which propagate, inside both fluid layers, towards the down-stream extremity of the channel. The required is to calculate the produced waves and also to determine the motion that should be assigned to the wave maker in order to generate internal waves only in the channel. In the mathematical model, the problem is assumed to be a two dimensional one, the flow is considered irrotational and the free surface and the interface are assumed to remain always near their positions at rest. Furthermore, the motion of the wave maker is considered slow and bounded and the problem is studied in the frame of the non-linear shallow water theory of motion. A fixed rectangular system of reference is used for the description of the motion of the fluid. The origin O is taken in the free surface at rest with horizontal x-axis pointing along the direction of propagation of waves. The y-axis is directed vertically upwards (Fig. 1). Notation The following notation is used throughout this paper: g the acceleration of gravity h1 ; h2 (h D h1 C h2 ) the constant depths of the upper and lower layers of the fluid, respectively P(x; y) the pressure S the stocks parameter t the time D x D " f (y; "t) the horizontal displacement of the wave maker (D " f (1) (y; "t) in the upper layer and " f (2) (y; "t) in the lower layer)

˚ ( j) (x; y; t)

the velocity potentials: j D 1; 2 for the upper and lower layers, respectively y D (1) (x; t) the equation of the free surface y D h1 C (2) (x; t) the equation of the interface (x; y) Cartesian coordinates of a point ın;m the Kronecker delta D 1 if n D m; 0 if n ¤ m " a small parameter

1 ; 2 the constant densities of the upper and lower layers, respectively ( 1 < 2 ) an operator of total differentiation Dz w.r.t. z @x;y;:::z an operator of partial differentiation w.r.t. x; y; : : : ; and z Superscripts 1 upper layer 2 lower layer 0 ; 00 first and second derivatives w.r.t. the argument of the superscripted function Subscripts e i

the barotropic (external) mode the baroclinic (internal) mode

Equations of Motion The general system of equations and simplifying conditions describing water wave motion are expressed in terms of the velocity potential in the case of irrotational motion for both homogeneous and stratified fluids (Boussinesq [9], Stoker [33], Wehausen and Laitone [35]).

Non-linear Internal Waves

As in Abou-Dina and Helal [2,3,4], we start with the set of distorted variables: b x;b y;b t defined in terms of x, y, t as: b x D "x ;

b yD y; b t D "t ;

(1)

where " is a small parameter. b ( j) ; b Let us denote by ˚ P ( j) and b  ( j) the functions x/";b y;b t/"), P( j) (b x/";b y;b t/") and ( j) (b x/";b t/") respec˚ ( j) (b tively. According to the physical and simplifying conditions adopted here, the system of equations and conditions governing the problem is written in terms of the distorted variables b x;b y;b t as follows (Abou-Dina and Helal [2,3,4], Abou-Dina and Hassan [1]): (I)

In the fluid layers with constant densities: b ( j) C @ ˚ b ( j) D 0 ; ˚ "2 @b b xb x yb y

(2)

  2 ( j) ( j) 1 b b ( j) b b P (b x;b y; t ) D j " @bt ˚ C "2 @b ˚ x 2   2  b ( j) C gb y ; (3) ˚ C @b y

(II)

where j hereafter stands for both 1 and 2. On the free surface which is impermeable and isobaric: (1) b (1) D "2 @ b b (1) C " @ b @b ˚ ˚  (1) @b b bt  y x x   x;b t ; at b y Db  (1) b

(4)

  2  2  b (1) C 1 "2 @ ˚ b (1) C @ ˚ b (1) " @bt ˚ b x y b 2   x;b t : (5) C gb y D 0 at b y Db  (1) b (III)

At the interface, the impermeability gives: (2) b ( j) D "2 @ b b ( j) C "@ b @b  (2) @b ˚ ˚ y x x b bt    x;b t ; at b y D h1 C b  (2) b

(IV)

On the horizontal bottom: b (2) D 0 @b ˚ y

(V) (VI)

Also the compatibility condition for the pressure across this boundary implies:

  b (1)

1 g h1 C b  (2) C " @bt ˚   1 2  b (1) 2  b (1) 2 ˚ ˚ C C @ " @b b x y 2

  b (2) D 2 g h1 C b  (2) C " @bt ˚   1 2  b (2) 2  b (2) 2 C C @ " @b ˚ ˚ b x y 2   x;b t : (7) at b y D h1 C b  (2) b

(8)

The radiation condition implies that no wave comes from infinity. The initial conditions: At the initial instant of time (b t D 0), the fluid is at rest with horizontal free surface at b y D 0 and interface at b y D h1 . This iniy;b t) tial condition assumes that the functions f ( j) (b have initial vanishing values and implies at b t D 0 that  inside the fluid layers b ( j) (b x;b y; 0) D 0 ; ˚

(9a)

b ( j) (b ˚ x;b y; 0) D 0 ; @b x

(9b)

b ( j) (b @b x;b y; 0) D 0 ; ˚ y

(9c)

 at the free surface b  (1) (b x; 0) D 0

(9d)

and  at the interface x; 0) D 0 ; b  (2) (b

(9e)

provided that y; 0) D 0 f ( j) (b

for j D 1; 2 :

(9f)

(VII) On the wave maker:    @ ( j)  b ( j) b b x;b y;b t D" y;b t @b f ˚ x @b t at b x D0;

(6)

at b y D h :

h C h2 ı j;1  b y  h1 ı j;2 :

(10)

to the system of equations and conditions governing the problem in the non-distorted (physical) space, we have to replace " by unity and omit the hats (^ ) over the symbols in Eqs. (2) to (10). The Shallow Water Theory In the frame work of the shallow water theory, the nonlinear system of Eqs. (2) to (10) will be solved using the double series representation:   1 X 1   X mb x b ( j) b x;b y;b t D "n exp  ˚ hj " nD1 mD1   ( j) b n;m b x;b y;b t ; (11a) ˚

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Non-linear Internal Waves

  1 X 1   X  mb x ( j)  x;b t D x;b t : b ( j) b "n exp  b n;m b hj " nD1 mD1 (11b) The validity of using the above representation has been studied by Germain [16,17]. The small parameter " represents the ratio of the water depth to the wave length. Hence, S  1 and we are dealing with the non-linear acoustic analogy. According to the theory under consideration, at each order (n,m), the above system of equations and conditions must be verified. For simplification in the following, the hats (^ ) over the symbols will be omitted.

obtain local oscillations along with progressive waves of two different modes: (i) External (or barotropic) modes corresponding to the wave speed C1 . (ii) Internal (or baroclinic) modes corresponding to the wave speed C2 . The corresponding expressions of the free surface and the interface are given, to the third order, by: (1) (x; t) D ("/g) @ t ˚ (1) (x; y; t) at y D 0 ; (14a)   " (2) (x; t) D @t g ( 2  1 ) n o  1 ˚ (1) (x; y; t)  2 ˚ (2) (x; y; t) at y D h1 :

Verification of the Homogeneous Equations

(14b)

The above procedure, when applied to the homogeneous Eqs. (2)–(8) along with the radiation condition (V), can lead after some manipulations to the following expression for the total velocity potential up to the second order of the small parameter ": h i ˚ ( j) (x; y; t) D " A( j) (x  C1 t) C B( j) (x  C2 t) i h C "2 R( j) (x  C1 t) C S ( j) (x  C2 t)   1 X m x ( j) C "2 A m (t) exp  hj " mD1   m  cos (y C ı j 2 h) C O("3 ) ; hj (12) where, o p gn C 2j D h  (1) j (h1  h2 )2 C 4 ( 1 / 2 ) h1 h2 : 2 (13a)

Complete Determination of the Solution To complete the determination of the velocity potentials ˚ ( j) (x; y; t), we verify the initial conditions (9) and the condition on the wave maker (10) which yield, together with expression (12) for the velocity potential functions, the following relations:  In the upper layer: 0

  cos

A(1) (x  C1 t) D 1 A(2) (x  C1 t) (1)

(2)

B (x  C2 t) D 2 B (x  C2 t)

(13b) (13c)

and,

j D

g h2 : C 2j  g h1

(13d)

The functions R( j) and S( j) are arbitrary functions combined by similar relations to those in Eq. (12) in terms of A( j) and B( j) respectively. Equations (11)–(13) indicate the existence of two different wave speeds C1 and C2 (C1 > C2 ). Accordingly, we

1  X mA(1) m (t) h1 mD1

 m @ (1) y D f (y; t) ; h1 @t

 h1  y  0 :

(15a)

 In the lower layer: 0

0

A(2) (C1 t) C B(2) (C2 t) 

( j)

The functions A( j) , B( j) and Am are arbitrary functions to be determined, with:

0

A(1) (C1 t) C B(1) (C2 t) 

  cos

1  X (1)m mA(2) m (t) h2 mD1

 m @ (2) (y C h1 ) D f (y; t) ; h2 @t

 h  y  h1 :

(15b)

Relations (15a) and (15b) together with the initial conditions (9) lead, after some manipulations, to the following expressions: C1 A(2) (x  C1 t) D

1  2 " Z   x  C1 t

2 h1 (2) y;  dy  f h2 h C1   # Z 0 1 x  C1 t (1)  f y;  dy (16a) h1 h1 C1

Non-linear Internal Waves

B(2) (x  C2 t) D

C2

2  1 " Z  

1 h1 (2) x  C2 t  f y;  dy h2 h C2   # Z 0 t 1 x  C 2  f (1) y;  dy h1 h1 C2 (16b)

 @ (1) f (y; t) @t h   1 m y dy  cos h1  Z h1

@ (2) 2 (1) A m (t) D f (y; t) m  h @t   m  cos (y C h1 ) dy h2 A(1) m (t) D

2 m

Z

0

(16c)

(16d)

The functions A(1) (xC1 t) and B(1) (xC2 t) are given in terms of the functions A(1) (x  C1 t) and B(1) (x  C2 t) by relations (13b) and (13c). Free Surface and Interface Elevations of Different Modes The contributions of the progressive waves of external and internal modes to the free surface elevation, (1) e (x; t) and (x; t), are given using (12) and (14a) as (1) i (1) e (x; t) D

"2 C1 1 g ( 1  2 ) " Z   @ 2 h1 (2) x  C1 t  f y;  dy @t h2 h C1   # Z 0 1 x  C1 t (1)  f y;  dy ; h1 h1 C1 (17a)

(1) i (x; t) D

"2 C2 2 g ( 2  1 ) " Z   @ 1 h1 (2) x  C2 t  f y;  dy @t h2 h C2   # Z 0 1 x  C2 t (1)  f y;  dy : h1 h1 C2

(1) terms of (1) e (x; t) and i (x; t) using (12)–(14) as

(2) e (x; t) D

2  1 1 (1)  (x; t) ;

1 ( 2  1 ) e

(18a)

(2) i (x; t) D

2  2 1 (1)  (x; t) :

2 ( 2  1 ) i

(18b)

It is well known, in the studies dealing with stratified fluids, that the external mode is dominant in the neighborhood of the free surface and has a negligible contribution on the interface, while the internal mode is dominant in the neighborhood of the interface and has a negligible contribution on the free surface. If the motion of the wave maker is such that the function (2) e (x; t) given by (17a) vanishes, the major contribution of both modes is localized in the neighborhood of the interface and in such cases we say that the wave maker generates internal waves only in the channel. Secular Term A suitable form of the double series representation (11) was used in studying certain problems in the case of homogeneous fluids (Abou-Dina and Helal [3], Abou-Dina and Hassan [1]). It has been shown that this procedure leads to a secular term of the third order in " in the expression of the velocity potential. This secular term, which increases indefinitely with the increase of the variable x, vanishes at x D 0. The same result can be shown in the case of stratified fluids, according to the analysis of Abou-Dina and Helal [2]. Hence, although expressions (12)–(18) are valid at x D 0, they are not adequate for the description of the propagation of waves far from the wave maker. This unacceptable result is due to certain aspects of the mathematical procedure used (see Abou-Dina and Helal [3], Abou-Dina and Hassan [1] for the justification) and needs to be remedied. Our main interest, in the remaining part of the present paper, is to modify the mathematical procedure used above in order to describe the propagation of waves generated by the wave maker in going towards the down-stream extremity of the channel. Multiple Scale Transformation of Variables We use the set of variables u, v and y defined in terms of the distorted set x, y, t by (cf. Benneye [8] and Temperville [34]):

(17b) uDxCt; The contributions of external and internal modes to (2) the interface elevation, (2) e (x; t) and i (x; t), are given in

v D "2 x ;

where C is a real constant to be precised.

(19)

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Non-linear Internal Waves

Equations (2), (4) up to (8) can be written in terms of u, v and y as: (i) In the fluid mass: 6

" @vv ˚

( j)

4

C 2 " @uv ˚

following representations for the functions ˚ ( j) (u; v; y) and ( j) (u; v) (cf. Temperville [34]): ˚ ( j) (u; v; y) D

( j)

2

C " @uu ˚

( j)

C @y y ˚

( j)

( j)

"2n1 ˚2n1 (u; v; y) ;

(24a)

nD1

D 0: (20)

1 X

( j) (u; v) D

1 X

( j)

"2n 2n (u; v) :

(24b)

nD1

(ii) On the free surface: @ y ˚ (1) D "6 @v (1) @v ˚ (1) n o C "4 @u (1) @v ˚ (1) C @v (1) @u ˚ (1) C "2 @u (1) @u ˚ (1)  " C @u (1) at y D (1) (u; v) ;

(21a)

n o2 1 g (1)  " C @u ˚ (1) C "2 @u ˚ (1) C "2 @v ˚ (1) 2  C (@ y ˚ (1) )2 D 0 at y D (1) (u; v) : (21b)

It can be shown from the above equations that for n D 1, ( j) the functions ˚1 (u; v; y), j D 1; 2 are independent of the variable y and that C (25) @u ˚1(1) (u; v; y) ; at y D 0 ; g h C

2 @u ˚1(2) (u; v; y) (2) 2 (u; v) D g ( 2  1 ) i  1 @u ˚1(1) (u; v; y) ; at y D h1 : (26) (1) 2 (u; v) D

Also, for n D 2, the values of the constant C appearing in (19) can be obtained as

(iii) On the interface: @ y ˚ ( j) D "6 @v (2) @v ˚ ( j) n o C "4 @u (2) @v ˚ ( j) C @v (2) @u ˚ ( j) C "2 @u (2) @u ˚ ( j)  " C @u (2) at y D (1) (u; v) for j D 1; 2

(22a)

(

 

1 g h1 C (2)  " C @u ˚ (1) )

o2  2  1 2n C " @u ˚ (1) C "2 @v ˚ (1) C @ y ˚ (1) 2 (   D 2 g h1 C (2)  " C @u ˚ (2) )

o2  2  1 2n (2) 2 (2) (2) C C @y ˚ " @u ˚ C " @v ˚ 2 (2)

at y D h1 C  (u; v) : (22b)

CD

C1 for the barotropic (external) mode ; C2 for the baroclinic (internal) mode ;

(27)

where Cj is given by (13a) for j D 1; 2. It can also be shown that the results up to n D 3 contain no secular terms, and that the function (1) 2 (u; v), characterizing the free surface elevation, satisfies the following KdV equation: (1) (1) (1) L @uuu (1) 2 C M 2 @u 2 C N @v 2 D 0 ;

(28)

where    C2 1 3 h1 2h C h13  3hh12 C  ; 6 2 g 3 2(1  )( 2  1 ) M D 3 C ;

( 2  1 )

LD

N D 2 (h2 C h1 )

(29a) (29b) (29c)

and

(iv) On the bottom: @ y ˚ (2) D 0 at y D h :

Derivation of the KdV Equation

(23)

In the regions where we are interested in far from the wave maker, the contribution of the local disturbance on the motion of the fluid layers can be neglected and we use the

D

C2

g h2 :  g h1

(29d)

The initial condition at v D o (or x D 0 and u D C t), needed for the solution of the KdV Eq. (28) is obtained from (17)in the form:

Non-linear Internal Waves

 Barotropic mode: (1) e (u; 0) D

"2 C12 1 g ( 1  2 ) " Z   u @ 2 h1 (2) y;  dy (30a)  f @u h2 h C1   # Z 0 1 u (1)  y;  dy f h1 h1 C1

 Baroclinic mode: (1) i (u; 0) D

"2 C22 2 g ( 2  1 ) " Z   u @ 1 h1 (2) y;  dy (30b)  f @u h2 h C2   # Z 0 1 u (1)  f y;  dy h1 h1 C2

The following relation is also obtained between (u; v) and (1) (2) 2 2 (u; v): (2) 2 (u; v) D

2  1 (1)  (u; v) :

( 2  1 ) 2

(31)

Conclusions A multiple scale transformation of variables and a single series representations for the velocity potentials and the free surface and the interface elevations show that these elevations up to the second order satisfy the KdV equation. The initial conditions needed for the solution of this equation are obtained at the position of the wave maker using another technique depending on a double series representation of the unknown functions. The particular choice of the motion of the wave maker can minimize the elevation of the free surface and hence lead to dominant nonlinear internal waves only in the channel. Future Directions For a future work following the present one, it is intended to investigate the following items:  Study the possibility of generating nonlinear internal waves by different types of the motion of the plane wave maker. This will need analytical and numerical work as well.  Study the inverse problem, precisely the possibility of the recuperation of the water waves’ energy and employing it in producing a controlled solid body motion.

 Study the possibility of generating nonlinear internal waves in model problems with different and more complicated geometry, to simulate the actual physical situations.

Bibliography Primary Literature 1. Abou-Dina MS, Hassan FM (2006) Generation and propagation of nonlinear tsunamis in shallow water by a moving topography. Appl Math Comput 177:785–806 2. Abou-Dina MS, Helal MA (1990) The influence of submerged obstacle on an incident wave in stratified shallow water. Eur J Mech B/Fluids 9(6):545–564 3. Abou-Dina MS, Helal MA (1992) The effect of a fixed barrier on an incident progressive wave in shallow water. Il Nuovo Cimento 107B(3):331–344 4. Abou-Dina MS, Helal MA (1995) The effect of a fixed submerged obstacle on an incident wave in stratified shallow water (Mathematical Aspects). Il Nuovo Cimento B 110(8): 927–942 5. Barthelemy E, Kabbaj A, Germain JP (2000) Long surface wave scattered by a step in a two-layer fluid. Fluid Dyn Res 26: 235–255 6. Benjamin TB (1966) Internal waves of finite amplitude and permanent form. J Fluid Mech 25:241–270 7. Benjamin TB (1967) Internal waves of permanent form of great depth. J Fluid Mech 29:559–592 8. Benney DJ, LIN CC (1960) On the secondary motion induced by oscillations in a shear flow. Phys Fluids 3:656–657 9. Boussinesq MJ (1871) Théorie de l’intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. Acad Sci Paris, CR Acad Sci 72:755–759 10. Chen Y, Liu PL-F (1995) Modified Boussinesq equations and associated parabolic models for water wave propagation. J Fluid Mech 288:351–381 11. Choi W, Camassa R (1996) Weakly nonlinear internal waves in a two-fluid system. J Fluid Mech 313:83–103 12. Choi W, Camassa R (1999) Fully nonlinear internal waves in a two-fluid system. J Fluid Mech 396:1–36 13. Craig W, Guyenne P, Kalisch H (2005) Hamiltonian long-wave expansions for free surfaces and interfaces. Commun Pure Appl Math 18:1587–1641 14. Davis RE, Acrivos A (1967) Solitary internal waves in deep water. J Fluid Mech 29:593–607 15. Garett C, Munk W (1979) Internal waves in the ocean. Ann Rev Fluid Mech 11:339–369 16. Germain JP (1971) Sur le caractère limite de la théorie des mouvements des liquides parfaits en eau peu profonde. CR Acad Sci Paris Série A 273:1171–1174 17. Germain JP (1972) Théorie générale d’un fluide parfait pesant en eau peu profonde de profondeur constante. CR Acad Sci Paris Série A 274:997–1000 18. Gobbi MF, Kirby JT, Wei G (2000) A fully nonlinear Boussinesq model for surface waves-Part 2. Extension to O(kh)4. J Fluid Mech 405:181–210 19. Helal MA, Moline JM (1981) Nonlinear internal waves in shallow water: A theoretical and experimental study. Tellus 33:488–504

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20. Kabbaj A (1985) Contribution a l’etude du passage des ondes des gravite sur le talus continental et a la generation des ondes internes. These de doctorat d’etat, IMG Université de Grenoble 21. Keulegan GH (1953) Hydrodynamical effects of gales on Lake Erie. J Res Natl Bur Std 50:99–109 22. Kubota T, Ko DRS, Dobbs LD (1978) Propagation of weakly nonlinear internal waves in a stratified fluid of finite depth. AIAA J Hydrodyn 12:157–165 23. LeBlond PH, Mysak LA (1978) Waves in Ocean. Elsevier, Amsterdam 24. Liu C-M, Lin M-C, Kong C-H (2008) Essential properties of Boussinesq equations for internal and surface waves in a twofluid system. Ocean Eng 35:230–246 25. Long RR (1956) Solitary waves in one- and two-fluid systems. Tellus 8:460–471 26. Lynett PJ, Liu PL-F (2002) A two-dimensional depth-integrated model for internal wave propagation over variable bathymetry. Wave Motion 36:221–240 27. Madsen PA, Schaffer HA (1998) Higher-order Boussinesq-type equations for surface gravity waves: derivation and analysis. Philos Trans R Soc Lond A 356:3123–3184 28. Matsuno Y (1993) A unified theory of nonlinear wave propagation in two-layer fluid systems. J Phys Soc Jpn 62:1902– 1916 29. Nwogu O (1993) Alternative form of Boussinesq equations for

30. 31. 32. 33. 34.

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nearshore wave propagation. J Waterways Port Coast Ocean Eng ASCE 119:618–638 Ono H (1975) Algebraic solitary waves in stratified fluids. J Phys Soc Jpn 39:1082–1091 Peters AS, Stoker JJ (1960) Solitary waves in liquid having nonconstant density. Comm Pure Appl Math 13:115–164 Robinson RM (1969) The effect of a vertical barrier on internal waves. Deep-Sea Res 16:421–429 Stoker JJ (1957) Water waves. Interscience, New York Temperville A (1985) Contribution à la théorie des ondes de gravité en eau peu profonde. Thèse de doctorat d’état, IMG Université de Grenoble Wehausen JV, Laitone EV (1960) Surface waves. In: Handbuch der Physik 9. Springer, Berlin Wei G, Kirby JT, Grilli ST, Subramanya R (1995) A fully nonlinear Boussinesq model for surface waves, Part 1. Highly nonlinear, unsteady waves. J Fluid Mech 294:71–92

Books and Reviews Germain JP, Guli L (1977) Passage d’une onde sur une barrier mince immergée en eau peu profonde. Ann Hydrog 5(746):7–11 Pinettes M-J, Renouard D, Germain J-P (1995) Analytical and experimental study of the oblique passing of a solitary wave over a shelf in a two-layer fluid. Fluid Dyn Res 16:217–235

Non-linear Ordinary Differential Equations and Dynamical Systems, Introduction to

Non-linear Ordinary Differential Equations and Dynamical Systems, Introduction to FERDINAND VERHULST Mathematisch Instituut, University of Utrecht, Utrecht, The Netherlands An ordinary differential equation (ODE) is called linear if it can be written in the form d y/dx D f (x)y C g(x) with x a real or complex variable and y an n-dimensional (finite) real or complex vector function. Non-linear ODEs are then n-dimensional ODEs of the form dy/dx D F(x; y) that are not linear. Jean Mawhin famously compared this distinction to a division of the animal world into ‘elephants’ and ‘non-elephants’, but even admitting that the distinction between linear and non-linear ODEs is artificial and of relatively recent date, it makes a little bit more sense than it looks like at first sight. The reason is, that in many problem formulations in classical physics, linear equations are quite common. Also, when considering non-linear ODEs, but linearizing around particular solutions, a number of fundamental theorems can be used to characterize the particular solutions starting with the features of the linearized equation. This will become clear in a number of the articles that follow. On the other hand, right from the beginning of the development of classical physics, the analysis of differential equations with their fully non-linear behavior has been necessary and essential. Think of celestial mechanics that started in the 18th century and for instance the analysis of solitons in fluid mechanics. The great scientists of the 18th century, among which Newton, Euler, Lagrange and Laplace, were all concerned with the formulation and first analysis steps of non-linear ODEs. This work was continued by mathematicians like Jacobi and Painlevé who devoted much of their attention to transformation methods and the analysis of special cases. A new stimulus came in the second half of the 19th century with the insights and fundamental ideas of Henri Poincaré. In fact his ideas are still fully alive and active today. Poincaré realized that non-linear ODEs can in general not be solved explicitly, i. e. expressed in terms of elementary functions, so that calculations should be supplemented by qualitative theory. Poincaré developed both quantitative and qualitative methods and his approach has shaped the analysis of non-linear ODEs in the period that followed up till now, becoming part of the topic of dynam-

ical systems. This volume of the Encyclopedia reflects to a large part these new developments. Basic questions on existence and uniqueness of solutions are discussed by Gianne Derks (see  Existence and Uniqueness of Solutions of Initial Value Problems) with attention to recent extensions of the notion of an ODE. Most of this is classical material. The article by Carmen Chicone (see  Stability Theory of Ordinary Differential Equations) deals with stability theory. It is concerned with the mathematical formulation and the basic results, but also includes the stability question in the context of conservative systems and the part played by the KAM theorem. The stability of periodic orbits gets special attention and in addition the stability of the orbit structure as a whole. This is usually referred to as structural stability. The theory of periodic solutions of non-autonomous ODEs displays a number of striking differences with the autonomous case. Jean Mawhin (see  Periodic Solutions of Non-autonomous Ordinary Differential Equations) presents classical and new results in this field. A more general technique to study equilibria, periodic solutions and their bifurcations is the Lyapunov–Schmidt method which is intimately connected to conditions of the implicit function theorem in its abstract form. André Vanderbauwhede (see  Lyapunov–Schmidt Method for Dynamical Systems) explains this basic method for equilibria and for periodic solutions with extensions to infinite dimensions. Center manifolds are manifolds associated with the critical part of the spectrum of equilibria and periodic solutions. They arise naturally in theory and applications. George Osipenko (see  Center Manifolds) discusses the theory with special attention to the corresponding (and very effective) reduction principle. A number of special topics have been getting a lot of attention, both in mathematical and in applied research. Relaxation oscillations are described by Johan Grasman (see  Relaxation Oscillations), starting with the classical example of the Van der Pol-oscillator and going on to more complicated systems. This also involves the discussion of canards in geometric singular perturbation theory which continues to produce activity in dynamical systems research and applications in mathematical biology. Periodic solutions of Hamiltonian systems merit a special treatment in the article by Luca Sbano (see  Periodic Orbits of Hamiltonian Systems). Natural problems are the continuation of orbits and the part played by symmetries. Variational methods have become important in this field, enabling us to consider the topology and geometry of periodic paths in sufficient generality.

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Another classical topic that is fully alive is the dynamics of parametric excitation. Alan Champneys (see  Dynamics of Parametric Excitation) introduces the necessary Floquet theory, corresponding bifurcation diagrams and a number of applications. Non-linear ODEs has grown into the more general field of dynamical systems. Beginning with hyperbolic behavior and examples, Araújo and Viana (see  Hyperbolic Dynamical Systems) discuss attractors and physical measures with the idea of stochastic stability. This leads naturally to the formulation of obstructions to hyperbolicity, partial and non-uniform hyperbolicity.

To handle dynamical systems in practice, one needs quantitative methods. Meijer, Dercole and Oldeman (see  Numerical Bifurcation Analysis) consider numerical bifurcation analysis for systems depending on parameters. This involves continuation and detection of bifurcations, complications like branch switching and orbit connection and a guide to possible software environments. The literature on non-linear ODEs grows at a fast pace. The articles are presenting a tour of the relevant literature, and even more importantly, they point out future directions of research in their respective fields.

Non-linear Partial Differential Equations, Introduction to

Non-linear Partial Differential Equations, Introduction to ITALO CAPUZZO DOLCETTA Dipartimento di Matematica, Sapienza Universita’ di Roma, Rome, Italy A large number of nonlinear phenomena in fundamental sciences (physics, chemistry, biology . . . ), in technology (material science, control of nonlinear systems, ship and aircraft design, combustion, image processing . . . ) as well in economics, finance and social sciences are conveniently modeled by nonlinear partial differential equations (NLPDE, in short). Let us mention, among the most important examples for the applications and from the historical point of view, the Euler and Navier–Stokes equations in fluid dynamics and the Boltzmann equation in gas dynamics. Other fundamental models, just to mention a few of them, are reaction-diffusion, porous media, nonlinear Schrödinger, Klein–Gordon, eikonal, Burger and conservation laws, nonlinear wave Korteweg–de Vries . . . The above list is by far incomplete as one can easy realize by looking at the current scientific production in the field as documented, for example, by the American Mathematical Society database MathSciNet. Despite an intense mathematical research activity going on since the second half of the XXth century, the extremely diversified landscape of the area of NLPDE is still largely unexplored and a number of difficult problems are still open. Generally (and therefore rather vaguely) speaking, one of the main difficulties induced by the nonlinear (as opposed to linear) structure of a partial differential equation is that such equations, especially first-order ones, do not in general possess smooth solutions. The analytical approach therefore requires the adoption of appropriate generalized notions of solutions (weak solutions in Sobolev spaces, entropy solutions, viscosity solutions, re-normalized solutions . . . ) in order to handle the classical Hadamard well-posedness approach to the rigorous validation of the model (existence, uniqueness and stability of solutions). Another fundamental difference with respect to the linear theory lies in the fact that no explicit representation formulas for solutions are in general available, even for simplified models. Fortunately enough however, good theory is (almost) as useful as exact formulas, as stated by L.C. Evans in the preface of his book Partial Differential Equations, Graduate Studies in Mathematics (Volume

19, American Mathematical Society, Providence Rhode Island, 1998). It is clear also, in this respect, that progress in NLPDE is necessarily strongly connected with the development of numerical methods of simulations. A further “pathology” due to nonlinearity is that solutions of initial value problems for nonlinear evolution equations may exist only for a small lap of time (the blowup phenomenon). These and other unavoidable features imply the need to develop new, sophisticated and often ad hoc mathematical methods for investigating different classes of NLPDE. The nonlinear theory (rather, theories) has been widely developed for different classes of equations, mainly in the direction of existence (via monotonicity, compactness, critical points methods . . . ), regularity of generalized solutions, qualitative analysis of solutions, limit problems (e. g. large time asymptotic, homogenization, scaling limits . . . ). Quoting P.L. Lions, new tools are required and discovered [at this purpose] in connection with all fields of analysis (functional, real, global, complex, numerical, Fourier . . . ), see On Some Challenging Problems in Nonlinear Partial Differential Equations (Mathematics: Frontiers and Perspectives 2000, International Mathematical Union). The 11 papers comprised in this section offer deep insight and up to date information on some of the various aspects mentioned above. A common feature of the papers is that their focus is on nonlinear partial differential equations arising in “real world” applications. The paper  Hyperbolic Conservation Laws by A. Bressan reports on very recent important advances in the understanding of firstorder nonlinear partial differential equations (or systems) describing the time evolution of certain basic quantities in physical models such as mass, momentum, energy, electric charge. The main mathematical difficulty is related to the strong linearity and the absence of dissipative terms in the equations; the evolution of smooth initial data may then produce discontinuities (shocks) in finite time, thus requiring the introduction of appropriate notion of weak entropic solutions. Many important NLPDE, e. g. Hamilton–Jacobi– Bellman and Isaacs equations arising in optimal control and game theory, mean-curvature flow and 1-Laplace equations do not have a divergence structure. In this case, the notion of weak solution in the distribution sense is not applicable. S. Koike’s  Non-linear Partial Differential Equations, Viscosity Solution Method in provides an account of ideas and results of the theory of viscosity solutions. This generalized notion of solution, which is intimately related to the maximum principle for elliptic equations, has proved to be most appropriate to implement the

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well-posedness program to large classes of second-order fully NLPDE not treatable by variational methods. First-order NLPDE of Hamilton–Jacobi-type have a central relevance both in classical and quantum mechanics as well in calculus of variations and optimal control theory. Lack of regularity occurs here at the level of the smoothness of the gradient of solutions for reasons which are somewhat similar to those indicated for hyperbolic conservation laws. The contribution by A. Siconolfi  Hamilton–Jacobi Equations and Weak KAM Theory reports on current research on the impact of the notion of the Aubry–Mather set, imported from dynamical systems theory, in the understanding of uniqueness, extra smoothness properties and large time behavior of viscosity solutions. In a physical model, dispersion can be understood as the decrease with time of the size of some relevant quantity such as matter or energy over a given volume. P. D’Ancona’s  Dispersion Phenomena in Partial Differential Equations examines in detail analytical techniques to evaluate, in suitable norms, the rate of dispersion in some very fundamental physical models such as the nonlinear wave and Schrödinger equations; new tools in Fourier and harmonic analysis play a key role in this respect. In connection with a general remark made above, it is worth pointing out that sharp dispersion (or Stricharz type) estimates have relevant applications to the issue of existence of solutions that are global in time. The interaction of nonlinearity with randomness in a PDE model is the leading theme of  Non-linear Stochastic Partial Differential Equations by G. Da Prato. This is a definitely new and rapidly developing area of investigation. The paper presents a general nonlinear semigroup approach to the study, mainly focused at global existence and well-posedness, of a large class of stochastic dynamical systems on Hilbert spaces. The large time behavior of solutions is also analyzed, relying on the notion of invariant measure. The application of general abstract results to specific models of interest, such as Ornstein–Uhlenbeck, reaction-diffusion, Burger’s, 2D Navier– Stokes equations perturbed by noise is also discussed in the paper. The Navier–Stokes equations of fluid dynamics are presently a fundamental model for simulations in several branches of applied sciences (e. g. meteorology, oceanography, biology . . . ) and design in industry (airplane, car, oil . . . ). The text by G.P. Galdi  Navier–Stokes Equations: A Mathematical Analysis is an extended and fairly complete review of mathematical tools (appropriate function spaces, a priori estimates, fixed point theorems), results

(concerning in particular the well-posedness of initial and boundary value problems) and open questions (including the famous one concerning the global regularity of solutions in 3D) pertaining to this important topic. Stokes equations, which are obtained as a formal limit from Navier–Stokes equations, are central in the article  Biological Fluid Dynamics, Non-linear Partial Differential Equations by A. De Simone, F. Alouges, and A. Lefebvre. A mathematical model for the swimming of a micro-organism at a low Reynolds number regime is proposed. Methods from sub-Riemannian geometry as well as numerical ones for the quantitative optimization of the strokes of micro-swimmers are discussed. Control theory is concerned with the issue of influencing, by way of an external action, the evolution of a system. Many applications, both from the scientific and technological point of view, involve distributed control systems, that is systems whose evolution is governed by partial differential equations. Important examples range from control of diffusion processes to nonlinear elasticity, from control of biological systems (see above) to traffic flows on networks. The text by Alabau-Boussouira and P. Cannarsa  Control of Non-linear Partial Differential Equations is a wideranging overview of the state-of-the-art in the field, covering among others the issues of controllability, stabilization and optimal control. Traffic flows on networks are discussed in  Vehicular Traffic: A Review of Continuum Mathematical Models by B. Piccoli and A. Tosin. The paper reviews in particular macroscopic and kinetic models. Macroscopic modeling of traffic flows stems from the Euler and Navier–Stokes equations while kinetic models rely on principles of statistical mechanics. Since macroscopic models take often the form of nonlinear hyperbolic conservation laws, special attention is given in the paper to the analysis of those models by nonlinear hyperbolic equation techniques (see also the paper by Bressan in this respect). An important issue in modeling nonlinear physical systems is the reduction of complexity. One way to realize this is to use scaling limit procedures allowing a more manageable description of the system itself through macroscopic coordinates. The paper  Scaling Limits of Large Systems of Non-linear Partial Differential Equations contributed by D. Benedetto and M. Pulvirenti illustrates ideas and methods to treat large classical and quantum particle systems in the weak coupling regime. It is also shown that those systems can be conveniently described by macroscopic quantities whose evolution is governed by the Fokker–Planck–Landau and the Boltzmann equations for the classical and quantum case, respectively.

Non-linear Partial Differential Equations, Viscosity Solution Method in

Non-linear Partial Differential Equations, Viscosity Solution Method in SHIGEAKI KOIKE Department of Mathematics, Saitama University, Saitama, Japan Article Outline Glossary Definition of the Subject Introduction Examples Comparison Principle Existence Results Boundary Value Problems Asymptotic Analysis Other Notions Future Directions Bibliography

Ellipticity/parabolicity General second order PDEs under consideration are F(x; u(x); Du(x); D2 u(x)) D 0 in ˝ :

(E)

Here, u : ˝ ! R is the unknown function, ˝ Rn an open set, F : ˝  R  Rn  S n ! R a given (continuous) function, Sn the set of n  n symmetric matrices with the standard ordering, Du(x) D ((@u)/(@x1 )(x); : : : ; (@u)/(@x n )(x)), and D2 u(x) 2 S n whose (i; j)th entry is (@2 u)/(@x i @x j )(x). According to the early literature in viscosity solution theory, (E) is called elliptic if X  Y H) F(x; r; p; X)  F(x; r; p; Y) for (x; r; p; X; Y) 2 ˝  R  Rn  S n  S n . It should be remarked that the opposite order has been also used. When F does not depend on the last variables (i. e. first order PDEs), it is automatically elliptic. Thus, the above notion has been called degenerate elliptic. The evolution version of general PDEs is u t (x; t) C F(x; t; u(x; t); Du(x; t); D2 u(x; t)) D 0

Glossary Weak solutions In the study of mth order partial differential equations (abbreviated, PDEs), a function is informally called a classical solution of the PDE if (a) it is m-times differentiable, and (b) the PDE holds at each point of the domain by putting its derivatives there. However, it is not easy to find such classical solutions of PDEs in general except for some special cases. The standard strategy to find a classical solution is first to look for a candidate of solutions, which becomes a classical solution if the property (a) holds for it. Here, such a candidate is called a weak solution of the PDE. Viscosity solutions In 1981, for first order PDEs of non-divergence type, M.G. Crandall and P.-L. Lions in [15,16] (also [17]) introduced the notion of weak solutions, which are called viscosity solutions. The definition of those is the property which the limit of approximate solutions via the vanishing viscosity method admits. Afterwards, the notion was extended to fully nonlinear second order elliptic/parabolic PDEs. For general theory of viscosity solutions, [3,5,7,18,20] are recommended for the interested readers. Throughout this article, to minimize the references, [18] will be often referred to instead of the original papers except for some pioneering works or those which appeared after [18].

in Q T :D ˝  (0; T] :

(P)

Here, u : Q T ! R is the unknown function, F : ˝  (0; T]  R  Rn  S n ! R a given function, T > 0, and u t (x; t) D (@u)/(@t)(x; t). In (P), the notations Du(x; t) and D 2 u(x; t) do not contain derivatives with respect to t. Similarly, (P) is called parabolic if F(; t; ; ; ) is elliptic for each t 2 (0; T]. Uniform ellipticity/uniform parabolicity Denoted by n :D fA 2 S n jI  A  Ig for fixed 0 <   , S; the Pucci operators P ˙ : S n ! R are defined by P C (X) :D max ftrace(AX)g n A2S ;

and

P  (X) :D min ftrace(AX)g : n A2S ;

Then, the PDE (E) is called uniformly elliptic if P  (X Y)  F(x; r; p; X) F(x; r; p; Y)  P C (X Y)

for (x; r; p; X; Y) 2 ˝  R  Rn  S n  S n . This is a fully nonlinear version of the standard uniform ellipticity. For the theory of second order uniformly elliptic PDEs, [24] is the standard text book. Similarly, (P) is called uniformly parabolic if F(; t; ; ; ) is uniformly elliptic for each t 2 (0; T].

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Dynamic programming principle In stochastic control problems, the value function is determined by minimizing given cost functionals. The dynamic programming principle (abbreviated, DPP), which was established as the Bellman’s principle of optimality, is a formula which the value function satisfies. The DPP indicates that the value function is a viscosity solution of the associated Hamilton–Jacobi–Bellman (abbreviated, HJB) equation. Definition of the Subject In order to investigate phenomena in both natural and social sciences, it is important to analyze solutions of PDEs derived from certain minimization principles such as calculus of variations. Since it is hard to find classical solutions of PDEs in general, the first strategy is to look for weak solutions of those. For PDEs of divergence type, the most celebrated notion of weak solutions is that in the distribution sense, which is formally derived through the integration by parts. On the other hand, before viscosity solutions were introduced, there were several notions of weak solutions for PDEs of non-divergence type such as generalized solutions for first order PDEs by S.N. Kružkov. For second order elliptic/parabolic PDEs of non-divergence type, it has turned out through much research that the notion of viscosity solutions is the most appropriate one by several reasons: (i) (ii) (iii)

(iv)

(v)

Viscosity solutions admit well-posedness (i. e. existence, uniqueness, stability), The notion of viscosity solutions is naturally derived from the vanishing viscosity method, In deterministic/stochastic control problems, the expected solutions are the unique viscosity solutions of the associated HJB equations in principle, The viscosity solution method can deal with first and second order elliptic/parabolic PDEs by a unified manner, The viscosity solution method can be applied to various non-divergent PDEs for which there were no good notions of weak solutions such as HJB equations, mean curvature flow equations, 1-Laplace equation etc.

Introduction In what follows, elliptic PDEs (E) will be mainly discussed since one can treat (P) with some modifications. For the sake of simplicity, the variables of the unknown functions and their derivatives in the PDEs will be suppressed if there is no confusion.

For U Rn , which will be ˝, ˝ or @˝ later, C 1 (U) (resp., C 2 (U)) denotes the set of real-valued functions  in U such that  and D (resp., , D and D2 ) are continuous in U. The definition of viscosity solutions presented in [17] has been widely used although the original one in [15,16] is different but equivalent. Definition 1 u : ˝ ! R is called a viscosity subsolution (resp., supersolution) of (E) in ˝ if F(x; u(x); D(x); D2 (x))  0 (resp.,  0)

(1)

whenever (u  )(x) D sup˝ (u  ) (resp., inf˝ (u  )) for  2 C 2 (˝) and x 2 ˝. Finally, u is called a viscosity solution of (E) in ˝ if it is a viscosity sub- and supersolution of (E) in ˝. Remark 2 In order to study boundary value problems in the viscosity sense in Sect. “Boundary Value Problems”, it is necessary to modify the above definition by replacing ˝ by ˝. To this end, ˝ in the semi-jets, the equivalent definition and Ishii’s lemma etc. below should be also replaced by ˝. The above definition indicates that if the so-called test function  2 C 2 (˝) touches u from above (resp., below) at x 2 ˝, by noting 0 D (u)(x)  (resp., )(u)(y) for y 2 ˝, the one-sided inequality (1) is required, instead of the Eq. (E), with the derivatives of  at x 2 ˝. It is worth mentioning that if u 2 C 2 (˝) is a viscosity subsolution (resp., supersolution) of (E) in ˝, then it is a classical subsolution (resp., supersolution) of (E) in ˝; F(x; u(x); Du(x); D 2 u(x))  0

(resp.,  0) for each x 2 ˝ :

On the contrary, if u 2 C 2 (˝) is a classical subsolution (resp., supersolution) of (E) in ˝, then it is a viscosity subsolution (resp., supersolution) of (E) in ˝ whenever (E) is elliptic. One can observe that if u is a viscosity subsolution (resp., supersolution) of (E) in ˝, then u is a viscosity supersolution (resp., subsolution) of F(x; u; Du; D2 u) D 0

in ˝ :

If F is independent of X (i. e. first order PDEs), then it is possible to replace C 2 (˝) by C 1 (˝) in the definition. For the definition to (P), ˝ and C 2 (˝), respectively, are changed by QT and C 2;1 (Q T ) in the above, where Q T D ˝  (0; T]. Here, C 2;1 (Q T ) denotes the set of real-

Non-linear Partial Differential Equations, Viscosity Solution Method in

valued functions  in QT such that , D, D2  and  t are continuous in QT . An equivalent definition of viscosity solutions will be utilized to establish the comparison principle for second order PDEs. To this end, it is necessary to prepare some notation: for u : ˝ ! R, the semi-jets J 2;˙ u(x) of order 2 at x 2 ˝ are defined by 8 < 2;C J˝ u(x) D (p; X) 2 Rn  S n : 9 ˇ ˇ ˇ 1 u(y)  u(x)  hp; y  xi 2 = ˇ  hX(y  x); y  xi C o(jx  yj ) ; ˇ 2 ; ˇ for y 2 ˝ as y ! x 8 < 2; J˝ u(x) D (p; X) 2 Rn  S n : 9 ˇ ˇ ˇ 1 u(y)  u(x)  hp; y  xi 2 = ˇ  hX(y  x); y  xi C o(jx  yj ) : ˇ 2 ; ˇ for y 2 ˝ as y ! x In order to state a key lemma in Sect. “Comparison Principle”, it is also needed to introduce a sort of closures of 2;˙ J˝ u(x); for x 2 ˝, 8 < 2;˙ J ˝ u(x) D (p; X) 2 Rn  S n : 9 ˇ ˇ 9(p k ; X k ) 2 J 2;˙u(x k ) for 9x k 2 ˝; = ˇ ˝ ˇ : such that (x k ; u(x k ); p k ; X k ) ˇ ; ˇ ! (x; u(x); p; X) as k ! 1 Definition 3 (Equivalent definition) u is a viscosity subsolution (resp., supersolution) of (E) in ˝ if and only if F(x; u(x); p; X)  0

u1 (x) D jxj  1 ; ( minfx C 1; xg u2 (x) D x1

2;C

(for x 2 [1; 1/2]) ; (for x 2 (1/2; 1])

etc :

However, it can be proved that u0 is the unique viscosity solution of (2) under u0 (˙1) D 0. It is worth mentioning that there are no test functions  2 C 1 (˝) touching u0 from below at x D 0. Thus, no inequalities for viscosity supersolutions at x D 0 are required. Moreover, if u" is a (classical) solution of the approximate equation via the vanishing viscosity method, " 4 u" C jDu" j D 1

in ˝1

(" > 0) ;

under u" (˙1) D 0, then one can verify that lim"!0 u" D u0 in ˝ 1 . The next example, to which the viscosity solution method is applicable, is a fully nonlinear PDE arising in stochastic control problems. For a set of parameters A ¤ ;, F is expressed by F(x; r; p; X) D maxftrace(A a (x)X)  hg a (x); pi a2A

C c a (x)r  f a (x)g ;

(resp.,  0) 2;

provided (p; X) 2 J ˝ u(x) (resp., (p; X) 2 J ˝ u(x)) for x 2 ˝. 2;˙

2;˙ One can replace J ˝ in the above by J˝ .

where A a : ˝ ! S n , g a : ˝ ! Rn , c a ; f a : ˝ ! R are given functions for each a 2 A. One can see that the mapping (r; p; X) ! F(x; r; p; X) is convex in this case. The corresponding PDE is called the HJB equation: ˚ max  trace(A a (x)D2 u)  hg a (x); Dui

Examples

a2A

Since the viscosity solution method is available only for second order elliptic/parabolic PDEs up to now, all the examples below are elliptic/parabolic. To explain how the viscosity solution method is useful, a typical example is the eikonal equation: for ˝1 D (1; 1) R, jDuj D 1 in ˝1

under the Dirichlet condition u(˙1) D 0. Since (2) is of non-divergence type, the notion of weak solutions in the distribution sense is not available to (2). It is also known that there are no classical solutions u 2 C(˝ 1 ) \ C 1 (˝1 ) of (2). In fact, from an optimal control and a geometric view points, the expected solution is the distance function from the boundary @˝1 ; u0 (x) D dist(x; @˝1 ) D 1  jxj. If one employs Lipschitz continuous functions satisfying (2) almost every point of ˝ 1 as weak solutions of (2), then the uniqueness of such weak solutions fails. Indeed, there exist many other weak solutions in this sense; e. g.

(2)

C c a (x)u  f a (x) D 0 :

(3)

If A a  0 in ˝ for a 2 A, then (3) is elliptic. From a stochastic control point of view, A a (x) is naturally represented by 1/2 a (x) at (x) for an n  m matrix  a (x) with some integer m (for a 2 A and x 2 ˝). Thus, the property A a  0 in ˝ holds true naturally in applications.

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Non-linear Partial Differential Equations, Viscosity Solution Method in

A fully nonlinear PDE for which the mapping (r; p; X) ! F(x; r; p; X) is neither convex nor concave arises in stochastic differential games: ˚

min max  trace(A a;b (x)D2 u)  hg a;b (x); Dui b2B a2A C c a;b (x)u  f a;b (x) D 0 ; where A a;b : ˝ ! S n , g a;b : ˝ ! Rn , c a;b ; f a;b : ˝ ! R are given for (a; b) 2 A  B. Here, B is another set of parameters with a different purpose. This is called the Hamilton–Jacobi–Bellman–Isaacs (abbreviated, HJBI) equation, for which the viscosity solution method works when A a;b  0 in ˝ for (a; b) 2 A  B. In principle, any fully nonlinear second order PDEs can be represented by HJBI equations with suitable choices of A a;b , g a;b , c a;b , f a;b , A and B though those HJBI equations might become complicated. It is trivial to derive parabolic versions of HJB and HJBI equations by adding ut together with allowing t-dependence on the given functions. There are more PDEs, to which the viscosity solution method has been applied. (Monge-Ampère equation)

2;C

there are X; Y 2 S n such that (D x (xˆ ; yˆ); X) 2 J ˝ u(xˆ ), 2; (D y (xˆ ; yˆ); Y) 2 J ˝ v( yˆ ), and 

I  ( C kAk) O

hD2 uDu; Dui

(mean curvature flow)

ut  4 u C

(1-Laplace equation)

 hD2 uDu; Dui D 0;

(Pucci equations)

P ˙ (D 2 u)  f (x) D 0:

jDuj2





X  O

The corresponding references are [26] for the Monge– Ampère equation, [23] for the mean curvature flow equation, [2,30] for 1-Laplace equation, and [12,13] for Pucci equations.

  AC

  3

I O

  O X  I O

O Y



  3

I I

Lemma 4 (Ishii’s lemma) For u; v 2 C(˝) and  2 C 2 (˝  ˝), if (x; y) ! u(x)  v(y)  (x; y) takes its maximum at (xˆ ; yˆ) 2 ˝  ˝, then for each  > 1,

I I

 : (5)

In order to establish the comparison principle, the following two hypotheses on F are sufficient. The first one is monotonicity in the second variable: >0

such that is non-decreasing

(6)

for (x; p; X) 2 ˝  Rn  S n . It should be noted that one may take D 0 in (6) if F is uniformly elliptic. The next one is called the structure condition on F: by setting M :D f! : [0; 1) ! [0; 1)j! is continuous such that !(0) D 0g, 8 ˆ 1x; y 2 ˝; r 2 R; X; Y 2 S n satisfying (5) :

Comparison Principle The comparison principle has been one of the main issues in the study of viscosity solutions since it implies their uniqueness. The original idea to prove the comparison principle for first order PDEs is to double the number of variables with a clever choice of a penalization term. However, for second order PDEs, this technique did not directly work. The breakthrough was achieved by Jensen in [29]. The following lemma with matrix inequalities was originally formulated by Ishii.

1 2 A ; (4) 

The proof of this lemma consists of twice differentiability of convex functions by Aleksandrov, Aleksandrov– Bakelman–Pucci type maximum principle for semi-convex functions by Jensen, and the so-called magic properties of sup-convolutions. For a typical (x; y) D 2 jx  yj2 , the above matrix inequalities become

r ! F(x; r; p; X)  r D 0;

O Y

where A D D2 (xˆ ; yˆ) 2 S 2n .

there exists  det(D2 u)  f (x) D 0;

O I

(SC) Some observations related to (SC) below indicate that the condition (SC) is sufficiently reasonable. Sufficient/Necessary Conditions for (SC): (i) If F satisfies (SC), then F is elliptic. (ii) If F is uniformly elliptic, and there is ! 2 M such that jF(x; r; p; X)F(y; r; p; X)j  !(jx  yj(1CjpjCkXk)) for (x; y; r; p; X) 2 ˝  ˝  R  Rn  S n , then F satisfies (SC).

Non-linear Partial Differential Equations, Viscosity Solution Method in

(iii) In the case of HJB equations (3), if there are M0  0 and !0 2 M such that 8 (a) maxfk a (x)k; jg a (x)j; j f a (x)j; jc a (x)jg  M0 ; ˆ ˆ ˆ ˆ ˆ ˆ <(b) maxfk a (x)   a (y)k; jg a (x)  g a (y)jg  M0 jx  yj; ˆ ˆ ˆ ˆ (x)  f (y)j; jc (x)  c a (y)jg (c) maxfj f a a a ˆ ˆ :  !0 (jx  yj) (7) for (x; y; a) 2 ˝  ˝  A, then F satisfies (SC). It is obvious to give the sufficient condition corresponding to (7) for HJBI equations. Due to the comparison principle below, it follows the uniqueness of viscosity solutions of (E) in ˝ such that u D h on @˝, where h 2 C(@˝). Comparison Principle: Under hypotheses (6) and (SC), if u 2 C(˝) and v 2 C(˝) are, respectively, a viscosity subsolution and a viscosity supersolution of (E) in ˝, then u  v on @˝ yields u  v in ˝. In order to understand how Ishii’s lemma is used for the proof of the comparison principle, a simple second order PDE with variable coefficients to the second derivatives will be chosen. Doubling the Number of Variables: Lipschitz continuous functions  j D ( i j ) : ˝ ! Rn ( j D 1; : : : ; m) are given; j j (x)   j (y)j  M0 jx  yj (x; y 2 ˝). By setting P A(x) D 12 m kD1  i k (x) jk (x), the following simple linear PDE is chosen as a typical example: trace(A(x)D2 u) C u D 0 in ˝ : Here ˝ is a bounded domain for simplicity. It is easy to verify that this PDE satisfies (6) and (SC). A typical proof in viscosity solution theory is by contradiction. If :D sup˝ (u  v) > 0, and xˆ 2 ˝ is the point such that (u  v)(xˆ ) D , then xˆ 2 ˝ because u  v on @˝. Here, a perturbation may allow xˆ to be the strict maximum. Now, the key idea is to consider the function (x; y) 2 ˝  ˝ ! u(x)  v(x)  2 jx  yj2 for  > 1. For (x ; y ) 2 ˝  ˝ being a maximum of this function, one can first observe that ˆ ; and lim (x ; y ) D (xˆ ; x)

!1

lim jx  y j2 D 0 :

!1

Due to Ishii’s lemma, there exist X and Y 2 S n such that     O X I I  3 ; O Y I I

2;C

((x  y ); X  ) 2 J ˝ u(x ) and ((x  y ); Y ) 2 2; J ˝ v(x ). By setting  j :D (1 j (x ); : : : ; n j (x )) t and  j :D (1 j (y ); : : : ; n j (y )) t ( j D 1; : : : ; m), the above matrix inequality implies 

X O

O Y

     j  ; j  3j j   j j2 j j  3M02 jx  y j2 :

Thus, by summing these over j D 1; : : : ; m, it follows that trace(A(x )X  )trace(A(y )Y ) 

3 mM02 jx y j2 : 2

Therefore, since the mappings x ! u(x)  v(y )  2 jx  y j2 and y ! v(y)  u(x ) C 2 jy  x j2 , respectively, take their maximum and minimum at x and y 2 ˝ for large  > 1, the definition yields u(x )  v(y )  trace(A(x )X  )  trace(A(y )Y ) 3  mM02 jx  y j2 ; 2 which gives  0 in the limit as  ! 1. This is a contradiction. Existence Results There are two basic ways to show the existence of viscosity solutions. The first one is motivated by the classical Perron’s method for harmonic functions. For Perron’s method below, it is necessary to introduce some relaxations of the notion. In the definition of viscosity subsolutions (resp., supersolution), u is replaced by u (resp., u ), which is the upper (resp., lower) semicontinuous envelope of u; for u : ˝ ! R and x 2 ˝,   u (x) D lim sup u resp., u (x) D lim inf u : "!0 ˝\B (x) "

"!0 ˝\B " (x)

Here and later, B" (x) denotes the open ball with its center x 2 Rn and its radius " > 0. For the reader’s convenience, the definition of viscosity sub- and supersolutions is restated. Definition 5 u : ˝ ! R is called a viscosity subsolution (resp., supersolution) of (E) in ˝ if F(x; u (x); D(x); D2 (x))  0 (reps., F(x; u (x); D(x); D2 (x))  0) whenever (u  )(x) D sup˝ (u  ) (resp., (u  ) (x) D inf˝ (u  ) for  2 C 2 (˝) and x 2 ˝.

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It should be remarked that for the function u, no hypotheses are assumed in this definition. Remark 6 Ishii’s lemma holds for u 2 U SC(˝) and v 2 LSC(˝), where for U Rn , USC(U) and LSC(U) are, respectively, the sets of upper and lower semicontinuous functions in U. With these terminologies, u and v are, respectively, replaced by u and v in the comparison principle. It is important to note that if the comparison principle holds for (E), and a viscosity solution u is continuous on @˝, then u is continuous in ˝ even though u is not initially supposed to belong to C(˝). Ishii (cf. [27]) established Perron’s method to show the existence of viscosity solutions. Perron’s Method: If there is a viscosity subsolution  2 U SC(˝) and a viscosity supersolution  2 LSC(˝) of (E) in ˝ such that    in ˝, by setting u(x) :D ˇ 

ˇ v 2 U SC(˝) is a viscosity subsolution ; sup v(x) ˇˇ of (E) in ˝ such that   v   in ˝ then u is a viscosity solution of (E) in ˝; u and u are, respectively, a viscosity subsolution and a viscosity supersolution of (E) in ˝. The above u does not belong to U SC(˝) [ LSC(˝) in general. The proof of Perron’s method consists of two parts. One of them should be separately stated since this property is often useful. Proposition 7 For a non-empty set S U SC(˝) (resp., LSC(˝)) of viscosity subsolutions (resp., supersolutions) of (E) in ˝, if u(x) :D supfv(x)jv 2 Sg (resp., :D inffv(x)jv 2 Sg) (x 2 ˝) is locally bounded from above (resp., from below), then u is a viscosity subsolution (resp., supersolution) of (E) in ˝. Remark 8 Thanks to Perron’s method, for the existence of viscosity solutions, it is enough to construct a viscosity subsolution  and a viscosity supersolution  of (E) in ˝ satisfying    in ˝. For instance, concerning the Dirichlet boundary value problem, for a given h 2 C(@˝), if there are  and  satisfying    in ˝, and  D  D h on @˝, then the u constructed by Perron’s method is a viscosity solution of (E) in ˝ under u D h on @˝. To find such a couple (; ) of viscosity sub- and supersolutions, when F is uniformly elliptic and @˝ satisfies the so-called uniform exterior cone condition, it is possible to build them by using a barrier function at each x0 2 @˝.

For HJB/HJBI equations, there is the other approach for the existence result. Concerning HJB Eqs. (3) for instance, it is known that the expected solution is the value function associated with a stochastic control problem. To avoid the boundary condition, the domain ˝ for (3) is fixed by Rn . For a given (progressively) measurable function ˛ : [0; 1) ! A, under assumption (7), it is known that one can solve the stochastic differential equation (abbreviated, SDE) dX(t) D g˛(t) (X(t))dt C ˛(t) (X(t))dW(t) for

t > 0;

under X(0) D x 2 Rn , where W is the m-dimensional Brownian motion in a probability space. By denoting the solution of the above by X(; x; ˛), a cost functional is given by

Z

1

J(x; ˛) D E

e

Rt 0

c ˛(s) (X(s;x;˛))ds

0

 ˇ ˇ  f˛(t) (X(t; x; ˛))dt ˇ X(0; x; ˛) D x ; where E is the expectation. Then, the value function is defined by u(x) D inf J(x; ˛) ;

(8)

˛

where the infimum is taken over all measurable controls. The DPP is a formula which the value function satisfies when inffc a (x)ja 2 A; x 2 Rn g > 0. Dynamic Programming Principle: For  > 0, u(x) D 2 Z inf E 4 ˛

0



e

Rt 0

Ce

c ˛(s) (X(s;x;˛))ds

R 0

f˛(t) (X(t; x; ˛))dt

c ˛(t) (X(t;x;˛))dt u(X(; x; ˛))

3 ˇ ˇ ˇ ˇ X(0; x; ˛) D x 5 : ˇ

Due to this DPP under (7), it is shown that u is a viscosity solution of (3) in Rn via the Itô formula. Boundary Value Problems In the study of degenerate elliptic/parabolic PDEs via viscosity solution method, there is a unified way to formulate boundary value problems of ( F(x; u; Du; D2 u) D 0 in ˝; (9) B(x; u; Du) D 0 on @˝ ;

Non-linear Partial Differential Equations, Viscosity Solution Method in

where B : @˝  R  Rn ! R is a given (continuous) function. Typical examples of B in the literature of PDEs are ( B(x; r; p) D r  h(x) (Dirichlet condition); B(x; r; p) D hp; (x)i  h(x)

(oblique condition),

where h 2 C(@˝), and : @˝ ! Rn with hn(x); (x)i > 0 (for x 2 @˝). Here, n(x) is the outward unit normal at x 2 @˝. When D n, the oblique condition is called the Neumann condition. To formulate the boundary value problem (9) in the viscosity sense, by setting ( F(x; r; p; X) (for x 2 ˝); G(x; r; p; X) D B(x; r; p) (for x 2 @˝) in the definition of viscosity subsolutions (resp., supersolutions), ˝ and F are replaced, respectively, by ˝ and G (resp., G  ). Here, G  and G are, respectively, an upper and lower semicontinuous envelopes of G with respect to all the variables. When F is continuous, G  (x; r; p; X) D G (x; r; p; X) D F(x; r; p; X) for (x; r; p; X) 2 ˝  R  Rn  S n . Moreover, if F is continuous up to @˝ in the first variable, and B is continuous, then one can verify that at x 2 @˝, G (x; r; p; X) D minfF(x; r; p; X); B(x; r; p)g ; G  (x; r; p; X) D maxfF(x; r; p; X); B(x; r; p)g for (r; p; X) 2 R  Rn  S n . Under this setting, the comparison principle means that if u and v are, respectively, a viscosity subsolution and a viscosity supersolution of (9) in the above sense, then u  v in ˝. To prove the comparison principle for (9), the key tool is often a good perturbation of the penalization term to avoid the boundary condition. If the boundary condition is imposed on the whole of @˝, then it would be an over-determined (i. e. unsolvable) problem when (E) is only elliptic. However, in the above setting, the given boundary condition, B(x; u; Du) D 0 on @˝, is required only on a part of @˝, which is not a priori known. This formulation has been successful particularly for oblique boundary value problems. For more information on the boundary value problems, [18] and its references are suggested. State Constraint Problems: There is a sort of boundary condition arising in optimal control problems. However, it seems that this boundary condition is not derived from physical applications.

When the infimum in (8) is taken over all controls ˛ such that X(t; x; ˛) 2 ˝ (when ˝ ¤ Rn ) for all t  0, it follows that the value function u is a viscosity supersolution of (3) in ˝. Thus, to formulate the state constraint problem of (E), the supersolution property on @˝ is regarded as a boundary condition; u : ˝ ! R is called a viscosity supersolution (resp., subsolution) of the state constraint problem of (E) in ˝ if it is a viscosity supersolution (resp., subsolution) of F(x; u; Du; D2 u) D 0 in ˝

(resp., ˝) :

The comparison principle for the state constraint problem was proved in [42] for first order PDEs.

Asymptotic Analysis Besides the existence and uniqueness of viscosity solutions, the remaining property in the well-posedness is the stability of those, which justifies that when the PDE is perturbed a little bit, the solution may change only a little. In order to state the stability of viscosity solutions, Barles–Perthame in [8] introduced the half relaxed limits. For functions u" : ˝ ! R for " > 0 and x 2 ˝, u(x) D lim"!0 sup u" (x) (resp., u(x) D lim"!0 inf u" (x)) are defined by lim supfuı (y)j0 < ı < "; y 2 B" (x)g "!0  ˚  resp., lim inf uı (y)j0 < ı < "; y 2 B" (x) : "!0

For given F" : ˝  R  Rn  S n ! R (" > 0), the stability of viscosity solutions of F" (x; u; Du; D2 u) D 0 in ˝

(10)

is stated in the following manner. Stability: If u" are viscosity subsolutions (resp., supersolutions) of (10) in ˝ for " > 0, and u (resp., u) is finite in ˝, then it is a viscosity subsolution (resp., supersolution) of F(x; u; Du; D2 u) D 0 (resp., F(x; u; Du; D2 u) D 0) in ˝ ; where F(x; r; p; X) (resp., F(x; r; p; X)) is given by

(11)

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Non-linear Partial Differential Equations, Viscosity Solution Method in

lim inf Fı (y; s; q; Y) "!0 ˇ  ˇ y 2 B" (x); jr  sj < "; q 2 B" (p); ˇ ˇ kX  Yk < "; 0 < ı < " 

resp., lim sup Fı (y; s; q; Y) "!0 ˇ  ˇ y 2 B" (x); jr  sj < "; q 2 B" (p); ˇ : ˇ kX  Yk < "; 0 < ı < " Thus, if F D F, and the (6) and (SC) holds for F, then u D u on @˝ yields u D u in ˝, which implies the convergence of u" to u D u. In order to understand how this stability is powerful, it is efficient to consider the singular perturbation of (E) with the vanishing viscosity term:  " 4 u C F(x; u; Du; D2 u) D 0

in ˝ ;

(12)

where " > 0, under some boundary condition. When F is not uniformly elliptic, it is often easier to solve (12) than (E). If u" is a (classical) solution of (12) in ˝ for " > 0, and the uniform bound of u" is known, then the above stability result together with the comparison principle for (E) implies the convergence of u" . On the other hand, if one follows the standard argument, then it is necessary to get higher a priori estimates on u" so that u" has a limit (along a subsequence if necessary). In applications from science and engineering, there are various singular perturbation problems where F ¤ F. One typical example is the following. Homogenization: When the PDE is formulated by F

x "

 ; u" ; Du" ; D2 u" D 0

in Rn ;

where the mapping y ! F(y; r; p; X) satisfies the periodicity F(y C z; r; p; X) D F(y; r; p; X) for (y; z; r; p; X) 2 Rn  Zn  R  Rn  S n , this PDE corresponds to the case when it has highly oscillating coefficients in a periodic medium. Under certain hypothesis, e. g. coercivity for first order PDEs, uniform ellipticity for second order ones, or more general cases, one may show that a subsequence of u" converges to u, which is a viscosity solution of the effective PDE: ˆ Du; D2 u) D 0 in ˝ ; F(u;

which is called the cell problem. It is important to characterize Fˆ in the homogenization theory since this new PDE may contain macroscopic information. While there is extensive literature on homogenization for PDEs of divergence type, less was known for those of non-divergence type. The so-called perturbed test function method was proposed to study homogenization for non-divergent PDEs (see [16]). Now, even for PDEs of non-divergence type, the study of homogenization has developed in periodic/almost periodic environments. Recently, the research of homogenization in ergodic media has started in [14,37]. There are also some singular perturbation problems in phase transitions. Section 5 in [5] is a good survey for those and their front propagations (see also [9]). Other Notions There are some other notions of viscosity solutions. Indeed, although the viscosity solution method can be applied to various PDEs in a unified manner, one might lose some information in each specific problem. Semicontinuous Viscosity Solutions: In the study of first order PDEs with convex Hamiltonian with respect to Du, H(x; u; Du) D 0

in ˝ ;

(13)

where H : ˝  R  Rn ! R is continuous, and p ! H(x; r; p) is convex for (x; r) 2 ˝  R, Barron and Jensen proposed the notion of semicontinuous viscosity solutions. In fact, it sometimes happens that the expected solution is discontinuous while the standard comparison principle implies its continuity. In such a problem, it is necessary to propose a better notion of weak solutions under which one guarantees the uniqueness of discontinuous solutions. Definition 9 u 2 U SC(˝) is called a semicontinuous viscosity solution of (12) in ˝ if H(x; (x); D(x)) D 0 whenever sup˝ (u  ) D (u  )(x) for  2 C 1 (˝) and x 2 ˝. It should be noted that the equality holds at x instead of the inequality in the standard definition. It is known that under general setting, the value function in deterministic optimal control problems satisfies this definition. Moreover, the uniqueness of semicontinuous viscosity solutions is proved in [6]. Lp -viscosity Solutions: In the study of (linear) uniformly elliptic/parabolic PDEs of divergence type, there are two

Non-linear Partial Differential Equations, Viscosity Solution Method in

basic regularity theories; the Schauder estimates and Lp estimates. Initiated by Caffarelli in [11], the regularity theory of viscosity solutions for fully nonlinear uniformly elliptic/parabolic PDEs was developed. Afterwards, when F is uniformly elliptic but the mapping x ! F(x; r; p; X) is merely measurable, the notion of Lp -viscosity solutions of (E) in ˝ was introduced in [13], roughly speaking, 2;p by using Wloc (˝) instead of C 2 (˝) in the definition for p > n/2. Definition 10 u 2 C(˝) is an Lp -viscosity subsolution (resp., supersolution) of (E) in ˝ if ess lim inf F(y; u(y); D(y); D2 (y))  0 y!x   resp., ess lim sup F(y; u(y); D(y); D2 (y))  0 y!x

whenever (u  )(x) D sup˝ (u  ) (resp., inf˝ (u  )) 2;p for  2 Wloc (˝) and x 2 ˝. As usual, u 2 C(˝) is called an Lp -viscosity solution of (E) in ˝ if it is an Lp -viscosity sub- and supersolution of (E) in ˝. A remarkable fact is that there are counter-examples even for linear uniformly elliptic PDEs for which uniqueness fails when the coefficient of the second derivatives is merely bounded. Thus, one may not expect the comparison principle to hold in general. Here, several other notions are briefly mentioned. Though the vanishing viscosity method works well for scalar conservation law, it is not clear if the viscosity solution method can be applied to it. There is a recent trial in the study of first order PDEs of divergence/non-divergence type in [22]. The main idea for the new notion is to consider the associated level set equations, and to use test surfaces instead of test functions in the definition. There are recent works on stochastic PDEs by introducing a stochastic version of viscosity solutions in [35,36]. Future Directions A list of areas where more research is expected via viscosity solution method is given. Venttsel condition: There is a boundary condition where the second derivatives are involved; B contains D2 u-variables. In the case when X ! B(x; r; p; X) is elliptic, (9) with this B is called the Venttsel boundary value problem. From a view point of viscosity solution theory, this case corresponds to that of discontinuous coefficients.

More regularity theory: For fully nonlinear uniformly elliptic/parabolic PDEs, there are still many open questions on regularity. The convexity (or concavity) assumption in the D2 u-variables on F seems necessary to show that viscosity solutions are classical ones. Indeed, some counter-examples have been discovered in [39,40] without convexity assumption. On the contrary, for a simple uniformly elliptic HJBI equation, it is shown that a viscosity solution is indeed a classical one in [10]. For degenerate elliptic but specified PDEs, there must be some criterion for regularity. For example, [41] for 1-Laplace equation etc. Qualitative properties: One may expect qualitative properties, which classical (or even strong) solutions possess, to hold for viscosity solutions such as Aleksandrov–Bakelman–Pucci maximum principle, Liouville theorem, Hadamard theorem, Phragmén–Lindelöf principle, symmetry etc. There have appeared several researches but there are still gaps to be fulfilled. Concerning on the other qualitative properties, the most important one among those must be the eigenvalue problems but only a few results have appeared (e. g. [34]). Also, the blow-up phenomenon for fully nonlinear parabolic PDEs is an attractive topic but almost nothing is known except [21]. Degenerate equations from geometry: There have been several results which treat degenerate second order PDEs arising in differential geometry; e. g. Heisenberg Laplacian in [4,38]. Dynamical systems: The weak KAM theory is an active area nowadays in viscosity solution theory. The part “Hamilton–Jacobi equations and weak KAM theory” in this section is recommended for more information. System of PDEs/nonlocal operators: Viscosity solution method can naturally apply to monotone systems of PDEs in optimal control problems and differential games such as weakly coupled systems and switching games. Also, in applications, PDEs involving non-local operators such as impulsive control problems have been studied. There is a general treatment for PDEs with non-local operators in [28]. Recently, [1] investigated Hamilton–Jacobi equations with some special non-local operator which comes from a dislocation dynamics. One should seek new systems of PDEs, and PDEs with other non-local operators in various applications. Applications to mathematical finance: There are lots of fully nonlinear PDEs appearing in mathematical finance. However, many of those cannot be applied di-

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rectly by viscosity solution method since the PDEs contain unbounded coefficients, the domain is often unbounded, etc. Since the goal in this field is to determine (approximate) optimal policies, it is necessary to study the regularity of viscosity solutions in each problem. There have also been several books in this field (e. g. [31]). Differential games: Although HJBI equations appeared as examples from differential games, no details are mentioned. One interesting question in differential games is how to formulate the boundary condition for the state constraint problems. There was a trial [32] but it is not clear if this formulation covers practical problems. Control problems in 1-dimensional spaces: If (stochastic) PDEs govern the underlying state equation in place of (SDE) to define the value function in (8), then the domain of the resulting HJB equations becomes an 1-dimensional space (typically, Hilbert space). The viscosity solution theory for 1-dimensional spaces has been studied. However, much more research should be done for specific PDEs such as the Navier–Stokes equation (e. g. in [25]). Again [18] is suggested to find more information. To finish this section, the reader should recall the words in the “User’s Guide” [18]. “the reader should not restrict his imagination to the borders we drew above.” Bibliography Primary Literature 1. Alvarez O, Hoch P, Le Bouar Y, Monneau R (2006) Dislocation dynamics: short time existence and uniqueness of the solution. Arch Ration Mech Anal 85:371–414 2. Aronsson G, Crandall MG, Juutinen P (2004) A tour of the theory of absolutely minimizing functions. Bull Amer Math Soc 41:439–505 3. Bardi M, Capuzzo Dolcetta I (1997) Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston 4. Bardi M, Mannuci P (2006) On the Dirichlet problem for non-totally degenerate fully nonlinear elliptic equations. Comm Pure Appl Anal 5:709–731 5. Bardi M, Crandall MG, Evans LC, Soner HM, Souganidis PE (1997) Viscosity Solutions and Applications. Springer, Berlin 6. Barles G (1993) Discontinuous viscosity solutions of first-order Hamilton–Jacobi equations: A guided visit. Nonlinear Anal 20:1123–1134 7. Barles G (1994) Solutions de Viscosité des Équations de Hamilton–Jacobi. Springer, Berlin

8. Barles G, Perthame B (1987) Discontinuous solutions of deterministic optimal stopping-time problems. Model Math Anal Num 21:557–579 9. Barles G, Souganidis PE (1998) A new approach to front propagation problems: theory and applications. Arch Ration Mech Anal 141:237–296 10. Cabré X, Caffarelli LA (2003) Interior C 2,˛ regularity theory for a class of nonconvex fully nonlinear elliptic equations. J Math Pures Appl 83:573–612 11. Caffarelli LA (1989) Interior a priori estimates for solutions of fully non-linear equations. Ann Math 130:180–213 12. Caffarelli LP, Cabré X (1995) Fully Nonlinear Elliptic Equations. Amer Math Soc, Providence 13. Caffarelli LA, Crandall MG, Kocan M, S´ wie¸ch A (1996) On viscosity solutions of fully nonlinear equations with measurable ingredients. Comm Pure Appl Math 49:365–397 14. Caffarelli LA, Souganidis PE, Wang L (2005) Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm Pure Appl Math 58:319–361 15. Crandall MG, Lions P-L (1981) Condition d’unicité pour les solutions generalisées des équations de Hamilton–Jacobi du premier ordre. CR Acad Sci Paris Sér I Math 292:183–186 16. Crandall MG, Lions P-L (1983) Viscosity solutions of Hamilton– Jacobi equations. Tran Amer Math Soc 277:1–42 17. Crandall MG, Evans LC, Lions P-L (1984) Some properties of viscosity solutions of Hamilton–Jacobi equations. Trans Amer Math Soc 282:487–435 18. Crandall MG, Ishii H, Lions PL (1992) User’s guide to viscosity solutions of second order partial differential equations. Bull Amer Math Soc 27:1–67 19. Evans LC (1992) Periodic homogenization of certain fully nonlinear partial differential equations. Proc Roy Soc Edinb 120:245–265 20. Fleming WH, Soner HM (1993) Controlled Markov Processes and Viscosity Solutions. Springer, Berlin 21. Friedman A, Souganidis PE (1986) Blow-up of solutions of Hamilton–Jacobi equations. Comm Partial Differ Equ 11:397– 443 22. Giga Y (2002) Viscosity solutions with shocks. Comm Pure Appl Math 55:431–480 23. Giga Y (2006) Surface Evolutions Equations: A Level Set Approach. Birkäuser, Basel 24. Gilbarg D, Trudinger NS (1983) Elliptic Partial Differential Equations of Second Order. Springer, New York 25. Gozzi F, Sritharan SS, S´ wie¸ch A (2005) Bellman equations associated to optimal control of stochastic Navier–Stokes equations. Comm Pure Appl Math 58:671–700 26. Gutiérrez CE (2001) The Monge–Ampère Equation. Birkhäuser, Boston 27. Ishii H (1987) Perron’s method for Hamilton–Jacobi equations. Duke Math J 55:369–384 28. Ishii H, Koike S (1993/94) Viscosity solutions of functional-differential equations. Adv Math Sci Appl 3:191–218 29. Jensen R (1988) The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch Rat Mech Anal 101:1–27 30. Jensen R (1993) Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient. Arch Ration Mech Anal 123:51–74

Non-linear Partial Differential Equations, Viscosity Solution Method in

31. Karatzas I, Shreve SE (1998) Methods of Mathematical Finance. Springer, New York 32. Koike S (1995) On the state constraint problem for differential games. Indiana Univ Math J 44:467–487 33. Koike S, S´ wiech ˛ A (2004) Maximum principle for fully nonlinear equations via the iterated comparison function method. Math Ann 339:461–484 34. Lions P-L (1983) Bifurcation and optimal stochastic control. Nonlinear Anal 7:177–207 35. Lions PL, Souganidis PE (1998) Fully nonlinear stochastic partial differential equations. CR Acad Sci Paris 326:1085–1092; 326:753–741 36. Lions PL, Souganidis PE (2000) Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations. CR Acad Sci Paris 331:783–790 37. Lions PL, Souganidis PE (2003) Correctors for the homogenization of Hamilton–Jacobi equations in the stationary ergodic setting. Comm Pur Appl Math 56:1501–1524 38. Manfredi JJ (2002) A version of the Hopf–Lax formula in the Heisenberg group. Comm Partial Differ Equ 27:1139–1159 39. Nadirashvili N, Vl˘adu¸t S (2007) Nonclassical solutions of fully nonlinear elliptic equations. Geom Funct Anal 17:1283–1296 40. Nadirashvili N, Vl˘adu¸t S (2008) Singular viscosity solutions to

fully nonlinear elliptic equations. J Math Pures Appl 89(9):107– 113 41. Savin O (2005) C 1 regularity for infinity harmonic functions in two dimensions. Arch Ration Mech Anal 176:351–361 42. Soner MH (1986) Optimal control with state-space constraint I. SIAM J Control Optim 24:552–562

Books and Reviews Bellman R (1957) Dynamic Programming. Princeton University Press, Princeton Evans LC, Gangbo W (1999) Differential Equations Methods for the Monge–Kantrovich Mass Transfer Problem. Amer Math Soc, Providence Fleming WH, Rishel R (1975) Deterministic and Stochastic Optimal Control. Springer, New York Koike S (2004) A Beginner’s Guide to the Theory of Viscosity Solutions. Math Soc Japan, Tokyo. Corrections: http://133.38.11.17/ lab.jp/skoike/correction.pdf Lions P-L (1982) Generalized Solutions of Hamilton–Jacobi Equations. Pitman, Boston Maugeri A, Palagachev DK, Softova LG (2000) Elliptic and Parabolic Equations with Discontinuous Coefficients. Wiley, Berlin

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Noise Let W(t) be a cylindrical Wiener process in a Hilbert space H. A noise is an expression of the form BW(t), where B 2 L(H). If B D I (the identity operator in H) the noise is called white. Stochastic dynamical system This is a system governed by a partial differential evolution equation perturbed by noise.

Non-linear Stochastic Partial Differential Equations GIUSEPPE DA PRATO Scuola Normale Superiore, Pisa, Italy Article Outline

Definition of the Subject Several nonlinear partial differential evolution equations can be written as abstract equations in a suitable Hilbert space H (we shall denote by j  j the norm and by h; i the scalar product in H) of the following form [78]: 8 dX(t) ˆ < D AX(t) C b(X(t)) ; t  0 ; dt (1) ˆ : X(0) D x 2 H ;

Glossary Definition of the Subject Introduction General Problems and Results Specific Equations Future Directions Bibliography Glossary Evolution equations Let H be a Hilbert (or Banach) space, T > 0 and f a mapping from [0; T]  H into H. An equation of the form u0 (t) D f (t; u(t)) ;

t 2 [0; T] ;

( )

is called an abstract evolution equation. If H is finitedimensional we call ( ) an ordinary evolution equation, whereas if H is infinite-dimensional and f is a differential operator we call ( ) a partial differential evolution equation. Continuous stochastic process Let (˝; F ; P ) be a probability space. A continuous stochastic process (with values in H) is a family of (H-valued) random variables (X(t) D X(t; !)) t0 (! 2 ˝) such that X(; !) is continuous for P -almost all ! 2 ˝. Brownian motion A real Brownian motion B D (B(t)) t0 on (˝; F ; P ) is a continuous real stochastic process in [0; C1) such that (1) B(0) D 0 and for any 0  s < t; B(t)  B(s) is a real Gaussian random variable with mean 0 and covariance t  s, and (2) if 0 < t1 <    < t n ; the random variables, B(t1 ); B(t2 )  B(t1 ); : : : ; B(t n )  B(t n1 ) are independent. Cylindrical Wiener process A cylindrical Wiener process in a Hilbert space H is a process of the form W(t) D

1 X

e k ˇ k (t) ;

t  0;

kD1

where (e k ) is a complete orthonormal system in H and (ˇ k ) a sequence of mutually independent standard Brownian motions in a probability space (˝; F ; P ).

where A: D(A) H ! H is the infinitesimal generator of a strongly continuous semigroup e tA of linear bounded operators in H and b : D(b) H ! H is a nonlinear mapping. In order to take into account unpredictable random perturbations, one is led to add to (1) a term of the form  (X(t))dW(t), where  : D( ) H ! L(H) (L(H) is the space of all linear operators from H into itself) and W(t) a cylindrical Wiener process in H: Then (1) is replaced by the following stochastic partial differential equation (SPDEs) ( dX(t) D (AX(t) C b(X(t)))dt C (X(t))dW(t) ; (2) X(0) D x 2 H ; whose precise meaning will be explained later. The importance of the SPDEs is due to the fact that in the modelization of several phenomena, (2) is often more realistic than (1). The study of SPDEs started in the 1970s and 1980s and interest is still growing. Among the early contributions we mention [7] for Navier–Stokes equations, [66] for SPDEs of monotone type, [79] for compactness methods, [57] for variational methods and [31]. Two approaches (essentially equivalent) are used for studying SPDEs: the first is based on the variational theory of partial differential equations and the second is based on the theory of strongly continuous semigroups of operators. In this article we are concerned with the latter [26,27]. For a recent introduction to the variational method see [71]. It is possible to consider perturbations of a partial differential equation by stochastic processes different from Brownian motion as jump processes, but we shall not consider this subject here.

Non-linear Stochastic Partial Differential Equations

O of R N with regular boundary @O:

Introduction In this section we are concerned with (2) in the Hilbert space H, that is, with a partial differential equation of the form (1) perturbed by the noise (X(t))dW(t), where W(t) is a cylindrical Wiener process in H. We recall that W(t) can be defined as follows: W(t) D

1 X

e k ˇ k (t) ;

(3)

kD1

where ek is a complete orthonormal system in H (often the system of eigenfunctions of A) and ˇ k is a sequence of mutually independent standard Brownian motions in a probability space (˝; F ; P ). Intuitively W(t) represents a random (white) perturbation which has the same intensity in all direction of the orthonormal basis (e k ). Remark 1 Definition (3) is formal since E[jW(t)j2 ] D

1 X

8 ˆ <@ t X(t; ) D  X(t; ) C p(X(t; )) ;  2 O; t > 0 ; X(t; ) D 0 ; t > 0;  2 @O ; ˆ : X(0; ) D x() ;  2 O; x 2 H ; (4) where  is the Laplace operator and p is a polynomial of degree d > 1 with negative leading coefficient. Let us write problem (4) as an abstract equation of the form (1) in the Hilbert space H D L2 (O) (the space of all equivalence classes of square integrable real functions on O with respect to the Lebesgue measure). For this purpose, we denote by A the realization of the Laplace operator with Dirichlet boundary conditions (other boundary conditions such as Neumann or Ventzell could be considered as well), ( Ax D  x ; x 2 D(A) ; D(A) D H 2 (O) \ H01 (O) ;

t D C1 8 t > 0 :

kD1

However, it is possible to define rigorously W(t) in a suitable space larger than H [26]. We shall see in the following how to deal with W(t). The main problems we shall consider are the following: (i) Existence and uniqueness of solutions X(t) of (2). (ii) Asymptotic behavior of X(t) as t ! 1. For the sake of brevity we shall limit ourselves to stochastic perturbations of the special form (X(t))dW(t) D BdW(t), where B 2 L(H). In this case we say that the noise involved in (2) is additive. The general case (multiplicative noise) is important and we shall give some references later. In particular a multiplicative noise is required when for physical reasons the solution X(t) must belong to some convex subset of H (for instance, X  0) [6]. In Sect. “General Problems and Results” we shall present an overview of the main problems concerning SPDEs. Section “Specific Equations” is devoted to some significant applications. We will not give complete proofs of all assertions but only a sketch in order to give an idea of the tools which are involved. The interested reader can look at the corresponding references. We shall concentrate on some SPDEs only. We present in the following three examples the corresponding deterministic problems. Example 2 (reaction–diffusion equations). Consider the following partial differential equation in a bounded subset

and define b(x)() D p(x()) ;

x 2 D(b) D L2d (O) :

For k D 1; 2 we denote by H k (O) the Sobolev space consisting of all functions which belong to L2 (O) together with their distributional derivatives of order up to k. H01 (O) is the subspace of those functions in H 1 (O) which vanish at the boundary @O of O: Example 3 (Burgers equation) Consider the equation in [0; 2] with periodic boundary conditions 8 ˆ @ t X(t; ) D @2 X(t; ) C 12@ (X 2 (t; )); ˆ ˆ ˆ <  2 [0; 2]; t > 0; ˆ X(t; 0) D X(t; 2) ; @ X(t; 0) D @ X(t; 2); t >0; ˆ ˆ ˆ : X(0; ) D x() ;  2 [0; 2] : (5) Problem (5) can be written in the abstract form (1) setting H D L2 (0; 2), Ax D D2 x ;

x 2 D(A) D H#2 (0; 2) ;

and b(x) D

1 2

D (x 2 ) ;

x 2 D(b) D H#1 (0; 2) ;

where H#1 (0; 2) D fx 2 H 1 (0; 2) : x(0) D x(2)g

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Non-linear Stochastic Partial Differential Equations

and H#2 (0; 2) D fx 2 H 2 (0; 2) : x(0) D x(2); @ x(0) D @ x(2)g: Example 4 (2D Navier–Stokes equation). Consider the equation in the square O :D [0; 2]  [0; 2]. 8 dZ D ( Z  Z C D Z  Z)dt C r pdt ˆ ˆ ˆ ˆ ˆ  2 O; t > 0; ˆ < div Z D 0  2 O ; t > 0 ; ˆ ˆ ˆ ˆ Z(t; ) is periodic with period 2 ; ˆ ˆ : Z(0; ) D z()  2 O :

 Mathematical biology and Fleming–Viot model [48, 77].  Mathematical finance [64].  Nonlinear Schrödinger equations [32,33,34,35].  Porous media equations [4,30,73].  Stochastic quantization [2,23,25,55,59,62].  Wave equations [5,18,19,62,63,69,75]. General Problems and Results

(6)

The unknown Z D (Z1 ; Z2 ) represents the velocity and p the pressure of the fluid; div Z is the divergence of Z. We denote by H the space of all square integrable divergence free vectors,

Well-Posedness of a Stochastic Partial Differential Equation We are here concerned with (2). The first problem is to prove the existence and uniqueness of solutions (to be defined in an appropriate way). We notice that it is not convenient to look for existence of strong solutions X(t) (as for the ordinary stochastic equations) such that X(t) D x C

Z th

i AX(s) C b(X(s)) ds C BW(t) ;

t  0:

0

H D fZ D (Z1 ; Z2 ) 2 (L2 (0; 2))2 : div Z D 0g ; (L2 (0; 2))2

onto H. and by P the orthogonal projector of Then setting X(t; x) D P Z(t; x), we obtain the following problem (where the pressure p has disappeared) 8 ˆ 0 ; X(t; ) is periodic with period 2 ; t > 0 ˆ : X(0; ) D x  2 O : Let us define the Stokes operator A: D(A) ! H setting Ax D P ( x  x) ;

x 2 D(A) D H#2 (O)

and the nonlinear operator b setting b(x) D P (D x  x) ;

x 2 H#1 (O)

(b is more commonly written b(x) D P (x  rx)). Here H#k (O), k D 1; 2 are Sobolev spaces with periodic boundary conditions. Now, problem (5) can be written in the form (1). We have to say that many other interesting equations have been studied recently. We mention among them the following (without claiming to present an exhaustive list).  Cahn–Hilliard equations [21,38].  Second-order linear SPDEs and Filtering equations; see [56,74] and references therein.  Ginzburg–Landau equations [51,52,53].  Kortweg–de Vries equation [36,37].

This definition will require too much regularity for the solution, which does not belong to D(A) \ D(b) in general. The concept of a weaker solution has to be defined for each specific problem; see Sect. “Specific Equations”. Assume now that the existence of a unique solution X(t, x) (in a suitable sense) of (2) has been proved for any x 2 H. Then, given a real function ' in H, one is interested in the evolution in time of E['(X(t; x))] (the expectation of '(X(t; x))). ' can be interpreted as an observable; its physical meaning could be, for instance, the mean velocity, pressure or temperature of a fluid. This leads to introduce the concept of transition semigroup Pt '(x) D E['(X(t; x))] ;

t  0; x 2 H; ' 2 B b (H) ; (7)

where Bb (H) is the space of all mappings ' : H ! R which are bounded and Borel. It is a Banach space with the supremum norm k'k1 D sup j'(x)j ;

' 2 B b (H) :

h2H

We denote by Cb (H) the closed subspace of Bb (H) of all uniformly continuous and bounded functions. It is not difficult to see that Pt enjoys the semigroup property PtCs D Pt Ps ; t; s  0, thanks to the Markovianity of the solution X(t, x) to (2). However, Pt is not a strongly continuous semigroup on Cb (H) in general (even when H is finite-dimensional and (2) is an ordinary differential equation).

Non-linear Stochastic Partial Differential Equations

Formally, the function u(t; x) D Pt '(x) is the unique solution to the following (Kolmogorov) parabolic equation: 8 du ˆ < (t; x) D K0 u(t; x) ; dt (8) ˆ :u(0; x) D '(x) ; where K 0 is the linear differential operator K0 '(x) D

1 2

h i Tr BB D2x '(x) C hAx C b(x); D x '(x)i ; x 2 D(A) \ D(b) ; (9)

T > 0 there is a control u such that x(T) D 0). 8 dx(t) ˆ < D Ax(t) C b(x(t)) C Bu(t) ; t  0 ; dt ˆ :x(0) D x 2 H ; where u is a control; see, e. g Sects. 7.3 and 7.4 in [27]. Finally, we denote by  t (x; d y) the law of X(t, x), so that the following integral representation for Pt , Pt '(x) D Z '(y) t (x; dy) ; 8t > 0 ; x 2 H ; ' 2 Bb (H) ; H

(10) holds.

where 1 X ˝ 2 ˛ D x '(x)B e k ; B e k ; Tr BB D2x '(x) D



h;kD1

and (ek ) is a complete orthonormal system on H. Notice that K0 ' is only defined for ' of class C2 such that the series above is convergent. Though the semigroup Pt is not strongly continuous one can define its infinitesimal generator K, following [72], K'(x) D lim

h!0

1 (Pt '(x)  '(x)) ; h

x2H;

provided the limit exists and the following condition holds sup h2(0;1]

1 kPt '  'k1 < C1 : h

Then an interesting problem consists in clarifying the relationship between the abstract operator K and the concrete differential operator K 0 . Let us list some important concepts related to Pt .  Pt is called Feller if Pt ' 2 Cb (H) for any ' 2 Cb (H) and any t  0.  Pt is called strong Feller if Pt ' 2 Cb (H) for any ' 2 Bb (H) and any t > 0. The strong Feller property is related to a smoothing effect of the transition semigroup Pt and to the hypoellipticity of the Kolmogorov operator K 0 (when H is finitedimensional) [75].  Pt is called irreducible if Pt 1I (x) > 0 for all x 2 H and all open sets I of H, where 1I is the characteristic function of I (1I (x) D 1 if x 2 I, 1I (x) D 0 if x … I). Irreducibility of Pt is related to the null-controllability of the deterministic system (that is, if for each x 2 H and

Asymptotic Behavior of the Transition Semigroup In order to study the asymptotic behavior of Pt ', where ' 2 Bb (H), the concept of invariant measure is crucial. We say that a Borel probability measure  on H is invariant for Pt if Z Z Pt '(x)(dx) D '(x)(dx) ; 8 ' 2 Cb (H) : H

H

Notice that if  is invariant and  is a random variable whose law is  then the solution X(t; ) to (2) with initial datum  is stationary. In order to prove the existence of invariant measures an important tool is the following Krylov–Bogoliubov theorem; see, e. g., Theorem 3.1.1 in [27]. Theorem 5 Assume that Pt is Feller and that for some x 2 H the family of probability measures ( t (x; dy)) t0 is tight (that is, for any > 0 there exists a compact set K in H such that  t (x; K ) > 1  for all t  0). Then there exists an invariant measure for Pt . Assume now that  is an invariant measure for Pt and let ' 2 Cb (H). Then Pt can be uniquely extended to a contraction semigroup in L2 (H; ). In fact, by (10) and the Hölder inequality we have [Pt '(x)]2  Pt (' 2 )(x) ;

t  0; x 2 H:

Integrating this inequality with Rrespect to  over H 2 and R 2taking into account that H Pt (' )(x)(dx) D H ' (x)(dx) by the invariance of  yields Z Z [Pt '(x)]2 (dx)  ' 2 (x)(dx) : H

H

Since Cb (H) is dense in L2 (H; ), this inequality can be extended to any function of L2 (H; ). This proves that Pt

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Non-linear Stochastic Partial Differential Equations

can be uniquely extended to a contraction semigroup in L2 (H; ): Moreover, the following Von Neumann theorem holds [70]. Theorem 6 Let  be an invariant measure for Pt . Then for any ' 2 L2 (H; ) there exists the limit Z T 1 Pt '(x)dt :D ˘ ' ; lim T!C1 T 0 where ˘ is a projection operator on ˙ :D f' 2 L2 (H; ) : Pt ' D '; 8 t  0g : If moreover Z Z T 1 lim Pt '(x)dt D '(y)(d y) T!C1 T 0 H

in L2 (H; );

then  is called ergodic. As in the deterministic case, ergodicity is a very important property of a stochastic dynamical system because it allows us to compute averages with respect to t in terms of averages with respect to x 2 H (“temporal” averages with “spatial” averages). When the more stringent condition Z '(y)(dy) ; 8 x 2 H ; lim Pt '(x)dt D T!C1

H

holds, we say that the measure  is strongly mixing. In this direction we recall the following Doob’s theorem; see e. g., Theorem 4.2.1 in [27]. Theorem 7 Assume that Pt is irreducible and strongly Feller. Then Pt possesses at most one invariant measure which is ergodic and strongly mixing. Other Important Problems Here we mention some additional problem which we cannot treat for space reasons. The first one concerns the so-called small noise. Often one is interested in studying stochastic perturbations of (1) of the form ( dX(t) D (AX(t) C b(X(t)))dt C dW(t) ; (11) X(0) D x 2 H ; where > 0. In fact it can happens that (11) has a unique invariant measure " , whereas the corresponding deterministic system (1) has more. Then studying the limit points of " as " ! 0 will help to select the “significant” invariant measures of (2) [39]. This problem also leads to study large deviations problems. Concerning Smoluchowski–Kramers approximations for parabolic SPDEs we mention [17].

Specific Equations Ornstein–Uhlenbeck Equations We are here concerned with the following stochastic differential equation: (

dX(t) D AX(t)dt C BdW(t) ;

t0

X(0) D x 2 H ;

(12)

where A: D(A) H ! H is the infinitesimal generator of a strongly continuous semigroup etA and B : H ! H is linear-bounded. Formally the solution of (12) is given by the variation of constants formula X(t; x) D e tA x C WA (t) ;

t 0;

(13)

where the process W A (t), called stochastic convolution, is given by Z

t

WA (t) D

e(ts)A BdW(s) ;

t 0:

0

By a formal computation we see that EjWA (t)j D 2

1 Z X

t

je(ts)A Be k j2 ds D Tr Q t ;

kD1 0

where Z

t

Qt x D



esA CesA xds ; x 2 H; t  0 ;

0

and C D BB . Consequently, though the definition (3) of the cylindrical Wiener process W(t) is formal (see Remark 1), the stochastic convolution defined by the series WA (t) D

1 Z X

t

e(ts)A Be k dˇ k (s)

(14)

kD1 0

is meaningful and convergent in L2 (˝; F ; P ; H) for all t  0, provided Tr Q t < C1 for all t > 0. We shall assume that this hypothesis holds from now on. It is easy to see that W A (t) is a Gaussian random variable N Q t in H. (A Borel probability measure  in H is said to be Gaussian with mean x 2 H and covariance Q, where ˆ of  Q 2 L(H) is of trace class, if the Fourier transform  1 ˆ is given by (h) D e ihx;hi 2 hQ h;hi for all h 2 H: In this case we set  D N x;Q and  D N Q if x D 0.) Remark 8 The limit case when B D I (white noise) [80], is important in some application in physics, because it means

Non-linear Stochastic Partial Differential Equations

that the noise acts uniformly in all directions. In this case the assumption Tr Q t < C1 is equivalent to Z

t

i h  Tr esA esA ds < C1 :

(15)

By a mild solution of problem (16) on [0; T] we mean a stochastic process X 2 CW ([0; T]; H) such that Z t X(t) D e tA x C e(ts)A b(X(s))ds C WA (t) ; P -a.s. 0

0

Assume, for instance, that A is the Laplacian in an open bounded set O Rd with Dirichlet boundary conditions. Let Ae k D ˛ k e k , where (e k ) k2N d is the complete orthonormal system of eigenfunctions of A and (˛ k ) k2N d are the corresponding eigenvalues. Then we have Tr Q t D

X 1 (1  e2˛k t ) ; ˛ k d

t > 0:

k2N

It is well known that ˛ k jkj2 and so (15) is fulfilled only if d D 1. To deal with the case d > 1 one introduces the so-called renormalization [68,76]. Now by (13) it follows that the law of X(t, x) is given by Net A x;Q t , so the corresponding transition semigroup looks like Z Pt '(x) D '(y)Net A x;Q t (dy) ; ' 2 Bb (H) : H

t 0;

and for some M; ! > 0 it follows that Z '(y)N Q 1 (dy) ; lim Pt '(x) D t!C1

H

Smooth Perturbations of System (12) Let us consider the following stochastic differential equation dX(t) D (AX(t) C b(X(t))dt C BdW(t) ;

t2[0;T]

It is called the space of all mean square continuous adapted processes on [0; T] taking values on H. Equation (17) can be solved easily by a standard fixedpoint argument in the Banach space CW ([0; T]; H). Thus we have the result Proposition 9 X(; x).

Equation (17) has a unique solution

hDPt '(x); hi D 

Z t 1 hD x X(s; x)  h; dW(s)i : E '(X(t; x)) t 0

and so  :D N Q 1 is the unique invariant measure for Pt which is in addition ergodic and strongly mixing (for a necessary and sufficient condition for existence and uniqueness of an invariant measure for Pt see [26]).

(

where W A is the stochastic convolution defined by (14). By CW ([0; T]; H) we mean the Banach space consisting of all continuous mappings Y : [0; T] ! L2 (˝; F ; P ; H) which are adapted to W (that is, such that for all t 2 [0; T], Y(t) is F t -measurable, where F t is the  -algebra generated by fW(s); s 2 [0; t]g), endowed with the norm, !1/2   2 : kYkC W ([0;T];H)) D sup E jY(t)j

One checks easily that the corresponding transition semigroup Pt defined by (7) is Feller. When B D I one can show that Pt is strongly Feller by using the Bismut– Elworthy formula [10,40] for the derivative of D x Pt '(x) which reads as follows:

If e tA is stable, ke tA k  Me!t ;

(17)

t 0;

In this case one can also show that Pt is irreducible [20]. So, in view of Doob’s theorem there is at most one invariant measure. Existence of an invariant measure can be proved under the assumption hAx; xi C hb(x); xi  a  bjxj2 ;

where a; b > 0. In fact in this case one finds easily by Itô’s formula an estimate of the second moment of X(t, x) independent of t of the form EjX(t; x)j2  c(1 C jxj2 ) :

X(0) D x 2 H ; (16) where A and B are as before and b : H ! H is of class C1 and Lipschitz continuous. The condition on b is very strong; however, this case is important because it can be used for approximating equations with irregular coefficients.

8 x 2 D(A) ;

(18)

By (18) the tightness of ( t (x; )) t0 follows for any fixed x 2 H. We have in fact for any R > 0 Z 1 jyj2  t (x; dy)  t (x; B cR )  2 R H c 1 D 2 EjX(t; x)j2  2 (1 C jxj2) ; R R

1131

1132

Non-linear Stochastic Partial Differential Equations

where B cR is the complement of the ball BR of center 0 and radius R. So, the existence of an invariant measure follows from the Krylov–Bogoliubov theorem. Reaction–Diffusion Equations Perturbed by Noise We consider here a stochastic perturbation of problem (4) from Example 2, but for the sake of simplicity we take O D [0; 1] and B D I. So, we have H D L2 (0; 1) and (4) becomes h i 8 ˆ dX(t; ) D D2 X(t; ) C p(X(t; )) dt C dW(t; ) ; ˆ ˆ ˆ <  2 [0; 1] ; ˆ ˆ X(t; 0) D X(t; 1) D 0 ; t  0 ; ˆ ˆ : X(0; ) D x() ;  2 [0; 1]; x 2 H ; (19) where p is a polynomial of degree d > 1 with negative leading coefficient. For generalizations to systems of reaction–diffusion equations in bounded subsets of Rn and for equations with multiplicative noise see [15,16] and references therein. For problems with reflexion see [65,82]. Now we write (19) in the mild form tA

Z

t

X(t) D e x C

e(ts)A p(X(s))ds C WA (t) ;

t  0;

0

(20) where W A (t) is the stochastic convolution defined by (14). Moreover, Ax D D2 x ;

x 2 D(A) D H 2 (0; 1) \ H01 (0; 1)

and b(x)() D p(x()) ;

x 2 D(b) D L2d (0; 1) :

We notice that, since the leading coefficient of p is negative, b enjoys the basic dissipativity property hb(x)  b(y); x  yi  c1 jx  yj2 ;

8 x; y 2 L2d (0; 1) ;

where c1 D sup p0 () :

Moreover Ae k D  2 k 2 e k ;

Therefore, we have hAx; xi   2 jxj2 ; for all x 2 H. Let us give the definition of the mild solution of (19). We say that X 2 CW ([0; T]; H) (CW ([0; T]; H) was defined in Sect. “Smooth Perturbations of System (12)” is a mild solution of problem (19) if X(t) 2 L2d (0; 1) for all t  0 and fulfills (20). We notice that the condition X(t) 2 L2d (0; 1) is necessary in order for the integrand in (20) be meaningful. The basic existence and uniqueness result is the following [3,26,41]. Theorem 10 For any x 2 L2d (O), there exists a unique mild solution X(; x) of problem (19). Proof First we consider a regular approximating problem, ( dX ˛ (t) D (AX ˛ (t) C b˛ (X ˛ (t))dt C dW(t) ; (21) X ˛ (0) D x 2 H ; where for any ˛ > 0; b˛ is of class C1 , b˛ (x) ! p(x) for all x 2 L2d (0; 1) as ˛ ! 0 and the basic dissipativity property, hb˛ (x)b˛ (y); xyi  c2 jxyj2 ;

e k () D (2/)1/2 sin() ;

0

Taking advantage of the dissipativity of b˛ it is possible to find suitable estimates for X˛ and to show that X˛ converge, as ˛ ! 0, to the unique solution X of (19).  By Theorem 10 we can construct a transition semigroup Pt on Bb (L2d (0; 1)). However, it is useful to extend this semigroup to Bb (L2 (0; 1)). For this we need a weaker notion of solution of (19) when x 2 H D L2 (0; 1). Let x 2 H: We say that X 2 CW ([0; T]; H) is a generalized solution of problem (19) if there exists a sequence (x n ) L2d (O); such that lim x n D x

 2 [0; 1]; k 2 N :

8 x; y 2 H; ˛ > 0;

is fulfilled, where c2 is a constant independent of ˛. By Proposition 9 we know that problem (21) has a unique mild solution X˛ , that is, Z t tA X˛ (t) D e x C e(ts)A b˛ (X(s))ds C WA (t); P -a.s.

2R

The operator A is self-adjoint and possesses a complete orthonormal system (e k ) of eigenfunctions, namely,

k2N:

n!1

in H ;

and lim X(; x n ) D X(; x) in CW ([0; T]; H) :

n!1

Non-linear Stochastic Partial Differential Equations

It is easy to see, using again dissipativity of p, that this definition does not depend on the choice of the sequence (x n ) and to prove the following result. Corollary 11 For any x 2 H, there exists a unique generalized solution X(; x) of problem (19) Now we can define the transition semigroup Pt '(x) D E['(X(t; x))] ;

Theorem 12 The semigroup Pt has a unique invariant measure ; which is ergodic and strongly mixing and, Z t!C1

'(y)(d y) ;

x2H:

(22)

H

Proof Let us consider the reaction–diffusion equation starting from s where s > 0, h i 8 ˆ dX(t; ) D D2 X(t; ) C p(X(t; )) dt CdW(t; ) ; ˆ ˆ ˆ <  2 [0; 1] ; ˆ ˆ X(t; 0) D X(t; 1) D 0 ; t  0 ; ˆ ˆ : X(s; ) D x() ;  2 [0; 1] ; x 2 H : (23) Obviously we shall extend W(t) to negative times setting W(t) D W(t) if t  0 : Let us denote by X(t; s; x) the solution of (23). Then, using the dissipativity of p, one can show that there exists  2 L2 (˝; F ; P ; H) such that lim X(t; s; x) D 

s!C1

We are concerned with the Burgers equation in [0; 2] perturbed by white noise   8 ˆ dX(t; ) D @2 X(t; ) C 12 @ (X 2 (t; )) dt ˆ ˆ ˆ < CdW(t; ) ;  2 [0; 2]; t > 0 ; ˆ ˆ X(t; 0) D X(t; 2) ; t>0 ˆ ˆ : X(0; ) D x() ;  2 [0; 2] : (24)

for all ' 2 B b (H): We finally discuss invariant measures of Pt assuming for simplicity that p0 ()  0 so that p is dissipative. In this case we can show the following result [26].

lim Pt '(x) D

Burgers Equations Perturbed by Noise

in L2 (˝; F ; P ; H) ;

x 2 H;

where  is independent of t. Now, using the fact that the law of X(t; s; x) coincides with that of X(t C s; x), it follows that the law  of  is an invariant measure for Pt . Moreover, one can show that Pt is irreducible and strongly Feller, so the uniqueness of  as well as (22) follow from Doob’s theorem. 

2 (0; 2)

and give the following definition. We set H D L A mild solution in [0; T] of (24) is a process X 2 CW ([0; T]; H) such that Z t 1 X(t) D e tA x C @ e(ts)A (X 2 (s))ds C WA (t) ; 2 0 t  0 ; (25) where the stochastic convolution W A (t) is defined by (14). We note that (25) is meaningful thanks to the estimate ˇ ˇ 3 ˇ (ts)A ˇ y ˇ   t  4 jxj1 ; x 2 L1 (0; 2) ˇ@ e (which can be proved by an elementary argument of harmonic analysis). Now we state, following [29], an existence and uniqueness result. Theorem 13 For any x 2 H and T > 0 there exists a unique mild solution X 2 CW ([0; T]; H) of (24). Moreover, there exists a unique invariant measure which is ergodic and strongly mixing. Proof By the contraction principle it is easy to show existence and uniqueness of a local solution of (25) in a stochastic interval [0; (!)) (since (25) is a semilinear equation with a locally Lipschitz nonlinearity). In order to show global existence one needs to find an a priori estimate for P -almost all ! 2 ˝. For this we first reduce (25) to a deterministic equation (more precisely to a family of deterministic equations indexed by ! 2 ˝), setting Y(t) D X(t)  WA (t) : Y(t) fulfills in fact the equation 8 dY(t) ˆ < D AY(t) C b(Y(t) C WA (t)) ; dt ˆ :Y(0) D x :

t 2 [0; T] ;

An a priori estimate for Y(t) can be found by some manipulations using the basic property of b, hb(x); xi D 0 ;

8 x 2 D(b) :

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Non-linear Stochastic Partial Differential Equations

For details and for the existence of an invariant measure see [29], for the uniqueness see [27].  Another approach for studying the Burgers equation can be found in [9].

Existence of invariant measures is not difficult. It can be obtained, as in the previous examples, by the Krylov– Bogoliubov theorem. Uniqueness is more delicate. When the noise BdW(t) is not too degenerate it was proved in [45]. Otherwise one has to use the coupling argument [54,58,60,81].

2D Navier–Stokes Equation Perturbed by Noise We are here concerned with the equation

Future Directions

8 dZ D ( Z  Z C D Z  Z)dt C r pdt C B1 dW(t) ˆ ˆ ˆ ˆ ˆ in [0; C1)  O ; ˆ < div Z D 0 in [0; C1)  O ; ˆ ˆ ˆ ˆ Z(t; ) is periodic with period 2 ; ˆ ˆ : Z(0; ) D z in O ; (26) where Z D (Z1 ; Z2 ) belongs to the space H of all square integrable divergence free vectors and B1 2 L(H). Several papers have been devoted to stochastic 2D Navier–Stokes equations. We mention, besides the pioneering work [1,7,13,22]. Concerning 3D Navier–Stokes equations (which we do not consider here) one can look at [13,14,23,42,43,44,46,47]. We write (26) in the mild form X(t) D e tA x C

Z

t

e(ts)A b(X(s))ds C WA (t) ;

(27)

0

where the operators A and b were defined in Example 4 and the stochastic convolution W A (t) is given by (14) with B D P B1 . In order to study (27) it is convenient to introduce the mapping Z

t

 (f) D

e(ts)A b( f (s))ds; f 2 L4 ((0; T)O); t  0;

0

and then to write (27) as tA

X(t) D e x C  (X)(t) C WA (t) : In fact one can show [27] that  maps L4 ((0; T)  O) into itself and for any f ; g 2 L4 ((0; T)  O) :D L4 we have j ( f )   (g)jL 4  4(j f j L 4 C jgj L 4 ) j f  gj L 4 : Theorem 14 For any x 2 H there exists a unique mild solution X(; x) of (26). Proof One reduces (27) to a family of deterministic equations as in the case of the Burgers equation. Then one uses a fixed-point argument in the space L4 (0; T; L4 ) [27]. 

The number of papers devoted to nonlinear SPDEs is increasing. In fact, several models arising in the application are more realistic when a random perturbation is taken into account. Among new directions of research we mention the following.  The direct study of the Kolmogorov equation (8) (considered as a parabolic equation with infinitely many variables). This can give important information about the properties of system (2) even in the case when the problem is not well posed. A result in this direction can be found in [28]. For a solution of (8) in the case of the 3D Navier–Stokes equation (with a failed attempt to prove uniqueness in law) see [24].  Hamilton–Jacobi equations such as 8 du(t; x) ˆ ˆ D K0 u(t; x) C H (u x (t; x)) ; < dt ˆ ˆ : u(0; x) D '(x) ;

x 2 H; t  0 ;

where H is a Hamiltonian. This is an important subject related to stochastic optimal control problems for SPDEs. For early results see Chap. 12 in [28]. Recent results are also connected with backward equations. Here the pioneering finite-dimensional result in [67] has been extended in infinite dimensions [50]. We believe also that an extension of the following topics to infinite dimensions would be interesting:  Homogenization and averaging. In finite dimension see, e. g., [8,49]  Measures solutions of Kolmogorov equations. In finite dimension this is a well-known subject, see, e. g., [11,12].

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64. Musiela M, Rutkowski M (1997) Martingale methods in financial modelling. In: Applications of Mathematics (New York), vol 36. Springer, Berlin 65. Nualart D, Pardoux E (1992) White noise driven quasilinear SPDEs with reflection. Prob Theory Rel Fields 93:77–89 66. Pardoux E (1975) Èquations aux derivées partielles stochastiques nonlinéaires monotones. Thèse, Université Paris XI 67. Pardoux E, Peng SG (1990) Adapted solution of a backward stochastic differential equation. Syst Control Lett 14(1):55–61 68. Parisi G, Wu YS (1981) Sci Sin 24:483–490 69. Peszat S, Zabczyk J (2000) Nonlinear stochastic wave and heat equations. Probab Th Rel Fields 116(3):421–443 70. Petersen K (1983) Ergodic Theory. Cambridge, London 71. Prevot C, Röckner M (2007) A concise course on stochastic partial differential equations, Monograph 2006. In: Lecture Notes in Mathematics. Springer, Berlin 72. Priola E (1999) On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math 136:271–295 73. Ren J, Röckner M, Wang FY (2007) Stochastic generalized porous media and fast diffusions equations. J Diff Equ 238(1):118–152 74. Rozovskii BL (2003) Linear theory and applications to nonlinear filtering (mathematics and its applications). Kluwer, Dordrecht 75. Sanz-Solé M (2005) Malliavin calculus with applications to stochastic partial differential equations. In: Fundamental sciences. EPFL Press, Lausanne; distributed by CRC Press, Boca Raton 76. Simon B (1974) The P()2 Euclidean (quantum) field theory. Princeton University Press, Princeton 77. Stannat W (2000) On the validity of the log-Sobolev inequality for symmetric Fleming–Viot operators. Ann Probab 28(2):667– 684 78. Temam R (1988) Infinite-dimensional dynamical systems in mechanics and physics. Springer, New York 79. Viot M (1976) Solution faibles d’équations aux dérivées partielles non linéaires. Thèse, Université Pierre et Marie Curie, Paris 80. Walsh JB (1986) An introduction to stochastic partial differential equations. In: cole d’t de probabilits de Saint-Flour, XIV—1984. Lecture Notes in Math, vol 1180. Springer, Berlin, pp 265–439 81. Weinan E, Mattingly JC, Sinai YG (2001) Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Commun Math Phys 224:83–106 82. Zambotti L (2001) A reflected stochastic heat equation as a symmetric diynamic with respect to 3-d Bessel Bridge. J Funct Anal 180(1):195–209

Nonsmooth Analysis in Systems and Control Theory

Nonsmooth Analysis in Systems and Control Theory FRANCIS CLARKE Institut universitaire de France et Université de Lyon, Lyon, France Article Outline Glossary Definition of the Subject Introduction Elements of Nonsmooth Analysis Necessary Conditions in Optimal Control Verification Functions Dynamic Programming and Viscosity Solutions Lyapunov Functions Stabilizing Feedback Future Directions Bibliography Glossary Generalized gradients and subgradients These terms refer to various set-valued replacements for the usual derivative which are used in developing differential calculus for functions which are not differentiable in the classical sense. The subject itself is known as nonsmooth analysis. One of the best-known theories of this type is that of generalized gradients. Another basic construct is the subgradient, of which there are several variants. The approach also features generalized tangent and normal vectors which apply to sets which are not classical manifolds. The article contains a summary of the essential definitions. Pontryagin Maximum Principle The main theorem on necessary conditions in optimal control was developed in the 1950s by the Russian mathematician L. Pontryagin and his associates. The Maximum Principle unifies and extends to the control setting the classical necessary conditions of Euler and Weierstrass from the calculus of variations, as well as the transversality conditions. There have been numerous extensions since then, as the need to consider new types of problems continues to arise. Verification functions In attempting to prove that a certain control is indeed the solution to a given optimal control problem, one important approach hinges upon exhibiting a function having certain properties implying the optimality of the given control. Such a function is termed a verification function. The approach

becomes widely applicable if one allows nonsmooth verification functions. Dynamic programming A well-known technique in dynamic problems of optimization is to solve (in a discrete context) a backwards recursion for a certain value function related to the problem. This technique, which was developed notably by Bellman, can be applied in particular to optimal control problems. In the continuous setting, the recursion corresponds to the Hamilton–Jacobi Equation. This partial differential equation does not generally admit smooth classical solutions. The theory of viscosity solutions uses subgradients to define generalized solutions, and obtains their existence and uniqueness. Lyapunov function In the classical theory of ordinary differential equations, global asymptotic stability is most often verified by exhibiting a Lyapunov function, a function along which trajectories decrease. In that setting, the existence of a smooth Lyapunov function is both necessary and sufficient for stability. The Lyapunov function concept can be extended to control systems, but in that case it turns out that nonsmooth functions are essential. These generalized control Lyapunov functions play an important role in designing optimal or stabilizing feedback. Definition of the Subject The term nonsmooth analysis refers to the body of theory which develops differential calculus for functions which are not differentiable in the usual sense, and for sets which are not classical smooth manifolds. There are several different (but related) approaches to doing this. Among the better-known constructs of the theory are the following: generalized gradients and Jacobians, proximal subgradients, subdifferentials, generalized directional (or Dini) derivates, together with various associated tangent and normal cones. Nonsmooth analysis is a subject in itself, within the larger mathematical field of differential (variational) analysis or functional analysis, but it has also played an increasingly important role in several areas of application, notably in optimization, calculus of variations, differential equations, mechanics, and control theory. Among those who have participated in its development (in addition to the author) are J. Borwein, A. D. Ioffe, B. Mordukhovich, R. T. Rockafellar, and R. B. Vinter, but many more have contributed as well. In the case of control theory, the need for nonsmooth analysis first came to light in connection with finding proofs of necessary conditions for optimal control, notably in connection with the Pontryagin Maximum Prin-

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Nonsmooth Analysis in Systems and Control Theory

ciple. This necessity holds even for problems which are expressed entirely in terms of smooth data. Subsequently, it became clear that problems with intrinsically nonsmooth data arise naturally in a variety of optimal control settings. Generally, nonsmooth analysis enters the picture as soon as we consider problems which are truly nonlinear or nonlinearizable, whether for deriving or expressing necessary conditions, in applying sufficient conditions, or in studying the sensitivity of the problem. The need to consider nonsmoothness in the case of stabilizing (as opposed to optimal) control has come to light more recently. It appears in particular that in the analysis of truly nonlinear control systems, the consideration of nonsmooth Lyapunov functions and discontinuous feedbacks becomes unavoidable. Introduction

converge to 0 (for all values of ˛) in a suitable sense. The most common approach to designing the required stabilizing feedback uses the technique that is central to most of applied mathematics: linearization. In this case, one examines the linearized system x˙ (t) D Ax(t) C Bu(t) where A : D f x (0; 0) ;

The basic object in the control theory of ordinary differential equations is the system x˙ (t) D f (x(t); u(t)) a.e. ;

This rather vaguely worded goal is often more precisely expressed as that of finding a stabilizing feedback control k(x); that is, a function k with values in U (this is often referred to as a closed-loop control) such that all solutions of the differential equation    x˙ (t) D f x(t); k x(t) ; x(0) D ˛ (3)

0tT;

(1)

where the (measurable) control function u() is chosen subject to the constraint u(t) 2 U a.e. ;

(2)

where U is a given set in an Euclidean space, and where the ensuing state x() ( a function with values in Rn ) is subject to certain conditions, including most often an initial one of the form x(0) D x0 , and perhaps other constraints, either throughout the interval (pointwise) or at the terminal time. A control function u of this type is referred to as an open loop control. This indirect control of x() via the choice of u() is to be exercised for a purpose, of which there are two principal sorts:  positional: x(t) is to remain in a given set in Rn , or approach that set;  optimal: x(), together with u(), is to minimize a given functional. The second of these criteria follows directly in the tradition of the calculus of variations, and gives rise to the subject of optimal control, in which the dominant issues are those of optimization: necessary conditions for optimality, sufficient conditions, regularity of the optimal control, sensitivity. We discuss below the role of nonsmooth analysis in optimal control; this was the setting of many of the earliest applications. In contrast, a prototypical control problem of purely positional sort would be the following: Find a control u() such that x() goes to 0 :

B : D f u (0; 0) :

If the linearized system satisfies certain controllability properties, then classical linear systems theory provides well-known and powerful tools for designing (linear) feedbacks that stabilize the linearized system. Under further mild hypotheses, this yields a feedback that stabilizes the original nonlinear system (1) locally; that is, for initial values x(0) sufficiently near 0. This approach has been feasible in a large number of cases, and in fact it underlies the very successful role that control theory has played in a great variety of applications. Still, linearization does require that a certain number of conditions be met:  The function f must be smooth (differentiable) so that the linear system can be constructed;  The linear system must be a ‘nondegenerate’ approximation of the nonlinear one (that is, it must be controllable);  The control set U must contain a neighborhood of 0, so that near 0 the choice of controls is unconstrained, and all state values near 0 must be acceptable (no state constraints);  Both x and u must remain small so that the linear approximation remains relevant (in dealing with errors or perturbations, the feedback is operative only when they are sufficiently small). It is not hard to envisage situations in which the last two conditions fail, and indeed such challenging problems are beginning to arise increasingly often. The first condition fails for simple problems involving electrical circuits in which a diode is present, for example (see [14]). A famous (smooth) mechanical system for which the second condition fails is the nonholonomic integrator, a term which

Nonsmooth Analysis in Systems and Control Theory

refers to the following system, which is linear (separately) in the state and in the control variables: 8 ˆ < x˙1 D u1 x˙2 D u2 ˆ : x˙3 D x1 u2  x2 u1

k(u1 ; u2 )k  1 :

(Thus n D 3 here, and U is the closed unit ball in R2 .) Here, the linearized system is degenerate, since its third component is x˙3 D 0. As discussed below, there is in fact no continuous feeback law u D k(x) which will stabilize this system (even locally about the origin), but certain discontinuous stabilizing feedbacks do exist. This illustrates the moral that when linearization is not applicable to a given nonlinear system (for whatever reason), nonsmoothness generally arises. (This has been observed in other contexts in recent decades: catastrophes, chaos, fractals.) Consider for example the issue of whether a (control) Lyapunov function exists. This refers to a pair (V; W) of positive definite functions satisfying notably the following Infinitesimal Decrease condition: inf hrV(x); f (x; u)i  W(x)

u2U

x ¤0:

The existence of such (smooth) functions implies that the underlying control system is globally asymptotically controllable (GAC), which is a necessary condition for the existence of a stabilizing feedback (and also sufficient, as recently proved [21]). In fact, exhibiting a Lyapunov function is the principal technique for proving that a given system is GAC; the function V in question then goes on to play a role in designing the stabilizing feedback. It turns out, however, that even smooth systems that are GAC need not admit a smooth Lyapunov function. (The nonholonomic integrator is an example of this phenomenon.) But if one extends in a suitable way the concept of Lyapunov function to nonsmooth functions, then the existence of a Lyapunov function becomes a necessary and sufficient condition for a given system to be GAC. One such extension involves replacing the gradient that appears in the Infinitesimal Decrease condition above by elements of the proximal subdifferential (see below). How to use such extended Lyapunov functions to design a stabilizing feedback is a nontrivial topic that has only recently been successfully addressed. In the next section we define a few basic constructs of nonsmooth analysis; knowledge of these is sufficient for interpreting the statements of the results discussed in this article.

Elements of Nonsmooth Analysis This section summarizes some basic notions in nonsmooth analysis, in fact a minimum so that the statements of the results to come can be understood. This minimum corresponds to three types of set-valued generalized derivative (generalized gradients, proximal subgradients, limiting subgradients), together with a notion of normal vector applicable to any closed (not necessarily smooth or convex) set. (See [24] for a thorough treatment and detailed references.) Generalized Gradients For smooth real-valued functions f on Rn we have a wellknown formula linking the usual directional derivative to the gradient: f 0 (x; v) : D lim t#0

f (x C tv)  f (x) D hr f (x); vi : t

We can extend this pattern to Lipschitz functions: f is said to be Lipschitz on a set S if there is a constant K such that j f (y)  f (z)j  Kky  zk whenever y and z belong to S. A function f that is Lipschitz in a neighborhood of a point x is not necessarily differentiable at x, but we can define a generalized directional derivative as follows: f ı (x; v) : D lim sup y!x; t#0

f (y C tv)  f (y) : t

Having done so, we proceed to define the generalized gradient: @C f (x) : D f 2 Rn : f ı (x; v)  h; vi 8v 2 Xg : It turns out that @C f (x) is a compact convex nonempty set that has a calculus reminiscent of the usual differential calculus; for example, we have @( f )(x) D @ f (x). We also have @( f C g)(x) @ f (x) C @g(x); note that (as is often the case) this is an inclusion, not an equation. There is also a useful analogue of the Mean Value Theorem, and other familiar results. In addition, though, there are new formulas having no smooth counterpart, such as one for the generalized gradient of the pointwise maximum of locally Lipschitz functions (@fmax1in f i (x)g : : : ). A very useful fact for actually calculating @C f (x) is the following Gradient Formula:

 @C f (x) D co lim r f (x i ) : x i ! x; x i …  ; i!1

where  is any set of measure zero containing the local points of nondifferentiability of f . Thus the generalized gradient is ‘blind to sets of measure zero’.

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 0 2 @P f (x 0 ) and  00 2 @P g(x 00 ) such that

Generalized gradients and their calculus were defined by Clarke [12] in 1973. The theory can be developed on any Banach space; the infinite-dimensional context is essential in certain control applications, but for our present purposes it suffices to limit attention to functions defined on Rn . There is in addition a corresponding theory of tangent and normal vectors to arbitrary closed sets; we give some elements of this below.

A sequential closure operation applied to @P f gives rise to the limiting subdifferential, useful for stating results:

Proximal Subgradients

@L f (x) : D flim  i :  i 2 @P f (x i ); x i ! x; f (x i ) ! f (x)g :

We now present a different approach to developing nonsmooth calculus, one that uses the notion of proximal subgradient. Let f : Rn ! (1; 1] be a given function (note that the value C1 is admitted here), and let x be a point where f (x) is finite. A vector  in Rn is said to be a proximal subgradient of f at x provided that there exist a neighborhood ˝ of x and a number   0 such that f (y)  f (x) C h; y  xi  ky  xk2 8y 2 ˝ : Thus the existence of a proximal subgradient  at x corresponds to the possibility of approximating f from below (thus in a one-sided manner) by a function whose  graph is a parabola. The point x; f (x) is a contact point between the graph of f and the parabola, and  is the slope of the parabola at that point. Compare this with the usual derivative, in which the graph of f is approximated (in a two-sided way) by an affine function. The set of proximal subgradients at x (which may be empty, and which is not necessarily closed, open, or bounded but which is convex) is denoted @P f (x), and is referred to as the proximal subdifferential. If f is differentiable at x, then we have @P f (x) f f 0 (x)g; equality holds if f is of class C 2 at x. As a guide to understanding, the reader may wish to carry out the following exercise (in dimension n D 1): the proximal subdifferential at 0 of the function f1 (x) : D jxj is empty, while that of f2 (x) : D jxj is the interval [1; 1]. The proximal density theorem asserts that @P f (x) is nonempty for all x in a dense subset of dom f : D fx : f (x) < 1g : Although it can be empty at many points, the proximal subgradient admits a very complete calculus for the class of lower semicontinuous functions: all the usual calculus rules that the reader knows (and more) have their counterpart in terms of @P f . Let us quote for example Ioffe’s fuzzy sum rule: if  2 @P ( f C g)(x), then for any > 0 there exist x 0 and x 00 within of x, together with points

 2  0 C  00 C B : The Limiting Subdifferential

If f is Lipschitz near x, then @L f (x) is nonempty, and (for any lower semicontinuous g finite at x) we have @L ( f C g)(x) @L f (x) C @L g(x) : When the function f is Lipschitz near x, then both the approaches given above (generalized gradients, proximal subdifferential) apply, and the corresponding constructs are related as follows: @C f (x) D co @L f (x) : Normal Vectors Given a nonempty closed subset S of Rn and a point x in S, we say that  2 X is a proximal normal (vector) to S at x if there exists   0 such that  2 ˝ 0 ˛ ; x  x   x 0  x  8x 0 2 S : (This is the proximal normal inequality.) The set (convex cone) of such , which always contains 0, is denoted N SP (x) and referred to as the proximal normal cone. We apply to N SP (x) a sequential closure operation in order to obtain the limiting normal cone: ˚ N SL (x) : D lim  i :  i 2 N SP (x i ); x i ! x; x i 2 S : These geometric notions are consistent with the analytical ones, as illustrated by the formulas @P I S (x) D N SP (x); @L I S (x) D N SL (x) ; where I S denotes the indicator of the set S: the function which equals 0 on S and C1 elsewhere. They are also consistent with the more traditional ones: When S is either a convex set, a smooth manifold, or a manifold with boundary, then both N SP (x) and N SL (x) coincide with the usual normal vectors (a cone, space, or half-space respectively).

Nonsmooth Analysis in Systems and Control Theory

Viscosity Subdifferentials We remark that the viscosity subdifferential of f at x (commonly employed in pde’s, but not used in this article) corresponds to the set of  for which we have f (y)  f (x) C h; y  xi C (jy  xj) ; where is a function such that lim t#0 (t)/t D 0. This gives a potentially larger set than @P f (x); however, the viscosity subdifferential satisfies the same (fuzzy) calculus rules as does @P f . In addition, the sequential closure operation described above gives rise to the same limiting subdifferential @L f (x) (see [24]). In most cases, therefore, it is equivalent to work with viscosity or proximal subgradients. Necessary Conditions in Optimal Control The central theorem on necessary conditions for optimal control is the Pontryagin Maximum Principle. (The literature on necessary conditions in optimal control is now very extensive; we cite some standard references in the bibliography.) Even in the somewhat special (by current standards) smooth context in which it was first proved [47], an element of generalized differentiability was required. With the subsequent increase in both the generality of the model and weakening of the hypotheses (all driven by real applications), the need for nonsmooth analysis is all the greater, even for the very statement of the result. We give here just one example, a broadly applicable hybrid maximum principle taken from [11], in order to highlight the essential role played by nonsmooth analysis as well as the resulting versatility of the results obtained. The Problem and Basic Hypotheses We consider the minimization of the functional Z b `(x(a); x(b)) C F(t; x(t); u(t)) dt a

subject to the boundary conditions (x(a); x(b)) 2 S and the standard control dynamics x˙ (t) D f (t; x(t); u(t)) a.e. The minimization takes place with respect to (absolutely continuous) state arcs x and measurable control functions u : [a; b] ! Rm . Note that no explicit constraints are placed upon u(t); if such constraints exist, they are accounted for by assigning to the extended-valued integrand F the value C1 whenever the constraints are violated. It is part of our intention here to demonstrate the

utility of indirectly taking account of constraints in this way. As basic hypotheses, we assume that F is measurable and lower semicontinuous in (x; u); ` is taken to be locally Lipschitz and S closed. For the theorem below, we require that f be measurable in t and continuously differentiable in (x; u) (but see the remarks for a weakening of the smoothness). The Growth Conditions The principal hypothesis here is that f and F satisfy the following: for every bounded subset X of Rn , there exist a constant c and a summable function d such that, for almost every t, for every (x; u) 2 dom F(t; ; ) with x 2 X, we have kD x f (t; x; u)k  c fj f (t; x; u)j C F(t; x; u)g C d(t) ; and for all (; ) in @P F(t; x; u) (if any) we have jj (1 C kDu f (t; x; u)k)  c fj f (t; x; u)j 1Cj j CF(t; x; u)g C d(t) : In the following, ( ) denotes evaluation at (x(t); u(t)). Theorem 1 Let the control function u give rise to an arc x which is a local minimum for the problem above. Then there exist an arc p on [a; b] such that the following transversality condition holds (p(a); p(b)) 2 @L `(x(a); x(b)) C N SL (x(a); x(b)) ; and such that p satisfies the adjoint inclusion: p˙(t) belongs almost everywhere to the set ˚ co !: (! CDx f (t; )p(t); Du f (t; )p(t)) 2 @L F(t; ) ; as well as the maximum condition: for almost every t, for every u in dom F(t; x (t); ), one has hp(t); f (t; x (t); u)i  F(t; x (t); u)  hp(t); f (t; )i  F(t; ) : We proceed to make a few remarks on this theorem, beginning with the fact that it can fail in the absence of the growth condition, even for smooth problems of standard type [30]. The statement of the theorem is not complete, since in general the necessary conditions may hold only in abnormal form; we do not discuss this technical point here for reasons of economy. The phrase ‘local minimum’ (which we have also not defined) can be interpreted very generally, see [11]. The ‘maximum condition’ above is of

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course what leads to the name ‘maximum principle’, while the exotic-looking adjoint inclusion reduces to more familiar conditions in a variety of special cases (see below). The transversality condition illustrates well the utility of couching in nonsmooth analysis terms the conclusion, since the given form encapsulates simultaneously a wide variety of conclusions obtainable in special cases. To give but one example, consider an optimal control problem in which x(a) is prescribed, x(b) does not appear in the cost, but is subject to an inequality constraint g(x(b))  0 for a certain smooth scalar function g. Then the transversality condition of the theorem (interpreted via standard facts in nonsmooth analysis) asserts that p(b) is of the form r g(x (b)) for some nonpositive number  (a Lagrange multiplier). To illustrate the versatility of the theorem, we look at some special cases.

The Calculus of Variations When we take f (t; x; u) D u, the problem reduces to the problem of Bolza in the calculus of variations. The first growth condition is trivially satisfied, and the second coincides with the generalized Tonelli–Morrey growth condition introduced in [11]; in this way we recover the stateof-the-art necessary conditions for the generalized problem of Bolza, which include as a special case the multiplier rule for problems with pointwise and/or isoperimetric constraints. Differential Inclusions When we specialize further by taking F(t; ) to be the indicator of the graph of a multifunction M(t; ), we obtain the principal necessary conditions for the differential inclusion problem. These in turn lead to necessary conditions for generalized control systems [11].

The Standard Problem The first case we examine is that in which for each t, F(t; x; u) D I U (t) (u) ; the indicator function which takes the value 0 when u 2 U(t) and C1 otherwise. This simply corresponds to imposing the condition u(t) 2 U(t) on the admissible controls u; this is the Mayer form of the problem (no integral term in the cost, obtained by reformulation if necessary). Note that in this case the second growth condition is trivially satisfied (since  D 0). The first growth condition is active only on U(t), and certainly holds if f is smooth in (t; x; u) and U(t) is uniformly bounded. The hybrid adjoint inclusion immediately implies the standard adjoint equation  p˙(t) D D x f (t; x (t); u (t)) p(t) ; and we recover the conclusions of the classical Maximum Principle of Pontryagin. (An extension is obtained when U(t) not bounded, see [10].) When f is not assumed to be differentiable, but merely locally Lipschitz with respect to x, there is a variant of the theorem in which the adjoint inclusion is expressed as follows:  p˙(t) 2 @C hp(t); f (t; ; u (t))i (x (t)) ; where the generalized gradient @C (see Sect. “Elements of Nonsmooth Analysis”) is taken with respect to the x variable (note the connection to the standard adjoint equation above). This is an early form of the nonsmooth maximum principle [13].

Mixed Constraints Consider again the optimal control problem in Mayer form (no integral term in the cost), but now in the presence of mixed state/control pointwise constraints of the form (x(t); u(t)) 2 ˝ a.e. for a given closed set ˝. Obtaining general necessary conditions for such problems is a well-known challenge in the subject; see [33,34,46]. We treat this case by taking F(t; ) D I˝ (). Then the second growth condition reduces to the following geometric assumption: for every (x; u) 2 ˝, for every (; ) 2 P (x; u), one has N˝ jj (1 C kDu f (t; x; u)k)  c j f (t; x; u)j C d(t) : j j By taking a suitable representation for ˝ in terms of functional equalities and/or inequalities, sufficient conditions in terms of rank can be adduced which imply this property, leading to explicit multiplier rules (see [10] for details). With an appropriate ‘transversal intersection’ condition, we can also treat the case in which both the constraints (x(t); u(t)) 2 ˝ and u(t) 2 U(t) are present. Sensitivity The multiplier functions p that appear in necessary conditions such as the ones above play a central role in analyzing the sensitivity of optimal control problems that depend on parameters. To take but one example, consider the presence of a perturbation ˛() in the dynamics of the problem: x˙ (t) D f (t; x(t); u(t)) C ˛(t) a.e.

Nonsmooth Analysis in Systems and Control Theory

Clearly the minimum in the problem depends upon ˛; we denote it V(˛). The function V is an example of what is referred to as a value function. Knowledge of the derivative of V would be highly relevant is studying the sensitivity of the problem to perturbations (errors) in the dynamics. Generally, however, value functions are not differentiable, so instead one uses nonsmooth analysis; the multipliers p give rise to estimates on the generalized gradient of V. We refer to [15,16] for examples and references. Verification Functions We consider now a special case of the problem considered in the preceding section: to minimize the integral cost functional Z b J(x; u) : D F(t; x(t); u(t)) dt a

subject to the prescribed boundary conditions x(a) D A ;

x(b) D B

and the standard control dynamics x˙ (t) D f (t; x(t); u(t)) a.e.; u(t) 2 U(t) a.e. Suppose now that a candidate (x  ; u ) has been identified as a possible solution to this problem (perhaps through a partial analysis of the necessary conditions, or otherwise). An elementary yet powerful way (traceable to Legendre) to prove that (x ; u ) actually does solve the problem is to exhibit a function (t; x) such that: F(t; x; u)   t (t; x) C h x (t; x); f (t; x; u)i 8t 2 [a; b]; (x; u) 2 Rn  U(t) ; with equality along (t; x (t); u (t)). The mere existence of such a function verifies that (x ; u ) is optimal, as we now show. For any admissible state/control pair (x; u), we have ˙ F(t; x(t); u(t))   t (t; x(t)) C h x (t; x(t)); x(t)i D

d dt

Z

b

) J(x; u) : D a

f(t; x(t))g

ˇ tDb F(t; x(t); u(t)) dt  (t; x(t))ˇ tDa D (b; B)  (a; A) :

But this lower bound on J holds with equality when (x; u) D (x ; u ); which proves that (x ; u ) is optimal. In this argument, we have implicitly supposed that the verification function ' is smooth. It is a fact that if

we limit ourselves to smooth verification functions, then there may not exist such a ' (even when the problem itself has smooth data). However, if we admit nonsmooth (locally Lipschitz) verification functions, then (under mild hypotheses on the data) the existence of a verification function ' becomes necessary and sufficient for (x ; u ) to be optimal. An appropriate way to extend the smooth inequality above uses the generalized gradient @C  of ' as follows: F(t; x; u)  C h; f (t; x; u)i 8 ( ; ) 2 @C (t; x) ; together with the requirement J(x ; u ) D (b; B)  (a; A) : It is a feature of this approach to proving optimality that it extends readily to more general problems, for example those involving unilateral state constraints, boundary costs, or isoperimetric conditions. It can also be interpreted in terms of duality theory, as shown notably by Vinter. We refer to [26] for details. The basic inequality that lies at the heart of this approach is of Hamilton–Jacobi type, and the fact that we are led to consider nonsmooth verification functions is related to the phenomenon that Hamilton–Jacobi Equations may not admit smooth solutions. This is one of the main themes of the next section. Dynamic Programming and Viscosity Solutions The Minimal Time Problem By a trajectory of the standard control system x˙ (t) D f (x(t); u(t)) a.e.; u(t) 2 U a.e. we mean a state function x() corresponding to some choice of admissible control function u(). The minimal time problem refers to finding a trajectory that reaches the origin as quickly as possible. Thus we seek the least T  0 admitting a control function u() on [0; T] having the property that the resulting trajectory x satisfies x(T) D 0. We proceed now to describe the well-known dynamic programming approach to solving the problem. We begin by introducing the minimal time function T(), defined on Rn as follows: T(˛) is the least time T  0 such that some trajectory x() satisfies x(0) D ˛; x(T) D 0 : An issue of controllability arises here: Is it always possible to steer ˛ to 0 in finite time? When such is not

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the case, then in accord with the usual convention we set T(˛) D C1. The principle of optimality is the dual observation that if x() is any trajectory, then we have, for s < t, 

   T x(t)  T x(s)  s  t :

  t 7! T x(t) C t is increasing. Furthermore, if x is optimal, then the same function is constant. Let us explain this in other terms: if x() is an optimal trajectory joining ˛ to 0, then  T x(t) D T(˛)  t

for

0  t  T(˛) ;

since an optimal trajectory from the point  x(t) is furnished by the truncation of x() to the interval t; T(˛) . If x() is any trajectory, then the inequality   T x(t)  T(˛)  t is a reflection of the fact that in going to the point x(t) from ˛ (in time t), we may have acted optimally (in which case equality holds) or  not(then inequality holds). Since t 7! T x(t) C t is increasing, we expect to have ˝   ˛ rT x(t) ; x˙ (t) C 1  0 ; with equality when x() is an optimal trajectory. The possible values of x˙ (t) for a trajectory being precisely the ele ments of the set f x(t); U , we arrive at ˝ ˛ min rT(x); f (x(t); u) C 1 D 0 : u2U

(4)

We define the (lower) Hamiltonian function h as follows: h(x; p) : D min hp; f (x; u)i : u2U

In terms of h, the partial differential equation obtained above reads   h x; rT(x) C 1 D 0 ;

u2U

Then, if we construct x() via the initial-value problem    x˙ (t) D f x(t); k x(t) ) ; x(0) D ˛ ; (7)

Equivalently, the function



following procedure: for each x, let k(x) be a point in U satisfying ˝ ˛ ˝ ˛ min rT(x); f (x; u) D rT(x); f (x; k(x)) D 1 : (6)

(5)

a special case of the Hamilton–Jacobi Equation. Here is the first step in the dynamic programming heuristic: use the Hamilton–Jacobi Equation (5), together with the boundary condition T(0) D 0, to find T(). How will this help us find the optimal trajectory? To answer this question, we recall that an optimal trajectory is such that equality holds in (4). This suggests the

we will have a trajectory that is optimal (from ˛)! Here is why: Let x() satisfy (7); then x() is a trajectory, and   ˝   ˛ d ˙ dt T x(t) D rT x(t) ; x (t) ˝     ˛ D rT x(t) ; f x(t); k x(t) D 1 : Integrating, we find   T x(t) D T(˛)  t ; which implies that at t D T(˛), we must have x D 0. Therefore x() is an optimal trajectory. Let us stress the important point that k() generates the optimal trajectory from any initial value ˛ (via (7)), and so constitutes what can be considered the ultimate solution for this problem: an optimal feedback synthesis. There can be no more satisfying answer to the problem: If you find yourself at x, just choose the control value k(x) to approach the origin as fast as possible. This goes well beyond finding a single open-loop optimal control. Unfortunately, there are serious obstacles to following the route that we have just outlined, beginning with the fact that T is nondifferentiable, as simple examples show, even when it is finite everywhere (which it generally fails to be). We will therefore have to examine anew the argument that led to the Hamilton–Jacobi Equation (5), which, in any case, will have to be recast in some way to accommodate nonsmooth solutions. Having done so, will the generalized Hamilton–Jacobi Equation admit T as the unique solution? The next step (after characterizing T) offers fresh difficulties of its own. Even if T were smooth, there would be in general no continuous function k() satisfying (6) for each x. The meaning and existence of a trajectory x() generated by k() via the differential Eq. (7), in which the righthand side is discontinuous in the state variable, is therefore problematic in itself. The intrinsic difficulties of this approach to the minimal-time problem have made it a historical focal point of activity in differential equations and control, and it is only recently that fully satisfying answers to all the questions raised above have been found. We begin with generalized solutions of the Hamilton–Jacobi Equation.

Nonsmooth Analysis in Systems and Control Theory

Subdifferentials and Viscosity Solutions We shall say that ' is a proximal solution of the Hamilton– Jacobi Equation (5) provided that h(x; @P (x)) D 1 8x 2 Rn ;

(8)

a ‘multivalued equation’ which means that for all x, for all  2 @P (x) (if any), we have h(x; ) D 1. (Recall that the proximal subdifferential @P  was defined in Sect. “Elements of Nonsmooth Analysis”.) Note that the equation holds automatically at a point x for which @P (x) is empty; such points play an important role, in fact, as we now illustrate. Consider the case in which f (x; U) is equal to the unit ball for all x, in dimension n D 1. Then h(x; p)  jpj (and the equation is of eikonal type). Let us examine the functions 1 (x) : D jxj and 2 (x) : D jxj; they both satisfy h(x; r(x)) D 1 at all points x ¤ 0, since (for each of these functions) the proximal subdifferential at points different from 0 reduces to the singleton consisting of the derivative. However, we have (see the exercise in the summary of nonsmooth analysis) @P 1 (0) D ;; @P 2 (0) D [1; 1] ; and it follows that 1 is (but 2 is not) a proximal solution of the Hamilton–Jacobi Equation (8). A lesson to be drawn from this example is that in defining generalized solutions we need to look closely at the differential behavior at specific and individual points; we cannot argue in an ‘almost everywhere’ fashion, or by ‘smearing’ via integration (as is done for linear partial differential equations via distributional derivatives). Proximal solutions are just one of the ways to define generalized solutions of certain partial differential equations, a topic of considerable interest and activity, and one which seems to have begun with the Hamilton–Jacobi Equation in every case. The first ‘subdifferential type’ of definition was given by the author in the 1970s, using generalized gradients and for locally Lipschitz solutions. While no uniqueness theorem holds for that solution concept, it was shown that the value function of the associated optimal control problem is a solution (hence existence holds), and is indeed a special solution: it is the maximal one. In 1980 A. I. Subbotin defined his ‘minimax solutions’, which are couched in terms of Dini derivates rather than subdifferentials, and which introduced the important feature of being ‘two-sided’. This work featured existence and uniqueness in the class of Lipschitz functions, the solution being characterized as the value of a differential game. Subsequently, M. Crandall and P.-L. Lions incorporated both subdifferentials and two-sidedness in their viscosity solutions, a theory which they developed for merely

continuous functions. In the current context, and under mild hypotheses on the data, it can be shown that minimax, viscosity, and proximal solutions all coincide [19,24]. Recall that our goal (within the dynamic programming approach) is to characterize the minimal time function. This is now attained, as shown by the following (we omit the hypotheses; see [63], and also the extensive discussion in Bardi and Capuzzo-Dolcetta [4]): Theorem 2 There exists a unique lower semicontinuous function  : Rn ! (1; C1] bounded below on Rn and satisfying the following: [HJ equation] h(x; @P (x)) D 18 x ¤ 0; [Boundary condition] (0) D 0 and h(0; @P (0))  1: That unique function is T(). The proof of this theorem is based upon proximal characterizations of certain monotonicity properties of trajectories related to the inequality forms of the Hamilton–Jacobi Equation (see Sect. 4.7 of [24]). The fact that monotonicity is closely related to the solution of the minimal time problem is already evident in the following elementary assertion: a trajectory x joining ˛ to 0 is optimal if the rate of change of the function t 7! T(x(t)) is 1 a.e. The characterization of T given by the theorem can be applied to verify the validity of a general conjecture regarding the time-optimal trajectories from arbitrary initial values. This works as follows: we use the conjecture to calculate the supposedly optimal time T(˛) from any initial value ˛, and then we see whether or not the function T constructed in this way satisfies the conditions of the theorem. If so, then (by uniqueness) T is indeed the minimal time function and the conjecture is verified (the same reasoning as in the preceding section is in fact involved here). Otherwise, the conjecture is necessarily incorrect (but the way in which it fails can provide information on how it needs to be modified; this is another story). Another way in which the reasoning above is exploited is to discretize the underlying problem as well as the Hamilton–Jacobi Equation for T. This gives rise to the backwards recursion numerical method developed and popularized by Bellman under the name of dynamic programming; of course, the approach applies to problems other than minimal time. With respect to the goal of finding an optimal feedback synthesis for our problem, we have reached the following point in our quest: given that T satisfies the proximal Hamilton–Jacobi Equation h(x; @P T(x)) D 1, which can be written in the form ˝ ˛ min ; f (x(t); u) D 1 8 2 @P T(x); 8 x ¤ 0 ; (9) u2U

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how does one proceed to construct a feedback k(x) having the property that any trajectory x generated by it via (7) is such that t 7! T(x(t)) decreases at a unit rate? There will also arise the issue of defining the very sense of (7) when k() is discontinuous, as it must be in general. This is a rather complex question to answer, and it turns out to be a special case of the issue of designing stabilizing feedback (where, instead of T, we employ a Lyapunov function). We shall address this below, and return there to the minimal time synthesis, which will be revealed to be a special case of the procedure. We need first to examine the concept of Lyapunov function. Lyapunov Functions In this section we consider the standard control system x˙ (t) D f (x(t); u(t)) a.e. ; u(t) 2 U a.e. ; under a mild local Lipschitz hypothesis: for every bounded set S in Rn there exists a constant K D K(S) such that j f (x; u)  f (x 0 ; u)j  Kjx  x 0 j 8x; x 0 2 S ; u 2 U : We also suppose that f (0; U) is bounded. A point ˛ is asymptotically guidable to the origin if there is a trajectory x satisfying x(0) D ˛ and lim t!1 x(t) D 0. When every point has this property, and when additionally the origin has the familiar local stability property known as Lyapunov stability, it is said in the literature to be GAC: (open loop) globally asymptotically controllable (to 0). A well-known sufficient condition for this property is the existence of a smooth (C 1 , say) pair of functions V : Rn ! R ;

W : Rn nf0g ! R

satisfying the following conditions: 1. Positive Definiteness: V(x) > 0 and W(x) > 0

8x ¤ 0; and V(0)  0 :

2. Properness: The sublevel sets fx : V(x)  cg are bounded 8c. 3. Weak Infinitesimal Decrease: inf hrV(x); f (x; u)i  W(x)

u2U

x ¤0:

The last condition asserts that V decreases in some available direction. We refer to V as a (weak) Lyapunov function; it is also referred to in the literature as a control Lyapunov function.

It is a fact, as demonstrated by simple examples (see [17] or [57]), that the existence of a smooth function V with the above properties fails to be a necessary condition for global asymptotic controllability; that is, the familiar converse Lyapunov theorems of Massera, Barbashin and Krasovskii, and Kurzweil (in the setting of a differential equation with no control) do not extend to this weak controllability setting, at least not in smooth terms. This may be a rather general phenomenon, in view of the following result [22], which holds under the additional hypothesis that the sets f (x; U) are closed and convex: Theorem 3 If the system admits a C 1 weak Lyapunov function, then it has the following surjectivity property: for every > 0, there exists ı > 0 such that f (B(0; ); U)  B(0; ı). It is not difficult to check that the nonholonomic integrator system (see Sect. “Introduction”) is GAC. Since it fails to have the surjectivity property described in the theorem, it cannot admit a smooth Lyapunov function. It is natural therefore to seek to weaken the smoothness requirement on V so as to obtain a necessary (and still sufficient) condition for a system to be GAC. This necessitates the use of some construct of nonsmooth analysis to replace the gradient of V that appears in the infinitesimal decrease condition. In this connection we use the proximal subgradient (Sect. “Elements of Nonsmooth Analysis”) @P V(x), which requires only that the (extendedvalued) function V be lower semicontinuous. In proximal terms, the Weak Infinitesimal Decrease condition becomes sup

inf h; f (x; u)i  W(x)

2@ P V (x) u2U

x ¤0:

Note that this last condition is trivially satisfied when x is such that @P V (x) is empty, in particular when V(x) D C1. (The supremum over the empty set is 1.) A general Lyapunov pair (V; W) refers to extended-valued lower semicontinuous functions V : Rn ! R [ fC1g and W : Rn nf0g ! R [ fC1g satisfying the positive definiteness and properness conditions above, together with proximal weak infinitesimal decrease. The following is proved in [24]. Theorem 4 Let (V; W) be a general Lyapunov pair for the system. Then any ˛ 2 dom V is asymptotically guidable to 0. It follows from the theorem that the existence of a lower semicontinuous Lyapunov pair (V ; W) with V everywhere finite-valued implies the global asymptotic guidability to 0 of the system. When V is not continuous, this does not

Nonsmooth Analysis in Systems and Control Theory

imply Lyapunov stability at the origin, however, so it cannot characterize global asymptotic controllability. An early and seminal result due to Sontag [56] considers continuous functions V, with the infinitesimal decrease condition expressed in terms of Dini derivates. Here is a version of that result in proximal subdifferential terms: Theorem 5 The system is GAC if and only if there exists a continuous Lyapunov pair (V; W). There is an advantage to being able to replace continuity in such a result by a stronger regularity property (particularly in connection with using V to design a stabilizing feedback, as we shall see). Of course, we cannot assert smoothness, as pointed out above. In [20] it was shown that certain locally Lipschitz value functions give rise to practical Lyapunov functions, that is, assuring stable controllability to arbitrary neighborhoods of 0. Building upon this, L. Rifford [49] was able to combine a countable family of such functions in order to construct a global locally Lipschitz Lyapunov function. This answered a long-standing open question in the subject. Rifford [51] also went on to show the existence of a semiconcave Lyapunov function, a stronger property (familiar in pde’s) whose relevance to feedback construction will be seen in the next section. The role of Lyapunov functions in characterizing various types of convergence to 0 (including guidability in finite time) is discussed in detail in [18]. Stabilizing Feedback We now address the issue of finding a stabilizing feedback control k(x); that is, a function k with values in U such that all solutions of the differential equation x˙ D g(x); x(0) D ˛ ; where g(x) : D f (x; k(x)) (10) converge to 0 (for all values of ˛) in a suitable sense. Here, the origin is supposed to be an equilibrium of the system; to be precise, we take 0 2 U and f (0; 0) D 0. When such a k exists, the system is termed stabilizable. A necessary condition for the system to be stabilizable is that it be GAC. A central question in the subject has been whether this is sufficient as well. An early observation of Sontag and Sussmann [58] showed that the answer is negative if one requires the feedback k to be continuous (which provides the easiest classical way of interpreting the differential equation that appears in (10)). Later, Brockett showed that the nonholonomic integrator fails to admit a continuous stabilizing feedback. One is therefore led to consider the use of discontinuous feedback, together with the attendant need to define an appropriate solution concept for a differential equation

in which the dynamics fail to be continuous in the state. The best-known solution concept in this regard is that of Filippov; it turns out, however, that the nonholonomic integrator fails to admit a (discontinuous) feedback which stabilizes it in the Filippov sense [32,55]. Clarke, Ledyaev, Sontag and Subbotin [21] gave a positive answer when the (discontinuous) feedbacks are implemented in the closedloop system sampling sense (also referred to as sample-andhold). We proceed now to describe the sample-and-hold implementation of a feedback. Let  D ft i g i0 be a partition of [0; 1), by which we mean a countable, strictly increasing sequence t i with t0 D 0 such that t i ! C1 as i ! 1. The diameter of , denoted diam (), is defined as sup i0 (t iC1  t i ). Given an initial condition x0 , the -trajectory x() corresponding to  and an arbitrary feedback law k : Rn ! U is defined in a step-by-step fashion as follows. Between t0 and t1 , x is a classical solution of the differential equation x˙ (t) D f (x(t); k(x0 )) ; x(0) D x0 ; t0  t  t1 : (In the present context we have existence and uniqueness of x, and blow-up cannot occur.) We then set x1 : D x(t1 ) and restart the system at t D t1 with control value k(x1 ): x˙ (t) D f (x(t); k(x1 )) ; x(t1 ) D x1 ; t1  t  t2 ; and so on in this fashion. The trajectory x that results from this procedure is an actual state trajectory corresponding to a piecewise constant open-loop control; thus it is a physically meaningful one. When results are couched in terms of -trajectories, the issue of defining a solution concept for discontinuous differential equations is effectively sidestepped. Making the diameter of the partition smaller corresponds to increasing the sampling rate in the implementation. We remark that the use of possibly discontinuous feedback has arisen in other contexts. In linear time-optimal control, one can find discontinuous feedback syntheses as far back as the classical book of Pontryagin et al. [47]; in these cases the feedback is invariably piecewise constant relative to certain partitions of state space, and solutions either follow the switching surfaces or cross them transversally, so the issue of defining the solution in other than a classical sense does not arise. Somewhat related to this is the approach that defines a multivalued feedback law [6]. In stochastic control, discontinuous feedbacks are the norm, with the solution understood in terms of stochastic differential equations. In a similar vein, in the control of certain linear partial differential equations, discontinuous feedbacks can be interpreted in a distributional sense. These cases are all unrelated to the one under

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discussion. We remark too that the use of discontinuous pursuit strategies in differential games [41] is well-known, together with examples to show that, in general, it is not possible to achieve the result of a discontinuous optimal strategy to within any tolerance by means of a continuous stategy (thus there can be a positive unbridgeable gap between the performance of continuous and discontinuous feedbacks). It is natural to say that a feedback k(x) (continuous or not) stabilizes the system in the sample-and-hold sense provided that for every initial value x0 , for all > 0, there exists ı > 0 and T > 0 such that whenever the diameter of the partition  is less than ı, then the corresponding -trajectory x beginning at x0 satisfies kx(t)k  8t  T : The following theorem is proven in [21]. Theorem 6 The system is open loop globally asymptotically controllable if and only if there exists a (possibly discontinuous) feedback k : Rn ! U which stabilizes it in the sample-and-hold sense. The proof of the theorem used the method of proximal aiming, which can be viewed as a geometric version of the Lyapunov technique. We now discuss how to define stabilizing feedbacks if one has in hand a sufficiently regular Lyapunov function. The Smooth Case We begin with the case in which a C 1 smooth Lyapunov function exists, where a very natural approach can be used. For x ¤ 0, we simply define k(x) to be any element u 2 U satisfying

But since a smooth Lyapunov function does not necessarily exist, we still require a way to handle the general case. It turns out that the smooth and general cases can be treated in a unified fashion through the notion of semiconcavity, which is a certain regularity property (not implying smoothness). Rifford has proven that any GAC system admits a semiconcave Lyapunov function; we shall see that this property permits a natural extension of the pointwise definition of a stabilizing feedback that was used in the smooth case. A function  : Rn ! R is said to be (globally) semiconcave provided that for every ball B(0; r) there exists

D (r)  0 such that the function x 7! (x)  jxj2 is (finite and) concave on B(0; r). (Hence ' is locally the sum of a concave function and a quadratic one.) Observe that any function of class C 2 is semiconcave; also, any semiconcave function is locally Lipschitz, since both concave functions and smooth functions have that property. (There is a local definition of semiconcavity that we omit for present purposes.) Semiconcavity is an important regularity property in partial differential equations (see for example [9]). The fact that the semiconcavity of a Lyapunov function V turns out to be useful in stabilization is a new observation, and may be counterintuitive: V often has an interpretation in terms of energy, and it may seem more appropriate to seek a convex Lyapunov function V. We proceed now to explain why semiconcavity is a highly desirable property, and why a convex V would be of less interest (unless it were smooth, but then it would be semiconcave too). Recall the ideal case discussed above, in which (for a smooth V) we select a function k(x) such that hrV(x); f (x; k(x))i  W(x)/2 :

hrV(x); f (x; u)i  W(x)/2 : Note that at least one such u does exist, in light of the Infinitesimal Decrease Condition. It is then elementary to prove [18] that the pointwise feedback k described above stabilizes the system in the sample-and-hold sense. We remark that Rifford [50] has shown that the existence of a smooth Lyapunov pair is equivalent to the existence of a locally Lipschitz one satisfying Infinitesimal Decrease in the sense of generalized gradients (that is, with @P V replaced by @C V), and that this in turn is equivalent to the existence of a stabilizing feedback in the Filippov (as well as sample-and-hold) sense. Semiconcavity We have seen that a smooth Lyapunov function generates a stabilizing feedback in a simple and natural way.

How might this appealing idea be adapted to the case in which V is nonsmooth? We cannot use the proximal subdifferential @P V (x) directly, since it may be empty for some values of x. We are led to consider the limiting subdifferential @L V(x) (see Sect. “Elements of Nonsmooth Analysis”), which is nonempty when V is locally Lipschitz. By passing to the limit, the Weak Infinitesimal Decrease Condition for proximal subgradients implies the following: inf h f (x; u); i  W(x)

u2U

8 2 @L V(x) ; 8x ¤ 0 :

Accordingly, let us consider the following idea: for each x ¤ 0, choose some element  2 @L V(x), then choose k(x) 2 U such that h f (x; k(x)); i  W(x)/2 :

Nonsmooth Analysis in Systems and Control Theory

Does this lead to a stabilizing feedback, when (of course) the discontinuous differential equation is interpreted in the sample-and-hold sense? When V is smooth, the answer is ‘yes’, as we have seen. But when V is merely locally Lipschitz, a certain ‘dithering’ phenomenon may arise to prevent k from being stabilizing. However, if V is semiconcave (locally on Rn nf0g), this does not occur, and stabilization is guaranteed. This explains the interest in finding a semiconcave Lyapunov function. When V is a locally Lipschitz Lyapunov function with no additional regularity (neither smooth nor semiconcave), then it can still be used for defining stabilizing feedback, but less directly. It is possible to regularize V: to approximate it by a semiconcave function through the process of inf-convolution (see [24]). This leads to practical semiglobal stabilizing feedbacks, that is, for any 0 < r < R, we derive a feedback which stabilizes all initial values in B(0; R) to B(0; r) [20]. Optimal Feedback The strategy described above for defining stabilizing feedbacks via Lyapunov functions can also be applied to construct (nearly) optimal feedbacks as well as stabilizing ones. The key is to use an appropriate value function instead of an arbitrary Lyapunov function. We obtain in this way a unification of optimal control and feedback control, at least at the mathematical level, and as regards feedback design. To illustrate, consider again the Minimal Time problem, at the point at which we had left it at the end of Sect. “Lyapunov Functions” (thus, unresolved as regards the time-optimal feedback synthesis). As pointed out there, T satisfies the Hamilton–Jacobi Equation: this yields Infinitesimal Decrease in subgradient terms. Consequently, if the Minimal Time function T happens to be finite everywhere as well as smooth or semiconcave, then we can use the same direct definition as above to design a feedback which (in the limiting sample-and-hold sense) produces trajectories along which T decreases at rate 1; that is, which are time-optimal. This of course yields the fastest possible stabilization (and in finite time). In general, however, T may lack such regularity, or (when the controllability to the origin is only asymptotic) not even be finite everywhere. Then it is necessary to apply an approximation (regularization) procedure in order to obtain a variant of T, and use that instead. When T is finite, we can obtain in this way an approximate time-optimal synthesis (to any given tolerance). The whole approach described here can be carried out for a variety of optimal control contexts [26,45], and also

for finding optimal strategies in differential games [25]. It also carries over to problems in which unilateral state constraints are imposed: x(t) 2 X, where X is a given closed set [26,28,29]. The issue of robustness, not discussed here, is particularly important in the presence of discontinuity; see [18,42,57]. Future Directions A lesson of the past appears to be that nonsmooth analysis is likely to be required whenever linearization is not adequate or is inapplicable. It seems likely therefore to accompany the subject of control theory as it sets out to conquer new nonlinear horizons, in ways that cannot be fully anticipated. Let us nonetheless identify a few directions for future work. The extensions of most of the results cited above to problems on manifolds, or of tracking, or with partial information (as in adaptive control) remain to be carried out to a great extent. There are a number of currently evolving contexts not discussed above in which nonsmooth analysis is highly likely to play a role, notably hybrid control; an example here is provided by multiprocesses [8,31]. Distributed control (of pde’s) is another area which requires development. There is also considerable work to be done on numerical implementation; in this connection see [36,37]. Bibliography 1. Artstein Z (1983) Stabilization with relaxed controls. Nonlinear Anal TMA 7:1163–1173 2. Astolfi A (1996) Discontinuous control of nonholonomic systems. Syst Control Lett 27:37–45 3. Astolfi A (1998) Discontinuous control of the Brockett integrator. Eur J Control 4:49–63 4. Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, Boston 5. Bardi M, Staicu V (1993) The Bellman equation for timeoptimal control of noncontrollable nonlinear systems. Acta Appl Math 31:201–223 6. Berkovitz LD (1989) Optimal feedback controls. SIAM J Control Optim 27:991–1006 7. Borwein JM, Zhu QJ (1999) A survey of subdifferential calculus with applications. Nonlinear Anal 38:687–773 8. Caines PE, Clarke FH, Liu X, Vinter RB (2006) A maximum principle for hybrid optimal control problems with pathwise state constraints. Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, 13–15 Dec 2006 9. Cannarsa P, Sinestrari C (2004) Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Birkhäuser, Boston 10. Clarke F (2005) The maximum principle in optimal control. J Cybern Control 34:709–722

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59. Subbotin AI (1995) Generalized Solutions of First-Order PDEs. Birkhäuser, Boston 60. Trélat E (2006) Singular trajectories and subanalyticity in optimal control and Hamilton–Jacobi theory. Rend Semin Mat Univ Politec Torino 64:97–109 61. Vinter RB (2000) Optimal Control. Birkhäuser, Boston 62. Warga J (1972) Optimal Control of Differential and Functional Equations. Academic Press, New York 63. Wolenski PR, Zhuang Y (1998) Proximal analysis and the minimal time function. SIAM J Control Optim 36:1048–1072

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Normal Forms in Perturbation Theory HENK W. BROER University of Groningen, Groningen, The Netherlands Article Outline Glossary Definition of the Subject Introduction Motivation The Normal Form Procedure Preservation of Structure Semi-local Normalization Non-formal Aspects Applications Future Directions Bibliography Glossary Normal form procedure This is the stepwise ‘simplification’ by changes of coordinates, of the Taylor series at an equilibrium point, or of similar series at periodic or quasi-periodic solutions. Preservation of structure The normal form procedure is set up in such a way that all coordinate changes preserve a certain appropriate structure. This applies to the class of Hamiltonian or volume preserving systems, as well as to systems that are equivariant or reversible with respect to a symmetry group. In all cases the systems may also depend on parameters. Symmetry reduction The truncated normal form often exhibits a toroidal symmetry that can be factored out, thereby leading to a lower dimensional reduction. Perturbation theory The attempt to extend properties of the (possibly reduced) normal form truncation, to the full system. Definition of the Subject Nonlinear dynamical systems are notoriously hard to tackle by analytic means. One of the few approaches that has been effective for the last couple of centuries, is Perturbation Theory. Here systems are studied, which in an appropriate sense, can be seen as perturbations of a given system with ‘well-known’ dynamical properties. Such ‘wellknown’ systems usually are systems with a great amount of symmetry (like integrable Hamiltonian systems [1]) or very low-dimensional systems. The methods of Perturbation Theory then try to extend the ‘well-known’ dynamical

properties to the perturbed system. Methods to do this are often based on the Implicit Function Theorem, on normal hyperbolicity [85,87] or on Kolmogorov–Arnold–Moser theory [1,10,15,18,38]. To obtain a perturbation theory set-up, normal form theory is a vital tool. In its most elementary form it amounts to ‘simplifying’ the Taylor series of a dynamical system at an equilibrium point by successive changes of coordinates. The lower order truncation of the series often belongs to the class of ‘well-known’ systems and by the Taylor formula the original system then locally can be viewed as a perturbation of this ‘well-known’ truncation, that thus serves as the ‘unperturbed’ system. More involved versions of normal form theory exist at periodic or quasi-periodic evolutions of the dynamical system. Introduction We review the formal theory of normal forms of dynamical systems at equilibrium points. Systems with continuous time, i. e., of vector fields, or autonomous systems of ordinary differential equations, are considered extensively. The approach is universal in the sense that it applies to many cases where a structure is preserved. In that case also the normalizing transformations preserve this structure. In particular this includes the Hamiltonian and the volume preserving case as well as cases that are equivariant or reversible with respect to a symmetry group. In all situations the systems may depend on parameters. Related topics are being dealt with concerning a vector field at a periodic solution or a quasi-periodic invariant torus as well as the case of a diffeomorphism at a fixed point. The paper is concluded by discussing a few non-formal aspects and some applications. Motivation The term ‘normal form’ is widely used in mathematics and its meaning is very sensitive for the context. In the case of linear maps from a given vector space to itself, for example, one may consider all possible choices of a basis. Each choice gives a matrix-representation of a given linear map. A suitable choice of basis now gives the well-known Jordan canonical form. This normal form, in a simple way displays certain important properties of the linear map, concerning its spectrum, its eigenspaces, and so on. Presently the concern is with dynamical systems, such as vector fields (i. e., systems of autonomous ordinary differential equations), or diffeomorphisms. The aim is to simplify these systems near certain equilibria and (quasiperiodic) solutions by a proper choice of coordinates. The

Normal Forms in Perturbation Theory

purpose of this is to better display the local dynamical properties. This normalization is effected by a stepwise simplification of formal power series (such as Taylor series). Reduction of Toroidal Symmetry In a large class of cases, the simplification of the Taylor series induces a toroidal symmetry up to a certain order, and truncation of the series at that order gives a local approximation of the system at hand. From this truncation we can reduce this toroidal symmetry, thereby also reducing the dimension of the phase space. This kind of procedure is reminiscent of the classical reductions in classical mechanics related to the Noether Theorem, compare with [1]. This leads us to a first perturbation problem to be considered. Indeed, by truncating and factoring out the torus symmetry we get a polynomial system on a reduced phase, and one problem is how persistent this system is with respect to the addition of higher order terms. In the case where the system depends on parameters, the persistence of a corresponding bifurcation set is of interest. In many examples the reduced phase space is 2-dimensional where the dynamics is qualitatively determined by a polynomial and this perturbation problem can be handled by Singularity Theory. In quite a number of cases the truncated and reduced system turns out to be structurally stable [67,86]. A Global Perturbation Theory When returning to the original phase space we consider the original system as a perturbation of the de-reduced truncation obtained so far. In the ensuing perturbation problem, several types of resonance can play a role. A classical example [2,38] occurs when in the reduced model a Hopf bifurcation takes place and we have reduced by a 1-torus. Then in the original space generically a Hopf– Ne˘ımark–Sacker bifurcation occurs, where in the parameter space the dynamics is organized by resonance tongues. In cases where we have reduced by a torus of dimension larger than 1, we are dealing with quasi-periodic Hopf bifurcation, in which the bifurcation set gets ‘Cantorized’ by Diophantine conditions, compare with [6,15,21,38]. Also the theory of homo- and heteroclinic bifurcations then is of importance [68]. We use the two perturbation problems as a motivation for the Normal Form Theory to be reviewed, for details, however, we just refer to the literature. For example, compare with  Perturbation Theory and references therein.

The Normal Form Procedure The subject of our interest is the simplification of the Taylor series of a vector field at a certain equilibrium point. Before we explore this, however, let us first give a convenient normal form of a vector field near a non-equilibrium point. Theorem 1 (Flow box [81]) Let the C 1 vector field X on Rn be given by x˙ D f (x) and assume that f (p) ¤ 0. Then there exists a neighborhood of p with local C 1 coordinates y D (y1 ; y2 ; : : : ; y n ), such that in these coordinates X has the form y˙1 D 1 ;

y˙ j D 0 ;

for 2  j  n. Such a local chart usually is called a flowbox and the above theorem the Flowbox Theorem. A proof simply can be given using a transversal local C 1 section that cuts the flow of the vector field transversally. For the coordinate y1 then use the time-parametrization of the flow, while the coordinates yj , for 2  j  n come from the section, compare with, e. g., [81]. Background, Linearization The idea of simplification near an equilibrium goes back at least to Poincaré [69], also compare with Arnold [2]. To be definite, we now let X be a C 1 vector field on Rn , with the origin as an equilibrium point. Suppose that X has the form x˙ D Ax C f (x); x 2 Rn , where A is linear and where f (0) D 0; D0 f D 0. The first idea is to apply successive C 1 changes of coordinates of the form Id C P, with P a homogeneous polynomial of degree m D 2; 3; : : :, ‘simplifying’ the Taylor series step by step. The most ‘simple’ form that can be obtained in this way, is where all higher order terms vanish. In that case the normal form is formally linear. Such a case was treated by Poincaré, and we shall investigate this now. We may even assume to work on C n , for simplicity assuming that the eigenvalues of A are distinct. A collection  D (1 ; : : : ; n ) of points in C is said to be resonant if there exists a relation of the form s D hr; i ; for r D (r1 ; : : : ; r n ) 2 Z n , with r k  0 for all k and with P r k  2. The order of the resonance then is the number P jrj D r k . The Poincaré Theorem now reads Theorem 2 (Formal linearization [2,69]) If the (distinct) eigenvalues 1 ; : : : ; n of A have no resonances, there exists a formal change of variables x D y C O(jyj2 ), transforming

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s P(x), while

the above vector field X, given by

X rj @x r x r  j x j D hr; ix r : Ax D @x xj

x˙ D Ax C f (x)

j

to y˙ D Ay : We include a proof [2], since this will provide the basis for almost all further considerations. Proof The formal power series x D y C O(jyj2 ) is obtained in an inductive manner. Indeed, for m D 2; 3; : : : a polynomial transformation x D y C P(y) is constructed, with P homogeneous of degree m, which removes the terms of degree m from the vector field. At the end we have to take the composition of all these polynomial transformations. 1. The basic tool for the mth step is the following. Let v be homogeneous of degree m, then if the vector fields x˙ D Ax C v(x) C O(jxj mC1 ) and y˙ D Ay are related by the transformation x D y C P(y) with P also homogeneous of degree m, then

x˙ D Ax C v m (x) C O(jxj mC1 ) ; we solve the homological equation adA(Pm ) D v m ; then carrying out the transformation x D y C Pm (y). This takes the above form to

D x PAx  AP(x) D v(x) : This relation usually is called the homological equation, the idea being to determine P in terms of v: by this choice of P the term v can be transformed away. The proof of this relation is straightforward. In fact, x˙ D (Id C D y P)Ay D (Id C D y P)A(x  P(x) C O(jxj

We conclude that the monomials x r e s are eigenvectors corresponding to the eigenvalues hr; i  s . We conclude that the operator adA is semisimple, since it has a basis of eigenvectors. Therefore, if ker adA D 0, the operator is surjective. This is exactly what the non-resonance condition on the eigenvalues of A amounts to. More precisely, the homological equation adA(P) D v can be solved for P for each homogeneous part v of degree m, provided that there are no resonances up to order m. 3. The induction process now runs as follows. For m  2, given a form

y˙ D Ay C w mC1 (y) C O(jyj mC2 ) : The composition of all the polynomial transformations then gives the desired formal transformation.  Remark

mC1

D Ax C fD x PAx  AP(x)g C O(jxj

))

mC1

);

where we used that for the inverse transformation we know y D x  P(x) C O(jxj mC1 ). 2. For notational convenience we introduce the linear operator adA, the so-called adjoint operator, by adA(P)(x) :D D x PAx  AP(x) ; then the homological equation reads adA(P) D v. So the question is reduced to whether v is in the image of the operator adA. It turns out that the eigenvalues of adA can be expressed in those of A. If x1 ; x2 ; : : : ; x n are the coordinates corresponding to the basis e1 ; e2 ; : : : ; e n , again it is a straightforward computation to show that for P(x) D x r e s , one has adA(P) D (hr; i  s )P : Here we use the multi-index notation x r D x1r 1 x2r 2    x nr n . Indeed, for this choice of P one has AP(x) D

 It is well-known that the formal series usually diverge. Here we do not go into this problem, for a brief discussion see below.  If resonances are excluded up to a finite order N, we can linearize up to that order, so obtaining a normal form. y˙ D Ay C O(jyj NC1 ) : In this case the transformation can be taken as a polynomial.  If the original problem is real, but with the matrix A having non-real eigenvalues, we still can keep all transformations real by also considering complex conjugate eigenvectors. We conclude this introduction by discussing two further linearization theorems, one due to Sternberg and the other to Hartman–Grobman. We recall the following for a vector field X(x) D Ax C f (x), x 2 Rn , with 0 as an equilibrium point, i. e., with f (0) D 0; D0 f D 0. The equilibrium 0 is hyperbolic if the matrix A has no purely imaginary eigenvalues. Sternberg’s Theorem reads

Normal Forms in Perturbation Theory

Theorem 3 (Smooth linearization [82]) Let X and Y be C 1 vector fields on Rn , with 0 as a hyperbolic equilibrium point. Also suppose that there exists a formal transformation (Rn ; 0) ! (Rn ; 0) taking the Taylor series of X at 0 to that of Y. Then there exists a local C 1 -diffeomorphism ˚ : (Rn ; 0) ! (Rn ; 0), such that ˚ X D Y. We recall that ˚ X(˚(x)) D D x ˚ X(x). This means that X and Y are locally conjugated by ˚ , the evolution curves of X are mapped to those of Y in a time-preserving manner. In particular Sternberg’s Theorem applies when the conclusion of Poincaré’s Theorem holds: for Y just take the linear part Y(x) D Ax(@/@x). Combining these two theorems we find that in the hyperbolic case, under the exclusion of all resonances, the vector field X is linearizable by a C 1 -transformation. The Hartman–Grobman Theorem moreover says that the nonresonance condition can be omitted, provided we only want a C0 -linearization. Theorem 4 (Continuous linearization e. g., [67]) Let X be a C 1 vector field on Rn , with 0 as a hyperbolic equilibrium point. Then there exists a local homeomorphism ˚ : (Rn ; 0) ! (Rn ; 0), locally conjugating X to its linear part. Preliminaries from Differential Geometry Before we develop a more general Normal Form Theory we recall some elements from differential geometry. One central notion used here is that of the Lie derivative. For simplicity all our objects will be of class C 1 . Given a vector field X we can take any tensor  and define its Lie-derivative L X  with respect to X as the infinitesimal transformation of  along the flow of X. In this way L X  becomes a tensor of the same type as . To be more precise, for  a real function f one so defines ˇ d ˇˇ L X f (x) D X( f )(x) D d f (X)(x) D f (X t (x)) ; dt ˇtD0 i. e., the directional derivative of f with respect to X. Here X t denotes the flow of X over time t. For  a vector field Y one similarly defines ˇ d ˇˇ L X Y(x) D (X t ) Y(x) dt ˇtD0 1 D lim f(X t ) Y(x)  Y(x)g t!0 t 1 D lim fY(x)  (X h ) Y(x)g ; h!0 h and similarly for differential forms, etc. Another central notion is the Lie-brackets [X; Y] defined for any two vector

fields X and Y on Rn by [X; Y]( f ) D X(Y( f ))  Y(X( f )) : Here f is any real function on Rn while, as before, X(f ) denotes the directional derivative of f with respect to X. We recall the expression of Lie-brackets in coordinates. If X is given by the system of differential equations x˙ j D X j (x), with 1  j  n, then the directional derivaPn tive X(f ) is given by X( f ) D jD1 X j @ f /@x j . Then, if [X; Y] is given by the system x˙ j D Z j (x), for 1  j  n, of differential equations, one directly shows that Zj D

n  X kD1

Xk

@Yj @X j  Yk @x k @x k

 :

Here Y j relates to Y as X j does to X. We list some useful properties. Proposition 5 (Properties of the Lie-derivative [81]) 1. 2. 3. 4.

L X (Y1 C Y2 ) D L X Y1 C L X Y2 (linearity over R) L X ( f Y) D X( f )  Y C f  L X Y (Leibniz rule)

[Y; X] D [X; Y] (skew symmetry) [[X; Y]; Z] C [[Z; X]; Y] C [[Y; Z]; X] D 0 (Jacobi identity) 5. L X Y D [X; Y] 6. [X; Y] D 0 , X t ı Ys D Ys ı X t Proof The first four items are left to the reader. 5. The equality L X Y D [X; Y] can be proven by observing the following. Both members of the equality are defined intrinsically, so it is enough to check it in any choice of (local) coordinates. Moreover, we can restrict our attention to the set fxjX(x) ¤ 0g. By the Flowbox Theorem 1 we then may assume for the component functions of X that X 1 (x) D 1; X j (x) D 0 for 2  j  n. It is easy to see that both members of the equality now are equal to the vector field Z with components Zj D

@Yj : @x1

6. Remains the equivalence of the commuting relationships. The commuting of the flows, by a very general argument, implies that the bracket vanishes. In fact, fixing t we see that X t conjugates the flow of Y to itself, which is equivalent to (X t ) Y D Y. By definition this implies that L X Y D 0. Conversely, let c(t) D ((X t ) Y)(p). From the fact that L X Y D 0 it then follows that c(t)  c(0). Observe that the latter assertion is sufficient for our purposes, since it

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implies that (X t ) Y D Y and therefore X t ı Ys D Ys ı X t . Finally, that c(t) is constant can be shown as follows. 1 fc(t C h)  c(t)g h!0 h 1 D lim f((X tCh ) Y)(p)  ((X t ) Y)(p)g h!0 h 1 D (X t ) lim f((X h ) Y)(X t (p))  Y(Xt (p))g h!0 h D (X t ) L X Y(X t (p))

c 0 (t) D lim

D (X t ) (0)

We now quote a theorem from Sect. 7.6.1. in Dumortier et al. [16]. Although its proof is very similar to the one of Theorem 2, we include it here, also because of its format. Theorem 6 (‘Simple’ in terms of Gm [74,82]) Let X be a C 1 vector field, defined in the neighborhood of 0 2 Rn , with X(0) D 0 and D0 X D A. Also let N 2 N be given and, for m 2 N, let Bm and Gm be such that B m ˚ G m D H m (Rn ). Then there exists, near 0 2 Rn , an analytic change of coordinates ˚ : Rn ! Rn , with ˚ (0) D 0, such that

D0:  ‘Simple’ in Terms of an Adjoint Action We now return to the setting of Normal Form Theory at equilibrium points. So given is the vector field x˙ D X(x), with X(x) D Ax C f (x); x 2 Rn , where A is linear and where f (0) D 0, D0 f D 0. We recall that it is our general aim to ‘simplify’ the Taylor series of X at 0. However, we have not yet said what the word ‘simple’ means in the present setting. In order to understand what is going on, we reintroduce the adjoint action associated to the linear part A, defined on the class of all C 1 vector fields on Rn . To be precise, this adjoint action adA is defined by the Lie-bracket

˚? X(y) D Ay C g2 (y) C    C g N (y) C O(jyj NC1 ) ; with g m 2 G m , for all m D 2; 3; : : : ; N. Proof We use induction on N. Let us assume that X(x) D Ax C g2 (x) C    C g N1 (x) C f N (x) C O(jxj NC1 ) ; with g m 2 G m , for all m D 2; 3; : : : ; N  1 and with f N homogeneous of degree N. We consider a coordinate change x D y C P(y), where P is polynomial of degree N, see above. For any such P, by substitution we get (Id C D y P) y˙ D A(y C P(y)) C g2 (y) C    C g N1 (y) C f N (y) C O(jyj NC1 ) ;

adA: Y 7! [A; Y] ; or where A is identified with the linear vector field x˙ D Ax. It is easily seen that this fits with the notation introduced in Theorem 2. Let H m (Rn ) denote the space of polynomial vector fields, homogeneous of degree m. Then the Taylor series of X can be viewed as an element of the product Q1 m n mD1 H (R ). Also, it directly follows that adA induces a linear map H m (Rn ) ! H m (Rn ), to be denoted by adm A. Let B m :D im adm A ; the image of the map adm A in H m (Rn ). Then for any complement Gm of Bm in H m (Rn ), in the sense that B m ˚ G m D H m (Rn ) ; we define the corresponding notion of ‘simpleness’ by requiring the homogeneous part of degree m to be in Gm . In the case of the Poincaré Theorem 2, since B m D H m (Rn ), we have G m D f0g.

y˙ D (Id C D y P)1 (A(y C P(y)) C g2 (y) C    C g N1 (y) C f N (y) C O(jyj NC1 )) D Ay C g2 (y) C    C g N1 (y) C f N (y) C AP(y)  D y PAy C O(jyj NC1 ) ; using that (Id C D y P)1 D Id  D y P C O(jyj N ). We conclude that the terms up to order N  1 are unchanged by this transformation, while the Nth order term becomes f N (y)  adN A(P)(y) : Clearly, a suitable choice of P will put this term in GN . This is the present version of the homological equation as introduced before.  Remark For simplicity the formulations all are in the C 1 -context, but obvious changes can be made for the case of finite differentiability. The latter case is of importance for applications of Normal Form Theory after reduction to a center manifold [85,87].

Normal Forms in Perturbation Theory

Torus Symmetry As a special case let the linear part A be semisimple, in the sense that it is diagonizable over the complex numbers. It then directly follows that also ad m A is semisimple, which implies that

So, the eigenvalues of A are ˙i and the transformations exp tA s ; t 2 R, form the rotation group SO(2; R). From this, we can arrive at once at the general format of the normal form. In fact, the normalized, Nth order part of ˚? X is rotationally symmetric. This implies that, if we pass to polar coordinates (r; '), by

im ad m A ˚ ker adm A D H m (Rn ) : The reader is invited to provide the eigenvalues of ad m A in terms of those of A, compare with the proof of Theorem 2. In the present case the obvious choice for the complementary space defining ‘simpleness’ is G m D ker adm A. Moreover, the fact that the normalized, viz. simplified, terms g m are in Gm by definition means that

This, in turn, implies that N-jet of ˚? X, i. e., the normalized part of ˚? X, by Proposition 5 is invariant under all linear transformations t2R

generated by A. For further reading also compare with Sternberg [82], Takens [84], Broer [7,8] or [12]. More generally, let A D A s C A n be the Jordan canonical splitting in the semisimple and nilpotent part. Then one directly shows that adm A D adm A s C adm A n is the Jordan canonical splitting, whence, by a general argument [89] it follows that im adm A C ker adm A s D H m (Rn ) ; so where the sum splitting in general no longer is direct. Now we can choose the complementary spaces Gm such that G m ker ad m A s ; ensuring equivariance of the normalized part of ˚? X, with respect to all linear transformations exp tA s ;

G

the normalized truncation of ˚? X obtains the form '˙ D f (r2 ) r˙ D rg(r2 ) ;

Remark  A more direct, ‘computational’ proof of this result can be given as follows, compare with the proof of Theorem 2 and with [16,84,87]. Indeed, in vector field notation we can write @ @ C x1 A D x2 @x1 @x2   @ @ D i z  z¯ ; @z @¯z where we complexified putting z D x1 C ix2 and use the well-known Wirtinger derivatives 1 @ D @z 2

m

n

 ker ad m A s n im adm A n  H (R ) ;

compare with Van der Meer [88]. For further discussion on the choice of Gm , see below. Example (Rotational symmetry [84]) Consider the case n D 2 where   0 1 A D As D : 1 0



@ @ i @x1 @x2



@ 1 ; D @¯z 2



@ @ Ci @x1 @x2

 :

Now a basis of eigenvectors for adm A can be found directly, just computing a few Lie-brackets, compare the previous subsection. In fact, it is now given by all monomials z k z¯ l

t 2R:

The choice of Gm can be further restricted, e. g., such that m

y2 D r sin ' ;

for certain polynomials f and g, with f (0) D 1 and g(0) D 0.

[A; g m ] D 0 :

exp tA;

y1 D r cos ';

@ ; @z

and

z k z¯ l

@ ; @¯z

with k C ` D m. The corresponding eigenvalues are i(k  `  1) viz. i(k  ` C 1). So again we see, now by a direct inspection, that adm A is semisimple and that we can take G m D ker adm A. This space is spanned by   @ @ ; (z z¯)r z ˙ z¯ @z @¯z with 2r C 1 D m  1, which indeed proves that the normal form is rotationally symmetric.

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 A completely similar case occurs for n D 3, where 0 1 0 1 0 A D A s D @1 0 0A : 0 0 0 In this case the normalized part in cylindrical coordinates (r; '; z), given by y1 D r cos ';

y2 D r sin ';

y3 D z ;

Since the truncated system of r˙ j - and z˙` -equations is independent of the angles ' j , this can be studied separately. A similar remark holds for the earlier examples of this section. As indicated in the introduction, this kind of ‘reduction by symmetry’ to lower dimension can be of great importance when studying the dynamics of X : as it enables us to consider X, viz. ˚? X, as an N-flat perturbation of the normalized part, which is largely determined by this lower dimensional reduction. For more details see below.

in general gets the axially symmetric form On the Choices of the Complementary Space and of the Normalizing Transformation

'˙ D f (r2 ; z) r˙ D rg(r2 ; z) z˙ D h(r2 ; z) ; for suitable polynomials f ; g and h. As a generalization of this example we state the following proposition, where the normalized part exhibits an m-torus symmetry. Proposition 7 (Toroidal symmetry [84]) Let X be a C 1 vector field, defined in the neighborhood of 0 2 Rn , with X(0) D 0 and where A D D0 X is semisimple with the eigenvalues ˙i!1 ; ˙i!2 ; : : : ; ˙i! m and 0. Here 2m  n. Suppose that for given N 2 N and all integer vectors (k1 ; k2 ; : : : ; k m ), 1

m X jD1

jk j j  N C 1 )

m X

kj!j ¤ 0 ;

(1)

In the previous subsection we only provided a general (symmetric) format of the normalized part. In concrete examples one has to do more. Indeed, given the original Taylor series, one has to compute the coefficients in the normalized expansion. This means that many choices have to be made explicit. To begin with there is the choice of the spaces Gm , which define the notion of ‘simple’. We have seen already that this choice is not unique. Moreover observe that, even if the choice of Gm has been fixed, still P usually is not uniquely determined. In the semisimple case, for example, P is only determined modulo the kernel G m D ker adm A. Remark To fix thoughts, consider the former of the above examples, on R2 , where the normalized truncation has the rotationally symmetric form

jD1

(i. e., there are no resonances up to order N C 1). Then there exists, near 0 2 Rn , an analytic change of coordinates ˚ : Rn ! Rn , with ˚(0) D 0, such that ˚? X, up to terms of order N has the following form. In suitable generalized cylindrical coordinates ('1 ; : : : ; 'm ; r12 ; : : : ; r2m ; z n2mC1 ; : : : ; z n ) it is given by '˙ j D f j (r12 ; : : : ; r2m ; z n2mC1 ;    ; z n ) r˙ j D r j g j (r12 ; : : : ; r2m ; z n2mC1 ;    ; z n ) z˙` D h` (r12 ; : : : ; r2m ; z n2mC1 ;    ; z n ) ; where f j (0) D ! j and h` (0) D 0 for 1  j  m; n  2m C 1  `  n. Proofs can be found, e. g., in [16,84,85,87]. In fact, if one introduces a suitable complexification, it runs along the same lines as the above remark. For the fact that finitely many non-resonance conditions are needed in order to normalize up to finite order, also compare a remark following Theorem 2.

'˙ D f (r2 ) r˙ D rg(r2 ) : Here g(r2 ) D cr2 C O(jrj4 ). The coefficient c dynamically is important, just think of the case where the system is part of a family that goes through Hopf-bifurcation. The computation of (the sign of) c in a concrete model can be quite involved, as it appears from, e. g., Marsden and McCracken [56]. Machine-assisted methods largely have taken over this kind of work. One general way to choose Gm is the following, compare with e. g., Sect. 7.6 in [16] and Sect. 2.3 in [87], : G m :D ker(adm AT ) : Here AT is the transpose of A, defined by the relation hAT x; yi D hx; Ayi, where h:; :i is an inner product on Rn . A suitable choice for an inner product on H m (Rn ) then directly gives that G m ˚ im(ad m A) D H m (Rn ) ;

Normal Forms in Perturbation Theory

as required. Also here the normal form can be interpreted in terms of symmetry, namely with respect to the group generated by AT . In the semisimple case, this choice leads to exactly the same symmetry considerations as before. The above algorithms do not provide methods for computing the normal form yet, i. e., for actually solving the homological equation. In practice, this is an additional computation. Regarding the corresponding algorithms we give a few more references for further reading, also referring to their bibliographies. General work in this direction is Bruno [35,36], Sect. 7.6 in Dumortier et al. [16], Takens [84] or Vanderbauwhede [87], Part 2. In the latter reference also a brief description is given of the sl(2; R)-theory of Sanders and Cushman [39], which is a powerful tool in the case where the matrix A is nilpotent [84]. For a thorough discussion on some of these methods we refer to Murdock [60]. Later on we shall come back to these aspects.

X t :D (Pt )? X : [X t ; P] D adm A(P) C O(jyj mC1 ) and

compare with Sect. “Preliminaries from Differential Geometry”. In fact, exactly this choice h D P1 was taken by Roussarie [74], also see the proof of Takens [84]. Notice that ˚ D h N ı h N1 ı    ı h3 ı h2 . Moreover notice that if the vector field P is in a given Lie-algebra of vector fields, its time 1 map P1 is in the corresponding Lie-group. In particular, if P is Hamiltonian, Pt is canonical or symplectic, and so on. For the validity of this set-up for a more general Liesubalgebra of the Lie-algebra of all C 1 vector fields, one has to study how far the grading 1 Y

Preservation of Structure

@X t D [X t ; P]; @t

H m (Rn )

mD1

It goes without saying that Normal Form Theory is of great interest in special cases where a given structure has to preserved. Here one may think of a symplectic or a volume form that has to be respected. Also a given symmetry group can have this rôle, e. g., think of an involution related to reversibility. Another, similar, problem is the dependence of external parameters in the system. A natural language for preservation of such structures is that of Lie-subalgebra’s of general Lie-algebra of vector fields, and the corresponding Lie-subgroup of the general Lie-group of diffeomorphisms.

of the formal power series, as well as the splittings

The Lie-Algebra Proof

The Volume Preserving and Symplectic Case On Rn , resp. R2n , we consider a volume form or a symplectic form, both denoted by  . We assume, that

Fortunately the setting of Theorem 6 is almost completely in terms of Lie-brackets. Let us briefly reconsider its proof. Given is a C 1 vector field X(x) D Ax C f (x); x 2 Rn , where A is linear and where f (0) D 0; D0 f D 0. We recall that in the inductive procedure a transformation h D Id C P ; with P a homogeneous polynomial of degree m D 2; 3; : : :, is found, putting the homogeneous mth degree part of the Taylor series of X into the ‘good’ space Gm . Now, nothing changes in this proof if instead of h D Id C P, we take h D P1 , the flow over time 1 of the vector field P : indeed, the effect of this change is not felt until the order 2m  1. Here we use the following formula for

B m ˚ G m D H m (Rn ) ; are compatible with the Lie-algebra at hand. This issue was addressed by Broer [7,8] axiomatically, in terms of graded and filtered Lie-algebra’s. Moreover, the methods concerning the choice of Gm briefly mentioned at the end of the previous section, all carry over to a more general Liealgebra set-up. As a consequence there exists a version of Theorem 6 in this setting. Instead of pursuing this further, we discuss its implications in a few relevant settings.

 D dx1 ^    ^ dx n ;

resp.

D

n X

dx j ^ dx jCn :

jD1

In both cases, let X denote the Lie-algebra of  -preserving vector fields, i. e., vector fields X such that L X  D 0. Here L again denotes the Lie-derivative, see Sect. “The Normal Form Procedure”. Indeed, one defines ˇ d ˇˇ L X  (x) D (X t )  (x) dt ˇ tD0 1 D lim f(X t )  (x)   (x)g : t!0 t

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Properties, similar to Proposition 5, hold here. Since in both cases  is a closed form, one shows by ‘the magic formula’ [81] that L X (x) D d( X ) C  X d

D d( X ) : Here  X  denotes the flux-operator defined by  X (Y) D  (X; Y). In the volume-preserving case the latter expression denotes div(X) and we see that preservation of  exactly means that div(X) D 0: the divergence of X vanishes. In the Hamiltonian case we conclude that the 1-form  X  is closed and hence (locally) of the form dH, for a Hamilton function H. In both cases, the fact that for a transformation h the fact that h  D  implies that with X also h X is -preserving. Moreover, for a -preserving vector field P and h D P1 one can show that indeed h  D . One other observation is, that by the homogeneity of the above expressions for , the homogeneous parts of the Taylor series of -preserving vector fields are again -preserving. This exactly means that H m (R(2)n ) \ X ; m D 1; 2; : : :, grades the formal power series corresponding to X . Here, notice that H 1 (R(2)n ) \ X D sl(n; R);

resp.

sp(2n; R) ;

the special- resp. the symplectic linear algebra. In summary we conclude that both the symplectic and the volume preserving setting are covered by the axiomatic approach of [7,8] and that an appropriate version of Theorem 6 holds here. Below we shall illustrate this with a few examples. External Parameters A C 1 family X D X  (x) of vector fields on Rn , with a multi-parameter  2 R p , can be regarded as one C 1 vector field on the product space Rn  R p . Such a vector field is vertical, in the sense that it has no components in the -direction. In other words, if  : Rn  R p ! R p is the natural projection on the second component, X is tangent to the fibers of . It is easily seen that this property defines a Lie-subalgebra of the Liealgebra of all C 1 vector fields on Rn  R p . Again, by the linearity of this projection, the gradings and splittings are compatible. The normal form transformations ˚ preserve the parameter , i. e., ˚ ı  D ˚ . When studying a bifurcation problem, we often consider systems X D X  (x) locally defined near (x; ) D (0; 0), considering series expansions both in x and in . Then, in the Nth order normalization ˚? X, the normalized part consists of a polynomial in y and , while the remainder term is of the form O(jyj NC1 C jj NC1).

As in the previous case, we shall not formulate the present analogue of Theorem 6 for this case, but illustrate its meaning in examples. The Reversible Case In the reversible case a linear involution R is given, while for the vector fields we require R? X D X. Let XR denote the class of all such reversible vector fields. Also, let C denote the class of all X such that R? X D X. Then, both XR and C are linear spaces of vector fields. Moreover, C is a Lie-subalgebra. Associated to C is the group of diffeomorphisms that commute with R, i. e., the R-equivariant transformations. Also it is easy to see that for each of these diffeomorphisms ˚ one has ˚? (XR ) XR . The above approach applies to this situation in a straightforward manner. The gradings and splittings fit, while we have to choose the infinitesimal generator P from the set C . For details compare with [29]. Remark In the case with parameters, it sometimes is possible to obtain an alternative normal form where the normalized part is polynomial in y alone, with coefficients that depend smoothly on . A necessary condition for this is that the origin y D 0 is an equilibrium for all values of  in some neighborhood of 0. To be precise, at the Nth order we van achieve smooth dependence of the coefficients on  for  2  N , where N is a neighborhood of  D 0, that may shrink to f0g as N ! 1. So, for N ! 1 only the formal aspect remains, as is the case in the above approach. This alternative normal form can be obtained by a proper use of the Implicit Function Theorem in the spaces H m (Rn ); for details e. g., see Section 2.2 in Vanderbauwhede [87]. For another discussion on this topic, cf. Section 7.6.2 in Dumortier et al. [16]. In Sect. “The Normal Form Procedure” the role of symmetry was considered regarding the semisimple part of the matrix A. A question is how this discussion generalizes to Lie-subalgebra’s of vector fields. Example (Volume preserving, parameter dependent axial symmetry [7,8]) On R3 consider a 1-parameter family X  of vector fields, preserving the standard volume  D dx1 ^ dx2 ^ dx3 . Assume that X 0 (0) D 0 while the spectrum of D0 X 0 consists of the eigenvalues ˙i and 0. For the moment regarding  as an extra state space coordinate, we obtain a vertical vector field on R4 and we apply a combination of the above considerations. The ‘generic’ Jordan normal form then is 0 1 0 1 0 0 B1 0 0 0C C ADB (2) @0 0 0 1A ; 0 0 0 0

Normal Forms in Perturbation Theory

with an obvious splitting in semisimple and nilpotent part. The considerations of Sect. “The Normal Form Procedure” then directly apply to this situation. For any N this yields a transformation ˚ : R4 ! R4 , with ˚(0) D 0, preserving both the projection to the 1-dimensional parameter space and the volume of the 3-dimensional phase space, such that the normalized, Nth order part of ˚? X(y; ), in cylindrical coordinates y1 D r cos '; y2 D r sin '; y3 D z, has the rotationally symmetric form '˙ D f (r2 ; z; ) r˙ D rg(r2 ; z; ) z˙ D h(r2 ; z; ) ; again, for suitable polynomials f ; g and h. Note, that in cylindrical coordinates the volume has the form  D rdr^ d' ^ dz. Again the functions f , g and h have to fit with the linear part. In particular we find that h(r2 ; z; ) D CazC    , observing that for  ¤ 0 the origin is no equilibrium point. Remark If A D A s C A n is the canonical splitting of A in H 1 (Rn ) D gl(n; R), then automatically both As and An are in the subalgebra under consideration. In the volume preserving setting this can be seen directly. In general the same holds true as soon as the corresponding linear Liegroup is algebraic, see [7,8] and the references given there. The Hamiltonian Case The Normal Form Theory in the Hamiltonian case goes back at least to Poincaré [69] and Birkhoff [5]. Other references are, for instance, Gustavson [47], Arnold [1,2,3], Sanders, Verhulst and Murdock [75], Van der Meer [88], Broer, Chow and Kim [17]. In Sect. “Preservation of Structure” we already saw that the axiomatic Lie algebra approach of [7,8] applies here, especially since the symplectic group SP(2n; R) is algebraic. The canonical form here usually goes with the name Williamson, compare Galin [44], Koçak [54] and Hoveijn [49]. We discuss how the Lie algebra approach compares to the literature. The Lie algebra of Hamiltonian vector fields can be associated to the Poisson-algebra of Hamilton functions as follows, even in an arbitrary symplectic setting. As before, the symplectic form is denoted by . We recall, that for any Hamiltonian H the corresponding Hamiltonian vector field X H is given by dH D (X H ; :), Now, let H and K be Hamilton functions with corresponding vector fields X H resp. X K . Then X fH;Kg D [X H ; X K ] ;

implying that the map H 7! X H is a morphism of Lie algebra’s. By definition this map is surjective, while its kernel consists of the (locally) constant functions. This implies, that the normal form procedure can be completely rephrased in terms of the Poisson-bracket. We shall now demonstrate this by an example, similar to the previous one. Example (Symplectic parameter dependent rotational symmetry [17]) Consider R4 with coordinates (x1 ; y1 ; x2 ; y2 ) and the standard symplectic form  D dx1 ^ dy1 C dx2 ^ dy2 , considering a C 1 family of Hamiltonian functions H  , where  2 R is a parameter. The fact that dH D  (X H ; :) in coordinates means x˙ j D

@ @y j ; H

y˙ j D 

@ @x j ; H

for j D 1; 2. We assume that for  D 0 the origin of R4 is a singularity. Then we expand as a Taylor series in (x; y; ) H  (x; y) D H2 (x; y; ) C H3 (x; y; ) C : : : ; where the H m is homogeneous of degree m in (x; y; ). It follows for the corresponding Hamiltonian vector fields that X H m 2 H m1 (R4 ) ; in particular X H 2 2 sp(4; R) H 1 (R4 ). Let us assume that this linear part X H 2 at (x; y; ) D (0; 0; 0) has eigenvalues ˙i and a double eigenvalue 0. One ‘generic’ Willamson’s normal form then is 0 1 0 1 0 0 B1 0 0 0C C ADB (3) @ 0 0 0 1A ; 0 0 0 0 compare with (2), corresponding to the quadratic Hamilton function H2 (x; y; ) D I C

1 2 y2 ; 2

where I :D 12 (x1 2 C y1 2 ). It is straightforward to give the generic matrix for the linear part in the extended state space R5 , see above, so we will leave this to the reader. The semisimple part A s D X I now can be used to obtain a rotationally symmetric normal form, as before. In fact, for any N 2 N there exists a canonical transformation ˚(x; y; ), which keeps the parameter fixed, and a polynomial F(I; x2 ; y2 ; ), such that (H ı ˚ 1 )(x; y; ) D F(I; x2 ; y2 ; ) C O(I C x22 C y22 C 2 )(NC1)/2 :

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Instead of using the adjoint action adA on the spaces H m1 (R5 ), we also may use the adjoint action adH2 : f 7! fH2 ; f g ; where f is a polynomial function of degree m in (x1 ; y1 ; x2 ; y2 ; ). In the vector field language, we choose the ‘good’ space G m1 ker adm1 A s , which, in the function language, translates to a ‘good’ subset of ker adI. Whatsoever, the normalized part of the Hamilton function H ı ˚ 1 , viz. the vector field ˚? X H D X Hı˚ 1 , is rotationally symmetric. The fact that the Hamilton function F Poisson-commutes with I exactly amounts to invariance under the action generated by the vector field X I , in turn implying that I is an integral of X F . Indeed, if we define a 2-periodic variable ' as follows: p p x1 D 2I sin '; y1 D 2I cos ' ; then  D dI^d'Cdx2 ^dy2 , implying that the normalized vector field X F has the canonical form I˙ D 0; x˙2 D

'˙ D 

@F ; @y2

@F ; @I

y˙2 D 

@F : @x2

Notice that ' is a cyclic variable, making the fact that I is an integral clearly visible. Also observe that '˙ D 1 C    . As before, and as in, e. g., the central force problem, this enables a reduction to lower dimension. Here, the latter two equations constitute the reduction to 1 degree of freedom: it is a family of planar Hamiltonian vector fields, parametrized by I and . Remark  This example has many variations. First of all it can be simplified by omitting parameters and even the zero eigenvalues. The conclusion then is that a planar Hamilton function with a nondegenerate minimum or maximum has a formal rotational symmetry, up to canonical coordinate changes.  It also can be easily made more complicated, compare with Proposition 7 in Sect. “The Normal Form Procedure”. Given an equilibrium with purely imaginary eigenvalues ˙i!1 ; ˙i!2 ; : : : ; ˙i! m and with 2(n  m) zero eigenvalues. Provided that there are no resonances up to order N C 1, see (1), also here we conclude that Hamiltonian truncation at the order N, by canonical transformations can be given the form F(I1 ; : : : ; I m ; x mC1 ; y mC1 : : : ; x n ; y n ) ;

where I j :D 12 (x j 2 C y j 2 ). As in Proposition 7 this normal form has aptoroidal symmetry. Writing x j D p 2I j sin ' j ; y j D 2I j cos ' j , we obtain the canonical system of equations I˙j D 0; x˙` D

'˙ j D 

@F ; @y l

@F ; @I j

y˙` D 

@F ; @x l

1  j  m; m C 1  l  n. Note that '˙ j D ! j C    . In the case m D n we deal with an elliptic equilibrium. The corresponding result usually is named the Birkhoff normal form [1,5,47]. Then, the variables (I j ; ' j ); 1  j  m, are a set of action-angle variables for the truncated part of order N, [1].  Returning to algorithmic issues, in addition to Subsect. “On the Choices of the Complementary Space and of the Normalizing Transformation”, we give a few further references in cases where a structure is being preserved: Broer [7], Deprit [40,41], Meyer [57] and Ferrer, Hanßmann, Palacián and Yanguas [43]. It is to be noted that this kind of Hamiltonian result also can be obtained, where the coordinate changes are constructed using generating functions, compare with, e. g., [1,3,77], see also the discussion in Broer, Hoveijn, Lunter and Vegter [20] and in particular Section 4.4.2 in [23]. For another, computationally very effective normal form algorithm, see Giorgilli and Galgani [45]. Also compare with references therein.

Semi-local Normalization This section roughly consists of two parts. To begin with, a number of subsections are devoted to related formal normal form results, near fixed points of diffeomorphisms and near periodic solutions and invariant tori of vector fields. A Diffeomorphism Near a Fixed Point We start by formulating a result by Takens: Theorem 8 (Takens normal form [83]) Let T : Rn ! Rn be a C 1 diffeomorphism with T(0) D 0 and with a canonical decomposition of the derivative D0 T D S C N in semisimple resp. nilpotent part. Also, let N 2 N be given, then there exists diffeomorphism ˚ and a vector field X, both of class C 1 , such that S? X D X and ˚ 1 ı T ı ˚ D S ı X 1 C O(jyj NC1 ) :

Normal Forms in Perturbation Theory

Here, as before, X 1 denotes the flow over time 1 of the vector field X. Observe that the vector field X necessarily has the origin as an equilibrium point. Moreover, since S? X D X, the vector field X is invariant with respect to the group generated by S. The proof is a bit more involved than Theorem 6 in Sect. “The Normal Form Procedure”, but it has the same spirit, also compare with [37]. In fact, the Taylor series of T is modified step-by-step, using coordinate changes generated by homogeneous vector fields of the proper degree. After a reduction to center manifolds the spectrum of S is on the complex unit circle and Theorem 8 especially is of interest in cases where this spectrum consists of roots of unity, i. e., in the case of resonance. Compare with [32,37,83]. Again, the result is completely phrased in terms of Lie algebra’s and groups and therefore bears generalization to many contexts with a preserved structure, compare Sect. “Preservation of Structure”. The normalizing transformations of the induction process then are generated from the corresponding Lie algebra. For a symplectic analogue see Moser [59]. Also, both in Broer, Chow and Kim [17] and in Broer and Vegter [14], symplectic cases with parameters are discussed, where S D Id, the Identity Map, resp. S D Id, the latter involving a period-doubling bifurcation. Remark Let us consider a symplectic map T of the plane, which means that T preserves both area and orientation. Assume that T is fixing the origin, while the eigenvalues of S D D0 T are on the unit circle, without being roots of unity. Then S generates the rotation group SO(2; R), so the vector field X, which has divergence zero, in this case is rotationally symmetric. Again this result often goes with the name of Birkhoff. Near a Periodic Solution The Normal Form Theory at a periodic solution or closed orbit has a lot of resemblance to the local theory we met before. To fix thoughts, let us consider a C 1 vector field of the form x˙ D f (x; y) y˙ D g(x; y) ;

(4)

with (x; y) 2 T 1  Rn . Here T 1 D R/(2Z). Assuming y D 0 to be a closed orbit, we consider the formal Taylor series with respect to y, with x-periodic coefficients. By Floquet Theory, we can assume that the coordinates (x; y)

are such that f (x; y) D ! C O(jyj);

g(x; y) D ˝ y C O(jyj2 ) ;

where ! 2 R is the frequency of the closed orbit and ˝ 2 gl(n; R) its Floquet matrix. Again, the idea is to ‘simplify’ this series further. To this purpose we introduce a grading as before, letting H m D H m (T 1  Rn ) be the space of vector fields n

Y(x; y) D L(x; y)

X @ @ M j (x; y) ; C @x @y j jD1

with L(x; y) and M(x; y) homogeneous in y of degree m  1 resp. m. Notice, that this space H m is infinite-dimensional. However, this is not at all problematic for the things we are doing here. By this definition, we have that A :D !

@ @ C ˝y @x @y

is a member of H 1 and with this normally linear part we can define an adjoint representation adA as before, together with linear maps adm A : H m ! H m : Again we assume to have a decomposition G m ˚ Im(adm A) D H m ; where the aim is to transform the terms of the series successively into the Gm , for m D 2; 3; 4; : : :. The story now runs as before. In fact, the proof of Theorem 6 in Sect. “The Normal Form Procedure”, as well as its Lie algebra versions indicated in Sect. “Preservation of Structure”, can be repeated almost verbatim for this case. Moreover, if ˝ D ˝s C ˝n is the canonical splitting in semisimple and nilpotent part, then !

@ @ C ˝s y y @x @

gives the semisimple part of adm A, as can be checked by a direct computation. From this computation one also deduces the non-resonance conditions needed for the present torus-symmetric analogue of Proposition 7 in Sect. “The Normal Form Procedure”. There are different cases with resonance either between the imaginary parts of the eigenvalues of ˝ (normal resonance) or between the latter and the frequency ! (normal-internal resonance). All of this extends to the various settings with preservation of structure as discussed before.

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In all cases direct analogues of the Theorems 6 and 8 are valid. General references in this direction are Arnold [2,3], Bruno [35,36], Chow, Li and Wang [37], Iooss [50], Murdock [60] or Sanders, Verhulst and Murdock [75]. Remark  This approach also is important for non-autonomous systems with periodic time dependence. Here the normalization procedure includes averaging. As a special case of the above form, we obtain a system x˙ D ! ;

y˙ D g(x; y) ;

so where x 2 T 1 is proportional to the time. Apart from the general references given above, we also refer to, e. g., Broer and Vegter [14] and to Broer, Roussarie and Simó [9,19]. The latter two applications also contain parameters and deal with bifurcations. Also compare with Verhulst  Perturbation Analysis of Parametric Resonance.  A geometric tool that can be successfully applied in various resonant cases is a covering space, obtained by a Van der Pol transformation (or by passing to corotating coordinates). This involves equivariance with respect to the corresponding deck group. This setting (with or without preservation of structure) is completely covered by the general Lie algebra approach as described above. For the Poincaré map this deck group symmetry directly yields the normal form symmetry of Theorem 8. For applications in various settings, with or without preservation of structure, see [14,24,32]. This normalization technique is effective for studying bifurcation of subharmonic solutions. In the case of period doubling the covering space is just a double cover. In many cases Singularity Theory turns out to be useful. Near a Quasi-periodic Torus The approach of the previous subsection also applies at an invariant torus, provided that certain requirements are met. Here we refer to Braaksma, Broer and Huitema [15], Broer and Takens [12] and Bruno [35,36]. Let us consider a C 1 -system x˙ D f (x; y) ;

y˙ D g(x; y)

(5)

as before, with (x; y) 2 T m  Rn . Here T m D Rm / (2Z)m . We assume that f (x; y) D ! C O(jyj), which implies that y D 0 is an invariant m-torus, with on it a constant vector field with frequency-vector !. We also assume that g(x; y) D ˝ y C O(jyj2 ), which is the present analogue of the Floquet form as known in the periodic case,

with ˝ 2 gl(n; R), independent of x. Contrarious to the situations for m D 1 and n D 1, in general reducibility to Floquet form is not possible. Compare with [10,15,18,38] and references therein. For a similar approach of a system that is not reducible to Floquet form, compare with [22]. Presently we assume this reducibility, expanding in formal series with respect to the variables y, where the coefficients are functions on T m . These coefficients, in turn, can be expanded in Fourier series. The aim then is, to ‘simplify’ this combined series by successive coordinate changes, following the above procedure. As a second requirement it then is needed that certain Diophantine conditions are satisfied on the pair (!; ˝). Below we give more details on this. Instead of giving general results we again refer to [12,15,26,29,35,36]. Moreover, to fix thoughts, we present a simple example with a parameter. Here again a direct link with averaging holds. Example (Toroidal symmetry with small divisors [38]) Given is a family of vector fields x˙ D X(x; ) with (x; ) 2 T m  R. We assume that X D X(x; ) has the form X(x; ) D ! C f (x; ) ; with f (x; 0)  0. It is assumed that the frequency vector ! D (!1 ; !2 ; : : : ; !n ) has components that satisfy the Diophantine non-resonance condition jh!; kij  jkj ;

(6)

for all k 2 Z n f0g. Here > 0 and  > n  1 are prescribed constants. We note that  > n  1 implies that this condition in Rn D f!g excludes a set of Lebesgue measure O( ) as # 0, compare with [38]. It follows, that by successive transformations of the form h : (x; ) 7! (x C P(x; ); ) the x-dependence of X can be pushed away to ever higher order in , leading to a formal normal form ˙ D ! C g() ; with g(0) D 0. Observe that in this case ‘simple’ means x- (or -) independent. Therefore, in a proper formalism, x-independent systems constitute the spaces Gm . Indeed, in the induction process we get X(x; ) D ! C g2 () C    C g N1 () C f N (x; ) C O(jj NC1 ) ; compare the proof of Theorem 6 in Sect. “The Normal Form Procedure”. Writing  D x C P(x; ), with

Normal Forms in Perturbation Theory

˜ ) N , we substitute P(x; ) D P(x; ˙ D (Id C D P)x˙ D ! C g2 () C    C g N1 () C f N (x; ) C D x P(x; ) ! C O(jj NC1 ) ; Where we express the right-hand side in x. So we have to satisfy an equation D x P(x; )! C f N (x; )  c N modO(jj NC1 ) ; for a suitable constant c. Writing f N (x; ) D f˜N (x; ) N , this amounts to ˜ )! D  f˜N (x; ) C c ; D x P(x; which is the present form of the homological equation. If X a k ()e ihx;ki ; f˜N (x; ) D k2Z n

then c D a0 , i. e., the m-torus average Z 1 a0 () D f˜N (x:)dx : (2)m T m Moreover, ˜ ) D P(x;

X a k () e ihx;ki : ih!; ki k¤0

This procedure formally only makes sense if the frequencies (!1 ; !2 ; : : : ; !m ) have no resonances, which means that for k ¤ 0 also h!; ki ¤ 0. In other words this means that the components of the frequency vector ! are rationally independent. Even then, the denominator ih!; ki can become arbitrarily small, so casting doubt on the convergence. This problem of small divisors is resolved by the Diophantine conditions (6). For further reference, e. g., see [2,10,15,18,38]: for real analytic X, by the Paley– Wiener estimate on the exponential decay of the Fourier coefficients, the solution P again is real analytic. Also in the C 1 -case the situation is rather simple, since then the coefficients in both cases decay faster than any polynomial. Remark  The discussion at the end of Subsect. “Near a Periodic Solution” concerning normalization at a periodic solution, largely extends to the present case of a quasi-periodic torus. Assuming reducibility to Floquet form, we now have a normally linear part !

@ @ C ˝y ; @x @y

with a frequency vector ! 2 Rn . As before ˝ 2 gl(m; R). For the corresponding KAM Perturbation

Theory the Diophantine conditions (6) on the frequencies are extended by including the normal frequencies, i. e., the imaginary parts !1N ; : : : ; !sN of the eigenvalues of ˝. To be precise, for > 0 and  > n  1, above the conditions (6), these extra Melnikov conditions are given by jh!; ki C ! jN j  jkj jh!; ki C 2! jN j  jkj jh!; ki C

! jN

C

!`N j

 jkj



(7) ;

for all k 2 Z n nf0g and for j; ` D 1; 2; : : : ; s with ` ¤ j. See below for the description of an application to KAM Theory (and more references) in the Hamiltonian case.  In this setting normal resonances can occur between the ! jN and normal-internal resonances between the ! jN and !. Certain strong normal-internal resonances occur when one of the left-hand sides of (7) vanishes. For an example see below. Apart from these now also internal resonances between the components of ! come into play. The latter generally lead to destruction of the invariant torus.  As in the periodic case also here covering spaces turn out to be useful for studying various resonant bifurcation scenarios often involving applications of both Singularity Theory and KAM Theory. Compare with [15,25,31,33]. Non-formal Aspects Up to this moment (almost) all considerations have been formal, i. e., in terms of formal power series. In general, the Taylor series of a C 1 -function, say, Rn ! R will be divergent. On the other hand, any formal power series in n variables occurs as the Taylor series of some C 1 -function. This is the content of a theorem by É. Borel, cf. Narasimhan [61]. We briefly discuss a few aspects regarding convergence or divergence of the normalizing transformation or of the normalized series. We recall that the growth rate of the formal series, including the convergent case, is described by the Gevrey index, compare with, e. g., [4,55,72,73]. Normal Form Symmetry and Genericity For the moment we assume that all systems are of class C 1 . As we have seen, if the normalization procedure is carried out to some finite order N, the transformation ˚ is a real analytic map. If we take the limit for N ! 1, we ˆ but, by the Borel Theorem, only get formal power series ˚, 1 a ‘real’ C -map ˚ exists with ˚ˆ as its Taylor series.

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Normal Forms in Perturbation Theory

Let us discuss the consequences of this, say, in the case of Proposition 7 in Sect. “The Normal Form Procedure”. Assuming that there are no resonances at all between the ! j , as a corollary, we find a C 1 -map y D ˚(x) and a C 1 vector field p D p(y), such that:  The vector field ˚? X  p, in corresponding generalized cylindrical coordinates has the symmetric C 1 -form '˙ j D f j (r12 ; : : : ; r2m ; z n2mC1 ; : : : ; z n ) r˙ j D r j g j (r12 ; : : : ; r2m ; z n2mC1 ; : : : ; z n ) z˙` D h` (r12 ; : : : ; r2m ; z n2mC1 ; : : : ; z n ) ; where f j (0) D ! j and h` (0) D 0 for 1  j  m; n  2m C 1  `  n.  The Taylor series of p identically vanishes at y D 0. Note, that an 1-ly flat term p can have component func2 tions like e1/y1 . We see, that the m-torus symmetry only holds up to such flat terms. Therefore, this symmetry, if present at all, also can be destroyed again by a generic flat ‘perturbation’. We refer to Broer and Takens [12], and references therein, for further consequences of this idea. The main point is, that by a Kupka–Smale argument, which generically forbids so much symmetry, compare with [67]. Remark The Borel Theorem also can be used in the reversible, the Hamiltonian and the volume preserving setting. In the latter two cases we exploit the fact that a structure preserving vector field is generated by a function, resp. an (n  2)-form. Similarly the structure preserving transformations have such a generator. On these generators we then apply the Borel Theorem. Many Lie algebra’s of vector fields have this ‘Borel Property’, saying that a formal power series of a transformation can be represented by a C 1 map in the same structure preserving setting.

whether there exists a local biholomorphic transformation ˚ : (C; 0) ! (C; 0) such that ˚ ıF D˚ : We say that ˚ linearizes F near its fixed point 0. A formal solution as in Sect. “The Normal Form Procedure” generally exists for all  2 C n f0g, not equal to a root of unity. The elliptic case concerns  on the complex unit circle, so of the form  D e2 i˛ , where ˛ … Q, and where the approximability of ˛ by rationals is of importance, cf. Siegel [76] and Bruno [34]. Siegel introduced a sufficient Diophantine condition related to (6), which by Bruno was replaced by a sufficient condition on the continued fraction approximation of ˛. Later Yoccoz [92] proved that the latter condition is both necessary and sufficient. For a description and further comments also compare with [2,10,42,58]. Remark  In certain real analytic cases Ne˘ıshtadt [63], by truncating the (divergent) normal form series appropriately, obtains a remainder that is exponentially small in the perturbation parameter. Also compare with, e. g., [2,3,4,78]. For an application in the context of the Bogdanov–Takens bifurcations for diffeomorphisms, see [9,19]. It follows that chaotic dynamics is confined to exponentially narrow horns in the parameter space.  The growth of the Taylor coefficients of the usually divergent series of the normal form and of the normalizing transformation can be described using Gevrey symptotics [4,55,70,71,72,73]. Apart from its theoretical interest, this kind of approach is extremely useful for computational issues also compare with [4,46,78, 79,80]. Applications

On Convergence The above topological ideas also can be pursued in many real analytic cases, where they imply a generic divergence of the normalizing transformation. For an example in the case of the Hamiltonian Birkhof normal form [5,47,77] compare with Broer and Tangerman [13] and its references. As an example we now deal with the linearization of a holomorphic germ in the spirit of Sect. “The Normal Form Procedure”. Example (Holomorphic linearization [34,58,92]) A holomorphic case concerns the linearization of a local holomorphic map F : (C; 0) ! (C; 0) of the form F(z) D z C f (z) with f (0) D f 0 (0) D 0. The question is

We present two main areas of application of the Normal Form Theory in Perturbation Theory. The former of these deals with more globally qualitative aspects of the dynamics given by normal form approximations. The latter class of applications concerns the Averaging Theorem, where the issue is that solutions remain close to approximating solutions given by a normal form truncation. ‘Cantorized’ Singularity Theory We return to the discussion of the motivation in Sect. “Motivation”, where there is a toroidal normal form symmetry up to a finite or infinite order. To begin with let us consider the quasi-periodic Hopf bifurcation [6,15], which is the analogue of the Hopf bifurcation for equilibria and

Normal Forms in Perturbation Theory

the Hopf–Ne˘ımark–Sacker bifurcation for periodic solutions, in the case of quasi-periodic tori. For a description comparing the differences between all these cases we refer to [38]. For a description of the resonant dynamics in the resonant gaps see [30] and references therein. Apart from this, a lot of related work has been done in the Hamiltonian and reversible context as well, compare with [25,26,29,48]. To be more definite, we consider families of systems defined on T m  Rm  R2p D fx; y; zg, endowed with the P 2 symplectic form  D m jD1 dx j ^dy j Cdz . We start with ‘integrable’, i. e., x-independent, families of systems of the form x˙ D f (y; z; ) ;

y˙ D g(y; z; ) ;

z˙ D h(y; z; ) ; (8)

compare with (5), to be considered near an invariant m-torus T m  fy0 g  fz0 g, meaning that we assume g(y0 ; z0 ) D 0 D h(y0 ; z0 ). The general interest is with the persistence of such a torus under nearly integrable perturbations of (8), where we include  as a multiparameter. This problem belongs to the Parametrized KAM Theory [10,15,18] of which we sketch some background now. For y near y0 and  near 0 consider ˝(; y) D Dz h(y; z0 ; ), noting that ˝(; y) 2 sp(2p; R). Also consider the corresponding normal linear part !(; y)

@ @ C ˝(; y)z : @x @z

As a first case assume that the matrix ˝(0; y0 ) has only simple non-zero eigenvalues. Then a full neighborhood of ˝(0; y0 ) in sp(2p; R) is completely parametrized by the eigenvalues of the matrices, where – in this symplectic case – we have to refrain from ‘counting double’. We roughly quote a KAM Theorem as this is known in the present circumstances [15]. As a nondegeneracy condition assume that the product map (; y) 7! (!(; y); ˝(; y)) is a submersion near (; y) D (0; y0 ). Also assume all Diophantine conditions (6), (7) to hold. Then the parametrized system (8) is quasi-periodically stable, which implies persistence of the corresponding Diophantine tori near T m  fy0 g  fz0 g under nearly integrable perturbations. Remark  A key concept in the KAM Theory is that of Whitney smoothness of foliations of tori over nowhere dense ‘Cantor’ sets. In real analytic cases even Gevrey regularity holds, and similarly when the original setting

is Gevrey; compare with [71,90]. For general reference see [10,15,18] and references therein.  The gaps in the ‘Cantor sets’ are centered around the various resonances. Their union forms an open and dense set of small measure, where perturbation series diverge due to small divisors. In each gap the considerations mentioned at the end of Subsects. “Near a Periodic Solution” and “Near a Quasi-Periodic Torus” apply. In [25,31,33,38] the differences between period and the quasi-periodic cases are highlighted. Example (Quasi-periodic Hamiltonian Hopf bifurcation [26]) As a second case we take p D 2, considering the case of normal 1 : 1 resonance where the eigenvalues of ˝(0; y0 ) are of the form ˙i0 , for a positive 0 . For simplicity we only consider the non-semisimple Williamson normal form 1 0 0 0 0 1 B0 0 0 1 C C; ˝(0; y0 ) B @0 0 0 0 A 0 0 0 0 where denotes symplectic similarity. The present format of the nondegeneracy condition regarding the product map (; y) 7! (!(; y); ˝(; y) is as before, but now it is required that the second component (; y) 7! ˝(; y) is a versal unfolding of the matrix ˝(0; y0 ) in sp(4; R) with respect to the adjoint SP(4; R)-action, compare with [2,26,27,29,44,49,54]. It turns out that a standard normalization along the lines of Sect. “Preservation of Structure” can be carried out in the z-direction, generically leading to a Hamiltonian Hopf bifurcation [88], characterized by its swallowtail geometry [86], which ‘governs’ families of invariant m-, (m C 1)- and (m C 2)-tori (the latter being Lagrangean). Here for the complementary spaces, see Sect. “The Normal Form Procedure”, the sl(2; R)-Theory is being used [39,60]. As before the question is what happens to this scenario when perturbing the system in a non-integrable way? In that case we need quasi-periodic Normal Form Theory, in the spirit of Sect. “Semi-Local Normalization”. Observe that by the 1 : 1 resonance, difficulties occur with the third of the three Melnikov conditions (7). In a good set of Floquet coordinates the resonance can be written in the form ! jN C !`N D 0. Nevertheless, another application of Parametrized KAM Theory [10,15,18,27] yields that the swallowtail geometry is largely preserved, when leaving out a dense union of resonance gaps of small measure. Here perturbation series diverge due to small divisors. What remains is a ‘Cantorized’ version of the swallowtail [26,27]. For a reversible analogue see [29,31].

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Remark  The example concerns a strong resonance and it fits in some of the larger gaps of the ‘Cantor’ set described in the former application of Parametrized KAM Theory. Apart from this, the previous remarks largely apply again. It turns out that in these and many other cases there is an infinite regression regarding the resonant bifurcation diagram.  The combination of KAM Theory with Normal Form Theory generally has been very fruitful. In the example of Sect. “Applications” it implies that each KAM torus corresponds to a parameter value 0 that is the Lebesgue density point of such quasi-periodic tori. In the real analytic case by application of [63], it can be shown that the relative measure with non-KAM tori is exponentially small in   0 . Similar results in the real analytic Hamiltonian KAM Theory (often going by the name of exponential condensation) have been obtained by [51,52,53], also compare with [35,36],  Diagrammatic Methods in Classical Perturbation Theory and with [3,10] and references therein. On the Averaging Theorem Another class of applications of normalizing-averaging is in the direction of the Averaging Theorem. There is a wealth of literature in this direction, that is not in the scope of the present paper, for further reading compare with [1,2,3,75] and references therein. Example (A simple averaging theorem [1]) Given is a vector field x˙ D !(y) C " f (x; y) ;

y˙ D "g(x; y)

with (x; y) 2 T 1  Rn , compare with (4). Roughly following the recipe of the normalization process, a suitable near-identity transformation (x; y) 7! (x; ) of T 1  Rn yields the following reduction, after truncating at order O("2 ): Z 2 1 ˙ D " g¯(); where g¯() D g(x; )dx : 2 0 We now compare y D y(t) and  D (t) with coinciding initial values y(0) D (0) as t increases. The Averaging Theorem asserts that if !() > 0 is bounded away from 0, it follows that, for a constant c > 0, one has jy(t)  (t)j < c";

for all t

1 with 0  t  : "

This theory extends to many classes of systems, for instance to Hamiltonian systems or, in various types of systems, in the immediate vicinity of a quasi-periodic torus. Further normalizing can produce sharper estimates that are polynomial in ", while in the analytic case this even extends over exponentially long time intervals, usually known under the name of Nekhorochev estimates [64,65,66], for a description and further references also see [10],  Nekhoroshev Theory. Another direction of generalization concerns passages through resonance, which in the example implies that the condition on !() is no longer valid. We here mention [62], for further references and descriptions referring to [2,3] and to [75].

Future Directions The area of research in Normal Form Theory develops in several directions, some of which are concerned with computational aspects, including the nilpotent case. Although for smaller scale projects much can be done with computer packages, for the large scale computations, e. g., needed in Celestial Mechanics, single purpose formula manipulators have to be built. For an overview of such algorithms, compare with [60,78,79,80] and references therein. Here also Gevrey asymptotics is of importance. Another direction of development is concerned with applications in Bifurcation Theory, often in particular Singularity Theory. In many of these applications, certain coefficients in the truncated normal form are of interest and their computation is of vital importance. For an example of this in the Hamiltonian setting see [11], where the Giorgili–Galgani [45] algorithm was used to obtain certain coefficients at all arbitrary order in an efficient way. For other examples in this direction see [28,32]. Related to this is the problem how to combine the normal form algorithms as related to the present paper, with the polynomial normal forms of Singularity Theory. The latter have a universal (i. e., context independent) geometry in the product of state space and parameter space. A problem of relevance for applications is to pull-back the Singularity Theory normal form back to the original system. For early attempts in this direction see [20,23], which, among other things, involve Gröbner basis techniques.

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Books and Reviews Braaksma BLJ, Stolovitch L (2007) Small divisors and large multipliers (Petits diviseurs et grands multiplicateurs). Ann l’institut Fourier 57(2):603–628 Broer HW, Levi M (1995) Geometrical aspects of stability theory for Hill’s equations. Arch Rat Mech An 131:225–240 Gaeta G (1999) Poincaré renormalized forms. Ann Inst Henri Poincaré 70(6):461–514 Martinet J, Ramis JP (1982) Problèmes des modules pour les èquations différentielles non linéaires du premier ordre. Publ IHES 5563–164 Martinet J Ramis JP (1983) Classification analytique des équations différentielles non linéaires résonnantes du premier ordre. Ann Sci École Norm Suprieure Sér 416(4):571–621 Vanderbauwhede A (2000) Subharmonic bifurcation at multiple resonances. In: Elaydi S, Allen F, Elkhader A, Mughrabi T, Saleh M (eds) Proceedings of the mathematics conference, Birzeit, August 1998. World Scientific, Singapore, pp 254–276

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Numerical Bifurcation Analysis HIL MEIJER1 , FABIO DERCOLE2 , BART OLDEMAN3 1 Department of Electrical Engineering, Mathematics and Computer Science, University of Twente, Enschede, The Netherlands 2 Department of Electronics and Information, Politecnico di Milano, Milano, Italy 3 Department of Computer Science and Software Engineering, Concordia University, Montreal, Canada

branches) of solutions for a defining system while one or more parameters are varied. Test function A function designed to have a regular zero at a bifurcation. During continuation a test function can be monitored to detect bifurcations. Branch switching Several branches of different codimension can emanate from a bifurcation point. Switching from the computation of one branch to an other requires appropriate procedures. Definition of the Subject

Article Outline Glossary Definition of the Subject Introduction Continuation and Discretization of Solutions Normal Forms and the Center Manifold Continuation and Detection of Bifurcations Branch Switching Connecting Orbits Software Environments Future Directions Bibliography Glossary Dynamical system A rule for time evolution on a state space. The term system will be used interchangeably. Here a system is a family given by an ordinary differential equation (ODE) depending on parameters. Equilibrium A constant solution of the system, for given parameter values. Limit cycle An isolated periodic solution of the system, for given parameter values. Bifurcation A qualitative change in the dynamics of a dynamical system produced by changing its parameters. Bifurcation points are the critical parameter combinations at which this happens for arbitrarily small parameter perturbations. Normal form A simplified model system for the analysis of a certain type of bifurcation. Codimension The minimal number of parameters needed to perturb a family of systems in a generic manner. Defining system A set of suitable equations so that the zero set corresponds to a bifurcation of a certain type or to a particular solution of the system. Also called defining function or equation. Continuation A numerical method suited for tracing one-dimensional manifolds, curves (here called

The theory of dynamical systems studies the behavior of solutions of systems, like nonlinear ordinary differential equations (ODEs), depending upon parameters. Using qualitative methods of bifurcation theory, the behavior of the system is characterized for various parameter combinations. In particular, the catalog of system behaviors showing qualitative differences can be identified, together with the regions in parameter space where the different behaviors occur. Bifurcations delimit such regions. Symbolic and analytical approaches are in general infeasible, but numerical bifurcation analysis is a powerful tool that aids in the understanding of a nonlinear system. When computing power became widely available, algorithms for this type of analysis matured and the first codes were developed. With the development of suitable algorithms, the advancement in the qualitative theory has found its way into several software projects evolving over time. The availability of software packages allows scientists to study and adjust their models and to draw conclusions about their dynamics. Introduction Nonlinear ordinary differential equations depending upon parameters are ubiquitous in science. In this article methods for numerical bifurcation analysis are reviewed, an approach to investigate the dynamic behavior of nonlinear dynamical systems given by x˙ D f (x; p) ; Rn

x 2 Rn ; p 2 Rn p ; Rn p

Rn

(1)

where f :  ! is generic and sufficiently smooth. In particular, x(t) represents the state of the system at time t and its components are called state (or phase) variables, x˙ (t) denotes the time derivative of x(t), while p denotes the parameters of the system, representing experimental control settings or variable inputs. In many instances, solutions of (1), starting at an initial condition x(0), appear to converge as t ! 1 to equilibria (steady states) or limit cycles (periodic orbits). Bounded solutions can also converge to more complex at-

Numerical Bifurcation Analysis

tractors, like tori (quasi-periodic orbits) or strange attractors (chaotic orbits). The attractors of the system are invariant under time evolution, i. e., under the application of the time-t map ˚ t , where ˚ denotes the flow induced by the system (1). Solutions attracting all nearby initial conditions, are said to be stable, and unstable if they repel some initial conditions. Generally speaking, it is hard to obtain closed formulas for ˚ t as the system is nonlinear. In some cases, one can compute equilibria analytically, but this is often not the case for limit cycles. However, numerical simulations of (1) easily give an idea of how solutions look, although one never computes the true orbit due to numerical errors. One can verify stability conditions by linearizing the flow around equilibria and cycles. In particular, an equilibrium x0 is stable if the eigenvalues of the linearization (Jacobi) matrix A D f x (x0 ; p) (where the subscript denotes differentiation) all have a negative real part. Similarly, for a limit cycle x0 (t) with period T, one defines the Floquet multipliers (or simply multipliers) as the eigenvalues of the monodromy matrix M D ˚xT (x0 (0)). The cycle is stable if the nontrivial multipliers, there is always one equal to 1, are all within the unit circle. Equilibria and limit cycles are called hyperbolic if the eigenvalues and nontrivial multipliers do not have a zero real part or modulus one, respectively. For any given parameter combination p, the state space representation of all orbits constitutes the phase portrait of the system. In practice, one draws a set of strategic orbits (or finite segments of them), from which all other orbits can be intuitively inferred, as illustrated in Fig. 1. Points X 0 ; X1 ; X2 are equilibria, of which X 0 and X 1 are unstable and X 2 is stable. In particular, X 0 is a repel-

Numerical Bifurcation Analysis, Figure 1 Phase portrait of a two-dimensional system with two attractors (the equilibrium X 2 and the limit cycle  ), a repellor (X 0 ), and a saddle (X 1 )

lor, i. e., nearby orbits do not tend to remain close to X 0 , while X 1 is a saddle, i. e., almost all nearby orbits go away from X 1 except two, which tend to X 1 and lie on the socalled stable manifold; the two orbits emanating from X 1 compose the unstable manifold. There are therefore two attractors, the equilibrium X 2 and the limit cycle  , whose basins of attraction consist of the initial conditions in the shaded and white areas, respectively. Note that while attractors and repellors can be easily obtained through simulation, forward and backward in time, saddles can be hard to find. The analysis of system (1) becomes even more difficult if one wants to follow the phase portrait under variation of parameters. Generically, by perturbing a parameter slightly the phase portrait changes slightly as well. Namely, if the new phase portrait is topologically equivalent to the original one, then nothing changed from a qualitative point of view, i. e., all attracting, repelling, and saddle sets are still present with unchanged stability properties, though slightly perturbed. By contrast, the critical points in parameter space where arbitrarily small parameter perturbations give rise to nonequivalent phase portraits are called bifurcation points, where bifurcations are said to occur. Bifurcations therefore result in a partition of parameter space into regions: parameter combinations in the same region correspond to topologically equivalent dynamics, while nonequivalent phase portraits arise for parameter combinations in neighboring regions. Most often, this partition is represented by means of a two-dimensional bifurcation diagram, where the regions of a parameter plane are separated by so-called bifurcation curves. Bifurcations are said to be local if they occur in an arbitrarily small neighborhood of the equilibrium or cycle; otherwise, they are said to be global. Although one might hope to detect bifurcations by simulating system (1) for various parameter combinations and initial conditions, a “brute force” simulation approach is hardly effective and accurate in practice, because bifurcations of equilibria and cycles are associated with a loss of hyperbolicity, e. g., stability, so that one should dramatically increase the length of simulations while approaching the bifurcation. In particular, saddle sets are hard to find by simulation, but play a fundamental role in bifurcation analysis, since they, together with attracting and repelling sets, form the skeleton of the phase portrait. This is why numerical bifurcation analysis does not rely on simulation, but rather on continuation, a numerical method suited for computing (approximating through a discrete sequence of points) one-dimensional manifolds (curves, “branches” in regular) implicitly defined as the zero set of a suitable defining function.

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The general idea is to formulate the computation of equilibria and their bifurcations as a suitable algebraic problem (AP) of the form F(u; p) D 0 ;

(2)

where u 2 Rn u is composed of x and possibly other variables characterizing the system, see, e. g., defining functions as in Sect. “Continuation and Detection of Bifurcations”. Here, however, for simplicity of notation, u will be considered as in Rn , but the actual dimension of u will always be clear from the context. Similarly, limit cycles and their bifurcations are formulated in the form of a boundary-value problem (BVP) 8 u˙  f (u; p) D 0 ; < g(u(0); u(T); p) D 0 ; (3) : RT h(u(t); p)dt D 0 ; 0 with nb boundary conditions, i. e., g : Rn  Rn  Rn p ! Rn b , ni integral conditions, i. e., h : Rn  Rn p ! Rn i , and u in a proper function space. In other words, a list of defining functions is formulated, in the form (2) or (3), to cover all cases of interest. For example, u D x and F(x; p) D f (x; p) is the AP defining equilibria of (1). The commonly used cycle BVP, with the time-rescaling t D T, is 8 x˙  T f (x; p) D 0 ; < x(0)  x(1) D 0 ; (4) :R1 >x k1 ()d D 0 ; ˙ x() 0 where from here on > denotes the transpose and, for simplicity, ˙ for d/d is used. The integral condition is the so-called phase condition, which ensures that x() is the 1 periodic solution of (1) closest to the reference solution x k1 () (typically known from the previous point along the continuation), among time-shifted orbits x(  0 ); 0 2 [0; 1]. As will be discussed in Sect. “Discretization of BVPs”, a proper time-discretization of u() allows one to approximate any BVP by a suitable AP. Thus, equilibria, limit cycles and their bifurcations can all be represented by an algebraic defining function like (2), and numerical continuation allows one to produce one-dimensional solution branches of (2) under the variation of strategic components of p, called free parameters. With this approach, equilibria and cycles can be followed without further difficulty in parameter regimes where these are unstable. Then, during continuation, the stability of equilibria and cycles is determined through linearization. Moreover, the characterization of nearby solutions of (1) can be done using normal forms, i. e., the simplest canonical models to which the

system, close to a bifurcation, can be reduced on a lowerdimensional manifold of the state space, the so-called center manifold. While a branch of equilibria or cycles is followed, bifurcations can be detected as the zero of suitable test functions. Upon detection of a bifurcation, the defining function can be augmented by this test function or another appropriate function, and the new defining function can then be continued using one more free parameter. An analytical bifurcation study is feasible for simple systems only. Numerical bifurcation analysis is one of the few but also very powerful tools to understand and describe the dynamics of systems depending on parameters. Some basic steps while performing bifurcation analysis will be outlined and software implementations of continuation and bifurcation algorithms discussed. First a few standard and often used approaches for the computation and continuation of zeros of a defining function are reviewed in Sect. “Continuation and Discretization of Solutions”. The presentation starts with the most obvious, but also naive, approaches to contrast these with the methods employed by software packages. In Sect. “Normal Forms and the Center Manifold” several possible scenarios for the loss of stability of equilibria and limit cycles are discussed. Not all bifurcations are characterized by linearization and for the detection and analysis of these bifurcations, codimension 1 normal forms are mentioned and a general method for their computation on a center manifold is presented. Then, a list of suitable test functions and defining systems for the computation of bifurcation branches is discussed in Sect. “Continuation and Detection of Bifurcations”. In particular, when a system bifurcates new solution branches appear. Techniques to switch to such new branches are described in Sect. “Branch Switching”. Finally, the computation and continuation of global bifurcations characterized by orbits connecting equilibria is presented, in particular homoclinic orbits, in Sect. “Connecting Orbits”. This review concludes with an overview of existing implementations of the described algorithms in Sect. “Software Environments”. Previous reviews [7,9,22,43] have similar contents. This review however, focuses more on the principles now underlying the most frequently used software packages for bifurcation analysis and the algorithms being used. Continuation and Discretization of Solutions The continuation of a solution u of (2) with respect to one parameter p is a fundamental application of the Implicit Function Theorem (IFT). Generally speaking, to define one-dimensional solution manifolds (branches), the number of unknowns in (2)

Numerical Bifurcation Analysis

should be one more than the number of equations, i. e., n p D 1. However, during continuation it is better not to distinguish between state variables and parameters as will become apparent in Sects. “Pseudo-Arclength Continuation” and “Moore–Penrose Continuation”. Therefore, write y D (u; p) 2 Y D RnC1 for the continuation variables in the continuation space Y and consider the continuation problem F(y) D 0 ; RnC1

(5) Rn .

! with F : Let F be at least continuous differentiable, y0 D (u0 ; p0 ) be a known solution point of (5), and the matrix F y (y0 ) D [Fu (u0 ; p0 )jF p (u0 ; p0 )] be full rank, i. e., rank(F y (y0 )) D 8 < (i) rank(Fu (u0 ; p0 )) D n ; or n , (ii) rank(Fu (u0 ; p0 )) D n  1 and : F p (u0 ; p0 ) … R(Fu (u0 ; p0 )) ;

(6)

where R(Fu ) denotes the range of F u . Then the IFT states that there exists a unique solution branch of (5) locally to y0 . Introducing a scalar coordinate s parametrizing the branch, e. g., the arclength positively measured from y0 in one of the two directions along the solution branch, then one can represent the branch by y(s) D (u(s); p(s)) and the IFT guarantees that F(y(s)) D 0 for jsj is sufficiently small. Moreover, y(s) is continuous differentiable and the vector (s) D y s (s) D (us (s); p s (s)) D (v(s); q(s)), tangent to the solution branch at y(s), exists and is the unique solution of 8 < F y (y(s))(s) D Fu (u(s) ; p(s))v(s) C F p (u(s) ; p(s))q(s) D 0 ; : (s)> (s) D v(s)> v(s) C q(s)2 D 1 : (7) In other words, the matrix F y (y0 ) has a one-dimensional nullspace N (F y (y0 )) spanned by (0) and y0 is said to be a regular point of the continuation space Y. Below, several variants of numerical continuation will be described. The aim is to produce a sequence of points y k ; k  0 that approximate the solution branch y(s) in one direction. Starting from y0 , the general idea is to make a suitable prediction y10 , typically along the tangent vector, from which the Newton method is applied to find the new point y1 . The predictor-corrector procedure is then iterated. First, the simplest implementation is presented and it is shown where it might fail. Many continuation packages for bifurcation theory use an alternative implementation of which two variants are discussed. Many more advanced predictor-corrector schemes have been designed, see [1,18,46] and references therein.

Parameter Continuation Parameter continuation assumes that the solution branch of (5) can be parameterized by the parameter p 2 R. Indeed, if F u has full rank, i. e., case (i) in (6), then this is possible by the IFT. Starting from (u0 ; p0 ) and perturbing the parameter a little, with a stepsize h, the new parameter is p1 D p0 C h and the most simple predictor for the state variable is given by u10 D u0 . Application of Newton’s method to find u1 satisfying (5) leads to jC1

u1

j

j

j

D u1  Fu (u1 ; p1 )1 F(u1 ; p1 ) ;

j D 0; 1; 2; : : :

The iterations are stopped when a certain accuracy is jC1 j achieved, i. e., kuk D ku1  u1 k < "u and/or j kF(u1 ; p1 )k < "F . In practice, also the maximum number of Newton steps is bounded, in order to guarantee termination. If this maximum is reached before convergence, the computation is restarted with a smaller (typically halved) stepsize. However, in case of quick convergence, e. g., after only a few iterations, the stepsize is multiplied (1.3 is a typical factor). In any case, the stepsize is varied between two assigned limits hmin and hmax , so that continuation cannot proceed when convergence is not reached even with minimum stepsize. When h is chosen too small, too much computational work may be performed, while for h that is chosen too large, little detail of the solution branch is obtained. As a first improved predictor, note that the IFT suggests to use the tangent prediction for the state variables u10 D u0 C hv0 , where v0 is obtained from (7) with s D 0 and as v(0)/q(0). Indeed, the tangent vector can be approximated by the difference v k D (u k u k1 )/h k or, even better, computed at negligible cost, since the numerical decomposition of the matrix Fu (u k ; p k ) is known from the last Newton iteration. These methods are illustrated in Fig. 2. Note in particular the folds in the sketch. Here parameter continuation does not work, since exactly at the fold Fu (u; p) is singular such that Newton’s method does not converge and beyond, for larger p, there is no local solution to (5). Pseudo-Arclength Continuation Near folds, the solution branch is not well parameterized by the parameter, but one can use a state variable for the parametrization. In fact, the fold is a regular point (case (ii) in (6)) at which the tangent vector  D (v; q) has no parameter component, i. e., q D 0. So, without distinguishing between parameters and state variables, one takes the tangent prediction y10 D y0 C h0 , as long as the starting solution y0 is a regular point. Since now both p and u

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Numerical Bifurcation Analysis, Figure 2 Parameter continuation without (a) and with (b) tangent prediction. The dotted lines indicate subspaces where solutions are searched

are corrected, one more constraint is needed. Pseudo-arclength continuation uses the stepsize h as an approximation of the required distance, in arclength, between y0 and the next point y1 . This leads to the so-called pseudo-arclength equation 0> (y1  y0 ) D h. In this way, solution branches can be followed past folds. The idea for this continuation method is due to Keller [50]. The Newton iteration, applied to

F(y1 ) D 0 ; 0> (y1  y0 )  h D 0 ; is given by jC1 y1

D

j y1

! !1 j j cF y (y1 ) cF(y1 ) ;  0 0> j D 0; 1; 2; : : : ; (8) jC1

j

where y D y1  y1 is forced to lie in the hyperplane orthogonal to the tangent vector, as illustrated in Fig. 3a. Upon convergence, the new tangent vector  1 is obtained by solving (7) at y1 . Moore–Penrose Continuation This continuation method is based on optimization. Starting with the tangent prediction y10 D y0 C h0 , a point y1 with F(y1 ) D 0 nearest to y10 is searched, so the following is optimized minfky1  y10 kjF(y1 ) D 0g : y1

Each correction is therefore required to be orthogonal to j the nullspace of F y (y1 ), i. e., ( jC1 F(y1 ) D 0 ; j > jC1 j (1 ) (y1  y1 ) D 0 :

Starting with 10 D 0 the Newton iterations are given by 8 ! !1 j j ˆ ˆ (y ) F jC1 j ) F(y y 1 ˆy 1 ˆ ; D y1  j ˆ < 1 0 (1 )> (9) ! ˆ jC1 1   ˆ ˆ jC1 0 F y (y1 ) ˆ ˆ ; j D 0; 1; 2; : : : : 1 D j 1 (1 )> As illustrated in Fig. 3b, the Moore–Penrose continuation can be interpreted as a variant of Keller’s method in which the tangent vector is updated at every Newton step. When the new point y1 is found, the tangent vector  1 is immejC1 jC1 diately obtained as 1 /k1 k from the last Newton itj eration, since 1 does not necessarily have unit length. Finally, for both the pseudo-arclength and MoorePenrose continuation methods one can prove that they converge (with superlinear convergence), provided that y0 is a regular point and the stepsize is sufficiently small. Discretization of BVPs In this section, orthogonal collocation [3,15] is described, a discretization technique to approximate the solution of a generic BVP (3) by a suitable AP. Let u be at least in the space C 1 ([0; 1]; Rn ) of continuous differentiable vectorvalued functions defined on [0; 1]. For BVPs the rescaled time t D T is used, so that the period T becomes a parameter and in the sequel T will be addressed as such. Introduce a time mesh 0 D 0 < 1 < : : : <  N D 1 and, on each interval [ j1 ;  j ], approximate the function u by a vector-valued polynomial } j of degree m, j D 1; : : : ; N. The polynomials } j are determined by imposing the ODE in (3) at m collocation points z j;i , i D 1; : : : ; m, i. e., }˙ j (z j;i ) D f (} j (z j;i ); p) ; j D 1; : : : ; N ;

i D 1; : : : ; m :

(10)

Numerical Bifurcation Analysis

Numerical Bifurcation Analysis, Figure 3 Pseudo-arclength (a) and Moore–Penrose (b) continuation. Searching a solution in hyperplanes without (a) and with (b) updating the tangent vector. The open dots correspond to Newton iterations, full dots to points on the curve. The dotted lines indicate subspaces where solutions are searched

One usually chooses the so-called Gauss points as the collocation points, the roots of the mth order Legendre polynomials. Moreover, }1 (0) and } N (1) must satisfy the boundary conditions and the whole piecewise polynomial must satisfy the integral conditions. Counting the number of unknowns, the discretization of (3) leads to nN polynomials, each with (m C 1) degrees of freedom, plus np free parameters, so there are nN(m C 1) C n p continuation variables. These variables are matched by nmN collocation equations from (10), n(N  1) continuity conditions at the mesh points, nb boundary conditions, and ni integral conditions, for a total of nN(m C 1) C nb C n i  n algebraic equations. Thus, in order for these equations to compose an AP, the number of free parameters is generically n p D nb C n i  n C 1 and, typically, one is the period T. The collocation method yields high accuracy with superconvergence at the mesh points [15]. The mesh can also be adapted during continuation, for instance to minimize the local discretization error [68]. The equations can be solved efficiently by exploiting the particular sparsity structure of the Jacobi matrix in the Newton iteration. In particular, a few full but essentially smaller systems are solved instead of one sparse but large system. During this process one finds two nonsingular (nn) -submatrices M 0 and M 1 such that M0 u(0) C M1 u(1) D 0, i. e., the monodromy matrix M D M11 M0 is found as a by-product. In the case of a periodic BVP (u(1) D u(0)) the Floquet multipliers are therefore computed at low computational cost. Normal Forms and the Center Manifold Local bifurcation analysis relies on the reduction of the dynamics of system (1) to a lower-dimensional center mani-

fold H 0 near nonhyperbolic equilibria or limit cycles, i. e., when they bifurcate at some critical parameter p0 . The existence of H 0 follows from the Center Manifold Theorem (CMT), see, e. g., [11], while the reduction principle is shown in [72]. The reduced ODE has the same dimension as H 0 given by the number nc of critical eigenvalues or nontrivial multipliers (counting multiplicity), and is transformed to a normal form. The power of this approach is that the bifurcation scenario of the normal form is preserved in the original system. The normal form for a specific bifurcation is usually studied only up to a finite order, i. e., truncated, and many diagrams for bifurcations with higher codimension are in principle incomplete due to global phenomena, such as connecting orbits. Also, H 0 is not necessarily unique or smooth [75], but fortunately, one can still draw some useful qualitative conclusions. The codimension (codim) of a bifurcation is the minimal number of parameters needed to encounter the bifurcation and to unfold the corresponding normal form generically. Therefore, in practice, one finds codim 1 phenomena when varying a single parameter, and continues them as curves in two-parameter planes. Codim 2 phenomena are found as isolated points along codim 1 bifurcation curves. Still, codim 2 bifurcations are important as they are the roots of codim 1 bifurcations, in particular of global phenomena. For this reason they are called organizing centers as, around these points in parameter space, one-parameter bifurcation scenarios change. For parameter-dependent systems the center manifold H 0 can be extended to a parameter-dependent invariant manifold H(p); H0 D H(p0 ), so that the bifurcation scenario on H(p) is preserved in the original system for kp  p0 k sufficiently small.

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In the following, the normal forms for all codim 1 bifurcations of equilibria and limit cycles are presented and their bifurcation scenarios discussed. Then a general computational method for the approximation, up to a finite order, of the parameter-dependent center manifold H(p) is presented. The method gives, as a by-product, explicit formulas for the coefficients of a given normal form in terms of the vector field f of system (1). Normal Forms Bifurcations can be defined by certain algebraic conditions. For instance, an equilibrium is nonhyperbolic if <() D 0 holds for some eigenvalue. The simplest possibilities are  D 0 (limit point bifurcation or branch point, though the latter is nongeneric, see Sect. “Branch Switching”) and 1;2 D ˙i!0 ; !0 > 0 (Hopf bifurcation). Bifurcations of limit cycles appear if some of the nontrivial multipliers cross the unit circle. The three simplest possibilities are  D 1 (limit point of cycles),  D 1 (period-doubling) or 1;2 D e˙i 0 ; 0 < 0 <  (Neimark– Sacker). At the bifurcation (p D p0 ), linearization of system (1) near the equilibrium x0 or around a limit cycle, does not result in any stability information in the center manifold. In this case, nonlinear terms are also necessary to obtain such knowledge. This is provided by the critical normal form coefficients as discussed below. The state variable in the normal form will be denoted by w and the unfolding parameter by ˛ 2 R, with w D 0 at ˛ D 0 being a nonhyperbolic equilibrium. Bifurcations are labeled in accordance with the scheme of [38]. Codimension 1 Bifurcations of Equilibria Limit point bifurcation (LP): The equilibrium has a simple eigenvalue  D 0 and the restriction of (1) to a one-dimensional center manifold can be transformed to the normal form w˙ D ˛ C aLP w 2 C O(jwj3 ) ;

w 2R;

(11)

where generically aLP ¤ 0 and O denotes higher order terms in state variables depending on parameters too. When the unfolding parameter ˛ crosses the critical value (˛ D 0), two equilibria, one stable and one unstable in the center manifold, collide and disappear. This bifurcation is also called saddle-node, fold or tangent bifurcation. Note that this bifurcation occurs at the folds in Figs. 2 and 3. Hopf bifurcation (H): The equilibrium has a complex pair of eigenvalues 1 D 2 D i!0 and the restriction of (1) to the two-dimensional center manifold is given by w˙ D (i!0 C ˛)w C cH w 2 w¯ C O(jwj4 ) ;

w 2 C ; (12)

where generically the first Lyapunov coefficient dH D <(cH ) ¤ 0. When ˛ crosses the critical value, a limit cycle is born. It is stable (and present for ˛ > 0) if dH < 0 and unstable if dH > 0 (and present for ˛ < 0). The case dH < 0 is called supercritical or “soft”, while dH > 0 is called subcritical or “hard” as there is no (local) attractor left after the bifurcation. This bifurcation is most often called Hopf, but also Poincaré–Andronov–Hopf as this was also known to the first two. Codimension 1 Bifurcations of Limit Cycles Bifurcations of limit cycles are theoretically very well understood using the notion of a Poincaré map. To define this map, choose a (n  1)-dimensional smooth cross-section ˙ transversal to the cycle and introduce a local coordinate z 2 Rn1 such that z D Z(x) is defined on ˙ and invertible. For example, one chooses a coordinate plane x j D 0 such that f j (x)j x j D0 ¤ 0. Let x0 (t) be the cycle with period T, so that z0 D Z(x0 (0)) is the cycle intersection with ˙ , where z D 0 can always be assumed without loss of generality. Denote by T(z) the return time to ˙ defined by the flow ˚ with T(z0 ) D T. Now, the Poincaré map P : Rn1 ! Rn1 maps each point close enough to z D 0 to the next return point on ˙ , i. e., P : z 7! Z(˚ T(z) (Z 1 (z))). Thus, bifurcations of limit cycles turn into bifurcations of fixed points of the Poincaré map which can be easily described using local bifurcation theory. Moreover, it can be shown that the n  1 eigenvalues of the linearization Pz (0) are the nontrivial eigenvalues of the monodromy matrix M D ˚xT (x0 (0)), which also has a trivial eigenvalue equal to 1 (the vector f (x0 (0); p), tangent to the cycle at x0 (0), is mapped by M to itself). The eigenvalues of Pz (0) are therefore the nontrivial multipliers of the cycle. Although the Poincaré map and its linearization can also be computed numerically through suitably organized simulations (so-called shooting techniques [17]), it is better to handle both the cycle multipliers and normal form computations associated to nonhyperbolic cycles using BVPs [9,23,57]. Here, however, the normal forms on a Poincaré section are presented, where w D 0 at ˛ D 0 is the fixed point of the Poincaré map corresponding to a nonhyperbolic limit cycle. Limit point of cycles (LPC): The fixed point has one simple nontrivial multiplier  D 1 on the unit circle and the restriction of P to a one-dimensional center manifold has the form w 7! ˛ C w C aLPC w 2 C O(w 3 ) ;

w 2 R;

where aLPC ¤ 0. As for the LP bifurcation two fixed points collide and disappear when ˛ crosses the critical value,

Numerical Bifurcation Analysis

provided aLPC ¤ 0. This implies the collision of two limit cycles of the original vector field f . Period-doubling (PD): The fixed point has one simple multiplier  D 1 on the unit circle and the restriction of P to a one-dimensional center manifold can be transformed to the normal form w 7! (1 C ˛)w C bPD w 3 C O(w 4 ) ;

w 2R;

where bPD ¤ 0. When the parameter ˛ crosses the critical value and bPD ¤ 0, a cycle of period 2 for P bifurcates from the fixed point corresponding to a limit cycle of period 2T for the original system (1). This phenomenon is also called the flip bifurcation. If bPD is positive [negative], the bifurcation is supercritical [subcritical] and the double period cycle is stable [unstable] (and present for ˛ > 0[˛ < 0]). Neimark–Sacker (NS): The fixed point has simple critical multipliers 1;2 D e˙i 0 and no other multipliers on the unit circle. Assume that ei k 0 ¤ 1 for k D 1; 2; 3; 4, i. e., there are no strong resonances. Then, the restriction of P to a two-dimensional center manifold can be transformed to the normal form w 7! e i (˛) (1 C ˛)w C cNS w 2 w¯ C O(jwj4 ) ;

w2C;

where cNS is a complex number and (0) D 0 . Provided dNS D <(ei 0 cNS ) ¤ 0, a unique closed invariant curve for P appears around the fixed point, when ˛ crosses the critical value. In the original vector field, this corresponds to the appearance of a two-dimensional torus with (quasi-) periodic motion. This bifurcation is also called secondary Hopf or torus bifurcation. If dNS is negative [positive], the bifurcation is supercritical [subcritical] and the invariant curve (torus) is stable [unstable] (and present for ˛ > 0[˛ < 0]). Center Manifolds Generally speaking, the CMT allows one to restrict the dynamics of (1) to a suspended system w˙ D G(w; ˛); G : Rn c  Rn p ! Rn c ;

(13)

on the center manifold H and here np is typically 1 or 2 depending on the codimension of the bifurcation. Although the normal forms (13) to which one can restrict the system near nonhyperbolic equilibria and cycles are known, these results are not directly applicable. Thus, efficient numerical algorithms are needed in order to verify the nondegeneracy conditions in the normal forms listed above. Here, a powerful normalization method due to Iooss and coworkers is reviewed, see [9,14,29,37,55,59]. This method assumes very little a priori information, actually

only the type of bifurcation such that the form, i. e., the nonzero coefficients of G, is known. This fits very well in a numerical bifurcation setting where one computes families of solutions and monitors and detects the occurrence of bifurcations with higher codimension during the continuation. Without loss of generality it is assumed that x0 D 0 at the bifurcation point p0 D 0. Expand f (x; p) in Taylor series f (x; p) D Ax C 12 B(x; x) C 16 C(x; x; x) C J1 p C A1 (x; p) C : : : ;

(14)

parametrize, locally to (x; p) D (0; 0), the parameter-dependent center manifold by H : Rn c  Rn p ! Rn ;

x D H(w; ˛) ;

(15)

and define a relation p D V (˛) between the original and unfolding parameters. The invariance of the center manifold can be exploited by differentiating this parametrization with respect to time to obtain the so-called homological equation f (H(w; ˛); V(˛)) D Hw (w; ˛)G(w; ˛) :

(16)

To verify nondegeneracy conditions, only an approximation to the solution of the homological equation is required. To this end, G; H and V are expanded in Taylor series: X 1 G(w; ˛) D g w  ˛  ; !! jjCjj1

H(w; ˛) D

X jjCjj1

1 h w  ˛  ; !!

(17)

V(˛) D v10 ˛1 C v01 ˛2 C O(k˛k2 ) ; where g are the desired normal form coefficients and ;  are multi-indices. For a multi-index  one has  D (1 ; 2 ; : : : ; n ) for nonnegative integers i , ! D ˜   if 1 !2 ! : : : n !; jj D 1 C 2 C : : : C n ,  n ˜i   i for all i D 1; : : : ; n and w  D w11 : : : w  n . When dealing with just the critical coefficients, i. e., ˛ D 0, the index  is omitted. Substitution of this ansatz into (16) gives a formal power series in w and ˛. As both sides should be equal for all w and ˛, the coefficients of the corresponding powers should be equal. For each vector h , (16) gives linear systems of the form L h D R ;

(18)

where L D A   I n (  is a weighted sum of the critical eigenvalues) and R involves known quantities

1179

1180

Numerical Bifurcation Analysis

Numerical Bifurcation Analysis, Table 1 Critical normal form coefficients for generic codim 1 bifurcations of equilibria and fixed points. Here, A, B and C refer to the expansion (14) for equilibria, while for fixed points they refer to (19) Eigenvectors Av D 0 LP

H

A> w D 0

aLP D 12 w > B(v; v) cH D

A> w D i!0 w

h11

A> w D w

aLPC D 12 w > B(v; v)

1 > w C(v; v; v) C 3B(v; h2 ) 6 A> w D w h2 D (In1  A)1 B(v; v) i0 Av D e v  1 cNS D w¯ > C(v; v; v¯ ) C 2B(v; h11 ) C B(¯v; h20 ) 2 A> w D ei0 w 1 2i0 ¯ In1  A)1 B(v; v) eik0 ¤ 1; k D 1; 2; 3; 4 h11 D (In1  A) B(v; v); h20 D (e Av D v

PD

NS

1 > w¯ C(v; v; v¯ ) C 2B(v; h11 ) C B(¯v; h20 ) 2 D A1 B(v; v¯ ); h20 D (2i!0 In  A)1 B(v; v)

Av D i!0 v Av D v

LPC

Critical normal form coefficients

bPD D

of G and H of order less than or equal to jj C jj. This leads to an iterative procedure, where, either system (18) is nonsingular, or the required coefficients g are obtained by imposing solvability, i. e., R lies in the range of L and is therefore orthogonal to the eigenvectors of L>  associated to the zero eigenvalue. In the second case, the solution of (18) is not unique, and one typically selects the h without components in the nullspace of L . However, the nonuniqueness of the center manifold does not affect qualitative conclusions. The parameter transformation, i. e., the v , is obtained by imposing certain conditions on some normal form coefficients, leading to a solvable system. One can perform an analogous procedure for the Poincaré map by using a Taylor expansion P(z; p) D Az C 12 B(z; z) C 16 C(z; z; z) C : : :

(19)

and the homological equation for maps P(H(w; ˛); V (˛)) D H(G(w; ˛); ˛) :

(20)

The detailed derivation of the formulas for all codim 1 and 2 cases for equilibria and cycles can be found in [9,55, 56,59,60]. The formulas for the critical normal form coefficients for codim 1 bifurcations are presented in Table 1. Note once more that for limit cycles a numerical more appropriate method exists [57] based on periodic normal forms [47,48]. Continuation and Detection of Bifurcations Along a solution branch one generically passes through bifurcation points of higher codimension. To detect such an

event, a test function ' is defined, where the event corresponds to a regular zero. If at two consecutive points y k1 ; y k along the branch the test function changes sign, i. e., '(y k )'(y k1 ) < 0, then the zero can be located more precisely. Usually, a one-dimensional secant method is used to find such a point. Now, if system (1) has a bifurcation at y0 D (x0 ; p0 ), then there is generically a curve y D y(s) where the system displays this bifurcation. In order to find this curve, one starts with a known point y 0 and formulates a defining system and then continue that solution in one extra free parameter. Test Functions for Codimension 1 Bifurcations An equilibrium may lose stability through a limit point, a Hopf bifurcation or in a branch point. At a limit or branch point bifurcation the Jacobi matrix A D f x (x0 ; p0 ) has an algebraically simple eigenvalue  D 0 (see Sect. “Branch Switching” for branch points), while at a Hopf point there is a pair of complex conjugate eigenvalues  D ˙i!0 ; !0 ¤ 0 and only one such pair. The simplest way of detecting the passage through a bifurcation during continuation, is to monitor the eigenvalues of the Jacobi matrix. For large systems and stiff problems this is prohibitive as it is numerically expensive and not always accurate. Instead, one can base test functions on determinants. Test Functions for Limit Point Bifurcations Along an equilibrium curve the product of the eigenvalues changes sign at a limit point. Recall that the determinant of A is the product of its eigenvalues. Therefore, the following test

Numerical Bifurcation Analysis

function can be computed 'LP D det( f x (x; p))

(21)

without computing the eigenvalues explicitly. For the LP bifurcation the pseudo-arclength or Moore–Penrose continuation methods provide an excellent test function as a by-product of the continuation. Note that while passing through the fold, the last component of the tangent vector  changes sign as the continuation direction in the parameter reverses. The test function is therefore defined as

Test Functions for Codimension 1 Cycle Bifurcations Recall that the nontrivial multipliers 1 ; : : : ; n1 determine the stability of the cycle and can be efficiently computed as the nontrivial multipliers of the monodromy matrix M, see Sect. “Discretization of BVPs”. Now the following two sets of test functions can be used to detect LPC, PD and NS bifurcations 'LPC D

n1 Y

( i  1) ;

'LPC D  p ;

iD1

'PD D

n1 Y

( i C 1) ;

'PD D det(M C I n ) ;

iD1

'LP D  nC1 :

(22) 'NS D

Test Functions for Hopf Bifurcations Denote the eigenvalues of A by  i (x; p) ; i D 1 : : : ; n and consider the following product 'H D

Y ( i (x; p) C  j (x; p)) : i< j

It can be shown that this product has a regular zero at a simple Hopf point [9], but it should be checked that this zero corresponds to an imaginary pair and not to the neutral saddle case  i D  j ;  i 2 R. Also here one can compute this product without explicit computation of the eigenvalues using the bi-alternate product [34,42,45,56]. The bi-alternate product of two (n  n)-matrices A and B, denoted by A ˇ B, is a (m  m)-matrix C (m D n(n  1)/2) with row index (i; j) and column index (k; l) and elements C(i; j)(k;l ) where

1 D 2

ˇ ˇ ˇ) (ˇ ˇa a ˇ ˇb b ˇ ˇ ik il ˇ ˇ ik il ˇ ˇ ˇCˇ ˇ ˇ bjk bjl ˇ ˇ ajk ajl ˇ

i D 2; 3; : : : ; n; j D 1; 2; : : : i  1 ; k D 2; 3; : : : ; n; l D 1; 2; : : : k  1 :

Let A be an n  n-matrix with eigenvalues 1 ; : : : ; n , then [73]  A ˇ A has eigenvalues  i  j ,  2A ˇ I n has eigenvalues  i C  j . The test function can now be expressed as 'H D det(2 f x (x; p) ˇ I n ) :

(23)

For higher dimensional systems, this matrix becomes very large and one should use precondition or subspace methods, see [36].

n1 Y 1Di< j

( i  j  1) ;

'NS D det(M ˇ M  I n(n1)/2 ) :

where  p denotes the parameter component of the tangent vector similar to (22). It should also be checked that a zero of 'NS corresponds to nonreal multipliers e i 0 , similar to the test function to detect the Hopf bifurcation. There are alternatives for these test functions. One can define bordered systems using the monodromy matrix [9,42] or a BVP formulation [23]. Defining Systems for Codimension 1 Bifurcations of Equilibria To compute curves of codim 1 equilibria bifurcations, first a defining system of the form (2) needs to be formulated to define the bifurcation curve, and then a second parameter for the continuation must be freed, so that now p 2 R2 . This is done by adding to the equilibrium equation f (x; p) D 0 appropriate equations that characterize the bifurcation. Defining systems come in two flavors, fully (also standard) and minimally augmented systems. The first computes all relevant eigenspaces, while the latter exploits the rank deficiency of the Jacobi matrix and adds only a few strategic equations to regularize the continuation problem. The evaluation of such equations requires the eigenspaces, but these can be computed separately. As the names suggest, the difference is in the dimension of the defining system leading to differently sized problems. In particular, the advantage of minimally augmented systems is that of solving several smaller linear problems, instead of a big one, which is known to be better in terms of both accuracy and computational time. For small phase dimension n there is little difference in computational effort. Both minimally and fully extended defining systems for both limit point and Hopf bifurcations are presented. The regularity of these systems is also known, e. g., see [42].

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Defining Systems for Limit Point Bifurcations The first defining system is minimally extended adding the test function (21)

f (x; p) D 0 ; det( f x (x; p)) D 0 :

the LP case. Adding the test function (23) creates a minimally extended system

(24)

f (x; p) D 0 ; det(2 f x (x; p) ˇ I n ) D 0 ;

(27)

while the fully extended system is given by This system consists of n C 1 equations in n C 2 unknowns (x; p). One problem is that the computation of the determinant can lose accuracy for large systems. This can be avoided in two ways, by augmenting the system with the eigenspaces or using a bordering technique. Fully extended systems include the eigenvectors and for a LP bifurcation this leads to 8 < f (x; p) D 0 ; f x (x; p)v D 0 ; (25) : > v0 v  1 D 0 ; where v0 is a vector not orthogonal to N ( f x (x; p)). This system consists of 2n C 1 equations in 2n C 2 unknowns (x; p; v). The bordering technique uses the bordering lemma [41]. Let A 2 Rnn be a singular matrix and let B; C 2 Rnm such that the system 

A C>

    V 0 B D nm 0m Im g

(26)

is nonsingular (V 2 Rnm , g 2 Rmm ). Typically B and C are associated to the eigenspaces of A> and A corresponding to the zero eigenvalue, respectively or, during continuation, approximated by their values computed at the previous point along the branch. It follows from the bordering lemma that A has rank deficiency m if and only if g has rank deficiency m. With A D f x (x; p) and m D 1, one has g D 0 if and only if det( f x (x; p)) D 0. A modified and minimally extended system for limit points is thus given by

f (x; p) D 0 ; g(x; p) D 0 ;

where g is defined by (26) with A> B D AC D 0 at a previously computed point. During the continuation the derivatives of g w.r.t. to x and p are needed. They can either be approximated by finite differences, or explicitly (and efficiently) obtained from the second-derivatives of the vector field f , see [42]. Defining Systems for Hopf Bifurcations Defining systems for Hopf bifurcations are formulated analogously to

8 f (x; p) ˆ ˆ ˆ ˆ < f x (x; p)v1 C !v2 f x (x; p)v2  !v1 ˆ ˆ ˆ w1> v1 C w2> v2  1 ˆ : w1> v2  w2> v1

D0; D0; D0; D0; D0;

(28)

where w D w1 C iw2 is not orthogonal to the eigenvector v D v1 C iv2 corresponding to the eigenvalue i!. The vector w D v k1 computed at the previous point is a suitable choice during continuation. System (28) is expressed using real variables and has 3n C 2 equations for 3n C 3 unknowns (x; p; v1 ; v2 ; !). A reduced defining system can be obtained from (28) by noting that the matrix f x (x; p)2 C I n has rank deficiency two at a Hopf bifurcation point with  D ! 2 [67]. An alternative to (28) is now formulated as 8 ˆ ˆ <

f (x; p) f x (x; p)2 C I n v v> v  1 ˆ ˆ : w> v

D0; D0; D0; D0;

(29)

where w is not orthogonal to the two-dimensional real eigenspace of the eigenvalues ˙i!. It has 2n C 2 equations for 2n C 3 unknowns (x; p; v; ). However, w needs to be updated during continuation, e. g., as the solution  of [ f x (x; p)2 C I n ]> w; v > w D (0; 0) computed at the previous continuation point. A further reduction is obtained exploiting the rank deficiency. Consider the system 

f x (x; p)2 C I n C>

B 02

    V 0 D n2 I2 g

and it follows from the bordering lemma that g vanishes at Hopf points and any two components of g, e. g., g 11 and g 22 , see [42], can be taken to augment Eq. (2) to obtain the following minimally augmented system 8 < f (x; p) D 0 ; g11 D 0 ; : g22 D 0 ;

(30)

which has n C 2 equations for n C 3 unknowns (x; p; ).

Numerical Bifurcation Analysis

Defining Systems for Codimension 1 Bifurcations of Limit Cycles In principle, to study bifurcations of limit cycles one can compute numerically the Poincaré map and study bifurcations of fixed points. If system (1) is not stiff, then the Poincaré map and its derivatives may be obtained with satisfactory accuracy. In many cases, however, continuation using BVP formulations is much more efficient. Suppose a cycle x bifurcates at p D p0 , then the BVP (4) defining the limit cycle must be augmented with suitable extra functions. As for codim 1 branches of equilibria, one can define either fully extended systems by including the relevant eigenfunctions in the computation [9], or minimally extended systems using bordered BVPs [23,39]. The regularity of these defining systems is also discussed in these references. Since the discretization of the cycle leads to large APs, here the minimally extended approach can lead to faster results even though some more algebra is involved, see the comparison in [57]. Below only the equations are presented, which are added to the defining system (4) for the continuation of limit cycles. Fully Extended Systems The following equations can be used to augment (4) and continue codim 1 bifurcations of limit cycles. The eigenfunctions v need to be discretized in a similar way to that in Sect. “Discretization of BVPs”. The previous cycle x k1 and eigenfunction v k1 are assumed to be known. LPC: For the limit point of cycles bifurcation, the BVP (4) is augmented with the equations 8 v˙()  T f x (x(); p)v()   f (x(); p) ˆ ˆ < v(1)  v(0) R1 > v () x˙ k1 ()d ˆ ˆ 0 R1 > : k1 d C  k1 0 v ()v()

D0; D0; D0; D1;

for the variables (x; p; T; v;  ). Note that 8 ˆ v˙()  T f x (x(); p)v()  T f p (x(); p)q ˆ ˆ ˆ ˆ  f (x(); p) < v(1)  v(0) R1 > ˆ ˆ ˆ v () x˙ k1 ()d ˆ 0 ˆ : R 1 v > ()v() k1 d C q k1 q C  k1  0

(31)

D0; D0; D0; D1;

defines the tangent vector  D (v; q;  ) to the solution branch, so that (31) simply imposes q D 0, i. e., the limit point. Together with (4), they compose a BVP with 2n ODEs, 2n boundary conditions, and 2 integral conditions, i. e., n p D 2n C 2  2n C 1 D 3, namely T and two free parameters. Similar dimensional considerations hold for the PD and NS cases below.

PD: For the period-doubling bifurcation, the extra equations augmenting (4) are 8 < v˙()  T f x (x(); p)v() D 0 ; v(1) C v(0) D 0 ; (32) R1 > : k1 ()d D 1 ; v ()v 0 for the variables (x; p; T; v). Here v is the eigenfunction of the linearized ODE associated with the multiplier  D 1. In fact, the second equation in (32) imposes v(1) D Mv(0) D v(0), where M is the monodromy matrix, while the third equation scales the eigenfunction against the previous continuation point. NS: For the Neimark–Sacker bifurcation, the BVP (4) is augmented with the equations 8 < v˙()  T f x (x(); p)v() D 0 ; v(1)  e i v(0) D 0 ; (33) R1 > : k1 ()d D 1 ; ¯ 0 v ()v for the variables (x; p; T; v; ) with v 2 C 1 ([0; 1]; C n ). Here v is the eigenfunction of the linearized ODE associated with the multiplier  D e i . Of course, the real formulation should be used in practice. Minimally Extended Systems For limit cycle continuation the discretization of the fully extended BVP (4) with (31), (32) or (33) may lead to large APs to be solved. In [39] a minimally extended formulation is proposed to augmenting (4) with a function g with only a few components. The corresponding function g is defined using bordered systems. LPC: For this bifurcation, one uses suitable bordering functions v1 ; w1 and vectors v2 ; w2 ; w3 such that the following system linear in (v; ; g) is regular 8 v˙()  T f x (x(); p)v  f (x(); p) C w1 g D 0 ; ˆ ˆ < v(1)  v(0) C w2 g D 0 ; R1 (34) f (x(); p)> v()d C w3 g D 0 ; ˆ ˆ 0 R1 > : 0 v1 v()d C v2  D 1 : The function g D g(x; T; p) vanishes at a LPC point. The bordering functions v1 ; w1 and vectors v2 ; w2 ; w3 can be updated to keep (34) nonsingular, in particular, v1 D v k1 and v2 D  k1 from the previously computed point are used. It is convenient to introduce the Dirac operator ı i f D f (i) and the integral operator Intv() f D R1 > 0 v() f ()d and to rewrite (34) in operator form 0 1 0 1 D  T f x (x(); p)  f (x(); p) w1 0 1 c0 v B C B ı  ı 0 w 0C 0 1 2C @ A B C  DB @ Int f (x();p) @0A: 0 w3 A g Intv 1 () v2 0 1 (35)

1183

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Numerical Bifurcation Analysis

PD: The same notation as for the minimally extended LPC defining system is used and suitable bordering functions v1 ; w1 and vector w2 are chosen such that the following system is regular 0 1 0 1 D  T f x (x(); p) w1   0 v @ D @0A : (36) ı0 C ı 1 w2 A g Intv 1 () 0 1 At a PD bifurcation g(x; T; p) defined by (36) vanishes. NS: Let ˆ D cos( ) denote the real part of the nonhyperbolic multiplier and choose bordering functions v1 ; v2 ; w11 ; w12 and vectors w21 ; w22 such that the following system is nonsingular and defines the four components of g 1 0 D  T f x (x(); p) w11 w12     B ı2  2ˆ ı1 C ı0 w21 w22 C v 0n2 C B : (37) D @ Intv 1 () 0 0 A g I2 0 0 Intv 2 () At a NS bifurcation the four components of g(x; T; p) defined by (37) vanish, and, similar to the Hopf bifurcation, the BVP (4) can be augmented with any two components of g. Test Functions for Codimension 2 Bifurcations

the zero-Hopf (ZH, also called Gavrilov-Guckenheimer, a simple zero eigenvalue and a simple imaginary pair), and the double Hopf (HH, two distinct imaginary pairs), while higher order degeneracies lead to cusp (CP, aLP D 0 in the normal form (11)) or generalized Hopf (GH, dH D 0 in the normal form (12), also called Bautin or degenerate Hopf). For cycles, there are strong resonances (R1– R4), fold-flip (LPPD), fold-Neimark–Sacker (LPNS), flipNeimark–Sacker (PDNS), and double Neimark–Sacker (NSNS) among those involving linear terms, while higher order degeneracies lead to cusp (CP), degenerate flip (GPD), and Chenciner (CH) bifurcations. Naturally the normal form coefficients, see [9,56], are a suitable choice for the corresponding test functions. In Tables 2 and 3, test functions are given which are defined along the corresponding codim 1 branches of equilibrium and limit cycle bifurcations, respectively. The functions refer to the corresponding defining system and to Table 1. Upon detecting and locating a zero of a test function it may be neces-

Numerical Bifurcation Analysis, Table 2 Test functions along limit point and Hopf bifurcation curves. The matrix Ac for the test function of the double Hopf bifurcation can be obtained as the orthogonal complement in Rn of the Jacobi matrix A w.r.t. the two-dimensional eigenspace associated with the computed branch of Hopf bifurcations

During the continuation of codim 1 branches, one meets generically codim 2 bifurcations. Some of which arise through extra instabilities in the linear terms, while other codim 2 bifurcations are defined through degeneracies in the normal form coefficients. For equilibria, codim 2 bifurcations of the first type are the Bogdanov–Takens (BT, two zero eigenvalues with only one associated eigenvector),

cusp generalized Hopf Bogdanov–Takens zero-Hopf double Hopf

Label CP GH BT ZH HH

LP aLP

H

dH >v wLP LP  'H 'LP det(2Ac ˇ In2 )

Numerical Bifurcation Analysis, Table 3 Test functions along LPC, PD and NS bifurcation curves. The matrix Mc along the Neimark–Sacker bifurcation curve is defined similarly as Ac in Table 2 along the Hopf bifurcation as the orthogonal complement of the monodromy matrix M w.r.t. the two-dimensional eigenspace associated with the computed branch of Neimark–Sacker bifurcations

cusp degenerate flip Chenciner resonance 1:1 resonance 1:2 resonance 1:3 resonance 1:4 fold-flip fold-Neimark–Sacker flip-Neimark–Sacker double Neimark–Sacker

Label CP GPD CH R1 R2 R3 R4 LPPD LPNS PDNS NSNS

LPC aLPC

PD

NS

bPD >v wLP LP

'PD 'NS

>v wPD PD

dNS ˆ  1 ˆ C 1 ˆ C 12 ˆ

'LP 'NS

'LP 'PD det(Mc ˇ Mc  I(n2)(n3)/2 )

Numerical Bifurcation Analysis

Numerical Bifurcation Analysis, Figure 4 (a) Projection of two solution branches of (2), intersecting at yBP , on the null-space of Fy (yBP ) (N ), close to yBP (planar representation in coordinates (˛1 ; ˛2 ) with respect to a given basis). The two solution branches are approximated by the straight lines in N spanned by their tangent vectors at yBP (thick vectors). (b) and (c) Projection on N , close to yBP , of the two solution branches of the perturbed problem (see (43) for b > 0 and b < 0, respectively)

sary to check that a bifurcation is really involved, similar to the Hopf case where neutral saddles are excluded. For details about the dynamics and the bifurcation diagrams at codim 2 points, see [2,44,56]. Branch Switching This section considers points in the continuation space from which several solution branches of interest, with the same codimension, emanate. At these points suitable “branch switching” procedures are required to switch from one solution branch to another. First, the transversal intersection of two solution branches of the same continuation problem is considered, which occurs at so-called Branch Points (BP) (also called “singular” or “transcritical” bifurcation points). Branch points are nongeneric, in the sense that arbitrarily small perturbations of F in (2) turn the intersection into two separated branches, which come close to the “ghost” of the (disappeared) intersection but then fold (LP) and leave as if they follow the other branch (see Fig. 4). BPs, however, are very common in applications due to particular symmetries of the continuation problem, like reflections in state space, conserved quantities or the presence of trivial solutions. This is why BP detection and continuation recently received attention [16,25,28]. Then, the switch from a codim 0 solution branch to that of a different continuation problem at codim 1 bifurcations is examined. In particular, the equilibrium-to-cycle switch at a Hopf bifurcation and the period-1-to-period-2 cycle switch at a flip bifurcation are discussed. Finally, various switches between codim 1 solution branches of different continuation problems at codim 2 bifurcations are addressed.

Branch Switching at Simple Branch Points Simple BPs are points y BP D (uBP ; pBP ), encountered along a solution branch of (2), at which the nullspace N (F y (y)) of F y (y) is two-dimensional, i. e., the nullspace is spanned by two independent vectors 1 ; 2 2 Y, with  i>  i D 1, i D 1; 2. Generically, two solution branches of (2) pass through y BP , with transversal tangent vectors given by suitable combinations of  1 and  2 . In the following, only the case of the AP (2), i. e., Y D RnC1 (u 2 Rn and p 2 R) and F : RnC1 ! Rn is considered. Similar considerations hold for the BVP (3) (see [16] for details), though, loosely speaking, results for APs can be applied to BVPs after time discretization. A BP is not a regular point, since rank (F y (y BP )) D n  1. Distinguishing between state and parameters, there are two possibilities 8 (i) dimN (Fu (y BP )) D 1; F p (y BP ) 2 R(Fu (y BP )) ˆ ˆ < H) 1 D (v1 ; 0); 2 D (v2 ; q2 ); (ii) dimN (Fu (y BP )) D 2; F p (y BP ) … R(Fu (y BP )) ˆ ˆ : H) 1 D (v1 ; 0); 2 D (v2 ; 0) ; (38) for suitably chosen v1 ; v2 ; q2 . In particular, in the first case, of Fu (y BP ) and  2 is determined v1 spans the   nullspace BP by solving Fu (y )v2 C F p (y BP )q2 ; v1> v2 ; v2> v2 C q22 D (0; 0; 1). BPs can be detected by means of the following test function 

 F y (y) 'BP D det ; (39) > where  is the tangent vector to the solution branch during continuation, which indeed vanishes when (2) admits

1185

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Numerical Bifurcation Analysis

a second independent tangent vector. Note from (38) that test function (21) also vanishes at BPs, so that test function (22) is more appropriate for LPs. The vectors tangent to the two solution branches intersecting at y BP can be computed as follows. Parametrize one of the two solution branches by a scalar coordinate s, e. g., the arclength, so that y(s) and y s (s) denote the branch and its tangent vector locally to y(0) D y BP . Then, F(y(s)) is identically equal to zero, so taking twice the derivative w.r.t. s one obtains Fyy (y(s))[y s (s); y s (s)] C F y (y(s))yss (s) D 0, which at y BP reads Fyy (y BP )[y s (0); y s (0)] C F y (y BP )yss (0) D 0 ;

(40)

with y s (0) D ˛1 1 C ˛2 2 . Let 2 Rn span the nullspace of F y (y BP )> with > D 1. Since the range of F y (y BP ) is orthogonal to the nullspace of F y (y BP )> , one can eliminate yss (0) in (40) by left-multiplying both sides by > , thus obtaining >

Fyy (y BP )[˛1 1 C ˛2 2 ; ˛1 1 C ˛2 2 ] D 0 :

(41)

Equation (41) is called the algebraic branching equation [50] and is often written as c11 ˛12 C 2c12 ˛1 ˛2 C c22 ˛22 D 0 ;

(42)

with cij D > Fyy (y BP )[ i ;  j ]; i; j D 1; 2. At BP de2 is generically negatection, the discriminant c11 c22  c12 tive (otherwise the BP would be an isolated solution point of (2)), so that two distinct pairs (˛1 ; ˛2 ) and (˛˜ 1 ; ˛˜ 2 ), uniquely defined up to scaling, solve (42) and give the directions of the two emanating branches. Once the two directions are known, one can easily perform branch switching by an initial prediction from y BP along the desired direction. This, however, requires the second-order derivatives of F w.r.t. all continuation variables. Though good approximations can often be achieved by finite differences, an alternative and computationally cheap prediction can be taken in the nullspace of F y (y BP ) along the direction orthogonal to y s (0). The vector y s (s) is in fact known at each point during the continuation of the solution branch up to BP detection, so that the cheap prediction for the other branch spans the (one-dimensional) nullspace of 

F y (y BP ) : y s (0)> Branch Point Continuation Generic Problems Several defining systems have been proposed for BP continuation, see [16,25,28,63,64,65].

Among fully extended formulations, the most compact one characterizes BPs as points at which the range of F y (y) has rank defect 1, i. e., the nullspace of F y (y)> is one-dimensional. BP continuation is therefore defined by 8 ˆ ˆ <

F(u; p) Fu (u; p)> > ˆ ˆ F p (u; p) : > 1

D0; D0; D0; D0:

Counting equations, 2n C 2, and variables u; 2 Rn , p 2 R, i. e., 2n C 1 scalar variables, it follows that two extra parameters generically need to be freed. In other words, BPs are codim 2 bifurcations, which are not expected along generic solution branches of (2). Non-generic Problems BP continuation can be performed in a single extra free parameter for nongeneric problems characterized by symmetries that persist for all parameter values. In such cases, the continuation problem (2) is perturbed into F(y) C bub D 0 ;

(43)

where b 2 R and ub 2 Rn are new variables of the defining system. The idea is that ub “breaks the symmetry”, in the sense that problem (43) has no BP for small b ¤ 0, and BP continuation can be performed in two extra free parameters, one of which, b, remains zero during the continuation. The choice of ub is not trivial. Geometrically, ub must be such that small values of b perturb Fig. 4a into Fig. 4b, say for b > 0, and into Fig. 4c for b < 0. It turns out (see, e. g., [16]) that ub D is a good choice, i. e., perturbations not in the range of F y (y BP ) break the symmetry, since close to the BP, they must be balanced by the nonlinear terms of the expansion of F in (43), and this implies significant deviations of the perturbed solution branch y from the unperturbed y(s). BP continuation for nongeneric problems is therefore defined by 8 F(x; p) C b ˆ ˆ < Fx (x; p)> ˆ F p (x; p)> ˆ : > 1

D0; D0; D0; D0:

(44)

This defining system is also useful for accurately computing BPs. In fact, the basin of convergence of the Newton iterations in (8) or (9) shrinks at BPs (recall F y (y) does not have full rank at BPs), while system (44), in the 2nC2 variables (u; p; b; ), has a unique solution (uBP ; pBP ; 0; ) close to the BP. Thus, when the BP test function (39)

Numerical Bifurcation Analysis

changes sign along a solution branch of (2), Newton corrections can be applied to (44), starting from the best possible prediction, i. e., with b D 0 and as the eigenvector of Fu (u; p)> associated with the real eigenvalue closest to zero.

The periodic solution branch of the limit cycle BVP (4) can therefore be followed, provided the phase R 1 condition (see the last equation in (4)) is replaced by 0 x > v˙d D 0 at the first Newton correction. Otherwise, x would be undetermined among time-shifted solutions.

Minimally Extended Formulation A minimally extended defining system for BP continuation requires two scalar conditions, g1 (u; p) D 0, g2 (u; p) D 0, to be added to the unperturbed or perturbed problem (2) or (43) for generic and nongeneric problems, respectively. These functions g 1 and g 2 are defined in [28] by solving

Branch Switching at Flip Points

8 F y (y)1 C g1 k1 ˆ ˆ ˆ ˆ F y (y)2 C g2 k1 ˆ ˆ < (1k1 )> 1  1 (2k1 )> 2  1 ˆ ˆ ˆ ˆ ˆ (1k1 )> 2 ˆ : (2k1 )> 1

D0; D0; D0; D0; D0; D0;

in the unknowns 1 ; 2 ; g1 ; g2 , while ing

F y (y)>

is updated by solv-

C g 1 1 C g 2 2 D 0 ; > 1 D0;

in the unknowns gence.

, g 1 , g 2 , after each Newton conver-

Branch Switching at Hopf Points At a Hopf bifurcation point y H D (x H ; pH ), one typically wants to start the continuation of the emanating branch of limit cycles. For this, one might think of using the branch switching procedure described above to switch from a constant to a periodic solution branch of the limit cycle BVP (4). Unfortunately, y H is not a simple BP for problem (4), since the period T is undetermined along the constant solution branch, so that, formally, an infinite number of branches emanate from y H . Thus, a prediction in the proper direction, i. e., along the vector  D (v; q) tangent to the periodic solution branch, is required. Let y(s) represent the periodic solution branch, with y(0) D y H . Then, x and v are period-1 vector-valued functions in C 1 ([0; 1]; Rn ), p 2 R and T are the free parameters, and q D (p s ; Ts ). The Hopf bifurcation theorem [56] ensures that p s D Ts D 0 and that v is the unit-length solution of the linearized, time-independent equation v˙ D T(0) f x (x H ; pH )v, i. e., v() D sin(2)wr C cos(2)w i , > where w D wr C iw i (wr> wr C w > i w i D 1; w r w i D 0) H H is the complex eigenvector of f x (x ; p ) associated to the eigenvalue i!, ! D 2/T(0).

At a flip bifurcation point y PD D (x PD ; pPD ), where x PD 2 C 1 ([0; 1]; Rn ), x PD (1) D x PD (0), one typically wants to start the continuation of the emanating branch of “period2” limit cycles, i. e., those which close to y PD have approximately the double of the period of the bifurcating cycle. For this, branch switching at simple BPs can be used. In fact, two solution branches of the limit cycle BVP (4) transversely intersect at y PD if one considers T as the doubled period: the branch of interest and the branch along which the corresponding period-1 cycle is traced twice. In other words, one can see the period-doubling bifurcation in the period-1 branch as the “period-halving” bifurcation in the period-2 branch. Alternatively, the vector  D (v; q) tangent to the period-2 branch at y PD is given by the flip theorem [56] and does not need to be computed by solving the algebraic branching Eq. (41). In particular, the initial solution of the period-2 BVP is x(t) D x PD (2t), p D pPD , T D 2T PD , while q D (p s ; Ts ) D 0 and ( w(t) ; 0 t <1; v(t) D w(t  1) ; 1  t < 2 ; where w(t) is the unit-length eigenfunction of the linearized (time-dependent) ODE associated with the multiplier  1, i. e., 8 PD PD PD < w˙  T f x (x ; p )w D 0 ; w(1) C w(0) D 0 ; R1 : > 0 w() w() d D 1 : Branch Switching at Codimension 2 Equilibria Assume that y2 D (x2 ; p2 ), x2 2 Rn , p2 2 R2 , identifies a codim 2 equilibrium bifurcation point of system (1), either cusp (CP), Generalized Hopf (GH), Bogdanov– Takens, (BT), zero-Hopf (ZH), or double Hopf (HH). Then, according to the analysis of the corresponding normal form (13) with two parameters and critical dimension n c D 1; 2; 3; 4 at CP, BT and GH, ZH, HH points, respectively, several curves of codim 1 bifurcations of equilibria and limit cycles emanate from p2 in the parameter plane [56]. Here, the problem of how the continuation of such curves can be started from y2 is discussed, by restrict-

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ing the attention to equilibria bifurcations (an example of a cycle bifurcation is given at the end, and see [9,61]). In general, the normal form analysis also provides an approximation of each emanating codim 1 bifurcation, in the form of a parameterized expansion X 1 X 1 (45) w  " ; ˛ D ˛ " ; wD ! ! 1 1 for small " > 0 and up to some finite order. Then, the (parameter-dependent) center manifold (15) maps such an approximation back to the solution branch of the original system (1), locally to y2 D (H(0; 0); V (0)), and allows one to compute the proper prediction from y2 along the direction of the desired codim 1 bifurcation. However, as will be concluded in the following, such approximations are not needed to start equilibria bifurcations and are therefore not derived (see [9] for all available details). Switching at a Cusp Point At a generic cusp point y CP D (uCP ; pCP ), two fold bifurcation curves terminate tangentially. Generically, the cusp point is a regular point for the fold defining systems (24) and (25), where the tangent vector has zero p-components. The cusp geometrically appears once the fold solution branch is projected in the parameter plane. Thus, the continuation of the two fold bifurcations can simply be started as forward and backward fold continuation from y CP . Since the continuation direction is uniquely defined, neither a tangent prediction nor a nonlinear expansion of the desired solution branch are necessary. Switching at a Generalized Hopf Point At a Hopf point with vanishing Lyapunov coefficient y GH D (x GH ; pGH ), a LPC bifurcation terminates tangentially to a Hopf bifurcation, which turns from super- to subcritical, and vice versa. Thus, y GH is a regular point for the Hopf defining systems (27)–(30). Switching at a Bogdanov–Takens Point At a generic Bogdanov–Takens point y BT D (uBT ; pBT ), a Hopf and a (saddle) homoclinic bifurcation terminate tangentially to a fold bifurcation, along which a real nonzero eigenvalue of the Jacobi matrix f x (x; p) changes sign at y BT . Generically, y BT is a regular point for the fold defining systems (24) and (25) and for the Hopf defining systems (27), (29), and (30). However, y BT is a simple BP for the Hopf defining system (28). In fact, the fold and Hopf branches are both solution branches of the continuation problem (28), where ! D 0 and v1 D v2 , with v1> v1 D v2> v2 D 1/2, along the fold branch. The branch switch procedures described in this section readily apply in this case.

Switching at Zero-Hopf and Double Hopf Points Generically, zero-Hopf and double Hopf points are regular points for codim 1 equilibria bifurcations (fold and Hopf), so that the proper initial prediction is uniquely defined. For limit cycle bifurcations and connecting orbits, nonlinear expansions of the fold and Hopf branches are needed to derive initial predictions for the emanating branches. For cycles the switching procedure can be set up using the center manifold [61]. However, initial predictions for homoclinic and heteroclinic bifurcations for both zero-Hopf and double Hopf cases are not available in general, but see [35]. The double Hopf bifurcation appears when two different branches of Hopf bifurcations intersect. Several bifurcation curves are rooted at the double Hopf point. In particular, it is known that there are generically two branches, two half-lines in the parameter plane, emanating from this point along which a Neimark–Sacker bifurcation of limit cycles occurs [56]. Here it is discussed how to initialize the continuation of a Neimark–Sacker branch (using (37)) from a double Hopf point after continuation of a Hopf branch. The initialization requires approximations of the cycle x, the period T, the parameters p and the real part of the multiplier ˆ . These can be obtained by reducing the dynamics of (1) to the center manifold. On the center manifold, the dynamics near a HH bifurcation point is governed by the following normal form 

 w˙1 D w˙2   (i!1 (˛) C ˛1 )w1 C g2100 w1 jw1 j2 C g1011 w1 jw2 j2 (i!2 (˛) C ˛2 )w2 C g1110 w2 jw1 j2 C g0021 w2 jw2 j2 C O(k (w1 ; w2 )k4 ) ; (46)

where (w1 ; w2 ) 2 C 2 . In polar coordinates, w1 D 1 e i 1 ; w2 D 2 e i 2 , the asymptotics from the normal form as in (45) for the nonhyperbolic cycle on one branch are given by ( 1 ; 2 ; ˛1 ; ˛2 )   D "; 0; Re(g2100 )"2 ; <(g1110 )"2 ;

(47)

with 1 2 [0; 2]; 2 D 0. Although high-dimensional, the computation of the coefficients and center manifold vectors is relatively straightforward. Introduce Av j D i! j v j ; A> w j D i! j w j ¯> with v¯> j vj D w j v j D 1 and let  D (10); (01) and introduce the standard basis vectors e10 D (1; 0); e01 D (0; 1). Using the expansion (14), the cubic critical normal form coefficients and the parameter dependence are calculated

Numerical Bifurcation Analysis

equilibrium points x and xC in the ODE system (1) is an orbit for which

from g2100 D w¯ 1> [C(v1 ; v1 ; v¯1 ) C B(h2000 ; v¯1 ) ı C 2B(h1100 ; v1 )] 2 ;

lim x(t) D x

t!1

g1011 D w¯ 1> [C(v1 ; v2 ; v¯2 ) C B(h1010 ; v¯2 ) C B(h1001 ; v2 ) C B(h0011 ; v1 )]; g1110 D w¯ 2> [C(v2 ; v1 ; v¯1 ) C B(h1100 ; v2 ) C B(h1010 ; v¯1 ) C B( h¯ 1001 ; v1 )] ; g0021 D w¯ 2> [C(v2 ; v2 ; v¯2 ) C B(h0020 ; v¯2 ) C 2B(h0011 ; v2 )]/2 ;  j; D

p¯> j [A1 (v j ; e )

ı  B(v j ; A1 J1 e )] 2 ;

(48)

where j D 1; 2 and h2000 D (2i!1 I n  A)1 B(v1 ; v1 ) ; h0020 D (2i!2 I n  A)1 B(v2 ; v2 ) ; h1100 D A1 B(v1 ; v¯1 ) ;

(49)

h0011 D A1 B(v2 ; v¯2 ) ; h1010 D (i(!1 C !2 )I n  A)1 B(v1 ; v2 ) ; h1001 D (i(!1  !2 )I n  A)1 B(v1 ; v¯2 ) ;

where h are the vectors in the expansion of the center manifold (17). Now, to construct a cycle for (46), a mesh for 1 is defined and the asymptotics (47) are inserted into the polar coordinates. This cycle is mapped back to the original system by using (17) with (47), (48) and (49). The transformation between free system and unfolding parameters is given by p(") D R(1 2 )1 (˛1 ; ˛2 )> using (47). Finally, approximating formulas for the period and the real part of the multiplier are given by TD

2 ; !1 C d!1 "2

ˆ D cos(T(!2 C d!2 "2 )) ;

(d!1 ; d!2 ) D =(1 2 )> (<(1 2 )> )1 <(g2100 ; g1110 )> C =(g2100 ; g1110 ) ; where d!1 ; d!2 indicate the change in rotation in the angles 1 ; 2 for " ¤ 0. This construction is done up to second order in " and leads to an initial approximation for the continuation of a Neimark–Sacker bifurcation curve starting from a double Hopf point. A similar set up can be defined for the other branch. Connecting Orbits Connecting orbits, such as homoclinic and heteroclinic orbits, can be continued using a variety of techniques. To fix some terminology: a heteroclinic orbit that connects two

and

lim x(t) D xC :

t!1

A homoclinic orbit is an orbit connecting an equilibrium point to itself, that is, if xC D x . Similarly there exist heteroclinic connecting orbits between equilibrium points and periodic orbits, and homoclinic and heteroclinic orbits connecting periodic orbits to periodic orbits. Traditionally, homoclinic orbits to equilibrium points were computed indirectly using numerical shooting or by continuing a periodic solution with a large enough but fixed period, that is close enough to a homoclinic orbit [24]. More modern and robust techniques compute connecting orbits directly using projection boundary conditions. Computing connections with and between periodic orbits is subject to current research [26,27,54]. For a more detailed description of the methods described here, see [9]. Formulation as a BVP A heteroclinic orbit can be expressed as a BVP in the following way: x˙ (t) D f (x(t); p) ; lim x(t) D x ; t!1 Z 1 (x(t)  x0 (t))> x˙0 (t)dt D 0 ; lim x(t) D xC ;

t!1

1

where the integral condition is with respect to a reference solution x0 (t) and fixes the phase, similarly to the phase condition for periodic orbits. This BVP, however, operates on an infinite interval, while, numerically, one can only operate on a finite, truncated interval [T ; TC ]. In this case the problem can be reformulated as 8 ˆ ˆ <

˙  f (x(t); p) x(t) Ls (p)(x(T )  x (p)) Lu (p)(x(TC )  xC (p)) ˆ ˆ : R TC > T (x(t)  x 0 (t)) x˙0 (t)dt

D0; D0; D0; D0;

where the equations involving Ls (p) and Lu (p) form the projection boundary conditions. Here Ls (p) is an ns  n matrix where the rows span the ns -dimensional stable eigenspace of A> (x ), and similarly Lu (p) is an nu  n matrix where the rows span the nu -dimensional unstable eigenspace of A> (xC ), where A(x) denotes the Jacobi matrix of (1) at x. The projection boundary conditions then ensure that the starting point x(T ) lies in the unstable eigenspace of x and that the end point x(TC ) lies in the stable eigenspace of xC (see Fig. 5).

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Numerical Bifurcation Analysis

strip. Homoclinics to a saddle-node, that is, where one of the eigenvalues of the equilibrium is zero, can be detected and followed similarly, by constructing appropriate test functions. For details, see [9] and the references mentioned therein. Homoclinic Branch Switching

Numerical Bifurcation Analysis, Figure 5 Projection boundary conditions in two dimensions: the orbit homoclinic to x 0 is approximated by the truncated orbit from x(T ) on the unstable eigenspace T u to x(TC ) on the stable eigenspace T s . The unstable and stable manifolds are denoted by W u and W s , respectively: this figure shows that the homoclinic orbit is also approximated in parameter space, because the two manifolds do not coincide

In parallel, one must also continue of the equilibrium points x (p) and xC (p), unless they are fixed. Then the eigenspaces can be determined by doing a Schur decomposition of the corresponding Jacobi matrices. These must be subsequently scaled to ensure continuity in the parameter p [6]. Alternatively, one can construct smooth projectors using an algorithm for the continuation of invariant subspaces which includes the Ricatti equation in the defining system, see [33] for this new method. All these conditions taken together give a BVP with two free parameters, since, in general, a homoclinic or heteroclinic connection is a codim 1 phenomenon. Detecting Homoclinic Bifurcations It is then possible to detect codim 2 bifurcations of homoclinic orbits by setting up test functions and monitoring those, detecting when they cross zero. For the continuation of these codim 2 bifurcations in three parameters such test functions can then be kept constant equal to zero, which provides an extra boundary condition. Simple test functions involve the values of leading eigenvalues of the equilibrium point, i. e., the stable and unstable eigenvalues closest to the imaginary axis. Some other bifurcations such as the inclination flip involve solving the so-called adjoint variational equation, which can detect whether the flow around the orbit is orientable or twisted like a Möbius

It is sometimes desirable to do branch switching from a homoclinic orbit to an n-homoclinic orbit, that is, a homoclinic orbit that goes through some neighborhood of the related equilibrium point n  1 times before finally converging to the equilibrium. Such n-homoclinic orbits arise in a number of situations involving the eigenvalues of the equilibrium and the orientation of the flow around the orbit. Suppose that the orbit is homoclinic to a saddle with complex conjugate eigenvalues (a saddlefocus). Let the saddle-quantity  be the sum of the real parts of the leading stable and the leading unstable eigenvalue. If this quantity is positive, then a so-called Shil’nikov snake exists, which implies the existence of infinitely many n-periodic and n-homoclinic orbits for nearby parameters. These n-homoclinic orbits also arise from certain codim 2 bifurcations: 1. Belyakov bifurcations: Either the saddle-quantity of the saddle-focus goes through zero, or there is a transition between a saddle-focus and a real saddle (here the equilibrium has a double leading eigenvalue). 2. Homoclinic flip bifurcations: The inclination flip and the orbit flip, where the flow around the orbit changes between orientable and twisted. 3. The resonant homoclinic doubling: A homoclinic orbit for which the flow is twisted connected to a real saddle where the saddle quantity  goes through zero. Case 1. produces infinitely many n-homoclinic orbits, whereas cases 2. and 3. only produce a two-homoclinic orbit. By breaking up a homoclinic orbit globally into two parts where the division is in a cross-section away from the equilibrium, somewhere “half-way”, and then gluing pieces together, it is possible to construct n-homoclinic orbits from 1-homoclinic orbits. The gaps between tobe-glued pieces can be well-defined using Lin’s method, which leaves the gaps in the direction of the adjoint vector. This vector can be found by solving the adjoint variational equation mentioned above at the “half-way” point. Combining gap distances and times taken for the flow in pieces, one can construct a well-posed BVP. This BVP can then be continued in those quantities so that if the gap sizes go to zero, one converges to an n-homoclinic orbit [66].

Numerical Bifurcation Analysis

Lin’s method was also applied in [54] to compute point-to-cycle connections by gluing a piece from the equilibrium to a cross-section to a piece from the cross section to the cycle. Software Environments This review has outlined necessary steps and suitable methods to perform numerical bifurcation analysis. Summarizing, the following subsequent tasks can be recognized. Initial procedure

1. Compute the invariant solution 2. Characterize the linearized behavior Continuation 3. Variation of parameters 4. Monitor dynamical indicators Automated analysis 5. Detect special points and compute normal forms 6. Switch branches

Ideally, these computations are automatically performed by software and indeed, many efforts have been spent on implementing the algorithms mentioned in this review and related ones. With the appearance of computers at research institutes the first codes and noninteractive packages were developed. For a recent historical overview of these and their capabilities and algorithms, see [38]. Here it is worthwhile to mention AUTO86 [24], LINBLF [51] and DSTOOL [4] as these are the predecessors to the packages AUTO-07P [21], CONTENT [58], MATCONT [19], P yDSTOOL [13] discussed here. In particular, AUTO86 is very powerful and in use to date, but not always easy to handle. Therefore, several attempts have been made to make an interactive graphical user interface. One example is XPPAUT [32] which is a standard tool in neuroscience. The latest version of AUTO is AUTO-07P and written in Fortran and supports parallelization. It uses pseudo-arclength continuation and performs a limited bifurcation analysis of equilibria, fixed points, limit cycles and connecting orbits using HOMCONT [12]. A recent addition is the continuation of branch points [16]. It uses fully extended systems, but has specially adapted linear solvers so that it is still quite fast. An interactive package written in C++ is CONTENT, where the Moore–Penrose continuation was first implemented. It supports bifurcation analysis of equilibria up to the continuation of codim 2 and detection of some codim 3 bifurcations. It uses a similar procedure as AUTO to continue limit cycles and detects codim 1 bifurcations of limit cycles, but does not continue these. It also handles bifurcations of cycles of maps up to the detection of codim 2 bifurcations. Normal forms for all codim 1 bifurcations

are computed. Interestingly, both fully and minimally extended systems are implemented. A new project MATCONT emerged out of CONTENT. It is written in MATLAB, a widely used software tool in modeling. In contrast to AUTO it uses Moore–Penrose continuation and minimally extended systems to compute equilibria, cycles and connecting orbits. Since MATLAB is an interpreted language, it is slower than the other packages, although a considerable speedup is obtained as the code for the Jacobi matrices for limit cycles and their bifurcations are written in C-codes and compiled. MATCONT has, however, much more functionality. It computes normal form coefficients up to codim 2 bifurcations of equilibria [20] and for codim 1 of limit cycles normal form coefficients are computed using a BVP-algorithm [28]. A recent addition is to switch branches from codim 2 bifurcation points of equilibria to codim 1 bifurcations of cycles [61]. Finally P yDSTOOL supports similar functionality for ODEs as CONTENT. Future Directions This review focuses on methods for equilibria and limit cycles to gain insight into the dynamics of a nonlinear ODE depending on parameters, but, generally speaking, other characteristics play a role too. For instance, a Neimark– Sacker bifurcation leads to the presence of tori. The computation and continuation of higher dimensional tori has been considered in [49,70,71]. The generalization of the methods for equilibria and cycles is, however, not straightforward. Stable and unstable manifolds are another aspect. In particular their visualization can hint at the presence of global bifurcations. A review of methods for computing such manifolds is presented in [52]. Connecting orbits can be calculated, but initializing such a continuation is a nontrivial task. This may be started from certain codim 2 bifurcation points as in [8], but good initial approximations are not available for other cases. This review is also restricted to ODEs, but one can define other classes of dynamical systems. For instance, if the system is given by an explicitly defined map, i. e., not implicitly as a Poincaré map, the described approach can also be carried out [10,37,40,59]. Another important and related class is given by delay equations, and the algorithms for equilibria and periodic orbits in the ODE case can be applied with suitable modifications, see [5,74] and the software implementations DDE-BIFTOOL [31] and PDDECONT . The computation of normal forms and connecting orbits for this class is not yet thoroughly investigated or supported.

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For large systems, e. g., discretizations of a partial differential equation (PDE), the algebra for some algorithms in this review becomes quite involved and numerically expensive. These problems need special treatment, see LOCA [69], PDE-CONT [30,62] and related references. These packages focus on computing (periodic) solutions and bifurcation curves. Good algorithms for analyzing bifurcations of PDEs will be a major research topic. Finally, there are also slow-fast (stiff) systems or systems with a particular structure such as symmetries. For these classes, many questions remain open. While on most of the mentioned topics pioneering work has been done, the methods are far from as “complete” as for ODEs. One overview of current research topics in dynamical systems can be found in [53].

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33. Friedman M, Govaerts W, Kuznetsov YA, Sautois B (2005) Continuation of homoclinic orbits in MATLAB. In: Sunderam VS, van Albada GD, Sloot PMA, Dongarra J (eds) Proceedings ICCS 2005, Atlanta, Part I, Springer Lecture Notes in Computer Science Vol. 3514. Springer, Berlin, pp 263–270 34. Fuller AT (1968) Conditions for a matrix to have only characteristic roots with negative real parts. J Math Anal Appl 23: 71–98 35. Gaspard P (1993) Local birth of homoclinic chaos. Phys D 62:94–122 36. Govaerts W, Guckenheimer J, Khibnik A (1997) Defining functions for multiple Hopf bifurcations. SIAM J Numer Anal 34:1269–1288 37. Govaerts W, Khoshsiar Ghaziani R, Kuznetsov YA, Meijer HGE (2007) Numerical methods for two-parameter local bifurcation analysis of maps. SIAM J Sci Comput 29:2644–2667 38. Govaerts W, Kuznetsov YA (2007) Numerical continuation tools. In: Krauskopf B, Osinga HM, Galen-Vioque J (eds) Numerical continuation methods for dynamical systems: Path following and boundary value problems. Springer-Canopus, Dordrecht, pp 51–75 39. Govaerts W, Kuznetsov YA, Dhooge A (2005) Numerical continuation of bifurcations of limit cycles in matlab. SIAM J Sci Comp 27:231–252 40. Govaerts W, Kuznetsov YA, Sijnave B (1999) Bifurcations of maps in the software package content. In: Ganzha VG, Mayr EW, Vorozhtsov EV (eds) Computer algebra in scientific computing–CASC’99 (Munich). Springer, Berlin, pp 191–206 41. Govaerts W, Pryce JD (1993) Mixed block elimination for linear systems with wider borders. IMA J Num Anal 13:161–180 42. Govaerts WJF (2000) Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia 43. Guckenheimer J (2002) Numerical analysis of dynamical systems. In: Fiedler B (ed) Handbook of dynamical systems, Vol. 2. Elsevier Science, North-Holland, pp 346–390 44. Guckenheimer J, Holmes P (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York 45. Guckenheimer J, Myers M, Sturmfels B (1997) Computing Hopf bifurcations I. SIAM J Numer Anal 34:1–21 46. Henderson Me (2007) Higher-dimensional continuation. In: Krauskopf B, Osinga HM, Galen-Vioque J (eds) Numerical continuation methods for dynamical systems: Path following and boundary value problems. Springer-Canopus, Dordrecht, pp 77–115 47. Iooss G (1988) Global characterization of the normal form for a vector field near a closed orbit. J Diff Eqs 76:47–76 48. Iooss G, Adelmeyer M (1992) Topics in Bifurcation Theory and Applications. World Scientific, Singapore 49. Jorba A (2001) Numerical computation of the normal behaviour of invariant curves of n-dimensional maps. Nonlinearity 14:943–976 50. Keller HB (1977) Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Rabinowitz PH (ed) Applications of bifurcation theory. Proc Advanced Sem, Univ Wisconsin, Madison, (1976), Publ Math Res Center, No. 38. Academic Press, New York pp 359–384 51. Khibnik AI (1990) LINBLF: A program for continuation and bifurcation analysis of equilibria up to codimension three. In: Roose D, de Dier D, Spence A (eds) Continuation and bifurcations: numerical techniques and applications. Leuven, (1989),

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74. Szalai R, Stépán G, Hogan SJ (2006) Continuation of bifurcations in periodic delay-differential equations using characteristic matrices. SIAM J Sci Comput 28:1301–1317 75. van Strien SJ (1979) Center manifolds are not C1 . Math Zeitschrift 166:143–145

Books and Reviews Devaney RL (2003) An introduction to chaotic dynamical systems. Westview Press, Boulder Doedel EJ, Keller HB, Kernevez JP (1991) Numerical Analysis and

Control of Bifurcation Problems (I): Bifurcation in Finite Dimensions. Int J Bif Chaos 1:493–520 Doedel EJ, Keller HB, Kernevez JP (1991) Numerical Analysis and Control of Bifurcation Problems (II): Bifurcation in Infinite Dimensions. Int J Bif Chaos 1:745–772 Hirsch MW, Smale S, Devaney RL (2004) Differential equations, dynamical systems, and an introduction to chaos, 2nd edn. Pure and Applied Mathematics, vol 60 Elsevier/Academic Press, Amsterdam Murdock J (2003) Normal Forms and Unfoldings for Local Dynamical Systems. Springer, New York

Observability (Deterministic Systems) and Realization Theory

Observability (Deterministic Systems) and Realization Theory JEAN-PAUL ANDRÉ GAUTHIER Department of Electrical Engineering, University of Toulon, Toulon, France

including the standard linear Kalman filter but also nonlinear filtering theory. Realization of some input-output behavior covers the practical idea of modeling systems by differential equations on the basis of input-output experiments (identification). Introduction

Article Outline Glossary Definition of the Subject Introduction Preliminaries Linear Systems Observability of Nonlinear Systems Realization Theory Observers Future Directions Bibliography Glossary Observability An observed (and eventually controlled) dynamical system is observable if two distinct initial conditions can be distinguished (via the observations) by choosing the control function. Universal inputs A universal input is a control function allowing to distinguish between all initial conditions. Observer An observer system is a device, given in general under the guise of a differential equation (or a differences equation in the discrete case), allowing to track asymptotically the state trajectory of the system, using only the controls and the observations. Input-output map An input-output map is a mapping (for fixed initial condition) which to “control functions” associates “output functions”. It is in general assumed to be “causal” in some sense. Realization A realization of an input-output map is a (controlled) nonlinear system realizing the given input-output map. A realization (system) is said to be minimal if it is controllable and observable.

In this article, we discuss the basic concepts and methods in observability, observation and realization theories. The area is so large that there are thousands of contributions. We provide a nonexhaustive limited list of references which is certainly far from complete, but corresponds to our taste: an entirely subjective selection. We focus on the continuous finite dimensional case, but there are very important developments for systems governed by PDE’s, and for discrete time systems. In this continuous, finite dimensional context, we chose the geometric setting, however there are other possibilities (algebraic setting, formal power series, Volterra series, . . . ). For more details, we provide a list of books of significant interest dealing with the topics. After setting the general definitions, we consider briefly linear systems for which the theory has been well established for a long time, the pioneers being Kalman and Luenberger. Then we state some important results from the geometric nonlinear observability theory, the most significant contributions being undoubtedly those of Hermann and Krener [12] and Sussmann [25,26,27]. Also, contrarily to the case of linear systems, the observability of a system depends on the control applied to it. The existence of universal controls is a very important point, clarified by Sussmann [28]. We state the main result. Concerning observability in an analytic-geometry setting, there are also interesting and important results by Bartoziewicz. The next part of the paper is devoted to realization theory, where mostly two problems may be considered:

Definition of the Subject

1. Given a nonlinear system, find a minimal realization; 2. Given some input-output mapping, find a realization of it (it will be minimal by construction).

Observability analysis, design of nonlinear observers and realization of input-output maps are subjects of central interest in control theory and systems analysis. Related to the synthesis of observer systems is the very important question of “dynamic output stabilization”: usually in practice a stabilizing feedback law is applied to the system via the estimation of the state provided by some observer device. Also, the topic is strongly connected with filtering theory,

The most important contribution in this setting is that of Jakubczyk [14,15]. In fact, it follows a basic idea of Kalman, first for finite automata and second for linear systems. We like Jakubczyk’s approach since in particular, it contains very naturally the linear case. To our knowledge, this natural approach has not been used (in the nonlinear framework) for practical identification of nonlinear systems. However, it is rather clear that interesting develop-

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ments are possible. Moreover, it is not so hard to show complete equivalence between this geometric approach and the formal power series approach. The contribution of Crouch [4] about realization of finite Volterra series is also important, original and involves a lot of geometric considerations. We just refer to the original paper. Realizing or approximating a system by a bilinear or state linear one is an important question in view of the observer synthesis problem. We state some results on the subject. In particular, there is an important geometric representation theorem (by bilinear systems) due to Fliess and Kupka [10], that we explain. After these theoretical considerations, we go to a more practical topic: observers. Besides the linear case, there are several contributions on nonlinear observers synthesis (sliding modes, high gain, . . . ). Here, we focus on two natural generalizations of the linear results:

Some initial condition x0 2 X being fixed, such a system ˙ defines (via Cauchy existence and uniqueness Theorem) an input-output mapping P˙ : L1 [U] ! AC[R p ], u() ! y(), where L1 [U] is the set of functions defined on semi-open intervals [0; Tu [ (depending on the control u()). Possibly Tu D C1. Here AC[R p ] denotes the set of absolutely continuous functions over some interval [0; Ty [ possibly depending on the output function y(). Moreover, Ty D inffTu ; e(u; x0 )g, where e(u; x0 ) is the explosion time of the solution of (1) associated with the initial condition x0 , and the control u(). Particular cases of systems under consideration are the usual linear systems (L), bilinear systems (B) or state-linear systems (LX):

1. The output injection method (the equivalent for observability of feedback linearization) due mostly to Isidori, Krener, Respondek [19,20]; 2. The use of the deterministic version of the linear Kalman’s filter: it applies to bilinear systems, that are popular also by several approximation results (Fliess and Jacob in particular [13]).

(2)

Preliminaries Surprisingly in the nonlinear case controllability plays a role in the observability properties of a system. It is the reason for the title of the next section. Nonlinear Systems Under Consideration and Controllability We consider nonlinear systems (˙ ) of the usual form: ( x˙ D f (x; u) ; u 2 U ; (˙ ) (1) y D h(x) : Here, the state x lives either in Rn or more generally in some n-dimensional differentiable manifold X. The set U of values of control u is some arbitrary set (for simplicity, we assume a closed subset of R l , may be finite). The observation function h takes values in R p . To simplify, we will consider the analytic case only, i. e. f and h are realanalytic w.r.t. x. In the special cases where U has some analytic structure (i. e. U D R l for instance) we assume joint real analyticity w.r.t. (x; u). If W is an open subset of X, we denote by ˙ jW the system ˙ restricted to W.

8 ˆ <(L)

(B) ˆ : (LX)

x˙ D Ax C Bu; y D Cx; X D Rn ; U D R l ; x˙ D Ax C Bx ˝ u; y D Cx; X D Rn ; U D R l ; x˙ D A(u)x; y D Cx; X D Rn :

In these formulas, A; B; C; A(u) are linear. Of course, in the case of a linear system (L), with initial condition x0 D 0, the input-output mapping PL is a linear mapping. Our system ˙ is said to be “controllable” if the Lie algebra Lie(˙ ) of smooth vector fields on X generated by the vector fields f u , u 2 U (where f u (x) D f (x; u)) has dimension n at each point of X. Also, we say that a system ˙ is symmetric if 8 u 2 U, 9 v 2 U s. t. fv D  f u , and ˙ is complete if all the vector fields f u , u 2 U, are complete. The following fact is standard, for analytic systems. A system is controllable iff: 1. The accessibility set A(x0 ) of x0 2 X, i. e. the set of points that can be reached from x0 by some trajectory of ˙ , in positive time, has open interior in X, whatever x0 2 X. 2. The orbit O(x0 ) of x0 2 X, i. e. the set of points that can be joined to x0 by some continuous curve which is a concatenation of trajectories of ˙ in positive or negative time, is equal to X; whatever x0 2 X. Moreover in 1, 2 above, it is enough to restrict to piecewise constant control functions. Also, if ˙ is symmetric, O(x0 ) D A(x0 ), 8 x0 2 X. Definition and Characterization of Observability, Minimal Systems Here, C ! (X) denotes the vector space of real analytic functions over X. First, let  C ! (X) denote the “observation space of ˙ ”, i. e. the smallest vector subspace

Observability (Deterministic Systems) and Realization Theory

of C ! (X) containing the p components h i () of the output function h and closed under Lie derivation L f u in the direction of the vector fields f u , u 2 U. Then,  is also closed under Lie derivation in the direction of vector fields in Lie(˙ ) and  is generated as a real vector space by the functions (L f u r ) k r (L f u r1 ) k r1    (L f u1 ) k 1 h i . Definition 1 The observability distribution  of ˙ is the distribution ker( d) formed by the kernel of the oneforms d ; 2 . The system ˙ is said to be rank-observable if the distribution  is trivial. This fact is also called the “observability rank condition”. The important fact relating the observability and controllability properties is that the observability distribution  has no singularities as soon as ˙ is controllable: the rank of  is preserved along trajectories of vector fields f u . Moreover, it is clear that  is involutive, hence integrable by Frobenius’s Theorem. Leaves of  are levels of . Definition 2 (Indistinguishability and weak indistinguishability relations) Let I be the binary relation over X defined by x01 Ix02 if for any (piecewise constant) control u() : [0; Tu [! U such that e(u; x01 ) D e(u; x02 ) D Tu , then the corresponding output functions y1 (t); y2 (t) from both initial conditions x01 ; x02 are equal, t 2 [0; Tu [. The relation I is called the indistinguishability relation for ˙ . If V is an open subset of X, we denote by I V (V-indistinguishability relation) the indistinguishability relation for the restriction ˙ jV . The weak-indistinguishability relation, denoted by I w is the equivalence relation associated with the foliation of X generated by . The indistinguishability relation is an equivalence relation as soon as ˙ is complete. It is not an equivalence relation in general. Hence in general, V-indistinguishability also is not equivalence over V. Definition 3 The system ˙ is said to be observable if the relation I is the trivial relation. It is said to be weakly observable if for all x0 2 X, there is a neighborhood W of x0 such that for each neighborhood V of x0 , V W, I V (x0 ) D x0 . Then weak observability means that locally, we can find inputs such that the initial conditions are distinguished by the observations, in arbitrarily short time. Observability means just that distinct initial conditions can be distinguished by observations. The system ˙ being observable, analytic, this can be done in arbitrary short time. In view of realization theory, we say that ˙ is minimal if it is both controllable and observable. We say that it is weakly minimal if it is controllable and weakly observable.

Definition 4 A universal input for ˙ is an input u(), that distinguishes among any pair of distinct states in arbitrarily short time. Observers For a system ˙ of the form (1) (that we assume to be observable) an observer is a system of the form: ( z˙ D F(z; y; u) ; (3) xˆ D H(z; u) ; where z 2 Z, some manifold. The observer system is fed by y(t) and u(t), the output and input of ˙ . The mapping H : ZU ! X, and we require that, for a large set of initial condition z0 for z, the output xˆ (t) tracks asymptotically the state x(t) of the system, i. e. at least, lim d(xˆ (t); x(t)) D 0 ;

t!C1

(4)

where d is some (Riemannian) metric over X. In general, there are additional requirements on the rate of convergence to zero of the estimation error "(t) D d(xˆ (t); x(t)) (such as exponential convergence, with arbitrary exponential rate). Of course even without such additional requirements, this definition is very vague and not serious at all. It has to be made more precise, depending on the context. There are mostly two types of problems:  This definition depends on the metric d. It may happen that "(t) goes to zero for some Riemannian metric d, although it goes to infinity for some other metric d 0 . Also, the state variables z or x may explode in finite time. Therefore, in general it is reasonable to require (4) only for trajectories of ˙ that remain in a given compact subset of X for all positive times. In that case, the usual convergence requirements becomes independent of the Riemannian metric d.  One cannot expect to observe unobservable systems. Therefore, one has to require convergence for “good” inputs only. Abstract Definition of an Input-Output Map We define the topological group G (resp. the topological semi group S) of extended (resp. positive time) piecewise constant controls as follows: typical elements of G and S are words of the form: ˇ ˇt ) D (u k ; t k )    (u1 ; t1 ) ; u(

(5)

where u i 2 U and t i 2 R (resp. RC ). The operation over G and S is the concatenation of words. We consider

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ˇ ˇt )" D "u( ˇ ˇt ) D u( ˇ ˇt ). We also the neutral element " : u( define the equivalence relation over G and S as being generated by the relations: ( (u; 0) " ; (6) (u; s)(u; ) (u; s C ) :

Remark 6 Given a pointed nonlinear system (˙; x0 ) where ˙ is of the form (1) and x0 2 X, it is clear that the associated input-output mapping defines an abstract input-output mapping, the rank of which is the dimension n of the state space.

We consider the quotient spaces G :D G/ , S :D S/ . Both are embedded with the topology co-induced by the maps:

Linear Systems

ˇ : Rk ! G u()

(resp. (RC ) k ! S) :

ˇ ˇt ) D (u k ; t k )    (u1 ; t1 ) 2 S, we For 2 RC and u( define ˇ ˇt ) D (urC1 ;  r )(ur ; tr )    (u1 ; t1 ) u( for 2 [r ; rC1 [ ;  r D t1 C    C t r ; ˇ ˇt ) D u( ˇ ˇt ) for   k : u( A real mapping: P : D  G ! R (resp. S ! R) with ˇ ˇt ) 2 D, open domain D is said to be analytic if, for all u( ˇ ˇ ˇ ˇ the mapping t ! P(u( t )) is analytic at t as a mapping R k ! R. The domain D of P : D  S ! R is said to be “starshaped” if a 2 D for all 2 RC and a 2 D. ˇ s ) D ((bˇ m (ˇs m ); : : : ; bˇ 1 (ˇs1 )) 2 G m (resp. Sm ), Denote B(ˇ with     bˇ i (ˇs i ) D b i n i ; s i n i    b i 1 ; s i 1 ; b i j 2 U; and set: ˇ



ˇ



ˇ

b m (ˇs m ) b 1 (ˇs 1 ) B(ˇs ) D u( ; : : : ; u( u( ; ˇ ˇt ) ˇ ˇt ) ˇ ˇt ) ˇ

b i (ˇs i ) ˇ ˇt )) : D P(bˇ i (ˇs i )u( u( ˇ ˇt )

The rank of P is defined as rank(P) D

sup

ˇ

B(ˇs ) rank D ˇt u( ; ˇ ˇt )

ˇ s );u( ˇ ˇt ) k; B(ˇ

where D ˇt means the differential w.r.t. ˇt 2 R k , and all the arguments belong to the possible domain defined by the domain D of P. Definition 5 An (abstract) input-output mapping P is an analytic mapping, from some open and star-shaped subset D S, with finite rank. An extension PC of an analytic mapping P is an analytic mapping such that dom(P) dom(PC ) S and P D PC jdom(P) (restriction of PC to dom(P)).

The simplest case for observability, design of observers and realization theory is the linear case. Given a linear system (L) from (2) the following results are standard and more or less obvious:  The observability property is independent of the control u() applied to the system, i.e (L) is observable iff it is observable for some fixed arbitrary control u().  The observability distribution  is a field of constant i planes, given by  D \n1 iD1 ker(CA ). Then ˙ is observable iff rank() D 0. This condition is known as the observability rank condition.  If (L) is observable the following device (Luenberger observer): (

z˙ D (A  ˝C)z C ˝ y C Bu ; ˝ : Rn ! R p ;

z 2 Rn ;

(7)

is an arbitrary exponential rate observer, i. e. the matrix ˝ can be chosen in such a way that the matrix A  ˝C has arbitrary spectrum, which implies: k"(t)k D kz(t)  x(t)k  k(˛)e ˛t kz0  x0 k D k(˛)e ˛t k"0 k ;

(8)

where ˛ > 0 is arbitrary, and k is some polynomial in ˛, independent of ˝.  Any linear system restricts to a controllable one on some subspace, and is mapped to an observable one, by the canonical projection ˘ : Rn ! Rn /I (where I is the indistinguishability relation from Definition 2).  Let Y(t) denote an “impulse response” (Y(t) : R l ! R p ; t  0). The input-output map is the causal linear mapping P : u() ! y() D Y u, where denotes the convolution of (positive time) signals. Then assume P1 tk that (as a formal power series) Y(t) D kD1 G k k! , and let H denote the infinite block-Hankel matrix constructed from the sequence of blocks G1 ; G2 ; : : : ; Gk ; : : : Then, Y(t) is the impulse response of a linear system (L) iff H has finite rank n.

Observability (Deterministic Systems) and Realization Theory

Observability of Nonlinear Systems What is clear is that if a system is rank-observable, then it is weakly observable. This is due to a Baker–Campbell– Hausdorf like formula, valid for piecewise constant conˇ ˇt ): trols u( r X t kk    t1r 1 rk r1 : y(t) D (L f u k )    (L f u 1 ) h(x0 ) r k !    r1 !

(9)

Indeed by real analyticity, if y1 (t) D y2 (t), all the terms (L f u k )r k    (L f u 1 )r 1 h(x01 ) and (L f u k )r k    (L f u 1 )r 1 h(x02 ) are equal, which contradicts the rank assumption, for x01 ; x02 close enough. Conversely, assume that ˙ is controllable and not rank-observable. Then, the observability distribution  is constant rank, integrable, nontrivial. Leaves of  are levels of . By the same formula (9) points of such leaves are indistinguishable. Therefore ˙ is not weakly observable. Then, the following theorem holds: Theorem 7 A controllable system ˙ is weakly observable iff it is rank-observable. The other important result (Sussmann [28]) is: Theorem 8 If ˙ is observable, there is a universal input. Moreover, the set of universal inputs is generic. Realization Theory Minimal Realizations Given a Realization We are given a realization i. e. a pointed system (˙; x0 ), x0 2 X. In fact, the results follow from the Sussmann’s theorem on quotient manifolds: a closed equivalence relation R differentiably passes to the quotient (i. e. quotient is a manifold and canonical projection is submersive) if there are enough complete vector fields that respect R. We apply this theorem to the indistinguishability relation R D I in the case of a complete and controllable system. Then all vector fields of Lie(˙ ) respect I. This is exactly Sussmann’s requirement, so that not only there is a quotient manifold and canonical mapping is submersive, but moreover vector fields of Lie(˙ ) pass to the quotient. Also, the elements of  obviously pass to the quotient. If ˙ is not controllable, then, as a first step, we can use the standard Hermann–Nagano Theorem to restrict to a (controllable) leaf of the distribution Lie(˙ ). Then, we have a similar theorem to the one of the linear case. Theorem 9 If ˙ is complete, then we can restrict to the leaf of Lie(˙ ) containing x0 2 X to get a controllable system. Passing to the quotient manifold by the indistinguishability relation I, we get a minimal realization. Moreover, two

minimal realizations are unique up to a diffeomorphism of the state spaces. For complete systems, there is an interesting refinement of this theorem. A realization is said to be weakly-minimal if it is controllable, weakly observable. It turns out that the equivalence relation I w associated to  meets also Sussmann’s conditions. It follows that the system goes to the quotient, and we get a weakly minimal realization ˙ˆ with ˆ We can apply the previous Theorem 9 to ˙ˆ state space X. to get again the (unique) minimal realization ˙m of ˙ , with state space X m . The following theorem is almost obvious. Theorem 10 Xˆ is a covering space of X m . Moreover, any covering space of X m determines a weakly-minimal realization of ˙ , by a trivial lifting procedure. In particular, there is (up to diffeomorphisms) a single simply-connected weakly-minimal realization. Note that in fact the relation I w is the same relation as: x01 I w x02 if there is a continuous curve : [0; 1] ! X connecting x01 to x02 and for r; s 2 [0; 1], (r)I (s). Minimal Realizations Given an Abstract Input-Output Map The set of controls U being given, we consider an abstract input-output map defined over the whole group G (domain D D G). Note that this is the case in particular for the input-output mappings determined by a complete symmetric system. In that case we have the following theorem, due to Jakubczyk [14]. Theorem 11 An abstract input-output mapping with domain G has a unique minimal realization, which is complete. Remark 12 The finite rank assumption for the input-output mapping is a generalization of the finite rank assumption of the Hankel matrix of the linear case. It is also the analog of certain finite rank assumptions appearing in the formal power series approach of Fliess, or in the Volterrakernels approach. Remark 13 There is one ugly detail in this theory: in general, we do not get a paracompact manifold as the state space X. The idea for the proof of the theorem is very simple: we consider the subgroup H of G defined by H D fa 2 GjP(ca) D P(c); 8c 2 Gg. Then, the state space will just be X D G/H. The finite rank condition implies that X has the structure of a Hausdorff analytic manifold. The output

1199

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Observability (Deterministic Systems) and Realization Theory

function h is defined by h(gH) D P(g). The vector-field f u is defined via its one parameter group: exp(t f u )(gH) D ˘ ((u; t)g), where ˘ : G ! G/H is the canonical projection. A more practical result is the following: if we assume that the set U of values of the control is a finite set, then the following global result (containing a local one) can be proven. Theorem 14 Assume U is finite, then, a necessary and sufficient condition for P to have a realization (weakly-minimal) is that P has an extension P C with star-shaped domain DC . The state space X of this realization is Hausdorff, paracompact. In the general analytic case with infinite U, existence of certain local realizations only, can be proved.

Besides the fact that this result allows one to solve the observer problem for such systems, “truncating” in some manner the observation space is a way to approximate systems by state-linear ones, and to get approximate observers. An interesting particular case where this theorem applies is the case of systems with polynomial observation h and state-linear dynamics: ( x˙ D A(u)x ; y D P(x) ; where P is some polynomial mapping. It is clear that  is finite-dimensional. More generally, if we start with a system with state-linear dynamics, we can uniformly approximate h on compact sets by a polynomial mapping to get a state-linear realization (and later on, an approximate observer device).

Bilinear or State-Linear Realization This point will be extremely important for the problem of constructing observer systems (Sect. “Observers”). A system is said to be control affine if the vector fields f u form an affine family w.r.t. u. The single control case (l D 1) is just the case f (x; u) D f (x) C g(x)u where f and g are two vector fields on X. Note that a bilinear system is just a state linear system, which is moreover affine in the controls. A state linear realization (LX; x0 ) from Eq. (2) is said to be minimal if it is observable and controllable in the following sense: the orbit of x0 is not contained in a strict subspace of Rn (the smallest such subspace would be automatically invariant under all the operators A(u); u 2 U). First, it is rather simple to show that any pointed statelinear system (LX; x0 ) has a minimal state-linear realization. Of course, the additional property to be bilinear is hereditary. Note that for state-linear systems, the observation space is a (finite-dimensional) vector space of linear forms over X. It turns out that this finite dimensionality condition is in fact a necessary and sufficient condition. This is a very important result from Fliess and Kupka [10]: Theorem 15 Assume that ˙ has a finite dimensional observation space . Then, ˙ is embeddable in a state-linear system. In other terms (˙; x0 ) has a state linear (minimal) realization. The proof is very easy. It is enough to take:    

X D   (dual space of ), For ' 2   ; C i ' D '(h i ), i D 1; : : : ; p, A(u) D (L f u ) (transpose of L f u ), The initial state xˆ0 meets xˆ0 (') D '(x0 ) for ' 2 .

State-Linear Skew-Adjoint Realization Here, for the sake of simplicity in the exposition we limit ourselves to the single output case p D 1. This section describes some particular cases and some generalizations of the results of the previous section, in view of synthesis of observers with a method presented in Subsect. “Observers for Skew-Adjoint State Linear Systems”. For some reason that will be made clear in the Subsect. “Observers for Skew-Adjoint State Linear Systems” we would like to know when it is possible to embed a system (or to have a realization of a system) into a skew-symmetric, or more generally skew-adjoint, state-linear one. This means that all the matrices A(u) are skew-symmetric w.r.t. the usual scalar product over the state space Rn of the realization. By Theorem 15, a necessary condition for the nonlinear system ˙ (minimal and complete) to be embeddable is that  be finite dimensional and hence the group of diffeomorphisms of X generated by the vector fields f u be a Lie group G. One could think that a necessary condition is that G be a compact Lie group. This is not the case as the following example shows: ( x˙ D u ; x; u 2 R ; y D cos(x) C cos(˛x) ;

where ˛ is irrational ;

since the group of diffeomorphisms is R, while a skew symmetric embedding does exist. The proper condition is given by the following theorem: Theorem 16 The system ˙ (complete, minimal) can be embedded into a state-linear skew symmetric system iff:

Observability (Deterministic Systems) and Realization Theory

1. dim() < 1 (from which it follows that G is a Lie group), 2. The observation function h(x) lifts over G into h˜ (in a natural way), an almost periodic function over G.

Theorem 17 The system ( g˙ D A(u)g ; (˙n ) y D h m (g) ;

Recall that an almost periodic function over G is a function that prolongs into a continuous function over the Bohr compactification G [ of G [5]. The two conditions of Theorem 16 are equivalent to the fact that G is a Lie group and h˜

has a (infinite dimensional) skew-adjoint state linear realization on a separable complex Hilbert space H , i.e: ( ˙ D A(u) ;

is a finite linear combination of coefficients1 of unitary irreducible finite dimensional representations of G. If G is “embeddable in a compact group”, i. e. if G is the semi-direct product of a compact group by a finite dimensional real vector space then, any h can be approximated in some sense by an almost periodic one. Actually, a special interesting case is the following: the system ˙ is such that X D G, a compact Lie group, and the vector fields f u are right invariant vector fields over G. We can take h as any continuous function h : G ! R, and consider the abstract Fourier transform hˆ of h. In fact, by Peter–Weyl’s Theorem [3], h is a uniform limit over G of finite linear combinations of the form X h(g) D ˛ i ˚ i (g) ; i

where ˚ i (g) is a coefficient of an irreducible (hence finite dimensional) unitary representation of G. This means that h has approximations hm that converge uniformly to h over G, such that each system ( g˙ D A(u)g ; (˙m ) y D h m (g) ; has a state-linear minimal realization of the form: ( x˙ D A m (u)x ; x 2 C n ; A m (u) is skew-adjoint ; y D Cm x : Hence the input-output mapping of any right invariant system over a compact group can be approximated by the one of a skew-adjoint state-linear one. Now, let us consider again a (complete, minimal) system ˙; with finite dimensional Lie algebra, but the group G is not compact. In that case h˜ (a lift of h over G) can be approximated uniformly on any compact subset of G by a function hm , which is a finite linear combination of “positive type” functions over G. This approximation result is known as the Gelfand–Raikov Theorem [5]. As a consequence we have the theorem: 1 A coefficient of a representation is a coefficient of the matrix representing the representation operator in certain orthonormal basis.

y D h; i : Here ;  2 H and h; i is the scalar product over H . All the operators A(u) are densely defined, essentially skew-adjoint operators, infinitesimal generators of strongly continuous one parameter groups of unitary operators over H . With this result, in Subsect. “Observers for Skew-Adjoint State Linear Systems”, we will be able to construct reasonable approximate observers for ˙ . Observers Kalman’s Observer for State-Linear Systems This is just the deterministic version of the linear timedependant Kalman filter. Therefore, inputs being known, it applies to state-linear systems (LX) from (2). Contrarily to linear systems, observability for those systems is not a property independent of the inputs: for some input u() it might be observable, for others it might not be. Clearly, if we want the observer to have some asymptotic property of convergence of the estimation error, it is reasonable to require that the input under consideration keeps a certain minimum level of observability when the time grows to infinity. It is natural to consider inputs living in the space U D L1 [0;1[;R p of measurable U-valued bounded functions. For an input u 2 U and for a real a  0, set u a (t) D u(t C a). We denote by ˚u (t) the matrix resolvent of the linear equation ˚˙ u (t) D A(u(t))˚u (t). Then for T > 0, the Gramm-observability matrix: Z T ˚u (t) C  C˚u (t) dt ; (10) G u;T D 0 

where stands for adjoint operator, measures observability in the following sense: the system is observable for u : [0; T] ! U iff Gu;T is positive definite. Hence there are several types of assumptions that are possible to express that u : [0; C1[! U keeps a certain level of observability when time passes. The most simple one is the following: There are ˛; T; T0 > 0 such that for all  T0 , Gu  ;T  ˛:Id n , where Id n is the identity matrix. This condition means intuitively that, from time T 0 on, the input u

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has minimum observability level ˛ on all time intervals of length T. Such an input could be called a “persistent-excitation” for ˙ . Then, the following theorem is just a restatement of the classical results about the deterministic version of the linear time-dependant Kalman’s filter: Theorem 18 ([18]) The matrices Q and R being positive definite symmetric matrices with adequate dimensions, the Riccati system: ( (1) S˙ D A(u(t))0 S(t)S(t)A(u(t))CC  R1 CSQS; (2) z˙ D A(u(t))z  S 1 C  R1 (Cz  y(t)) ; (11) is an asymptotic observer for persistent-excitations u(). Convergence of the estimation error is exponential. The matrices S(t) (as soon as the same holds for the initial condition S0 ) live in the open cone of positive definite symmetric matrices. Observers for Systems that are Injectable in a State-Linear One Of course, the technique of the previous section applies stricto-sensu to such systems from Subsect. “Bilinear or State-Linear Realization”. The Output-Injection Idea It turns out that both the Luenberger observer (7) for linear systems and the Kalman observer (11) for statelinear systems can be applied in more general nonlinear situations. Assume that ˙ is linear “up to output injection”, i. e. 

x˙ D Ax C '(y; u) ; (12) (˙ ) y D Cx or respectively that ˙ is state-linear (or bilinear) up to output injection, i. e.

 x˙ D A(u)x C '(y; u) (˙ ) ; (13) y D Cx where ' (the output injection) is some nonlinear term depending on the output and input only. Then there are easy modifications of the Luenberger observer (resp. the Kalman’s observer) that provide exactly the same results of convergence of the estimation error as for the corresponding systems without the output-injection term. For case (12) we take the observer under the Luenberger-modified form: z˙ D (A  ˝C)z C '(y; u) C ˝(y  Cz) ;

while for case (13) we take: ( S˙ D A(u(t))0 S(t)  S(t)A(u(t)) C C  R1 C  SQS z˙ D A(u(t))z C '(y; u)  S 1 C  R1 (Cz  y(t)) : To check the result it is enough to write the estimation error equation and to see that it is exactly the same as in the situation without output-injection. For that reason, it is important to characterize systems that can be put under the form of a linear or statelinear system up to output-injection. There is an industry around this question. It starts with the works of Isidori, Krener, Respondek [19,20]. The first result of this type is in the uncontrolled case. For an uncontrolled system ( x˙ D f (x) (˙ ) y D h(x) ; with single output (p D 1), consider the vector fields X i defined by L X1 (L f ) i1 h D ı i;n ;

i D 1; : : : ; n ;

where ı is the Kronecker symbol ; X j D  [ f ; X j1 ] ;

j D 2; : : : ; n ;

The system ˙ can be linearized up to a diffeomorphism and an output injection iff the two following conditions are met [19]: 1. The family fdh; dL f h; : : : ; d(L f )n1 hg has full rank n at all points of X. 2. [X k ; X m ] D 0 for 1  k, m  n. Of course, this is a “local almost everywhere result” only. There is also a lot of results on the problem of characterizing systems that are diffeomorphic to or embeddable in state-linear systems up to output injection. A significant result to the problem of embedding up to output injection is the one of Jouan [16]. Observers for Skew-Adjoint State Linear Systems Again, to simplify the exposition we consider the single output case p D 1 only. In this case we have a (minimal) state-linear realization which is also skew-adjoint, there is a construction of an observer which is much simpler than Kalman’s one [no Riccati equation besides the prediction-correction Eq. (11), (2)]. Moreover this construction extends to infinite-dimensional realizations, a fact which allows it to treat any (complete minimal) system with finite dimensional Lie algebra.

Observability (Deterministic Systems) and Realization Theory

To start, consider some skew-symmetric state linear system: ( (LX)

x˙ D A(u)x ; A(u) skew-symmetric 8 u 2 U ; y D Cx ; (14)

We consider the following candidate observer system: z˙ D A(u)z  rC  (Cz  y) ;

bounded, such that the translated inputs u n : [0; T] ! R l converge to u in the weak–* topology of L1 (which [0;T];R l  topology is precompact over bounded sets) and u is a universal input for ˙ on [0; T]. This means also that a certain level of observability is preserved, on regularly spaced time intervals, while the time increases. In that case, of course the result is weaker than in the finite dimensional case. We have only:

(15) weak– lim "(t) D 0 :

in which r > 0 is a parameter. The estimation error is " D z  x:

Future Directions

"˙ D (A(u)  rC  C)" : Then it is not so hard to show that, if u : [0; 1[! U is a “persistent excitation” of ˙ in some sense (for instance in the sense of Subsect. “Kalman’s Observer for State-Linear Systems”, then we have: lim k"(t)k D 0 :

t!C1

As a consequence, the systems with compact group G of diffeomorphisms (or with G semidirect product of compact group by vector space), admit also approximate observers, using the results of Subsect. “State-Linear Skew-Adjoint Realization”. It turns out that this method can be extended in a reasonable way to systems with (infinite dimensional) skewadjoint state-linear realization. In particular, it is possible to construct approximate observers for all (complete minimal) systems with finite dimensional Lie algebra. Consider a skew adjoint realization from Subsect. “State-Linear Skew-Adjoint Realization”: ( ˙ D A(u) ; y D h; i on the (separable) Hilbert space H . Then, the candidate observer device is: ˙ D A(u)  r(h; i  y(t)) :

t!C1

(16)

In fact, the persistency assumption cannot be of the same type as in the finite dimensional case. The reason is that the Gramm observability matrix G u;T is a compact operator in that case. Hence it cannot satisfy an inequality of the type Gu;T  ˛IdH since H is infinite dimensional. Hence, the definition of a persistent excitation has to be replaced by one of the following type: there is a time T > 0 and a real sequence n , n ! C1, with nC1  n

For observability and synthesis of observers, besides the improvement of the current methods (including sliding modes, high gain, . . . ) several directions have to be investigated more deeply, namely infinite dimensional systems, delay and hybrid systems. For realization theory, and as a consequence identification theory, almost no “practical result” is known in the nonlinear context. However, we think interesting and consistent developments are possible, even starting from the apparently abstract theory outlined there. This is clearly the challenge for the future.

Bibliography Primary Literature 1. Bartosiewicz Z (1995) Local observability of nonlinear systems. Syst Control Lett 25(4,1):295–298 2. Bartosiewicz Z (1998) Real analytic geometry and local observability. Proc Sympos Pure Math. Am Math Soc 64:65–72 3. Barut AO, Raczca R (1986) Theory of Group Representations and Applications. World Scientific, Singapore 4. Crouch PE (1981) Dynamical realizations of finite volterra series. SIAM J Control Opt 19:177–202 5. Dixmier J (1964) Les C  algébres et leurs représentations. Cahiers Scientifiques, fasc. XXIX. Gauthier-Villars, Paris 6. Fliess M (1981) Fonctionnelles causales non linéaires et indéterminées non commutatives. Bull Soc Math France 109: 3–40 7. Fliess M (1983) Réalisation locale des systèmes non linéaires, algèbres de Lie filtrés transitives et séries génératrices non commutatives. Inventiones Mathematicae 71:521–537. Springer 8. Fliess M (1973) Sur la realization des systemes dynamiques bilineaires. CR Acad Sc Paris A 277:923–926 9. Fliess M (1980) Nonlinear realization theory and abstract transitive Lie algebras. Bull Amer Math Soc (NS) 2:444–446 10. Fliess M, Kupka I (1983) A finiteness criterion for nonlinear input-output differential systems. SIAM J Control Optim 21:721–728

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11. Grizzle JW, Moraal PE (1995) Observer design for nonlinear systems with discrete-timemeasurements. IEEE Trans Autom Control 40(3):395–404 12. Hermann R, Krener AJ (1977) Nonlinear controllability and observability. IEEE Trans Autom Control 22(5):728–740 13. Jacob G, Hespel C (1991) Approximation of nonlinear dynamic systems by rational series. Theor Comp Sci Arch 79(1):151– 162 14. Jakubczyk B (1986) Local realizations of nonlinear causal operators. SIAM J Control Optim 24(2):230–242 15. Jakubczyk B (1984) Réalisations locales des opérateurs causals non linéaires. Comptes rendus des séances de l’Académie des sciences. Série A 299(15):787–793 16. Jouan P (2003) Immersion of nonlinear systems into linear systems modulo output injection. SIAM J Control Optim 41(6):1756–1778 17. Kalman RE (1963) Mathematical description of linear dynamical systems. SIAM J Control Optim 1(2):152–192 18. Kalman RE, Bucy RS (1961) New results in linear filtering and prediction theory. J Basic Eng Trans ASME 83:95–108 19. Krener AJ, Isidori A (1983) Linearization by output injection and nonlinear observers. Syst Control Lett 3(1):47–52 20. Krener AJ, Respondek W (1985) Nonlinear observers with linearizable error dynamics. SIAM J Control Optim 23:197– 216 21. Luenberger D (1966) Observers for multivariable systems. IEEE Trans Autom Control 11(2):190–197 22. Luenberger D (1971) An introduction to observers. IEEE Trans Autom Control 16(6):596–602 23. Sontag E (1979) On the observability of polynomial systems, I: Finite-time problems. SIAM J Control Optim 17(1):139–151

24. Sussmann HJ (1973) Orbits of families of vector fields and integrability of distributions. Trans Amer Math Soc 180:171–188 25. Sussmann HJ (1975) A generalization of the closed subgroup theorem to quotients of arbitrary manifolds. J Diff Geom 10:151–166 26. Sussmann HJ (1974) On quotients of manifolds: a generalization of the closed subgroup theorem. Bull Amer Math Soc 80:573–575 27. Sussmann HJ (1976) Existence and uniqueness of minimal realizations of nonlinear systems. J Theor Comput Syst 10(1):263– 284, 371–393 28. Sussmann HJ (1978) Single-input observability of continuoustime systems. J Theor Comput Syst 12(1):371–393 29. Yamamoto Y (1981) Realization theory of infinite-dimensional linear systems, Part I. Math Syst Theor 15:55–77

Books and Reviews Isidori A (1995) Nonlinear Control Systems. Springer Communications and Control Engineering Series. Springer, New York Kalman RE, Falb PL, Arbib MA (1969) Topics in Mathematical System Theory. Mc Graw Hill, New York Khalil H (2002) Nonlinear systems, 2nd edn. Prentice Hall, Upper Slade River Nijmeijer H, van der Schaft A (1990) Nonlinear Dynamical Control Systems. Springer Rugh WJ (1981) Nonlinear System Theory, The Volterra/Wiener Approach. The Johns Hopkins University Press, Baltimore Sontag ED (1998) Mathematical Control Theory: Deterministic Finite Dimensional Systems, 2nd edn. Springer, New York Utkin VI (1992) Sliding Modes in Control and Optimization. Springer

Partial Differential Equations that Lead to Solitons

Partial Differential Equations that Lead to Solitons ˘ DO GAN KAYA Department of Mathematics, Firat University, Elazig, Turkey

Article Outline Definition of the Subject Introduction Some Nonlinear Models that Lead to Solitons Future Directions Bibliography Definition of the Subject In this part, we introduce the reader to a certain class of nonlinear partial differential equations which are characterized by solitary wave solutions of the classical nonlinear equations that lead to solitons. The classical nonlinear equations of interest show the existence of special types of traveling wave solutions which are either solitary waves or solitons. In this study, we will review a few solutions arising from the analytic work of the Korteweg–de Vries (KdV) equations, the generalized regularized longwave RLW equation, Kadomtsev–Petviashvili (KP) equation, the Klein–Gordon (KG) equation, the Sine-Gordon (SG) equation, the Boussinesq equation, Pochhammer– Chree (PC) equation and the nonlinear Schrödinger (NLS) equation, the Fisher equation, Burgers equation, the Korteweg–de Vries Burgers’ equation (KdVB), the two-dimensional Korteweg-deVries Burgers’ (tdKdVB), the potential Kadomtsev–Petviashvili equation, the Kawahara equation, Generalized Zakharov-Kuznetsov (gZK) equation, the Sharma-Tasso-Olver equation, and the Cahn– Hilliard equation. Introduction Nonlinear phenomena play a crucial role in applied mathematics and physics. Calculating exact and numerical solutions, in particular, traveling wave solutions, of nonlinear PDEs in mathematical physics plays an important role in soliton theory. Moreover, these equations are mathematical models of complex physical occurrences that arise in engineering, chemistry, biology, mechanics, and physics. In this work, we give a brief history of the abovementioned nonlinear equations and how this type of equation has led to the soliton solutions; we then present an introduction to the theory of solitons. Soliton theory is an

important branch of applied mathematics and mathematical physics. In the last decade this topic has become an active and productive area of research, and applications of the soliton equations in physical cases have been considered. These have important applications in fluid mechanics, nonlinear optics, ion plasma, classical and quantum fields’ theories etc. The best introduction to the soliton is that contained in J. Scott Russell’s (a Scottish naval engineer) seminal 1844 report to the Royal Society. Scott Russell’s report titled “Report on Waves” [62] was presented to the Royal Society in the 18th Century and he wrote: “I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel . . . ” Scott Russell’s experimental observations in 1834 were followed by the theoretical scientific work of Lord Rayleigh and Joseph Boussinesq around 1870 [60] and finally, two Dutchmen, Korteweg and de Vries, developed a nonlinear partial differential equation to model the propagation of shallow water waves applicable to the situation in 1895 [47]. This work was really what Scott Russell fortuitously witnessed. This famous classical equation is known simply as the KdV equation. Korteweg and de Vries published a theory of shallow water waves which reduced Russell’s observations to its essential features. The nonlinear classical dispersive equation was formulated by Korteweg and de Vries in the form @u @u @u @3 u Cc C" 3 C u D0 @t @x @x @x where c; "; are physical parameters. This equation plays a key role in soliton theory. In the late 1960s, Zabusky and Kruskal numerically studied and then analytically solved the KdV equation [79]. They came to the result that stable pulse-like

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waves could exist in a problem described by the KdV equation from their numerical study. A remarkable aspect is that the discovered solitary waves retain their shapes and speeds after collision. Zabusky and Kruskal called these waves solitons as they resemble particles in nature. In 1967, this numerical development was placed on a settled mathematical basis with Gardner et al.’s discovery of the inverse-scattering-transform method [20]. The soliton concept is related with solutions for nonlinear partial differential equations. The soliton solution of a nonlinear equation usually is used a single wave. If there are several soliton solutions, these solutions are called solitons. On the other hand, if a soliton is separated infinitely from another soliton, this soliton is called a single wave. Besides, a single wave solution can’t be sec h2 function for equations other than nonlinear equations, such as the KdV equation. But this solution can be sech or tan1 (e˛x ). At this stage, we could ask what the definition of the soliton solutions is? It is not easy to define the soliton concept. Wazwaz [69] describes solitons as solutions of nonlinear differential equations as follows: (i)

A long and shallow water wave should not lose its permanent form; (ii) A long and shallow water wave of the solution is localized, meaning either the solutions decay exponentially to zero such as in the solitons admitted by the KdV equation, or approach a constant at infinity such as solitons provide by the SG equation; (iii) A long and shallow water wave of the solution can interact with other solitons preserving its character. There is also a more formal definition of this concept, but these definitions require substantial mathematics [17]. On the other hand, the phenomena of the solitons doesn’t always follow these three properties. For example, there is the concept of “light bullets” in the subject of nonlinear optics which are often called solitons despite losing energy during interaction. This idea can be found at an internet web page of the Simon Fraser University British Columbia, Canada [51]. Some Nonlinear Models that Lead to Solitons In this section, we will deal with the fundamental ideas and some nonlinear partial differential equations that lead to solitons. These equations are the result of a huge amount of research work and physical implementations which focus on the most diverse and active areas of applied mathematics and mathematical physics. Example 1 In various fields of science and engineering, nonlinear evaluation equations, as well as their an-

alytic and numerical solutions, are of fundamental importance. One of the most attractive and surprising wave phenomena is the creation of solitary waves or solitons. It was approximately two centuries ago, that an adequate theory for solitary waves was developed, in the form of a modified wave equation known as the KdV equation [14,15,16,47,69,75]. The third-order KdV equation is a classical nonlinear partial differential equation originally formulated to model shallow water flow [47]. Herein, we are particularly interested in the original third-order KdV equation [47] given by u t C u ux C ˛ ux x x D 0 ; respectively, and the generalized form u t C u p ux C ˛ ux x x D 0 ; where p models the dispersion and subscripts in t and x denote partial derivatives with respect to these independent variables. The study of long waves, tides, solitary waves, and related phenomena, leads to an equation referred to as the generalized KdV equation. If p D 0; p D 1, and p D 2, this equation becomes the linearized KdV, nonlinear KdV, and modified KdV equations, respectively [14,15,16,69,75]. The modified KdV equation has also found applications in many areas of physical situations, for instance the equation describes nonlinear acoustic waves in an anharmonic lattice [78]. Example 2 Consider the solution u(x; t) of the generalized RLW equation u t C u x C ˛ ux  ˇ u x x t D 0 ; where  is a positive integer, and ˛ and ˇ are positive constants. The generalized RLW equation was first put forward as a model for small-amplitude long waves on the surface of water in a channel by Peregrine [37,57] and later by Benjamin et al. [2]. In physical situations such as unidirectional waves propagating in a water channel, longcrested waves in near-shore zones, and many others, the generalized RLW equation serves as an alternative model to the KdV equation [4,5]. Example 3 The nonlinear KG equations in one-dimensional space u t t  ux x C

dV D0; du

where V D V (u) is a general nonlinear function of u, but not its derivatives. This equation is the most natural nonlinear generalization of the wave equation and first

Partial Differential Equations that Lead to Solitons

arose in a mathematical context, with V (u) D exp(u), in the theory of surfaces of constant curvature. In addition, this equation appears in many different fields of application. For instance, a polynomial nonlinearity can be used as a model field theory, while a cos u term yields the sine-Gordon equation and so on [14,15,16,69,75]. We will consider a particular case of equation KG, the so-called sine-Gordon nonlinear hyperbolic equation, which has the form u t t  cu x x C  sin(u) D 0 ;

1 < x < 1 ; t > 0 ;

where c and  are constants. The sine-Gordon equation is firstly used in the work of differential geometry and for investigation of the propagation of a ‘slip’ dislocation in crystals [15]. The generalized one-dimensional KG equation u t t  ku x x C b1 u C b2 u mC1 C b3 u2mC1 D 0 ; is given in [80]. The KG equation represents a nonlinear model of longitudinal wave propagation of elastic rods when m D 1 [12]. Example 4 In this example, we consider the generalized Boussinesq-type equation [7] u t t  u x x C ıu x x x x D ( (u))x x ; 1< x <1; t >0; where ı  0 is constant, (u) D juj˛1 u and ˛ > 1. This equation represents a generalization of the classical Boussinesq equation which arises in the modeling of nonlinear strings. The Boussinesq equation describes in the continuous limit the propagation of waves in a one-dimensional nonlinear lattice and the propagation of waves in shallow water [6,7,15,77]. A proof of the local wellposedness of the Boussinesq equation has been showed by Bona and Sachs [6]. In [13] the authors noticed that the Boussinesq equation admits solitary solutions and the existence of solitary wave solutions illustrates the perfect balance between dispersion and the nonlinearity of the Boussinesq equation. For the ˛ integer, solitary-wave solutions have been shown to be stable under some restrictions on the wave speed by Bona and Sachs [6]. Scott Russell’s study [62] of solitary water waves motivated the development of nonlinear partial differential equations for the modeling of wave phenomena in fluids, plasmas, elastic bodies, etc. The Boussinesq equation is an important model that approximately describes the propagation of long waves on shallow water like the other Boussinesq equations (with uxxtt , instead of uxxxx ). This

equation was first deduced by Boussinesq [7]. In the case ı > 0 this equation is linearly stable and governs small nonlinear transverse oscillations of an elastic beam [15]. It is called the “good” Boussinesq equation, while the equation with ı < 0 received the name “bad” Boussinesq equation since it possesses linear instability. Example 5 Consider the PC equation ut t 

1 ` u  u x x  u t tx x D 0 ; p xx

where ` indicates different material with different values of it. The PC equation is applied as a nonlinear model of longitudinal wave propagation of elastic rods [15,16,75]. In the work of Bogolubsky [75], the author obtained exact solitary wave solutions to the PC equation ` D 2; 3; 5; respectively [15]. Example 6 In this paper, we consider the Burgers equation u t C "uu x  u x x D 0 where " and  are parameters and the subscripts t and x denote differentiation. The Burgers’ equation is a model of flow through a shock wave in a viscous fluid [13] and in the Burgers’ model of turbulence [9]. Example 7 The Fisher equation  u u t  u x x D ku 1  ;  is well known in population dynamics, where v > 0 is the diffusion constant, k > 0 is the linear growth rate, and  > 0 is the carrying capacity  environment. The  of the right-hand side function ku 1  u represents a nonlinear growth rate [15]. This well-known equation was first proposed by Fisher [18] for a model of the advancement of a mutant gene in an infinite one-dimensional habitat. In recent years, the equation has been used as a basis for a wide variety of models for the spatial spread of genes in a population and for chemical wave propagation [18]. Example 8 Let us consider the (1 C 1)-dimensional NLS equation with two higher-order nonlinear terms in the form iu t C 12 u x x C ˛ juj u C ˇ juj2 u D 0 ; where  is a positive constant, ˛ and ˇ are constant parameters [15,16,69]. u D u(x; t) is a complex function that represents the complex amplitude of the wave form, the variable t should be interpreted as the normalized propagation distance, x the normalized transverse coordinate

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that represents a retarded time. The NLS equation, especially that with lower  values, appears in various branches of contemporary physics [18,27,61,67,80,81] and has been extensively investigated for the cases of  D 1 and  D 2, its various solutions have been obtained. The NLS equation also arises in some form of ˛ D 0 and  D 1 in some other physical systems such as nonlinear optic [3,26,38], hydro magnetic and plasma waves [30,63] and the propagation of solitary waves in piezoelectric semiconductors [55]. Example 9 In applied mathematics, the KP equation [36] is a nonlinear partial differential equation. It is also sometimes called the Kadomtsev–Petviashvili-Boussinesq equation. The KP equation is usually written as: (u t  6uu x x C u x x x )x C 32 u y y D 0 ; where  D ˙1. This equation is implemented to describe slowly varying nonlinear waves in a dispersive medium [58]. The above written form shows that the KP equation is a generalization form of the KdV equation which is like the KdV equation; the KP equation is completely integrable. The KP equation was first discovered in 1970 by Kadomtsev and Petviashvili when they relaxed the restriction that the waves be strictly one-dimensional. Example 10 Consider the solution u(x; t) of the nonlinear partial differential equation u t C "uu x  u x x C u x x x D 0 where "; ( and  are positive parameters. This equation is called the Korteweg–de Vries Burgers’ equation (KdVB) which is derived by Su and Gardner [65]. KdVB is a model equation for a wide class of nonlinear systems in the weak nonlinearity and long wavelength approximations because it contains both damping and dispersion. KdVB has been constructed when including electron inertia effects in the description of weak nonlinear plasma waves. The KdVB equation possesses a steady-state solution which has been demonstrated to model weak plasma shocks propagating perpendicular to a magnetic field [24]. There are some other implementation places to use the KdVB equation. Examples are its use in the study of wave propagation through a liquid-filled elastic tube [34] and for a description of shallow water waves on a viscous fluid [21,35]. Example 11 The nonlinear partial differential equation (u t C uu x  qu x x C pu x x x )x C ru y y D 0 ; where p; q; r are real parameters. This equation is called the two-dimensional Korteweg–deVries Burgers’

(tdKdVB) equation. The tdKdVB equation is a model equation for a wide class of nonlinear wave models of fluids in an elastic tube, liquids with small bubbles and turbulence [52,53]. Example 12 In this example we consider in the (2 C 1)-dimension the potential Kadomtsev–Petviashvili equation, u x t C 32 u x u x x C 14 u x x x x C 34 u y y D 0 ; where the initial conditions u(x; 0; t) and u y (x; 0; t) are given. Nonlinear phenomena play a crucial role in applied mathematics and physics. Obtaining the exact or approximate solutions of PDEs in physics and mathematics is important and it is still a hot spot to seek new methods to obtain new exact or approximate solutions [14,15,16]. Different methods have been put forward to seek various exact solutions of multifarious physical models described by nonlinear PDEs. Example 13 We consider the numerical solution to a problem involving a nonlinear partial differential equation of the form u t C u ux C ux x x  ux x x x x D 0 ; this is called the Kawahara equation. The Kawahara equation occurs in the theory of magneto-acoustic waves in plasmas [39] and in the theory of shallow water waves with surface tension [29]. Example 14 A Generalized Zakharov–Kuznetsov (gZK) equation [14,15,16,48,75] is proposed to understand the physical and scientific mechanisms in different physical and engineering problems. The gZK equation is as follows u t C b1 u u x C b2 u2 u x C b3 u x y C b4 u x x x C b5 u x y y D 0; where b1 ; b2 ; b3 ; b4 ¤ 0; b5 ¤ 0 and ¤ 0 are constant parameters [48]. When b1 D 6, b2 D b3 D 0, b4 D b5 D 1 and D 1 the equation gZK is said to be Zakharov–Kuznetsov (ZK) which can be derived for Alfvën waves in a magnetized plasma at a special, critical angle to the magnetic field by means of an asymptotic multi-scale technique [8,56]. If b1 D 0, b2 D 1, b3 D 0, b4 D b5 D 1 and D 1, then the equation gZK is said to be modified Zakharov–Kuznetsov (mZK) which represents an anisotropic two-dimensional generalization of the KdV equation and can be derived in a magnetized plasma for a small amplitude Alfvën wave at a critical angle to the undisturbed magnetic field. The radially symmetric positive solutions for the mZK equation have been computed [54,82].

Partial Differential Equations that Lead to Solitons

Example 15 Let’s consider the Sharma–Tasso–Olver equation (STO) with its fission and fusion [76] u t C 3˛u2x C 3˛u2 u x C 3˛uu x x C ˛u x x x D 0 : Attention has been focused on the STO equation in [50,68] and the references therein due to its appearance in scientific applications [74]. Example 16 Consider the Cahn–Hilliard equation   u t C u x x x x D u3  u x x C ˇ u x : The Cahn–Hilliard equation is related to a number of interesting physical phenomena like spinodal decomposition, phase separation and phase-ordering dynamics. On the other hand this equation is very hard and difficult to solve. In this paper by considering a modified extended tanh method, we found some exact solutions of the Cahn– Hilliard equation. This equation is very crucial in materials science [10,11,25]. Many articles have focused on mathematical and numerical studies of this equation [1,19,49]. Future Directions Nonlinear phenomena play a crucial role in applied mathematics and physics. Furthermore, when an original nonlinear equation is directly calculated, the solution will preserve the actual physical characters of solutions. Explicit solutions to the nonlinear equations are of fundamental importance. Various effective methods have been developed to understand the mechanisms of these physical models, to help physicians and engineers and to ensure knowledge for physical problems and its applications. Many explicit exact methods have been introduced in the literature [14,15,16,31,46,69,70,71,72,73,75]. These include the Bäcklund transformation, Hopf–Cole transformation, Generalized Miura Transformation, the Inverse scattering method, Darboux transformation, Painleve method, homogeneous balance method, similarity reduction method, tanh method, Exp-function method, sine– cosine method and so on. There are also many numerical methods implemented for these equations [22,23,28, 32,33,40,41,42,43,44,45,59,64,66]. These include the Finite elements method, finite difference methods and some approximate methods such as the Adomian decomposition method, Homotopy perturbation method, Variational perturbation method, Sinc-Galerkin method and so on. There is still much work to be done by researchers in this field involving applications of nonlinear equations and exact and numerical implementations.

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74. Wazwaz AM (2007) New solitons and kinks solutions to the Sharma–Tasso–Olver equation. Appl Math Comp 188:1205– 1213 75. Whitham GB (1974) Linear and Nonlinear Waves. Wiley, New York 76. Yan Z (2003) Integrability for two types of the (2 C 1)dimensional generalized Sharma–Tasso–Olver integro-differential equations. MM Res 22:302–324 77. Zabusky NJ (1967) Nonlinear Partial Differential Equations. Academic Press, New York 78. Zabusky NJ (1967) A synergetic approach to problems of nonlinear dispersive wave propagations and interaction. In: Ames WF (ed) Proc. Symp. on Nonlinear Partial Differential equations. Academic Press, Boston, pp 223–258 79. Zabusky NJ, Kruskal MD (1965) Interactions of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243 80. Zhang W, Chang Q, Fan E (2003) Methods of judging shape of solitary wave and solution formulae for some evolution equations with nonlinear terms of high order. J Math Anal Appl 287:1–18 81. Zhou CT, He XT, Chen SG (1992) Basic dynamic properties of the high-order nonlinear Schrödinger equation. Phys Rev A 46:2277 82. Munro S, Parker EJ (1997) The stability of solitary-wave Solutions to a modified Zakharov–Kuznetsov equation. J Plasma Phys 64:411–426

Books and Reviews The following, referenced by the end of the paper, is intended to give some useful for further reading. For another obtaining of the KdV equation for water waves, see Kevorkian and Cole (1981); one can see the work of the Johnson (1972) for a different water-wave application with variable depth, for waves on arbitrary shears in the work of Freeman and Johnson (1970) and Johnson (1980) for a review of one and two-dimensional KdV equations. In addition to these; one can see the book of Drazin and Johnson (1989) for some numerical solutions of nonlinear evolution equations. In the work of the Zabusky, Kruskal and Deam (F1965) and Eilbeck (F1981), one can see the motion pictures of soliton interactions. See a comparison of the KdV equation with water wave experiments in Hammack and Segur (1974) For further reading of the classical exact solutions of the nonlinear equations can be seen in the works: the Lax approach is described in Lax (1968); Calogero and Degasperis (1982, A.20), the Hirota’s bilinear approach is developed in Matsuno (1984), the Bäckland transformations are described in Rogers and Shadwick (1982); Lamb (1980, Chap. 8), the Painleve properties is discussed by Ablowitz and Segur (1981, Sect. 3.8), In the book of Dodd, Eilbeck, Gibbon and Morris (1982, Chap. 10) can found review of the many numerical methods to solve nonlinear evolution equations and shown many of their solutions.

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Periodic Orbits of Hamiltonian Systems LUCA SBANO Mathematics Institute, University of Warwick, Warwick, UK Article Outline Glossary Definition Introduction Periodic Solutions Poincaré Map and Floquet Operator Hamiltonian Systems with Symmetries The Variational Principles and Periodic Orbits Further Directions Acknowledgments Bibliography Glossary Hamiltonian are called all those dynamical systems whose equations of motion form a vector field X H defined on a symplectic manifold (P ; !), and X H is given by i X H ! D dH, where H : P ! R is the Hamiltonian function. Poisson systems These are dynamical systems whose vector field X H can be described through a Poisson structure (Poisson brackets) defined on the ring of differentiable functions on a given manifold that is not necessarily symplectic (see “Hamiltonian Equations”). Note that on any symplectic manifold there is a natural Poisson structure such that any Hamiltonian system admits a Poisson formulation, but the contrary is false. The Poisson formulation of the dynamics is a generalization of the Hamiltonian one. A periodic orbit (:) is a solution of the equations of motion that repeats itself after a certain time T > 0 called a period, that is, (t C T) D (t) for every t. Poincaré section/map Given a periodic orbit (:) a Poincaré section is a hyperplane S intersecting the curve f(t) : t 2 [0; T)g transversely. The associated Poincaré map ˘ maps neighborhoods of S into itself by following the orbit (:) (see Definition 10). A Hamiltonian system with symmetry is a Hamiltonian system in which there is a group G acting on P , i. e., there is a map ˚ : G  P 7! P , with ˚ preserving the Hamiltonian and the symplectic form. Relative periodic orbit Let G be a symmetry group for the dynamics. A path (:) is a relative periodic orbit

if solves the equations of motion and repeats itself up to a group action after a certain time T > 0, that is, (t C T) D ˚ g ((t)) for every t and for some g 2 G. Continuation Continuation is a procedure based on the implicit function theorem (IFT) that allows one to extend the solution of an equation for different values of the parameters. Let f (x; ) D 0 be an equation in x 2 Rn where f is differentiable and  0 a parameter. Assume that f (x0 ; 0) D 0; a curve x( ) is called a continued solution if x(0) D x0 and f (x( ); ) D 0 for some  0. In general x( ) exists whenever the IFT can be applied, that is, if D x f (x; ) is invertible at (x0 ; 0). Liapunov–Schmidt reduction Let f be a function on a Banach space. Liapunov–Schmidt reduction is a procedure that allows one to study f (x; ) D 0 under the condition that the kernel of Df is not empty but it is finite-dimensional. Variational principles The principles which aim to translate the problem of solving the equations of motion of a dynamical system (e. g., Hamiltonian systems) into the problem of finding the critical points of certain functionals defined on spaces of all possible trajectories of the given system. Definition The study of periodic motions is very important in the investigations of natural phenomena. In particular the Hamiltonian formulation of the laws of motion has been able to formalize and solve many fundamental problems in mechanics and dynamical systems. This paper is focused on a selection of results in the study of periodic motions in Hamiltonian systems. We shall consider local problems (e. g., stability and continuation/bifurcation) and also the application of variational methods to study the existence in the large. Throughout the paper we give some details of the methods and proofs. Introduction Periodic motions and behaviors in Nature have always been of interest to mankind. All phenomena that have some cyclic nature have captured our attention because they are a sign and a clue for regularity. Therefore, they are indications of the possibility of understanding the laws of Nature. Since the second century BC, Greek philosophers and astronomers have looked into the possibility of describing the motions of celestial bodies through the theory of epicycles, which are combinations of circular periodic motions. Notably from a modern point of view this theory can be interpreted as a clever geometrical applica-

Periodic Orbits of Hamiltonian Systems

tion of Fourier expansions of the observed motions [23]. The development of mechanics, the discovery that the laws of Nature can be written in the language of calculus and that laws of motion can be described in terms of differential equations opened up the study of periodic solutions of equations of motion. In particular, since Newton and then Poincaré [46], the main interest has been the understanding of the planetary motions and the solution of the so-called N-body problem, the reader should refer to  n-Body Problem and Choreographies in this encyclopedia. The interest in periodic motions has not been restricted to celestial mechanics but became a sort of paradigm in all areas where mechanics was successfully applied. In this article we shall illustrate some general aspects of the results regarding the theory of periodic orbits in Hamiltonian systems. Such systems are the modern formulation of those mechanical systems which are described by second-order differential equations and have an energy function. As an example the reader could think of Newton’s equations for a point mass in potential field. The equations read ¨ D rV(x(t)) ; with x(t) 2 R3 m x(t) for every t ;

m is the mass

(1)

V(x) is the potential and r D (@/@x1 ; @/@x2 ; @/@x3 ). The energy function ED

m kx˙ (t)k2 C V(x(t)) 2

is conserved along the trajectories solving (1). It is important to say that most of the systems of interest in physics can be naturally written in Hamiltonian form. The plan of this article is as follows. First we introduce the Hamiltonian formulations of the equations of motion for a classical mechanical system. It is well known that in many applications Hamiltonian systems derive from a Lagrangian formulation; therefore, this is also presented. Furthermore we introduce the Poisson formulation that is essentially the first generalization to the Hamiltonian point of view. Then we turn to the study of the properties of periodic solutions. In particular we focus on their “local properties”, persistence and stability. In this analysis the main tool will be the implicit function theorem (IFT). In order to emphasize the utility of the IFT we present some of the proofs that contain typical calculations often scattered in the literature. Then we consider the problem of periodic orbit for Hamiltonian systems with symmetries, where we introduce the notion of relative equilibrium and relative periodic orbit. Inevitably we have also a short excursion about symmetry reduction, which is the natural theoretical setting to study systems with symmetries. The second part of

the article is devoted to the exposition of the study of periodic orbits by variational methods. The discovery that the equations of motion of mechanical systems can be derived by a variational principle, the so-called least action principle, is usually attributed to Maupertuis (eighteenth century). According to this principle, the motions are critical points of a functional called the action defined in a suitable space of paths. Variational methods turned out to be one of the most effective methods to prove the existence of periodic orbits; notable is the case of the N-body problem (see [2] and  n-Body Problem and Choreographies). The valuable feature of the variational methods is the possibility to study the existence problem by looking at the topology and geometry of the space of periodic paths without further restrictions. In the presentation of the results some elements of the proofs are illustrated in order to clarify the main ideas. In the final section there are some open problems and further directions of investigation, in particular a simple example of the so-called multisymplectic structures that has extended the possibility of applying the finite-dimensional Hamiltonian approach to multiperiodic problems for a large class of partial differential equations is presented. For centuries the study of periodic orbits has been one of the main centers of mathematical investigations and developments and still presents challenges and the capacity for producing new interesting mathematical ideas to understand the complexity of Nature. Hamiltonian Equations A Hamiltonian system is given by specifying a symplectic manifold (P ; !), where P is a differentiable manifold of even dimension, ! is a closed differential two-form and a function H : P ! R is called a Hamiltonian. In the language of the differential forms the Hamiltonian vector X H field on P is written as i X H ! D dH :

(2)

In the case P D R2n , the symplectic structure is !0 D Pn iD1 dx i ^ dy i and the Hamiltonian vector field is : X H (z) D J rz H(z) ;

(3)

where z D (x; y) 2 R2n and J is the symplectic matrix  JD

0 id n

idn 0

 :

(4)

The Hamiltonian equations are then dz(t) D X H (z(t)) : dt

(5)

1213

1214

Periodic Orbits of Hamiltonian Systems

In the case P D R2n (5) reads

Poisson Formulation Let F (P ) be the space of differentiable functions on (P ; !). On F (P ) can be introduced   z˙(t) D (x˙ (t); y˙(t)) D r y H(x(t); y(t)); rx H(x(t); y(t)) : a product f:; :g [1,5,16]. The Poisson brackets

For a mechanical systems described by (1) the Hamiltonian function is HD

kyk2 2m

C V(x) ;

where (x; y) 2 R6 ;

f f ; gg is bilinear with respect to f and g ; f f ; gg D fg; f g

Note that the first equation corresponds to the classical definition of momentum in mechanics (here denoted with y) and the Hamiltonian function H coincides with the energy E. For more details the reader could consult [1,5]. Lagrangian Formulation In many applications in physics and in particular in mechanics Hamiltonian systems arise from the Lagrangian description. In such a setting a mechanical system is described by prescribing a differentiable manifold M (the configuration space) and a Lagrangian function L defined on the tangent bundle T M. Let L : T M ! R be a Lagrangian on a manifold M of dimension n. If L is hyperregular (i. e., rank(Dv2q L(v q ; q)) D n), then the Hamiltonian function is naturally constructed on the cotangent bundle T  M by using the Legendre transform [1,5] as follows: @L ; @v qi

H(p; q) D

(6) v qi (p) q i

(9)

The Poisson brackets satisfy the following properties. For all f ; g 2 F (P )

8 y(t) ˆ <x˙ (t) D m ˆ : y˙(t) D rV(x(t)) :

n X

(8)

In terms of the Poisson brackets the Hamiltonian equations can be written as a derivation acting on F (P ): X H ( f ) D f f ; Hg for f 2 F (P ) :

and its Hamiltonian equations read

pi D

!(X f ; X g ) D f f ; gg for f ; g 2 F (P ) :

 L(v q (p; q); q) :

iD1

The Hamiltonian system is then defined on the cotangent bundle of M that is P D T  M, which is endowed with the canonical symplectic form ! D d , where D Pn iD1 p i dq i . In the Lagrangian description the equations of motion are d @L(v q (t); q(t)) @L(v q (t); q(t))  D0; dt @v q i @q i i D 1; : : : ; n where v q i (t) D q˙i (t) ;

(7)

i D 1; : : : ; n :

Note that (7) contains second order time-derivatives. For more details see [1,5].

f f g; hg D f fg; hg C gf f ; hg ff f ; gg; hg C ffh; f g; gg C ffg; hg; f g D 0 Jacobi identity (10) An easy consequence of (9) and (10) is Proposition 1 The Hamiltonian function H is a constant of motion. A manifold P endowed with the brackets f:; :g is called a Poisson manifold. Any symplectic manifold is a Poisson manifold [1,27] but the contrary is false. In fact Poisson brackets on a symplectic manifold are always nondegenerate, namely, the condition fk; f g D 0 for all f 2 F (P ) implies that k is identically zero. In a general Poisson manifold there might exists nonvanishing k, which are then called Casimir functions. This is related to the fact that symplectic manifolds are always even-dimensional, whereas Poisson manifolds can be odd-dimensional. We can look at Poisson manifolds as a useful generalization of Hamiltonian systems; in fact in order to define Poisson brackets it is sufficient to have the ring of functions F (P ). For a general and complete description of the Poisson structure the reader could see [1,5,16,27]. Periodic Solutions Given a Hamiltonian vector field X H (z) on (P ; !), one can consider the following Cauchy problem: 8 dz(t) ˆ < D X(z(t)) dt (11) ˆ :z(0) D z : 0 Equation (11) is meant to be defined on a local chart in P . Definition 1 We call flow or integral flow the map t 7! (t; z0 ) where z(t) D (t; z0 ) solves problem (11).

Periodic Orbits of Hamiltonian Systems

Some simple consequences follow [36]. Remark 1 If X H is complete, then the flow is defined for all t 2 R. Remark 2 If the Hamiltonian vector field is autonomous, (that is, not explicitly dependent on time), then (:; z0 ) satisfies the composition property (t; (s; z0 )) D (t C s; z0 ) and (:; :) is called Hamiltonian flow. Let us now introduce the main object of this exposition. Definition 2 A flow  : R ! P , z(t) D (t; z0 ) is said to be a T-periodic solution of (11) if there exists T > 0 such that (t C T; z0 ) D (t; z0 ) for all t. Remark 3 Note that if (:; z0 ) is a T-periodic solution, then (:; z0 ) is n T-periodic for any n 2 N. In fact from the definition (T; z0 ) D z0 and using Remark 2, one can iterate (t C n T; z0 ) D (t C (n  1)T; (T; z0 )) D (t C (n  1)T; z0 ) and find (t C nT; z0 ) D (t; z0 ): Definition 3 T is called a minimal period of (t; z0 ) if T D min2RC f : (t C ; z0 ) D (t; z0 ) for all tg. Definition 4 A point z 2 P such that J rz H(z ) D 0 is called an equilibrium solution. Obviously any equilibrium solution can be seen as a periodic solution with T D 0. One can easily show

Definition 7 (Linearly stable) A periodic orbit (t; z0 ) is linearly stable if it is spectrally stable and DX H (z0 ) can be diagonalized. One can show that spectral stability is implied by either linear stability or Liapunov stability. A natural notion of stability can be introduced by using the Poincaré map and will be presented in Sect. “Poincaré Map and Floquet Operator”. The following results describe the structure of linear Hamiltonian systems, namely, systems whose Hamiltonian function is H(z) D 12 hz; A zi and the equations of motion read z˙(t) D J A z(t). Theorem 1 ([36]) Let A be time-independent. The characteristic polynomial of J A is even and if  is an eigenvalue then so are ; ; . A consequence of the previous result is that linear stability is equivalent to spectral stability for linear autonomous Hamiltonian systems. Linear systems can depend on parameters; it is therefore interesting to have a notion of stability with respect to parametric changes. Definition 8 A linear stable Hamiltonian system H(z) D 1 2 hz; A zi is said to be parametrically stable if for every symplectic matrix B such that kA  Bk < the system z˙ D J B z is linearly stable. We finally give a characterization of linear Hamiltonian systems which are parametrically stable.

Lemma 1 (t; z0 ) is periodic of period T if and only if (T; z0 ) D z0 .

Theorem 2 ([36]) If the Hamiltonian H is positive (or negative) definite or all the eigenvalues are simple, then A is parametrically stable.

Stability of Periodic Orbits

Continuation and Bifurcation of Equilibrium Solutions

Given a periodic orbit the first natural question is to study its stability. There are three possible stability criteria [1,36]:

Let H(z; ) be a Hamiltonian function depending on a parameter  0. Let

Definition 5 (Liapunov-stable) A periodic orbit (:; z0 ) is Liapunov-stable if for all > 0 there is ı(z0 ; ) such that kz0  z0 k  ı(z0 ; ) implies that k(t; z0 )  (t; z0 )k  for all t  0. The previous definition is natural for a non-Hamiltonian system but in a Hamiltonian context it is very strong, since in the Hamiltonian systems periodic orbits are not isolated. It is useful though to compare Liapunov stability with the following weaker notions. Definition 6 (Spectrally stable) A periodic orbit (t; z0 ) is spectrally stable if the eigenvalues of DX H (z0 ) lie all on the unit circle.

Z0 D fz : JrH(z ; 0) D 0g be the set of equilibrium positions. An interesting problem is to study how Z0 is modified when > 0. The local properties of Z0 depend on the spectrum of J D2 H(z ; 0). If z 2 Z0 is such that JD2 H(z ; 0) has nonzero eigenvalues, then by the IFT there exists a curve z ( ) of equilibrium points for " positive and sufficiently small. If J D2 H(z ; 0) has at least one zero eigenvalue then bifurcation can occur, but the interesting point is that bifurcations are possible also when J D2 H(z ; 0) is not degenerate. In fact in Hamiltonian systems generically the spectrum of the linearization of the vector field contains couples of complex conjugated eigenvalues [5]. In such a case there is a theorem

1215

1216

Periodic Orbits of Hamiltonian Systems

due to Liapunov which shows the existence of periodic orbits – so-called nonlinear normal modes in a neighborhood of z . Let us now present Liapunov’s theorem. This result describes how a nondegenerate equilibrium can be continued into a periodic orbit. The types of orbits are called nonlinear normal modes. Theorem 3 (Liapunov’s center theorem [1,18]) If the Hamiltonian system has a nondegenerate equilibrium at which the linearized vector field has eigenvalues ˙i!; 3 ; : : : ; n with  k /! … Z then there exists a oneparameter family of periodic orbits emanating from z . The period tends to 2/! when the orbit radius tends to zero and the nontrivial multipliers tend to exp(2 k /!) with k D 3 : : : n. Proof Without loss of generality the nondegenerate equilibrium can be fixed at the origin. In a neighborhood of the origin the Hamiltonian vector field can be written as follows: z˙(t) D JA z(t) C r(z) ; where kr(z)k D o(kzk); that is, kr(z)k is infinitesimal with respect to kzk. The spectrum of JA is Spec(JA) D f˙i!; 3 ; : : : ; n g. Let y D z with 2 [0; 1], then

with r(y; ) infinitesimal for ! 0 uniformly for kyk bounded. For D 0 the system becomes (12)

and admits a periodic solution with period T D 2/!: y0 (t) D exp(t JA) y0 ; JA y0 D v0 y0 : Equation (12) is linear, and thereby coincides with its linearization. The Floquet multipliers are therefore (1; 1; exp(2 3 /!); : : : ; exp(2 n /!)) with exp(2  k /!) ¤ 1 by hypothesis. Now we look for a solution of   d  JA y(t) D r(y(t); ) : (13) dt On the space C 1 ([0; T]; R2n ) with periodic boundary condition the operator d/dt  JA has a two-dimensional kernel. Therefore, one could solve (13) by looking for a solution in the form y(t; ) D exp(t JA)y0 C u(t) ;

y(t; ) D exp(t JA)(1 C o(1)) y0 ; that is, z(t; ) D exp(t JA)(1Co(1)) y0 with lim kz(t; )k D 0: !0

 Remark 4 The analysis of the operator d/dt  JA used to prove the previous result is known as Liapunov–Schmidt reduction. In Sect. “Continuation of Periodic Orbits as Critical Points” we shall give an application of it in the study of critical points. Normal Form Analysis Near Equilibrium Points In [55] Liapunov’s theorem was generalized to cases where the condition /! 2 Z might hold. This corresponds to the so-called resonance condition. Definition 9 (Resonance) The set of eigenvalues f! l g klD1 of the linearization DX H (z0 ) are said to be resonant if R(! i ) D 0 for all i and there exist fn l g klD1 Z such that k X

y˙(t) D JA y(t) C r(y; ) ;

y˙(t) D JA y(t)

where u(t) 2 rank(d/dt  JA). By the IFT one can show that ku(t)k D o(1) and therefore the solution can be continued for small as

!l nl D 0 :

l D1

In [55] it is shown that around z0 there are at least n periodic orbits. Because of the possible presence of resonance not all the orbits are a continuation of periodic orbits of the linearized vector field. In order to study this case one has to go beyond the linear approximation and analyze the Hamiltonian system in a neighborhood of the equilibrium taking into account the structure of terms of order higher than 2 in the canonical coordinates. To achieve this objective there is a general method, the normal form theory, to expand the Hamiltonian function H using suitable coordinates in an -neighborhood of z0 . A given Hamiltonian H can be expanded as X m H m (z) ; (14) H(z) D H0 (z) C mD1

where H0 (z) D

X !i jD1

D

2

X !i jD1

2

kz j  z0; j k2 [(x j  x0; j )2 C (y j  y0; j )2 ]

Periodic Orbits of Hamiltonian Systems

and H m (z) are polynomials of degree m C 2 in the canonical coordinates. Normal form theory allows us to classify the possible form of expansions (14) according to the resonance condition. This classification is also very important in the study of small perturbations of integrable Hamiltonian systems (KAM theory) where it is still the resonance condition that causes the main difficulties [5]. For a detailed account of normal form theory and its applications the reader could refer to [49, 54] for an introduction. Once the Hamiltonian is in normal form, one can study the bifurcation occurring when the resonance holds. Particularly interesting is to understand what conditions the frequencies have to fulfill in order for a system to have a number of periodic orbits exceeding the estimation given in [55]. There have been further generalizations to the results given in [55] and the reader could refer to [40,41].

Periodic Orbits of Hamiltonian Systems, Figure 1 Sa is Poincaré section for (t; z0 ), in this case a D XH (z0 ). The continuation produces a new orbit with initial data z00 close to z0

Poincaré Map and Floquet Operator To study a periodic orbit one could consider looking at it in a hyperplane which is transverse to its direction. This can be done locally and by constructing the Poincaré map. Definition 10 Let (:; z0 ) be a T-periodic orbit. A Poincaré cross section of (:; z0 ) is S a D fz 2 P : ha; (zz0 )i D 0; with ha; JrH(z0 )i ¤ 0g: Let U S a , a neighborhood of z0 sufficiently small such that the first return time is T : U ! RC ; (T (z); z) 2 S a with T (z0 ) D T :

The Poincaré map ˘ : U ! S a is defined as : U 3 z ! ˘ (z) D (T (z); z) 2 S a : Using the regularity properties of (t; z0 ), one can show : Proposition 2 ([36]) Let (t; z) D ha; (t; z)  zi be defined for z 2 S a . For U S a sufficiently small the return times T : U ! RC defined by the implicit equation  (T (z); z) D 0 and ˘ : U ! S a are smooth functions. Using this result, one can think of characterizing periodic orbits as fixed points of ˘ (z) D (T (z); z). In fact if there exists z 2 S a such that ˘ (z ) D z , then the flow (t; z ) is a periodic orbit with period T (z ). An example of a Poincaré section is given in Fig. 1. Definition 11 Let (t; z0 ) be a periodic solution of (5). The Floquet operator : @(t; z0 ) V(t) D @z0

solves the variational equation ˙ V(t) D Dz X H ((t; z0 )) V (t)

(15)

with V(0) D id. The matrix V(T) is called a monodromy matrix and its eigenvalues are termed Floquet multipliers. For a general discussion about Floquet theory one can refer to [36,49,59]. In general, the construction of the Poincaré map is not explicit. But knowledge of the monodromy matrix and Proposition 2 with the equation  (T (z); z) D 0 allow us to compute the Taylor expansion of ˘ (z) in a neighborhood of z0 in Sa . We shall examine this idea in the next section, where we also see the consequences of the fact that periodic orbits in Hamiltonian systems are not isolated. In fact the following result holds true: Proposition 3 ([36]) Periodic solutions are never isolated and C1 is always a multiplier whose eigenvector is JrH(z). If F : P ! R is a first integral, then rF is a left eigenvector of the monodromy matrix with eigenvalue C1. Note that a consequence is that since in a Hamiltonian system the Hamiltonian H is always conserved, the multiplier C1 has algebraic multiplicity of at least 2. In the presence of an integral of motion like the Hamiltonian, the Poincaré map can be restricted to the level set of H. The map ˘ turns out to be symplectic [5,36]. The stability of a periodic orbit can now be defined in terms of the stability of the fixed points of the Poincaré map. Let ˘ n (z) denote the nth iteration of the Poincaré map applied to z 2 S a . We can define

1217

1218

Periodic Orbits of Hamiltonian Systems

Definition 12 ([1,36,49]) A periodic orbit (:; z0 ) is stable for all > 0 if there exists ı( ; z0 ) such that kz z0 k < ı implies k˘ n (z) z0 k <

for all

n > 0:

This stability criterion is difficult to verify. A weaker criterion is obtained by considering the linearization of ˘ (z) at z D z0 : Definition 13 ([1,36]) A periodic orbit (:; z0 ) is spectrally stable if the associated Poincaré map ˘ , which is restricted to the manifold defined by the integrals of motion, has a linearization Dz ˘ (z0 ) with a spectrum on the unit circle. By a local change of coordinates one can show that the eigenvectors of Dz ˘ (z0 ) are equal to the eigenvectors of V(T0 ) different from X H (z0 ) D JrH(z0 ) [36]. The Poincaré map removes the degeneracy of the monodromy matrix V(T). To illustrate this point let us consider x 2 Rn and an autonomous system x˙ (t) D f (x(t)) Rn

(16)

Rn

7! differentiable. Let (t; x0 ) be a T-periwith f : odic solution of (16) emanating from x0 . To study the trajectories near x0 one can construct a local diffeomorphism h : Rn 7! Rn , y D h(x) such that y0 D h(x0 ) with (16) in the form y˙(t) D Dh(h1 (y(t))) f (h1 (y(t))) D f˜(y(t)) (17) with f˜(y0 ) D (1; 0; : : : ; 0) : In y coordinates the periodic orbit (t; x0 ) reads (t; y0 ) D h((t; h1 (y0 ))) and we can define a map  as (t; y) D h f˜(y0 ); (t; y0 )  yi :

(18)

The form of f˜(y0 ) implies (y; t) D 1 (t; y)  y0;1 D h1 ((t; h1 (y0 )))  y0;1 . An easy calculation shows that ˇ ˇ n ˇ X @ ˇˇ @h1 ˇ D f (x) D1; ˇ l ˇ @t ˇ(0;y0 ) @x l l D1

(0;y 0 )

which implies the existence of a return time (y) satisfying ((y); y) D 0 for y in a sufficiently small neighborhood of y0 . Now we define the map ˘ (y) D ((y); y). By applying the IFT, we can compute X @h1 @ l @x m @ D ı1 j  @y j @x l @x m @y j

f (z 0 )

V(T0 ) D exp(T0 JA) ; where T0 D 2/jj, with  an imaginary eigenvalue of JA. Now let H D 12 hz; Azi C h(z) be a Hamiltonian with h(z) D o(kzk2 ) near z D 0. The equation for the monodromy matrix is (15). Let 0 (t) be another periodic orbit, then (15) can be written as follows: T dV (s) DX H ( D ds T0 where t D T0 s/T and can show

0 (s)) V 0 (s)

(s) ;

D 0 (T0 s/T ). Then one

Proposition 4 ([39]) The Floquet operator V (t) is C 1 in and T . Corollary 1 ([39]) If 0 (t) is sufficiently close to z D 0, then V (T ) is arbitrarily close to exp(T0 JA). Corollary 1 is interesting because it allows us to use linearized dynamics. Now the analysis of the nonlinear stability can be carried out by using Krein’s theory. For this typical references are [21,59]. Here we recall

(19)

(20)

By combining Corollary 1 and Theorem 4, we could show that a solution 0 (t) close enough to z D 0 is spectrally stable.

and also the Jacobian of ˘ :

l ;m

f (z 0 )

into S f˜(z 0 ) and its linearization has no eigenvectors in the direction of f˜(y0 ). The map ˘ f˜(z 0 ) describes the stability of the periodic orbit (t; x0 ). A similar construction can be carried out when f is a Hamiltonian vector field. In that case it is necessary to take into account the existence of integrals of motion and construct the Poincaré map on the manifold where the integral of motions are fixed. A more complete study of the Poincaré map will be presented in the next section. Let us now present some properties of the Floquet operator. Let us consider a Hamiltonian system with imaginary multipliers. In the linear time-independent case the monodromy matrix is given by

Theorem 4 (Krein) Let V(T) be a spectrally stable monodromy matrix. Let Q be a quadratic form Q (v) D hV (T) v; J vi where v belongs to the  eigenspace of V(T). Then V(T) is in an open set of spectrally stable matrices if and only if the quadratic Q has definite sign.

l ;m

X @y i @ l @x m @˘ i (y) @ D f˜i (y) C : @y j @y j @x l @x m @y j

Using (19), one can easily say that the matrix @˘ i (y)/@y j at y D y0 has the first column equal to f˜(y0 ) D (1; 0; : : : ; 0). The diffeormophism h allowed us to “isolate” the direction of the vector field f at x0 . This implies that the map ˘S f˜(z ) , the restriction of ˘ on the Poincaré section 0 D fy : h f˜(y0 ); y  y0 i D 0g ' Rn1 , maps S ˜ S˜

Periodic Orbits of Hamiltonian Systems

Initially let us consider a non-Hamiltonian dynamical system in Rn :

Continuation of Periodic Orbits in Hamiltonian Systems In general it is difficult to prove the existence of and then construct periodic orbits. Sometimes, for certain specific values of the parameters characterizing the system it is possible to find particular solutions. Typically these are the equilibria and relative equilibria. In these cases the continuation method can be a useful approach. The idea is to look at how a given periodic orbit changes according to a small modification of the parameters. The method reduces the research of periodic orbits to the problem of finding fixed points of the Poincaré map that can be continued as a function of the parameters; see Fig. 1. Definition 14 (Continuation of an orbit) Given a dynamical system and 0 (t) one of its orbits, we say that 0 (t) can be continued if there exists a family of orbits (t; ˛) smoothly dependent on parameters ˛’s and such that (t; 0) D 0 (t). Let us consider a Hamiltonian vector field X H (z; ˛) where z 2 P and ˛ 2 R k are k parameters. We now write the equations of motion in a form where the period T appears explicitly. After t ! t/T the equations read z˙(t) D T J rH(z(t); ˛) D T X H (z(t); ˛) ;

Definition 15 Let (t; z; T; ˛) be a solution of (21). The map : R(z; T; ˛) D (1; z; T; ˛)  z (22) is called a return map. The orbit (t; z0 ; ˛0 ) is T 0 -periodic if R(z0 ; T0 ; ˛0 ) D 0. Now we are interested to see what is the fate of the orbit when z0 , T 0 and ˛0 are varied; therefore, it is useful to determine the local behavior of the map R. This is collected in the following proposition. Proposition 5 ([42]) Let 0 (t; z0 ) be a periodic orbit with period T 0 and ˛ D 0, then the following relations hold    

Dz R(z0 ; T0 ; 0) D V(1)  id, X H (z0 ; 0) 2 ker(V(1)  id), D T R(z0 ; T0 ; 0) D X H (z0 ; 0), D˛ R(z0 ; T0 ;R0) 1 D T0 V(T) 0 V 1 (s)D˛ X H (0 (s; z0 ))ds,

together with the differential map DR(z0 ; T0 ; 0)(; T; ˛) D (V(1)  id)   C T X H (z0 ; 0) C D˛ R(z0 ; T0 ; 0)  ˛ ; where (; T; ˛) 2 R2nCkC1 .

(23)

We now illustrate how the IFT is used to construct a new solution from a given one, i. e., by continuation. We present a detailed proof for the reader’s convenience. Proposition 6 ([3,42]) Let 0 D f T0 (t; x0 ); t  0g be ˙ D T0 f (x(t); ) for D a periodic orbit of period 1 of x(t) 0. If 0 is an eigenvalue of D x R(x0 ; T0 ; 0) with multiplicity 1, then orbit 0 can be continued. Proof Let us consider the map G : Rn  RC  [0; 1] ! RnC1 defined by G(x; T; ) D (R(x; T; ); h f (x0 ; 0); x  x0 i) :

(24)

Note that h f (x0 ); x  x0 i D 0 is the equation of the Poincaré section S f (x 0 ) . Since 0 is a periodic orbit with period T 0 emanating from x0 , then G(x0 ; T0 ; 0) D 0. As already stated the strategy is to employ the IFT to derive the continuation of 0 . Thus, it is necessary to compute D x;T G at (x0 ; T0 ; 0): 

(21)

with t 2 [0; 1].

2R:

x˙ (t) D f (x(t); ) ;

D x;T G(x0 ; T0 ; 0) D

D x R(x0 ; T0 ; 0) f (x0 ; 0)

f (x0 ; 0) 0

 :

In order to show that D x;T G(x0 ; T0 ; 0) is invertible one notes that the equation D x;T G(x0 ; T0 ; 0)(X; a) D (0; 0) is equivalent to the system ( D x R(x0 ; T0 ; 0)(X) C f (x0 ; 0) a D 0 h f (x0 ; 0); Xi D 0 :

(25)

Assume X ¤ 0. The multiplicity of the 0 eigenvalue of D x R(x0 ; T0 ; 0) is 1 and D x R(x0 ; T0 ; 0) f (x0 ; 0) D 0 : Now from (25) we derive (D x R(x0 ; T0 ; 0))2 X D 0. Therefore, D x R(x0 ; T0 ; 0)(X) C f (x0 ; 0) a D 0 would be satisfied only for a D 0 and X D c f (x0 ; 0) with c 2 R. But this would imply c h f (x0 ; 0); f (x0 ; 0)i D 0, which is possible only for c D 0. The kernel of D x;T G is empty at (x0 ; T0 ; 0) and therefore the map is invertible and the application of the IFT provides the existence of T( ) and x( ) for in a neighborhood of zero such that x(0) D x0 , T(0) D T0 and G(x( ); T( ); ) D 0. This corresponds to the existence of a new periodic orbit close to 0 . Upon

1219

1220

Periodic Orbits of Hamiltonian Systems

Finally we present how to construct the continuation of nontrivial periodic orbits in Hamiltonian systems.

the assumption of sufficient regularity for the vector field, the IFT provides also the possibility of approximating x( ) and T( ) by constructing a Taylor expansion in . Let x( ) D x0 C ( ) and T( ) D T0 C ( ), then (25) can be evaluated along the continuation curve defined by (( ); ( ); ): 8 ˆ D x R(x0 ; T0 ; 0)( 0 ( )) C f (x0 ; 0)  0 ( ) ˆ ˆ Z 1 < d f ( T0 (s; x0 ); 0) CT0 V(1) V(s) ds D 0 ˆ d ˆ 0 ˆ :h f (x ; 0);  0 ( )i D 0 : 0

Theorem 5 ([42]) Let 0 D f T0 (t; z0 ) : t 2 [0; 1)g be an orbit of the Hamiltonian system H 0 on the energy level e0 . Let the algebraic multiplicity of eigenvalue 0 of V(1)  id be 2. Let H D H0 C H1 be a smooth perturbation of H 0 , then there exists a two-dimensional family of normal periodic orbits (t; z0 ; ; e) where (t; z0 ; e0 ; 0) D (t; z0 ).

(26)

G(x; T; e; ) D (R(z; T; ); hX H (z0 ; 0); zz0 i; H (z)e):

This set of equations allows us to compute an approximation for ( ) and ( ). 

(28)

Rn

For a general dynamical system in (23) the possibility of constructing a Poincaré section is related to the notion of a nondegenerate periodic orbit. Definition 16 A periodic orbit (t; z0 ) is called nondegenerate if rank(V(1)  id) ˚ R f (z0 ) D Rn : Now for Hamiltonian systems the time evolution is contained in the level set determined by the the Hamiltonian function (the energy) and all integrals of motion fF i g kiD1 X H (F i ) D fH; F i g D 0 ;

i D 1; : : : ; k :

This requires a modification of the notion of nondegeneracy. In fact note that the set of integrals of motions (F1 ; : : : ; F k ) define a map F : P 7! R k and span W D frF i ; i D 1 : : : kg. Now we have W ? D ker(Dz F(z)). One can show rank(V (1)  id) ker(Dz F(z)) and X H (F) D 0. The last condition is equivalent to X H (z) 2 ker(Dz F(z)). Suppose that dim W ? D 2n  k, if dim(ker(V(1)  id)) D k then dim(rank(V (1)  id) D 2n  k and hence rank(V (1)  id) D W ? . Therefore, in the Hamiltonian case the natural notion of nondegeneracy has to involve the presence of integrals of motion. This is obtained by using the following definition. Definition 17 ([42]) A periodic orbit (t; z0 ) is called normal if rank(V (1)  id) ˚ R X H (z0 ) D W ? ;

(27)

where all the gradients frH; rF1 ; : : : ; rF k g are linearly independent. Lemma 2 ([42]) If the algebraic multiplicity of the zero eigenvalue of V(1)  id is m a D k C 1, then condition (27) is satisfied.

Proof Let us consider the map G : R2n  RC  [0; 1] ! R2nC1 defined by

Since 0 is a periodic orbit with period T 0 emanating from z0 then G(z0 ; T0 ; e0 ; 0) D 0. The strategy is always to employ the IFT to derive the continuation of 0 . Thus, it is necessary to compute Dz;T G at (z0 ; T0 ; e0 ; 0). A straightforward calculation gives 1 0 Dz R(z0 ; T0 ; 0) X H (z0 ; 0) A: X H (z0 ; 0) 0 Dz;T G(z0 ; T0 ; 0) D @ 0 rH0 (z0 ) In order to show that Dz;T G(z0 ; T0 ; h0 ; 0) is invertible one notes that the equation Dz;T G(z0 ; T0 ; h0 ; 0)(X; a; 0) D (0; 0; 0) is equivalent to the system 8 ˆ < Dz R(z0 ; T0 ; e0 ; 0)(X) C X H (z0 ; 0) a D 0 hX H (z0 ; 0); Xi D 0 ˆ : hrH0 (z0 ); Xi D 0 :

(29)

Since X H (z0 ) 2 ker Dz R(z0 ; T0 ; e0 ; 0) then from (29) we derive (Dz R(z0 ; T0 ; e0 ; 0))2 (X) D 0 and therefore a D 0. Now Lemma 2 implies that X 2 W ? ; therefore, the third equation in (29) is satisfied and the first implies X D b X H (z0 ; 0) for some b 2 R. But this would contradict b hX H (x0 ; 0); X H (x0 ; 0)i D 0 unless b D 0. This implies that that kernel of D x;T G is empty at (x0 ; T0 ; e0 ; 0) and therefore the map G is invertible. The application of the IFT provides the existence of T(e; ) and z(e; ) for in a neighborhood of 0 and e in a neighborhood of e0 such that z(e0 ; 0) D z0 , T(e0 ; 0) D T0 . The functions T(e; ) and z(e; ) satisfy G(z(e; ); T(e; ); e; ) D 0. This corresponds to the existence of a new periodic orbit close to  . Upon the assumption of sufficient regularity for the vector field, the IFT provides also the possibility of approximating z(e; ) and T(e; ) by following the same line of argument seen in Proposition 6. This concludes the Proof. 

Periodic Orbits of Hamiltonian Systems

Numerical Studies The analysis of periodic orbits is very important for its concrete applications, hence for the construction of numerical algorithms to construct periodic orbits and their continuation. Here we do not consider explicitly this problem but the reader is invited to consult, for example, [31,32]. Moreover there is some free and open-source software available, for example, AUTO (see http://indy.cs.concordia.ca/auto/ and [19,20]). Example On P D R4 with coordinates z D (p; q) and the standard symplectic form, we consider the Hamiltonian HD

1 1  1 kpk2 C V0 (q); V0 (q) D kpk2  kqk2 C kqk4 2 2 2 4 (30)

with  > 0. The reader could check that V0 (q) has the shape of a “Mexican-hat.” The Hamilton equationHamiltonians of motion read ( q˙ D p (31) p˙ D (  kqk2 )q : Let e(t) be a unit vector with e(t C 2/0 ) D e(t), then there is a periodic orbit of the form q0 (t) D A0 e(t); p0 (t) D A0 e˙(t). This orbit is a relative equilibrium (see below for a formal definition). The initial conditions determine A0 ; 0 and in particular the energy E, which in turn can be used to parameterize A0 ; 0 : A20 D 23 ( C p p 2 C 3E); 02 D 13 ( C 2 C 3E); here E  0. Note that an easy calculation shows that the admissible values of the energy are E  minq fV0 (q)g D 2 /4. Let (t; z0 ) D (p0 (t); q0 (t)) be the periodic solution, then the Floquet operator is V(t) D

@(t; z0 ) @z0

and the linearized equations can be written as dV (t) D M(t)V(t) ; where M(t) is a 4  4 matrix: dt   0 id : M(t) D D 2 V0 (q0 (t)) 0 Now if we perform the transformation   R(t) 0 X(t) W(t) D S(t) V(t) D 0 R(t)   cos 0 t  sin 0 t where R(t) D cos 0 t sin 0 t

we obtain dW(t) ˆ D MW(t) ; dt where ˙ S 1 (t) C S(t) M(t) S 1 (t) : ˆ D S(t) M ˆ is not time-dependent. In fact M ˆ is the following maM trix:   ˝ id ˆ D M ; R T (t)D2 V0 (q0 (t)R(t) ˝ where  ˝D

0 0

0 0

 and

 R T (t)D2 V0 (q0 (t)R(t) D



3A20 C  0

0 A20 C 

 :

Now at t D T0 ˆ T0 ) : V(T0 ) D S 1 (T0 ) W(T0 ) D exp( M ˆ has spectrum The matrix M 

q ˆ D 0; 0; ˙i 6A2  4 Spec( M) 0 and the corresponding multipliers are:

  q ˆ D 1; 1; exp ˙i T0 6A2  4 exp(Spec( M)) 0 n  o p 4 D 1; 1; exp ˙i T0 2 C 3E ; where T0 D 2/0 . The matrix V(T0 )id has a zero eigenvalue with multiplicity 2, and the periodic orbit 0 (t; z0 ) can be continued by using Theorem 5. Remark 5 Note that using the expression for 0 , one can check that for E D E n , where

 2 n4 En D 1 C 2 with n 2 Z ; 3 (n  12)2 the third and the fourth multipliers coalesce to 1 and therefore q0 (:) loses its linear stability in the directions transverse to itself. Hamiltonian Systems with Symmetries In many applications there are Hamiltonian systems with symmetries, the best known example of which is the N-body problem (see [1,5] and  n-Body Problem

1221

1222

Periodic Orbits of Hamiltonian Systems

and Choreographies). A simpler example is (31), which is symmetric with respect the linear action of SO(2). For such systems the Hamiltonian function is invariant under the action of a Lie group G. We shall see that symmetries can greatly simplify the study of the dynamics. A Hamiltonian system with symmetry is a quadruple (P ; !; H; G) [1,5,48], where  (P ; !) is a symplectic manifold,  H : P ! R is a Hamiltonian function,  G is a Lie group that acts smoothly on P according to G  P 3 (g; z) 7! ˚ g (z) 2 P . The map ˚ g preserves the Hamiltonian (G-invariance) that is H(˚ g (z)) D H(z).  The action ˚ is semisymplectic, namely, ˚ g ! D (g)!, with (g) D ˙1. (:) is called temporal character. In the sequel we consider symplectic actions whereby (g) D 1 for all g 2 G. One can show that for a system (P ; !; H; G) the vector field X H (z) is equivariant [1], that is, X H (˚ g (z)) D Dz ˚ g (z) X H (z) :

(32)

With any element  in the Lie algebra g of G we can associate a infinitesimal generator P (z) of the action defined by P (z) D

ˇ d˚exp( t) (z) ˇ ˇ : ˇ dt tD0

(33)

Remark 6 In many applications Hamiltonian systems are constructed from Lagrangian systems; therefore, a symmetry appears usually as a group action on the configuration space M. The Hamiltonian symmetry is then the lifted action to the cotangent bundle T  M. For instance, if SO(2) acts linearly on M D R2 by R (q) D R q with R 2 SO(2), then its lifted action on T  M ' R4 is ˚R (q; p) D (R q; R T q), where RT is the transpose of R. For more details see [1,36,48]. Symmetry and Reduction Given a symplectic action of a group G there is a map J : P ! g defined by

momentum map is its encoding of the conserved quantities associated with the G action. This is the content of the famous Noether’s theorem that reads in modern formulation as follows. Theorem 6 (Noether [1,5]) Let H be a G-invariant Hamiltonian on P with momentum map J. Then J is conserved on the trajectories of the Hamiltonian vector field XH . For instance, in a system like (30) with a SO(2) symmetry action the momentum map is the classical angular momentum J(p; q) D p ^ q : The momentum map J has also the property of being equivariant with respect to the coadjoint action associated with G. In the case of G being semisimple or compact the result is as follows. Theorem 7 (Souriau [27]) J(˚ g (z)) D Adg J(z) : The level sets of the momentum map are invariant with respect to the Hamiltonian flow; thus, it is natural to restrict the motion to J1 (). The construction of the dynamics reduced to the manifold defined by the conserved quantities is the origin of the theory of symmetry reduction. The first important result is the following. Theorem 8 (Marsden–Weinstein) Let (P ; !; H; G) be a Hamiltonian system with a symplectic action of the Lie group G, then the triple (P ; ! ; H ) is called a reduced Hamiltonian system where  : P ! P D J1 ()/G ;   ! D ! ; H D H ı  and G D fg 2 G : Adg () D g : The manifold P is symplectic with symplectic form ! . The Hamiltonian flow on P is induced by the Hamiltonian vector field X H defined by i X H ! D dH :

(34)

Remark 7 Note that the reduced vector field X H is now defined on a manifold whose dimension is dim P D dim P  dim G  dim G .

The map J is called a momentum map. This always exists locally, and its global existence requires conditions on G and the topology of P [27]. The crucial property of the

For a detailed discussion of this theorem the reader should refer to [1,16,38]. In many cases it can be more useful to perform the reduction through a dual approach using the description of the dynamics in terms of functions on P ,

hdJ(v(z)); i D !(v(z); P (z)) for all

z 2 P ; v 2 Tz P

and  2 g :

Periodic Orbits of Hamiltonian Systems

Periodic Orbits of Hamiltonian Systems, Figure 2 A simple example: the reduction of S1 group action on the plane is singular. There are two strata r D 0 and r 2 (0; 1)

namely, through the Poisson approach. This permits us to consider also cases where P is not a smooth manifold but rather a stratified space. A very simple example is shown in Fig. 2. The theory of singular reduction can be found in [4]. A recent exposition and generalization is given in [45]. The reader could, in particular, consider the expositions given in [16,17], where it is shown through several examples that invariant theory and algebraic methods can be applied to describe reduced dynamics on spaces with singularities. The singular reduction can be summarized in the following result: Theorem 9 (Singular reduction [45]) Let (P ; f:; :g) be a Poisson manifold and let ˚ : G  P ! P be a smooth proper action preserving the Poisson bracket. Then the following holds: The pair (F (P /G); f:; :gP /G ) is a Poisson algebra, where the Poisson bracket f:; :gP /G is characterized by f f ; ggP /G D f f ı ; g ı g; for any f ; g 2 F (P /G);  : P ! P /G denotes the canonical smooth projection. (ii) Let h be a G-invariant function on M. The Hamiltonian flow (t; :) of X h commutes with the G-action, so it induces a flow  P /G (:; :) on P /G which is a Poisson flow and is characterized by (t; z) D  P /G (t; (z)). (iii) The flow  P /G (:; :) is the unique Hamiltonian flow defined by the function [h] 2 F (P /G) defined by [h]((z)) D h(z). We will call HP /G D [h] the reduced Hamiltonian. (i)

Relative Equilibria, Relative Periodic Orbits and their Continuation In a Hamiltonian system with symmetry there are orbits that originated just from the symmetry invariance. These are the relative equilibria.

Periodic Orbits of Hamiltonian Systems, Figure 3 A relative periodic orbit. After a relative period T the orbit z(:) closes onto the group orbit. The projection of z(:) on P /G is closed

Definition 18 (Relative equilibria) A curve ze (t) in P is a relative equilibrium of (P ; !; H; G) if z e (t) D ˚ g(t) (w e ), where g(t) is a curve in G and we is such that X H (z e ) D z˙e (t). Note that if g(t) D g(t C T) then ze (t) becomes a T-periodic orbit. For instance in the system (30) there is z e (t) D R(t) w e D (A0 e(t); A0 e˙(t)) with R 2 SO(2). There are orbits that can be considered closed up to a G-action, i. e., they are closed on P /G . These are the relative periodic orbits. Definition 19 (Relative periodic orbit) A curve z(t) in P is a relative periodic orbit of (P ; !; H; G) with period T if z(t C T) D ˚ g (z(t)), where g 2 G and g ¤ id. An illustration is given in Fig. 3. The reduction theory allows us to redefine the relative equilibrium. Let  2 g and z 2 P be an equilibrium for X H . Then z gives rise to a relative equilibrium in P , for a closed curve g(t) in G and define w(t) D ˚ g(t) (w e ) with w e 2 1 (z). If z(t) is a relative equilibrium then (z(t)) is an equilibrium of the reduced flow [1]. In general there are two possible ways to study the relative equilibrium. The first approach is to construct the reduction of the Hamiltonian system and then to study X H (z) D 0

(35)

in Hamiltonian form and fH G ; f gP /G (z) D 0

for all

f 2 F (P /G)

(36)

1223

1224

Periodic Orbits of Hamiltonian Systems

in the Poisson description. The second approach is to observe that if z(t) D ˚ g(t) (z ) is a relative equilibrium, then the condition z˙(t) D X H (z(t)) implies 

a suitable decomposition of the motion. In [39] following decomposition was introduced: Tz P D W ˚ X ˚ Y ˚ Z ;

(39)



dH(z )  dhJ(z ); i D 0 ; with; P (z ) D

ˇ d˚ g(t) (z ) ˇ ˇ : ˇ dt tD0

where (37)

On the three formulations, (35), (36) and (37), one can apply all the standard continuation techniques based on the IFT. The possible difficulty in the study of the continuation of relative equilibria and relative periodic orbits can be seen by looking at the linearization of (37). The symmetry contributes to the degeneracy of the linearized equations, in fact one can show Proposition 7 ([37]) Let X(z) be an equivariant vector field with respect to a Lie group G. Then a relative equilibrium has multiplier C1 with multiplicity at least equal to dim g; and a relative periodic orbit has multiplier C1 with multiplicity at least equal to dim g C 1. In [39] relative periodic orbits are defined by looking at the fixed points of the action of G  S 1 on the space of T-periodic paths. This reads (g; )  z(t) D ˚ g (z(t C T)) :

(38)

Remark 8 Note that (38) depends on T and it is easy to verify that T/k is a minimal period if the intersection of the isotropy subgroup of the action G  S 1 with S1 is equal to Z k . Theorem 10 ([39]) Let (P ; !; H; G) be a symmetric Hamiltonian system. Let z e 2 P such that 1. d2 H(z e ) is a nondegenerate quadratic form 2. d2 H(z e ) is positive-definite if restricted to V , which is the real part of the eigenspace associated with the eigenvalue i ,  2 R. Then for every isotropy subgroup  of the G  S 1 -action on V and sufficiently small there exist at least Fix(; V ) periodic trajectories with period near 2/jj, a symmetry group contained in  and jH(z(t))  H(z e )j D 2 . This result has been proved using a combination of a variational approach and the IFT, and will be considered in “Hamiltonian Viewpoint”. We have seen that the Poincaré section is a useful construction to analyze the local structure of a periodic orbit. A very similar approach can be introduced for relative periodic orbits. A relative periodic orbit can be seen as a combination of motions in P and in the symmetry group G. It is therefore natural to look for

W D ker Dz J(z) \ Tz (G  z); G  z D f˚ g (z) : g 2 Gg ; X D Tz (G  z)/W ; X D ker Dz J(z)/W ; Z D Tz P /(ker Dz J(z) C Tz (G  z)) : Let Gz D fg 2 : ˚ g (z) D zg and G D fg 2 G : Ad g () D g be the isotropy subgroups. It turns out that  ker Dz J(z) and Tz (G  z) are !-orthogonal,  ! restricted ker Dz J(z) C Tz (G  z) is singular and W is the kernel,  ! induces a Gz -invariant symplectic form ! X on X and !Y on Y,  ! defines a Gz -invariant isomorphism between W and Z the dual of Z. The splitting (39) allows us to decompose the vector field X H and to analyze the motion along the group orbit and along the transverse directions. This is a tool used in [39] to study the Floquet operator in a neighborhood of the relative periodic orbit. For applications in the study of nonlinear normal modes and stability see [40,41]. In [58] it is shown that such a construction can be used to decompose the Poincaré section in a part which is tangent to the conserved momentum, another part which is tangent to shape and a part parameterizing the momentum. This approach has also been applied to the study of the geometry of mechanical systems defined on the cotangent bundle of a differentiable manifold; for this see [50] and references therein. The Variational Principles and Periodic Orbits The idea behind variational principles is to transform a problem in differential equations into a question about critical points of a certain functional called action, whose domain is formed by trajectories of interest. Now in the study of T-periodic orbits we are interested in finding trajectories  solving the equations of motion and satisfying (t) D (t C T). This is a periodicity condition. The variational approach is particularly useful in the study of periodic problems because the periodicity condition is included in the definition of the space where the action is defined. Furthermore the method enables us to prove results not restricted to small perturbations. Let A :  7! R be

Periodic Orbits of Hamiltonian Systems

a functional on a space of loops usually modeled on a Banach or Hilbert space. We shall see that the condition of vanishing of the first derivative of A is equivalent, in an appropriate sense, to solving the equations of motion. In order to describe the properties of A let us introduce Definition 20 (Critical points) Let A[:] be a differentiable functional on . A path q(:) is a critical point of A[:] if DA[q](v) D 0 for all v(:) 2 T : Definition 21 (Critical set) The set K D fq(:) 2  : DA[q] D 0g is the critical set of A[:]. Definition 22 (Critical value) A real number c is called : a critical value if K c D A1 [c] \ K ¤ ;. Lagrangian Viewpoint A typical way to write Newton’s equations in a variational form is by using Lagrange’s equations which are formulated through the least action principle. This can be achieved for all mechanical systems that have potential forces. Consider a mechanical system whose configuration space is a Riemannian manifold M of dimension n. We denote with (q; v q ) the local coordinates in T M and with L : T M ! R the Lagrangian functionLagrangian. In particular in the case of the so-called natural mechanical system [1] the Lagrangian has the form 1 L(q; v q ) D hv q ; v q i  V(q) ; 2

(40)

where h:; :i is the metric on T M and V : M ! R is the potential. Given a path q : [0; T] ! M with integrable time derivative one can define Definition 23 (Action functional in Lagrangian form) AL [q] D

Z

T

˙ dt L(q(t); q(t)) :

(41)

is equivalent to Newton’s equations with q(0) D q(T). In particular the equations of motion in local coordinates are the Euler–Lagrange equations ˙ ˙ d @L(q(t); q(t)) q(t)) @L(q(t);  D 0; dt @q˙ i @q i

Remark 9 For historical reasons the condition DAL [q] (v) D 0 is called least action although it is only a condition for stationary points of AL [:] in . In general the Lagrangian contains a quadratic form in vq ; in turn the action has a time integral of a quadratic form ˙ in q(t). This essentially shows that the natural domain for AL [:] is the Sobolev space H 1 ([0; T]; Rn ). Actually the action is defined on H 1 ([0; T]; M), which is a Hilbert manifold [8,29]. This is locally described by absolutely continuous functions with the time derivative in the Lebesgue space L2 [28]. The interest in variational methods is related to the possibility of using critical point theory to find critical points corresponding to certain type of trajectories and then to show that such trajectories are solutions of Newton’s equations [2,21,33]. In particular this approach has been very successful in studying the problem of periodic orbits. Here is a general result in the Lagrangian setting: Theorem 11 ([8]) Let M be compact Riemannian manifold and let L : T M ! R be a Lagrangian of the form (40) with V 2 C 1 (M; R). Then there exists a periodic orbit in any free homotopy class. We illustrate the ideas of the proof by considering the special case where M is the n-dimensional torus T n ' Rn /Z n . The problem is now to show that the action a critical point, a minimum, in ( M) D ˚AL [:] attains q(:) 2 H 1 ([0; T]; M) : q(:) is not null homotopic . Let us consider the sublevel of the action A k D fq(:) 2 (M) : AL [q]  kg. Now since V is smooth there exists minM V (q) D m > 1 and therefore AL [q] 

0

In what follows we will be mostly interested in closed paths. Let C 2 ([0; T]; Rn ) be the space of a closed path of period T with two continuous time derivatives. One can easily show that Proposition 8 ([1,5]) Let L be a smooth Lagrangian function on M of the form (40). Let AL be defined over  D C 2 ([0; T]; M). Then DAL [q](v) D 0

2

for every v(:) 2 T ' C ([0; T]; T M)

i D 1; : : : ; n :

1 2

Z

T

2 ˙ dt kq(t)k m:

0

Since M is compact, the norm of q(:) in (M) is equiva˙ 2 . This allows us to show that the action A[:] is lent to kqk coercive, namely: Definition 24 (Coercivity) A :  ! R is coercive in  if for all q n (:) such that limn!1 kq n k D 1. Then lim n!1 A[q n ] D 0. Moreover in Ak necessarily we have ˙ 22 D kqk

Z

T 0

2 ˙ dt kq(t)k  2(k C m) :

1225

1226

Periodic Orbits of Hamiltonian Systems

This condition guarantees that Ak is weakly compact in the topology of  [2,33,53] and using the coercivity, one can obtain the existence of a minimizer q (:). The minimizer q (:) is attained in Ak and it is a weak solution of the equations of motion. Indeed A[q ] D min A[q] and

L : T M ! R and consider the problem of finding relative periodic paths (:) as critical points of the action AL [:]. We need to study the topology of  g (M) D f (:) 2 H 1 ([0; T]; M) : (t C T) D g: (t)g : (45)

DA[q ](v) D 0 8v(:) 2 (TRn ) ; namely, Z

T

dt

n  X @L(q˙ (t); q (t))

0

@q i

iD1

v i (t)

 @L(q˙ (t); q (t)) ˙ v i (t) D 0 : C @q˙ i

(42)

Using the fact that q (:) is absolutely continuous and that L is regular, one can integrate by parts Z

T

dt

n  X @L(q˙ (t); q (t))

0

@q˙ i

iD1

Z 



t

ds

(43)

@L(q˙ (s); q (s)) v˙i (t) D 0 ; @q i

and obtain @L(q˙ (t); q (t))  @q˙ i

Z

t

ds

@L(q˙ (s); q (s)) D ci ; @q i

where c i is constant in L2 ([0; T]; Rn ) : By differentiating with respect to t, we obtain the Euler– Lagrange equations: d @L(q˙ (t); q (t)) @L(q˙ (t); q (t))  D 0 a.e. 8i : (44) dt @q˙ i @q i It turns out that the equality in (44) holds for all times t because L is smooth. The reader should observe that the derivations of (42) and (44) are part of the proof of Proposition 8. It is necessary to note that constant paths are not in (M); indeed the number of minimizers is bounded from below by the Lusternik–Schnirelman category Cat(M) D n C 1 [15,33]. Thus, we exclude possible minimizers which would be trivial periodic orbits. In this very simple example we can therefore appreciate the role of the definition of the space of paths (M) and its topology. More interesting cases can be found in the study of the N-body problem and in particular in the article  n-Body Problem and Choreographies in this encyclopedia. The result in [8] can be generalized to Lagrangian systems with symmetries. Assume there is a group action on the configuration space G  M 7! M – denoted by (g; m) 7! g:m – which preserves the Lagrangian

In fact if the action AL [:] is bounded from below on  g (M), then each connected component would contain at least a minimum that is a critical point and therefore a periodic orbit. The following analysis was presented in [35]. In what follows for notational simplicity we denote a path and its homotopy class by the same symbol and use to denote both concatenation of paths and the induced operations on homotopy classes. Assume that M is connected. Choose a base point m 2 M and let : g m (M) D f 2  g (M) : (0) D mg ; the space of continuous paths from m to gm. Let : m (M) D id m (M) denote the space of continuous loops based at m. Note that the space of connected components of m (M) is the fundamental group of M: 0 (m (M)) D g 1 (M; m). Fix a particular path ! 2 m (M). The map 1 ˚! ( ) D ! is a bijection g

˚! : 0 (m (M)) ! 0 (m (M)) D 1 (M; m) ; where ! 1 is the path obtained by traversing ! “backwards.” This bijection depends (only) on the homotopy g class of ! in m (M). For any ˛ 2 m (M) let g:˛ be the loop in  gm (M) obtained by applying the diffeomorphism g to ˛ and define an automorphism of 1 (M; m) by ˛ 7! ˛ g D ! 1 g:˛ ! : Again this depends (only) on the homotopy class of ! in g m (M). Now define the g-twisted action of 1 (M; m) on itself by ˛  ˇ D ˛ g ˇ ˛ 1

˛; ˇ 2 1 (M; m) : (46)

The number of connected components of the relative loop space is given by the following result: Theorem 12 The map ˚! induces a bijection g

0 ( g (M)) Š 1 (M; m) ; g

where 1 (M; m) is the set of orbits of the g-twisted action of 1 (M; m) on itself. Remark 10 If g is homotopic to the identity then  g (M) : is homotopy-equivalent to the loop space (M) D

Periodic Orbits of Hamiltonian Systems

id (M) and the g-twisted action of 1 (M; m) on itself is just conjugation. This is therefore the well-known result that the connected components of the loop space map bijectively to the conjugacy classes of the fundamental group [29]. Remark 11 The g-twisted action of 1 (M; m) on itself induces an affine action of the first homology group H1 (M) on itself: h˛i  hˇi D g:h˛i  h˛i C hˇi ; where the brackets h:i denote the homology class and g:h˛i denotes the natural action of g on H1 (M). When 1 (M; m) is Abelian this is the same as the action of 1 (M; m) on itself. More generally it is easier to calculate than the 1 (M; m) action and in typical applications may be used to describe relative periodic orbits in terms of winding numbers. The analysis gives explicit results for systems whose configuration space has the property that all its homotopy groups except the fundamental group are trivial. In this case M is said to be K(; 1); for more details see [12,57]. Examples of K(; 1) spaces include tori, the plane R2 with N points removed, and the configuration spaces of planar N-body problems. Theorem 13 Assume M is a K(; 1). Then for any 2 g m (M) the connected component of  g (M) containing , g denoted  (M), is also a K(; 1) with g

g

1 ( (M)) Š Z1 (M) (˚! ( )); where g

Z1 (M) (˚! ( )) : D f˛ 2 1 (M) : ˛ g ˚! ( ) ˛ 1 D ˚! ( )g i. e., the isotropy subgroup (or centralizer) at ˚! ( ) of the g-twisted action of 1 (M; m) on itself. We note that all K(; 1)’s with isomorphic fundamental groups are homotopy-equivalent to each other [12,57], and so this result determines the homotopy types of connected components of relative loop spaces. The homology groups can be computed algebraically as the homology groups of the fundamental group [14].

numbers. Since T 1 is a K(; 1), Theorem 13 says that each component of relative loop space is also a K(; 1) with a fundamental group isomorphic to Z, and therefore has the homotopy type of a circle. Now consider  g (T 1 ) where g : T 1 ! T 1 is a reflection. Choose one of the two fixed points of the reflection to be the base point m. We may choose ! to be the trivial path from m to g:m. Then for each ˛ 2 1 (T 1 ; m) Š Z we have ˛ g D ˛ and so the “g-twisted action” (46) is the translation ˛:ˇ D ˇ  2˛: g

This has two orbits, 1 (T 1 ) Š Z2 , and the isotropy subgroups are trivial. It follows from Theorems 12 and 13 that the space of relative loops  g (T 1 ) has two components, each of which is contractible. Numerical Studies In many interesting problems, typically in celestial mechanics, the action functional is bounded from below and therefore the expected critical points are minimizers. In recent years, in connection with the discovery of the so-called choreographic periodic orbits, Simó [52] developed an algorithm to study how to perform a numerical minimization of the action in classes of loops with a definite symmetry type. The idea is based on the description of the orbit in terms of its Fourier coefficients and defining the action AL [:] as a function on the Fourier space. The action AL [:] is then minimized on the space of Fourier coefficients. The interested reader should consult  n-Body Problem and Choreographies in this encyclopedia. It would be very interesting to generalize this method to different actions and to see whether one can impose constraints not only on the symmetry type but also on the topology of the space of loops. Hamiltonian Viewpoint The Hamilton principle gives a variational characterization to the Hamiltonian equation. For Hamiltonian systems in R2n the formulation of the principle is very simple. Let H : R2n ! R be the Hamiltonian function, then the action functional is Definition 25 (Action functional in Hamiltonian form) :

A Simple Example Let M D T 1 , the circle, and consider first the loop space (T 1 ). The “g-twisted action” of 1 (T 1 ) on itself is just conjugation, and since 1 (T 1 ) Š Z is Abelian this is trivial. So 0 ((M)) Š Z, the homotopy classes of loops being specified precisely by their winding

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AH [ ] D

Z

T

dt 0

n X

p i (t) q˙i (t)

iD1

Z

T

dt H(q(t); p(t)) ;

 0

where (t) D (q(t); p(t)).

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1227

1228

Periodic Orbits of Hamiltonian Systems

It is worth recalling that if the Lagrangian function is hyperregular, then the system can be transformed through the Legendre transform into a Hamiltonian system on the cotangent bundle of the configuration space. In any case, given AH [:] one can show [1,5] Proposition 9 Let H a smooth Hamiltonian function on R2n . Let AH be defined over C 2 ([0; T]; R2n ); then DAH [q; p](v) D 0 for every v(:) 2 C 2 ([0; T]; R2n ) is equivalent to Hamiltonian equations with q(0) D q(T), p(0) D p(T).

but then it would become multivalued. In fact in M one has to introduce a 1-form ˛ such that ! D d˛. The 1-form ˛ is not unique and depends on the co-homology of M. Although AH is multivalued the differential DAH is single-valued [34]. Let '(s) be a finite variation with d'(s) ds jsD0 D , then ˇ dAH ['(s) ˇˇ DAH [z]() D ˇ ds sD0 Z Z T D L (!)  dthrH(z); i ; ˙

0

Recall that in the Hamiltonian context p and q are independent variables and therefore to prove the preceding proposition it is necessary to compute the variations with respect to q’s and p’s independently. From the analytical point of view the functional AH [:] is a difficult object to study since it is unbounded from below and above, namely, it is indefinite [33]. It is not difficult to give an example, consider n D 1 and a Hamiltonian like H D p2 /2. For indefinite functionals a whole theory has been developed to apply the mountain pass theorem [47]. There is also a generalization of the Legendre transform [21]. The new transform [21] can be applied directly to the action functional AH [:] rather than to H. This allows us to work with a convex functional. We do not enter into the details but we want to recall one result which describes the conditions for the existence of at least n periodic orbits.

where L is the Lie derivative with respect to . Now since d! D 0 one can show that Z T Z T i i X H !  dthrH(z); i ; DAH [z]() D

Theorem 14 ([21]) Suppose that H 2 C 1 (R2n ; R) and for some ˇ > 0, assume that C D fz 2 R2n : H(z)  ˇg is strictly convex, with boundary @C D fz 2 R2n : H(z) D ˇg satisfying hz; rH(z)i > 0 for allpz 2 @C. Suppose that for r; R > 0 with r < R < 2 r there are two balls Br (0); B R (0) centered at the origin of R2n such that Br (0) al lowbreakC B R (0), then there are at least n distinct periodic solutions on @C of the Hamilton system z˙ D J rH(z).

where R is the so-called Routhian and ˇ is a 2-form dependent on the conserved momentum . The construction of the Routhian and of the reduced action AR [:] can be found in [34].

The functional AH [:] has been defined for Hamiltonian systems on R2n , but it admits a generalization to symplectic manifolds which are not tangent bundles: Definition 26 Let (P ; !) be a symplectic manifold and H : P ! R a smooth Hamiltonian function, let : P ! R be a closed path which is the boundary of a two-dimensional connected region ˙ , then Z Z T : AH [ ] D ! dt H(z(t)) : (49) ˙

0

The functional (49) is a very interesting object. It is naturally defined on closed paths which bound two-dimensional regions. The functional can be defined on “paths”

0

0

which is single-valued. The theory for such multivalued functionals has been developed by many authors; here we would like to cite [22,43,56]. Note that functionals of the form (49) can result from symmetry reduction. In fact as shown in [34] a Lagrangian system with a non-Abelian symmetry G has a reduced dynamics determined by a variational principle of the form : AR [ ] D

Z

Z

T

˙ dt R(q(t); q(t))  0

˙

˙ ˇ (q(t); q(t)) ; (50)

Fixed-Energy Problem, Hill’s Region Let us consider Hamiltonian systems (P ; !; H) where the phase space a cotangent bundle P D T  M ' R2n . The symplectic form is then given by ! D d , where is the canonical form, and the Hamiltonian is H(p; q) D

1 hp; pi C V(q) 2

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h:; :i is a Riemannian metric on Tq M. The Hamiltonian flow preserves the Hamiltonian function; therefore, a natural problem is to restrict the dynamics to the manifold defined by a fixed constant value of the Hamiltonian that physically corresponds to fixing the energy. Letting E be the energy value, one can define this natural constraint by constructing the following submanifold of the phase space P : ˙E D f(p; q) 2 P : H(p; q) D Eg :

Periodic Orbits of Hamiltonian Systems

From ˙E it is possible to construct a new manifold that corresponds to the image of the projection of ˙E onto the configuration space M. To obtain such a projection it is sufficient to look at (51) and observe that the norm of p on Tq M cannot be negative. From this, the new space, Hill’s region, turns out to be defined as follows: Definition 27 (Hill’s region) : PE D fq 2 M : E  V (q)  0g Remark 12 The manifold PE depends on the values of E and may have boundaries. For instance, consider the Hamiltonian

Periodic Orbits of Hamiltonian Systems, Figure 4 Example of a compact nonsimply connected Hill region with two brake orbits brake and one internal orbit internal that cannot be deformed into a point

1  1 kpk2  kqk2 C kqk4 with  > 0 ; 2 2 4

q(t) 2 PE n@PE for all t 2 (0; T/2) and q(0) 2 @PE , q(T/2) 2 @PE .

which has a Hill region defined by

  1 : PE D q 2 M : E C kqk2  kqk4  0 : 2 4

Definition 29 (Internal periodic orbit) A solution q(:) of the Hamilton equation is an internal periodic orbit of period T if q(t C T) D q(t) and q(t) 2 PE n@PE for all t.

HD

First note that PE is a subset of following cases:     

R2

and that there are the

PE D ; for E < 2 /4, PE is a circle for E D 2 /4, PE is an annulus for 2 /4 < E < 0, PE is a disk minus its center for E D 0, PE is a disk for E > 0.

Note that if PE is not empty, then the boundary of @PE is given by

  1 : @PE D q 2 M : E C kqk2  kqk4 D 0 2 4 and is not empty. The boundary corresponds to the set of points where all momenta p vanish. The Hill region has a topology which changes according to the values of the energy; hence, it is a natural problem to search for periodic orbits in it. What are the possible orbits in PE ? Assuming there is a generic Hill region with a nonempty boundary, there are two possible types of orbits: (A) Orbits joining two points of the boundary, the brake orbits, (B) Orbits not intersecting the boundary, the internal periodic orbits. In general given a Hamiltonian system with a nonempty Hill region PE we define Definition 28 (Brake orbit) A solution q(:) of the Hamilton equation is a brake orbit of period T if q(t C T) D q(t),

Remark 13 In principle one could think that brake orbits could have more than two intersections with the boundary @PE . This is not possible in systems with Hamiltonian (51) because the velocity on @PE is zero and the Hamiltonian equations are “reversible”, that is, if q(t) is a solution then q(t) is a solution. These two properties imply that any solution q(t) such that q(t i ) 2 @PE follows the trajectory q(t) for t > t i with reversed velocity [24,30]. An example of Hill’s region with brake orbits and internal periodic orbit is given in Fig. 4. Jacobi Metric and Tonelli Functional The general approach to study the periodic orbits of type A or B is to use a variational principle which is naturally defined on paths in PE . Let us consider Definition 30 (Jacobi metric)   Z  dq(s) 2 1 1  JE [q] D ds(E  V(q(s))   ds  : 2 0

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: The domain of JE [:] is (PE ) D fq(:) 2 H 1 ([0; 1]; Rn ) : q(s C 1) D q(s) q(s) 2 PE g, and the following result holds: Proposition 10 Let q(:) be a path in (PE ) where JE [:] is differentiable and let q(:) C v(:) 2 (PE n@PE ) for all sufficiently small, then DJE [q](v) D 0

for all v(:) 2 T(PE )

is equivalent to dq(t) D rV(q(t)) dt

1229

1230

Periodic Orbits of Hamiltonian Systems

after a time rescaling defined by    dq(s)  dt 1  D  ds  p E  V (q(s)) : ds

Remark 14 Observe that in both the Jacobi and the Tonelli formulation there is a reparameterization of the time and therefore one can always restrict the consideration to trajectories with unit period.

The proof of this proposition is quite standard and can be found in [1,8,11,30,51]. Let us observe that in the regions where E  V(q) > 0 the functional JE [:] can be derived from the Riemannian metric X ı i j dq i ˝ dq j (53) d2 ` D (E  V(q))

In variational methods we aim to show that the critical set is not empty and, if possible, to estimate its cardinality. This is certainly affected by the topology of the space, where the paths are defined. Let us now consider the Jacobi metric JE [:]. This functional depends on the energy E, which affects the properties of PE and (PE ). There are four possible cases:

i; j

and indeed from this its name originated. From the Jacobi metric a length can be defined Z 1 q : 2: ˙ ds (E  V(q(s))kq(s)k `[q] D 0

The Jacobi metric (53) transformed the study of the classical Newton equations into the study of the geodesics on PE . It is crucial to note that the Jacobi metric is vanishing on @PE and therefore cannot be complete in PE whenever the boundary is not empty. There is another functional that can be used to study periodic orbits in PE . This is defined as follows: Definition 31 (Tonelli functional)  Z 1  Z  dq(s) 2 1 1   : T E [q] D ds(E  V (q(s)) ds  2 0 ds  0 Tonelli’s functional is a product of two integrals and therefore does not have a natural geometrical interpretation like (53). One can observe that (by Schwartz inequality) the Tonelli functional is bounded from below by Jacobi length: (`[q])2  2TE [q]. Note that also TE [:] vanishes on @PE . The functional TE [:] is defined on (PE ) and provides another possible variational description of Newton’s equations. Indeed one can easily show [2]. Proposition 11 Let q(:) be a path in (PE ) where TE [:] is differentiable and let q(:) C v(:) 2 (PE n@PE ) for all sufficiently small, then DTE [q](v) D 0 for all v(:) 2 T(PE ) is equivalent to dq(t) D rV(q(t)) dt after time rescaling defined by  Z 1   dq(s) 2    2 ds  dt ds  0 D Z 1 : ds ds(E  V(q(s)) 0

   

PE is compact and @PE D ;, PE is compact and @PE ¤ ;, PE is not compact and @PE D ;, PE is not compact and @PE ¤ ;.

In the next section we shall illustrate some results regarding the preceding four cases. We shall see that the main approaches have much in common with Riemannian geometry. For more details on variational methods the reader is encouraged to consult [2,15,28,33,53]. Orbits in Compact Hill’s Regions Without a Boundary Let us consider a region PE without a boundary for E > sup q2M V(q). If M is compact, then the previous condition can be satisfied for some finite E. In this case PE ' M and the Jacobi metric JE [:] becomes a Riemannian metric on M; therefore, the problem of periodic orbits in M translates into the problem of closed geodesics in the Riemannian manifold M. This is solved by Theorem 15 (Lusternik and Fet) Each compact Riemannian manifold contains a closed geodesic. For a proof see [28,29]. The main tool for proving the theorem is the so-called curve shortening. The manifold PE is Riemannian with metric (53). A base for the topology is given by Br (q) D fq0 2 M : d(q; q0 ) < rg, where d(q; q0 ) D inf f`[ ]; (:) is a piecewise smooth curve such that (0) D q and (1) D q0 : Remark 15 The reader may observe that if PE has a nonempty boundary then d(:; :) turns into a pseudometric because d(q; q0 ) D 0 for all q; q0 2 @PE . Certainly d(:; :) is still a metric restricted to the open set PE n@PE . Let us now give a sketch of the curve-shortening method. A standard result in Riemannian geometry guarantees that given a point q0 and a neighborhood Bı (q0 ), there exists ı such that any point in @Bı (q0 ) can be joined to q0 by a unique geodesic [28]. Since M is compact one

Periodic Orbits of Hamiltonian Systems

can use a finite family of such neighborhoods to join any two points q0 ; q1 2 M with a piecewise geodesic (:). Now consider on M the space of closed paths of class H 1 ([0; 1]; M) and consider a sequence 0  t0(k) < t1(k) < (k) 2    < t (k) n1 < t n  1 such that t iC1  t i < ı /(2c). Take 1 a curve (:) 2 H ([0; 1]; M) with JE [ ]  c then define 8 ˆ is a piecewise geodesic curve ˆ ˆ ˆ < where S (k) ( )(t) is restricted to t (k) ; t (k) ; i iC1 S (k) ( ) D (k) (k) ˆ to

t is a geodesic joining

t ˆ i iC1 ˆ ˆ : for i D 1 : : : n : Now clearly JE [S (k) ( )]  JE [ ]; `[S (k) ( )]  `[ ]. The map S (kC1) (:) is constructed by taking a new partition of [0; 1] such that t (k) < t (kC1) < t (k) i i iC1 . Since one can easily verify `[S (kC1) ( )]  `[S (k) ( )]

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the iteration of S (k) (:) is called curve shortening. In [24,28], it is shown that S (k) ( ) converges uniformly to a geodesic. The limit could be just a point curve. To show that this cannot be the case one uses the fact that on a compact manifold of dimension n there is always a map h : S d 7! M (with 1  d  n) which is not homotopic to a constant map. There have been many generalizations of Lusternik– Fet theorem and the reader is invited to refer to [29]. Brake Orbits in Hill’s Regions with a Boundary Let PE be a region with boundary for inf q2M V(q) < E < inf q2M V(q). In this case there is the following result: Theorem 16 ([11]) Suppose that PE is compact and there are no equilibrium positions in @PE . Then the number of brake orbits in PE is at least equal to the number of generators of 1 (PE n@PE ). Since the Jacobi metric is singular on @PE it is necessary to analyze the geodesic motion near the boundary. This type of analysis goes back to earlier works [9,10,51]. The main point is to construct a new region PE PE on which JE [:] is positive-definite. The next step is to construct a geodesic joining two points on the new boundary @PE . Finally the construction on PE is used to show that the geodesic q (:) becomes a brake orbit in the limit ! 0. In the case of noncompact PE , the existence of a brake orbit was proved in [44]. This result is based on two assumptions: (i) @PE is not empty and it is formed by two connected components A1 and A2 such that if x n 2 A1 and y n 2

A2 with kx n k ! 1, ky n k ! 1 then kx n  y n k ! 1. (ii) Let Rı D fq 2 M : E  ı < V(q) < Eg for ı > 0. If either A1 or A2 is not compact, there is a number r > 0 with the following property: if r0 > r then there exist r1 > r0 and ı > ı  > 0 such that for every q 2 Rı \ fq : kqk > r1 g and (q; p) 2 H 1 (E) implies that the Hamiltonian flow (t; q; p) stays in fq : kqk > r1 g for all t  0 where defined. Theorem 17 ([44]) Suppose that PE is connected, inf q2@PE krV(q)k > 0 and conditions i and ii hold true. Then there exists a periodic solution which is a brake orbit. In this result the lack of compactness of PE does not allow us to use the curve shortening. For this reason in [44] the direct minimization was employed. First a new region PEı PE is defined for ı > 0. Such region now has a boundary with two different connected components, Aı1 and Aı2 , respectively. In [44] the following minimization problem was studied min JE [q] where q(:) 2 H 1 and boundary conditions q(0) 2 Aı1 ; q(0) 2 Aı1 : Condition i is able to control the lack of compactness of PE and to show that the minimization problem admits a solution qı (:). Condition ii allows us to take the limit ı ! 0 to obtain a brake orbit in PE . Orbits in Closed, Nonsimply Connected Hill’s Regions with Boundaries Let us now consider a general Hill region which is not necessarily compact, with a boundary and with nontrivial homotopy group 1 (PE ). The natural problem is to determine under which condition it is possible to prove the existence of a internal periodic orbit within a specific homotopy class of PE . A partial answer is given by Theorem 18 ([6,7]) Suppose that PE is closed and bounded and that rV(q) ¤ 0 for all q 2 @PE . Then there exists a periodic orbit q(s) 2 PE for all s. The orbit q(:) can be either a brake orbit or an internal periodic orbit. As we have already seen, the Jacobi metric (but also the Tonelli functional) is degenerate on the boundary @PE , i. e., JE [ ] is identically zero on every closed path (:) such that (s) 2 @PE for every s 2 [0; 1]. In order to avoid this problem, in [6,7], a modified functional was introduced: :

J [q] D JE [q] C

Z

1 0

ds U [q(s)] :

1231

1232

Periodic Orbits of Hamiltonian Systems

The functional J [:] is called penalized. If q n (:) is a sequence of curves in (PE ) (the closure of (PE )) weakly converging to a curve q (:) intersecting the boundary, then the form of U (:) implies that J ( ) [q n ] ! C1 whenever > 0. This type of penalization is similar to the so-called strong force potential used in the N-body problem (see [2,26] and also  n-Body Problem and Choreographies). In this case there are two kinds of difficulties: PE is no longer compact and the boundary @PE is not empty. The strategy is as follows: prove that the critical level sets of J [:] are precompact. This is obtained by a generalization of the Palais–Smale condition. This prevents the sequences from converging on the boundary, and it is realized by taking a functional : C 1 ((PE )) ! RC such that if q n (:) converges to a path intersecting the boundary, then (q n ) ! C1. This allows us to introduce: Definition 32 (Weighted Palais–Smale condition [6]) The action J [:] satisfies the weighted Palais–Smale condition if any sequence q n (:) fulfills one of the following alternatives: 1. (q n ) and J [q n ] are bounded and DJ [q n ] ! 0 and qn has a convergent subsequence, 2. J [q n ] is convergent and (q n ) ! C1 and there exists  > 0 such that kDJ [q n ]k   kD (q n )k for n sufficiently large. The verification of the weighted Palais–Smale condition guarantees that the set K c \ fq : (q)  Mg is compact for every M > 0. The next step in the proof is to construct a mini-max structure [47] that allows us to identify the critical values. This is achieved by constructing two manifolds Q and S such that: (i) S \ @Q D ;, (ii) If u 2 C 0 (Q; (PE )) such that u(q(:)) D q(:) for every q(:) 2 @Q then h(Q) \ S ¤ ;. The manifolds Q and S are defined such that the critical level

c D inf sup J (h(q)) u2U q2Q

C 0 (Q; (PE )) :

u(q(:)) D is finite. Here U D fu 2 q(:) if J [q]  0g. Then the weighted Palais–Smale condition allows us to construct a gradient flow and to prove the existence of a critical point q (:). In [7] it is shown also that there exists ˛  ˇ independent of such that



˛  J [q ]  ˇ : This uniform estimates make it possible to prove that for ! 0 the path q (:) converges uniformly to a closed path

: q0 (:) such that the set I D ft 2 [0; 1] : q0 (t) 2 PE n@PE g is not empty. Since q0 (:) is continuous then the interval I has connected components. Note that q0 (:) has the same homotopy type as q (:), but in general this is no longer true for the solution of the equations of motion. In fact q0 (:) is a solution only on the connected component of I. Let  be one connected component of I. On  the path q0 (:) solves the equations of motion. Therefore, the topological characterization of the periodic orbit depends on . If  [0; 1] then q0 (:) is a brake orbit, if  D [0; 1] then q0 (:) is an internal periodic orbit. The problem of finding internal periodic orbits in any prescribed homotopy class is still open. Continuation of Periodic Orbits as Critical Points Variational methods can also be very useful to study the problem of continuation of periodic orbits. Here we restrict ourselves to a few examples. Lagrangian Variational Principle Let us consider a dynamical system in Rn that can be written in the Lagrangian form: L(v q ; q) D

1 kv q k  V (q) : 2

Let us assume (i) There exists a > 0 such that V( q) D a V(q) for any  > 0, (ii) There exists a 1-periodic orbit q1 (:). The R 1 orbit q1 (:) is a critical point of AL [q] D ˙ 0 dsL(q(s); q(s)). Now let us consider another potential term W(q) such that W( q) D b W(q)

with b > a :

Let us look for periodic orbits of period T for the system whose Lagrangian is L(v q ; q) D

1 kv q k  V (q)  W(q) : 2

This suggests searching for critical points x(:) of the functional  Z T

kx˙ (t)k2 T AL [x] D dt  V(x(t))  W(x(t)) : (55) 2 0 The scaling properties of V and W can be used to construct a perturbation argument. Define x(t) D T p q(t/T) ; where q(s) is defined for s 2 [0; 1] ; and p D 2/(a C 2) :

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Periodic Orbits of Hamiltonian Systems

The dynamical equations for x(:) are equivalent to the Lagrangian equation for  Z 1

2 ˙ kq(s)k AL [q; ] D ds  V(q(s))  W(q(s)) ; 2 0 (57) where : 2(ba) D T aC2 :

tion when the Hamiltonian is perturbed. In order to use the formulation in the loop space we can introduce the Liapunov–Schmidt reduction. In fact, in general, Hamiltonian vector fields at a periodic orbit have a linearization with a nonempty kernel. The construction we now present is a very well known approach in the infinite-dimensional setting. Liapunov–Schmidt Reduction for Hamiltonian Systems [3,25] Let (H ; P ; ˝) be a Hamiltonian system with P D C 1 ([0; 1]; R2n ). Let X : P ! Y be the Hamiltonian vector field such that z0 (:) is a zero of X (:) and

The scaling properties of the potential allow us to transform the given problem into a perturbation problem. Note that there is the following correspondence: a small perturbation for the scaled action (57) corresponds to large periods for (55). Now AL [:; 0] D A1L [:] and in general the critical point q1 (:) is not isolated and therefore D2 AL [q1 ; 0] has some degeneracy. If the manifold of critical points is degenerate along normal directions (normal degeneracy), then it is possible to continue the critical point q1 (:) into a T-periodic orbit by decomposing the continuation procedure. This approach is the so-called Liapunov–Schmidt reduction; it is used to generalize the IFT to situations where it cannot be applied directly. An example will be illustrated in the last part of this section in the study of Hamiltonian systems. For details about normal degeneracy the reader could consult [15].

Condition i implies that it is possible to make the following decomposition

Hamiltonian Variational Principle Hamiltonian equations can be thought of as a vector field on a space of loops. The geometrical construction was described in [56]. Let us consider the space C 1 ([0; 1]; Rn ) of differentiable loops. We can define

where ˘ : Y 7! rank(L). The first equation in (61) can be solved with respect to w by the IFT. The second equation becomes the so-called bifurcation equation:

:

X (z) D 

dz(s) JrH(z(s)); z(:) 2 C 1 ([0; 1]; R2n ): (58) ds

The zero set of X (z) is formed by loops z(:) such that 

dz(s) D JrH(z(s)) : ds

This corresponds to a periodic orbit z(:) with period 2/. It turns out that X (:) is a Hamiltonian vector field whose Hamiltonian on C 1 ([0; 1]; R2n ) [25,56] is given by Z 2 1 H [z] D ds fh J z˙(s); z(s)i  H(z(s))g (59) 2 0 and with symplectic form Z 2 1 ˝[z; w] D dshJz(s); w(s)i : 2 0

(60)

Indeed a simple calculation shows that ˝(X (z); w) D dH [z](w) : Let us now suppose there is a periodic orbit z0 (:). We want to illustrate how to study the continuation and bifurca-

(i) L D DX (z0 ) is Fredholm, (ii) J(ker(L)) D ker(L).

1. P D ker(L) ˚ W, W closed subspace, 2. Y D rank(L) ˚ N, N closed subspace. In fact any z can be written as z D k C w with k 2 ker(L) and w 2 W. This allows us to reduce X (z) D 0 to solving the following system of equations: ( ˘ X (k C w) D 0 2 rank(L) (61) (I  ˘ ) X (k C w) D 0 2 N

X g (k) D (I  ˘ )X (k C w(k)) D 0 :

(62)

Note that since L is Fredholm dim ker(L) < 1, (62) is a finite-dimensional problem. Now condition ii implies that X g can be thought of as a map from ker(L) into itself. Furthermore this allows us to show that X g is a Hamiltonian vector field with Hamiltonian g(k) D H (k C w(k)) and symplectic form ˝(:; :). If condition ii does not hold a further condition has to be included in order to guarantee that X g is a Hamiltonian vector field. The bifurcation equation becomes therefore ˝(X g (k); u)  dH (k C w(k))(u) D 0 8u :

(63)

In [56] the loop space approach and the reduction were used to show that if a Hamiltonian system has a manifold ˙ of periodic orbits whose tangent space coincides with the kernel of D2 AH (nondegeneracy condition) then small perturbations of the Hamiltonian cannot destroy all the periodic orbits:

1233

1234

Periodic Orbits of Hamiltonian Systems

Theorem 19 ([56]) Let ˙ be a compact, nondegenerate manifold of periodic orbits for a Hamiltonian system (P ; !; H). Given any neighborhood U of ˙ there exists 0 > 0 such that for j j < 0 , the number of periodic orbits in U for (P ; !; H C H1 ) is no less than the Lusternik–Schnirelman category Cat(˙ /S 1 ). If ˙ satisfies the additional condition that the algebraic multiplicity of 1 as an eigenvalue of the Poincaré map is uniformly equal to the dim ˙ , then Cat(˙ /S 1 )  (1 C dim ˙ )/2. In [25] Liapunov–Schmidt reduction was used to prove the Liapunov center theorem and also the Hamiltonian Hopf bifurcation. Further Directions This article has focused on periodic orbits in Hamiltonian systems. This problem can be seen as an organizing center in the history of the development of modern mathematics. In fact it is related to many theoretical aspects: bifurcation theory, symmetry reduction, variational methods and topology of closed curves on manifolds. Still, open problems remain, for example, one would like to prove the existence of internal periodic orbits in every homotopy class of a Hill region. Another interesting direction is the analytical study of the action functional that results from symmetry reduction. A further generalization of the Hamiltonian formalism is the study of the so-called multiperiodic patterns. Here is an example. Let ; p be two functions defined on the two-dimensional torus T 2 and valued in R. The functions ; p are 2-periodic because they satisfy p(x C 1 ; y C 2 ) D p(x; y), (x C 1 ; y C 2 ) D (x), where (x; y) C (1 ; 2 ) ' (x; y) in T 2 . One can pose the problem to solve a couple of partial differential equations of the form @ V(; p; x; y) D 0 and @ @ r 2 p C V(; p; x; y) D 0 : @p

(64)

Here V : R2 7! R is a smooth potential function. Note that the periodicity of  and p is partial, that is why the term “multiperiodic patterns” is used. It has been shown that Eqs. (64) admits a description in terms of finite-dimensional multi-symplectic structure and the equations of motion can be cast into a general Hamiltonian variational principle. In fact one can define Z D (; u1 ; u2 ; p) defined as functions on T 2 and the Hamiltonian 1 2 (u C u22 ) C V(; p; x; y) : 2 1

DL(Z) D 0 ;

(65)

where L is the action functional   Z 1 Z 2 1 @Z @Z L(Z) D dx dy J1 C J2 ; Z 2 @x @y 0 0 !  S(Z; x; y)

(66)

defined on 2-periodic maps Z : T 2 7! R4 . The matrices J 1 and J 2 are two symplectic structures and form a multisymplectic structure. In this example J 1 and J 2 read 0 1 0 1 0 0 B1 0 0 0 C C J1 D B @0 0 0 1A and 0 0 1 0 (67) 0 1 0 0 1 0 B0 0 0 1C C: J2 D B @1 0 0 0A 0

1

0

0

For Eqs. (64) one can pose the problem of finding solutions as “multiperiodic” critical points of the action L, which is a generalization of the Hamiltonian action functional (48). An interesting reference for this problem is [13]. Acknowledgments The author would like to thank Heinz Hanßmann for his critical and thorough reading of the manuscript and Ferdinand Verhulst for his useful suggestions. Bibliography

r 2 C

S(Z; x; y) D

Then Eqs. (64) turn out to be equivalent to the variational equation

1. Abraham R, Marsden J (1978) Foundations of Mechanics. Benjamin-Cummings, Reading 2. Ambrosetti A, Coti V (1994) Zelati Periodic Solutions of singular Lagrangian Systems. Birkhäuser, Boston 3. Ambrosetti A, Prodi G (1993) A primer on Nonlinear Analysis. CUP, Cambridge UK 4. Arms JM, Cushman R, Gotay MJ (1991) A universal reduction procedure for Hamiltonian group actions. In: Ratiu TS (ed) The Geometry of Hamiltonian Systems. Springer, New York, pp 33– 51 5. Arnold VI (1989) Mathematical Methods of Classical Mechanics. Springer, New York 6. Benci V (1984) Normal modes of a Lagrangian system constrained in a potential well. Annales de l’IHP section C tome 5(1):379–400 7. Benci V (1984) Closed geodesics for the Jacobi metric and periodic solutions of prescribed energy of natural Hamiltonian systems. Annales de l’IHP section C tome 5(1):401–412

Periodic Orbits of Hamiltonian Systems

8. Benci V (1986) Periodic solutions for Lagrangian systems on a compact manifold. J Diff Eq 63:135–161 9. Birkhoff GD (1927) Dynamical Systems with two degrees of freedom. Trans Am Math Soc 18:199–300 10. Birkhoff GD (1927) Dynamical Systems. Colloq. Publ. 9. Amer. Math. Soc., Providence RI 11. Bolotin SV, Kozlov VV (1978) Libration in systems with many degrees of freedom. J Appl Math Mech 42:256–261 12. Bott R, Tu W (1995) Differential forms in algebraic topology. Springer, New York 13. Bridges T (2006) Canonical multi-symplectic structure on the total exterior algebra bundle. Proc R Soc A 462:1531–1551 14. Brown KS (1982) Cohomology of groups. Springer, New York 15. Chang K (1991) Infinite dimensional Morse theory and multiple solution problems. Birkhäuser, Boston 16. Cushman R, Bates L (1997) Global aspects of classical integrable systems. Birkhäuser, Boston 17. Cushmann R, Sadowski D, Efstathiou K (2005) No polar coordinates. In: Montaldi J, Ratiu T (eds) Geometric Mechanics and Symmetry: the Peyresq Lectures. Cambridge University Press, Cambridge UK 18. Dell Antonio G (1995) Agomenti scelti di meccanica SISSA/ISAS Trieste. ref. ILAS/FM-19/1995 19. Doedel EJ, Keller HB, Kernvez JP (1991) Numerical analysis and control of bifurcation problems: (I) Bifurcation in finite dimensions. Int J Bifurcat Chaos 3(1):493–520 20. Doedel EJ, Keller HB, Kernvez JP (1991) Numerical analysis and control of bifurcation problems: (II) Bifurcation in infinite dimensions. Int J Bifurcat Chaos 4(1):745–772 21. Ekeland I (1990) Convexity methods in Hamiltonian mechanics. Springer, Berlin 22. Farber M (2004) Topology of Closed One-Forms. AMS, Providence 23. Gallavotti G (2001) Quasi periodic motions from Hipparchus to Kolmogorov. Rend Mat Acc Lincei s. 9, 12:125–152 24. Gluck H, Ziller W (1983) Existence of periodic motions of conservative systems. In: Bombieri E (ed) Seminar on Minimal Submanifolds. Princeton University Press, Princeton NJ 25. Golubitsky M, Marsden JE, Stewart I, Dellnitz M (1995) The constrained Liapunov–Schmidt procedure and periodic orbits. Fields Institute Communications 4. Fields Institute, Toronto, pp 81–127 26. Gordon W (1975) Conservative dynamical systems involving strong force. Trans Am Math Soc 204:113–135 27. Guillemin V, Sternberg S (1984) Symplectic techniques in Physics. Cambridge University Press, Cambridge UK 28. Jost J (1995) Riemannian geometry and geometric analysis. Springer, Berlin 29. Klingenberg W (1978) Lectures on closed geodesics. Springer, Berlin 30. Kozlov VV (1985) Calculus of variations in the large and classical mechanics. Russ Math Surv 40:37–71 31. Kuznetsov YA (2004) Elements of Applied Bifurcation Theory. Springer, Berlin 32. Leimkuhler B, Reich S (2005) Simulating Hamiltonian Dynamics. CUP, Cambridge UK 33. Mawhin J, Willhelm M (1990) Critical point theory and Hamiltonian systems. Springer, Berlin 34. Marsden J, Ratiu TS, Scheurle J (2000) Reduction theory and the Lagrange–Routh equations. J Math Phys 41:3379– 3429

35. McCord C, Montaldi J, Roberts M, Sbano L (2003) Relative Periodic Orbits of Symmetric Lagrangian Systems. Proc. Equadiff. World Scientific, Singapore 36. Meyer K, Hall GR (1991) Introduction to Hamiltonian Systems and the N-Body Problem. Springer, Berlin 37. Meyer K (1999) Periodic solutions of the N-body problem. LNM, vol 1719. Springer, Berlin 38. Montaldi J, Buono PL, Laurent F (2005) Poltz Symmetric Hamiltonian Bifurcations. In: Montaldi J, Ratiu T (eds) Geometric Mechanics and Symmetry: the Peyresq Lectures. Cambridge University Press, Cambridge UK 39. Montaldi J, Roberts M, Stewart I (1988) Periodic solutions near equilibria of symmetric Hamiltonian systems. Phil Trans R Soc Lon A 325:237–293 40. Montaldi J, Roberts M, Stewart I (1990) Existence of nonlinear normal modes of symmetric Hamiltonian systems. Nonlinearity 3:695–730 41. Montaldi J, Roberts M, Stewart I (1990) Stability of nonlinear normal modes of symmetric Hamiltonian systems. Nonlinearity 3:731–772 42. Munoz–Almaraz FJ, Freire E, Galan J, Doedel E, Vanderbauwhede (2003) A Continuation of periodic orbits in conservative and Hamiltonian systems. Physica D 181:1–38 43. Novikov SP (1982) The Hamiltonian formalism and a multivalued analogue of Morse theory. Russ Math Surv 37:1–56 44. Offin D (1987) A Class of Periodic Orbits in Classical Mechanics. J Diff Equ 66:9–117 45. Ortega J, Ratiu T (1998) Singular Reduction of Poisson Manifolds. Lett Math Phys 46:359–372 46. Poincaré H (1956) Le Méthodes Nouvelles de la Mécanique Céleste. Gauthiers-Villars, Paris 47. Rabinowitz PH (1986) Minimax Methods in Critical Point Theory with Applications to Differential Equations. In: CBMS Reg. Conf. Ser. No. 56, Amer. Math. Soc., Providence, RI 48. Ratiu T, Sousa Dias E, Sbano L, Terra G, Tudora R (2005) A crush course. In: Montaldi J, Ratiu T (eds) geometric mechanics in Geometric Mechanics and Symmetry: the Peyresq Lectures. Cambridge University Press, Cambridge UK 49. Sanders J, Verhulst F, Murdock J (2007) Averaging Methods in Nonlinear Dynamical Systems. Applied Mathematical Sciences, vol 59. Springer, Berlin 50. Schmah T (2007) A cotangent bundle slice theorem. Diff Geom Appl 25:101–124 51. Seifert H (1948) Periodische Bewegungen mechanischer Systeme. Math Z 51:197–216 52. Simó C (2001) New families of solutions in N-body problems. In: European Congress of Mathematics, vol I (Barcelona, 2000), pp 101–115, Progr. Math., 201. Birkhäuser, Basel 53. Struwe M (1990) Variational methods. Springer, Berlin 54. Verhulst F (1990) Nonlinear Dynamical Equations and Dynamical Systems. Springer, Berlin 55. Weinstein A (1973) Normal Modes for Nonlinear Hamiltonian Systems. Inventiones Math 20:47–57 56. Weinstein A (1978) Bifurcations and Hamilton Principle. Math Z 159:235–248 57. Whitehead GW (1995) Elements of homotopy theory, 3rd edn. Springer, Berlin 58. Wulf C, Roberts M (2002) Hamiltonian Systems Near Relative Periodic Orbits. Siam J Appl Dyn Syst 1(1):1–43 59. Yakubovich VA, Starzhinsky VM (1975) Linear Differential Equations with Periodic coefficients, vol 1:2. Wiley, UK

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Periodic Solutions of Non-autonomous Ordinary Differential Equations

Periodic Solutions of Non-autonomous Ordinary Differential Equations JEAN MAWHIN Département de Mathématique, Université Catholique de Louvain, Maryland, USA Article Outline Glossary Definition of the Subject Introduction Poincaré Operator and Linear Systems Boundedness and Periodicity Fixed Point Approach: Perturbation Theory Fixed Point Approach: Large Nonlinearities Guiding Functions Lower and Upper Solutions Direct Method of the Calculus of Variations Critical Point Theory Future Directions Bibliography Glossary Banach fixed point theorem If M is a complete metric space with distance d, and f : M ! M is contractive, i. e. d( f (u); f (v))  ˛d(u; v) for some ˛ 2 [0; 1) and all u; v 2 M, then f has a unique fixed point u and u D lim k!1 f k (u0 ) for any u0 2 M. Brouwer degree An integer d B [ f ; ˝] which ‘algebraically’ counts the number of zeros of any continuous mapping f : ˝ Rn ! Rn such that 0 62 f (@˝), and is invariant for sufficiently small perturbations of f . If f is of class C 1 and its zeros are non degenerate, P then d B [ f ; ˝] D x2 f 1 (0) sign det f 0 (x). Brouwer fixed point theorem Any continuous mapping f : B ! B, with B is homeomorphic to the closed unit ball in Rn , has at least one fixed point. Leray–Schauder degree The extension d LS [I  g; ˝] of the Brouwer degree, where ˝ is an open bounded subset of the Banach space X, and g : ˝ ! X is continuous, g(˝) is relatively compact and 0 62 (I  g)(@˝). Leray–Schauder–Schaefer fixed point theorem If X is a Banach space, g : X ! X is a continuous mapping taking bounded subsets into relatively compact ones, and if the set of possible fixed points of "g (" 2 [0; 1]) is bounded independently of ", then g has at least one fixed point. Ljusternik–Schnirelmann category The Ljusternik– Schnirelmann category cat(M) of a metric space M

into itself is the smallest integer k such that M can be covered by k sets contractible in M. Lower and upper solutions A lower (resp. upper) solution of the periodic problem u00 D f (t; u), u(0) D u(T), u0 (0) D u0 (T) is a function ˛ (resp. ˇ) of class 00 0 C 2 such  that ˛ (t)  f (t; ˛(t)), ˛(0) D ˛(T), ˛ (0)  ˛ 0 (T)  resp:ˇ 00 (t)  f (t; ˇ(t)); ˇ(0) D ˇ(T); ˇ 0 (0)  ˇ 0 (T) . Palais–Smale condition for a C 1 function ' : X ! R Any sequence (u k ) k2N such that ('(u k )) k2N is bounded and lim k!1 ' 0 (u k ) D 0 contains a convergent subsequence. Poincaré operator The mapping defined in Rn by PT : y 7! p(T; y), where p(t; y) is the unique solution of the Cauchy problem x 0 D f (t; x); x(0) D y. Schauder fixed point theorem If C is a closed bounded convex subset of a Banach spaxe X, any continuous mapping g : C ! C such that g(C) is relatively compact has at least one fixed point. Sobolev inequality For any function u 2 L2 (0; T) such RT that u0 2 L2 (0; T) and 0 u(t)dt D 0, one has p RT maxt2[0;T] ju(t)j  (T 1/2 /2 3)[ 0 ju0 (t)j2 dt]1/2 . Wirtinger inequality For any function u 2 L2 (0; T) such RT that u0 2 L2 (0; T) and 0 u(t)dt D 0, one has  2 ı 2R T 0 2 RT 2 4 0 ju(t)j dt  T 0 ju (t)j dt. Definition of the Subject Many phenomena in nature can be modeled by systems of ordinary differential equations which depend periodically upon time. For example, a linear or nonlinear oscillator can be forced by a periodic external force, and an important question is to known if the oscillator can exhibit a periodic response under this forcing. This question originated from problems in classical and celestial mechanics, before receiving important applications in radioelectricity and electronics. Nowadays, it also plays a great role in mathematical biology and population dynamics, as well as in mathematical economics, where the considered systems are often subject to seasonal variations. The general theory originated with Henri Poincaré’s work in celestial mechanics, at the end of the XIXth century, and has been constantly developed since. Introduction To motivate the problem and its difficulties, let us start with the simple linear oscillator with forcing (or input) h 2 L2 (0; 2) L u :D u00  u D h(t) ;

(1)

Periodic Solutions of Non-autonomous Ordinary Differential Equations

where  2 R. We are interested in discussing the existence or non-existence, and the uniqueness or multiplicity of a 2-periodic solution u of (1). Let h(t) c0 C u(t) a0 C

1 X

1  [dist(; ˙ )]2

[a k cos kt C b k sin kt] ;

with u(t) D ˛ C

1 2

u00

Z

2

(k D 1; 2; : : :)

0

1 1X 2 [c k C d 2k ] ; 2

h2 (t)dt D c02 C

(4)

1 X 1 [c k cos kt C d k sin kt] (˛ 2 R) k2

if j D 0 and u(t) D ˛ cos jt C ˇ sin jt  C

1 X kD1 k¤ j

c0 j2

1 [c k cos kt C d k sin kt] k 2  j2

(˛; ˇ 2 R) if j ¤ 0. For the nonlinear oscillator

kD1

L2 (0; 2),

P1

u00 (t)

and, as 2  kD1 k 2 [a k cos kt C b k sin kt]. Therefore, finding the 2-periodic solutions of (1) is equivalent to solving the infinite-dimensional linear system in l 2 with unknowns (a0 ; a1 ; b1 ; : : :)  a0 D c0 ;

kD1

kD1

and similarly for u, be the Fourier series of h and u. If u is a possible 2-periodic solution of (1), u is of class C 1 and u00 2 L2 (0; 2), so that Fourier series of u and u0 converge uniformly on [0; 2] to u and u0 . Furthermore, from Parseval equality khk22 :D

# 1 1X 2 2 C (c k C d k ) 2

Now, if  D j2 2 ˙ (resonance) and if c j ¤ 0 or d j ¤ 0, then (2) has no solution and (1) has no 2-periodic solution. If c j D d j D 0, then (1) has the infinite family of 2-periodic solutions

kD1

Z 2 1 h(t)dt ; c0 D 2 0 



Z 2 1 cos kt ck dt h(t) D sin kt dk  0

c02

1 D khk22 : [dist(; ˙ )]2

[c k cos kt C d k sin kt] ;

kD1 1 X

"

(k 2  )a k D c k ; (k 2  )b k D d k

(k D 1; 2; : : :) :

(2)

Letting

 u00 D g(u) C h(t)

(5)

where g : R ! R is continuous, a nonlinear version of the non-resonant linear situation can be obtained. Indeed, assume that there exist numbers 0 < ˛  ˇ such that, for all u ¤ v 2 R one has ˛

g(u)  g(v) ˇ; uv

[˛; ˇ] \ ˙ D ;

(6)

which means that there exists j2 2 ˙ such that

2

˙ :D fk : k D 0; 1; 2; : : :g ; we see that if  62 ˙ (non-resonance), system (2) has the unique solution a0 D 

c0 ck dk ; ak D 2 ; bk D 2 (k D 1; 2; : : :);  k  k 

which gives the unique 2-periodic solution of (1) u(t) D (L1  h)(t)

j2 < ˛  ˇ < ( j C 1)2 :

(7)

If :D (˛ C ˇ)/2, so that 62 ˙ , we can write (5) in the equivalent form  u00  u D g(u)  u C h(t) ;

(8)

and, if we define F : L2 (0; 2) ! L2 (0; 2) by [F (u)] (t) D g(u)  u(t) C h(t), it follows from (6) that

1

D

c0 X 1 C [c k cos kt C d k sin kt]:  k2  

kF (u)  F (v)k2  ıku  vk2 ;

(9)

kD1

Furthermore, using Parseval equality, kuk22 D kL1 hk22 D

1 1 c02 1 X C [c 2 C d 2k ] (3) 2 2 (k 2  )2 k kD1

with ı D (ˇ  ˛)/2. Furthermore, finding the 2-periodic solutions of (8) is equivalent to solving the equation in L2 (0; 2) u D L1  F (u) :

(10)

1237

1238

Periodic Solutions of Non-autonomous Ordinary Differential Equations

Now, using estimates (3), (7) and (9), we get, for all u; v 2 L2 (0; 2), kL1  [F (u)  F (v)]k2 

ı ku  vk2 ; dist( ; ˙ )

with ı/dist( ; ˙ ) < 1. Banach fixed point theorem implies the existence of a unique fixed point u of L1  F , and hence of a unique 2-periodic solution of (5). Such a result, proved in 1949 by Dolph [14] for Dirichlet boundary conditions and in 1976 in [31] for the periodic problem, has been extended in various directions. For example, it was proved in [8], using sophisticated topological Rmethods and delicate a priori estimates, that u if G(u) D 0 g(s)ds, the forced nonlinear oscillator (5) has at least one 2-periodic solution for each continuous h : [0; 2] ! R if g is odd, ug(u)  G(u) > 0 for some  1 and all large juj, and if # " 2G(u) 2G(u) ; lim sup ¤ fk 2 g lim inf 2 u!C1 u 2 u!C1 u for any positive integer k. However, it is still an open problem to know if the natural generalization of (6)

of (11) can be continued as a solution defined over R and such that x(t) D x(t C T) for all t 2 R. Poincaré already observed at the end of the XIXth century [43] that p(t; y) is a T-periodic solution of (11) if and only if y 2 Rn is such that C (y) > T and y D p(T; y), i. e. y is a fixed point of Poincaré operator PT defined by PT (y) D p(T; y) for y 2 Rn such that C (y) > T. A simple application is the case of a linear system x 0 D A(t)x

where A: [0; T] ! L(Rn ; Rn ) is a continuous (n  n)matrix-valued function. Given y 2 Rn , the solution of (12) such that x(0) D y can be written x(t) D X(t)y for some n  n matrix such that X(0) D I, the fundamental matrix of (12). The corresponding Poincaré operator, given by PT (y) D X(T)y, has non-trivial fixed points if and only if I  X(T) is singular. If h : [0; T] ! Rn is continuous, the solution of the forced linear system x 0 D A(t)x C h(t)

(14)

0

2G(u) 2G(u) j < lim inf  lim sup < ( j C 1)2 2 juj!C1 u 2 juj!C1 u

the corresponding Poincaré operator is

is sufficient for the existence of a 2-periodic solution to (5). The problem of obtaining nonlinear versions of the resonance situation is more difficult and also requires more sophisticated tools. It will be considered in Sects. “Lower and Upper Solutions” to “Critical Point Theory” Poincaré Operator and Linear Systems Let T > 0 be fixed, f : R  Rn ! Rn ; (t; x) 7! f (t; x) be locally Lipschitzian in x and continuous. For each y 2 Rn , there exists a unique solution x(t) D p(t; y) of the Cauchy problem x(0) D y ;

Z

T

PT (y) D X(T)y C

X(T)X 1 (s)h(s)ds :

(15)

0

Consequently, if I  X(T) is non singular, i. e. if (12) only has the trivial T-periodic solution x(t)  0, (15) has the unique fixed point Z T X(T)X 1 (s)h(s)ds y D [I  X(T)]1 0

and, by inserting this value of y in (14), (13) has the unique T-periodic solution Z T G(t; s)h(s)ds ; (16) x(t) D 0

defined and continuous on a maximal open set G D f(t; y) 2 R  Rn :  (y) < t < C (y)g, for some 1   (y) < 0 < C (y)  C1. A T-periodic solution of the differential system x 0 D f (t; x)

(13)

such that x(0) D y being given by Z t X(t)X 1 (s)h(s)ds ; p(t; y) D X(t)y C

2

x 0 D f (t; x) ;

(12)

(11)

is a solution of (11) defined at least over [0; T] and such that x(0) D x(T). If we assume in addition that f (t; x) D f (t C T; x) for all t 2 R and x 2 Rn , a T-periodic solution

where G(t; s) is the Green matrix explicitly given by G(t; s) ( if 0  s  t  T X(t)[I  X(T)]1 X 1 (s) D 1 1 X(t)[I  X(T)] X(T)X (s) if 0  t < s  T (17) The situation is more complicated when I  X(T) is singular, which always happens in the simple case where

Periodic Solutions of Non-autonomous Ordinary Differential Equations

A(t)  0, to which weRrestrict ourself. In this case, X(t)  T I, and PT (y) D y C 0 h(s)ds. It has fixed points if and only if h has mean value zero, namely h :D

1 T

Z

T

h(s)ds D 0 ; 0

in which case (13) with A(t)  0 has the family of Tperiodic solutions Z

t

x(t) D y C

h(s)ds

(y 2 Rn ) :

0

Boundedness and Periodicity When n D 1 and f : R  R ! R is T-periodic with respect to t, the local uniqueness assumption implies that two solutions p(t; y) and p(t; z) with y < z are such that p(t; y) < p(t; z) for all t were they are defined. Hence, if p(t; y) is defined for all t  0, p(t C nT; y) is the solution of (11) equal to y n D p(nT; y) at t D 0 for any integer n  1. If y1 D y, p(t; y) is T-periodic; if, say, y1 < y, then p(t C T; y) D p(t; y1 ) < p(t; y) for all t  0, and hence, p(t C (n C 1)T; y) < p(t C nT; y) for all t  0. Thus, for any t  0, the sequence (p(t C nT; y))n2N is monotone. If p(t; y) is bounded in the future, i. e. if there exists M > 0 such that jp(t; y)j  M for all t  0, the sequence above, monotone, bounded and equicontinuous (as (p0 (t C nT; y))n2N is bounded), converges uniformly on each bounded interval to a continuous function (t). It follows from the identity Z

t

p(t C nT; y) D p(nT; y) C

f (s; p(s C nT; y))ds 0

that (t) is a solution of (11) defined for t  0 and that (T) D lim p((n C 1)T; y) D lim p(nT; y) D (0) : n!1

n!1

This gives a result proved by Massera [29] in 1950: if n D 1 and f is T-periodic in t, locally Lipschitzian in x, and continuous, then (11) admits a T-periodic solution if and only if it admits a solution bounded in the future. Of course, the same statement holds if we replace ‘in the future’ by ‘in the past’. The result needs not to be true for n  2, but, using delicate arguments of two-dimensional topology, Massera [29] has proved that if n D 2, f is Tperiodic in t, locally Lipschitzian in x, continuous, if all solutions of (11) exist in the future and if one of them is bounded in the future, then (11) admits a T-periodic solution. Again, the same statement holds if we replace ‘in the future’ by ‘in the past’. An important consequence of

Massera’s results is that the absence of T-periodic solutions in a one or two-dimensional T-periodic system implies the unboundedness of all its solutions in the past and in the future. By reinforcing Massera’s conditions, one can find criteria for the existence of T-periodic solutions valid for any n. A set G is a positively invariant set for (11) if, for each y 2 G, p(t; y) 2 G for all t 2 [0; C (y)]. In particular, if G is bounded, then C (y) D C1. Now, if G is invariant, and homeomorphic to the unit closed ball B[0; 1] in Rn , Poincaré operator maps continuously G into itself, and Brouwer fixed point theorem implies that (11) has at least one T-periodic solution with values in G. Such a result, which can be traced to Lefschetz [24] and to Levinson [25] in the early 1940s has been widely used to study the periodic solutions of the periodically forced Liénard differential equation y 00 C h(y)y 0 C g(y) D e(t);

(18)

written as a two-dimensional first order system, under various conditions upon the friction coefficient h(y), the restoring force g(y) and the forcing term e(t). Some of Levinson’s results have been at the origin of the theory of chaos. Fixed Point Approach: Perturbation Theory Formula (16) suggests another approach for finding periodic solutions, which is independent of Cauchy’s problem and requires less regularity. Consider the nonlinear differential system x 0 D A(t)x C f (t; x)

(19)

where A: [0; T] ! L(Rn ; Rn ) and f : [0; T]  Rn ! Rn are continuous, and the corresponding linear system x 0 D A(t)x only has the trivial T-periodic solution. Using formula (16), finding the T-periodic solutions of (19) is equivalent to solving the nonlinear integral equation Z x(t) D

T

G(t; s) f (s; x(s))ds :D [H (x)](t)

0

(t 2 [0; T]) ;

(20)

with G(t; s) defined in (17), in the Banach space C #T of continuous functions such that x(0) D x(T), i. e. to finding the fixed points of the nonlinear operator H : C #T ! C #T defined in (20). Consider now the family of problems x 0 D A(t)x C " f (t; x) ;

x(0) D x(T) (" 2 R) ; (21)

where x 0 D A(t)x only has the trivial T-periodic solution. Solving (21) is equivalent to solving the equation in C #T K (x; ") :D x  "H (x) D 0 :

1239

1240

Periodic Solutions of Non-autonomous Ordinary Differential Equations

Trivially, K (x; 0) D 0 if and only if x D 0. If f x0 (t; x) exists and is continuous, the Fréchet derivative Kx0 (x; ") at x 2 C #T is given by Kx0 (x; ") D I  "H 0 (x), so that Kx0 (0; 0) D I is invertible. It follows from the implicit function theorem in Banach spaces that for some "0 > 0 and each j"j  "0 , (21) has a unique solution x" such that x" ! 0 as " ! 0. Such a result can be traced to Poincaré [43], who proved it using the associated Cauchy problem. From the proof of the implicit function theorem or from Banach fixed point theorem, it follows that x" can be obtained as the limit of the sequence of successive approximations given by x"0 D 0 ;

x"kC1 (t) D "

Z

T 0

G(t; s) f (s; x"k (s))ds (k D 0; 1; 2; : : :) :

The situation is more complicated when x 0 D A(t)x has nontrivial T-periodic solutions, and again we restrict ourself to the important special case where A(t)  0, i. e. to the family of problems x 0 D " f (t; x) ;

x(0) D x(T) (" 2 R) ;

(22)

which admits, for " D 0, the n-parameter family of Tperiodic solutions x(t) D c (c 2 Rn ). To reduce (22) to a fixed point problem, one can first notice that, as easily verified, the linear problem Lx :D x 0 C x(0) D h(t) ;

N (x; 0) D 0 is equivalent to x(t) D x(0) C f (; x()), and hence to x(t)  c, with c 2 Rn such that Z 1 T f (s; c)ds D 0: (25) F(c) :D T 0

If c0 is a solution of (25) satisfying condition " Z # 1 T 0 Jac F(c0 ) :D det f (s; c0 )ds ¤ 0; T 0 x

the Fréchet derivative Nx0 (c0 ; 0) is invertible, and the implicit function theorem in Banach spaces implies the existence of "0 > 0 such that, for each j"j  "0 , equation N (x; ") D 0 has a unique solution x" such that x" ! c0 as " ! 0, and the same conclusion holds for (22). This result is also a consequence of other perturbation methods for periodic solutions based upon Poincaré, averaging, Lyapunov–Schmidt or Cesari–Hale methods, and described in some of the books given in the Bibliography. The proof by iteration of the implicit function theorem implies that x" can be obtained as the limit of the sequence of successive approximations given by x"0 D c0 ; x"kC1 D c C "[I  M0x (c0 ; 0)]1 h i  M(x"k ; ")  M(c0 ; 0)  M0x (c0 ; 0)(x"k  c) (k D 0; 1; 2; : : :) :

x(0) D x(T)

has, for each continuous h : [0; T] ! Rn , the unique solution Z t [h(s)  h]ds : (23) [L1 h](t) D h C 0

In particular, L1 h D h. Now, for " ¤ 0, problem (22) is equivalent to x 0 D (1  ") f (; x()) C " f (t; x) ;

x(0) D x(T) ; (24)

which, using (23), is equivalent to the fixed point problem in C #T x(t) D x(0) C f (; x()) Z t C" [ f (s; x(s))  f (; x())]ds 0

(26)

For example, the search of positive periodic solutions of Verhulst equation with seasonal variations in population dynamics y 0 D "[a(t)y  b(t)y 2 ] (" > 0) ;

(27)

where a; b : R ! (0; C1) are continuous and have period T, is equivalent, through the transformation y D eu , to the problem u0 D "[a(t)  b(t)e u ] ;

u(0) D u(T)

for which F(c) D a  bec . This equation has the unique solution c0 D log a/b, and F 0 (c0 ) D a ¤ 0. Hence, for sufficiently small " > 0, Verhulst equation with seasonal variations (27) has at least one positive T-periodic solution y" such that y" ! a/b when " ! 0.

:D [M(x; ")](t) introduced by Mawhin in 1969 [30], i. e. to the equation in C #T N (x; ") :D x  M(x; ") D 0 :

Fixed Point Approach: Large Nonlinearities The use of more sophisticated fixed point theorems provides existence results which are not of perturbation type. If f is continuous, H defined in (20) is continuous, and,

Periodic Solutions of Non-autonomous Ordinary Differential Equations

using Arzelá-Ascoli theorem, H takes bounded sets into relatively compact sets, i. e. H is completely continuous on C #T . Using Schauder fixed point theorem, H has at least one fixed point if it maps a closed ball of C #T into itself. It is the case in particular when f satisfies the growth condition ((t; x) 2 [0; T]  Rn ) ;

k f (t; x)k  ˛kxk C ˇ

RT with ˇ  0 and j˛j 0 jG(t; s)jds < 1 for all t 2 [0; T], and, in particular, when f is bounded on [0; T]  Rn . More general existence conditions can be deduced from Leray–Schauder–Schaefer fixed point theorem, which, applied to H , implies that (19) has at least one Tperiodic solution if there exists R > 0 such that any possible solution x of the family of problems x 0 (t) D A(t)x C " f (t; x) ;

x(0) D x(T) (" 2 [0; 1])

is such that maxt2[0;T] kx(t)k < R. Special cases of this result, for particular equations, can be traced to Stoppelli [49], and the general form was first given by Reissig [46] and Villari [50]. For example, coming back to the equivalent form of the problem of positive periodic solutions of Verhulst equation with seasonal variations u0 D a(t)  b(t)eu ;

u(0) D u(T) ;

(28)

where a; b : R ! (0; C1) are continuous and have period T, we associate to (28) the family of problems u0 C u D "[a(t) C b(t)e u C u];

u(0) D u(T); (" 2 [0; 1]) ;

(29)

which reduces to (28) for " D 1, and only has the trivial solution for " D 0. If, for some " 2 [0; 1], a possible solution u of (29) reaches a positive maximum at , then 0 D u0 () D "[a()  b()e u() ]  (1  ")u() ; so that "[a()  b()eu() ] D (1  ")u()  0 and u()  log

a() max[0;T] a  log : b() min[0;T] b

Similarly, if u reaches a negative minimum at  0 , then u( 0 )  log

a( 0 ) b( 0 )

 log

min[0;T] a : max[0;T] b

The existence of at least one T-periodic solution u for (27) follows from the previous theorem with   min[0;T] a max[0;T] a ;  log ; R > max log min[0;T] b max[0;T] b

and it gives the positive solution y(t) D eu(t) for the original Verhulst equation with seasonal variations y 0 D a(t)y  b(t)y 2 , which reduces to the positive equilibrium y(t)  ba in the non-seasonal case where a and b are constant. In the case of system (22), Schauder or Leray– Schauder–Schaefer fixed point theorems may be difficult to apply to M(x; ") because of the presence of the x(0) term. A generalization is required, based upon the concept of topological degree of some mappings F defined on the closure ˝ of a bounded open set ˝ of a Banach space X, and such that F (x) ¤ 0 for x 2 @˝. This topological degree, an integer counting ‘algebraically’ the number of zeros of F in ˝, is equal, for F D I to 1 or 0 according to 0 2 ˝ or 0 62 ˝, implies the existence of a zero of F in ˝ when it is nonzero, and the topological degree of F (; ") is independent of " when F (x; ") ¤ 0 for all (x; ") 2 @˝  [0; 1]. Applied to the family of mappings I  M(x; ") introduced in (22), Leray–Schauder and Brouwer degree theory implies the following continuation theorem, proved by Mawhin in 1969 [30] : if one can find an open bounded set ˝ C #T such that for each " 2 (0; 1], problem (22) has no solution on @˝, (ii) system F(c) D 0 defined in (25) has no solution on @˝ \ Rn , with Rn identified with constant functions in C #T , (iii) the Brouwer degree d B [F; ˝ \ Rn ] is different from zero, (i)

then system (11) has at least one T-periodic solution in ˝. If condition (i) is dropped, the conclusion remains valid for (22) with " sufficiently small. Notice that in the conditions of the perturbation result described above, condition (ii) holds for ˝ a sufficient small open neighborhood of the zero c0 of F, and condition (26) implies that d B [F; ˝ \ Rn ] D sign Jac F(c0 ) D ˙1. For example, consider the complex-valued Riccatitype equation z Cb z2 z0 D p(t) C q(t)z C r(t)b

(30)

where b z denotes the complex conjugate of z, and p; q; r : [0; T] ! C are continuous. It is nothing but a concise writing for a system of two real differential equations. If " 2 (0; 1] and z(t) is a possible T-periodic solution of z Cb z 2] ; z0 D "[p(t) C q(t)z C r(t)b

(31)

1241

1242

Periodic Solutions of Non-autonomous Ordinary Differential Equations

then, multiplying each member of (31) by z2 and integrating over [0; T], we get Z  0 Z 1 T 2 1 T z3 dt D z (t)z0 (t)dt 0D T 0 3 T 0 ( Z 1 T D" p(t)z2 (t) C q(t)z3 (t) T 0 ) 2 4 Cr(t)b z(t)z (t) C jz(t)j dt :

kuk p D

1 T

Z

#1

p

T

p

ju(t)j dt

;

0

kuk1 D max ju(t)j ; t2[0;T]

and using Hölder inequality, we obtain kzk44



kpk2 kzk24

x 0 D (1")g(x)C" f (t; x);

C [kqk4 C krk4 ]kzk34

so that kzk4  R1 , where R1 is the positive root of equation r2 D [kqk4 C krk4 ]r C kpk2 . From (31) follows then that kz0 k1  kpk1 C [kqk4/3 C krk4/3 ]R1 C R12 D R2 : Inequalities upon kzk4 and kz0 k1 imply that kzk1 < R for some R D R(R1 ; R2 ). Now, for c 2 C, F(c) D p C qc C rb c Cb c2 and hence, if F(c) D 0, we have jcj2  (kqk1 Ckrk1 )jcjCkpk1  (kqk4 Ckrk4 )jcjCkpk2 so that jcj  R1 < R. Finally, d B [F; B(0; R)] D 2. Hence all conditions of the second existence theorem hold for ˝ D B(0; R), and (30) has at least one T-periodic solution, a result first proved, using a different approach, by Srzednicki [48] in 1994, the present proof being given in [35]. Notice that this result could not have been deduced from Leray–Schauder–Schaefer theorem, whose assumptions imply that the associated Leray–Schauder degree has absolue value one, although, for (30), its has absolue value two. The perturbed problem z0 D "[p(t) C q(t)z C r(t)z C z2 ] ;

z(0) D z(T)

satisfies conditions (ii) and (iii) of Mawhin’s continuation theorem for ˝ D B(0; R) with R sufficiently large, and has at least one T-periodic solution for j"j sufficiently small. But Lloyd [28] and Campos–Ortega [6] have shown by examples that the problem z0 D p(t) C q(t)z C r(t)z C z2 ;

Guiding Functions A useful variant of Mawhin’s continuation theorem given in Sect. “Fixed Point Approach: Large Nonlinearities” has been proved in 1992 by Capietto, Mawhin and Zanolin [6], with another proof in [5] based upon degree theory for S 1 -equivariant mappings. It states, for g : Rn ! Rn and f : [0; T]  Rn ! Rn continuous, that if there exists an open bounded set ˝ C #T such that the family of problems

Hence letting, for p  1, "

in contrast to (30), may have no T-periodic solution for some choice of p; q; r. Finally, notice that the examples developed here illustrate two fundamental approaches to obtain a priori bounds for the possible T-periodic solutions : maximum principle-type argument and integral estimates.

z(0) D z(T) ;

x(0) D x(T) (" 2 [0; 1])

has no solution on @˝, and if d B [g; ˝ \ Rn ] ¤ 0, then the problem x 0 D f (t; x) ;

x(0) D x(T)

(32)

has at least one solution in ˝. The proof is based upon the fact that jd LS [I  M; ˝]j D jd B [g; ˝ \ Rn ]j, with M the fixed point operator associated to the autonomous system x 0 D g(x). Now a guiding function on G Rn for (32) is a function V : Rn ! R of class C 1 such that, with (ujv) the inner product of u and v in Rn , kV 0 (x)k ¤ 0

and (V 0 (x)j f (t; x))  0 when

(t; x) 2 [0; 1]  G :

This is an extension due to Mawhin–Ward [37] of a slight generalization given in [32] of a concept introduced in 1958 by Krasnosel’skii and Perov (see [22]) in the case where G D Rn n B(0; ) for some > 0. For example V(x) D (1/2)kxk2 is a guiding function on @B(0; r) if (xj f (t; x))  0 for (t; x) 2 [0; T]  @B(0; r). We set V r D fx 2 Rn : V(x) < rg. It is proved in [37] that if there exists a C 1 function V : Rn ! R such that V 0 is non-empty and bounded, V is a guiding function on V 1 (f0g) for (32), and d B [V 0 ; V 0 ] ¤ 0, then problem (32) has at least one solution with values in V0 . The proof can be based upon the continuation theorem mentioned above with g(x) D V 0 (x) and ˝ D fx 2 C #T : x(t) 2 G (t 2 [0; T])g, and its essential ingredient consists in showing that, for  2 [0; 1), the family of problems x 0 D (1  ")V 0 (x) C " f (t; x) ;

x(0) D x(T) ;

Periodic Solutions of Non-autonomous Ordinary Differential Equations

has no solution on @˝, which is done by studying the maximum of V(x(t)). In particular, if one takes G D Rn n B(0; ) for some > 0, and V coercive, i. e. limkxk V (x) D C1, one can show that d B [V 0 ; V 0 ] D 1, and the existence follows. This generalizes the result about positively invariant sets mentioned in Sect. “Boundedness and Periodicity”. The following generalization of the concept of guiding function is also given in [37]. An averaged guiding function on G Rn for (32) is a function V : G ! R of class C 1 such that Z T (V 0 (x(t))j f (t; x(t))dt)  0 kV 0 (x)k ¤ 0 and 0

C #T

when x 2 and x(t) 2 G for all t 2 [0; T]. It is proved in [37] that problem (32) has at least a solution if there exists a C 1 function V : Rn ! R such that V 0 is nonempty, V r is bounded for

r D max T

max

u2V 1 (f0g)

Z max

u2V 1 (f0g) 0

kV 0 (u)k2 ; )

T

0

j(V (u)j f (t; u))jdt

;

V is an averaged guiding function on Rn n V 0 for (32) and d B [V 0 ; V 0 ] ¤ 0. Of course, any guiding function is an averaged guiding function, but the converse is false. On the other hand, any V such that (V 0 (x)j f (t; x))  ˛(t) for some nonnegative ˛ 2 L1 (0; T), all (t; x) 2 [0; T]  Rn , and such that Z

T

0

lim sup(V 0 (x)j f (t; x)))dt < 0 kxk!1

is an averaged guiding function. In particular, taking x , V(x) D kxk  log(1 C kxk), so that V 0 (x) D 1Ckxk it is easy to see that (32) has at least one solution if   x j f (t; x) kxk

Z

 ˛(t) and 

T

lim sup 0

kxk!1

 x j f (t; x) dt < 0 : kxk

Lower and Upper Solutions A powerful way of proving the existence of T-periodic solutions for first or second order scalar differential equations is the method of lower and upper solutions (or sub- and supersolutions), introduced by Scorza Dragoni [47] in 1931 for Dirichlet boundary conditions and by Knobloch [21] in 1967 for periodic solutions. We describe

it here, following the approach initiated in [33], for problem u00 D f (t; u) ;

u0 (0) D u0 (T) ; (33)

u(0) D u(T) ;

where f : [0; T]  R ! R is continuous. The function ˛ (resp. ˇ) 2 C 2 ([0; T]) is a lower (resp. upper) solution for (33) if ˛ 00 (t)  f (t; ˛(t))

(t 2]0; T[) ;

˛(0) D ˛(T) ;

˛ 0 (0)  ˛ 0 (T)

00

(resp. ˇ (t)  f (t; ˇ(x)) (t 2]0; T[) ; ˇ(0) D ˇ(T) ;

ˇ 0 (0)  ˇ 0 (T)) :

The method of lower and upper solutions reduces the existence of a T-periodic solution to that of an ordered couple of lower and upper solutions: if there exists a lower solution ˛ and an upper solution ˇ for (33), such that ˛(t)  ˇ(t) for all t 2 [0; T], then (33) has as least one solution u such that ˛(t)  u(t)  ˇ(t) for all t 2 [0; T]. To prove such a result, one first introduce a modified problem whose solutions are solutions of (33), namely u00  u D f (t; (t; u))  (t; u) ; u(0) D u(T) ;

u0 (0) D u0 (T) ;

(34)

where (t; u) D ˛(t); u or ˇ(t) according to u < ˛(t); ˛(t)  u  ˇ(t) or u > ˇ(t). Notice that (33) and (34) coincide when ˛(t)  u  ˇ(t), that its linear part only has the trivial T-periodic solution, and that its right-hand member is continuous and bounded everywhere. A result in Sect. “Fixed Point Approach: Large Nonlinearities” implies that (34) has at least one solution, say e u. A contradiction argument, based upon simple characterizations of a maximum or a minimum, implies that ˛(t)  e u(t)  ˇ(t) for all t 2 [0; T], so that e u is a solution of (33). A simple consequence is an elegant necessary and sufficient existence condition first proved by Kazdan and Warner [20] in 1975 in the Dirichlet case : if f (t; c) is nondecreasing in c for each fixed t, problem (33) has at least RT one solution if and only if T1 0 f (t; c)dt D 0 for some c 2 R. Indeed, if (33) has a solution u, with u m :D min[0;T] u, u M :D max[0;T] u, integrating both members of (33) over [0; T] and using the monotonicity of f (t; ) gives Z T Z T f (t; u m )dt  0  f (t; u M )dt ; 0

0

RT and the necessity follows. For the sufficiency, if 0 f (t; c) dt D 0, and v(t) is the unique solution of the linear problem v 00 D f (t; c) ;

v(0) D 0 D v(T) ;

v 0 (0) D v 0 (T) ;

1243

1244

Periodic Solutions of Non-autonomous Ordinary Differential Equations

then ˛(t) :D c  max[0;T] v C v(t)  c  ˇ(t) :D c  min[0;T] v C v(t) are ordered lower and upper solutions for (33). In particular, the problem u00 C g(u) D h(t) ;

u0 (0) D u0 (T)

u(0) D u(T) ;

with g : R ! R continuous non increasing and h : [0; T] ! R continuous, has at least one solution if and only if h 2 g(R). This is a first nonlinear generalization of resonance at the first eigenvalue zero. For example, the problem u00  arctan u D h(t) ;

u(0) D u(T) ;

u0 (0) D u0 (T)

has at least one solution if and only if  2 < h < the problem u00  uC D h(t) ;

u(0) D u(T) ;

 2,

and

u0 (0) D u0 (T)

has at least one solution if and only if h  0. This last problem is an example of differential equation with asymmetric or jumping nonlinearities. Since the pioneering work of Fuˇcik [18] in 1978, the study of problems of the type

˛  1 and ˇ  1 shows that the method of lower and upper solutions fails in general in the case of unordered lower and upper solutions. However, Amann, Ambrosetti and Mancini [2] have proved in 1978, for some elliptic boundary value problems, that existence may still hold under further conditions, using the following approach. For each e h2e C #T D fh 2 C #T : h D 0g, the linear problem e h(t) ; Lu :D u00 D e

u(0) D 0 D u(T) ;

u0 (0) D u0 (T)

has the unique solution u(t) D [e L1 h](t) D

Z

T

K(t; s)e h(s)ds ;

0

where K(t; s) D s/T(T  t) if 0  s  t  T and K(t; s) D t/T(T  s) if 0  t < s  T. Hence u is a Tperiodic solution of (33) if and only if u 2 C #T is a solution of Z T K(t; s)[ f (s; u(s))  f (; u())]ds (36) u(t)  u(0) D 0

f (; u()) D 0 :

(37)

u00 C ˛uC  ˇu C g(u) D h(t) ; u(0) D u(T) ;

u0 (0) D u0 (T)

where the linear part in the nonlinear oscillator is replaced by a piecewise linear one, has been intensively studied. See for example [17] for references. Combined with degree arguments, the method of lower and upper solutions also allows to obtain multiplicity results for T-periodic solutions. For example, Fabry, Mawhin and Nkashama [16] have proved in 1986 for the problem (with c 2 R; s 2 R) u00 C cu0 C g(u) D h(t) C s ; u(0) D u(T) ;

u0 (0) D u0 (T)

(35)

with limjuj!1 g(u) D C1, the existence of s0 2 R such that problem (35) has no solution for s < s0 , at least one solution for s D s0 and at least two solutions for s > s0 . Such a result is generally refered as an Ambrosetti– Prodi problem, after the pioneering work of Ambrosetti and Prodi [4], in 1969, for elliptic boundary value problems. Ortega has studied the stability of the T-periodic solutions when c > 0 (see [40]). The example of u00 C u D sin t ;

u(0) D u(2) ;

u0 (0) D u0 (2)

which has no solution (resonance at the second eigenvalue 1) and has the (unordered) lower and upper solution

Letting c D u(0); y(t) D u(t)  u(0), so that y 2 b C T# D fx 2 C #T : x(0) D 0g, the first equation in (36) becomes Z T K(t; s)[ f (s; c C y(s))  f (; c C y())]ds y(t) D 0

:D [R(y; c)](t) ;

(38)

with R is completely continuous in b C #T  R. Assuming now in addition that j f j is bounded on [0; T]  R, say by M, Leray–Schauder degree theory applied to (38) as a fixed point problem in y with c as a parameter implies that the set (y; c) 2 b C #T  R satisfying (38) contains a continuum C whose projection on b C #T is contained in a ball B[0; R] with R depending only upon T and M, and whose projection on R is R. On C , we have, differentiating (38), y 00 D f (t; c C y)  f (; c C y()) ; y(0) D y(T) ;

y 0 (0) D y 0 (T) :

(39)

Hence, if there exists (c; y) 2 C such that f (; c C y()) D 0, the second equation in (36) is also satisfied and c C y is a solution of (33). If not, by connexity, either f (; c C y()) < 0 or f (; c C y()) > 0 for all (c; y) 2 C . Assume now that (33) has a lower solution ˛ and an upper solution ˇ which are not ordered, and consider, say the case where f (; c C y()) < 0 on C , the other one being

Periodic Solutions of Non-autonomous Ordinary Differential Equations

similar. For each c 2 R, it follows from (39) that c C y is a lower solution for (33) whenever (c; y) 2 C . Taking c such that c C y(t)  ˇ(t) for all t 2 [0; T] gives a couple of ordered lower and upper solution, and hence a T-periodic solution of (33). Proceeding like in the ordered case, we deduce from this result that if f is bounded and f (t; ) nonincreasing for each fixed t 2 [0; T], then (33) has at least RT one solution if and only if 1/T 0 f (t; c)dt D 0 for some c 2 R. In particular, the problem u00 C g(u) D h(t) ;

u(0) D u(T) ;

u0 (0) D u0 (T)

with g : R ! R continuous, bounded and non decreasing, and h : [0; T] ! R continuous, has at least one solution if and only if h 2 g(R). This is another nonlinear generalization of resonance at the first eigenvalue 0. For example, the problem u00 C arctan u D h(t) ;

u(0) D u(T) ;

u0 (0) D u0 (T)

as at least one solution if and only if  2 < h < the problem u00 C

uC D h(t) ; 1 C juj

u(0) D u(T) ;

 2,

and

u0 (0) D u0 (T)

has at least one solution if and only if 0  h < 1. Notice that the boundedness condition upon f can be replaced by suitable linear growth conditions. The case of constant lower and upper solutions gives a simple but useful existence condition: if f (t; ˛)  0  f (t; ˇ) for some ˛  ˇ and all t 2 [0; T], problem (33) has at least one solution u with ˛  u(t)  ˇ for all t 2 [0; T]. The same conclusion holds for f bounded and f (t; ˇ)  0  f (t; ˛) for all t 2 [0; T]. For example, taking ˇ D R D ˛ for sufficiently large R > 0, the problem u00 D p(u) C h(t) ;

u(0) D u(T) ;

u0 (0) D u0 (T)

has at least one solution for each continuous h, when p is a real polynomial of odd order whose highest order term has a positive coefficient. This result can be traced to Lichtenstein [26], who proved it in 1915 using a variational method, and can be applied to the original Duffing’s equation (with a > 0)

u00 C a sin u D h(t) ;

u(0) D u(T) ;

u0 (0) D u0 (T) (40)

a necessary condition for the existence of a solution is a  h  a, as follows from integrating both members over [0; T]. To obtain a necessary and sufficient existence condition for (40), Dancer [12] has used in 1982 an approach similar to the one described for the case of unordered lower and upper solutions. Writing h D h C e h, he has proved, for each e h, the existence of a (possibly degenerate) closed interval Ie h [a; a] such that (40) has at least one solution if and only if h 2 Ie h . In 1984, using degree theory, Mawhin and Willem [38] have proved the existence of at least two solutions for (40) when h 2 int I . The same authors have proved that the set of e h for which Ie has a non-empty interior is open and dense in the space h of continuous functions with mean value zero, but it is still an open problem to know if there exists or not some e h such that Ie is a singleton. See [36] for a survey of the various h results related to the forced pendulum equation. Direct Method of the Calculus of Variations When a differential equation or system can be written as the Euler–Lagrange equations of a problem of the calculus of variations, the direct method can be used to prove the existence of periodic solutions. Its fundamental result is that if X is a reflexive Banach space and if a weakly lower semi-continuous (w.l.s.c.) function ' : X ! (1; C1] has a bounded minimizing sequence, then it has a minimum on X. Recall that ' is weakly lower semi-continuous at a 2 X if lim inf k!1 '(u k )  '(u) whenever u k * u, and that (u k ) k2N is a minimizing sequence for ' if '(u k ) ! inf X '. Another result, valid for an arbitrary Banach space X, is that a function ' : X ! R of class C 1 , bounded from below and satisfying the Palais–Smale condition has a minimum on X. The Palais–Smale condition was introduced by Palais and Smale [42] in 1964. For simplicity, only the case of Lagrangian systems of differential equations of the type u00 D Fu0 (t; u) C e h(t) ;

u(0) D u(T) ;

u0 (0) D u0 (T) (41)

[0; T]  Rn

  u3 u00 C a u  D h(t) ; 6 u(0) D u(T) ;

However, for the exact pendulum equation

u0 (0) D u0 (T)

introduced in 1918 by Duffing [15] as a nonlinear approximation to the forced pendulum equation.

will be considered, where F : ! R is continuh : [0; T] ! Rn ous together with Fu0 : [0; T]  Rn ! Rn , e RT h(t)dt D 0. We denote by H 1T the is continuous with 0 e Sobolev space fu 2 L2 ([0; T]; Rn ) : u has a weak derivative u0 2 L2 ([0; T]; Rn ); u(0) D u(T)g;

1245

1246

Periodic Solutions of Non-autonomous Ordinary Differential Equations

where h : [0; T] ! R is continuous,R g : R ! R is continu uous, bounded and, with G(u) D 0 g(s)ds, is such that

with the inner product and norm Z

T

hu; vi D

[(u(t); v(t)) C (u0 (t); v 0 (t))]dt ;

b g(˙1) :D

0 1 2

kuk1;1 D hu; ui : It is easy to show that the action functional associated to (41) given by Z

T



ku0 (t)k2

'(u) :D

2

0

 C F(t; u(t)) C (e h(t); u(t)) dt (42)

Z

T



D 0

 ku0 (t)k2 C F(t; u(t)) C (e h(t);e u (t)) dt (43) 2

' 0 (u)[v] Z Th i D h(t); v(t)) dt (u0 (t); v 0 (t)) C (Fu0 (t; u(t)) C e 0

for all u; v 2 H 1T . Furthermore, if u is a critical point of ', i. e. if ' 0 (u) D 0, then u 2 C 2 ([0; T]; Rn ) and satisfies (41). As a first application, following Mawhin–Willem [39], assume that kFu0 k is bounded, say by M, on [0; T]  Rn and that F satisfied the condition Z 1 T F(t; u)dt D C1 ; (44) lim kuk!1 T 0 first introduced by Ahmad–Lazer–Paul [1] in 1976 for elu, and liptic boundary value problems. Writing u D u C e using Wirtinger’s inequality, we obtain Z T Z T 0 ku (t)k2 F(t; u)dt dt C '(u) D 2 0 0 Z TZ 1 C (Fu0 (t; u C se u(t)) C e h(t);e u (t))dt 0

T



0

ku0 (t)k2 2

0

Z C

3

M0T 2 dt  2

"Z 0

T

ku0 (t)k2 dt 2

# 12

T

F(t; u)dt 0

with M 0 D M C ke hk1 , which implies that '(u) ! C1 as kuk1;1 ! 1. Hence all minimizing sequences for ' are bounded and the minimum of ' is reached at some u 2 H 1T , and is a solution of (41). For example, consider the problem u00 C g(u) D h(t) ;

u(0) D u(T) ;

(45)

exist. For this problem Z lim

juj!1 0

T



 G(u) F(t; u)dt D lim u h  D C1 u juj!1

if b g(C1) < h < b g(1). Such a condition, first introduced by Alonso and Ortega [2] in 1996, generalizes the one introduced by Lazer and Leach [23] in 1969 (with g(u) instead of G(u)/u in (45)), but usually refered as a Landesman–Lazer condition. For example the problem u00  arctan u C a sin u D h(t) ;

is well defined, w.l.s.c. and of class C 1 on H 1T , with

Z

G(u) u!˙1 u lim

u0 (0) D u0 (T) ;

u(0) D u(T) ;

u0 (0) D u0 (T)

with a 2 R has at least one solution if  2 < h < 2 . As a second application, due to Willem [51], consider the case of a spatially periodic F, i. e. such that F(t; u C Ti e i ) D F(t; u) (1  i  n)

(46)

for some Ti > 0 and all (t; u) 2 [0; T]  Rn , where the e i denote the unit vectors in Rn (1  i  n). As F is bounded over [0; T]  Rn , it is easy to see, using Sobolev inequality, that Z 1 T 0 '(u)  ku (t)k2 dt 2 0 # 12 "Z T 0 2  C1 ku (t)k dt  C2 0

for some C1 ; C2  0, which implies the existence of C3 > 0 such that any minimizing sequence (u k ) k2N for ' satRT isfies 0 ku k (t)k2 dt  C3 . On the other hand, it follows from (46) that ' is such that '(u C Ti e i ) D '(u)

(u 2 H 1T ; 1  i  n)

(47)

and hence (u k C v k ) k2N with v k;i D k i Ti , (k i 2 Z; 1  i  n) is also a minimizing sequence. So there is always a minimizing sequence u k such that 0  u k;i  Ti (1  i  n), and hence a bounded minimizing sequence in H 1T , implying the existence of a T-periodic solution of (41) when (46) holds. In particular, this result implies that the periodic problem for the forced pendulum equation (40) has at least one solution for each a 2 R and each h with h D 0, so that 0 2 Ie h . Such a result, already proved

Periodic Solutions of Non-autonomous Ordinary Differential Equations

by Hamel [19] in 1922 using calculus of variations, was independently rediscovered by Willem [51] in 1981 and Dancer [12] in 1982. Notice that, for n D 1, Dancer and Ortega [13] have proved in 2004 that if the minimum is isolated as a critical point of the action functional, then it is unstable in the sense of Lyapunov. In this case of a periodic potential, multiplicity results can be proved using more sophisticated tools of the calculus of variations. Property (47) shows that one can cone 1 , where T n desider naturally ' on the manifold T n  H T e 1 the subspace of H 1 of functions notes the n-torus and H T T with mean value zero. In such a case, a result of Palais [41] implies that if ' is bounded from below and satisfies the Palais–Smale condition, then ' has at least cat(T n  e H T1 ) critical points. In this statement, cat(M) denotes the Lusternik–Schnirelmann category of the set M in itself, i. e. the least integer k such that M can be covered by k contractible subsets in M, a concept introduced by Ljusternik and Schnirelmann in 1934 [27]. Now it can be shown that cat(T n  e H T1 ) D cat(T n ) D n C 1 ; and hence, under condition (46), system (41) has at least n C 1 geometrically distinct T-periodic solutions. In particular, when h D 0, the forced pendulum equation (40) has at least two T-periodic solutions, a result first obtained in 1984 by Mawhin and Willem [38], using another variational approach. For other results in the spirit of Lusternik–Schnirelmann category, see [9,34,45]. Using, instead of Ljusternik–Schnirelmann category, another variational technique, namely Morse theory (see [10]), one can prove the existence of at least 2 n geometrically distinct Tperiodic solutions when they are non-degenerate. In the case of a spatially periodic Hamiltonian system Ju0 D H u0 (t; u) ;

u(0) D u(T) ;

! R, Hu0 : [0; T]  R2n ! R2n are where H : [0; T]  R2n  continuous, and J D

0 I

I 0

is the symplectic matrix,

a similar result, namely the existence of at least 2n C 1 Tperiodic solutions, has been proved by Conley and Zehnder [11] in 1984 using some finite-dimensional reduction and Conley’s index. It solves a conjecture of Arnold in symplectic geometry (see e. g. [10]).

functionals defined on a Banach space or a Banach manifold have been developed in the last thirty years, and have found many applications in the study of periodic solutions of differential systems having a variational structure, like Lagrangian or Hamiltonian systems. One such result, both simple and useful, is the saddle point theorem proved in 1978 by Rabinowitz [44]. Let ' : X ! R be of class C 1 on a Banach space X which can be splitted as a direct sum of closed subspaces X  ˚ X C , with X C finite-dimensional. Assume that, for some R > 0, sup@B[0;R]\X  ' < inf X C ', and let M :D fg : B[0; R] \ X  ! X  ; continuous; g(s) D s on @B[0; R] \ X  g : If ' satisfies a Palais–Smale condition on X, ' has at least one critical point u such that '(u) D c D inf g2M sups2B[0;R]\X  '(g(s)). As a simple application, we return to problem (41) with Fu0 bounded over [0; T]  Rn , and we now assume instead of (44) that Z lim

kuk!1 0

T

F(t; u)dt D 1:

(48)

It is easy to see that ' is unbounded from below on the subspace of constant functions and bounded from below but unbounded from above on the subspace of functions with mean value zero. Hence, in particular, infe H T1 ' > 1, and, for all sufficiently large R > 0, sup@B[0;R]\H 1 ' < T 1 infe H T1 ', where H T is the subspace of constant functions, 1 so that H 1T D H T ˚ e H T1 . The verification of the Palais– Smale condition follows from getting first a bound on e uk using the boundedness of Fu0 and then a bound for u k using condition (48). Hence (41) has at least one T-periodic solution. In particular, the problem u00 C g(u) D h(t) ;

u(0) D u(T) ;

u0 (0) D u0 (T) ;

where h : [0; T] ! R is continuous and g : R ! R is continuous, bounded and such that (with b g(˙1) defined g(C1), has at least one solution. in (45)) b g(1) < h < b For example, problem u00 C arctan u C a sin u D h(t) ;

Critical Point Theory A function ' unbounded from below and from above needs not to have a minimum or a maximum, but may have a critical point of saddle point type. Many existence theorems for critical points of saddle point type of some

u(0) D u(T) ;

u0 (0) D u0 (T) ;

has a solution if a 2 R and  2 < h < 2 . This is another example of nonlinear generalization of the resonance condition at zero eigenvalue.

1247

1248

Periodic Solutions of Non-autonomous Ordinary Differential Equations

Using topological degree, variational methods, or symplectic techniques, existence results have also been obtained in the case of resonance at a nonzero eigenvalue. For example, the problem u00 C k 2 u C g(u) D h(t) ; u(0) D u(2) ;

u0 (0) D u0 (2)

with g : R ! R continuous and bounded and k a positive integer, has at least one solution for all continuous h : [0; 2] ! R such that the real function  g(C1)  b g(1)  ˚( ) :D 2 b

Z

2

h(t) sin k(t C )dt 0

has no zero or more than two zeros, all simple, in [0; 2/ k[ . See [17] for the proof and references to earlier contributions of Lazer–Leach, Dancer, and Fabry–Fonda.

An enormous gap still exists between the methods of approach and the results about what seems to be the natural generalization of periodic solutions, namely the almost periodic solutions. It is a paradoxal fact that some statements are true both for periodic solutions and for solutions bounded on the whole real line, but false for the intermediate case of almost periodic solutions! Finally, like the equilibria in autonomous systems, the study of periodic solutions of non-autonomous equations will remain the unavoidable first step in trying to understand the complexity of the set of all solutions, and much remains to be done in this direction. Poincaré’s famous sentence ‘What renders these periodic solutions so precious is that they are, so to speak, the only breach through which we may try to penetrate a stronghold previously reputed to be impregnable’ keeps its full significance in the beginning of the XXIth century.

Future Directions In the whole XXth century, the study of periodic solutions of non-autonomous ordinary differential equations has been highly influential in the creation and development of fundamental parts of present mathematics, like functional analysis (operator theory, iterative methods,: : :), algebraic topology (fixed point theorems, topological degree, Conley index,: : :), variational methods (dual action principle, minimax theorems, Morse theory,: : :), symplectic techniques (Poincaré-Birkhoff-type fixed point theorems,: : :). One can expect that further topological tools will be useful or developed in searching new existence and multiplicity theorems for periodic solutions. The difficult problem of discussing the stability of those periodic solutions, still in infancy, is fundamental for the applications and must be developed. Most earlier studies of special differential equations have been devoted to models coming from mechanics and electronics, which are far from being completely understood, but the recent applications to biology, demography and economy will introduce new classes of differential equations and systems with periodic time dependence, requiring the use of new analytical and topological tools. In this respect, the study of periodic solutions of nonlinear nonautonomous difference equations is of increasing importance. Even if periodic solutions are not the easiest objects to be found numerically with a large degree of certitude, numerical methods and computers will play an increasing role in detecting the presence of periodic solutions. Some efficient softwares have already been developed in this respect.

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Periodic Solutions of Non-autonomous Ordinary Differential Equations

13. Dancer EN, Ortega R (1994) The index of Lyapunov stable fixed points in two dimensions. J Dyn Differ Equations 6:631–637 14. Dolph CL (1949) Nonlinear integral equations of Hammerstein type. Trans Amer Math Soc 66:289–307 15. Duffing G (1918) Erzwungene Schwingungen bei veränderlicher Eigenfrequenz. Vieweg, Braunschweig 16. Fabry C, Mawhin J, Nkashama M (1986) A multiplicity result for periodic solutions of forced nonlinear second order differential equations. Bull London Math Soc 18:173–180 17. Fabry C, Mawhin J (2000) Oscillations of a forced asymmetric oscillator at resonance. Nonlinearity 13:493–505 18. Fuˇcik S (1976) Boundary value problems with jumping nonlinˇ earities. Casopis Pest Mat 101:69–87 19. Hamel G (1922) Ueber erzwungene Schwingungen bei endlichen Amplituden. Math Ann 86:1–13 20. Kazdan JL, Warner FW (1975) Remarks on some quasilinear elliptic equations. Comm Pure Appl Math 28:567–597 21. Knobloch HW (1963) Eine neue Methode zur Approximation periodischer Lösungen von Differentialgleichungen zweiter Ordnung. Math Z 82:177–197 22. Krasnosel’skii MA (1968) The operator of translation along the trajectories of differential equations. Amer Math Soc, Providence 23. Lazer AC, Leach DE (1969) Bounded perturbations for forced harmonic oscillators at resonance. Ann Mat Pura Appl 82(4):49–68 24. Lefschetz S (1943) Existence of periodic solutions of certain differential equations. Proc Nat Acad Sci USA 29:29–32 25. Levinson N (1943) On the existence of periodic solutions for second order differential equations with a forcing term. J Math Phys 22:41–48 26. Lichtenstein L (1915) Ueber einige Existenzprobleme der Variationsrechnung. J Reine Angew Math 145:24–85 27. Ljusternik L, Schnirelmann L (1934) Méthodes topologiques dans les problèmes variationnels. Hermann, Paris 28. Lloyd NG (1975) On a class of differential equations of Riccati type. J London Math Soc 10(2):1–10 29. Massera JL (1950) The existence of periodic solutions of systems of differential equations. Duke Math J 17:457–475 30. Mawhin J (1969) Équations intégrales et solutions périodiques de systèmes différentiels non linéaires. Bull Cl Sci Acad Roy Belgique 55(5):934–947 31. Mawhin J (1976) Contractive mappings and periodically perturbed conservative systems. Arch Math (Brno) 12:67–73 32. Mawhin J (1979) Topological Degree and Nonlinear Boundary Value Problems. CBMS Conf Math No 40. Amer Math Soc, Providence 33. Mawhin J (1983) Points fixes, points critiques et problèmes aux limites. Sémin Math Supérieures No 92, Presses Univ de Montréal, Montréal 34. Mawhin J (1989) Forced second order conservative systems with periodic nonlinearity. Ann Inst Henri Poincaré Anal non linéaire 6:415–434 35. Mawhin J (1994) Periodic solutions of some planar nonautonomous polynomial differential equations. Differ Integral Equations 7:1055–1061 36. Mawhin J (2004) Global results for the forced pendulum equation. In: Cañada A, Drabek P, Fonda A (eds) Handbook of Differential Equations. Ordinary Differential Equations, vol 1. Elsevier, Amsterdam, pp 533–590

37. Mawhin J, Ward JR (2002) Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discret Continuous Dyn Syst 8:39–54 38. Mawhin J, Willem M (1984) Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J Differ Equations 52:264–287 39. Mawhin J, Willem M (1989) Critical point theory and Hamiltonian systems. Springer, New York 40. Ortega R (1995) Some applications of the topological degree to stability theory. In: Granas A, Frigon M (eds) Topological methods in differential equations and inclusions, NATO ASI C472. Kluwer, Amsterdam, pp 377–409 41. Palais R (1966) Ljusternik–Schnirelmann theory on Banach manifolds. Topology 5:115–132 42. Palais R, Smale S (1964) A generalized Morse theory. Bull Amer Math Soc 70:165–171 43. Poincaré H (1892) Les méthodes nouvelles de la mécanique céleste, vol 1. Gauthier–Villars, Paris 44. Rabinowitz P (1978) Some minimax theorems and applications to nonlinear partial differential equations. In: Cesari L, Kannan R, Weinberger H (eds) Nonlinear Analysis, A tribute to E. Rothe. Academic Press, New York 45. Rabinowitz P (1988) On a class of functionals invariant under a Zn -action. Trans Amer Math Soc 310:303–311 46. Reissig R (1964) Ein funktionenanalytischer Existenzbeweis für periodische Lösungen. Monatsber Deutsche Akad Wiss Berlin 6:407–413 47. Scorza Dragoni G (1931) Il problema dei valori ai limite studiate in grande per le equazione differenziale del secondo ordine. Math Ann 105:133–143 48. Srzednicki R (1994) On periodic planar solutions of planar polynomial differential equations with periodic coefficients. J Differ Equations 114:77–100 49. Stoppelli F (1952) Su un’equazione differenziale della meccanica dei fili. Rend Accad Sci Fis Mat Napoli 19(4):109–114 50. Villari G (1965) Contributi allo studio dell’esistenzadi soluzioni periodiche per i sistemi di equazioni differenziali ordinarie. Ann Mat Pura Appl 69(4):171–190 51. Willem M (1981) Oscillations forcées de systèmes hamiltoniens. Publications du Séminaire d’Analyse non linéaire, Univ Besançon

Books and Reviews Bobylev NA, Burman YM, Korovin SK (1994) Approximation procedures in nonlinear oscillation theory. de Gruyter, Berlin Bogoliubov NN, Mitropolsky YA (1961) Asymptotic methods in the theory of nonlinear oscillations. Gordon and Breach, New York Cesari L (1971) Asymptotic behavior and stability problems for ordinary differential equations, 3rd edn. Springer, Berlin Coddington EA, Levinson N (1955) Ordinary differential equations. McGraw-Hill, New York Cronin J (1964) Fixed points and topological degree in nonlinear analysis. Amer Math Soc, Providence Ekeland I (1990) Convexity methods in hamiltonian mechanics. Springer, Berlin Farkas M (1994) Periodic motions. Springer, New York Gaines RE, Mawhin J (1977) Coincidence degree and nonlinear differential equations. Lecture Notes in Mathematics, vol 568. Springer, Berlin

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Hale JK (1969) Ordinary differential equations. Wiley Interscience, New York Mawhin J (1993) Topological degree and boundary value problems for nonlinear differential equations. Lecture Notes in Mathematics, vol 1537. Springer, Berlin, pp 74–142 Mawhin J (1995) Continuation theorems and periodic solutions of ordinary differential equations. In: Granas A, Frigon M (eds) Topological methods in differential equations and inclusions, NATO ASI C472. Kluwer, Amsterdam, pp 291–375 Pliss VA (1966) Nonlocal problems of the theory of oscillations. Academic Press, New York Rabinowitz P (1986) Minimax methods in critical point theory with

applications to differential equations. CBMS Reg Conf Ser Math No 65. Amer Math Soc, Providence Reissig R, Sansone G, Conti R (1963) Qualitative Theorie nichtlinearer Differentialgleichungen. Cremonese, Roma Reissig R, Sansone G, Conti R (1974) Nonlinear differential equations of higher order. Noordhoff, Leiden Rouche N, Mawhin J (1980) Ordinary differential equations. Stability and periodic solutions. Pitman, Boston Sansone G, Conti R (1964) Non-linear differential equations. Pergamon, Oxford Verhulst F (1996) Nonlinear differential equations and dynamical systems, 2nd edn. Springer, Berlin

Perturbation Analysis of Parametric Resonance

Perturbation Analysis of Parametric Resonance FERDINAND VERHULST Mathematisch Instituut, University of Utrecht, Utrecht, The Netherlands Article Outline Glossary Definition of the Subject Introduction Perturbation Techniques Parametric Excitation of Linear Systems Nonlinear Parametric Excitation Applications Future Directions Acknowledgment Bibliography Glossary Coexistence The special case when all the independent solutions of a linear, T-periodic ODE are T-periodic. Hill’s equation A second order ODE of the form x¨ C p(t)x D 0, with p(t) T-periodic. Instability pockets Finite domains, usually intersections of instability tongues, where the trivial solution of linear, T-periodic ODEs is unstable. Instability tongues Domains in parameter space where the trivial solution of linear, T-periodic ODEs is unstable. Mathieu equation An ODE of the form x¨ C (a C b cos(t))x D 0. Parametric resonance Resonance excitation arising for special values of coefficients, frequencies and other parameters in T-periodic ODEs. P Quasi-periodic A function of the form niD1 f i (t) with f i (t) Ti -periodic, n finite, and the periods Ti independent over R. Sum resonance A parametric resonance arising in the case of at least three frequencies in a T-periodic ODE. Definition of the Subject Parametric resonance arises in mechanics in systems with external sources of energy and for certain parameter values. Typical examples are the pendulum with oscillating support and a more specific linearization of this pendulum, the Mathieu equation in the form x¨ C (a C b cos(t))x D 0 :

The time-dependent term represents the excitation. Tradition has it that parametric resonance is usually not considered in the context of systems with external excitation of the form x˙ D f (x) C (t), but for systems where timedependence arises in the coefficients of the equation. Mechanically this means usually periodically varying stiffness, mass or load, in fluid or plasma mechanics one can think of frequency modulation or density fluctuation, in mathematical biology of periodic environmental changes. The term ‘parametric’ refers to the dependence on parameters and certain resonances arising for special values of the parameters. In the case of the Mathieu equation, the parameters are the frequency ! (a D ! 2 ) of the equation without time-dependence and the excitation amplitude b; see Sect. “Parametric Excitation of Linear Systems” for an explicit demonstration of resonance phenomena in this two parameters system. Mathematically the subject is concerned with ODEs with periodic coefficients. The study of linear dynamics of this type gave rise to a large amount of literature in the first half of the 20th century and this highly technical, classical material is still accessible in textbooks. The standard equations are Hill’s equation and the Mathieu equation (see Subsect. “Elementary Theory”). We will summarize a number of basic aspects. The reader is also referred to the article  Dynamics of Parametric Excitation by Alan Champneys in this Encyclopedia. Recently, the interest in nonlinear dynamics, new applications and the need to explore higher dimensional problems has revived the subject. Also structural stability and persistence problems have been investigated. Such problems arise as follows. Suppose that we have found a number of interesting phenomena for a certain equation and suppose we embed this equation in a family of equations by adding parameters or perturbations. Do the ‘interesting phenomena’ persist in the family of equations? If not, we will call the original equation structurally unstable. A simple example of structural instability is the harmonic equation which shows qualitative different behavior on adding damping. In general, Hamiltonian systems are structurally unstable in the wider context of dissipative dynamical systems. Introduction Parametric resonance produces interesting mathematical challenges and plays an important part in many applications. The linear dynamics is already nontrivial whereas the nonlinear dynamics of such systems is extremely rich and largely unexplored. The role of symmetries is essential, both in linear and in nonlinear analysis. A classical exam-

1251

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Perturbation Analysis of Parametric Resonance

ple of parametric excitation is the swinging pendulum with oscillating support. The equation of motion describing the model is   x¨ C !02 C p(t) sin x D 0 ;

(1)

where p(t) is a periodic function. Upon linearization – replacing sin x by x – we obtain Hill’s equation (Subsect. “Elementary Theory”):   x¨ C !02 C p(t) x D 0 : This equation was formulated around 1900 in the perturbation theory of periodic solutions in celestial mechanics. If we choose p(t) D cos ! t, Hill’s equation becomes the Mathieu equation. It is well-known that special tuning of the frequency ! 0 and the period of excitation (of p(t)) produces interesting instability phenomena (resonance). More generally we may study nonlinear parametric equations of the form   x¨ C k x˙ C !02 C p(t) F(x) D 0 ;

(2)

where k > 0 is the damping coefficient, F(x) D x C bx 2 C cx 3 C    and time is scaled so that p(t) is a -periodic function with zero average. We may also take for p(t) a quasi-periodic or almost-periodic function. The books [48] cover most of the classical theory, but for a nice introduction see [38]. In [36], emphasis is placed on the part played by parameters, it contains a rich survey of bifurcations of eigenvalues and various applications. There are many open questions for Eqs. (1) and (2); we shall discuss aspects of the classical theory, recent theoretical results and a few applications. As noted before, in parametric excitation we have an oscillator with an independent source of energy. In examples, the oscillator is often described by a one degree of freedom system but of course many more degrees of freedom may play a part; see for instance in Sect. “Applications” the case of coupled Mathieu-equations as studied in [33]. In what follows, " will always be a small, positive parameter.

Poincaré–Lindstedt Series One of the oldest techniques is to approximate a periodic solution by the construction of a convergent series in terms of the small parameter ". The method can be used for equations of the form x˙ D f (t; x) C "g(t; x) C "2    ; with x 2 Rn and (usually) assuming that the ‘unperturbed’ problem y˙ D f (t; y) is understood and can be solved. Note that the method can also be applied to perturbed maps and difference equations. Suppose that the unperturbed problem contains a periodic solution, under what conditions can this solution be continued for " > 0? The answer is given by the conditions set by the implicit function theorem, see for formulations and theorems [30] and [44]. Usually we can associate with our perturbation problem a parameter space and one of the questions is then to find the domains of stability and instability. The common boundary of these domains is often characterized by the existence of periodic solutions and this is where Poincaré-Lindstedt series are useful. We will demonstrate this in the next section. Averaging Averaging is a normalization method. In general, the term“normalization” is used whenever an expression or quantity is put in a simpler, standardized form. For instance, a n  n-matrix with constant coefficients can be put in Jordan normal form by a suitable transformation. When the eigenvalues are distinct, this is a diagonal matrix. Introductions to normalization can be found in [1, 15,19] and [13]. For the relation between averaging and normalization in general the reader is referred to [34] and [44]. For averaging in the so-called standard form it is assumed that we can put the perturbation problem in the form x˙ D "F(t; x) C "2    ;

Perturbation Techniques

and that we have the existence of the limit Z 1 T lim F(t; x)dt D F 0 (x) : T!1 T 0

In this section we review the basic techniques to handle parametric perturbation problems. In the case of Poincaré–Lindstedt series which apply to periodic solutions, the expansions are in integer powers of ". It should be noted that in general, other order functions of " may play a part; see Subsect. “Elementary Theory” and [46].

The analysis of the averaged equation y˙ D F 0 (y) produces asymptotic approximations of the solutions of the original equation on a long timescale; see [34]. Also, under certain conditions, critical points of the averaged equation correspond with periodic solutions in the original system. The choice to use Poincaré-Lindstedt series or the averaging

Perturbation Analysis of Parametric Resonance

method is determined by the amount of information one wishes to obtain. To find the location of stability and instability domains (the boundaries), Poincaré-Lindstedt series are very efficient. On the other hand, with somewhat more efforts, the averaging method will also supply this information with in addition the behavior of the solutions within the domains. For an illustration see Subsect. “Elementary Theory”.

Here F(x; ; t) : Rm R p R ! Rm is C 1 in x and  and T-periodic in the t-variable. We assume that x D 0 is a solution, so F(0; ; t) D 0 and, moreover assume that the linear part of the vectorfield D x F(0; 0; t) is time-independent for all t 2 R. We will write L0 D D x F(0; 0; t). Expanding F(x; ; t) in a Taylor series with respect to x and  yields the equation: x˙ D L0 x C

Resonance

(3)

with x 2 Rn , A a constant n  n-matrix; f (t; x; ") can be expanded in a Taylor series with respect to " and in homogeneous vector polynomials in x starting with quadratic terms. Normalization of Eq. (3) means that by successive transformation we remove as many terms of Eq. (3) as possible. It would be ideal if we could remove all the nonlinear terms, i. e. linearize Eq. (3) by transformation. In general, however, some nonlinearities will be left and this is where resonance comes in. The eigenvalues 1 ; : : : ; n of the matrix A are resonant if for some i 2 f1; 2; : : : ; ng one has: n X

Fn (x; ; t) C O(j(x; )j kC1 ) ;

(6)

nD2

Assume that x D 0 is a critical point of the differential equation and write the system as: x˙ D Ax C f (t; x; ") ;

k X

where the Fn (x; ; t) are homogeneous polynomials in x and  of degree n with T-periodic coefficients. Theorem 1 Let k 2 N. There exists a (parameter- and time-dependent) transformation x D xˆ C

k X

Pn (xˆ ; ; t) ;

nD2

where Pn (xˆ ; ; t) are homogeneous polynomials in x and  of degree n with T-periodic coefficients, such that Eq. (6) takes the form (dropping the hat): x˙ D L0 x C

k X

F˜n (x; ; t) C O(j(x; )j kC1 ) ;

nD2

(7)

˙ D0:  m j  j D i ;

(4)

jD1

with m j  0 integers and m1 C m2 C    C m n  2. If the eigenvalues of A are non-resonant, we can remove all the nonlinear terms and so linearize the system. However, this is less useful than it appears, as in general the sequence of successive transformations to carry out the normalization will be divergent. The usefulness of normalization lies in removing nonresonant terms to a certain degree to simplify the analysis. Normalization of Time-Dependent Vectorfields In problems involving parametric resonance, we have time-dependent systems such as equations perturbing the Mathieu equation. Details of proofs and methods to compute the normal form coefficients in such cases can be found in [1,18] and [34]. We summarize some aspects. Consider the following parameter and time dependent equation: x˙ D F(x; ; t) ; with x 2 Rm and the parameters  2 R p .

(5)

The truncated vectorfield: x˙ D L0 x C

k X

˜ ; t) ; F˜n (x; ; t) D F(x;

(8)

nD2

which will be called the normal form of Eq. (5), has the following properties: 1.

d L ˜ L 0 t x; ; t) 0 t F(e dt e

D 0, for all (x; ) 2 RmCp ;

t 2 R. 2. If Eq. (5) is invariant under an involution (i. e. SF(x; ; t) D F(Sx; ; t) with S an invertible linear operator such that S 2 D I), then the truncated normal form (8) is also invariant under S. Similarly, if Eq. (5) is reversible under an involution R (i. e. RF(x; ; t) D F(Rx; ; t)), then the truncated normal form (8) is also reversible under R. For a proof, see [18]. The theorem will be applied to situations where L0 is semi-simple and has only purely imaginary eigenvalues. We take L0 D diagfi1 ; : : : ; im )g. In our applications, m D 2l is even and  l C j D  j for j D 1; : : : ; l. The variable x is then often written as x D (z1 ; : : : ; z l ; z¯1 ; : : : ; z¯l ).

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Perturbation Analysis of Parametric Resonance

Assume L0 D diagfi1 ; : : : ; im )g then:  x1 1

2  : : : x mn e i T k t

is in the jth component of the  A term ˜ ; t) if: Taylor–Fourier series of F(x;  j C

2 k C 1 1 C    C m m D 0 : T

(9)

This is known as the resonance condition.  Transforming the normal form through x D eL 0 t w leads to an autonomous equation for w: w˙ D

k X

F˜n (w; ; 0) :

(10)

nD2

 An important result is this: If Eq. (5) is invariant (respectively reversible) under an involution S, then this also holds for Eq. (10).  The autonomous normal form (10) is invariant under the action of the group G D fgjgx D e jL 0 T x; j 2 Zg, generated by e L 0 T . Note that this group is discrete if the ratios of the  i are rational and continuous otherwise. For a proof of the last two statements see [31]. By this procedure we can make the system autonomous. This is very effective as the autonomous normal form (10) can be used to prove the existence of periodic solutions and invariant tori of Eq. (5) near x D 0. We have: Theorem 2 Let " > 0, sufficiently small, be given. Scale ˆ w D "w.

 Chaotic attractors. Various scenarios were found, see [31,32,42].  Strange attractors without chaos, see [27]. The natural presence of various forcing periods in real-life models make their occurrence quite plausible.  Attracting tori. These limit sets are not difficult to find; they arise for instance as a consequence of a Neimark– Sacker bifurcation of a periodic solution, see [19].  Attracting heteroclinic cycles, see [20]. A large number of these phenomena can be studied both by numerics and by perturbation theory; using the methods simultaneously gives additional insight. Parametric Excitation of Linear Systems As we have seen in the introduction, parametric excitation leads to the study of second order equations with periodic coefficients. More in general such equations arise from linearization near T-periodic solutions of T-periodic equations of the form y˙ D f (t; y). Suppose y D (t) is a T-periodic solution; putting y D (t)Cx produces upon linearization the T-periodic equation x˙ D f x (t; (t))x :

(11)

This equation often takes the form x˙ D Ax C "B(t)x ;

(12)

1. If wˆ 0 is a hyperbolic fixed point of the (scaled) Eq. (10), then Eq. (5) has a hyperbolic periodic solution x(t) D "wˆ 0 C O(" kC1 ). 2. If the scaled Eq. (10) has a hyperbolic closed orbit, then Eq. (5) has a hyperbolic invariant torus.

in which x 2 Rm ; A is a constant m  m-matrix, B(t) is a continuous, T-periodic m  m-matrix, " is a small parameter. For elementary studies of such an equation, the Poincaré-Lindstedt method or continuation method is quite efficient. The method applies to nonlinear equations of arbitrary dimension, but we shall demonstrate its use for equations of Mathieu type.

These results are related to earlier theorems in [3], see also the survey [45]. Later we shall discuss normalization in the context of the so-called sum-resonance.

Elementary Theory

Remarks on Limit Sets In studying a dynamical system the behavior of the solutions is for a large part determined by the limit sets of the system. The classical limit sets are equilibria and periodic orbits. Even when restricting to autonomous equations of dimension three, we have no complete classification of possible limit sets and this makes the recognition and description of non-classical limit sets important. In parametrically excited systems, the following limit sets, apart from the classical ones, are of interest:

Floquet theory tells us that the solutions of Eq. (12) can be written as: x(t) D ˚ (t; ")eC(")t ;

(13)

with ˚ (t; ") a T-periodic m  m-matrix, C(") a constant m  m-matrix and both matrices having an expansion in order functions of ". The determination of C(") provides us with the stability behavior of the solutions. A particular case of Eq. (12) is Hill’s equation: x¨ C b(t; ")x D 0 ;

(14)

Perturbation Analysis of Parametric Resonance

which is of second order; b(t; ") is a scalar T-periodic function. A number of cases of Hill’s equation are studied in [23]. A particular case of Eq. (14) which arises frequently in applications is the Mathieu equation: x¨ C (! 2 C " cos 2t)x D 0; ! > 0 ;

(15)

which is reversible. (In [23] one also finds Lamé’s, Ince’s, Hermite’s, Whittaker–Hill and other Hill equations.) A typical question is: for which values of ! and " in (! 2 ; ")-parameter space is the trivial solution x D x˙ D 0 stable? Solutions of Eq. (15) can be written in the Floquet form (13), where in this case ˚ (t; ") will be -periodic. The eigenvalues 1 , 2 of C, which are called characteristic exponents and are "-dependent, determine the stability of the trivial solution. For the characteristic exponents of Eq. (12) we have: Z n X 1 T i D Tr(A C "B(t))dt ; (16) T 0

Perturbation Analysis of Parametric Resonance, Figure 1 Floquet tongues of the Mathieu Eq. (15); the instability domains are shaded

iD1

see Theorem 6.6 in [44]. So in the case of Eq. (15) we have: 1 C 2 D 0 :

(17)

The exponents are functions of ", 1 D 1 ("), 2 D 2 (") and clearly 1 (0) D i!, 2 (0) D i!. As 1 (") D 2 ("), the characteristic exponents, which are complex conjugate, are purely imaginary or real. The implication is that if ! 2 ¤ n2 , n D 1; 2; : : : the characteristic exponents are purely imaginary and x D 0 is stable near " D 0. If ! 2 D n2 for some n 2 N, however, the imaginary part of exp(C(")t) can be absorbed into ˚(t; ") and the characteristic exponents may be real. We assume now that ! 2 D n2 for some n 2 N, or near this value, and we shall look for periodic solutions of x(t) of Eq. (15) as these solutions define the boundaries between stable and unstable solutions. We put: ! 2 D n2  "ˇ ;

(18)

with ˇ a constant, and we apply the Poincaré-Lindstedtj method to find the periodic solutions; see Appendix 2 in [44]. We find that periodic solutions exist for n D 1 if: 1 ! 2 D 1 ˙ " C O("2 ) : 2

Higher Order Approximation and an Unexpected Timescale The instability tongue of the Mathieu equation at n D 1 can be determined with more precision by Poincaré expansion. On using averaging, one also characterizes the flow outside the tongue boundary and this results in a surprise. Consider Eq. (15) in the form x¨ C (1 C "a C "2 b C " cos 2t)x D 0; where we can choose a D ˙ 12 to put the frequency with first order precision at the tongue boundary. The eigenvalues of the trivial solution are from first order averaging 1;2

In the case n D 2, periodic solutions exist if: 1 ! D 4  "2 C O("4 ) ; 48 5 2 ! D 4 C "2 C O("4 ) : 48

The corresponding instability domains are called Floquet tongues, instability tongues or resonance tongues, see Fig. 1. On considering higher values of n, we have to calculate to a higher order of ". At n D 1 the boundary curves are intersecting at positive angles at " D 0, at n D 2 (! 2 D 4) they are tangent; the order of tangency increases as n  1 (contact of order n), making instability domains more and more narrow with increasing resonance number n.

2

(19)

1 D˙ 2

r

1  a2 ; 4

which agrees with Poincaré expansion; a2 > 14 gives stability, the < inequality instability. The transition value a2 D 1 4 gives the tongue location. Take for instance the C sign. Second order averaging, see [46], produces for the eigen-

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Perturbation Analysis of Parametric Resonance

for " small. To find an instability domain we have to decrease the damping, for instance if n D 2 we have to take  D " 2 0 .

values of the trivial solution v  3 1  u 11 1 u  4 32 C b " C 64 C 2 b t 7  1;2 D ˙ :  64  12 b "4 1 32

1 32

C b > 0 we have stability, if C b < 0 instaSo, if 1 bility; at b D  32 we have the second order approximation of this tongue boundary. Note that near this bound3 ary the solutions are characterized by eigenvalues of O(" 2 ) 3 and accordingly the time-dependence by timescale " 2 t. The Mathieu Equation with Viscous Damping In reallife applications there is always the presence of damping. We shall consider the effect of its simplest form, small viscous damping. Eq. (15) is extended by adding a linear damping term: x¨ C  x˙ C (! 2 C " cos 2t)x D 0 ;

Coexistence Linear periodic equations of the form (12) have m independent solutions and it is possible that all the independent solutions are periodic. This is called ‘coexistence’ and one of the consequences is that the instability tongues vanish. An example is Ince’s equation: (1 C a cos t)x¨ C  sin t x˙ C (! 2 C " cos t)x D 0 ; see [23]. An interesting question is whether this phenomenon persists under nonlinear perturbations; we return to this question in Subsect. “Coexistence Under Nonlinear Perturbation”. More General Classical Results

a;  > 0 :

(20)

We assume that the damping coefficient is small,  D "0 , and again we put ! 2 D n2  "ˇ to apply the PoincaréLindstedt method. We find periodic solutions in the case n D 1 if: r 1 2 2 (21) "  2 : ! D1˙ 4 Relation (21) corresponds with the curve of periodic solutions, which in (! 2 ; ")-parameter space separates stable and unstable solutions. We observe the following phenomena. If 0 <  < 12 ", we have an instability domain which by damping has been lifted from the ! 2 -axis; also the width has shrunk. If  > 12 " the instability domain has vanished. For an illustration see Fig. 2. Repeating the calculations for n  2 we find no instability domains at all; damping of O(") stabilizes the system

The picture presented by the Mathieu equation resulting in resonance tongues in the !; "-parameter space, stability and instability intervals as parametrized by ! shown in Fig. 1, has been studied for more general types of Hill’s equation. The older literature can be found in [39], see also [43]. Consider Hill’s equation in the form x¨ C (! 2 C " f (t))x D 0 ;

(22)

with f (t) periodic and represented by a Fourier series. Along the ! 2 -axis there exist instability intervals of size L m , where m indicates the mth instability interval. In the case of the Mathieu equation, we have from [16] L m D O(" m ) : The resonance tongues become increasingly narrow. For general periodic f (t) we have weak estimates, like L m D O("), but if we assume that the Fourier series is finite, the estimates can be improved. Put f (t) D

s X

f j cos 2 jt ;

jD0

so f (t) is even and -periodic. From [22] we have the following estimates:  If we can write m D sp with p 2 N, we have   j f s "j p 8s 2 Lm D C O(" pC1 ) : ((p  1)!)2 8s 2 Perturbation Analysis of Parametric Resonance, Figure 2 First order approximation of instability domains without and with damping for Eq. (20) near !2 D 1

 If we cannot decompose m like this and sp < m < s(p C 1), we have L m D O(" pC1 ) :

Perturbation Analysis of Parametric Resonance

In the case of Eq. (22) we have no dissipation and then it can be useful to introduce canonical transformations and Poincaré maps. In this case, for example, put x˙ D y; y˙ D 

@H ; @x

with Hamiltonian function H(x; y; t) D

1 2 1 2 y C (! C " f (t))x 2 : 2 2

 The instability tongues become thinner and recede into the !-axis as  increases.  High-order resonance tongues seem to be more affected by dissipation than low-order ones producing a dramatic loss of ‘fine detail’, even for small .  The results of varying the parameter  certainly needs more investigation. Parametrically Forced Oscillators in Sum Resonance

We can split H D H0 C "H1 with H0 D 12 (y 2 C ! 2 x 2 ) and apply canonical perturbation theory. Examples of this line of research can be found in [6] and [10]. Interesting conclusions can be drawn with respect to the geometry of the resonance tongues, crossings of tongues and as a possible consequence the presence of so-called instability pockets. In this context, the classical Mathieu equation turns out to be quite degenerate. Hill’s equation in the case of damping was considered in [35]; see also [36] where an arbitrary number of degrees of freedom is discussed.

In applications where more than one degree of freedom plays a part, many more resonances are possible. For a number of interesting cases and additional literature see [36]. An important case is the so-called sum resonance. In [17] a geometrical explanation is presented for the phenomena in this case using ‘all’ the parameters as unfolding parameters. It will turn out that four parameters are needed to give a complete description. Fortunately three suffice to visualize the situation. Consider the following type of differential equation with three frequencies

Quasi-Periodic Excitation

which describes a system of two parametrically forced coupled oscillators. Here A is a 44 matrix, containing parameters, and with purely imaginary eigenvalues ˙i!1 and ˙i!2 . The vector valued function f is 2-periodic in !0 t and f (0; !0 t; ) D 0 for all t and . Equation (24) can be resonant in many different ways. We consider the sum resonance

Equations of the form x¨ C (! 2 C "p(t))x D 0 ;

(23)

with parametric excitation p(t) quasi-periodic or almostperiodic, arise often in applications. Floquet theory does not apply in this case but we can still use perturbation theory. A typical example would be two rationally independent frequencies: p(t) D cos t C cos t ; with irrational. As an interesting example, in [7], D p 1 5) was chosen, the golden number. It will be no 2 (1 C surprise that many more complications arise for large values of ", but for " small (the assumption in this article), the analysis runs along similar lines producing resonance tongues, crossings of tongues and instability pockets. See also extensions in [11]. Detailed perturbation expansions are presented in [49] with a comparison of Poincaré expansion, the harmonic balance method and numerics; there is good agreement between the methods. Real-life models contain dissipation which inspired the authors of [49] to consider the equation x¨ C 2x˙ C (! 2 C "(cos t C cos t))x D 0;

irrational. They conclude that

 >0;

z˙ D Az C " f (z; !0 t; );

z 2 R4 ;

 2 Rp ;

(24)

! 1 C !2 D !0 ; where the system may exhibit instability. The parameter  is used to control detuning ı D (ı1 ; ı2 ) of the frequencies (!1 ; !2 ) from resonance and damping  D (1 ; 2 ). We summarize the analysis from [17].  The first step is to put Eq. (24) into normal form by normalization or averaging. In the normalized equation the time-dependence appears only in the higher order terms. But the autonomous part of this equation contains enough information to determine the stability regions of the origin. The linear part of the normal form is z˙ D A(ı; )z with  A(ı; ) D

B(ı; ) 0

0 B(ı; )

 ;

(25)

and  B(ı; ) D

iı1  1 ˛2

˛1 iı2  2

 :

(26)

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Perturbation Analysis of Parametric Resonance

Since A(ı; ) is the complexification of a real matrix, it commutes with complex conjugation. Furthermore, according to the normal form Theorem 1 and if ! 1 and ! 2 are independent over the integers, the normal form of Eq. (24) has a continuous symmetry group.  The second step is to test the linear part A(ı; ) of the normalized equation for structural stability i. e. to answer the question whether there exist open sets in parameter space where the dynamics is qualitatively the same. The family of matrices A(ı; ) is parametrized by the detuning ı and the damping . We first identify the most degenerate member N of this family and then show that A(ı; ) is its versal unfolding in the sense of [1]. The family A(ı; ) is equivalent to a versal unfolding U() of the degenerate member N.  Put differently, the family A(ı; ) is structurally stable for ı;  > 0, whereas A(ı; 0) is not. This has interesting consequences in applications as small damping and zero damping may exhibit very different behavior, see Sect. “Rotor Dynamics”. In parameter space, the stability regions of the trivial solution are separated by a critical surface which is the hypersurface where A(ı; ) has at least one pair of purely imaginary complex conjugate eigenvalues. This critical surface is diffeomorphic to the Whitney umbrella, see Fig. 3 and for references [17]. It is the singularity of the Whitney umbrella that causes the discontinuous behavior of the stability diagram in Sect. “Rotor Dynamics”. The structural stability argu-

ment guarantees that the results are ‘universally valid’, i. e. they qualitatively hold for generic systems in sum resonance. Nonlinear Parametric Excitation Adding nonlinear effects to parametric excitation strongly complicates the dynamics. We start with adding nonlinear terms to the (generalized) Mathieu equation. Consider the following equation that includes dissipation: x¨ C  x˙ C (! 2 C "p(t)) f (x) D 0 ;

(27)

where  > 0 is the damping coefficient, f (x) D x C bx 2 C cx 3 C    , and time is scaled so that: X p(t) D a2l e2i l t ; a0 D 0; a2l D a¯2l ; (28) l 2Z

is an even -periodic function with zero average. As we have seen in Sect. “Parametric Excitation of Linear Systems”, the trivial solution x D 0 is unstable when  D 0 and ! 2 D n2 , for all n 2 N. Fix a specific n 2 N and assume that ! 2 is close to n2 . We will study the bifurcations from the solution x D 0 in the case of primary resonance, which by definition occurs when the Fourier expansion of p(t) contains nonzero terms a2n e2i nt and a2n e2i nt . The bifurcation parameters in this problem are the detuning  D ! 2  n2 , the damping coefficient  and the Fourier coefficients of p(t), in particular a2n . The Fourier coefficients are assumed to be of equal order of magnitude. The Conservative Case,  D 0 An early paper is [21] in which Eq. (27) for  D 0 is associated with the Hamiltonian Z x !2 2 1 f (s)ds : x C p(t) H(x; x˙ ; t) D x˙ 2 C 2 2 0

Perturbation Analysis of Parametric Resonance, Figure 3 The critical surface in (C ;  ; ıC ) space for Eq. (24). C D 1 C 2 ,  D 1  2 , ıC D ı1 C ı2 . Only the part C > 0 and ıC > 0 is shown. The parameters ı1 ; ı2 control the detuning of the frequencies, the parameters 1 ; 2 the damping of the oscillators (vertical direction). The base of the umbrella lies along the ıC -axis

After transformation of the Hamiltonian, Lie transforms are implemented by MACSYMA to produce normal form approximations to O("2 ). A number of examples show interesting bifurcations. A related approach can be found in [5]; as p(t) is even, the equation is time-reversible. After construction of the Poincaré (timeperiodic) map, normal forms are obtained by equivariant transformations. This leads to a classification of integrable normal forms that are approximations of the family of Poincaré maps, a family as the map is parametrized by ! and the coefficients of p(t). Interestingly, the nonlinearity ˛x 3 is combined with the quasi-periodic Mathieu equation in [50] where global phenomena are described like resonance bands and chaos.

Perturbation Analysis of Parametric Resonance

Adding Dissipation,  > 0

Applications

Again time-periodic normal form calculations are used to approximate the dynamics; see [31], also [32] and the monograph [42]. The reflection symmetry in the normal form equations implies that all fixed points come in pairs, and that bifurcations of the origin will be symmetric (such as pitchfork bifurcations). We observe that the normal form equations show additional symmetries if either f (x) in Eq. (27) is odd in x or if n is odd. The general normal form can be seen as a non-symmetric perturbation of the symmetric case. One finds pitchfork and saddle-node bifurcations, in fact all codimension one bifurcations; for details and pictures see Chap. 9 in [42].

There are many applications of parametric resonance, in particular in engineering. In this section we consider a number of significant applications, but of course without any attempt at completeness. See also [36] and the references in the additional literature.

Coexistence Under Nonlinear Perturbation A model describing free vibrations of an elastica is described in [26]: 

1

 " cos 2t x¨ C " sin 2t x˙ C cx C "˛x 2 D 0 : 2

For ˛ D 0, the equation shows the phenomenon of coexistence. It is shown by second order averaging in [26] that for ˛ ¤ 0 there exist open sets of parameter values for which the trivial solution is unstable. An application to the stability problem of a family of periodic solutions in a Hamiltonian system is given in [29]. Other Nonlinearities In applications various nonlinear terms play a part. In [25] one considers x¨ C (! 2 C " cos(t))C "(Ax 3 C Bx 2 x˙ C Cx x˙ 2 C D x˙ 3 ) D 0 ; where averaging is applied near the 2 : 1-resonance. If B; D < 0 the corresponding terms can be interpreted as progressive damping. It turns out that for a correct description of the bifurcations second-order averaging is needed. Nonlinear damping can be of practical interest. The equation x¨ C (! 2 C " cos(t)) C j x˙ j x˙ D 0 ; is studied with  also a small parameter. A special feature is that an acceptable description of the phenomena can be obtained in a semi-analytical way by using Mathieu-functions as starting point. The analysis involves the use of Padé-approximants, see [28].

The Parametrically Excited Pendulum Choosing the pendulum case f (x) D sin(x) in Eq. (27) we have x¨ C  x˙ C (! 2 C "p(t)) sin(x) D 0 : It is natural, because of the sin periodicity, to analyze the Poincaré map on the cylindrical section t D 0 mod 2Z. This map has both a spatial and a temporal symmetry. As we know from the preceding section, perturbation theory applied near the equilibria x D 0; x D , produces integrable normal forms. For larger excitation (larger values of "), the system exhibits the usual picture of Hamiltonian chaos; for details see [12,24]. The inverted case is intriguing. It is well-known that the upper equilibrium of an undamped pendulum can be stabilized by vertical oscillations of the suspension point with certain frequencies. See for references [8,9] and [37]. In [8] the genericity of the classical result is studied for (conservative) perturbations respecting the symmetries of the equation. In [9] genericity is studied for (conservative) perturbations where the spatial symmetry is broken, replacing sin x by more general 2-periodic functions. Stabilization is still possible but the dynamics is more complicated. Rotor Dynamics When adding linear damping to a system there can be a striking discontinuity in the bifurcational behavior. Phenomena like this have already been observed and described in for instance [48] or [40]. The discontinuity is a fundamental structural instability in linear gyroscopic systems with at least two degrees of freedom and with linear damping. The following example is based on [41] and [33]. Consider a rigid rotor consisting of a heavy disk of mass M which is rotating with rotationspeed ˝ around an axis. The axis of rotation is elastically mounted on a foundation; the connections which are holding the rotor in an upright position are also elastic. To describe the position of the rotor we have the axial displacement u in the vertical direction (positive upwards), the angle of the axis of rotation with respect to the z-axis and around the z-axis.

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and putting t D , we obtain:   1 C ˛2 v 00 C C 4" cos 2 v D0; 2

(32)

where the prime denotes differentiation with respect to . By writing down the real and imaginary parts of this equation, we get two identical Mathieu equations. We conclude that the trivial p solution is stable for " small enough, providing that 1 C ˛ 2 is not close to n, for some n D 1; 2; 3; : : : . The first-order interval of instability, n D 1; arises if: p 1 C ˛2   : (33) If condition (33) is satisfied, the trivial solution of Eq. (32) is unstable. Therefore, the trivial solution of system (29) is also unstable. Note that this instability arises when: !1 C !2 D 2 ; Perturbation Analysis of Parametric Resonance, Figure 4 Rotor with diskmass M, elastically mounted with axial (u) and lateral directions

Instead of these two angles we will use the projection of the center of gravity motion on the horizontal (x; y)-plane, see Fig. 4. Assuming small oscillations in the upright (u) position, frequency 2, the equations of motion become: x¨ C 2˛ y˙ C (1 C 4"2 cos 2t)x D 0 ; y¨  2˛ x˙ C (1 C 4"2 cos 2t)y D 0 :

(29)

System (29) constitutes a system of Mathieu-like equations, where we have neglected the effects of damping. Abbreviating P(t) D 42 cos 2t, the corresponding Hamiltonian is: HD

1 1 1 (1C˛ 2 C"P(t))x 2 C p2x C (1C˛ 2 C"P(t))y 2 2 2 2 1 C p2y C ˛x p y  ˛ y p x ; 2

where p x ; p y are the momenta. The natural frequencies of p 2 the unperturbed p system (29), " D 0; are !1 D ˛ C 1 C 2 ˛ and !2 D ˛ C 1  ˛. By putting z D x C iy, system (29) can be written as: z¨  2˛i z˙ C (1 C 4"2 cos 2t)z D 0 :

(30)

Introducing the new variable: v D ei˛t z ;

(31)

i. e. when the sum of the eigenfrequencies of the unperturbed system equals the excitation frequency 2. This is known as a sum resonance of first order. The domain of instability can be calculated as in Subsect. “Elementary Theory”; we find for the boundaries: p (34) b D 1 C ˛ 2 (1 ˙ ") C O("2 ) : The second order interval of instability of Eq. (32), n D 2, arises when: p 1 C ˛ 2  2 ; (35) i. e. !1 C !2  . This is known as a sum resonance of second order. As above, we find the boundaries of the domains of instability:   p 1 2 D 1 C ˛ 2 1 C "2 C O("4 ) ; 24 (36)   p 5 2 4 2 2 D 1 C ˛ 1  " C O(" ) : 24 Higher order combination resonances can be studied in the same way; the domains of instability in parameter space continue to narrow as n increases. It should be noted that the parameter ˛ is proportional to the rotation speed ˝ of the disk and to the ratio of the moments of inertia. Instability by Damping We add small linear damping to system (29), with positive damping parameter  D 2". This leads to the equation: z¨  2˛i z˙ C (1 C 4"2 cos 2t)z C 2" z˙ D 0 :

(37)

Perturbation Analysis of Parametric Resonance

Because of the damping term, we can no longer reduce the complex Eq. (37) to two identical second order real equations, as we did in the previous section. In the sum resonance of the first order, we have !1 C !2  2 and the solution of the unperturbed (" D 0) equation can be written as: (38) z(t) D z1 e i!1 t C z2 ei!2 t ; z1 ; z2 2 C ; p p with !1 D ˛ 2 C 1 C ˛; !2 D ˛ 2 C 1  ˛. Applying variation of constants leads to equations for z1 and z2 :  i"   2 i!1 z1  i!2 z2 ei(!1 C!2 )t z˙1 D !1 C !2   C 42 cos 2t z1 C z2 ei(!1 C!2 )t ; (39)  i"   i(!1 C!2 )t z˙2 D 2 i!1 z1 e  i!2 z2 !1 C !2   C 42 cos 2t z1 e i(!1 C!2 )t C z2 : To calculate thep instability interval around the value 0 D 1 (! C ! ) D ˛ 2 C 1, we apply perturbation theory to 1 2 2 find for the stability boundary: s ! p 2 2 2 b D 1 C ˛ 1 ˙ " 1 C ˛  2 C    ; 0 0 1 s  2 p  C A : D 1 C ˛ 2 @1 ˙ (1 C ˛ 2 )"2  0 (40) It follows that the domain of instability actually becomes larger when damping is introduced. The most unusual aspect of the above expression for the instability interval, however, is that there is a discontinuity at  D 0. If  ! 0, then the boundaries of the instability domain tend to the p p limits b ! 1 C ˛ 2 (1 ˙ " 1 C ˛ 2 ) which p differs from the result we found when  D 0 : b D 1 C ˛ 2 (1 ˙ "). In mechanical terms, the broadening of the instabilitydomain is caused by the coupling between the two degrees of freedom of the rotor in lateral directions which arises in the presence of damping. Such phenomena are typical for gyroscopic systems and have been noted earlier in the literature; see [2,4] and [36]. The explanation of the discontinuity and its genericity in [17], see Subsect. “Parametrically Forced Oscillators in Sum Resonance”, is new. For hysteresis and phase-locking phenomena in this problem, the reader is referred to [33]. Autoparametric Excitation In [42], autoparametric systems are characterized as vibrating systems which consist of at least two consisting

Perturbation Analysis of Parametric Resonance, Figure 5 Two coupled oscillators with vertical oscillations as Primary System and parametric excitation of the coupled pendulum (Secondary System)

subsystems that are coupled. One is a Primary System that can be in normal mode vibration. In the instability (parameter) intervals of the normal mode solution in the full, coupled system, we have autoparametric resonance. The vibrations of the Primary System act as parametric excitation of the Secondary System which will no longer remain at rest. An example is presented in Fig. 5. In actual engineering problems, we wish sometimes to diminish the vibration amplitudes of the Primary System; sometimes this is called ‘quenching of vibrations’. In other cases we have a coupled Secondary System which we would like to keep at rest. As an example we consider the following autoparametric system studied in [14]:   4 x 00 C x C " k1 x 0 C 1 x C a cos 2 x C x 3 C c1 y 2 x 3 D0   4 y 00 C y C " k2 y 0 C 2 y C c2 x 2 y C y 3 D 0 3 (41) where  1 and  2 are the detunings from the 1 : 1-resonance of the oscillators. In this system, y(t) D y˙(t) D 0 corresponds with a normal mode of the x-oscillator. The system (41) is invariant under (x; y) ! (x; y), (x; y) ! (x; y), and (x; y) ! (x; y). Using the method of averaging as a normalization procedure we investigate the stability of solutions of system (41). To give an explicit example we follow [14] in more detail. Introduce the usual variation of constants transformation: x D u1 cos  Cv1 sin  ;

x 0 D u1 sin  Cv1 cos  (42)

y D u2 cos  Cv2 sin  ;

y 0 D u2 sin  Cv2 cos  (43)

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After rescaling  D "2 ˜ the averaged system of (41) becomes:     1 u10 D k1 u1 C 1  a v1 C v1 u12 C v12 2 1 3 2 1 2 C c1 u 2 v1 C c1 v2 v1 C c1 u2 v2 u1 4 4 2     1 0 v1 D k1 v1  1 C a u1  u1 u12 C v12 2 3 1 1  c1 u22 u1  c1 v22 u1  c1 u2 v2 v1 4 4 2 (44)  2  1 0 2 2 u2 D k2 u2 C 2 v2 C v2 u2 C v2 C c2 u1 v2 4 3 2 1 C c2 v1 v2 C c2 u1 v1 u2 4 2   3 v20 D k2 v2  2 u2  u2 u22 C v22  c2 u12 u2 4 1 2 1  c2 v1 u2  c2 u1 v1 v2 : 4 2 This system is analyzed for critical points, periodic and quasi-periodic solutions, producing existence and stability diagrams in parameter space. The system also contains a sequence of period-doubling bifurcations leading to chaotic solutions, see Fig. 6. To prove the presence of chaos involves an application of higher dimensional Melnikov theory developed in [47]. A rather technical analysis in [14] shows the existence of a Šilnikov orbit in the averaged equation, which implies chaotic dynamics, also for the original system.

Future Directions Ongoing research in dynamical systems includes nonlinear systems with parametric resonance, but there are a number of special features as these systems are non-autonomous. This complicates the dynamics from the outset. For instance a two degrees of freedom system with parametric resonance involves at least three frequencies, producing many possible resonances. The analysis of such higher dimensional systems with many more combination resonances, has begun recently, producing interesting limit sets and invariant manifolds. Also the analysis of PDEs with periodic coefficients will play a part in the near future. These lines of research are of great interest. In the conservative case, the association with Hamiltonian systems, KAM theory etc. gives a natural approach. This has already produced important results. In real-life modeling, there will always be dissipation and it is important to include this effect. Preliminary results suggest that the impact of damping on for instance quasi-periodic systems, is quite dramatic. This certainly merits more research. Finally, applications are needed to solve actual problems and to inspire new, theoretical research. Acknowledgment A number of improvements and clarifications were suggested by the editor, Giuseppe Gaeta. Additional references were obtained from Henk Broer and Fadi Dohnal. Bibliography Primary Literature

Perturbation Analysis of Parametric Resonance, Figure 6 The strange attractor of the averaged system (44). The phaseportraits in the (u2 ; v2 ; u1 )-space for c2 < 0 at the value 2 D 5:3. The Kaplan–Yorke dimension for 2 D 5:3 is 2.29

1. Arnold VI (1983) Geometrical methods in the theory of ordinary differential equations. Springer, New York 2. Banichuk NV, Bratus AS, Myshkis AD (1989) Stabilizing and destabilizing effects in nonconservative systems. PMM USSR 53(2):158–164 3. Bogoliubov NN, Mitropolskii Yu A (1961) Asymptotic methods in the theory of nonlinear oscillations. Gordon and Breach, New York 4. Bolotin VV (1963) Non-conservative problems of the theory of elastic stability. Pergamon Press, Oxford 5. Broer HW, Vegter G (1992) Bifurcational aspects of parametric resonance, vol 1. In: Jones CKRT, Kirchgraber U, Walther HO (eds) Expositions in dynamical systems. Springer, Berlin, pp 1–51 6. Broer HW, Levi M (1995) Geometrical aspects of stability theory for Hill’s equation. Arch Rat Mech Anal 131:225–240 7. Broer HW, Simó C (1998) Hill’s equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena. Bol Soc Brasil Mat 29:253–293 8. Broer HW, Hoveijn I, Van Noort M (1998) A reversible bifurcation analysis of the inverted pendulum. Physica D 112:50–63

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9. Broer HW, Hoveijn I, Van Noort M, Vegter G (1999) The inverted pendulum: a singularity theory approach. J Diff Eqs 157:120– 149 10. Broer HW, Simó C (2000) Resonance tongues in Hill’s equations: a geometric approach. J Differ Equ 166:290–327 11. Broer HW, Puig J, Simó C (2003) Resonance tongues and instability pockets in the quasi-periodic Hill-Schrödinger equation. Commun Math Phys 241:467–503 12. Broer HW, Hoveijn I, Van Noort M, Simó C, Vegter G (2005) The parametrically forced pendulum: a case study in 1 12 degree of freedom. J Dyn Diff Equ 16:897–947 13. Cicogna G, Gaeta G (1999) Symmetry and perturbation theory in nonlinear dynamics. Lecture Notes Physics, vol 57. Springer, Berlin 14. Fatimah S, Ruijgrok M (2002) Bifurcation in an autoparametric system in 1:1 internal resonance with parametric excitation. Int J Non-Linear Mech 37:297–308 15. Golubitsky M, Schaeffer D (1985) Singularities and groups in bifurcation theory. Springer, New York 16. Hale J (1963) Oscillation in nonlinear systems. McGraw-Hill, New York, 1963; Dover, New York, 1992 17. Hoveijn I, Ruijgrok M (1995) The stability of parametrically forced coupled oscillators in sum resonance. ZAMP 46:383– 392 18. Iooss G, Adelmeyer M (1992) Topics in bifurcation theory. World Scientific, Singapore 19. Kuznetsov Yu A (2004) Elements of applied bifurcation theory, 3rd edn. Springer, New York 20. Krupa M (1997) Robust heteroclinic cycles. J Nonlinear Sci 7:129–176 21. Len JL, Rand RH (1988) Lie transforms applied to a non-linear parametric excitation problem. Int J Non-linear Mech 23:297– 313 22. Levy DM, Keller JB (1963) Instability intervals of Hill’s equation. Comm Pure Appl Math 16:469–476 23. Magnus W, Winkler S (1966) Hill’s equation. Interscience-John Wiley, New York 24. McLaughlin JB (1981) Period-doubling bifurcations and chaotic motion for a parametrically forced pendulum. J Stat Phys 24:375–388 25. Ng L, Rand RH (2002) Bifurcations in a Mathieu equation with cubic nonlinearities. Chaos Solitons Fractals 14:173–181 26. Ng L, Rand RH (2003) Nonlinear effects on coexistence phenomenon in parametric excitation. Nonlinear Dyn 31:73–89 27. Pikovsky AS, Feudel U (1995) Characterizing strange nonchaotic attractors. Chaos 5:253–260 28. Ramani DV, Keith WL, Rand RH (2004) Perturbation solution for secondary bifurcation in the quadratically-damped Mathieu equation. Int J Non-linear Mech 39:491–502 29. Recktenwald G, Rand RH (2005) Coexistence phenomenon in autoparametric excitation of two degree of freedom systems. Int J Non-linear Mech 40:1160–1170 30. Roseau M (1966) Vibrations nonlinéaires et théorie de la stabilité. Springer, Berlin 31. Ruijgrok M (1995) Studies in parametric and autoparametric resonance. Thesis, Utrecht University, Utrecht 32. Ruijgrok M, Verhulst F (1996) Parametric and autoparametric resonance. Prog Nonlinear Differ Equ Their Appl 19:279–298 33. Ruijgrok M, Tondl A, Verhulst F (1993) Resonance in a rigid rotor with elastic support. ZAMM 73:255–263

34. Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems, rev edn. Appl Math Sci, vol 59. Springer, New York 35. Seyranian AP (2001) Resonance domains for the Hill equation with allowance for damping. Phys Dokl 46:41–44 36. Seyranian AP, Mailybaev AA (2003) Multiparameter stability theory with mechanical applications. Series A, vol 13. World Scientific, Singapore 37. Seyranian AA, Seyranian AP (2006) The stability of an inverted pendulum with a vibrating suspension point. J Appl Math Mech 70:754–761 38. Stoker JJ (1950) Nonlinear vibrations in mechanical and electrical systems. Interscience, New York, 1950; Wiley, New York, 1992 39. Strutt MJO (1932) Lamé-sche, Mathieu-sche und verwandte Funktionen. Springer, Berlin 40. Szemplinska-Stupnicka W (1990) The behaviour of nonlinear vibrating systems, vol 2. Kluwer, Dordrecht 41. Tondl A (1991) Quenching of self-excited vibrations. Elsevier, Amsterdam 42. Tondl A, Ruijgrok M, Verhulst F, Nabergoj R (2000) Autoparametric resonance in mechanical systems. Cambridge University Press, New York 43. Van der Pol B, Strutt MJO (1928) On the stability of the solutions of Mathieu’s equation. Phil Mag Lond Edinb Dublin 7(5):18–38 44. Verhulst F (1996) Nonlinear differential equations and dynamical systems. Springer, New York 45. Verhulst F (2005) Invariant manifolds in dissipative dynamical systems. Acta Appl Math 87:229–244 46. Verhulst F (2005) Methods and applications of singular perturbations. Springer, New York 47. Wiggins S (1988) Global Bifurcation and Chaos. Appl Math Sci, vol 73. Springer, New York 48. Yakubovich VA, Starzhinskii VM (1975) Linear differential equations with periodic coefficients, vols 1 and 2. Wiley, New York 49. Zounes RS, Rand RH (1998) Transition curves for the quasi-periodic Mathieu equation. SIAM J Appl Math 58:1094–1115 50. Zounes RS, Rand RH (2002) Global behavior of a nonlinear quasi-periodic Mathieu equation. Nonlinear Dyn 27:87–105

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Perturbation of Equilibria in the Mathematical Theory of Evolution

Perturbation of Equilibria in the Mathematical Theory of Evolution ANGEL SÁNCHEZ1,2 1 Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de Matemáticas, Universidad Carlos III de Madrid, Madrid, Spain 2 Instituto de Biocomputación y Física de Sistemas Complejos (BIFI), Universidad de Zaragoza, Zaragoza, Spain Article Outline Glossary Definition of the Subject Introduction Evolution on a Fitness Landscape Stability of Equilibria on a Fitness Landscape Perturbation of Equilibria on a Fitness Landscape Frequency Dependent Fitness: Game Theory Equilibria in Evolutionary Game Theory Perturbations of Equilibria in Evolutionary Game Theory Future Directions Bibliography Glossary Evolutionarily stable equilibria (ESS) An ESS is a set of frequencies of different types of individuals in a population that can not be invaded by the evolution of a single mutant. It is the evolutionary counterpart of a Nash equilibrium. Fitness landscape A metaphorical description of fitness as a function of individual’s genotypes or phenotypes in terms of a multivariable function that does not depend on any external influence. Genetic locus The position of a gene on a chromosome. The different variants of the gene that can be found at the same locus are called alleles. Nash equilibrium In classical game theory, a Nash equilibrium is a set of strategies, one for each player of the game, such that none of them can improve her benefits by unilateral changes of strategy. Scale free network A graph or network such that the degrees of the nodes are taken from a power-law distribution. As a consequence, there is not a typical degree in the graph, i. e., there are no typical scales. Small-world network A graph or network of N nodes such that the mean distance between nodes scales as log N. It corresponds to the well-known “six degrees of separation” phenomenon.

Definition of the Subject The importance of evolution can hardly be overstated. As the Jesuit priest Pierre Teilhard de Chardin put it, Evolution is a general postulate to which all theories, all hypotheses, all systems must hence forward bow and which they must satisfy in order to be thinkable and true. Evolution is a light which illuminates all facts, a trajectory which all lines of thought must follow – this is what evolution is. Darwin’s evolution theory is based on three fundamental principles: reproduction, mutation and selection, which describe how populations change over time and how new forms evolve out of old ones. Starting with W. F. R. Weldon, whom at the beginning of the 20th century realized that “the problem of animal evolution is essentially a statistical problem”, and blooming in the 30’s with Fisher, Haldane and Wright, numerous mathematical descriptions of the resulting evolutionary dynamics have been proposed, developed and studied. Deeply engraved in these frameworks are the mathematical concepts of equilibrium and stability, as descriptions of the observed population compositions and their lifetimes. Many results have been obtained regarding the stability of equilibria of evolutionary dynamics in idealized circumstances, such as infinite populations or global interactions. In the evolutionary context, stability is peculiar, in the sense that it is entangled with collective effects arising from the interaction of individuals. Therefore, perturbations of the idealized mathematical framework representing more realistic situations are of crucial importance to understand stability of equilibria. Introduction The idea of evolution is a simple one: Descent with modification acted upon by natural selection. Descent with modification means that we consider a population of replicators, entities capable of reproducing themselves, in which reproduction is not exact and allows for small differences between parents and offspring. Natural selection means that different entities reproduce in different quantities because their abilities are also different: some are more resistant to external factors, some need less resources, some simply reproduce more... While in biology these replicators are, of course, living beings, the basic ingredients of evolution have by now trascended the realm of biology into the kingdom of objects such as computer codes (thus giving rise, e. g., to genetic algorithms). The process of evolution is often described in short as “the survival of the fittest”, meaning that those replicators

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that succeed and thrive are more fit than those which progressively disappear. This statement is not very appropriate, because evolution does not imply making organisms or entities more fit; it is simply a consequence of differential reproduction in the face of selection pressure. On the other hand, it leads to a tautology: The question of which are the fittest organisms is answered by saying that they are those that survive. It is then clear that a correct use of the concept of fitness is at the crux of any attempt to formalize mathematically evolution theory. In this article we are going to discuss two manners to deal mathematically with the concept of fitness. The first and simplest one is to resort to a fitness landscape, whose basic feature is that the fitness of a given individual depends only on the individual’s characteristics and not on external factors. We will present this approach in Sects. “Evolution on a Fitness Landscape” and “Stability of Equilibria on a Fitness Landscape” below, to subsequently discuss the effect of perturbations on the equilibria described by this picture in Sect. “Perturbation of Equilibria on a Fitness Landscape”. To go beyond the fitness landscape picture one has to introduce frequency-dependent selection, i. e., to remove the independence of the fitness from external influences. In Sects. “Frequency Dependent Fitness: Game Theory” and “Equilibria in Evolutionary Game Theory” we consider the evolutionary game theory approach to this way to model evolution and, as before, analyze the perturbations of its equilibria in Sec “Perturbations of Equilibria in Evolutionary Game Theory”. Sect. “Future Directions” summarizes the questions that remain open in this field. Evolution on a Fitness Landscape The metaphor of evolution on a “fitness landscape” reaches back at least to [26]: Drawing on the connection between fitness and adaptation, fitness is defined as the expected number of offspring of a given individual that reach adulthood, and thus represents a measure of its adaptation to the environment. In this context, fitness landscapes are used to visualize the relationship between genotypes (or phenotypes) and reproductive success. It is assumed that every genotype has a well defined fitness, in the sense above, and that this fitness is the “height” of the landscape. Genotypes which are very similar are said to be “close” to each other, while those that are very different are “far” from each other. The two concepts of height and distance are sufficient to form the concept of a “landscape”. The set of all possible genotypes, their degree of similarity, and their related fitness values is then called a fitness landscape.

Perturbation of Equilibria in the Mathematical Theory of Evolution, Figure 1 Sketch of a fitness landscape

Fitness landscapes are often conceived of as ranges of mountains. There exist local peaks (points from which all paths are downhill, i. e. to lower fitness) and valleys (regions from which most paths lead uphill). A fitness landscape with many local peaks surrounded by deep valleys is called rugged. If all genotypes have the same replication rate, on the other hand, a fitness landscape is said to be flat. A sketch of such a fitness landscape, showing the dependence of the fitness on two different “characteristics” or “genes”, is shown in Fig. 1. Of course, the true fitness landscape would need a highly multidimensional space for its representation, as it would depend on all the characteristics of the organism, even those it still does not show. Therefore, the sketch is an extreme oversimplification, only to suggest the structure of a rugged fitness landscape. Given the immense complexity of the genotype-fitness mapping, theoretical models have to make a variety of simplifying assumptions. Most models in biological literature focus on the effect of one or a few genetic loci on the fitness of individuals in a population, assuming that each of the considered loci can be occupied by a limited number of different alleles that have different effects on the fitness, and that the rest of the genome is part of the invariant environment. This approximation, the first attempt to obtain analytical results for changes in the gene pool of a population under the influence of inheritance, selection and mutation is the pioneering work of Fisher, Haldane and Wright, who founded the field of population genetics. Their method of randomly drawing the genes of the daughter population from the pool of parent genes, with weights proportional to the fitness, proved to be very successful at calculating the evolution of allele frequencies from one generation to the next, or the chances of a new mutation to spread through a population, even taking into account various patterns of mating, dominance effects,

Perturbation of Equilibria in the Mathematical Theory of Evolution

nonlinear effects between different genes, etc. Population genetics has since then developed into a mature field with a sophisticated mathematical apparatus, and with wideranging applications. Stability of Equilibria on a Fitness Landscape The simplified pictures we have just described lead to a description in terms of dynamical systems, and therefore the stability of its equilibria can be studied by means of standard techniques. In principle, one can envisage the evolution of a population on a fitness landscape in the usual frame of particle dynamics on a potential. Every individual is a point in the space of phenotypes or genotypes, and evolves towards the maxima of the fitness; in the potential picture, the potential is given by minus the fitness. Furthermore, the dynamics is overdamped, i. e., there are no oscillations around the equilibria. Maxima of the fitness are therefore the equilibria of the evolutionary process. That this is so is a consequence of Fisher’s theorem [3], whose original derivation is very general but quite complicated. Following [2] and [4], we prefer here to present two simpler situations: an asexual population, and a sexually reproducing population where the fitness is determined by a single gene with two alleles. For an asexually reproducing population, the derivation of Fisher’s theorem is straightforward: Let yi be the number of genes i in the population, and y the total number of genes; then p i  y i /y is the frequency of genotype i in the population. If W i is gene i’s fitness, sticking to the interpretation of fitness in terms of offspring, the number of individuals carrying gene i in the next generation is Wi y i and, subsequently, the change in the frecuency pi from one generation to the next is ¯ W ¯ ; p i D p i (Wi  W)/ leading to a change in mean fitness P P ¯ 2) ¯ p i (Wi2  W Wi p i W ; D i D i 2 ¯ ¯ ¯ W W W which is proportional to the genetic variance in fitness. If the fitness changes from one generation to the next are small, this becomes an equation which states that the rate of change in fitness is identical to the genetic variance in fitness. When reproduction is sexual, we note that pi is the frequency of gene i. Assuming for simplicity that only two alleles are possible at the genetic locus of interest, the fitness of type 1 is w11 p1 C w12 p2 , where wij is the fitness of an individual carrying alleles i and j, and hence the number of 1 genes will be y1 (w11 p1 Cw12 p2 ). We then have the

differential equation y˙1 D y1 (w11 p1 C w12 p2 ) :

(1)

It is enough then to differentiate the identity ln p1 D ln y1  ln y and use a little algebra to show that y1 obeys the replicator equation: ¯ p˙ 1 D p1 (w1  w)

(2)

Fisher’s theorem then states that fitness increases along trajectories of this equation: Indeed, by noting that the average fitness is w¯ D

X

wi j pi p j ;

(3)

i; jD1;2

differentiating and using the replicator equation we arrive at the final result X ˙¯ D 2 ¯ 2: w p i (w i  w) (4) i

Fisher’s theorem thus means that an evolving population will typically climb uphill in the fitness landscape, by a series of small genetic changes, until a local optimum is reached. This is due to the fact that the average fitness of the population always increases, as we have just shown; hence the analogy with overdamped dynamics on a (inverted) potential function. Furthermore, because of this result, the population remains there, at the equilibrium point, because it cannot reduce its fitness. We then realize that all equilibria in a fitness landscape within the interpretation of fitness as the reproductive success are stable. Perturbation of Equilibria on a Fitness Landscape In order to understand the possible breakdowns of stability in the fitness landscape picture, one has to look carefully at the hypothesis of Fisher’s theorem. We have not stated it in a formal manner, hence it is important to summarize here the main ones:  Population is infinite.  There are no mutations (i. e., the only source of every gene or species is reproduction).  There is only one population (i. e., there are no population fluxes or migrations between separate groups).  Fitness depends only on the individual’s genotype and not on the other individuals. It is then clear that, even if it is mathematically true, the applicability of Fisher’s theorem is a completely different story, and as a consequence, the conclusion that maxima

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of the fitness landscape are stable may be wrong when discussing real systems. A detailed discussion of all these issues can be found in [2], and we refer the reader to her paper for a thorough discussion of all these factors. For our present purposes, namely to show that these perturbations can change the stability of the equilibria, it will suffice to present a few ideas about the case of finite population size. Afterwards, the rest of the paper will proceed along the idea of fitness depending on other individuals, giving up the paradigm of a fixed fitness landscape. The subject of finite size populations is the subject of fluctuations and its main consequences, genetic drift and stochastic escape. Regarding the first concept, as compared to natural selection, i. e., to the tendency of beneficial alleles to become more common over time (and detrimental ones less common), genetic drift is the fundamental tendency of any allele to vary randomly in frequency over time due to statistical variation alone, so long as it does not comprise all or none of the distribution. In other words, even when individuals face the same odds, they will differ in their success. A rare succession of chance events can thus bring a trait to predominance, causing a population or species to evolve (in fact, this idea is at the core of the neutral theory of evolution, first proposed by [10]). On the other hand, stochastic escape refers to the situation in which a population of individuals placed at a maximum of the fitness landscape may leave this maximum due to fluctuations. Obviously, both genetic drift and stochastic escape affect the stability of the maxima as predicted by Fisher’s theorem. One consequence of finite population sizes and fluctuations in the composition of a population is that genes get lost from the gene pool. If there is no new genetic input through mutation or migration, the genetic variability within a population decreases with time. After sufficiently many generations, all individuals will carry the same allele of a given gene. This allele is said to have become fixed. In the absence of selection, the probability that a given allele will become fixed is proportional to the number of copies in the initial population. Thus, if a new mutant arises that has no selective advantage or disadvantage, this mutant will spread through the entire population with a probability 1/M, M being the population size. If the individuals of the population are diploid, each carries two sets of genes, and M must be taken as the number of sets of genes, i. e., as twice the population size. On the other hand, it can be shown [2] that the probability that a mutant that conveys a small fitness increase by a factor 1 C s has as probability of the order s to spread through a population. In populations of sizes much smaller than 1/s, this selective advantage is not felt, because mutations that carry no advantage

become fixed at a similar rate. In the same manner, a mutation that decreases the fitness of its carrier by a factor 1  s, is not felt in a population much smaller than 1/s. An interesting consequence of these results is that the rate of neutral (or effectively neutral) substitutions is independent of the population size. The reason is that the probability that a new mutant is generated in the population is proportional to M, while its probability of becoming fixed is 1/M. Frequency Dependent Fitness: Game Theory In the preceding sections we have considered the case when the fitness depends on the genotype, but is independent of the composition of the population, i. e., the presence of inviduals of the same genotype or of other genotypes does not change the fitness of the focal one. This assumption, that allows for an intuitive picture in terms of a fitness landscape, is clearly an over-simplification, as was already mentioned above. For instance, consider an homogeneous population in a closed environment. The population will grow at a pace given by the fitness of its individuals until it eventually exhausts the available resources or even physically fills the environment. Therefore, even if the individuals are all equal, their fitness will not be the same if there are only a few of them or if there are very many. Another trivial example is the effect on the fitness of the presence or absence of predators of the species of interest; clearly, predators will reduce the fitness (understood as above in a reproductive sense) of their prey. Therefore, individuals will evolve subject not only to external influences but also to their mutual competition, both intra-specific and inter-specific. This leads us to consider frequency-dependent selection, which can be described by very many, different theoretical approaches. These include game theory as well as discrete and continuous genetic models, and the concepts of kin selection, group selection, and sexual selection. Among the possible dynamical patterns arising, there are single fixed points, lines of fixed points, runaway, limit cycles, and chaos. A review of all these descriptions, whose use to model evolution depends on the specific issues one is interested in, is clearly far beyond the scope of this article and, hence, we have focused on evolutionary game theory as a particularly suited case study to show the effects of perturbations on equilibria. Before going into the study of evolutionary game theory, we need to summarize briefly a few key concepts about its originating theory, namely (classical) game theory. Pioneered in the early XIX century by the economist Cournot, game theory was introduced by the brilliant,

Perturbation of Equilibria in the Mathematical Theory of Evolution

multi-faceted mathematician John von Neumann in 1928, and it was first presented as a specific subject in von Neumann’s book (with Oskar Morgenstern) Theory of Games and Economic Behavior in 1944. Since then, game theory has been used to model strategic situations, i. e., situations in which actors or agents follow different strategies (meaning that they choose among different possible actions or behaviors) to maximize their benefit, usually referred to as payoff . These arise in very many different contexts, from biology and psychology to philosophy through politics, economics or computer science. The central concept of game theory is the Nash equilibrium [14], introduced by the mathematician John Nash in 1955, awarded with a Nobel Prize in Economics almost forty years later for this work. A set of strategies, one for each participant in the game, is a Nash equilibrium if every strategy is the best response (in terms of maximizing the player’s payoff) to the subset of the strategies of the rest of the players. In this case, if all players use strategies belonging to a Nash equilibrium, none of them will have any incentive to change her behavior. In this situation we indeed have the equivalent of the traditional concept of equilibrium in dynamical systems: players keep playing the same strategy as, given the behavior of the others, they follow the optimal strategy (note that this does not mean the strategy is optimal in absolute terms: it is only optimal in view of the actions of the rest). Equilibria in Evolutionary Game Theory In the seventies, game theory, which as proposed by von Neumann and Nash was to be used to understand economic behavior, entered the realm of biology through the pioneering work of John Maynard-Smith and George Price [12], who introduced the evolutionary version of the theory. The key contribution of their work was a new interpretation of the general framework of game theory in terms of populations instead of individual players. While traditional game theoretical players behaved following some strategy and could change it to improve their performance, in the picture of Maynard-Smith and Price individuals had a fixed strategy, determined by their genotype, and different strategies were represented by subpopulations of individuals. In this representation, changes of strategies correspond to the replacement of the individuals by their offspring, possibly with mutations. Payoffs obtained by individuals in the game are accordingly understood as fitness, the reproductive rate that governs how the replacement occurs. There is a large degree of arbitrariness as to the evolutionary dynamics of the populations. All we have said

so far is that fitness, obtained through the game, determines the composition of the population at the next time step (or instant, if we think of continuous time). Probably the most popular choice (but by no means the only one, see [18] for different evolutionary proposals and their relationships, see also [9] for other dynamics) is to use the replicator equation we have previously found to describe the evolution of the frequency yi of strategists of type i: ¯ : y˙ i D y i (w i  w)

(5)

It is important to note that the steps leading to the derivation of this equation are the same as above, and therefore for it to be applicable in principle one must keep in mind the same hypotheses. The difference is that now the fitness is not a constant but rather it is determined by a game, which enters the equation in the following way. Let us call A the payoff matrix of the game (for simplicity, we will consider only symmetric games), whose entries aij are the payoffs to an individual using strategy i facing another using strategy j. Assuming the frequencies yi (t) are differentiable functions, if individuals meet randomly and then engage in the game, and this takes place very many (infinite) times, then (Ay) i is the expected payoff for type i individuals in a population described by the vector y, whose components are the frequencies of each type. By the same token, the average payoff in the population is w¯ D yT Ay, so substituting in (5) we are left with  y˙ i D y i (Ay) i  yT Ay ;

(6)

where we now see explicitly how the game affects the evolution. Nevertheless, it is also clear that this rule is arbitrary, and there are many other options one can use to postulate how the population evolves. We will come back to this issue when considering perturbations of the equilibria. If Nash equilibrium is the key concept in game theory, evolutionarily stable strategy is the relevant one in its evolutionary counterpart. [11] defined evolutionarily stable strategy (ESS) as a strategy such that, if every individual in the population uses it, no other (mutant) strategy could invade the population by natural selection. It is trivial to show that, in terms of the payoffs of the game, for strategy i to be an ESS, one of the following two conditions must hold: a i i > a i j 8 j ¤ i ; or a i i D a i j for some j and

ai j > a j j :

(7) (8)

If the first condition is fulfilled, we speak of strict ESS. It is important to realize that this concept is absolutely general and, in particular, it does not depend on the evolutionary

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dynamics of choice (in so far as it favors the strategies that receive the best payoffs). Of course, the two concepts, Nash equilibrium and ESS, are related. This is in fact one of the reasons why evolutionary game theory ended up appealing to the economists, who faced the question as to how individuals ever get to play the Nash equilibrium strategies: They now had a dynamical way that might precisely describe that process and, furthermore, to decide which Nash equilibrium was selected if there were more than one. To show the connection, one must decide on a dynamical rule, for which we will stay within the framework of the replicator dynamics. For this specific evolutionary dynamics, it can be rigorously shown that (see, e. g., [7,9]) 1. if y0 is a Nash equilibrium, it is a rest point (a zero of the rhs of (5); 2. if y0 is a strict Nash equilibrium, it is asymptotically stable; 3. if y0 is a rest point and is the limit of an interior orbit for t ! 1, then it is a Nash equilibrium; and 4. if y0 is a stable rest point, it is a Nash equilibrium. This means that there indeed is a relationship between Nash equilibria and ESS, but more subtle that could appear at first. Probably, the most important non trivial aspect of this result is that not all ESS are Nash equilibria, as stability is required in addition. Perturbations of Equilibria in Evolutionary Game Theory The evolutionary viewpoint on game theory allows to study Nash equilibria/ESS within the standard framework of dynamical systems theory, by using the concepts of stability, asymptotical stability, global stability and related notions. In fact, one can do more than that: the problem of invasion by a mutant, the biological basis of the ESS concept of Maynard-Smith, can always be formulated in terms of a dynamical coupling of the mutant and the incumbent species and hence studied in terms of the stability of a rest point of a dynamical system. In principle, the same idea can be generalized to simultaneous invasion by more than one mutant and, although the problem may be technically much more difficult, the basic procedure remains the same. As we did with the fitness landscape concept, when considering perturbations of equilibria, our interest goes beyond this traditional stability ideas, and once again, we need to focus on the deviations from the framework that allows to derive the replicator equation. There are a number of such deviations. The simplest ones are the

inclusion of mutations or migrations, leading to the socalled replicator-mutator equation [18], that can be subsequently studied as a dynamical system. Other deviations affect much more, and in a way more difficult to aprehend, to the evolutionary dynamics and its equilibria, such as considering finite size populations, alternative learning/reproduction dynamics, or the non-universality of interactions among individuals. In this section we will choose this last point as our specific example, and analyze the consequences of relaxing the hypothesis that every player plays every other one. This hypothesis is needed to substitute the payoff earned by a player by what she would have obtained facing the average player of the population (an approach that has been traditionally used in physics under the name of mean-field approximation). However, interactions may not be universal after all, either because of spatial or temporal limitations. We will address both in what follows. The reader is referred to [15] for discussions of other perturbations. Spatial Perturbations One of the reasons why maybe not all individuals interact with all others is that they could not possibly meet. In biological terms this may occur because the population is very sparsely distributed and every individual meets only a few others within its living range, or else in a very numerous population where it is impossible in practice to meet all individuals. In social terms, an alternative view is the existence of a social network or network or contacts that prescribes who interacts with whom. This idea was first introduced in a famous paper by [16] on the evolutionary dynamics of the Prisoner’s Dilemma on a square lattice. In the Prisoner’s Dilemma two players simultaneously decide cooperate or to defect. Cooperation results in a benefit b to the recipient but incurs costs c to the donor (b > c > 0). Thus, mutual cooperation pays a net benefit of R D b  c whereas mutual defection results in P D 0. However, unilateral defection yields the highest payoff T D b and the cooperator has to bear the costs S D c. It immediately follows that it is best to defect regardless of the opponents decision. For this reason defection is the evolutionarily stable strategy even though all individuals would be better of if all would cooperate (mutual cooperation is better than mutual defection because R > P). Mutual defection is also the only Nash equilibrium of the game. All this translates into the following payoff matrix: C D

 C R T

D  S : P

(9)

Perturbation of Equilibria in the Mathematical Theory of Evolution

For this matrix to correspond to a Prisoner’s Dilemma game, the ordering of payoffs must be T > R > P > S. As we will see below, other orderings define different games. What Nowak and May did was to set the individuals on the nodes of a square lattice, where they played the game only with their nearest and next-nearest neighbors (Moore neighborhood). They ran simulations with the following dynamics: every individual played the game with her neighbors and collected the corresponding payoff, and afterwards she updated her strategy by imitating that of her most successful (in terms of payoff) neighbor. In their simulations, Nowak and May found that if they started with a population with a majority of cooperators, a large fraction of them remained cooperators instead of changing their behavior towards the ESS, namely, defection. The reason is that the structured interaction allowed cooperators to do well and avoid exploitation by defectors by grouping into clusters, inside which they interacted mostly with other cooperators, whereas defectors at the boundaries of those clusters, interacting mostly with other defectors, did not fare as well and therefore did not induce cooperators to defect. This result is partly due to the imitation dynamics, which, if postulated to rule the evolution of a population of individuals that interact with all others, does not lead to the replicator equation. As we mentioned in the preceding section, the update rule for the strategies is arbitrary and can be chosen at will (preferrably with some specific modelling in my mind). To reproduce the behavior of the replicator equation, a probabilistic rule has to be used [4], and with this rule the equilibrium is not changed and the population evolves to full defection. However, [16] opened the door to a number of more detailed studies that considered also different dynamical rules including the one corresponding to the replicator dynamics in which it was shown that the structure of a population definitely had a strong influence on the game equilibria. One such study, perhaps the most systematic to date, was carried out by [5], who compared the equilibrium frequencies of cooperators and defectors in populations with and without spatial structuring (square lattices), finding two important results: First, including spatial extension has indeed significant effects on the equilibrium frequencies of cooperators and defectors. In some parameter regions spatial extension promotes cooperative behavior while inhibiting it in others; and, second, differences in the initial frequencies of cooperators are readily leveled out and hardly affect the equilibrium frequencies except for T < 1, S < 0. This choice is not the Prisoner’s Dilemma anymore, it corresponds to the socalled Stag Hunt game [22], and in the replicator dynam-

ics is a bistable system where the initial frequencies determine the long term behavior, a feature that is generally preserved for the spatial setting. Of course, [5] also observed that the size of the neighborhood obviously affects the spreading speed of successful strategies. Interestingly, although the message seems to be that the strategy of cooperation is favored over the strategy of defection by the presence of a spatial structure, this is not always the case, and in games where the equilibrium population has a certain percentage of both types of strategists, the network of interactions makes the frequency of cooperators decrease [6]. Therefore, the effect of this perturbation is not all trivial and needs careful consideration. All the results discussed so far correspond to a square lattice as substrate to define the interaction pattern, but this is certainly a highly idealized setup that can hardly correspond to any real, natural system. Recent studies have shown that the results also depend on the type of graph or network used. Thus, [21] have shown that in more realistic, heterogeneous populations, modeled by random graphs of different types, the sustainability of cooperation (implying the departure of the equilibrium predicted by the replicator equation) is simpler to achieve than in homogeneous populations, a result which is valid irrespective of the dilemma or game adopted as a metaphor of cooperation. Therefore, heterogeneity constitutes a powerful mechanism for the emergence of cooperation (and consequently an important perturbation of the dynamics), since even for mildly heterogeneous populations it leads to sizeable effects in the evolution of cooperation. The overall enhancement of cooperation obtained on single-scale and scale-free graphs [1] may be understood as resulting from the interplay of two mechanisms: The existence of many long-range connections in random and small-world networks [25], which precludes the formation of compact clusters of cooperators, and the heterogeneity exhibited by these networks, which opens a new route for cooperation to emerge and contributes to enhance cooperation (which increases with heterogeneity), counteracting the previous effect. This result depends also on the intricate ties between individuals, even for the same class of graphs, features absent in the replicator dynamics. We have thus seen that removing the hypothesis of universal interaction is a strong perturbation to equilibrium as understood from the replicator dynamics. There have been many other studies following those we can possibly review here; a recent, very comprehensive summary can be found in [23]. It must be realized that, intermixed with the effect of the network of interactions, the different dynamical rules one can think of have also a relevant influ-

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ence on the equilibria. While we have not considered them as perturbations of the replicator dynamics here, because they do not behave as described by that equation even in the presence of universal interactions, it must be kept in mind that they do affect the equilibria and therefore they must be properly specified in any serious study of evolutionary game theory. Time Scales Let us now come back to the situation in which there is no spatial structure, and every agent can in principle play the game against every other one. Afterwards, reproduction proceeds according to the payoff earned during the game stage. As we have already said, for large populations, this amounts to saying that every player gains the payoff of the game averaged in the current distribution of strategies. In terms of time scales, such an evolution corresponds to a regime in which reproduction-selection events take place at a much slower rate than the interaction between agents. However, these two time scales need not be different in general and, in fact, for many specific applications they can arguably be of the same order [7]. To study different rates of selection we can consider the following new dynamics [19,20,24]. There is a population with N players. A pair of individuals is randomly selected for playing, earning each one an amount of fitness according to the rules of the game. This game act is repeated s times, choosing a new random pair of players in each occasion. Afterwards, selection takes place. Following [17], we have chosen Moran dynamics [13] as the most suitable to model selection in a finite population. This is necessary because the replicator equation is posed for continuous values of the populations and here we need to consider discrete values, i. e., individual by individual, in order to pinpoint the existing time scales. However, it can be shown [19] that the equilibria of Moran dynamics are the same as those of the replicator equation, and in fact, the whole evolution is the same except for a rescaling of time. Moran dynamics is defined as follows: One individual among the population of N players is chosen for reproduction proportionally to its fitness, and its offspring replaces a randomly chosen individual. As the fitness of all players is set to zero before the following round of s games, the overall result is that all players have been replaced by one descendant, but the player selected for reproduction has had a reproductive advantage of doubling its offspring a the expense of the randomly selected player. It is worth noting that the population size N is therefore constant along the evolution.

The parameter s controls the time scales of the model, i. e. reflects the relation between the rate of selection and the rate of interaction. For s  N selection is very fast and very few individuals interact between reproduction events. Higher values of s represent proportionally slower rates of selection. Thus, when s  N selection is very slow and population is effectively well-mixed and we recover the behavior predicted by the replicator equation. The most striking example of the influence of the selection is the so-called Harmony game, a trivial one that has henceforth never been studied, and that is determined by R > T > S > P. The only Nash equilibrium or ESS of this game is mutual cooperation, as it is obvious from the payoffs: The best option for both players is to cooperate, which yields the maximum payoff for each one. Let us denote by 0  n  N the number of cooperators present in the population, and look at the probability xn of ending up in state n D N (i. e., all players cooperate) when starting in state n < N. For s D 1 and s ! 1, an exact, analytical expression for xn can be obtained [19]. For arbitrary values of s, such a closed form cannot be found; however, it is possible to carry out a combinatorial analysis of the possible combinations of rounds and evaluate, numerically but exactly, xn . In Fig. 2a, we show that the rationally expected outcome of a population consisting entirely of cooperators is not achieved for small and moderate values of s, our selection rate parameter. For the smallest values, only when starting from a population largely formed by cooperators there is some chance to reach full cooperation; most of the times, defectors will eventually prevail and invade the whole population. This counterintuitive result may arise even for values of s comparable to the population size, by choosing suitable payoffs. Interestingly, the main result that defection is selected for small values of s does not depend on the population size N; only details such as the shape of the curves (cf. Fig. 2b) are modified by N. In the preceding paragraph we have chosen the Harmony game to discuss the effect of the rate of selection, but this effect is very general and appears in many other games. Consider the example of the already mentioned Stag-Hunt game [22], with payoffs R > T > P > S. This is the paradigmatic situation of game with two Nash equilibria in pure strategies, mutual cooperation and mutual defection, each one with its own basis of attraction in the replicator equation framework (in general, which of these equilibria is selected has been the subject of a long argument in the past, and rationales for both of them can be provided [22]). As Fig. 3 (left) shows, simulation results for finite s are largely different from the curve obtained for s ! 1: Indeed, we see that for s D 1, all agents become defectors

Perturbation of Equilibria in the Mathematical Theory of Evolution

Perturbation of Equilibria in the Mathematical Theory of Evolution, Figure 2 Probability of ending up with all cooperators starting from n cooperators, x n for different values of s. a For the smallest values of s, full cooperation is only reached if almost all agents are initially cooperators. Values of s of the order of 10 show a behavior much more favorable to cooperators. In this plot, the population size is N D 100. b Taking a population of N D 1000, we observe that the range of values of s for which defectors are selected does not depend on the population size, only the shape of the curves does. Parameter choices are: Number of games between reproduction events, s, as indicated in the plots; payoffs for the Harmony game, R D 11, S D 2, T D 10, P D 1. The dashed line corresponds to a probability to reach full cooperation equal to the initial fraction of cooperators and is shown for reference

Perturbation of Equilibria in the Mathematical Theory of Evolution, Figure 3 Left: Same as Fig. 2 for the Stag-Hunt game. The probability of ending up with all cooperators starting from n cooperators, x n , is very low when s is small, and as s increases it tends to a quasi-symmetric distribution around 1/2. Payoffs for the Stag-Hunt game, R D 6, S D 1, T D 5, P D 2. Right: Same as Fig. 2 for the Snowdrift game. The probability of ending up with all cooperators starting from n cooperators is almost independent of n except for very small or very large values. Small s values lead once again to selection of defectors, whereas cooperators prevail more often as s increases. Payoffs for the Snowdrift game, R D 1, S D 0:35, T D 1:65, P D 0. Other parameter choices are: Population, N D 100; number of games between reproduction events, s, as indicated in the plot

except for initial densities close to 1. Even for values of s as large as N evolution will more likely lead to a population entirely consisting of defectors. Yet another example of the importance of the selection rate is provided by the Snowdrift game, with payoffs T > R > S > P. Figure 3 (right) shows that for small val-

ues of s defectors are selected for almost any initial fraction of cooperators. When s increases, we observe an intermediate regime where both full cooperation and full defection have nonzero probability, which, interestingly, is almost independent of the initial population. And, for large enough s, full cooperation is almost always achieved.

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With these examples, it is clear that considering independent interaction and selection time scales may lead to highly non-trivial, counter-intuitive results. [19] showed that out of the 12 possible different games with two players, six were severely changed by the introduction of the game time scale, whereas the other six remained with the same equilibrium structure. Of course, the extent of the modifications of the replicator dynamics picture depends on the structure of the unperturbed phase space. Thus, rapid selection perturbations show up in changes of the asymptotically selected equilibria, i. e., of the asymptotically stable one, to changes of the basins of attraction of equilibria, or to suppression of long-lived metastable equilibria. As in the case of spatial perturbations, we are thus faced with a most relevant influence on the equilibria of the evolutionary game.

Future Directions As we have seen in this necessarily short excursion, the simplest mathematical models of evolution allow for a detailed, analytical study of their equilibria (which are supposed to represent stable states of populations) but, when leaving aside some of the hypothesis involved in the derivation of those simple models, the structure of equilibria may be seriously modified and highly counter-intuitive results may arise. We have not attempted to cover all possible perturbations but we believe we have provided evidence enough that their effect is certainly very relevant. When trying to bridge the gap between simple models and reality, other hypotheses will be broken, maybe more than one simultaneously, and subsequently the equilibria will be severely affected. In the future, we believe that this line of research will undergo very interesting developments, particularly in the framework of evolutionary game theory, as the fitness landscape picture seems to be rather well understood and, on the other hand, is felt to be a much too simple model. In the case of evolutionary games, while some of the results we have collected here are analytical, there are many others which are only numerical, in particular when perturbations depending on space (evolutionary game theory on graphs) are considered. A lot of research needs to be devoted in the next few years to understand this problem analytically, more so because there is now a “zoo” of results that are even hard to classify or interpret within a common basis. This will probably require a combined effort from different mathematical disciplines, ranging from discrete mathematics to dynamical systems through, of course, graph theory.

On the other hand, our examples have consisted of two-player, two-strategy, symmetric games, i. e., the simplest possible scenario. There are practically no results about games with more than two strategies or, even worse, with more than two players. In fact, even the classification of the phase portraits within the replicator equations for those situations is far from understood, the more so the higher the dimensionality of the problem. Much remains to be done in this direction. Asymmetrical games are a different story; for those, the replicator equation is not (6) anymore but rather one has to take into account that payoffs when playing as player 1 or player 2 are not the same, and the corresponding equation is more complicated. Again, this line of research is still in its infancy and awaiting for dedicated work. Finally, a very interesting direction is the application of the results to problems in social or biological contexts. The evolutionary game theory community has been relying strongly on the predictions from the replicator equation which we now see may not agree with reality or at least with what occurs when some of its hypotheses are not fulfilled. This has led to a number of conundrums, particularly prominent among those being the problem of the emergence of cooperation. A recent study [24] have shown that, in a scenario described by the so-called Ultimatum game, taking into consideration the possible separation of time scales leads to results compatible with the experimental observations on human subjects, observations that the replicator equation is not able to reproduce. We envisage that similar results will be ubiquitous when trying to match the predictions of the replicator equation with actual systems or problems. Understading the effect of perturbations in a comprehensive manner will then be the key to the fruitful development of the theory as a “natural” or “physical” one. Bibliography Primary Literature 1. Barabási A-L, Albert R (1999) Emergence of scaling in random networks. Science 286:509–512 2. Drossel B (2001) Biological evolution and statistical physics. Adv Phys 50:209–295 3. Fisher RA (1958) The Genetical Theory of Natural Selection, 2nd edn. Dover, New York 4. Gintis H (2000) Game Theory Evolving. Princeton University, Princeton 5. Hauert C (2002) Effects of space in 2  2 games. Int J Bifurc Chaos 12:1531–1548 6. Hauert C, Doebeli M (2004) Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428:643–646 7. Hendry AP, Kinnison MT (1999) The pace of modern life:

Perturbation of Equilibria in the Mathematical Theory of Evolution

8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19.

Measuring rates of contemporary microevolution. Evolution 53:1637–1653 Hofbauer J, Sigmund K (1998) Evolutionary Games and Population Dynamics. Cambridge University, Cambridge Hofbauer J, Sigmund K (2003) Evolutionary game dynamics. Bull Am Math Soc 40:479–519 Kimura M (1983) The Neutral Theory of Molecular Evolution. Cambridge University, Cambridge Maynard-Smith J (1982) Evolution and the Theory of Games. Cambridge University, Cambridge Maynard-Smith J, Price GR (1973) The logic of animal conflict. Nature 246:15–18 Moran PAP (1962) The Statistical Processes of Evolutionary Theory. Clarendon, Oxford Nash JF (1950) Equilibrium points in n-person games. Proc Natl Acad Sci USA 36:48–49 Nowak MA (2006) Evolutionary Dynamics: Exploring the Equations of Life. Harvard Univ Press, Harvard Nowak MA, May R (1992) Evolutionary games and spatial chaos. Nature 359:826–829 Nowak MA, Sasaki A, Taylor C, Fudenberg D (2004) Emergence of cooperation and evolutionary stability in finite populations. Nature 428:646–650 Page K, Nowak MA (2002) A unified evolutionary dynamics. J Theor Biol 219:93–98 Roca CP, Cuesta JA, Sánchez A (2006) Time scales in evolutionary dynamics. Phys Rev Lett 97: art. no. 158701

20. Roca CP, Cuesta JA, Sánchez A (2007) The importance of selection rate in the evolution of cooperation. Eur Phys J Special Topics 143:51–58 21. Santos FC, Pacheco JM, Lenaerts T (2006) Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc Natl Acad Sci USA 103:3490– 3494 22. Skyrms B (2003) The Stag Hunt and the Evolution of Social Structure. Cambridge University, Cambridge 23. Szabó G, Fáth G (2007) Evolutionary games on graphs. Phys Rep 446:97–216 24. Sánchez A, Cuesta JA (2005) Altruism may arise by individual selection. J Theor Biol 235:233–240 25. Watts DJ, Strogatz SH (1998) Collective dynamics of “Small World” Networks. Nature 393:440–442 26. Wright S (1932) The roles of mutation, inbreeding, crossbreeding and selection in evolution. Proc 6th Int Cong Genet 1:356– 366

Books and Reviews Nowak M, Sigmund K (2004) Evolutionary dynamics of biological games. Science 303:793–799 Taylor PD, Jonker L (1978) Evolutionarily stable strategies and game dynamics. J Math Biosci 40:145–156 von Neumann J, Morgenstern O (1944) Theory of Games and Economic Behavior. Princeton University Press, Princeton

1275

1276

Perturbation of Systems with Nilpotent Real Part

in ˝. By the Stirling formula one may replace ˛! by j˛j! , j˛jj˛j or  ( j˛j), where  (z) stands for the Euler Gamma function cf. the book of Rodino [41] for more details on the Gevrey spaces. One associates also Gevrey index to formal power series, namely, given a (formal) power series X f (x) D f˛ x ˛

Perturbation of Systems with Nilpotent Real Part TODOR GRAMCHEV Dipartimento di Matematica e Informatica, Università di Cagliari, Cagliari, Italy Article Outline

˛

Glossary Definition of the Subject Introduction Complex and Real Jordan Canonical Forms Nilpotent Perturbation and Formal Normal Forms of Vector Fields and Maps Near a Fixed Point Loss of Gevrey Regularity in Siegel Domains in the Presence of Jordan Blocks First-Order Singular Partial Differential Equations Normal Forms for Real Commuting Vector Fields with Linear Parts Admitting Nontrivial Jordan Blocks Analytic Maps near a Fixed Point in the Presence of Jordan Blocks Weakly Hyperbolic Systems and Nilpotent Perturbations Bibliography Glossary Perturbation Typically, one starts with an “initial” system S0 , which is usually simple and/or well understood. We perturb the system by adding a (small) perturbation R so that the new object becomes S0 C R. In our context the typical examples for S0 will be systems of linear ordinary differential equations with constant coefficients in Rn or the associated linear vector fields. Nilpotent linear transformation Let A: Kn 7! Kn be a linear map, where K D R or K D C. We call A nilpotent if there exists a positive integer r such that the rth iteration Ar become the zero map, in short Ar D 0. Gevrey spaces Let ˝ be an open domain in Rn and let   1. The Gevrey space G  (˝) stands for the set of all functions f 2 C 1 (˝) such that for every compact subset K

˝ one can find C D C K; f > 0 such that sup j@˛x f (x)j  C j˛jC1 ˛!

(1)

x2K

n , ˛! D ˛ ! : : : ˛ !, for all ˛ D (˛1 ; : : : ; ˛n ) 2 ZC 1 n j˛j :D ˛1 C : : : C ˛n . If  D 1 we recapture the space of real analytic functions in ˝ while the scale G  (˝),  > 1, serves as an intermediate space between the real analytic functions and the set of all C 1 functions

this is in the formal Gevrey space G f (Kn ) if there exist C > 0 and R > 0 such that j f˛ j  C j˛jC1 j˛j!1

(2)

n . for all ˛ 2 ZC In fact, one can find in the literature another definition of the formal Gevrey spaces G f of index , namely replacing   1 by  (see e. g. Ramis [40]).

Definition of the Subject The main goal of this article is to dwell upon the influence of the presence (explicit and/or hidden) of nontrivial real nilpotent perturbations appearing in problems in Dynamical Systems, Partial Differential Equations and Mathematical Physics. Under the term nilpotent perturbation we will mean, broadly speaking, a classical linear algebra type setting: we start with an object (vector field or map near a fixed point, first-order singular partial differential equations, system of evolution partial differential equations) whose “linear part” A is semisimple (diagonalizable) and we add a (small) nilpotent part N. The problems of interest might be summarized as follows: are the “relevant properties” (in suitable functional framework) of the initial “object” stable under the perturbation N. If not, to classify, if possible, the novel features of the perturbed systems. Broadly speaking, the cases when the instabilities occur are rare, they form some kind of exceptional sets. However, they appear in important problems (both in mathematical and physical contests) when degeneracies (bifurcations) occur. Introduction We will focus our attention on topics where the presence of nontrivial Jordan blocks in the linear parts changes the properties of the original systems (i. e., instabilities occurs unless additional restrictions are imposed): (i)

Convergence/divergence issues for the normal form theory of vector fields and maps near a singular (fixed) point in the framework of spaces of analytic functions and Gevrey classes.

Perturbation of Systems with Nilpotent Real Part

(ii) (Non)solvability for singular partial differential equations near a singular point. (iii) Cauchy problems for hyperbolic systems of partial differential equations with multiple characteristics. Some basic features of the normal form theory for vector fields near a point will be recalled with an emphasis on the difficulties appearing in the presence of nontrivial Jordan blocks in the classification and computational aspects of the normal forms. For more details and various aspects of perturbation theory in Dynamics we refer to other articles in the Perturbation Theory Section of this Encyclopedia: cf. Bambusi  Perturbation Theory for PDEs, Gaeta  Non-linear Dynamics, Symmetry and Perturbation Theory in, Gallavotti  Perturbation Theory, Broer  Normal Forms in Perturbation Theory, Broer and Hanssmann  Hamiltonian Perturbation Theory (and Transition to Chaos), Teixeira  Perturbation Theory for Non-smooth Systems, Verhulst  Perturbation Analysis of Parametric Resonance, Walcher  Perturbative Expansions, Convergence of. We start by outlining some motivating examples. Consider the nilpotent planar linear system of ordinary differential equations  x˙ D

0 0

" 0

 x;

x(0) D x 0 2 R2 ;

where " 2 R. The explicit solution is given by x1 (t) D x10 C x20 "t, x2 (t) D x20 and clearly the equilibrium (0; 0) is not stable if " ¤ 0. On the other hand, if U(x1 ) is a smooth real valued analytic function, satisfying U(0) D 0, U(x1 ) > 0 for x1 ¤ 0, then it is well known that for " > 0 the Newton equation  x˙ D

0 0

" 0



 x

0 U 0 (x1 )



is stable at (0; 0). Another example, which enters in the framework considered here, is given by a conservative (i. e. Hamiltonian) dynamical system perturbed by a friction term. Next, we illustrate the influence of the real nilpotent perturbations in the realm of the normal form theory and in the general theory of singular partial differential equations. Consider the linear PDE (x1 C "x2 )@x 1 u C x2 @x 2 u 

p 2x3 @x 3 u  u D f (x) ;

where " 2 R, and f stands for a convergent power series in a neighborhood of the origin of R3 , at least quadratic at x D 0. Such equations appear in the so-called homological

equations for the reduction to Poincaré–Dulac linear normal form of systems of analytic ODEs having an equilibrium at the origin. It turns out that in the semisimple case " D 0 we can solve the equation in the space of convergent power series while for " ¤ 0 (i. e., when a nontrivial Jordan block appears) the equation is solvable only formally, namely, divergent solutions appear. One is led in a natural way to study the Gevrey index of divergent solutions. Complex and Real Jordan Canonical Forms We start by revisiting the notion of complex and real canonical Jordan forms. Recall that each linear map is decomposed uniquely into the sum of a semisimple map and a nilpotent one. We state a classical result in linear algebra Lemma 1 Let A be an n  n matrix with real or complex entries. Then it is uniquely decomposed as A D As C Anil ;

(3)

where As is semisimple (i. e., diagonalizable over C) and Anil is nilpotent, i. e., Arnil D 0nn for some positive integer r. Here 0nn stands for the zero n  n matrix. Denote by spec(A) D f1 ; : : : ; n g C the set of all eigenvalues of A counted with their multiplicity. The complex Jordan canonical form (JCF) is defined by the following assertion cf. Gantmacher [21]: Theorem 2 Let A be an n  n matrix over K, K D C or K D R. Then there exist positive integers m; k1 ; : : : ; k m , m  n, k1 C : : : C k m D n and a matrix S 2 GL(n; C) such that S 1 AS D J A :D 0 A 1 I k 1 C N k 1 0 k2 k1 B B :: @ : 0 k m k1

0 k1 k2 2A I k2 C N k2 :: :

0 k m k2

::: ::: :: : :::

1

0 k1 k p 0 k2 k3 C C :: A : AI m km C Nkm

(4) where 1A ; : : : ; Am are the eigenvalues of A, which need not all be distinct, and N r , when r  2, stands for the square r  r matrix 0 B B B B B @

0 0 :: : 0 0

1 0 :: : 0 0

0 1 :: : 0 0

::: ::: :: : ::: :::

0 0 :: : 0 0

0 0 :: : 1 0

1 C C C C C A

(5)

1277

1278

Perturbation of Systems with Nilpotent Real Part

with the convention N1 D 0. Moreover, if Anil ¤ 0, i. e., k j  2 for at least one j 2 f1; : : : ; mg, then for every " 2 C n 0 the matrix S(") 2 GL(n; C) D diagfS1 ("); : : : ; S m (")g, S j (") D 1 if k j D 1, S j (") D diagf1; : : : ; " k j 1 g, provided k j  2, j D 1; : : : ; m, satisfies the identity S 1 (")AS(") D J A (") D 0 A

satisfying k1 C    C k p C 2(`1 C    C `q ) D n, and S 2 SL(n; R), such that   J AR 0 k2` S 1 AS D (7) 02`k J AC with

1

1 I k1 C "N k1 0 k1 k2 ::: 0 k1 k p 0 k2 k1 2A I k2 C "N k2 : : : 0 k2 k3 C B C: B :: :: :: :: A @ : : : : A 0 k m k2 : : :  m I k m C "N k m 0 k m k1

(6)

0

J AR D

1A I k1 C "1A N k1 0 k2 k1 B B :: @ : 0 k p k1

0 k1 k2 C "2A N k2 :: : 0 k p k2

2A I k2

::: ::: :: : :::

1

0 k1 k p 0 k2 k3 C C; :: A : A A  p Ik p C "p Nk p

(8) One may define in an obvious way another JCF (lower triangular) replacing J by its transposed J T . Note that Ak , k D 1; : : : ; m, are not necessarily distinct. If  is an eigenvalue of A with algebraic multiplicity d, i. e., it is a zero of multiplicity d of the characteristic polynomial PA () D jA  Ij ; one can have different Jordan block structures. For example, a 33 matrix with a triple eigenvalue  can be reduced to one of the three JCF:  the matrix I3 , i. e.,  does not admit nontrivial Jordan blocks; 0 1  1 0  @ 0  1 A; 0 0  0 1  0 0  @ 0  1 A 0 0  For higher dimensions the description of all JCF becomes more involved, cf. [3,21]. Remark 3 We can choose j"j arbitrarily small but never zero if the nilpotent part is nonzero. The columns of the conjugating matrix S are formed by eigenvectors and generalized eigenvectors. The smallness of j"j leads in a natural way to view the presence of nontrivial nilpotent parts as a perturbation. Next, if A is a real matrix, using the real and the imaginary parts of the eigenvectors for the complex eigenvalues, one introduces the real JCF. Theorem 4 Let A 2 M nn (R). Then there exist nonnegative integers p; q, 1  p C 2q  n, p (if p  1) positive integers k1 ; : : : ; k p , q (if q  1) positive integers `1 ; : : : ; `q

for some  j 2 R, j D 1; : : : ; p, and 0 02`1 2`2 : : : D1A B 0 D2A ::: B 2`2 2`1 C B JA D B :: :: :: @ : : : 02`q 2`1 02`q 2`2 : : :

02`1 2`q 02`2 2`q :: : D qA

1 C C C; C A

(9)

A being 2` 2` matrices, written as ` ` block with D     matrices of 2 2 matrices of the following 2 2 block matrix form:

A D D



˛ ˇ

ˇ ˛



 I` (2)C

A A ı

A ı

A

 N (2); (10)

for some ˛ ; ˇ 2 R, ˇ ¤ 0, with ˛ k ˙ ˇ k i 2 spec(A). Here, I k (2) D diagfI2; : : : ; I2 g denotes the 2k  2k matrix written as a k  k matrix with 2  2 block matrices while N k (2) stands for the following 2k  2k nilpotent matrix written as k  k matrix with 2  2 block matrices as entries: 1 0 I2 022 : : : 022 022 B 022 022 I2 : : : 022 C C B B :: :: C :: :: :: N k (2) D B : (11) : C : : : C B A @ 022 022 022 : : : I2 022 022 022 : : : 022 and 0rs stands for the zero r  s matrix. The smallness of the parameter " and the explicit form of the conjugating matrices S(") are instrumental in showing some useful estimates for the study of the dynamics of the linear maps A which are not semisimple. Let r(A) :D maxfjj :  2 spec(A)g (the spectral radius of A). Then the following estimate, useful in different branches of Dynamical Systems, holds (cf. [27])

Perturbation of Systems with Nilpotent Real Part

Lemma 5 For every  > 0 there exists a norm in Rn such that kAk  r(A) C . Moreover, if f 2 spec(A) : jj D r(A)

do not admit Jordan blocksg

one has kAk D r(A) for some norm. In particular, if A is semisimple, the last conclusion holds. Next, consider the linear autonomous systems of ordinary differential equations x˙ D Ax :

(12)

where A 2 M n (R). We recall that if x(0) D  2 Rn , then the unique solution is defined by x(t) D exp(tA) :D

1 k k X t A ; k!

(13)

kD0

e. g., cf. Arnold [3], Coddington and Levinson [14]. We exhibit an assertion where the structure of the real nilpotent perturbation Anil in the linear part plays a crucial role for the stability for t  0 of the solutions of the linear system (12). We recall that the origin is stable if for every " > 0 one can find ı > 0 such that kk < ı implies k exp(tA)k < " for t  0. Proposition 6 The zero solution of (12) is stable for t  0 if and only if the following two conditions hold: i) spec(A) f 2 C; Re   0g; ii) if  2 spec(A) and Re  D 0 then  does not admit a nontrivial Jordan block. On the other hand, the origin is asymptotically stable for t ! C1 if and only if spec(A) f 2 C; Re  < 0g : The proof is straightforward in view of the explicit formula for the exponent of the matrix exp(tA) by means of the Jordan canonical form. In particular, we get

analysis of dynamical systems near an equilibrium (singular) point x0 for autonomous systems of ODE x˙ D X(x)

(14)

or the associated vector field e X(x) D hX(x); @x i D

n X

X j (x)@x j

(15)

jD1

(see [3,7,9,12,19,20,43], and the references therein). Without loss of generality (after a translation) one may assume that x0 coincides with the origin and write X(x) D Ax C R(x) ;

A D r X(0) ;   R(x) D O jxj2 ; jxj ! 0 :

(16)

Denote by spec(A) D f1 ; : : : ; n g the spectrum of A. The basic idea, going back to Poincaré, is to find a (formal) change of the coordinates defined as a (at least) quadratic perturbation of the identity x D u(y) D y C v(y) ;

(17)

which transforms e X into a new vector field e Y(y) which has a “simpler” form (Poincaré–Dulac normal form). The original idea of Poincaré regards the possibility to linearize e X, i. e., e Y(y) D Ay. Straightforward calculations show that the linearization of e X means that v(y) satisfies (at least formally) a system of first order semilinear partial differential equations, called the system of the homological (difference) equations L A v(y) D R(y C v(y))

(18)

In fact, it is the system above where the first substantial technical difficulty appears if the nilpotent part Anil is not zero, i. e., the matrix A is not diagonalizable. Indeed, if A is diagonalizable and we choose (after a linear change of the variables in C n ) A D diagf1 ; : : : ; n g, then the system (18) is written as n X

 k @ y k v j (y)   j v j D R j (y C v(y)) ;

j D 1; : : : ; n

kD1

Corollary 7 Let A be a real matrix such that all eigenvalues lie on the pure imaginary axis. Then the zero solution of (12) is stable if and only if A is semisimple (i. e., Anil D 0). Nilpotent Perturbation and Formal Normal Forms of Vector Fields and Maps Near a Fixed Point Normal form theory (originating back to Poincaré’s thesis) has proven to be one of the most useful tools for the local

(19) We recall that e X (or specA) is said to be in the Poincaré domain (respectively, Siegel domain) if the convex hull of f1 ; : : : ; n g in the complex plane does not contain (respectively, contains) 0. Further, spec(A) is called nonresonant iff h; ˛i   j ¤ 0 ;

n j D 1; : : : ; n; ˛ 2 ZC (2) ;

(20)

1279

1280

Perturbation of Systems with Nilpotent Real Part

Pn n where h; ˛i D jD1  j ˛ j , ZC (2) :D f˛ D (˛1 ; : : : ; n ˛ n ) 2 ZC : j˛j :D ˛1 C    C ˛n  2g. By the Poincaré–Dulac theorem, if spec(A) is in the Poincaré domain, then there are at most finitely many resonances and there exists a convergent transformation (in some neighborhood of the singular point) which reduces e X to a (finitely resonant) normal form. Theorem 8 Let the linear part A of the complex (respectively, real) field above be nonresonant. Then the vector field is formally linearizable by a complex (respectively, real) transformation.

The description and the computation of nilpotent normal forms, based on an algebraic approach and the systematic use of the theory of invariance, with particular emphasis on the equivalence between different normal forms, has been developed in a body of papers (see [16,36,37], and the references therein). Finally, we mention that nilpotent perturbations appear in the classification and Casimir invariants of Lie– Poisson brackets that are formed by Lie algebra extensions for physical systems admitting Hamiltonian structure to such brackets (e. g. cf. Thiffeault and Morison [49]).

Denote by n ResAj D f˛ 2 ZC (2) :  j D h; ˛ig ;

j D 1; : : : ; n :

Clearly the nonresonance hypothesis is equivalent to ResAj D ; for j D 1; : : : ; n. Additional technical complications appear if we consider real vector fields. Theorem 9 Every formal vector field with a singular point at the origin is transformed by a formal complex change of variables to a field of the form hJ A z; @z i C

n X X

q j;˛ z˛ @z j

(21)

jD1 ˛2Res A j

The coefficients q j;˛ may be complex even though the original vector field is real. We note that if the linear part is nilpotent, i. e., spec(A) D f0g, then the theorem above gives no simplification. In that case Belitskii [7] has classified completely the formal normal forms. Let A be a nilpotent matrix (i. e. the semisimple part As is the zero matrix). The Poincaré–Dulac NF does not provide any information. The following theorem is due to Belitskii [7] (see also Arnold and Ilyashenko [4]). Theorem 10 Let A be a nilpotent matrix and let X(x) be a formal vector field with a linear part given by Ax. Then X is transformed by a formal complex change of variables x 7! z to a field of the form hJ A z; @z i C hB(z); @z i

(22)

with B(z) being at least quadratic near the origin, where the nonlinear vector field hB(z); @z i commutes with hJ A z; @z i. (Here  stands for the Hermitian conjugation). We point out that another important problem is the computation of the normal form. Here the presence of the Jordan blocks leads to substantial difficulties.

Loss of Gevrey Regularity in Siegel Domains in the Presence of Jordan Blocks The convergence question in the Siegel domain is more difficult since small divisors appear. In a fundamental paper Bruno [9] succeeded in proving a deep result of the following type: a formal normal form is convergent under an (optimal) arithmetic condition on the small divisors jh; ˛i   j j1 and a condition on the formal normal form, called the A condition. It should be pointed out that while in the original paper [9] the condition A allows in some cases nontrivial Jordan blocks of the linear part, in the subsequent works the linear part A is required to be semisimple (diagonalizable). We recall that for the finitely smooth and C 1 local normal forms the presence of the real nilpotent part does not influence the convergence: e. g., in the Sternberg theorem (cf. [45]) the small divisors and the presence of Jordan blocks play no role (cf. [7], where many other references can be found). Little is known about convergence–divergence problems in the analytic category if spec(A) is in the Siegel domain and A is not semisimple. Even in the cases where the presence of the nilpotent part does not influence the assertions, the proofs become more involved. Here we outline various aspects of normal forms when nilpotent perturbations are present. We stress especially the real case. The combined influence of the Jordan blocks and the small divisors on the convergence of the formal linearizing transformation for analytic vector fields in R3 has been studied in Gramchev [23] (see also [53] for some examples). Let n D 3 and consider a nonsemisimple linear part A 2 GL(3; K) satisfying spec(A) D f; ; g,  ¤ . This means that we can reduce A to the " Jordan normal form 0

 A" D @ 0 0

0  0

1 0 " A; 

" ¤0:

(23)

Perturbation of Systems with Nilpotent Real Part

We recall that one can make j"j arbitrarily small by linear change of the variables (but never 0). Then spec(A) is nonresonant and in the Siegel domain iff

:D

 < 0; 

62 Q :

(24)

The typical example is a real vector field with a linear part given by 0 1

0 0 (25) A0" D @ 0 1 " A ; " ¤ 0 : 0 0 1 One observes that such real vector fields are nonresonant and hyperbolic and therefore, by the Chen theorem, linearizable by smooth transformations. We recall that an irrational number is said to be diophantine of order  > 0, and write 2 D(), if there exist C > 0 such that C min jq C pj   ; q p2Z

q2N:

(26)

By a classical result in number theory D() ¤ ; iff   1. An irrational number is called Liouville iff it is not diophantine. Given an irrational number we set 0 D 0 ( ) D inff > 0 : such that 2 D()g ; before

(27)

with the convention 0 D C1 if is a Liouville number. Theorem 11 Let spec(A) be in the Siegel domain. Then LA is not solvable in the space of convergent power series, namely, we can find RHS f which is analytic but the unique formal power series solution is divergent. Moreover, we can always find a convergent RHS f such that the unique formal solution u satisfies [ G f (K3 ) (28) u 62 1<2C0

In particular, if 0 D C1, then u 62 G  ;

 1:

(29)

In case a diophantine condition is satisfied, estimates on the Gevrey character of the divergent power series are derived. The nonsolvability of the LHE in the spaces of the convergent power series does note exclude a priori that the vector field is linearizable by analytic transformations.

However, using a fundamental result of Pérez Marco [39] one can state the following assertion for hyperbolic vector fields which are not in the Poincaré domain. Theorem 12 Let A be a real n  n matrix which is hyperbolic, not semisimple and is not in the Poincaré domain. Let R(x) D (R1 (x); : : : ; R(x)) be a real valued analytic function near the origin, at least quadratic near x D 0, i. e. R(0) D 0, rR(0) D 0. Then there exists 0 > 0 such that for almost all  2]0; 0 ] in the sense of the Lebesgue measure, the vector field defined by X" (x) D Ax C R(x)

(30)

is not linearizable by convergent transformations. Remark 13 In fact, the result can be made more precise, using the notion of capacity instead of the Lebesgue measure and allowing polynomial dependence on  (cf. [39]). Another direction where the presence of real nilpotent perturbations in the linear parts presents challenging obstacles is the study of the dynamics in a neighborhood of a fixed point carried out via a normalization up to finite order and the issue of optimal truncation X u˛ x ˛ j˛jN opt

of normal form transformations for analytic vector fields. In an impressive work Iooss and Lombardi [33] demonstrate in particular that for large classes of real analytic vector fields with semisimple linear parts one can truncate the formal normal form transformation for jxj  ı, ı0 > 0 arbitrarily small, in such a way that the reminder in the normal form R(x) satisfies the following estimates  w (31) sup jR N opt (x)j  Mı 2 exp  b ı jxjı where b D 1 C . Here either  > n  1, in which case  ¤ 0 is the diophantine index of the small divisor type estimates modulo the resonance set, namely for some > 0 the following estimates hold for the eigenvalues 1 ; : : : ; n of A jh; ˛i   j j 

; j˛j

if h; ˛i   j ¤ 0

(32)

n (2), or  D 0 and  satisfies the for j D 1; : : : ; n, ˛ 2 ZC nonresonant type estimates

jh; ˛i   j j  ;

if h; ˛i   j ¤ 0

n (2). for j D 1; : : : ; n, ˛ 2 ZC

(33)

1281

1282

Perturbation of Systems with Nilpotent Real Part

The question of the validity of such results for analytic vector fields with nonsemisimple linearization is far more intricate. Iooss and Lombardi [33] give two examples of non-semisimple linearizations (nilpotent perturbations of size 2 and 3) for which the result is still true. The question remains totally open for other non-semisimple linearizations. First-Order Singular Partial Differential Equations This section deals with the study of formal power series solutions to singular linear first-order partial differential equations with analytic coefficients of the form d X

a j (x)@x j u(x) C b(x)u(x) D f (x) ;

(34)

jD1

where a j (x) (with j D 1; : : : ; d), b(x) and the right-hand side f (x) are analytic in a neighborhood of the origin of Cd , and a j (0) D 0 for j D 1; : : : ; d. In an interesting paper Hibino [29] allows a Jacobian matrix r a(0) without a Poincaré condition. More precisely, the Jacobi matrix at the origin can be reduced via a conjugation with a nonsingular matrix S to the following form: for some nonnegative integers, m, d, p, and d positive integers k j  2 (with j D 1; : : : ; d), if d  1, such that m C r1 C : : : rd C p D n, one can write as follows: 0 S

1

B B B r a(0)S D B B @

1

A Nr1

C C C C C A

:: : Nr d

(35)

0 pp

rD1

mD0

Theorem 14 Under the conditions (35)–(37) for every RHS f (x) D

X

f˛ x ˛

n ˛2ZC

which converges in a neighborhood of the origin the equation (34) has a unique formal solution which belongs to the formal Gevrey space G f (Kn ) with 8 ˆ <20 D 2 ˆ : 1

if

d1

if

d D 0; p  1 :

if

dDpD0

(38)

The Gevrey index is determined by a Newton polyhedron, a generalization of the notion of the Newton polygon for singular ordinary differential equations. The arguments of the proof rely on subtle Gevrey combinatorial estimates. Extensions for first-order quasilinear singular partial differential equations are done by Hibino [31]. It should be pointed out that the loss of Gevrey regularity comes not only from the nilpotent Jordan blocks, but from the nonlinear (at least quadratic) terms in a(x) as well. Indeed, for the equation (1  x12 @x 1  x22 @x 2 )u(x) D x1 C x2

where N r , r  2, stands for the r  r nilpotent Jordan block (5) and A is an m  m satisfying the Poincaré condition (the convex hull in C of the eigenvalues 1 ; : : : ; m of A does not contain the origin, provided m  1). One observes that the hypothesis d  1 implies that r a(0) does not satisfy the Poincaré condition. The fundamental hypothesis on the zero order term b(x) reads as follows ˇ m ˇ ˇX ˇ ˇ ˇ m ˛r C b(0)ˇ ¤ 0; ˛ 2 ZC if m  1 (36) ˇ ˇ ˇ jb(0)j ¤ 0 if

they outline some interesting features in the presence of nontrivial Jordan blocks even in the lack of small divisors phenomena. Set 0 D max`D1;:::;d r` if d  1. Then the main result in [29] reads as follows

(37)

In particular, by (37) one gets that necessarily b(0) ¤ 0. It should be stressed that the classes of singular partial differential equations above do not capture the systems of homological equations when small divisors occur, but

the unique formal solution is defined as follows u(x) D

1 X

j

j

( j  1)!(x1 C x2 )

jD0

cf. [29]. For further investigations on the loss of Gevrey regularity for solutions of singular ordinary differential equations of irregular type see Gramchev and Yoshino [26]. For characterizations of the Borel summability of a divergent formal power series solution of classes of firstorder linear singular partial differential equation of nilpotent type see [30] and the references therein. Solvability in classical Sobolev spaces and Gevrey spaces for linear systems of singular partial differential equations with real coefficients in Rn with nontrivial real Jordan blocks are derived by Gramchev and Tolis [24].

Perturbation of Systems with Nilpotent Real Part

Normal Forms for Real Commuting Vector Fields with Linear Parts Admitting Nontrivial Jordan Blocks The main goal of this section is to exhibit the influence of nontrivial linear nilpotent parts for the simultaneous reduction to convergent normal forms of commuting vector fields with a common fixed point. Consider a family of commuting n  n matrices A1 ; : : : ; A d . If all matrices are semisimple, then they can be simultaneously diagonalized over C or put into a blockdiagonal form over R, if A1 ; : : : ; A d are real, by a linear transformation S (e. g., [12,21]). This property plays a crucial role in the study of the normal forms of commuting vector fields with semisimple linear parts (cf. [46,47]). However, if the matrices have nontrivial nilpotent parts, then it is not possible to transform simultaneously A1 ; : : : ; A d in Jordan canonical forms. This is a consequence of the characterization of the centralizer of a matrix in a JCF (see [3,21]). Apparently the first examples of simultaneous reduction to (formal) normal forms of commuting vector fields with nonsemisimple linear parts are due to Cicogna and Gaeta [12]) for two commuting vector fields using the set up of the symmetries. More precisely, first, the definition of Semisimple Joint Normal Form (SJNF) is introduced: let X D h f (x); @x i ;

Y D hg(x); @x i ;

\

Ker(Bs ) ; and

g 2 Ker(As )

\

Ker(Bs ) :

Clearly in Y-JNF the role of f and g is reversed. The main assertion is: Theorem 15 Let [X; Y] D 0. Then X and Y can be reduced to a SJNF by means of a formal change of the variables. They can also be reduced (formally) to X-Joint or Y-Joint NF. Example 16 0

 AD@ 0 0

0  0

1

0 " A 

0

p BD@ r 0

Next, we outline recent results on simultaneous reductions to normal forms of commuting analytic vector fields admitting nontrivial Jordan blocks in their linear parts following Yoshino and Gramchev [54]. Let K be K D C or K D R, and B D 1, B D ! or B D k for some k > 0. Let GBn denote a d-dimensional Lie algebra of germs at 0 2 Kn of CB vector fields vanishing at 0. Let be a germ of singular infinitesimal Kd -actions of class C B (d  2)

: Kd ! GBn :

(40)

Denote by Act B (Kd : Kn ) the set of germs of singular infinitesimal Kd -actions of class CB at 0 2 Kn . By choosing a basis e1 ; : : : ; e d 2 Kn , the infinitesimal action can be identified with a d-tuple of germs at 0 of commuting vector fields X j D (e j ), j D 1; : : : ; d (cf. [18,46,47,55]). We can define, in view of the commutativity relation, the action e

: Kd  Kn ! Kn ;

A D r f (0), B D r g(0) and f (x) D Ax C F(x), g(x) D Bx C G(x). Then X and Y are said to be in SJNF if both F T and G belong to Ker(As ) Ker(Bs ). Here As stands for the homological operator associated to the semisimple part As of A. Next, X; Y are in X-Joint NF iff F 2 Ker(AC )

where ; "; p; q; r; s; t 2 C n0. Then A and B commute but it is impossible, in general, to reduce B to the Jordan block structure of A, preserving that of A. We point out that the problem of finding conditions guaranteeing simultaneous reduction of commuting matrices to Jordan normal forms is related to questions in the Lie group theory (e. g., see Chap. IV of [12] and the references therein).

0 s 0

d

s D (s1 ; : : : ; s d ) ; for all permutations  D (1 ; : : : ; d ) of f1; : : : ; dg, where j X t denotes the flow of X j . We denote by lin the linear action formed by the linear parts of the vector fields defining . A natural question is to investigate necessary and sufficient conditions for the linearization of (allowing nilpotent perturbations in the linear parts) namely, whether there exists a CB diffeomorphism g preserving 0 such that g conjugates e

and f lin e

(s; g(z)) D g( f lin (s; z))

(39)

(s; z) 2 Kd  Kn :

(42)

It is well known (e. g., cf. [35]) that there exists a positive integer m  n such that Kn is decomposed into a direct sum of m linear subspaces invariant under all A` D r X` (0) (` D 1; : : : ; d): Kn D I s1 C    C I s m ;

1

q t A; s

e

(s; z) D X s11 ı    ı X sdd (z) D X s1 ı    X sd (z) ; (41) 1

dim I s j D s j ;

j D 1; : : : ; m ;

s1 C    C s m D n : (43)

1283

1284

Perturbation of Systems with Nilpotent Real Part

The matrices A1 ; : : : ; Ad can be simultaneously brought into upper triangular form, and we write again A` for the matrices 0

A`1

::: ::: :: : :::

0s 1 s 2 A`2 :: : 0s m s 2

B 0s 2 s 1 B A` D B :: @ : 0s m s 1

0s 1 s m 0s 2 s m :: : A`m

1

If K D C, the matrix 0

`j

A`j;12

B 0 B A`j D B B :: @ : 0

`j :: : 0

(44)

B B B ` Aj D B B @

 [m ] D 9 8 m m = <X X t j Ej 2 Kd ; t j  0; j D 1; : : : ; m; tj ¤ 0 : ; : jD1

jD1

(52)

A`j;1s j

:::

A`j;2s j :: : `j

::: :: : :::

1 C C C; C A

R2 (`j ; `j )

A12 `; j

:::

0 :: : 0

R2 (`j ; `j )

::: :: : :::

(45)

:: : 0

1e sj A` j 2e sj A` j

:: :

R2 (`j ; `j )

` D 1; : : : ; d ;

1 C C C C; C A (46)

Definition 17 A Kd -action is called a Poincaré morphism if there exists a basis m Km such that  [m ] is a proper cone in Km , namely it does not contain a straight real line. If the condition is not satisfied, then, we say that the Kd -action is in a Siegel domain. Note that the definition is invariant under the choice of the basis m . Remark 18 The geometric definition above is equivalent to the notion of Poincaré morphism given by Stolovitch (Definition 6.2.1 in [46]). Next, we need to introduce the notion of simultaneous resonance. For ˛ D (˛1 ; : : : ; ˛m ) 2 Km , ˇ D (ˇ1 ; : : : ; Pm ˇ m ) 2 Km , we set h˛; ˇi D D1 ˛ ˇ . For a positive m m (k) D f˛ 2 ZC ; j˛j  kg. Put integer k we define ZC ! j (˛) D

d X

jhe  ; ˛i  j j ;

j D 1; : : : ; m ;

 R2 (; ) :D

 

 

 ;

;  2 R ;

(47)

and Ars are appropriate real matrices. `j Following the decomposition (45) (respectively, (46)) we define ej by k ) 2 Km ; ek D t (1k ; : : : ; m

k D 1; : : : ; d :

(48)

Then we assume are linearly independent in Km :

(49)

One can easily see that (49) is invariantly defined. By (44) we define Ej D t (1j ;    ; dj ) 2 Kd ;

j D 1; : : : ; m ;

(50)

(53)

D1

!(˛) D minf!1 (˛); : : : ; !m (˛)g :

where

fd e1 ;    ; 

(51)

is given by

with `j ; A`j; 2 C, ` D 1; : : : ; d, j D 1; : : : ; m. On the other hand, if K D R, then we have, for every 1  j  m two possibilities: firstly, all A`j (` D 1; : : : ; d) are given by (45) with `j 2 R. Secondly, s j D 2e s j is even and A`j is s j square block matrix given by ae sj  e 0

m :D fE1 ; : : : ; Em g : We define the cone  [m ] by

C C C; A

` D 1; : : : ; d : A`j

and

(54)

Definition 19 The cone m is called simultaneously nonresonant (or, in short is simultaneously nonresonant), if !(˛) ¤ 0 ;

m (2) : 8˛ 2 ZC

(55)

If (55) does not hold, then m is said to be simultaneously resonant. Clearly, the simultaneously nonresonant condition (55) is invariant under a change of the basis m . The next assertion provides a geometrically invariant condition guaranteeing that the simultaneous reduction to normal form does not depend on (small) nilpotent perturbation of the linear part. Theorem 20 Let be a Poincaré morphism. Then is conjugated to a polynomial action by a convergent change of variables.

Perturbation of Systems with Nilpotent Real Part

Remark 21 As a corollary of Theorem 20 for vector fields having linear parts with nontrivial Jordan blocks one obtains generalizations of results for the existence of convergent normal forms for analytic vector fields admitting symmetries cf. [5,12,13]. Example 22 Let be a R2 -action in Rn , n  4 with m D 3. Choose a basis 2 of R3 such that 2 D

˚t

(1; 1; ); t (0; 1; ) ;

;  2 R :

(56)

By (52),  [2 ] is generated by the set of vectors f(1; 0); (1; 1); (; )g. Hence the action is a Poincaré morphism if and only if these vectors generate a proper cone, namely (; ) is not in the set f(; ) 2 R2 ;     0g. We note that the interesting case is  <   0, where every generator in (56) is in a Siegel domain. Next, given a two-dimensional Lie algebra, choose a basis X 1 , X 2 with linear parts A j 2 GL(4; C) satisfying spec(A1 ) D f1; 1; ; g and spec(A2 ) D f0; 1; ; g, respectively, where  <  < 0, (; ) 62 Q2 , and 0

1 B 0 A1 D B @ 0 0

0 1 0 0

0 0  0

1 0 0 C C; 0 A 

0

0 B 0 A2 D B @ 0 0

0 1 0 0

0 0  0

1 0 0 C C; " A  (57)

where " ¤ 0. We show a refinement of the divergence result in Gevrey classes in [54] for the solution v of the overdetermined systems of linear homological equations L j v :D rv(x)A j x  A j v D f j ( j D 1; 2), with the compatibility conditions for the RHS (see [54] for more details). Theorem 23 Let 1/2  0 < 1. Then there exists

Analytic Maps near a Fixed Point in the Presence of Jordan Blocks As for the real analytic local diffeomorphisms preserving the origin in Rn , one has in fact to deal necessarily with hyperbolic maps (cf. [4]). Recall first the complex analytic case cf. [3,4]. Let ˚(x) be a biholomorphic map of C n preserving the origin and ˚ 0 (0) the Jacobian matrix at the origin. Denote by spec(˚ 0 (0)) D f1 ; : : : ; n g the spectrum of ˚ 0 (0). Clearly  j ¤ 0 for all j D 1; : : : ; n. We define the set of all resonance multiindices of ˚ (actually it depends only on spec(˚ 0 (0))) as follows: Res[1 ; : : : ; n ] D

Res j [1 ; : : : ; n ]

jD1 j

Res [1 ; : : : ; n ] D f˛ 2

n ZC (2) :

(59)

˛

   j D 0g ;

n n where ZC (k) :D f˛ 2 ZC : j˛j  kg,  D (1 ; : : : ; n ) ˛n ˛1 n ˛ and  D 1 : : : n for ˛ D (˛1 ; : : : ; ˛n ) 2 ZC . Given  2 GL(n; C) define O[] as the germ of all local complex analytic diffeomorphisms (biholomorphic maps) ˚(x) of C n with 0 as a fixed point and such that ˚ 0 (0) is in the GL(n; C)-conjugacy class of . We stress that if  is diagonalizable (semisimple) then the germ is determined by the spectrum of , i. e., by n nonzero complex numbers 1 ; : : : ; n . We will say that ˚(x) is formally linearizable if there P exists a formal series u(x) D x C ˛2Zn (2) u˛ x ˛ such C that

u1 ı ˚ ı u D ˚ 0 (0) formally in

(C[x])n ;

(60)

where C[x] stands for the set of all formal power series with complex coefficients. It is well known (the Poincaré– Dulac theorem, cf. [3]) that under the nonresonance hypothesis Res[1 ; : : : ; n ] D ;, i. e., ˛   j ¤ 0 ;

E0 f(; ) 2 (R n Q) ;  <  < 0 ;

n [

n ˛ 2 ZC (2) ;

j D 1; : : : ; n ;

(61)

2

 does not satisfy the Bruno conditiong with the density of continuum such that for every (; ) 2 E0 , there exists analytic f D t ( f 1 ; f 2 ) 2 (C24 fxg)2 , satisfying the compatibility condition for the overdetermined system and such that the unique formal solution v(x) is not in S  4 1<2C G (C ). Moreover, for every analytic f we can find C > 0 such that the unique formal solution satisfies the anisotropic Gevrey estimates jv˛ j  C j˛jC1 (˛3 C ˛4 )(1C)˛4 ;

˛ 2 Z4C (2) ;

(58)

˚ is formally linearizable. In fact, (61) is a necessary and sufficient condition in order that every holomorphic ˚(x) with spec˚ 0 (0) D f1 ; : : : ; n g is formally linearizable. We refer to Gramchev and Walcher [25] for formal and algebraic aspects of normal forms of maps. The formal solution of (60) involves expressions of the form (˛   j )1 ; so when inf˛ j˛   j j D 0 for some j 2 f1; : : : ; ng, the convergence of u becomes a subtle question. One of the main problems, starting from the pioneering work of Siegel [44], has been (and still is) to find general conditions which guarantee that u(x) converges,

1285

1286

Perturbation of Systems with Nilpotent Real Part

i. e., that ˚(x) is linearizable. We recall the state of the art of this subject. If the linear part ˚ 0 (0) is semisimple (i. e., ˚ 0 (0) has no nontrivial Jordan blocks), then it is well known that for convergence we need arithmetic (Diophantine) conditions on !(m) :D

min

2j˛jm;1 jn

j˛   j j ;

m 2 ZC (2) : (62)

We refer for the history and references to the survey paper by Herman [28]. The best condition that implies linearizability for all maps with a given semisimple linear part is due to Bruno [9], and can be expressed as (following Herman, p. 143 in [28]) 1 X

2k ln(! 1 (2 kC1 )) < 1 :

(63)

kD1

If one assumes the Poincaré condition max j j j < 1

1 jn

or

min j j j > 1 ;

1 jn

Theorem 24 Let A be a real n  n matrix such that its eigenvalues are nonresonant and lie outside the unit circle in C, and A is neither expansive nor contractive (i. e., the Poincaré condition (64) is not satisfied). Let R(x) D (R1 (x); : : : ; R(x)) be a real valued analytic function near the origin, at least quadratic near x D 0, i. e. R(0) D 0, rR(0) D 0. Then there exists 0 > 0 such that for almost all  2]0; 0 ] in the sense of the Lebesgue measure, the local analytic diffeomorphism defined by ˚(x) D Ax C R(x)

(64)

then ˚ is always analytically equivalent to its linear part ˚ 0 (0), provided the nonresonance hypothesis holds. More generally, (64) implies that there are finitely many resonances and, according to the Poincaré–Dulac theorem, we can find a local biholomorphic change of the variables u bringing ˚ to normal form (u1 ı ˚ ı u)(x) D ˚ 0 (0)x C Pres (x) ;

As a corollary from results of Delatte and Gramchev [17] on nonsolvability of the LHE in the presence of Jordan blocks in the linear parts of biholomorphic maps preserving the origin of C 3 and the fundamental results of Pérez Marco [39] one gets readily the following assertion for hyperbolic analytic maps preserving the origin in Rn and having nondiagonalizable linear parts at the origin.

(65)

where the remainder Pres (x) is a polynomial map containing only resonant terms. Little is known about the (non)linearizability of ˚ in the analytic category if the Poincaré condition doesn’t hold and the matrix ˚ 0 (0) has at least one nontrivial Jordan block. When n D 2 and A has a double eigenvalue 1 D 2 , j1 j D 1 and a nontrivial Jordan block, then in general ˚ is not linearizable. This result is contained in Proposition 3, p. 143 in [28], which is a consequence of results of Ilyashenko [32] and Yoccoz (the latter proved in 1978; for published proofs we refer to Appendix, pp. 86– 87 in [52]). It is not difficult to extend this negative result to C n , n  3. In a recent paper of DeLatte and Gramchev [17] biholomorphic maps in C n , n D 3 and n D 4 having a single nontrivial Jordan block of the linear part have been studied. We mention also the paper of Abate [1], where nondiagonalizable discrete holomorphic dynamical systems have been investigated using geometrical tools. One observes that in the real case, as the matrix ˚ 0 (0) is real, the nonresonance condition excludes eigenvalues on the unit circle, i. e., the map is hyperbolic.

(66)

is not linearizable by convergent transformations. Weakly Hyperbolic Systems and Nilpotent Perturbations The presence of nondiagonalizable matrices appear as a challenging problem in the framework of the well-posedness of the Cauchy problem for evolution partial differential equations. In order to illustrate the main features we focus on linear hyperbolic systems with constant coefficients in space dimension one @ t u C A@x u C Bu D 0 ; u(0; x) D u0 (x)

t > 0;

x2R

(67) (68)

where A and B are real m  m matrices, u D (u1 ; : : : ; u m ) stands for a vector-valued smooth function. One is interested in the C 1 well-posedness of the Cauchy problem (67), (68), namely, for every initial data u0 2 (C 1 (R))m there exists a unique solution u 2 C 1 (R2 ) of (67), (68). We recall that the system is hyperbolic if the characteristic equation jA  Ij D 0

(69)

has m real solutions 1 ; : : : ; m . If the roots are distinct, the system is called strictly hyperbolic. The strict hyperbolicity implies that A is diagonalizable, which in turn implies that, after a linear change of variables, one can assume A to be diagonal and reduce essentially the problem to m first-order scalar equations. In that case the Cauchy problem is well-posed in the classical sense, namely for every

Perturbation of Systems with Nilpotent Real Part

smooth initial data u0 (it is enough u0 2 C 1 (R)) there exists a unique smooth solution u(t; x) to the Cauchy problem. In fact, it is enough to require that A is semisimple (i. e., allowing multiple eigenvalues but excluding nilpotent parts) in order to have well-posedness (e. g., cf. the book of Taylor [48] and the references therein for more details on strictly hyperbolic systems with variable coefficients, and more general set-up in the framework of pseudodifferential operators). If the system is hyperbolic but not strictly hyperbolic (it is called also weakly hyperbolic), it means that the matrix A has multiple eigenvalues. Here the influence of the Jordan block structure is decisive and one has non existence theorems in the C 1 category unless one imposes additional restrictions on the lower-order term B. In many applications one encounters weakly hyperbolic systems. One example is in the so-called water waves problem, concerning the motion of the free surface of a body of an incompressible irrotational fluid under the influence of gravity (see [15] and the references therein), where for the linearized system one encounters a 2  2 matrix of the type   c ~ AD ; 0 c where c stands for the velocity, and ~ is a nonzero real number. The assertions, even in the seemingly simple model cases, are not easy to state in simple terms. The first results on such systems are due to Kajitani [34]. The classification problem was completely settled for the so-called hyperbolic systems of constant multiplicities by Vaillant [50] using subtle linear algebra arguments with multiparameter dependence if the space dimension is greater then 1. We illustrate the assertions for m D 2 and m D 3, where the influence of the nilpotent part is somewhat easier to describe in details. In what follows we rewrite the results in [34,50] by means of the Jordan block structures. The case m D 2 is easy. Proposition 25 Let   0 1 AD ; 0 0

(70)

for some 0 2 R. Then the Cauchy problem is C 1 well posed iff b21 D 0, with   b11 b12 : BD b21 b22 Let the 3  3 real matrix A have a triple eigenvalue 0 and be not semisimple. Then we are reduced (modulo conjugation with an invertible matrix) to two possibilities: either

the JCF of A is a maximal Jordan block 0

0 @ 0 0

1 0 0

1 0 1 A; 0

(71)

or the degenerate case of one 22 and one 11 elementary Jordan blocks 0

0 AD@ 0 0

1 0 0

1 0 0 A; 0

(72)

Proposition 26 Let n D 3. Then the following assertions hold: (i) let A be defined by (71). Then the Cauchy problem is well-posed in C 1 iff the entries of the matrix 0

b11 B D @ b21 b31

b12 b22 b32

1 b13 b23 A b33

satisfy the identities b31 D b21 C b32 D b11  b13 D 0 ;

(73)

(ii) suppose that A is given by (72). The well-posedness in C 1 holds iff b21 D b21 b32 D 0 :

(74)

In an interesting work, Petkov [38], using real Jordan block structures depending on parameters and reduction to normal forms of matrices depending on parameters (cf. [2]), derived canonical microlocal forms for the full symbol of a pseudodifferential system with real characteristics of constant multiplicity and applies them to study the propagation of singularities of solutions of certain systems. More generally, the Cauchy problem for hyperbolic systems with multiple characteristics have been studied by various authors where (implicitly) conditions on the nilpotent perturbations and the lower order term are imposed (e. g., cf. [8,34,51], and the references therein). It would be interesting to write down such conditions in terms of conditions on the nilpotent perturbations on the principal part and the lower-order terms. Finally, we mention also the work of Ghedamsi, Gourdin, Mechab, and Takeuchi [22], concerned with the Cauchy problem for Schrödinger-type systems with characteristic roots of multiplicity two admitting nontrivial Jordan blocks.

1287

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Bibliography 1. Abate M (2000) Diagonalization of nondiagonalizable discrete holomorphic dynamical systems. Amer J Math 122:757–781 2. Arnold VI (1971) Matrices depending on parameters. Uspekhi Mat Nauk 26:101–114 (in Russian); Russ Math Surv 26:29–43 3. Arnold VI (1983) Geometrical methods in the theory of ordinary differential equations. Springer, New York 4. Arnold VI, Ilyashenko YU (1988) In: Anosov DV, Arnold VI (eds) Encyclopedia of Math Sci, vol 1. Dynamical Systems I. Springer, New York, pp 1–155 5. Bambusi D, Cicogna G, Gaeta G, Marmo G (1998) Normal forms, symmetry and linearization of dynamical systems. J Phys A 31:5065–5082 6. Belitskii GR (1978) Equivalence and normal forms of germs of smooth mappings. Uspekhi Mat Nauk 33:95–155, 263 (in Russian); Russ Math Surv 33:107–177 7. Belitskii GR (1979) Normal forms, invariants, and local mappings. Naukova Dumka, Kiev (in Russian) 8. Bove A, Nishitani T (2003) Necessary conditions for hyperbolic systems. II. Jpn J Math (NS) 29:357–388 9. Bruno AD (1971) The analytic form of differential equations. Tr Mosk Mat O-va 25:119–262; (1972) 26:199–239 (in Russian); See also (1971) Trans Mosc Math Soc 25:131–288; (1972) 26:199– 239 10. Bruno AD, Walcher S (1994) Symmetries and convergence of normalizing transformations. J Math Anal Appl 183:571–576 11. Chen KT (1965) Diffeomorphisms: C 1 -realizations of formal properties. Amer J Math 87:140–157 12. Cicogna G, Gaeta G (1999) Symmetry and perturbation theory in nonlinear dynamics. Lecture Notes in Physics. New Series m: Monographs, vol 57. Springer, Berlin 13. Cicogna G, Walcher S (2002) Convergence of normal form transformations: the role of symmetries. (English summary) Symmetry and perturbation theory. Acta Appl Math 70:95– 111 14. Coddington EA, Levinson N (1955) Theory of ordinary differential equations. McGraw-Hill, New York 15. Craig W (1987) Nonstrictly hyperbolic nonlinear systems. Math Ann 277:213–232 16. Cushman R, Sanders JA (1990) A survey of invariant theory applied to normal forms of vector fields with nilpotent linear part. In: Stanton D (ed) Invariant theory and tableaux. IMA Vol Math Appl, vol 19. Springer, New York, pp 82–106 17. DeLatte D, Gramchev T (2002) Biholomorphic maps with linear parts having Jordan blocks: Linearization and resonance type phenomena. Math Phys Electron J 8(2):1–27 18. Dumortier F, Roussarie R (1980) Smooth linearization of germs of R2 -actions and holomorphic vector fields. Ann Inst Fourier Grenoble 30:31–64 19. Gaeta G, Walcher S (2005) Dimension increase and splitting for Poincaré-Dulac normal forms. J Nonlinear Math Phys 12(1):327–342 20. Gaeta G, Walcher S (2006) Embedding and splitting ordinary differential equations in normal form. J Differ Equ 224:98–119 21. Gantmacher FR (1959) The theory of matrices, vols 1, 2. Chelsea, New York 22. Ghedamsi M, Gourdin D, Mechab M, Takeuchi J (2002) Équations et systèmes du type de Schrödinger à racines caractéristiques de multiplicité deux. Bull Soc Roy Sci Liège 71:169– 187

23. Gramchev T (2002) On the linearization of holomorphic vector fields in the Siegel Domain with linear parts having nontrivial Jordan blocks. In: Abenda S, Gaeta G, Walcher S (eds) Symmetry and perturbation theory, Cala Gonone, 16–22 May 2002. World Sci. Publ., River Edge, pp 106–115 24. Gramchev T, Tolis E (2006) Solvability of systems of singular partial differential equations in function spaces. Integral Transform Spec Funct 17:231–237 25. Gramchev T, Walcher S (2005) Normal forms of maps: formal and algebraic aspects. Acta Appl Math 85:123–146 26. Gramchev T, Yoshino M (2007) Normal forms for commuting vector fields near a common fixed point. In: Gaeta G, Vitolo R, Walcher S (eds) Symmetry and Perturbation theory, Oltranto, 2–9 June 2007. World Sci Publ, River Edge, pp 203–217 27. Hasselblatt B, Katok A (2003) A first course in dynamics: with a panorama of recent developments. Cambridge University Press, Cambridge 28. Herman M (1987) Recent results and some open questions on Siegel’s linearization theorem of germs of complex analytic diffeomorphisms of C n near a fixed point. VIIIth international congress on mathematical physics, Marseille 1986. World Sci Publ, Singapore, pp 138–184 29. Hibino M (1999) Divergence property of formal solutions for first order linear partial differential equations. Publ Res Inst Math Sci 35:893–919 30. Hibino M (2003) Borel summability of divergent solutions for singular first order linear partial differential equations with polynomial coefficients. J Math Sci Univ Tokyo 10:279–309 31. Hibino M (2006) Formal Gevrey theory for singular first order quasi–linear partial differential equations. Publ Res Inst Math Sci 42:933–985 32. Il’yashenko Y (1979) Divergence of series reducing an analytic differential equation to linear form at a singular point. Funct Anal Appl 13:227–229 33. Iooss G, Lombardi E (2005) Polynomial normal forms with exponentially small remainder for vector fields. J Differ Equ 212:1–61 34. Kajitani K (1979) Cauchy problem for non–strictly hyperbolic systems. Publ Res Inst Math 15:519–550 35. Katok A, Katok S (1995) Higher cohomology for Abelian groups of toral automorphisms. Ergod Theory Dyn Syst 15:569–592 36. Murdock J (2002) On the structure of nilpotent normal form modules. J Differ Equ 180:198–237 37. Murdock J, Sanders JA (2007) A new transvectant algorithm for nilpotent normal forms. J Differ Equ 238:234–256 38. Petkov VM (1979) Microlocal forms for hyperbolic systems. Math Nachr 93:117–131 39. Pérez Marco R (2001) Total convergence or small divergence in small divisors. Commun Math Phys 223:451–464 40. Ramis J-P (1984) Théorèmes d’indices Gevrey pour les équations différentielles ordinaires. Mem Amer Math Soc 48:296 41. Rodino L (1993) Linear partial differential operators in Gevrey spaces. World Science, Singapore 42. Sanders JA (2005) Normal form in filtered Lie algebra representations. Acta Appl Math 87:165–189 43. Sanders JA, Verhulst F, Murdock J (2007) Averaging methods in nonlinear dynamical systems, 2nd edn. Applied Mathematical Sciences, vol 59. Springer, New York 44. Siegel CL (1942) Iteration of analytic functions. Ann Math 43:607–614

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45. Sternberg S (1958) The structure of local homeomorphisms. II, III. Amer J Math 80:623–632, 81:578–604 46. Stolovitch L (2000) Singular complete integrability. Publ Math IHES 91:134–210 47. Stolovitch L (2005) Normalisation holomorphe d’algèbres de type Cartan de champs de vecteurs holomorphes singuliers. Ann Math 161:589–612 48. Taylor M (1981) Pseudodifferential operators. Princeton Mathematical Series, vol 34. Princeton University Press, Princeton 49. Thiffeault J-L, Morison PJ (2000) Classification and Casimir invariants of Lie–Poisson brackets. Physica D 136:205–244 50. Vaillant J (1999) Invariants des systèmes d’opérateurs différentiels et sommes formelles asymptotiques. Jpn J Math (NS) 25:1–153 51. Yamahara H (2000) Cauchy problem for hyperbolic systems in

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Perturbation Theory GIOVANNI GALLAVOTTI Dipartimento di Fisica and I.N.F.N., sezione Roma-I, Università di Roma I “La Sapienza”, Roma, Italy Article Outline Glossary Definition of the Subject Introduction Poincaré’s Theorem and Quanta Mathematics and Physics. Renormalization Need of Convergence Proofs Multiscale Analysis A Paradigmatic Example of PT Problem Lindstedt Series Convergence. Scales. Multiscale Analysis Non Convergent Cases Conclusion and Outlook Future Directions Bibliography Glossary Formal power series a power series, giving the value of a function f (") of a parameter ", that is derived assuming that f is analytic in ". Renormalization group method for multiscale analysis and resummation of formal power series. Usually applied to define a systematic collection of terms to organize a formal power series into a convergent one. Lindstedt series an algorithm to develop formal power series for computing the parametric equations of invariant tori in systems close to integrable. Multiscale problem any problem in which an infinite number of scales play a role. Definition of the Subject Perturbation Theory: Computation of a quantity depending on a parameter " starting from the knowledge of its value for " D 0 by deriving a power series expansion in ", under the assumption of its existence, and if possible discussing the interpretation of the series. Perturbation theory is very often the only way to get a glimpse of the properties of systems whose equations cannot be “explicitly solved” in computable form. The importance of Perturbation Theory is witnessed by its applications in Astronomy, where it led not only to the discovery of new planets (Neptune) but also to the discovery of Chaotic motions, with the completion of the

Copernican revolution and the full understanding of the role of Aristotelian Physics formalized into uniform rotations of deferents and epicycles (today Fourier representation of quasi periodic motions). It also played an essential role in the development of Quantum Mechanics and the understanding of the periodic table. The successes of Quantum Field Theory in Electrodynamics first, then in Strong interactions and finally in the unification of the elementary forces (strong, electromagnetic, and weak) are also due to perturbation theory, which has also been essential in the theoretical understanding of the critical point universality. The latter two themes concern the new methods that have been developed in the last fifty years, marking a kind of new era for perturbation theory; namely dealing with singular problems, via the techniques called, in Physics, “Renormalization Group” and, in Mathematics, “Multiscale Analysis”. Introduction Perturbation theory, henceforth PT, arises when the value of a function of interest is associated with a problem depending on a parameter, here called ". The value has to be a simple, or at least explicit and rigorous, computation for " D 0 while its computation for " ¤ 0, small, is attempted by expressing it as the sum of a power series in " which will be called here the “solution”. It is important to say since the beginning that a real PT solution of a problem involves two distinct steps: the first is to show that assuming that there is a convergent power series solving the problem then the coefficients of the nth power of " exist and can be computed via finite computation. The resulting series will be called formal solution or formal series for the problem. The second step, that will be called convergence theory, is to prove that the formal series converges for " small enough, or at least find a “summation rule” that gives a meaning to the formal series thus providing a real solution to the problem. None of the two problems is trivial, in the interesting cases, although the second is certainly the key and a difficult one. Once Newton’s law of universal gravitation was established it became necessary to develop methods to find its implications. Laplace’s “Mécanique Céleste” [19], provided a detailed and meticulous exposition of a general method that has become a classic, if not the first, example of perturbation theory, quite different from the parallel analysis of Gauss which can be more appropriately considered a “non perturbative” development. Since Laplace one can say that many applications along his lines followed. In the XIX century wide attention was dedicated to extend Laplace’s work to cover various astro-

Perturbation Theory

nomical problems: tables of the coefficients were dressed and published, and algorithms for their construction were devised, and planets were discovered (Neptune, 1846). Well known is the “Lindstedt algorithm” for the computation of the nth order coefficients of the PT series for the non resonant quasi periodic motions. The algorithm provides a power series representation for the quasi periodic motions with non resonant frequencies which is extremely simple: however it represents the nth coefficient as a sum of many terms, some of size of the order of a power on n!. Which of course is a serious problem for the convergence. It became a central issue, known as the “small denominators problem” after Poincaré’s deep critique of the PT method, generated by his analysis of the three-body problem. It led to his “non-integrability theorem” about the generic nonexistence of convergent power series in the perturbation parameter " whose sum would be a constant of motion for a Hamiltonian H" , member of a family of Hamiltonians parametrized by " and reducing to an integrable system for " D 0. The theorem suggested (to some) that even the PT series of Lindstedt (to which Poincaré’s theorem does not apply) could be meaningless even though formally well defined [23]. A posteriori, it should be recognized that PT was involved also in the early developments of Statistical Mechanics in the XIX century: the virial theorem application to obtain the Van der Waals equation of state can be considered a first-order calculation in PT (although this became clear only a century later with the identification of " as the inverse of the space dimension). Poincaré’s Theorem and Quanta With Poincaré begins a new phase: the question of convergence of series in " becomes a central one in the Mathematics literature. Much less, however, in the Physics literature where the new discoveries in the atomic phenomena attracted the attention. It seems that in the Physics research it was taken for granted that convergence was not an issue: atomic spectra were studied via PT and early authoritative warnings were simply disregarded (for instance, explicit by Einstein, in [1], and clear, in [3], but “timid” being too far against the mainstream, for his young age). In this way quantum theory could grow from the original formulations of Bohr, Sommerfeld, Eherenfest relying on PT to the final formulations of Heisenberg and Schrödinger quite far from it. Nevertheless, the triumph of quantum theory was quite substantially based on the technical development and refinement of the methods of formal PT: the calculation of the Compton scattering, the

Lamb shift, Fermi’s weak interactions model and other spectacular successes came in spite of the parallel recognition that some of the series that were being laboriously computed not only could not possibly be convergent but their very existence, to all orders n, was in doubt. The later Feynman graphs representation of PT was a great new tool which superseded and improved earlier graphical representations of the calculations. Its simplicity allowed a careful analysis and understanding of cases in which even formal PT seemed puzzlingly failing. Renormalization theory was developed to show that the convergence problems that seemed to plague even the computation of the individual coefficients of the series, hence the formal PT series at fixed order, were, in reality, often absent, in great generality, as suspected by the earlier treatments of special (important) cases, like the higher-order evaluations of the Compton scattering, and other quantum electrodynamics cross sections or anomalous characteristic constants (e. g. the magnetic moment of the muon). Mathematics and Physics. Renormalization In 1943 the first important result on the convergence of the series of the Lindstedt kind was obtained by Siegel [25]: a formal PT series, of interest in the theory of complex maps, was shown to be convergent. Siegel’s work was certainly a stimulus for the later work of Kolmogorov who solved [18], a problem that had been considered not soluble by many: to find the convergence conditions and the convergence proof of the Lindstedt series for the quasi periodic motions of a generic analytic Hamiltonian system, in spite of Poincaré’s theorem and actually avoiding contradiction with it. Thus, showing the soundness of the comments about the unsatisfactory aspects of Poincaré’s analysis that had been raised almost immediately by Weierstrass, Hadamard and others. In 1956 not only Kolmogorov theorem appeared but also convergence of another well known and widely used formal series, the virial series, was achieved in an unnoticed work by Morrey [21], and independently rediscovered in the early 1960’s. At this time it seems that all series with well-defined terms were thought to be either convergent or at least asymptotic: for most Physicists convergence or asymptoticity were considered of little interest and matters to be left to Mathematicians. However, with the understanding of the formal aspects of renormalization theory the interest in the convergence properties of the formal PT series once again became the center of attention.

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On the one hand mathematical proofs of the existence of the PT series, for interesting quantum fields models, to all orders were investigated settling the question once and for all (Hepp’s theorem [16]); on the other hand it was obvious that even if convergent (like in the virial or Meyer expansions, or in the Kolmogorov theory) it was well understood that the radius of convergence would not be large enough to cover all the physically interesting cases. The sum of the series would in general become singular in the sense of analytic functions and, even if admitting analytic continuation beyond the radius of convergence, a singularity in " would be eventually hit. The singularity was supposed to correspond to very important phenomena, like the critical point in statistical mechanics or the onset of chaotic motions (already foreseen by Poincaré in connection with his non convergence theorem). Thus, research developed in two direction. The first aimed at understanding the nature of the singularities from the formal series coefficients: in the 1960s many works achieved the understanding of the scaling laws (i. e. some properties of the divergences appearing at the singularities of the PT series or of its analytic continuation, for instance in the work of M. Fisher, Kadanoff, Widom and may others). This led to trying to find resummations, i. e. to collect terms of the formal series to transform them into convergent series in terms of new parameters, the running couplings. The latter would be singular functions of the original " thus possibly reducing the study of the singularity to the singularities of the running couplings. The latter could be studied by independent methods, typically by studying the iterations of an auxiliary dynamical system (called the beta function flow). This was the approach or renormalization group method of Wilson [10,29]. The second direction was dedicated to finding out the real meaning of the PT series in the cases in which convergence was doubtful or a priori excluded: in fact already Landau had advanced the idea that the series could be just illusions in important problems like the paradigmatic quantum field theory of a scalar field or the fundamental quantum electrodynamics [4,15]. In a rigorous treatment the function that the series were supposed to represent would be in fact a trivial function with a dependence on " unrelated to the coefficients of the well defined and non trivial but formal series. It was therefore important to show that there were at least cases in which the perturbation series of a nontrivial problem had a meaning determined by its coefficients. This was studied in the scalar model of quantum field theory and a proof of “non triviality” was achieved after the ground-breaking work of Nelson on two-dimensional

models [22,26]: soon followed by similar results in two dimensions and the difficult extension to three-dimensional models by Glimm and Jaffe [14], and generating many works and results on the subject which took the name of “constructive field theory” [6]. But Landau’s triviality conjecture was actually dealing with the “real problem”, i. e. the 4-dimensional quantum fields. The conjecture remains such at the moment, in spite of very intensive work and attempts at its proof. The problem had relevance because it could have meant that not only the simple scalar models of constructive field theory were trivial but also the QED series which had received strong experimental support with the correct prediction of fine structure phenomena could be illusions, in spite of their well-defined PT series: which would remain as mirages of a non existing reality. The work of Wilson made clear that the “triviality conjecture” of Landau could be applied only to theories which, after the mentioned resummations, would be controlled by a beta function flow that could not be studied perturbatively, and introduced the new notion of asymptotic freedom. This is a property of the beta function flow, implying that the running couplings are bounded and small so that the resummed series are more likely to have a meaning [29]. This work revived the interest in PT for quantum fields with attention devoted to new models that had been believed to be non renormalizable. Once more the apparently preliminary problem of developing a formal PT series played a key role: and it was discovered that many Yang–Mills quantum field theories were in fact renormalizable in the ultraviolet region [27,28], and an exciting period followed with attempts at using Wilson’s methods to give a meaning to the Yang–Mills theory with the hope of building a theory of the strong interactions. Thus, it was discovered that several Yang–Mills theories were asymptotically free as a consequence of the high symmetry of the model, proving that what seemed to be strong evidence that no renormalizable model would have asymptotic freedom was an ill-founded belief (that in a sense slowed down the process of understanding, and not only of the strong interactions). Suddenly understanding the strong interactions, until then considered an impossible problem became possible [15], as solutions could be written and effectively computed in terms of PT which, although not proved to be convergent or asymptotic (still an open problem in dimension d D 4) were immune to the argument of Landau. The impact of the new developments led a little later to the unification of all interactions into the standard model for the theory of elementary particles (including the elec-

Perturbation Theory

tromagnetic and weak interactions). The standard model was shown to be asymptotically free even in the presence of symmetry breaking, at least if a few other interactions in the model (for instance the Higgs particle self interaction) were treated heuristically while waiting for the discovery of the “Higgs particle” and for a better understanding of the structure of the elementary particles at length scales intermediate between the Fermi scale ( 1015 cm (the weak interactions scale)) and the Planck scale (the gravitational interaction scale, 15 orders of magnitude below). Given that the very discovery of renormalizability of Yang–Mills fields and the birth of a strong interactions theory had been firmly grounded on experimental results [15], the latter “missing step” was, and still is, considered an acceptable gap. Need of Convergence Proofs The story of the standard model is paradigmatic of the power of PT: it should convince anyone that the analysis of formal series, including their representation by diagrams, which plays an essential part, is to be taken seriously. PT is certainly responsible for the revival and solution of problems considered by many as hopeless. In a sense PT in the elementary particles domain can only, so far, partially be considered a success. Different is the situation in the developments that followed the works of Siegel and Kolmogorov. Their relevance for Celestial Mechanics and for several problems in applied physics (particle accelerator design, nuclear fusion machines for instance) and for statistical mechanics made them too the object of a large amount of research work. The problems are simpler to formulate and often very well posed but the possibility of existence of chaotic motions, always looming, made it imperative not to be content with heuristic analysis and imposed the quest of mathematically complete studies. The lead were the works of Siegel and Kolmogorov. They had established convergence of certain PT series, but there were other series which would certainly be not convergent even though formally well defined and the question was, therefore, which would be their meaning. More precisely it was clear that the series could be used to find approximate solutions to the equations, representing the motion for very long times under the assumption of “small enough” ". But this could hardly be considered an understanding of the PT series in Mechanics: the estimated values of " would have to be too small to be of interest, with the exception of a few special cases. The real question was what could be done to give the PT series the status of exact solution.

As we shall see the problem is deeply connected with the above-mentioned asymptotic freedom: this is perhaps not surprising because the link between the two is to be found in the “multiscale analysis” problems, which in the last half century have been the core of the studies in many areas of Analysis and in Physics, when theoretical developments and experimental techniques became finer and able to explore nature at smaller and smaller scales. Multiscale Analysis To illustrate the multiscale analysis in PT it is convenient to present it in the context of Hamiltonian mechanics, because in this field it provides us with nontrivial cases of almost complete success. We begin by contrasting the work of Siegel and that of Kolmogorov: which are based on radically different methods. The first being much closer in spirit to the developments of renormalization theory and to the Feynman graphs. Most interesting formal PT series have a common feature: namely their nth order coefficients are constructed as sums of many “terms” and the first attempt to a complete analysis is to recognize that their sum, which gives the uniquely defined nth coefficient is much smaller than the sum of the absolute values of the constituent terms. This is a property usually referred to as a “cancellation” and, as a rule, it reflects some symmetry property of the problem: hence one possible approach is to look for expressions of the coefficients and for cancellations which would reduce the estimate of the nth order coefficients, very often of the order of a power of n!, to an exponential estimate O(%n ) for some % > 0 yielding convergence (parenthetically in the mentioned case of Yang–Mills theories the reduction is even more dramatic as it leads from divergent expressions to finite ones, yet of order n!). The multiscale aspect becomes clear also in Kolmogorov’s method because the implicit functions theorem has to be applied over and over again and deals with functions implicitly defined on smaller and smaller domains [5,7]. But the method purposedly avoids facing the combinatorial aspects behind the cancellations so much followed, and cherished [28], in the Physics works. Siegel’s method was developed to study a problem in which no grouping of terms was eventually needed, even though this was by no means clear a priori [24]; and to realize that no cancellations were needed forced one to consider the problem as a multiscale one because the absence of rapid growth of the nth order coefficients became manifest after a suitable “hierarchical ordering” of the terms generating the coefficients. The approach estab-

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lishes a strong connection with the Physics literature because the technique to study such cases was independently developed in quantum field theory with renormalization, as shown by Hepp in [16], relying strongly on it. This is very natural and, in case of failure, it can be improved by looking for “resummations” turning the power series into a convergent series in terms of functions of " which are singular but controllably so. For details see below and [8]. What is “natural”, however, is a very personal notion and it is not surprising that what some consider natural is considered unnatural or clumsy or difficult (or the three qualifications together) by others. Conflict arises when the same problem can be solved by two different “natural” methods and in the case of PT for Hamiltonian systems close to integrable ones (closeness depending on the size of a parameter "), the so-called “small denominators” problem, the methods of Siegel and Kolmogorov are antithetic and an example of the just mentioned dualism. The first method, that will be called here “Siegel’s method” (see below for details), is based on a careful analysis of the structure of the various terms that occur at a given PT order achieving a proof that the nth order coefficient which is represented as the sum of many terms some of which might have size of order of a power of n! has in fact a size of O(%n ) so that the PT series is convergent for j"j < %. Although strictly speaking the original work of Siegel does not immediately apply to the Hamiltonian Mechanics problems (see below), it can nevertheless be adapted and yields a solution, as made manifest much later in [2,8,24]. The second method, called here “Kolmogorov’s method”, instead does not consider the individual coefficients of the various orders but just regards the sum of the series as a solution of an implicit function equation (a “Hamilton–Jacobi” equation) and devises a recursive algorithm approximating the unknown sum of the PT series by functions analytic in a disk of fixed radius % in the complex "-plane [5,7]. Of course the latter approach implies that no matter how we achieve the construction of the nth order PT series coefficient there will have to be enough cancellations, if at all needed, so that it turns out bounded by O(%n ). And in the problem studied by Kolmogorov cancellations would be necessarily present if the nth order coefficient was represented by the sum of the terms in the Lindstedt series. That this is not obvious is supported by the fact that it was considered an open problem, for about thirty years, to find a way to exhibit explicitly the cancellation mechanism in the Lindstedt series implied by Kolmogorov’s work. This was done by Eliasson [2], who proved that the

coefficients of the PT of a given order n as expressed by the construction known as the “Lindstedt algorithm” yielded coefficients of size of O(%n ): his argument, however, did not identify in general which term of the Lindstedt sum for the nth order coefficient was compensated by which other term or terms. It proved that the sum had to satisfy suitable relations, which in turn implied a total size of O(%n ). And it took a few more years for the complete identification [8], of the rules to follow in collecting the terms of the Lindstedt series which would imply the needed cancellations. It is interesting to remark that, aside from the example of Hamiltonian PT, multiscale problems have dominated the development of analysis and Physics in recent time: for instance they appear in harmonic analysis (Carleson, Fefferman), in PDE’s (DeGiorgi, Moser, Caffarelli–Kohn–Ninberg), in relativistic quantum mechanics (Glimm, Jaffe, Wilson), in Hamiltonian Mechanics (Siegel, Kolmogorov, Arnold, Moser), in statistical mechanics and condensed matter (Fisher, Wilson, Widom) . . . Sometimes, although not always, studied by PT techniques [10]. A Paradigmatic Example of PT Problem It is useful to keep in mind an example illustrating technically what it means to perform a multiscale analysis in PT. And the case of quasi periodic motions in Hamiltonian mechanics will be selected here, being perhaps the simplest. Consider the motion of ` unit masses on a unit circle and let ˛ D (˛1 ; : : : ; ˛` ) be their positions on the circle, i. e. ˛ is a point on the torus T ` D [0; 2]` . The points interact with a potential energy " f (˛) where " is a strength parameter and f is a trigonometric even polynomial, of deP i˛ ; f D f gree N : f (˛) D   2 R, 2Z` ;jjN f e ` where Z denotes the lattice of the points with integer P components in R` and jj D j j j j. Let t ! ˛0 C !0 t be the motion with initial data, ˙ at time t D 0, ˛(0) D ˛0 ; ˛(0) D !0 , in which all particles rotate at constant speed with rotation velocity !0 D (!01 ; : : : ; !0` ) 2 R` . This is a solution for the equations of motion for " D 0 and it is a quasi periodic solution, i. e. each of the angles ˛ j rotates periodically at constant speed !0 j , j D 1; : : : ; `. The motion will be called non resonant if the components of the rotation speed !0 are rationally independent: this means that !0   D 0 with  2 Z` is possible only if  D 0. In this case the motion t ! ˛0 C !0 t covers, 8˛0 , densely the torus T ` as t varies. The PT problem that we consider is to find whether there is a family of motions “of the same kind” for each ", small enough, solving the

Perturbation Theory

equations of motion; more precisely whether there exists a function a" ('); ' 2 T ` , such that setting ˛(t) D ' C !0 t C a" (' C !0 t) ;

for ' 2 T `

(1)

one obtains, 8' 2 T ` and for " small enough, a solution of the equations of motion for a force "@˛ f (˛): i. e. ¨ D "@˛ f (˛(t)) : ˛(t)

(2)

By substitution of Eq. (1) in Eq. (2), the condition becomes (!0  @' )2 a(' C !0 t) D @˛ f (' C !0 t). Since !0 is assumed rationally independent ' C !0 t covers densely the torus T ` as t varies: hence the equation for a" is (!0  @' )2 a" (') D " @˛ f (' C a" (')) :

(3)

Applying PT to this equation means to look for a solution a" which is analytic in " small enough and in ' 2 T ` . In colorful language one says that the perturbation effect is slightly deforming a nonresonant torus with given frequency spectrum (i. e. given !0 ) on which the motion develops, without destroying it and keeping the quasi periodic motion on it with the same frequency spectrum. Lindstedt Series As it follows from a very simple special case of Poincaré’s work, Eq. (3) cannot be solved if also !0 is considered variable and the dependence on "; !0 analytic. Nevertheless if !0 is fixed and non resonant and if a" is supposed analytic in " small enough and in ' 2 T ` , then there can be at most one solution to the Eq. (3) with a" (0) D 0 (which is not a real restriction because if a" (') is a solution also a" (' C c" ) C c" is a solution for any constant c" ). This so because the coefficients of the power series in ", P1 n nD1 " a n ('), are uniquely determined if the series is convergent. In fact they are trigonometric polynomials of order  nN which will be written as X ˛n; e i' : (4) ˛n (') D 0<jjNn

It is convenient to express them in terms of graphs. The graphs to use to express the value an; are (i) trees with n nodes v1 ; : : : ; v n , (ii) one root r, (iii) n lines joining pairs (v 0i ; v i ) of nodes or the root and one node, always one and not more, (r; v i ); (iv) the lines will be different from each other and distinguished by a mark label, 1; : : : ; n attached to them. The connections between the nodes that the lines generate have to be loopless, i. e. the graph formed by the lines must be a tree.

(v) The tree lines will be imagined oriented towards the root: hence a partial order is generated on the tree and the line joining v to v 0 will be denoted v 0 v and v 0 will be the node closer to the root immediately following v, hence such that v 0 > v in the partial order of tree. The number of such trees is large and exactly equal to n n1 , as an application of Cayley’s formula implies: their collection will be denoted Tn0 . To compute an; consider all trees in Tn0 and attach to each node v a vector  v 2 Z` , called “mode label”, such that fv ¤ 0, hence j v j  N. To the root we associate one of the coordinate unit vectors  r  er . We obtain a set T n of decorated trees (with  (2N C 1)`n n n1 elements, by the above counting analysis). Given 2 Tn and  D (v 0 ; v) 2 we define the def current on the line  to be the vector ()  (v 0 ; v) D P wv  w : i. e. we imagine that the node vectors  v i represent currents entering the node v i and flowing towards the root. Then () is, for each , the sum of the currents which entered all the nodes not following v , i. e. current accumulated after passing the node v . P The current flowing in the root line  D v  v will be denoted ( ). Let Tn be the set trees in T n in which all lines carry a non zero current () ¤ 0. A value Val( ) will be defined, for 2 Tn , by a product of node factors and of line factors over all nodes and Val( ) D

i(1)n Y fv n! v2

Y D(v 0 ;v)

v 0  v : (5) (!0  (v 0 ; v))2

The coefficient an; will then be an; D

X

Val( )

(6)

2Tn ( )D

and, when the coefficients are imagined to be conP n structed in this way, the formal power series 1 nD1 " P a is called the “Lindstedt series”. Eq. (5) and jjNn n; its graphical interpretation in Fig. 1 should be considered the “Feynman rules” and the “Feynman diagrams” of the PT for Eq. (3) [9,10]. Convergence. Scales. Multiscale Analysis The Lindstedt series is well defined because of the non resonance condition and the nth term is not even a sum of def too many terms: if F D max j f j, each of them can be Q n bounded by F /n! 2 N 2 /(!0  ()2 ); hence their sum

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Perturbation Theory

Perturbation Theory, Figure 1 A tree  with mv0 D 2; mv1 D 2; mv2 D 3; mv3 D 2; mv4 D 2; mv12 D 1 lines entering the nodes v i , k D 13. Some labels or decorations explicitly marked (on the lines 0 ; 1 and on the nodes v1 ; v2 ); the number labels, distinguishing the branches, are not shown. The arrows represent the partial ordering on the tree

can be bounded, if G is such that ((2N C1)`n n n1 F n )/n!  Q G n , by G n 2 N 2 /(!0  ()2 ). Thus, all an are well defined and finite but the problem is that j()j can be large (up to Nn at given order n) and therefore !0  () although never zero can become very small as n grows. For this reason the problem of convergence of the series is an example of what is called a small denominators problem. And it is necessary to assume more than just non resonance of !0 in order to solve it in the present case: a simple condition is the Diophantine condition, namely the existence of C;  > 0 such that j!0  j 

1 ; C jj

80 ¤  2 Z` :

(7)

But this condition is not sufficient in an obvious way: because it only allows us to bound individual tree-values by n! a for some a > 0 related to ; furthermore it is not difficult to check that there are single graphs whose value is actually of “factorial” size in n. Although non trivial to see (as mentioned above) this was only apparently so in the earlier case of Siegel’s problem but it is the new essential feature of the terms generating the nth order coefficient in Eq. (6). A resummation is necessary to show that the tree-values can be grouped so that the sum of the values of each group can be bounded by %n for some % > 0 and 8n, although the group may contain (several) terms of factorial size. The terms to be grouped have to be ordered hierarchically according to the sizes of the line factors 1/(!0 ())2 , which are called propagators in [8,12].

A similar problem is met in quantum field theory where the graphs are the Feynman graphs: such graphs can only have a small number of lines that converge into a node but they can have loops, and to show that the perturbation series is well defined to all orders it is also necessary to collect terms hierarchically according to the propagators sizes. The systematic way was developed by Hepp [16,17], for the PT expansion of the Schwinger functions in quantum field theory of scalar fields [6]. It has been used on many occasions later and it plays a key role in the renormalization group methods in Statistical Mechanics (for instance in theory of the ground state of Fermi systems) [10,11]. However, it is in the Lindstedt series that the method is perhaps best illustrated. Essentially because it ends up in a convergence proof, while often in the field theory or statistical mechanics problems the PT series can be only proved to be well defined to all orders, but they are seldom, if ever, convergent so that one has to have recourse to other supplementary analytic means to show that the PT series are asymptotic (in the cases in which they are such). The path of the proof is the following. (1) Consider only trees in which no two lines C and  , with C following  in the partial order of the tree, have the same current  0 . In this case the maximum Q of the  1/(!0  ())2 over all tress 2 Tn can be bounded by G1 n for some G1 . This is an immediate consequence and the main result in Siegel’s original work [25], which dealt with a different problem with small denominators in its formal PT solution: the coefficients of the series could also be represented by tree graphs, very similar too the ones above: but the only allowed  2 Z` were the non zero vectors with all components  0. The latter property automatically guarantees that the graphs contain no pair of lines C ;  following each other as above in the tree partial order and having the same current. Siegel’s proof also implies a multiscale analysis [24]: but it requires no grouping of the terms unlike the analogue Lindstedt series, Eq. (6). (2) Trees which contain lines C and  , with C following  , in the partial order of the tree, and having the same current  0 can have values which have size of order O(n! a ) with some a > 0. Collecting terms is therefore essential. A line  of a tree is said to have scale k if 2k1  1/Cj!0  j < 2k . The lines of a tree 2 Tn can then be collected in clusters.1 1 The scaling factor 2 is arbitrary: any scale factor > 1 could be used.

Perturbation Theory

of the trees obtained by changing simultaneously sign to the  v of the nodes inside the self energy clusters have also to be added together. After collecting the terms in the described way it is possible to check that each sum of terms so collected for some %0 (which can also be is bounded by %n 0 estimated explicitly). Since the number of addends left is not larger than the original one the bound on P n n ` n 2n  ja n; j becomes  (F (2N C 1) n N )/n! %0  %n , for suitable %0 ; %, so that convergence of the formal series for a" (') is achieved for j"j < %, see [8]. Perturbation Theory, Figure 2 An example of three clusters symbolically delimited by circles, as visual aids, inside a tree (whose remaining branches and clusters are not drawn and are indicated by the bullets); not all labels are explicitly shown. The scales (not marked) of the branches increase as one crosses inward the circles boundaries: recall, however, that the scale labels are integers  1 (hence typically  0). The  labels are not drawn (but must be imagined). If the  labels of (v4 ; v5 ) add up to 0 the cluster T 00 is a self-energy graph. If the  labels of (v2 ; v4 ; v5 ; v6 ) add up to 0 the cluster T 0 is a self-energy graph and such is T if the  labels of (v1 ; v2 ; v3 ; v4 ; v5 ; v6 ; v7 ) add up to 0. The cluster T 0 is maximal in T

A cluster of scale p is a maximal connected set of lines of scale k  p with at least one line of scale p. Clusters are connected to the rest of the tree by lines of lower scale which can be incoming or outgoing with respect to the partial ordering. Clusters also contain nodes: a node is in a cluster if it is an extreme of a line contained in a cluster; such nodes are said to be internal to the cluster. Of particular interest are the self energy clusters. These are clusters with only one incoming line and only one outgoing line which furthermore have the same current  0 . To simplify the analysis the Diophantine condition can be strengthened to insure that if in a tree graph the line incoming into a self energy cluster and ending in an internal node v is detached from the node v and reattached to another node internal to the same cluster which is not in a self-energy subcluster (if any) then the new tree nodes are still enclosed in the same clusters. Alternatively the definition of scale of a line can be modified slightly to achieve the same goal. (3) Then it makes sense to sum together all the values of the trees whose nodes are collected into the same families of clusters and differ only because the lines entering the self energy clusters are attached to a different node internal to the cluster, but external to the inner self energy subclusters (if any). Furthermore, the value

Non Convergent Cases Convergence is not the rule: very interesting problems arise in which the PT series is, or is believed to be, only asymptotic. For instance in quantum field theory the PT series are well defined but they are not convergent: they can be proved, in the scalar ' 4 theories in dimension 2 and 3 to be asymptotic series for a function of " which is Borel summable: this means in particular that the solution can be in principle recovered, for " > 0 and small, just from the coefficients of its formal expansion. Other non convergent expansions occur in statistical mechanics, for example in the theory of the ground state of a Fermi gas of particles on a lattice of obstacles. This is still an open problem, and a rather important one. Or occur in quantum field theory where sometimes they can be proved to be Borel summable. The simplest instances again arise in Mechanics in studying resonant quasi periodic motions. A paradigmatic case is provided by Eqs. (1),(2) when !0 has some vanishing components: !0 D (!1 ; : : : ; !r ; 0; : : : ; 0) D (e !0 ; 0) with 1 < r < `. If one writes ˛ D (e ˛; e ˇ) 2 T r  T `r and looks motions like Eq. (1) of the form e ˛(t) D e 'C! e0 t Ce a" (e 'C! e0 t) e e 'C! e0 t) ˇ(t) D ˇ 0 C b" (e

(8)

where e a" (e '); e b" (e ') are functions of e ' 2 T r , analytic in " and e '. In this case the analogue of the Lindstedt series can be devised provided ˇ 0 is chosen to be a stationary point for R de ˛ the function e f (e ˇ) D f (e ˛; e ˇ) (2) e0 satr , and provided ! isfies a Diophantine property je !0  e  j > 1/(Cje  j ) for all 0 ¤e  2 Zr and for ; C suitably chosen. This time the series is likely to be, in general, non convergent (although there is not a proof yet). And the terms of the Lindstedt series can be suitably collected to improve the estimates. Nevertheless, the estimates cannot be improved enough to obtain convergence. Deeper resummations are needed to show that in some cases the terms of

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the series can be collected and rearranged into a convergent series. The resummation is deeper in the sense that it is not enough to collect terms contributing to a given order in " but it is necessary to collect and sum terms of different order according to the following scheme. (1) The terms of the Lindstedt series are first “regularized” so that the new series is manifestly analytic in " with, however, a radius of convergence depending on the regularization. For instance one can consider only terms with lines of scale  M. (2) Terms of different orders in " are then summed together and the series becomes a series in powers of functions  j ("; M) of " with very small radius of convergence in ", but with an M-independent radius of convergence % in the  j ("; M). The labels j D 0; 1; : : : ; M are scale labels whose value is determined by the order in which they are generated in the hierarchical organization of the collection of the graphs according to their scales. (3) One shows that the functions  j ("; M) (“running couplings”) can be analytically continued in " to an M-independent domain D containing the origin in its closure and where they remain smaller than % for all M. Furthermore,  j ("; M) !  j ("), for " 2 D. M!1

(4) The convergent power series in the running couplings admits an asymptotic series in " at the origin which coincides with the formal Lindstedt series. Hence in the domain D a meaning is attributed to the sum of Lindstedt series. (5) One checks that the functions e a" ; e b" thus defined are such that Eq. (8) satisfies the equations of motion Eq. (2). The proof can be completed if the domain D contains real points ". If e ˇ 0 is a maximum point the domain D contains a circle tangent to the origin and centered on the positive real axis. So in this case thee a" ; e b" are constructed in D \ RC , def

RC D (0; C1).

If instead e ˇ 0 is a minimum point the domain D exists but D \ RC touches the positive real axis on a set of points with positive measure and density 1 at the origin. Soe a" ; e b" are constructed only for " in this set which is a kind of “Cantor set”[13]. Again the multiscale analysis is necessary to identify the tree values which have to be collected to define  j ("; M). In this case it is an analysis which is much closer to the similar analysis that is encountered in quantum field theory in the “self energy resummations”, which involve

collecting and summing graph values of graphs contributing to different orders of perturbation. The above scheme can also be applied when r D `, i. e. in the case of the classical Lindstedt series when it is actually convergent: this leads to an alternative proof of the Kolmogorov theorem which is interesting as it is even closer to the renormalization group methods because it expresses the solution in terms of a power series in running couplings [Chaps. 8, 9 in 12]. Conclusion and Outlook Perturbation theory provides a general approach to the solution of problems “close” to well understood ones, “closeness” being measured by the size of a parameter ". It naturally consists of two steps: the first is to find a formal solution, under the assumption that the quantities of interest are analytic in " at " D 0. If this results in a power series with well-defined coefficients then it becomes necessary to find whether the series thus constructed, called a formal series, converges. In general the proof that the formal series exists (when it really does) is nontrivial: typically in quantum mechanics problems (quantum fields or statistical mechanics) this is an interesting and deep problem giving rise to renormalization theory. Even in classical mechanics PT of integrable systems it has been, historically, a problem to obtain (in wide generality) the Lindstedt series (of which a simple example is discussed above). Once existence of a PT series is established, very often the series is not convergent and at best is an asymptotic series. It becomes challenging to find its meaning (if any, as there are cases, even interesting ones, on which conjectures exist claiming that the series have no meaning, like the quantum scalar field in dimension 4 with “' 4 interaction” or quantum electrodynamics). Convergence proofs, in most interesting cases, require a multiscale analysis: because the difficulty arises as a consequence of the behavior of singularities at infinitely many scales, as in the case of the Lindstedt series above exemplified. When convergence is not possible to prove, the multiscale analysis often suggest “resummations”, collecting the various terms whose sums yields the formal PT series (usually the algorithms generating the PT series give its terms at given order as sums of simple but many quantities, as in the discussed case of the Lindstedt series). The collection involves adding together terms of different order in " and results in a new power series, the resummed series, in a family of parameters  j (") which are functions of ", called the “running couplings”, depending on a “scale index” j D 0; 1; : : :.

Perturbation Theory

The running couplings are (in general) singular at " D 0 as functions of " but C 1 there, and obey equations that allow one to study and define them independently of a convergence proof. If the running couplings can be shown to be so small, as " varies in a suitable domain D near 0, to guarantee convergence of the resummed series and therefore to give a meaning to the PT for " 2 D then the PT program can be completed. The singularities in " at " D 0 are therefore all contained in the running couplings, usually very few and the same for various formal series of interest in a given problem. The idea of expressing the sum of formal series as sum of convergent series in new parameters, the running couplings, determined by other means (a recursion relation denominated the beta function flow) is the key idea of the renormalization group methods: PT in mechanics is a typical and simple example. On purpose attention has been devoted to PT in the analytic class: but it is possible to use PT techniques in problems in which the functions whose value is studied are not analytic; the techniques are somewhat different and new ideas are needed which would lead quite far away from the natural PT framework which is within the analytic class. Of course there are many problems of PT in which the formal series are simply convergent and the proof does not require any multiscale analysis. Greater attention was devoted to the novel aspect of PT that emerged in Physics and Mathematics in the last half century and therefore problems not requiring multiscale analysis were not considered. It is worth mentioning, however, that even in simple convergent PT cases it might be convenient to perform resummations. An example is Kepler’s equation ; ` 2 T 1 D [0; 2]

` D   " sin  ;

(9)

which can be (easily) solved by PT. The resulting series has a radius of convergence in " rather small (Laplace’s limit): however if a resummation of the series is performed transforming it into a power series in a “running coupling” 0 (") (only 1, because no multiscale analysis is needed, the PT series being convergent) given by [Vol. 2, p. 321 in 20] p

2

" e 1" ; 0 D p 1 C 1  "2 def

(10)

then the resummed series is a power series in 0 with radius of convergence 1 and when " varies between 0 and 1 the parameter 0 corresponding to it goes from 0 to 1. Hence in terms of 0 it is possible to invert by power series

the Kepler equation for all " 2 [0; 1), i. e. in the entire interval of physical interest (recall that " has the interpretation of eccentricity of an elliptic orbit in the 2-body problem). Resummations can improve convergence properties. Future Directions It is always hard to indicate future directions, which usually turn to different paths. Perturbation theory is an ever evolving subject: it is a continuous source of problems and its applications generate new ones. Examples of outstanding problems are understanding the triviality conjectures of models like quantum ' 4 field theory in dimension 4 [6]; or a development of the theory of the ground states of Fermionic systems in dimensions 2 and 3 [11]; a theory of weakly coupled Anosov flows to obtain information of the kind that it is possible to obtain for weakly coupled Anosov maps [12]; uniqueness issues in cases in which PT series can be given a meaning, but in a priori non unique way like the resonant quasi periodic motions in nearly integrable Hamiltonian systems [12]. Bibliography 1. Einstein A (1917) Zum Quantensatz von Sommerfeld und Epstein. Verh Dtsch Phys Ges 19:82–102 2. Eliasson LH (1996) Absolutely convergent series expansions for quasi-periodic motions. Math Phys Electron J (MPEJ) 2:33 3. Fermi E (1923) Il principio delle adiabatiche e i sistemi che non ammettono coordinate angolari. Nuovo Cimento 25:171–175. Reprinted in Collected papers, vol I, pp 88–91 4. Frölich J (1982) On the triviality of 'd4 theories and the approach to the critical point in d ( 4) dimensions. Nucl Phys B 200:281–296 5. Gallavotti G (1985) Perturbation Theory for Classical Hamiltonian Systems. In: Fröhlich J (ed) Scaling and self similarity in Physics. Birkhauser, Boston 6. Gallavotti G (1985) Renormalization theory and ultraviolet stability for scalar fields via renormalization group methods. Rev Mod Phys 57:471–562 7. Gallavotti G (1986) Quasi integrable mechanical systems. In: Phenomènes Critiques, Systèmes aleatories, Théories de jauge (Proceedings, Les Houches, XLIII (1984) vol II, pp 539–624) North Holland, Amsterdam 8. Gallavotti G (1994) Twistless KAM tori. Commun Math Phys 164:145–156 9. Gallavotti G (1995) Invariant tori: a field theoretic point of view on Eliasson’s work. In: Figari R (ed) Advances in Dynamical Systems and Quantum Physics. World Scientific, pp 117–132 10. Gallavotti G (2001) Renormalization group in Statistical Mechanics and Mechanics: gauge symmetries and vanishing beta functions. Phys Rep 352:251–272 11. Gallavotti G, Benfatto G (1995) Renormalization group. Princeton University Press, Princeton

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12. Gallavotti G, Bonetto F, Gentile G (2004) Aspects of the ergodic, qualitative and statistical theory of motion. Springer, Berlin 13. Gallavotti G, Gentile G (2005) Degenerate elliptic resonances. Commun Math Phys 257:319–362. doi:10.1007/s00220-0051325-6 14. Glimm J, Jaffe A (1981) Quantum Physics. A functional integral point of view. Springer, New York 15. Gross DJ (1999) Twenty Five Years of Asymptotic Freedom. Nucl Phys B (Proceedings Supplements) 74:426–446. doi:10.1016/S0920-5632(99)00208-X 16. Hepp K (1966) Proof of the Bogoliubov–Parasiuk theorem on renormalization. Commun Math Phys 2:301–326 17. Hepp K (1969) Théorie de la rénormalization. Lecture notes in Physics, vol 2. Springer, Berlin 18. Kolmogorov AN (1954) On the preservation of conditionally periodic motions. Dokl Akad Nauk SSSR 96:527–530 and in: Casati G, Ford J (eds) (1979) Stochastic behavior in classical and quantum Hamiltonians. Lecture Notes in Physics vol 93. Springer, Berlin 19. Laplace PS (1799) Mécanique Céleste. Paris. Reprinted by Chelsea, New York 20. Levi-Civita T (1956) Opere Matematiche, vol 2. Accademia Nazionale dei Lincei and Zanichelli, Bologna 21. Morrey CB (1955) On the derivation of the equations of hydro-

22.

23. 24.

25. 26. 27.

28. 29.

dynamics from Statistical Mechanics. Commun Pure Appl Math 8:279–326 Nelson E (1966) A quartic interaction in two dimensions. In: Goodman R, Segal I (eds) Mathematical Theory of elementary particles. MIT, Cambridge, pp 69–73 Poincaré H (1892) Les Méthodes nouvelles de la Mécanique céleste. Paris. Reprinted by Blanchard, Paris, 1987 Pöschel J (1986) Invariant manifolds of complex analytic mappings. In: Osterwalder K, Stora R (eds) Phenomènes Critiques, Systèmes aleatories, Théories de jauge (Proceedings, Les Houches, XLIII (1984); vol II, pp 949–964) North Holland, Amsterdam Siegel K (1943) Iterations of analytic functions. Ann Math 43:607–612 Simon B (1974) The P(')2 Euclidean (quantum) field theory. Princeton University Press, Princeton ’t Hooft G (1999) When was asymptotic freedom discovered? or The rehabilitation of quantum field theory. Nucl Phys B (Proceedings Supplements) 74:413–425. doi:10.1016/S09205632(99)00207-8 ’t Hooft G, Veltman MJG (1972) Regularization and renormalization of gauge fields, Nucl Phys B 44:189–213 Wilson K, Kogut J (1973) The renormalization group and the "-expansion. Phys Rep 12:75–199

Perturbation Theory in Celestial Mechanics

Perturbation Theory in Celestial Mechanics ALESSANDRA CELLETTI Dipartimento di Matematica, Università di Roma Tor Vergata, Roma, Italy Article Outline Glossary Definition of the Subject Introduction Classical Perturbation Theory Resonant Perturbation Theory Invariant Tori Periodic Orbits Future Directions Bibliography Glossary KAM theory Provides the persistence of quasi-periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i. e. a non-degeneracy of the integrable system and a diophantine condition of the frequency of motion. It yields a constructive algorithm to evaluate the strength of the perturbation ensuring the existence of invariant tori. Perturbation theory Provides an approximate solution of the equations of motion of a nearly-integrable system. Spin-orbit problem A model composed of a rigid satellite rotating about an internal axis and orbiting around a central point-mass planet; a spin-orbit resonance means that the ratio between the revolutional and rotational periods is rational. Three-body problem A system composed by three celestial bodies (e. g. Sun-planet-satellite) assumed to be point-masses subject to the mutual gravitational attraction. The restricted three-body problem assumes that the mass of one of the bodies is so small that it can be neglected. Definition of the Subject Perturbation theory aims to find an approximate solution of nearly-integrable systems, namely systems which are composed by an integrable part and by a small perturbation. The key point of perturbation theory is the construction of a suitable canonical transformation which removes the perturbation to higher orders. A typical exam-

ple of a nearly-integrable system is provided by a two-body model perturbed by the gravitational influence of a third body whose mass is much smaller than the mass of the central body. Indeed, the solution of the three-body problem greatly stimulated the development of perturbation theories. The solar system dynamics has always been a testing ground for such theories, whose applications range from the computation of the ephemerides of natural bodies to the development of the trajectories of artificial satellites. Introduction The two-body problem can be solved by means of Kepler’s laws, according to which for negative energies the pointmass planets move on ellipses with the Sun located in one of the two foci. The dynamics becomes extremely complicated when adding the gravitational influence of another body. Indeed Poincaré showed [34] that the three-body problem does not admit a sufficient number of prime integrals which allow to integrate the problem. Nevertheless, the so-called restricted three-body problem deserves special attention, namely when the mass of one of the three bodies is so small that its influence on the others can be neglected. In this case one can assume that the primaries move on Keplerian ellipses around their common barycenter; if the mass of one of the primaries is much larger than the other (as it is the case in any Sunplanet sample), the motion of the minor body is governed by nearly-integrable equations, where the integrable part represents the interaction with the major body, while the perturbation is due to the influence of the other primary. A typical example is provided by the motion of an asteroid under the gravitational attraction of the Sun and Jupiter. The small body may be taken not to influence the motion of the primaries, which are assumed to move on elliptic trajectories. The dynamics of the asteroid is essentially driven by the Sun and perturbed by Jupiter, since the Jupiter-Sun mass-ratio amounts to about 103 . The solution of this kind of problem stimulated the work of many scientists, especially in the XVIII and XIX centuries. Indeed, Lagrange, Laplace, Leverrier, Delaunay, Tisserand and Poincaré developed perturbation theories which are the basis of the studies of the dynamics of celestial bodies, from the computation of the ephemerides to the recent advances in flight dynamics. For example, on the basis of perturbation theory Delaunay [16] developed a theory of the Moon, providing very refined ephemerides. Celestial Mechanics greatly motivated the advances of perturbation theories as witnessed by the discovery of Neptune: its position was theoretically predicted by John Adams and by Jean Urbain Leverrier on the basis of perturbative com-

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putations; following the suggestion provided by the theoretical investigations, Neptune was finally discovered on 23 September 1846 by the astronomer Johann Gottfried Galle. The aim of perturbation theory is to implement a canonical transformation which allows one to find the solution of a nearly-integrable system within a better degree of approximation (see Sect. “Classical Perturbation Theory” and references [3,6,20,24,32,37,38]. Let us denote the frequency vector of the system by ! (see “Normal Forms in Perturbation Theory”, “Kolmogorov– Arnol’d“Moser (KAM Theory)”), which we assume to belong to Rn , where n is the number of degrees of freedom of the system. Classical perturbation theory can be implemented provided that the frequency vector satisfies a nonresonant relation, which means that there does not exist P a vector m 2 Zn such that !  m  njD1 ! j m j D 0. In case there exists such commensurability condition, a resonant perturbation theory can be developed as outlined in Sect. “Resonant Perturbation Theory”. In general, the three-body problem (and, more extensively, the N-body problem) is described by a degenerate Hailtonian system, which means that the integrable part (i. e., the Keplerian approximation) depends on a subset of the action variables. In this case a degenerate perturbation theory must be implemented as explained in Subsect. “Degenerate Perturbation Theory”. For all the above perturbation theories (classical, resonant and degenerate) an application to Celestial Mechanics is given: the precession of the perihelion of Mercury, orbital resonances within a three-body framework, the precession of the equinoxes. Even if the non-resonance condition is satisfied, the quantity !  m can become arbitrarily small, giving rise to the so-called small divisor problem; indeed, these terms appear in the denominator of the series defining the canonical transformations necessary to implement perturbation theory and therefore they might prevent the convergence of the series. In order to overcome the small divisor problem, a breakthrough came with the work of Kolmogorov [26], and was later extended to different mathematical settings by Arnold [2] and Moser [33]. The overall theory is known as the acronym KAM theory. As far as concrete estimates on the allowed size of the perturbation are concerned, the original versions of the theory gave discouraging results, which were extremely far from the physical measurements of the parameters involved in the proof. Nevertheless the implementation of computerassisted KAM proofs allowed one to obtain results which are in good agreement with reality. Concrete estimates with applications to Celestial Mechanics are reported in Sect. “Invariant Tori”.

In the framework of nearly-integrable systems, a very important role is provided by periodic orbits, which might be used to approximate the dynamics of quasi-periodic trajectories; for example, a truncation of the continued fraction expansion of an irrational frequency provides a sequence of rational numbers, which are associated to periodic orbits eventually approximating a quasi-periodic torus. A classical computation of periodic orbits using a perturbative approach is provided in Sect. “Periodic Orbits”, where an application to the determination of the libration in longitude of the Moon is reported. Classical Perturbation Theory The Classical Theory Consider a nearly-integrable Hamiltonian function of the form H(I; ') D h(I) C " f (I; ') ;

(1)

where h and f are analytic functions of I 2 V (V is an open set of Rn ) and ' 2 T n (Tn is the standard n-dimensional torus), while " > 0 is a small parameter which measures the strength of the perturbation. The aim of perturbation theory is to construct a canonical transformation, which allows to remove the perturbation to higher orders in the perturbing parameter. To this end, let us look for a canonical change of variables (i. e., with symplectic Jacobian matrix) C : (I; ') ! (I 0 ; ' 0 ), such that the Hamiltonian (1) takes the form H 0 (I 0 ; ' 0 ) D H ı C (I; ')  h0 (I 0 ) C "2 f 0 (I 0 ; ' 0 ) ;

(2)

where h0 and f 0 denote the new unperturbed Hamiltonian and the new perturbing function, respectively. To achieve such a result we need to proceed along the following steps: build a suitable canonical transformation close to the identity, perform a Taylor series expansion in the perturbing parameter, require that the unknown transformation removes the dependence on the angle variables up to second order terms, and expand in a Fourier series in order to get an explicit form of the canonical transformation. The change of variables is defined by the equations I D I0 C " 0

' D' C"

@˚ (I 0 ; ') @' @˚ (I 0 ; ') @I 0

(3) ;

where ˚(I 0 ; ') is an unknown generating function, which is determined so that (1) takes the form (2). Decompose the perturbing function as f (I; ') D f0 (I) C f˜(I; ') ;

Perturbation Theory in Celestial Mechanics

where f 0 is the average over the angle variables and f˜ is the remainder function defined through f˜(I; ')  f (I; ')  f0 (I). Define the frequency vector ! D !(I) as !(I) 

@h(I) : @I

Inserting the transformation (3) in (1) and expanding in a Taylor series around " D 0 up to the second order, one gets h I0 C "

@˚ (I 0 ; ')

! C "f

@'

D h(I 0 ) C !(I 0 )  "

I0 C "

@˚ (I 0 ; ') @'

@˚ (I 0 ; ') @'

! ;'

C " f0 (I 0 )

We stress that this algorithm is constructive in the sense that it provides an explicit expression for the generating function and for the transformed Hamiltonian. We remark that (6) is well defined unless there exists an integer vector m 2 I such that !(I 0 )  m D 0 : On the contrary, if ! is rationally independent, there are no zero divisors in (6), though these terms can become arbitrarily small with a proper choice of the vector m. This problem is known as the small divisor problem, which can prevent the implementation of perturbation theory (see “Normal Forms in Perturbation Theory”, “Kolmogorov– Arnol’d–Moser (KAM Theory)”, “Perturbation Theory”).

C " f˜(I 0 ; ') C O("2 ) : The Precession of the Perihelion of Mercury The new Hamiltonian is integrable up to O("2 ) provided that the function ˚ satisfies: !(I 0 ) 

@˚ (I 0 ; ') @'

C f˜(I 0 ; ') D 0 :

(4)

In such case the new integrable part becomes h0 (I 0 ) D h(I 0 ) C " f0 (I 0 ) ; which provides a better integrable approximation with respect to (1). The solution of (4) yields the explicit expression of the generating function. In fact, let us expand ˚ and f˜ in Fourier series as X ˚ˆ m (I 0 )eim' ; ˚(I 0 ; ') D m2Z n nf0g

f˜(I 0 ; ') D

X

(5)

fˆm (I 0 )eim' ;

m2I

where I denotes the set of integer vectors corresponding to the non-vanishing Fourier coefficients of f˜. Inserting the above expansions in (4) one obtains X X !(I 0 )  m˚ˆ m (I 0 )eim' D  fˆm (I 0 )eim' ; i m2I

m2Z n nf0g

As an example of the implementation of classical perturbation theory we consider the computation of the precession of the perihelion in a (restricted, planar, circular) three-body model, taking as a sample the planet Mercury. The computation requires the introduction of Delaunay action-angle variables, the definition of the three-body Hamiltonian, the expansion of the perturbing function and the implementation of classical perturbation theory (see [7,39]). Delaunay Action-Angle Variables We consider two bodies, say P0 and P1 with masses, respectively, m0 , m1 ; let M  m0 C m1 and let  > 0 be a positive parameter. Let r be the orbital radius and ' be the longitude of P1 with respect to P0 ; let (I r ; I' ) be the momenta conjugated to (r; '). In these coordinates the two-body problem Hamiltonian takes the form ! 2 I 1 M ' I2 C 2  : (7) H2b (I r ; I' ; r; ') D 2 r r r On the orbital plane we introduce the planar Delaunay action-angle variables (; ; ; ) as follows [12]. Let E denote the total mechanical energy; then: s

which provides fˆm (I 0 ) : ˚ˆ m (I 0 ) D  i!(I 0 )  m

Ir D

Casting together the above formule, the generating function is given by ˚(I 0 ; ') D i

X m2I

fˆm (I 0 ) im' e : !(I 0 )  m

(6)

2E C

22 M I'2  2 : r r

Since p (7) does not depend on ', setting  D I' and  D (3 M 2 )/(2E), we introduce a generating function of the form Z r 4 M 2 22 M  2 F(; ; r; ') D  C  2 dr C ' : 2  r r

1303

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Perturbation Theory in Celestial Mechanics

From the definition of  the new Hamiltonian H2D becomes 3 M 2 ; H2D (; ; ; ) D  22 where (;  ) are the Delaunay action variables; by Kepler’s laws one finds that (;  ) are related to the semimajor axis a and to the eccentricity e of the Keplerian orbit of P1 around P0 by the formula: p  D  Ma;

p  D  1  e2 :

Concerning the conjugated angle variables, we start by introducing the eccentric anomaly u as follows: build the auxiliary circle of the ellipse, draw the line through P1 perpendicular to the semi-major axis whose intersection with the auxiliary circle forms at the origin an angle u with the semi-major axis. By the definition of the generating function, one finds Z 4 M 2 @F q dr D D 4 2 2 2 @ 3   M2 C 2r M  r 2

three-body problem becomes 1 H3b (; ; ; ; t) D  2 2 0 B C " @r1 cos('  t)  q

1 1

1 C r12  2r1 cos('  t)

C A;

where r1 is the distance between P0 and P1 . The first term of the perturbation comes out from the choice of the reference frame, while the second term is due to the interaction with the external body. Since ' t D f C t, we perform the canonical change of variables LD

`D

GD

g D t;

which provides the following two degrees-of-freedom Hamiltonian H3D (L; G; `; g) D 

1  G C "R(L; G; `; g) ; 2L2

(8)

where

D u  e sin u ; R(L; G; `; g) which defines the mean anomaly  in terms of the eccentric anomaly u. In a similar way, if f denotes the truepanomaly related to the eccentric anomaly by tan f /2 D (1 C e)/(1  e) tan u/2, then one has: Z  @F q dr D'

D 4 M2 2 2 @  r2  2 C 2r M  r 2

 r1 cos('  t)  q

1

(9)

1 C r12  2r1 cos('  t)

where r1 and '  t must be expressed in terms of the Delaunay variables (L; G; `; g). Notice that when " D 0 one obtains the integrable Hamiltonian function h(L; G)  1/(2L2 )  G with associated frequency vector ! D (@h/ @L; @h/@G) D (1/L3 ; 1).

D'f ; which represents the argument of the perihelion of P1 , i. e. the angle between the perihelion line and a fixed reference line. The Restricted, Planar, Circular, Three-Body Problem Let P0 , P1 , P2 be three bodies with masses m0 , m1 , m2 , respectively. We assume that m1 is much smaller than m0 and m2 (restricted problem) and that the motion of P2 around P0 is circular. We also assume that the three bodies always move on the same plane. We choose the free parameter  as   1/m2/3 0 , so that the two-body Hamiltonian becomes H2D D 1/(22 ), while we introduce the perturbing parameter as "  m2 /m2/3 0 [12]. Set the units of measure so that the distance between P0 and P2 is one and so that m0 C m2 D 1. Taking into account the interaction of P2 on P1 , the Hamiltonian function governing the

Expansion of the Perturbing Function We expand the perturbing function (9) in terms of the Legendre polynomials Pj obtaining RD

1 1 X 1 Pj (cos('  t)) j : r1 r jD2

1

The explicit expressions of the first few Legendre polynomials are: 1 3 C cos 2('  t) 4 4 3 5 P3 (cos('  t)) D cos('  t) C cos 3('  t) 8 8 9 5 P4 (cos('  t)) D C cos 2('  t) 64 16 35 C cos 4('  t) 64 P2 (cos('  t)) D

Perturbation Theory in Celestial Mechanics

P5 (cos('  t)) D

15 35 cos('  t) C cos 3('  t) 64 128 63 C cos 5('  t) : 128

We invert Kepler’s equation ` D u  e sin u to the second order in the eccentricity as u D ` C e sin ` C

e2 2

Computation of the Precession of the Perihelion We identify the three bodies P0 , P1 , P2 with the Sun, Mercury, and Jupiter, respectively. Taking " as perturbing parameter, we implement a first order perturbation theory, which provides a new integrable Hamiltonian function of the form h0 (L0 ; G 0 ) D 

sin(2`) C O(e 3 ) ;

From Hamilton’s equations one obtains

from which one gets g˙ D 5 '  t D g C ` C 2e sin ` C e 2 sin 2` C O(e 3 ) 4   1 1 r1 D a 1 C e 2  e cos `  e 2 cos 2` C O(e 3 ) : 2 2 Then, up to inessential constants the perturbing function can be expanded as R D R00 (L; G) C R10 (L; G) cos ` C R11 (L; G) cos(` C g) C R12 (L; G) cos(` C 2g) C R22 (L; G) cos(2` C 2g) C R32 (L; G) cos(3` C 2g) C R33 (L; G) cos(3` C 3g) C R44 (L; G) cos(4` C 4g) C R55 (L; G) cos(5` C 5g)

@h0 (L0 ; G 0 ) @R00 (L0 ; G 0 ) D 1 C " ; @G 0 @G 0

neglecting O(e 3 ) in R00 and recalling that g D  t, one has

˙ D "

3 @R00 (L0 ; G 0 ) D "L02 G 0 : @G 0 4

Notice that to the first order in " one has L0 D L; G 0 D G. The astronomical data are m0 D 2  1030 kg, m2 D 1:9  1027 kg, which give " D 9:49  104 ; setting to one the Jupiter–Sun distance one has a D 0:0744, while e D 0:2056. Taking into account that the orbital period of Jupiter amounts to about 11.86 years, one obtains

C ::: ; (10) where the coefficients Rij are given by the following exp pressions (recall that e D 1  G 2 /L2 ):   L4 3 9 1 C L4 C e 2 C : : : ; 4 16 2   4 L e 9 D 1 C L4 C : : : 2 8   3 5 D  L6 1 C L4 C : : : ; 8 8  L4 e  9 C 5L4 C : : : D 4   L4 5 D 3 C L4 C : : : ; 4 4 3 4 D  L e C ::: 4   5 6 7 D  L 1 C L4 C : : : ; 8 16 35 8 D  L C ::: 64 63 10 D L C ::: : 128

R00 D  R10 R11 R12 R22 R32 R33 R44 R55

1  G 0 C "R00 (L0 ; G 0 ) : 2L02

˙ D 154:65

arcsecond ; century

which represents the contribution due to Jupiter to the precession of perihelion of Mercury. The value found by Leverrier on the basis of the data available in the year 1856 was of 152.59 arcsecond/century [14]. Resonant Perturbation Theory The Resonant Theory Let us consider an Hamiltonian system with n degrees of freedom of the form H(I; ') D h(I) C " f (I; ')

(11)

and let ! j (I) D (@h(I))/(@I j ) ( j D 1; : : : ; n) be the frequencies of the motion, which we assume to satisfy `, ` < n, resonance relations of the form !  mk D 0

for k D 1; : : : ; ` ;

for suitable rational independent integer vectors m1 , . . . , m` . A resonant perturbation theory can be implemented to eliminate the non-resonant terms. More precisely, the aim is to construct a canonical transformation

1305

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Perturbation Theory in Celestial Mechanics

C : (I; ') ! (J 0 ; # 0 ) such that the transformed Hamilto-

nian takes the form H 0 (J 0 ; # 0 ) D h0 (J 0 ; #10 ; : : : ; #`0 ) C "2 f 0 (J 0 ; # 0 ) ;

(12)

j D 1; : : : ; `

#k D m k  '

k D ` C 1; : : : ; n ;

where the first ` angle variables are the resonant angles, while the latter n  ` angle variables are defined as suitable linear combinations so to make the transformation canonical together with the following change of coordinates on the actions J 2 Rn : Ij D mj  J

j D 1; : : : ; `

Ik D mk  J

k D ` C 1; : : : ; n :

The aim is to construct a canonical transformation which removes (to higher order) the dependence on the short-period angles (#`C1 ; : : : ; # n ), while the lowest order Hamiltonian will necessarily depend upon the resonant angles. Let us decompose the perturbation as f (J; #) D f (J) C f r (J; #1 ; : : : ; #` ) C f n (J; #) ;

n X kD1

where h0 depends only on the resonant angles #10 ; : : : ; #`0 . To this end, let us first introduce the angles # 2 T n as #j D mj  '

provided that ! k0

@˚ D  f n (J 0 ; #) ; @# k

where ! k0 D ! k0 (J 0 )  (@h(J 0 ))/(@J 0k ). The solution of (15) gives the generating function, which allows one to reduce the Hamiltonian to the required form (12); as a consequence, the conjugated action variables, say J`C10 , . . . , J n0 , are constants of the motion up to the second order in ". We conclude by mentioning that using the new frequencies ! k0 , the resonant relations take the form ! k0 D 0 for k D 1; : : : ; `. Three-Body Resonance We consider the three-body Hamiltonian (8) with perturbing function (10)–(11) and let !  (!` ; ! g ) be the frequency of motion. We assume that the frequency vector satisfies the resonance relation !` C 2! g D 0 : According to the theory described in the previous section we perform the canonical change of variables

(13) #1 D ` C 2g

where f is the average of the perturbation over the angles, f r is the part depending on the resonant angles and f n is the non-resonant part. In analogy to the classical perturbation theory, we implement a canonical transformation of the form @˚ 0 (J ; #) @# @˚ # 0 D # C " 0 (J 0 ; #) ; @J J D J0 C "

#2 D 2`

kD1

0

C " f n (J ; #) C O("2 ) : Equating same orders of " one gets that h0 (J 0 ; #10 ; : : : ; #`0 ) D h(J 0 ) C " f (J 0 ) C " f r (J 0 ; #10 ; : : : ; #`0 ) ; (14)

1 G 2 1 1 J2 D L  G : 2 4

J1 D

In the new coordinates the unperturbed Hamiltonian becomes h0 (J)  

such that the new Hamiltonian takes the form (12). Taking into account (13) and developing up to the second order in the perturbing parameter, one obtains:   @˚ C " f (J 0; #) C O("2 ) h J0 C " @# n X @h @˚ D h(J 0 ) C " C " f (J 0 ) C " f r (J 0 ; #1 ; : : : ; #` ) @J k @# k

(15)

1  2J1 ; 2(J1 C 2J2 )2

with frequency vector ! 0  takes the form

@h 0 (J) , @J

while the perturbation

  1 #2 R(J1 ; J2 ; #1 ; #2 )  R00 (J) C R10 (J) cos 2   1 1 #1 C #2 C R12 (J) cos(#1 ) C R11 (J) cos 2 4   1 C R22 (J) cos #1 C #2 C R32 (J) cos(#1 C #2 ) 2   3 3 #1 C #2 C R44 (J) cos(2#1 C #2 ) C R33 (J) cos 2 4   5 5 C R55 (J) cos #1 C #2 C : : : 2 4 with the coefficients Rij as in (11). Let us decompose the perturbation as R D R(J) C Rr (J; #1 ) C R n (J; #),

Perturbation Theory in Celestial Mechanics

where R(J) is the average over the angles, Rr (J; #1 ) D R12 (J) cos(#1 ) is the resonant part, while Rn contains all the remaining non-resonant terms. We look for a canonical transformation close to the identity with generating function ˚ D ˚(J 0 ; #) such that @˚ (J 0 ; #) D R n (J 0 ; #) ; ! 0 (J 0 )  @# which is well defined since ! 0 is non-resonant for the Fourier components appearing in Rn . Finally, according to (14) the new unperturbed Hamiltonian is given by h0 (J 0 ; #10 )  h(J 0 ) C "R00 (J 0 ) C "R12 (J 0 ) cos #10 :

while ˚ is determined solving the equation d X @h @˚ C f˜(I 0 ; ') D 0 : @I k @' k kD1

Expanding ˚ and f˜ in Fourier series as in (5) one obtains P that ˚ is given by (6) where !  m D dkD1 m k ! k , being ! k D 0 for k D d C 1; : : : ; n. The generating function is well defined provided that !  m 6D 0 for any m 2 I , which is equivalent to requiring that d X

m k ! k 6D 0

for m 2 I :

kD1

Degenerate Perturbation Theory

The Precession of the Equinoxes

A special case of resonant perturbation theory is obtained when considering a degenerate Hamiltonian function with n degrees of freedom of the form

An example of the application of the degenerate perturbation theory in Celestial Mechanics is provided by the computation of the precession of the equinoxes. We consider a triaxial rigid body moving in the gravitational field of a primary body. We introduce the following reference frames with a common origin in the barycen(i) (i) ter of the rigid body: (O; i (i) 1 ; i 2 ; i 3 ) is an inertial ref(b) (b) erence frame, (O; i (b) 1 ; i 2 ; i 3 ) is a body frame oriented along the direction of the principal axes of the ellipsoid, (s) (s) (O; i (s) 1 ; i 2 ; i 3 ) is the spin reference frame with the vertical axis along the direction of the angular momentum. Let (J; g; `) be the Euler angles formed by the body and spin frames, and let (K; h; 0) be the Euler angles formed by the spin and inertial frames. The angle K is the obliquity (representing the angle between the spin and inertial vertical axes), while J is the non-principal rotation angle (representing the angle between the spin and body vertical axes). This problem is conveniently described in terms of the following set of action-angle variables introduced by Andoyer in [1] (see also [17]). Let M 0 be the angular momentum and let M0  jM 0 j; the action variables are defined as

H(I; ') D h(I1 ; : : : ; I d ) C " f (I; ');

d < n ; (16)

notice that the integrable part depends on a subset of the action variables, being degenerate in I dC1 , . . . , I n . In this case we look for a canonical transformation C : (I; ') ! (I 0 ; ' 0 ) such that the transformed Hamiltonian becomes 0 H 0 (I 0 ; ' 0 ) D h0 (I 0 )C "h10 (I; 'dC1 ; : : : ; 'n0 )C "2 f 0 (I 0 ; ' 0 );

(17) where the part h0 C "h10 admits d integrals of motion. Let us decompose the perturbing function in (16) as f (I; ') D f (I) C f d (I; 'dC1 ; : : : ; 'n ) C f˜(I; ') ; (18) where f is the average over the angle variables, f d is independent on ' 1 , . . . , ' d and f˜ is the remainder. As in the previous sections we want to determine a near-to-identity canonical transformation ˚ D ˚(I 0 ; ') of the form (3), such that in view of (18) the Hamiltonian (16) takes the form (17). One obtains h(I10 ; : : : ; I d0 ) C "

d X @h @˚ C " f (I 0 ) @I k @' k kD1

C " f d (I ; 'dC1 ; : : : ; 'n ) C " f˜(I 0 ; ') C O("2 ) 0

D h0 (I 0 ) C "h10 (I 0 ; 'dC1 ; : : : ; 'n ) C O("2 ) ; where h0 (I 0 ) D h(I10 ; : : : ; I d0 ) C " f (I 0 ) h10 (I 0 ; 'dC1 ; : : : ; 'n ) D f d (I 0 ; 'dC1 ; : : : ; 'n ) ;

G  M 0  i (s) 3 D M0 L  M 0  i (b) 3 D G cos J H  M 0  i (i) 3 D G cos K ; while the corresponding angle variables are the quantities (g; `; h) introduced before. We limit ourselves to consider the gyroscopic case in which I1 D I2 < I3 are the principal moments of inertia of the rigid body E around the primary S; let mE and mS be their masses and let jEj be the volume of E. We assume that E orbits on a Keplerian ellipse around S with semimajor axis a and eccentricity e, while E and r E denote the

1307

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Perturbation Theory in Celestial Mechanics

longitude and instantaneous orbital radius (due to the assumption of Keplerian motion E and r E are known functions of the time). The Hamiltonian describing the motion of E around S is given by [15] H (L; G; H; `; g; h; t)

D

I1  I3 2 G2 C L C V(L; G; H; `; g; h; t) ; 2I1 2I1 I3

where the perturbation is implicitly defined by Z ˜ Gm S m E dx ; V  E jr E C xj jEj G˜ being the gravitational constant. Setting r E D jr E j and x D jxj, we can expand V using the Legendre polynomials as " ˜ S m E Z dx Gm 1 xr V D 1 2E C 2 rE rE 2r E E jEj #    3 ! x (x  r E )2 2  3 x : CO 2 rE rE We further assume that J D 0 (i. e. G D L) so that E rotates around a principal axis. Let G0 and H 0 be the initial values of G and H; if ˛ denotes the angle between r E and i (b) 3 , the perturbing function can be written as VD

G 2 (1  e cos E )3 3 cos2 ˛ ! 0 2 H0 (1  e2 )3

Neglecting first order terms in the eccentricity, we approximate (1  e cos E )3 /(1  e 2 )3 with one. A first order degenerate perturbation theory provides that the new unperturbed Hamiltonian is given by G2 3 G2 G2  H2 C ! 0 : 2I3 2 H0 2G 2

Therefore the average angular velocity of precession is given by G2 H 3 @K (G; H) D  ! 0 2 : h˙ D @H 2 H0 G At t D 0 it is 3 3 h˙ D  ! D  ! 2y !d1 cos K ; 2 2

Invariant Tori Invariant KAM Surfaces We consider an n-dimensional nearly-integrable Hamiltonian function H(I; ') D h(I) C " f (I; ') ; defined in a 2n-dimensional phase space M  V  T n , where V is an open bounded region of Rn . A KAM torus associated to H is an n-dimensional invariant surface on which the flow is described parametrically by a coordinate 2 T n such that the conjugated flow is linear, namely 2 T n ! C ! t where ! 2 Rn is a Diophantine vector, i. e. there exist > 0 and  > 0 such that j!  mj 

˜ S )/a3 I3 H0 /G 2 . Elewith  D (I3  I1 )/I3 and ! D (Gm 0 mentary computations show that s H2 cos ˛ D sin(E  h) 1  2 : G

K (G; H) D

where we used ! D ! 2y !d1 cos K with ! y being the frequency of revolution and ! d the frequency of rotation. In the case of the Earth, the astronomical measurements show that  D 1/298:25, K = 23.45°. The contribution due to the Sun is thus obtained by inserting ! y D 1 year, !d D 1 day in (19), which yields h˙ (S) D 2:51857  1012 rad/sec, corresponding to a retrograde precessional period of 79 107.9 years. A similar computation shows that the contribution of the Moon amounts to h˙ (L) D 5:49028  1012 rad/sec, corresponding to a precessional period of 36 289.3 years. The total amount is obtained as the sum of h˙ (S) and h˙ (L) , providing an overall retrograde precessional period of 24 877.3 years.

(19)

; jmj

8m 2 Zn nf0g :

Kolmogorov’s theorem [26] (see also “Kolmogorov– Arnol’d–Moser (KAM Theory)”) ensures the persistence of invariant tori with diophantine frequency, provided " is sufficiently small and provided the unperturbed Hamiltonian is non-degenerate, i. e. for a given torus fI 0 g  T n

M

det h00 (I 0 )  det



 @2 h (I 0 ) ¤0: @I i @I j i; jD1;:::;n

(20)

The condition (20) can be replaced by the isoenergetic non-degeneracy condition introduced by Arnold [2]  00  h (I0 ) h0 (I0 ) det ¤0; (21) 0 h0 (I0 ) which ensures the existence of KAM tori on the energy level corresponding to the unperturbed energy h(I 0 ), say M0  f(I; ') 2 M : H(I; ') D h(I 0 )g. In the context of the n-body problem Arnold [2] addressed the question of the existence of a set of initial conditions with positive measure such that, if the initial position and velocities of

Perturbation Theory in Celestial Mechanics

the bodies belong to this set, then the mutual distances remain perpetually bounded. A positive answer is provided by Kolmogorov’s theorem in the framework of the planar, circular, restricted three-body problem, since the integrable part of the Hamiltonian (8) satisfies the isoenergetic non-degeneracy condition (21); denoting the initial values of the Delaunay’s action variables by (L0 ; G0 ), if " is sufficiently small, there exist KAM tori for (8) on the energy level M0  fH3D D 1/(2L20 )  G0 g. In particular, the motion of the perturbed body remains forever bounded from the orbits of the primaries. Indeed, a stronger statement is also valid: due to the fact that the two-dimensional KAM surfaces separate the three dimensional energy levels, any trajectory starting between two KAM tori remains forever trapped in the region between such tori. In the framework of the three-body problem, Arnold [2] stated the following result: “If the masses, eccentricities and inclinations of the planets are sufficiently small, then for the majority of initial conditions the true motion is conditionally periodic and differs little from Lagrangian motion with suitable initial conditions throughout an infinite interval time 1 < t < 1”. Arnold provided a complete proof for the case of three coplanar bodies, while the spatial three-body problem was investigated by Laskar and Robutel in [27,35] using Poincaré variables, the Jacobi’s “reduction of the nodes” (see, e. g., [11]) and Birkhoff’s normal form [3,4,38]. The full proof of Arnold’s theorem was provided in [19], based on Herman’s results on the planetary problem; it makes use of Poincaré variables restricted to the symplectic manifold of vertical total angular momentum. Explicit estimates on the perturbing parameter ensuring the existence of KAM tori were given by M. Hénon [25]; he showed that direct applications of KAM theory to the three-body problem lead to analytical results which are much smaller than the astronomical observations. For example, the application of Arnold’s theorem to the restricted three-body problem is valid provided the mass-ratio of the primaries is less than 10333 . This result can be improved up to 1048 by applying Moser’s theorem, but it is still very far from the actual Jupiter-Sun mass-ratio which amounts to about 103 . In the context of concrete estimates, a big improvement comes from the synergy between KAM theory and computer-assisted proofs, based on the application of interval arithmetic which allows to keep rigorously track of the rounding-off and propagation errors introduced by the machine. Computer-assisted KAM estimates were implemented in a number of cases in Celestial Mechanics, like the three-body problem and the spin-orbit model as briefly recalled in the following subsections.

Another interesting example of the interaction between the analytical theory and the computer implementation is provided by the analysis of the stability of the triangular Lagrangian points; in particular, the stability for exponentially long times is obtained using Nekhoroshev theory combined with computer-assisted implementations of Birkhoff normal form (see, e. g., [5,13,18,21,22,23,28,36]). Rotational Tori for the Spin-Orbit Problem We study the motion of a rigid triaxial satellite around a central planet under the following assumptions [8]: i) ii) iii) iv)

The orbit of the satellite is Keplerian, The spin-axis is perpendicular to the orbital plane, The spin-axis coincides with the smallest physical axis, External perturbations as well as dissipative forces are neglected.

Let I1 < I2 < I3 be the principal moments of inertia; let a, e be the semi-major axis and eccentricity of the Keplerian ellipse; let r and f be the instantaneous orbital radius and the true anomaly of the satellite; let x be the angle between the longest axis of the triaxial satellite and the periapsis line. The equation of motion governing the spinorbit model is given by: x¨ C

3 I 2  I 1  a 3 sin(2x  2 f ) D 0 : 2 I3 r

(22)

Due to assumption i, the quantities r and f are known functions of the time. Expanding the second term of (22) in Fourier–Taylor series and neglecting terms of order 6 in the eccentricity, setting y  x˙ one obtains that the equation of motion corresponds to Hamilton’s equations associated to the Hamiltonian H(y; x; t) 

  e e3 5 5 y2 "  C  e cos(2x  t) 2 4 32 768   1 5 2 13 4  e C e cos(2x  2t) C 2 4 32   123 3 489 5 7 e e C e cos(2x  3t) C 4 32 256   17 2 115 4 C e  e cos(2x  4t) 4 12   845 3 32525 5 C e  e cos(2x  5t) 96 1536 533 4 e cos(2x  6t) C 32  228347 5 e cos(2x  7t) ; C 7680 (23)

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where "  3/2(I2  I1 )/I3 and we have chosen the units so that a D 1, 2/Trev D 1, where Trev is the period of revolution. Let p, q be integers with q 6D 0; a p : q resop nance occurs whenever hx˙ i D q , meaning that during q orbital revolutions, the satellite makes on average p rotations. Since the phase-space is three-dimensional, the twodimensional KAM tori separates the phase-space into invariant regions, thus providing the stability of the trapped p orbits. In particular, let P ( q ) be the periodic orbit associated to the p : q resonance; its stability is guaranteed by the existence of trapping rotational tori with frequencies T (!1 ) and T (!2 ) with !1 < p/q < !2 . For example, one can consider the sequences of irrational rotation numbers (p/q)

k



p 1  ; q kC˛

(p/q)

k



p 1 C ; q kC˛

k 2 Z; k  2 p with ˛  ( 5  1)/2. In fact, the continued fraction expansion of 1/(k C ˛) is given by 1/(k C ˛) D [0; k; 11 ]. (p/q) (p/q) Therefore, both  k and  k are noble numbers (i. e. with continued fraction expansion definitely equal to one); by number theory they satisfy the diophantine condition p and bound q from below and above. As a concrete sample we consider the synchronous spin-orbit resonance (p D q D 1) of the Moon, whose physical values of the parameters are "  3:45  104 and e D 0:0549. The stability of the motion is guaranteed by the existence of the surfaces T (40(1) ) and T ((1) 40 ), which is obtained by implementing a computer-assisted KAM theory for the realistic values of the parameters. The result provides the confinement of the synchronous periodic orbit in a limited region of the phase space [8].

where !  !(") is the frequency of the harmonic oscillator, while h(I) and R(I; '; t) are suitable functions, precisely polynomials in the action (of order of the square of the action). Then apply a Birkhoff normal form (see – Normal Forms in Pertubation Theory–) up to the order k (k D 5 in [9]) to obtain the following Hamiltonian: H k (I 0 ; ' 0 ; t) D !I 0 C "h k (I 0 ; ") C " kC1 R k (I 0 ; ' 0 ; t) : Finally, implementing a computer-assisted KAM theorem one gets the following result: consider the Moon–Earth case with "obs D 3:45104 and e D 0:0549; there exists an invariant torus around the synchronous resonance corresponding to a libration of 8:79ı for any "  "obs /5:26. The same strategy applied to different samples, e. g. the Rhea– Saturn pair, allows one to prove the existence of librational invariant tori around the synchronous resonance for values of the parameters in full agreement with the observational measurements [9]. Rotational Tori for the Restricted Three-Body Problem The planar, circular, restricted three-body problem has been considered in [12], where the stability of the asteroid 12 Victoria has been investigated under the gravitational influence of the Sun and Jupiter. On a fixed energy level invariant KAM tori trapping the motion of Victoria have been established for the astronomical value of the Jupiter– Sun mass-ratio (about 103 ). After an expansion of the perturbing function and a truncation to a suitable order (see [12]), the Hamiltonian function describing the motion of the asteroid is given in Delaunay’s variables by 1  G  " f (L; G; `; g) ; 2L2 p where setting a  L2 , e D 1  G 2 /L2 , the perturbation is given by H(L; G; `; g)  

Librational Tori for the Spin-Orbit Problem The existence of invariant librational tori around a spinorbit resonance can be obtained as follows [9]. Let us consider the 1:1 resonance corresponding to Hamilton’s equations associated to (23). First one implements a canonical transformation to center around the synchronous periodic orbit; after expanding in Taylor series, one diagonalizes the quadratic terms, thus obtaining a harmonic oscillator plus higher degree (time-dependent) terms. Finally, it is convenient to transform the Hamiltonian using the action-angle variables (I; ') of the harmonic oscillator. After these symplectic changes of variables one is led to a Hamiltonian of the form H(I; '; t)  !I C "h(I) C "R(I; '; t); I 2 R; ('; t) 2 T 2 ;

9 4 3 2 2 a2 C a C a e f (L; G; `; g) D 1 C 4 64 8   1 9 2 2  C a a e cos ` 2 16   3 3 15 5 C a C a cos(` C g) 8 64   9 5 2 2 C a a e cos(` C 2g)  4 4   5 4 3 2 a C a cos(2 ` C 2 g) C 4 16 3 2 C a e cos(3 ` C 2 g) 4

Perturbation Theory in Celestial Mechanics



 5 3 35 5 a C a cos(3 ` C 3 g) 8 128 35 4 a cos(4 ` C 4 g) C 64 63 5 C a cos(5` C 5g) : 128 C

For the asteroid Victoria the orbital elements are a V ' 2:334 AU, e V ' 0:220, which give the observed values of the Delaunay’s action variables as LV D 0:670, GV D 0:654. The energy level is taken as 1  GV ' 1:768 ; 2LV2 ˝ ˛   f (LV ; GV ; `; g) ' 1:060 ;

EV(0)   EV(1)

EV (")  EV(0) C "EV(1) : The osculating energy level of the Sun–Jupiter–Victoria model is defined as EV  EV (" J ) D EV(0) C " J EV(1) ' 1:769 : We now look for two invariant tori bounding the observed values of LV and GV . To this end, let L˜ ˙ D LV ˙0:001 and let ! 1 ˜˙ D ; 1  (˛˜ ˙ ; 1) : ! L˜ 3 ˙

To obtain diophantine frequencies, the continued fraction expansion of ˛˜ ˙ is modified adding a tail of ones after the order 5; this procedure gives the diophantine numbers ˛˙ which define the bounding frequencies as ! ˙ D (˛˙ ; 1). By a computer-assisted KAM theorem, the stability of the asteroid Victoria is a consequence of the following result [12]: for j"j  103 the unperturbed tori can be analytically continued into invariant KAMtori for the perturbed system on the energy level H 1 EV (")), keeping fixed the ratio of the frequencies. Therefore the orbital elements corresponding to the semi-major axis and to the eccentricity of the asteroid Victoria stay forever "-close to their unperturbed values. Planetary Problem The dynamics of the planetary problem composed by the Sun, Jupiter and Saturn is investigated in [29,30,31]. In [29] the secular dynamics of the following model is studied: after the Jacobi’s reduction of the nodes, the 4-dimensional Hamiltonian is averaged over the fast angles and its series expansion is considered up to the second order in the masses. This procedure provides a Hamiltonian function with two degrees of freedom, describing

the slow motion of the parameters characterizing the Keplerian approximation (i. e., the eccentricities and the arguments of perihelion). Afterwards, action-angle coordinates are introduced and a partial Birkhoff normalization is performed. Finally, a computer-assisted implementation of a KAM theorem yields the existence of two invariant tori bounding the secular motions of Jupiter and Saturn for the observed values of the parameters. The approach sketched above is extended in [31] so to include the description of the fast variables, like the semi-major axes and the mean longitudes of the planets. Indeed, the preliminary average on the fast angles is now performed without eliminating the terms with degree greater or equal than two with respect to the fast actions. The canonical transformations involving the secular coordinates can be adapted to produce a good initial approximation of an invariant torus for the reduced Hamiltonian of the three-body planetary problem. This is the starting point of the procedure for constructing the Kolmogorov’s normal form which is numerically shown to be convergent. In [30] the same result of [31] has been obtained for a fictitious planetary solar system composed by two planets with masses equal to 1/10 of those of Jupiter and Saturn. Periodic Orbits Construction of Periodic Orbits One of the most intriguing conjectures of Poincaré concerns the pivotal role of the periodic orbits in the study of the dynamics; more precisely, he states that given a particular solution of Hamilton’s equations one can always find a periodic solution (possibly with very long period) such that the difference between the two solutions is small for an arbitrary long time. The literature on periodic orbits is extremely wide (see, e. g., [4,7,24,38,39] and references therein); here we present the construction of periodic orbits implementing a perturbative approach (see [10]) as shown by Poincaré in [34]. We describe such a method taking as an example the spin-orbit Hamiltonian (23) that we write in a compact form as H(y; x; t)  y 2 /2  " f (x; t) for a suitable function f D f (x; t); the corresponding Hamilton’s equations are x˙ D y y˙ D " f x (x; t) :

(24)

A spin-orbit resonance of order p : q is a periodic solution of period T D 2 q (q 2 Znf0g), such that x(t C 2 q) D x(t) C 2 p y(t C 2 q) D y(t) :

(25)

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From (24) the solution can be written in integral form as Z t y(t) D y(0) C " f x (x(s); s)ds 0 Z tZ  x(t) D x(0) C y(0)t C " f x (x(s); s)ds d 0 0 Z t D x(0) C y(s)ds ;

The previous computation of the p : q periodic solution can be used to evaluate the libration in longitude of the Moon. More precisely, setting p D q D 1 one obtains x0 D 0 y0 D 1 x1 (t) D 0:232086t  0:218318 sin(t)

0

combining the above equations with (25) one obtains Z 2 q f x (x(s); s)ds D 0 0 2 q

Z

The Libration in Longitude of the Moon

 6:36124  103 sin(2t)  3:21314  104 sin(3t)  1:89137  105 sin(4t)  1:18628  106 sin(5t) (26)

y1 (t) D 0:232086  0:218318 cos(t)  0:0127225 cos(2t)  9:63942  104 cos(3t)  7:56548  105 cos(4t)

y(s)ds  2 p D 0 : 0

 5:93138  106 cos(5t)

Let us write the solution as the series

x1 D 0

x(t)  x C yt C "x1 (t) C : : :

(27)

y(t)  y C "y1 (t) C : : : ;

where x(0) D x and y(0) D y are the initial conditions, while x1 (t), y1 (t) are the first order terms in ". Expanding the initial conditions in power series of ", one gets: 2

x D x 0 C "x 1 C " x 2 C : : :

(28)

2

y D y 0 C "y 1 C " y 2 C : : :

y 1 D 0:232086 ; where we used e D 0:05494, " D 3:45104 . Therefore the synchronous periodic solution, computed up to the first order in ", is given by x(t) D x 0 C y 0 t C "x1 (t) D t  7:53196  105 sin(t)  2:19463  106 sin(2t)  1:10853  107 sin(3t)  6:52523  109 sin(4t)  4:09265  1010 sin(5t)

Inserting (27) and (28) in (24), equating same orders in " and taking into account the periodicity condition (26), one can find the following explicit expressions for x1 (t), y1 (t), y0 , y 1 : Z t f x (x 0 C y 0 s; s)ds y1 (t) D y1 (t; y; x) D 0 Z t x1 (t) D x1 (t; y; x) D y1 (s)ds

y(t) D y 0 t C "y1 (t) D 1  7:53196  105 cos(t)  4:38926  106 cos(2t)  3:3256  107 cos(3t)  2:61009  108 cos(4t)  2:04633  109 cos(5t) : It turns out that the libration in longitude of the Moon, provided by the quantity x(t)  t, is of the order of 7  105 in agreement with the observational data.

0

Future Directions

p y0 D q 1 y1 D  2 q

Z

2 q 0

Z 0

t

f x (x 0 C y0 s; s)dsdt :

Furthermore, x 0 is determined as a solution of Z 2 q f x (x 0 C y0 s; s)ds D 0 ; 0

while x 1 is given by x 1 D  R 2 q 0

1

f x0x dt Z 2 q Z  y1 t f x0x dt C

0

2 q 0

 f x0x x1 (t)dt ;

where, for shortness, we have written f x0x D f x x (x 0 C y 0 t; t).

The last decade of the XX century has been greatly marked by astronomical discoveries, which changed the shape of the solar system as well as of the entourage of other stars. In particular, the detection of many small bodies beyond the orbit of Neptune has moved the edge of the solar system forward and it has increased the number of its population. Hundreds objects have been observed to move in a ring beyond Neptune, thus forming the so-called Kuiper’s belt. Its components show a great variety of behaviors, like resonance clusterings, regular orbits, scattered trajectories. Furthermore, far outside the solar system, the astronomical observations of extrasolar planetary systems have opened new scenarios with a great variety of dynamical behaviors. In these contexts classical and resonant perturbation theories will deeply contribute to provide a fundamental insight of the dynamics and will play

Perturbation Theory in Celestial Mechanics

a prominent role in explaining the different configurations observed within the Kuiper’s belt as well as within extrasolar planetary systems. Bibliography 1. Andoyer H (1926) Mécanique Céleste. Gauthier-Villars, Paris 2. Arnold VI (1963) Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi Mat Nauk 6 18(114):91–192 3. Arnold VI (1978) Mathematical methods of classical mechanics. Springer, Berlin 4. Arnold VI (ed) (1988) Encyclopedia of Mathematical Sciences. Dynamical Systems III. Springer, Berlin 5. Benettin G, Fasso F, Guzzo M (1998) Nekhoroshev-stability of L4 and L5 in the spatial restricted three-body problem. Regul Chaotic Dyn 3(3):56–71 6. Boccaletti D, Pucacco G (2001) Theory of orbits. Springer, Berlin 7. Brouwer D, Clemence G (1961) Methods of Celestial Mechanics. Academic Press, New York 8. Celletti A (1990) Analysis of resonances in the spin-orbit problem. In: Celestial Mechanics: The synchronous resonance (Part I). J Appl Math Phys (ZAMP) 41:174–204 9. Celletti A (1993) Construction of librational invariant tori in the spin-orbit problem. J Appl Math Phys (ZAMP) 45:61–80 10. Celletti A, Chierchia L (1998) Construction of stable periodic orbits for the spin-orbit problem of Celestial Mechanics. Regul Chaotic Dyn (Editorial URSS) 3:107–121 11. Celletti A, Chierchia L (2006) KAM tori for N-body problems: a brief history. Celest Mech Dyn Astron 95 1:117–139 12. Celletti A, Chierchia L (2007) KAM Stability and Celestial Mechanics. Mem Am Math Soc 187:878 13. Celletti A, Giorgilli A (1991) On the stability of the Lagrangian points in the spatial restricted problem of three bodies. Celest Mech Dyn Astron 50:31–58 14. Chebotarev AG (1967) Analytical and Numerical Methods of Celestial Mechanics. Elsevier, New York 15. Chierchia L, Gallavotti G (1994) Drift and diffusion in phase space. Ann l’Inst H Poincaré 60:1–144 16. Delaunay C (1867) Mémoire sur la théorie de la Lune. Mém l’Acad Sci 28:29 (1860) 17. Deprit A (1967) Free rotation of a rigid body studied in the phase space. Am J Phys 35:424–428 18. Efthymiopoulos C, Sandor Z (2005) Optimized Nekhoroshev stability estimates for the Trojan asteroids with a symplectic mapping model of co-orbital motion. MNRAS 364(6):253–271 19. Féjoz J (2004) Démonstration du “théorème d’Arnold” sur la stabilité du système planétaire (d’après Michael Herman). Ergod Th Dynam Syst 24:1–62

20. Ferraz-Mello S (2007) Canonical Perturbation Theories. Springer, Berlin 21. Gabern F, Jorba A, Locatelli U (2005) On the construction of the Kolmogorov normal form for the Trojan asteroids. Nonlinearity 18:1705–1734 22. Giorgilli A, Skokos C (1997) On the stability of the trojan asteroids. Astron Astroph 317:254–261 23. Giorgilli A, Delshams A, Fontich E, Galgani L, Simó C (1989) Effective stability for a Hamiltonian system near an elliptic equilibrium point, with an application to the restricted three-body problem. J Diff Eq 77:167–198 24. Hagihara Y (1970) Celestial Mechanics. MIT Press, Cambridge 25. Hénon M (1966) Explorationes numérique du problème restreint IV: Masses egales, orbites non periodique. Bull Astron 3(1, fasc 2):49–66 26. Kolmogorov AN (1954) On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Dokl Akad Nauk SSR 98:527–530 27. Laskar J, Robutel P (1995) Stability of the planetary threebody problem I Expansion of the planetary Hamiltonian. Celest Mech and Dyn Astron 62(3):193–217 28. Lhotka Ch, Efthymiopoulos C, Dvorak R (2008) Nekhoroshev stability at L4 or L5 in the elliptic restricted three body problem-application to Trojan asteroids. MNRAS 384:1165–1177 29. Locatelli U, Giorgilli A (2000) Invariant tori in the secular motions of the three-body planetary systems. Celest Mech and Dyn Astron 78:47–74 30. Locatelli U, Giorgilli A (2005) Construction of the Kolmogorov’s normal form for a planetary system. Regul Chaotic Dyn 10:153– 171 31. Locatelli U, Giorgilli A (2007) Invariant tori in the Sun–Jupiter– Saturn system. Discret Contin Dyn Syst-Ser B 7:377–398 32. Meyer KR, Hall GR (1991) Introduction to Hamiltonian dynamical systems and the N-body problem. Springer, Berlin 33. Moser J (1962) On invariant curves of area-preserving mappings of an annulus. Nach Akad Wiss Göttingen. Math Phys Kl II 1:1 34. Poincarè H (1892) Les Méthodes Nouvelles de la Mécanique Céleste. Gauthier-Villars, Paris 35. Robutel P (1995) Stability of the planetary three-body problem II KAM theory and existence of quasi-periodic motions. Celest Mech Dyn Astron 62(3):219–261 36. Robutel P, Gabern F (2006) The resonant structure of Jupiter’s Trojan asteroids – I Long-term stability and diffusion. MNRAS 372(4):1463–1482 37. Sanders JA, Verhulst F (1985) Averaging methods in nonlinear dynamical systems. Springer, Berlin 38. Siegel CL, Moser JK (1971) Lectures on Celestial Mechanics. Springer, Heidelberg 39. Szebehely V (1967) Theory of orbits. Academic Press, New York

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Perturbation Theory, Introduction to GIUSEPPE GAETA Dipartimento di Matematica, Università di Milano, Milan, Italy The idea behind Perturbation Theory is that when we are not able to determine exact solutions to a given problem, we might be able to determine approximate solutions to our problem starting from solutions to an approximate version of the problem, amenable to exact treatment. Thus, in a way, we use exact solutions to an approximate problem to get approximate solutions to an exact problem. It goes without saying that many mathematical problems met in realistic situations, in particular as soon as we leave the linear framework, are not exactly solvable– either for an inherent impossibility or for our insufficient skills. Thus, Perturbation Theory is often the only way to approach realistic nonlinear systems. It is implicit in the very nature of perturbation theory that it can only worke once a problem which is both solvable–one also says “integrable”–and in some sense “near” to the original problem can be identified (it should be mentioned in this respect that the issue of “how near is near enough” is a delicate one). Quite often, the integrable problem to be used as a starting point is a linear one–maybe obtained as the firstorder expansion around a trivial or however known solution–and nonlinear corrections can be computed term by term via a recursive procedure based on expansion in a small parameter (usually denoted as " by tradition); the point is that at each stage of this procedure one should only solve linear equations, so that the procedure can–at least in principle–be carried over up to any desired order. In practice, one is limited by time, computational power, and the increasing dimension of the linear systems to be solved. But limitations are not only due to the limits of the humans–or the computers–performing the actual computations: in fact, some delicate points arise when one considers the convergence of the " series involved in the computations and in the expression of the solutions obtained by Perturbation Theory. These points–i. e. the power of Perturbation Theory, its basic features and tools, and its limitations in particular with regard to convergence issues–are discussed in the article  Perturbation Theory by Gallavotti. This article also stresses the role which problems originating in Physics had in the development of Perturbation Theory; and this not only in historical terms (the computation of planetary or-

bits), but also in more recent times through the work of Poincaré first and then via Quantum Theory. The modern setting of Perturbation Theory was laid down by Poincaré, and goes through the use of what is today known as Poincaré normal forms; these are a cornerstone of the whole theory and hence, implicitly or explicitly, of all the articles presented in this section of the Encyclopedia. But, they are also discussed in detail, together with their application, in the article  Normal Forms in Perturbation Theory by Broer. The latter deals with the general problem, i. e. with evolution differential equations (Dynamical Systems) with no special structure; or, in applications originating from Physics or Engineering one is often dealing with systems that (within a certain approximation) preserve Energy and can be written in Hamiltonian form. In this case, as emphasized by Birkhoff, one can more efficiently consider perturbations of the Hamiltonian rather than of the equations of motion (the advantage originating in the fact that the Hamiltonian is a scalar function, while the equations of motion are a system of 2n equations in 2n dimensions). The normal form approach for Hamiltonian systems, and more generally Hamiltonian perturbation theory, is discussed in the article  Hamiltonian Perturbation Theory (and Transition to Chaos) by Broer and Hanßmann. This also discusses the problem of transition–as some control parameter, often the Energy, is varied–from the regular behavior of the unperturbed system to the chaotic (“turbulent” if we deal with fluid motion) behavior displayed by many relevant Hamiltonian as well as non-Hamiltonian systems. As mentioned above, in all the matters connected with Perturbation Theory and its applications, convergence issues play an extremely important role. They are discussed in the article  Perturbative Expansions, Convergence of by Walcher, both in the general case and for Hamiltonian systems. The interplay between perturbations–and more generally changes in some relevant parameter characterizing the system within a more general family of system–and qualitative (not only quantitative) changes in its behavior is of course of general interest not only in the “extreme” case of transition from integrable to chaotic behavior, but also when the qualitative change in the behavior of the system is somehow more moderate. Such a change is also known as a bifurcation. Albeit there is no article specifically devoted to these, the reader will note the concept of bifurcation appears in many, if not most, of the articles. The behavior of a “generically perturbed” system depends on what is meant by “generically”. In particular if we deal with an unperturbed system which has some degree of

Perturbation Theory, Introduction to

symmetry, this may be an “accidental” feature–maybe due to the specially simple nature of integrable systems such as the one chosen as an unperturbed one–but might also correspond to a requirement by the very problem we are modeling; this is often the case when we deal with problems of physical or engineering origin, just because the fundamental equations of Physics have some degree of symmetry. The presence of symmetry can be quite helpful–e. g. reducing the effective degrees of freedom of a given problem–and should be taken into account in the perturbative expansion. Moreover, the perturbative expansions can be made to have some degree of symmetry which can be used in the solution of the resulting equations. These matters are discussed at length in the article  Non-linear Dynamics, Symmetry and Perturbation Theory in by Gaeta. A special–but widely applicable and very interesting– framework for the occurrence of bifurcation is provided by systems exhibiting parametric resonance. This is, for example, the case for an ample class of coupled oscillator systems, which would per se suffice to guarantee the phenomenon is of special interest in applications, beside its theoretical appeal. The analysis of parametric resonance from the point of view of Perturbation Theory is discussed in the article  Perturbation Analysis of Parametric Resonance by Verhulst. As mentioned above, the transition from fully regular (integrable) to chaotic behavior is discussed in general terms in the article  Hamiltonian Perturbation Theory (and Transition to Chaos) by H. Broer and H. Hanßmann. However, quite remarkably, in some cases a perturbation will only moderately destroy the integrable behavior. This should be meant in the following sense: integrable behavior is characterized by the fact that whatever the initial conditions of the system, we are able to predict its behavior after an arbitrary long time. It may happen that albeit this is not true, we are still able to predict either (a) the arbitrarily long time behavior for a dense subset of all the possible initial conditions; or (b) the exponentially long time behavior for a subset of full measure of possible initial conditions (usually, those “sufficiently near” to the exactly integrable case). In the first case, the meaning of the statement is that any possible initial condition is “near” to an initial condition leading to an integrable-type behavior over all times (which does not imply its behavior will be near to integrable over arbitrary times, but only for sufficiently small– albeit this “small” could be extremely long on human scale–times). This kind of situation is investigated by the KAM theory (named after the initials of Kolmogorov, Arnold, and Moser), discussed in the article  Kolmogorov–Arnold–Moser (KAM) Theory by Chierchia.

In the second case, the statements about stable behavior are valid only for a finite (albeit exponentially long, hence again often extremely long on human scale) time, but apply to an open set of initial conditions. This approach was taken by Nekhoroshev, and is presently known–together with the results obtained in this direction–as Nekhoroshev theory; this is the subject of the article  Nekhoroshev Theory by Niederman. As mentioned above, the problem of planetary motions was historically at the origin of Perturbation Theory, since Ancient Greece; actually more recent results, including those due to the work of Poincaré–and those embodied in KAM and Nekhoroshev theories–also have roots in Celestial Mechanics (albeit then being used in completely different fields, e. g. in the study of electron motion in a crystal). The application of Perturbation Theory in Celestial Mechanics is a very active field of research, and the subject of the article  Perturbation Theory in Celestial Mechanics by Celletti. In this context, one often considers reduced problems where not all the planets are taken into account; this is the origin of the “three-body problem,” the three bodies being, for example the Sun, Jupiter, and the Earth; or the Earth, the Moon, and an artificial satellite. Much effort has been recently devoted to the study of special solutions for the N-body problem (that is, N bodies mutually attracting via potential forces) after the discovery of remarkable special solutions–termed “choreographies”–in which the bodies move along one or few common trajectories. This theory has not yet found applications in concrete physical systems or technology, but on the one hand these special solutions provide an organizing center for general nonlinear dynamics, and on the other the applicative potential of such collective motions (say in micro-devices) is rather obvious. The N-body problem and these special solutions are discussed in the article  n-Body Problem and Choreographies by Terracini. The reader has probably noted that all these problems correspond to smooth conservative systems, or at least to perturbation of such systems. When this is relaxed–which is not appropriate in studying the motion of planetary objects, but may be appropriate in a number of contexts–the situation is both different and less well understood. The article  Perturbation of Systems with Nilpotent Real Part by Gramchev studies perturbation of linear systems with a nilpotent linear part, which is the case for dissipative unperturbed systems. The article  Perturbation Theory for Non-smooth Systems by Teixeira discusses the case of nonsmooth systems, which is the case–quite relevant in real-world engineering applications–of systems with impacts.

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Perturbation Theory, Introduction to

It was also mentioned that Quantum Theory was another major source of motivation for the development of Perturbation Theory, both historically and still in recent times. As for the latter, one should remark how a standard tool in quantum perturbation theory, i. e. the technique of Feynman diagrams–or more generally, diagrammatic expansions–was incorporated into classical perturbation theory only in relatively recent times. The use of diagrammatic expansions in classical perturbation theory is discussed in the article  Diagrammatic Methods in Classical Perturbation Theory by Gentile. Apart from this, knowledge of the perturbation-theoretic techniques developed in the framework of quantum theory–both in general and for the study of atoms and molecules–is of general interest, both as a source of inspiration for tackling problems in different contexts and for the intrinsic interest of microscopic systems; while well known to physicists, this theory is maybe less known to mathematicians and engineers. The articles provided in this section of the Encyclopedia can be an excellent entry point for those not familiar with this theory. In the article  Perturbation Theory in Quantum Mechanics by Picasso, Bracci, and D’Emilio, the general setting and results are described, together with some selected special topics; the role of symmetries–and hence degeneracies–within quantum perturbation theory is paramount and also discussed here. The article  Perturbation Theory and Molecular Dynamics by Panati focuses instead on the specific aspects of the perturbative approach to the quantum dynamics of molecules; this is a remarkable example of how taking into account the separation between slow and fast degrees of freedom allows one to deal with seemingly intractable problems. A bridge between quantum and classical Perturbation Theory is provided by the semiclassical case, corresponding to taking into account the smallness of the energy scale set by Planck’s constant h with respect to the energy scale involved in many (macro- or meso-scopic) problems. This is discussed in the article  Perturbation Theory, Semiclassical by Sacchetti. The quantum framework is also very interesting in connection with Bifurcation Theory; in this framework the “qualitative changes in the dynamics” which characterize bifurcations corresponds to qualitative changes in the spectrum. This in turn is related to monodromy on the mathematical side; and to the problem of an atom in crossed magnetic and electric fields on the physical side. These matters, strongly related to several of those mentioned above, are discussed in the article  Quantum Bifurcations by Zhilinskii.

From the mathematical point of view, in the quantum case one deals with a Partial Differential Equation– the Schrödinger equation–rather than with a system of Ordinary Differential Equations (it should be noted that when dealing with the spectrum only, one is actually not requiring to study the full set of solutions to the concerned PDE). Needless to say, this is not the only case where one has to deal with PDEs in the applications, continuum mechanics providing a classical framework where one is obliged to deal with PDEs. Rigorous results in Perturbation Theory for PDEs are not at the same level as for ODEs (and the insight provided by the quantum case is henceforth specially valuable); research in this direction is very active, and faces rather difficult problems despite the progresses obtained in recent years. Some of these results, together with an overview of the field, are described in the article  Perturbation Theory for PDEs by Bambusi. Quite appropriately, this article ends up stating that in several applied fields a sound understanding of PDEs rigorous Perturbation Theory would be relevant for applications, mentioning in particular the water wave problem, quantum mechanics, electromagnetic theory and magnetohydrodynamics, and elastodynamics. This could also be a convenient way to conclude this Introduction, but in this case the reader would unavoidably remain with the impression that Perturbation Theory is mainly dealing with nonlinear problems originating in Physics or Engineering. While this is historically the origin of the most striking developments of the theory–and the realm in which it proved most successful–such characterization is by no means a built-in restriction. Perturbation Theory can also deal effectively with problems originating in different fields and having a rather different mathematical formulation, such as those arising in certain fields of Biology (beside fields where the mathematical formulation is anyway in terms of Dynamical Systems). This is shown concretely in the article  Perturbation of Equilibria in the Mathematical Theory of Evolution by Sanchez; in this case the problem is formulated in terms of Evolutionary Game Theory. It should be noted that this is interesting not only for the intrinsic interest of the Darwin theory of Evolution, but also because Game Theory is increasingly used in rather diverse contexts. Finally, I would like to warmly thank all the Authors for providing the remarkable articles making up this section of the Encyclopedia, as well as the Referees who checked them anonymously and in several cases gave suggestions leading to improvements.

Perturbation Theory and Molecular Dynamics

Perturbation Theory and Molecular Dynamics GIANLUCA PANATI Dipartimento di Matematica, Università di Roma “La Sapienza”, Roma, Italy Article Outline Glossary Perturbation Theory, Introduction to Introduction The Framework The Leading Order Born–Oppenheimer Approximation Beyond the Leading Order Future Directions Bibliography Glossary Adiabatic decoupling In a complex system (either classical or quantum), the dynamical decoupling between the slow and the fast degrees of freedom. Adiabatic perturbation theory A mathematical algorithm which exploits the adiabatic decoupling of degrees of freedom in order to provide an approximated (but yet accurate) description of the slow part of the dynamics. In the framework of QMD, it is used to approximately describe the dynamics of nuclei, the perturbative parameter " being related to the small electron/nucleus mass ratio. Electronic structure problem The problem consisting in computing, at fixed positions of thenuclei, the energies (eigenvalues) and eigenstates corresponding to the electrons. An approximate solution is usually obtained numerically. Molecular dynamics The dynamics of the nuclei in a molecule. While a first insight in the problem can be obtained by using classical mechanics (Classical Molecular Dynamics), a complete picture requires quantum mechanics (Quantum Molecular Dynamics)  Perturbation Theory in Quantum Mechanics. This contribution focuses on the latter viewpoint. Definition of the Subject In the framework of Quantum Mechanics the dynamics of a molecule is governed by the (time-dependent) Schrödinger equation, involving nuclei and electrons coupled through electromagnetic interactions. While the equation is mathematically well-posed, yielding the exis-

tence of a unique solution, the complexity of the problem makes the exact solution unattainable. Even for small molecules, the large number of degrees of freedom prevents from direct numerical simulation, making an approximation scheme necessary. Indeed, one may exploit the smallness of the electron/nucleus mass ratio to introduce a convenient computational scheme leading to approximate solutions of the original time-dependent problem. In this article we review the standard approximation scheme (dynamical Born– Oppenheimer approximation) together with its ramifications and some recent generalizations, focusing on mathematically rigorous results. The success of this approximation scheme is rooted in a clear separation of time-scales between the motion of electrons and nuclei. Such separation provides the prototypical example of adiabatic decoupling between the fast and the slow part of a quantum dynamics. More generally, adiabatic separation of time-scales plays a fundamental role in the understanding of complex system, with applications to a wide range of physical problems. Introduction Through the discovery of the Schrödinger equation the theoretical physics and chemistry community attained a powerful tool for computing atomic spectra, either exactly or in perturbation expansion. Born and Oppenheimer [2] immediately strived for a more ambitious goal, namely to understand the excitation spectrum of molecules on the basis of the new wave mechanics. They accomplished to exploit the small electron/nucleus mass ratio as an expansion parameter, which then leads to the stationary Born–Oppenheimer approximation. Since then it has become a standard and widely used tool in quantum chemistry, now supported by rigorous mathematical results [4,5,18,19,27]. Beyond molecular structure and excitation spectra, dynamical processes have gained in interest. Examples are the scattering of molecules, chemical reactions, or the decay of an excited state of a molecule through tunneling processes. Such problems require a dynamical version of the Born–Oppenheimer approximation, which is the topic of this article. At the leading order, the electronic energy at fixed positions of the nuclei serves as an effective potential between the nuclei. We call this the zeroth order Born–Oppenheimer approximation. The resulting effective Schrödinger equation can be used for both statical and dynamical purposes. The input is an electronic structure calculation, which for the purpose of our article we regard as given by other means.

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While there are many physical and chemical properties of molecules explained by the zeroth order Born– Oppenheimer approximation, there are cases where higher order corrections are required. Famous examples are the dynamical Jahn–Teller effect and the dynamics of singled out nuclear degrees of freedom near the conical intersection of two energy surfaces. The first order Born– Oppenheimer approximation involves geometric phases, which are of great interest also in other domains of Quantum Mechanics ([41],  Quantum Bifurcations). Finally, we mention that some dynamical processes can be modeled as scattering problems. In such cases it is convenient to combine the Born–Oppenheimer scheme together with scattering theory, a topic which goes beyond the purpose of this contribution (see [28,29] and references therein). A complete overview of the vast literature on the subject of the dynamical Born–Oppenheimer approximation is provided in [24].

where eZ k , for Z k 2 Z, is the electric charge of the kth nucleus. In some cases, to obtain rigorous mathematical results one needs to slightly smear out the charge distribution of the nuclei. This is in agreement with the physical picture that nuclei are not point like but extended objects. Hereafter we will assume, for sake of a simpler notation, that all the nuclei have the same mass M. The subsequent discussion is still valid in the general case, with the appropriate choice of the adiabatic parameter indicated below. As mentioned above, the large number of degrees of freedom makes convenient to elaborate an approximation scheme, exploiting the smallness of the parameter r me " :D (3) D 102    103 : M p In the general case, one has to choose " D maxf me /M k : 1  k  Kg. By introducing atomic units („ D 1; me D 1) and making explicit the role of the adiabatic parameter ", the Hamiltonian Hmol reads (up to a change of energy scale)

The Framework We consider a molecule consisting of K nuclei, whose positions are denoted as x D (x1 ; : : : ; x K ) 2 R3K D: X, and N electrons, with positions y D (y1 ; : : : ; y N ) 2 R3N D: Y. The wavefunction of the molecule is therefore a square-integrable function  depending on all these coordinates. Molecular dynamics is described through the Schrödinger equation ı„

d s D Hmol s ; ds

(1)

where s denotes time measured in microscopic units and the Hamiltonian operator is given by Hmol D 

K N X X „2 „2 x k  y 2M k 2me i iD1

kD1

C Ve (y) C Vn (x) C Ven (x; y) :

(2)

Here „ is the Planck constant, me is the mass of the electron and M k the mass of the kth nucleus, and the interaction terms are explicitly given by Vn (x) D

K X K X e2 Z k Z l ; jx k  x l j

Ve (y) D

kD1 l ¤k

N N X X iD1 j¤i

e2 ; jy i  y j j

H" D 

kD1

C

Ven (x; y) D

kD1 iD1

e2 Z k  ; jx k  y i j

N X

1   y i C Ve (y) C Ven (x; y) : 2 iD1 ƒ‚ … „

(4)

H el (x)

Notice that, for each fixed nuclei configuration x D (x1 ; : : : ; x K ) 2 X, the operator Hel (x) is an operator acting on the Hilbert space Hel corresponding to the electrons alone. If the kinetic energies of the nuclei and the electrons are comparable, as it happens in the vast majority of physical situations in view of energy equipartition, then the velocities scale as r me jv n j  (5) jve j D "jve j ; M where vn and ve denotes respectively the typical velocity of nuclei and electrons. Therefore, in order to observe a nontrivial dynamics for the nuclei, one has to wait a microscopically long time, namely a time of order O("1 ). This scaling fixes the macroscopic time scale t, together with the relation t D "s, where s is the microscopic time appearing in Eq. (1). We are therefore interested in the behavior of the solutions of the equation

and K X N X

K X "2 x k C Vn (x) 2

ı"

d  (t) D H"  (t) dt

in the limit " ! 0.

(6)

Perturbation Theory and Molecular Dynamics

We assume as an input a solution of the electronic structure problem, i. e. that for every fixed configuration of the nuclei x D (x1 ; : : : ; x K ) one knows the solution of the eigenvalue problem He (x) j (x) D E j (x) j (x) ;

(7)

with E j (x) 2 R and  j (x) 2 Hel . Since electrons are fermions, one has Hel D Sa L2 (R3N ) with Sa projecting onto the antisymmetric wave functions. The eigenvectors in Eq. (7) are normalized as Z j (x; y) ` (x; y)dy D ı j` h j (x) ; ` (x)iHel  Y

with respect to the scalar product in Hel . Note that the eigenvectors are determined only up to a phase # j (x). Generically, in addition to the bound states, He (x) has continuous spectrum. We label the eigenvalues in Eq. (7) as E1 (x)  E2 (x)     ;

(8)

including multiplicity. The graph of Ej is called the jth energy surface or energy band, see Fig. 1. Generically, in realistic examples such energy bands cross each other, and the possible structures of band crossing have been classified [22,35]. Figure 2 illustrates a realistic example of energy bands, showing in particular the typical conical intersection of two energy surfaces.

Let (x) be a nucleonic wave function, 2 H n , the Hilbert space corresponding to the nuclei. For simplicity we take H n D L2 (X), remembering that to impose the physically correct statistics for the nuclei requires extra considerations [33]. States  of the molecules with the property that the electrons are precisely in the jth eigenstate are then of the form  (x; y) D

(x)  j (x; y) :

(9)

We can think of  either as a wave function in the total Hilbert space H D H n ˝ Hel , or as a wave function for the electrons (i. e. an element of Hel ) depending parametrically on x. In the common jargon, a state in the form Eq. (9) is said a state concentrated on the jth band. We denote as Pj the projector on the subspace consisting of states of the form Eq. (9); since the f j (x)g j2N are orthonormal, Pj is indeed an orthogonal projection in H n ˝ Hel . The Leading Order Born–Oppenheimer Approximation We focus now on an a specific energy band, say En , assuming that it is globally isolated from the rest of the spectrum (the behavior of the wavefunction at the crossing points will be addressed later). Under such assumption, a state  0 which is initially concentrated on the nth band will stay localized in the same band up to errors of order O("): more specifically, one shows that   (1  Pn ) eıH " t/" Pn 0  D O(") : (10) H The space Ran Pn , consisting of wavefunctions in the form Eq. (9), is usually called the adiabatic subspace corresponding to the nth band. Since Ran Pn is approximately invariant under the dynamics, one may wonder whether there is a simple and convenient way to approximately describe the dynamics inside such subspace. Indeed, one may argue that for an initial state in Ran Pn the dynamics of the nuclei is governed by the reduced Hamiltonian Pn H" Pn D 

K "2 X x k C Vn (x) C E n (x) C O(") (11) 2 kD1

Perturbation Theory and Molecular Dynamics, Figure 1 A schematic representation of energy bands. At each fixed nuclei configuration x D (x1 ; : : : ; xK ) the electronic Hamiltonian Hel (x) exhibits point spectrum (corresponding to states with all the electrons bound to the nuclei) and continuous spectrum (corresponding to states in which one or more electrons are quasi-free, i. e. the molecule is ionized)

acting in Ran Pn Š H n D L2 (X). The dynamical Born– Oppenheimer approximation consists, at the leading order, in replacing the original Hamiltonian Eq. (4) by the Hamiltonian HBO D 

K "2 X x k C Vn (x) C E n (x) 2 kD1

(12)

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Perturbation Theory and Molecular Dynamics, Figure 2 The first two energy bands for the hydrogen quasi-molecule H3 , i. e. the system consisting of three protons and three electrons. The picture shows the restriction of such bands over a 2-dimensional subspace of the configuration space. Two hydrogen nuclei are located on the x-axis with a fixed separation of 1.044 Angstrom Å, and the energy bands are plotted as a function of the relative position (x; y) of the third nucleus. Notice the conical intersection between the ground and the first excited state, which appears at equilateral triangular geometries. (© Courtesy of Eckart Wrede, Durham University. The plot is generated using the analytic representation of the H3 energy bands obtained in [40])

acting in L2 (X), getting thus an impressive dimensional reduction. In other words: let  (t; x; y) be the solution of Eq. (6) with initial datum 0 (x; y) D '0 (x)n (x; y); then  (t; x; y) D '(t; x)n (x; y) C O(") where '(t; x) is the solution of the effective equation d ı '(t) D HBO '(t) dt

D ieiH " t/" (13)

with initial datum ' 0 . To prove mathematically the previous claim, one has to bound the difference 

yields   iH t/" e "  eiPn H " Pn t/" Pn Z t/" iH " t/" D ie ds eiH " s (Pn H" Pn  H" ) eiPn H " Pn s Pn

 eiH " t/"  eiPn H " Pn t/" Pn :

A proof of this fact is not immediate as one might expect. Indeed, the Duhamel method (consisting essentially in rewriting a function as the integral of its derivative)

D ieiH " t/"

Z Z

0

t/" 0 t/" 0

ds eiH " s (Pn H" Pn  H" ) Pn eiPn H " Pn s ds eiH " s [Pn ; H" ] Pn eiPn H " Pn s : „ ƒ‚ … O(")

The commutator appearing in the last line is estimated as

 "2 [Pn ; H" ]Pn D jn (x)ihn (x)j;  x Pn D O(") ; 2 (14) but the integration interval diverges as O("1 ). Thus the naïve approach fails. A rigorous proof has been provided

Perturbation Theory and Molecular Dynamics

in [39], elaborating on [26], exploiting the fact that the integral in Eq. (14) is, roughly speaking, an oscillatory operator integral. A more direct approach, based on the evolution of generalized Gaussian wavepackets, has been followed in the pioneering papers by Hagedorn [17,20]. Beyond the Leading Order The dynamics of a state initially concentrated on an isolated energy band is described, up to errors of order O("), by the Born–Oppenheimer dynamics Eq. (13). It is physically interesting to find an effective dynamics which approximates the original dynamics with an higher degree of accurancy. At first sight one might think that this goal can be simply reached by expanding the operator Pn H" Pn to the next order in ". (Notice that the first term appearing in Eq. (11) does contribute as a term of O(1), since we are considering states such that the kinetic energy of the nuclei is not vanishing, i. e. k  "2 x  k D O(1), in agreement with the mentioned energy equipartition). However such naïf expansion has no physical meaning since it makes no sense to compute the operator appearing in Eq. (11) with greater accuracy if the space Ran Pn itself is invariant only up to terms of order O("). To get a deeper insight in the problem, one has to investigate the origin of the O(") term appearing in the Eq. (10): either (a) there is a part of the wavefunction of order O(") which is scattered in all the directions in the Hilbert space, or (b) still there is a subspace invariant up to smaller errors, which is however tilted with respect to Ran Pn by a term of order O("). Therefore, two natural questions arise: (i) Almost-invariant subspace: is there a subspace of H D H n ˝ Hel which is invariant under the dynamics up to errors " N , for N > 1 ? (ii) Intra-band dynamics: in the affirmative case, is there any simple and convenient way to accurately describe the dynamics inside this subspace? As for the first question, one may show that to any globally isolated energy band En corresponds a subspace of the Hilbert space which is almost-invariant under the dynamics. More precisely, one constructs an orthogonal projector ˘n; " 2 B(H ), with ˘n; " D Pn C O("), such that for any N 2 N there exists CN such that   (1  ˘n; " ) eıH " t/" ˘n; " 0   C N " N (1 C jtj)(1 C E )k0 k :

(15)

Here E denotes a cut-off on the kinetic energy of the nuclei, which corresponds to the physical assumption that

Perturbation Theory and Molecular Dynamics, Figure 3 A schematic illustration of the superadiabatic subspace Ran ˘n;" , tilted by a correction of order " with respect to the usual adiabatic subspace Ran Pn

the kinetic energies of nuclei and electrons are comparable. Equation (15) shows that if the molecule is initially in a state 0 2 Ran ˘n; " , then after a macroscopic time t the molecule is in a state which is still in Ran ˘n; " up to an error smaller than any power of ", with the error scaling linearly with respect to time and to the kinetic energy cut-off. For this reason the space Ran ˘n; " is called superadiabatic subspace or almost-invariant subspace. We emphasize that the adiabatic decoupling, as formulated in Eq. (15), holds on a long time-scale, as opposed to the semiclassically limit which is known to hold on a time scale of order O(ln "). Indeed the adiabatic decoupling is a pure quantum phenomenon, conceptually and mathematically independent from the semiclassical limit. The previous result is based on a long history of mathematical research, starting with pioneering ideas of Sjöstrand [3,9,31,32,34,37,38]. It has been formulated in the form above in [36]. As for question (ii), one has to face the problem that there is no natural identification between the super-adiabatic subspace Ran ˘n; " and H n Š L2 (X), and therefore no evident reduction in the number of degrees of freedom. This difficulty can be circumvented by constructing a unitary operator which intertwines the previous two spaces, namely U n; " : Ran ˘n; " ! H n Š L2 (X) :

(16)

Notice that such a unitary operator is not unique. With the help of U n; " we can map the intraband dynamics to the nuclei Hilbert space, obtaining an effective Hamilto1 acting in L2 (X). It ˆ eff; " :D U n; " ˘n; " H" ˘n; " U n; nian H " follows from Eq. (15) that for every N 2 N there exist CN such that    ˆ eff;" t/"   iH " t/" 1 i H  U n; U n; " ˘n; " 0   e " e H

 C N " N (1 C jtj)(1 C E )k0 k :

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Perturbation Theory and Molecular Dynamics

However, with an arbritary choice of U n; " the efˆ eff; " does not appear simpler than fective Hamiltonian H ˘n; " H" ˘n; " . On the other side, the non-uniqueness of U n; " can be conveniently exploited to simplify the structure of the effective Hamiltonian. It has been proved in [36] that the unitary operator U n; " can be explicitly conˆ eff; " has a simple structure, structed in such a way that H namely it is (close to) the "-Weyl quantization of a function Heff;" : X  R3K ! R ;

(q; p) 7! Heff;" (q; p) ;

are an active field of research [10,11,12], see  Quantum Bifurcations and references therein. As for the second order correction h2 , the first term completes the square (p  An (x))2 showing that the dynamics involves a covariant derivative; the second term is known as the Born– Huang term; the last term contains the reduced resolvent (i. e. the resolvent in the orthogonal complement of Ran Pn ) and is due to the fact that the superadiabatic subspace Ran ˘n; " is tilted with respect to Ran Pn . The third term in h2 , namely ˝

defined over the classical phase space. We recall that the "Weyl quantization maps a (smooth) function over X R3K into a (possibly unbounded) operator acting in L2 (X). The correspondence is such that any function f (q) is mapped into the multiplication operator times f (x), and any function g(p) is mapped into g(ı"rx ); for a generic function f (q; p) the ordering ambiguity is fixed by choosing eı˛q eıˇ p 7! eı(˛xCˇ (ı"rx )) : For readers interested in the mathematical structure of Weyl quantization, we recommend [15]. Equipped with this terminology, we come back to the effective Hamiltonian. It turns out that, with the appropriˆ eff; " is the "-Weyl quantization of the ate choice of U n; " , H function Heff;" (q; p) D h0 (q; p) C "h1 (q; p) C "2 h2 (q; p) C O("3 ) (17)

M(q; p) D p  rn (q); (Hel (q)  E n (q))1

˛  (1  Pn (q)) p  rn (q) H ;

appeared firstly in [36], as a consequence of the rigorous adiabatic perturbation theory developed there. This term is responsible for an O("2 )-correction to the effective mass of the nuclei. Indeed, since different quantization schemes differ by a term of order O("), we may replace the Weyl quantization with the simpler symmetric quantization, namely we consider b )(x) D (M

3K X 1 m`k (x)(i"@x ` )(i"@x k ) 2 `;kD1  C (i"@x ` )(i"@x k )m`k (x) (x) ;

(20)

where m is the x-dependent matrix ˝ m`k (x) D @` n (x); (He (x)  E n (x))1

˛  (1  Pn (x)) @ k n (x) H :

where h0 (q; p) D

(19)

el

el

1 2 2p

C E n (q) C Vn (q) ˝ ˛ (18) h1 (q; p) D ıp  n (q); rq n (q) D: p  An (q)

and ˝ h2 (q; p) D 12 A2n (q) C 12 rq n (q); (1  Pn (q)) ˛  rq n (q) H el ˝  p  rq n (q) ; 1    1  Pn (q) Hel (q)  E n (q) ˛  p  rq n (q) H :

It is clear from Eq. (20) that this term induces a correction of order O("2 ) to the Laplacean, i. e. to the inertia of the nuclei. Finally, we point out that the effective Hamiltonian Hˆ eff; " can be conveniently truncated at any order in ", getting corresponding errors in the effective quantum dynamics: if we pose ˆ eff; " D H

N X

" j hˆ j C O(" NC1 ) D: hˆ (N);" C O(" NC1 ) ; (21)

jD0

el

The Weyl quantization of h0 provides the leading order Born–Oppenheimer Hamiltonian Eq. (13). The term h1 has a geometric origin, involving the Berry connection An (x), a quantity appearing in a variety of adiabatic problems [41]; this term is responsible for the screening of magnetic fields in atoms [42]. Geometric effects in molecular systems (and more generally in adiabatic systems)

then there exist a constant C˜ N such that      iH " t/" 1 i hˆ (N);" t/"  U n; U n; " ˘n; " 0   e "e

H

 C˜ N " NC1 (1 C jtj) (1 C E )k0 k : The determination of the effective Hamiltonian, here described following [36,40], has been investigated earlier

Perturbation Theory and Molecular Dynamics

in [31,41] with different but related techniques. The result in [36] is based on an iterative algorithm inspired by classical perturbation theory (see  Kolmogorov–Arnold– Moser (KAM) Theory and  Normal Forms in Perturbation Theory). Future Directions Generically energy surfaces cross each other (see Fig. 2), and a globally isolated energy band is just a mathematical idealization. On the other side, if the initial datum 0 (x; y) D '0 (x)n (x; y) contains a nucleonic wavefunction ' 0 localized far away from the crossing points, the adiabatic approximation is still valid, up to the time when the wavefunction becomes relevant in a neighborhood of p radius " of the crossing points. This “hitting-time” can be estimated semiclassically, as done for example in [39]. When the wavefunction reaches the region around the crossing point a relevant part of it might undergo a transition to the other crossing band. (The simultaneous crossing of more than two bands is not generic, see [35], so we focus on the crossing of two bands). The understanding of the dynamics near a conical crossing is a very active field of research. The first step is a convenient classification of the possible structures of band crossings. Since the early days of Quantum Mechanics [35], it has been realized that eigenvalue crossings occurs on submanifolds of various codimension, according to the symmetry of the problem. In the case of a molecular Hamiltonian in the form Eq. (2), generic crossings of bands with the minimal multiplicity allowed by the symmetry group have been classified in [22]. The second step consist in an analysis of the propagation of the wavefunction near the conical crossing, assuming that the initial state is concentrated on a single band, say the nth band. A pioneering work [23] shows that the qualitative picture is the following: for crossings of codimension 1 the wavefunction follows, at the leading order, the analytic continuation of the nth band, as if there was no crossing. In the higher codimension case, a part of the wavefunction of order O(1) undergoes a transition to the other band. More recently, propagation through conical crossings has been investigated with new techniques [6,7,8,13,14] opening the way to future research. Alternatively, one may consider a family of energy bands which cross each other, but which are separated by an energy gap from the rest of the spectrum. Indeed, in a molecular collision or in excitations through a laser pulse only a few energy surfaces take part in the subsequent dynamics. Thus we take a set I of adjacent energy surfaces

and call PI D

X

Pj

(22)

j2I

the projection onto the relevant subspace (or subspace of physical interest). To ensure that other bands are not involved, we assume them to have a spectral gap of size agap > 0 away from the energy surfaces in I, i. e. sup jE i (x)  E j (x)j  agap

for all j 2 I ; i 2 I c : (23)

x2X

Also the continuous spectrum is assumed to be at least agap away from the relevant energy surfaces. Under such assumption, the multiband adiabatic theory assures that the subspace Ran PI is adiabatically protected against transitions, i. e. k(1  PI ) ei H " t/" PI 0 kH D O(") : Analogously to the case of a single band, one may construct the corresponding superadiabatic projector. The effective Hamiltonian b H eff;" corresponding to a family of m bands becomes, in this context, the "-Weyl quantization of matrix-valued function over the classical phase space [36,40]. A deeper understanding of nuclear dynamics near conical crossings and a further developments of multiband adiabatic perturbation theory are, in the opinion of the author, two of the main directions for future research. Bibliography Primary Literature 1. Berry MV, Lim R (1990) The Born–Oppenheimer electric gauge force is repulsive near degeneracies. J Phys A 23:L655–L657 2. Born M, Oppenheimer R (1927) Zur Quantentheorie der Molekeln. Ann Phys (Leipzig) 84:457–484 3. Brummelhuis R, Nourrigat J (1999) Scattering amplitude for Dirac operators. Comm Partial Differ Equ 24(1–2):377–394 4. Combes JM (1977) The Born–Oppenheimer approximation. Acta Phys Austriaca 17:139–159 5. Combes JM, Duclos P, Seiler R (1981) The Born–Oppenheimer approximation. In: Velo G, Wightman A (eds) Rigorous Atomic and Molecular Physics. Plenum, New York, pp 185–212 6. de Verdière YC (2004) The level crossing problem in semi-classical analysis. II. The Hermitian case. Ann Inst Fourier (Grenoble) 54(5):1423–1441 7. de Verdière YC, Lombardi M, Pollet C (1999) The microlocal Landau–Zener formula. Ann Inst H Poincaré Phys Theor 71:95– 127 8. de Verdière YC (2003) The level crossing problem in semi-classical analysis. I. The symmetric case. Proceedings of the international conference in honor of Frédéric Pham (Nice, 2002). Ann Inst Fourier (Grenoble) 53(4):1023–1054

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9. Emmrich C, Weinstein A (1996) Geometry of the transport equation in multicomponent WKB approximations. Commun Math Phys 176:701–711 10. Faure F, Zhilinskii BI (2000) Topological Chern indices in molecular spectra. Phys Rev Lett 85:960–963 11. Faure F, Zhilinskii BI (2001) Topological properties of the Born– Oppenheimer approximation and implications for the exact spectrum. Lett Math Phys 55:219–238 12. Faure F, Zhilinskii BI (2002) Topologically coupled energy bands in molecules. Phys Lett 302:242–252 13. Fermanian–Kammerer C, Gérard P (2002) Mesures semi-classiques et croisement de modes. Bull Soc Math France 130:123– 168 14. Fermanian–Kammerer C, Lasser C (2003) Wigner measures and codimension 2 crossings. J Math Phys 44:507–527 15. Folland GB (1989) Harmonic analysis in phase space. Princeton University Press, Princeton 16. Hagedorn GA (1980) A time dependent Born–Oppenheimer approximation. Commun Math Phys 77:1–19 17. Hagedorn GA (1986) High order corrections to the time-dependent Born–Oppenheimer approximation. I. Smooth potentials. Ann Math (2) 124(3):571–590 18. Hagedorn GA (1987) High order corrections to the time-independent Born–Oppenheimer approximation I: Smooth potentials. Ann Inst H Poincaré Sect. A 47:1–16 19. Hagedorn GA (1988) High order corrections to the timeindependent Born–Oppenheimer approximation II: Diatomic Coulomb systems. Comm Math Phys 116:23–44 20. Hagedorn GA (1988) High order corrections to the time-dependent Born–Oppenheimer approximation. II. Coulomb systems. Comm Math Phys 117(3):387–403 21. Hagedorn GA (1989) Adiabatic expansions near eigenvalue crossings. Ann Phys 196:278–295 22. Hagedorn GA (1992) Classification and normal forms for quantum mechanical eigenvalue crossings. Méthodes semiclassiques, vol 2 (Nantes, 1991). Astérisque 210(7):115– 134 23. Hagedorn GA (1994) Molecular propagation through electron energy level crossings. Mem Amer Math Soc 111:1–130 24. Hagedorn GA, Joye A (2007) Mathematical analysis of Born– Oppenheimer approximations. Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, In: Proc Sympos Pure Math 76, Part 1, Amer Math Soc, Providence, RI, pp 203–226 25. Herrin J, Howland JS (1997) The Born–Oppenheimer approximation: straight-up and with a twist. Rev Math Phys 9:467–488 26. Kato T (1950) On the adiabatic theorem of quantum mechanics. Phys Soc Jap 5:435–439 27. Klein M, Martinez A, Seiler R, Wang XP (1992) On the Born– Oppenheimer expansion for polyatomic molecules. Commun Math Phys 143:607–639 28. Klein M, Martinez A, Wang XP (1993) On the Born–

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Books and Reviews Bohm A, Mostafazadeh A, Koizumi A, Niu Q, Zwanziger J (2003) The geometric phase in quantum systems. Texts and monographs in physics. Springer, Heidelberg Teufel S (2003) Adiabatic perturbation theory in quantum dynamics. Lecture notes in mathematics, vol 1821. Springer, Berlin

Perturbation Theory for Non-smooth Systems

MARCO ANTÔNIO TEIXEIRA Department of Mathematics, Universidade Estadual de Campinas, Campinas, Brazil

are the main sources of motivation for our study concerning the dynamics of those systems that emerge from differential equations with discontinuous right-hand sides. We understand that non-smooth systems are driven by applications and they play an intrinsic role in a wide range of technological areas.

Article Outline

Introduction

Glossary Definition of the Subject Introduction Preliminaries Vector Fields near the Boundary Generic Bifurcation Singular Perturbation Problem in 2D Future Directions Bibliography

The purpose of this article is to present some aspects of the geometric theory of a class of non-smooth systems. Our main concern is to bring the theory into the domain of geometry and topology in a comprehensive mathematical manner. Since this is an impossible task, we do not attempt to touch upon all sides of this subject in one article. We focus on exploring the local behavior of systems around typical singularities. The first task is to describe a generic persistence of a local theory (structural stability and bifurcation) for discontinuous systems mainly in the twoand three-dimensional cases. Afterwards we present some striking features and results of the regularization process of two-dimensional discontinuous systems in the framework developed by Sotomayor and Teixeira in [44] and establish a bridge between those systems and the fundamental role played by the Geometric Singular Perturbation Theory (GSPT). This transition was introduced in [10] and we reproduce here its main features in the two-dimensional case. For an introductory reading on the methods of geometric singular perturbation theory we refer to [16,18,30]. In Sect. “Definition of the Subject” we introduce the setting of this article. In Sect. “Introduction” we survey the state of the art of the contact between a vector field and a manifold. The results contained in this section are crucial for the development of our approach. In Sect. “Preliminaries” we discuss the classification of typical singularities of non-smooth vector fields. The study of non-smooth systems, via GSPT, is presented in Sect. “Vector Fields near the Boundary”. In Sect. “Generic Bifurcation” some theoretical open problems are presented. One aspect of the qualitative point of view is the problem of structural stability, the most comprehensive of many different notions of stability. This theme was studied in 1937 by Andronov–Pontryagin (see [3]). This problem is of obvious importance, since in practice one obtains a lot of qualitative information not only on a fixed system but also on its nearby systems. We deal with non-smooth vector fields in R nC1 having a codimension-one submanifold M as its discontinuity set. The scheme in this work toward a systematic classification of typical singularities of non-smooth systems follows the

Perturbation Theory for Non-smooth Systems

Glossary Non-smooth dynamical system Systems derived from ordinary differential equations when the non-uniqueness of solutions is allowed. In this article we deal with discontinuous vector fields in Rn where the discontinuities are concentrated in a codimension-one surface. Bifurcation In a k-parameter family of systems, a bifurcation is a parameter value at which the phase portrait is not structurally stable. Typical singularity Are points on the discontinuity set where the orbits of the system through them must be distinguished. Definition of the Subject In this article we survey some qualitative and geometric aspects of non-smooth dynamical systems theory. Our goal is to provide an overview of the state of the art on the theory of contact between a vector field and a manifold, and on discontinuous vector fields and their perturbations. We also establish a bridge between two-dimensional nonsmooth systems and the geometric singular perturbation theory. Non-smooth dynamical systems is a subject that has been developing at a very fast pace in recent years due to various factors: its mathematical beauty, its strong relationship with other branches of science and the challenge in establishing reasonable and consistent definitions and conventions. It has become certainly one of the common frontiers between mathematics and physics/engineering. We mention that certain phenomena in control systems, impact in mechanical systems and nonlinear oscillations

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ideas developed by Sotomayor–Teixeira in [43] where the problem of contact between a vector field and the boundary of a manifold was discussed. Our approach intends to be self-contained and is accompanied by an extensive bibliography. We will try to focus here on areas that are complimentary to some recent reviews made elsewhere. The concept of structural stability in the space of nonsmooth vector fields is based on the following definition: Definition 1 Two vector fields Z and Z˜ are C0 equivalent if there is an M-invariant homeomorphism h : R nC1 ! ˜ R nC1 that sends orbits of Z to orbits of Z. A general discussion is presented to study certain unstable non-smooth vector fields within a generic context. The framework in which we shall pursue these unstable systems is sometimes called generic bifurcation theory. In [3] the concept of kth-order structural stability is also presented; in a local approach such setting gives rise to the notion of a codimension-k singularity. In studies of classical dynamical systems, normal form theory has been well accepted as a powerful tool in studying the local theory (see [6]). Observe that, so far, bifurcation and normal form theories for non-smooth vector fields have not been extensively studied in a systematic way. Control Theory is a natural source of mathematical models of these systems (see, for instance, [4,8,20,41,45]). Interesting problems concerning discontinuous systems can be formulated in systems with hysteresis ([41]), economics ([23,25]) and biology ([7]). It is worth mentioning that in [5] a class of relay systems in Rn is discussed. They have the form: X D Ax C sgn(x1 )k where x D (x1 ; x2 ; : : : ; x n ), A 2 M R (n; n) and k D (k1 ; k2 ; : : : ; k n ) is a constant vector in Rn . In [28,29] the generic singularities of reversible relay systems in 4D were classified. In [54] some properties of non-smooth dynamics are discussed in order to understand some phenomena that arise in chattering control. We mention the presence of chaotic behavior in some non-smooth systems (see for example [12]). It is worthwhile to cite [17], where the main problem in the classical calculus of variations was carried out to study discontinuous Hamiltonian vector fields. We refer to [14] for a comprehensive text involving nonsmooth systems which includes many models and applications. In particular motivating models of several nonsmooth dynamical systems arising in the occurrence of impacting motion in mechanical systems, switchings in electronic systems and hybrid dynamics in control systems are presented together with an extensive literature

on impact oscillators which we do not attempt to survey here. For further reading on some mathematical aspects of this subject we recommend [11] and references therein. A setting of general aspects of non-smooth systems can be found also in [35] and references therein. Our discussion does not focus on continuous but rather on non-smooth dynamical systems and we are aware that the interest in this subject goes beyond the approach adopted here. The author wishes to thank R. Garcia, T.M. Seara and J. Sotomayor for many helpful conversations. Preliminaries Now we introduce some of the terminology, basic concepts and some results that will be used in the sequel. Definition 2 Two vector fields Z and Z˜ on Rn with ˜ are germ-equivalent if they coincide on some Z(0) D Z(0) neighborhood V of 0. The equivalent classes for this equivalence are called germs of vector fields. In the same way as defined above, we may define germs of functions. For simplicity we are considering the germ notation and we will not distinguish a germ of a function and any one of its representatives. So, for example, the notation h : R n ; 0 ! R means that the h is a germ of a function defined in a neighborhood of 0 in Rn . Refer to [15] for a brief and nice introduction of the concepts of germ and k-jet of functions. Discontinuous Systems Let M D h1 (0), where h is (a germ of) a smooth function h : R nC1 ; 0 ! R having 0 2 R as its regular value. We assume that 0 2 M. Designate by (n C 1) the space of all germs of Cr vector fields on R nC1 at 0 endowed with the Cr -topology with r > 1 and large enough for our purposes. Call -(nC1) the space of all germs of vector fields Z in R nC1 ; 0 such that ( X(q) ; for h(q) > 0 ; Z(q) D (1) Y(q) ; for h(q) < 0 ; The above field is denoted by Z D (X; Y). So we are considering -(n C 1) D (n C 1)  (n C 1) endowed with the product topology. Definition 3 We say that Z 2 -(n C 1) is structurally stable if there exists a neighborhood U of Z in -(n C 1) such that every Z˜ 2 U is C0 -equivalent with Z. To define the orbit solutions of Z on the switching surface M we take a pragmatic approach. In a well characterized open set O of M (described below) the solution of Z

Perturbation Theory for Non-smooth Systems

Z through p follows, for t  0, the orbit of a vector field tangent to M. Such system is called sliding vector field associated with Z and it will be defined below. Definition 4 The sliding vector field associated to Z D (X; Y) is the smooth vector field Zs tangent to M and defined at q 2 M3 by Z s (q) D m  q with m being the point where the segment joining q C X(q) and q C Y(q) is tangent to M.

Perturbation Theory for Non-smooth Systems, Figure 1 A discontinuous system and its regularization

through a point p 2 O obeys the Filippov rules and on M  O we accept it to be multivalued. Roughly speaking, as we are interested in studying the structural stability in -(n C 1) it is convenient to take into account all the leaves of the foliation in R nC1 generated by the orbits of Z (and also the orbits of X and Y) passing through p 2 M. (see Fig. 1) The trajectories of Z are the solutions of the autonomous differential system q˙ D Z(q). In what follows we illustrate our terminology by presenting a simplified model that is found in the classical electromagnetism theory (see for instance [26]): x¨  « x C ˛signx D 0 : with ˛ > 0. So this system can be expressed by the following objects: h(x; y; z) D x and Z D (X; Y) with X(x; y; z) D (y; z; z C ˛) and Y(x; y; z) D (y; z; z  ˛). For each X 2 (n C 1) we define the smooth function X h : R nC1 ! R given by X h D X  r h where  is the canonical scalar product in R nC1 . We distinguish the following regions on the discontinuity set M: M 1 is the sewing region that is represented by h D 0 and (X h)(Y h) > 0; (ii) M 2 is the escaping region that is represented by h D 0, (X h) > 0 and (Y h) < 0; (iii) M 3 is the sliding region that is represented by h D 0, (X h) < 0 and (Y h) > 0. S We set O D iD1;2;3 M i . Consider Z D (X; Y) 2 -(n C 1) and p 2 M3 . In this case, following Filippov’s convention, the solution (t) of

It is clear that if q 2 M3 then q 2 M2 for Z and then we define the escaping vector field on M 2 associated with Z by Z e D (Z)s . In what follows we use the notation ZM for both cases. We recall that sometimes ZM is defined in an open region U with boundary. In this case it can be Cr extended to a full neighborhood of p 2 @U in M. S When the vectors X(p) and Y(p), with p 2 M2 M3 are linearly dependent then Z M (p) D 0. In this case we say that p is a simple singularity of Z. The other singularities of Z are concentrated outside the set O. We finish this subsection with a three-dimensional example: Let Z D (X; Y) 2 -(3) with h(x; y; z) D z, X D (1; 0; x) and Y D (0; 1; y). The system determines four quadrants around 0, bounded by  X D fx D 0g and Y D fy D 0g. They are: Q1C D fx > 0; y > 0g, Q1 D fx < 0; y < 0g, Q2 D fx < 0; y > 0g (sliding region) and Q3 D fx > 0; y < 0g (escaping region). ObS serve that M1 D Q1C Q1 . The sliding vector field defined in Q2 is expressed by:   yCx ;0 : Z s (x; y; z) D (y  x)1 x C y; 8 Such a system is (in Q2 ) equivalent to G(x; y; z) D (x C y; yCx 8 ; 0)). In our terminology we consider G a smooth extension of Zs , that is defined in a whole neighborhood of 0. It is worthwhile to say that G is in fact a system which is equivalent to the original system in Q2 . In [50] a generic classification of one-parameter families of sliding vector fields is presented.

(i)

Singular Perturbation Problem A singular perturbation problem is expressed by a differential equation z0 D ˛ (z; ") (refer to [16,18,30]) where z 2 R nCm , " is a small non-negative real number and ˛ is a C 1 mapping.  Let z D x; y 2 R nCm and f : R mCn ! R m ; g : R mCn ! R n be smooth mappings. We deal with equa-

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tions that may be written in the form ( x 0 D f (x; y; ") x D x(); y D y() : y 0 D "g(x; y; ")

(2)

An interesting model of such systems can be obtained from the singular van der Pol’s equation 00

2

0

"x C (x C x)x C x  a D 0 :

(3)

The main trick in the geometric singular perturbation (GSP) is to consider the family (2) in addition to the family ( "x˙ D f (x; y; ") x D x(t); y D y(t) (4) y˙ D g(x; y; ") obtained after the time rescaling t D ". Equation (2) is called the fast system and (4) the slow system. Observe that for " > 0 the phase portrait of fast and slow systems coincide. For " D 0, let S be the set of all singular points of (2). We call S the slow manifold of the singular perturbation problem and it is important to notice that Eq. (4) defines a dynamical system on S called the reduced problem. Combining results on the dynamics of these two limiting problems (2) and (4), with " D 0, one obtains information on the dynamics for small values of ". In fact, such techniques can be exploited to formally construct approximated solutions on pieces of curves that satisfy some limiting version of the original equation as " goes to zero. Definition 5 Let A; B R nCm be compact sets. The Hausdorff distance between A and B is D(A; B) D maxz 1 2A;z 2 2B fd(z1 ; B); d(z2 ; A)g. The main question in GSP-theory is to exhibit conditions under which a singular orbit can be approximated by regular orbits for " # 0, with respect to the Hausdorff distance. Regularization Process An approximation of the discontinuous vector field Z D (X; Y) by a one-parameter family of continuous vector fields will be called a regularization of Z. In [44], Sotomayor and Teixeira introduced the regularization procedure of a discontinuous vector field. A transition function is used to average X and Y in order to get a family of continuous vector fields that approximates the discontinuous one. Figure 1 gives a clear illustration of the regularization process. Let Z D (X; Y) 2 -(n C 1). Definition 6 A C 1 function ' : R ! R is a transition function if '(x) D 1 for x  1, '(x) D 1 for x  1

and ' 0 (x) > 0 if x 2 (1; 1). The -regularization of Z D (X; Y) is the one-parameter family X" 2 C r given by     1 1 '" (h(q)) '" (h(q)) C X(q)C  Y(q): Z" (q) D 2 2 2 2 (5) with h given in the above Subsect. “Discontinuous Systems” and '" (x) D '(x/"), for " > 0. As already said before, a point in the phase space which moves on an orbit of Z crosses M when it reaches the region M 1 . Solutions of Z through points of M 3 , will remain in M in forward time. Analogously, solutions of Z through points of M 2 will remain in M in backward time. In [34,44] such conventions are justified by the regularization method in dimensions two and three respectively. Vector Fields near the Boundary In this section we discuss the behavior of smooth vector fields in R nC1 relative to a codimension-one submanifold (say, the above defined M). We base our approach on the concepts and results contained in [43,53]. The principal advantage of this setting is that the generic contact between a smooth vector field and M can often be easily recognized. As an application the typical singularities of a discontinuous system can be further classified in a straightforward way. We say that X; Y 2 (n C 1) are M-equivalent if there exists an M-preserving homeomorphism h : R nC1 ; 0 ! R nC1 ; 0 that sends orbits of X into orbits of Y. In this way we get the concept of M-structural stability in (n C 1). We call 0 (n C 1) the set of elements X in (n C 1) satisfying one of the following conditions: 0) X h(0) ¤ 0 (0 is a regular point of X). In this case X is transversal to M at 0. 1) X h(0) D 0 and X 2 h(0) ¤ 0 (0 is a 2-fold point of X;) 2) X h(0) D X 2 h(0) D 0, X 3 h(0) ¤ 0 and the set fDh(0); DX h(0); DX 2 h(0)g is linearly independent (0 is a cusp point of X;) ... n) X h(0) D X 2 h(0) D    D X n h(0) D 0 and X nC1 h(0) ¤ 0. Moreover the set fDh(0); DX h(0); DX 2 h(0); : : : ; DX n h(0)g is linearly independent, and 0 is a regular point of the mapping X hjM . We say that 0 is an M-singularity of X if h(0) D X h(0) D 0. It is a codimension-zero M-singularity provided that X 2 0 (n C 1). We know that 0 (n C 1) is an open and dense set in (n C 1) and it coincides with the M-structurally stable vector fields in (n C 1) (see [53]).

Perturbation Theory for Non-smooth Systems

Denote by  X M the M-singular set of X 2 (nC1); this set is represented by the equations h D X h D 0. It is worthwhile to point out that, generically, all two-folds constitute an open and dense subset of  X . Observe that if X(0) D 0 then X 62 0 (n C 1). The M-bifurcation set is represented by 1 (n C 1) D (n C 1)  0 (n C 1) Vishik in [53] exhibited the normal forms of a codimension-zero M-singularity. They are: I) Straightened vector field X D (1; 0; : : : ; 0) and h(x) D x1kC1 C x2 x1k1 C x3 x1k2 C    C x kC1 ;

k D 0; 1; : : : ; n

or II) Straightened boundary

We define the smooth mapping X : R n ; 0 ! R n ; 0 by

X (p) D p˜. This mapping is a powerful tool in the study of vector fields around the boundary of a manifold (refer to [21,42,43,46,53]). We observe that  X coincides with the singular set of X . The late construction implements the following method. If we are interested in finding an equivalence between two vector fields which preserve M, then the problem can be sometimes reduced to finding an equivalence between X and Y in the sense of singularities of mappings. We recall that when 0 is a fold M-singularity of X then associated to the fold mapping X there is the symmetric diffeomorphism ˇ X that satisfies X ı ˇ X D X . Given Z D (X; Y) 2 -(n C 1) such that X and Y are fold mappings with X 2 h(0) < 0 and Y 2 h(0) > 0 then the composition of the associated symmetric mappings ˇ X and ˇY provides a first return mapping ˇ Z associated to Z and M. This situation is usually called a distinguished foldfold singularity, and the mapping ˇ Z plays a fundamental role in the study of the dynamics of Z.

h(x) D x1 and X(x) D (x2 ; x3 ; : : : ; x k ; 1; 0; 0; : : : ; 0) We now discuss an important interaction between vector fields near M and singularities of mapping theory. We discuss how singularity-theoretic techniques help the understanding of the dynamics of our systems. We outline this setting, which will be very useful in the sequel. The starting point is the following construction. A Construction Let X 2 (n C 1). Consider a coordinate system x D (x1 ; x2 ; : : : ; x nC1 ) in R nC1 ; 0 such that M D fx1 D 0g

Codimension-one M-Singularity in Dimensions Two and Three Case n D 1 In this case the unique codimensionzero M-singularity is a fold point in R2 ; 0. The codimension-one M-singularities are represented by the subset 1 (2) of 1 (2) and it is defined as follows. Definition 7 A codimension-one M-singularity of X 2 1 (2) is either a cusp singularity or an M-hyperbolic critical point p in M of the vector field X. A cusp singularity (illustrated in Fig. 2) is characterized by X h(p) D X 2 h(p) D 0, X 3 h(p) ¤ 0. In the second case this means that p is a hyperbolic critical point (illustrated in Fig. 3) of X with distinct eigenvalues and with invariant manifolds (stable, unstable and strong stable and strong unstable) transversal to M.

and X D (X 1 ; X 2 ; : : : ; X nC1 ) Assume that X(0) ¤ 0 and X 1 (0) D 0. Let N 0 be any transversal section to X at 0. By the implicit function theorem, we derive that: for each p 2 M; 0 there exists a unique t D t(p) in R; 0 such that the orbit-solution t 7! (p; t) of X through p meets N 0 at a point p˜ D (p; t(p)).

Perturbation Theory for Non-smooth Systems, Figure 2 The cusp singularity and its unfolding

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Perturbation Theory for Non-smooth Systems

Theorem 11 The following statements hold: (i) 1 (3) D 1 (a) [ 2 (b) is a codimension-one submanifold of (3). (ii) 1 (3) is open and dense set in 1 (3) in the topology induced from 1 (3). (iv) For a residual set of smooth curves : R; 0 ! (3);

meets 1 (3) transversally. Perturbation Theory for Non-smooth Systems, Figure 3 The saddle point in the boundary and its unfolding

In this subsubsection we consider a coordinate system in R2 ; 0 such that h(x; y) D y. The next result was proved in [46]. It presents the normal forms of the codimension-one singularities defined above. Theorem 8 Let X 2 1 (2). The vector field X is M-structurally stable relative to 1 (2) if and only if X 2 1 (2). Moreover, 1 (2) is an embedded codimension-one sub manifold and dense in 1 (2). We still require that any oneparameter family X  , ( 2 ("; ")) in (1) transverse to 1 (2) at X 0 , has one of the following normal forms: 0.1: 0.2: 1.1: 1.2: 1.3: 1.4:

X  (x; y) D (1; 0) (regular point); X  (x; y) D (1; x) (fold singularity); X  (x; y) D (1;  C x 2 ) (cusp singularity); X  (x; y) D (ax; x C by C ); a D ˙1; b D ˙2; X  (x; y) D (x; x  y C ); X  (x; y) D (x C y; x C y C ).

Case n D 2 Definition 9 A vector field X 2 (3) belongs to the set 1 (a) if the following conditions hold: (i) X(0) D 0 and 0 is a hyperbolic critical point of X; (ii) the eigenvalues of DX(0) are pairwise distinct and the corresponding eigenspaces are transversal to M at 0; (iii) each pair of non complex conjugate eigenvalues of DX(0) has distinct real parts. Definition 10 A vector field X 2 (3) belongs to the set 1 (b) if X(0) ¤ 0; X h(0) D 0; X 2 h(0) D 0 and one of the following conditions hold: (1) X 3 h(0) ¤ 0; rankfDh(0); DX h(0); DX 2 h(0)g D 2 and 0 is a non-degenerate critical point of X hjM . (2) X 3 h(0) D 0; X 4 h(0) ¤ 0 and 0 is a regular point of X hjM . The next results can be found in [43].

Throughout this subsubsection we fix the function h(x; y; z) D z. Lemma 12 (Classification Lemma) The elements of 1 (3) are classified as follows: (a11 ) Nodal M-Singularity: X(0) D 0, the eigenvalues of DX(0); 1 ; 2 ; and3 , are real, distinct, 1  j > 0; j D 2; 3 and the eigenspaces are transverse to M at 0; (a12 ) Saddle M-Singularity: X(0) D 0, the eigenvalues of DX(0); 1 ; 2 and 3 , are real, distinct, 1  j < 0; j D 2 or 3 and the eigenspaces are transverse to M at 0; (a13 ) Focal M-Singularity: 0 is a hyperbolic critical point of X, the eigenvalues of DX(0) are 12 D a˙ ib; 3 D c, with a; b; c distinct from zero and c ¤ a, and the eigenspaces are transverse to M at 0. (b11 ) Lips M-Singularity: presented in Definition 8, item 1, when Hess(F h/S (0)) > 0: (b12 ) Bec to Bec M-Singularity: presented in Definition 8, item 1, when Hess(F h/S (0)) < 0; (b13 ) Dove’s Tail M-Singularity: presented in Definition 8, item 2. The next result is proved in [38]. It deals with the normal forms of a codimension-one singularity. Theorem 13 i) (Generic Bifurcation and normal forms) Let X 2 (3). The vector field X is M-structurally stable relative to 1 (3) if and only if X 2 1 (3). ii) (Versal unfolding) In the space of one-parameter families of vector fields X ˛ in (3); ˛ 2 ("; ") an everywhere dense set is formed by generic families such that their normal forms are:  X˛ 2 0 (3) 0.1: X ˛ (x; y; z) D (0; 0; 1) 0.2: X ˛ (x; y; z) D (z; 0; ˙x) 0.3: X ˛ (x; y; z) D (z; 0; x 2 C y)  X0 2 1 (3) 3x 2 Cy 2 C˛ 1.1: X ˛ (x; y; z) D (z; 0; ) 2

3x 2 y 2 C˛ ) 2 4ı x 3 CyC˛x D (z; 0; ), with ı D ˙1 2 axCb yCcz 2 C˛ D (axz; byz; ), with 2

1.2: X˛ (x; y; z) D (z; 0; 1.3: X ˛ (x; y; z)

1.4: X ˛ (x; y; z) (a; b; c) D ı(3; 2; 1), ı D ˙1

Perturbation Theory for Non-smooth Systems

axCb yCcz 2 C˛

1.5: X ˛ (x; y; z) D (axz; byz; ), with 2 (a; b; c) D ı(1; 3; 2), ı D ˙1 axCb yCcz 2 C˛ 1.6: X ˛ (x; y; z) D (axz; byz; ), with 2 (a; b; c) D ı(1; 2; 3), ı D ˙1 xC2ycz 2 C˛ ) 1.7: X ˛ (x; y; z) D (xz; 2yz; 2 1.8: X ˛ (x; y; z) D ((x C y)z; (x  y)z; 3xyCz 2 C˛ ) 2

Perturbation Theory for Non-smooth Systems, Figure 4 M-critical point for X, M-regular for Y and its unfolding

Generic Bifurcation Let Z D (X; Y) 2 -r (n C 1). Call by ˙0 (n C 1) (resp. ˙1 (nC1)) the set of all elements that are structurally stable in -r (n C 1) (resp. -1r (n C 1) D -r (n C 1)n˙0 (nC1) ) in -r (n C 1). It is clear that a pre-classification of the generic singularities is immediately reached by: If Z D (X; Y) 2 ˙0 (n C 1) (resp. Z D (X; Y) 2 ˙1 (nC1)) then X and Y are in 0 (nC1) (resp. X 2 0 (nC 1) and Y 2 1 (n C 1) or vice versa). Of course, the case when both X and Y are in 1 (n C 1) is a-codimension-two phenomenon. Two-Dimensional Case The following result characterizes the structural stability in -r (2). Theorem A (see [31,44]): ˙0 (2) is an open and dense set of -r (2). The vector field Z D (X; Y) is in ˙0 (2) if and only one of the following conditions is satisfied: i) Both elements X and Y are regular. When 0 2 M is a simple singularity of Z then we assume that it is a hyperbolic critical point of ZM . ii) X is a fold singularity and Y is regular (and vice-versa). The following result still deserves a systematic proof. Following the same strategy stipulated in the generic classification of an M-singularity, Theorem 11 could be very useful. It is worthwhile to mention [33] where the problem of generic bifurcation in 2D was also addressed. Theorem B (Generic Bifurcation) (see [36,43]) ˙1 (2) is an open and dense set of -1r (2). The vector field Z D (X; Y) is in ˙1 (2) provided that one of the following conditions is satisfied: Both elements X and Y are M-regular. When 0 2 M is a simple singularity of Z then we assume that it is a codimension-one critical point (saddle-node or a Bogdanov–Takens singularity) of ZM . ii) 0 is a codimension-one M-singularity of X and Y is M-regular. This case includes when 0 is either a cusp i)

M-singularity or a critical point. Figure 4 illustrates the case when 0 is a saddle critical point in the boundary. iii) Both X and Y are fold M-singularities at 0. In this case we have to impose that 0 is a hyperbolic critical point of the Cr -extension of ZM provided that it is in the boundary of M2 [ M3 (see example below). Moreover when 0 is a distinguished fold-fold singularity of Z then 0 is a hyperbolic fixed point of the first return mapping ˇ Z . Consider in a small neighborhood of 0 in R2 , the system Z D (X; Y) with X(x; y) D (1  x 3 C y 2 ; x), Y(x; y) D (1 C x C y; x C x 2) and h(x; y) D y. The point 0 is a foldfold-singularity of Z with M2 D fx < 0g and Z s (x; 0) D (2x  x 2 )1 (2x  x 4 C x 5 ). Observe that 0 is a hyperbolic critical point of the extended system G(x; y) D 2x  x 4 C x5. The classification of the codimension-two singularities in -r (2) is still an open problem. In this direction [51] contains information about the classification of codimension-two M-singularities. Three-Dimensional Case Let Z D (X; Y) 2 -r (3). The most interesting case to be analyzed is when both vector fields, X and Y are fold singularities at 0 and the tangency sets  X and  Y in M are in general position at 0. In fact they determine (in M) four quadrants, two of them are M 1 -regions, one is an M 3 -region and the other is an M 2 -region (see Fig. 5). We emphasize that the sliding vector field ZM can be Cr -extended to a full neighborhood of 0 in M. Moreover, Z M (0) D 0. Inside this class the distinguished fold-fold singularity (as defined in Subsect. “A Construction”) must be taken into account. Denote by A the set of all distinguished fold-fold singularities Z 2 -r (3). Moreover, the eigenvalues of Dˇ Z (0) are p  D a ˙ (a 2  1). If  2 R we say that Z belongs to As . Otherwise Z is in A e . Recall that ˇ Z is the first return mapping associated to Z and M at 0 as defined in Subsect. “A Construction”.

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Perturbation Theory for Non-smooth Systems, Figure 5 The distinguished fold-fold singularity

It is evident that the elements in the open set A e are structurally unstable in -r (3). It is worthwhile to mention that in A e we detect elements which are asymptotically stable at the origin [48]. Concerning As few things are known. We have the following result: Theorem C The vector field Z D (X; Y) belongs to ˙0 (3) provided that one of the following conditions occurs: Both elements X and Y are regular. When 0 2 M is a simple singularity of Z then we assume that it is a hyperbolic critical point of ZM . ii) X is a fold singularity at 0 and Y is regular. iii) X is a cusp singularity at 0 and Y is regular. iv) Both systems X and Y are of fold type at 0. Moreover: a) the tangency sets  X and  Y are in general position at 0 in M; b) The eigenspaces associated with ZM are transverse to  X and  Y at 0 2 M and c) Z is not in A. Moreover the real parts of non conjugate eigenvalues are distinct. i)

Definition 14 Let U R2 be an open subset and "  0. A singular perturbation problem in U (SP-Problem) is a differential system which can be written as x0 D

dx  D f (x; y; ") ; d

y0 D

dy  D "g(x; y; ") (6) d

or equivalently, after the time re-scaling t D " "x˙ D "

dx t D f (x; y; ") ; d

y˙ D

dy t D g(x; y; ") ; (7) d

with (x; y) 2 U and f ; g smooth in all variables. Our first result is concerned with the transition between non-smooth systems and GSPT.

Singular Perturbation Problem in 2D

Theorem D Consider Z 2 -r (2); Z" its '-regularization, and p 2 M. Suppose that ' is a polynomial of degree k in a small interval I  (1; 1) with 0 2 I. Then the trajectories of Z" in V" D fq 2 R2 ; 0 : h(q)/" 2 Ig are in correspondence with the solutions of an ordinary differential equation z0 D ˛(z; "), satisfying that ˛ is smooth in both variables and ˛(z; 0) D 0 for any z 2 M. Moreover, if ((X  Y)h k )(p) ¤ 0 then we can take a C r1 -local coordinate system f(@/@x)(p); (@/@y)(p)g such that this smooth ordinary differential equation is a SP-problem.

Geometric singular perturbation theory is an important tool in the field of continuous dynamical systems. Needless to say that in this area very good surveys are available (refer to [16,18,30]). Here we highlight some results (see [10]) that bridge the space between discontinuous systems in -r (2) and singularly perturbed smooth systems.

The understanding of the phase portrait of the vector field associated to a SP-problem is the main goal of the geometric singular perturbation-theory (GSP-theory). The techniques of GSP-theory can be used to obtain information on the dynamics of (6) for small values of " > 0, mainly in searching minimal sets.

We recall that bifurcation diagrams of sliding vector fields are presented in [50,52].

Perturbation Theory for Non-smooth Systems

System (6) is called the fast system, and (7) the slow system of the SP-problem. Observe that for " > 0 the phase portraits of the fast and the slow systems coincide. Theorem D says that we can transform a discontinuous vector field in a SP-problem. In general this transition cannot be done explicitly. Theorem E provides an explicit formula of the SP-problem for a suitable class of vector fields. Before the statement of such a result we need to present some preliminaries. Consider C D f : R2 ; 0 ! Rg with  2 C r and L() D 0 where L() denotes the linear part of  at (0; 0). Let -d -r (2) be the set of vector fields Z D (X; Y) in -r (2) such that there exists  2 C that is a solution of r(X  Y) D ˘ i (X  Y) ;

(8)

where r is the gradient of the function and ˘ i denote the canonical projections, for i D 1 or i D 2. Theorem E Consider Z 2 -d and Z" its '-regularization. Suppose that ' is a polynomial of degree k in a small interval I R with 0 2 I. Then the trajectories of Z" on V" D fq 2 R2 ; 0 : h(q)/" 2 Ig are solutions of a SP-problem. We remark that the singular problems discussed in the previous theorems, when " & 0, defines a dynamical system on the discontinuous set of the original problem. This fact can be very useful for problems in Control Theory. Our third theorem says how the fast and the slow systems approximate the discontinuous vector field. Moreover, we can deduce from the proof that whereas the fast system approximates the discontinuous vector field, the slow system approaches the corresponding sliding vector field. Consider Z 2 -r (2) and : R2 ; 0 ! R with (x; y) being the distance between (x; y) and M. We denote by b Z the vector field given by b Z(x; y) D (x; y)Z(x; y). In what follows we identify b Z " and the vector field on Z(x; y; ") D (b Z " (x; y); ffR2 ; 0g n M  Rg R3 given by b 0). Theorem F Consider p D 0 2 M. Then there exists an open set U R2 ; p 2 U, a three-dimensional manifold M, a smooth function ˚ : M ! R3 and a SP-problem W on M such that ˚ sends orbits of Wj˚ 1 (U (0;C1)) in orbits of b Zj(U (0;C1)) .

non-regular point is (0; 0). In this case we may apply Theorem E.   2. Let Z" (x; y) D y/"; 2x y/"  x . The associated partial differential equation (refer to Theorem E) with i D 2 given above becomes 2(@/@x) C 4x(@/@y) D 4x. We take the coordinate change x D x; y D y  x 2 . The trajectories of X " in these coordinates are the solutions of the singular system "x˙ D y C x 2 ;

y˙ D x :

3. In what follows we try, by means of an example, to present a rough idea on the transition from nonsmooth systems to GSPT. Consider X(x; y) D (3y 2  y2; 1), Y(x; y) D (3y 2  yC2; 1) and h(x; y) D x. The regularized vector field is   1 1 x Z" (x; y) D C ' (3y 2  y  2; 1) 2 2 "   1 1 x C  ' (3y 2  y C 2; 1) : 2 2 " After performing the polar blow up coordinates ˛ : [0; C1)  [0; ]  R ! R3 given by x D r cos and " D r sin the last system is expressed by: r ˙ D  sin (y C '(cot )(3y 2  2)) ;

y˙ D '(cot ) :

So the slow manifold is given implicitly by '(cot ) D y/(3y 2  2) which defines two functions y1 ( ) D (1 C p 2 p1 C 24' (cot ))/(6'(cot )) and y2 ( ) D (1  1 C 24' 2 (cot ))/(6'(cot )). The function y1 ( ) is increasing, y1 (0) D 1; lim !/2 y1 ( ) D C1; lim !/2C y1 ( ) D 1 and y1 () D 1. The function y2 ( ) is increasing, y2 (0) D 2/3; lim !/2 y2 ( ) D 0 and y2 () D 2/3. We can extend y2 to (0; ) as a differential function with y2 (/2) D 0. The fast vector field is ( 0 ; 0) with 0 > 0 if ( ; y) belongs to h   [   0;  (y2 ( ); y1 ( )) ;   (y2 ( ); C1) 2 2 i [   ;   (1; y1 ( )) 2

Examples

and with 0 < 0 if ( ; y) belongs to h   [  0; 0;  (y1 ( ); C1)  (1; y2 ( )) 2 2 i [   ;   (y1 ( ); y2 ( )) : 2

1. Take X(x; y) D (1; x); Y(x; y) D (1; 3x), and h(x; y) D y. The discontinuity set is f(x; 0) j x 2 Rg. We have X h D x; Y h D 3x, and then the unique

The reduced flow has one singular point at (0; 0) and it takes the positive direction of the y-axis if y 2 ( 23 ; 0) [ (1; 1) and the negative direction of the y-axis if y 2 (1; 1) [ (0; 23 ).

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2.

3. 4.

5.

Perturbation Theory for Non-smooth Systems, Figure 6 Example of fast and slow dynamics of the SP-Problem

One can see that the singularities ( ; y; r) D (0; 1; 0) and ( ; y; r) D (0; 1; 0) are not normally hyperbolic points. In this way, as usual, we perform additional blow ups. In Fig. 6 we illustrate the fast and the slow dynamics of the SP-problem. We present a phase portrait on the blowing up locus where a double arrow over a trajectory means that the trajectory belongs to the fast dynamical system, and a simple arrow means that the trajectory belongs to the slow dynamical system. Future Directions Our concluding section is devoted to an outlook. Firstly we present some open problems linked with the setting that point out future directions of research. The main task for the future seems to bring the theory of non-smooth dynamical systems to a similar maturity as that of smooth systems. Finally we briefly discuss the main results in this text.

6.

How about the isochronicity of such a center? c) When does a polynomial perturbation of such a system in -(2) produce limit cycles? The articles [9,13,21,22,47] can be useful auxiliary references. Let -(N) be the set of all non-smooth vector fields on a two-dimensional compact manifold N having a codimension-one compact submanifold M as its discontinuity set. The problem is to study the global generic bifurcation in -(N). The articles [31,33,40,46] can be useful auxiliary references. Study of the bifurcation set in -r (3). The articles [38,40,43,50] can be useful auxiliary references. Study of the dynamics of the distinguished fold-fold singularity in -r (n C 1). The article [48] can be a useful auxiliary reference. In many applications examples of non-smooth systems where the discontinuities are located on algebraic varieties are available. For instance, consider the system x¨ C xsign(x) C sign(x˙ ) D 0. Motivated by such models we present the following problem. Let 0 be a non-degenerate critical point of a smooth mapping h : R nC1 ; 0 ! R; 0. Let ˚ (n C 1) be the space of all vector fields Z on R nC1 ; 0 defined in the same way as -(n C 1). We propose the following. i) Classify the typical singularities in that space. ii) Analyze the elements of ˚(2) by means of “regularization processes” and the methods of GSPT, similarly to Sect. “Vector Fields near the Boundary”. The articles [1,2] can be very useful auxiliary references. In [27,29] classes of 4D-relay systems are considered. Conditions for the existence of one-parameter families of periodic orbits terminating at typical singularities are provided. We propose to find conditions for the existence of such families for n-dimensional relay systems.

Conclusion

In connection to this present work, some theoretical problems remain open:

In this paper we have presented a compact survey of the geometric/qualitative theoretical features of non-smooth dynamical systems. We feel that our survey illustrates that this field is still in its early stages but enjoying growing interest. Given the importance and the relevance of such a theme, we have pointed above some open questions and we remark that there is still a wide range of bifurcation problems to be tackled. A brief summary of the main results in the text is given below.

1. The description of the bifurcation diagram of the codimension-two singularities in -(2). In this last class we find some models (see [37]) where the following questions can also be addressed. a) When is a typical singularity topologically equivalent to a regular center? b)

1. We firstly deal with two-dimensional non-smooth vector fields Z D (X; Y) defined around the origin in R2 , where the discontinuity set is concentrated on the line fy D 0g. The first task is to characterize those systems which are structurally stable. This characterization is

Some Problems

Perturbation Theory for Non-smooth Systems

a starting point with which to establish a bifurcation theory as indicated by the Thom–Smale program. 2. In higher dimension the problem becomes much more complicated. We have presented here sufficient conditions for the three-dimensional local structural stability. Any further investigation on bifurcation in this context must pass through a deep analysis of the so called foldfold singularity. 3. We have established a bridge between discontinuous and singularly perturbed smooth systems. Many similarities between such systems were observed and a comparative study of the two categories is called for.

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Perturbation Theory for PDEs

Perturbation Theory for PDEs DARIO BAMBUSI Dipartimento di Matematica, Università degli Studi di Milano, Milano, Italia Article Outline Glossary Definition of the Subject Introduction The Hamiltonian Formalism for PDEs Normal Form for Finite Dimensional Hamiltonian Systems Normal Form for Hamiltonian PDEs: General Comments Normal Form for Resonant Hamiltonian PDEs and its Consequences Normal Form for Nonresonant Hamiltonian PDEs Non Hamiltonian PDEs Extensions and Related Results Future Directions Bibliography Glossary Perturbation theory The study of a dynamical systems which is a perturbation of a system whose dynamics is known. Typically the unperturbed system is linear or integrable. Normal form The normal form method consists of constructing a coordinate transformation which changes the equations of a dynamical system into new equations which are as simple as possible. In Hamiltonian systems the theory is particularly effective and typically leads to a very precise description of the dynamics. Hamiltonian PDE A Hamiltonian PDE is a partial differential equation (abbreviated PDE) which is equivalent to the Hamilton equation of a suitable Hamiltonian function. Classical examples are the nonlinear wave equation, the Nonlinear Schrödinger equation, and the Kortweg–de Vries equation. Resonance vs. Non-Resonance A frequency vector f! k gnkD1 is said to be non-resonant if its components are independent over the relative integers. On the contrary, if there exists a non-vanishing K 2 Z n such that ! K D 0 the frequency vector is said to be resonant. Such a property plays a fundamental role in normal form theory. Non-resonance typically implies stability. Actions The action of a harmonic oscillator is its energy divided by its frequency. It is usually denoted by I. The typical issue of normal form theory is that in nonresonant systems the actions remain approximatively un-

changed for very long times. In resonant systems there are linear combinations of the actions with such properties. Sobolev space Space of functions which have weak derivatives enjoying suitable integrability properties. Here we will use the spaces H s , s 2 N of the functions which are square integrable together with their first s weak derivatives. Definition of the Subject Perturbation theory for PDEs is a part of the qualitative theory of differential equations. One of the most effective methods of perturbation theory is the normal form theory which consists of using coordinate transformations in order to describe the qualitative features of a given or generic equation. Classical normal form theory for ordinary differential equations has been used all along the last century in many different domains, leading to important results in pure mathematics, celestial mechanics, plasma physics, biology, solid state physics, chemistry and many other fields. The development of effective methods to understand the dynamics of partial differential equations is relevant in pure mathematics as well as in all the fields in which partial differential equations play an important role. Fluidodynamics, oceanography, meteorology, quantum mechanics, and electromagnetic theory are just a few examples of potential applications. More precisely, the normal form theory allows one to understand whether a small nonlinearity can change the dynamics of a linear PDE or not. Moreover, it allows one to understand how the changes can be avoided or forced. Finally, when the changes are possible it allows to predict the behavior of the perturbed system. Introduction The normal form method was developed by Poincaré and Birkhoff between the end of the 19th century and the beginning of the 20th century. During the last 20 years the method has been successfully generalized to a suitable class of partial differential equations (PDEs) in finite volume (in the case of infinite volume dispersive effects appear and the theory is very different. See e. g. [58]). In this article we will give an introduction to this recent field. We will almost only deal with Hamiltonian PDEs, since on the one hand the theory for non Hamiltonian systems is a small variant of the one we will present here, and on the other hand most models are Hamiltonian. We will start by a generalization of the Hamiltonian formalism to PDEs, followed by a review of the classical theory and by the actual generalization of normal form theory to PDEs.

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Perturbation Theory for PDEs

In the next section we give a generalization of the Hamiltonian formalism to PDEs. The main new fact is that in PDEs the Hamiltonian is usually a smooth function, but the corresponding vector field is nonsmooth (it is an operator extracting derivatives). So the standard formalism has to be slightly modified [10,29,45,49,52,61]. Here we will present a version of the Hamiltonian formalism which is enough to cover the models of interest for local perturbation theory. To clearly illustrate the situation we will start the article with an introduction to the Lagrangian and Hamiltonian formalism for the wave equation. This will lead to the introduction of the paradigm Hamiltonian which is usually studied in this context. This will be followed by a few results on the Hamiltonian formalism that are needed for perturbation theory. Subsequently, we shortly present the standard Birkhoff normal form theory for finite dimensional systems. This is useful since all the formal aspects are equal in the classical case and the case of PDEs. Then we come to the generalization of normal form theory to PDEs. In the present paper we will concentrate almost only on the case of 1-dimensional semilinear equations. This is due to the fact that the theory of higher dimensional and quasilinear equations is still quite unsatisfactory. In PDEs one essentially meets two kinds of difficulties. The first one is related to the existence of non smooth vector fields. The second difficulty is due to the fact that in the infinite dimensional case there are small denominators which are much worse than in the finite dimensional one. We first present the theory for completely resonant systems [10,14] in which the difficulties related to small denominators do not appear. It turns out that it is quite easy to obtain a normal form theorem for resonant PDEs, but the kind of normal form one gets is usually quite poor. In order to extract dynamical informations from the normal form one can only compute and study it explicitly. Usually this is very difficult. Nevertheless in some cases it is possible and leads to quite strong results. We will illustrate such a situation by studying a nonlinear Schrödinger equation [2,25]. For the general case there is a theorem ensuring that a generic system admits at least one family of “periodic like trajectories” which are stable over exponentially long times [15]. We will give its statement and an application to the nonlinear wave equation u t t  u x x C 2 u C f (u) D 0 ;

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Then we turn to the case of nonresonant PDEs. The main difficulty is that small denominators accumulate to zero already at order 3. Such a problem has been overcome in [4,6,9,12,43] by taking advantage of the fact that the nonlinearities appearing in PDEs typically have a special form. In this case one can deduce a very precise description of the dynamics and also some interesting results of the kind of almost global existence of smooth solutions [46]. To illustrate the theory we will make reference to the nonlinear wave Eq. (1) with almost any , and to the nonlinear Schrödinger equation. Another aspect of the theory of close to integrable Hamiltonian PDEs concerns the extension of KAM theory to PDEs. We will not present it here. We just recall the most celebrated results which are those due to Kuksin [48], Wayne [60], Craig–Wayne [34], Bourgain [24,26], Kuksin–Pöschel [50], Eliasson–Kuksin [39]. All these results ensure the existence of families of quasiperiodic solutions, i. e. solutions lying on finite dimensional manifolds. We also mention the papers [21,57] where some Cantor families of full dimension tori are constructed. We point out that in the dynamics on such 1dimensional tori the amplitude of oscillation of the linear modes decreases super exponentially with their index. A remarkable exception is provided by the paper [27] where the tori constructed are more “thick” (even if of course they lie on Cantor families). On the contrary, the results of normal form theory describe solutions starting on opens subsets of the phase space, and do not have particularly strong localizations properties with respect to the index. The price one has to pay is that the description one gets turns out to be valid only over long but finite times. Finally we point out a related research stream that has been carried on by Bourgain [21,22,23,25] who studied intensively the behavior of high Sobolev norms in close to integrable Hamiltonian PDEs (see also [28]). The Hamiltonian Formalism for PDEs The Gradient of a Functional Definition 1 Consider a function f 2 C 1 (Us ; R), Us

H s (T ) open, s  0 a fixed parameter and T :D R/2Z is the 1-dimensional torus. We will denote by r f (u) the gradient of f with respect to the L2 metric, namely the unique function such that hr f (u); hiL 2 D d f (u)h ; where

with  D 0 and the Dirichlet boundary conditions on a segment [55].

hu; vi L 2 :D

Z

8h 2 H s

(2)



u(x)v(x)dx 

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Perturbation Theory for PDEs

is the L2 scalar product and d f (u) is the differential of f at u. The gradient is a smooth map from H s to H s (see e. g. [3]). Example 2 Consider the function Z  2 ux f (u) :D dx ;  2

 ˙ L 2  L(u; u) ˙ uD H(v; u) :D hv; ui : ˙ ˙ u(u;v) (4)

which is differentiable as a function from H s ! R for any s  1. One has Z

Z



d f (u)h D 

u x h x dx D



u x x hdx D hu x x ; hiL 2

and therefore in this case one has r f (u) D u x x . Example 3 Let F : R2 ! R be a smooth function and define Z  f (u) D F (u; u x )dx (6) 

then the gradient of f coincides with the so called functional derivative of F : ıF @F @ @F : :D  ıu @u @x @u x

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Definition 7 Let H(v; u) be a Hamiltonian function, then the corresponding Hamilton equations are given by v˙ D ru H ;

u˙ D rv H :

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Until Subsect. “Basic Elements of Hamiltonian Formalism for PDEs” we will work at a formal level, without specifying the function spaces and the domains. ˙ be a Lagrangian function, then Definition 4 Let L(u; u) the corresponding Lagrange equations are given d ru˙ L D 0 dt

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where ru L is the gradient with respect to u only, and sim˙ ilarly ru˙ is the gradient with respect to u. Example 5 Consider the Lagrangian  Z   2 u˙ u2 u2 ˙ :D L(u; u)  x  2  F(u) dx : 2 2 2 

As in the finite dimensional case, one has that the Lagrange equations are equivalent to the Hamilton equation of H. An elementary computation shows that for the wave equation one has v D u˙ and Z





H(v; u) D 

 v 2 C u2x C 2 u2 C F(u) dx 2

(9)

then the corresponding Lagrange equations are given by (1) with f D F 0 . Given a Lagrangian system with a Lagrangian function L, one defines the corresponding Hamiltonian system as follows.

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Canonical Coordinates Consider a Lagrangian system and let e k be an orthonorP P ˙k e k , ˙ D mal basis of L2 , write u D k q k e k and u kq then one has the following proposition. Proposition 8 The Lagrange Eqs. (8) are equivalent to d @L @L  D0 @q k dt @q˙ k

Lagrangian and Hamiltonian Formalism for the Wave Equation

ru L 

(10)



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rf 

Definition 6 Consider the momentum v :D ru˙ L conju˙ then gated to u; assume that L is convex with respect to u, the Hamiltonian function associated to L is defined by

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Proof Taking the scalar product of (8) with e k one gets he k ; ru LiL 2 

d he k ; ru˙ LiL 2 D 0 dt

@L but one has he k ; ru Li D @q and similarly for the other k term. Thus the thesis follows. 

This proposition shows that, once a basis has been introduced, the Lagrange equations have the same form as in the finite dimensional case. In the Hamiltonian case exactly the same result holds. P Precisely, denoting v :D k p k e k one has the following proposition. Proposition 9 The Hamilton equations of a Hamiltonian function H are equivalent to p˙ k D 

@H ; @q k

q˙ k D

@H : @p k

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In the case of the nonlinear wave equation, in order to get a convenient form of the equations, one can choose the

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in the space of functions with zero mean value. The conserved energy is given by

Fourier basis. Such a basis is defined by 8 1 p cos kx ˆ k>0 ˆ <  1 kD0 eˆk :D p2 ˆ ˆ : p1 sin kx k < 0

(15)

Thus the Hamiltonian (12) takes the form ! X p2 C ! 2 q2 Z  X k k k H(p; q) D F q k eˆk (x) dx ; C 2 

k2

C 2 .

:D For the forthcoming developments where it is worth to rescale the variables by defining pk p0k :D p ; !k

q0k :D

p !k qk ;

(17)

so that, omitting primes, the Hamiltonian takes the form H(p; q) D

X

!k

k

p2k C q2k C H P (p; q) 2

(18)

where H P has a zero of order higher than 2. In the following we will always study systems of the form (18). Moreover, by relabeling the variables and the frequencies it is possible to reduce the problem to the case where k varies in N  f1; 2; 3 : : : g. This is what we will assume in developing the abstract theory. Example 10 An example of a different nature in which the Hamiltonian takes the form (18) is the nonlinear Schrödinger equation i˙ D

xx

C f (j j2)

;

(19)

where f is a smooth function. Eq. (19) has the conserved energy functional Z    2 H( ; ¯ ) :D (20) j j C F(j j2) dx ; 

where F is such that F 0 D f . Introduce canonical coordinates (p k ; q k ) by D

X p k C iq k eˆk ; p 2 k2Z

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then the energy takes the form (18) with ! k D k 2 and the NLS is equivalent to the corresponding Hamilton equations. Example 11 Consider the Kortweg–de Vries equation u t C u x x x C uu x D 0 ;

(22)

u2x u3 C 2 6

 dx ;

(23)

which again is also the Hamiltonian of the system. Canonical coordinates are here introduced by Xp k(p k eˆk C q k eˆk ) ; (24) u(x) D k>0

k

(16) ! k2



 



k2Z

Z H(u) D

in which the Hamiltonian takes the form (18) with ! k D k3 . Remark 12 It is also interesting to study some of these equations with Dirichlet boundary conditions (DBC), typically on [0; ]. This will always be done by identifying the space of the functions fulfilling DBC with the space of the function fulfilling periodic boundary conditions on [; ] which are skew symmetric. Similarly, Neumann boundary conditions will be treated by identifying the corresponding functions with periodic even functions. In some cases (e. g. in Eq. (1) with DBC and an f which does not have particular symmetries) the equations do not extend naturally to the space of skew symmetric and this has some interesting consequences (see [7,13]). Basic Elements of Hamiltonian Formalism for PDEs A suitable topology in the phase space is given by a Sobolev like topology. For any s 2 R, define the Hilbert space `2s of the sequences x  fx k g k1 with x k 2 R such that kxk2s :D

X

jkj2s jx k j2 < 1

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k

and the phase spaces Ps :D `2s ˚ `2s  z 3 (p; q)  (fp k g; fq k g). In Ps we will sometimes use the scalar product h(p; q); (p1 ; q1 )is :D hp; p1 i`2s C hq; q1 i`2s :

(26)

In the following we will always assume that j! k j  Cjkjd

(27)

for some d. Remark 13 Defining the operator A0 : D(A0 ) ! Ps by A0 (p; q) D (! k p k ; ! k q k ) one can write H0 D 12 hA0 z; zi0 , D(A0 )  PsCd . Given a smooth Hamiltonian function  : Ps  Us ! R, Us being an open neighborhood of the origin, we define

Perturbation Theory for PDEs

the corresponding Hamiltonian vector field X : Us 7! Ps by   @ @ ; : (28) X   @q k @p k Remark 14 Corresponding to a function  as above we will denote by r its gradient with respect to the `2  `20 metric. Defining the operator J by J(p; q) :D (q; p) one has X D Jr. Definition 15 The Poisson Bracket of two smooth functions 1 , 2 is formally defined by f1 ; 2 g :D d1 X2  hr1 ; Jr2 i0 :

(29)

P P Remark 16 As the example 1 D k kq k , 2 :D k kp k shows, there are cases where the Poisson Bracket of two functions is not defined. For this reason a crucial role is played by the functions whose vector field is smooth. Definition 17 A function  2 C 1 (Us ; Ps ), Us Ps open, is said to be of class Gens , if the corresponding Hamiltonian vector field X is a smooth map from Us ! Ps . In this case we will write  2 Gens Proposition 18 Let 1 2 Gens . If 2 2 C 1 (Us ; R) then f1 ; 2 g 2 C 1 (Us ; R). If 2 2 Gens then f1 ; 2 g 2 Gens .

Theorem 22 (Birkhoff) For any positive integer r  0, there exist a neighborhood U(r) of the origin and a canonical transformation Tr : R2n  U(r) ! R2n which puts the system (18) in Birkhoff Normal Form up to order r, namely s.t. H (r) :D H ı Tr D H0 C Z (r) C R(r)

(31)

where Z (r) is a polynomial of degree rC2 which is in normal form, R(r) is small, i. e. jR(r) (z)j  Cr kzkrC3 ;

8z 2 U(r) ;

(32)

moreover, one has kz  Tr (z)k  Cr0 kzk2 ;

8z 2 U(r) :

(33)

An inequality identical to (33) is fulfilled by the inverse transformation Tr1 . If the frequencies are nonresonant at order rC2, namely if !  K 6D 0 ;

8K 2 Z n ;

0 < jKj  r C 2

(34)

the function Z (r) depends on the actions I j :D

p2j C q2j 2

only.

Definition 19 A smooth coordinate transformation T : Ps  Us ! Ps is said to be canonical if for any Hamiltonian function H one has X HıT D T  X H  dT 1 X H ı T , i. e. it transforms the Hamilton equations of H into the Hamilton equations of H ı T .

Remark 23 If the nonlinearity is analytic and the frequencies are Diophantine, i. e. there exist > 0 and  such that

Proposition 20 Let 1 2 Gens , and let ˚t 1 be the corresponding time t flow (which exists by standard theory). Then ˚t 1 is a canonical transformation. Normal Form for Finite Dimensional Hamiltonian Systems

then one can compute the dependence of the constant Cr (cf. Eq. (32)) on r and optimize the value of r as a function of kzk. This allows one to improve (32) and to show that there exists and ropt such that (see e. g. [40])   c jR(r opt ) (z)j  C exp  : (36) kzk1/(C1)

Consider a system of the form (18), but with finitely many degrees of freedom, namely a system with a Hamiltonian of the form (18) with

In turn, such an estimate is the starting point for the proof of the celebrated Nekhoroshev’s theorem [53].

H0 (p; q) D

n X kD1

!k

p2k C q2k ; 2

!k 2 R

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and H P which is a smooth function having a zero of order at least 3 at the origin. Definition 21 A polynomial Z will be said to be in normal form if fH0 ; Zg  0.

j!  Kj 

; jKj

8K 2 Z n  f0g ;

(35)

The idea of the proof is to construct a canonical transformation putting the system in a form which is as simple as possible, namely the normal form. More precisely one constructs a canonical transformation T (1) pushing the non normalized part of the Hamiltonian to order four followed by a transformation T (2) pushing it to order five and so on, thus getting Tr D T (1) ı T (2) ı    ı T (r) . Each of the transformations T ( j) is constructed as the time one

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flow of a suitable auxiliary Hamiltonian function say j (Lie transform method). It turns out that j is determined as the solution of the Homological equation Z j :D f j ; H0 g C H

( j)

(37)

where H ( j) is constructed recursively and Zj has to be determined together with j in such a way that fZ j ; H0 g D 0 and (37) holds. In particular H (1) coincides with the first non vanishing term in the Taylor expansion of H P . The algorithm of solution of the Homological Eq. (37) involves division by the so called small denominators i!  K, where K 2 Zn  f0g, fulfills jKj  j C 2 and !  K 6D 0. The above construction is more or less explicit: provided one has at disposal enough time, he can explicitly compute Z (r) up to any given order. In the case of nonresonant frequencies this is not needed if one wants to understand the dynamics over long times. Indeed its features are an easy consequence of the fact that Z (r) depends on the actions only. A precise statement will be given in the case of PDEs. It has to be noticed that the normal form can be used also as a starting point for the construction of invariant tori through KAM theory. To this end however one has to verify a nondegeneracy condition and this requires the explicit computation of the normal form. In the resonant case the situation is more complicated, however, it is often enough to compute the first non vanishing term of Z (r) in order to get relevant information on the dynamics. This usually requires only the ability to compute the function Z1 , defined by (37) with H (1) coinciding with the first non vanishing term of the Taylor expansion of H P . For a detailed analysis we refer to other sections of the Encyclopedia. A particular case where one can use a coordinate independent formula for the computation of Zj and j is the one in which the frequencies are completely resonant. Assume that there exists  > 0 and integer numbers `1 ; ::; `n such that ! k D ` k

8k D 1; : : : ; n :

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Lemma 25 Let f be smooth function, defined in neighborhood of the origin. Define

Z(z)  h f i (z) :D 1 (z) :D T

Z

T

1 T

Z

T

f ( t (z))dt ;

0

(40)



t f ( t (z))  Z( t (z)) dt ;

0

then such quantities fulfill the equation fH0 ; g C f D Z.

Normal Form for Hamiltonian PDEs: General Comments As anticipated in the introduction there are two problems in order to generalize Birkhoff’s theorem to PDEs: (1) the existence of nonsmooth vector fields and (2) the appearance of small denominators accumulating at zero already at order 3. There are two reasons why (1) is a problem. The first one is that if the vector field of 1 were not smooth then it would be nontrivial to ensure that it generates a flow, and thus that the normalizing transformation exists. The second related problem is that, if a transformation could be generated, then the Taylor expansion of the transformed Hamiltonian would contain a term of the form fH1 ; 1 g D dH1 X1 , which is typically not smooth if X  is not smooth. Thus one has to show that the construction involves only functions which are of class Gens for some s (see Definition 17). The difficulty related to small denominators is the following: In the finite dimensional case, f!  K 6D 0; jKj  r C 2g implies j!  Kj  > 0. Thus division by !  K is a harmless operation in the finite dimensional case. In the infinite dimensional p case this is no longer true. For example, when ! k D k 2 C 2 one already has inf

06DjKj3

j!  Kj D 0 :

t

the flow of the linear system Remark 24 Denote by with Hamiltonian H 0 , then one has  tCT D  t ;

T :D

2 ; 

t 2R:

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Moreover in this case one has !  K 6D 0 H) j!  Kj   > 0, so there are no small denominators. In this case one has an interesting coordinate independent formula for the solution of the homological Eq. (37).

In order to solve such a problem one has to take advantage of a property of the nonlinearity which typically holds in PDEs and is called having localized coefficients. By also assuming a suitable nonresonance property for the frequency vector, one can deduce a normal form theorem identical to Theorem 22. The main difficulty consists in verifying the assumptions of the theorem. We will show how to verify such assumptions by applying this method to some typical examples.

Perturbation Theory for PDEs

Normal Form for Resonant Hamiltonian PDEs and its Consequences In the case of resonant frequencies and smooth vector field it is possible to obtain a normal form up to an exponentially small remainder. Consider the system (18) in the phase space Ps with some fixed s. Assume that the frequencies are completely resonant, namely that (38) holds (with k 2 N); assume that H P 2 Gens and that its vector field extends to a complex analytic function in a neighborhood of the origin. Finally assume that H P has a zero of order n  3 at the origin. Then we have the following theorem. Theorem 26 ([10,14]) There exists a neighborhood of the origin Us Ps and an analytic canonical transformation T : Us ! Ps with the following properties: T is close to identity. Namely, it satisfies kz  T (z)ks  Ckzksn1 :

(41)

the case of Dirichlet boundary conditions, thus the function will always be assumed to be skew symmetric with respect to the origin. Define  Z  1 s :D j@s 2  x

(0)

(x)j

2

1/2 ;

(44)

i. e., the H s norm of the initial datum (0) , s  0, and denote by I k (0) the initial value of the linear actions. Theorem 28 ([2]) Fix N  1, then there exists a constant  ; with the property that, if the initial datum (0) is such that X 1 <  ; I k (0)2  C 14 121/N ; (45) kNC1

then along the corresponding solution of (19) one has X jI k (t)  I k (0)j2  C 0 14C1/N (46) k1

T puts the Hamiltonian in resonant normal form up to an

exponentially small remainder, namely one has H ı T D H0 C hH P i C Z2 C R

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where hH P i is the average (defined by (40)) of H P with respect to the unperturbed flow; Z 2 is in normal form, namely fZ2 ; H0 g  0, and has a zero of order 2n  2 at the origin; R is an exponentially small remainder whose vector field is estimated by   C : kX R (z)ks  Ckzksn1 exp  kzksn2 Example 27 The nonlinear Schrödinger equation (19). Here one has ` k D k 2 and  D 1. The Sobolev embedding theorems ensure that the vector field of the nonlinearity is analytic if f is analytic in a neighborhood of the origin. Thus Theorem 26 applies to the NLS. To deduce dynamical consequences it is convenient to explicitly compute hH P i. Assuming f (0) D 0 and f 0 (0) D 1 this was done in [2] using formula (40) which gives

for the times t fulfilling 

 jtj  C exp 1 00

1/N :

This result in particular allows one to control the distance in energy norm of the solution from the torus given by the intersection of the level surfaces of the actions taken at the initial time. Theorem 29 ([25]) Fix an arbitrarily large r, then there exists sr such that for any s  sr there exists s , such that the following holds true: most of the initial data with s < s give rise to solutions with k (t)ks  C s ;

8jtj 

C : sr

(47)

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For the precise meaning of “most of the initial data,” we refer to the original paper. The result is based on the proof that the considered solutions remain close in the H s topology to an infinite dimensional torus. In particular the uniform estimate of the Sobolev norm is relevant for applications to numerical analysis [30,44].

where I k D (p2k C q2k )/2 are the linear actions. Thus one has that H0 C hH P i is a function of the actions only, and thus it is an integrable system. It is thus natural to study the system (42) as a perturbation of such an integrable system. This was done in [2] and [25] obtaining the results we are going to state. For simplicity we will concentrate here on

Example 30 Consider the nonlinear wave Eq. (1) with  D 0. Here the frequencies are given by jkj and thus they are completely resonant. Again the smoothness of the nonlinearity is ensured by Sobolev embedding theorem. In the case of DBC in order to ensure smoothness one has also to assume that the nonlinearity is odd, namely that f (u) D  f (u), then Theorem 26 applies. However

1 X hH P i(z) D Ik 2 k

!2 

1X jI k j2 8 k

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Perturbation Theory for PDEs

in this case the computation of hH P i is nontrivial. It has been done (see [55]) in the case of f (u) D ˙u3 C O(u4 ) and Dirichlet boundary conditions. The result however is that the function hH P i does not have a particularly simple structure, and thus it is not easy to extract informations on the dynamics. In order to extract information on the dynamics, consider the simplified system in which the remainder is neglected, namely the system with Hamiltonian H S :D H0 C hH P i C Z2 :

(48)

Such a system has two integrals of motion, namely H 0 and hH P i C Z2 . Let be the set of the z’s at which hH P i C Z2 is restricted to the surface S :D fz : H0 (z) D 2 g has an extremum, say a maximum. Then is an invariant set for the dynamics of H S . By the invariance under the flow of H 0 , one has that is the union of one dimensional closed curves, but generically it is just a single closed curve. In such a situation it is also a stable periodic orbit of (48) (see [35]). Actually it is very difficult to compute (hH P i C Z2 )jS , but a maximum of such a function can be easily constructed by applying the implicit function theorem to a non degenerate maximum of hH P ijS . The addition of the remainder then modifies the dynamics only after an exponentially long time. We are now going to state the corresponding theorem. First remark that a critical point of hH P ijS1 is a solution za of the system  a A0 z a C rhH P i(z a ) D 0 ;

H0 (z a ) D 1

(49)

where we used the notations of Remarks 13 and 14. Here a is clearly the Lagrange multiplier. The closed curve S t

a :D t  (z a ) is (the trajectory of) a periodic solution of H0 C hH P i. Consider now the linear operator B a :D d(rhH P i)(z a ). Definition 31 The critical point za is said to be non degenerate if the system  a A0 h C B a h D 0 ;

hA0 z a ; hi0 D 0

(50)

has at most one solution. Under the assumptions of Theorem 32 below it is easy to prove that a is ˇa smooth curve and that its tangent vector d  t (z a )ˇ tD0 is a solution of (50). h a :D dt Theorem 32 ([14,15]) Assume that H P 2 Gens for any s large enough, assume also that there exists a non degenerate maximum za of hH P ijS1 , then there exists a constant  , such that the following holds true: consider a solution z(t)

of the Hamilton equation of (18) with initial datum z0 ; if there exists <  , such that d E ( a ; z0 )  C n ;

(51)

then one has d E ( a ; z(t))  C 0 n ;

(52)

C exp for all times t with jtj  r1 distance in the energy norm.

  n1

. Here dE is the

Such a theorem does not ensure that there exist periodic orbits of the complete system, but just a family of closed curves with the property that starting close to it one remains close to it for exponentially long times. Some results concerning the existence of true periodic orbits close to such periodic like trajectories can also be proved (see e. g. [16,19,20,42,51]). Example 33 In the paper [55] it has been proved that the non degeneracy assumption (50) of Theorem 32 holds for the Eq. (1) with f (u) D ˙u3 Chigher order terms and Dirichlet boundary conditions. In the case of such an equation an extremum of hH P ijS1 is given by u(x) D Vm sn(wx; m) ;

v(x)  0 ;

with V m , w and m suitable constants, and sn the Jacobi elliptic sine. Therefore the curve a is the phase space trajectory of the solution of the linear wave equation with such an initial datum. There are no other extrema of hH P ijS1 . Thus the Theorem 32 ensures that solutions starting close to a rescaling of such a curve remain close to it for very long times. Normal Form for Nonresonant Hamiltonian PDEs A Statement We turn now to the nonresonant case. The theory we will present has been developed in [6,9,43], and is closely related to the one of [4,11,37]. First we introduce the class of equations to which the theory applies. To this end it is useful to treat the p’s and the q’s exactly on an equal footing ¯ :D Z  f0g the set of so we will denote by z  (z k ) k2Z¯ , Z all the variables, where zk :D p k ;

z k :D q k

k  1:

Given a polynomial function f : P1 ! R of degree r one can decompose it as follows X f (z) D f k 1 ;:::;k r z k 1 : : : z k r : (53) k 1 ;:::;k r

Perturbation Theory for PDEs

We will assume suitable localization properties for the coefficients f k 1 ;:::;k r as functions of the indexes k1 ; : : : ; kr . Definition 34 Given a multi-index k  (k1 ; : : : ; kr ), let (k i 1 ; k i 2 ; k i 3 : : : ; k i r ) be a reordering of k such that jk i 1 j  jk i 2 j  jk i 3 j      jk i r j : We define (k) :D jk i 3 j and S(k) :D (k) C jjk i 1 j  jk i 2 jj :

(54)

Definition 35 Let f : P1 ! R be a polynomial of degree r. We say that f has localized coefficients if there exists  2 [0; C1) such that 8N  1 there exists CN such that for any choice of the indexes k1 ; : : : ; kr , the following inequality holds: ˇ ˇ fk

1 ;:::;k r

ˇ ˇ  CN

(k)CN S(k) N

:

(55)

It is easy to see that under this condition one can solve the homological equation and that, if the known term of the equation has localized coefficients, then the solution also has localized coefficients. Theorem 40 ([9,12]) Fix r  1, assume that the frequencies fulfill the nonresonance condition (r-NR); assume that H P has localized coefficients. Then there exists a finite sr , a neighborhood U(r) s r of the origin in Ps r , and a canonical transformation T : U(r) s r ! Ps r which puts the system in normal form up to order r C 3, namely H (r) :D H ı T D H0 C Z (r) C R(r)

where Z (r) has localized coefficients and is a function of the actions I k only; R(r) has a small vector field, i. e. kX R(r) (z)ks r  CkzkrC2 ; sr

Some important properties of functions with localized coefficients are given by Theorem 37. Theorem 37 Let f : P1 ! R be a polynomial of degree r with localized coefficients, then there exists s0 such that for any s  s0 the vector field X f extends to a smooth map from Ps to itself; moreover the following estimate holds: kX f (z)k  Ckzks kzksr2 : 0

(56)

In particular it follows that a function with localized coefficients is of class Gens for any s  s0 . Theorem 38 The Poisson Bracket of two functions with localized coefficients has localized coefficients. In order to develop perturbation theory we also need a quantitative nonresonance condition. Definition 39 Fix a positive integer r. The frequency vector ! is said to fulfill the property (r-NR) if there exist

> 0; and ˛ 2 R such that for any N large enough one has ˇ ˇ ˇ ˇ ˇX ˇ

ˇ ! k K k ˇˇ  ˛ ; (57) ˇ ˇ ˇ N

8z 2 U(r) sr ;

(59)

8z 2 U(r) sr :

(60)

thus, one has kz  Tr (z)ks r  Ckzk2s r ;

Definition 36 A function f 2 Gens for any s large enough, is said to have localized coefficients if all the terms of its Taylor expansion have localized coefficients.

(58)

An inequality identical to (60) is fulfilled by the inverse transformation Tr1 . Finally for any s  s r there exists (r) a subset U(r) s Us r open in Ps such that the restriction of the canonical transformation to U(r) s is analytic also as a map from Ps ! Ps and the inequalities (59) and (60) hold with s in place of sr . This theorem allows one to give a very precise description of the dynamics. Proposition 41 Under the same assumptions of Theorem 37, 8s  s r there exists s such that, if the initial datum fulfills :D kz0 ks < s , then one has X kz(t)ks  4 ; k 2s jI k (t)  I k (0)j  C 3 (61) k

for all the times t fulfilling jtj  r . Moreover there exists a smooth torus T0 such that, 8M  r ds (z(t); T0 )  C (MC3)/2 ;

for jtj 

1 rM

(62)

where ds (:; :) is the distance in Ps . A generalization to the resonant or partially resonant case is easily obtained and can be found in [11]. Verification of the Property of Localization of Coefficients

k1

for any K 2 Z1 , fulfilling 0 6D jKj :D P k>N jK k j  2.

P

k jK k j  r C 2,

The property of localization of coefficients is quite abstract. We illustrate via a few examples some ways to verify it.

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Perturbation Theory for PDEs

Example 42 Consider the nonlinear wave Eq. (1) with Neumann boundary conditions on [0; ]. We recall that the corresponding space of functions will be considered as a subset of the space of periodic functions. Consider the TaylorR expansion of the nonlinearity, i. e. P  write F(u) D r3 cr  u r . Then one has to prove that R r the functions f r (u)   u have localized coefficients. The coefficients f k 1;:::;k r are given by Z  cos(k1 x) cos(k2 x) : : : cos(kr x)dx : (63) f k 1 ;:::;k r D 

One could compute and estimate such a quantity directly, but it is easier to proceed in a different way: to show that f 3 has localized coefficients and then to use Theorem 38 to show that each f r has localized coefficients for any r. This is the path we will follow. Consider Z  cos(k1 x) cos(k2 x) cos(k3 x)dx : (64) f k 1 ;k 2 ;k 3 D

Lemma 44 Let S be as above, then, for any N  0 one has ˇ ˇL k

1 ;k 2

ˇ ˇ˝ ˛ˇ ˇ D ˇ L' k ; ' k ˇ  1 2

j k 1

ˇ˝ ˛ˇ 1 ˇ L N 'k ; 'k ˇ 1 2 N  k2 j (68)

where  k j is the eigenvalue of S corresponding to ' k j . Then, in order to conclude the verification of the property of localization of the coefficients, one has just to compute LN and to estimate the scalar product product of the r.h.s. All the computations can be found in [6]. Example 45 A third example where the verification of the property of localization of coefficients goes almost in the same way is the nonlinear Schrödinger equation i˙ D

xx

CV

C

@F( ; ¯ ) @¯

(69)



Since the estimate (55) is symmetric with respect to the indexes, we can assume that they are ordered, k1  k2  k3 , so that (64)D ı kk12 Ck 3 /2, (k) D k3 , S(k) D k3 C k1  k2 from which one immediately sees that (55) holds with  D 0.R As a consequence one also gets that the function  g3 :D  vu2 has localized coefficients. Since fg3 ; f r g D r f rC1 , by induction Theorem 38 ensures that f r has localized coefficients for any r. Often it is impossible to explicitly compute the coefficients f k 1 ;:::;k r , so one needs a different way to verify the property. Example 43 Consider the nonlinear wave equation u t t  u x x C Vu D f (u)

(65)

with Neumann boundary conditions. Here V is a smooth, even, periodic potential. The Hamiltonian reduces to the form (18) by introducing the variables qk by u(x) D P k q k ' k (x) where ' k (x) are the eigenfunctions of the Sturm Liouville operator @x x C V, and similarly for v. In such a case one has Z  f k 1 ;k 2 ;k 3 D ' k 1 (x)' k 2 (x)' k 3 (x)dx : (66) 

Here the idea is to consider (66) as the matrix element L k 1 ;k 2 of the operator L of multiplication by ' k 3 (x) on the basis of the eigenfunctions of the operator S :D @x x C V. The key idea is to proceed as follows. Let L be a linear operator which maps D(S r ) into itself for all r  0, and define the sequence of operators L N :D [S; L N1 ] ;

L0 :D L :

(67)

with Dirichlet Boundary conditions on [0; ]. Here one has to assume that V is a smooth even potential and that F is smooth and fulfills F( ;  ¯ ) D F( ; ¯ ) (this is required in order to leave the space of skew symmetric functions invariant, see Remark 12). Here the variables (p; q) are introduced by D

X p k C iq k p 'k ; 2 k2Z

(70)

where k are the eigenfunctions of S with Dirichlet boundary conditions. Here the Taylor expansion of the nonlinearity has only even terms. Thus the building block for the proof of the property of localization of coefficients is the operator L of multiplication by ' k 3 ' k 4 . Then, mutatis mutandis the proof goes as in the previous case. Verification of the Nonresonance Property Finally in order to apply Theorem 40 one has to verify the nonresonance property (r-NR). As usual in dynamical systems, this is done by tuning the frequencies using parameters. In the case of the nonlinear wave Eq. (1) one can use the mass . Theorem 46 ([4,12,36]) There exists a zero measure set S R such that, if  2 R  S, then the frequencies ! k D p k 2 C 2 , k  1 fulfill the condition (r-NR) for any r. Thus the Theorem 40 applies to the Eq. (1) with almost any mass. A similar result holds for the Eq. (65), where the role of the mass is played by the average of the potential.

Perturbation Theory for PDEs

The situation of the nonlinear Schrödinger is more difficult. Here one can use the Fourier coefficients of the potential as parameters. Fix  > 0 and, for any positive R define the space of the potentials, by ( VR :D

V(x) D

X

v k cos kx j v 0k :D v k R1 e k

k1



1 1 2  ; 2 2

for k  1

(71)

Endow such a space with the product normalized probability measure. Theorem 47 ([12], see also [21]) For any r there exists a positive R and a set S VR such that property (r-NR) holds for any potential V 2 S and jVR  Sj D 0. So, provided the potential is chosen in the considered set, Theorem 40 also applies to the Eq. (69). We point out that the proof of Theorem 46 and of Theorem 47 essentially consists of two steps. First one proves that for most values of the parameters one has ˇ ˇ ˇ ˇ ˇX ˇ

ˇ ˇ ! K ; k kˇ  ˇ ˛ ˇ k1 ˇ N

8K 2 Z

with 0 6D jKj :D

X

1

;

jK k j  r C 2 :

(72)

k

Then one uses the asymptotic of the frequencies, namely ! k ak d with d  1, in order to get the result. Non Hamiltonian PDEs In this section we will present some results for the non Hamiltonian case. It is useful to complexify the phase space. Thus, in this section we will always denote the space of the complex sequences fz k g whose norm (defined by (25)) is finite by Ps . In the space Ps consider a system of differential equations of the form z˙k D  k z k C Pk (z) ;

k 2 Z  f0g

Example 49 Consider the following nonlinear heat equation with periodic boundary conditions on [; ]: u t D u x x  V(x)u C f (u) :

)



Example 48 A system of the form (18) with H P having a vector field which is analytic. The corresponding Hamilton equations have the form (73) with  k D k D i! k , k  1.

(73)

where k are complex numbers and P(z)  fPk (z)g has a zero of order at least 2 at the origin. Moreover we will assume P to be a complex analytic map from a neighborhood of the origin of Ps to Ps . The quantities k are clearly the eigenvalues of the linear operator describing the linear part of the system, and for this reason they will be called “the eigenvalues”.

(74)

If f is analytic then it can be given the form (73) by introducing the basis of the eigenfunctions k of the Sturm Liouville operator S :D @x x C V , i. e. denoting u D P k z k ' k . In this case the the eigenvalues k are the opposite of the periodic eigenvalues of S. Thus in particular one has  k 2 R and  k k 2 . In this context one has to introduce a suitable concept of nonresonance: Definition 50 A sequence of eigenvalues is said to be resonant if there exists a sequence of integer numbers K k  0 and an index i such that X k K k  i D 0 : (75) k

In the finite dimensional case the most celebrated results concerning systems of the form (73) are the Poincaré theorem, the Poincaré–Dulac theorem and the Siegel theorem [1]. The Poincaré theorem is of the form of Birkhoff’s Theorem 22, while the Poincaré–Dulac and Siegel theorems guarantee (under suitable assumptions) that there exists an analytic coordinate transformation reducing the system to its normal form or linear part (no remainder!). At present there is not a satisfactory extension of the Poincaré–Dulac theorem to PDEs (some partial results have been given in [41]). We now state a known extension of the Siegel theorem to PDEs. Theorem 51 ([54,63]) Assume that the eigenvalues fulfill the Diophantine type condition ˇ ˇ ˇX ˇ

ˇ ˇ k K k  i ˇ  ; 8i; K with 2  jKj ; (76) ˇ ˇ ˇ jKj k

where > 0 and  2 R are suitable parameters; then there exists an analytic coordinate transformation defined in a neighborhood of the origin, such that the system (73) is transformed into its linear part z˙k D  k z k :

(77)

The main remark concerning this theorem is that the condition (76) is only exceptionally satisfied. If  2 C n then condition (76) is generically satisfied only if  > (n  2)/2.

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1348

Perturbation Theory for PDEs

Nevertheless some examples where such an equation is satisfied are known [54]. The formalism of Sect. “Normal Form for Hamiltonian Nonresonant PDEs” can be easily generalized to the non Hamiltonian case giving rise to a generalization of Poincaré’s theorem, which we are going to state. Given a polynomial map P : P1 ! P1 one can expand it on the canonical basis e k of P0 as follows: X Pki 1 ;::;k r z k 1 : : : z k r e i ; Pki 1 ;::;k r 2 C : (78) P(z) D k 1 ;::;k r ;i

Definition 52 A polynomial map P is said to have localized coefficients if there exists  2 [0; C1) such that 8N  1 there exists CN such that for any choice of the indexes k1 ; : : : ; kr ; i following the inequality holds: ˇ ˇ (k; i)CN ˇ i ˇ ; (79) ˇPk ;:::;k i ˇ  C N 1 r S(k; i) N where (k; i) D (k1 ; ::; kr ; i). A map is said to have localized coefficients if for any s large enough, it is a smooth map from Ps to itself and each term of its Taylor expansion has localized coefficients. Definition 53 The eigenvalues are said to be strongly nonresonant at order r if for any N large enough, any K D (K k 1 ; : : : ; K k r ) and any index i such that jKj  r and there are at most two of the indexes k1 ; ::; kr ; i larger than N the following inequality holds: ˇ ˇ ˇ ˇX

ˇ ˇ k K k  i ˇ  ˛ : (80) ˇ ˇ N ˇ k

Theorem 54 Assume that the nonlinearity has localized coefficients and that the eigenvalues are strongly nonresonant at order r, then there exists an sr and an analytic coordinate transformation Tr : Us r ! Ps r which transforms the system (73) to the form z˙k D  k z k C R k (z) ;

(81)

where the following inequality holds kR(z)ks r  Ckzkrs r :

(82)

Extensions and Related Results The theory presented here applies to quite general semilinear equations in one space dimension. At present a satisfactory theory applying to quasilinear equations and/or equations in more than one space dimensions is not available. The main difficulty for the extension of the theory to semilinear equations in higher space dimension is related to the nonresonance condition. The general theory can be

easily extended to the case where the differences ! k  ! l between frequencies accumulate only at a set constituted by isolated points. Example 55 Consider the nonlinear wave equation on the d-dimensional sphere u t t   g u C 2 u D f (x; u) ;

x 2 Sd

(83)

with g the Laplace Beltrami operator; the frequencies are p given by ! k D k(k C d  1) C 2 and their differences accumulate only at integers. A version of Theorem 40 applicable to (83) was proved in [9]. As a consequence in particular one has a lower bound on the existence time t of the small amplitude solutions of the form jtj  r , where " is proportional to a high Sobolev norm of the initial datum. An extension to u t t   g u C Vu D f (x; u) is also known. Example 56 A similar result was proved in [11] for the nonlinear Schrödinger equation  i ˙ D  C V

C f (j j2 ) ;

x 2 T d ; (84)

where the star denotes convolution. The only general result available at present for quasilinear system is the following theorem. Theorem 57 ([5]) Fix r  1, assume that the frequency vector fulfills condition (72) and that there exists d1 such that the vector field of H P is a smooth map from PsCd 1 to Ps for any s large enough. Then the same result of Theorem 40 holds, but the functions do not necessarily have localized coefficients and, for any s large enough the remainder is estimated by kX R(r) (z)ks  CkzkrC2 sCd r ;

8z 2 U(r) s ;

(85)

where dr is a large positive number. The estimate (85) shows that the remainder is small only when considered as an operator extracting a lot of derivatives. In particular it is non trivial to use such a theorem in order to extract information on the dynamics. Following the approach of [8] and [5] this can be done using the normal form to construct approximate solutions and suitable versions of the Gronwall lemma to compare it with solutions of the true system. This however allows to control the dynamics only over times of the order of 1 , " being again a measure of the size of the initial datum. Such a theory has been applied to quasilinear wave equations in [5] and to the Fermi Pasta Ulam problem in [17]. Among the large number of papers containing related results we recall [41,47,56,59]. A stronger result for the nonlinear wave equation valid over times of order 2 can be found in [36].

Perturbation Theory for PDEs

Future Directions Future directions of research include both purely theoretical aspects and applications to other sciences. From a purely theoretical point of view, the most important open problems pertain to the validity of normal form theory for equations in which the nonlinearity involves derivatives, and for general equations in more than one space dimension. These results would be particularly important since the kind of equations appearing in most domains of physics are quasilinear and higher dimensional. Concerning applications, we would like to describe a few of them which would be of interest.  Water wave problem. The problem of description of the free surface of a fluid has been shown to fit in the scheme of Hamiltonian dynamics [62]. Normal form theory could allow one to extract the relevant informations on the dynamics in different situations [31,38], ranging from the theory of Tsunamis [32] to the theory of fluid interface, which is relevant e. g. to the construction of oil platforms [33].  Quantum mechanics. A Bose condensate is known to be well described by the Gross Pitaevskii equation. When the potential is confining, such an equation is of the form (18). Normal form theory has already been used for some preliminary results [18], but a systematic investigation could lead to interesting new results.  Electromagnetic theory and magnetohydrodynamics. The equations have a Hamiltonian form; normal form theory could help to describe some instability arising in plasmas.  Elastodynamics. Here one of the main theoretical open problems is the stability of equilibria which are a minimum of the energy. The problem is that in higher dimensions the conservation of energy does not ensure enough smoothness of the solution to ensure stability. Such a problem is of the same kind as the one solved in [9] when dealing with the existence times of the nonlinear wave equation. Bibliography 1. Arnold V (1984) Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Mir, Moscow 2. Bambusi D (1999) Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equation. Math Z 130:345– 387 3. Bambusi D (1999) On the Darboux theorem for weak symplectic manifolds. Proc Amer Math Soc 127(11):3383– 3391 4. Bambusi D (2003) Birkhoff normal form for some nonlinear PDEs. Comm Math Phys 234:253–283

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Perturbation Theory in Quantum Mechanics

Perturbation Theory in Quantum Mechanics LUIGI E. PICASSO1,2 , LUCIANO BRACCI 1,2 , EMILIO D’EMILIO1,2 1 Dipartimento di Fisica, Università di Pisa, Pisa, Italy 2 Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Pisa, Italy Article Outline Glossary Definition of the Subject Introduction Presentation of the Problem and an Example Perturbation of Point Spectra: Nondegenerate Case Perturbation of Point Spectra: Degenerate Case The Brillouin–Wigner Method Symmetry and Degeneracy Problems with the Perturbation Series Perturbation of the Continuous Spectrum Time Dependent Perturbations Future Directions Bibliography Glossary Hilbert space A Hilbert space H is a normed complex vector space with a Hermitian scalar product. If '; 2 H the scalar product between ' and is written as ('; )  ( ; ') and is taken to be linear in and antilinear in ': if a; b 2 C, the scalar product between a ' and b pis a b('; ). The norm of is defined as k k  ( ; ). With respect to the norm k  k, H is a complete metric space. In the following H will be assumed to be separable, that is any complete orthonormal set of vectors is countable. States and observables In quantum mechanics the states of a system are represented as vectors in a Hilbert space H , with the convention that proportional vectors represent the same state. Physicists mostly use Dirac’s notation: the elements of H are represented by j  i (“ket”) and the scalar product between j ' i and j i is written as h ' j i (“braket”). The observables, i. e. the physical quantities that can be measured, are represented by linear Hermitian (more precisely: self-adjoint) operators on H . The eigenvalues of an observable are the only possible results of the measurement of the observable. The observables of a system are generally the same of the corresponding classical system: energy, angular momentum, etc., i. e. they are of the

form f (q; p), with q  (q1 ; : : : ; q n ); p  (p1 ; : : : ; p n ) the position and momentum canonical variables of the system: qi and pi are observables, i. e. operators, which satisfy the commutation relations [q i ; q j ]  q i q j  q j q i D 0, [p i ; p j ] D 0, [q i ; p j ] D i„ ı i j , with „ the Planck’s constant h divided by 2. Representations Since separable Hilbert spaces are isomorphic, it is always possible to represent the elements of H as elements of l2 , the space of the sequences fu i g; u i 2 C, with the scalar product P (v; u)  i v i u i . This can be done by choosing an orthonormal basis of vectors ei in H : (e i ; e j ) D ı i j and defining u i D (e i ; u); with Dirac’s notations j A i ! fa i g; a i D h e i j A i. Linear operators  are then represented by f i j g;  i j D (e i ;  e j )  h e i j  j e j i. The  ij are called “matrix elements” of  in the representation ei . If   is the Hermitian-conjugate of , then (  ) i j D  ji . If the ei are eigenvectors of  then the (infinite) matrix  ij is diagonal, the diagonal elements being the eigenvalues of . Schrödinger representation A different possibility is to represent the elements of H as elements of L2 [Rn ], the space of the square-integrable functions on Rn , where n is the number of degrees of freedom of the system. This can be done by assigning how the operators qi and pi act on the functions of L2 [Rn ]: in the Schrödinger representation the qi are taken to act as multiplication by xi and the pi as i„@/@x i : if j A i ! A (x1 ; : : : ; x n ), then q i jAi ! x i

A (x1 ; : : : ; x n )

p i jAi ! i„@

;

A (x1 ; : : : ; x n )/@x i

:

Schrödinger equation Among the observables, the Hamiltonian H plays a special role. It determines the time evolution of the system through the time dependent Schrödinger equation i„

@ DH @t

;

and its eigenvalues are the energy levels of the system. The eigenvalue equation H D E is called the Schrödinger equation. Definition of the Subject In the investigation of natural phenomena a crucial role is played by the comparison between theoretical predictions and experimental data. Those practicing the two arts of the trade continuously put challenges to one another either presenting data which ask for an explanation or proposing new experimental verifications of a theory. Celestial

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Perturbation Theory in Quantum Mechanics

mechanics offers the first historical instance of this interplay: the elliptical planetary orbits discovered by Kepler were explained by Newton; when discrepancies from the elliptical paths definitely emerged it was necessary to add the effects of the heavier planets to the dominant role of the sun, until persistent discrepancies between theory and experiment asked for the drastic revision of the theory of gravitation put forth by Einstein, a revision which in turn offered a lot of new effects to observe, some of which have been verified only recently. In this dialectic interaction between theory and experiment only the simplest problem, that of a planet moving in the field of the sun within Newton’s theory, can be solved exactly. All the rest was calculated by means of perturbation theory. Generally speaking, perturbation theory is the technique of finding an approximate solution to a problem where to a dominant factor, which allows for an exact solution (zeroth order solution), other “perturbing” factors are added which are outweighed by the dominant factor and are expected to bring small corrections to the zeroth order solution. Perturbation theory is ever-pervasive in physics, but an area where it plays a major role is quantum mechanics. In the early days of this discipline, the interpretation of atomic spectra was made possible only by a heavy use of perturbation theory, since the only exactly soluble problem was that of the hydrogen atom without external fields. The explanation of the Stark spectra (hydrogen in a constant electric field) and of the Zeeman spectra (atom in a magnetic field) was only possible when a perturbation theory tailored to the Schrödinger equation, which rules the atomic world, was devised. As for heavier atoms, in no case an exact solution for the Schrödinger equation is available: they could only be treated as a perturbation of simpler “hydrogenoid” atoms. Most of the essential aspects of atomic and molecular physics could be explained quantitatively in a few years by recourse to suitable forms of perturbation theory. Not only did it explain the position of the spectral lines, but also their relative intensities, and the absence of some lines which showed the impossibility of the corresponding transitions (selection rules) found a convincing explanation when symmetry considerations were introduced. When later more accurate measurements revealed details in the hydrogen spectrum (the Lamb shift) that only quantum field theory was able to explain, perturbation theory gained a new impetus which sometimes resulted in the anticipation of theory (quantum electrodynamics) over experiment as to the accuracy of the effect to be measured. An attempt to describe all the forms that perturbation theory assumes in the various fields of physics would be

vain. We will limit to illustrate its role and its methods in quantum mechanics, which is perhaps the field where it has reached its most mature development and finds its widest applications. Introduction An early example of the use of perturbation theory which clearly illustrates its main ideas is offered by the study of the free fall of a body [29]. The equation of motion is E C˝ E  (Er  ˝) E vE˙ D gE C 2E v˝

(1)

E the anwhere gE is the constant gravity acceleration and ˝ gular velocity of the rotation of the earth about its axis. ˝ is the parameter characterizing the perturbation. If we wish to find the eastward deviation of the trajectory to first order in ˝ we can neglect the third term in the RHS of Eq. (1), whose main effect is to cause a southward deviation (in the northern hemisphere). The ratio of the second term to the first one in the RHS p of Eq. (1) (the effective perturbation parameter) is ˝ h/g ' 104 for the fall from a height h 100 m, so we can find the effect of ˝ by writing vE D vE0 C vE1 in Eq. (1), where vE0 is the zeroth order solution (E v0 D gEt if vE0 (0) D 0) and vE1 obeys E D 2t gE  ˝ E: vE˙1 D 2E v0  ˝

(2)

E The eastward The solution is Er D hE C 12 gEt 2 C 13 t 3 gE  ˝. E and its deviation is the deviation in the direction of gE  ˝ value is ı D 13 ¯t 3 g˝ cos , where is the latitude and t¯ the p zeroth order time of fall, ¯t D 2gh. While the above example is a nice illustration of the main features of perturbation theory (identification of a perturbation parameter whose powers classify the contributions to the solution, existence of a zeroth order exact solution) the beginning of modern perturbation theory can be traced back to the work of Rayleigh on the theory of sound [33]. In essence, he wondered how the normal modes of a vibrating string

(x)

@2 v @2 v D ; @t 2 @x 2

v(0; t) D v(; t) D 0

(3)

are modified when passing from a constant density D 1 to a perturbed density C (x). To solve this problem he wrote down most of the formulae [10,33] which are still in use to calculate the first order correction to non-degenerate energy levels in quantum mechanics. The equation for the normal modes is u00 (x) C  (x)u(x) D 0 ;

u(0) D u() D 0 :

(4)

Perturbation Theory in Quantum Mechanics

p Let u(0) 2/ sin nx be the unperturbed solution for n  (1) the nth mode, n D n2 , and u(0) n C u n the perturbed solution through first order, corresponding to a frequency n C n . By writing the equation for u(1) n d2 u(1) n (0) (0) C n u(1) n C n u n C n u n D 0 ; dx 2 (1) u(1) n (0) D u n () D 0

(5)

after multiplying by u(0) r and using Green’s theorem he found Z   2 n D n (x) u(0) dx ; (6) n 0

Z



a rn  0

(1) u(0) r u n dx D

n r   n

Z

 0

(0) u(0) r u n dx

(r ¤ n) Z 0



(1) u(0) n u n dx D 0 :

order energy the perturbation is diagonal in the basis of the parabolic eigenfunctions, thus circumventing the intricacies of the degenerate case. Later, he used the spherical coordinates, which entails a non diagonal perturbation matrix and calls for the full machinery of the perturbation theory for degenerate eigenvalues. It is of no use to repeat here Schrödinger’s calculations, since the methods which they use are at the core of modern perturbation theory, which is referred to as the Rayleigh–Schrödinger (RS) perturbation theory. It rapidly superseded other approaches (as that by Born, Heisenberg and Jordan [7], who worked in the framework of the matrix quantum mechanics), and will be presented in the following sections. Presentation of the Problem and an Example

(7) (8)

As an application Rayleigh found the position /2 C ıx  /2 C  of the nodal point of the perturbed mode n D 2 when the perturbation to the density is  Dp ı(x  /4). The vanishing of u2(0) C u2(1) determines 2 2/ D u2(1) (/2). By Eq. (7) the function u2(1) has an expansion P 4 u2(1) D n¤2 a n2 u(0) n , a n2 D n 2 4 sin n/4. The result for  is   2 1 1 1 1 1  D p 1C   C C   3 5 7 9 11  2  D : 2 R1 (TheR series in brackets is equal to 0 (1 C x 2 )/(1 C x 4 )dx D 1 1/2 0 (1 C x 2 )/(1 C x 4 )dx, which can be calculated by contour integration.) Perturbation theory was revived by Schrödinger, who introduced it into quantum mechanics in a pioneering work of 1926 [44]. There, he applied the concepts and methods which Rayleigh had put forth to the case where the zeroth order problem was a partial differential equation with non-constant coefficients, and he wrote down, in the language of wave mechanics, all the relevant formulae which yield the correction to the energy levels and to the wave functions for the case of both non-degenerate and degenerate energy levels. As an application he calculated the shift of the energy levels of the hydrogen atom in a constant electric field by two different methods. First he observed that in parabolic coordinates the wave equation is separable also with a constant electric field, which implies that in the subspace of the states with equal zeroth

The most frequent application of perturbation theory in quantum mechanics is the approximate calculation of point spectra. The Hamiltonian H is split into an exactly solvable part H 0 (the unperturbed Hamiltonian) plus a term V (the perturbation) which, in a sense to be specified later, is small with respect to H 0 : H D H0 C V . In many cases the perturbation contains an adjustable parameter which depends on the actual physical setting. For example, for a system in an external field this parameter is the field strength. For weak fields one expects the spectrum of H to differ only slightly from the spectrum of H 0 . In these cases it is convenient to single out the dependence on a parameter by setting H()  H0 C V :

(9)

Accordingly we will write the Schrödinger equation as H() () D E() () :

(10)

We will retain the form Eq. (9) of the Hamiltonian even when H does not contain a variable parameter, thereby understanding that the actual eigenvalues and eigenvectors are the values at  D 1. The basic idea of the RS perturbation theory is that the eigenvalues and eigenvectors of H can be represented as power series () D

1 X 0

n

(n)

;

E() D

1 X

n n ;

(11)

0

whose coefficients are determined by substituting expansions Eq. (11) into Eq. (10) and equating terms of equal order in . Generally, only the first few terms of the series can be explicitly computed, and the primary task of the

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Perturbation Theory in Quantum Mechanics

RS perturbation theory is their calculation. The practicing scientist who uses perturbation theory never has to tackle the mathematical problem of the convergence of the series. This problem, however, or more generally the connection between the truncated perturbation sums and the actual values of the energy and the wave function, is fundamental for the consistency of perturbation theory and will be touched upon in a later section. Before expounding the technique of the RS perturbation theory we will consider a simple (two-dimensional) problem which can be solved exactly, since in its discussion several features of perturbation theory will emerge clearly, concerning both the behavior of the energy E() and the behavior of the Taylor expansion of this function. From the physical point of view a system with two-dimensional Hilbert space C 2 can be thought of as a particle with spin 1/2 when the translational degrees of freedom are ignored. Let us write the Hamiltonian H D H0 C V in a representation where H 0 is diagonal:  0    E1 0 V11 V12 HD C  V21 V22 0 E20  0  E1 C V11 V12 D : (12)  V12 E20 C V22 We consider first the case E10

E20 ; V12

¤ ¤ 0. The exact eigenvalues E1;2 () of H are found by solving the secular equation: E1;2 ()



   p 1  0 E1 C V11 C E20 C V22 ˙ () ; 2 (13)   0 2  0 2 2 ()  E1 C V11  E2 C V22 C4 jV12 j : (14) D

E2 () D E20 C V22  2 C 3

jV12 j2 E10  E20

jV12 j2 (V11  V22 ) C O(4 ) :  0 2 E1  E20

(18)

At order 1 only the diagonal matrix elements of V contribute to E1;2 . The validity of the approximation requires jV12 j  jE10  E20 j. If this condition is not satisfied, that is if the eigenvalues E10 ; E20 are “quasi-degenerate”, all terms of the expansion can be numerically of the same order of magnitude and no approximation of finite order makes sense. Note that, within the first order approximation, “level crossing” (E1 () D E2 ()) occurs at   ¯ D  E 0  E 0 / (V11  V22 ) :  (19) 1 2 On the other hand Eq. (13) shows that level-crossing is impossible, unless V12 D 0, in which case the first order approximation yields the exact result. If V12 ¤ 0 the behavior ¯ is shown in Fig. 1: the of the levels E1 () and E2 () near  two levels “repel” each other [49]. At first order the eigenvectors 1;2 () are    0 [1] 0 (20) 1 D 1; V21 / E2  E1 [1] 2

    D V12 / E10  E20 ; 1 :

(21)

The expectation value ( 1[1] ; H 1[1] )/( Hamiltonian over 1[1] , for example, is E10 C V11 C 2

[1] 1 ;

[1] 1 )

of the

jV12 j2 jV12 j2 (V11  V22 )  3  2 0 0 E1  E2 E10  E20  4 

jV12 j4 E10  E20

5 3 C O( )

(22)

The corresponding eigenvectors, in the so called intermediate normalization defined by ( (0); ()) D 1, are !   p ()  E10  E20   (V11  V22 ) (15) 1 () D 1; 2V12 !   p ()  E10  E20   (V11  V22 ) ;1 : 2 () D  2V21 (16) Expanding E1;2 () through order 3 we get: E1 () D E10 C V11 C 2  3

jV12 j2 E10  E20

jV12 j2 (V11  V22 ) C O(4 )  0 2 0 E1  E2

(17)

Perturbation Theory in Quantum Mechanics, Figure 1 The behavior of the exact eigenvalues E1;2 () when V12 D 0 (blue lines) and when V12 ¤ 0 (red lines)

Perturbation Theory in Quantum Mechanics

which agrees with E1 () up to the 3 terms (the correct fourth order term contains also jV12 j2 (V11  V22 )2 /(E10  E20 )3 ). This is an example of Wigner’s (2n C 1)-theorem [52], see Subsect. “Wigner’s Theorem”. The power expansions ofpE1;2 () and 1;2 () converge in the disk jj < jE10  E20 j/ (V11  V22 )2 C 4jV12 j2 . The denominator is just twice the infimum over a of the operator norm of V  aI. Since adding to V a multiple of the identity does not affect the convergence properties of the Taylor’s series of E(), we see that the convergence domain always contains the disk jj < jE10  E20 j/2kVk, a property which holds true for any bounded perturbation in Hilbert space (see Sect. “Problems with the Perturbation Series”). If H 0 is degenerate, that is E10 D E20  E 0 , then the eigenvalues are obtained by diagonalizing V. The degeneracy is removed and the corrections to the eigenvalues are of first order in : 1  E1;2 () D E 0 C  V11 C V22 2  p ˙ (V11  V22 )2 C 4jV12 j2 (23) while the eigenvectors are  independent. The infinite dimensional case is much more involved. In particular, in most cases the perturbation series does not converge at all, that is its radius of convergence vanishes. However, we shall meet again the three situations discussed above: the case of non-degenerate eigenvalues E 0n such that jE 0n  E 0m j  jVnm j, the case of degenerate eigenvalues and finally the case of “quasi-degenerate” eigenvalues, i. e. groups of eigenvalues E 0n i such that jE 0n i  E 0n j j . jVn i n j j. As discussed above, in this last case H n i n j must be diagonalized exactly prior to applying perturbation theory. Perturbation of Point Spectra: Nondegenerate Case In this section we consider an eigenvector 0 of H 0 belonging to a non-degenerate eigenvalue E0 and apply the RS theory to determine the power expansions Eq. (11) such that Eq. (10) is satisfied, the Hamiltonian H() being given by Eq. (9). The case of a degenerate eigenvalue will be considered in Sect. “Perturbation of Point Spectra: Degenerate Case”. For both cases the starting point is the substitution of the expansions Eq. (11) into Eq. (10), which, upon equating terms with equal powers, yields the following system of equations (H0  E0 )

(n)

CV

(n1)

D

n1 X

(k)

nk ;

kD0

n D 1; 2; : : :

(24)

A perturbative calculation of the energy and the wave function through order h amounts to calculating n and (n) up to n D h and truncating the series in Eq. (11) at n D h. Corrections to the Energy and the Eigenvectors In the following let k , Ek be the normalized eigenvectors and the eigenvalues of H 0 , and let E k0  E k  E0 , Vhk  ( h ; V k ). The correction n is recursively defined in terms of the lower order corrections to the energy and the wave function: by left multiplying Eq. (24) by 0 we find n D



(n1)

0; V





n1  X

(h)

0;



nh :

(25)

hD1

Similarly, the components ( k ; (n) ); k ¤ 0, are found by left-multiplying by k ; k ¤ 0: 

(n)

k;



D C



(n1)

k; V

n1  X

k;



(h)



E k0 1 nh E k0 1 :

(26)

hD1

Note that, even if the functions (k) ’s for k < n were known, still ( 0 ; (n) ) is intrinsically undefined, since to any solution of Eq. (24) we are allowed to add any multiple of 0 . The reason of this indeterminacy is that Eq. (10) defines () only up to a multiplicative factor ˛(). Even the normalization condition ( (); ()) D 1 leaves () undetermined by a phase factor exp(i'()); '() 2 R. On the contrary, the corrections n as well as all the expectation values (up to order n) are unaffected by these modifications of the wave function () [16]. We can turn to our advantage the indeterminacy of ( 0 ; (n) ) by requiring that in the expression of n , Eq. (25), the dependence on the values of ( 0 ; (k) ), k  n  1, disappears. For example, after writing 

0; V

(n1)

D V00





0;

(n1)



C

X

V0h



h;

(n1)



h¤0

the independence of ( 0 ; (n1) ) implies 1 D V00 . Next, requiring n to be independent of ( 0 ; (n2) ) determines 2 and so on, until finally Eq. (25) gives n . As an example we carry through this procedure for n D 3. Start-

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Perturbation Theory in Quantum Mechanics

ing from 3 D V00



0;

(2)



C

X

V0k



k;

k¤0

 1



(2)

0;



(2)

 2

 

0;

(1)



we first find 1 D V00 :

(27)

Next, from Eq. (26) for n D 2, we get X jV0k j2  3 D  E k0 k¤0



X h;k¤0

(1)

0;

k;

while the value of ( k ; (n1) ) can be read immediately in the expression of n : ( k ; (n1) ) is obtained from n by omitting in each term the factor V0k and the sum over k. For example the wave function [2]  0 C (1) C2 (2) in the intermediate normalization by Eqs. (28) and (29) is



[2]

D

0



X

k

kD1

 V0k Vk h E k0

X V0k  C E k0

Using the intermediate normalization the expression of n is   (n1) n D ; (33) 0; V

(1)

h;

(1)





X

C 2

k

Vk h Vh0 E k0 E h0

k

Vk0 : E k0 2

h;kD1

1  2



0;

(1)



2

  1

:

X kD1

Vk0 E k0

(34)

k¤0

The independence of ( 2 D 

0;

(1) )

In order to calculate expectation values, transition probabilities and so on one needs the normalized wave function

implies

X jV0k j2 : E k0

(28)

k¤0

Finally, by using Eq. (26) for n D 1 we find 3 D

X h;k¤0

X jV0k j2 V0k Vk h Vh0  1 : E k0 E h0 E k0 2

(29)

k¤0

Note that if n is required, the lower order corrections being known, one can use a simplified version of Eqs. (25) and (26) where the terms ( 0 ; (k) ) are omitted since the beginning. Once the values of k ; k  n, have been determined, Eq. (26) yields ( k ; (n) ). For example, for the first order correction to the wave function we have   Vk0 (1) : (30) D k; E k0 By suitably choosing the arbitrary factor ˛() we referred to after Eq. (26) we can impose ( 0 ; ()) D 1. With this choice (known as the “intermediate normalization”, since () is not normalized) we have ( 0 ; (k) ) D 0 for any k > 0. As a result, for the wave function through order n we find [n]



0C

n X

k

(k)



0

C ın

(31)

kD1

with 

0;

 [n]

[n] N

D N 1/2 (

(32)

C ın )

(35)

with N 1 D 1 C (ın ; ın ). N can be chosen real. Note that the wave function [1] is correctly normalized up to first order. From the above equations one sees in which sense the perturbation V must be small with respect to the unperturbed Hamiltonian H 0 : the separation between the unperturbed energy levels must be large with respect to the matrix elements of the perturbation between those levels and the total correction ıE to E0 should be small with respect to jE i  E0 j, Ei standing for any other level of the spectrum of H 0 . Wigner’s Theorem From Eq. (24) it follows that H

[n]

D E [n]

[n]

  C O nC1 ;

whence one should infer that, if E is the exact energy, E  ( N[n] ; H N[n] ) D O(nC1 ). It is therefore remarkable Wigner’s result that the knowledge of [n] allows the calculation of the energy up to order 2n C 1 (Wigner’s 2n C 1 theorem) [52]. Indeed, he proved that, if E is the exact energy, 

D1:

0

E 

[n] ; H [n] ;

[n] [n]



   D O 2nC2 :

Perturbation Theory in Quantum Mechanics

which is a special case of the Feynman–Hellmann theorem [17,23]:   @E @H D (); () : (38) @ @

To this purpose, let (nC1) D

[n]

 q

[n] ;

[n]



where is the normalized exact wave function, H D E . Then   (nC1) D O nC1 ;     ; (nC1) C (nC1) ;     D  (nC1) ; (nC1) D O 2nC2 : As a consequence  [n]    ; H [n]    ( ; H ) D O 2nC2 : [n] ; [n]

D q

C  (1)  1 C 2 (1) ;

[1] N ;H

[1] N



(1)

;

2 2 C 3 3   1 C 2 (1) ; (1)   D E0 C  1 C 2 2 C 3 3 C O 4 :

D E0 C  1 C

The Feynman–Hellmann Theorem The RS perturbative expansion rests on the hypothesis that both the eigenvalues E() and the corresponding eigenvectors () admit a power series expansion, in short, that they are analytic functions of  in a neighborhood of the origin. As we shall see in Sect. “Problems with the Perturbation Series”, as a rule it is not so and the perturbative expansion gives rise only to a formal series. For this reason it is advisable to derive the various terms of the perturbation expansion without assuming analyticity. If we need E() and () through order n it is sufficient to assume that, as functions of , they are C nC1 , that is continuously differentiable (n C 1) times. The procedure consists in taking the derivatives of Eq. (10) [15,28]: at the first step we get H

0



k;

0



D

X kD1

0

() C V () D E 0 () () C E()

(39)

whence 1 D V00 , in agreement with Eq. (27). Next, after left multiplying Eq. (36) by k and taking  D 0 we get

E 00 (0) D 2

by using Eq. (30) and recalling Eq. (28) and Eq. (29) we have 

E 0 (0) D V00 ;

Vk0 E k0

(40)

which, again, agrees with Eq. (30). Taking now the derivative of (37) at  D 0 and using Eq. (40) we obtain

We make explicit this point with an example. Since [1] N

For  D 0 we find

0

()

(36)

and by left multiplication by (), with ( (); ()) D 1, we get   (37) E 0 () D (); V () ;

V0k



k;

0



D 2

X jV0k j2 E k0

(41)

kD1

whence 2 D 12 E 00 (0), in agreement with Eq. (28). It is clear that the procedure can be pursued to any allowed order, and that the results for the energy corrections, as well as for the wave functions, are the same we obtained earlier by the RS technique. However, the conceptual difference, that no analyticity hypothesis is required, is important since in many cases this hypothesis is not satisfied. As to the relation of E [n]  E0 C  1 C   C n n with E() we recall that, since by assumption E() is C nC1 , we can write Taylor’s formula with a remainder:

E() D

n X E (p) p E (nC1) ( ) nC1 ;  C  p! (n C 1)! 0

0 < < 1: (42)

As observed in [28], since for small  the sign of the remainder is the sign of E (nC1) (0)nC1 , Eq. (42) allows to establish whether the sum in Eq. (42) underestimates or overestimates E(). Moreover, if two consecutive terms, say q and q C 1, have opposite sign, then (for sufficiently small ) E() is bracketed between the partial sums including and excluding the qth term. It is a pity that no one can anticipate how small such a  should be. (Of course these remarks apply to the RS truncated series as well.) Perturbation of Point Spectra: Degenerate Case The case when the unperturbed energy E0 is a degenerate eigenvalue of H 0 , i. e. in the Hilbert space there exists

1357

1358

Perturbation Theory in Quantum Mechanics

a subspace W 0 generated by a set f 0(i) g; 1  i  n0 , of orthogonal normalized states, such that each 0 in W 0 obeys (H0  E0 ) 0 D 0, deserves a separate treatment. The main problem is that, if () is an eigenstate of the exact Hamiltonian H D H0 C V, we do not know beforehand which state of W 0 (0) is. In order to use a more compact notation it is convenient to introduce the projection P0 onto the subspace W 0 and its complement Q0 D

P0

n0 X

(i) 0



(i) 0 ;



;

Q0  I  P0 ;

(43)

iD0 (i) 0 ;

where 1  i  n0 , is any orthonormal basis of W 0 . The Hamiltonian H D H0 C V can be written as H D (P0 C Q0 )(H0 C V )(P0 C Q0 )

By substituting Eq. (49) into Eq. (46) we have (E0 C VPP )P0

D EP0

C Q0 H 0 Q0 ;

(51)

The energy shifts E  E  E0 appear as eigenvalues of an operator A(E) acting in W 0 A(E)  VPP C 2 VPQ (E  H Q Q )1 VQ P

(52)

which however still depends on the unknown exact energy E. A calculation of the energy corrections up to a given order is possible, starting from Eq. (52), provided we expand the term (E  H Q Q )1 as far as is necessary to include all terms of the requested order.

(44)

The contributions i are extracted from Eq. (51) by expanding E D E0 C  1 C 2 2 C   

where VPP D P0 V P0 ;

VPQ D P0 V Q0 ;

VQ P D Q0 V P0 ;

VQ Q D Q0 V Q0 :

(45)

After projecting the Schrödinger equation onto W 0 and its orthogonal complement W0? , we find (E0 C VPP )P0 VQ P P0

C VPQ Q0

C Q0 H 0 Q0

D EP0

C VQ Q Q0

(46) D EQ0

: (47)

Letting H Q Q D Q0 HQ0 D Q0 H0 Q0 C VQ Q

(48)

can be extracted from Eq. (47): Q0

:

Corrections to the Energy and the Eigenvectors

D E0 P0 C VPP C VPQ C VQ P C VQ Q

Q0

C 2 VPQ (E  H Q Q )1 VQ P P0

D (E  H Q Q )1 VQ P P0

:

(49)

Note that in Eq. (47) the operator H QQ acts on vectors of W0? and that E  H Q Q does possess an inverse in W0? . Indeed, the existence of a vector  in W0? such that (H Q Q  E) D 0

(50)

contradicts the assumptions which perturbation theory is grounded in: the separation between E() and E(0) should be negligible with respect to the separation between different eigenvalues of H 0 . Actually, if k is such that H0 k D E k k ; E k ¤ E0 , by left multiplying Eq. (50) by k we would find (E  E k )(

k ; )

D (

k ; V)

where the LHS is of order 0 in , whereas the RHS of order 1.

P0

D '0 C '1 C 2 '2 C   

and equating terms of equal order. At the first order, since the second term in the LHS of Eq. (51) is of order 2 or larger, we have VPP '0 D 1 '0 :

(53)

The first order corrections to the energy are the eigenvalues of the matrix V PP and the corresponding zeroth order wave function is the corresponding eigenvector. In the most favorable case the eigenvalues of V PP are simple, and the degeneracy is completely removed since the first order of perturbation theory. In this case, in order to get the higher order corrections, we can avail ourselves of the arbitrariness in the way of splitting the exact Hamiltonian into a solvable unperturbed Hamiltonian plus a perturbation by putting H D (H0 C VPP ) C (V  VPP )  H00 C V 0 : (54) The eigenvectors of H00 are the solutions of Eq. (53), with eigenvalues E0 C 1(i) ; 1  i  n0 , plus the eigenvectors j of H 0 with eigenvalues E j ¤ E0 . Since the eigenvalues E0 C  1(i) are no longer degenerate, the formalism of non-degenerate perturbation theory can be applied, but a warning is in order. When in higher perturbation orders a denominator E k0 occurs with the index k referring to another vector of the basis of W 0 , this denominator is of order  and consequently the order of the term containing this denominator is lower than the naive V-counting would imply. In each such term, the effective order is

Perturbation Theory in Quantum Mechanics

the V-counting order minus the number of these denominators. As shown below, this situation occurs starting from terms of order 4 in the perturbation V. Note that, also in the case of non-complete removal of the degeneracy, the procedure outlined above, with obvious modifications, can be applied to search the higher order corrections to those eigenvalues which at first order turn out to be non-degenerate. If a residual degeneracy still exists, i. e. an eigenvalue 1 of Eq. (53) is not simple, we must explore the higher order corrections until the degeneracy, if possible, is removed. First of all we must disentangle the contributions of different order in  from (E  H Q Q )1 . Since E  H Q Q D (E  Q0 H0 Q0 )   1  (E  Q0 H0 Q0 )1 VQ Q ; we have (E  H Q Q )1 D

1 X

 n n (E  Q0 H0 Q0 )1 VQ Q (55)

As the energy E still contains contributions of any order, the operator (E  Q0 H0 Q0 )1 must in turn be expanded into a series in . To make notations more readable, we define

Q0 VQ P '0 D 2 '0 C 1 '1 : VPP '1 C VPQ a

(57)

Let P0(i) be the projections onto the subspaces W0(i) of W 0 corresponding to the eigenvalues 1(i) : P0(i) ;

VPP D 

X

1(i) P0(i) ; (58)

i

P1  P0(1) ;

1  1(1) :

By projecting onto W1  W0(1) and recalling that ' 0 is in W 1 we get P1 VPQ

Q0 VQ P '0 D 2 '0 ; a

Q0 Q0 VQ P '1  1 VPQ 2 VQ P '0 a a Q0 Q0 C VPQ VQ Q VQ P '0 D 1 '2 C 2 '1 C 3 '0 : a a (62)

We want to convert this equation into an eigenvalue problem for 3 . In analogy with Eq. (58) we have

Q0 Q0 VQ P P1 D P1 V V P1 : a a

P2 

X

P1(i) ;

i P1(1)

V1 D

X

2(i) P1(i) ; (63)

;

2 

2(1)

:

Since P2 VPP D 1 P2 , first we eliminate ' 2 by applying P2 to Eq. (62): Q0 Q0 VQ P '1  1 P2 VPQ 2 VQ P '0 a a Q0 Q0 C P2 VPQ VQ Q VQ P '0 D 3 P2 '0 C 2 P2 '1 : a a (64)

P2 VPQ

P Writing '1 D i P0(i) '1 , since P2 V1 D 2 P2 the contribution with i D 1 of the first term in the LHS of Eq. (64) is P2 V1 '1 D 2 P2 '1 . Hence, Eq. (64) reads X

(59)

P2 VPQ

i¤1

whence 2 is an eigenvalue of the operator V1  P1 VPQ

(61)

The vectors which make V 1 diagonal belong to non-degenerate eigenvalues of H000 , hence the non-degenerate theory can be applied. If, on the contrary, the eigenvalue 2 of V 1 is still degenerate, the above procedure can be carried out one step further, with the aim of removing the residual degeneracy. We work out the calculation for 3 , since a new aspect of degenerate perturbation theory emerges: a truly third order term which is the ratio of a term of order 4 in the potential and a term of first order (see Eq. (67) below). From Eq. (51) and (55) we extract the contribution of order 3:

(56)

The second order terms from Eqs. (51) and (55) give

X

 H000 C V 00 :

P1 D

Q0  (E0  Q0 H0 Q0 )n : an

P0 D

H D (H0 C VPP C V1 ) C (V  VPP  V1 )

VPP '2 C VPQ

0

 (E  Q0 H0 Q0 )1 :

Again, if the eigenvalue 2 is non-degenerate, we can use the previous theory by splitting the Hamiltonian as

C P2 VPQ

Q0 Q0 VQ P P0(i) '1  1 P2 VPQ 2 VQ P '0 a a

X (i) Q0 Q0 P0 '1 : VQ Q VQ P '0 D 3 P2 '0 C 2 a a i¤1

(60)

(65)

1359

1360

Perturbation Theory in Quantum Mechanics

Finally, P0(i) '1 , i ¤ 1, is extracted from Eq. (57) by projecting with P0(i) ; i ¤ 1, and recalling that P0(i) '0 D 0 if i ¤ 1:   Q0 P0(i) '1 D P0(i) VPQ VQ P '0 / 1  1(i) ; a

i ¤ 1 : (66)

Substituting into Eq. (65) we see that 3 is defined by the eigenvalue equation for the operator

Following [4], we define a  dependent operator U in this way: U P0

(67)

Despite the presence of four factors in the potential, the last term is actually a third order term due to the denominators 1  1(i) . The procedure outlined above, which essentially embodies the Rayleigh–Schrödinger approach, can be pursued until the degeneracy is (if possible, see below Sect. “Symmetry and Degeneracy”) completely removed, after which the theory for the non-degenerate case can be used. Rather than detailing the calculations, we present an alternative iterative procedure due to Bloch [4] which allows a more systematic calculation of the corrections to the energy and the wave function. Bloch’s Method

B()P0

k ()

D E k P0

k ()

:

(68)

First of all, note that the vectors P0 k () are a basis for the subspace W 0 . Indeed, it is implicit in the assumption that perturbation theory does work that the perturbing potential should produce only slight modifications of the unperturbed eigenvectors of the Hamiltonian, so that the vectors P0 k () are linearly independent (although not orthogonal). Since their number equals the dimension of W 0 , they are a basis for this subspace.

k ()

U D U P0 ; k ()

;

U Q0 D 0 :

(69)

P0 U D P0 ;

D

k ()

(70)

:

(71)

The former of Eq. (70) follows immediately from the definition of U. Hence P0 U D P0 U P0 , which implies the latter of Eq. (70). Equation (71) is verified by applying the former of Eq. (70) to k (). Let B()  P0 V U :

(72)

We verify that, if E k  E k  E0 , then B()P0

k ()

D E k P0

k ()

:

(73)

Indeed, by Eq. (69) we have P0 VU P0 Writing Eq. (10) as (H0  E0 C V )

k ()

D E k

k ()

D P0 V

k ().

k ()

and multiplying by P0 we find P0 V

In equations Eq. (51) and (52) we have seen that the energy corrections E and the projections onto W 0 of the vectors k () are eigenvalues and eigenvectors of an operator acting in W 0 . This observation is not immediately useful since the operator depends on the unknown exact energy E(). However, it is possible to produce an operator B(), which can be calculated in terms of known quantities and has the property that, if E k (); k () are eigenvalues and eigenvectors of Eq. (10) such that E k (0) D E0 , then

D

As a consequence we have

U Q0 Q0 Q0 V V P2  1 P2 V 2 V P2 V2  P2 V a a a (i) X Q0 Q0 P0 C P2 V V V V P2 : a 1  (i) a 1 i¤1

k ()

k ()

D E k P0

k ()

;

(74)

hence Eq. (73) is satisfied. A practical use of Eq. (73) requires an iterative definition of U in terms of known quantities. From Eqs. (69) and (70) we have U D P0 U C Q0 U D P0 C Q0 U P0 :

(75)

We calculate the latter term of Eq. (75) on the vectors P0 k (). Since (V  E k )

k

D (E0  H0 )

k

;

recalling Eq. (69) we have Q0 U P0

k ()

D Q0 k () Q0 D (V  E k ) k () a Q0 Q0 D  VU k ()  E k U k () a a Q0 Q0 D  VU k ()  E k U P0 k () : a a

Perturbation Theory in Quantum Mechanics

By Eq. (74) Q0 U P0

k ()

Q0 Q0 V U k ()   U P0 V k () a a Q0 Q0 D  V U k ()   U P0 VU k () a a  Q0  VU  UV U P0 k () : D a D

As a consequence the desired iterative equation for U is U D P0 C 

Q0 (VU  UV U) : a

(76)

Equation (76) in turn allows an iterative definition of the operator B() of Eq. (72) depending only on quantities which can be computed in terms of the known spectral representation of H 0 . Knowing U through order n1 gives P B[n] ()  niD1 B(i) (), whose eigenvalues are the energy P (i) corrections through order n. In fact, if B D 1 iD1 B and P1 s P0 k D sD0  's , the order r contribution to Eq. (73) is r X

B(i) 'ri D

iD1

r X

i 'ri :

The last two terms represent off-diagonal blocks which can be omitted for the calculation of  1 C 2 2 , since the lowest order contribution to the eigenvalues of a matrix X from the off-diagonal terms X ij is jX i j j2 /(X i i  X j j ). For a second order expansion as B[2] this yields third order contributions of the type 3 P1 V

B[n] ' [n] D E [n] ' [n] C O(nC1 ) :

(78)

Once P0 k[n] () has been found, Eq. (69) gives the component of k () in W0? through order n C 1. As an example, for n D 3 we have Q0 Q0 Q0 V P0 C 2 V V P0 a a a Q0  2 2 V P0 V P0 ; a

B[3] D P0 V P0 C2 P0 V

Q0 V P1 a Q0 Q0 Q0 C2 P10 V V P10 C2 P1 V V P10 C2 P10 V V P1 : a a a

B[2] () D  1 P1 C P10 V P10 C 2 P1 V

(77)

iD1

P P Defining P0 k[n]  0n r 'r  ' [n] , E [n]  1n r r , we see that the sum of Eq. (77) for values of r through n gives

U [2] D P0 C 

has the dimension of W 0 . However, as noted above, for n > 1 the eigenvalues of B[n] () are different from  1 C 2 2 C  Cn n by terms of order at least nC1. Similarly, the eigenvectors of Eq. (78) differ from the component in W 0 of [n] D 0 C (1) C  Cn (n) by terms of order larger than n. It is instructive to reconsider the calculation of 2 and 3 in the light of Bloch’s method. If P0 D P1 C P10 , then P0 V P0 D 1 P1 C P10 V P10 and

(79)

Q0 Q0 Q0 V P0 C3 P0 V V V P0 a a a Q0  3 P0 V 2 V P0 V P0 : (80) a

If W 0 is one dimensional, Eq. (80) gives for  1 C 2 2 C 3 3 the same result as Eqs. (27), (28) and (29). The main difference between the RS perturbation theory and Bloch’s method is that within the former the energy corrections through order n are calculated by means of a sequential computation starting from 1 , with the consequence that at each step the dimension of the matrix to be diagonalized is smaller. Conversely, within Bloch’s method one has to diagonalize the matrix B[n] (), which

Q0 Q0 P10 V V V P1 : a 1  10 a

These are just the contributions to 3 which we met in the RS approach: the expression of V 2 given in Eq. (67) combines the block-diagonal term of order 3 with the off-diagonal terms of order 2 giving a third order contribution. The Quasi–Degenerate Case There are cases, in both atomic and molecular physics, where the energy levels of H 0 present a multiplet structure: the energy levels are grouped into “multiplets” whose separation E is large compared to the energy separation ıE between the levels belonging to the same multiplet. For instance, in atomic physics this is the case of the fine structure (due to the the so called spin–orbit interaction) or of the hyperfine structure (due to the interaction of the nuclear magnetic moment with the electrons); in molecular physics typically this is the case of the rotational levels associated with the different and widely separated vibrational levels. If a perturbation V is such that its matrix elements between levels of the same multiplet are comparable to ıE, while being small with respect to E, then naive perturbation theory fails because of the small energy denominators pertaining to levels belonging to the same multiplet. To solve this problem, named the problem of quasi-degenerate levels, once again we can exploit the arbitrariness in the way of splitting the Hamiltonian H into an unperturbed

1361

1362

Perturbation Theory in Quantum Mechanics

Substituting into the expression of A the expansion Eq. (55) for (E  H Q Q )1 and noting that, if fE k g is the spectrum of H 0 ,

Hamiltonian and a perturbation. Let E0(1)  E0 C ıE (1) ;

E0(2)  E0 C ıE (2) ; E0(n)

::: ;

 E0 C ıE (n) ;



be the unperturbed energies within a multiplet, with E0 any value close to the E0(i) ’s (for instance their mean value), and P0(i) the projections onto the corresponding eigenspaces. Let H00

 H0 

X

ıE (i) P0(i)

;

e  V C V

i

X

0 ; VPQ (E

 Q0 H0 Q0 )1 VQ P

;

(81)

We consider as the unperturbed Hamiltonian and e as the perturbation. From the physical point of view V this procedure, if applied to all multiplets, is just the inclusion into the perturbation of those terms of H 0 that are responsible for the multiplet structure. With the splitting of the Hamiltonian as in Eq. (81) we can apply the methods of degenerate perturbation theory. The most efficient of these techniques is Bloch’s method, which yields a simple prescription for the calculation of the corrections of any order. If for example we are content with the lowest order, we must diagonalize the matrix P0 e V P0 , or equivalently P0 HP0 , that is the energies through first order are the eigenvalues of the equation ;

(82)

P

(i) where P0 D i P0 is the projection onto W 0 , the 0 eigenspace of H0 corresponding to the eigenvalue E0 . These eigenvalues are algebraic functions of , and no finite order approximation is meaningful, since all terms can be numerically of the same order, due to the occurrence of small denominators (ıE (i)  ıE ( j) )n .

VQ P

0



 Q0 H0 Q0 )1 VQ Q (E  Q0 H0 Q0 )1 X D V0k (E  E k )1 Vk h (E  E h )1 Vh0

and so on, we find the following implicit expression for the exact energy E: E D E0 C (

Equations (52) and (55) yield an alternative approach to the calculation of the energy shift E due to a perturbation to a non-degenerate energy level E0 , the so called Brillouin–Wigner method [9,22,52]. In this case W 0 , the space spanned by the unperturbed eigenvector 0 , is onedimensional. The correction E obeys the equation 0 ; A(E) 0 ) ;

where the operator A(E) is defined in Eq. (52).

(83)

0; V

0)

C 2

X jV0k j2 E  Ek k¤0

X

C 3

V0k Vk h Vh0 C    : E  Ek E  Eh

k;h¤0

(84)

Consistently with the assumption that perturbation theory does work, the denominators in Eq. (84) are nonvanishing. The equation can be solved by arresting the expansion to a given power n in the potential and searching a solution iteratively starting with E D E0 . However, the result differs from the energy E [n] D E0 C  1 C 2 2 C   Cn n , calculated by means of the RS perturbation theory, by terms of order n C 1 in the potential. The result of the RS perturbation theory can be recovered from the Brillouin–Wigner approach by substituting in the denominators E D E0 C  1 C 2 2 C    C n n , expanding the denominators in powers of  k k /E0 and equating terms of equal orders in both sides of Eq. (84). As for the perturbed wave function, if the intermediate normalization is used, by Eq. (49) we have: D

0

C Q0

D

0

C (E  H Q Q )1 VQ P

0

: (85)

Vk h Vh0 C  : E  Ek E  Eh

(86)

Again, using the expansion Eq. (55) we find D

The Brillouin–Wigner Method

E D (

X jV0k j2 ; E  Ek

0 ; VPQ (E

H00

D EP0

D

k¤0



so that

P0 HP0



k;h¤0

ıE (i) P0(i)

i

e: H D H00 C V

0

0

C

X k k¤0

C 2

Vk0 E  Ek

X k;h¤0

k

As for the energy, if we arrest this expression to order n and substitute for E the value calculated by using Eq. (84), the result will differ from the one of Rayleigh–Schrödinger perturbation theory by terms of order n C 1. A major drawback of the Brillouin–Wigner method is its lack of size-consistency: for a system consisting of non-

Perturbation Theory in Quantum Mechanics

interacting subsystems, the perturbative correction to the energy of the total system is not the sum of the perturbative corrections to the energies of the separate subsystems through any finite order. This is best illustrated by the simple case of two systems a, b with unperturbed eigenvectors, energies and interactions 0a ; E0a ; V a and 0b ; E0b ; V b respectively. If for example the expansion Eq. (84) is arrested at order 2, by noting that the matrix elements V0;i j between the unperturbed state and the states ia jb are V0;i j 



  a b V a C Vb i j    D 0a ; V a ia ı0 j C 0b ; V b

a 0

b 0;



b j



ı0i ;

for the second order equation defining E we find

ED

E0a

C

E0b

C

 1a

C

 1b

2

C

X

ˇ ˇ2 ˇ bˇ ˇV0 j ˇ

E  E0a  E bj ˇ a ˇ2 ˇV ˇ X 0i C 2 : (87) b a E  E 0  Ei i j

On the other hand, for the energy of each system at second order we find ˇ ˇ X ˇV a ˇ2 0i a a a 2 ; E D E0 C  1 C  E a  E ia i ˇ ˇ2 (88) ˇ ˇ X ˇV0bj ˇ E b D E0b C  1b C 2 : Eb  E bj j It is apparent that the sum of the expression reported in Eq. (88) does not equal the expression of the energy reported in Eq. (87). This pathology is absent in the RS perturbation theory, where for non-interacting systems E() D E a () C E b (), hence, for any j, j D (1/ j!)D j E()jD0 D aj C bj . Symmetry and Degeneracy In Sect. “Perturbation of Point Spectra: Degenerate Case” we applied perturbation theory to the case of degenerate eigenvalues with special emphasis on the problem of the removal of the degeneracy at a suitable order of perturbation theory. The main problem is to know in advance whether the degeneracy can be removed completely, or a residual degeneracy is to be expected. The answer is given by group theory [21,51,53]. The very existence of degenerate eigenvalues of a Hamiltonian H is intimately connected with the symmetry properties of this operator. Generally speaking,

a group G is a symmetry group for a physical system if there exists an associated set fT(g)g of transformations in the Hilbert space of the system such that j(T(g)'; T(g) )j2 D j('; )j2 ; g 2 G [53]. It is proven that the operators T(g) must be either unitary or antiunitary [2,53]. We will consider the most common case that they are unitary and can be chosen in such a way that T(g1 )T(g2 ) D T(g1 g2 ) ;

g1 ; g2 2 G ;

(89)

so that the operators fT(g)g are a representation of G. A system described by a Hamiltonian H is said to be invariant under the group G if the time evolution operator commutes with T(g). Under fairly wide hypotheses this implies 

H; T(g) D 0 ;

g2G:

(90)

A consequence is that, for any g 2 G, H

DE

) HT(g)

D ET(g)

;

(91)

that is the restrictions T(g)jW of the operators T(g) to the space W corresponding to a given energy E are a representation of G. Given an orthonormal basis f i g in W, we have X t ji (g) j (92) T(g) i D j

and the vectors i are said to transform according to the representation of G described by the matrices tji . This representation, apart from the occurrence of the so called accidental degeneracy (which in most cases actually is a consequence of the invariance of the Hamiltonian under additional transformations) is irreducible: no subspace of W is invariant under all the transformations of the group. As a consequence, knowing the dimensions dj of the irreducible representations of G allows to predict the possible degree of degeneracy of a given energy level, since the dimension of W must be equal to one of the numbers dj . If the group of invariance is Abelian, all the irreducible representations are one dimensional, and degeneracy can only be accidental. Two irreducible representations are equivalent if there are bases which transform with the same matrix t ji (g). Otherwise they are inequivalent. The following orthogonality theorems hold. If a and b are inequivalent representations and i(a) , ' (b) j transform according these representations, then   (a) (b) ; ' D0 (93) i j

1363

1364

Perturbation Theory in Quantum Mechanics

while, if ar and as are equivalent, for the basis vectors s) ' (a we have j 

(a r ) s) ; ' (a i j



(a) D K rs ıi j :

(a r ) , i

(94)

Moreover, if Ars is a matrix which commutes with all the matrices t (a) i j of an irreducible representation b, then A rs D aırs (Schur’s lemma). Symmetry and Perturbation Theory If H D H0 C V , let G0 be the group under which H 0 is invariant. Although it is not the commonest case, we start with assuming that also the perturbation V commutes with T(g) for any element g of G0 . As a rule W 0 , the space of eigenvectors of H 0 with energy E0 , hosts an irreducible representation T(g) of G0 . In this case the degeneracy cannot be removed at any order of perturbation theory. While this follows from general principles (for any value of , () and T(g) () are eigenvectors of H(), and by continuity the eigenspace W  will have the same dimension as W 0 ), it is interesting to understand how the symmetry properties affect the mechanism of perturbation theory. If f i0 g is a basis of W 0 transforming according to an irreducible representation a of G0 , then the matrix Vi j D ( i0 ; V 0j ) commutes with all the matrices t (a) ji (g) and, according to Schur’s lemma, ( i0 ; V 0j ) D vı i j D 1 ı i j . No splitting occurs at the level of first order perturbation theory, neither can it occur at any higher order. Indeed, when V commutes with the operators T(g), then Bloch’s operator U, and consequently the operator B() of Eq. (72), both commute with the T(g)’s too. Again by Schur’s lemma, the operator B() is a multiple of the identity. At any order of perturbation the degeneracy of the level is not removed. In most of the cases, however, the perturbation V does not commute with all the operators T(g). The set ˚  G D g : g 2 G0 ; T(g); V D 0 is a subgroup G of G0 and the group of invariance for the Hamiltonian H is reduced to G. W 0 generally contains G-irreducible subspaces Wi ; 1  i  n: the operators T(g)jW0 are a reducible representation of G. The decomposition into irreducible representations of G is unique up to equivalence. The crucial information we gain from group theory is the following: the number of energy levels which the energy E0 is split into is the number of irreducible representations of G which the representation of G0 in W 0 is

split into. The degrees of degeneracy are the dimensions of these representations. What is relevant is that we only need to study the eigenspace W 0 of H 0 , which is known by hypothesis. In fact, let W() be the space spanned by the eigenvectors k () of H() such that k (0) 2 W0 . W() is invariant under the operators T(g); g 2 G, since Bloch’s operator U commutes with the operators T(g), g 2 G. W() can be decomposed into G-irreducible subspaces Wk (), and in each of them by Schur’s lemma the Hamiltonian H() is represented by a matrix E k ()I Wk () . The projections P0 Wk () span the space W 0 and transform with the same representation of G as Wk (), since P0 commutes with T(g) for any g in G0 , hence for any g in G. Thus, the space W 0 hosts as many irreducible representations of G as W(). Assuming that the eigenvalues E k () are different from one another, we see that the decomposition of the representation of G0 in W 0 into irreducible representations of G determines the number and the degeneracy of the eigenvalues of H() such that the corresponding eigenvectors () are in W 0 for  D 0. The possibility that some of the E k () are equal will be touched upon in the next subsection. Examples where the above mechanism is at work are common in atomic physics. When an atom, whose unperturbed Hamiltonian H 0 is invariant under O(3), is subjected to a constant electric field EE D E zˆ (Stark effect), the invariance group G of its Hamiltonian is reduced to the rotations about the z axis (SO(2)) and the reflections with respect to planes containing the z axis. The irreducible representations of this group have dimension at most 2, and the G-irreducible subspaces of W 0 (the space generated by the eigenvectors E 0 l m of H 0 corresponding to the energy E0 ) are generated by E 0 l 0 (one dimensional representation) and E 0 l ˙m (two dimensional representations). Hence, the level E0 is split into l C1 levels, the states with m and m remaining degenerate since reflections transform a vector with a given m into the vector with opposite m. Instead, if the atom is subjected to a constant magnetic field BE D Bˆz (Zeeman effect), the surviving invariance group G consists of SO(2) plus the reflections with respect to planes z D z0 . G being Abelian, the degeneracy is completely removed and this occurs at the first order of perturbation theory. In the rather special case that W 0 contains subspaces transforming according to inequivalent representations of G0 , also a G0 -invariant perturbation V can separate in energy the states belonging to inequivalent representations. For example, the spectrum of alkali atoms can be calculated by considering in a first approximation an electron in the field of the unit charged atomic rest, which is

Perturbation Theory in Quantum Mechanics

treated as pointlike. In this problem the obvious invariance group of the Hamiltonian of the optical electron is O(3), the group of rotations and reflections, and the space W 0 corresponding to the principal quantum number n > 1 contains n inequivalent irreducible representations which are labeled by the angular momentum l  n1. When the finite dimension of the atomic rest is taken into account as a perturbation, its invariance under O(3) splits the levels with given n and different l into n sublevels. A more careful consideration, however, shows that also the Lenz vector commutes with the unperturbed Hamiltonian [5], and that the space W 0 is irreducible under a larger group, the group SO(4) [18], which is generated by the angular momentum and the Lenz vector. As a consequence the l degeneracy is by no means accidental: a space irreducible under a given group can turn out to be reducible with respect to one of its subgroups. Group theory is a valuable tool in degenerate perturbation theory to search the correct vectors k (0) which make the operator P0 V P0 diagonal. In fact, let i(a) be vectors which reduce the representation T of G in W 0 into its irreducible components T (a) . The vectors i(a) and V i(a) transform according to the same irreducible representation T (a) . Hence, by Eqs. (93) and (94) we find that the P0 V P0 is a diagonal block matrix with respect to inequivalent representations: 

(a r ) ;V i

(b s ) j



(a) D K rs ı i j ı ab ;

(95)

with ı ab D 1 if representations a and b are equiva(a) are generally lent, ı ab D 0 otherwise. The matrices K rs much smaller than the full matrix of the potential. Thus, the operation of diagonalizing V is made easier by finding the G-irreducible subspaces W a . Conversely, the reduction of an irreducible representation of a group G0 in a space W 0 into irreducible representations of a subgroup G can be achieved by the following trick: find an operator V whose symmetry group is just G and interpret W 0 as the degeneracy eigenspace of a Hamiltonian H 0 . The G-irreducible subspaces of W 0 are the eigenspaces of P0 V P0 . Level Crossing As shown in the foregoing section, the existence of a nonAbelian group of symmetry for the Hamiltonian entails the existence of degenerate eigenvalues. The problem naturally arises as to whether there are cases when, on the contrary, the degeneracy is truly “accidental”, that is it cannot be traced back to symmetry properties. The problem was discussed by J. Von Neumann and E.P. Wigner [49],

who showed that for a generic n  n Hermitian matrix depending on real parameters 1 ; 2 ; : : : , three real values of the parameters have to be adjusted in order to have the collapse of two eigenvalues (level crossing). When passing to infinite dimension, arguments valid for finite dimensional matrices might fail. Moreover, often the Hamiltonian is not sufficiently “generic” so that level crossing may occur. As a consequence, we look for necessary conditions in order that, given the Hamiltonian H() D H0 CV , two eigenvalues collapse for some (real) ¯ D E2 () ¯  E. ¯ of the parameter : E1 () ¯ In this value  ¯ and 2 () ¯ are any two orthonormal eigencase, if 1 () ¯ belonging to the eigenvalue ¯ D H0 C V vectors of H() ¯ the matrix E,    ¯  Hi j 

i

    ¯ ; H0 C V ¯ 

j

  ¯ ; 

i; j D 1; 2

must be a multiple of the identity:     ¯ D H22  ¯ ; H11 

(96)

  ¯ D0: H12 

(97)

Equations (96) and (97) are three real equations for the ¯ hence, except for special cases, level crossing unknown ; cannot occur. The condition expressed by Eq. (97) is satisfied if the states corresponding to the eigenvalues E1 () and E2 () possess different symmetry properties, that is if they belong to inequivalent representations of the invariance group of the Hamiltonian or, equivalently, if they are eigenvectors with different eigenvalues of an operator which for any  commutes with the Hamiltonian H() (hence it commutes with both H 0 and V). In this case H12 D 0 and the occurrence of level crossing depends on whether Eq. (96) has a real solution. This explains the statement that level crossing can occur only for states with different symmetry, while states of equal symmetry repel each other. Indeed, if Eq. (97) is not satisfied, the behavior of two close eigenvalues as functions of  is illustrated in Fig. 1 (Sect. “Presentation of the Problem and an Example”). Figure 2 illustrates the behavior of the quasi-degenerate energy levels 2p1/2 ; 2p3/2 of the lithium atom in the E In the absence of presence of an external magnetic field B. the magnetic field they are split by the spin-orbit interaction, with a separation ıE  E3/2 E1/2 D 0:4104 eV, to be compared with the separation in excess of 1 eV from the adjacent 2s and 3s levels. This justifies treating the effect of the magnetic field by means of the first order perturbation theory for quasi-degenerate levels.

1365

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Perturbation Theory in Quantum Mechanics

Perturbation Theory in Quantum Mechanics, Figure 2 The effect of a magnetic field on the doublet 2p1/2 ; 2p3/2 of the lithium whose degeneracies, in the absence of the magnetic field, are respectively 2 and 4.  is the magnetic moment of the electron and B/ıE D 1 for B  1:4 T

When the magnetic field is present, the residual symmetry is the (Abelian) group of rotations about the diE Hence, the Hamiltonian commutes with the rection of B. component of the angular momentum along the direcE whose eigenvalues are denoted with m. In Fig. 2 tion of B, the energies of states with equal symmetry, that is with the same value of m, are depicted with the same color. No crossing occurs between states with equal m, while the level with m D 3/2 does cross both the levels with m D 1/2 and with m D 1/2 which the 2p1/2 level is split into. Problems with the Perturbation Series So far we have assumed that all the power expansions appearing in the calculations were converging for jj  1, that is we assumed analyticity in  of E(). Actually, it is only for rather special cases that analyticity can be proved. For most of the cases of physical interest, even if the terms of the perturbation series can be shown to exist, the series does not converge, or, when it converges, the limit is not E(). In spite of this, special techniques have been devised to extract a good approximation to E() from the (generally few) terms of the perturbation series which can be computed. We will outline the main results existing in the field, without delving into mathematical details, for which we refer the reader to the books of Kato [25] and Reed–Simon [34] and the references therein. The most favorable case is that of the so called regular perturbations [36,37,38,39], where the perturbation series does converge to E(). More precisely, if E0 is a nondegenerate eigenvalue of H 0 , for  in a suitable neighborhood of

 D 0 the Hamiltonian H D H0 C V has a nondegenerate eigenvalue E() which is analytic in  and equals E0 for  D 0. The same property holds for the eigenvector (). A sufficient condition for this property to hold is expressed by the Kato–Rellich theorem [26,36,37,38,39], which essentially states that if the perturbation V is H 0 -bounded, in the sense that constants a; b exist such that kV k  akH0 k C bk k

(98)

for any in the domain of V (which must include the domain of H 0 ) then the perturbation is regular. A lower bound to the radius r such that the perturbation series converges to the eigenvalue E() for jj < r can be given in terms of the parameters a; b appearing in Eq. (98) and the distance ı of the eigenvalue E0 from the rest of the spectrum of H 0 . We have



 1 ı 2 rD aC : (99) b C a jE0 j C ı 2 It must be stressed, however, that the constants a and b are not uniquely determined by V and H 0 . If the perturbation V is bounded (a D 0; b D kV k) condition Eq. (99) reads r D ı/(2kV k), which implies that the perturbation series for H D H0 C V with V bounded converges if kV k < ı/2 (Kato bound [26]). The analysis of the twolevel system (Sect. “Presentation of the Problem and an Example”) shows that the figure 1/2 cannot be improved. Still, Kato bound is only a lower bound to r. A similar statement holds for degenerate eigenvalues [34]: if E0 has multiplicity m there are m single valued analytic functions E k (); k D 1; : : : ; m such that

Perturbation Theory in Quantum Mechanics

E k (0) D E0 and, for  in a neighborhood of 0, E k () are eigenvalues of H() D H0 C V. Some of the functions E k () may be coincident, and in a neighborhood of E0 there are no other eigenvalues of H(). Regular perturbations are in fact exceedingly rare, a notable case being that of helium-like atoms [47]. Actually, there are cases where, although on physical grounds H0 C V does possess bound states, the relationship between E() and the RS expansion is far more complicated than for regular perturbations. As pointed out by Kramers [27], with an argument similar to an observation by Dyson [12] for quantum electrodynamics, the quartic anharmonic oscillator with Hamiltonian H D H0 C V 

p2 m !2 x2 m2 ! 3 4 C C x (100) 2m 2 „

is such an example. In fact, on the one hand bound states exist only for   0; on the other hand, if a power series converges for  > 0, then the series should converge also for negative values of . But for  < 0 no bound state exists. Still worse, by estimating the coefficients of the RS expansion it has been proved that the series has vanishing radius of convergence [3]. In spite of this negative result, in this case it has been proved [46] that the perturbation series is an asymptotic P series. This means that, for each n, if 0n k  k is the sum through order n of the perturbation series, then Pn lim

!0

0

k  k  E() D0: n

(101)

We recall the difference between an asymptotic and an absolutely converging series, such as occurs with regular perturbations. For the latter one, given any  in the P convergence range of the series, the distance j 0n k  k  E()j can be made arbitrarily small provided n is sufficiently large (so that a converging series is also an asymptotic series). On the contrary, for an asymptotic series P j 0n k  k  E()j is arbitrarily small only if  is sufficiently near 0, but for a definite value of  the quantity P j 0n k  k  E()j might decrease to a minimum, attained for some value N, and then it could start to oscillate for n > N (this is indeed the case for the anharmonic oscillator). As a consequence, for asymptotic series it is not expedient to push the calculation of the terms of the series beyond the limit where wild oscillations set in. Any C 1 function has an asymptotic series, as can be seen by inspection of the Taylor’s formula with a remainder (see Eq. (42)). By this means Krieger [28] argued that, if k (or equivalently the kth derivative of E()) exists for any k, the RS series is asymptotic. However, generally there

is not a range where the series converges to E(), that is E() is not analytic. An asymptotic series may fail to converge at all for  ¤ 0, as noted for the anharmonic oscillator. The asymptotic series of a function, if it exists, is unique, but the converse is not true. For example, for the C 1 function defined for real x as f (x) D exp(1/x 2 ) if x ¤ 0, f (0) D 0, the asymptotic series vanishes. There are also cases when the perturbation series is asymptotic for arg  lying in a range [˛; ˇ]. This occurs for example for the generalized anharmonic oscillator with perturbation V / x 2n . It has been proved that its perturbation series is asymptotic for j arg j  <  [46] (note that the domain does not include negative values of ). The result was later extended to multidimensional anharmonic oscillators [19]. General theorems stating sufficient hypotheses for the perturbation series to be asymptotic can be found in the literature. As a rule, however, they do not cover most of the cases of physical interest. Even in the felicitous case when the perturbation series is asymptotic, it is only known that a partial sum approaches E() as much as desired provided  is sufficiently small. This is not of much help to the practicing scientist, who generally is confronted with a definite value of the parameter , which can always be considered  D 1 by an appropriate rescaling of the potential V. Recalling that different functions can have the same asymptotic series, it seems hopeless to try to recover the function E() from its asymptotic series, but this is possible for the so called strong asymptotic series. A function E() analytic in a sectorial region (0 < jj < B; j arg j < /2 C ı) is said to P k have strong asymptotic series 1 0 a k  if for all  in the sector ˇ ˇ n ˇ ˇ X ˇ kˇ a k  ˇ  C nC1 jj nC1 (n C 1)! ˇE()  ˇ ˇ

(102)

0

for some constants C; . For strong asymptotic series it is proved that the function E() is uniquely determined by the series. Conditions that ensure that the RS series is a strong asymptotic series have been given [34]. The problem of actually recovering the function E() from its asymptotic series can be tackled by several methods. The most widely used procedure is the Borel summation method [6], which amounts to what follows. Given P k the strongly asymptotic series 1 0 a k  , one considers P1 k the series F()  0 (a k /k!) . This is known as the Borel transform of the initial series, which, by the hypothesis of strong asymptotic convergence, can be proved to have a non-vanishing radius of convergence and to possess an analytic continuation to the positive real axis. Then the

1367

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Perturbation Theory in Quantum Mechanics

function E() is given by Z 1 E() D F(x) exp(x) dx :

(103)

0

The above statement is Watson’s theorem [50]. Roughly speaking, it yields the function E() as if the following exchange of the series with the integral were allowed: E()

1 X

Z 1 X ak 1 exp(x)x k dx k k! 0 0 Z 1 1 X ak D exp(x) (x) k dx k! 0 0 Z 1 F(x) exp(x)dx : D

a k k D

0

0

A practical problem with perturbation theory is that, apart from a few classroom examples, one is able to calculate only the lower order terms of the perturbation series. Although in principle it is impossible to divine the rest of a series by knowing its terms through a given order, a technique which in some cases turned out to work is that of Padé approximants [1,32]. A Padé [M; N] approximant to a series is a rational function R M N (z) D

PM (z) Q N (z)

(104)

whose power expansion near z D 0 is equal to the first M C N terms of the series. It has been proved [31] that the Padé [N; N] approximants converge to the true eigenvalue of the anharmonic oscillator with x4 or x6 perturbation. The Padé [M; N] approximant to a function f (z) is unique, but its domain of analyticity is generally larger. Even for asymptotic series whose first terms are known one can write the Padé approximants. One can either use directly the Padé approximant as the value of E() for the desired value of , or can insert it into the Borel summation method. For the case of the quartic anharmonic oscillator (Eq. (100)) both methods have been proved to work (at the cost of calculating some tens of terms of the series). Another approach to the problem is the method of self-similar approximants [55], whereby approximants to the function E() for which the asymptotic series is known are sought by means of products f2p () D

p Y

(1 C A i )n i :

(105)

iD1

The 2p parameters A i ; n i ; 1  i  p, are determined by equating the Taylor expansion of f2p () with the asymptotic series through order 2p (a0 D 1 can be assumed, with

no loss of generality, see [55]). Also, odd order approximants f2pC1 are possible. For the anharmonic oscillator (Eq. (100)) the calculations exhibit a steady convergence to the correct value of the energy of both the even order and the odd order approximants also for  D 200. The problem with the above approaches is that their efficiency seems limited to toy models as the anharmonic oscillator. For realistic problems it is difficult to establish in advance that the method converges to the correct answer. Perturbation of the Continuous Spectrum In this section we consider the effect of a perturbing potential V on states belonging to the continuous spectrum. Since the problem is interesting mainly for the theory of scattering, we will assume that the unperturbed Hamiltonian H 0 is the free Hamiltonian of a particle of mass m. Also, assuming that the potential V(Er ) vanishes at infinity, the spectrum of the free Hamiltonian H 0 and the continuous spectrum of the exact Hamiltonian H D H0 C V are equal and consist of the positive real semi-axis. Given an energy E D „2 k 2 /2m, the problem is how the potential V affects that particular eigenfunction 0 of H 0 which would represent the state of the system if the interaction potential were absent. Letting D 0 C ı , the Schrödinger equation reads (E  H0 )ı

D V(

0

Cı ):

(106)

In the spirit of the perturbation approach, ı can be calculated by an iterative process provided we are able to find the solution of the inhomogeneous equation (E  H0 )ı

D

(107)

in the form ı

De G0 ;

(108)

e G 0 being the Green’s function of Eq. (107). Assuming that e G 0 is known, we find ı

D e G0V e0 V D G

C e G 0 Vı e0 V(e D e G 0 V 0 C G G 0 V 0 C e G0V ı ) 2 2 e0 V 0 C  e e0 V G D e G0V 0 C  G G 0 Vı G0V e 0

D  ; (109) that is ı is written as a power expansion in  in terms of the free wave function 0 .

Perturbation Theory in Quantum Mechanics

Scattering Solutions and Scattering Amplitude One has to decide which eigenfunction of H 0 must be inserted into the above expression, and which Green function e G 0 must be used, since, of course, the solution of Eq. (107) is not unique. The questions are strongly interrelated, and the answers depend on which solution of the exact Schrödinger equation one wishes to find. Since the study of the perturbation of the continuous spectrum is relevant mainly for the theory of potential scattering, we will focus on this aspect. In the theory of scattering it is shown [24] that, for a potential V (Er) vanishing faster than 1/r for r ! 1, a wave function which in the asymptotic region is an eigenfunction of the momentum operator plus an outgoing wave   exp(ikr) ! exp i kE  Er C f Ek ( ; ') r

r!1

(110)

( ; ' being the polar angles with respect to the kE axis) is suitable for describing the process of diffusion of a beam of free particles with momentum Ek which impinge onto the interaction region and are scattered according the amplitude f Ek ( ; '). (The character of outgoing wave of the second term in Eq. (110) is apparent when the time factor exp(iEt) is taken into account.) The differential cross section d/ d˝ is the ratio of the flux of the probability current density due to the outgoing wave to the flux due to the impinging plane wave. One finds ˇ ˇ2 d ˇ ˇ D ˇ f Ek ( ; ')ˇ : d˝

(111)

In conclusion, we require that 0 is a plane wave, and that the Green function e G 0 has to be chosen in such a way as to yield an outgoing wave for large r. Thus we need to solve the equation       Ek 2 C  ı D  2m V exp i kE  Er C ı 2 „       (112)  U Er exp i kE  Er C ı with the asymptotic condition ı ! f Ek exp(ikr)/r for r ! 1. In terms of the Green’s function G0 (Er ; Er 0 ), which satisfies the equation       (113)  C kE 2 G0 Er ; Er 0 D ı Er  Er 0 ; the solution of Eq. (112) can be written as ı

  Er D

Z

    G0 Er ; Er 0 U Er 0  h  exp i kE  Er 0 C ı

 0 i 0 Er dEr ;

(114)

which is a form of the Lippmann–Schwinger equation [30]. The integral operator G0 with kernel G0 (Er ; Er 0 ) is connected to the operator e G 0 of Eq. (108) by the equation e G 0 D 2mG0 /„2 . The leading term of ı for r ! 1 is determined by the leading term of G0 (Er; rE 0 ), so we look for a solution of Eq. (113) with the behavior of outgoing wave for r ! 1. Due to translation and rotation invariance (if both the incoming beam and the scattering potential are translated or rotated by the same amount, the scattering amplitude f Ek ( ; ') is unchanged), we require for the solution a dependence only on jEr  Er 0 j. Recalling that 1/r D 4ı(Er), we look for a solution of Eq. (113) with Er 0 D 0 of the form F(r)/(4 r), with F(0) D 1. The function G0 (Er ; Er 0 ) then will be ˇ ˇ  0 F ˇEr  Er 0 ˇ ˇ : G0 Er ; Er D ˇ (115) ˇEr  Er 0 ˇ The equation for F(r) is F 00 C k 2 F D 0 ;

(116)

whose solutions are exp(˙ikr) (outgoing and incoming wave respectively). In conclusion for G0 we find ˇ  ˇ 1 exp ik ˇEr  Er 0 ˇ ˇ ˇ G0 (Er ; rE ) D  : ˇrE  Er 0 ˇ 4 0

(117)

The solution of the Schrödinger equation with the Green function given in Eq. (117) is denoted as EC and obeys k the integral equation known as the Lippmann–Schwinger equation [30]: C Ek

ˇ  ˇ Z     exp ik ˇEr  Er 0 ˇ 1 E ˇ ˇ Er D exp i k  Er  ˇEr  Er 0 ˇ 4  0 C  0 Er dEr 0 : (118) U Er E k

The behavior for r ! 1 can be easily checked to be as in Eq. (110) by inserting the expansion 0 ˇ ˇ ˇEr  Er 0 ˇ D r  Er  Er C O(1/r) r

(119)

into the Green function G0 . We find (ˆr  Er /r) ˇ  ˇ 1 exp ik ˇEr  Er 0 ˇ ˇ ˇ  ˇrE  Er 0 ˇ 4 

   1 exp ik r  rˆ  Er 0 rE  Er 0 r!1 ; !  1C 2 4 r r

(120)

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Perturbation Theory in Quantum Mechanics

C (Er ) Ek

which yields for C Ek

( kE f  kˆr)

    r!1 1 exp(ikr) Er ! exp i kE  Er  4 r Z      0 C 0 0 E Er dEr 0 : exp i k f  Er U Er E k

The solutions exp(i kE  Er ):  C ; E k

C Ek 0



Inserting the above expansion into Eq. (124), for the scattering amplitude f Ek ( ; ') we find f Ek ( ; ') D

  D (2)3 ı kE  Ek 0 :

(122)

In addition, they are orthogonal to any possible bound state solution of the Schrödinger equation with the Hamiltonian H D H0 C V. Together with the bound state solutions they constitute a complete set. On a par with the solutions EC (Er ) one can also envisage solutions E (Er ) with k k asymptotic behavior of incoming wave. They are obtained using for H (see Eq. (116)) the solution exp(ikr). The normalization and orthogonality properties of the functions E (Er) are the same as for the EC (Er ) functions. k k From Eq. (121) we derive an implicit expression for the scattering amplitude f Ek ( ; '): f Ek ( ; ') D 

1 4

Z

  exp i kE f  Er 0     U Er 0 EC Er 0 dEr 0 k

where the unknown function

C (Er ) Ek

k

is obtained by substituting EC in Eq. (124) with the conk tribution of order n  1 of the expansion Eq. (125). The term of order 1 is called the Born approximation [8], and is given by   1 2m f EB ( ; ')  f E(1) ( ; ') D  k k 4 „2 Z  i   h  exp i Ek  Ek f  Er 0 V Er 0 dEr 0 : (127) The term of order 2 is f E(2) ( ; ') k

1 D 4



2m „2

2 (' f ; VG0 V'Ek )

1 D 4



2m „2

n (' f ; VG0 V    G0 V'Ek ) (n times V ) :

(123)

k

Equations (118) and (124) are the starting point to obtain the expression of the exact wave function EC (Er ) and the k scattering amplitude f Ek ( ; ') as a power series in , in the spirit of the perturbation approach. Recalling that U D (2m/„2 )V (see Eq. (112)), if 'Ek  exp(i kE  Er) for EC (Er ) we k find 2m 2m 2m D 'Ek CG0 2 V 'Ek C2 G0 2 VG0 2 V'Ek C   „ Z „ „    0  2m  0  V Er D exp i kE  Er C  G0 Er; Er „2 Z Z     2m exp i kE  Er 0 dEr0 C 2 dEr 0 dEr 00 G0 Er; Er 0 2 „    0   0 00  2m  00  V Er G0 Er ; Er V Er exp i kE  Er 00 C    : „2 (125)

(129)

The scattering amplitude through order n is f E[n] ( ; ') D

(124)

(128)

and the general term of order n is f E(n) ( ; ') k

The Born Series and its Convergence

C Ek

(126)

k

where f E(n) , the contribution of order n in  to f Ek ( ; '),

still appears. Letting

' f  exp (i kE f  Er ), Eq. (123) can also be written as  1  f Ek ( ; ') D  ' f ; U EC : k 4

f E(n) ( ; ')

nD1

(121)

C (Er ) are normalized as the plane waves Ek

1 X

n X iD1

f E(i) ( ; ')

(130)

k

P (i) with f E(1) ( ; ')  f EB ( ; '), and the series 1 1 f Ek ( ; ') k k is known as the Born series [8]. Of course, when using Eq. (130) for calculating the differential cross section d / d˝ only terms of order not exceeding n should be consistently retained. For a discussion of the range of validity and the convergence of the expansions Eqs. (125) and (126) it is convenient to pose the problem in the framework of integral equations in the Hilbert space L2 [40,54], which provides a natural notion of convergence. To this purpose, since C (Er ) is not square integrable, we start assuming that the E k

potential V (Er) is summable Z ˇ  ˇ ˇV Er ˇ dEr < 1

(131) 1

and multiply Eq. (118) by jV(Er )j 2 [20,41,45]. Letting V (Er )  V(Er )/jV(Er )j ( V (Er )  0 if V (Er) D 0) and defining   ˇ  ˇ 1   (132) EC Er  ˇV Er ˇ 2 EC Er k

k

Perturbation Theory in Quantum Mechanics

ˇ1   ˇ1 ˇ   ˇ 2m  K 0 rE; Er 0   2 G0 Er; rE 0 ˇV (Er)ˇ 2 ˇV(Er 0 )ˇ 2 v Er 0 ; „ (133)

we see that, if 2

m2 4 2 „4

Z

Z dEr

Eq. (118) reads: EC k

  ˇ1   ˇ Er D ˇV(Er )ˇ 2 exp i kE  Er Z     C  K 0 Er ; Er 0 EC Er 0 dEr 0 : k

dEr 0

ˇ ˇ ˇ  ˇ ˇV (Er)ˇ ˇV Er 0 ˇ <1; ˇ ˇ ˇEr  Er 0 ˇ2

the condition kKˆ 0 k < 1 is satisfied and consequently the inverse operator (I  Kˆ 0 )1 exists. In conclusion, a sufficient condition for the convergence of the expansion (134)

EC D 0 C Kˆ 0 0 C 2 K 02 0 C   

(142)

k

Now the function in front of the integral is square integrable and the kernel K 0 (Er ; Er 0 ) is square integrable too Z

Z dEr

ˇ ˇ2 dEr 0 ˇK 0 (Er ; Er 0 )ˇ Z D

Z dEr

dEr 0

ˇ  ˇ ˇ  0 ˇ ˇV Er ˇ ˇV Er ˇ <1 ˇ ˇ ˇEr  Er 0 ˇ2

(141)

(135)

provided the potential V(Er ) obeys the additional condition ˇ  0 ˇ Z ˇ E ˇ 0 V r (136) dEr ˇ ˇ <1: ˇEr  Er 0 ˇ2

in the L2 norm is that Eq. (141) holds [43]. Since kKˆ 0 k4  Tr(Kˆ 0  Kˆ 0 Kˆ 0  Kˆ 0 ), by the Riemann–Lebesgue lemma it is possible to prove [56] that for any given  the condition kKˆ 0 k < 1 is satisfied provided the energy „2 k 2 /2m is sufficiently large. The implications for the convergence of the expansion Eq. (126) of the scattering amplitude are immediate, once Eq. (124) is written in the form m f Ek ( ; ') D  2„2 ˇ  ˇ 1     ˇ  ˇ 1 ˇV Er ˇ 2 ' f Er V rE ; ˇV Er ˇ 2

C Ek

  Er :

(143)

Equation (134) can be formally written as EC k

D

0 C Kˆ 0 EC ; k

  ˇ ˇ1 0  ˇV(Er )ˇ 2 exp i kE  Er ; (137)

Kˆ 0 being the integral operator with kernel K 0 given in Eq. (133). The function EC is formally given as k

1  0 EC D I  Kˆ 0 k

(138)

where the inverse operator (I  Kˆ 0 )1 exists except for those values of  (singular values) for which I  Kˆ 0 has the eigenvalue 0. Since by Eq. (135) Kˆ 0 is a compact operator, the singular values are isolated points which obey the inequality jj  kKˆ 0 k1 , since the spectrum of an operator is contained in the closed disc of radius equal to the norm of the operator. Thus, when kKˆ 0 k < 1 the inverse operator (I  Kˆ 0 )1 exists and is given by the Neumann series 

I  Kˆ 0

1

The Born series converges whenever the iterative solution of Eq. (137) converges, that is if Eq. (135) is satisfied. As noted above, for any given  this happens for sufficiently large energy. An additional useful result is that the Born approximation f EB ( ; ') (or the expansion f E[n] ( ; ') k k through any n) converges to the exact scattering amplitude f Ek ( ; ') when the energy grows to infinity. More precisely [48], if Eq. (131) holds then ˇ k!1 ˇ ˇ ˇ [n] ˇ f Ek ( ; ')  f E ( ; ')ˇ ! 0 : k

If kKˆ 0 k  1 the Neumann series Eq. (139) does not converge and the perturbation approach is no longer viable. However, if  is not a singular value Eq. (137) can be solved by reducing it to an integral equation with a kernel D of norm less than 1 plus a problem of linear algebra [54]. In fact, for any positive value L the operator Kˆ 0 can be approximated by a finite rank operator F

0 D I C Kˆ 0 C 2 Kˆ 02 C     I C RK ; (139)

F D

which is clearly norm convergent. By the inequality    0 2 Kˆ   2 Tr Kˆ 0  Kˆ 0 Z Z ˇ  ˇ2 D 2 dEr dEr 0 ˇK 0 Er ; Er 0 ˇ

(144)

n X

       ˛ i rE ˇ i Er 0 ;  Er 0

(145)

iD1

such that, if D  Kˆ 0  F, kDk < 1/L. Equation (137) then reads (140)

(I  D)EC D 0 C FEC : k

k

(146)

1371

1372

Perturbation Theory in Quantum Mechanics

Since for jj < L we have kDk < 1, EC can be written

Substituting expansion Eq. (151) into Eq. (148) we find

in terms of the appropriate Neumann series I C RD (see Eq. (139)):

i„

k

X

a˙ n (t)'n (t) C

n

EC D 0 C RD 0 C FEC C RD FEC : k

k

k

(147)

The unknown quantities (ˇ i ; EC ) which appear in the RHS k of Eq. (147) are determined by solving the linear-algebraic problem obtained by left-multiplying both sides of the equation by ˇr ; 1  r  n. The values of  in the range jj < L for which the algebraic problem is not soluble are the singular values of Eq. (137) in that range. Thus, Eq. (137) can be solved for any non-singular value.

A rather different problem is presented by the case that a time independent Hamiltonian H 0 , for which the spectrum and the eigenfunctions are known, is perturbed by a time dependent potential V(t). This occurs, for example, when an atom or a molecule interacts with an external electromagnetic field. For the total Hamiltonian H D H0 C V (t) stationary states no longer exist, and the relevant question is how the perturbation affects the time evolution of the system. We assume that the state is known at a given time, which can be chosen as t D 0, and search for (t). Obviously, any time t0 prior to the setting on of the perturbation V (t) could be chosen instead of t D 0. The time dependent Schrödinger equation reads

a n (t)E n 'n (t) C 

X

n

a n (t)V (t)'n (t) :

n

By left multiplying by ' k (t), for the coefficients a k (t) we find the system of equations X i„ a˙ k (t) D  a n (t)(' k (t); V (t)'n (t)) n



X

I Vkn (t)a n (t) ;

(152)

n

I (t) D (' k (t); V (t)'n (t)) : Vkn

(153)

I (t) are the matrix elements of The matrix elements Vkn I an operator V (t) between the time independent vectors  k ; n . Indeed, since

'n (t) D n exp(iE n t/„) D exp(iH0 t/„)n ;

(154)

Eq. (153) can be written as I Vkn (t) D (exp (iH0 t/„)  k ; V (t) exp (iH0 t/„) n )     k ; V I (t)n ;

(155) where the operator V I (t) is defined as follows:

(148)

At any t, (t) can be expanded in the basis of the eigenfunctions 'n (t) of H 0 : H0 'n (t) D E n 'n (t)

X

a n (t)E n 'n (t)

n

where we have defined

Time Dependent Perturbations

@ i„ D H (t) D H0 (t) C V (t) (t) : @t

D

X

(149)

'n (t) D 'n (0) exp(iE n t/„)  n exp(iE n t/„) : (150) For the sake of simplicity we treat H 0 as if it only had discrete spectrum, but the presence of a continuous component of the spectrum does not create any problem. We can write [11,42] X (t) D a n (t)'n (t) : (151) n

Note that the basis vectors 'n (t) are themselves time dependent (by the phase factor given in Eq. (150)), whereas the vectors  n are time independent. The isolation of the contribution of H 0 to the time evolution as a time dependent factor allows a simpler system of equations for the unknown coefficients a n (t).

V I (t)  exp(iH0 t/„)V(t) exp(iH0 t/„) :

(156)

System Eq. (152) must be supplemented with the appropriate initial conditions, which depend on the particular problem. The commonest application of time dependent perturbation theory is the calculation of transition probabilities between eigenstates of H 0 . Thus, we assume that at t D 0 the system is in an eigenstate of the unperturbed Hamiltonian H 0 , say the state ' 1 . In this case a1 (0) D 1; a n (0) D 0 if n ¤ 1. In the spirit of perturbation theory, each an is expanded into powers of  X r a1(r) (t) ; a1 (t) D 1 C a n (t) D

X

rD1

r a(r) n (t)

(157) n ¤1;

rD1

and terms of equal order are equated. For a (r) k we find X I (r1) Vkn an ; r >0: (158) i„ a˙(r) k D n

Perturbation Theory in Quantum Mechanics

By Eq. (157), for any r > 0, a(r) n (0) D 0. As a consequence, for r D 1 we have Z i t I (1) ak D V (t1 )dt1 „ 0 k1 Z t i D ( k ; V (t1 )1 ) exp (iE k1 t1 /„) dt1 ; (159) „ 0 where E k1  E k  E1 . For r D 2 we find X I Vkn (t)a (1) i„ a˙(2) n (t) k D n

Z t i X I I V (t) Vn1 (t1 )dt1 D „ n kn 0

 

D

i „ i „

2 Z

Z

t

t2

dt2

dt1

0

2 Z

0

Z

t

t1  t2      t r :

(164)

 If (i/„)r ( k ; T V I (t p 1 )    V I (t p r ) 1 ) is integrated over the r-cube, then each of the r! subdomains defined by Eq. (163) yields the same contribution. As a consequence Eq. (161) can be written as  Z Z t Z t i r t dtr dtr1    dt2 „ 0 0 0 Z t    dt1  k ; T V I (tr )V I (tr1 )    V I (t2 )V I (t1 ) 1 : 1 r!



0

(165)

I I Vkn (t2 )Vn1 (t1 )

n t2

dt2 0

X

 T V I (t p 1 )    V I (t p r )  V I (tr )    V I (t1 ) ;

a (r) k (t) D

whose solution is a(2) k (t) D

operators V I (t p 1 ); V I (t p 2 ); : : : ; V I (t p r ) is introduced according to the definition

0

X dt1 ( k ; V (t2 )n ) n

exp(iE kn t2 /„)(n ; V(t1 )1 ) exp(iE n1 t1 /„) :

The amplitudes a k (t) can then be written as 

 Z t   i V I (t 0 )dt 0 1 (166) a k (t) D  k ; T exp „ 0

(160) It is clear how the calculation proceeds for higher values of r. The general expression is a(r) k (t) D

 Z

i „

r Z 0

t2

X

Z

tr

t3

dtr1   

dtr dt1

0

Z

t

0

dt2 0

I (t )VnIr n r1 (tr1 )    Vkn r r

VnI3 n 2 (t2 )VnI2 1 (t1 ) :

(161)



 Z Z tr Z t3 i r t dtr dtr1    dt2 „ 0 0 0 Z t2   dt1  k ; V I (tr )V I (tr1 )    V I (t2 )V I (t1 )1 : 0

(162) It is customary to write Eq. (162) in a different way. The r-dimensional cube 0  t i  t; 1  i  r, can be split into r! subdomains 0  t p 1  t p 2      t p r1  t p r  t ;

(163)

with fp1 ; p2 ; : : : ; p r1 ; p r g a permutation of f1; 2; : : : ; r  1; rg. The time ordered product of r (non-commuting)





i U (t; t0 )  T exp „ I

By the completeness of the vectors  n , the sums over the intermediate states can be substituted by the identity and the expression of a(r) k is simplified into a (r) k (t) D

with obvious significance of the T-exponential: each monomial in the V I operators which appear in the expansion of the exponential is to be time ordered according to the T-prescription. If the initial state is given at time t0 the integral appearing in the T-exponential should start at t0 . We define Z

t

I

0

V (t )dt

0

 :

(167)

t0

The expansion of the T-exponential into monomials in the V I operators is called the Dyson series [13,14]. It is extensively employed in perturbative quantum field theory. From Eqs. (166) and (167) it is easy to derive an expression for the time evolution operator U(t; t0 ) such that U(t; t0 ) (t0 ) D

(t) :

(168)

Indeed, (t) and (t0 ) can be expanded in the basis of the vectors 'n (t) and 'n (t0 ) respectively as in Eq. (151). By the linearity of the Schrödinger equation it suffices to determine (' k (t); U(t; t0 )'n (t0 )), which we already know to be ( k ; U I (t; t0 )n ). We have (' k (t); U(t; t0 )'n (t0 )) D (exp(iH0 t/„) k ; U(t; t0 ) exp(iH0 t0 /„)n ) D ( k ; exp(iH0 t/„)U(t; t0 ) exp(iH0 t0 /„)n ) :

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Perturbation Theory in Quantum Mechanics

As a consequence we find the equation

Bibliography Primary Literature

U (t; t0 )





i D exp (iH0 t/„) T exp „

Z

t

  V t 0 dt 0 I



t0

exp (iH0 t0 /„) ;

(169)

which provides the perturbation expansion of the evolution operator U(t; t0 ) in powers of . It can be proved that if the operator function V(t) is strongly continuous and the operators V(t) are bounded, then the expansion which defines the T-exponential is norm convergent to a unitary operator, as expected [35]. The restriction to bounded operators V(t) does not detract from the range of applications of Eqs. (166) and (169), since time dependent perturbation theory is almost exclusively used for treating interactions of a system with external fields, which generate bounded interactions.

Future Directions The long and honorable service of perturbation theory in every sector of quantum mechanics must be properly acknowledged. Its future is perhaps already in our past: the main achievement is its application to quantum field theory where, just to quote an example, the agreement between the measured value and the theoretical prediction of the electron magnetic moment anomaly to ten significant digits has no rivals. Despite its successes, still perturbation theory is confronted with fundamental questions. In most of realistic problems it is unknown whether the perturbation series is convergent or at least asymptotic. In non-relativistic quantum mechanics this does not represent a practical problem since only a limited number of terms can be calculated, but in quantum field theory, where higher order terms are in principle calculable, this calls for dedicate investigations. There, in particular, conditions for recovering the exact amplitudes from the first terms of the series by such techniques as the Padé approximants or the self similar approximants, and the estimate on the bound of the error, deserve further investigation. Somewhat paradoxically, it can be said that the future of perturbation theory is in the non-perturbative results (analyticity domains, large coupling constant behavior, tunneling effect . . . ) – an issue where much work has already been done – since they have proved to be complementary to the use of perturbation theory.

1. Baker GA, Graves-Morris P (1996) Padé approximants. Cambridge Univ. Press, Cambridge 2. Bargmann V (1964) Note on Wigner’s theorem on symmetry operations. J Math Phys 5:862–868 3. Bender CM, Wu TT (1969) Anharmonic oscillator. Phys Rev 184:1231–1260 4. Bloch C (1958) Sur la théorie des perturbations des états liées. Nucl Phys 6:329–347 5. Böhm A (1993) Quantum mechanics, foundations and applications. Springer, New York, pp 208–215 6. Borel E (1899) Mémoires sur le séries divergentes. Ann Sci École Norm Sup 16:9–136 7. Born M, Heisenberg W, Jordan P (1926) Zur Quantenmechanik, II. Z Phys 35:557–615 8. Born M (1926) Quantenmechanik der Stossvorgänge. Z Phys 38:803–827 9. Brillouin L (1932) Perturbation problem and self consistent field. J Phys Radium 3:373–389 10. Courant R, Hilbert D (1989) Methods of mathematical physics, vol I. Wiley, New York, pp 343–350 11. Dirac PAM (1926) On the theory of quantum mechanics. Proc Roy Soc A112:661–677 12. Dyson FJ (1952) Divergence of perturbation theory in quantum electrodynamics. Phys Rev 85:631–632 13. Dyson FJ (1949) The radiation theories of Tomonaga, Schwinger and Feynman. Phys Rev 75:486–502 14. Dyson FJ (1949) The S-matrix in quantum electrodynamics. Phys Rev 75:1736–1755 15. Epstein ST (1954) Note on perturbation theory. Amer J Phys 22:613–614 16. Epstein ST (1968) Uniqueness of the energy in perturbation theory. Amer J Phys 36:165–166 17. Feynman RP (1939) Forces in molecules. Phys Rev 56:340–343 18. Fock VA (1935) Zur Theorie des Wasserstoffatoms. Z Phys 98:145–154 19. Graffi S, Grecchi V, Simon B (1970) Borel summability: Application to the anharmonic oscillator. Phys Lett B32:631–634 20. Grossman A (1961) Schrödinger scattering amplitude I. J Math Phys 3:710–713 21. Hamermesh M (1989) Group theory and its application to physical problems. Dover, New York, pp 32–114 22. Hannabuss K (1997) Introduction to quantum theory. Clarendon, Oxford, pp 131–136 23. Hellmann H (1937) Einführung in die Quantenchemie. Deuticke, Leipzig 24. Joachain J (1983) Quantum collision theory. North–Holland, Amsterdam 25. Kato T (1966) Perturbation theory for linear operators. Springer, New York 26. Kato T (1949) On the convergence of the perturbation method I. Progr Theor Phys 4:514–523 27. Kramers HA (1957) Quantum mechanics. North-Holland, Amsterdam, pp 198–202 28. Krieger JB (1968) Asymptotic properties of perturbation theory. J Math Phys 9:432–435 29. Landau LD, Lifshitz EM (1960) Mechanics. Pergamon Press, Oxford, p 129

Perturbation Theory in Quantum Mechanics

30. Lippmann BA, Schwinger J (1950) Variational principles for scattering processes I. Phys Rev 79:469–480 31. Loeffel J, Martin A, Wightman A, Simon B (1969) Padé approximants and the anharmonic oscillator. Phys Lett B 30:656– 658 32. Padé H (1899) Sur la représentation approchée d’une fonction pour des fonctions rationelles. Ann Sci Éco Norm Sup Suppl 9(3):1–93 33. Rayleigh JW (1894–1896) The theory of sound, vol I. Macmillan, London, pp 115–118 35. Reed M, Simon B (1975) Methods of modern mathematical physics, vol II. Academic Press, New York, pp 282–283 34. Reed M, Simon B (1978) Methods of modern mathematical physics, vol IV. Academic Press, New York, pp 10–44 36. Rellich F (1937) Störungstheorie der Spektralzerlegung I. Math Ann 113:600–619 37. Rellich F (1937) Störungstheorie der Spektralzerlegung II. Math Ann 113:677–685 38. Rellich F (1939) Störungstheorie der Spektralzerlegung III. Math Ann 116:555–570 39. Rellich F (1940) Störungstheorie der Spektralzerlegung IV. Math Ann 117:356–382 40. Riesz F, Sz.-Nagy B (1968) Leçons d’analyse fonctionelle. Gauthier-Villars, Paris, pp 143–188 41. Rollnik H (1956) Streumaxima und gebundene Zustände. Z Phys 145:639–653 42. Sakurai JJ (1967) Advanced quantum mechanics. AddisonWesley, Reading, pp 39–40 43. Scadron M, Weinberg S, Wright J (1964) Functional analysis and scattering theory. Phys Rev 135:B202-B207 44. Schrödinger E (1926) Quantisierung als Eigenwertproblem. Ann Phys 80:437–490 45. Schwartz J (1960) Some non-self-adjoint operators. Comm Pure Appl Math 13:609–639 46. Simon B (1970) Coupling constant analyticity for the anharmonic oscillator. Ann Phys 58:76–136

47. Simon B (1991) Fifty years of eigenvalue perturbation theory. Bull Am Math Soc 24:303–319 48. Thirring W (2002) Quantum mathematical physics. Springer, Berlin, p 177 49. von Neumann J, Wigner E (1929) Über das Verhalten von Eigenwerten bei adiabatischen Prozessen. Phys Z 30:467–470 50. Watson G (1912) A theory of asymptotic series. Philos Trans Roy Soc Lon Ser A 211:279–313 51. Weyl H (1931) The theory of groups and quantum mechanics. Dover, New York 52. Wigner EP (1935) On a modification of the Rayleigh– Schrödinger perturbation theory. Math Natur Anz (Budapest) 53:477–482 53. Wigner EP (1959) Group theory and its application to the quantum mechanics of atomic spectra. Academic Press, New York 54. Yosida K (1991) Lectures on differential and integral equations. Dover, New York, pp 115–131 55. Yukalov VI, Yukalova EP (2007) Methods of self similar factor approximants. Phys Lett A 368:341–347 56. Zemach C, Klein A (1958) The Born expansion in non-relativistic quantum theory I. Nuovo Cimento 10:1078–1087

Books and Reviews Hirschfelder JO, Byers Brown W, Epstein ST (1964) Recent developments in perturbation theory. In: Advances in quantum chemistry, vol 1. Academic Press, New York, pp 255–374 Killingbeck J (1977) Quantum-mechanical perturbation theory. Rep Progr Phys 40:963–1031 Mayer I (2003) Simple theorems, proofs and derivations in quantum chemistry. Kluwer Academic/Plenum Publishers, New York, pp 69–120 Morse PM, Feshbach H (1953) Methods of theoretical physics, part 2. McGraw-Hill, New York, pp 1001–1106 Wilcox CH (1966) Perturbation theory and its applications in quantum mechanics. Wiley, New York

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Perturbation Theory, Semiclassical ANDREA SACCHETTI Dipartimento di Matematica Pura ed Applicata, Universitá di Modena e Reggio Emilia, Modena, Italy Article Outline Glossary Definition of the Subject Introduction The WKB Approximation Semiclassical Approximation in Any Dimension Propagation of Quantum Observables Future Directions Bibliography Glossary Agmon metric In the classically forbidden region where the potential energy V(x) is larger than the total energy E, i. e. V(x) > E, we introduce a notion of distance based on the Agmon metric defined as [V(x)  E]dx 2 , where dx 2 is the usual Riemann metric. We emphasize that such an Agmon metric is the “semiclassical” equivalent of the “classical” Jacobi metric [E V (x)]dx 2 introduced in classical mechanics in the classically permitted region where V(x) < E. Asymptotic series The notion of asymptotic series goes back to Poincaré. We say that a formal power series P r is taken to be the asymptotic power series for r ar z a function f (z), as jzj ! 1 in a given infinite sector S D fz 2 C : ˛  arg z  ˇg, if for any fixed N the remainder term R N (z) D f (z) 

N1 X

ar zr

rD0

is such that R N (z) D O(zN ), that is ˇ ˇ jR N (z)j  C N ˇzN ˇ ; 8z 2 S ; for some positive constant CN depending on N. Classically allowed and forbidden regions – turning points We distinguish two different regions: the region V (x) < E where the classical motion is allowed and the region V(x) > E where the classical motion is forbidden. The points that separate these two regions, that is such that V(x) D E, will play a special role and they are named turning points.

Semiclassical limit Cornerstone of Quantum Mechanics is the time-dependent Schrödinger equation i„

@ „2 D  @t 2m

C V (x)

;

(1)

where V(x) is assumed to be a real-valued function, usually it represents the potential energy, and m is the mass of the particle. The solution of this equation defines the density of probability j (x; t)j2 to find the particle in some region of space. The parameter ¯ in Eq. (1) is related to Planck’s constant h „D

h D 1:0545  1027 erg sec 2 D 6:5819  1022 MeV sec :

According to the correspondence principle, when Planck’s constant can be considered small with respect to the other parameters, such as masses and distances, then quantum theory approaches classical Newton theory. Thus, roughly speaking, we expect that classical mechanics is contained in quantum mechanics as a limiting form (i. e. „ ! 0). The limit of small ¯, when compared with the other parameters, is the socalled semiclassical limit. We should emphasize that making Planck’s constant small in Eq. (1) is a rather singular limit and many difficult mathematical problems occur. Stokes lines There is some misunderstanding in the literature concerning the name of the curves = (z) D 0, where =() denotes the imaginary part of (z) D

i „

Zz p

E  V(q)dq ;

z2C;

xE

and x E 2 R is a turning point (i. e. V(x E ) D E), the potential V is assumed to be an analytic function in the complex plane. As usually physicists do, and in agreement with Stokes’ original treatment, we adopt here the convention to name as Stokes line any path coming from the turning point xE such that =[(z)] D 0; reserving the name of anti-Stokes lines, or regular lines, for the curves such that <[(z)] D 0, where <() denotes the real part of  (we should emphasize that most mathematicians adopt the opposite rule, that is they call Stokes lines the paths such that <[(z)] D 0). Tunnel effect At a real (simple) turning point xE where E D V(x) (and V 0 (x) ¤ 0), classical particles incident from an accessible region [where E > V (x)] reverse their velocity and return back; the adjacent region [where E < V(x)] is forbidden to these particles according to classical mechanics. In fact, quantum

Perturbation Theory, Semiclassical

mechanically exponentially growing or damped waves can exist in forbidden regions and a quantum particle can pass through a potential barrier. This is the socalled tunnel effect. WKB The WKB method, named after the contributions independently given by Wentzel, Kramers and Brillouin, consists of connecting the approximate solutions of the time independent Schrödinger equation across turning points. Definition of the Subject Several kinds of perturbation methods are commonly used in quantum mechanics  Perturbation Theory and Molecular Dynamics,  Perturbation Theory in Quantum Mechanics,  Perturbation Theory. This chapter deals with semiclassical approximations where expressions for energy levels and wave functions are obtained in the limiting case of small values of Planck’s constant. We emphasize that wave functions are highly singular as the parameter ¯ goes to zero. In fact the semiclassical limit is a singular perturbation problem; namely, Eq. (1) suffers a reduction of order setting ¯ equal to zero and the resulting equation is not a differential equation and does not give the classical limit correctly. Therefore, ordinary perturbation methods, which usually give energy levels and wave-functions as convergent power series of the small parameter, cannot be applied. The main goal of semiclassical methods consists of obtaining asymptotic series, in the limit of small ¯, for the quantum-mechanical quantities. For instance, semiclassical methods permit us to solve the eigenvalues problem 

„2  C V(x) 2m

DE

;

x 2 Rn ; n  1 ; (2)

where the energy levels E and the wave-functions (x) are given by means of asymptotic series in the limit of vanishing ¯, or to obtain a direct link between the time evolution of a quantum observable and the Hamiltonian flux of the associated classical quantity.

JWKB method to acknowledge that the approximate connection formula was previously discovered by Jeffrey. Actually, oscillatory and exponential approximation formulas, obtained far from turning points, were independently used by Green (1837) and Liouville (1837) and they can be already found in an investigation on the motion of a planet in an unperturbed elliptic orbit by the Italian astronomer Carlini (1817). For historical notes on the WKB-method we refer to [7,11,23]. The WKB method can be also applied to three-dimensional problems only under some particular circumstances; for instance when the potential is spherically symmetric and the radial differential equation can be separated. In general WKB approximation is not suitable for problems in dimension n higher than 1. Actually, semiclassical methods in higher dimension require new sophisticated tools such as the Agmon metric, microlocal calculus and ¯-pseudodifferential operators. These tools have been developed by Agmon, Hörmander and Maslov in the 1970s and since then a large number of mathematicians have contributed to this subject. In Sects. “Semiclassical Approximation in Any Dimension” and “Propagation of Quantum Observables” we briefly introduce the reader to the basic concepts of these theories and we resume the most important results such as the exponential decay of wave-functions and the Egorov Theorem, which asymptotically describes the quantum evolution of an observable by means of the classical evolution of its classical counterpart. Actually, these methods may be applied to many other fields, where ¯ is a small quantity not related to the Planck constant but it may represent a different small physical quantity such as the adiabatic parameter in adiabatic problems, the (square root of the) inverse of the heavy mass in the Born–Oppenheimer approximation, etc. Notation Hereafter, for the sake of definiteness, let us fix 2m D 1. The WKB Approximation

Introduction The first contributions to this theory, in the framework of Quantum Mechanics, go back to Wentzel (1926), Kramers (1926) and Brillouin (1926). In their papers they independently developed the determination of connection formulas linking exponential and oscillatory approximations of the solution of Eq. (2) in dimension n D 1 across a turning point. This method, explained in Sect. The WKB Approximation, is usually called WKB approximation, or also

If the potential V(x) does not have a very simple form then the solution of the time-independent Schrödinger equation even in one dimension  „2

00

C V(x)

DE

;

x 2 (x1 ; x2 ) ;

(3)

is a quite complicated problem which requires the ı use of a sort of approximation method (hereafter 0 D d dx denotes the derivative with respect x); here (x1 ; x2 ) is a given finite or infinite one-dimensional interval.

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Perturbation Theory, Semiclassical

We distinguish different regions: the classically allowed regions where V(x) < E and the classically forbidden regions where V(x) > E. Approximation formulas for the solution of Eq. (3) in these separate regions have been studied since Carlini, Green, Liouville and Jacobi. The problem to connect these approximated solutions across the turning points was raised in the framework of Quantum Mechanics and it was independently solved by Jensen, Wentzel, Kramers and Brillouin. For the general treatment of semiclassical approximation in dimension one and for physical applications we refer to [2,3,10,21,24,25,33]. Semiclassical Solutions in the Classically Allowed Region The basic idea is quite simple: if V D const. is a constant smaller than E then Eq. (3) has solutions e˙i kx/„ for a suitable real-valued constant k. This fact suggests to us that if the potential, while no longer constant, varies only slowly with x and it is such that V(x) < E for any x 2 (x1 ; x2 ), we might try a solution of the form (x) D a(x)e

i S(x)/„

;

(4)

except that the amplitude a(x) is not constant and the phase S(x) is not simply proportional to x. Substituting (4) into (3) and separating the real and imaginary parts of the coefficients of e i S(x)/„ , then we get a system of equations for the x-dependent real-valued phase S(x) and amplitude a(x):

2 u  p2 (x) D „2 a00 /a (a 2 u)0 D 0 where we set u(x) D S 0 (x) and

p(x) D

p

E  V(x) :

Hence, a(x) D C[u(x)]1/2 for some constant C. Since the complex conjugation a(x)ei S(x)/„ of (4) is a solution of the same equation and the Wronskian between these two solutions is not zero then the general solution of Eq. (3) can be written as (x) D bC ˙ (x)

Dp

C (x) C b  (x) R ˙i/„ u(x)dx

1 e u(x)

:

; (5)

Here, b˙ are arbitrary constants and u D u(x; „) is a realvalued solution, depending on the variable x and on the semiclassical parameter ¯, of the nonlinear ordinary differential equation F (u) D u2  p2  „2 u f (u) D 0

(6)

where

 00 f (u) D u1/2 u1/2 :

The fact that (6) is a nonlinear equation, whereas the Schrödinger equation (3) is linear, would be usually regarded as a drawback, but we shall take advantage of the nonlinearity to develop a simple approximation method for solving (6). Taking the limit of ¯ small in (6) then it turns out that the leading order of u is simply given by p. Actually, it is possible to prove that (3) has twice continuously differentiable solutions ˙ satisfying the following asymptotic behavior ˙ (x)

Dp

1 p(x)

e

˙i/„

Rx

p(q)dq

a

 1 C ı˙ (x) ; x 2 (x1 ; x2 ) ;

(7)

where a is an arbitrary point in (x1 ; x2 ) and where ˇx ˇR

jı˙ (x)j  e

1 ˇ 2 „ˇ

a

ˇ ˇ

j f [p(q)]jdq ˇˇ

 1 D O(„) :

Henceforth, the dominant term of the solution (5) has an oscillating behavior of the form

(x) D bC

e

i/„

Rx p

EV (q)dq

a

p 4

E  V(x) C b

e

i/„

Rx p

EV (q)dq

a

p 4

E  V(x)

C O(„) :

In order to compute also the other terms of the asymptotic expansion of the solutions ˙ (x) we formally solve the nonlinear Eq. (6) by means of the formal power series in „2 : 1 X „2n u2n (x) : (8) u D u(x; „) D nD0

The functions u2n (x) are determined explicitly, order by order, by formally substituting (8) into (6) and requiring that the coefficients of the same terms „2n are zero for any n  0. In such a way and assuming that the potential V(x) admits derivatives at any order then we obtain that u0 (x) D p(x) u2 (x) D 

1 p00 (x) 3 [p0 (x)]2 C 2 4 p (x) 8 p3 (x)

 2 1 p0000 (x) 5 p000 (x)p0 (x) 13 p00 (x) u4 (x) D   16 p4 (x) 8 p5 (x) 32 p5 (x)   4 2 00 0 0 99 p (x) p (x) 297 p (x) C  ; 32 p6 (x) 128 p7 (x)

Perturbation Theory, Semiclassical

and so on. Thus, the asymptotic behavior (7) can be improved up to any order. In particular, let N;˙ (x) D p

1

˙i/„

p N (x)

e

Rx a

p N (q) dq

;

x 2 (x1 ; x2 ) ;

where a 2 (x1 ; x2 ) is arbitrary and fixed and p N (x) D PN 2n nD0 u2n (x)„ , in particular p0 (x) D p(x), let us introduce the error-control function N (x) D





1 „p N (x)

F p N (x) D O(„2NC1 )

where 0 (x) D „ f [p(x)]; then the two functions N;˙(x) approximate the solutions ˙ (x) of Eq. (3) up to the order 2N C 1, that is ˇ1 2 0 ˇ x 3 ˇ ˇZ ˇ ˇ 1 j ˙ (x)  N;˙ (x)j  4exp @ ˇˇ j N (q)j dqˇˇA  15 2ˇ ˇ a

D O(„

2NC1

) ; 8x 2 (x1 ; x2 ) :

We underline that this result is valid whether or not x1 and x2 are finite, also whether or not V(x) is bounded: it suffices that the error-control function is absolutely integrable: N 2 L1 (x1 ; x2 ). Semiclassical Solutions in the Classically Forbidden Region The previous asymptotic computation of the solutions ˙ applies also in the classically forbidden region where V(x) > E, provided that p(x) p is one of the two purely imaginary determination of E  V(x). By assuming that =p(x) < 0 then Eq. (3) has twice continuously differentiable solutions of the form Rx

˙1/„ jp(q)jdq  1 a 1 C ı˙ (x) ; e ˙ (x) D p jp(x)j

x 2 (x1 ; x2 ) ;

(9)

where a is an arbitrary point in (x1 ; x2 ) and where the remainder terms ı˙ are bounded as follows ˇ

jı˙ (x)j  e

ˇ Rx 1 ˇ 2 „ˇˇ a˙

ˇ ˇ

j f [p(q)]jdq ˇˇˇ

 1 D O(„)

where aC D x1 and a D x2 . In particular, if V(x) > E for any x 2 (1; x2 ), that is x1 D 1 (and similarly we can consider the case where x2 D C1), and the errorcontrol function f [p(x)] 2 L1 (1; x2 ) then the above asymptotic estimate holds for any x 2 (1; x2 ). We also

emphasize that the asymptotic behavior (9) could be also improved up to any order, as done for the approximated solutions in the classically allowed regions. We underline that the two solutions ˙ play here a different role. Indeed, solution  (x) is an exponentially decreasing function as x > a grows, while C (x) is exponentially increasing. The first solution is usually called recessive solution and it is uniquely determined by the asymptotic power expansion as „ ! 0. The other solution is called dominant solution and it is not uniquely determined by the asymptotic expansion; in fact, C (x) C c  (x), x > a, is still a solution of Eq. (3) which has the same asymptotic behavior of C (x) for any c 2 C. Connection Formula It is not possible to use the semiclassical solutions (7) and (9) at turning points xE since the term 1/p diverges and there is no guarantee that the same combination of simple semiclassical solutions will fit the same particular solution on both sides of the turning point. This is the so-called connection problem: if the values b˙ of the semiclassical solution are known in a given region then the problem consists of computing the values of b˙ in a different region separated from the first region by one (or more) turning points. The main two methods used to treat this problem are the following ones:  The complex method, where the two regions surrounding the turning point are joined by a path in the complex plane which is sufficiently far from the turning points. In such a case the semiclassical solutions in different regions are connected by means of holomorphic extension arguments. Thus, in order to apply this method we have to assume that the potential V(x) can be holomorphically extended to the complex plane.  The second method employs the technique of the uniform approximation, where the required solution is mapped on the solution of a simpler and suitable equation which has the same disposition of turning points as the original one. As well as enabling the connection problem to be solved, this method also provides the wave function in the neighborhood of the turning points, which was bypassed in the complex method. We are going now to explain these methods in detail. The Complex Method By assuming that V (x) is the restriction on the real axis of a holomorphic function V(z), z 2 C, we turn now to the approximate solution of the

1379

1380

Perturbation Theory, Semiclassical

differential equation „2

 d2 (z) D E  V (z) (z) dz2

in a complex domain D in which V(z) is holomorphic and p2 (z) D E  V (z) does not vanish. Then, this equation has two holomorphic solutions ˙ (z) D p

1 p(z)

e

˙i/„

Rz

p(q)dq

a

 1 C O(„) ; z 2 D ; (10)

where a is an arbitrary point; in particular, for our purposes it is convenient to choose it coinciding with the turning point, i. e.: a D x E . Actually, this asymptotic behavior holds under a technical assumption: given a reference point aC (respectively a ) then (10) is true for C (z) (resp.  (z)) for any z such that there exists a progressive (resp. regressive) path C (resp.  ) contained in D and connecting z with aC (resp. a ); that is as q passes along C (resp.  ) from aC (resp. a ) to z then <[(z)] is nondecreasing (resp. nonincreasing) where i (z) D „

Zz p(q)dq : xE

Now, we denote as a Stokes line any path coming from the turning point xE such that =[(z)] D 0; we denote as an anti-Stokes line, or principal line, any path coming from the turning point xE such that <[(z)] D 0. Classically forbidden (resp. allowed) regions are particular cases of Stokes (resp. anti-Stokes) lines, which lie on the real axis. Let us consider now the case, as in Fig. 1, where the turning point xE is simple (that is V 0 (x E ) ¤ 0), p2 (x) < 0 for x < x E and p2 (x) > 0 for x > x E . Since the turning point xE is simple then in a neighborhood of the turning 1 point we have that p(z)  c [z  x E ] 2 , for some c 2 RC , 3 and (z)  2i3„c [z  x E ] 2 . Thus, Stokes lines are three different paths coming from the turning point and with asymptotic directions arg(z  x E ) D 13 ; ; 53 ; while the asymptotic directions of the anti-Stokes lines are given by arg(z  x E ) D 0; 23 ; 43 . Along the three anti-Stokes lines 1 , 2 and 3 the solution can be written as (z) D bC; j

C; j (z)Cb; j

; j (z);

z 2 j ; j D 1; 2; 3;

where ˙; j have the asymptotic behavior given by (10) and where the coefficients b˙; j are asymptotically fixed along the anti-Stokes lines. Let F j be the 22 matrix which connects the coefficients:     bC; j bC; jC1 D Fj : b; jC1 b; j

Perturbation Theory, Semiclassical, Figure 1 Stokes and anti-Stokes lines in a neighborhood of a simple turning point x E where the left-hand side of the turning point is classically forbidden and the right-hand side is classically allowed. The wavy line denotes the cut of the multi-valued function p(z)

In the region I, with boundaries given by the antiStokes lines 1 and 2 , the solution C is dominant and thus the coefficient bC cannot change in this region, similarly in the region III with boundaries 3 and the cut. In contrast, in the region II, with boundaries given by the anti-Stokes lines 2 and 3 , the solution  is dominant and thus the coefficient b cannot change in this region. Then the matrices F j can be written as  F1 D  and

F3 D

1 r

0 1

1 t

0 1



 ;

F2 D



1 0

s 1



where r, s and t have to be determined. To this end let us consider the closed anti-clockwise path  surrounding the simple turning point xE ; along H this path the exponential term gains a phase  D „1  p(z)dz and the argument of p(z), which appears in (10), decreases of . Hence, the holomorphic extension of ˙ , around this closed path, are thus proportional to the solution  : C

! ie i



and



! iei

C

:

Perturbation Theory, Semiclassical

Therefore, the numbers r, s and t are such that   1 C st s F1 F2 F3 D r C (rs C 1)t rs C 1   0 iei ie i 0

from the turning point to the classically forbidden region, and   1 ˛ cos jj C    (x) ! p 4 jp(x)j

from which it follows that s D iei ;

r D t D ie i : R In particular, since p(z)dz does not diverge at xE then we can take  in a small neighborhood of xE and  0. Thus, the Stokes’ rule follows: the connection between the two coefficients from one Stokes line to the (anti-clockwise) adjacent one follows the following rule bdominant ! bdominant

and

brecessive ! brecessive C ibdominant :

from the turning points to the classically allowed region. Thus, we have established a connection between oscillatory and exponentially increasing and decreasing solutions. Clearly, this entire approach breaks down if the energy is too close to a value corresponding to a stationary point of the potential. In this case a different approximation must be implemented. For instance, in the case of a turning point with multiplicity 2 then the approximate solution of the Schrödinger equation (3) is given by means of Weber parabolic cylinder functions. Bound States for a Single Well Potential

The Method of Comparison Equations With this method we obtain a local approximate solution, in a neighborhood of the turning points, in terms of the known solutions of an equivalent equation d2  C q(y)(y) D 0 ; dy 2

(11)

where q(y) is chosen to be similar in some way to p2 (x), but simpler in order to have an explicit solution. For the sake of definiteness, we restrict here our analysis to the case of a simple turning point xE . In order to get an approximate solution to Eq. (3) in a small neighborhood of a simple zero xE of p2 (x) we introduce the following changes of variable

We apply now the previous techniques in order to compute the stable states, that is normalized solutions of the Eq. (2), when the potential V (x) has a single well shape; that is it has a simple minimum point xmin , with minimum value V min such that V(x) > Vmin for any x ¤ xmin and lim infjxj!C1 V(x) > Vmin . Then, for E > Vmin close enough to the bottom of the well, equation p(x) D 0 has only two simple solutions x1 < xmin < x2 and the typical picture of the Stokes lines appears as in Fig. 2. Here,



 23 Zx 3 i x ! y(x) D ; (x) D p(q)dq : (x) 2 „ xE

Equation (3) takes the form of the approximate Eq. (11) where q(y)  y in a neighborhood of the turning point, which admits solutions given by the Airy functions Ai(y) and Bi(y). Then, the approximate solution of the Schrödinger equation (3) in a neighborhood of the turning point has the form

(x) ˛

 2 y(x) p2 (x)

 14 fcos()Ai[y(x)] C sin()Bi[y(x)]g

where ˛ and ˇ are constants. The Airy functions are well understood and exact connection formulas can be established obtaining that h i ˛ (x) ! p cos()ejj C 2 sin()ejj 2 jp(x)j

Perturbation Theory, Semiclassical, Figure 2 Stokes and anti-Stokes lines in a neighborhood of the bottom of a single well

1381

1382

Perturbation Theory, Semiclassical

˙;x 1 (resp. ˙;x 2 ) denotes the fundamental solutions with asymptotic behaviors (7) and (9) in a neighborhood of x1 (resp. x2 ). Since we are looking for normalized solutions then in the classically forbidden region x > x2 , contained in the region I, the dominant term C;x 2 of the I exactly zero, that solution should have coefficient bC;x 2 is

(z) D

;

;x 2 (z)

z 2I;

where the coefficients b;x 2 of the recessive solution ;x 2 is chosen equal to 1. Turning around the turning point x2 and crossing the anti-Stokes line it follows that the coeffiII II (x)Cb;x (x) cients of the solution D bC;x 2 ;x 2 2 C;x 2 in the region II take the form 

II bC;x 2 II b;x 2



 D F2

0 1

 ;

F2

1 0

i 1

We consider now the case of a symmetric double well potential; that is V(x) D V(x) and it has two simple minima at xC > 0 and x D xC separated by a barrier, we assume also that the minimum value Vmin < lim infjxj!1 V (x). For instance, V(x) D x 4  2˛x 2 for some ˛ > 0; in this case the shape of the potential has two p symmetric wells where x˙ D ˙ ˛ and Vmin D ˛ 2 < 0, the two wells are separated by an energy barrier with top Vmax D 0 at x D 0. The semiclassical double well model is not only a very enlightening pedagogical problem, but it is also the basic argument explaining many relevant physical questions, see, e. g., [5,12,16,32,34]. In the interval (Vmin ; VM ), where

 VM D min Vmax ; lim inf V (x) ; jxj!1

the stable states appear as doublets E˙ whose distance !, named splitting, is quite small. The associated eigenvectors are even and odd (real-valued) wave-functions, that is

where 

Double Well Model: Estimate of the Splitting and the “Flea of the Elephant”



˙ (x)

according to the Stokes’ rule. In order to study the matching condition around the other turning point x1 we have to transport the origin of integration at x1 obtaining (z) D

II bC;x 1

C;x 1 (z)

C

II b;x 1

II C;x 2 (x1 )bC;x 2

and

;x 1 (z)

II b;x D 1

II ;x 2 (x1 )b;x 2 :

On the other hand, also in region III the dominant III term C;x 1 of the solution should have coefficient bC;x 1 exactly zero; from this fact and from the Stokes’ rule applied to the turning point x1 it follows that the equation for the dominant term of the bound states is C;x 2 (x1 )

C

;x 2 (x1 )

D0;

which implies 13 2 0x Z2 1 cos 4 @ p(x)dx C O(„)A5 D 0 : „ x1

Hence, we obtain the well known Bohr–Sommerfeld rule

Zx 2 p(x)dx D „ x1

˙ (x)

:

(12)

The splitting ! D E  EC can be computed as ! D „2 1 R

0 C (0)  (0)  (x) C (x)dx

0

where II D bC;x 1

D˙

 1  C n C O(„2 ) ; n 2 N : 2

and it turns out to be exponentially small as „ ! 0. More precisely, if EC is the ground state then ! D O(eS 0 /„ ) where S0 D

Zx Cp

V(x)  Vmin dx

x

is the Agmon distance between the two wells. We would emphasize that the eigenvectors ˙ are asymptotically localized on both wells because of the symmetry property (12). The effect on the ground state of a small perturbation W(x) that breaks the symmetry is worth mentioning. In such a case the property (12) does not work and, even if the small perturbation W(x) is supported only on one side and far from the bottom of the well, the ground state, instead of being asymptotically supported on both wells, may be localized on just one well. According to Barry Simon we may state that the perturbation W(x) is a small flea on the elephant V (x). The flea does not change the shape of the elephant – in the sense that the splitting is still exponentially small – but it can irritate the elephant enough so that it shifts its weight – in the sense that the ground state is localized on just one well.

Perturbation Theory, Semiclassical

Semiclassical Approximation in Any Dimension Here we consider the eigenvalue problem (2) in any dimension n  1 where the potential V is assumed to be a multi-well potential [6,17,18,31]. More precisely, we assume that

with normalized eigenvectors („)n/4

n Y 

2˛ j ˛ j !

1/2

jD1

1/8 j

p

e

 j x 2j /2„



Vmin :D inf V(x) < V1 D lim inf V(x) jxj!C1

and for any E 2 (Vmin ; V1 ) we can write the decomposition of V 1 ((1; E]) D [ NjD1 U j as the union of N disjoint, compact and connected sets U j . Inside these sets, named wells, the classical motion is allowed; outside the classical motion is forbidden. From well-known results we have that for energies in the interval (Vmin ; V1 ) the eigenvalue problem (2) admits only discrete spectrum, that is we have only isolated eigenvalues with finite multiplicity. In order to compute these eigenvalues we’ll consider, at first, the solutions of Eq. (2) inside any single well and then we’ll take into account the tunneling effect among the adjacent wells as a perturbation. Actually, it is also possible to consider the case where E > V1 . In such a case we don’t expect to have isolated eigenvalues but, under some circumstances, resonant states [13,19,20]. However, we don’t fix our attention here on this problem. Semiclassical Eigenvalues at the Bottom of a Well Let x0 be a local nondegenerate minimum for the potential V(x). For the sake of definiteness we assume that x0 D 0 and the local minimum is such that  D0  V (0)ı D rV(0) and the Hessian matrix Hess D @2 V(0) @x i @x j i; jD1:::n is positively defined with eigenvalues 2 j > 0, j D 1; : : : ; n; furthermore, we choose the system of coordinates such that the Hessian matrix has diagonal form, i. e. Hess D diag(2 j ). In order to compute the eigenvalues at the bottom of the well containing the minimum x0 we approximate the potential by means of the n-dimensional harmonic oscillator. Thus, Eq. (2) takes the form # " n 2 X 2@ 2 C j x j  E D0 „ @x 2j jD1 which has exact eigenvalues 3 2 n q X  j (2˛ j C 1)5 ; ˛ D (˛1 ; : : : ; ˛n ) 2 N n ; „4 jD1

  1/2 H˛ j 1/4 x ; j „

where H m is the Hermite polynomial of degree m. It is clear that when ¯ is small enough then the first energy level of the harmonic oscillator is very close to the bottom of the potential and thus we expect that such an approximation gives the leading term of the first energy level and wave-function of (2). In fact, there exist two formal power series E(„)

1 X

„jej

and

a(x; „)

1 X

„r ar (x) ; (13)

rD0

jD0

p where e0 D Vmin D 0, „e1 D „ jD1  j is the first eigenvalue of the associated harmonic oscillator and a0 D  Qn 2 1/8 , such that the function jD1  j / Pn

(x; „) D „n/4 a(x; „)e'(x)/„

(14)

is such that  „2  C V (x)  E(„)

D O(„1 )e'(x)/„ p P in a neighborhood of x0 D 0, where '(x) D 12 njD1  j x 2j CO(jxj2 ) is the positive solution of the Eikonal equation jr'j2 D V  e0 in a neighborhood of x0 D 0. The way to obtain this result essentially consists of inserting the formal power series (14) in Eq. (2) and expanding in powers of ¯ the coefficients of e'(x)/„ , then of requiring the cancelation of the coefficients of „ j for any j 2 N. These approximate solutions are valid in a neighborhood of the nondegenerate minimum and they are the natural candidates to become the true eigenvalue and eigenfunction of the problem (2). In fact, we’ll extend this single well approximate solution (14) to a larger domain and then we’ll solve the connection problem. To this end we fix an open, sufficiently small, regular bounded set ˝ containing the minimum point x0 ; then Eq. (2) with Dirichlet boundary condition on ˝, that is ˝ j@˝

D0;

k

˝ kL 2 (˝)

D1;

(15)

admits one simple eigenvalue E˝ („) at the bottom of the well, close to the first eigenvalue of the harmonic oscillator e0 C „e1 , which admits the asymptotic expansion (13).

1383

1384

Perturbation Theory, Semiclassical

Actually, such an eigenvalue will depend on the choice of the domain ˝. More precisely, if E˝ 0 („) denotes the first eigenvalue of Eq. (2) with Dirichlet boundary condition on ˝ 0 for a different domain ˝ 0 containing x0 , then E˝ 0 („) differs from E˝ („) for an exponentially small term. Furthermore, it is also possible to prove that, modulo an exponentially small error, the spectrum of the Schrödinger equation (2) in some interval depending on ¯ is the same as the spectrum of the direct sum of the Dirichlet single well problems in the same interval. Agmon Metric In order to study the exponential behavior of the solution (14) when x is far from the minimum x0 we introduce now the Agmon distance between two points x1 , x2 2 Rn defined as Z p dE (x1 ; x2 ) D inf [V (x)  E]C dx ; 



where

be an eigenvalue of Eq. (2) with Dirichlet boundary condition (15) with associated eigenfunction ˝ . Then, the Agmon theorem enables us to obtain the following decay estimate for the eigenfunction ˝ : for any fixed and positive then  h  r edE (x;U )/„

˝

i  

L 2 (˝)

  C edE (x;U )/„

 

˝

L 2 (˝)

 C e /„ for some positive constant C depending on (of course we expect that C will grow as goes to zero). That is we have a good a priori estimate of the wavefunction ˝ in the weighted H 1 (˝) space with weight edE (x;U )/„ . Since d(x; U) D 0 for any x belonging to the well U then it follows that the solution ˝ is exponentially decreasing outside the classical allowed region, as already seen in the one-dimensional problems. Tunneling Between Wells In order to consider the tunneling effect among the wells fU j g NjD1 for any fixed E 2 (Vmin ; V1 ) we define

[V(x)  E]C D maxfV(x)  E; 0g ; d E (U i ; U j ) D

is any regular path connecting the two points x1 and x2 and E  Vmin is fixed. We underline that the Agmon distance defines a (pseudo)-metric on the Euclidean space Rn . Indeed, it satisfies the triangle inequality dE (x1 ; x2 )  d E (x1 ; x3 )CdE (x3 ; x2 ); 8x1 ; x2 ; x3 2 Rn ; but it is degenerate on each well U where V < E, in particular dE (x1 ; x2 ) D 0 for any couple of points x1 and x2 belonging to the same well and the diameter of each well U is zero with respect to this metric. This kind of (pseudo-)metric has been used to obtain precise exponential decay of wave-functions of Schrödinger operators. Indeed, the exponential term '(x) in (14) is simply given by '(x) D d e 0 (x0 ; x) for any x in a neighborhood of x0 . Furthermore, by means of the Agmon metric we are also able to give an estimate of the exponential decay of wavefunctions not only for x in a neighborhood of the minimum x0 , but also for x far from this point. In particular, let Vmin < E < V1 be fixed and let U be one of the wells, we consider now the eigenvalue problem for the Schrödinger equation with Dirichlet boundary conditions on a regular open set ˝ containing the well U. Let E˝ < E

inf

x2U i ;y2U j

dE (x; y)

as the Agmon distance between the two wells U i and U j , by construction it turns out that this distance is strictly positive for any i ¤ j. A special role will be played by the minimal distance among these wells S0 D min dE (U i ; U j ) : i¤ j

In fact, the discrete spectrum of the Dirichlet realization of the Schrödinger equation on the boundary of the wells give, up to error of the order eS 0 /„ , the discrete spectrum of the original Schrödinger equation (2). In order to be more definite, let M S; be the open set containing the well j U j defined as S;

Mj

˚ D x 2 Rn : dE (x; U j ) < S

and dE (x; U k ) > ; k ¤ j

S;

that is M j is, essentially, a ball (with respect to the Agmon pseudo-metric) large enough centered in the well U j where we have eliminated the points from the other wells. S; If necessary we can regularize the boundary of M j . In the following we take  > 0 small enough and S large enough, in particular we require that S > 2S0 . Let  j be the discrete spectrum of the Schrödinger equation (2) with

Perturbation Theory, Semiclassical

S;

Dirichlet condition on M j and let  be the discrete spectrum of the Schrödinger equation (2). Then, for any interval I(„) D [˛(„); ˇ(„)], where ˛(„), ˇ(„) ! e0 as „ ! 0, there exists a bijection h i b :  \ I(„) ! [ NjD1  j \ I(„) such that for any < S0  2 jb()  j  C e /„

(16)

for some C > 0. Actually, this result could be improved by assuming some regularity properties on the potential V and estimate (16) can be replaced by the precise asymptotic behavior. If we consider now the symmetric double well problem where N D 2 and with symmetric potential, e. g. V(x1 ; x2 ; : : : ; x n ) D V (x1 ; x2 ; : : : ; x n ), then, by symmetry, we have that the two Dirichlet realizations coincide, up to the inversion x1 ! x1 . Then 1 D 2 . If we denote by E1 and E2 the first two eigenvalues of  then E1 < E2 (in fact, the first eigenvalue is always nondegenerate) and the splitting between them can be estimated as jE2  E1 j  jE2  b(E2 )j C jb(E2 )  b(E1 )j C jb(E1 )  E1 j  C e /„ ; since b(E1 ) D b(E2 ), for any < S0 since  > 0 is arbitrary. Propagation of Quantum Observables So far we have restricted our analysis to the semiclassical computation of the stable states of the time-independent Schrödinger equation (2). However, there exist some relevant results for what concerns the time evolution operator ei tH/„ , which is the formal solution of the time-dependent Schrödinger equation (1) with Hamiltonian H D „2  C V. In other words, this result is connected to the time-evolution of quantum observables [8,29]. The main tool is the ¯-pseudodifferential calculus we briefly review below. Brief Review of ¯-Pseudodifferential Calculus Here, we briefly review some basic results of the semiclassical pseudodifferential (also called ¯-pseudodifferential) calculus. We refer to the books by Folland [9], Grigis and Sjöstrand [15], Martinez [22] and Robert [28] for a detailed treatment. In this brief review, in order to avoid some technicalities, we restrict ourselves to the simpler case of bounded

potentials; however, some of the following results hold in the general case of unbounded potentials too (see, e. g. [28]). To this end we consider symbols a(x; y; p) 2 S3n (hpim ) for some positive m, that is the function a defined on R3n depends smoothly on p and for any ˛ D (˛1 ; : : : ; ˛n ) 2 N n one has ˇ ˇ   ˇ ˛ ˇ (17) ˇr p a(x; y; p)ˇ D O hpim uniformly. That is ˇ ˇ  m/2 ˇ ˇ ˛ ˇr p a(x; y; p)ˇ  Chpim D C 1 C jpj for some positive constant C independent of x, y and p, where r p˛ D (@˛p 11 ; : : : ; @˛p nn ) and jpj D jp1 j C : : : C jp n j. We associate to a symbol a 2 S3n (hpim ) the semiclassical pseudodifferential operator of degree m defined for u 2 C01 (Rn ) as the Fourier integral operator  1 Op„ (a)u (x) D [2„]n Z  e i(xy)p/„ a(x; y; p)u(y)dydp : R n R n

We notice that it is formally self-adjoint on the Hilbert space L2 (Rn ) when the symbol a is such that a(x; y; p) D a(y; x; p). The above pseudodifferential operator can be extended in a unique way to a linear continuous operator on the space of smooth rapidly decreasing functions S(Rn ) and its dual space of tempered distributions S0 (Rn ). Furthermore, when the symbol a 2 S3n (1) then the associate pseudodifferential operator is continuous on L2 (Rn ) (this result is the so-called Calderón–Villancourt theorem). It is easy to see that this class of pseudodifferential operators contains the usual differential operators. For instance, in the particular case where the symbol a has the form X a(x; y; p) D b˛ (x)p˛ j˛jm

then its associated operator is the differential operator given by "  X  #  ˛ Op„ (a)u (x) D Op„ b˛ (x)p u (x) D

X

j˛jm

b˛ (x) (i„rx )˛ :

j˛jm

The class of pseudodifferential operator is closed with respect to the composition. That is: given two symbols

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0

a 2 S3n (hpim ) and b 2 S3n (hpim ) then the composition of the associated pseudodifferential operators is still a pseudodifferential operator: Op„ (a) ı Op„ (b) D Op„ (c)   0 where the symbol c 2 S3n hpimCm depends on ¯ and it is given by the Moyal product 1 c(x; y; p) D (a]b) (x; y; p) D [2„]n Z  e i(xz)(p)/„ a(x; z; )b(z; y; p)dz d R n R n



X „j˛j  ˇ rz˛ r˛ a(x; z; )b(z; y; p) ˇzDx;Dp j˛j i ˛!

j˛j0

as ¯ goes to zero; we should underline that the above asymptotic formula becomes an exact formula when the symbol a is polynomial with respect to p since the sum becomes finite. As a result it follows that for any elliptic symbol a 2 S3n hpim , that is such that ja(x; y; p)j  Chpim for some positive constant C, then its associate pseudodifferential operator is invertible in the sense that there exists  a symbol b 2 S3n hpim , depending on ¯, where Op„ (a) ı Op„ (b) D 1 C Op„ (r)

We emphasize that the Weyl quantization is particularly important in quantum mechanics because when the classical observable a is a real valued function then the associate pseudodifferential operator Op„W (a) is formally selfadjoint on L2 (Rn ). We close this review by recalling the following important result which connects the commutator [A; B] D

AB  BA of the pseudodifferential operators A D Op„ (a)

and B D Op„ (b), where a, b are two classical observables, and the Poisson bracket of a and b: let c be the unique sym

bol such that Op„ (c) D [A; B], then c D i„ fa; bg C O(„2 ) : Egorov Theorem Now, we are ready to compare the classical evolution of a classical observable with the quantum evolution of its quantum counterpart. With more details let h(x; p) be a classical Hamiltonian where x 2 Rn denotes the spatial variable and p 2 Rn denotes the momentum. Let  t : R2n ! R2n be the (classical) Hamiltonian flux. Thus, for any classical observable function b D b(x; p) 2 C 1 (R2n ; R) let   b t (x; p) D b ı  t (x; p) D b[ t (x; p)]

and Op„ (b) ı Op„ (a) D 1 C Op„ (s) ; P with r; s D O(„1 ) in S3n (1). The symbol b j „ j b j is iteratively defined where b0 D a1 . Now, we are ready to define the quantization of a classical observable. Classical observables are functions a(x; p) of the position x 2 Rn and of the momenta p 2 Rn , if we assume that a is a smooth function such that ˇ ˇ ˇ ˇ ˛ ˇr p a(x; p)ˇ  Chpim then, for any 2 [0; 1], it follows that    a (x; y; z) D a (1  )x C y; p 2 S3n hpim : We define the quantization of the observable a as: Op „ (a) D Op„ (a ) where the values D 0; 12 ; 1 will play a special role; in particular, for D 0 we have the standard (also called left) quantization, for D 1 we have the right quantization and for D 12 we have the Weyl quantization 1/2 Op„W (a) :D Op1/2 „ (a) D Op„ (a ) :

be the classical evolution of this observable. It is well known that we can associate to any real-valued classical observable b(x; p) a symmetric ¯-pseudodifferential linear operator denoted by Op„W (b) by means of the semiclassical (Weyl) quantization rule formally defined as  W 1 Op„ (b)u (x) :D [2„]n   Z xCy i(xy)p/„  e b ; p u(y)dydp 2 R n R n

P where (x  y)  p D niD1 (x i  y i )p i . In order to properly define this integral operator on the Hilbert space L2 (Rn ) we require that estimate (17) holds; actually, to the present purposes it is sufficient to assume the following weaker condition on the observable b: ˇ ˇ  m/2 ˇ ˇ ˛ (18) ˇrx;p b(x; p)ˇ  C 1 C jxj2 C jpj2 for some m  0 and any ˛ 2 N 2n . For instance, if h0 is the Hamiltonian associated to a harmonic oscillator, then it is a quadratic function with

Perturbation Theory, Semiclassical

is a bounded operator such that kR(t)k D O(„) in the sense that the norm of the operator R(t) is bounded

respect to both position and momentum variables h0 (x; p) D

n h X

p2j C ! 2j x 2j

i

(19)

jD1

and the associated Hamiltonian operator (let n D 1 for the sake of simplicity) takes the usual form  [H0 u] (x) D Op„W (h0 )u (x) " 2 #  Z 1 i(xy)p/„ 2 2 xCy e p C! D 2„ 2 RR

 u(y)dydp d2 D „2 2 C ! 2 x 2 dx

kR(t)k  C„ for some C > 0 and for any t  0. Furthermore, it is possible to extend this asymptotic result to any order N  1 for a suitable choice of p n;t , that is the classical and quantum evolution coincides up to a term of order „ NC1 in the semiclassical limit:   e i tH/„ Op„W (b) ei tH/„  Op„W b ı  t 

N X

  „ n Op„W p n;t D O(„ NC1 )

nD1

R

by integrating by parts twice and since 1/(2„) R e i(xy)p/„ dp D ı(x  y). In such a way it is possible to associate a classical observable b to a quantum operator B. If we denote by B D Op„W (b) and H D Op„W (h) the operators associated to the classical observable b and to the Hamiltonian h then the time quantum evolution B t D e i tH/„ Bei tH/„ of the observable B solves the Heisenberg equation dB t i  D H; B t dt „ and it is, in some sense, related with the classical evolution bt as we are going to explain. In the very particular case where the Hamiltonian h0 is given by the harmonic oscillator (19) then we have that the quantum evolution Bt of the observable B is a ¯-pseudodifferential operator with symbol bt given by means of the classical flux Hamiltonian; that is   e i tH 0 /„ Op„W (b) ei tH 0 /„ D Op„W b ı  t where  t is the Hamiltonian flux generated by the Hamiltonian (19). In other words, for the harmonic oscillator Hamiltonian then quantum evolution and Weyl quantization commute. This property is specific for this problem and it comes from the fact that the flux Hamiltonian is a linear function with respect to (x; p). This fact is not true in a general case. However, it is possible to see that a generalization of such a result holds when H 0 is replaced by any symmetric ¯-pseudodifferential operator H; this result is the so-called Egorov theorem (see [8] for the original statement, see also [29] for a detailed review). In particular, under some assumptions, it is possible to prove that the zeroth order remainder term   R(t) :D e i tH/„ Op„W (b) ei tH/„  Op„W b ı  t

in the norm sense. Future Directions Semiclassical methods are a field of research where theoretical results are rapidly evolving. Just to name some active research topics: ¯-pseudodifferential operators, Weyl functional calculus, frequency sets, semiclassical localization of eigenfunctions, semiclassical resonant states, Born–Oppenheimer approximation, stability of matter and Scott conjecture, semiclassical Lieb–Thirring inequality, Peierls substitution rule, etc. Furthermore, semiclassical methods have been also successfully applied in different contests such as superfluidity and statistical mechanics. Looking forward, we see new emerging research fields in the area of semiclassical methods: numerical WKB interpolation techniques and semiclassical nonlinear Schrödinger equations. Indeed, the recent researches in the area of nanosciences and nanotechnologies have opened up new fields where models for semiconductor devices of increasingly small size and electric charge transport along nanotubes cannot be fully understood without considering their quantum nature. Since the oscillating behavior of the solutions of the Schrödinger equation induces serious difficulties for standard numerical simulations, then new numerical approaches based on WKB interpolation are required [1,4,26]. Although the nonlinear Schrödinger equation has been an argument of theoretical research since the 1970s, only in the last few years, with the successful experiments on Bose–Einstein condensate states, has an increasing interest been shown. When we add a nonlinear term to the time-dependent Schrödinger equation (1) then the dynamics of the model drastically changes and new pecu-

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liar features, such as the blow-up effect and stability of stationary states, appear. Semiclassical arguments applied to nonlinear Schrödinger equations justify their reduction to finite dimensional dynamical systems and thus it is possible to obtain an approximate solution, at least for nonlinear time-dependent Schrödinger equations in small dimensional spaces, typically for n D 1 and n D 2. The extension of this technique to the case n > 2 and the validity of such an approximation for large times is still an open problem [14,27,30].

15.

16. 17.

18. 19.

Bibliography 1. Ben Abdallah N, Pinaud O (2006) Multiscale simulation of transport in an open quantum system: Resonances ans WKB interpolation. J Comp Phys 213:288–310 2. Berezin FA, Shubin MA (1991) The Schrödinger equation. Kluwer, Dordrecht 3. Berry MV, Mount KE (1972) Semiclassical approximation in wave mechanics. Rep Prog Phys 35:315–397 4. Bonnaillie-Noël V, Nier F, Patel Y (2006) Computing the steady states for an asymptotic model of quantum transport in resonant heterostructures. J Comp Phys 219:644–670 5. Claviere P, Jona Lasinio G (1986) Instability of tunneling and the concept of molecular structure in quantum mechanics: The case of pyramidal molecules and the enantiomer problem. Phys Rev A 33:2245–2253 6. Dimassi M, Sjöstrand J (1999) Spectral asymptotics in the semiclassical limit. In: London Math Soc Lecture Note Series 268. Cambridge University Press, Cambridge 7. Dingle RB (1973) Asymptotic expansion: Their derivation and interpretation. Academic, London 8. Egorov YV (1971) Canonical transformation of pseudodifferential operators. Trans Moscow Math Soc 24:1–24 9. Folland G (1988) Harmonic analysis in phase space. Princeton University Press, Princeton 10. Fröman N, Fröman PO (1965) JWKB approximation. North Holland, Amsterdam 11. Fröman N, Fröman PO (2002) Physical problems solved by the phase integral methods. Cambridge University Press, Cambridge 12. Graffi S, Grecchi V, Jona-Lasinio G (1984) Tunneling instability via perturbation theory. J Phys A: Math Gen 17:2935–2944 13. Grecchi V, Martinez A, Sacchetti A (1996) Splitting instability: The unstable double wells. J Phys A: Math Gen 29:4561–4587 14. Grecchi V, Martinez A, Sacchetti A (2002) Destruction of the

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beating effect for a non-linear Schrödinger equation. Comm Math Phys 227:191–209 Grigis B, Sjöstrand J (1994) Microlocal analysis for differential operators. An introduction. In: London Math. Soc. Lecture Note Series 196. Cambridge University Press, Cambridge Harrell EM (1980) Double wells. Commun Math Phys 75:239–261 Helffer B (1988) Semi-classical analysis for the Schrödinger operator and applications. Lecture Notes in Mathematics 1336. Springer, Berlin Helffer B, Sjöstrand J (1984) Multiple wells in the semiclassical limit I. Comm Part Diff Eq 9:337–408 Helffer B, Sjöstrand J (1986) Resonances en limite semi-classique. Mém Soc Math France (N.S.) 24–25 Hislop P, Sigal IM (1996) Introduction to spectral theory. In: Appl Math Sci, vol. 113. Springer, New York Landau LD, Lifshitz EM (1959) Quantum mechanics. Course of theoretical physics. Pergamon, Oxford Martinez A (2002) An introduction to semiclassical and microlocal analysis. Springer, New York McHugh JAM (1971) An historical survey of ordinary linear differential equations with a large parameter and turning points. Arch Hist Exact Sci 7:277–324 Merzbacher E (1970) Quantum mechanics, 2nd edn. Wiley, New York Olver FWJ (1974) Asymptotics and Special Functions. Academic, New York Presilla C, Sjöstrand J (1996) Transport properties in resonant tunneling heterostructures. J Math Phys 37:4816–4844 Raghavan S, Smerzi A, Fantoni S, Shenoy SR (1999) Coherent oscillations betwene two weakly coupled Bose–Einstein condensates: Josephson effects,  oscillations, and macroscopic quantum self-trapping. Phys Rev A 59:620–633 Robert D (1987) Autour de l’Approximation Semiclassique. Birkhäuser, Basel Robert D (1988) Semi-classical approximation in quantum mechanics. A survey of old and recent mathematical results. Helv Phys Acta 71:44–116 Sacchetti A (2005) Nonlinear double well Schrödinger equations in the semiclassical limit. J Stat Phys 119:1347–1382 Simon B (1983) Semiclassical limit of low lying Eigenvalues I: Non degenerate minima. Ann H Poincaré 38:295–307 Simon B (1985) Semiclassical limit of low lying Eigenvalues IV: The flea of the elephant. J Funct Anal 63:123–136 Voros A (1982) Spectre de l’Équation de Schrödinger et Méthode BKW. Publications Mathmatiques d’Orsay 81.09 Wilkinson M, Hannay JH (1987) Multidimensional tunneling between excited states. Phys D: Nonlin Phenom 27:201–212

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Perturbative Expansions, Convergence of SEBASTIAN W ALCHER Lehrstuhl A für Mathematik, RWTH Aachen, Aachen, Germany Article Outline Glossary Definition of the Subject Introduction Poincaré–Dulac Normal Forms Convergence and Convergence Problems Lie Algebra Arguments NFIM and Sets of Analyticity Hamiltonian Systems Future Directions Bibliography Glossary Resonant eigenvalues Let B be a linear endomorphism of C n , with eigenvalues 1 ; : : : ; n (counted according to multiplicity). One calls these eigenvalues resonant P d j  2, and some if there are integers d j  0, k 2 f1; : : : ; ng such that d1 1 C    C d n n   k D 0 : If B is represented by a matrix in Jordan canonical then the associated vector monomial x1d 1 : : : x nd n e k will be called a resonant monomial. (Here e1 ; : : : ; e n denote the standard basis, and the xi are the corresponding coordinates.) Poincaré–Dulac normal form Let f D B C : : : be a formal or analytic vector field about 0, and let Bs be the semisimple part of B. Then one says that f is in Poincaré–Dulac normal form (PDNF) if [Bs ; f ] D 0. An equivalent characterization, if B is in Jordan form, is to say that only resonant monomials occur in the series expansion. Normalizing transformations and convergence A relatively straightforward argument shows that any formal vector field f D B C : : : can be transformed to a formal vector field in PDNF via a formal power series transformation. But for analytic vector fields, the existence of a convergent transformation is not assured. There are two obstacles to convergence: First, the possible existence of small denominators (roughly, this means that the eigenvalues satisfy “near-resonance conditions”); and second, “algebraic” obstructions due to the particular form of the normalized vector field.

Lie algebras of vector fields The vector space of analytic vector fields on an open subset U of C n , with the bracket [p; q] defined by [p; q](x) :D Dq(x) p(x)  Dp(x) q(x) becomes a Lie algebra, as is well known. Mutatis mutandis, this also holds for local analytic and for formal vector fields. As noted previously, PDNF is most naturally defined via this Lie bracket. Moreover, the “algebraic” obstructions to convergence are most appropriately discussed within the Lie algebra framework. Normal form on invariant manifolds While there may not exist a convergent transformation to PDNF for a given vector field f , one may have a convergent transformation to a “partially normalized” vector field, which admits a certain invariant manifold and is in PDNF when restricted to this manifold. This observation is of some practical importance. Definition of the Subject It seems appropriate to first clarify what types of perturbative expansions are to be considered here. There exist various types of such expansions in various settings (a very readable introduction is given in Verhulst’s monograph [41]), but for many of these settings the question of convergence is not appropriate or irrelevant. Therefore we restrict attention to the scenario outlined, for instance, in the introductory chapter of the monograph [13] by Cicogna and Gaeta, which means consideration of normal forms and normalizing transformation for local analytic vector fields. Normal forms are among the most important tools for the local analysis and classification of vector fields and maps near a stationary point. (See  Normal Forms in Perturbation Theory.) Convergence problems arise here, and they turn out to be surprisingly complex. While the first contributions date back more than a century, some very strong and very deep results are just a few years old, and this remains an active area of research. Clearly convergence questions are relevant for the analytic classification of local vector fields, but they are also of practical relevance in applications, e. g. for stability questions and for the existence of particular types of solutions. Introduction The theory of normal forms was initiated by Poincaré [35], and later extended by Dulac [17], and by Birkhoff [5] to Hamiltonian vector fields. There exist various types of normal forms, depending on the specific problem one

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wants to address. Bruno (see [6,7]) in the 1960s and 1970s performed a comprehensive and deep investigation of Poincaré–Dulac normal forms, which are defined with respect to the semisimple part of the linearization. Such normal forms are very important in applications, and moreover they have certain built-in symmetries, which allows a well-defined reduction procedure. We will first give a quick review of normalization procedures and normal forms, and then discuss convergence problems (which mostly refers to convergence or divergence of normalizing transformations). We will present fundamental convergence and divergence results due to Poincaré, Siegel, Bruno and others. We then proceed to discuss the relevance of certain Lie algebras of analytic vector fields for these matters, including results by Cicogna and the author of this article on the influence of symmetries, and the far-reaching generalization of Bruno’s theorems (among others) due to Stolovitch. Variants of normal forms which guarantee convergence on certain subsets (due to Bibikov and Bruno) are then discussed, and applications are mentioned. Finally, the Hamiltonian setting, which deserves a discussion in its own right, is presented, starting with results of Ito and recently culminating in Zung’s convergence theorem. For Hamiltonian systems there is also work due to Perez–Marco on divergence of normal forms. Within the space limitations of this contribution, and in view of some very intricate and space-consuming technical questions and conditions, the author tried to find an approach that, for some problems, should provide some insight into a result, the arguments in its proof or its relevance, without exhibiting all the technicalities, or without giving the most general statement. The author hopes to have been somewhat successful in this, and apologizes to the creators of the original theorems for presenting just “light” versions. Poincaré–Dulac Normal Forms We will start with a coordinate-free approach to normal form theory. Our objects are local ordinary differential equations (over K D R or C) x˙ D F(x) ;

F(0) D 0

with F analytic, thus we have a convergent series expansion F(x) D Bx C

X

f j (x) D Bx C f2 (x) C f3 (x) C   

j2

near 0 2 Kn :

Here B D DF(0) is linear, and each f j is homogeneous of degree j. Our objective is to simplify the Taylor expansion of F. For this purpose, take an analytic “near-identity” map H(x) D x C h2 (x) C    Since H is locally invertible, there is a unique F  (x) D Bx C

X

f j (x)

j2

such that the identity DH(x)F  (x) D F(H(x))

(R)

holds. H “preserves solutions” in the sense that parametrized solutions of x˙ D F  (x) are mapped to parametrized solutions of x˙ D F(x) by H. It is convenient to introduce the following abbreviation: H

F  ! F

if (R) holds :

Given the expansion F(x) D Bx C f2 (x) C    C f r1 (x) C f r (x) C    ; assume that f2 ; : : : ; f r1 are already deemed “satisfactory” (according to some specified criterion). Then the ansatz H(x) D x C hr (x) C : : : yields F  (x) D Bx C f2 (x) C    C f r1 (x) C f r (x) C    (with terms of degree < r unchanged), and at degree r one obtains the so-called homological equation: [B; hr ] D f r  f r (Here [p; q](x) D Dq(x)p(x)  Dp(x)q(x) denotes the usual Lie bracket of vector fields). The space Pr of homogeneous vector polynomials of degree r is finite dimensional, and adB D [B; ] sends Pr to Pr . Thus the homological equation poses a linear algebra problem on a finite dimensional vector space: Given B and f r , determine f r so that the equation can be solved and let hr be a solution. How can f r be chosen? Let W be any subspace of Pr such that image (adB) C W D Pr : Then one may choose f r 2 W . If the sum is direct then f r 2 W is uniquely determined by f r . Generally, the type of normal form is specified by the choice of a subspace Wr for each degree r such that

Perturbative Expansions, Convergence of

image (adB) C Wr D Pr . The Poincaré–Dulac choice is as follows: Given the decomposition B D Bs C Bn into semisimple and nilpotent part, then ad B D adBs C adBn is known to be the corresponding decomposition on Pr . Choose Wr D Ker(ad Bs ). By linear algebra Wr C image(ad B) D Pr ;

and the sum is direct if B is semisimple. In any case we have [Bs ; f r ] D 0. Definition 1 The vector field F  is in Poincaré–Dulac normal form if [Bs ; f j ] D 0 for all j; equivalently if [Bs ; F  ] D 0. If one sets aside convergence questions for the moment, an immediate consequence of the considerations above is: Proposition 1 For any F D B C    there are formal power series H(x) D x C    , F  (x) D Bx C : : : such H that F  ! F and F  is in Poincaré–Dulac normal form. One should note that the homological equation is of relevance beyond the setting of Poincaré–Dulac, and forms the foundation for various other (or more refined) types of normal form. See also the entry in this section of the Encyclopedia by T. Gramchev on normal forms with respect to a nilpotent linear part. Now let us turn to the standard approach via suitable coordinates. Given F be as above, let 1 ; : : : ; n be the eigenvalues of B. Complexify, if necessary, and assume that B is in Jordan canonical form with respect to the given coordinates x1 ; : : : ; x n (with corresponding basis e1 ; : : : ; e n of Kn ). In particular, Bs D diag (1 ; : : : ; n ) : In the following we will use xi and ei to denote eigencoordinates, resp. eigenbasis elements, and reserve 1 ; : : : ; n for the eigenvalues of B, without explicitly saying so in every instance. Lemma 1 The “vector monomial” p(x) D x1m1 : : : x nm n e j satisfies [Bs ; p] D (m1 1 C    C m n n   j )  p Thus, the vector monomials form an eigenbasis of ad Bs on the space Pr , with eigenvalues m1 1 C    C m n n   j (mj P nonnegative integers, m j D r). The eigenvalues play a crucial role both for the classification of formal normal forms and for convergence issues. The following distinction is pertinent here:

Definition 2 One calls (1 ; : : : ; n ) resonant if there are P integers d j  0, d j  2, and some k 2 f1; : : : ; ng such that d1 1 C    C d n n   k D 0 : In this case, the vector monomial x1d 1 : : : x nd n e k is also called a resonant monomial. One calls (1 ; : : : ; n ) nonresonant otherwise. Given eigencoordinates, one can characterize a Poincaré– Dulac normal form by the property that only resonant monomials occur in the series expansion. Moreover, evaluation of the homological equation shows that the nonzero eigenvalues of ad B will occur as denominators in its solution. To iterate the normalization, one may proceed degree by degree with a series of transformations of the form exp(hr ), using the solution of the homological equation; see [44]. One may also choose a different iterative approach to compute a normalizing transformation, such as the important“distinguished transformation” of Bruno [6]. In any case this will yield coefficients whose denominators are products of nonzero terms m1 1 C    C m n n   j . This is the source of convergence problems caused by small denominators: H There are formal series H and F  such that F  ! F  and F is in Poincaré–Dulac normal form, but does there exist a convergent H? (Here “convergent” means: convergent in some neighborhood of 0.) To answer this question, it is necessary to specify a particular type of transformation: Transformations to Poincaré–Dulac normal form are not necessarily unique, even if one stipulates a near-identity transformation H(x) D x C    . Non-uniqueness occurs whenever the eigenvalues of B are resonant, because then some homological equation will not have a unique solution: In eigencoordinates of Bs , the series of the transformation is fixed only up to resonant monomial terms. In particular, if F itself is in normal form then any formal power series H(x) D x C    that contains only resonant monomials will provide a normal form F  , and a suitable nontrivial choice of H will even force F  D F. Thus there may be divergent transformations sending an equation in normal form to itself. Convergence and Convergence Problems Let us note at the start that for given analytic F there may not exist any convergent transformation to normal form; thus the convergence question has no simple answer. An early positive convergence result is due to Poincaré [35], with a later improvement due to Dulac [17]:

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Perturbative Expansions, Convergence of

Theorem 1 (Poincaré–Dulac) If 1 ; : : : ; n lie in an open half-plane in C which does not contain 0 then there exists a convergent transformation. For example, one may think of the open left half-plane. The proof is relatively straightforward, employing natural P majorants. (Due to the hypothesis, the j niD1 m i  i   j j P are unbounded for m i ! 1.) Poincaré’s condition does not preclude the existence of resonant monomials but it ensures that there are at most finitely many of these. One main technical difficulty in proving convergence was to replace the direct majorant arguments by more efficient, and more sophisticated, tools, so that small denominator problems could be tackled. The following result, due to C.L. Siegel [38], may be seen as the start of the “modern phase” for convergence results. Characteristically, this result goes back to a mathematician who also worked in analytic number theory. Siegel assumed that the eigenvalues satisfy a certain arithmetic condition. Condition S: The eigenvalues are pairwise different and there are constants C > 0,  > 0 such that for all nonnegaP tive integer tuples (m i ), m i > 1, the following inequality holds: ˇX ˇ ˇ n ˇ  ˇ ˇ m    i i j ˇ  C  (m1 C    C m n ) ˇ iD1

Theorem 2 (Siegel) If Condition S holds then there is a convergent transformation to normal form. Proof A very rough sketch of the proof is as follows. One works in eigencoordinates. By scaling one may assume that all coefficients in the expansion of F are absolutely bounded by some constant M  1. For the transformation one writes  X X ˛m1 ;:::;m n ;k  x1m1    x nm n e k H(x) D x C r2

where the sum inside the bracket extends over all nonnegP ative integer tuples with m i D r and 1  k  n. (Since Condition S precludes resonances, the coefficients of the series are uniquely determined.) Now set A m1 ;:::;m n :D

X

j˛m1 ;:::;m n ;k j :

where the summation on the right hand side extends over all tuples (d i;1 ; : : : ; d i;n ) that add up to (m1 ; : : : ; m n ). From this one may obtain an estimate for A m1 ;:::;m n , and invoking Condition S one eventually arrives at the conP clusion that A m1 ;:::;m n x1m1 : : : x nm n is majorized by the series of x1 C    C x n ; 1  K  (x1 C    C x n )

some K > 0 : 

Example Let x˙ D Bx C    be given in dimension two, and assume that the eigenvalues 1 , 2 of B are nonresonant, and are algebraic irrational numbers. (This is the case when the entries of B are rational but the characteristic polynomial is irreducible over the rationals.) Then 2 /1 is algebraic but not rational, and (1 ; 2 ) satisfies Condition S, due to a celebrated number-theoretic result of Thue, Siegel and Roth. Thus there exists a convergent transformation to normal form. While Siegel’s convergence proof uses majorizing series, the approach is not as straightforward as in the Poincaré setting. Siegel’s result is strong in the sense that Condition S is satisfied by Lebesgue – almost all tuples (1 ; : : : ; n ) 2 C n . But the condition forces the normal form to be uninteresting: One necessarily has F  D B D Bs . In the same paper [38], Siegel also notes that divergence is possible, even for a set of “eigenvalue vectors” that is everywhere dense in n-space. In the resonant case there is a second source of obstacles to convergence. An early example for this is due to Horn (about 1890); see [6]: Example The system x˙1 D x12 x˙2 D x2  x1 (with eigenvalues (0; 1) for the linear part) admits no convergent transformation to normal form. A detailed proof for this can be found in [14]. The underlying reason is that the ansatz for a transformation – unavoidably – leads to the differential equation x 2  y0 D y  x ;

k

From the homological equations one finds by recursion ˇ ˇX ˇ ˇ ˇ  j˛m ;:::;m ;k j  M ˇ m    i i k 1 n ˇ ˇ i



X

A d 1;1 ;:::;d 1;n    A d s;1 ;:::;d s;n

(which goes back to Euler) with divergent solution P k k1 (k  1)! x . There are no small denominators here. The problem actually lies within the normal form, which can be computed as y˙1 D y12 y˙2 D y2 :

Perturbative Expansions, Convergence of

The single nonlinear term is sufficient to obstruct convergence. Pliss [34] showed that Siegel’s theorem still holds if there are no such nonlinear obstructions in the normal form: Theorem 3 (Pliss) Assume that: P (i) The nonzero elements among the niD1 m i  i   j satisfy Condition S. (ii) Some formal normal form of F is equal to B D Bs . Then there exists a convergent transformation to normal form. While it seemingly extends Siegel’s result only to a rather narrow special setting, Pliss’ theorem proved to be quite important for future developments. Pliss uses a different approach to proving convergence, via a generalized Newton method. Fundamental insights into normal forms, and in particular into convergence and divergence problems were achieved by Bruno, starting in the 1960s; see [6,7]. His results included or surpassed much of the earlier work. Let us take a closer look at Bruno’s conditions. As above, let B be in Jordan form, with eigenvalues 1 ; : : : ; n . For k  1 set ˇ nˇX ˇ ˇ ! k :D min ˇ m i  i   j ˇ 6D 0 : o X mi < 2k 1  j  n; m i 2 ZC ; Bruno introduced two arithmetic conditions: Condition !: 

1 X ln ! k <1 2k kD1

Condition !: lim sup  k

ln ! k <1 2k

Condition ! can be shown to imply Condition S. Clearly Condition ! implies Condition !. Possibly in view of Pliss’ theorem, Bruno introduced the following algebraic condition on the normal form: Condition A: Some formal normal form is of the type F  D   Bs with  a scalar formal power series. Pliss’ condition corresponds to Condition A with  D 1. Moreover, one can show that Condition A (as well as Pliss’ condition) is satisfied by every (formal) normal form if it is satisfied by one.

Now let us turn to Bruno’s main theorems. The convergence theorem here may be expected in view of the previous results, but the divergence theorem – as well as its proof – conquers new ground. (We do not state the most general version of the divergence theorem.) Theorem 4 (Bruno’s convergence theorem) If Condition ! and Condition A are satisfied then a convergent normal form transformation exists. Theorem 5 (Bruno’s divergence theorem) Assume that 1 ; : : : ; n do not lie in a complex half-plane with 0 in its boundary (in particular they do not satisfy the Poincaré condition). Moreover assume that an analytic vector field F  in normal form does not satisfy a weaker version of Condition A, or that Condition ! is not satisfied. Then there exists an analytic F with normal form F  such that no transformation of F to (any) normal form converges. As for the “weaker version of Condition A”, see Remark (c) below. Bruno also discusses the scenario when all eigenvalues are contained in some complex half-plane with 0 in its boundary. Example There exists an analytic vector field 0 p

2 F(x) D @ 0 0

0 1 0

1 0 1 A C 1

which admits no convergent transformation to normal form (which is just the linear part). The arithmetic conditions on the eigenvalues are satisfied, but the nontrivial Jordan block violates Condition A. For the related problem of normalizing local analytic diffeomorphisms (rather than vector fields), see e. g. DeLatte and Gramchev [16], and Gramchev [21], for similar divergence results. Remark (a) Bruno’s divergence theorem cannot be applied directly to prove divergence of all normalizing transformations for a specific given vector field. (Therefore any particular example still has to be worked “by hand”.) Rather, the divergence theorem provides a generic result. Since the arithmetic conditions (both ! and !) are very weak, one sees that in absence of the Poincaré condition, the algebraic obstructions from the formal normal form are mostly responsible for divergence. Remark (b) Bruno’s divergence theorem starts from the assumption that a convergent normal form exists for a given vector field; thus he deals with convergence or divergence of normalizing transformations. Little seems to be known about analytic vector fields that admit only divergent normal forms.

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Remark (c) The version of Condition A stated here is taken from Bruno’s monograph (see Chapter III, § in [7]). The original paper (p. 140 ff. in [6]). contains somewhat different versions. In the interesting case when the hypothesis of Theorem 5 holds, Bruno’s original requirement on the normal form in eigencoordinates is as follows:   x˙ j D x j  j  C  j  ;

1 jn;

with scalar series  and . This condition seems to have been stated with special regard to real systems, because otherwise complex conjugation plays no distinguished role. To illustrate this point, consider the complexification of a real system, and multiply this through by some scalar exp(i ) which is neither real nor purely imaginary. Convergence or divergence of normalizing transformations is not affected by this, but the shape of the condition above changes considerably. (The appropriate setting for this seems to be the one developed by Stolovitch [39]; see below.) In later publications, notably in [7], Bruno himself mostly used Condition A in the simple form we stated above. Lie Algebra Arguments Bruno’s theorems set the standard against which later results are measured. However, they do not directly address this basic question: Given a particular local analytic vector field F, characterize properties that are necessary (and, ideally, also sufficient) for the existence of a convergent normal form transformation. (Obviously, Condition A is not generally necessary for the existence of a convergent normalizing transformation.) To answer this basic question and related problems, it turns out helpful to consider Lie algebras of analytic vector fields. A possible approach is based on the earlier observation that normal forms admit symmetries. (See also the entry on Symmetry and Perturbation Theory in this section.) The coordinate-free approach proves to be quite suitable here. First, let us formalize the observation: Lemma 2 If F admits a convergent normalizing transformation then there exists a nontrivial G (i. e., G 62 KF) such that [G; F] D 0. Proof There exist a convergent  and a convergent F  in normal form such that 

F  ! F Now  sends Bs to some analytic G D Bs C : : : , and [Bs ; F  ] D 0 implies [G; F] D 0. If F  ¤ Bs , we are done.

If F  D Bs then take some linear map C 62 K  Bs that 

commutes with Bs , and define G by C ! G.



In other words, if there is a convergent normalizing transformation then there exists a nontrivial infinitesimal symmetry at the stationary point. One can try to turn this necessary condition around and thus obtain sufficient convergence criteria. This has been done, with some success, since the early 1990s. Among the relevant contributions are those by Markhashov [29], Bruno and Walcher [9], Cicogna [11,12], Bambusi et al. [3], and, from a different starting point, Stolovitch [39] (see below and see also Gramchev and Yoshino [24] for maps). The survey paper [14] by Cicogna and Walcher collects the development up to 2001. Here we will present a few results to give the reader an impression of the arguments employed. The objects to deal with are Cfor (F) and Can (F), i. e., the formal, respectively analytic, centralizer of F, which by definition consist of all formal, respectively analytic, vector fields H such that [H; F] D 0. The first result is due to Cicogna [11,12] and Walcher [45]. Note that Pliss’ theorem (and thus Condition A) plays a crucial role in the proof. Theorem 6 Given the analytic vector field F with formal normal form b F, assume that dim Cfor (b F) D k < 1 : If the eigenvalues of B satisfy Condition ! and dim Can (F) D k then there exists a convergent transformation to normal form. Proof We have dim Cfor (F) D dim Cfor (b F), so dim Can (F) D k implies Can (F) D Cfor (F). Given a formal trans formation  with b F ! F, there exists an analytic H such

 that B ! H and [H; F] D 0, since B 2 Cfor (b F). Note that H D BC   , so B is a normal form of H. Due to Pliss’ theo˚ ˚ F ! F rem, there is a convergent ˚ with B ! H. Now e for some e F, and [B; e F] D 0, so e F is in normal form. 

The dimension of the formal centralizer is computable in many cases. The requirement on Condition ! can be relaxed; see [11,45] and [14]. The next result is due to Markhashov [29] for the non-resonant case, and to Bruno and Walcher [9] in the resonant case. One may base a proof on the observation that dim Cfor (F) is infinite only if Condition A holds, and use Theorem 6 otherwise. Theorem 7 In dimension n D 2, there is a convergent transformation of F to normal form if and only if F admits a nontrivial commuting vector field in 0.

Perturbative Expansions, Convergence of

In dimension two, there are other, very precise, characterizations of resonant vector fields (for 2 /1 a negative rational number) admitting a convergent normalization, which can be drawn from the work of Martinet and Ramis [30]. Beyond this, building on work of Ecalle [18,19] and Voronin [42], Martinet and Ramis succeeded in giving an analytical classification of germs of such vector fields, and those admitting a convergent normalizing transformation can be characterized by the vanishing of infinitely many analytical invariants. In this sense, the convergence problem was settled earlier, at least for the interesting cases. But Theorem 7 approaches the question from a different perspective, gives an algebraic characterization and provides structural insight that is not directly available from [30]. Lie algebra arguments also play a fundamental role, from a different perspective, in the work of Stolovitch [39]. We will here give a simplified and incomplete account of his important results; a full presentation would take up much more space. To motivate Stolovitch’s approach, note that in many cases there are natural decompositions B D B1 C    C B` , with all [B i ; B j ] D 0 which come from eigenvalues splitting up into groups of pairwise commensurable ones, with pairwise incommensurability between P the groups. The resonance conditions m j  j  k D 0 then split up into conditions involving only the groups. We illustrate this with a simple example for such a decomposition: diag (1; 1;

p

p 2;  2) D

diag (1; 1; 0; 0) C diag (0; 0;

p

p 2;  2) :

The setting considered by Stolovitch is as follows: Given (complex) analytic vector fields F1 D B1 C    ;

:::;

F` D B` C   

that commute pairwise, thus all [F i ; F j ] D 0, ask about simultaneous analytic normalization of these vector fields. There are sensible notions of normal form of a vector field with respect to a linear Lie algebra, and in particular there is a natural extension of Poincaré–Dulac for abelian Lie algebras of diagonal matrices; see [39], Sect. “Convergence and Convergence Problems”: Each semisimple linear part B i;s commutes with each F j in such a normal form. To avoid trivial scenarios, Stolovitch requires the semisimple parts B i;s to be linearly independent. To formulate diophantine conditions extending Bruno’s Condition !, one may proceed as follows: Let  i;1 ; : : : ;  i;n denote the eigenvalues of Bi . For nonnegative integers m1 ; : : : ; m n ,

and for 1  d  n set

m1 ;:::;m n ;d

ˇ ˇ ˇ X ˇˇ X ˇ :D m j  i; j  d ˇ ˇ ˇ ˇ i

j

and (for instance; Stolovitch gives a more general formulation) n ! k (B1 ; : : : ; B` ) :D inf m1 ;:::;m n ;d 6D 0 : o X 1  d  n; 2  m j  2k : This leads to an appropriate diophantine condition which extends Bruno’s Condition !. Condition ! # : 

1 X ! k (B1 ; : : : ; B` ) <1 2k kD1

We remark briefly that there appear to be some issues of well-definedness here. For instance, the choice of the F i , and hence of the Bi , is not unique, and one has to verify that the important notions, like Condition ! # , do not depend on these choices. Stolovitch [39] works in an invariant setting from the start; as a consequence, no such questions arise. Finally, Stolovitch introduces the notion of formal complete integrability. Disregarding technical subtleties (even though these are relevant and of interest), one may informally characterize this property as follows: The system F1 ; : : : ; F` is formally completely integrable if it has as many formal integrals as admissible by the semisimple linear parts B1;s ; : : : ; B`;s . To cast at least some light on this, we note that every formal integral of the system in normal form is also a simultaneous first integral of the B i;s . See Walcher [43] for the case ` D 1 and Stolovitch [39], Sect. “Lie Algebra Arguments” for the general case. This notion of complete integrability is the appropriate extension of Bruno’s Condition A. The following is a simplified representative of the results in [39]. The system B1 ; : : : ; B` is said to have small divisors if 0 is a limit point of the m1 ;:::;m n ;d 6D 0. Theorem 8 (Stolovitch) In the presence of small divisors, if Condition ! # holds then every formally completely integrable system F1 ; : : : ; F` admits a convergent transformation to normal form. The principal value of Stolovitch’s results is that they open up a unified approach to a number of applications. One almost immediate application is a recovery of Bruno’s convergence theorem. (See also Remark (c) following Theorem 5.) We give two more applications. For the first one, compare also Bambusi et al. [3].

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Corollary 1 (Linearization) In the presence of small divisors, assuming that Condition ! # holds, every formally linearizable system F1 ; : : : ; F` admits a convergent linearization. The second application is a short and clear proof of a theorem due to Vey [40]: Corollary 2 (Volume-preserving vector fields) Assume that ` D n  1 and that F1 ; : : : ; Fn1 are commuting volume-preserving vector fields, with diagonal and linearly independent linear parts B i D B i;s . Then the F i are simultaneously analytically normalizable. Remark (a) For similar results see Zung [46]. Zung uses a different approach based on a convergence result (a precursor of [47]) for Hamiltonian vector fields; see also the section on Hamiltonian vector fields below. His argument seems somewhat sketchy. Remark (b) There is a nontrivial overlap of Stolovitch’s results with the approach to convergence via symmetries outlined above: Some results can be proven by either method, as the references indicate. We finish this section with a third aspect of Lie algebra arguments in convergence proofs. The following is taken from Walcher [44]: Theorem 9 Let L be a finite dimensional Lie algebra of polynomial vector fields which is graded in the sense that it contains all homogeneous parts of each of its elements, and let F 2 L and F(0) D 0. Then F admits a convergent transformation to normal form F  , and one may take F  2 L. The basic idea for the proof is to take suitable solutions hr of the homological equations, which can be chosen in L, and transformations exp(hr ); see [44], Prop. 2.7. Due to finite dimension of L, finitely many such transformations suffice. There are several interesting Lie algebras among the finite dimensional graded ones: Example (a) Every projective vector field, as well as every conformal vector field in dimension  3, admits a convergent transformation to Poincaré–Dulac normal form. Example (b) Matrix Riccati equations: Every matrix differential equation of the form x˙ D x ax C bx C xc ; with x, a, b, c matrices of appropriate sizes, admits a convergent transformation to normal form.

NFIM and Sets of Analyticity As we have seen, convergence problems for normalizing transformations are unsurmountable in many instances. But there are sensible strategies to achieve convergence by relaxing the requirements on the normalized vector field. Moreover, such strategies frequently provide interesting information, e. g. on stability or on the existence of periodic solutions. There are two related, but not equivalent, approaches to be discussed here: Bibikov’s normal form on an invariant manifold (NFIM), see [4]; and Bruno’s sets of analyticity, see [7]. We will first discuss Bibikov’s work, including a coordinate-free approach proposed in [44]. Definition 3 Let C be a semisimple linear map. A vector subspace of Kn is called strongly C-stable if it is invariant for every vector field F D B C    , with Bs D C, in normal form. Strongly C-stable spaces are in particular C-stable. A coordinate-dependent characterization is as follows. Lemma 3 Assume that C is in diagonal form, and let 1 < r < n. Then the subspace U :D Ke1 C    C Ke r is strongly C-stable if and only if m1 1 C    C mr r   j 6D 0 for all nonnegative integers mi with fr C 1; : : : ; ng.

P

m i > 0, and all j 2

Here, the choice of indices 1; : : : ; r is just for the sake of convenience. The proof is simple: The condition ensures that no monomial x1m1 : : : x rm r e j ;

j>r

will occur in the normal form, and therefore the subspace U, characterized by x rC1 D    D x n D 0, is invariant for any normal form F D C C Bn C    . Examples include the stable, unstable and center subspaces of C. Now one can introduce the notion of NFIM: Definition 4 Assume that U D Ke1 C    C Ke r is strongly C-stable. A vector field F D B C    with Bs D C is said to be in normal form on the invariant manifold U (NFIM on U) if U is invariant for F and furthermore [Bs jU ; FjU ] D 0 : Example The two-dimensional vector field     2 0 x1  2x1 x2 C 3x13 ; with Bx D ; FD x2 (1 C x1 C x22 ) x2 is in NFIM on U D Ke1 .

Perturbative Expansions, Convergence of

Bibikov also introduced a refined version of NFIM, which he calls quasi-normal form (QNF). The above example is not in quasi-normal form; in a QNF the first entry would contain only functions of x1 . Because normal forms are, in particular, in NFIM on every strongly Bs -stable subspace, it is obvious that formal transformations to NFIM exist. But there is more freedom to construct such transformations, which may be utilized to force convergence. To give a flavor of Bibikov’s results, we present a weakened version of one of his theorems. Theorem 10 (Bibikov) Given a vector field F D B C    , assume that the subspace U :D Ke1 C    C Ke r is strongly Bs -stable, and moreover that: (i) There is an > 0 such that jm1 1 C    C mr r   j j  P for all nonnegative integers mi , m i > 0, and for all j > r. (ii) Some formal normal form b FjU satisfies the Pliss condition on U. Then there exists a convergent transformation to NFIM on U. Remark In Bibikov’s original theorems (see [4], Theorem 3.2 and Theorem 10.2), one finds a more general (quite technical) condition instead of (ii). Thus the range of Bibikov’s theorems is wider than our statement indicates. Condition (i), or some related condition, cannot be discarded completely: There are examples of strongly C-stable subspaces which do not correspond to analytic invariant manifolds for certain vector fields. (This is another incarnation of the small denominator problem.) Applications of Bibikov’s theorems include the existence of analytic stable and unstable manifolds; stability of the stationary point in case 1 D 0 when all other  i have negative real parts, and b FjU =0 on U D Ke1 (“transcendental case”); and the existence of certain periodic solutions. Let us now turn to Bruno’s method of analytic invariant sets; see Bruno Part I, Ch. III, and Part II in [7]. To motivate the approach, one may use the following observation: For an analytic vector field b F D B C : : : in normal form the set ˚

AD z :b F(z) and Bs z linearly dependent in Kn



F] D 0: is invariant for b F. (This is a consequence of [Bs ; b The set of points for which a vector field and an infinitesimal symmetry “point in the same direction” is invariant.) It is clearly possible to write down analytic functions

such that A is their common zero set: For instance, take suitable 2  2-determinants. One may now introduce the notion that a vector field F˜ is normalized on A: In the coordinate version this means that for each entry of the righthand side the sum of all nonresonant parts lies in the defining ideal of A. If F is not in normal form but some formal normal form b F satisfies Condition A then – assuming some mild diophantine conditions – A is a whole neighborhood of the stationary point, according to Bruno’s convergence theorem. Bruno now refines this by investigating whether the generally “formal” set A is analytic (or at least certain subsets are), and what can be said about the solutions on such subsets. To be more precise: Given a not necessarily convergent normal form b F of F, one can still write down formal power series whose “common zero set” defines A, and it makes sense to ask what of this can be salvaged for analyticity. The following is a sample of Bruno’s results (see Part I, Ch. III, Theorem 2 and Theorem 4 in [7]). As above, we do not write down the technical conditions completely. Theorem 11 (Bruno) Let the analytic vector field F D B C    be given. (a) If the eigenvalues 1 ; : : : ; n of B are commensurable (i. e., pairwise linearly dependent over the rationals) then the set A is analytic and there is a convergent transformation to a vector field F˜ that is normalized on A. (b) Generally, there exists a (formal) subset B of A which is analytic, and for B the same conclusion as above holds. For applications see Bruno [7], Part II, and the recent papers by Edneral [20], and Bruno and Edneral [8], on existence of periodic solutions for certain equations. Hamiltonian Systems Hamiltonian systems have a special position among differential equations (see e. g. [2] for an overview, and  Hamiltonian Perturbation Theory (and Transition to Chaos)); in view of their importance it is appropriate to give them particular attention. When discussing normal forms of Hamiltonian systems (where Poincaré–Dulac becomes Birkhoff; see [5]) it is natural to consider canonical transformations only. In view of the correspondence between integrals of F and Hamiltonian vector fields commuting with F, it is furthermore natural to consider integrals in the Hamiltonian setting. As for convergence results, we first state two theorems by H. Ito [26,27], from around 1990:

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Theorem 12 (Ito; non-resonant case) Let F D B C   be Hamiltonian and let (!1 ; !1 ; : : : ; !r ; !r ) be the eigenvalues of B. Moreover assume that !1 ; : : : ; !r are non-resP onant, thus m j ! j D 0 for integers m1 ; : : : ; mr implies m1 D    D mr D 0. If F possesses r independent integrals in involution (i. e. with vanishing Poisson brackets) then there exists a convergent canonical transformation of f to Birkhoff normal form. This condition is also necessary in the non-resonant case: If there is a convergent transformation to analytic normal form b F then there are r linearly independent linear Hamiltonian vector fields that commute with b F, and these, in turn, correspond to r independent quadratic integrals of b F. For the “single resonance” case one has: Theorem 13 (Ito; a simple-resonance case) Let F D B C    be Hamiltonian and let (!1 ; !1 ; : : : ; !r ; !r ) be the eigenvalues of B. Moreover assume that there are nonzero integers n1 ; n2 such that n1 !1 C n2 !2 D 0, but there are no further resonances. If F possesses r independent integrals in involution then there exists a convergent canonical transformation to Birkhoff normal form. Again, the condition is also necessary. Ito’s proofs are quite long and intricate. Kappeler, Kodama and Nemethi [28] proved a generalization of Theorem 13 for more general single-resonance cases. Moreover, they showed that there is a natural obstacle to further generalizations, since there exist non-integrable (polynomial) Hamiltonian systems in normal form. Thus, a complete integrability condition is not generally necessary for convergence. Nguyen Tien Zung [47] recently succeeded in a farreaching generalization of Ito’s theorems. Considering the existence of non-integrable normal forms, this seems the best possible result on integrability and convergence. Theorem 14 (Zung) Any analytically integrable Hamiltonian system near a stationary point admits a convergent transformation to Birkhoff normal form. One remarkable feature of Zung’s proof is its relative shortness, compared with the proofs by Ito. Finally, turning to divergent normal forms (rather than normalizing transformations) of Hamiltonian systems, Perez–Marco [33] recently established a theorem about convergence or generic divergence of the normal form in the non-resonant scenario. Although numerical computations indicate the existence of analytic Hamiltonian vector fields which admit only divergent normal forms, there still seems to be no example known. Perez– Marco showed that if there is one example then divergence is generic.

Future Directions There are various ways to extend the approaches and results presented above, and clearly there are unresolved questions. In the following, some of these will be listed, respectively, recalled. The convergence problem for normal forms and normalizing transformations is part of the much bigger problem of analytic classification of germs of vector fields. Except for dimension two (see the references [18,19,30] mentioned above) little seems to be known in the case of nontrivial formal normal forms. Going beyond analyticity, an interesting extension would be towards Gevrey spaces. Some work on this topic exists already; see e. g. [22]. Passing from vector fields to maps, matters turn out to be much more complicated in the case of nontrivial formal normal forms, and even one-dimensional maps show very rich behavior; see [31,32]. A brief introduction is given in the survey [23] mentioned above. A complete (“algebraic”) understanding of Bruno’s Condition A would be desirable; this could also provide an approach to a non-Hamiltonian version of Zung’s Theorem 14. It seems well possible that all the necessary ingredients for this endeavor are contained in Stolovitch’s work [39]. An extension or refinement of Bibikov’s and Bruno’s results on the existence of certain invariant sets (in the case of non-convergence) would obviously be interesting. There seems to be a natural guideline here: Check what invariant sets are forced onto a system in PDNF and see which ones can be salvaged. (The existence of a commuting vector field, for instance, has more consequences than those exploited by Bruno in the arguments leading up to Theorem 11.) Finally, one could turn to more refined versions of normal forms, such as normal forms with respect to a nilpotent linear part (see [15]), and quite general constructions such as presented by Sanders [36,37]. It seems that little attention has been paid to convergence questions for such types of normal forms. For normal forms with respect to a nilpotent linear part, there are obviously no small denominator problems, but algebraic obstructions abound. There exists a precise algebraic characterization for such normal forms, involving the representation theory of sl(2) (see [15]). Thus there may be some hope for an algebraic characterization of convergently normalizable vector fields. Bibliography 1. Arnold VI (1982) Geometrical methods in the theory of ordinary differential equations. Springer, Berlin

Perturbative Expansions, Convergence of

2. Arnold VI, Kozlov VV, Neishtadt AI (1993) Mathematical aspects of classical and celestial mechanics. In: Arnold VI (ed) Dynamical Systems III, Encyclop Math Sci, vol 3, 2nd edn. Springer, New York 3. Bambusi D, Cicogna G, Gaeta G, Marmo G (1998) Normal forms, symmetry and linearization of dynamical systems. J Phys A 31:5065–5082 4. Bibikov YN (1979) Local theory of nonlinear analytic ordinary differential equations. Lecture Notes in Math 702. Springer, New York 5. Birkhoff GD (1927) Dynamical systems, vol IX. American Mathematical Society, Colloquium Publications, Providence, RI 6. Bruno AD (1971) Analytical form of differential equations. Trans Mosc Math Soc 25:131–288 7. Bruno AD (1989) Local methods in nonlinear differential equations. Springer, New York 8. Bruno AD, Edneral VF (2006) The normal form and the integrability of systems of ordinary differential equations. (Russian) Programmirovanie 3:22–29; translation in Program Comput Software 32(3):139–144 9. Bruno AD, Walcher S (1994) Symmetries and convergence of normalizing transformations. J Math Anal Appl 183:571–576 10. Chow S-N, Li C, Wang D (1994) Normal forms and bifurcations of planar vector fields. Cambridge Univ Press, Cambridge 11. Cicogna G (1996) On the convergence of normalizing transformations in the presence of symmetries. J Math Anal Appl 199:243–255 12. Cicogna G (1997) Convergent normal forms of symmetric dynamical systems. J Phys A 30:6021–6028 13. Cicogna G, Gaeta G (1999) Symmetry and perturbation theory. In: Nonlinear Dynamics, Lecture Notes in Phys 57. Springer, New York 14. Cicogna G, Walcher S (2002) Convergence of normal form transformations: The role of symmetries. Acta Appl Math 70:95–111 15. Cushman R, Sanders JA (1990) A survey of invariant theory applied to normal forms of vectorfields with nilpotent linear part, IMA vol Math Appl 19. Springer, New York, pp 82–106 16. DeLatte D, Gramchev T (2002) Biholomorphic maps with linear parts having Jordan blocks: linearization and resonance type phenomena. Math Phys Electron J 8(2):27 (electronic) 17. Dulac H (1912) Solutions d’un systéme d’équations différentielles dans le voisinage de valeurs singuliéres. Bull Soc Math Fr 40:324–383 18. Ecalle J (1981) Sur les fonctions résurgentes I. Publ Math d’Orsay 81(5):1–247 19. Ecalle J (1981) Sur les fonctions résurgentes II. Publ Math d’Orsay 81(6):248–531 20. Edneral VF (2005) Looking for periodic solutions of ODE systems by the normal form method. In: Differential equations with symbolic computation. Trends Math, Birkhäuser, Basel, pp 173–200 21. Gramchev T (2002) On the linearization of holomorphic vector fields in the Siegel domain with linear parts having nontrivial Jordan blocks SPT. In: Symmetry and perturbation theory (Cala Gonone). World Sci Publ, River Edge, NJ, pp 106–115 22. Gramchev T, Tolis E (2006) Solvability of systems of singular partial differential equations in function spaces. Integral Transforms Spec Funct 17:231–237

23. Gramchev T, Walcher S (2005) Normal forms of maps: Formal and algebraic aspects. Acta Appl Math 87(1–3):123–146 24. Gramchev T, Yoshino M (1999) Rapidly convergent iteration method for simultaneous normal forms of commuting maps. Math Z 231:745–770 25. Iooss G, Adelmeyer M (1992) Topics in Bifurcation Theory and Applications. World Scientific, Singapore 26. Ito H (1989) Convergence of Birkhoff normal forms for integrable systems. Comment Math Helv 64:412–461 27. Ito H (1992) Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case. Math Ann 292:411–444 28. Kappeler T, Kodama Y, Nemethi A (1998) On the Birkhoff normal form of a completely integrable system near a fixed point in resonance. Ann Scuola Norm Sup Pisa Cl Sci 26:623–661 29. Markhashov LM (1974) On the reduction of an analytic system of differential equations to the normal form by an analytic transformation. J Appl Math Mech 38:788–790 30. Martinet J, Ramis J-P (1983) Classification analytique des équations différentielles non linéaires résonnantes du premier ordre. Ann Sci Ecole Norm Sup 16:571–621 31. Perez Marco R (1995) Nonlinearizable holomorphic dynamics having an uncountable number of symmetries. Invent Math 119:67–127 32. Perez–Marco R (1997) Fixed points and circle maps. Acta Math 179:243–294 33. Perez–Marco R (2003) Convergence and generic divergence of the Birkhoff normal form. Ann Math 157:557–574 34. Pliss VA (1965) On the reduction of an analytic system of differential equations to linear form. Differ Equ 1:153–161 35. Poincaré H (1879) Sur les propriétés des fonctions deéfinies par les équations aux differences partielles. These, Paris 36. Sanders JA (2003) Normal form theory and spectral sequences. J Differential Equations 192:536–552 37. Sanders JA (2005) Normal form in filtered Lie algebra representations. Acta Appl Math 87:165–189 38. Siegel CL (1952) Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung. Nachr Akad Wiss Göttingen, Math-Phys Kl, pp 21–30 39. Stolovitch L (2000) Singular complete integrability. IHES Publ Math 91:133–210 40. Vey J (1979) Algébres commutatives de champs de vecteurs isochores. Bull Soc Math France 107(4):423–432 41. Verhulst F (2005) Methods and applications of singular perturbations Boundary layers and multiple timescale dynamics. In: Texts in Applied Mathematics, vol 50. Springer, New York 42. Voronin SM (1981) Analytic classification of germs of conformal mappings (C; 0) ! (C; 0). Funct Anal Appl 15:1–13 43. Walcher S (1991) On differential equations in normal form. Math Ann 291:293–314 44. Walcher S (1993) On transformations into normal form. J Math Anal Appl 180(2):617–632 45. Walcher S (2000) On convergent normal form transformations in presence of symmetries. J Math Anal Appl 244:17–26 46. Zung NT (2002) Convergence versus integrability in Poincaré– Dulac normal form. Math Res Lett 9(2–3):217–228 47. Zung NT (2005) Convergence versus integrability in Birkhoff normal form. Ann Math (2) 161(1):141–156

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Phase Transitions on Fractals and Networks DIETRICH STAUFFER Institute for Theoretical Physics, Cologne University, Köln, Germany Article Outline Glossary Definition of the Subject Introduction Ising Model Fractals Diffusion on Fractals Ising Model on Fractals Networks Future Directions Bibliography

be observed is periodic with very few lattice faults. Other phase transitions like the boiling of water, or the liquidvapor critical point where the density difference between a liquid and its vapor vanishes, happen in a continuum without any underlying lattice structure. Nevertheless, the critical exponents of the Ising model on a simple-cubic lattice agree well with those of liquid-vapor experiments. Impurities, which are either fixed (“quenched dilution”) or mobile (“annealed dilution”), are known to change these exponents slightly, e. g. by a factor 1  ˛, if the specific heat diverges in the undiluted case at the critical point, i. e. if the specific heat exponent ˛ is positive. In this review we deal neither with regular lattices nor with continuous geometry, but with phase transitions on fractal and other networks. We will compare these results with the corresponding phase transitions on infinite periodic lattices like the Ising model. Ising Model

Glossary Cluster Clusters are sets of occupied neighboring sites. Critical exponent At a critical point or second-order phase transition, many quantities diverge or vanish with a power law of the distance from this critical point; the critical exponent is the exponent for this power law. Diffusion A random walker decides at each time step randomly in which direction to proceed. The resulting mean square distance normally is linear in time. Fractals Fractals have a mass varying with some power of their linear dimension. The exponent of this power law is called the fractal dimension and is smaller than the dimension of the space. Ising model Each site carries a magnetic dipole which points up or down; neighboring dipoles “want” to be parallel. Percolation Each site of a large lattice is randomly occupied or empty. Definition of the Subject At a phase transition, as a function of a continuously varying parameter (like the temperature), a sharp singularity happens in infinitely large systems, where quantities (e. g. the density) jump, vanish, or diverge. Introduction Some phase transitions, like the ferromagnetic Curie point where the spontaneous magnetization vanishes, happen in solids, and experiments often try to grow crystals very carefully such that the solid in which the transition will

Ernst Ising in 1925 (then pronounced EEsing, not EYEsing) published a model which is, besides percolation, one of the simplest models for phase transitions. Each site i is occupied by a variable S i D ˙1 which physicists often call a spin but which may also be interpreted as a trading activity [10] on stock markets, as a “race” or other ethnic group in the formation of city ghettos [26], as the type of molecule in binary fluid mixtures like isobutyric acid and water, as occupied or empty in a lattice-gas model of liquid-vapor critical points, as an opinion for or against the government [29], or whatever binary choice you have in mind. Also models with more than two choices, like S i D 1, 0 and 1 have been investigated both for atomic spins as well as for races, opinions, : : : Two spins i and k interact with each other by an energy JS i S k which is J if both spins are the same and CJ if they are the opposite of each other. Thus 2J is the energy to break one bond, i. e. to transform a pair of equal spins to a pair of opposite spins. The total interaction energy is thus E D J

X

Si Sk ;

(1a)



with a sum over all neighbor pairs. If you want to impress your audience, you call this energy a Hamiltonian or Hamilton operator, even though most Ising model publications ignore the difficulties of quantum mechanics except for assuming the discrete nature of the Si . (If instead of these discrete one-dimensional values you want to look at vectors rotating in two- or three-dimensional space, you should investigate the XY or Heisenberg models instead of the Ising model.)

Phase Transitions on Fractals and Networks

Different configurations in thermal equilibrium at absolute temperature T appear with a Boltzmann probability proportional to exp(E/kB T), and the Metropolis algorithm of 1953 for Monte Carlo computer simulations flips a spin with probability exp(E/kB T), where kB is Boltzmann’s constant and E D Eafter  Ebefore the energy difference caused by this flip. If one starts with a random distribution of half the spins up and half down, using this algorithm at positive but low temperatures, one sees growing domains [28]. Within each domain, most of the spins are parallel, and thus a computer printout shows large black domains coexisting with large white domains. Finally, one domain covers the whole lattice, and the other spin orientation is restricted to small clusters or isolated single spins within that domain. This self-organization (biologist may call it “emergence”) of domains and of phase separation appears only for positive temperatures below the critical temperature T c and only in more than one dimension. For T > Tc (or at all positive temperatures in one dimension) we see only finite domains which no longer engulf the whole lattice. This phase transition between long-range order below and short-range order above T c is called p the Curie or critical point; we have J/kB Tc D 12 ln(1 C 2) on the square lattice and 0.221655 on the simple cubic lattice with interactions to the z nearest lattice neighbors; z D 4 and 6, respectively. The mean field approximation becomes valid for large z and gives J/kB Tc D 1/z. Near T D Tc the difference between the number of up and down spins vanishes as (Tc  T)ˇ with ˇ D 1/8 in two, ' 0:32 in three, and 1/2 in six and more dimensions and in mean field approximation. We may also influence the Ising spins though an external field h by adding X Si (1b) h i

to the energy of Eq. (1a). This external field then pushes the spins to become parallel to h. Thus we no longer have emergence of order from the interactions between the spins, but imposition of order by the external field. In this simple version of the Ising model there is no sharp phase transition in the presence of this field; instead the spontaneous magnetization (fraction of up spins minus fraction of down spins) smoothly sinks from one to zero if the temperature rises from zero to infinity. Fractals Fractals obey a power law relating their mass M to their radius R: M / RD

(2)

where D is the fractal dimension. An exactly solved example are random walks (= polymer chains without interaction) where D D 2 if the length of the walk is identified with the mass M. For self-avoiding walks (= polymer chains with excluded volume interaction), the Flory approximation gives D D (d C 2)/3 in d  4 dimensions (D(d  4) D 2 as for random walks), which is exact in one, two and four dimensions, and too small by only about two percent in three dimensions. We now discuss the fractal dimension of the Ising model. In an infinite system at temperatures T close to T c , the difference M between the number of up and down spins varies as (Tc  T)ˇ while the correlation length  varies as jT  Tc j . Thus, M /  ˇ / . The proportionality factor varies as the system size Ld in d dimensions since all spins are equivalent. In a finite system right at the critical temperature T c we replace  by L and thus have M / Ldˇ / D L D with the fractal dimension DDd

ˇ 

(d  4) :

(3a)

Warning: one should not apply these concepts to spin clusters if clusters are simply defined as sets of neighboring parallel spins; to be fractals at T D Tc the clusters have to be sets of neighboring parallel spins connected by active bonds, where bonds are active with probability 1  exp(2J/kB T). Then the largest cluster at T D Tc is a fractal with this above fractal dimension. This warning is superfluous for percolation theory (see separate reviews in this encyclopedia) where each lattice site is occupied randomly with probability p and clusters are defined as sets of neighboring occupied sites. For p > pc one has an infinite cluster spanning from one side of the sample to the other; for p < pc one has no such spanning cluster; for p D pc one has sometimes such spanning clusters, and then the number of occupied sites in the largest or spanning cluster is M / LD ;

DDd

ˇ 

(d  6)

(3b)

with the critical exponents ˇ;  of percolation instead of Ising models. These were probabilistic fractal examples, as opposed to deterministic ones like the Sierpinski carpets and gaskets, which approximate in their fractal dimensions the percolation problem. We will return to them in the Sect. “Ising Model on Fractals”. Now, instead of asking how phase transitions produce fractals we ask what phase transitions can be observed on these fractals.

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Diffusion on Fractals Unbiased Diffusion The most thoroughly investigated phase transitions on fractals are presumably random walkers on percolation clusters [17], particularly at p D pc . This research was started by Brandt [7] but it was the later Nobel laureate de Gennes [16] who gave it the catchy name “ant in the labyrinth”. The anomalous diffusion [5,14,22] then made it famous a few years later and may have also biological applications [13]. We put an ant onto a randomly selected occupied site in the middle of a large lattice, where each site is permanently occupied (randomly with probability p) or empty (1 p). At each time step, the ant selects randomly a neighbor direction and moves one lattice unit in this direction if and only if that neighbor site is occupied. We measure the mean distance ˝ ˛1/2 R(t) D r(t)2

or

D hr(t)i ;

(4)

where r is the vector from the starting point of the walk and the present position, and r D jrj its length. The average h: : : i goes over many such walking ants and disordered lattices. These ants are blind, that means they do not see from their old place whether or not the selected neighbor site is accessible (occupied) or prohibited (empty). (Also myopic ants were grown which select randomly always an occupied neighbor since they can see over a distance of one lattice unit.) The squared distance r2 is measured by counting how often the ant moved to the left, to the right, to the top, to the bottom, to the front, or to the back on a simple cubic lattice. The problem is simple enough to be given to students as a programming project. They then should find out by their simulations that for p < pc the above Rpremains finite while for p > pc is goes to infinity as t, for sufficiently long times t. But even for p > pc it may happen that for a single ant the distance remains finite: If the starting point happened to fall on a finite cluster, then R(t ! 1) measures the radius of that cluster. Let  be the exponent for the conductivity if percolation is interpreted as a mixture of electrically conducting and insulating sites. Then right at p D pc , instead of a constant or a squareroot law, we have anomalous diffusion: R / tk ;

kD

  ˇ/2 ; 2 C   ˇ

Phase Transitions on Fractals and Networks, Figure 1 Log-log plot for unbiased diffusion at (middle curve), above (upper data) and below (lower data) the percolation threshold pc . We see the phase transition from limited growth at pc  0:01 to diffusion at pc C 0:01, separated by anomalous diffusion at pc . Average over 80 lattices with 10 walks each

(5)

for sufficiently long times. This exponent k is close to but not exactly 1/3 in two and 1/5 in three dimensions. ˇ and  are the already mentioned percolation exponents. If we

always start the ant walk on the largest cluster at p D pc instead of on any cluster, the formula for the exponent k simplifies to /(2 C   ˇ). The theory is explained in detail in the standard books and reviews [8,17]. We see here how the percolative phase transition influences the random walk and introduces there a transition between diffusion for p > pc and finite motion for p < pc , with the intermediate “anomalous” diffusion (exponent below 1/2) at p D pc . Figure 1 shows this transition on a large cubic lattice.

Biased Diffusion Another type of transition is seen in biased diffusion, also for p > pc . Instead of selecting all neighbors randomly, we do that only with probability 1  B, while with probability B the ant tries to move in the positive x-direction. One may think of an electron moving through a disordered lattice in an external electric field. For a long time experts discussed whether for p > pc one has a drift behavior (distance proportional to time) for small B, and a slower motion for larger B, with a sharp transition at some p-dependent Bc . In the drift regime one may see log-periodic oscillations / sin(const log t) in the approach towards the long-time limit, Fig. 2. Such oscillations have been predicted for stock markets [19], where they could have made us rich, but for diffusion they hamper the analysis. They come presumably from sections of occupied sites which allow motion in the biased direction and then end in prohibited sites [20].

Phase Transitions on Fractals and Networks

Phase Transitions on Fractals and Networks, Figure 2 Log-periodic oscillation in the effective exponent k for biased diffusion; p D 0:725; B D 0:98. The limit k D 1 corresponds to drift. 80 lattices with 10 walks each

Phase Transitions on Fractals and Networks, Figure 4 Biased diffusion at p D pc (middle curve) and p D pc ˙ 0:01 (upper and lower data) for bias B D 0:8; 80 lattices with 10 walks each

Ising Model on Fractals

Phase Transitions on Fractals and Networks, Figure 3 Difficulties at transition from drift (small bias, upper data) to slower motion (large bias, lower data); 80 lattices with 10 walks each

Even in a region without such oscillations, Fig. 3 shows no clear transition from drift to no drift; that transition could only be seen by a more sophisticated analysis which showed for the p of Fig. 3 that the reciprocal velocity, plotted vs. log(time), switches from concave to convex shape at Bc ' 0:53. Fortunately, only a few years after these simulations [11] the transition was shown to exist mathematically [6]. These simulations were made for p > pc ; at p D pc with a fractal largest cluster, drift seems impossible, and for a fixed B the distance varies logarithmically, with a stronger increase slightly above pc and a limited distance slightly below pc , Fig. 4.

What happens if we set Ising spins onto the sites of a fractal? In particular, but also more generally, what happens to Ising spins on the occupied site of a percolation lattice, when each site is randomly occupied with probability p? In this case one expects three sets of critical exponents describing how the various quantities diverge or vanish at the Curie temperature T c (p). For p D 1 one has the standard Ising model with the standard exponents. If pc is the percolation threshold where an infinite cluster of occupied sites starts to exist, then one has a second set of exponents for pc < p < 1, where 0 < Tc (p) < Tc (p D 1). Finally, for zero temperature as a function of p  pc one has the percolation exponents as a third set of critical exponents. (If p D pc and the temperature approaches Tc (pc ) D 0 from above, then instead of powers of T  Tc exponential behavior is expected.) In computer simulations, the second set of critical exponents is difficult to observe; due to limited accuracy the effective exponents have a tendency to vary continuously with p. The behavior at zero temperature is in principle trivial: each cluster of occupied neighbors has parallel spins, the spontaneous magnetization is given by the largest cluster while the many finite clusters cancels each other in their magnetization. However, the existence of several infinite clusters at p D pc disturbs this argument there; presumably the total magnetization (i. e. not normalized at magnetization per spin) is a fractal with the fractal dimension of percolation theory. Deterministic fractals, instead of the random “incipient infinite cluster” at the percolation threshold, may have a positive T c and then allow a more usual study of crit-

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ical exponents at that phase transition. Koch curves and Sierpinski structures have been intensely studied in that aspect since decades. To build a Sierpinski carpet we take a square, divide each side into three thirds such that the whole square is divided into nine smaller squares, and then we take away the central square. On each of the remaining eight smaller squares this procedure is repeated, diving each into nine squarelets and omitting the central squarelet. This procedure is repeated again and again, mathematically ad infinitum. Physicists like more to think in atoms of a fixed distance and would rather imagine each square to be enlarged in each direction by a factor three with the central square omitted; and then again and again this enlargement is repeated. In this way we grow a large structure built by squares of unit area. Unfortunately, the phase transitions on these fractals depend on details and are not already fixed if the fractal dimension is fixed. Also other properties of the fractals like their “ramification” are important [15]; see [3] for recent work. This is highly regrettable since modern statistical physics is not restricted to three dimensions. Models were studied also in seven and in minus two dimensions, in the limits of dimensionality going to infinity or to zero, or for non-integral dimensionality. (Similarly, numbers were generalized from positive counts to negative integers, to rational and irrational numbers, and finally to imaginary/complex numbers.) It would have been nice if these fractals would have been models for these non-integral dimensions, giving one and the same set of critical exponents once their fractal dimension is known. Regrettably, we had to give up that hope. Many other phase transitions, like those of Potts or voter models, were studied on such deterministic fractals, but are not reviewed here. We mention that also percolation transitions exist on Sierpinski structures [24]. Also, various hierarchical lattices different from the above fractals show phase transitions, if Ising spins are put on them; the reader is referred to [18,25] for more literature. As the to our knowledge most recent example we mention [21] that Ising spins were also thrown into the sandpiles of Per Bak, which show self-organized criticality. Networks Definitions While fractals were a big physics fashion in the 1980s, networks are now a major physics research field. Solid state physics requires nice single crystals where all atoms sit on a periodic lattice. In fluids they are ordered only over shorter distances but still their forces are restricted to their neighbors. Human beings, on the other hand, form a reg-

ular lattice only rarely, e. g. in a fully occupied lecture hall. In a large crowd they behave more like a fluid. But normally each human being may have contacts with the people in neighboring residences, with other neighbors at the work place, but also via phone or internet with people outside the range of the human voice. Thus social interactions between people should not be restricted to lattices, but should allow for more complex networks of connections. One may call Flory’s percolation theory of 1941 a network, and the later random graphs of Erdös and Rényi (where everybody is connected with everybody, albeit with a low probability) belong to the same “universality class” (same critical exponents) as Flory’s percolation. In Kauffman’s random Boolean network of 1969, everybody has K neighbors selected randomly from the N participants. Here we concentrate on two more recent network types, the small-world [31] and the scale-free [1] networks of 1998 and 1999 respectively (with a precursor paper of economics Nobel laureate Simon [27] from 1955). The small-world or Watts–Strogatz networks start from a regular lattice, often only a one-dimensional chain. Then each connection of one lattice site to one of its nearest neighbors is replaced randomly, with probability p, by a connection to a randomly selected other site anywhere in the lattice. Thus the limits p D 0 and 1 correspond to regular lattices and roughly random graphs, respectively. In this way everybody may have exactly two types of connections, to nearest neighbors and to arbitrarily far away people. This unrealistic feature of small-world networks is avoided by the scale-free networks of Barabási and Albert [1], defined only through topology with (normally) no geometry involved: We start with a small set of fully connected people. Then more people join the network, one after the other. Each new member selects connections to exactly m already existing members of the network. These connections are not random but follow preferential attachment: The more people have selected a person to be connected with in the past, the higher is the probability that this person is selected by the newcomer: the rich get richer, famous people attract more followers than normal people. In the standard Barabási–Albert network, this probability is proportional to the number of people who have selected that person. In this case, the average number of people who have been selected by k later added members varies as 1/k 3 . A computer program is given e. g. in [28]. These networks can be undirected (the more widespread version) or directed (used less often.) For the undirected or symmetric case, the connections are like friendships: If A selects B as a friend, then B also has A as a friend.

Phase Transitions on Fractals and Networks

For directed networks, on the other hand, if A has selected B as a boss, then B does not have A as a boss, and the connection is like a one-way street. Up to 108 nodes were simulated in scale-free networks. We will now check for phase transitions on both directed and undirected networks. Phase Transitions The Ising model in one dimension does not have a phase transition at some positive critical temperature T c . However, its small-world generalization, i. e. the replacement of small fraction of neighbor bonds by long-range bonds, produces a positive T c with a spontaneous magnetization proportional to (Tc  T)ˇ , and a ˇ smaller than the 1/8 of two dimensional lattices [4]. The Solomon network is a variant of the small-world network: Each person has one neighborhood corresponding to the workplace and another neighborhood corresponding to the home [23]. It was suggested and simulated by physicists Solomon and Malarz, respectively, before Edmonds [12] criticized physicists for not having enough “models which explicitly include actions and effects within a physical space as well as communication and action within a social space”. Even in one dimension a spontaneous magnetization was found. On Barabási–Albert (scale-free) networks, Ising models were found [2] for small m and millions of spins to have a spontaneous magnetization for temperatures below some critical temperature T c which increases logarithmically with the number N of spins: kB Tc /J ' 2:6 ln(N) for m D 5. Here we had undirected networks with symmetric couplings between spins: actio = –reactio, as required by Newton. Ising spins on directed networks, on the other hand, have no well-defined total energy, though each single spin may be influenced as usual by its m neighbor spins. If in an isolated pair of spins i and k we have a directed interaction in the sense that spin k tries to align spin i into the direction of spin k, while i has no influence on k, then we have a perpetuum mobile: Starting with the two spins antiparallel, we first flip i into the direction of k, which gives us an energy 2J. Then we flip spin k which does not change the energy. Then we repeat again and again these two spin flips, and gain an energy 2J for each pair of flips: too nice to be true. The violations of Newton’s symmetry requirement makes this directed network applicable to social interactions between humans, but not to forces between particles in physics. On this directed Barabási–Albert network, the ferromagnetic Ising spins gave no spontaneous magnetization, but the time after which the magnetization becomes zero

(starting from unity) becomes very long at low temperatures, following an Arrhenius law [30]: time proportional to exp(const/T). Also on a directed lattice such Arrhenius behavior was seen while for directed random graphs and for directed small-world lattices a spontaneous magnetization was found [30]. A theoretical understanding for these directed cases is largely lacking. Better understood is the percolative phase transition on scale-free networks (see end of Sect. “Introduction” for definition of percolation). If a fraction 1  p of the connections in an undirected Barabási–Albert network is cut randomly, does the remaining fraction p keep most of the network together? It does, for large enough networks [9], since the percolation threshold pc below which no large connected cluster survives, goes to zero as 1/log(N) where N counts the number of nodes in the network. This explains why inspite of the unreliability of computer connections, the internet still allows most computers to reach most other computers in the word: If one link is broken, some other link may help even though it may be slower [1]. (For intentional [9] cuts in hierarchical networks one may have a finite percolation threshold [32].) Future Directions We reviewed here a few phase transitions, and ignored many others. At present most interesting for future research seem to be the directed networks, since they have been investigated with methods from computational physics even though they are not part of usual physics, not having a global energy. A theoretical (i. e. not numerical) understanding would help. We thank K. Kulakowski for comments. Bibliography 1. Albert R, Barabási AL (2002) Rev Mod Phys 74:47; Boccalettia S, Latora V, Moreno Y, Chavez M, Hwang D-U (2006) Phys Repts 424:175 2. Aleksiejuk A, Hołyst JA, Stauffer D (2006) Physica A 310:260; Dorogovtsev SN, Goltsev AV, Mendes JFF (2002) Phys Rev E 66:016104. arXiv:0705.0010 [cond-mat.stat-mech]; Bianconi G (2002) Phys Lett A 303:166 3. Bab MA, Fabricius G, Albano EV (2005) Phys Rev E 71:036139 4. Barrat A, Weigt M (2000) Eur Phys J B 13:547; Gitterman M (2000) J Phys A 33:8373; Pe¸kalski A (2001) Phys Rev E 64:057104 5. Ben-Avraham D, Havlin S (1982) J Phys A 15:L691 6. Berger N, Ganten N, Peres Y (2003) Probab Theory Relat Fields 126:221 7. Brandt WW (1975) J Chem Phys 63:5162 8. Bunde A, Havlin S (1996) Fractals and Disordered Systems. Springer, Berlin 9. Cohen R, Erez K, ben-Avraham D, Havlin S (2000) Phys Rev Lett 85:4626; Cohen R, Erez K, ben-Avraham D, Havlin S (2001)

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10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Phys Rev Lett 86:3682; Callaway DS, Newman MEJ, Strogatz SH, Watts DJ (2000) Phys Rev Lett 85:5468 Cont R, Bouchaud J-P (2000) Macroecon Dyn 4:170 Dhar D, Stauffer D (1998) Int J Mod Phys C 9:349 Edmonds B (2006) In: Billari FC, Fent T, Prsakwetz A, Scheffran J (eds) Agent-based computational modelling. Physica-Verlag, Heidelberg, p 195 Frey E, Kroy K (2005) Ann Physik 14:20 Gefen Y, Aharony A, Alexander S (1983) Phys Rev Lett 50:77 Gefen Y, Mandelbrot BB, Aharony A (1980) Phys Rev Lett 45:855 de Gennes PG (1976) Rech 7:916 Havlin S, Ben Avraham D (1987) Adv Phys 36:395; Havlin S, Ben Avraham D (2002) Adv Phys 51:187 Hinczewski M, Berker AN (2006) Phys Rev E 73:066126 Johansen A, Sornette D (1999) Int J Mod Phys C 10:563 Kirsch A (1999) Int J Mod Phys C 10:753 Koza Z, Ausloos M (2007) Physica A 375:199 Kutner R, Kehr K (1983) Phil Mag A 48:199

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Malarz K (2003) Int J Mod Phys C 14:561 Monceau P, Hsiao PY (2004) Phys Lett A 332:310 Rozenfeld HD, Ben-Abraham D (2007) Phys Rev E 75:061102 Schelling TC (1971) J Math Sociol 1:143 Simon HA (1955) Biometrika 42:425 Stauffer D, Moss de Oliveira S, de Oliveira PMC, Sá Martins JS (2006) Biology, Sociology, Geology by Computational Physicists. Elsevier, Amsterdam Sznajd-Weron K, Sznajd J (2000) Int J Mod Phys C 11:1157 Sánchez AD, López JM, Rodríguez MA (2002) Phys Rev Lett 88:048701; Sumour MA, Shabat MM (2005) Int J Mod Phys C 16:585; Sumour MA, Shabat MM, Stauffer D (2006) Islamic Univ J (Gaza) 14:209; Lima FWS, Stauffer D (2006) Physica A 359:423; Sumour MA, El-Astal AH, Lima FWS, Shabat MM, Khalil HM (2007) Int J Mod Phys C 18:53; Lima FWS (2007) Comm Comput Phys 2:522 and (2008) Physica A 387:1545; 3503 Watts DJ, Strogatz SH (1998) Nature 393:440 Zhang Z-Z, Zhou S-G, Zou T (2007) Eur Phys J B 56:259

Philosophy of Science, Mathematical Models in

Philosophy of Science, Mathematical Models in Z OLTAN DOMOTOR University of Pennsylvania, Philadelphia, USA Article Outline Glossary Definition of the Subject Introduction Mathematical Models: What Are They? Philosophical and Mathematical Structuralism Three Approaches to Applying Mathematical Models Validating Mathematical Models Future Directions Bibliography Glossary Philosophy of science Broadly understood, philosophy of science is a branch of philosophy that studies and reflects on the presuppositions, concepts, theories, arguments, methods and aims of science. Philosophers of science are concerned with general questions which include the following: What is a scientific theory and when can it be said to be confirmed by its predictions? What are mathematical models and how are they validated? In virtue of what are mathematical models representations of the structure and behavior of their target systems? In sum, a major task of philosophy of science is to analyze and make explicit common patterns that are implicit in scientific practice. Mathematical model Stated loosely, models are simplified, idealized and approximate representations of the structure, mechanism and behavior of real-world systems. From the standpoint of set-theoretic model theory, a mathematical model of a target system is specified by a nonempty set – called the model’s domain, endowed with some operations and relations, delineated by suitable axioms and intended empirical interpretation. No doubt, this is the simplest definition of a model that, unfortunately, plays a limited role in scientific applications of mathematics. Because applications exhibit a need for a large variety of vastly different mathematical structures – some topological or smooth, some algebraic, order-theoretic or combinatorial, some measure-theoretic or analytic, and so forth, no useful overarching definition of a mathematical model is known even in the edifice of modern category theory. It is difficult to come up with a workable

concept of a mathematical model that is adequate in most fields of applied mathematics and anticipates future extensions. Target system There are many definitions of the concept of ‘system’. Here by a target system we mean an effectively isolated (physical, biological, or other empirical) part of the universe – made to function or run by some internal or external causes, whose interactions with the universe are strictly delineated by a fixed (input and output) interface, and whose structure, mechanism, or behavior are the objects of mathematical modeling. Changes produced in the target system are presumed to be externally detectable via measurements of the system’s characterizing quantitative properties. Definition of the Subject Models are indispensable scientific tools in generating information about the world. Concretely, scientists rely on models for explanation and prediction. Recent years have seen intensive attempts to extend the art of modeling and simulation to a vast range of application areas. Fostered by inexpensive computers and software, and the accelerating growth of mathematical knowledge, in contemporary applied research reference to models is far more frequent than reference to theories or principles. In applications, the term ‘model’ is used in a wide variety of senses, including mathematical, physical, mental, and computational models. Our focus here is on mathematical models, their structure, representational role, and validation. Currently, many philosophers of science are interested in two major aspects of mathematical models: their nature and representational role. The question “What are mathematical models?” is answered in two fundamentally different but closely related ways: 1. In a seemingly natural way, by viewing mathematical models as families of equations of some kind, accompanied with certain empirical interpretations that link them to their target systems. It turns out that this simple conception, called the received view or the syntactic approach, presents several troubling representational and interpretational problems. 2. The so-called structuralist or semantic answer takes mathematical models of target systems to be suitable set-theoretic structures or generally objects in a specific dynamical (or other) category. In order to pursue this approach, model builders and users need to be able to understand how complex notions, relevant

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to modeling, are defined in terms of the model’s structure. One of the most useful general results on the nature of mathematical models is the following. If a given semantic formulation of a model involves also a specification of the solution space of some equations, then in this case the equational and structural conceptions of models are formally equivalent. Indeed, equations uniquely characterize their solution spaces and these in turn determine the associated semantic model. Conversely, if the semantic model specifies an abstract solution space, then the latter delineates a system of characterizing equations. This type of Galois correspondence between equations and their solutions has been established in many linear and even in some nonlinear settings. There is a major counter to this syntactic-semantic dilemma. Structuralists are quick to point out that the mathematical models of exchange economy, n-person games, probability and decision, and so forth, are autonomous set-theoretic structures that are not associated with any system of equations. For example, recall that a classical probability model of a statistical experiment is defined by a triple, consisting of a set together with a designated field of its subsets – forming an underlying measurable space, and a probability measure thereon. Remarkably, even though in this case there are no equations to consider, thanks to the powerful Stone–Gelfand duality result, every probability model has an associated algebraic counterpart model, given by a linear space (thought of as the space of bounded random variables on the model’s underlying measurable space) and a positive linear functional on it (viewed as the expectation functional induced by the measure). Statisticians in particular (e. g., [21]) prefer to build their models of experiments in a ‘dual’, computationally stronger, algebraic setting. It happens quite often that autonomous mathematical models arise in ‘adjoint’ or paired geometric/algebraic formulations. Characteristically, models of this nature tend to form impressively versatile ‘mathematical universes’ for all of the mathematics that model users may need in a given area of application. The question “What is the role of mathematical models in scientific practice?” is answered by describing how models are conceived, constructed, and used in various applications. Since models are mathematical constructs, in addition to being objects of a formal inquiry, they are also involved in epistemic relations, expressing intended uses. Models do not and need not match reality in all of its aspects and details to be adequate. A mathematical

model is usually developed for a specific class of target systems, and its validity is determined relative to its intended applications. A model is considered valid within its intended domain of applicability provided that its predictions in that domain fall within an acceptable range of error, specified prior to the model’s development or identification. To construct a mathematical model of a target system, it is typically necessary to formalize the system’s decisive causal mechanisms and behavior with special regards to the model’s structural stability and mutability into a larger network of models that includes not only additional causeeffect relations but also a broad range of deterministic and stochastic perturbations. In the next section we consider a simple class of dynamical models in physics that is also relevant to modeling in other disciplines. Introduction The subject of mature mathematical models in the form of equations has its roots in post-Newtonian developments of classical mechanics, hydrodynamics, electromagnetism, and kinetic theory of gases. It came on the scene of applied mathematics gradually, during the analytic period before 1880, thanks to the innovative efforts of great scientists, including, among many others, the Swiss mathematician Leonhard Euler (1707– 1783), Italian–French mathematician Louis Joseph Lagrange (1736–1813), French astronomer-physicist Pierre Simon de Laplace (1749–1827), Scottish physicist James Clerk Maxwell (1831–1879), English physicist Lord John William Strutt Rayleigh (1842–1919), and the Austrian physicist Ludwig Boltzmann (1844–1906). It was the genius of the French mathematician Henri Poincaré (1854– 1912) that generated many of our current topological and differential methods of mathematical modeling in the world of dynamical systems. Over the past 100 years or so, mathematical models have evolved to become the basic tools in a wide variety of disciplines, including not only most of physical sciences, but also chemical kinetics, population dynamics, economics, sociology, and psychology. The many different theoretical areas of natural and social sciences have led to the development of a large assortment of mathematical models, including but not limited to descriptive vs. normative, static vs. dynamic, phenomenological vs. process-based, discrete vs. continuous, deterministic vs. stochastic, linear vs. nonlinear, finite- vs. infinite-dimensional, difference vs. differential, and topological vs. measure-theoretic models. Using category theory, efforts have been made to construct general theories of

Philosophy of Science, Mathematical Models in

mathematical models of which models of logical systems and dynamical systems are special cases. There has been a renewed interest also among philosophers of science in the problems of structure and function of abstract models. (See, for example, [3,5,10,13,25], and [33].) Prime questions about abstract models good many philosophers ask and attempt to answer include the following: (i)

Ontology of models: What, precisely, are mathematical models and how are they used in science? Are they structures in the sense of classical set theory, modern category theory, or something else, belonging, e. g., to the nebulous world of fictional entities or human constructs? (ii) Semantics of models: In virtue of what are mathematical models representations of the structure, causal mechanisms and behavioral regimes of their target systems? Is it in virtue of some (possibly partial) ‘isomorphisms’ holding between mathematical and physical domains or by reason of certain designated ‘similarities’, analogies, or resemblance relations between models and aspects of the world, or because of ‘empirical interpretations’ of (parts of) the representing model’s mathematical vocabulary – implying quantitative claims about the world that can be corroborated by empirical data, or in virtue of yet something else, such as homology or physical instantiation? Since models appear to be the main vehicles in the pursuit of scientific knowledge, philosophers are also interested in analyzing the truth conditions of semantic relations of a more general nature, such as “Researcher R uses model M to represent target system S for purpose ˘ ”. (iii) Validation of models: How are mathematical models validated? Is validation just a procedure in which the model’s predictions are simply compared with a set of observations within the model’s domain of applicability, or is it a comprehensive, all-out testing to determine the degree of agreement between the model and its target system in terms of internal structure, cause-effect relationships, and predictions? Needless to add, these and many other questions to be examined below do not belong to science per se; they are about science. In this sense, philosophy of science is a second-order discipline, addressing the practices, methods and aims of the various sciences. However, it is clear from [24] that second-order questions about science are investigated also by scientists themselves.

Before considering the details of models of mathematical models, we begin by characterizing a particular use of the term mathematical model which, although not representative of the more careful formulations of the majority of philosophers of science, is nevertheless the most common practice encountered in physics, engineering and economics. Most physicists, engineers and mathematically oriented social scientists understand mathematical models to be systems of (algebraic, difference, differential, integral or other) equations (linking time- and spacetime-dependent quantities, and parameters), derived from first principles under various idealizing and simplifying scenarios of target systems or induced by available observational data and ‘empirical laws’. As an example illustrating the equational conception of mathematical models, we shall consider briefly the most familiar simple classical planar pendulum model, describing an undamped pendulum’s dynamical behavior. (Additional examples can be found in [12].) It is specified by the autonomous, deterministic second-order nonlinear differential equation d2 g C sin D 0 ; dt 2 ` in which the time-dependent indeterminate represents the target pendulum’s variable angle from its downward vertical to the pivoting rod, to which a bob of mass m is attached at its swinging end. Coefficient g denotes the homogeneous gravitational acceleration acting on the pendulum’s bob downward, and coefficient ` captures the length of the pendulum’s idealized ‘massless’ and perfectly stiff rod. These constant coefficients are needed for individuating the target pendulum in its classical gravitational ambience. Under the accompanying physical interpretation, fixed by the pendulum’s idealizing scenario (involving significant idealizations of friction, torque, resistance, and elasticity) and first-principle framework, the all-important observable quantity is the pendulum’s total energy (Hamiltonian)   d Ddf m`2 H ; dt

1 2



d dt

2

g  cos `

! :

The first term in the differential equation above encodes inertia and the second term stands for gravity. Recall that the pendulum’s rod is suspended from a pivot point, around which it oscillates or rotates in a vertical plane without surface resistance and forcing, so that there are no extra additive terms in the equation for the effects of

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friction and torque. Coarsely speaking, in general a model is a simplified representation of a real-world system of interest for designated scientific purposes. Although the target system has many important features, not all of them can and should be included in the model for reasons of tractability and limited epistemic import. And those that are included, often involve drastic idealizations, known to be empirically false. It has long been known that in general the foregoing differential equation does not have a closed-form solution in terms of traditional elementary functions. For general analytic solutions, Jacobi’s periodic elliptic functions are needed, with values knowable only with specified degrees of accuracy from mathematical tables or computations performed by special computer programs. The gap between theoretically granted solutions and their approximate variants has led Harald Atmanspacher and Hans Primas [2] to advocate a dichotomy between states of reality and states of knowledge. In a nutshell, derivation of highly theoretical equations of motion (providing maximal information) offers a so-called ontic (endophysical) view of modes of being of target systems, whereas (statistical) approximation and measurement procedures give an epistemic (exophysical) perspective on real-world systems, involving errors and updating. It is well known that the nonlinear equation above has several geometric types of solution that capture all sorts of swinging and rotating motions, and states of rest – for the most part knowable only approximately. Moving beyond the important ontic vs. epistemic dichotomy in modeling, note that because the foregoing pendulum equation’s nonlinear component is representable by the infinite series sin D  3 /(3!) C 5 /(5!)     and since for small deflections (less than 5ı ) of the pendulum’s rod from the vertical the values sin (t) of the angle quantity are very nearly equal to (t), we can substitute for sin in the equation and forget about the higher-order polynomial terms in the infinite series. So, at the cost of obvious approximation errors, we obtain the basic linear pendulum differential equation d2 /dt 2 Cg/` D 0 that is easy to solve. It is an elementary exercise to show that its smooth closed-form solutions are given by parametrized trigonometric position functions of the form (t) D (0) cos(! t) C

˙ (0) sin(! t) !

for all time instants t, with initial conditions ˇ p (0) and ˇ , and parameter ! Ddf ` : g, char ˙ (0) Ddf d (t) dt tD0 acterizing the pendulum’s frequency of oscillation. In the study of differential equations and their solutions it is stan-

dard to adopt the so-called flow point of view. Simply, instead of studying a particular solution for a given initial value, the entire solution space (

ˇ ) ˇ d2 g ˇ 2 C (T ) ˇ C D0 ˇ dt 2 ` 1

is studied, preferably in a parametrized form. Granted that the model is ‘correct’, this subspace of the space C1 (T ) of smooth real-valued functions on a designated continuum-time domain T provides the investigator with complete information about the target pendulum’s possible angular positions and dynamical behavior. Clearly, the foregoing solution space includes many relativistically forbidden solutions that encode physically impossible behaviors – individuated by the pendulum’s superluminal velocities. The key idea here is that in the absence of complete knowledge of the model’s empirical domain of applicability, a researcher may be in an epistemic danger of trying to physically interpret (in terms of behavior) some of the solutions – countenanced by the model, that are actually meaningless in the physical world. To remedy the situation, the foregoing classical simple pendulum model must be extended to the relativistic case (studied, e. g., in [11]), having the form d2 C 1 dt 2



` d c dt

2 !

g sin D 0 ; `

where parameter c denotes the speed of light. Because formal models usually possess limited empirical domains of applicability, in comprehensive treatments of target system behavior several different, closely related models may become necessary. Real-world systems tend to have many representing models and these models often possess a surplus content, which supports their mutations and extensions in unexpected ways. For example, in the presence of randomness or ‘noise’, a classical stochastic pendulum model provides the most appropriate representation of randomly perturbed motion. A typical model of the behavior of simple pendulums affected by ‘noise’ is presented by the stochastic differential equation d2 X C (1 C "W)! 2 sin X D 0 ; dt 2 where X denotes the stochastic position indeterminate, " > 0 is a parameter with small values, and W is a (e. g., Gaussian) stochastic process, capturing the pendulum’s random perturbation. Naturally, to extract information

Philosophy of Science, Mathematical Models in

from a stochastic pendulum model, the investigator has to calculate the moments of target pendulum’s positions or consider the transition probability density that allows to calculate the conditional probability of a future position of the pendulum’s bob, given its position at a designated starting time. In many applications, the same basic pendulum differential equation drops out of a wholly different idealizing scenario of, say, a coupled mass-spring system, consisting of a (point) mass attached to a spring at one end, where the other end of the spring is tied to a fixed frame. Other prominent examples of systems, whose dynamical behavior is also modeled by ‘pendulum equations’, include electric circuits, chemical reactions and interacting biological populations with oscillatory behaviors. Of course, in each application, the indeterminate and individuating parameters are interpreted differently. These examples illustrate clearly that mathematical models are characteristically generic or protean with respect to real-world systems, meaning that often the same mathematical model applies to different empirical situations, admits several different empirical interpretations, and is functioning both as an investigative instrument and as an object of mathematical inquiry. Although, broadly speaking, models can be representations of a particular token target system (canonical examples are cosmological models) or of a stereotype class of systems (e. g., pendulums), they are always representations of some particular structure of a phenomenon, mechanism or behavior. Needless to add, empirically meaningful deductions from the representing equations are guided not just by the free-standing equational structure of pure mathematics but also by the accompanying empirical interpretation, encapsulated in part by the target system’s idealizing scenario. Along these lines, in [23] Saunders Mac Lane asserts that “mathematics is protean science; its subject matter consists of those structures which appear (unchanged) in different scientific contexts”. On this picture, scientists construct formal models in the form of (differential, partial differential, stochastic differential, etc.) equations, proceeding in a simple top-down analytic fashion – using first principles (e. g., Newton’s, Kirchhoff’s and other laws) and idealizing scenarios or ‘empirical rules’ that are not part of any ambient theory, or in a more involved bottom-up, data-driven synthetic manner, starting from experimental (e. g., time series) data, the extant stage of knowledge, analogies with other systems, and background assumptions. In a total absence of any data or prior knowledge pertaining to the target system, it is inappropriate to consider mathematical models at all. This raises several additional foundational questions:

(i)

What is at stake, if anything, in conceiving mathematical models as empirically interpreted equations of some sort? (ii) How is it that the causal mechanisms and behaviors of real-world systems can be so effectively studied in terms of solutions of various equations that employ highly idealized assumptions, known to be false? (iii) Since most (nonlinear) equations arising in science are solvable only approximately and in a discrete range of values, what is their epistemic import? Recall that predictions from these equations may have to be obtained by brute-force numerical methods that require more computational power than the scientists are likely to have. Thus, only a Laplacean superscientist with unlimited cognitive capacities and computational resources could make full use of such equations. In contrast, although a real-world scientist does start with such “true-in-heaven” equations but then he or she quickly modifies them to make them computationally easier to extract predictions from them. We are now ready to start discussing some of the answers to these and other previously listed philosophical questions about mathematical models. Mathematical Models: What Are They? We begin, as a way of entering the subject of mathematical models by clarifying the dichotomy between equational (syntactic) and structural (semantic) conceptions of models. The standard view among most theoretical physicists, engineers and economists is that mathematical models are syntactic (linguistic) items, identified with particular systems of equations or relational statements. From this perspective, the process of solving a designated system of (algebraic, difference, differential, stochastic, etc.) equations of the target system, and interpreting the particular solutions directly in the context of predictions and explanations are primary, while the mathematical structures of associated state and orbit spaces, and quantity algebras – although conceptually important, are secondary. This is a good place to recall that, contrary to the above, philosophical structuralists (e. g., Landry [18]) defend the claim that mathematical models are structures and category theory is the correct framework for their study. Structuralists have two major arguments against the equational (syntactic) view of mathematical models. The first reason why the popular syntactic conception of models is troubling to philosophers of science is the following. Even though a stipulated system of equations of a target system admits infinitely many alternative formulations – linked by scale, coordinate, change-of-variable and

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other transformations, researchers do not presume of having different models presented by these inessentially different formulations. It is important to bear in mind that the structure of the associated solution space, serving as a storehouse for all model-based information about the target system’s causal mechanisms and behavior, remains basically the same. Each type of solution of a given system of differential (difference) equations is not defined by the system of equations per se, but generally by a much larger (prime) differential (difference) ideal of equations, generated by it. Since solutions and their mathematical semantics are more important than change-of-variable transformations, systems of equations are related to each other in a considerably deeper way by solution-preserving transformations than by, say, scale transformations. A case in point is the familiar physical equivalence of Hamiltonian and Lagrangean formulations of equations of motion in mechanics. Though the underlying language in each case is different, nevertheless the modeling results are the same. In brief, models in the sense suggested by the practice of science possess properties equations do not seem to have. It is well known that the basic linear second-order pendulum differential equation d2 /dt 2 Cg/` D 0 can be put in the form of two linear first-order differential equations (D

d dt

and

d( g C D0; dt `

illustrating the notion of an equation-based state space model. As an aside, we note that this type of transformation is general in that any system of higher-order differential equations can be rewritten as a first-order system of higher dimensionality. In this system of equations, the position-velocity pairs h (t); ((t)i are thought to encode the target pendulum’s dynamical state (or physical mode of being) of interest at time t. More importantly, the equations’ smooth, two-dimensional parametric solutions are now given by the position-velocity function pair h a;b (t); ( a;b (t)i   b D a cos(! t) C sin(! t); b cos(! t)  a! sin(! t) ; ! with parameters a D (0) and b D ((0). Thus, to specify a particular (position-velocity) state-space trajectory for the pendulum model, all we need is the initial state ha; bi, arrived at via first and second integration of the original equation. We mention in passing that if the differential equation characterizing a target system were of order n,

then generally we would need n independent parameter values for a unique individuation of any of its solutions in the form of specific time-functions. The point of this simple exercise in pendulum modeling is to make a headway in the direction of a powerful alternate structuralist formulation of the pendulum model that has the following smooth group action (smooth flow or smooth group representation) form h ;i

T  (R/2Z  R) ! R/2Z  R ; to be defined next. Here and below, T denotes a continuum-time domain, isomorphic to the group hR; 0; Ci of real numbers, and R/2Z  R denotes the two-dimensional cylinder-shaped state space, generated by the unit circle and intended to encode all physically relevant states in terms of values of the parametrized position-velocity function. The customary geometric interpretation of stateparametrized solutions consists of smooth curves, covering the cylinder state space and thereby forming the target system’s phase portrait. By a continuous group action of a topological group hT ; 0; Ci (interpreted as a continuum-time domain or time-group) on a topological space X (thought of as a state space) we mean a jointly continuous map of the form ı : T  X ! X (also known as a dynamical transition map) such that the following axioms of group action hold for all time instants t; t 0 2 T and for all states x in X: (i) Identity property: ı(0; x) D x, and (ii) Group property: ı(t; ı(t 0 ; x)) D ı(t C t 0 ; x): Because in what follows, group actions will be used mainly to represent the temporal evolution of target systems’ states, the above-introduced group action ı : T  X ! X is alternatively called a (deterministic) topological dynamical model and is symbolized more succinctly by the curved X. An impressively large class of group-action arrow T Õ ı deterministic dynamical models arises from autonomous systems of ordinary first-order differential equations by simple state-parametrizations of their solution spaces. If the time domain T and the state space X are both smooth manifolds, and if the transition map ı is also smooth (i. e., infinitely differentiable), then T Õ X is called a (determinı istic) smooth dynamical model. In particular, the structural variant of the earlier discussed smooth pendulum dynamÕ (R ical model has the form T ı; /2Z  R), obtained directly from the solutions of a pair of first-order pendulum equations. Because the time group’s operation is an action on its own underlying space, we automatically obtain the jT j. ‘clock’ dynamical model T Õ C

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Passage from equations to group action formulations of their solutions represents a significant change of viewpoint. From this new structuralist perspective, in formulating a state-based mathematical model or simply a dynamical model of a target system’s behavior, the modeler needs to specify the following two conceptual ingredients: a state space X (endowed with additional smooth, topological, or measurable structure, assumed to be respected by all maps on it) and a time-group action ı : T  X ! X, satisfying the identity and group properties. A defining property of a state space is its being the domain of realvalued, observable quantities. A prime example of an observable quantity in the pendulum model is the Hamiltonian, representing the pendulum’s energy levels. Structuralists argue that the particular system of equations (if any) that generates the group action, is of secondary concern. In a dynamical model, comprised of a state space and a time-group action on it, each statespace point is an abstract, information-bearing encoding of the target system’s physical mode of being at a given time. Assuming that the system under consideration has a physical state structure, its adequate state space model comes with a state space comprised of points that are presumed to contain complete information about the past history of the system, relevant to its future behavior. Thus, if the target system’s instantaneous physical state – encoded by a point in its representing state model, is known, then the system’s subsequent temporal evolution can be predicted with the help of the model’s time-group action, without any additional knowledge of what has happened to the system. The group action need not be continuous or smooth. It can also be measurable, computable, discrete, and local (i. e., only partially defined). The structuralist point of view of mathematical models also includes the so-called behavioral approach, proposed by Jan Willems [35]. It turns out to be just as effective as the group action approach. Recall from the basic pendulum model example above that the group action pair h ; (i has a function space transpose

b

h ;i

R/2Z  R ! (R/2Z  R)T ;

1

 defined by the˛ time function h ; (i(ha; bi) (t) Ddf ˝ a;b (t); ( a;b (t) for all time instants t and initial states ha; bi in R/2Z  R. Note that here the image function space h ; (i(R/2Z  R) is comprised of those time functions (pictured as state trajectories) in (R/2Z  R)T that satisfy the initially given pair of first-order pendulum equations.

1

In engineering applications, it is often conceptually advantageous to work directly with the solution space ) ( ˇ ˇ d2 g ˇ C D 0  C1 (T ) ˇ ˇ dt 2 ` of higher-order (linear) equations – interpreted as the subspace of behaviors, without any passage to a parametrizing state space. In this behavioral setting, a behavioral topological dynamical model is a triple hT ; X; Bi, consisting of a continuum-time domain T , a topological space X (interpreted as a signal space), and a function subspace B  X T of maps (encoding behaviors), satisfying the system’s dynamical equations. (Here and below, X T denotes the space of all continuous time-functions, i. e., continuous maps from T to X, endowed with the product topology.) This notion of a model is an outgrowth of the traditional input-output method, popular in engineering. The proponents of the behavioral approach in algebraic systems theory emphasize that in general the space of time-functions X T need not be just the space C1 (T ) of all smooth functions. Instead, it can also be the space of compactly supported smooth functions, locally integrable functions, distributions, and so forth. Polderman and Willems [27] provide a very readable introduction to the behavioral conception of mathematical models in systems theory. The foregoing lengthy digression into structuralist conceptions of models completes the first reason why the syntactic definition of mathematical models is unsatisfactory. We are now ready for the second reason. The second reason why the equational conception in science is inadequate is because equations are unable to deal fully and directly with intended empirical interpretations, representational power, denotation, model networks, and many other semantic and representational functions of models. To place the equations on a mathematically rigorous ground, the model builder must choose a host ring or field for the possible values of indeterminates and observable quantities. Beyond that, several other choices have to be made, including the ‘correct’ choice of a function space for indeterminates and quantities, and that of parameter spaces for constants. Depending on the interpretation, solutions can be everywhere defined and smooth, or they can be generalized functions (distributions), and so forth. There is also the problem of parameter identification and the question of how various quantities are to be measured. These functions of models are typically determined by the intended interpretation of solution spaces and phase portraits. Last but not least, parent models come with various structural enrichments and impoverishments – forming a model network. But of course

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in many applications, mathematical models involve both linguistic and structural (non-linguistic) elements. In representing natural systems, usually equations provide an all-important starting point. However, in view of various inevitable abstractions and idealizations, the corresponding group-action and behavioral models – having “more meat”, serve uniquely well as equation-world mediators or proxies, and stand-ins for the real-world target systems. Indeed, statements derived from well-confirmed equations are strictly true in the associated group-action and behavioral models, but they are only indirectly and approximately true of the actual system’s behavior. The applied mathematics literature usually leaves implicit many of the details needed for understanding the equations. So what is the merit of philosophical objections to the equational treatment of mathematical models? In their seminal work, Oberst [26], Röhrl [28], and Walcher [34] have established a natural Galois correspondence between the two (syntactic and structural) approaches, leading to a deeper understanding of the relations between the properties of (differential and difference) equations and the properties of their solutions, representing behavior. This fundamental relationship between the world of equations and that of solution spaces (or input-output behaviors) has been established for a large class of linear (and also for some nonlinear) cases, in the form of category-theoretic duality. In parallel with the above, classical models of statistical experiments, decision theory, game theory, and so forth, built over underlying smooth, topological, metric and measurable spaces, possess computationally convenient algebraic counterparts, granted by Stone–Gelfand duality results, discussed, e. g., in [17]. Having studied the ways in which equations lead to group-actions on state spaces and function spaces of trajectories representing behaviors, we now discuss how these and other types of models can be put together from more basic mathematical structures, and how certain universal constructions support the construction of complex models from simple ones. Philosophical and Mathematical Structuralism Since much of what is taken as distinctive of mathematical models and their connections to target systems is tied to mathematical structuralism, a brief description of the concept of structure is in order. We begin by reviewing the principal theses of philosophical structuralism. Within the current philosophical literature (e. g., [8]) on the status of scientific realism, two major positions can be discerned. The first, resurrected by Worrall [36], is epistemic structuralism that places a special restriction on scientific

knowledge. Its central thesis is that we can have knowledge of structures without knowledge of natures. Here structures are identified with relations – broadly understood, and natures are the intrinsic modes of being of objects that are related. This view is motivated by the need to handle the ontological discontinuity across theory-change in terms of a well-grounded knowledge of structures. For example, although conceptions of the nature of light have changed drastically through history (from a particle ontology to a wave ontology, and then again to a wave-particle duality), nevertheless, the old equation-based models describing the propagation of light have not been abandoned. Rather, at each stage of knowledge, they have been incorporated into more refined successor models. The second position, called ontic structuralism, initiated by Ladyman [16], holds that since structure is all there is to reality, what we can have at best is knowledge of the structural aspects of real-world systems. Inspired by the peculiar nature of quantum theory, ontic structuralism argues for the thesis that there are no objects. This object-free ontology leaves its proponents with the burden of showing how physico-chemical entities (e. g., superconductors and radioactive chemicals), endowed with astonishing causal powers, can be dissolved into structures, without anything left over. In the field of mathematical ontology, structuralism is regarded as one of the most successful approaches to the philosophy of mathematics. For example, Shapiro [31] states that mathematics is the study of independently existing structures, and not of collections of mathematical objects per se. This picture may seem to be a bit distorted to some, since not all of mathematics is concerned with structures. For example, the distribution of prime numbers in the set N of natural numbers and arguments as to why  must be a transcendental real are hardly structural matters. Be that as it may, from the standpoint of classical model theory, a mathematical structure is simply a list of operations and relations on a set, together with their required properties, commonly stated in terms of equational axioms. The most familiar examples of such structures are Bourbaki’s mother structures on sets: algebraic (e. g., time groups and quantity rings), topological (e. g., metric, measurable and topological state spaces), and order-theoretic (e. g., partially ordered sets of observables and event algebras). Of course, there are many vastly more elaborate composite types of set-theoretic structures, including ordered groups, topological groups, ordered topological groups, group actions, linear spaces, differentiable manifolds, bundles, sheaves, stacks, schemes, and so forth. The world of sets is granted by a fixed maximal (possibly Pla-

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tonist) mathematical ontology out of which all structures of mainstream mathematics are presumed to be built. The foregoing idea of mathematical structure is due mainly to the influence of Nicolas Bourbaki [7], who (together with other mathematicians around him) noted that mathematical structures of practical interest can be generated by three universal operations on sets: The (Cartesian) product XY of two sets X and Y, the power set P(X) of X, and the exponentiation-type function set Y X (consisting of all functions that map X into Y). For example, a group structure on a set X can be viewed as a designated element of the function set X (XX) with the usual equational properties of the group operation, together with a suitable element of X X (for the inverse operation) and a designated element 0 2 X. Similarly, a topological structure on a set X is given by a designated element of the iterated power set construct P(P(X)), satisfying the axioms of topology. Thus, Bourbaki’s concept of a mathematical structure is given by an underlying set together with some higher-order set-theoretic data, forming a string of designated elements of suitable product, power or function set constructions on it, and satisfying certain axioms. Unfortunately, Bourbaki’s definition of the concept of mathematical structure includes many pathological examples of no known utility. Furthermore, in general, Bourbaki does not consider the notion of structure-preserving mappings between mathematical structures, except isomorphisms that are always suitable functions, treated as subsets of the Cartesian product of their domains and codomains. The plurality of set theories with divergent properties in particular and that of universes of mathematical objects in general with wildly different and mutually inconsistent formulations have led to new ideas in philosophical structuralism. Among other things, these ideas suggest that in the presence of an abstract product ×, exponentiation ()() and other universal operations, any entity whatever can serve as a proxy for the underlying set-theoretic domain or carrier of a structure. This undermines the Platonist commitment to sets, believed to underly mathematical reality. The plurality of set theories has been replaced by a plurality of toposes – a special class of categories with products, function space constructions and subobject classification, in which most set-like mathematical constructions can be carried out. Category theory is seen by many as providing a structuralist framework for mathematics and hence also for mathematical models per se. Simply, structures are determined up to isomorphism, making the particular nature, individuality or constitution of their underlying domains irrelevant. Domains are only ‘positions’ in structured systems. For example, real numbers are fully specified by the structure of a complete Archimedean ordered

field that has many realizations. Mathematics is not about objects, either empirical or mathematical. It is about the axiomatic presentation of the structure of such objects in general, and of no objects in particular. If mathematics discusses objects, it does so only by construing them as ‘positions’ in structured systems, devoid of any special ontology. In sum, mathematical objects need not be sets and mappings need not be functions. However, mathematical structuralism does not always translate into physical structuralism. For example, the quantum structure of particles and the physical structure of liquids is empirically meaningless when viewed in isolation from their concrete physical carriers. Present-day mathematics uses structure-preserving mappings to specify different kinds of sets-with-structure. For example, instead of describing groups traditionally in terms of elements and higher-order data (i. e., operations satisfying the group axioms) on a set, mathematical practice demonstrates that it is far more effective to treat groups as abstract black-box type objects together with distinguished maps, emulating group homomorphisms between them. Upon engaging product and other operations in the construction of these abstractly given objects and maps, algebraists quickly arrive at a crucial universe of discouse, namely the category Grp of groups and group homomorphisms. For a thorough discussion of categories, see [22]. In [30] Dana Schlomiuk specifies the classical notion of a topological structure strictly in terms of abstract spaces and maps of a category Top – called the category of topological spaces, independently of the usual set-based means of specification, employing elements and closure operators, or algebras of open or closed subsets. Even far more sophisticated structures can be treated in this fundamentally structuralist manner. For example, a smooth structure of a differentiable manifold can be given categorytheoretically by specifying exactly which continuous maps must be smooth in the category of abstractly conceived smooth manifolds. Here the utility of category theory is in providing a uniform treatment of the concept of structure, solely in terms of holistically given ‘structure-preserving’ maps that serve remarkably well also the needs of empirical applications. Having now indicated the ways category theory specifies mathematical structures, we may profitably revisit the earlier discussed syntactic equational and structuralist approaches to mathematical models from the point of view we have just developed. Here we only give one illustration of this approach out of many possibilities, namely the category TopT of topological dynamical models in the form of a topological group-action on state spaces by a desig-

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nated time group hT ; 0; Ci, discussed in the previous Section. This category consists of topological dynamical models of the form T Õ X, serving as its objects, where the map ı ı : T  X ! X is a continuous action of time-group T on its codomain topological state space X. Next, maps between dynamical models of the form ' : (T Õ X) ! ı 0 ), called dynamorphisms, are given by continuous (T Õ X ı0 mappings ' : X ! X 0 between the underlying state spaces such that the diagram T ?X ? 1T ' y T  X0

ı

! ! ı0

X ? ?' y X0

commutes, i. e., the compositions of constituent maps along both paths in the diagram give the same map. In other words, we have the equality ' ı ı D ı 0 ı (1T  '). Because the identity maps of state spaces are trivially dynamorphisms and since the composition of two dynamorphisms is again a dynamorphism, by definition, TopT is indeed a category. X to T Õ X 0 is said to be A dynamorphism ' from T Õ ı ı0 an isomorphic dynamorphism or simply a conjugacy provided that ' : X ! X 0 is a homeomorphism. From the standpoint of mathematical systems theory, two conjugate dynamical models are structurally identical. Any dynamical property (i. e., a property defined in terms of a group action on its underlying state space) possessed by one is also possessed by the other. Concretely, a dynamical model TÕ X is said to be structurally stable provided that any dyı namical model T Õ X with transition map ı 0 sufficiently ı0 close to ı in a suitable topological sense is conjugated to X. TÕ ı Surjective (onto) dynamorphisms are called factor dynamorphisms. Each factor dynamorphism ' : X ! X 0 comes with its natural equivalence relation, given by x1  x2 iff '(x1 ) D '(x2 ), and induced quotient dynamical X . Injective (one-to-one) dynamorphims model T Õ ı' / specify dynamical submodels. For example, the trajectory ˚passing through a state x at time zero, defined by T (x) D ı(t; x) j t 2 T , can be viewed as a member of the family of the smallest nontrivial submodels of T Õ X. Many other ı notions and constructions available in the category Top of topological spaces automatically transfer to the category TopT of topological dynamical models and dynamorX)  (T Õ X 0 ) of phisms. For example, the product (T Õ ı ı0 Õ Õ 0 two dynamical models T ı X and T ı 0 X is defined in the same way as in topology, namely by (T Õ X)  (T Õ X 0 ) D T Õ (X  X 0 ) ; ı

ı0

df

ı; ı 0

where the diagonal action is given by [ı;ı 0 ](t; (x; x 0 )) D

˝

˛ ı(t; x); ı 0 (t; x 0 ) . Disjoint sum (coproduct) of two dynamical models is defined similarly. For more details regarding category-theoretic constructions, see [20] that includes an elaborate category-theoretic treatment of dynamical models. Similar constructions work with varying degrees of success in many other host categories. For example, if Top is replaced by the category Mes of measurable spaces and measurable maps between them, we obtain the much studied category MesT of measurable dynamical models. Another significant mutation of dynamical models arises, when the continuum-time domain is replaced by a discrete-time domain, isomorphic to the group hZ; 0; Ci of integers or the monoid hN; 0; Ci of natural numbers. Topological (smooth or measurable) dynamical models provide complete support also for the representation of perturbed, controlled, nonautonomous, and random dynamical systems, in terms of various skew-product constructions. In more detail, let T Õ X be a (topological, ı smooth or measurable) base dynamical model that represents a driving system (i. e., perturbation, control or environmental noise process), affecting the target system’s dynamical behavior. The model of this behavior, subject to the influence of added perturbation, control, and so on, is based on a so-called (continuous, differentiable or measurable) cocycle map ˛ : T  X  Y ! Y, acting on the driven target system’s principal state space Y and satisfying the following skew-product action (or cocycle) axioms for all t; t 0 in T , x 2 X, and y 2 Y: (i) Identity property: ˛(0; x; y) D y, and  (ii) Cocycle property: ˛(t C t 0 ; x; y) D ˛ t; ı(t 0 ; x);  ˛(t 0 ; x; y) . The space X  Y should be thought of, informally, as a trivial bundle, made up of fibers fxg  Y, indexed by points x 2 X and “glued together” by the topology. In the same way, for time instants t; t 0 , the cocycle map should be viewed as an indexed family ˛(t;;)

˛(t 0 ;ı(t;);)

fxgY ! fı(t; x)gY ! fı(t 0 Ct; x; )gY of maps between fiber state spaces. As the extant base state x at time t is shifted by the base model’s dynamics to ı(t; x), the restricted cocycle map ˛(t; x; ) moves each system state y in the fiber fxg  Y over x to the state ˛(t; x; y) belonging to the fiber fı(t; x)g  Y over ı(t; x). The skew-product (T Õ X)Ë ˛ Y of a base dynamical ı Õ model T ı X and a principal state space Y under cocycle ˛ acting on Y is defined by the dynamical model Õ (T Õ X)Ë ˛ Y Ddf T  (X  Y) ı on the trivial bundle X  Y, where  : T  (X  Y) !

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X  Y is specified by (t; (x; y)) Dd f hı(t; x); ˛(t; x; y)i for all time instants t and states x; y. It is easy to check that the transition map  is fiber-preseving and hence a bundle morphism. Since each group-action induces a skew-product action, product dynamical models are special cases of skew-product dynamical models. More importantly, because the second projection map Y : (T Õ X)Ë ˛ Y ! ı Õ (T ı X) is a factor dynamorphism, skew-products are best ÕX viewed as objects of the so-called slice category TopT //T ı that fuses topological (smooth, measurable, etc.) bundle theory with the theory of dynamical systems. Rohlin’s representation theorem states that the domains of factor morphisms in the category of probability spaces and maps are essentially skew-products. Thus, all stochastic representations of perturbed systems are tractable uniformly by skew-product constructions. Nonautonomous differential equations (describing nonautonomous systems) of the form dy/dt D f (t; y) with t 2 T and y 2 Y (involving an explicit time dependence), are known to have solutions that induce skewproduct dynamical models of the form (T Õ jT j)Ë ˛ Y over C a suitable cocycle ˛. Quite simply, the model enlarges the principal state space Y by the space jT j of time coordinates. Because solutions of stochastic differential equations have the form of cocycles over base measurable dynamical models, once again, smooth skew-products, called random dynamical models, provide the correct structuralist framework for them. For a thorough treatment of some of these topics, see [1] and [9].

disciplines) using only internal (i. e., internal to set theory) relations between, say, bodies and numbers. The defenders of the internal view treat the names of (say) planets as rigid designators, always picking the same object in every possible situation in which they exist at all. It is now easy to see how the intended empirical interpretation of mathematical models leads directly to claims about the physical world. It must be stressed, however, that there is a good basis for questioning this fairly popular approach. One obvious concern is this: For particles, bodies and other physical objects to form classical sets, it must be assumed that they can never be annihilated, split or changed in some other ways, leading to a loss of their identity. Under these types of implicit comprehensive idealizations, it is better to think of such ‘physical’ or ‘empirical sets’ as ordinary sets involving mathematical encodings. Simply, the actual physical objects under study are encodable in terms of suitable abstract elements (e. g., numbers) of a set, so that the informal semantic assumptions of the sort “B is a set of physical X is a dynambodies”, “X is a set of physical states”, “T Õ ı ical system”, and so forth, are treated as succinct abbreviations of the formal details of fallible mathematical encodings and their empirical interpretation. A more serious philosophical concern pertains to the nature of (e. g., differentiable) maps on various ‘empirical sets’, consisting of fluids and other objects of continuum mechanics.

Three Approaches to Applying Mathematical Models

Some philosophers suggest to draw a thick conceptual line between timeless mathematical models and actual physical systems. But then, for a model to be a model of something else, a relationship between a model and what it models is required. Specifically, applications of mathematical models involve certain external (preferably intensional) relations between models and real-world systems. In more detail, philosophical structuralists argue that the modelworld relation is best described by structure-preserving maps (usually isomorphisms or embeddings) between representing mathematical models and their intended physical (or generally empirical) situations. As a concrete illustration of this viewpoint, consider the representational model of weight measurement, studied in great detail by Kranz et al. in [15]. The empirical domain D of the ‘physical structure’ of weight measurement is a set of material objects. This domain is equipped with a binary ‘heavierthan’ weak-order relation , where the empirical statement d  d 0 – stating that object d is heavier than object d 0 , is operationally established by placing d and d 0 on the two pans of an equal-arm balance scale and then ob-

Since one of the most perplexing philosophical questions is how models relate to their target systems, in this Section we review three major approaches to the problem of model-world relationship. Internal Approach Set-theoretic models are applicable to real-world systems simply because physical objects can literally be members of legitimate sets and there are special functions between sets comprised of physical objects (e. g., particles and bodies) and pure mathematical objects (e. g., reals). As a classical example of a physical application of this nature, recall the application of Newton’s laws to the solar system. The proponents of the internal view (including, e. g., Steiner [32]) argue that planets together with the sun form a perfectly meaningful finite set on which real-valued mass and position functions can be defined in the usual way. In this manner, set-theoretic models are applied to physics (and other

External Approach

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serving which pan drops. In addition, there is a binary weakly commutative and weakly associative composition operation ˚ on D, where for objects d and d 0 the value d ˚ d 0 denotes the composite object obtained by placing both objects in the same pan with d beneath d 0 . Now, here the world-model relation is given by an algebraic homomorphism from the so-called physical extensive measurement structure hD; ; ˚i of weight measurement to the ordered additive structure hR; >; Ci of reals. Such homomorphisms are guaranteed to exist provided that the extensive measurement structure satisfies certain relatively simple axioms, similar to those used in the definitions of Archimedean-ordered semigroups. We see that for a structure-preserving world-tomodel map to exist, numerous idealizing restrictions must be placed on the target physical domain’s structure. For example, in the case of weight measurement, the Archimedean property forces the empirical domain D to be infinite, even though in applications experimenters always work with finitely many objects. Because idealizations typically involve known-to-be-false descriptions, ignorance of causes, and deliberate stipulations of objects that do not exist in the actual physical situation, a structure-preserving map can be guaranteed to exist only between the idealization itself and a representing mathematical model, but not necessarily between the actual realworld situation (enjoying unlimited degrees of freedom) and the model. Finally, one must question the ontological status of the structure-preserving maps (isomorphisms). If these maps go from a mathematical domain to a real-world physical domain or conversely, then their graphs, being subsets of a ‘hybrid set’ of physical-mathematical object pairs, take us back straight to the internal viewpoint, questioned earlier. A rival and equally popular approach to this general topic is to replace the isomorphism (or homomorphism) relation between models and the world by some other, technically less demanding devices. For example, Giere [13] proposes to solve the problem of structure-preserving maps by passing to considerably simpler graded context-dependent similarity relations between models and aspects of the real world. Thus, if the representing model is sufficiently similar to its target system, then the analysis of the model is also (indirectly) an analysis of the system. This solution comes at quite a price, since it is hard to see what exactly, if anything, is similar between the familiar Lotka–Volterra differential equations and the predator-prey two-species system of finitely many sharks and fishes in the Adriatic Sea. Another major concern is that under similarity anything can be a model of anything else, since any two things can always be claimed to be

similar in some respect and to some degree or other. In general, model builders cannot reduce the problem of application of models to similarity relations, because models usually come with falsifying idealizations that simply violate any similarity relation that may perhaps exist between intractable ‘perfect’ models and their systems. Last but not least, mathematical models tend to possess a ‘surplus structure’ and features that do not correspond to anything in particular, similar or not, in the actual system (e. g., recall the applications of complex analysis to the behavior of electric circuits or the role of undetectable measure-zero attractors in a dynamical model). Model Network Approach The relation between a mathematical model and its target system is not as straightforward as it is often thought to be. Models interact with other models, both globally and locally in complex ways, and their relationship to the world is not a static affair. If connections between models and their target systems were tractable by rigid relations, then a large bulk of scientific research would become redundant. However, as mathematical models are structurally mutated, their representational capacities often undergo unexpected refinements and changes. As a result, in subsequent applications new epistemic links between models and systems emerge, and accrue along the way. Thus, from the standpoint of scientific practice, an essential aspect of mathematical modeling is extendability, enrichment and open-endedness of model constructions. Model builders cannot realistically constrain models to be just single frozen structures. Instead, applied mathematical models are best viewed as complex networks of structures with distributed empirical interpretations that guide model users in the network-world interface to gather more informative data and make better predictions. (For related views but motivated by different considerations, see [4] and [5].) Applications of category theory to mathematical models has opened up the possibility of building new models from old ones via functorial constructions. In Sect. “Philosophical and Mathematical Structuralism”, we have given a brief introduction to some concepts, intended for the class of topological dynamical models. But there is more. For example, the study of deterministic dynamical systems operating in a chaotic regime – hard to understand in deterministic terms, calls for a passage to probabilistic dynamical models. These are obtained by introducing probability monads – special functors discussed by Giry [14], on suitable categories of topological and measurable spaces. In brief, the parent deterministic dy-

Philosophy of Science, Mathematical Models in

namical model T Õ X is used in conjunction with the ı associated logically higher-level probabilistic dynamical model T Õ D(X) with time-group action on the convex ı space D(X) of probability density functions, representing the target system’s stochastic states. Here the transition map ı  is defined by the so-called Perron–Frobenius integral equation, studied in great detail by Lasota and Mackey [19]. Mutation of T Õ X into nondeterministic, ı fuzzy and other dynamical models is also obtained by the monad approach. Since the time evolution of states of a target system is observed only indirectly via measurable real-valued quantities, defined on the representing model’s state space X, the parent model T Õ X is used in paralı lel with another associated higher-level dynamical model TÕ C(X) on the Banach algebra C(X) of (bounded) realı valued continuous functions on X. The induced timegroup action ı  , defined by the fundamental Gelfand duality result, characterizes the time evolution of observable quantities, crucial in time-series analysis. Because in applications the evaluation of the analytic parent model’s transition map tends to involve roundoff and other truncation operations that may have long-term pathological effects, and since generally the model’s time and space points cannot be identified with arbitrarily fine degrees of accuracy, several forms of (temporal, spatial and parametric) discretizations become necessary. Time discretizations of a continuum-time model T Õ X is solved by ı passing to discrete-time dynamical models ZÕ X, where ı Z D f   ; 2; ; 0; ; 2;    g is the group of discrete timesteps with sampling period  > 0. Spatial discretization of T Õ X is captured by a nested family of coarseı X , where the grained dynamical models of the form T Õ ı underlying state space X is comprised of a uniform grid (lattice) of finitely many points in X with mesh size D n1 , tractable with finitary resources. Spatially discretized dynamical models provide the formal meeting ground for measurement results and the parent model’s predictions. To understand the gap between ontologically and epistemologically motivated dynamical models, it is important to know how well a discretized space mimics the properties of the parent model’s state space. Remarkably, Hausdorff topological state spaces are known to be approximable by inverse limits of nested sequences of T 0 finite topological spaces. For more details on this subject, see [6]. Validating Mathematical Models Although many models are built for scientific research purposes, applications demand that models be validated iteratively by comparisons of their predictions with measurement data. Validation is intended to demonstrate that

the representing model generates predictions and claims in its domain of applicability that are consistent with its intended application – within an acceptable range of accuracy. Because it is not true that good predictions can only be obtained from a model that is causally sound and since data may not portray the target system accurately, full verification of a model requires various testing procedures for agreement with cause-effect relationships (operating in the target system) and analysis of inter-model relations in a model network to which it belongs. Validation tends to be limited by the available domain of measurement data – to be used in model-world comparisons, and generally is not sufficient to demonstrate that the model is adequate in its entire state space, modulo locally granted margins of error. As a matter of fact, mathematical models are seldom valid in all regions of their phase portraits. As we have seen, the most popular example is the Newtonian dynamical model of a simple pendulum that gradually fails as its angular velocity approaches the speed of light. In scientific practice, a dynamical model is typically valid only in limited regions of its underlying state space that are of particular interest. Problems arise when the model fails to be valid in most parts of the state space regions under consideration. Deficiencies in the model may be traced to a wrong choice of differential equations (that ignores higher degrees of nonlinearity), a crude choice of time and/or space discretization parameters, and reliance on data samples contaminated with gross errors. In many applications, it is far from sufficient to know that the model is statistically valid. The reason is that a statistically valid theoretical model may turn out to be dynamically (qualitatively) invalid, meaning that it may fail to represent correctly the dynamical invariants of the target system (including equilibrium states, periodic, aperiodic, chaotic or strange attractors, and their basins) under various choices of parameter values. Note that even if a dynamical model provides highly accurate predictions in its tested state space regions, it does not necessarily follow that the model is dynamically valid, since it may include spurious dynamics in untested state space regions that are also of interest. The spectrum of actual behaviors of the target system remains largely unknown from the perspective of its representing model. Further observation and theoretical research may be needed to obtain a better knowledge about the dynamical invariants and characteristics of the system. These considerations lead us to reason about the validation of dynamical models geometrically, in terms of suitable conditional geometric measures of adequacy. Given a parent dynamical model of a target system, sup-

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the adequacy of the given dynamical model by the ratio (Meas \ (Pred  Fail)) ; (Meas)

Philosophy of Science, Mathematical Models in, Figure 1 Geometry of model validation

pose the associated body of actual observation and measurement data, available at a given stage of knowledge and collected independently of the model under consideration, forms a subset Meas of a finite grid of the parent model’s underlying state space, as illustrated by a shaded circle in Fig. 1 below. The size of the set Meas is constrained by various resources, available measuring instruments, their accuracy, and research interests of the validating scientist. Of course, Meas varies with time and it can form a multiply connected or scattered subset of the state space. For simplicity, we assume that the discretization embodied in the grid captures the uniform admissible margin of error. Along related lines, let Pred be a subset of a finite grid of the parent model’s state space, given by all quantitative predictions calculated from the representing model at a given stage of research and depicted by the second shaded circle in Fig. 1 that overlaps with Meas. Naturally, the set Pred grows in time as brand-new predictions are generated by the model. It should be obvious that the intersection Meas \ Pred represents those predictions that have been checked by observation or measurement. Since the model of interest need not be perfect or fully reliable, it is likely to generate some predictions that falsify the model. In Fig. 1, the set of falsifiers is indicated by the ellipse Fail, forming a subset of Pred. Note that at the current stage of research, only a proper subset of incorrect predictions in Fail is comparable with measurement results. The other potentially falsifying predictions in Fail have not yet been verified. Finally, note also that subsequent discrete measurements of a state trajectory (indicated in Fig. 1 by a dotted curve segment) over a longer period of time may gradually diverge from the corresponding parent model-given continuous state trajectory, both generated by the same presumed initial state. We are now ready to use the geometric scheme developed above to measure the adequacy of dynamical models. Even though measurement results and predictions form finite sets, for the sake of simplicity we choose to measure

where  denotes the Lebesgue measure defined on the model’s underlying state space. The idea is that the ratio of the volume of fully verified correct predictions and the volume of all available measurement results is a good indicator of the model’s adequacy at a given stage of knowledge. In any case, it should be clear that a semantic approach to confirmation theory must directly engage the structure of dynamical models under consideration. Because adequacy and reliability should be judged according to the volume of phase portrait regions on which measurements and predictions agree, modulo admissible errors, measures of the sort displayed above are in the ballpark of measuring adequacy. Note, however, that just like many other measures in dynamical systems theory (including topological and Kolmogorov–Sinai entropy functions), the foregoing measure of model adequacy is largely conceptual and not readily implementable in all instances of real-life models. Testing models for adequacy is conceptually quite similar to testing statistical hypotheses. Recall that a given representative sample of data validates a statistical hypothesis about the target population only with a certain degree of confidence; the larger the sample, the higher the degree of confidence in the correctness of the hypothesis. Likewise, representative measurement data validate the dynamical model under consideration via its predictions pertaining to the target system’s addressed behavior only locally, specified by a region in its phase portrait. The larger the body of validating data in its phase portrait, the higher the degree of confidence in the model’s global adequacy. Future Directions We do not give a detailed list but briefly mention two major directions of current research. Galois Correspondence Between Equational and Structuralist Approaches There are several algebraic methods in the realm of linear differential equations that demonstrate a Galois correspondence between differential or difference ideals of equations (modules of differential operators) and group-action (behavioral) models. There are reasons to conjecture that these results remain valid also for a large class of nonlinear differential or difference equations. Although some examples of this are studied by Walcher [34], a more complete understanding of the equation-solution relationship

Philosophy of Science, Mathematical Models in

is needed. A similar Galois connection must be addressed also in the context of stochastic differential equations and random dynamical models or stochastic flows.

19.

Validation and Verification of Mathematical Models It is a problem of considerable interest to formalize the processes of model validation and testing. In recent years efforts have been made to come to grips with this problem in the field of validation of computer simulation models. (See, for example, the influential work of Sargent [29].) However, a substative theory of dynamical model validation that circumvents the inadequacies of classical confirmation theory is not yet available.

20. 21. 22. 23. 24. 25.

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1. Arnold L (1998) Random dynamical systems. Springer, New York 2. Atmanspacher H, Primas H (2003) Epistemic and ontic realities. In: Castell L and Ischebeck O (eds) Time, Quantum and Information. Springer, New York, pp 301–321 3. Bailer-Jones DM (2002) Scientists’ thoughts on scientific models. Perspectives Sci 10:275–301 4. Balzer W, Moulines CU, Sneed J (1987) An Architectonic for science: The structuralist program. Reidel, Dordrecht 5. Balzer W, Moulines CU (1996) Structuralist theory of science. Focal issues, new results. de Gruyter, New York 6. Batitsky V, Domotor Z (2007) When good theories make bad predictions. Synthese 157:79–103 7. Bourbaki N (1950) The architecture of mathematics. Am Math Mon 57:221–232 8. Chakravartty A (2004) Structuralism as a form of scientific realism. Int Stud Philos Sci 18:151–171 9. Colonius F, Kliemann W (1999) The dynamics of control. Birkhäuser, Boston 10. Da Costa NCA, French S (2003) Science and partial truth. A unitary approach to models and scientific reasoning. Oxford University Press, New York 11. Erkal C (2000) The simple pendulum: a relativistic revisit. Eur J Phys 21:377–384 12. Fulford G, Forrester P and Jones A (1997) Modelling with differential and difference equations. Cambridge University Press, New York 13. Giere R (2004) How models are used to represent reality. Philos Sci (Proc) 71:742–752 14. Giry M (1981) A categorical approach to probability theory. In: Banaschewski B (ed) Categorical Aspects of Topology and Analysis, Lecture Notes in Mathematics, vol 915. Springer-Verlag, New York, pp 68–85 15. Krantz DH, Luce RD, Suppes P, Tversky A (1971) Foundations of measurement. Academic Press, New York 16. Ladyman J (1998) What is structural realism? Stud Hist Philos Sci 29:409–424 17. Lambek J, Rattray BA (1979) A general Stone-Gelfand duality. Trans Am Math Soc 248:1–35 18. Landry E (1999) Category theory: The language of mathemat-

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ics. In: Howard DA (ed) Proceeedings of the 1998 Biennial Meeting of the Philosophy of Science Association. The University of Chicago Press, Chicago, pp S14–S26 Lasota A, Mackey MC (1994) Chaos, Fractals and Noise. Stochastic Aspects of Dynamics, 2nd edn. Springer-Verlag, New York Lawvere FW (2005) Taking categories seriously. Theor Appl Categ 8:1–24 Le Cam L (1986) Asymptotic Methods in Statistical Decision Theory. Springer-Verlag, New York Mac Lane S (1971) Categories for the working mathematician. Springer-Verlag, New York Mac Lane S (1996) Structure in Mathematics. Philosophia Mathematica 20:1–175 Moore C (1991) Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity 4:199–230 Morgan M (2005) Experiments versus models: New phenomena, inference and surprise. J Econ Methodol 12:317–329 Oberst U (1990) Multidimensional constant linear systems. Acta Appl Math 68:59–122 Polderman JW, Willems JC (1998) Introduction to mathematical systems theory: A behavioral approach. Texts in Applied Mathematics 26. Springer-Verlag, New York Röhrl H (1977) Algebras and differential equations. Nagoya Math J 68:59–122 Sargent RG (2003) Verification and validation of simulation models. In: Chick S, Sanchez PJ, Ferrin E, and Morrice DJ (eds) Proceedings of the 2003 Winter Simulation Conference. IEEE, Piscataway, pp 37–48 Schlomiuk DI (1970) An elementary theory of the category of topological spaces. Trans Am Math Soc 149:259–278 Shapiro S (1997) Philosophy of Mathematics. Structure and Ontology. Oxford University Press, Oxford Steiner M (1998) The applicability of mathematics as a philosophical problem. Harvard University Press, New York Suarez M (2003) Scientific representation: Against similarity and isomorphism. Int Stud Philos Sci 17:225–244 Walcher S (1991) Algebras and differential equations. Hadronic Press, Palm Harbor Willems JC (1991) Paradigms and puzzles in the theory of dynamical systems. IEEE Trans Automatic Control 36:259–294 Worrall J (1996) Structural realism: the best of both worlds? In: Papineau D (ed) The Philosophy of Science, Oxford University Press, New York, pp 139–165

Books and Reviews Brin M, Stuck G (2002) Introduction to Dynamical Systems. Cambridge University Press, New York Hellman G (2001) Three varieties of mathematical structuralism. Philosophia Mathematica 9:184–211 Margani L, Nersessian N, Thagard P (eds) (1999) Model-based Reasoning in Scientific Discovery. Kluwer, Dordrecht Margani L, Nersessian N (eds) (2002) Model-based Reasoning: Science, Technology, Values. Kluwer, Dordrecht Morgan M, Morrison M (1999) Models as Mediators. Perspectives on Natural and Social Science. Cambridge University Press, Cambridge Psillos S (1999) Scientific Realism: How Science Tracks Truth. Routledge, London Suppes P (2002) Representation and Invariance of Scientific Structures. CSLI Publications, Stanford University, Stanford

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Pressure and Equilibrium States in Ergodic Theory JEAN-RENÉ CHAZOTTES1 , GERHARD KELLER2 1 Centre de Physique Théorique, CNRS/École Polytechnique, Palaiseau, France 2 Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

a functional of the form F() D h() C h; i are called “equilibrium states” for . Pressure The maximum of the functional F() is denoted by P() and called the “topological pressure” of , or simply the “pressure” of . Gibbs state In many cases, equilibrium states have a local structure that is determined by the local properties of the potential . They are called “Gibbs states”. Sinai–Ruelle–Bowen measure Special equilibrium or Gibbs states that describe the statistics of the attractor of certain smooth dynamical systems.

Article Outline Glossary Definition of the Subject Introduction Warming Up: Thermodynamic Formalism for Finite Systems Shift Spaces, Invariant Measures and Entropy The Variational Principle: A Global Characterization of Equilibrium The Gibbs Property: A Local Characterization of Equilibrium Examples on Shift Spaces Examples from Differentiable Dynamics Nonequilibrium Steady States and Entropy Production Some Ongoing Developments and Future Directions Bibliography Glossary Dynamical system In this article: a continuous transformation T of a compact metric space X. For each x 2 X, the transformation T generates a trajectory (x; Tx; T 2 x; : : : ). Invariant measure In this article: a probability measure  on X which is invariant under the transformation T, i. e., for which h f ı T; i D h f ; i for each continuous Rf : X ! R. Here h f ; i is a short-hand notation for X f d. The triple (X; T; ) is called a measurepreserving dynamical system. Ergodic theory Ergodic theory is the mathematical theory of measure-preserving dynamical systems. Entropy In this article: the maximal rate of information gain per time that can be achieved by coarse-grained observations on a measure-preserving dynamical system. This quantity is often denoted h(). Equilibrium state In general, a given dynamical system T : X ! X admits a huge number of invariant measures. Given some continuous  : X ! R (“potential”), those invariant measures which maximize

Definition of the Subject Gibbs and equilibrium states of one-dimensional lattice models in statistical physics play a prominent role in the statistical theory of chaotic dynamics. They first appear in the ergodic theory of certain differentiable dynamical systems, called “uniformly hyperbolic systems”, mainly Anosov and Axiom A diffeomorphisms (and flows). The central idea is to “code” the orbits of these systems into (infinite) symbolic sequences of symbols by following their history on a finite partition of their phase space. This defines a nice shift dynamical system called a subshift of finite type or a topological Markov chain. Then the construction of their “natural” invariant measures and the study of their properties are carried out at the symbolic level by constructing certain equilibrium states in the sense of statistical mechanics which turn out to be also Gibbs states. The study of uniformly hyperbolic systems brought out several ideas and techniques which turned out to be extremely fruitful for the study of more general systems. Let us mention the concept of Markov partition and its avatars, the very important notion of SRB measure (after Sinai, Ruelle, and Bowen) and transfer operators. Recently, there was a revival of interest in Axiom A systems as models to understand nonequilibrium statistical mechanics. Introduction Our goal is to present the basic results on one-dimensional Gibbs and equilibrium states viewed as special invariant measures on symbolic dynamical systems, and then to describe without technicalities a sample of results they allow to obtain for certain differentiable dynamical systems. We hope that this contribution will illustrate the symbiotic relationship between ergodic theory and statistical mechanics, and also information theory. We start by putting Gibbs and equilibrium states in a general perspective. The theory of Gibbs states and equilibrium states, or Thermodynamic Formalism, is a branch of rigorous Statistical Physics. The notion of a Gibbs state

Pressure and Equilibrium States in Ergodic Theory

dates back to R.L. Dobrushin (1968–1969) [17,18,19,20] and O.E. Lanford and D. Ruelle (1969) [41] who proposed it as a mathematical idealization of an equilibrium state of a physical system which consists of a very large number of interacting components. For a finite number of components, the foundations of statistical mechanics were already laid in the nineteenth century. There was the wellknown Maxwell–Boltzmann–Gibbs formula for the equilibrium distribution of a physical system with given energy function. From the mathematical point of view, the intrinsic properties of very large objects can be made manifest by performing suitable limiting procedures. Indeed, the crucial step made in the 1960s was to define the notion of a Gibbs measure or Gibbs state for a system with an infinite number of interacting components. This was done by the familiar probabilistic idea of specifying the interdependence structure by means of a suitable class of conditional probabilities built up according to the Maxwell– Boltzmann–Gibbs formula [29]. Notice that Gibbs states are often called “DLR states” in honor of Dobrushin, Lanford, and Ruelle. The remarkable aspect of this construction is the fact that a Gibbs state for a given type of interaction may fail to be unique. In physical terms, this means that a system with this interaction can take several distinct equilibria. The phenomenon of nonuniqueness of a Gibbs measure can thus be interpreted as a phase transition. Therefore, the conditions under which an interaction leads to a unique or to several Gibbs measures turns out to be of central importance. While Gibbs states are defined locally by specifying certain conditional probabilities, equilibrium states are defined globally by a variational principle: they maximize the entropy of the system under the (linear) constraint that the mean energy is fixed. Gibbs states are always equilibrium states, but the two notions do not coincide in general. However, for a class of sufficiently regular interactions, equilibrium states are also Gibbs states. In the effort of trying to understand phase transitions, simplified mathematical models were proposed, the most famous one being undoubtedly the Ising model. This is an example of a lattice model. The set of configurations of d a lattice model is X :D AZ , where A is a finite set, which is invariant by “spatial” translations. For the physical interpretation, X can be thought, for instance, as the set of infinite configurations of a system of spins on a crystal lattice Zd and one may take A D f1; C1g, i. e., spins can take two orientations, “up” and “down”. The Ising model is defined by specifying an interaction (or potential) between spins and studying the corresponding (translationinvariant) Gibbs states. The striking phenomenon is that for d D 1 there is a unique Gibbs state (in fact a Markov

measure) whereas if d  2, there may be several Gibbs states although the interaction is very simple [29]. Equilibrium states and Gibbs states of one-dimensional lattice models (d D 1) played a prominent role in understanding the ergodic properties of certain types of differentiable dynamical systems, namely uniformly hyperbolic systems, Axiom A diffeomorphisms in particular. The link between one-dimensional lattice systems and dynamical systems is made by symbolic dynamics. Informally, symbolic dynamics consists of replacing the orbits of the original system by its history on a finite partition of its phase space labeled by the elements of the “alphabet” A. Therefore, each orbit of the original system is replaced by an infinite sequence of symbols, i. e., by an element of the set AZ or AN , depending on whether the map describing the dynamics is invertible or not. The action of the map on an initial condition is then easily seen to correspond to the translation (or shift) of its associated symbolic sequence. In general there is no reason to get all sequences of AZ or AN . Instead one gets a closed invariant subset X (a subshift) which can be very complicated. For a certain class of dynamical systems the partition can be successfully chosen so as to form a Markov partition. In this case, the dynamical system under consideration can be coded by a subshift of finite type (also called a topological Markov chain) which is a very nice symbolic dynamical system. Then one can play the game of statistical physics: for a given continuous, real-valued function (a “potential”) on X, construct the corresponding Gibbs states and equilibrium states. If the potential is regular enough, one expects uniqueness of the Gibbs state and that it is also the unique equilibrium state for this potential. This circle of ideas – ranging from Gibbs states on finite systems over invariant measures on symbolic systems and their (Shannon-)entropy with a digression to Kolmogorov–Chaitin complexity to equilibrium states and Gibbs states on subshifts of finite type – is presented in the next four sections. At this point it should be remembered that the objects which can actually be observed are not equilibrium states (they are measures on X) but individual symbol sequences in X, which reflect more or less the statistical properties of an equilibrium state. Indeed, most sequences reflect these properties very well, but there are also rare sequences that look quite different. Their properties are described by large deviations principles which are not discussed in the present article. We shall indicate some references along the way. In Sects. “Examples on Shift Spaces” and “Examples from Differentiable Dynamics” we present a selection of important examples: measure of maximal entropy, Markov measures and Hofbauer’s example of nonuniqueness of equilibrium state; uniformly expanding Markov

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maps of the interval, interval maps with an indifferent fixed point, Anosov diffeomorphisms and Axiom A attractors with Sinai–Ruelle–Bowen measures, and Bowen’s formula for the Hausdorff dimension of conformal repellers. As we shall see, Sinai–Ruelle–Bowen measures are the only physically observable measures and they appear naturally in the context of nonuniformly hyperbolic diffeomorphisms [71]. A revival of the interest to Anosov and Axiom A systems occurred in statistical mechanics in the 1990s. Several physical phenomena of nonequilibrium origin, like entropy production and chaotic scattering, were modeled with the help of those systems (by G. Gallavotti, P. Gaspard, D. Ruelle, and others). This new interest led to new results about old Anosov and Axiom A systems, see, e. g., [15] for a survey and references. In Sect. “Nonequilibrium Steady States and Entropy Production”, we give a very brief account of entropy production in the context of Anosov systems which highlights the role of relative entropy. This article is a little introduction to a vast subject in which we have tried to put forward some aspects not previously described in other expository texts. For readers willing to deepen their understanding of equilibrium and Gibbs states, there are the classic monographs by Bowen [6] and by Ruelle [58], the monograph by one of us [38], and the survey article by Chernov [15] (where Anosov and Axiom A flows are reviewed). Those texts are really complementary.

Warming Up: Thermodynamic Formalism for Finite Systems We introduce the thermodynamic formalism in an elementary context, following Jaynes [34]. In this view, entropy, in the sense of information theory, is the central concept. Incomplete knowledge about a system is conveniently described in terms of probability distributions on the set of its possible states. This is particularly simple if the set of states, call it X, is finite. Then the equidistribution on X describes complete lack of knowledge, whereas a probability vector that assigns probability 1 to one single state and probability 0 to all others represents maximal information about the system. A well-established measure of the amount of uncertainty represented by a probability distribution  D ((x))x2X is its entropy H() :D 

X x2X

which is zero if the probability is concentrated in one state and which attains its maximum value log jXj if  is the equidistribution on X, i. e., if (x) D jXj1 for all x 2 X. In this completely elementary context we will explore two concepts whose generalizations are central to the theory of equilibrium states in ergodic theory:  Equilibrium distributions – defined in terms of a variational problem.  The Gibbs property of equilibrium distributions. The only mathematical prerequisite for this section are calculus and some elements from probability theory. Equilibrium Distributions and the Gibbs Property Suppose that a finite system can be observed through a function U : X ! R (an “observable”), and that we are looking for a probability distribution  which maximizes entropy among all distributions  with a prescribed P expected value hU; i :D x2X (x)U(x) for the observable U. This means we have to solve a variational problem under constraints: H() D maxfH() : hU; i D Eg :

As the function  7! H() is strictly concave, there is a unique maximizing probability distribution  provided the value E can be attained at all by some hU; i. In order to derive an explicit formula for this  we introduce a Lagrange multiplier ˇ 2 R and study, for each ˇ, the unconstrained problem H(ˇ ) C hˇU; ˇ i D p(ˇU) :D max(H() C hˇU; i) : 

(2) In analogy to the convention in ergodic theory we call p(ˇU) the pressure of ˇU and the maximizer ˇ the corresponding equilibrium distribution (synonymously equilibrium state). The equilibrium distribution ˇ satisfies ˇ (x) D exp(p(ˇU) C ˇU(x))

for all x 2 X

(3)

as an elementary calculation using Jensen’s inequality for the strictly convex function t 7! t log t shows: H() C hˇU; i D

X

(x) log

x2X

 log

X x2X

(x) log (x) ;

(1)

D log

X

x2X

(x)

eˇ U (x) (x) eˇ U (x) (x)

eˇ U (x) ;

Pressure and Equilibrium States in Ergodic Theory

with equality if and only if eˇ U is a constant multiple of . The observation that  D ˇ is a maximizer proves at the P same time that p(ˇU) D log x2X eˇ U (x) . The equality expressed in (3) is called the Gibbs property of ˇ , and we say that ˇ is a Gibbs distribution if we want to stress this property. In order to solve the constrained problem (1) it remains to show that there is a unique multiplier ˇ D ˇ(E) such that hU; ˇ i D E. This follows from the fact that the map ˇ 7! hU; ˇ i maps the real line monotonically onto the interval (min U; max U) which, in turn, is a direct consequence of the formulas for the first and second derivative of p(ˇU) w.r.t ˇ: dp D hU; ˇ i ; dˇ

d2 p D hU 2 ; ˇ i  hU; ˇ i2 : (4) dˇ 2

As the second derivative is nothing but the variance of U under ˇ , it is strictly positive (except when U is a constant function), so that ˇ 7! hU; ˇ i is indeed strictly increasing. Observe also that dp/dˇ is indeed the directional derivative of p : RjAj ! R in direction U. Hence the first identity in (4) can be rephrased as: ˇ is the gradient at ˇU of the function p. A similar analysis can be performed for an Rd -valued observable U. In that case a vector ˇ 2 Rd of Lagrange multipliers is needed to satisfy the d linear constraints. Systems on a Finite Lattice We now assume that the system has a lattice structure, modeling its extension in space, for instance. The system can be in different states at different positions. More specifically, let Ln D f0; 1; : : : ; n  1g be a set of n positions in space, let A be a finite set of states that can be attained by the system at each of its sites, and denote by X :D ALn the set of all configurations of states from A at positions of Ln . It is helpful to think of X as the set of all words of length n over the alphabet A. We focus on observables U n which are sums of many local contributions in the P n1 sense that U n (a0 : : : a n1 ) D iD0 (a i : : : a iCr1 ) for some “local observable”  : Ar ! R. (The index i C r  1 has to be taken modulo n.) In terms of  the maximizing measure can be written as ˇ (a0 : : : a n1 ) D exp nP(ˇ) C ˇ

n1 X

! (a i : : : a iCr1 )

; (5)

iD0

where P(ˇ) :D n1 p(ˇU n ). A first immediate consequence of (5) is the invariance of ˇ under a cyclic shift of its argument, namely ˇ (a1 : : : a n1 a0 ) D ˇ (a0 : : :

a n1 ). Therefore, we can restrict the maximizations in (1) and (2) to probability distributions  which are invariant under cyclic translations which yields P(ˇ) D max(n1 H() C hˇ; i) 

D n1 H(ˇ ) C hˇ; ˇ i :

(6)

If the local observable  depends only on one coordinate, ˇ turns out to be a product measure: ˇ (a0 : : : a n1 ) D

n1 Y

exp (P(ˇ) C ˇ(a i )) :

iD0

Indeed, comparison with (3) shows that ˇ is the n-fold on A that maxproduct of the probability distribution loc ˇ imizes H() C ˇ() among all distributions  on A. It follows that n1 H(ˇ ) D H(loc ) so that (6) implies ˇ P(ˇ) D p(ˇ) for observables  that depend only on one coordinate. Shift Spaces, Invariant Measures and Entropy We now turn to shift dynamical systems over a finite alphabet A. Symbolic Dynamics We start by fixing some notation. Let N denote the set f0; 1; 2; : : : g. In the sequel we need  a finite set A (the “alphabet”),  the set AN of all infinite sequences over A, i. e., the set of all x D x0 x1 : : : with x n 2 A for all n 2 N,  the translation (or shift)  : AN ! AN , ( x)n D x nC1 , for all n 2 N,  a shift invariant subset X D  (X) of AN . With a slight abuse of notation we denote the restriction of  to X by  again. We mention two interpretations of the dynamics of  : it can describe the evolution of a system with state space X in discrete time steps (this is the prevalent interpretation if  : X ! X is obtained as a symbolic representation of another dynamical system), or it can be the spatial translation of the configuration of a system on an infinite lattice (generalizing the point of view from Subsect. “Systems on a Finite Lattice” above). In the latter case one usually looks at the shift on the two-sided shift space AZ , for which the theory is nearly identical. On AN one can define a metric d by d(x; y) :D 2N(x;y) where N(x; y) :D minfk 2 N : x k ¤ y k g :

(7)

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erty that, for all bounded measurable f : X ! R,

Hence d(x; y) D 1 if and only if x0 ¤ y0 , and d(x; x) D 0 upon agreeing that N(x; x) D 1 and 21 D 0. Equipped with this metric, AN becomes a compact metric space and  is easily seen to be a continuous surjection of AN . Finally, if X is a closed subset of AN , we call the restriction  : X ! X, which is again a continuous surjection, a shift dynamical system. We remark that d generates on AN the product topology of the discrete topology on A, just as many variants of d do. For more details  Symbolic Dynamics. As usual, C(X) denotes the space of real-valued continuous functions on X equipped with the supremum norm k  k1 .

i. e., for a set of x of -measure one. They are the indecomposable “building blocks” of all other measures in M (X),  Measure Preserving Systems or  Ergodic Theorems. The almost everywhere convergence in (8) is Birkhoff’s ergodic Theorem  Ergodic Theorems, the constant limit characterizes the ergodicity of .

Invariant Measures

Entropy of Invariant Measures

A probability distribution  (or simply distribution) on X is a Borel probability measure on X. It is unambiguously specified by its values [a0 : : : a n1 ] (n 2 N, a i 2 A) on cylinder sets [a0 : : : a n1 ] :D fx 2 X : x i D a i for all i D 0; : : : ; n  1g : Any bounded and measurable f : X ! R (in particular any f 2 C(X)) can be integrated by any distribution . To stress the linearity of the integral in both, the integrand and the integrator, we use the notation Z h f ; i :D

f d : X

In probabilistic terms, h f ; i is the expectation of the observable f under . The set M(X) of all probability distributions is compact in the weak topology, the coarsest topology on M(X) for which  7! h f ; i is continuous for all f 2 C(X),  Measure Preserving Systems, Subsect. “Existence of Invariant Measures”. (Note that in functional analysis this is called the weak-* topology.) Henceforth we will use both terms, “measure” and “distribution”, if we talk about probability distributions. A measure  on X is invariant if expectations of observables are unchanged under the shift, i. e., if h f ı ; i D h f ; i for all bounded measurable f : X ! R : The set of all invariant measures is denoted by M (X). As a closed subset of M(X) it is compact in the weak topology. Of special importance among all invariant measures  are the ergodic ones which can be characterized by the prop-

lim

n!1

n1 1X f ( k x) D h f ; i n kD0

for -a. e. (almost every) x ;

(8)

We give a brief account of the definition and basic properties of the entropy of an invariant measure . For details and the generalization of this concept to general dynamical systems we refer to  Entropy in Ergodic Theory or [37], and to [36] for an historical account. Let  2 M (X). For each n > 0 the cylinder probabilities [a0 : : : a n1 ] give rise to a probability distribution on the finite set ALn , see Sect. “Warming Up: Thermodynamic Formalism for Finite Systems”, so X H n () :D  [a0 : : : a n1 ] log [a0 : : : a n1 ] a 0 ;:::;a n1 2A

is well defined. Invariance of  guarantees that the sequence (H n ())n>0 is subadditive, i. e., H kCn ()  H k () C H n (), and an elementary argument shows that the limit h() :D lim

n!1

1 H n () 2 [0; log jAj] n

(9)

exists and equals the infimum of the sequence. We simply call it the entropy of . (Note that for general subshifts X many of the cylinder sets [a0 : : : a n1 ]  X are empty. But, because of the continuity of the function t 7! t log t at t D 0, we may set 0 log 0 D 0, and, hence, this does not affect the definition of H n ().) The entropy h() of an ergodic measure  can be observed along a “typical” trajectory. That is the content of the following theorem, sometimes called the “ergodic theorem of information theory”  Entropy in Ergodic Theory. Theorem (Shannon–McMillan–Breiman Theorem) lim

n!1

1 log [x0 : : : x n1 ] D h() n

for -a. e. x : (10)

Observe that (9) is just the integrated version of this statement. A slightly weaker reformulation of this theorem

Pressure and Equilibrium States in Ergodic Theory

(again for ergodic ) is known as the “asymptotic equipartition property”. Asymptotic Equipartition Property Given (arbitrarily small) > 0 and ˛ > 0, one can, for each sufficiently large n, partition the set An into a set Tn of typical words and a set En of exceptional words such that each a0 : : : a n1 2 Tn satisfies (11) en(h()C˛)  [a0 : : : a n1 ]  en(h()˛) P

and the total probability a 0 :::a n1 2En [a0 : : : a n1 ] of the exceptional words is at most ".

A Short Digression on Complexity Kolmogorov [40] and Chaitin [14] introduced the concept of complexity of an infinite sequence of symbols. Very roughly it is defined as follows: First, the complexity K(x0 : : : x n1 ) of a finite word in An is defined as the bit length of the shortest program that causes a suitable general purpose computer (say a PC or, for the mathematically minded reader, a Turing machine) to print out this word. Then the complexity of an infinite sequence is defined as K(x) :D lim supn!1 n1 K(x0 : : : x n1 ). Of course, the definition of K(x0 : : : x n1 ) depends on the particular computer, but as any two general purpose computers can be programmed to simulate each other (by some finite piece of software), the limit K(x) is machine independent. It is the optimal compression factor for long initial pieces of a sequence x that still allows complete reconstruction of x by an algorithm. Brudno [8] showed: If X  AN and  2 M (X) is ergodic, then K(x) D 1 log 2 h() for -a. e. x 2 X. Entropy as a Function of the Measure An important technical remark for the further development of the theory is that the entropy function h : M (X) ! [0; 1) is upper semicontinuous. This means that all sets f : h()  tg with t 2 R are closed and hence compact. In particular, upper semicontinuous functions attain their supremum. Indeed, suppose a sequence  k 2 M (X) converges weakly to some  2 M (X) and h( k )  t for all k so that also n1 H n ( k )  t for all n and k. As H n () is an expression that depends continuously on the probabilities of the finitely many cylinders [a0 : : : a n1 ] and as the indicator functions of these sets are continuous, n1 H n () D lim k!1 n1 H n ( k )  t, hence h()  t in the limit n ! 1.

A word of caution seems in order: the entropy function is rarely continuous. For example, on the full shift X D AN each invariant measure, whatever its entropy is, can be approximated in the weak topology by equidistributions on periodic orbits. But all these equidistributions have entropy zero. The Variational Principle: A Global Characterization of Equilibrium Usually, a dynamical systems model of a “physical” system consists of a state space and a map (or a differential equation) describing the dynamics. An invariant measure for the system is rarely given a priori. Indeed, many (if not most) dynamical systems arising in this way have uncountably many ergodic invariant measures. This limits considerably the “practical value” of Birkhoff’s ergodic theorem (8) or the Shannon–McMillan–Breiman theorem (10): not only do the limits in these theorems depend on the invariant measure , but also the sets of points for which the theorems guarantee almost everywhere convergence are practically disjoint for different  and  0 in M (X). Therefore, a choice of  has to be made which reflects the original modeling intentions. We will argue in this and the next sections that a variational principle with a judiciously chosen “observable” may be a useful guideline – generalizing the observations for finite systems collected in the corresponding section above. As announced earlier we restrict again to shift dynamical systems, because they are rather universal models for many other systems. Equilibrium States We define the pressure of an observable  2 C(X) as P() :D supfh() C h; i :  2 M (X)g :

(12)

Since M (X) is compact and the functional  7! h() C h; i is upper semicontinuous, the supremum is attained – not necessarily at a unique measure as we will see (which is remarkably different from what happens in finite systems). Each measure  for which the supremum is attained is called an equilibrium state for . Here the word “state” is used synonymously with “distribution” or “measure” – a reflection of the fact that in “well-behaved cases”, as we will see in the next section, this measure is uniquely determined by the constraint(s) under which it maximizes entropy, and that means by the macroscopic state of the system. (In contrast, the word “state” was used in the above section on finite systems to designate microscopic states.) As, for each  2 M (X), the functional  7! h() C h; i is affine on C(X), the pressure functional

1427

1428

Pressure and Equilibrium States in Ergodic Theory

P : C(X) ! R, which, by definition, is the pointwise supremum of these functionals, is convex. It is therefore instructive to fit equilibrium states into the abstract framework of convex analysis [32,38,45,68]. To this end recall the identities in (4) that identify, for finite systems, equilibrium states as gradients of the pressure function p : RjAj ! R and guarantee that p is twice differentiable and strictly convex. In the present setting where P is defined on the Banach space C(X), differentiability and strict convexity are no more guaranteed, but one can show: Equilibrium states as (sub)-gradients  2 M (X) is an equilibrium state for  if and only if  is a subgradient (or tangent functional) for P at , i. e., if P( C )P()  h ; i for all 2 C(X). In particular,  has a unique equilibrium state  if (13) and only if P is differentiable at  with gradient , i. e., if lim t!0 1t (P( C t )  P()) D h ; i for all 2 C(X). Let us see how equilibrium states on X D AN can directly be obtained from the corresponding equilibrium distributions on finite sets An introduced in Subsect. “Systems on a Finite Lattice”. Define  (n) : An ! R by  (n) (a0 : : : a n1 ) :D (a0 : : : a n1 a0 : : : a n1 : : : ), denote by U n the corresponding global observable on An , and let n be the equilibrium distribution on An that maximizes H() C hU n ; i. Then all weak limit points of the “approximative equilibrium distributions” n on An are equilibrium states on AN . This can be seen as follows: Let the measure  on AN be any weak limit point of the n . Then, given > 0 there exists k 2 N such that 1 h() C h; i  H k () C h; i  k 1  H k (n ) C h (n) ; n i  2 k for arbitrarily large n, because k   (n) k1 ! 0 as n ! 1 by construction of the  (n) . As the n are invariant under cyclic coordinate shifts (see Subsect. “Systems on a Finite Lattice”), it follows from the subadditivity of the entropy that 1 (H n (n ) C hU n ; n i)  2 n k  log jAj : n Hence, for each  2 M (X), 1 h() C h; i  (H n () C hU n ; i)  2 n k  log jAj ! h() C h; i  2 n h() C h; i 

as n ! 1, and we see that  is indeed an equilibrium state on AN . The Variational Principle In Subsect. “Equilibrium Distributions and the Gibbs Property”, the pressure of a finite system was defined as a certain supremum and then identified as the logarithm of the normalizing constant for the Gibbsian representation of the corresponding equilibrium distribution. We are now going to approximate equilibrium states by suitable Gibbs distributions on finite subsets of X. As a byproduct the pressure P() is characterized in terms of the logarithms of the normalizing constants of these approximating distributions. Let S n (x) :D (x) C ( x) C    C ( n1 x). From each cylinder set [a0 : : : a n1 ] we can pick a point z such that S n (z) is the maximal value of S n  on this set. We denote the collection of the jAj n points we obtain in this way by En . Observe that En is not unambiguously defined, but any choice we make will do. Theorem (Variational Principle for the Pressure) P() D lim sup n!1

1 Pn () n

where Pn () :D log

X

eS n (z) :

(14)

z2E n

To prove the “ ” direction of this identity we just have to show that n1 H n ()Ch; i  n1 Pn () for each  2 M (X) or, after multiplying by n, H n () C hS n ; i  Pn (). But Jensen’s inequality implies: H n () C hS n ; i X  [a0 : : : a n1 ] a 0 ;:::;a n1 2A

! ˚ sup eS n (x ) : x 2 [a0 : : : a n1 ] log [a0 : : : a n1 ] n o X  log sup eS n (x) : x 2 [a0 : : : a n1 ] a 0 ;:::;a n1 2A

D log

X

e S n (z) D Pn () :

z2E n

For the reverse inequality consider the discrete Gibbs distributions X ız exp(Pn () C S n (z)) n :D z2E n

on the finite sets En , where ız denotes the unit point mass in z. One might be tempted to think that all weak limit

Pressure and Equilibrium States in Ergodic Theory

points of the measures  n are already equilibrium states. But this need not be the case because there is no good reason that these limits are shift invariant. Therefore, one forces invariance of the limits by passing to measures n P k defined by h f ; n i :D h n1 n1 kD0 f ı  ;  n i. Weak limits of these measures are obviously shift invariant, and a more involved estimate we do not present here shows that each such weak limit  satisfies h() C h; i  P(). We note that the same arguments work for any other sequence of sets En which contain exactly one point from each cylinder. So there are many ways to approximate equilibrium states, and if there are more than one equilibrium state, there is generally no guarantee that the limit is always the same. Nonuniqueness of Equilibrium States: An Example Before we turn to sufficient conditions for the uniqueness of equilibrium states in the next section, we present one of the simplest nontrivial examples for nonuniqueness of equilibrium states. Motivated by the so-called Fisher– Felderhof droplet model of condensation in statistical mechanics [23,25], Hofbauer [31] studies an observable  on X D f0; 1gN defined as follows: Let (a k ) be a sequence of negative real numbers with lim k!1 a k D 0. Set s k :D a0 C    C a k . For k  1 denote M k :D fx 2 X : x0 D    D x k1 D 1; x k D 0g and M0 :D fx 2 X : x0 D 0g, and define (x) :D a k

for x 2 M k and (11 : : : ) D 0 :

Then  : X ! R is continuous, so that there exists at least one equilibrium state for . Hofbauer proves that there is P sk more than one equilibrium state if and only if 1 kD0 e D P1 s k 1 and kD0 (k C 1)e < 1. In that case P() D 0, so one of these equilibrium states is the unit mass ı11::: , and we denote the other equilibrium state by 1 , so h(1 ) C h; 1 i D 0. In view of (13) the pressure function is not differentiable at . What does the pressure function ˇ 7! P(ˇ) look like? As h(ı11::: ) C hˇ; ı11:::i D 0 for all ˇ, P(ˇ)  0 for all ˇ. Observe now that (x)  0 with equality only for x D 11 : : : This implies that h; i < 0 for all  2 M (X) different from ı11::: . From this we can conclude:  P(ˇ)  P() D 0 for ˇ > 1, so P(ˇ) D 0 for ˇ  1.  P(ˇ)  h(1 ) C hˇ; 1 i D h(1 ) C h; 1 i  (1  ˇ)h; 1 i D (1  ˇ)h; 1 i. It follows that, at ˇ D 1, the derivative from the right of P(ˇ) is zero, whereas the derivative from the left is at most h; 1 i < 0.

More on Equilibrium States In more general dynamical systems the entropy function is not necessarily upper semicontinuous and hence equilibrium states need not exist, i. e., the supremum in (12) need not be attained by any invariant measure. A well-known sufficient property that guarantees the upper semicontinuity of the entropy function is the expansiveness of the system, see, e. g., [53]: a continuous transformation T of a compact metric space is positively expansive, if there is a constant > 0 such that for any two points x and y from the space there is some n 2 N such that T n x and T n y are at least a distance apart. If T is a homeomorphism one says it is expansive, if the same holds for some n 2 Z. The previous results carry over without changes (although at the expense of more complicated proofs) to general expansive systems. The variational principle (14) holds in the d very general context where T is a continuous action of ZC on a compact Hausdorff space X. This was proved in [44] in a simple and elegant way. In the monograph [45] it is extended to amenable group actions. The Gibbs Property: A Local Characterization of Equilibrium In this section we are going to see that, for a sufficiently regular potential  on a topologically mixing subshift of finite type, one has a unique equilibrium state which has the “Gibbs property”. This property generalizes formula (5) that we derived for finite lattices. Subshifts of finite type are the symbolic models for Axiom A diffeomorphisms, as we shall see later on. Subshifts of Finite Type We start by recalling what is a subshift of finite type and refer the reader to  Symbolic Dynamics or [43] for more details. Given a “transition matrix” M D (M ab ) a;b2A whose entries are 0’s or 1’s, one can define a subshift X M as the set of all sequences x 2 AN such that M x i x iC1 D 1 for all i 2 N. This is called a subshift of finite type or a topological Markov chain. We assume that there exists some integer p0 such that M p has strictly positive entries for all p  p0 . This means that M is irreducible and aperiodic. This property is equivalent to the property that the subshift of finite type is topologically mixing. A general subshift of finite type admits a decomposition into a finite union of transitive sets, each of which being a union of cyclically permuted sets on which the appropriate iterate is topologically mixing. In other words, topologically mixing subshifts of finite type are the building blocks of subshifts of finite type.

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Pressure and Equilibrium States in Ergodic Theory

The Gibbs Property for a Class of Regular Potentials The class of regular potentials we consider is that of “summable variations”. We denote by vark () the modulus of continuity of  on cylinders of length k  1, that is, vark () :D supfj(x)  (y)j : x 2 [y0 : : : y k1 ]g : If var k () ! 0 as k ! 1, this means that  is (uniformly) continuous with respect to the distance (7). We impose the stronger condition 1 X

var k () < 1 :

(15)

kD1

We can now state the main result of this section. The Gibbs state of a summable potential Let X M be a topologically mixing subshift of finite type. Given a potential  : X M ! R satisfying the summability condition (15), there is a (probability) measure  supported on X M , that we call a Gibbs state. It is the unique -invariant measure which satisfies the following property: There exists a constant C > 0 such that, for all x 2 X M and for all n  1, C 1 

 [x0 : : : x n1 ]  C: exp(S n (x)  nP())

(“Gibbs property”) (16)

Moreover, the Gibbs state  is ergodic and is also the unique equilibrium state of , i. e., the unique invariant measure for which the supremum in (12) is attained. We now make several comments on this theorem.  The Gibbs property (16) gives a uniform control of the measure of all cylinders in terms of their “energy”. This strengthens considerably the asymptotic equipartition property (11) that we recover if we restrict (16) to the set of  measure 1 where Birkhoff’s ergodic Theorem (8) applies, and use the identity h;  i  P() D h( ).  Gibbs measures on topologically mixing subshifts of finite type are ergodic (and actually mixing in a strong sense) as can be inferred from Ruelle’s Perron– Frobenius Theorem see the next subsection.  Suppose that there is another invariant measure 0 satisfying (16), possibly with a constant C 0 different from C. It is easy to verify that 0 D f  for some -integrable function f by using (16) and the Radon– Nikodym Theorem. Shift invariance imposes that, a. e., f D f ı . Then the ergodicity of  implies that f is a constant -a. e., thus 0 D  ; see [6].

 One could define a Gibbs state by saying that it is an invariant measure  satisfying (16) for a given continuous potential . If one does so, it is simple to verify that such a  must also be an equilibrium state. Indeed, using (16), one can deduce that h; i C h()  P(). The converse need not be true in general, see Subsect. “More on Hofbauer’s Example” below. But the summability condition (15) is indeed sufficient for the coincidence of Gibbs and equilibrium states. A proof of this fact can be found in [58] or [38]. Ruelle’s Perron–Frobenius Theorem The powerful tool behind the theorem in the previous subsection is a far-reaching generalization of the classical Perron–Frobenius theorem for irreducible matrices. Instead of a matrix, one introduces the so-called transfer operator, also called the “Perron–Frobenius operator” or “Ruelle’s operator”, which acts on a suitable Banach space of observables. It is D. Ruelle [52] who first introduced this operator in the context of one-dimensional lattice gases with exponentially decaying interactions. In our context, this corresponds to Hölder continuous potentials: these are potentials satisfying var k ()  c k for some c > 0 and 2 (0; 1). A proof of “Ruelle’s Perron–Frobenius Theorem” can be found in [4,6]. It was then extended to potentials with summable variations in [67]. We refer to the book of V. Baladi [1] for a comprehensive account on transfer operators in dynamical systems. We content ourselves to define the transfer operator and state Ruelle’s Perron–Frobenius Theorem. Let L : C(X M ) ! C(X M ) be defined by (L f )(x) :D

X

e(y) f (y)

y2 1 x

D

X

e(ax) f (ax) :

a2A:M(a;x 0 )D1

(Obviously, ax :D ax0 x1 : : : ) Theorem (Ruelle’s Perron–Frobenius Theorem) Let X M be a topologically mixing subshift of finite type. Let  satisfy condition (15). There exist a number  > 0, h 2 C(X M ), and  2 M(X) such that h > 0, hh; i D 1, L h D h, L   D , where L  is the dual of L . Moreover, for all f 2 C(X M ), kn L n f  h f ; i  hk1 ! 0;

as n ! 1 :

By using this theorem, one can show that  :D h satisfies (16) and  D eP() .

Pressure and Equilibrium States in Ergodic Theory

Let us remark that for potentials which are such that (x) D (x0 ; x1 ) (i. e., potentials constant on cylinders of length 2), L can be identified with a jAj  jAj matrix and the previous theorem boils down to the classical Perron–Frobenius theorem for irreducible aperiodic matrices [63]. The corresponding Gibbs states are nothing but Markov chains with state space A (Chapter 3 in [29]). We shall take another point of view below (Subsect. “Markov Chains over Finite Alphabets”). Relative Entropy We now define the relative entropy of an invariant measure  2 M (X M ) given a Gibbs state  as follows. We first define H n (j ) :D

X

[a0 : : : a n1 ] log

a 0 ;:::;a n1 2A

[a0 : : : a n1 ]  [a0 : : : a n1 ] (17)

with the convention 0 log(0/0) D 0. Now the relative entropy of  given  is defined as h(j ) :D lim sup n!1

1 H n (j ) : n

(By applying Jensen’s inequality, one verifies that h(j )  0.) In fact the limit exists and can be computed quite easily using (16): h(j ) D P()  h; i  h() :

(18)

To prove this formula, we first make the following observation. It can be easily verified that the inequalities in (16) remain the same when S n  is replaced by 1 ˜ the R “locally averaged” energy  n :D ([x0 : : : x n1 ]) S (y)d(y) for any cylinder with [x : 0 :: [x 0 :::x n1 ] n x n1 ] > 0. Cylinders with  measure zero does not contribute to the sum in (17). We can now write that 1  log C n   1 1 1   H n (j ) C P()  hS n ; i  H n () n n n 1  log C : n To finish we use that hS n ; i D nh; i (by the invariance of ) and we apply (9) to obtain 1 1 lim H n (j ) D P()  h; i  lim H n () n!1 n n!1 n D P()  h; i  h() which proves (18).

The variational principle revisited We can reformulate the variational principle in the case of a potential satisfying the summability condition (15): h(j ) D 0

if and only if  D  ;

(19)

i. e., given  , the relative entropy h(j ), as a function on M (X M ), attains its minimum only at  . Indeed, by (18) we have h(j ) D P()  h; i  h(). We now use (12) and the fact that  is the unique equilibrium state of  to conclude. More Properties of Gibbs States Gibbs states enjoy very good statistical properties. Let us mention only a few. They satisfy the “Bernoulli property”, a very strong qualitative mixing condition [4,6,67]. The sequence of random variables ( f ı  n )n satisfies the central limit theorem [15,16,49] and a large deviation principle if f is Hölder continuous [21,38,39,70]. Let us emphasize the central role played by relative entropy in large deviations. (The deep link between thermodynamics and large deviations is described in [42] in a much more general context.) Finally, the so-called “multifractal analysis” can be performed for Gibbs states, see, e. g., [48]. Examples on Shift Spaces Measure of Maximal Entropy and Periodic Points If the observable  is constant zero, an equilibrium state simply maximizes the entropy. It is called measure of maximal entropy. The quantity P(0) D supfh() :  2 M (X)g is called the topological entropy of the subshift  : X ! X. When X is a subshift of finite type X M with irreducible and aperiodic transition matrix M, there is a unique measure of maximal entropy, see, e. g., [43]. As a Gibbs state it satisfies (16). By summing over all cylinders [x0 : : : x n1 ] allowed by M, it is easy to see that the topological entropy P(0) is the asymptotic exponential growth rate of the number of sequences of length n that can occur as initial segments of points in X M . This is obviously identical to the logarithm of the largest eigenvalue of the transition matrix M. It is not difficult to verify that the total number of periodic sequences of period n equals the trace of the matrix M n , i. e., we have the formula m X ˚ Card x 2 X M :  n x D x D tr(M n ) D ni ; iD1

where 1 ; : : : ; m are all the eigenvalues of M. Asymptotically, of course, Cardfx 2 X M :  n x D xg D e nP(0) C

1431

1432

Pressure and Equilibrium States in Ergodic Theory

O(j0 j n ), where 0 is the second largest (in absolute value) eigenvalue of M. The measure of maximal entropy, call it 0 , describes the distribution of periodic points in X M : one can prove [3,37] that for any cylinder B X M ˚ Card x 2 B :  n x D x D 0 (B) : ˚ lim n!1 Card x 2 X M :  n x D x In other words, the finite atomic measures that assign equal weights 1/ Cardfx 2 X M :  n x D xg to each periodic point in X M with period n weakly converges to 0 , as n ! 1. Each such measure has zero entropy while h(0 ) D P(0) > 0, so the entropy is not continuous on the space of invariant measures. It is, however, upper-semicontinuous (see Subsect. “Entropy as a Function of the Measure”). In fact, it is possible to approximate any Gibbs state  on X M in a similar way by finite atomic measures on periodic orbits, if one assigns weights properly (see, e. g., Theorem 20.3.7 in [37]).

is typical for a Gibbs distribution ˇ  where ˇ D ˇ(p) is such that h; ˇ  i D 2p  1. It follows that there is a constant C > 0 such that for each n 2 N and any two “spin patterns” a D a0 : : : a n1 and b D b0 : : : b n1 ˇ ˇ ˇ ˇ ˇlog ˇ  [a0 : : : a n1 ]  ˇ(N a  N b )ˇ  C ; ˇ ˇ ˇ  [b0 : : : b n1 ] where N a and N b are the numbers of identical adjacent spins in a and b, respectively. More on Hofbauer’s Example We come back to the example described in Subsect. “Nonuniqueness of Equilibrium States: An Example”. It is easy to verify that in that example varkC1 () D ja k j. For instance, if a k D 1/(k C 1)2 there is a unique Gibbs/ equilibrium state. If a k D 3 log ((k C 1)/k) for k  1 P 3 and a0 D  log 1 jD1 j , then from [31] we know that  admits more than one equilibrium state, one of them being ı11::: , which cannot be a Gibbs state for any continuous .

Markov Chains over Finite Alphabets

Examples from Differentiable Dynamics

Let Q D (q a;b ) a;b2A be an irreducible stochastic matrix over the finite alphabet A. It is well known (see, e. g., [63]) that there exists a unique probability vector  on A that defines a stationary Markov measure  Q on X D AN by Q [a0 : : : a n1 ] D  a 0 q a 0 a 1 : : : q a n2 a n1 . We are going to identify  Q as the unique Gibbs distribution  2 M (X) that maximizes entropy under the constraints [ab] D [a]q ab , i. e., h ab ; i D 0 (a; b 2 A), where  ab :D 1[ab]  q ab 1[a] . Indeed, as  is a Gibbs measure, there are ˇ ab 2 R (a; b 2 A) and constants P 2 R, C > 0 such that

In this section we present a number of examples to which the general theory developed above does not apply directly but only after a transfer of the theory from a symbolic space to a manifold. We restrict to examples where the results can be transferred because those aspects of the smooth dynamics we focus on can be studied as well on a shift dynamical system that is obtained from the original one via symbolic coding. (We do not discuss the coding process itself which is sometimes far from trivial, but we focus on the application of the Gibbs and equilibrium theory.) There are alternative approaches where instead of the results the concepts and (partly) the strategies of proofs are transferred to the smooth dynamical systems. This has led both to an extension of the range of possible applications of the theory and to a number of refined results (because some special features of smooth systems necessarily get lost by transferring the analysis to a completely disconnected metric space). In the following examples, T denotes a (possibly piecewise) differentiable map of a compact smooth manifold M. Points on the manifold are denoted by u and v. In all examples there is a Hölder continuous coding map  : X ! M from a subshift of finite type X onto the manifold which respects the dynamics, i. e., T ı D  ı . This factor map  is “nearly” invertible in the sense that the set of points in M with more than one preimage under  has measure zero for all T-invariant measures we are interested in. Hence ˜ on M correspond unambiguously to shift such measures 

C 1 

exp

P

[x0 : : : x n1 ]  C ab a;b2A ˇ ab  n (x)  nP

(20)

for all x 2 AN and all n 2 N. Let r ab :D exp(ˇ ab  P b 0 2A ˇ ab 0 q ab 0 P). Then the denominator in (20) equals r x 0 x 1 : : : r x n2 x n1 , and it follows that  is equivalent to the stationary Markov measure defined by the (nonstochastic) matrix (r ab ) a;b2A . As  is ergodic,  is this Markov measure, and as  satisfies the linear constraints [ab] D [a]q ab , we conclude that  D Q . The Ising Chain Here the task is to characterize all “spin chains” in x 2 f1; C1gN (or, more commonly, f1; C1gZ ) which are as random as possible with the constraint that two adjacent spins have a prescribed probability p ¤ 12 to be identical. With (x) :D x0 x1 this is equivalent to requiring that x

Pressure and Equilibrium States in Ergodic Theory

˜ ı  1 . Similarly observables ˜ invariant measures  D  on M and  D ˜ ı  on X are related. Uniformly Expanding Markov Maps of the Interval A transformation T on M :D [0; 1] is called a Markov map, if there are 0 D u0 < u1 <    < u N D 1 such that each restriction Tj(u i1 ;u i ) is strictly monotone, C 1Cr for some r > 0, and maps (u i1 ; u i ) onto a union of some of these N monotonicity intervals. It is called uniformly expanding if there is some k 2 N such that  :D inf x j(T k )0 (x)j > 1. It is not difficult to verify that the symbolic coding of such a system leads to a topological Markov chain over the alphabet A D f1; : : : ; Ng. To simplify the discussion we assume that the transition matrix M of this topological Markov chain is irreducible and aperiodic. ˜ repreOur goal is to find a T-invariant measure  sented by  2 M (X M ) which minimizes the relative entropy to Lebesgue measure on [0; 1] ˜ h(jm) :D lim

n!1

1 n

X

[a0 : : : a n1 ]

a 0 ;:::;a n1 2f1;:::;Ng

log

[a0 : : : a n1 ] ; n [a0 : : : a n1 ]

where n [a0 : : : a n1 ] :D jI a 0 :::a n1 j. (Recall that, without insisting on invariance, this would just be the Lebesgue measure itself.) The existence of the limit will be justified below – observe that m is not a Gibbs state as v is in Eq. (17). The argument rests on the simple observation (implied by the uniform expansion and the piecewise Hölder-continuity of T 0 ) that T has bounded distortion, i. e., that there is a constant C > 0 such that for all n 2 N, a0 : : : a n1 2 f1; : : : ; Ngn and u 2 I a 0 :::a n1 holds C 1  jI a 0 :::a n1 j  j(T n )0 (u)j  C ; or , equivalently , C 1

jI a 0 :::a n1 j  C;   ˜ exp S n (u) (21)

˜ where (u) :D  log jT 0 (u)j. (Observe the similarity between this property and the Gibbs property (16).) Assuming bounded distortion we have at once ! n1 X 1 k ˜ h(jm) D lim h ı  ; i H n ()  n!1 n kD0

D h()  h; i ; and minimizing this relative entropy just amounts to maximizing h() C h; i for  D  log jT 0 j ı . As the re-

sults on Gibbs distributions from Sect. “The Gibbs Property: A Local Characterization of Equilibrium” apply, we conclude that C 1 

[a0 : : : a n1 ] C jI a 0 : : : a n1 j

˜ for some C > 0. So the unique T-invariant measure  ˜ that minimizes the relative entropy h(jm) is equivalent to Lebesgue measure m. (The existence of an invariant probability measure equivalent to m is well known, also without invoking entropy theory. It is guaranteed by a “Folklore Theorem” [33].) Interval Maps with an Indifferent Fixed Point The presence of just one point x 2 [0; 1] such that T 0 (x) D 1 dramatically changes the properties of the system. A canonical example is the map T˛ : x 7! x(1C2˛ x ˛ ) if x 2 [0; 1/2[ and x 7! 2x  1 if x 2 [1/2; 1]. We have T 0 (0) D 1, i. e., 0 is an indifferent fixed point. For ˛ 2 [0; 1[ this map admits an absolutely continuous invariant probability measure d(x) D h(x)dx, where h(x) x ˛ when x ! 0 [66]. In the physics literature, this type of map is known as the “Manneville–Pomeau” map. It was introduced as a model of transition from laminar to intermittent behavior [50]. In [28] the authors construct a piecewise affine version of this map to study the complexity of trajectories (in the sense of Subsect. “A Short Digression on Complexity”). This gives rise to a countable state Markov chain. In [69] the close connection to the Fisher–Felderhof model and Hofbauer’s example (see Subsect. “Nonuniqueness of Equilibrium States: An Example”) was realized. We refer to [61] for recent developments and a list of references. Axiom A Diffeomorphisms, Anosov Diffeomorphisms, Sinai–Ruelle–Bowen Measures The first spectacular application of the theory of Gibbs measures to differentiable dynamical systems was Sinai’s approach to Anosov diffeomorphisms via Markov partitions [64] that allowed one to code the dynamics of these maps into a subshift of finite type and to study their invariant measures by methods from equilibrium statistical mechanics [65] that had been developed previously by Dobrushin, Lanford, and Ruelle [17,18,19,20,41]. Not much later this approach was extended by Bowen [2] to Smale’s Axiom A diffeomorphisms (and to Axiom A flows by Bowen and Ruelle [7]); see also [54]. The interested reader can consult, e. g., [71] for a survey, and either [6] or [15] for details.

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Both types of diffeomorphisms act on a smooth compact Riemannian manifold M and are characterized by the existence of a compact T-invariant hyperbolic set   M. Their basic properties are described in detail in the contribution  Ergodic Theory: Basic Examples and Constructions. Very briefly, the tangent bundle over  splits into two invariant subbundles – a stable one and an unstable one. Correspondingly, through each point of  there passes a local stable and a local unstable manifold which are both tangent to the respective subspaces of the local tangent space. The unstable derivative of T, i. e., the derivative DT restricted to the unstable subbundle, is uniformly expanding. Its Jacobian determinant, denoted by J (u) , is Hölder continuous as a function on . Hence the observable  (u) :D  log jJ (u)j ı  is Hölder continuous, and the Gibbs and equilibrium theory apply (via the symbolic coding) to the diffeomorphism T (modulo possibly a decomposition of the hyperbolic set into irreducible and aperiodic components, called basic sets, that can be modeled by topologically mixing subshifts of finite type). The main results are: Characterization of attractors The following assertions are equivalent for a basic set ˝  : ˝ is an attractor, i. e., there are arbitrarily small neighborhoods U  M of ˝ such that TU U. (ii) The union of all stable manifolds through points of ˝ is a subset of M with positive volume. (iii) The pressure PTj˝ ( (u) ) D 0. (i)

In this case the unique equilibrium and Gibbs state C of Tj˝ is called the Sinai–Ruelle–Bowen (SRB) measure of Tj˝ . It is uniquely characterized by the identity h Tj˝ (C ) D h (u); C i. (For all other T-invariant measures on ˝ one has “<” instead of “D”.)

cause of transitivity, property (ii) from the characterization of attractors is trivially satisfied, so there is always a unique SRB measure C . As T 1 is an Anosov diffeomorphism as well – only the roles of stable and unstable manifolds are interchanged – T 1 has a unique SRB measure  which is the unique equilibrium state of T 1 (and hence also of T) for  (s) :D log jJ (s) j. One can show: SRB measures for Anosov diffeomorphisms The following assertions are equivalent: (i) C D  . (ii) C or  is absolutely continuous w.r.t the volume measure on M. (iii) For each periodic point u D T n u 2 M, jJ(u)j D 1, where J denotes the determinant of DT. We remark that, similarly to the case of Markov interval maps, the unstable Jacobian of T n at u is asymptotically equivalent to the volume of the “n-cylinder” of the Markov partition around u. So the maximization of h() C h (u); i by the SRB measure C can again be interpreted as the minimization of the relative entropy of invariant measures with respect to the normalized volume, and the fact that P( (u) ) D 0 in the Anosov (or more generally attractor) case means that C is as close to being absolutely continuous as it is possible for a singular measure. This is reflected by the above properties (a) and (b). We emphasize the meaning of property (a) above: it tells us that the SRB measure C is the only physically observable measure. Indeed, in numerical experiments with physical models, one picks an initial point u 2 M “at random” (i. e., with respect to the volume or Lebesgue measure) and follows its orbit T k u, k  0.

Further properties of SRB measures Suppose PTj˝ ( (u) ) D 0 and let C be the SRB measure.

Bowen’s Formula for the Hausdorff Dimension of Conformal Repellers

(a) For a set of points u 2 M of positive volume we have:

Just as nearby orbits converge towards an attractor, they diverge away from a repeller. Conformal repellers form a nice class of systems which can be coded by a subshift of finite type. The construction of their Markov partitions is much simpler than that of Anosov diffeomorphisms, see, e. g., [72]. Let us recall the definition of a conformal repeller before giving a fundamental example. Given a holomorphic map T : V ! C where V C is open and J a compact subset of C, one says that (J; V; T) is a conformal repeller if

lim

n!1

n1 1X f (T k u) D h f ; C i : n kD0

(Indeed, because of (ii) of the above characterization, this holds for almost all points of the union of the stable manifolds through points of ˝.) (b) Conditioned on unstable manifolds, C is absolutely continuous to the volume measure on unstable manifolds. In the special case of transitive Anosov diffeomorphisms, the whole manifold is a hyperbolic set and ˝ D M. Be-

(i)

there exist C > 0, ˛ > 1 such that j(T n )0 (z)j  C˛ n for all z 2 J, n  1;

Pressure and Equilibrium States in Ergodic Theory

T (ii) J D n1 T n (V), and (iii) for any open set U such that U \ J ¤ ;, there exists n such that T n (U \ J)  J. From the definition it follows that T(J) D J and T 1 (J) D J. A fundamental example is the map T : z ! z2 C c, c 2 C being a parameter. It can be shown that for jcj < 14 there exists a compact set J, called a (hyperbolic) Julia set, such that (J; C; T) is a conformal repeller. Conformal repellers J are in general fractal sets and one can measure their “degree of fractality” by means of their Hausdorff dimension, dimH (J). Roughly speaking, one computes this dimension by covering the set J by balls with radius less than or equal to ı. If Nı (J) denotes the cardinality of the smallest such covering, then we expect that Nı (J) ı  dimH (J) ;

as ı ! 0 :

We refer the reader to  Ergodic Theory: Fractal Geometry or [22,46] for a rigorous definition (based on Carathéodory’s construction) and for more information on fractal geometry. Bowen’s formula relates dimH (J) to the unique zero ˜ where ˜ :D of the pressure function ˇ 7! P(ˇ ) (log jT 0 j)j J . It is not difficult to see that indeed this map has a unique zero for some positive ˇ. By property (i), S n ˜  const  n log ˛, which implies d ˜ D h; ˜ ˇ i   log ˛ < 0. As (by (13)) that dˇ P(ˇ ) P(0) equals the topological entropy of J, i. e., the logarithm of the largest eigenvalue of the matrix M associated to the Markov partition, P(0) is strictly positive. Therefore, (recall that the pressure function is continuous) there exists ˜ D 0. a unique number ˇ0 > 0 such that P(ˇ0 ) It turns out that this unique zero is precisely dimH (J): Bowen’s formula The Hausdorff dimension of J is the ˜ D 0, ˇ 2 R; in unique solution of the equation P(ˇ ) particular ˜ D0: P(dimH (J)) This formula was proven in [55] for a general class of conformal repellers after the seminal paper [5]. The main tool is a distortion estimate very similar to (21). A simple exposition can be found in [72]. Nonequilibrium Steady States and Entropy Production SRB measures for Anosov diffeomorphisms and Axiom A attractors have been accepted recently as conceptual

models for nonequilibrium steady states in nonequilibrium statistical mechanics. Let us point out that the word “equilibrium” is used in physics in a much more restricted sense than in ergodic theory. Only diffeomorphisms preserving the natural volume of the manifold (or a measure equivalent to the volume) would be considered as appropriate toy models of physical equilibrium situations. In the case of Anosov diffeomorphisms this is precisely the case if the “forward” and “backward” SRB measures C and  coincide. Otherwise, the diffeomorphism models a situation out of equilibrium, and the difference between C and  can be related to entropy production and irreversibility. Gallavotti and Cohen [26,27] introduced SRB measures as idealized models of nonequilibrium steady states around 1995. In order to have as firm a mathematical basis as possible they made the “chaotic hypothesis” that the systems they studied behave like transitive Anosov systems. Ruelle [56] extended their approach to more general (even nonuniformly) hyperbolic dynamics; see also his reviews [57,59] for more recent accounts discussing also a number of related problems; see by [51], too. The importance of the Gibbs property of SRB measures for the discussion of entropy production was also highlighted in [35], where it is shown that for transitive Anosov diffeomorphisms the relative entropy h(C j ) equals the average entropy production rate hlog jJj; C i of C where J denotes again the Jacobian determinant of the diffeomorphism. In particular, the entropy production rate is zero if, and only if, h(C j ) D 0, i. e., using coding and (19), if, and only if, C D  . According to Subsect. “Axiom A Diffeomorphisms, Anosov Diffeomorphisms, Sinai–Ruelle–Bowen Measures”, this is also equivalent to C or  being absolutely continuous with respect to the volume measure. Some Ongoing Developments and Future Directions As we saw, many dynamical systems with uniform hyperbolic structure (e. g., Anosov maps, axiom A diffeomorphisms) can be modeled by subshifts of finite type over a finite alphabet. We already mentioned in Subsect. “Interval Maps with an Indifferent Fixed Point” the typical example of a map of the interval with an indifferent fixed point, whose symbolic model is still a subshift of finite type, but with a countable alphabet. The thermodynamic formalism for such systems is by now well developed [24,30,60,61,62] and used, e. g., for multidimensional piecewise expanding maps [13]. An active line of research is related to systems admitting representations by symbolic models called “towers” constructed by using “inducing schemes”. The fundamental example is the class

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of one-dimensional unimodal maps satisfying the “Collet–Eckmann condition”. A first attempt to develop thermodynamic formalism for such systems was made in [10] where existence and uniqueness of equilibrium measures for the potential function ˜ˇ (u) D ˇ log jT 0 (u)j with ˇ close to 1 was established. Very recently, new developments in this direction appeared, see, e. g., [11,12,47]. A largely open field of research concerns a new branch of nonequilibrium statistical mechanics, the socalled “chaotic scattering theory”, namely the analysis of chaotic systems with various openings or holes in phase space, and the corresponding repellers on which interesting invariant measures exist. We refer the reader to [15] for a brief account and references to the physics literature. The existence of (generalized) steady states on repellers and the so-called “escape rate formula” have been observed numerically in a number of models. So far, little has been proven mathematically, except for Anosov diffeomorphisms with special holes [15] and for certain nonuniformly hyperbolic systems [9]. Bibliography 1. Baladi V (2000) Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics, vol 16. World Scientific, Singapore 2. Bowen R (1970) Markov partitions for Axiom A diffeomorphisms. Amer J Math 92:725–747 3. Bowen R (1974/1975) Some systems with unique equilibrium states. Math Syst Theory 8:193–202 4. Bowen R (1974/75) Bernoulli equilibrium states for Axiom A diffeomorphisms. Math Syst Theory 8:289–294 5. Bowen R (1979) Hausdorff dimension of quasicircles. Inst Hautes Études Sci Publ Math 50:11–25 6. Bowen R (2008) Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, vol 470, 2nd edn (1st edn 1975). Springer, Berlin 7. Bowen R, Ruelle D (1975) The ergodic theory of Axiom A flows. Invent Math 29:181–202 8. Brudno AA (1983) Entropy and the complexity of the trajectories of a dynamical system. Trans Mosc Math Soc 2:127–151 9. Bruin H, Demers M, Melbourne I (2007) Existence and convergence properties of physical measures for certain dynamical systems with holes. Preprint 10. Bruin H, Keller G (1998) Equilibrium states for S-unimodal maps. Ergodic Theory Dynam Syst 18(4):765–789 11. Bruin H, Todd M (2007) Equilibrium states for potentials with sup '  inf ' < htop (f ). Commun Math Phys doi:10.1007/ s00220-0-008-0596-0 12. Bruin H, Todd M (2007) Equilibrium states for interval maps: the potential t log jDf j. Preprint 13. Buzzi J, Sarig O (2003) Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps. Ergod Theory Dynam Syst 23(5):1383–1400 14. Chaitin GJ (1987) Information, randomness & incompleteness. Papers on algorithmic information theory. World Scientific Series in Computer Science, vol 8. World Scientific, Singapore

15. Chernov N (2002) Invariant measures for hyperbolic dynamical systems. In: Handbook of Dynamical Systems, vol 1A. NorthHolland, pp 321–407 16. Coelho Z, Parry W (1990) Central limit asymptotics for shifts of finite type. Isr J Math 69(2):235–249 17. Dobrushin RL (1968) The description of a random field by means of conditional probabilities and conditions of its regularity. Theory Probab Appl 13:197–224 18. Dobrushin RL (1968) Gibbsian random fields for lattice systems with pairwise interactions. Funct Anal Appl 2:292–301 19. Dobrushin RL (1968) The problem of uniqueness of a Gibbsian random field and the problem of phase transitions. Funct Anal Appl 2:302–312 20. Dobrushin RL (1969) Gibbsian random fields. The general case. Funct Anal Appl 3:22–28 21. Eizenberg A, Kifer Y, Weiss B (1994) Large deviations for Zd actions. Comm Math Phys 164(3):433–454 22. Falconer K (2003) Fractal geometry. Mathematical foundations and applications, 2nd edn. Wiley, San Francisco 23. Fisher ME, Felderhof BU (1970) Phase transition in one-dimensional clusterinteraction fluids: IA. Thermodynamics, IB. Critical behavior. II. Simple logarithmic model. Ann Phys 58:177–280 24. Fiebig D, Fiebig U-R, Yuri M (2002) Pressure and equilibrium states for countable state Markov shifts. Isr J Math 131:221–257 25. Fisher ME (1967) The theory of condensation and the critical point. Physics 3:255–283 26. Gallavotti G (1996) Chaotic hypothesis: Onsager reciprocity and fluctuation-dissipation theorem. J Stat Phys 84:899–925 27. Gallavotti G, Cohen EGD (1995) Dynamical ensembles in stationary states. J Stat Phys 80:931–970 28. Gaspard P, Wang X-J (1988) Sporadicity: Between periodic and chaotic dynamical behaviors. In: Proceedings of the National Academy of Sciences USA, vol 85, pp 4591–4595 29. Georgii H-O (1988) Gibbs measures and phase transitions. In: de Gruyter Studies in Mathematics, 9. de Gruyter, Berlin 30. Gurevich BM, Savchenko SV (1998) Thermodynamic formalism for symbolic Markov chains with a countable number of states. Russ Math Surv 53(2):245–344 31. Hofbauer F (1977) Examples for the nonuniqueness of the equilibrium state. Trans Amer Math Soc 228:223–241 32. Israel R (1979) Convexity in the theory of lattice gases. Princeton Series in Physics. Princeton University Press 33. Jakobson M, S´ wiatek ˛ (2002) One-dimensional maps. In: Handbook of Dynamical Systems, vol 1A. North-Holland, Amsterdam, pp 321–407 34. Jaynes ET (1989) Papers on probability, statistics and statistical physics. Kluwer, Dordrecht 35. Jiang D, Qian M, Qian M-P (2000) Entropy production and information gain in Axiom A systems. Commun Math Phys 214:389– 409 36. Katok A (2007) Fifty years of entropy in dynamics: 1958–2007. J Mod Dyn 1(4):545–596 37. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. Encyclopaedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge 38. Keller G (1998) Equilibrium states in ergodic theory. In: London Mathematical Society Student Texts, vol 42. Cambridge University Press, Cambridge 39. Kifer Y (1990) Large deviations in dynamical systems and stochastic processes. Trans Amer Math Soc 321:505–524

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40. Kolmogorov AN (1983) Combinatorial foundations of information theory and the calculus of probabilities. Uspekhi Mat Nauk 38:27–36 41. Lanford OE, Ruelle D (1969) Observables at infinity and states with short range correlations in statistical mechanics. Commun Math Phys 13:194–215 42. Lewis JT, Pfister C-E (1995) Thermodynamic probability theory: some aspects of large deviations. Russ Math Surv 50:279– 317 43. Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge 44. Misiurewicz M (1976) A short proof of the variational principle for a ZNC action on a compact space. In: International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), Astérisque, No. 40, Soc Math France, pp 147–157 45. Moulin-Ollagnier J (1985) Ergodic Theory and Statistical Mechanics. In: Lecture Notes in Mathematics, vol 1115. Springer, Berlin 46. Pesin Y (1997) Dimension theory in dynamical systems. Contemporary views and applications. University of Chicago Press, Chicago 47. Pesin Y, Senti S (2008) Equilibrium Measures for Maps with Inducing Schemes. J Mod Dyn 2(3):1–31 48. Pesin Y, Weiss (1997) The multifractal analysis of Gibbs measures: motivation, mathematical foundation, and examples. Chaos 7(1):89–106 49. Pollicott M (2000) Rates of mixing for potentials of summable variation. Trans Amer Math Soc 352(2):843–853 50. Pomeau Y, Manneville P (1980) Intermittent transition to turbulence in dissipative dynamical systems. Comm Math Phys 74(2):189–197 51. Rondoni L, Mejía-Monasterio C (2007) Fluctuations in nonequilibrium statistical mechanics: models, mathematical theory, physical mechanisms. Nonlinearity 20(10):R1–R37 52. Ruelle D (1968) Statistical mechanics of a one-dimensional lattice gas. Commun Math Phys 9:267–278 53. Ruelle D (1973) Statistical mechanics on a compact set with Z action satisfying expansiveness and specification. Trans Amer Math Soc 185:237–251 54. Ruelle D (1976) A measure associated with Axiom A attractors. Amer J Math 98:619–654 55. Ruelle D (1982) Repellers for real analytic maps. Ergod Theory Dyn Syst 2(1):99–107

56. Ruelle D (1996) Positivity of entropy production in nonequilibrium statistical mechanics. J Stat Phys 85:1–23 57. Ruelle D (1998) Smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics. J Stat Phys 95:393– 468 58. Ruelle D (2004) Thermodynamic formalism: The mathematical structures of equilibrium statistical mechanics. Second edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge 59. Ruelle D (2003) Extending the definition of entropy to nonequilibrium steady states. Proc Nat Acad Sc 100(6):3054– 3058 60. Sarig O (1999) Thermodynamic formalism for countable Markov shifts. Ergod Theory Dyn Syst 19(6):1565–1593 61. Sarig O (2001) Phase transitions for countable Markov shifts. Comm Math Phys 217(3):555–577 62. Sarig O (2003) Existence of Gibbs measures for countable Markov shifts. Proc Amer Math Soc 131(6):1751–1758 63. Seneta E (2006) Non-negative matrices and Markov chains. In: Springer Series in Statistics. Springer 64. Sinai Ja G (1968) Markov partitions and C-diffeomorphisms. Funct Anal Appl 2:61–82 65. Sinai Ja G (1972) Gibbs measures in ergodic theory. Russ Math Surv 27(4):21–69 66. Thaler M (1980) Estimates of the invariant densities of endomorphisms with indifferent fixed points. Isr J Math 37(4):303– 314 67. Walters P (1975) Ruelle’s operator theorem and g-measures. Trans Amer Math Soc 214:375–387 68. Walters P (1992) Differentiability properties of the pressure of a continuous transformation on a compact metric space. J Lond Math Soc (2) 46(3):471–481 69. Wang X-J (1989) Statistical physics of temporal intermittency. Phys Rev A 40(11):6647–6661 70. Young L-S (1990) Large deviations in dynamical systems. Trans Amer Math Soc 318:525–543 71. Young L-S (2002) What are SRB measures, and which dynamical systems have them? Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J Statist Phys 108(5–6):733–754 72. Zinsmeister M (2000) Thermodynamic formalism and holomorphic dynamical systems. SMF/AMS Texts and Monographs, 2. American Mathematical Society

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Quantum Bifurcations BORIS Z HILINSKIÍ Université du Littoral, Dunkerque, France Article Outline Glossary Definition of the Subject Introduction Simplest Effective Hamiltonians Bifurcations and Symmetry Imperfect Bifurcations Organization of Bifurcations Bifurcation Diagrams for Two Degree-of-Freedom Integrable Systems Bifurcations of “Quantum Bifurcation Diagrams” Semi-Quantum Limit and Reorganization of Quantum Bands Multiple Resonances and Quantum State Density Physical Applications and Generalizations Future Directions Bibliography Glossary Classical limit The classical limit is the classical mechanical problem which can be constructed from a given quantum problem by some limiting procedure. During such a construction the classical limiting manifold should be defined which plays the role of classical phase space. As soon as quantum mechanics is more general than classical mechanics, going to the classical limit from a quantum problem is much more reasonable than discussing possible quantizations of classical theories [73]. Energy-momentum map In classical mechanics for any problem which allows the existence of several integrals of motion (typically energy and other integrals often named as momenta) the Energy-Momentum (EM) map gives the correspondence between the phase space of the initial problem and the space of values of all independent integrals of motion. The energy-momentum map introduces the natural foliation of the classical phase space into common levels of values of energy and momenta [13,35]. The image of the EM map is the region of the space of possible values of integrals of motion which includes regular and critical values. The quantum analog of the image of the energy-momentum map is the joint spectrum of mutually commuting quantum observables.

Joint spectrum For each quantum problem a maximal set of mutually commuting observables can be introduced [16]. A set of quantum wave functions which are mutual eigenfunctions of all these operators exists. Each such eigenfunction is characterized by eigenvalues of all mutually commuting operators. The representation of mutual eigenvalues of n commuting operators in the n-dimensional space gives the geometrical visualization of the joint spectrum. Monodromy In general, the monodromy characterizes the evolution of some object after it makes a close path around something. In classical Hamiltonian dynamics the Hamiltonian monodromy describes for completely integrable systems the evolution of the first homology group of the regular fiber of the energy-momentum map after a close path in the regular part of the base space [13]. For a corresponding quantum problem the quantum monodromy describes the modification of the local structure of the joint spectrum after its propagation along a close path going through a regular region of the lattice. Quantum bifurcation Qualitative modification of the joint spectrum of the mutually commuting observables under the variation of some external (or internal) parameters and associated in the classical limit with the classical bifurcation is named quantum bifurcation [59]. In other words the quantum bifurcation is the manifestation of the classical bifurcation presented in the classical dynamic system in the quantum version of the same system. Quantum-classical correspondence Starting from any quantum problem the natural question consists of defining the corresponding classical limit, i. e. the classical dynamic variables forming the classical phase space and the associated symplectic structure. Whereas in simplest quantum problems defined in terms of standard position and momentum operators with commutation relation [q i ; p j ] D i„ı i j , [q i ; q j ] D [p i ; p j ] D 0 (i; j D 1 : : : n) the classical limit phase space is the 2n-dimensional Euclidean space with standard symplectic structure on it, the topology of the classical limit manifold in many other important for physical applications cases can be rather non-trivial [73,87]. Quantum phase transition Qualitative modifications of the ground state of a quantum system occurring under the variation of some external parameters at zero temperature are named quantum phase transitions [65]. For finite particle systems the quantum phase transition can be considered as a quantum bifurcation [60].

Quantum Bifurcations

Spontaneous symmetry breaking Qualitative modification of the system of quantum states caused by perturbation which has the same symmetry as the initial problem. Local symmetry of solutions decreases but the number of solutions increases. In the energy spectra of finite particle systems the spontaneous symmetry breaking produces an increase of the “quasidegeneracy”, i. e. formation of clusters of quasi-degenerate levels whose multiplicity can be much higher than the dimension of the irreducible representations of the global symmetry group [51]. Symmetry breaking Qualitative changes in the properties (dynamical behavior, and in particular in the joint spectrum) of quantum systems which are due to modifications of the global symmetry of the problem caused by external (less symmetrical than original problem) perturbation can be described as symmetry breaking effects. Typical effects consist of splitting of degenerate energy levels classified initially according to irreducible representation of the initial symmetry group into less degenerate groups classified according to irreducible representation of the subgroup (the symmetry group of the perturbation) [47]. Definition of the Subject Quantum bifurcations (QB) are qualitative phenomena occurring in quantum systems under the variation of some internal or external parameters. In order to make this definition a little more precise we add the additional requirement: The qualitative modification of the “behavior” of a quantum system can be described as QB if it consists of the manifestation for the quantum system of the classical bifurcation presented in classical dynamic systems which is the classical analog of the initial quantum system. Quantum bifurcations are typical elementary steps leading from the simplest in some way effective Hamiltonian to more complicated ones under the variation of external or internal parameters. As internal parameters one may consider the values of exact or approximate integrals of motion. The construction of an effective Hamiltonian is typically based on the averaging and/or reduction procedure which results in the appearance of “good” quantum numbers (or approximate integrals of motion). The role of external parameters can be played by forces of external champs, phenomenological constants in the effective Hamiltonians, particle masses, etc. In order to limit the very broad field of qualitative changes and of possible quantum bifurcations in particular, we restrict ourselves mainly to quantum systems whose classical limit is associated with compact phase space and is nearly integrable.

This means that for quantum problems the set of mutually commuting observables can be constructed within a reasonable physical approximation almost everywhere at least locally. Quantum bifurcations are supposed to be universal phenomena which appear in generic families of quantum systems and explain how relatively simple behavior becomes complicated under the variation of some physical parameters. To know these elementary bricks responsible for increasing complexity of quantum systems under control parameter modifications is extremely important in order to make the extrapolation to regimes unaccessible to experimental study. Introduction In order to better understand the manifestations of quantum bifurcations and their significance for concrete physical systems we start with the description of several simple model physical problems which exhibit in some sense the simplest (but nevertheless) generic behavior. Let us start with the harmonic oscillator. A one-dimensional harmonic oscillator has an equidistant system of eigenvalues. All eigenvalues can be labeled by consecutive integer quantum numbers which have the natural interpretation in terms of the number of zeros of eigenfunctions. The classical limit manifold (classical phase space) is a standard Euclidean 2-dimensional space with natural variables fp; qg. The classical Hamiltonian for the harmonic oscillator is an example of a Morse-type function which has only one stationary point p D q D 0 and all non-zero energy levels of the Hamiltonian are topological circles. If now we deform slightly the Hamiltonian in such a way that its classical phase portrait remains topologically the same, the spectrum of the quantum problem changes but it can be globally described as a regular sequence of states numbered consecutively by one integer and such description remains valid for any mass parameter value. Note, that for this problem increasing mass means increasing quantum state density and approaching classical behavior (classical limit). More serious modification of the harmonic oscillator can lead, for example, to creation of new stationary points of the Hamiltonian. In classical theory this phenomenon is known as fold bifurcation or fold catastrophe [3,31]. The phase portrait of the classical problem changes qualitatively. As a function of energy the constant level set of the Hamiltonian has different topological structure (one circle, two circles, figure eight, circle and a point, or simply point). The quantum version of the same problem shows the existence of three different sequences

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Quantum Bifurcations, Figure 1 Classical and quantum bifurcations for a one degree-of-freedom system. Situations before (a,b,e) and after (c,d,f) the bifurcation are shown. a Energy map for harmonic oscillator-type system. Inverse images of each point are indicated. b Quantum state lattice for harmonic oscillator-type system. c Energy map after the bifurcation. Inverse images of each point are indicated. d Quantum state lattice after bifurcation represented as composed of three regular parts glued together. e Phase portrait for harmonic oscillator-type system. Inverse images are S1 (generic inverse image) and S0 (inverse image for minimal energy value). f Phase portrait after bifurcation

of states which become clearly visible in the limit of the high density of states which can be reached by increasing the mass value parameter [32]. Such qualitative modification of the energy spectrum of the 1D-quantum Hamiltonian gives the simplest example of the phenomenon which can be described as a quantum bifurcation. Figure 1 shows a schematic representation of quantum bifurcations for a model system with one degree-of-freedom in parallel in quantum and classical mechanics. After looking for one simple example we can formulate a more general question which concerns the appearance in more general quantum systems of qualitative phenomena which can be characterized as quantum bifurcations. Simplest Effective Hamiltonians We turn now to several models which describe some specific classes of relatively simple real physical quantum systems formed by a finite number of particles (atoms, molecules, . . . ). Spectra of such quantum objects are studied nowadays with very high accuracy and this allows us to compare the behavior predicted by quantum bifurcations with the precise information about energy level structure found, for example, from high-resolution molecular spectroscopy.

Typically, the intra-molecular dynamics can be split into electronic, vibrational, and rotational ones due to important differences in characteristic energy excitations or in time scales. The most classical is the rotational motion and probably due to that the quantum bifurcations as a counterpart to classical bifurcations were first studied for purely rotational problems [59,61]. Effective rotational Hamiltonians describe the internal structure of rotational multiplets formed by isolated finite particle systems (atoms, molecules, nuclei) [36]. For many molecular systems in the ground electronic state any electronic and vibrational excitations are much more energy consuming as compared with rotational excitations. Thus, to study the molecular rotation the simplest physical assumption is to suppose that all electronic and all vibrational degrees-of-freedom are frozen. This means that a set of quantum numbers is given which have the sense of approximate integrals of motion specifying the character of vibrational and electronic motions in terms of these “good” quantum numbers. At the same time for a free molecule in the absence of any external fields due to isotropy of the space the total angular momentum J and its projection J z on the laboratory fixed frame are strict integrals of motion. Consequently, to describe the rotational motion for fixed values of J and J z it is sufficient to analyze the effective problem with only one degree-offreedom. The dimension of classical phase space in this case equals two and the two classical conjugate variables are: the projection of the total angular momentum on the body fixed frame and conjugate angle variable. The classical phase space is topologically a two-dimensional sphere, S2 . There is a one-to-one correspondence between the points on a sphere and the orientation of the angular momentum in the body-fixed frame. Such a representation gives a clear visualization of a classical rotational Hamiltonian as a function defined over a sphere [36,49]. In quantum mechanics the rotation of molecules is traditionally described in terms of an effective rotational Hamiltonian which is constructed as a series in rotational operators J x , J y , J z , the components of the total angular momentum J. In a suitably chosen molecular fixed frame the effective Hamiltonian has the form Heff D AJ x2 C BJ 2y C CJ z2 C

X

c˛ˇ J x˛ J ˇy J z C    ; (1)

where A, B, C and c˛ˇ are constants. To relate quantum and classical pictures we note that J2 and energy are integrals of Euler’s equations of motion for dynamic variables J x , J y , J z . The phase space of the classical rotational problem with constant jJj is S2 , the two-dimen-

Quantum Bifurcations

sional sphere, and it can be parametrized with spherical angles ( ; ) in such a way that the points on S2 define the orientation of J, i. e. the position of the axis and the direction of rotation. To get the classical interpretation of the quantum Hamiltonian we introduce the classical analogs of the operators J x , J y , J z 1 1 0 sin cos  Jx p J ! @ J y A D @ sin sin  A J(J C 1) cos Jz 0

(2)

and consider the rotational energy as a function of the dynamical variables ( ; ) and the parameter J. Thus, for an effective rotational Hamiltonian the corresponding classical symbol is a function E J ( ; ) defined over S2 and named usually the rotational energy surface [36]. Taking into account the symmetry imposed by the initial problem and the topology of the phase space the simplest rotational Hamiltonian can be constructed. In classical mechanics the simplest Hamiltonian can be defined (using Morse theory [55,89]) as a Hamiltonian function with the minimal possible number of non-degenerate stationary points compatible with the symmetry group action of the classical phase space. Morse theory in the presence of symmetry (or equivariant Morse theory) implies important restrictions on the number of minima, maxima, and saddle points. In the absence of symmetry the simplest Morse type function on the S2 phase space has one minimum and one maximum, as a consequence of Morse inequalities. In the presence of non-trivial symmetry group action the minimal number of stationary points on the sphere increases. For example, many asymmetric top molecules (possessing three different moment of inertia of the equilibrium configuration) have D2h symmetry group [47]. This group includes rotations over  around fx; y; zg axes, reflections in fx y; yz; zxg planes and inversion as symmetry operations. Any D2h invariant function on the sphere has at least six stationary points (two equivalent minima, two equivalent maxima, and two equivalent saddle points). This means that in quantum mechanics the asymmetric top has eigenvalues which form two regular sequences of quasi-degenerate doublets with the transition region between them. The correspondence between the quantum spectrum and the structure of the energy map for the classical problem is shown in Fig. 2. Highly symmetrical molecules which have cubic symmetry, for example, can be described by a simplest Morsetype Hamiltonian with 26 stationary points (6 and 8 minima/maxima and 12 saddle points). As a consequence, the

Quantum Bifurcations, Figure 2 a Schematic representation of the energy level structure for asymmetric top molecule. Vertical axis corresponds to energy variation. Quantum levels are classified by the symmetry group of the asymmetric top. Two fold clusters at two ends of the rotational multiplet are formed by states with different symmetry. b Foliation of the classical phase space (S2 sphere) by constant levels of the Hamiltonian given in the form of its Reeb graph. Each point corresponds to a connected component of the constant level set of the Hamiltonian (energy). c Geometric representation of the constant energy sections

corresponding quantum Hamiltonian shows the presence of six-fold and eight-fold quasi-degenerate clusters of rotational levels. As soon as the simplest classical Hamiltonian is characterized by the appropriate system of stationary points the whole region of possible classical energy values (in the case of dynamical systems with only one degree-offreedom the energy-momentum map becomes simply the energy map) appears to be split into different regions corresponding to different dynamical regimes, i. e. to different regions of the phase portrait foliated by topologically non-equivalent systems of classical trajectories. Accordingly, the energy spectrum of the corresponding quantum Hamiltonian can be qualitatively described as formed by regular sequences of states within each region of the classical energy map. Quantum bifurcations are universal phenomena which lead to a new organization of the energy spectrum into qualitatively different regions in accordance with corresponding qualitative modifications of the classical energy-momentum map under the variation of some control parameter.

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Simplest Hamiltonians for Two Degree-of-Freedom Systems When the quantum system has two or larger number of degrees-of-freedom the simplest dynamical regimes often correspond in classical mechanics to a quasi-regular dynamics which can be reasonably well approximated by an integrable model. The integrable model in classical mechanics can be constructed by normalizing the Hamiltonian and by passing to so-called normal forms [2,49]. The quantum counterpart of normalization is the construction of a mutually commuting set of operators which should not be mistaken with quantization of systems in normal form. Corresponding eigenvalues can be used as “good” quantum numbers to label quantum states. A joint spectrum of mutually commuting operators corresponds to the image of the energy-momentum map for the classical completely integrable dynamical problem. In this context the question about quantum bifurcations first of all leads to the question about qualitative classification of the joint spectra of mutually commuting operators. To answer this question we need to start with the qualitative description of foliations of the total phase space of the classical problem by common levels of integrals of motion which are mutually in involution [2,7]. One needs to distinguish the regular and the singular values of the energy-momentum map. For Hamiltonian systems the inverse images of the regular values are regular tori (one or several) [2]. A lot of different singularities are possible. In classical mechanics different levels of the classifications are studied in detail [7]. The diagram which represents the image of the classical EM map together with its stratification into regular and critical values is named the bifurcation diagram. The origin of such a name is due to the fact that the values of integrals of motion can be considered as control parameters for the phase portraits (inverse images of the EM map) of the reduced systems. For quantum problems the analog of the classical stratification of the EM map for integrable systems is the splitting of the joint spectrum of several commuting observables into regions formed by regular lattices of joint eigenvalues. Any local simply connected neighborhood of a regular point of the lattice can be deformed into part of the regular Zn lattice of integers. This means that local quantum numbers can be consistently introduced to label states of the joint spectrum. If the regular region is not simply connected it still can be characterized locally by a set of “good” quantum numbers. At the same time this is impossible globally. Likewise in classical mechanics the Hamiltonian monodromy is the simplest obstruction to the existence of the global action-angle variables [17,57], in

Quantum Bifurcations, Figure 3 Joint spectrum of two commuting operators together with the image of the classical EM map for the resonant 1 : (1) oscillator given by (3),(4). Quantum monodromy is seen as a result of transportation of the elementary cell of the quantum lattice along a close path through a non simply connected region of the regular part of the image of the EM map. Taken from [58]

quantum mechanics the analog notion of quantum monodromy [14,34,68,80] characterizes the global non-triviality of the regular part of the lattice of joint eigenvalues. Figure 3 demonstrates the effect of the presence of a classical singularity (isolated focus-focus point) on the global properties of the quantum lattice formed by joint eigenvalues of two commuting operators for a simple problem with two degrees-of-freedom, which is essentially the 1 : (1) resonant oscillator [58]. Two integrals of motions in this example are chosen as  1 2  1 2 p1 C q21  p2 C q22 ; 2 2 2 1 2 f 2 D p 1 q2 C p 2 q1 C p1 C q21 C p22 C q22 : 4

f1 D

(3) (4)

Locally in any simply connected region which does not include the classical singularity of the EM map situated at f1 D f2 D 0, the joint spectrum can be smoothly deformed to the regular Z2 lattice [58,90]. Such lattices are shown, for example, in Fig. 4. If somebody wants to use only one chart to label states, it is necessary to take care in respect of the multivaluedness of such a representation. There are two possibilities: (i) One makes a cut and maps the quantum lattice to a regular Z2 lattice with an appropriate solid angle removed from it (see Fig. 5 [58,68,90]). Points on the boundary of such a cut should be identified and a special matching rule explaining how to cross the path should be introduced. Similar constructions are quite popular in solid state physics in order to represent defects of lattices, like dislocations, disclinations, etc. We just note that the “monodromy defect” introduced in such a way is different from standard construction

Quantum Bifurcations

Quantum Bifurcations, Figure 4 Two chart atlas which covers the quantum lattice of the 1 : (1) resonant oscillator system represented in Fig. 3. Top plots show the choice of basis cells and the gluing map between the charts. Bottom plots show the transport of the elementary cell (dark gray quadrangles) in each chart. Central bottom panel shows closed path  and its quantum realization (black dots) leading to non-trivial monodromy (compare with Fig. 3). Taken from [58]

Quantum Bifurcations, Figure 5 Construction of the 1 : (1) lattice defect starting from the regular Z 2 lattice. The solid angle is removed from the regular Z 2 lattice and points on the so-obtained boundary are identified by vertical shifting. Dark gray quadrangles show the evolution of an elementary lattice cell along a closed path around the defect point. Taken from [58]

for dislocation and disclination defects [45,50]. The inverse procedure of the construction of the “monodromy defect” [90] from a regular lattice is represented in Fig. 5. Let us note that the width of the solid angle removed depends on the direction of the cut and the direction of the cut itself can be chosen in an ambiguous way. (ii) An alternative possibility is to make a cut in such a way that the width of the removed angle becomes equal to zero. For focus-focus singularities one such

direction always exists and is named an eigenray by Symington [75]. The same construction is used in some physical papers [10,11,85]. The inconvenience of such a procedure is the appearance of discontinuity of the slope of the constant action (quantum number) line at the cut, whereas the values of actions themselves are continued (see Fig. 6). This gives the wrong impression that this eigenray is associated with some special non-regular behavior of the initial problem, whereas there is no singularity except at one focusfocus point. Bifurcations and Symmetry The general mathematical answer about the possible qualitative modifications of a system of stationary points of functions depending on some control parameters can be found in bifurcation (or catastrophe) theory [3,31,33]. It is important that the answer depends on the number of control parameters and on the symmetry. Very simple classification of possible typical bifurcations of stationary points of a one-parameter family of functions under presence of symmetry can be formulated for dynamical systems with one degree-of-freedom. The situation is particularly simple because the phase space is two-dimensional and the complete list of local symmetry groups (which are the stabilizers of stationary points) includes only 2D-point groups [84]. It should be noted that the global symmetry

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Quantum Bifurcations, Figure 6 Representation of the quantum joint spectrum for the “Mexican hat” potential V(r) D ar4  br2 with the “cut” along the eigenray. For such a cut the left and the right limits at the cut give the same values of actions (good quantum numbers) but the lines of constant values of actions exhibit a “kink” at the cut (the discontinuity of the first derivative)

of the problem can be larger than the local symmetry of the bifurcating stationary points. In such a case the bifurcations occur simultaneously for all points forming one orbit of the global symmetry group [51,52]. We describe briefly here (see Table 1) the classification of the bifurcations of stationary points in the presence of symmetry for families of functions depending on one parameter and associated quantum bifurcations [59,61]. Their notation includes the local symmetry group and several additional indexes which specify creation/annihilation of stationary points and the local or non-local character of the bifurcation. The list of possible bifurcations includes: C1˙ A non-symmetrical non-local bifurcation resulting in the appearance (+) or disappearance () of a stableunstable pair of stationary points with the trivial local symmetry C1 . In the quantum problem this bifurcation

is associated with the appearance or disappearance of a new regular sequence of states glued at its end with the intermediate part of another regular sequence of quantum states [32,77]. C2L˙ A local bifurcation with the broken C2 local symmetry. This bifurcation results either in appearance of a triple of points (two equivalent stable points with C1 local symmetry and one unstable point with C2 local symmetry) instead of one stable point with C2 symmetry, or in inverse transformation. The number of stationary points in this bifurcation increases or decreases by two. For the quantum problem the result is the transformation of a local part of a regular sequence of states into one sequence of quasi-degenerate doublets. C2N˙ A non-local bifurcation with the broken C2 local symmetry. This bifurcation results in appearance (+) or disappearance () of two new unstable points with broken C2 symmetry and simultaneous transformation of the initially stable (for +) or unstable (for ) stationary point into an unstable/stable one. The number of stationary points in this bifurcation increases or decreases by two. For the quantum problem this means the appearance of a new regular sequence of states near the separatrix between two different regular regions. C nN (n D 3; 4) A non-local bifurcation corresponding to passage of n unstable stationary points through a stable stationary point with Cn local symmetry which is accompanied with the minimum $ maximum change for a stable point with the Cn local symmetry. The number of stationary points remains the same. For the quantum problem this bifurcation corresponds to transformation of the increased sequence of energy levels into a decreased sequence. C nL˙ (n  4) A local bifurcation which results in appearance (+) or disappearance () of n stable and n unstable stationary points with the broken Cn symmetry and a simultaneous minimum $ maximum change of a stable point with the Cn local symmetry. The num-

Quantum Bifurcations, Table 1 Bifurcations in the presence of symmetry. Solid lines denote stable stationary points. Dashed lines denote unstable stationary points. Numbers in parenthesis indicate the multiplicity of stationary points

Quantum Bifurcations

Quantum Bifurcations, Table 2 Molecular examples of quantum bifurcations in the rotational structure of individual vibrational components under the variation of the absolute value of angular momentum, J. Jc is the critical value corresponding to bifurcation Molecule Component Jc

Bifurcation type

SiH4

2 (C)

12 C2NC

SnH4

2 ()

10 C2NC ; C3N ; C4N ; C2N

CF4

2 (C)

50 C4LC

H2 Se

j0i

20 C2LC

ber of stationary points increases or decreases by 2n. In the quantum problem after bifurcation a new sequence of n-times quasi-degenerate levels appears/disappears. Universal quantum Hamiltonians which describe the qualitative modification of the quantum energy level system around the bifurcation point are given in [59,61]. The presence of symmetry makes it much easier to observe the manifestation of quantum bifurcations. Modification of the local symmetry of stable stationary points results in the modification of the cluster structure of energy levels, i. e. the number and the symmetry types of energy level forming quasi-degenerate groups of levels. This phenomenon is essentially the spontaneous breaking of symmetry [51]. Several concrete molecular systems which show the presence of quantum bifurcations in rotational structure under rotational excitation are cited in Table 2. Many other examples can be found in [9,23,59,67,71,88,89, 92,93] and references therein. In purely vibrational problems breaking dynamical SU(N) symmetry of the isotrope harmonic oscillator till finite symmetry group results in formation of so-called non-linear normal modes [23,54] or quasimodes [1], or local modes [9,25,39,40,43,46,48]. In the case of two degrees-of-freedom the analysis of the vibrational problem can be reduced to the analysis of the problem similar to the rotational one [36,66] and all the results about possible types of bifurcations found for rotational problems remain valid in the case of intra-molecular vibrational dynamics. Imperfect Bifurcations According to general results the possible types of bifurcations which are generically present (and persist under small deformations) in a family of dynamical systems strictly depend on the number of control parameters. In the absence of symmetry only one bifurcation of stationary points is present for a one-parameter family of Morse-

type functions, namely the formation (annihilation) of two new stationary points. This corresponds to saddle-node bifurcation for one degree-of-freedom Hamiltonian systems. The presence of symmetry increases significantly the number of possible bifurcations even for families with only one parameter [31,33]. From the physical point-of-view it is quite natural to study the effect of symmetry breaking on the symmetry allowed bifurcation. Decreasing symmetry naturally results in the modification of the allowed types of bifurcations but at the same time it is quite clear that at sufficient slight symmetry breaking perturbation the resulting behavior of the system should be rather close to the behavior of the original system with higher symmetry. In the case of a small violation of symmetry the socalled “imperfect bifurcations” can be observed. Imperfect bifurcations, which are well known in the classical theory of bifurcations [33] consist of the appearance of stationary points in the neighborhood of another stationary point which does not change its stability. In some way one can say that imperfect bifurcation mimics generic bifurcation in the presence of higher symmetry by the special organization of several bifurcations which are generic in the presence of lower symmetry. Naturally quantum bifurcations follow the same behavior under the symmetry breaking as classical ones. Very simple and quite natural examples of imperfect quantum bifurcations were demonstrated on the example of the rotational structure modifications under increasing angular momentum [91]. The idea of appearance of imperfect bifurcations is as follows. Let us suppose that some symmetrical molecule demonstrates under the variation of angular momentum a quantum rotational bifurcation allowed by symmetry. The origin of this bifurcation is due, say, to centrifugal distortion effects which depend strongly on J but are not very sensitive to small variation of masses even in the case of symmetry breaking isotopic substitution. In such a case a slight modification of the masses of one or several equivalent atoms breaks the symmetry and this symmetry violation can be made very weak due to the small ratio M/M under isotope substitution. In classical theory the effect of symmetry breaking can be easily seen through the variation of the position of stationary points with control parameter. For example, instead of a pitchfork bifurcation which is typical for C2 local symmetry, we get for the unsymmetrical problem (after slight breaking of C2 symmetry) a smooth evolution of the position of one stationary point and the appearance of two new stationary points in fold catastrophe (see Fig. 7). In associated quantum bifurcations the most important effect is the splitting of clusters. But one should be careful with this interpretation because in quantum mechanics of finite particle systems the clusters are always split

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Quantum Bifurcations, Figure 7 Imperfect bifurcations. a Position x of stationary points as a function of control parameter  during a pitchfork bifurcation in the presence of C 2 local symmetry. b Modifications induced by small symmetry perturbation of lower symmetry. Solid line: Stable stationary points. Dashed lines: Unstable stationary points

due to quantum mechanical tunneling between different equivalent regions of localization of quantum wave functions. Intercluster splitting increases rapidly approaching the region of classical separatrix. The behavior of quantum tunneling was studied extensively in relation to the quantum breathers problem [6,29]. Systematic application of quasi-classical methods to reproduce quantum energy level structure near the singularities of the energy-momentum maps where exponentially small corrections are important is possible but requires special efforts (see for example [12]) and we will not touch upon this problem here. Organization of Bifurcations The analysis of the quantum bifurcations in concrete examples of rotating molecules have shown that in some cases the molecule undergoes several consecutive qualitative changes which can be interpreted as a sequence of bifurcations which sometimes cannot even be separated into elementary bifurcations for the real scale of the control parameter [89]. One can imagine in principle that successive bifurcations lead to quantum chaos in analogy with classical dynamical systems where the typical scenario for the transition to chaos is through a sequence of bifurcations. Otherwise, the molecular examples were described with effective Hamiltonians depending only on one degree-offreedom and the result of the sequence of bifurcations was just the crossover of the rotational multiplets [64]. In some sense such a sequence of bifurcations can be interpreted as an imperfect bifurcation assuming initially higher dynamical symmetry, like the continuous SO(3) group. Later, a similar crossover phenomenon was found in a quite different quantum problem, like the hydrogen

atom in external fields [24,53,72]. The general idea of such organization of bifurcations is based on the existence of two different limiting cases of dynamical regimes for the same physical quantum system (often under presence of the same symmetry group) which are qualitatively different. For example, the number of stationary points, or their stability differs. If H 1 and H 2 are two corresponding effective Hamiltonians, the natural question is: Is it possible to transform H 1 into H 2 by a generic perturbation depending on only one parameter? And if so, what is the minimal number of bifurcations to go through? The simplest quantum system for which such a question becomes extremely natural is the hydrogen atom in the presence of external static electric (F) and magnetic (G) fields. Two natural limits – the Stark effect in the electric field and Zeeman effect in the magnetic field – show quite different qualitative structure even in the extremely low field limit [15,20,63,72,78]. Keeping a small field one can go from one (Stark) limit to another (Zeeman) and this transformation naturally goes through qualitatively different regimes [24,53]. In spite of the fact that the hydrogen atom (even without spin and relativistic corrections) is only a three degree-of-freedom system, the complete description of qualitatively different regimes in a small field limit is still not done and remains an open problem [24]. An example of clearly seen qualitative modifications of the quantum energy level system of the hydrogen atom under the variation of F/G ratio of the strengths of two parallel electric and magnetic fields is shown in Fig. 8. The calculations are done for a two degree-of-freedom system after the normalization with respect to the global action. In quantum mechanics language this means that only energy levels which belong to the same n-shell of the hy-

Quantum Bifurcations

Bifurcation Diagrams for Two Degree-of-Freedom Integrable Systems

Quantum Bifurcations, Figure 8 Reorganization of the internal structure of the n-multiplet of the hydrogen atom in small parallel electric and magnetic fields. Energies of stationary points of the classical Hamiltonian (red solid lines) are shown together with quantum energy levels (blue solid lines). The figure is done for n D 10 (there are n2 D 100 energy levels forming this multiplet). As the ratio F/G of electric F and magnetic G fields varies this two degree-of-freedom system goes through different zones associated with special resonance relations between two characteristic frequencies (shown by vertical dashed lines). Taken from [24]

drogen atom are treated and the interaction with other n0 shells is taken into account only effectively. The limiting classical phase space for this effective problem is the fourdimensional space S 2  S 2 , which is the direct product of two two-dimensional spheres. In the presence of axial symmetry this problem is completely integrable and the Hamiltonian and the angular momentum provide a complete set of mutually commuting operators. Energies of stationary points of classical Hamiltonian limit are shown on the same Fig. 8 along with quantum levels. When one of the characteristic frequencies goes through zero, the socalled collapse phenomena occurs. Some other non-trivial resonance relations between two frequencies are also indicated. These resonances correspond to special organization of quantum energy levels. At the same time it is not necessary here to go to joint spectrum representation in order to see the reorganization of stationary points of the Hamiltonian function on S 2  S 2 phase space under the variation of the external control parameter F/G. A more detailed treatment of qualitative features of the energy level systems for the hydrogen atom in low fields is given in [15,20,24].

Let us consider now the two degree-of-freedom integrable system with compact phase space as a bit more complex but still reasonably simple problem. Many examples of such systems possess EM maps with the stratification of the image formed by the regular part surrounded by the singular boundary. The most naturally arising examples of classical phase spaces, like S 2  S 2 , CP2 , are of that type. All internal points on the image of the EM map are regular in these cases. In practice, real physical problems, even after necessary simplifications and approximations lead to more complicated models. Some examples of fragments of images of the EM map with internal singular points are shown in Fig. 9. In classical mechanics the inverse images of critical values are singular tori of different kinds. Some of them are represented in Fig. 10. Inverse images of critical points situated on the boundary of the EM image have lower dimension. They can be one-dimensional tori (S1 circles), or zero-dimensional (points). The natural question now is to describe typical generic modifications of the Hamiltonian which lead to qualitative modifications of the EM map image in classical mechanics and to associated modifications of the joint spectrum in quantum mechanics. The simplest classical bifurcation leading to modification of the image of the EM map is the Hamiltonian Hopf bifurcation [79]. It is associated with the following modification of the image of the EM map. The critical value of the EM map situated on the boundary leaves the boundary and enters the internal domain of regular values (see Fig. 11). As a consequence, the toric fibration over the closed path surrounding an isolated singularity is nontrivial. Its non-triviality can be characterized by the Hamiltonian monodromy which describes the mapping from the fundamental group of the base space into the first ho-

Quantum Bifurcations, Figure 9 Typical images of the energy momentum map for completely integrable Hamiltonian systems with two degrees-of-freedom in the case of: a integer monodromy, b fractional monodromy, c non-local monodromy, and d bidromy. Values in the light shaded area lift to single 2-tori; values in the dark shaded area lift to two 2-tori. Taken from [69]

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Quantum Bifurcations, Figure 10 Two-dimensional singular fibers in the case of integrable Hamiltonian systems with two degrees-of-freedom (left to right): singular torus, bitorus, pinched and curled tori. Singular torus corresponds to critical values in Fig. 9c, d (ends of bitoris line). Bitorus corresponds to critical values in Fig. 9c, d, which belong to singular line (fusion of two components). Pinched torus corresponds to isolated focus-focus singularity in Fig. 9a. Curled torus is associated with critical values at singular line in Fig. 9b (fractional monodromy). Taken from [69]

Quantum Bifurcations, Figure 11 Qualitative modification of the image of the EM map due to Hamiltonian Hopf bifurcation. Left: Simplest integrable toric fibration over S2  S2 classical phase space. A, B, C, D: Critical values corresponding to singular S0 fibers. Regular points on the boundary correspond to S1 fibers. Regular internal points: Regular T 2 fibers. Right: Appearance of an isolated critical value inside the field of regular values. Critical value B corresponds to pinched torus shown in Fig. 10

mology group of the regular fiber [18]. A typical pattern of the joint spectrum around such a classical singularity is shown in Fig. 3. The joint spectrum manifests the presence of quantum monodromy. Its interpretation in terms of regular lattices is given in Figs. 4 and 5. Taking into account additional terms of higher order it is possible to distinguish different types of Hamiltonian Hopf bifurcations usually named as subcritical and supercritical [19,79]. New qualitative modification, for example, corresponds to transformation of an isolated singular value of the EM map into an “island”, i. e. the region of the EM image filled by points whose inverse images consist of two connected components. Integrable approximation for vibrational motion in the LiCN molecule shows the presence of such an island associated with the non-local quantum monodromy (see Fig. 12) [41]. The monodromy naturally coincides with the quantum monodromy of isolated focus-focus singularity which deforms continuously into the island monodromy. It is interesting to note that in

molecule HCN which is rather similar to LiCN, the region with two components in the inverse image of the EM map exists also but the monodromy cannot be defined due to impossibility to go around the island [22]. In the quantum problem the presence of “standard” quantum monodromy in the joint spectrum of two mutually commuting observables can be seen through the mapping of a locally regular part of the joint spectrum lattice to an idealized Z2 lattice. Existence of local actions for the classical problem which are defined almost everywhere and the multivaluedness of global actions from one side and the quantum-classical correspondence from another side allow the interpretation of the joint spectrum with quantum monodromy as a regular lattice with an isolated defect. Recently, the generalization of the notion of quantum (and classical) monodromy was suggested [21,58]. For quantum problems the idea is based on the possibility to study instead of the complete lattice formed by the joint spectrum only a sub-lattice of finite index. Such a transformation allows one to eliminate certain “weak line singularities” presented in the image of the EM map. The resulting monodromy is named “fractional monodromy” because for the elementary cell in the regular region the formal transformation after a propagation along a close path crossing “weak line singularities” turns out to be represented in a form of a matrix with fractional coefficients. An example of quantum fractional monodromy can be given with a 1 : (2) resonant oscillator system possessing two integrals of motion f 1 , f 2 in involution:  2!  2  ! 2 p1 C q21  p2 C q22 C R1 (q; p) ; 2 2  2 f2 D Im (q1 C i p1 ) (q2 C i p2 ) C R2 (q; p) :

f1 D

(5) (6)

Quantum Bifurcations

Quantum Bifurcations, Figure 12 Quantum joint spectrum for the quantum model problem with two degrees-of-freedom describing two vibrations in the LiCN molecule. The non-local quantum monodromy is shown by the evolution of the elementary cell of the quantum lattice around the singular line associated with gluing of two regular lattices corresponding in molecular language to two different isomers, LiCN and LiNC. Classical limit (left) shows the possible deformation of isolated focus-focus singularity for pendulum to non-local island singularity for LiNC model. In contrast to LiCN, the HCN model has an infinite island which cannot be surrounded by a close path. Taken from [41]

Quantum Bifurcations, Figure 13 Joint quantum spectrum for two-dimensional non-linear 1 : (2) resonant oscillator (5),(6). The singular line is formed by critical values whose inverse images are curled tori shown in Fig. 10. In order to get the unambiguous result of the propagation of the cell of the quantum lattice along a closed path crossing the singular line, the elementary cell is doubled. Taken from [58]

The corresponding joint spectrum for the quantum problem is shown in Fig. 13. It can be represented as a regular Z2 lattice with a solid angle removed (see Fig. 14). The main difference with the standard integer monodromy representation is due to the fact that even after gluing two sides of the cut we get the one-dimensional singular stratum which can be neglected only after going to a sub-lattice (to a sub-lattice of index 2 for 1 : 2 fractional singularity).

Another kind of generalization of the monodromy notion is related to the appearance of multi-component inverse images for the EM maps. We have already mentioned such a possibility with the appearance of non-local monodromy and Hamiltonian Hopf bifurcations (see Fig. 12). But in this case two components of the inverse image belong to different regular domains and cannot be joined by a path going only through regular values. Another possibility is suggested in [69,70] and is explained

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Quantum Bifurcations, Figure 14 Representation of a lattice with 1 : 2 rational defect by cutting and gluing. Left: The elementary cell goes through cut in an ambiguous way. The result depends on the place where the cell crosses the cut. Right: Double cell crosses the cut in an unambiguous way. Taken from [58]

b

a

c

b

Quantum Bifurcations, Figure 15 Schematic representation of the inverse images for a problem with bidromy in the form of the unfolded surface. Each connected component of the inverse image is represented as a single point. The path b0 ab00 starts and ends at the same point of the space of possible values of integrals of motion but it starts at one connected component and ends at another one. At the same time the path goes only through regular tori. Taken from [70]

schematically in Fig. 15. This figure shows that the arrangement of fibers can be done in such a way that one connected component can be deformed into another connected component along a path which goes only through regular tori. The existence of a quantum joint spectrum corresponding to such a classical picture was demonstrated on the example of a very well-known model problem with three degrees-of-freedom: Three resonant oscillators with 1 : 1 : 2 resonance, axial symmetry and with small detuning between double degenerate and non-degenerate modes [30,70]. The specific behavior of the joint spectrum for this model can be characterized as self-overlapping of a regular lattice. The possibility to propagate the initially chosen cell through a regular lattice from the region of self-overlapping of lattice back to the same region but to another component was named “bidromy”. More complicated construction for the same problem allows us to introduce the “bipath” notion. The bipath starts at a regular point of the EM image, and crosses the singular line

by splitting itself into two components. Each component belongs to its proper lattice in the self-overlapping region. Two components of the path can go back through the regular region only and fuse together. The behavior of quantum cells along a bipath is shown in Fig. 16. Providing a rigorous mathematical description of such a construction is still an open problem. Although the original problem has three degrees-of-freedom, it is possible to construct a model system with two degrees-of-freedom and with similar properties. Bifurcations of “Quantum Bifurcation Diagrams” We want now to stress some differences in the role of internal and external control parameters. From one pointof-view a quantum problem, which corresponds in the classical limit to a multidimensional integrable classical model, possesses a joint spectrum qualitatively described by a “quantum bifurcation diagram”. This diagram shows that the joint spectrum is formed from several parts of regular lattices through a cutting and gluing procedure. Going from one regular region to another is possible by crossing singular lines. The parameter defined along such a path can be treated as an internal control parameter. It is essentially a function of values of integrals of motion. To cross the singular line is equivalent to passing the quantum bifurcation for a family of reduced systems with a smaller number of degrees of freedom. On the other side we can ask the following more general question. What kinds of generic modifications of “bifurcation diagrams” are possible for a family of integrable systems depending on some external parameters? Hamiltonian Hopf bifurcation leading to the appearance of a new isolated singular value and as a consequence appearance of monodromy is just one of the possible effects of this kind. Another possibility is the transformation of an isolated focus-focus singular value into the island associated

Quantum Bifurcations

of an island, these two components can be smoothly deformed one onto another along a continuous path going only through regular values of the EM map. Examples of all such modifications were studied on simple models inspired by concrete quantum molecular systems like the H atom, CO2 , LiCN molecules and so on [24,30,41]. Semi-Quantum Limit and Reorganization of Quantum Bands

Quantum Bifurcations, Figure 16 Joint quantum spectrum for problem with bidromy. Quantum states are given by two numbers (energy, E, and polyad number, n) which are the eigenvalues of two mutually commuting operators. Inside the OAB curvilinear triangle two regular lattices are clearly seen. One can be continued smoothly through the OC boundary whereas another continues through the BC boundary. This means that the regular part of the whole lattice can be considered as a one self-overlapping regular lattice. The figure suggests also the possibility to define the propagation of a double cell along a “bipath” through the singular line BO which leads to splitting of the cell into two elementary cells fusing at the end into one cell defining in such a way the “bidromy” transformation associated with a bipath. Taken from [70]

with the presence of a second connected component of the inverse image of the EM map. It is also possible that such an island is born within the regular region of the EM map. In such a case naturally the monodromy transformation associated with a closed path surrounding the so-obtained island should be trivial (identity). The boundary of the image of the EM map can also undergo transformation which results in the appearance of the region with two components in the inverse image but, in contrast to the previous example of the appearance

Up to now we have discussed the qualitative modifications of internal structures of certain groups of quantum levels which are typically named bands. Their appearance is physically quite clear in the adiabatic approximation. The existence of fast and slow classical motions manifests itself in quantum mechanics through the formation of socalled energy bands. The big energy difference between energies of different bands correspond to fast classical variables whereas small energy differences between energy levels belonging to the same band correspond to classical slow variables. Typical bands in simple quantum systems correspond to vibrational structure of different electronic states, rotational structure of different vibrational states, etc. If now we have a quantum problem which shows the presence of bands in its energy spectrum, the natural generalization consists of putting this quantum system in a family, depending on one (or several) control parameters. What are the generic qualitative modifications which can be observed within such a family of systems when control parameters vary? Apart from qualitative modifications of the internal structure of individual bands which can be treated as the earlier discussed quantum bifurcations, another qualitative phenomenon is possible, namely the redistribution of energy levels between bands or more generally, the reorganization of bands under the variation of some control parameters [8,26,28,62,68]. In fact this phenomenon is very often observed in both the numerical simulations and the real experiments with molecular systems exhibiting bands. A typical example of molecular rovibrational energy levels classified according to their energy and angular momentum is shown in Fig. 17. It is important to note that the number of energy levels in bands before and after their “intersection” changes. The same phenomenon of the redistribution of energy levels between energy bands can be understood by the example of a much simpler quantum system of two coupled angular momenta, say orbital angular momentum and spin in the presence of a magnetic field interacting only with spin [62,68]. HD

1

Sz C (N  S) ; S NS

0 1:

(7)

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Quantum Bifurcations, Figure 17 System of rovibrational energy levels of 13 CF4 molecule represented schematically in E, J coordinates. The number of energy levels in each clearly seen band is 2JC1Cı, where ı is a small integer which remains constant for isolated bands and changes at band intersections. In the semi-quantum model ı is interpreted as the first Chern class, characterizing the non-triviality of the vector bundle formed by eigenfunctions of the “fast” subsystem over the classical phase space of the “slow” subsystem [27]

The Hamiltonian for such a system can be represented in the form of a one-parameter family (7) having two natural limits corresponding to uncoupled and coupled angular momenta. The interpolation of eigenvalues between these two limits is shown in Fig. 18 for different values of spin quantum number, S D 1/2; 1; 3/2. The quantum number of orbital momentum is taken to be N D 4. Although this value is not much larger than the S values, the existence of bands and their reorganization under the variation of the external parameter is clearly seen in the figure. Although the detailed description of this reorganization of bands will take us rather far away from the principal subject it is important to note that in the simplest situations there exists a very close relation between the redistribution phenomenon and the Hamiltonian Hopf bi-

furcations leading to the appearance of Hamiltonian monodromy [81]. In the semi-quantum limit when part of the dynamical variables are treated as purely classical and all the rest as quantum, the description of the complete system naturally leads to a fiber bundle construction [27]. The role of the base space is taken by the classical phase space for classical variables. A set of quantum wave-functions associated with one point of the base space forms a complex fiber. As a whole the so-obtained vector bundle with complex fibers can be topologically characterized by its rank and Chern classes [56]. Chern classes are related to the number of quantum states in bands formed due to quantum character of the total problem with respect to “classical” variables. Modification of the number of states in bands can occur only at band contact and is associated with the modification of Chern classes of the corresponding fiber bundle [26]. The simplest situation takes place when the number of degrees of freedom associated with classical variables is one. In this case only one topological invariant – the first Chern class is sufficient to characterize the non-triviality of the fiber bundle and the difference in Chern classes is equal to the number of energy levels redistributed between corresponding bands. Moreover, in the generic situation (in the absence of symmetry) the typical behavior consists of the redistribution of only one energy level between two bands. The generic phenomena become more complicated with increasing the number of degrees of freedom for the classical part of variables. The model problem with two slow degrees of freedom (described in classical limit by the CP2 phase space) and three quantum states was studied in [28]. A new qualitative phenomenon was found, namely, the modification of the number of bands due to formation of topologically coupled bands. Figure 19 shows the evolution of the system of energy levels along with the variation of control parameter . Three quantum bands (at  D 0) transform into two bands (in the  D 1) limit. One of these bands has rank one, i. e. it is associated with one quantum state. Another has rank two. It is associated with two quantum states. Both bands have non-trivial topology (non-trivial Chern classes). Moreover, it is quite important that the newly formed topologically coupled band of rank two can be split into two bands of rank one only if a coupling with the third band is introduced. The corresponding qualitative modifications of quantum spectra can be considered as natural generalizations of quantum bifurcations and probably should be treated as topological bifurcations. Thus, the description of possible “elementary” rearrangements of energy bands is a direct consequence of topological restrictions imposed by a fiber bundle structure of the studied problem.

Quantum Bifurcations

Quantum Bifurcations, Figure 18 Rearrangement of energy levels between bands for model Hamiltonian (7) with two, three, or four states for “fast” variable. Quantum energy levels are shown by solid lines. Classical energies of stationary points for energy surfaces are shown by dashed lines. Taken from [68]

effect (in physical terms). In fact, the mathematical basis of the quantum Hall effect is exactly the same fiber bundle construction which explains the redistribution of energy levels between bands in the above-mentioned simple quantum mechanical model. Multiple Resonances and Quantum State Density

Quantum Bifurcations, Figure 19 Rearrangement of three bands into two topologically non-trivially coupled bands. Example of a model with three electronic states and vibrational structure of polyads formed by three quasi-degenerate modes. At  D 0 three bands have each the same number of states, namely 15. In the classical limit each initial band has rank one and trivial topology. At  D 1 there are only two bands. One of them has rank 2 and non-trivial first and second Chern classes. Taken from [28]

It is interesting to mention here the general mathematical problem of finding proper equivalence or better to say correspondence between some construction made over real numbers and their generalizations to complex numbers and quaternions. This paradigm of complexification and quaternization was discussed by Arnold [4,5] on many different examples. The closest to the present subject is the example of complexification of the Wigner– Neumann non-crossing rule resulting in a quantum Hall

Rearrangement of quantum energy states between bands is presented in the previous section as an example of a generic qualitative phenomenon occurring under variation of a control parameter. One possible realization of bands is the sequence of vibrational polyads formed by a system of resonant vibrational modes indexed by the polyad quantum number. In the classical picture this construction corresponds to the system of oscillators reduced with respect to the global action. The reduced classical phase space is in such a case the weighted projective space. In the case of particular 1 : 1 : : : : : 1 resonance the corresponding reduced phase space is a normal complex projective space CP n . The specific resonance conditions impose for a quantum problem specific conditions on the numbers of quantum states in polyads. In the simplest case of harmonic oscillators with n1 : n2 : : : : : n k resonance the numbers of states in polyads are given by the generating function X 1 gD C N t N ; (8) D n n n (1  t 1 ) (1  t 2 )    (1  t k ) N where N is the polyad quantum number. Numbers CN are integers for integer N values, but they can be extended to arbitrary N values and represented in the form of a quasipolynomial, i.e, a polynomial in N with coefficients being

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a periodic function whose period equals the least common multiplier of n i ; i D 1; : : : ; k. Moreover, the coefficients of the polynomial can be expressed in terms of so-called Todd polynomials which indicates the possibility of topological interpretation of such information [52,89]. Physical Applications and Generalizations The most clearly seen physical applications of quantum bifurcations is the qualitative modification of the rotational multiplet structure under rotational excitation, i. e. under the variation of the absolute value of the angular momentum. This is related first of all with the experimental possibility to study high J multiplets (which are quite close to the classical limit but nevertheless manifest their quantum structure) and to the possibility to use symmetry arguments, which allow one to distinguish clusters of states before and after bifurcation just by counting the number of states in the cluster, which depends on the order of group of stabilizer. Nuclear rotation is another natural example of quantum rotational bifurcations [60]. Again the interest in corresponding qualitative modifications is due to the fact that rotational bands are extremely well studied up to very high J values. But in contrast to molecular physics examples, in nuclear physics it mostly happens that only ground states (for each value of J) are known. Thus, one speaks more often about qualitative changes of the ground state (in the absence of temperature) named quantum phase transitions [65]. Internal structure of vibrational polyads is less evident for experimental verifications of quantum bifurcations, but it gives many topologically non-trivial examples of classical phase spaces on which the families of Hamiltonians depending on parameters are defined [25,30,39, 42,44,46,66,76,77,86]. The main difficulty here is the small number of quantum states in polyads accessible to experimental observations. But this problem is extremely interesting from the point-of-view of extrapolation of theoretical results to the region of higher energy (or higher polyad quantum numbers) which is responsible as a rule for many chemical intra-molecular processes. Certain molecules, like CO2 , or acetylene (C2 H2 ) are extremely well studied and a lot of highly accurate data exist. At the same time the qualitative understanding of the organization of excited states even in these molecules is not yet completed and new qualitative phenomena are just starting to be discovered. Among other physically interesting systems it is necessary to mention model problems suggested to study the behavior of Bose condensates or quantum qubits [37,38, 74,83,84]. These models have a mathematical form which

is quite similar to rotational and vibrational models. At the same time their physical origin and the interpretation of results is quite different. This is not an exception. For example, the model Hamiltonian corresponding in the classical limit to a Hamiltonian function defined over S2 classical phase space is relevant to rotational dynamics, description of internal structure of vibrational polyads formed by two (quasi)degenerate modes, in particular to so-called local-normal mode transition in molecules, interaction of electromagnetic field with a two-level system, the Lipkin– Meshkov–Glick model in nuclear physics, entanglement of qubits, etc. Future Directions To date many new qualitative phenomena have been suggested and observed in experimental and numerical studies due to intensive collaboration between mathematicians working in dynamical system theory, classical mechanics, complex geometry, topology, etc., and molecular physicists using qualitative mathematical tools to classify behavior of quantum systems and to extrapolate this behavior from relatively simple (low energy regions) to more complicated ones (high energy regions). Up to now the main accent was placed on the study of the qualitative features of isolated time-independent molecular systems. Specific patterns formed by energy eigenvalues and by common eigenvalues of several mutually commuting observables were the principal subject of study. Existence of qualitatively different dynamical regimes for time-independent problems at different values of exact or approximate integrals of motion were clearly demonstrated. Many of these new qualitative features and phenomena are supposed to be generic and universal although their rigorous mathematical formulation and description is still absent. On the other side, the analysis of the time-dependent processes should be developed. This step is essential in order to realize at the level of quantum micro-systems the transformations associated with the qualitative modifications of dynamical regimes and to control such timedependent processes as elementary reactions, information data storage, and so on. From this global perspective the main problem of the future development is to support the adequate mathematical formulation of qualitative methods and to improve our understanding of qualitative modifications occurring in quantum micro-systems in order to use them as real micro-devices. Bibliography 1. Arnold VI (1972) Modes and quasimodes. Funct Anal Appl 6:94–101

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73. Simon B (1980) The classical limit of quantum partition functions. Commun Math Phys 71:247–276 74. Somma R, Ortiz G, Barnum H, Knill E, Viola L (2004) Nature and measure of entanglement in quantum phase transitions. Phys Rev A 70:042311-1–21 75. Symington M (2003) Four dimensions from two in symplectic topology. In: Athens GA (ed) Topology and geometry of manifolds. Proc Symp Pure Math, vol 71. AMS, Providence, pp 153–208 76. Tyng V, Kellman ME (2006) Bending dynamics of acetylene: New modes born in bifurcations of normal modes. J Phys Chem B 119:18859–18871 77. Uwano Y (1999) A quantum saddle-node bifurcation in a resonant perturbed oscillator with four parameters. Rep Math Phys 44:267–274 78. Uzer T (1990) Zeeman effect as an asymmetric top. Phys Rev A 42:5787–5790 79. Van der Meer JC (1985) The Hamiltonian Hopf bifurcation. Lect Notes Math, vol 1160. Springer, New York 80. Vu˜ Ngoc S (1999) Quantum monodromy in integrable systems. Comm Math Phys 203:465–479 81. Vu˜ Ngoc S (2007) Moment polytopes for symplectic manifolds with monodromy. Adv Math 208:909–934 82. Waalkens H, Dullin HR (2001) Quantum monodromy in prolate ellipsoidal billiards. Ann Phys NY 295:81–112 83. Wang J, Kais S (2004) Scaling of entanglement at a quantum phase transition for a two-dimensional array of quantum dots. Phys Rev A 70:022301-1-4 84. Weyl H (1952) Symmetry. Princeton University Press, Princeton 85. Winnewisser M, Winnewisser B, Medvedev I, De Lucia FC, Ross SC, Bates LM (2006) The hidden kernel of molecular quasi-linearity: Quantum monodromy. J Mol Struct V 798:1–26 86. Xiao L, Kellman ME (1990) Catastrophe map classification of the generalized normal-local transition in Fermi resonance spectra. J Chem Phys 93:5805–5820 87. Zhang W-M, Feng DH, Gilmore R (1990) Coherent states: Theory and some applications. Rev Mod Phys 62:867–927 88. Zhilinskií BI (1996) Topological and symmetry features of intramolecular dynamics through high resolution molecular spectroscopy. Spectrochim Acta A 52:881–900 89. Zhilinskií BI (2001) Symmetry, invariants, and topology, vol II. Symmetry, invariants, and topology in molecular models. Phys Rep 341:85–171 90. Zhilinskií BI (2006) Hamiltonian monodromy as lattice defect. In: Monastyrsky M (ed) Topology in condensed matter. Springer series in solid state sciences, vol 150. Springer, Berlin, pp 165–186 91. Zhilinskií BI, Kozin I, Petrov S (1999) Correlation between asymmetric and spherical top: Imperfect quantum bifurcations. Spectrochim Acta A 55:1471–1484 92. Zhilinskií BI, Pavlichenkov IM (1988) Critical phenomenon in the rotational spectra of water molecule. Opt Spectrosc 64:688–690 93. Zhilinskií BI, Petrov SV (1996) Nonlocal bifurcation in the rotational dynamics of an isotope-substituted A2 A 2 molecule. Opt Spectrosc 81:672–676

Reaction Kinetics in Fractals

Reaction Kinetics in Fractals EZEQUIEL V. ALBANO Facultad de Ciencias Exactas, UNLP, CONICET, Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA) CCT La Plata, La Plata, Argentina Article Outline Glossary Definition of the Subject Introduction Fractals and Some of Their Relevant Properties Random Walks Diffusion-limited Reactions Irreversible Phase Transitions in Heterogeneously Catalyzed Reactions Future Directions Bibliography Glossary Fractals Fractal geometry is a mathematical tool well suited to treating complex systems that exhibit scale invariance or, equivalently, the absence of any characteristic length scale. Scale invariance implies that objects are self-similar: if we take a part of the object and magnify it by the same magnification factor in all directions, the obtained object cannot be distinguished from the original one. Self-similar objects are often characterized by non-integer dimensions, a fact that led B. Mandelbrot, early in the 1980s, to coin the name “fractal dimension”. Also, all objects described by fractal dimensions are generically called fractals. Within the context of the present work, fractals are the underlying media where the reaction kinetics among atoms, molecules, or particles in general, is studied. Reaction kinetics The description of the time evolution of the concentration of reacting particles ( i , where i D 1; 2; : : : ; N identifies the type of particle) is achieved, far from a stationary regime, by formulating a kinetic rate equation ˙i (t) D F[ i (t)], where F is a function. This description, known in physical chemistry as the law of mass action, states that the rate of a chemical reaction is proportional to the concentration of reacting species and was formulated by Waage and Guldberg in 1864. Often, especially when dealing with reactions occurring in homogeneous media, F involves integer powers (also known as the reaction orders) of the concentrations, leading to classical reaction kinetics. However, as in most cases treated in this arti-

cle, if a reaction takes place in a fractal, one may also have kinetic rate equations involving non-integer powers of the concentration that lead to fractal reaction kinetics. Furthermore, it is usual to find that the slowest step involved in a kinetic reaction determines its rate, leading to a process-limited reaction, where e. g. the process could be diffusion or mass transport, adsorption, reaction, etc. Heterogeneously catalyzed reactions A reaction limited by at least one rate-limiting step could be prohibitively slow for practical purposes when, e. g., it occurs in a homogeneous media. The role of a good solid-state catalyst in contact with the reactants – in the gas or fluid phase – is to obtain an acceptable output rate of the products. Reactions occurring in this way are known as heterogeneously catalyzed. This type of reaction involves at least the following steps: (i) the first one comprises trapping, sticking and adsorption of the reactants on the catalytic surface. Particularly important, from the catalytic point of view, is that molecules that are stable in the homogeneous phase, e. g. H 2 , N2 , O2 , etc., frequently undergo dissociation on the catalyst surface. This process is essential in order to speed up the reaction rate. (ii) After adsorption, species may diffuse or remain immobile (chemisorbed) on the surface. The actual reaction step occurs between neighboring adsorbed species of different kinds. The result of the reaction is the formation of products that can either be intermediates of the reaction or its final output. (iii) The final step is the desorption of the products, which is essential not only for the practical purpose of collecting and storing the desired output, but also in order to regenerate the catalytically active surface sites. Definition of the Subject Processes involving reaction among atoms, molecules, and particles in general, are ubiquitous both in nature and in laboratories. After the introduction of the concept of fractals by B. Mandelbrot [89] in the 1980s, it has been realized that a wide variety of reactions take place in fractal media, leading to the observation of anomalous behavior, i. e. the so called “fractal reaction kinetics” [33,67,81]. The study of this subject has received considerable theoretical attention [38,43,76,77,103] and a huge numerical effort has been made to improve its understanding [13,17,18,20,34,80,92]. Furthermore, among the practical examples of fractal reaction kinetics one can quote exciton annihilation in composite materials, chemical reactions in membranes pores, charge recom-

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bination in colloids and clouds, coagulation, polymerization and growth of dendrites, [15,79,83,94] etc. Another scenery for the study of reaction kinetics in fractals is in the field of heterogeneous catalysis. In fact, it is well known that most catalysts are composed of small, catalytically active particles, supported by highly porous (fractal) substrates [21,22,23,30,75,95,106]. So, it is not surprising that this field of research has become particularly active due not only to its practical and technological relevance, but also to the occurrence of quite interesting and challenging phenomena such as irreversible phase transitions, oscillatory behavior, chaos and stochastic resonance, propagation and interference of chemical waves, interface coarsening, metastability and hysteretic effects, etc. [9,10,52,68,70,71,84,85,90,109]. Introduction The description of the kinetic behavior of reaction processes occurring in homogeneous media can be found in most textbooks on chemical physics. However, care must be taken considering reaction kinetics in fractals because conventional textbook equations are based on mean-field approaches that neglect not only the fractal structure of the underlying media, but also fluctuations in the concentration of the reactants, many-particle effects, etc. that may lead to the observation of anomalous behavior [81]. In fact, soon after the recognition of the relevance of the concept of fractals for the description of the structure and properties of physical objects by Mandelbrot [89], more than three decades ago, it was realized that random transport in low-dimensional and disordered media may be anomalous [33,67,105], as are diffusion-limited reactions occurring in those media [13,15,17,18,20,34,43,79, 80,81,83,92,94]. Therefore, the classical reaction kinetic approach becomes unsatisfactory in a wide variety of situations, such as when the reactants are spatially constrained by walls, phase boundaries or force fields. Due to these studies, early in the 1980s, it was realized that even elementary reactions (e. g. A C A ! inert, A C B ! products, etc.) can no longer be described by classical rate equations with integer exponents (the so-called “reaction order”) but instead, fractal orders were identified leading to fractal reaction kinetics [89]. The emerging new theory also predicts the occurrence of self-ordering and self-unmixing of reactants [38,76,77,103], as well as time-dependent rate “constants”, i. e. rate coefficients with temporal “memories” [89]. On the other hand, after the exhaustive study and characterization of the fractal nature of a wide variety of substrates used as support in most catalysts, due to Avnir et

al. [21,22,23,95], the study of heterogeneously catalyzed reactions in fractal media has also attracted growing attention [1,2,5,7,41,72,86,87]. These studies are focused on the understanding of irreversible phase transitions occurring between an active state with reaction and an inactive state where the reaction ceases irreversibly. This latter state is due to the poisoning of the catalyst, by the reactants and/or their subproducts, and is known as the absorbing state: a system trapped in the absorbing state can never escape from it [9,10,68,85,90]. In order to cover those topics, this article is organized as follows: in Sect. “Fractals and Some of Their Relevant Properties”, the basic properties of fractals are presented and discussed. Section “Random Walks” is devoted to the description of the behavior of random walks in fractal media, while in Sect. “Diffusion-limited Reactions” archetypical cases of diffusion limited reactions among random walks are considered. Irreversible phase transitions occurring in heterogeneously catalyzed reactions are addressed in Sect. “Irreversible Phase Transitions in Heterogeneously Catalyzed Reactions”. Finally, in Sect. “Future Directions” promising directions for future works are briefly outlined. Fractals and Some of Their Relevant Properties Classical Euclidean geometry is useful for describing the properties of regular objects such as circles, spheres, cones, etc. However, disordered objects such as clouds, dielectric breakdown patterns, coastlines, mountains, landscapes, etc., have not been so far satisfactorily described by using the Euclidean geometry. B. Mandelbrot overcome this shortcoming by introducing the fractal geometry as a suitable tool for the treatment of disordered or fractal media [89]. Along this article, fractals will be used as media where reactions of interest take place. A relevant property of fractal media is self-similarity or symmetry under dilatation [27,33,40,61,67]. Here, it is worth noting the main difference between regular Euclidean space and fractal geometry: whereas the former has translation symmetry, this type of symmetry is violated in the latter, which exhibits a new symmetry known as scale invariance or invariance under dilatation. Since these concepts may appear to be obscure, it is convenient to develop an intuitive understanding and outline some basic definitions. Fractals can either be deterministic or non-deterministic (often called random fractals). However, it is worth mentioning that many non-deterministic fractals may be obtained as a result of complex dynamic (non-stochastic) processes. From the large variety of known deterministic fractals, e. g. the Koch curve, the Julia set, Sierpinski gaskets, carpets,

Reaction Kinetics in Fractals

and sponges, etc., let us focus our attention on the construction of a Sierpinski Carpet (SC). In order to build up a generic SC(a,c), a square in d D 2 dimensions is segmented into ad subsquares and c of them are then removed. This square, of side a, is known as the generating cell. The segmentation process is then iterated on the remaining squares a number k of segmentation steps. Figure 1 shows the SC(4,4) obtained after k D 4 segmentation steps. These kinds of fractals have a lower cutoff length given by the side of the generating cell, but, in principle, they lack an upper cutoff length for k ! 1. However, in practice one performs a finite number of segmentation steps. Let us now remind the reader about the concept of dimension in regular systems. In this case, the dimension d characterizes the dependence of the mass M(L) as a function of the linear size L of the system. If we now consider a smaller part of the system of size bL(a < 1), then M(aL) will be decreased by a factor of ad , so that ˜ M(a; L) D (a)M(aL) D a d M(L) :

(1)

The solution of Eq. (1) gives M(L) D ALd , where A is a constant. So, for a wire one has d D 1, while for a thin plate d D 2 is obtained, etc. Intuitively, the SC(4,4) shown in Fig. 1 seems to be “denser” than a wire but also “sparser” than a thin plate. So, Eq. (1) has to be generalized for fractals, leading to M(aL) D a d f M(L) ;

(2)

Reaction Kinetics in Fractals, Figure 1 Sierpinski carpet obtained after k D 4 iterations on the generating cell shown in the top-left corner, i. e. the SC(4,4). The lattice side is L D 64 and the fractal dimension of this object is Df D log(12)/log(4) ' 1:7925

which yields the following solution M(L) D BLd f ;

(3)

where B is a constant, and df is a non-integer dimension known, after Mandelbrot, as the fractal dimension. By applying Eq. (2) to the SC(4,4) shown in Fig. 1, one has M(1/4 L) D1/12 M(L), so that df Dlog(12)/ log(4) ' 1:7925. In general, for an SC one has M(1/aL) D 1/(a2  c) M(L), which yields df Dlog(a2  c)/ log(a). The factor a in Eqs. (1) and (3) could be an arbitrary real number, leading to continuous scale invariance. However, many fractals exhibit discrete scale invariance (DSI) [99], which is a weak kind of scale invariance such that a is no longer an arbitrary real number, but it can only take specific discrete values of the form a n D (a1 )n , where a1 is a fundamental scaling ratio. Then, for the case of DSI, the solution of Eq. (1) yields   log(L) ; (4) M(L) D Ld f F log(a1 ) where F is a periodic function of period one. The measurement of soft oscillations in spatial domain [99] is a signature of spatial DSI. For instance, the SC(4; 4) shown in Fig. 1 exhibits DSI with a1 D 4. Deterministic fractals can be finitely ramified or infinitely ramified objects. A fractal is classified as finitely ramified if any bounded set of the fractal can be separated from the whole structure just by removing a finite number of bonds between the individual constituting entities. This is not possible for the case of the SC(4,4) shown in Fig. 1, so that it is an infinitely ramified fractal. However, other fractals such as the Sierpinski Gasket and the Koch curve are finitely ramified. One advantage of finitely ramified clusters is that various physical properties, such as the conductivity and the vibrational excitations, can be calculated exactly helping to understand the anomalous behavior of some observables. However, it is worth mentioning that classical critical phenomena, e. g. the Ising model, are not observed in finitely ramified fractals [58,59,60]. This is not the case of irreversible critical phenomena (e. g. the contact process [68]), criticality occurring at a trivial value of the control parameter, e. g. the coarsening without surface tension observed in the voter model at T D 0 [48], which is still present in finitely ramified fractals [102], etc. For a careful description of the building rules and relevant properties of various deterministic fractals see e. g. [40]. In the large variety of non-deterministic fractals, there are many examples of random fractals such as diffusion-limited aggregates [27,40,107], cluster-cluster aggregates [101], the incipient percolation cluster [101], etc. So,

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Reaction Kinetics in Fractals, Figure 2 Percolating cluster obtained at the critical threshold. The lattice side is L D 64 and the cluster contains 2100 particles. The fractal dimension of the infinite percolation cluster is Df ' 1:89

in order to acquaint the reader with this subject, let us briefly describe the percolation model in d D 2 dimensions. Considering the square lattice, one has that each site can be occupied randomly with probability p or left empty with probability (1  p). This model mimics, for example, the deposition of conducting metallic particles in an isolating substrate, so that p is the surface density of deposited particles. At lower densities, one has isolated clusters of particles and the films do not conduct electrical current between the edges of the sample. However, by properly increasing the concentration, the onset of electrical conductivity is abruptly observed at a certain critical value pc , known as the percolation threshold [101]. So, at criticality one has a spanning cluster that connects opposite edges of the sample, which is known as the Incipient Percolation Cluster (IPC), see Fig. 2. Equations (2) and (3) also hold for random fractals, but the mass M has to be averaged over different realizations of the fractal in order to achieve reliable statistics. The IPC is composed of several fractal substructures such as the backbone, blobs, red bonds, dangling ends, etc., so that one needs to define different fractal dimensions in order to achieve a better description. Several methods have been developed in order to measure the fractal dimension of non-deterministic fractals, the “sandbox” and the “box counting” methods being among the most extensively used, for further details see e. g., [40]. Random Walks The properties of random walks are useful for the understanding of a great variety of phenomena in virtually all

sciences, e. g. in Physics, Chemistry, Astronomy, Biology, Ecology, and even in Economics [105]. Throughout this article, we will focus our attention on simple random walks on lattices, either regular or fractal, as a model for diffusion. The simple discrete random walk is a stochastic process, such that the walk advances one step in unit time. The walker steps from its present position to another site of the lattice according to a specified random rule. Since the rule is independent of the history of the walk, the process is Markovian. Along this article, we will consider cases where the step is performed, with the same probability, to one of the nearest neighbor sites of the lattice. While that choice is always possible in regular lattices, it is no longer the case for random walks in fractals, because all nearest neighbor sites may not belong to the substrate. In this situation one often considers steps performed with equal probability to any of the nearest neighbor sites belonging to the substrate. Let us now consider the displacement of the random walk. After n steps (since one has discrete time scale, n D t holds) the net displacement (R(t)) is given by [33,67,105] R(t) D

n X

uj ;

(5)

jD1

where u j is a unitary vector pointing to a nearest-neighbor site, so that it represents the jth step of the walk. Due to the fact that hu j i D 0, the displacement averaged over a large number of realizations of the walk vanishes, i. e. hR(t)i D 0. So, a more interesting and useful kinetic observable of random walks is the rms displacement from the origin (R2 ), given by * n + n X 2 X 2 hR (t)i D uj hu j :u i i D t ; (6) D tC2 jD1

j>i

since hu j :u j i D 1 and hu j :u i i D 0 for i ¤ j because the steps are independent. So, the classical result for Euclidean space is obtained. A distinctive feature of transport in fractal media is that the linear time dependence of the rms displacement of the walk given by Eq. (6) has to be replaced by R2 / t 2/d w ;

(7)

where dw is the anomalous diffusion exponent. In most studied models the fractal dimension dw exceeds 2, due to the fact that disorder tends, on average, to slow down the diffusion of the walk moving on those media. Another interesting observable of a walk is the average number of distinct sites visited by a single random walk

Reaction Kinetics in Fractals

after N steps (SN ), which is also known as the exploration space of the walk [17,19,67,105] and references therein. Assuming that N / t, one has [96] S N / t d s /2 ;

t !1;

(8)

where ds is the spectral dimension and Eq. (8) holds for ds < 2 [14,96], which corresponds to low-dimensional and fractal media, leading to the so-called “anomalous diffusion” behavior [19,33,67]. Furthermore, d D 2 is the upper critical dimension, such as for a d-dimensional regular space one has ds D d > 2 and Eq. (8) becomes S N / t, leading to classical diffusion (for further details see also Eq. (17) below). Diffusion, or random walks, are also related to the density of states for harmonic excitations of the media [78] (h( )) through the probability that the random walk returns to the origin. So, one has df

h( ) / dw 1 

ds 2

1

;

(9)

where the energy and vibrational density of states are related through h( )d D g(!)d!. The relationship (9) is similar to the well-known density of states in Euclidean space, except that d has to be replaced by ds D 2

df ; dw

Reaction kinetics is influenced by the characteristic time of the processes involved. The rate of a heterogeneous reaction is often determined by the adsorption of the reactants from a fluid phase on the catalyst surface. For reactions occurring in a single phase the transport of the reactants and the time of reaction influence the overall reaction rate. Here, we focus on diffusion-limited reaction processes for which the transport time, given by the typical time required for the reactants to meet, is much longer than the reaction time. Since the diffusion of particles in fractals is often anomalous, it is expected that this behavior will affect diffusion-limited reactions. Furthermore, one also needs to account for density fluctuations of the reactants occurring at all length scales. For these reasons, as well as for the occurrence of many particle effects, the study of diffusionlimited reactions is difficult and often eludes straightforward mean-field approaches. In this section, simple and archetypical reactions occurring in fractal media will be discussed. For a more general overview of diffusion-limited reactions in homogeneous and disordered media, see e. g. [33,67]. One-Species Reactions

(10)

which is also known as the fracton dimensionality, since the vibration modes are called fractons instead of phonons. Very recently we have found evidence of discrete scale invariance in the time domain by measuring the relaxation of the magnetization in the Ising model on Sierpinski Carpets [25,26]. Subsequently, we have conjectured that physical processes characterized by an observable O(t), occurring in fractal media with DSI, and that develop a monotonically increasing time-dependent characteristic length (t), may also exhibit time DSI. In fact, by assuming  / t 1/z , where z is a dynamic exponent, it can be shown that O(t) has to obey time DSI according to [53]   log(t) O(t) D Ct ˛/z F  C ; (11) log(a1z ) where C and  are constants. So, the conjecture given by Eq. (11) implies the existence of a logarithmic periodic modulation of time observables characterized by a timescaling ratio  given by  D a1z ;

Diffusion-limited Reactions

(12)

see also Eq. (4). So, in the case of random walks on fractal media with DSI, Eqs. (7) and (8) have to be generalized according to Eq. (11) with dw D z.

So far, the most studied case is the one-species annihilation process, where species A diffuse and annihilate upon encounter [33,67], according to the reaction scheme ACA! 0;

(13)

such that the reaction is instantaneous. This example also includes the case in which the product 0 is some inert particle that does not influence the overall kinetics. A closely related reaction is the one-species coalescence process given by [33,67] ACA! A:

(14)

In the mean-field limit, which holds for the reactionlimited case, the rate equation for both reactions (13) and (14) is given by d A (t) D K AX (t) ; dt

(15)

where A (t) is the concentration of A-particles at time t and K is the reaction constant. Eq. (15), where X D 2 is the reaction order, is the typical second-order rate equation that can be solved yielding

A (t) D

1 ; kt C A1(0)

(16)

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so that in the long-time limit one has A (t)  t 1 . On the other hand, for the diffusion-controlled regime, it can be proved rigorously that the concentration decays, for both processes (13) and (14) according to [33,67]

A (t) /

1 t d/2

;

d < dc D 2 ;

(17)

where dc D 2 is the upper critical dimension, such that for d > dc the density decays according to the mean-field prediction, given by Eq. (16). So, for d < 2 the kinetics of the diffusion-controlled process is anomalous and just at dc one has logarithmic corrections to Eq. (17). So, let us focus our attention on heterogeneous chemical reactions occurring in media having a fractal structure with df < dc D 2. Typical examples for these structures are catalysts, porous glass [12], diffusion-limited aggregates [107], percolation clusters [79], zeolites, etc. Since, as already discussed, diffusion in such media is anomalous [14,96], so are expected to be diffusion-limited chemical reactions [13,15,17,18,45,80,92,103]. Then, the simple textbook elementary reaction given by Eq. (15) has to be modified. A straightforward way to understand the nature of the modifications can be envisioned just by considering that SN (see Eq. (8)) is the exploration space of the random walk, so that the rate constant K can be written in terms of the visitation efficiency [80] as follows K/ 

dS N / t d s /21 : dt

governed by the same exponent, given by the spectral dimension of the media where the physical process actually takes place. Furthermore, by replacing Eq. (20) into Eq. (19), it follows that d D k 1C2/d s ; dt

t ! 1;

(21)

and the reaction order for the elementary reaction given by Eq. (15) can be written as [15,17,45,81,92] X D1C

2 ; ds

(22)

which yields the textbook result (X D 2) for classical diffusion, but one has X > 2 for anomalous diffusion that is expected to occur in low-dimensional and fractal media for ds < 2, e. g. for d D 1 one has ds D 1 and the reaction order is X D 3. This relationship for the reaction order, based on the conjecture on the time dependence of the effective rate constant given by Eq. (18), was later derived rigorously by Clément et al. [16] based on calculations of the pair-correlation function and the macroscopic reaction law for Euclidean dimensions d  3 and self-similar fractal structures with spectral dimensions 1  ds  2. Figure 3a shows the time dependence of the decay of the density of random walks as a result of the reaction

(18)

Of course, for standard diffusion one has ds D 2, K is actually a constant and the classical reaction order X D 2 is recovered. However, it should be noticed that for anomalous diffusion one has ds < 2, so that K is no longer a constant but depends on time. In fact, by inserting Eq. (8) into Eq. (18) it follows that d / t d s /21 2 ; dt

t !1;

(19)

and the integrated rate equation that is obtained by using Eq. (19) reads d s /2 ;

1  1 o D kt

(20)

where o  (t D 0) is the initial density of random walks and, of course, Eqs. (19) and (20) are expected to hold in the low density limit since they are derived from the behavior of a single walker. Now, by comparing Eqs. (8) and (20), it follows that both the asymptotic (low-density) regime of the density of species in the diffusion-limited reaction and the exploration space of the random walk are

Reaction Kinetics in Fractals, Figure 3 a Log-log plots of the decay of the density of reacting species as obtained by means of Monte Carlo simulations in the SG(5,10) whose generating cell is shown in the upper-right corner. b Linear-log plot of the raw data shown in a but after proper subtraction and subsequent normalization by the corresponding power-law behavior fitted in a. In both panels, the dashed and full lines correspond to the reactions A C B ! 0 and A C A ! 0, respectively. More details in the text

Reaction Kinetics in Fractals

given by Eq. (20). The results correspond to the Sierpinski Gasket SG(5,10). A power-law decay with exponent ds /2 D 0:661 ˙ 0:003 is observed in agreement with Eq. (20). However, a careful inspection of the curve allows us to distinguish soft log-periodic oscillations modulating the power-law behavior. These oscillations become even more evident in Fig. 3b, which was obtained from the original data already shown in Fig. 3a after subtraction and subsequent normalization by the fitted power law. The presence of log-periodic oscillations is the signature of the occurrence of time DSI (see Eq. (11)) due to the coupling of the reaction kinetics and the spatial DSI of the substrate (see Eq. (4)), as discussed in Sect. “Fractals and Some of Their Relevant Properties” and “Random Walks”.

Two-Species Annihilation The study of the two-species diffusion-limited reaction of the type AC B ! 0

(23)

has received considerable attention due to the fact that, most likely, it is the simplest case where fluctuations in the local concentration of the reactants play a prevailing role in the emerging kinetic behavior [33,38,43,67, 76,103]. According to Eq. (23) the concentration difference of both species is conserved, so that A (t)  B (t) D

A (0)  B (0)  const. If the initial concentrations are different, the mean-field approach predicts an exponential decrease of the minority species [33,67]. However, for

A (0) D B (0) one expects an algebraic decay of the form

/ 1/t. Let us now briefly review the effect caused by spatial fluctuations in the concentrations [33,38,43,67,76,103]. For the sake of simplicity, let us assume that both species have the same diffusion constant (D) and that the initial concentrations are equal. For a random initial distribution of particles, a region of the space p of linear size l would initially have N A D A (0)l d ˙ A (0)l d particles of type A, and equivalently for B-particles. The second term accounts for the local fluctuation, at a scale of the order of l, in the number of particles. By assuming classical diffusion, one has that for a time t, such as Dt ' l 1/2 , all the particles within the considered region would have time enough to react with each other. In the absence of fluctuations, the concentrations of both species will vanish. However, at the end p of the reaction process one still has of the order of

(0)l d particles of the majority species. Then, the concentration of either A- or B-particles at time t is actually

given by

A;B /

q 

A;B (0)l d ld

/

p ( A;B (0)) ; Dt d/4

(24)

and the decay exponent is twice as small than in the case of the one-species processes (Eq. (17)). According to the above scaling reasoning, one would expect the formation of alternating domains of A- and B-particles. This segregation of the reactants, known as the Toussaint–Wilczek effect [103], causes the slowdown of the kinetics since the reaction actually takes place along the boundary between domains. A careful analysis reveals that for d  4, the domains are unstable, so that dc D 4 is the upper critical dimension above which the mean-field approach holds. So, Eq. (24) holds for d  4, and segregation has qualitatively been observed, e. g. in d D 2, by means of simulations [103] (see also Fig. 3). In order to gain a quantitative insight into the segregation of the reactants, it is useful to evaluate the excess density of either A- or B-particles within spatial domains of size ld , defined as  A;B (l) D

jN A  N B j ; ld

(25)

where N A (N B ) is the number of A- (B-) particles within the domain. Figure 4 shows log-log plots of the time dependence of the decay of the density of particles, as well as that of the excess density, as measured for different values of l. Results are obtained by means of Monte Carlo simulations

Reaction Kinetics in Fractals, Figure 4 Log-log plots of the density of species (o) and the excess density measured for different length scales as indicated. Results obtained for the A C B ! 0 reaction (Eq. (23)) in d D 2 dimensions by using a lattice of side L D 4096. Results are averaged over 100 different realizations. The double-arrow lines at the top indicate the three different time regimes discussed in the text. The length scales used to measure the excess density are identified by the corresponding symbols. The dashed line, with slope d/4 D 1/2 shows the expected behavior according to Eq. (24)

1463

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Reaction Kinetics in Fractals

in d D 2 dimensions. Three different regimes can be observed, namely: (i) For the short-time behavior (t  103 in Fig. 4) one observes deviations from Eq. (24), which are in agreement with the fact that at early times the segregation of the reactants, as evidenced by the measurements of the excess density, is almost negligible. (ii) During an intermediate time regime (103 < t  106 in Fig. 4) the Toussaint– Wilczek effect becomes dominant – the excess densities are maxima in all measured length scales – and the density decays according to Eq. (24) with exponent d/4 D 1/2. (iii) Finally, for long times (t > 106 in Fig. 4) one observes the fast decay of both the density and the excess density due to the finite size of the sample used. Let us now analyze the expected behavior for the case of reactions occurring in fractals. Here one has that the characteristic scaling time is of the order of t / l 1/d w , and according to Eq. (24) the residual concentration can be written as

A;B (t) /

l d f /2 t d f /2d w 1 / / d /4 ; d d /d w f f l t t s

(26)

where Eq. (10) has been used. Therefore, for fractals one has to replace d by the spectral dimension ds of the fractal in Eq. (24). Figure 5 shows log-log plots of the time dependence of the decay of the density of particles and the excess density, as in the case of Fig. 4, but obtained by means of Monte Carlo simulations on random-fractal substrates given by percolation clusters at the critical threshold (see

Reaction Kinetics in Fractals, Figure 5 Log-log plots of the density of species (o) and the excess density measured for different length scales as indicated. Results obtained for the A C B ! 0 reaction (Eq. (23)) in incipient percolation clusters in d D 2 dimensions by using a lattice of side L D 4096. Results are averaged over 100 different realizations. The length scales used to measure the excess density are identified by the corresponding symbols. The full line (slightly shifted up for the sake of clarity), with slope ds /4 D 0:3344, is the best fit of the data according to Eq. (26)

Fig. 2). In contrast to the d D 2 dimensional case discussed within the context of Fig. 4, the disordered nature of the fractal causes the occurrence of density fluctuations at all the spatial scales even for the initial configuration. So, the Toussaint–Wilczek effect starts to play a dominant role from the very beginning of the reaction. Consequently, the density decays according to Eq. (26) along the whole time interval measured. The best fit of the data yields ds /4 D 0:3344, in excellent agreement with reported values [20]. Of course, finite-size effects, which are not observed in Fig. 5, could be expected to occur at later times. It is also worth mentioning that Fig. 5 confirms that the excess density also decays with the same power-law behavior as the density, in agreement with the arguments outlined in order to obtain Eqs. (24) and (26). Figure 3a also shows the time dependence of the decay of the density of particles for the AC B ! 0 reaction given by Eq. (26). These results were obtained in the SG(5,10). The best fit of the data gives ds /4 D 0:331 ˙ 0:003, in excellent agreement with the value obtained for the annihilation reaction (Eq. (20)) given by ds /2 D 0:663 ˙ 0:003 (see also Fig. 3a). The soft log-periodic oscillation of the density that can be observed in Fig. 3a is enhanced in Fig. 3b. Again, this behavior is the fingerprint of the occurrence of time DSI, as discussed in Sect. “Fractals and Some of Their Relevant Properties” and “Random Walks” (see Eqs. (4) and (11), respectively). Summing up, for the simplest example of a bimolecular irreversible decay of the form A C B ! 0 the obtained universal kinetic law depends on the dimensionality, the initial concentration and the particle-conservation law that holds for the system. So far, the transient A C B ! 0 reaction is fundamentally different from the A C A ! 0 reaction due to the Toussaint–Wilczek effect. This effect represents a quite delicate balance that may become elusive to be properly identified, as discussed for the case of d D 2 dimensions (Fig. 4). Therefore, it is interesting to study its persistence under tiny perturbations, e. g. the steady feeding of reactants. Early simulations by Anacker et al. [93] have revealed that, in a cubic lattice, the Toussaint–Wilczek effect is destroyed by the addition of a steady source of walkers. Furthermore, in contrast to the transient result where the reaction order is X D 3, the steady-state A C B ! 0 reaction gives X D 2:00 ˙ 0:02, for steady-state densities as low as 0.001. Since in d D 2 and for the transient regime, the occurrence of the effect is confined to a narrow density window (see Fig. 4) careful measurements will be needed in order to test if segregation still remains relevant in d D 2 under steady-state conditions. Segregation of the reactants under steady-state conditions has also been

Reaction Kinetics in Fractals

observed to occur in percolation clusters [93,100]. Furthermore, dramatic segregation effects are observed in the Sierpinski Gasket under steady-state conditions [93,100]. In this case the value X D 2:00 ˙ 0:2 has been reported for the reaction order. All these results point out that steadystate segregation cannot simply be due to the Toussaint– Wilczek effect and the understanding of this behavior remains elusive, even after more than 20 years. Irreversible Phase Transitions in Heterogeneously Catalyzed Reactions Classical phase transitions, such as those observed in magnets, fluids, alloys, etc., which take place under equilibrium conditions, are reversible [28,35,111]. For example, one can change the phase of a magnet, from the ferromagnetic to the paramagnetic one, just by properly tuning the temperature. In contrast, irreversible phase transitions (IPTs) [9,10,68,85,90] occurring in reaction systems take place between an active state, characterized by a steady reaction among the reactants and the output of the products, and an inactive – also called poisoned or absorbing – state, where the reaction stops irreversibly. The inactive state is absorbing in the sense that after undergoing trapping, the system can never escape from it. Consequently, it is impossible to change from the absorbing state to the active one, just by tuning the control parameter, and the transition is irreversible. IPTs are typically studied by using lattice-gas reaction models of heterogeneously catalyzed reactions. Within this context IPTs observed upon the catalytic oxidation of CO, H 2 and NO have extensively been studied [42,85,88,108]. Further studies of IPTs comprise the generic monomer-monomer (A C B ! 0) reaction [90], as well as a wide variety of reaction processes such as epidemic propagation [84], forest-fire models [8,97], prey-predator systems [51], etc., for reviews see e. g. [68]. Since a wide variety of catalysts are composed by a solid-state fractal support containing catalytically active particles [21,22,23,95], most of the abovementioned reactions have also been studied in fractal media [1,2,5,7,41,72,86,87]. A brief overview of the state of the art in the characterization of IPTs in fractal substrates will be presented below. The ZGB Model for the Catalytic Oxidation of Carbon Monoxide It is well known that the catalytic oxidation of carbon monoxide, namely 2CO C O2 ! 2CO2 , which is likely the most studied reaction system, proceeds according to the Langmuir–Hinshelwood mechanism [50,65], i.e with

both reactants adsorbed on the catalyst surface CO(g) C S ! CO(a) ;

(27)

O2 (g) C 2S ! 2O(a) ;

(28)

CO(a) C O(a) ! CO2 (g) ;

(29)

where S is an empty site on the surface, while (a) and (g) refer to the adsorbed and gas phase, respectively. The reaction takes place with the catalyst, e. g. a transition metal surface such as Pt, in contact with a reservoir of CO and O2 whose partial pressures are PCO and PO2 , respectively. Equation (27) describes the irreversible molecular adsorption of CO on a single site of the catalyst surface. It is known that under suitable temperature and pressure reaction conditions, CO molecules diffuse on the surface. Furthermore, there is a small probability of CO desorption that increases as the temperature is raised [71]. Equation (28) corresponds to the irreversible adsorption of O2 molecules that involves the dissociation of such species and the resulting O atoms occupy two sites of the catalytic surface. Under reaction conditions, both the diffusion and the desorption of oxygen are negligible. Due to the high stability of the O2 molecule, the whole reaction does not occur in the homogeneous phase due to the lack of O2 dissociation. So, Eq. (28) dramatically shows the role of the catalyst that makes the rate-limiting step of the reaction feasible. Finally, Eq. (29) describes the formation of the product (CO2 ) that desorbs from the catalyst surface. This final step is essential for the regeneration of the catalytically active surface. For the practical implementation of the Ziff–Gulari– Barshad (ZGB) model [108], the catalyst surface is replaced by a lattice, so a lattice-gas reaction model is actually considered. The catalyst is assumed to be in contact with an infinitely large reservoir containing the reactants in the gas phase. Adsorption events are treated stochastically neglecting energetic interactions. Furthermore, diffusion and desorption of CO are also neglected. Then, on the two-dimensional square lattice the Monte Carlo simulation algorithm for the ZGB model is as follows: (i) CO or O2 molecules are selected at random with relative probabilities PCO and PO , respectively. These probabilities are the relative impingement rates of both species, which are proportional to their partial pressures in the gas phase in contact with the catalyst. Due to the normalization, PCO C PO D 1, the model has a single parameter, i. e. PCO . If the selected species is CO, one surface site is selected at random, and if that site is vacant, CO is adsorbed on it according to Eq. (27). Otherwise, if that site is occupied, the trial ends and a new molecule is selected. If the

1465

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Reaction Kinetics in Fractals

selected species is O2 , a pair of nearest-neighbor sites is selected at random and the molecule is adsorbed on them only if they are both vacant, as required by Eq. (28). (ii) After each adsorption event, the nearest-neighbors of the added molecule are examined in order to account for the reaction given by Eq. (29). If more than one [CO(a); O(a)] pair is identified, a single one is selected at random and removed from the surface. Assuming irreversible adsorption-reaction steps, as in the case of Eqs. (28)–(29), it may be expected that in the limit of large PCO and small PO2 (small PCO and large PO2 ) values, the catalyst surface would become saturated by CO- (O2 -) species and the reaction would stop. In fact, the catalyst surface fully covered by a single type of species, where further adsorption of the other species is no longer possible, corresponds to an inactive state of the system. This state is known as ‘poisoned’, in the sense that adsorbed species on the catalyst surface are the poison that causes the reaction to stop. Physicists used to call such a state (or configuration) ‘absorbing’ because a system becomes trapped by it forever, with no possibility of escape [49]. These concepts are clearly illustrated in Fig. 6a and b, which show plots of the rate of CO2 production (RCO2 ) and the surface coverage with CO and O2 ( CO and O , respectively), versus the partial pressure of CO (PCO ), as obtained by simulating the ZGB model [108]. Figure 6a corresponds to simulations performed in homogeneous, two-dimensional lattices. In this case, for PCO  P1CO ' 0:38975 the surface becomes irreversibly poisoned by O species with CO D 0, O D 1 and RCO2 D 0. In contrast, for PCO  P2CO ' 0:525 the catalyst is irreversibly poisoned by CO molecules with CO D 1, O D 0 and RCO2 D 0. However, as shown in Fig. 6a, between these absorbing states there is a reaction window, namely for P1CO < PCO < P2CO , such that a steady state with sustained production of CO2 is observed. As follows from Fig. 6a, when approaching the oxygen absorbing state from the reactive phase, all quantities of interest change smoothly until they adopt the values corresponding to the absorbing state. This behavior typically corresponds to a second-order IPT [9,10,68,85,90]. Remarkably, the behavior of the system is quite different upon approaching the CO absorbing state from the reactive regime (see Fig. 6a). In this case, all quantities of interest exhibit a marked discontinuity close to P2CO ' 0:525. This is a typical first-order IPT and P2CO is the coexistence point. Experimental results for the catalytic oxidation of carbon monoxide on Pt(210) and Pt(111) [3,36] are in qualitative agreement with simulation results of the ZGB model. A remarkable agreement is the (almost) linear in-

crease in the reaction rate observed when the CO pressure is raised and the abrupt drop of the reactivity when a certain ‘critical’ pressure is reached. However, two essential differences are worth discussing: (i) the oxygen-poisoned phase exhibited by the ZGB model within the CO lowpressure regime is not observed experimentally. (ii) The CO-rich phase exhibiting low reactivity found experimentally resembles the CO-poisoned state predicted by the ZGB model. However, in the experiments the non-vanishing CO-desorption probability prevents the system from entering into a truly absorbing state and the abrupt, ‘firstorder-like’ transition, observed in the experiments is actually reversible. Of course, these and other disagreements are not surprising since the lattice gas reaction model, with a single parameter, is a simplified approach to the actual catalytic reaction that is far more complex. In order to observe the influence caused by the fractal nature of the substrate on the phase diagram of the ZGB model, Fig. 6b shows results obtained by means of simulations performed in incipient percolating clusters at the critical threshold in two dimensions (see also Fig. 2). In this case, for PCO  P1CO ' 0:325 the surface becomes irreversibly poisoned by O species, while for PCO  P2CO ' 0:400 the catalyst is irreversibly poisoned by CO molecules [2]. Also, a reaction window between these absorbing states is found. So, by comparing Fig. 6a and b at least three main differences can be identified [2]: (i) The IPT observed at a low CO pressure in homogeneous lattices becomes largely shifted towards lower pressures in the case of the IPC. This effect is due to the constraint imposed by the disordered cluster on the adsorption of oxygen molecules that need two adjacent sites for deposition. (ii) The sharp first-order IPT characteristic of the homogeneous lattice becomes of second-order for the case of the fractal. (iii) The reaction window, which is of the order of PCO D P2CO  P1CO ' 0:135 in the homogeneous lattice, becomes narrowed up to PCO ' 0:085 for the case of the fractal. These results are in agreement with the findings of Casties et al. [41], who also studied the influence of CO diffusion on the location of the critical points. In particular, it is found that P2CO increases when the rate of CO-diffusion also increases. On the other hand, simulations of the ZGB model performed in disordered clusters below the critical threshold, i. e. in the so-called ‘lattice animals’ in percolation [101], show that poisoning can be caused either by CO- or O-species [41,91] and that due to finite-size effects the finite-width reaction window vanishes for lattices of side L ' 20 [41,91]. These observations are consistent with the fact that the characteristic lengths of the fluctuations in the coverages are larger than the sizes of the

Reaction Kinetics in Fractals

Reaction Kinetics in Fractals, Figure 6  Phase diagrams of the ZGB model showing the dependence of the surface coverage with CO ( CO , ) and Oxygen ( O , ), and the rate of CO2 production (RCO2 , ˘) on the partial pressure of CO (PCO ) in the gas phase. (a) Results obtained in d D 2 homogeneous lattices. The irreversible phase transition (IPT) occurring at P1CO ' 0:38975 (second-order) is shown by an arrow, while at P2CO ' 0:525 one has a sharp (first-order) IPT. (b) Results obtained by performing simulations in incipient percolation clusters of fractal dimension df ' 1:89 in d D 2 dimensions. Second-order IPTs occurring at P1CO ' 0:325 and P2CO ' 0:400 can clearly be observed. Notice that the rate of CO2 production has been amplified by a factor 25 for the sake of clarity

animals, playing a dominant role in the catalytic behavior of small aggregates, such as those used in actual catalysts [21,75,106]. The common feature of Monte Carlo simulations of the ZGB model performed in Sierpinski Carpets is that the first-order IPT characteristic of the homogeneous lattice is no longer observed, so that transition becomes of second-order [5,41]. In fact, even for the SC(5; 1) with fractal dimension df D log(24)/ log(25) ' 1:975 one observes a second-order IPT at high CO pressures [5]. Simulations performed in the SC(3; 1) with a fractal dimension df ' 1:89, which is very close to that of the IPC random fractal, yield P1CO ' 0:397 and P2CO ' 0:52 [41], i. e. two critical points that are very close to those measured in d D 2 homogeneous media (see e. g. Fig. 6a) but far from the results obtained for the IPC (see e. g. Fig. 6b). These findings point out that the location of the critical points depends on topological properties of the fractals that are not fully accounted for a single fractal dimension. Also, the finite-width reaction window, which lies between both second-order IPTs, becomes narrowed when decreasing df [5,41]. It has even been suggested that for Sierpinski Carpets the reaction window may become of zero width close to df  1:6 [5]. This statement is in agreement with the fact that Casties et al. [41] have reported that the reaction window of the ZGB model in an SC with df D 1:59 is very narrow, i. e. PCO ' 0:02 It is also worth mentioning that, in d D 1, the ZGB model lacks a finite-width reaction window. For second-order IPTs, as in the case of their reversible counterparts, it is possible to define an order parameter, which for the former is given by the concentration of mi-

nority species ( CO , in the case of the second-order IPT of the catalytic oxidation of CO). Furthermore, it is known that CO vanishes according to a power law upon approaching the critical point [64], so that CO / (PCO  P1CO )ˇ ;

(30)

where ˇ is the order parameter critical exponent and P1CO is the critical point. Careful studies performed in the d D 2 homogeneous lattice [63,73,104] have confirmed that this IPT belongs to the universality class of directed percolation [62,82] in d C 1 dimensions. Generic Lattice-Gas Reaction Models of Catalyzed Reactions The simplest reaction system studied proceeding according to the Langmuir–Hinshelwood mechanism [50,65] is the monomer-monomer (MM) model. In the MM model A- and B-species adsorb, react and desorb according to the following scheme Adsorption: A(g) C S ! A(a) ;

(31)

B(g) C S ! B(a) ;

(32)

Reaction: A(a) C B(a) ! AB(g) C 2S ;

(33)

Desorption: A(a) ! A(g) C S ;

(34)

B(a) ! B(g) C S ;

(35)

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Reaction Kinetics in Fractals

where S is an empty site on the surface, while (a) and (g) refer to the adsorbed and gas phase, respectively. Adsorption of A- and B-species takes place with probability pA and pB , respectively. So, by taking p A C p B D 1 one has a single parameter for the adsorption process, given by pad  p A D 1  p B . Also the rates of reaction and desorption of A- and B-species are kR , PdA and PdB , respectively. By properly selecting the parameters one can study different regimes of the MM model. Let us first consider the adsorption-limited regime that corresponds to kR  1 and PdA D PdB  0. Here, for pad ¤ 1/2 one has that any sample becomes irreversibly poisoned by the majority species and the final states are always absorbing. However, for pad D 1/2 a slow coarsening process with the formation of solid domains of A- and B-species on the surface of the catalyst is observed. By mapping the MM model into a kinetic Ising model, it can be proved that any finite sample of N s sites will ultimately become irreversibly poisoned due to fluctuations in the coverages after a time T P of the order of TP / Ns ln(Ns ) [32]. This exact result has early been predicted by means of numerical simulations by benAvraham et al. [4]. On the other hand, by considering desorption of a single species, say PdA  0 and PdB > 0, one has that the MM model exhibits a second-order IPT between an active state and a single poisoned state with A-species [6]. The critical points are found to be PdB D 0:5099 ˙ 0:0003, PdB D 0:6150 ˙ 0:0003 and PdB D 0:5562 ˙ 0:0003 for the case of d D 2, d D 1 homogeneous substrates and the IPC in d D 2 [54], respectively. Interesting results are also obtained by taking Pd  PdA D PdB , as proposed by Fichthorn et al. [46]. In fact, by means of numerical simulations, they have shown that for that choice of parameters the MM with desorption exhibits a noise-induced transition from monostability to bistability. This result was subsequently also obtained by means of a mean-field approach [44]. Later on, the occurrence of the transition was proved rigorously [32]. On the other hand, for a fractal lattice and in the limit of Pd ! 0 it has been conjectured that any finite sample of N s sites would become saturated by either A- or B-species when   Pdd s /2 Ns ' 1

(36)

where ds is the spectral dimension of the underlying fractal and  is a reduced parameter [55]. Furthermore, for an intermediate rate of desorption, a simple scaling behavior, which depends on the spectral dimension, is expected. In fact, for   1, a segregation regime with a reaction rate per site R(Pd ) given by R(Pd ) ' Pd1d s /2

(37)

has been predicted [55]. Also, for   1, one expects the saturation of any finite cluster due to a fluctuation-dominated regime, and the rate of reaction is expected to depend on the cluster size according to R(Pd ) D Ns Pd :

(38)

All these analytical results, namely equations (36), (37) and (38), have been validated by means of numerical simulations by using IPCs in d D 2 and d D 3 Euclidean dimensions [55] as fractal substrates. Another interesting generic system is the dimerdimer (DD) lattice-gas reaction model, which is described by the following Langmuir–Hinshelwood scheme Adsorption: A2 (g) C 2S ! 2A(a) ;

(39)

B2 (g) C 2S ! 2B(a) ;

(40)

Reaction: A(a) C B(a) ! AB(g) C 2S ;

(41)

where S is an empty site on the surface, while (a) and (g) refer to the adsorbed and gas phase, respectively. In this case we are interested in the reaction-controlled regime, so that the adsorption of either A2 - or B2 -species (see e. g. Eqs. (39) and (40)) takes place with the same probability and proceeds instantaneously on every pair of empty sites left behind by the reaction (Eq. (41)) [37]. It is worth mentioning that the DD model can be exactly mapped onto the so-called “voter” model [37]. The voter model is a spin system with two possible orientations, namely spin-up and spin-down. In the voter model, a randomly selected spin adopts the orientation of a randomly selected neighbor. Of course, if a spin is surrounded by equally oriented spins, it does not change its orientation. So, the voter dynamics is zero-temperature in nature. In fact, the voter model defines a broad universality class that describes coarsening without surface tension [48]. The voter model is also a simple model for opinion formation [57]: individuals with different opinions, labeled by A and B, respectively, change their opinions according to the voter dynamics. A remarkable feature of the DD model, or equivalently the voter model, is that it is one of the very few models that can be solved exactly in any integer dimension d. The order parameter of the voter model is the density of interfaces with different states of opinion (CAB ), which corresponds to the density of A-B interfaces in the DD model. Starting form a fully disordered configuration that maximizes the interfaces, it has been found that the den-

Reaction Kinetics in Fractals

sity of interface evolves according to [37] 8 ˛ d<2 2

(42)

where ˛ D 1  d/2. So, in the long-time regime on has that C AB ! 0 when d < 2, and the coarsening of domains is observed. However, for d > 2 single-species domains do not appear. The borderline is the two-dimensional case where coarsening proceeds logarithmically, so that dc D 2 is the upper critical dimension of the voter model. Monte Carlo simulations of the voter model performed in both Sierpinski Gaskets and Carpets with df < 2 have confirmed the coarsening of the domains. An interesting feature observed in the plots of C AB (t) versus t, as obtained for various fractals, is the presence of soft logperiodic modulations of the power-law decay predicted by Eq. (42) [53,102]. As already discussed in Sect. “Fractals and Some of Their Relevant Properties” and “Random Walks” (see Eqs. (4) and (11), respectively), these oscillations are the signature of time discrete scale invariance caused by the coupling of the voter dynamics with the spatial discrete scale invariance of the underlying fractal.

Patches consisting of 3–6 neighboring empty sites are frequently employed, but it is known that the asymptotic results are independent of the size of the initial patch. Such patch is the kernel of the subsequent epidemic. After the generation of the starting configuration, the time evolution of the system is followed, and during this dynamic process the following quantities are recorded: (i) the average number of empty sites (N(t)), (ii) the survival probability P(t), which is the probability that the epidemic is still alive at time t, and (iii) the average mean square distance, R2 (t), over which the empty sites have spread. Of course, each single epidemic stops if the sample is trapped in the poisoned state with N(t) D 0 and, since these events may happen after very short times (depending on the patch size), results have to be averaged over many different epidemics. It should be noticed that N(t) (R2 (t)) is averaged over all (surviving) epidemics. If the epidemic is performed just at criticality, a powerlaw behavior (scaling invariance) can be assumed and the following Ansätze are expected to hold, N(t) / t  ; P(t) / t

ı

(43) ;

(44)

and Epidemic Simulations for the Characterization of IPTs The study of second-order IPTs by using standard simulation methods is hindered by the fact that, due to large fluctuations occurring close to critical points, any finite system will ultimately become irreversibly trapped by the absorbing state. So, the measurements are actually performed within metastable states facing two competing constraints: on the one hand, the measurement time has to be long enough in order to allow the system to develop the corresponding correlations and, on the other hand, such time must be short enough to prevent poisoning of the sample. In view of these shortcomings, experience indicates that the best approach to second-order IPTs is to complement the standard approach, consisting in performing finite-size scaling studies of stationary quantities, with epidemic simulations. In fact, the application of the Epidemic Method (EM) to the study of IPTs has become a useful tool for the evaluation of critical points, dynamic critical exponents and eventually for the identification of universality classes [62,82,104]. The idea behind the EM is to initialize the simulation using a configuration very close to the absorbing state. Such a configuration can be achieved by removing some species from the center of an otherwise fully poisoned sample. Then a small patch of empty sites is left. In the case of the ZGB model, this can be done by filling the whole lattice with O, except for a small patch.

R2 (t) / t z epi ;

(45)

where , ı and zepi are dynamic critical exponents. Thus, at the critical point log-log plots of N(t), P(t) and R2 (t) will asymptotically show a straight line behavior, while off-critical points will exhibit curvature. This behavior allows the determination of the critical point, and from the slopes of the plots the critical exponents can also be evaluated quite accurately [62]. The validity of Eqs. (43), (44) and (45) for secondorder IPTs taking place in substrates of integer dimensions (1  d  3) is very well established. Furthermore, the observation of a power-law behavior for second-order IPTs is in agreement with the ideas developed in the study of equilibrium (reversible) phase transitions: scale invariance reflects the existence of a diverging correlation length at criticality. For the ZGB model in d D 2 dimensions the best reported values of the exponents are [63]:  D 0:2295 ˙ 0:001; ı D 0:4505 ˙ 0:001 and zepi D 1:1325 ˙ 0:001. These values confirm that the IPT belongs to the universality class of directed percolation in d C 1 dimensions, where the values reported by Grassberger [82] are  D 0:229 ˙ 0:003; ı D 0:451 ˙ 0:003 and zepi D 1:133 ˙ 0:002. Early epidemic studies of the ZGB model performed in both the IPC and the SC(3,1), with almost the same fractal

1469

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dimension of df ' 1:89, confirmed that the IPTs are of second-order [7]. The values of the critical exponents obtained for both types of fractals are almost the same, within error bars, and are given by   0:23 and ı ' 0:41. These values are also in agreement with epidemic calculations of the contact process [84] performed by Jensen [72], who has reported  D 0:235 ˙ 0:010 and ı ' 0:40 ˙ 0:01. Subsequently, Gao and Yang [110] have studied the influence of the lacunarity of the SC on the epidemic behavior of the ZGB model. The epidemic exponents are very different for lattices of the same fractal dimension but different lacunarity, suggesting that a single fractal dimension is not enough to fully specify the critical points and exponents of the critical process. This finding is in qualitative agreement with the same conclusion obtained by means of stationary measurements and discussed in the previous section. It is worth mentioning that the better statistics and the longer measurement time achieved by Gao et al. [110], as compared e. g. to references [7,72], allow us to observe a subtle log-periodic modulation of the power-law behavior of some dynamic observables. This effect is particularly clear for the case of the time dependence of N(t) and it is the signature of time discrete-scale invariance (DSI). In fact, since the underlying fractals used by Gao et al. [110] exhibit spatial DSI with a fundamental scaling ratio a1 D 4 (see also the discussion on Eq. (4)), we expect that the dynamics of any critical process occurring in those fractals would become coupled to the spatial DSI [53] through the growing correlation length  / t 1/z , where z is a dynamic exponent. This coupling would ultimately lead to time DSI characterized by a soft log-periodic modulation of the power laws, according to Eq. (11). Then the characteristic time-scaling ratio  is given by Eq. (12) with z D 2/zepi . Of course, the accurate evaluation of  by fitting the reported epidemic measurements is no longer possible because it will require a careful fit of the data. However, based on the fact that zepi ' 1 [110], we predict  ' 42 ' 16. The Constant Coverage Ensemble Applied to the Study of First-Order IPTs It is well known that first-order transitions are characterized by hysteresis effects, long-lived metastabilities, etc., which make difficult their study by means of conventional simulation methods. In order to avoid these shortcomings, Brosilow and Ziff [39] have proposed the constant coverage (CC) ensemble for the study of first-order IPTs. The method has early been applied to the ZGB model [39], and subsequently the same authors [11] have also performed CC simulations of the model proposed by Yaldran and Khan [88] for the NO C CO catalyzed reaction.

For the case of the ZGB model, the CC method can be implemented by means of the following procedure. First, a stationary configuration of the system is achieved using the standard algorithm, as described in Sect. “The ZGB Model for the Catalytic Oxidation of Carbon Monoxide”. For this purpose, one selects a value of the parameter close to the coexistence point, e. g. PCO D 0:51. After achieving the stationary state the simulation is actually switched to the CC method. So, in order to maintain the CO coverCC , age as constant as possible around a pre-fixed value CO only oxygen (CO) adsorption attempts take place whenCC , ( CC ever CO > CO CO < CO ). Let NCO and NOX be the number of carbon monoxide and oxygen attempts respectively. Then, the value of the “pressure” of CO in CC ) is determined just as the ratio the CC ensemble (PCO CC PCO D NCO /(NCO C NOX ). Subsequently, the coverage CC . A transient peis increased by an small amount, say  CO riod P is then disregarded for the proper relaxation of the CC CC value, and finally averages of PCO system to the new CO are taken over a certain measurement time M . The CC ensemble allows to determine spinodal points and investigate hysteresis effects at first-order IPTs [11,29,39], which are expected to be relevant as compared with their counterpart in equilibrium (reversible) conditions [9,10,85]. Future Directions In spite of the large effort involved in the study of the kinetics of reactions occurring in low-dimensional and fractal media, the subject still poses both theoretical and experimental challenges whose understanding would be of a wide interdisciplinary interest. It is largely recognized that the development and characterization of new materials and nanostructures is essential in order to boost the technology of the present century. Within this context, one has that reaction kinetics in nanosystems and confined materials exhibits a quite interesting physical behavior that is very different from that of the bulk. The average number of reactants in a nanosystem is typically very small and eventually it may be of the order of the fluctuations. So, mean-field-like treatments are far from being useful and new analytical tools, often guided by numerical simulations, need to be developed. Relevant experiments in this field are the study of the relaxation dynamics of photoexited charge carriers in semiconducting nanotubes and nanoparticles [56]. Also, the presence of catalytically active micro- and nanoparticles could lead to a considerable enhancement of the reaction rate of heterogeneously catalyzed reactions, the exhaustive understanding of this phenomena being an open challenge [30,75,106].

Reaction Kinetics in Fractals

During the 1990s physicists and mathematicians started the study of the properties of complex networks opening a new interdisciplinary branch of science that also involves biology, ecology, economics, social and computer sciences, and others. Early studies in the science of complex networks were aimed to discover general laws governing the creation and growth of such structures. Subsequently, growing interest has been addressed to the understanding of processes taking place in networks, such as e. g. the description of the behavior of random walkers, the characterization of qualitatively new critical phenomena, the study of the kinetics of reaction-diffusion processes, the discovery of the emergency of complex social behavior, etc. Due to the unique properties of the networks, which involve, among others, the combination of the compactness, their inherent inhomogeneity, their complex architectures, etc., reaction-diffusion processes in those media are essentially different from both classical and anomalous results already discussed. In fact, annihilation rates are abnormally high, no segregation is observed in the archetypical A C B ! 0 reaction, and depletion zones are absent in the annihilation A C A ! 0 reaction [69]. So, we expect that this will be an active field of interdisciplinary research in the study of reaction kinetics in complex media. The recently formulated conjecture that the coupling between the spatial discrete scale invariance of fractal media and the dynamics and kinetics of physical processes occurring in those media may lead to the observation of time discrete scale invariance [53], opens new theoretical perspectives for the study of diffusion-limited reactions. This kind of study is further stimulated by recent reports suggesting that spatial discrete scale invariance is no longer restricted to deterministic fractals, but it may also emerge as a consequence of different growing mechanisms in some disordered fractals, such as e. g. diffusionlimited aggregates, animals in percolation, etc. [66,74]. In order to achieve some progress in this subject, of course, extensive simulations and careful measurements would be necessary because the phenomenon is quite subtle and the standard averaging methods may destroy it as if was simple noise [66]. The propagation of fronts has attracted considerable attention since its understanding is relevant in many areas of research and technology. Among others, some examples of front propagation include: material growth and interfaces, forest-fire and epidemic propagation, particle-diffusion fronts, chemical pattern formation, biological invasion problems, displacement of an unstable phase by a stable one close to first-order phase transitions, etc. Within this context, the properties of reaction-diffusion fronts in simple reactions where the reactants are initially segre-

gated, such as A C B ! O, have extensively been studied [31,47]. However, most of the studies addressed reaction-diffusion fronts occurring in homogeneous media while less attention has been devoted to the case of, increasingly important inhomogeneous media [98]. Inhomogeneities can have different origin: i. e. physical, chemical, geometrical, etc. It is expected that the study of front propagation generated by diffusion-limited reactions in fractal media would also be a promising field of future research, where anomalous diffusion may lead to the occurrence of very interesting phenomena. In a related context, theoretical studies of the diffusion of A- and B-species in the bulk of a confined (homogeneous) media, leading to reaction at the surface with desorption of the product, predict the development of an annihilation catastrophe [98]. This phenomenon is due to the self-organized explosive growth of the surface concentration of species leading to an abrupt desorption peak. The influence of anomalous diffusion in fractal media leading to surface reaction will certainly be addressed in the near future, hopefully deserving the observation of interesting physical phenomena.

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78. Kittel C (1986) Introduction to solid state physics, 8th edn. Wiley, New York 79. Kopelman R (1976) In: Fong FK (ed) Radiationless processes in molecules and condensed phases, vol 15. Springer, Berlin, p 297 80. Kopelman R (1986) Rate processes on fractals: Theory, simulations, and experiments. J Stat Phys 42:185–200 81. Kopelman R (1988) Fractal reaction kinetics. Science 241:1620–1626 82. Krapivsky PL (1992) Kinetics of monomer-monomer surface catalytic reactions. Phys Rev A 45:1067–1072 83. Lee Koo Y-E, Kopelman R (1991) Space- and time-resolved diffusion-limited binary reaction kinetics in capillaries: experimental observation of segregation, anomalous exponents, and depletion zone. J Stat Phys 65:893–918 84. Liggett TM (1985) Interacting particle systems. Springer, New York 85. Loscar E, Albano EV (2003) Critical behaviour of irreversible reaction systems. Rep Prog Phys 66:1343–1382 86. Mai J, Casties A, von Niessen W (1992) A model for the catalytic oxidation of CO on fractal lattices. Chem Phys Lett 196:358–362 87. Mai J, Casties A, von Niessen W (1993) A Monte Carlo simulation of the catalytic oxidation of CO on DLA clusters. Chem Phys Lett 211:197–202 88. Maltz A, Albano EV (1992) Kinetic phase transitions in dimerdimer surface reaction models studied by means of meanfield and Monte Carlo methods. Surf Sci 277:414–428 89. Mandelbrot BB (1983) The fractal geometry of nature. Freeman, San Francisco 90. Marro J, Dickman R (1999) Nonequilibrium phase transitions and critical phenomena. Cambridge Univ Press, Cambridge 91. Meakin P, Scalapino DJ (1987) Simple models for heterogeneous catalysis: Phase transition-like behavior in nonequilibrium systems. J Chem Phys 87:731–741 92. Newhouse JS, Kopelman R (1985) Fractal chemical kinetics: Binary steady-state reaction on a percolating cluster. Phys Rev B 31:1677–1678 93. Newhouse JS, Kopelman R (1988) Steady-state chemical kinetics on surface clusters and islands: Segregation of reactants. J Phys Chem 92:1538–1541 94. Parus SJ, Kopelman R (1989) Self-ordering in diffusion-controlled reactions: Exciton fusion experiments and simulations on naphthalene powder, percolation clusters, and impregnated porous silica. Phys Rev B 39:889–892 95. Pfeifer P, Avnir D (1983) Chemistry in noninteger dimensions between two and three. I Fractal theory of heterogeneous surfaces. J Chem. Phys 79:3558–3565 96. Rammal R, Toulouse G (1983) Random walks on fractal structures and percolation clusters. J Physique Lett 44:L13–L22 97. Rozenfeld AF, Albano EV (2001) Critical and oscillatory behavior of a system of smart preys and predators. Phys Rev E 63:061907 98. Shipilevsky BM (2007) Formation of a finite-time singularity in diffusion-controlled annihilation dynamics. J Phys C Condens Matter 19:065106 99. Sornette D (1998) Discrete scale invariance and complex dimensions. Phys Rep 297:239–270 100. Stanley HE (1971) Introduction to phase transitions and critical phenomena. Oxford University Press, New York

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101. Stauffer D, Aharoni A (1992) Introduction to the percolation theory, 2nd edn. Taylor and Francis, London 102. Suchecki K, Hołyst JA (2006) Voter model on Sierpinski fractals. Physica A 362:338–344 103. Toussaint D, Wilczek F (1983) Particle-antiparticle annihilation in diffusive motion. J Chem Phys 78:2642–2647 104. Voigt CA, Ziff RM (1997) Epidemic analysis of the secondorder transition in the Ziff–Gulari–Barshad surface-reaction model. Phys Rev E 56:R6241–R6244 105. Weiss GH (1995) A primer of random walkology. In: Bunde A, Havlin S (eds) Fractals in science. Springer, Berlin, p 119 106. Wieckowski A, Savinova ER, Vayenas CG (2003) Catalysis and electrocatalysis at nanoparticle surfaces. Marcel Dekker, New York, pp 1–959 107. Witten TA, Sander LM (1983) Diffusion-limited aggregation. Phys Rev B 27:5686–5697 108. Yaldram K, Khan MA (1991) NO-CO reaction on square and hexagonal surfaces: A Monte Carlo simulation. J Catal 131:369–377 109. Zhdanov VP, Kasemo B (1994) Kinetic phase transitions in simple reactions on solid surfaces. Surf Sci Rep 20:111–189 110. Ziff RM, Brosilow BJ (1992) Investigation of the first-order phase transition in the A  B2 reaction model using a constant-coverage kinetic ensemble. Phys Rev A 46:4630–4633

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Books and Reviews Binder K (1997) Applications of Monte Carlo methods to statistical physiscs. Rep Prog Phys 60:487–560 Bunde A, Havlin S (1991) Fractals and disordered media. Springer, New York Cardy J (1997) Goddard P, Yeomas JM (eds) Scaling and renormalization in statistical physics. Cambridge University Press, Cambridge Christmann K (1991) Introduction to surface physical chemistry. Steinkopff, Darmstadt, p 1 Egelhoff WF Jr (1982) King DA, Woodruff DP (eds) Fundamental studies of heterogeneous catalysis, vol 4. Elsevier, Amsterdam Lindenberg K, Oshanin G Tachiya M (2007) Chemical kinetics beyond the texbook: Fluctuations, Many-particle effects and anomalous dynamics. Special issue of J Phys C Condens Matter 19 Walgraef D (1997) Spatio-temporal pattern formation. Springer, New York

Relaxation Oscillations

Relaxation Oscillations JOHAN GRASMAN Wageningen University and Research Centre, Wageningen, The Netherlands Article Outline Glossary Definition of the Subject Introduction Asymptotic Solution of the Van der Pol Oscillator Coupled Van der Pol-Type Oscillators Canards Dynamical Systems Approach Future Directions Acknowledgment Bibliography

are stable equilibria of "dx/dt D f (x; y0 ). Next the system will enter a quasi-stationary state and changes slowly within M (") . With singular perturbations separate approximate solutions are constructed for the two phases. They have the form of power series expansions in ". Integration constants are determined by a matching procedure. Relaxation oscillation A relaxation oscillation is a limit cycle of a singularly perturbed dynamical system. Within a cycle at least once the system leaves and returns to the manifold M (") . Canard If within a cycle the relaxation oscillation comes near a part of M (0) with unstable equilibrium points of "dx/dt D f (x; y0 ) then we have a so-called canard type of oscillation. In addition to the parameter " a second parameter a can be identified that passes a Hopf bifurcation point, where a stable equilibrium changes into a stable relaxation oscillation. Then just before the regular relaxation oscillation arises a canard appears. For very small " a canard is not easily detected.

Glossary Dynamical system At any time t the state of a n-dimensional dynamical system is determined by the state variables forming a vector x(t) in Rn . The change in time of this vector is given by the vector differential equation dx/dt D f (x). In the state space Rn solutions of this equation are the trajectories or orbits of the system. If the right-hand side of the equation also depends directly on t, dx/dt D f (t; x), then the system is forced and called nonautonomous. Oscillator An oscillator is a time periodic solution of the system forming a closed orbit in state space. If trajectories, close to a periodic solution, tend to this periodic solution for t ! ˙1, then the oscillator is called a limit cycle. If a nonautonomous system is periodically forced, f (t; x) D f (t C T; x), then the system may get entrained: Periodic solutions with the same period T or a period kT with some integer larger than 1 (subharmonic solution) may occur. Singular perturbations If a dynamical system contains a small parameter " and we let " ! 0 then the system may become degenerate meaning that it cannot satisfy generic initial- or periodicity conditions. In fact such a system has two time scales: The fast variables vector x and the slow variables vector y : "dx/dt D f (x; y), dy/dt D g(x; y). For " D 0 initial values outside the manifold M (0) : f (x0 ; y0 ) D 0 cannot be satisfied. For small positive " we see that in the initial phase x changes rapidly until a manifold M (") near M (0) is reached at a part where the points (x0 ; y0 )

Definition of the Subject A relaxation oscillation is a type of periodic behavior that occurs in physical, chemical and biological processes. To describe it mathematically, a system of coupled nonlinear differential equations is formulated. Such a system is studied with qualitative and quantitative methods of mathematical analysis. The well-known linear pendulum (harmonic oscillator) is not the appropriate system to model real life oscillations. Characteristic for a relaxation oscillator is its nonlinearity and the presence of phases in the cycle with different time scales: A phase of slow change is followed by a short phase of rapid change in which the system jumps to the next stage of slow variation. These oscillations belong to the class of nonlinear systems that give rise to a self-sustained oscillation meaning that the system goes alternately through phases in which energy dissipates and is taken up again. Van der Pol [66] studied such a type of oscillation in a triode circuit. For small values of a system parameter he found an almost sinusoidal oscillation, while for larger values the system exhibited the type of slow-fast dynamics described above. In the last case the period of the oscillation is almost proportional to that parameter. The name relaxation oscillation, introduced by Van der Pol, refers to this characteristic time constant of the system. In a next publication Van der Pol [67] points out that not only in electronics but in far more fields of science relaxation oscillations may be present. In this respect the physiology of nerve excitation and, more specifically, the heart beat

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was given special attention by him. In new generations of electronic and electromagnetic systems, such as transistor circuits, Josephson junctions and laser systems, relaxation oscillations are prominently present. Applications are also found in chemistry such as the Zhabotinskii reaction [56], and in geophysics with e. g. the irregular pattern of earthquakes at folds [70]. It is noted that the stick-slip model [71] in the form of a Brownian relaxation oscillator is often brought up in these earthquake studies. In addition to the physiological applications, relaxation oscillations are found in the biology of interacting populations such as in epidemiology [39] and in prey-predator systems, see [12,53,54]. Also in the humanities comparable phenomena are met such as business cycles in economics, see e. g. [31,69]. One way to handle mathematically relaxation oscillations is to exploit the presence of a small number. Dynamical systems with a small parameter " multiplying the time derivative of some of the state variables degenerate when this parameter is set to zero, as it is not possible anymore to satisfy all initial- or periodicity conditions. In the theory of singular perturbations the limit process " ! 0 is followed and the nonuniform convergence of the solution to some limit function is taken in consideration. At the point in time, where this limit solution is discontinuous, a new time scale is introduced by a stretching transformation. Then again the limit " ! 0 is taken leading to a new (locally valid) limit solution. Integration constants in both limit solutions are found from a matching procedure [15]. In a second approach, based on nonstandard analysis [55], it is not necessary to consider at each step of the computational process this limit procedure. The set of real numbers is extended with infinitesimal numbers being numbers that are nonzero and have an absolute value that is smaller than any real number. Then a solution of a differential equation with an infinitesimally small parameter lies infinitely close to the limit solution in an appropriate function space, see [58,79]. The nonstandard approach of a type of relaxation oscillations known as canards or “French ducks” [6] has given the method an important place in the literature. In a third type of approach the attention fully goes into the analysis of the vector field related to the dynamical system. The onset of a phase of fast change in the period is marked by the passage of a special point in state space (the fold point). By a blow up of the vector field at this point [14] the periodic trajectory can be rigorously described over a time interval containing this point and therefore a full description over the entire period is at hand. It has been worked out for relaxation oscillations [38] and is connected to the dynamical systems theory known as geometrical singular perturbations [19].

Introduction Periodic processes control our daily life. External influences such as the dynamics of sun, earth and moon play an important role in e. g. the seasons, the tides and our day–night rhythm. However, internally we also have our circadian clock as well as other autonomous periodic processes such as the regulation of our metabolic functions. Periodic electric activities in our brain and heart play a key role in the functioning of these organs. Through the work of Hodgkin and Huxley [34] we have a good understanding of the transport of electric pulses in nerve cells. Looking back we notice that the result of Balthasar van der Pol on periodicity in electric circuits is one of the scientific achievements forming the basis of this breakthrough. In Fig. 1 Van der Pol’s triode circuit is depicted. It produces a self-sustained oscillation due to the nonlinearity in the triode characteristic given by I a D V  1/3V 3 . Replacing V by a scaled potential x and scaling also the time variable we obtain from circuit theory the following differential equation d2 x dx C (x 2  1) Cx D0; d 2 d p p  D M/ LC  R C/L :

(1)

For  small a nearly sinusoidal oscillation is found with amplitude close to 2 and a period close to 2. For   1 an almost discontinuous solution appears for which the period is nearly proportional with this parameter, see Fig. 2. The best way to study this last type of oscillation is to rewrite the second-order differential Eq. (1) as a system of two coupled first-order differential equations and to introduce the small parameter " D 1/2 . Furthermore, a new time scale is introduced t D /. It leads to the socalled Lienard-type of representation of the system [41]: "

dx D y  F(x); dt

F(x) D 1/3 x 3  x ; 0 <"  1;

(2)

Relaxation Oscillations, Figure 1 Triode circuit giving rise to a self-sustained oscillation. L is a selfinductance, M a mutual inductance and R a resistance. I and V are, respectively, the current and the grid voltage

Relaxation Oscillations

which two trajectories move away from the unstable equilibrium at the origin. At the point where a stable branch becomes unstable the system jumps to the opposite stable branch. Clearly the amplitude of the discontinuous oscillation has the value 2 in the x variable. The period follows from Z2/3 T0 D 2 2/3

dt dy D 2 dy

Z1

1 0 F (x)dx D 3  2 ln 2 : (4) x

2

This discontinuous solution can be seen as a zero-order asymptotic approximation of the solution with respect to the small parameter ". In Sect. “Asymptotic Solution of the Van der Pol Oscillator” we will deal with higher-order approximations using the theory of singular perturbations. Canards Relaxation Oscillations, Figure 2 Periodic solution of Eq. (1) for a  D 0:1, b  D 1, c  D 10

dy D x C a ; dt

a D0:

(3)

Discontinuous Limit Solution In Fig. 3a it is seen how the periodic solution of (2) behaves in the limit " ! 0. It has the form of a discontinuous solution consisting of two time intervals in which the system follows alternately in state space two branches (DA and BC) of the graph of y D F(x) with jxj > 1, see Fig. 3b. These are the stable branches; they are rapidly approached because of a fast changing variable x if y ¤ F(x). In between the two branches there is the unstable branch AC at

If in formula (2) the parameter a is varied an intriguing type of Hopf bifurcation arises. When passing the values a D ˙1 for decreasing absolute value of a, a stable equilibrium turns unstable and a relaxation oscillation arises. The scenario of a common Hopf bifurcation is that a stable equilibrium changes into an unstable one at the bifurcation point and that a periodic solution branches off with an amplitude that grows in the beginning quadratically as a function of the distance of the parameter to the just passed bifurcation point. However, the above discontinuous periodic solution suggests that directly after the bifurcation point a fully developed relaxation oscillation arises. Using concepts of nonstandard analysis [55] French mathematicians [6] explain how the curious emergence of a relaxation type of oscillation at a Hopf bifurcation can be understood. Also by an intricate asymptotic analysis the phenomenon can be described with singular perturbation

Relaxation Oscillations, Figure 3 The discontinuous approximation of the solution of the Van der Pol equation for " ! 0. The parts AB and DC of the orbit are taken infinitely fast. a The function x(t) for " ! 0. b The closed orbit in the x; y-plane

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Relaxation Oscillations, Figure 4 A perturbation (dotted line) of the equilibrium ( ) above threshold triggers a cycle. During the cycle perturbations do not have a large effect (refractory period)

theory [16]. In Sect. “Canards” we sketch the result. Finally, as part of a dynamical systems approach (Sect. “Dynamical Systems Approach”) a geometrical singular perturbation analysis can be carried out as well [38]. Bonhoeffer–Van der Pol Equation or FitzHugh–Nagumo Equation When we take in (3) a D 1  ı; 0 < ı  1 with ı independent of ", a stable equilibrium (x, y) arises with x D 1  ı. A small perturbation of the x-component of the system, causing a deviation that stays below 2ı, will damp out quickly. A positive perturbation just above this threshold (dotted line) will trigger a cycle of the system as depicted in Fig. 4. FitzHugh [21] constructed a similar variant of the Van der Pol Equation (2) providing a mathematical model of a nerve excitation as described by Bonhoeffer [3]. Firing of a neuron occurs when an electric stimulation of a dendrite is above some threshold value triggering an electric pulse in the neuron itself which is passed through the axon to other neurons. Also the presence of a so-called refractory period can be understood from the above system. The refractory period is the time directly after an above threshold stimulation. During this time the neuron is insensitive to perturbations. Later on more refined models evolved [57]. The introduction of spatial structure and diffusion made it possible to consider traveling pulses [1,34]. Orbital Stability, Entrainment, Chaos, Quenching, and Noise In applications we typically meet nonlinear oscillations that are orbitally highly stable but rather easily speeded up or slowed down in their cycle. In a biological context or-

ganisms are provided in this way with a mechanism that helps them to adapt to external circumstances. Also in physics we meet such behavior in systems with strong energy exchange with the environment (lasers), as opposed to conservative systems without dissipation such as in celestial systems. In (2) entrainment is found if we let the parameter a depend periodically on time with a period say T. If the period T (0) of the autonomous system with a D 0 in (2) is sufficiently close to the forcing period T or if the amplitude of the forcing is sufficiently large, then the system will take over this period T. If T (0) is near a value nT, n D 2; 3; : : : a subharmonic solution with period nT may arise. It also may occur that two stable subharmonics with different n values co-exist. Then the starting value determines which one is chosen. Van der Pol and Van der Mark [68] and Littlewood [43,44] already concluded on respectively, experimental and theoretical grounds that in such a case also other “strange” (chaotic) solutions may be present in the Van der Pol-type oscillator with periodic forcing, see [4,40,42]. For autonomous systems chaotic dynamics can be found in systems consisting of at least three components [23]. A system with one fast and two slow variables already exhibits a wealth of interesting dynamical features, including chaos. Quenching of an oscillation can be achieved by extending the system with a set of differential equations including a feedback to the original system such that the amplitude of the oscillation is reduced by choosing appropriate values for the parameters. For an application to relaxation oscillations see [72,74]. Relaxation oscillations are met in a wide range of applications. Since in practice the action of random forces mostly cannot be excluded, stochastic oscillation forms an essential part of the theory of periodic phenomena in the description of natural processes as well as in the engineering sciences, see [27,33]. An additional reason to take stochasticity in account comes from the fact that the phase velocity of the relaxation oscillator is easily influenced. Higher-Order Systems and Coupling of Oscillators Making a generalization we consider the periodic solution of a system of differential equations of the form dy dx D f (x; y; "); D g(x; y; "); 0 < "  1 ; (5) dt dt where x and y are respectively k- and l-dimensional vector functions of time. The vector functions f and g remain bounded for " ! 0. Then the (slow) dynamics is governed by y 0 (t) D g(x; y; 0) with constraint 0 D f (x; y; 0) and for the fast dynamics we have "x 0 (t) D f (x; y; 0) with y constant. An extension of the Van der Pol-type dynamics to "

Relaxation Oscillations

such a higher-dimensional system is not difficult to conceive. However, formulating precisely the conditions for the system (5) to have a periodic solution and to work out a full proof for the existence of such a solution is a rather complex task [46]. As we mentioned before, chaotic solutions may occur too. Another way to arrive at higher-order systems is to couple relaxation oscillators [76]. For relaxation oscillators of type (2) with different autonomous periods we take "

dx i D y i  F(x i ) ; dt n X dy i ı pi j x j ; D c i x i C dt

(6) 0< "ı 1:

jD0 j¤i

For a system of n D 100 oscillators with a high degree of coupling, e. g. p i j ¤ 0 for all j ¤ i, and widely different frequencies, e. g. c i D 0:5 C i/n, we will observe (after a spin up) a spectrum of frequencies with distinct peaks. These peaks are related by a simple ratio of their frequencies such as 1:2 or 3:2 and so on, see [17,25]. Considering spatially distributed oscillators with nearest neighbor coupling we meet interesting phenomena such as traveling phase waves. The corresponding fronts may have a preferred direction given by the gradient in the autonomous periods (inherently faster oscillators are likely ahead in phase). Locally, oscillators tend to have an equal actual period (phase velocity) although their intrinsic periods are different. This phenomenon, called plateau behavior, has been investigated by Ermentrout and Kopell [18]. Wave fronts may also move in spirals or travel randomly breaking down when meeting each other as we see in the Zhabotinskii reaction [20] or during fibrillation of the heart [62]. Asymptotic Analysis with Respect to the Small Parameter By letting " ! 0 the Van der Pol oscillator (2) reduces to the system: (x02  1)

dx0 C x0 D 0 dt

(7)

having a discontinuous solution t D ln(x0 )  12 (x02  1) for  12 T0 < t < 0 ; t D ln(x0 )  12 (x02  1) C 12 T0

(8)

for 0 < t < 12 T0 (9)

with T 0 given by (4). This approximation, see Fig. 3a, is improved in Sect. “Asymptotic Solution of the Van der Pol Oscillator” by the construction of higher-order terms with

respect to the small parameter ". Contrary to regular perturbation problems, where a power series expansion with respect to " holds over the entire period, we have to consider here separate power series expansions for the stable branches. Moreover, we have to exclude a small neighborhood of the point t D 0 where the discontinuity occurs. Near (t; x) D (0; 1) a local approximation is made based on fractional powers of ". It is followed by an internal layer solution approximating the fast change in x from the value 1 to the starting value  2 at the next stable branch. Unknown (integration) constants are determined by matching the local solutions at t D 0 to the regular solutions at the two stable branches, see Sect. “Asymptotic Solution of the Van der Pol Oscillator”. Matching of local asymptotic solutions typically applies to differential equations with a small parameter multiplying the highest derivative such as in (1) with  and  replaced by t and ", see [15,50]. In the literature on relaxation oscillations the procedure has been carried out in different ways depending on whether one chooses to solve the problem in the x; t-plane [8], the x; x˙ -plane [13], or the Lienard plane. For a survey of this literature see [25]. For higher-order systems the construction of matched local asymptotic solutions that involve higher-order terms with respect to " turns out to be rather complicated, see [46]. A different type of approximation, based on power sep ries expansion with respect to the parameter  D 1/ ", was made possible by the use of computerized formula manipulation packages. In this way elements of a periodic solution such as period and amplitude are approximated also for large values of the parameter , see [2,10]. Topological Methods, Mappings and the Dynamical Systems Approach The existence of a periodic solution for (2) has been proved with the Poincaré–Bendixson theorem. This method, based on topological arguments, only applies to two-dimensional systems. If in the x; y-plane we can construct a domain that does not contain equilibrium points and all trajectories crossing the boundary are entering the domain, then the domain contains a limit cycle. If the domain is a narrow annulus enclosing the limit cycle the method also produces an approximation of the solution together with its period and amplitude [52]. Another important tool in the dynamical systems approach is the Poincaré map or transition map. For autonomous n-dimensional systems we may consider starting values at a bounded transversal (n  1)-dimensional surface and analyze how the trajectories intersect some other surface in the state space or the same surface (re-

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turnmap). In periodically forced systems we may consider the mapping of the full state space into itself after a time interval equal to the forcing period. Such a type of mapping may also apply to the phase of an oscillator or to the phases of a system of coupled oscillators. A periodic solution corresponds with a fixed point of the return map or, in the case of a forced system, with a fixed point of the map of the state space into itself after one or more periods of the forcing term. Chaotic solutions may occur due to the presence of a so-called horse-shoe map [61]. Levi [40] used a map of this type to describe chaotic solutions of a piecewise linear Van der Pol equation with a sinusoidal forcing term. Presently, relaxation oscillations are also analyzed with geometrical singular perturbation theory [64]. Geometrical singular perturbations deal with the structure of the slow manifold f (x; y; 0) D 0 containing trajectories of (5) with " D 0 that vary slowly in time [19]. A point of the slow manifold is an equilibrium of the fast system in the time scale  D t/" : dx/ d D f (x; y (0) ; 0). It is assumed that the eigenvalues of this system linearized at the equilibrium have real parts that are bounded away from zero (hyperbolicity condition). For relaxation oscillations this condition is not satisfied [73]. More details are given in Sect. “Dynamical Systems Approach”. Asymptotic Solution of the Van der Pol Oscillator From the different methods to approximate asymptotically the solution of (2) we choose the approach of Carrier and Lewis [8], who consider the solution in the t; x-plane, see Fig. 5. Writing (2) again as a second-order differential equation

we can expand the solution valid for the stable branch as x(t; ") D x0 (t) C "x1 (t) C "2 x2 (t) C    :

Substitution of (11) in (10) gives, after equating terms with equal powers in ", a recurrent system of differential equations for the coefficients x i (t) of (11) with x0 (t) given by (7) when taking the branch for t < 0. We see from p (8) that for t " 0 this first term behaves as x0  1 C t, so the second-order derivative cannot be neglected near the origin. To analyze the local behavior of the solution in more detail we apply a stretching transformation t D "2/3 ;

x D 1 C "1/3 v() :

d2 v 0 dv0 C 2v0 C 1 D 0 or d 2 d

dv0 C v02 C  D B : (13) d

In order p to match x0 for  ! 1 the solution must satisfy v0 ()   when taking this limit. The solution v0 () D 

Ai 0 () Ai()

(14)

with Ai(z) the Airy function complies with this matching condition. As  approaches ˛ D 2:33811 : : :, being the first zero of the Airy function, the solution will behave as v0 ()  1/( ˛). Consequently, the first two terms of (10) will be leading in the left hand side of the equation and make the solution enter the phase of fast change for which we choose the appropriate time variable  using the transformation t D ˛"2/3 C " ;

(10)

(12)

Substitution in (10) yields for the leading terms the equation

2

d x dx " 2 C (x 2  1) Cx D0; dt dt

(11)

(15)

so d2 w0 dw0 C (w02  1) D0 2 d d

or

1 dw0 D w0  w03 C D : d 3 (16)

Matching with v0 for  ! 1 yields D D 2/3  ˛"2/3 and integrating Eq. (16) we obtain   w0 C 2 C 1/3˛"2/3 1 1 C ln  D : (17) 1  w0 3 1  w0

Relaxation Oscillations, Figure 5 The solution in the t; x-plane with the fast change just after t D 0 starting with a specific local behavior near the point (0,1)

For  ! 1 the solution behaves as w0 D 2  1/3˛"2/3 C O(exp(3)). This must match the regular asymptotic solution at the stable branch for t > 0 for which the zeroorder approximation satisfies t D ln(x0 )  12 (x02  1) C E

(18)

Relaxation Oscillations

Relaxation Oscillations, Figure 6 Numerical solutions of period 2m with m depending on  and b

giving E D 3/2ln(2)C3/2˛"2/3 . At x0 D 1 the variable t is at the value 12 T0 , so T0  3  2 ln(2) C 3˛"2/3 :

(19)

The amplitude A follows from the minimal value of w0 which is approached in (17) for  ! 1: A  2 C 1/3˛"2/3 :

(20)

For a higher-order approximation of both the period and the amplitude we refer to Grasman [25]. In Grasman et al. [30] the Lyapunov exponents are approximated in a similar way. Sinusoidal Forcing We next consider the Van der Pol oscillator with a sinusoidal forcing term dx d2 x C (x 2  1) C x D (b C c) cos(k) ; d 2 d

prominent ones have period 2 m with m an odd number. The value of m depends on . In Fig. 6 we depicted the domains in the parameter plane where these solutions are found. Their boundaries follow from the construction of the asymptotic solution. The result is compared with the numerical solution of the system. Notice the overlap of these domains and the coexistence of different numerical solutions. Also chaotic solutions can be approximated asymptotically. They come with the phenomenon of dips and slices which also occurs in the general case for c ¤ 0 [28]. The main difference with the previous case is that instead of Airy functions parabolic cylinder functions turn up. In Fig. 7 we see how dips and slices arise. They occur when the trajectory stays close to the regular solution below the line x D 1, while switching from a stable branch (x > 1) to an unstable branch (x < 1) of this regular solution. It can be seen as a canard type of phenomenon as noted by Smolyan and Wechselberger [63].

  1: (21)

A first mathematical investigation of this problem was made by Cartwright and Littlewood [9] followed by Littlewood [43,44]. Subharmonic solutions were constructed. Solutions with different periods may coexist and also solutions that behave chaotically may turn up. Following the method of Carrier and Lewis for this nonautonomous Eq. (21) with c D 0 and k D 1 we go through the same sequence of asymptotic solutions: The regular approximation of the type (11), the local solution when the trajectory crosses the lines x D ˙1, giving rise to an expression with Airy functions, and a fast changing part directly after with a local approximation similar to (17), see [29]. We may construct various periodic solutions. The most

Relaxation Oscillations, Figure 7 A dip and a slice for slightly different initial values chosen such that upon arrival at the line x D 1 a point is passed where stable (x > 1) and unstable (x < 1) regular solutions meet

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Coupled Van der Pol-Type Oscillators The discontinuous periodic solution of (2) for " ! 0 is given by (7) and has the form of the vector fx0 (t); y0 (t)g. We next consider (2) with the parameter a replaced by a small amplitude piecewise continuous periodic function ıh(t) with period T: "

dx D y  F(x) ; dt

dy D cx C ıh(t) ; c D 1 ; dt 0 < "  ı  1 : (22)

For " ! 0 and ı sufficiently small the solution will take the same orbit as for ı D 0; only the velocity on the limit cycle is influenced. Consequently, a solution of Eq. (22) can be approximated by fx0 ((t)); y0 ((t))g :

(23)

Substitution in the Eq. (7) yields dy0 d D x0 ((t))Cıh(t) d dt

or

d ıh(t) D 1 : dt x0 ((t)) (24)

Starting with a phase (0) D ˛0 and integrating over one period T gives a phase shift ZT ı (˛0 ) D 0

ıh(t) dt : x0 ((t))

(25)

Thus, the dynamics of the oscillator is described in a stroboscopic way by the iteration map ˛ kC1 D ˛ k C T C ı (˛ k )(mod)T0 :

(26)

Besides a periodic solution of period T different types of subharmonic solutions can be found as well as chaotic solutions depending on the function (˛). An entrained solution corresponds with a stationary point ˛ s of the map (26). Such a point exists if for some m D 1; 2; : : :

the position to analyze different forms of mutual entrainment such as wave phenomena in systems of spatially distributed oscillators with nearest neighbor coupling as described in Sect. “Introduction”. A system of spatially distributed oscillators with nearest neighbor coupling can be seen as a discrete representation of a nonlinear diffusion problem with oscillatory behavior. A time delay in the coupling can be handled as well, see [7,26]. It is also allowed that the orbits and the autonomous period of the oscillators are different. As an example we take 25 piecewise linear oscillators with in (22) F(x) D x C 2

for x < 1 ;

F(x) D x  2

for x > 1 ;

F(x) D x

for jxj  1

and ı D 0:0025. For this example an analytic expression for (˛) can be derived. For oscillator i the parameter c is in (6) replaced by c i (ı) D 1  q i ı with qi being a random number generated by the normal distribution with expected value zero and standard deviation, so that the autonomous period is T0(i) D T0 (1 C ıq i ). As the forcing function of oscillator i we choose h D ˙ h i j (x j ) D ˙ x j with j ¤ i. The evolution of the dynamics with the oscillators starting in a uniformly distributed random phase has the form of an iteration map of the type (26) and is depicted in Fig. 8. It is observed that by the type of coupling the oscillators are slowed down considerably and that in the almost fully entrained state (t D 50T0 ) the inherently faster oscillators are slightly ahead in phase. Canards We return to the Van der Pol equation in the Lienardform (2–3) and apply a translation of the state variables x and y as well as the parameter a such that (x; y; a) D (1; 2/3; 1) moves to the origin (0; 0; 0): "

dx D y  F(x) ; dt

F(x) D 1/3x 3 C x 2 ; 0< " 1;

fı (˛)gmin < mT0  T < fı (˛)gmax :

(27)

Mapping of the phase in the above way is a well-known approach of modeling entrainment by the use of circle maps [51]. It is noted that here we go back further and relate this map to an oscillator given by a differential equation. If we consider a system of identical oscillators fx0 ( i (t)); y0 ( i (t))g, i D 1; 2; : : :, n with a forcing of oscillator i by the others of the form ı˙ h i j (x j ; y j ), we are in

dy D x C a : dt

(28) (29)

For 2 < a < 0 with a fixed and " ! 0 a relaxation oscillation is found as we analyzed in Sect. “Asymptotic Solution of the Van der Pol Oscillator” using the asymptotic method of Carrier and Lewis [8]. The computation was done for a D 1, but any value at this interval would have given the same result. For a < 2 or a > 0 a sta-

Relaxation Oscillations

Relaxation Oscillations, Figure 8 A system of coupled piecewise linear Van der Pol relaxation oscillators with autonomous period T0(i) D T0 (1 C ıqi ), i D 1; : : : ; 25. The phase of each oscillator runs from  12 T0 D  ln(3) to  12 T0 . The actual phase and period at times t D nT0 are depicted. The actual period Ti (n) is given by its deviation from T0 : ri D (Ti (n)  T0 )/ı. This expression compares with qi (for n D 0, the points are situated at the diagonal)

ble equilibrium turns up. In a small interval near either the point  2 or the point 0 the solution exhibits a drastic change. Eckhaus [16] analyzes that more explicitly near these values critical points exist, where in a small interval the solution rapidly changes from an equilibrium into a full grown relaxation oscillation. Near a D 0 we have the critical point ˛c (")g D 1/8" C    ;

(30)

where such a change takes place. The following stretching transformation reveals the continuous change of the limit

solution as a function of the parameter a: a D ˛c (") C  "3/2 exp(k 2 /") :

(31)

In Fig. 9 we sketch two periodic limit solutions with  having different signs. First notice that for  D o(1) a solution that is on the stable branch (x > 0) will keep following the unstable branch until the point where it meets the other stable branch (x D 2) and will jump then back to the stable branch where it started. For  > 0 we have a periodic solution that in the limit for  ! 1 transforms into an equilibrium solution (Fig. 9a). For  < 0 we have a pe-

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Relaxation Oscillations

Relaxation Oscillations, Figure 9 Two stages in the canard explosion with parameter values given by (31)

riodic solution having the shape of a duck that in the limit for  ! 1 takes the form of a fully developed relaxation oscillation (Fig. 9b). The phenomenon, that is revealed here, is called a “canard explosion”. It is basically due to the fact that a trajectory is very close to a stable branch and that at the moment the branch becomes unstable it keeps on going close to the unstable branch because the deviation from this branch does not grow rapidly, see also [24,60]. A similar phenomenon is met in “slow passage through bifurcation” [45]. Dynamical Systems Approach We start from the system (5) with vector functions x(t) and y(t) and consider the fast dynamics by making a transformation to the fast time scale  D t/" and keeping y D y (0) fixed: dx D f (x; y (0) ; 0) : d

(32)

In the dynamical systems approach trajectories of (5) with " D 0 exist, called fibers having y D y (0) for all t and with x given by (32). Let them move to the point (x (0) ; y (0) ) at the smooth manifold M (0) given by f (x; y; 0) D 0. Thus, the point (x (0) , y (0) ) at M (0) is a stable equilibrium of the fast system (32). The equilibria are the base points of the fibers. Then next at the manifold M (0) the slow dynamics is governed by dy D g(x; y; 0) with constraint dt

f (x; y; 0) D 0 : (33)

In the geometrical singular perturbation theory of Fenichel [19] a condition for the equilibrium (x (0) ; y (0) ) of (32) is formulated. Fenichel only considers parts of M (0) for which at all points (x (0) ; y (0) ) the Jacobian @ f (x; y; 0)/@x has eigenvalues with a real part that is bounded away from zero. This is called the hyperbolicity condition. It is noted that eigenvalues with positive real parts are allowed. Then the corresponding eigenvectors

span a subspace V (u) for which in the system (32) linearized at (x (0) ; y (0) ) all trajectories move away from the equilibrium. For (32) itself a manifold W (u) is defined with V (u) tangent to this manifold at (x (0) ; y (0) ). The manifold W (u) consists of the set of unstable fibers. Similarly the manifold W (s) is defined containing the set of stable fibers. It is proved that, for " positive and sufficiently small, near M (0) a slow manifold M (") exists satisfying (5). The fast dynamics is governed by the fibers keeping in mind that we move from one fiber to its neighboring one as described by the slow change of the fiber base point at M (") . When approximating asymptotically the solution, a transformation is preferred that differs slightly from the matched asymptotic expansions approach. We illustrate it with the following simple system with scalar functions x(t) and y(t): "

dx D yx; dt

dy D1: dt

(34)

Then for " D 0 we have fast fibers with y D constant moving towards the slow manifold M (0) : y D x, see Fig. 10. For small positive " the slow manifold is defined by M (") : y D x C ". The motion at this manifold (and the change of the base point of the actual fiber) is governed by dy/dt D 1. For the fast motion parallel to the fibers we make the transformation x D y  " C v, so that the new variable v represents the distance to the base point of the fiber. In the fast time scale we then obtain for the motion parallel to the fibers dv/ d D v. For more complex systems the expressions for M (") and for the solutions of the slow and fast systems take the form of power series expansions with respect to ". For more details we refer to [36,37]. For relaxation oscillations the hyperbolicity condition is not satisfied. This is easily verified from the Van der Pol Eq. (2) where M (0) : y D 1/3x 3  x, because at the fold points x D ˙1 the eigenvalue of the Eq. (2) linearized at these x values equals zero. To restore hyperbolicity a blow up technique [14] is used. It applies to the system (2) extended with the differential equation d"/dt D 0 for which the fold point (x; y; ") D (1; 2/3; 0) as well as (1; 2/3; 0) and their direct neighborhoods are put under a magni-

Relaxation Oscillations

Relaxation Oscillations, Figure 10 The dynamics of (34) for " D 0 (left) with the fast flow represented by the lines y is constant (fibers) and the slow dynamics by the line y D x being the manifold M(0) . For small positive " (right) the trajectories cross the fibers (given for " D 0) until they arrive at an exponentially small neighborhood of the slow manifold M(")

fying glass. We follow the exposition by Krupa and Szmolyan [38] and consider the neighborhood of a fold point situated at the origin for the following system in the fast time scale dx D y C x 2 C a(x; y; ") ; d d" D0; d

dy D " C b(x; y; ") ; d

(35) where the terms a(x; y; ") and b(x; y; ") can be neglected if we are sufficiently close to the origin, see Fig. 11a for the dynamics near the fold point. The blow up transformation (x; y; ") ! (x¯ ; y¯; "¯; r¯) takes the form x D r¯x¯ ;

y D r¯2 y¯ ;

" D r¯3 "¯ :

(36)

It is such that in R3 a ball B D S 2  [0; ] with sufficiently small is mapped onto R3 . By this transformation the new coordinates x¯ ; y¯ and "¯ indicate a point at a spherical surface S2 . In Fig. 11b a circle is depicted that corresponds with the fold point (¯r D 0) in a blow up. Inside the circle, where "¯ is positive and r¯ D 0, we are in S2 above the equator, while at the circle we are at the equator where "¯ D 0. Outside the circle we have "¯ D " D 0 but now with r¯ > 0. There transformed fibers of the limit system near the fold point are drawn. Note that the slow manifold consists of equilibria because of the fast time scale. The bold lines indicate the path of the orbit as it passes the fold point. Away from the equator the dynamics near the fold point is described by taking the fixed value "¯ D 1. We introduce a new time scale ¯ D r¯ and let r¯ ! 0. In that

way we arrive at the system dx¯ D  y¯ C x¯ 2 ; d¯

d y¯ D 1 ; d¯

(37)

having the same solution as (13). For more details we refer to Krupa and Szmolyan [38], who proceed with a similar analysis of a canard explosion. Furthermore, the method has been extended to relaxation oscillations in R3 [64], including canards, see [5,11,63]. In [75] the method is applied to a prey-predator system. Future Directions Over more than 75 years studies on relaxation oscillations have appeared in the literature. It is noticed that each time a new development takes place in physics or mathematics, relaxation oscillations turn out to be an interesting subject to be investigated with such a novel theory. In electronics we find the use of nonlinear devices starting with triodes in electric circuits [66]. In mathematics we see that relaxation oscillation is a rewarding subject to which many new theories can be applied. In chronological order we mention: Rigorous theories for estimating solutions of differential equations [9], singular perturbation theory and matched asymptotic expansions [8], nonstandard analysis [6], catastrophe theory [69], chaos theory [40], and geometrical singular perturbations [38]. The reason for this lies in the fact that a relaxation oscillation is a highly nonlinear phenomenon with internal mechanisms that are close to real life functions like adaptation by entrainment and threshold behavior in case of triggering. Possible new topics in the field of nonlinear oscillation can, in particular, be found in biology. Literature in

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Relaxation Oscillations, Figure 11 Dynamics near and in a fold point, see the text for a detailed description

which models of biological processes are formulated may function as a source of inspiration. In particular we mention Winfree [78] and Murray [47,48]. To be more explicit about new opportunities we bring up the following cases. The study of the dynamics of oscillations by circle maps or phase equations turned out to be very useful in particular for the description of entrainment of biological oscillators. In Sect. “Coupled Van der Pol-Type Oscillators” we made the connection between a differential equation model for state variables and the phase equations, see (22)–(26), see also [35]. However, the result only holds for the discontinuous solution as we see from (25), where the denominator of the integrand jumps over the value zero. How in the limit for " ! 0 we arrive at this result has not yet been worked out. We furthermore mention that spatial distribution of oscillators and the way they are coupled may give rise to interesting entrainment phenomena. In the first place there is the nearest neighbor coupling. Taking an appropriate limit in case of diffusive coupling we may model the dynamics by a nonlinear diffusion equation. Although studies in this direction exist, see e. g. [49], still a lot of work can be done. Furthermore, observations of rhythms in neural networks with long distance coupling show special forms of entrainment, see [65]. Mathematical analysis of such a network of coupled relaxation oscillators may help to understand more from these phenomena. Not only the distance can be varied also the type of coupling; we may consider excitatory and inhibitory forcing referring to the effect they have on an oscillator of the Bonhoeffer–Van der Pol-type, see [77]. Besides easy entrainment of coupled relaxation oscillators there is a second form of adaptation which is comparable with that of neural com-

puting. The coupling between two oscillators in a network may increase if they are entrained for a longer period of time and decrease if they are not. In this way the network is structured by the input and if the input changes the network will follow. The neural theory of cell assemblies [32] is based on such a mechanism known as plasticity of the network, see [22,59]. Effects due to a combination of long range coupling and plasticity may occur as well [79]. Studies on systems with complex interactions between many oscillators as we sketched above may take different directions. On one hand large-scale computer simulations may offer a wealth of results, but models that are more conceptual and are investigated (partly) analytically may give more insight into the fundamental properties of such systems. However, it is not easy to see at forehand that a study of the second type will lead to a meaningful result and is for that reason a more risky enterprise. Acknowledgment The author is grateful to Ed Veling (Technical University Delft, The Netherlands) for his valuable advice. Bibliography Primary Literature 1. Agüera y Arcas B, Fairhall AL, Bialek W (2003) Computation on a single neuron: Hodglin and Huxley revisited. Neural Comput 15:1715–1749 2. Andersen CM, Geer J (1982) Power series expansions for the frequency and period of the limit cycle of the Van der Pol equation. SIAM J Appl Math 42:678–693

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3. Bonhoeffer KF (1948) Activation of passive iron as a model for the excitation of nerve. J Gen Physiol 32:69–91 4. Braaksma B, Grasman J (1993) Critical dynamics of the Bonhoeffer–Van der Pol equation and its chaotic response to periodic stimulation. Physica D 68:265–280 5. Brons M, Krupa M, Wechselberger M (2006) Mixed mode oscillations due to the generalized canard phenomenon. Fields Inst Comm 49:39–63 6. Callot JL, Diener F, Diener M (1978) Le problème de la ‘chasse au canard’. CR Acad Sc Paris Sér A 286:1059–1061 7. Campbell SR, Wang DL (1998) Relaxation oscillators with time delay coupling. Physica D 111:151–178 8. Carrier GF, Lewis JA (1953) The relaxation oscillations of the Van der Pol oscillator. Adv Appl Mech 3:12–16 9. Cartwright M, Littlewood J (1947) On nonlinear differential equations of the second order II. Ann Math 48:472–494 10. Dadfar MB, Geer JF (1990) Resonances and power series solutions of the forced Van der Pol oscillator. SIAM J Appl Math 50:1496–1506 11. De Maesschalck P, Dumortier F (2005) Canard solutions at generic turning points. In: Dumortier F, Broer HW, Mawhin J, Vanderbauwhede A, Van Duyn Lunel S (eds) Proc Equadiff 2003. World Scientific, Singapore, pp 900–905 12. Deng B (2004) Food chain chaos with canard explosion. Chaos 14:1083–1092 13. Dorodnicyn AA (1947) Asymptotic solution of Van der Pol’s Equation. Akad Nauk SSSR Prikl Mat Mech 11:313–328, Russian (1962, Am Math Soc Transl Series 1 4:1–23) 14. Dumortier F (1993) Techniques in the theory of local bifurcations: Blow up, normal forms, nilpotent bifurcations, singular perturbations. In: Schlomiuk D (ed) Structures in bifurcations and periodic orbits of vector fields. Kluwer, Dordrecht, pp 19– 73 15. Eckhaus W (1973) Matched asymptotic expansions and singular perturbations. North-Holland Mathematics Studies 6. North Holland, Amsterdam 16. Eckhaus W (1983) Relaxation oscillations including a standard chase on French ducks. In: Verhulst F (ed) Asymptotic analysis II. Lecture Notes in Math, vol 985. Springer, Berlin, pp 449–494 17. Ermentrout GB (1985) Synchronization in a pool of mutually coupled oscillators with random frequencies. J Math Biol 22:1– 10 18. Ermentrout GB, Kopell N (1984) Frequency plateaus in a chain of weakly coupled oscillators I. SIAM J Math Anal 15:215–237 19. Fenichel N (1997) Geometrical singular perturbation theory for ordinary differential equations. J Differ Equ 31:53–98 20. Fife PC (1985) Understanding the patterns in the BZ reagent. J Stat Phys 39:687–703 21. FitzHugh R (1955) Mathematical models of threshold phenomena in the nerve membrane. Bull Math Biophys 17:257–278 22. Gerstner W, Kistler WM (2002) Spiking neuron models: Single neurons, populations, plasticity. Cambridge University Press, Cambridge 23. Ginoux JM, Rossetto B (2006) Differential geometry and mechanics applications to chaotic dynamical systems. Int J Bifurc Chaos 16:887–910 24. Gorelov GN, Sobolev VA (1992) Duck-trajectories in a thermal explosion problem. Appl Math Lett 5:3–6 25. Grasman J (1987) Asymptotic methods for relaxation oscillations and applications. Springer, New York 26. Grasman J, Jansen MJW (1979) Mutually synchronized relax-

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48. Murray JD (2003) Mathematical biology II: Spatial models and biomedical applications. Springer, New York 49. Nishiura Y, Mimura M (1989) Layer oscillations in reaction-diffusion systems. SIAM J Appl Math 49:481–514 50. O’Malley RE Jr (1991) Singular perturbation methods for ordinary differential equations. Springer, New York 51. Osipov CV, Kurths J (2001) Regular and chaotic phase synchronization of coupled circle maps. Phys Rev E 65:016216 52. Ponzo PJ, Wax N (1965) On certain relaxation oscillations: Asymptotic solutions. SIAM J Appl Math 13:740–766 53. Rinaldi S, Muratori S (1992) Slow-fast limit cycles in predatorprey models. Ecol Model 61:287–308 54. Rinaldi S, Muratori S, Kuznetsov YA (1993) Multiple attractors, catastrophes and chaos in seasonally perturbed predator-prey communities. Bull Math Biol 55:15–35 55. Robinson A (1966) Non-standard analysis. North-Holland, Amsterdam 56. Rössler OE, Wegmann K (1978) Chaos in the Zhabotinskii reaction. Nature 271:89–90 57. Rossoreanu C, Georgescu A, Giurgteanu N (2000) The FitzHugh–Nagumo model – bifurcation and dynamics. Kluwer, Dordrecht 58. Sari T (1996) Nonstandard perturbation theory of differential equations, at symposium on nonstandard analysis and its applications. ICMS, Edinburgh 59. Seliger P, Young SC, Tsimring LS (2002) Plasticity and learning in a network of coupled phase oscillators. Phys Rev E 65:041906 60. Shchepakina E, Sobolev V (2001) Integral manifolds, canards and black swans. Nonlinear Anal Ser A: Theory Meth 44:897– 908 61. Smale S (1967) Differentiable dynamical systems. Bull Am Math Soc 73:747–817 62. Strumiłło P, Strzelecki M (2006) Application of coupled neural oscillators for image texture segmentation and modeling of biological rhythms. Int J Appl Math Comput Sci 16:513–523 63. Szmolyan P, Wechselberger M (2001) Canards in R3 . J Differential Equ 177:419–453 64. Szmolyan P, Wechselberger M (2004) Relaxation oscillations in R3 . J Differential Equ 200:69–104 65. Traub RD, Whittington MA, Stanford IM, Jefferys JGR (1991) A mechanism for generation of long-range synchronous fast oscillators in the cortex. Nature 383:621–624

66. Van der Pol B (1926) On relaxation oscillations. Phil Mag 2:978– 992 67. Van der Pol B (1940) Biological rhythms considered as relaxation oscillations. Acta Med Scand Suppl 108:76–87 68. Van der Pol B, Van der Mark J (1927) Frequency demultiplication. Nature 120:363–364 69. Varian HR (1979) Catastrophe theory and the business cycle. Econ Inq 17:14–28 70. Vasconcelos GL (1996) First-order phase transition in a model for earthquakes. Phys Rev Lett 76:4865–4868 71. Vatta F (1979) On the stick-slip phenomenon. Mech Res Commun 6:203–208 72. Verhulst F (2005) Quenching of self-excited vibrations. J Engin Math 53:349–358 73. Verhulst F (2007) Periodic solutions and slow manifolds. Int J Bifurc Chaos 17:2533–2540 74. Verhulst F, Abadi (2005) Autoparametric resonance of relaxation oscillations. Z Angew Math Mech 85:122–131 75. Verhulst F, Bakri T (2007) The dynamics of slow manifolds. J Indon Math Soc 13:1–10 76. Wang DL (1999) Relaxation oscillators and networks. In: Webster JG (ed) Wiley encyclopedia of electrical and electronics engineering, vol 18. Wiley, Malden, pp 396–405 77. Wang DL, Terman D (1995) Locally excitatory globally inhibitory oscillator networks. IEEE Trans Neural Netw 6:283–286 78. Winfree AT (2000) The geometry of biological time, 2nd edn. Springer, New York 79. Womelsdorf T, Schoffelen JM, Oosterveld R, Singer W, Desimore R, Engel AK, Fries P (2007) Modulation of neural interactions through neural synchronization. Science 316:1609–1612 80. Zvonkin AK, Shubin MA (1984) Nonstandard analysis and singular perturbations of ordinary differential equations. Russ Math Surveys 39:69–131

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Robotic Networks, Distributed Algorithms for

Robotic Networks, Distributed Algorithms for FRANCESCO BULLO1 , JORGE CORTÉS2 , SONIA MARTÍNEZ2 1 Department of Mechanical Engineering, University of California, Santa Barbara, USA 2 Department of Mechanical and Aerospace Engineering, University of California, San Diego, USA

Article Outline Glossary Definition of the Subject Introduction Distributed Algorithms on Networks of Processors Distributed Algorithms for Robotic Networks Bibliographical Notes Future Directions Acknowledgments Bibliography Glossary Cooperative control In recent years, the study of groups of robots and multi-agent systems has received a lot of attention. This interest has been driven by the envisioned applications of these systems in scientific and commercial domains. From a systems and control theoretic perspective, the challenges in cooperative control revolve around the analysis and design of distributed coordination algorithms that integrate the individual capabilities of the agents to achieve a desired coordination task. Distributed algorithm In a network composed of multiple agents, a coordination algorithm specifies a set of instructions for each agent that prescribe what to sense, what to communicate and to whom, how to process the information received, and how to move and interact with the environment. In order to be scalable, coordination algorithms need to rely as much as possible on local interactions between neighboring agents. Complexity measures Coordination algorithms are designed to enable networks of agents achieve a desired task. Since different algorithms can be designed to achieve the same task, performance metrics are necessary to classify them. Complexity measures provide a way to characterize the properties of coordination algorithms such as completion time, cost of communication, energy consumption, and memory requirements.

Averaging algorithms Distributed coordination algorithms that perform weighted averages of the information received from neighboring agents are called averaging algorithms. Under suitable connectivity assumptions on the communication topology, averaging algorithms achieve agreement, i. e., the state of all agents approaches the same value. In certain cases, the agreement value can be explicitly determined as a function of the initial state of all agents. Leader election In leader election problems, the objective of a network of processors is to elect a leader. All processors have a variable “leader” initially set to unknown. The leader-election task is solved when only one processor has set the variable “leader” to true, and all other processors have set it to false. LCR algorithm The classic Le Lann–Chang–Roberts (LCR) algorithm solves the leader election task on a static network with the ring communication topology. Initially, each agent transmits its unique identifier to its neighbors. At each communication round, each agent compares the largest identifier received from other agents with its own identifier. If the received identifier is larger than its own, the agent declares itself a non-leader, and transmits it in the next communication round to its neighbors. If the received identifier is smaller than its own, the agent does nothing. Finally, if the received identifier is equal to its own, it declares itself a leader. The LCR algorithm achieves leader election with linear time complexity and quadratic total communication complexity, respectively. Agree-and-pursue algorithm Coordination algorithms for robotic networks combine the features of distributed algorithms for networks of processors with the sensing and control capabilities of the robots. The agree-and-pursue motion coordination algorithm is an example of this fusion. Multiple robotic agents moving on a circle seek to agree on a common direction of motion while at the same achieving an equallyspaced distribution along the circle. The agree-andpursue algorithm achieves both tasks combining ideas from leader election on a changing communication topology with basic control primitives such as “follow your closest neighbor in your direction of motion.” Definition of the Subject The study of distributed algorithms for robotic networks is motivated by the recent emergence of low-power, highlyautonomous devices equipped with sensing, communication, processing, and control capabilities. In the near future, cooperative robotic sensor networks will perform

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critical tasks in disaster recovery, homeland security, and environmental monitoring. Such networks will require efficient and robust distributed algorithms with guaranteed quality-of-service. In order to design coordination algorithms with these desirable capabilities, it is necessary to develop new frameworks to design and formalize the operation of robotic networks and novel tools to analyze their behavior. Introduction Distributed algorithms are a classic subject of study for networks composed of individual processors with communication capabilities. Within the automata-theoretic literature, important research topics on distributed algorithms include the introduction of mathematical models and precise specifications for their behavior, the formal assessment of their correctness, and the characterization of their complexity. Robotic networks have distinctive features that make them unique when compared with networks of static processors. These features include the operation under ad-hoc dynamically changing communication topologies and the complexity that results from the combination of continuous- and discrete-time dynamics. The spatiallydistributed character of robotic networks and their dynamic interaction with the environment make the classic study of distributed algorithms, typically restricted to static networks, not directly applicable. This chapter brings together distributed algorithms for networks of processors and for robotic networks. The first part of the chapter is devoted to a formal discussion about distributed algorithms for a synchronous network of processors. This treatment serves as a brief introduction to important issues considered in the literature on distributed algorithms such as network evolution, task completion, and complexity notions. To illustrate the ideas, we consider the classic Le Lann–Chang–Roberts (LCR) algorithm, which solves the leader election task on a static network with the ring communication topology. Next, we present a class of distributed algorithms called averaging algorithms, where each processor computes a weighted average of the messages received from its neighbors. These algorithms can be described as linear dynamical systems, and their correctness and complexity analysis has nice connections with the fields of linear algebra and Markov chains. The second part of the chapter presents a formal model for robotic networks that explicitly takes into account communication, sensing, control, and processing. The notions of time, communication, and space complexity in-

troduced here allow us to characterize the performance of coordination algorithms, and rigorously compare the performance of one algorithm versus another. In general, the computation of these notions is a complex problem that requires a combination of tools from dynamical systems, control theory, linear algebra, and distributed algorithms. We illustrate these concepts in three different scenarios: the agree-and-pursue algorithm for a group of robots moving on a circle, aggregation algorithms that steer robots to a common location, and deployment algorithms that make robots optimally cover a region of interest. In each case, we report results on the complexity associated to the achievement of the desired task. The chapter ends with a discussion about future research directions. Distributed Algorithms on Networks of Processors Here we introduce a synchronous network as a group of processors with the ability to exchange messages and perform local computations. What we present is a basic classic model studied extensively in the distributed algorithms literature. Our treatment is directly adopted with minor variations from the texts [57] and [74]. Physical Components and Computational Models Loosely speaking, a synchronous network is a group of processors, or nodes, that possess a local state, exchange messages among neighbors, and compute an update to their local state based on the received messages. Each processor alternates the two tasks of exchanging messages with its neighboring processors and of performing a computation step. Let us begin by providing some basic definitions. A directed graph [22], in short digraph, of order n is a pair G D (V; E) where V is a set with n elements called vertices (or sometimes nodes) and E is a set of ordered pair of vertices called edges. In other words, E  V  V. We call V and E the vertex set and edge set, respectively. For u; v 2 V, the ordered pair (u; v) denotes an edge from u to v. The vertex u is called an in-neighbor of v, and v is called an out-neighbor of u. A directed path in a digraph is an ordered sequence of vertices such that any two consecutive vertices in the sequence are a directed edge of the digraph. A vertex of a digraph is globally reachable if it can be reached from any other vertex by traversing a directed path. A digraph is strongly connected if every vertex is globally reachable. A cycle in a digraph is a non-trivial directed path that starts and ends at the same vertex. A digraph is acyclic if it contains no cycles. In an acyclic graph, every vertex with

Robotic Networks, Distributed Algorithms for

Robotic Networks, Distributed Algorithms for, Figure 1 From left to right, directed tree, chain, and ring digraphs

no in-neighbors is named source, and every vertex with no out-neighbors is named sink. A directed tree is an acyclic digraph with the following property: there exists a vertex, called the root, such that any other vertex of the digraph can be reached by one and only one path starting at the root. A directed spanning tree, or simply a spanning tree, of a digraph is a subgraph that is a directed tree and has the same vertex set as the digraph. A directed chain is a directed tree with exactly one source and one sink. A directed ring digraph is the cycle obtained by adding to the edge set of a chain a new edge from its sink to its source. Figure 1 illustrates these notions. The physical component of a synchronous network S is a digraph (I; Ecmm ), where (i) I D f1; : : : ; ng is called the set of unique identifiers (UIDs), and (ii) Ecmm is a set of directed edges over the vertices f1; : : : ; ng, called the communication links. The set Ecmm models the topology of the communication service among the nodes: for i; j 2 f1; : : : ; ng, processor i can send a message to processor j if the directed edge (i; j) is present in Ecmm . Note that, unlike the standard treatments in [57] and [74], we do not assume the digraph to be strongly connected; the required connectivity assumption is specified on a case by case basis. Next, we discuss the state and the algorithms that each processor possesses and executes, respectively. By convention, we let the superscript [i] denote any quantity associated with the node i. A distributed algorithm DA for a network S consists of the sets: (i)

A, a set containing the null element, called the communication alphabet; elements of A are called messages;

(ii) W [i] , i 2 I, called the processor state sets; (iii) W0[i]  W [i] , i 2 I, sets of allowable initial values;

and of the maps: (i) msg[i] : W [i]  I ! A, i 2 I, called message-generation functions; (ii) stf [i] : W [i] An ! W [i] , i 2 I, called state-transition functions. If W [i] D W, msg[i] D msg, and stf [i] D stf for all i 2 I, then DA is said to be uniform and is described by a tuple (A; W; fW0[i] g i2I ; msg; stf ). Now, with all elements in place, we can explain in more detail how a synchronous network executes a distributed algorithm. The state of processor i is a variable w [i] 2 W [i] , initially set equal to an allowable value in W0[i] . At each time instant ` 2 Z0 , processor i sends to each of its out-neighbors j in the communication digraph (I; Ecmm ) a message (possibly the null message) computed by applying the message-generation function msg[i] to the current values of its state w [i] and to the identity j. Subsequently, but still at time instant ` 2 Z0 , processor i updates the value of its state w [i] by applying the statetransition function stf[i] to the current value of its state w [i] and to the messages it receives from its in-neighbors. Note that, at each round, the first step is transmission and the second one is computation. We conclude this section with two sets of remarks. We first discuss some aspects of our communication model that have a large impact on subsequent development. We then collect a few general comments about distributed algorithms on networks. Remark 1 (Aspects of the communication model) (i)

The network S and the algorithm DA are referred to as synchronous because the communications between all processors takes place at the same time for all processors.

(ii) Communication is modeled as a so-called “point to point” service: a processor can specify different mes-

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sages for different out-neighbors and knows the processor identity corresponding to any incoming message. (iii) Information is exchanged between processors as messages, i. e., elements of the alphabet A; the message null indicates no communication. Messages might encode logical expressions such as true and false, or finite-resolution quantized representations of integer and real numbers. (iv) In some uniform algorithms, the messages between processors are the processors’ states. In such cases, the corresponding communication alphabet is A D W [fnullg and the message generation function msgstd (w; j) D w is referred to as the standard message-generation function. Remark 2 (Advanced topics: Control structures and failures) (i) Processors in a network have only partial information about the network topology. In general, each processor only knows its own UID, and the UID of its in- and out-neighbors. Sometimes we assume that the processor knows the network diameter. In some cases [74], actively running networks might depend upon “control structures,” i. e., structures that are computed at initial time and are exploited in subsequent algorithms. For example, routing tables might be computed for routing problems, “leader” processors might be elected and tree structures might be computed and represented in a distributed manner for various tasks, e. g., coloring or maximal independent set problems. We present some sample algorithms to compute these structures below. (ii) A key issue in the study of distributed algorithms is the possible occurrence of failures. A network might experience intermittent or permanent communication failures: along given edges a null message or an arbitrary message might be delivered instead of the intended value. Alternatively, a network might experience various types of processor failures: a processor might transmit only null messages (i. e., the msg function returns null always), a processor might quit updating its state (i. e., the stf function neglects incoming messages and returns the current state value), or a processor might implement arbitrarily modified msg and stf functions. The latter situation, in which completely arbitrary and possibly malicious behavior is adopted by faulty nodes, is referred to as a Byzantine failure in the distributed algorithms literature.

Complexity Notions Here we begin our analysis of the performance of distributed algorithms. We introduce a notion of algorithm completion and, in turn, we introduce the classic notions of time, space, and communication complexity. We say that an algorithm terminates when only null messages are transmitted and all processors states become constants. Remark 3 (Alternative termination notions) (i) In the interest of simplicity, we have defined evolutions to be unbounded in time and we do not explicitly require algorithms to actually have termination conditions, i. e., to be able to detect when termination takes place. (ii) It is also possible to define the termination time as the first instant when a given problem or task is achieved, independently of the fact that the algorithm might continue to transmit data subsequently. The notion of time complexity measures the time required by a distributed algorithm to terminate. More specifically, the (worst-case) time complexity of a distributed algorithm DA on a network S, denoted TC(DA; S), is the maximum number of rounds required by the execution of DA on S among all allowable initial states until termination. Next, we quantify memory and communication requirements of distributed algorithms. From an information theory viewpoint [35], the information content of a memory variable or of a message is properly measured in bits. On the other hand, it is convenient to use the alternative notions of “basic memory unit” and “basic message.” It is customary [74] to assume that a “basic memory unit” or a “basic message” contains log(n) bits so that, for example, the information content of a robot identifier i 2 f1; : : : ; ng is log(n) bits or, equivalently, one “basic memory unit.” Note that elements of the processor state set W or of the alphabet set A might amount to multiple basic memory units or basic messages; the null message has zero cost. Unless specified otherwise, the following definitions and examples are stated in terms of basic memory unit and basic messages. (i) The (worst-case) space complexity of a distributed algorithm DA on a network S, denoted by SC(DA; S), is the maximum number of basic memory units required by a processor executing the DA on S among all processors and among all allowable initial states until termination. (ii) The (worst-case) communication complexity of a distributed algorithm DA on a network S, denoted by

Robotic Networks, Distributed Algorithms for

CC(DA; S), is the maximum number of basic messages transmitted over the entire network during the execution of DA among all allowable initial states until termination. Remark 4 (Space complexity conventions) By convention, each processor knows its identity, i. e., it requires log(n) bits to represent its unique identifier in a set with n distinct elements. We do not count this cost in the space complexity of an algorithm. We conclude this section by discussing ways of quantifying time, space, and communication complexity. The idea, borrowed from combinatorial optimization, is to adopt asymptotic “order of magnitude” measures. Formally, complexity bounds will be expressed with respect to the Bachman–Laundau symbols O, ˝, and . Let us be more specific. In the following definitions, f denotes a function from N to R. We say that an algorithm has time complexity of order ˝( f (n)) over some network if, for all n, there exists a network of order n and initial processor values such that the time complexity of the algorithm is greater than a constant factor times f (n). (ii) We say that an algorithm has time complexity of order O( f (n)) over arbitrary networks if, for all n, for all networks of order n and for all initial processor values the time complexity of the algorithm is lower than a constant factor times f (n). (iii) We say that an algorithm has time complexity of order ( f (n)) if its time complexity is of order ˝( f (n)) over some network and O( f (n)) over arbitrary networks at the same time.

lower bound statement is “existential” rather than “global,” in the sense that the bound does not hold for all graphs. As discussed in [74], it is possible to define also “global” lower bounds, i. e., lower bounds over all graphs, or lower bounds over specified classes of graphs. (ii) The complexity notions introduced above focus on the worst-case situation. It is also possible to define expected or average complexity notions, where one might be interested in characterizing, for example, the average number of rounds required or the average number of basic messages transmitted over the entire network during the execution of an algorithm among all allowable initial states until termination. (iii) It is possible to define complexity notions for problems, rather than algorithms, by considering, for example, the worst-case optimal performance among all algorithms that solve the given problem, or over classes of algorithms or classes of graphs.

(i)

We use similar conventions for space and communication complexity. In many cases the complexity of an algorithm will typically depend upon the number of nodes of the network. It is therefore useful to present a few simple facts about these functions now. Over arbitrary digraphs S D (I; Ecmm ) of order n, we have diam(S) 2 (n);

jEcmm (S)j 2 (n2 )

and

radius(v; S) 2 (diam(S)) ;

where v is any node of S. Remark 5 Numerous variations of these definitions are possible. Even though we will not pursue them here, let us provide some pointers. (i)

In the definition of lower bound, consider the logic quantifier describing the role of the network. The

Leader Election We formulate here a classical problem in distributed networks and summarize its complexity measures. Problem 6 (Leader election) Assume that all processors of a network have a state variable, say leader, initially set to unknown. We say that a leader is elected when one and only one processor has the state variable set to true and all others have it set to false. Elect a leader. This is a task that is a bit more global in nature. We display here a solution that requires individual processors to know the diameter of the network, denoted by diam(S), or an upper bound on it. [Informal description] At each communication round, each agent sends to all its neighbors the maximum UID it has received up to that time. This is repeated for diam(S) rounds. At the last round, each agent compares the maximum received UID with its own, and declares itself a leader if they coincide, or a non-leader otherwise. The algorithm is called FLOODMAX: the maximum UID in the network is transmitted to other agents in an incremental fashion. At the first communication round, agents that are neighbors of the agent with the maximum UID receive the message from it. At the next communication round, the neighbors of these agents receive the message with the maximum UID. This process goes on for diam(S) rounds to ensure that every agent receives the maximum UID. Note that there are networks for which all agents receive

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the message with the maximum UID in fewer communication rounds than diam(S). The algorithm is formally stated as follows. Synchronous Network: S D (f1; : : : ; ng; Ecmm ) Distributed Algorithm: FLOODMAX Alphabet: A D f1; : : : ; ng [fnullg Processor State: w D (my-id, max-id, leader, round), where my-id 2 f1; : : : ; ng, initially: my-id[i] D i for all i max-id 2 f1; : : : ; ng, initially: max-id[i] D i for all i leader 2 ffalse; true; unknowng, initially: leader[i] D unknown for all i round 2 f0; 1; : : : ; diam(S)g, initially: round[i] D 0 for all i function msg(w; i) 1: if {round < diam(S)} then 2: return max-id 3: else 4: return null function stf(w; y) 1: new-id := maxfmax-id; largest identifier in yg 2: case 3: round < diam(S): new-lead :D unknown 4: round D diam(S) AND max-id D my-id: new-lead :D true 5: round D diam(S) AND max-id > my-id: new-lead :D false 6: return (my-id, new-id, new-lead, round C 1) Figure 2 shows an execution of the FLOODMAX algorithm. The properties of the algorithm are characterized in the following lemma. A complete analysis of this algo-

rithm, including modifications to improve the communication complexity, is discussed in [Section 4.1 in 1]. Lemma 7 (Complexity upper bounds for the FLOODMAX algorithm) For a network S containing a spanning tree, the FLOODMAX algorithm has communication complexity in O(diam(S)jEcmm j), time complexity equal to diam(S), and space complexity in (1). A simplification of the FLOODMAX algorithm leads to the Le Lann–Chang–Roberts (LCR) algorithm for leader election in rings, see [Chapter 3.3 in 1], that we describe next.1 . The LCR algorithm runs on a ring digraph and does not require the agents to know the diameter of the network. [Informal description] At each communication round, if the agent receives from its in-neighbor a UID that is larger than the UIDs received earlier, then the agent records the received UID and forwards it to the out-neighbor during the following communication round. (Agents do not record the number of communication rounds.) When the agent with the maximum UID receives its own UID from a neighbor, it declares itself the leader. The algorithm is formally stated as follows. Synchronous Network: ring digraph Distributed Algorithm: LCR Alphabet: A D f1; : : : ; ng [fnullg Processor State: w D (my-id, max-id, leader, snd-flag), where my-id 2 f1; : : : ; ng, initially: my-id[i] D i for all i max-id 2 f1; : : : ; ng, initially: max-id[i] D i for all i leader 2 ftrue; false; unknowng, initially: leader[i] D unknown for all i snd-flag 2 ftrue; falseg, initially: snd-flag [i] D true for all i 1 Note that the description of the LCR algorithm given here is slightly different from the classic one as presented in [57].

Robotic Networks, Distributed Algorithms for, Figure 2 Execution of the FLOODMAX algorithm. The diameter of the network is 4. In the leftmost frame, the agent with the maximum UID is colored in red. After 4 communication rounds, its message has been received by all agents

Robotic Networks, Distributed Algorithms for

Robotic Networks, Distributed Algorithms for, Figure 3 Execution of the LCR algorithm. In the leftmost frame, the agent with the maximum UID is colored in red. After 5 communication rounds, this agent receives its own UID from its in-neighbor and declares itself the leader

function msg(w; i) 1: if {snd-flag D true} then 2: return max-id 3: else 4: return null function stf(w; y) 1: case 2: (y contains only null msgs) OR (largest identifier in y < my-id): 3: new-id :D max-id 4: new-lead :D leader 5: new-snd-flag :D false 6: (largest identifier in y D my-id): 7: new-id :D max-id 8: new-lead :D true 9: new-snd-flag :D false 10: (largest identifier in y > my-id): 11: new-id :D largest identifier in y 12: new-lead :D false 13: new-snd-flag :D true 14: return (my-id, new-id, new-lead, new-snd-flag) Figure 3 shows an execution of the LCR algorithm. The properties of the LCR algorithm can be characterized as follows [57]. Lemma 8 (Complexity upper bounds for the LCR algorithm) For a ring network S of order n, the LCR algorithm has communication complexity in (n2 ), time complexity equal to n, and space complexity in (1). Averaging Algorithms This section provides a brief introduction to a special class of distributed algorithms called averaging algorithms. The synchronous version of averaging algorithms can be modeled within the framework of synchronous networks. In an averaging algorithm, each processor updates its state by

computing a weighted linear combination of the state of its neighbors. Computing linear combinations of the initial states of the processors is one the most basic computations that a network can implement. Averaging algorithms find application in optimization, distributed decision-making, e. g., collective synchronization, and have a long and rich history, see e. g., [29,30,45,65,97,98]. The richness comes from the vivid analogies with physical processes of diffusion, with Markov chain models, and the theory of positive matrices developed by Perron and Frobenius, see e. g., [19,20,44]. Averaging algorithms are defined by stochastic matrices. For completeness, let us recall some basic linear algebra definitions. A matrix A 2 Rnn with entries aij , i; j 2 f1; : : : ; ng, is (i)

nonnegative (resp., positive) if all its entries are nonnegative (resp., positive); (ii) row-stochastic (or stochastic for brevity) if it is nonP negative and njD1 a i j D 1, for all i 2 f1; : : : ; ng; in other words, A is row-stochastic if A1 n D 1n ;

where 1n D (1; : : : ; 1)T 2 Rn . (iii) doubly stochastic if it is row-stochastic and columnstochastic, where we say that A is column-stochastic if 1Tn A D 1Tn ; (iv) irreducible if, for any nontrivial partition J [ K of the index set f1; : : : ; ng, there exists j 2 J and k 2 K such that a jk ¤ 0. We are now ready to introduce the class of averaging algorithms. The averaging algorithm associated to a sequence of stochastic matrices fF(`) j ` 2 Z0 g Rnn is the discrete-time dynamical system w(` C 1) D F(`)  w(`) ;

` 2 Z0 :

(1)

In the literature, averaging algorithms are also often referred to as agreement algorithms or as consensus algorithms.

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Averaging algorithms are naturally associated with weighted digraphs, i. e., digraphs whose edges have weights. More precisely, a weighted digraph is a triplet G D (V; E; A) where V D fv1 ; : : : ; v n g and E are a dinn is a weighted adjacency magraph and where A 2 R0 trix with the following properties: for i; j 2 f1; : : : ; ng, the entry a i j > 0 if (v i ; v j ) is an edge of G, and a i j D 0 otherwise. In other words, the scalars aij , for all (v i ; v j ) 2 E, are a set of weights for the edges of G. Note that edge set is uniquely determined by the weighted adjacency matrix and it can be therefore omitted. The weighted out-degree matrix Dout (G) and the weighted in-degree matrix Din (G) are the diagonal matrices defined by Dout (G) D diag (A1 n ) ; and   Din (G) D diag AT 1 n : The weighted digraph G is weight-balanced if Dout (G) D Din (G). Given a nonnegative n  n matrix A, its associated weighted digraph is the weighted digraph with nodes f1; : : : ; ng, and weighted adjacency matrix A. The unweighted version of this weighted digraph is called the associated digraph. The following statements can be proven: (i)

if A is stochastic, then its associated digraph has weighted out-degree matrix equal to I n ; (ii) if A is doubly stochastic, then its associated weighted digraph is weight-balanced and additionally both indegree and out-degree matrices are equal to I n ; (iii) A is irreducible if and only if its associated weighted digraph is strongly connected. Next, we characterize the convergence properties of averaging algorithms. Let us introduce a useful property of collections of stochastic matrices. Given ˛ 2 ]0; 1], the set of non-degenerate matrices with respect to ˛ consists of all stochastic matrices F with entries f ij , for i; j 2 f1; : : : ; ng, satisfying f i i 2 [˛; 1] ;

and f i j 2 f0g [[˛; 1] for j ¤ i :

Additionally, the sequence of stochastic matrices fF(`) j ` 2 Z0 g is non-degenerate if these exists ˛ 2 ]0; 1] such that F(`) is non-degenerate with respect to ˛ for all ` 2 Z0 . We now state the following convergence result from [65]. Theorem 9 (Convergence for time-dependent stochastic matrices) Let fF(`) j ` 2 Z0 g Rnn be a non-degenerate sequence of stochastic matrices. For ` 2 Z0 , let G(`) be the unweighted digraph associated to F(`). The following statements are equivalent:

(i) the set diag(Rn ) is uniformly globally attractive for the associated averaging algorithm, that is, every evolution of the averaging algorithm at any time `0 , approaches the set diag(Rn ) in the following time-uniform manner: for all `0 2 Z0 , for all w0 2 Rn , and for all neighborhoods W of diag(Rn ), there exists a single 0 2 Z0 such that the evolution w : [`0 ; C1[! Rn defined by w(`0 ) D w0 , takes value in W for all times `  `0 C 0 . (ii) there exists a duration ı 2 N such that, for all ` 2 Z0 , the digraph G(` C 1) [    [ G(` C ı) contains a globally reachable vertex. Distributed Algorithms for Robotic Networks This section describes models and algorithms for groups of robots that process information, sense, communicate, and move. We refer to such systems as robotic networks. In this section we review and survey a few modeling and algorithmic topics in robotic coordination; earlier versions of this material were originally presented in [15,59,60,61]. The section is organized as follows. First, we present the physical components of a network, that is, the mobile robots and the communication service connecting them. We then present the notion of control and communication law, and how a law is executed by a robotic network. We then discuss complexity notions for robotic networks. As an example of these notions, we introduce a simple law, called the agree-and-pursue law, which combines ideas from leader election algorithms and from cyclic pursuit (i. e., a game in which robots chase each other in a circular environment). We then consider in some detail algorithms for two basic motion coordination tasks, namely aggregation and deployment. We briefly formalize these problems and provide some basic algorithms for these two tasks. Robotic Networks and Complexity The global behavior of a robotic network arises from the combination of the local actions taken by its members. Each robot in the network can perform a few basic tasks such as sensing, communicating, processing information, and moving according to it. The many ways in which these capabilities can be integrated make a robotic network a versatile and, at the same time, complex system. The following robotic network model provides the framework to formalize, analyze, and compare distinct distributed behaviors.

Robotic Networks, Distributed Algorithms for

We consider uniform networks of robotic agents defined by a tuple S D (I; R; Ecmm ) consisting of a set of unique identifiers I D f1; : : : ; ng, a collection of control systems R D fR[i] g i2I , with R[i] D (X; U; X 0 ; f ), and a map Ecmm from X n to the subsets of I  I called the communication edge map. Here, (X; U; X0 ; f ) is a control system with state space X Rd , input space U, set of allowable initial states X 0 X, and system dynamics f : X  U ! X. An edge between two identifiers in Ecmm implies the ability of the corresponding two robots to exchange messages. A control and communication law for S consists of the sets: (i) A, called the communication language, whose elements are called messages; (ii) W, set of values of some processor variables w [i] 2 W, i 2 I, and W0  W, subset of allowable initial values. These sets correspond to the capability of robots to allocate additional variables and store sensor or communication data; and the maps: (iii) msg : X  W  I ! A, called message-generation function; (iv) stf : X  W  An ! W, called state-transition function; (v) ctl : X  W  An ! U, called control function. To implement a control and communication law each robot performs the following sequence or cycle of actions. At each instant ` 2 Z0 , each robot i communicates to each robot j such that (i; j) belongs to Ecmm (x [1] ; : : : ; x [n] ). Each robot i sends a message computed by applying the message-generation function to the current values of x [i] and w [i] . After a negligible period of time, robot i resets the value of its logic variables w [i] by applying the state-transition function to the current value of w [i] , and to the messages y [i] (`) received at `. Between communication instants, i. e., for t 2 [`; ` C 1), robot i applies a control action computed by applying the control function to its state at the last sample time x [i] (`), the current value of w [i] , and to the messages y [i] (`) received at `. Remark 10 (Algorithm properties and congestion models) (i) In our present definition, all robots are identical and implement the same algorithm; in this sense the control and communication law is called uniform (or anonymous). If W D W0 D ;, then the control and communication law is static (or memoryless) and no state-transition function is defined. It is also possible for a law to be time-independent if the three relevant

maps do not depend on time. Finally, let us also remark that this is a synchronous model in which all robots share a common clock. (ii) Communication and physical congestion affect the performance of robotic networks. These effects can be modeled by characterizing how the network parameters vary as the number of robots becomes larger. For example, in an ad hoc networks with n uniformly randomly placed nodes, it is known [42] that the maximum-throughput communication range r(n) of each node decreases as the density of nodes increases; in d dimensions the appropriate scaling law is r(n) 2 ((log(n)/n)1/d ). As a second example, it is reasonable to assume that, as the number of robots increase, so should the area available for their motion. An alternative convenient approach is the one taken by [88], where robots’ safety zones decrease with decreasing robots’ speed. This suggests that, in a fixed environment, individual nodes of a large ensemble have to move at a speed decreasing with n, and in particular, at a speed proportional to n1/d . Next, we establish the notion of coordination task and of task achievement by a robotic network. Let S be a robotic network and let W be a set. A coordination task for S is a map T : X n  W n ! ftrue; falseg. If W is a singleton, then the coordination task is said to be static and can be described by a map T : X n ! ftrue; falseg. Additionally, let CC a control and communication law for S. (i) The law CC is compatible with the task T : X n  W n ! ftrue; falseg if its processor state take values in W , that is, if W [i] D W , for all i 2 I. (ii) The law CC achieves the task T if it is compatible with it and if, for all initial conditions x0[i] 2 X0 and w0[i] 2 W0 , i 2 I, there exists T 2 Z0 such that the network evolution ` 7! (x(`); w(`)) has the property that T (x(`); w(`)) D true for all `  T. In control-theoretic terms, achieving a task means establishing a convergence or stability result. Beside this key objective, one might be interested in efficiency as measured by required communication service, required control energy or by speed of completion. We focus on the latter notion. (i) The (worst-case) time complexity to achieve T with CC from (x0 ; w0 ) 2 X 0n  W0n is TC (T ; CC ; x0 ; w0 ) D inff` j T (x(k); w(k)) D true ; for all k  `g ;

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where ` 7! (x(`); w(`)) is the evolution of (S; CC ) from the initial condition (x0 ; w0 ); (ii) The (worst-case) time complexity to achieve T with CC is ˚ TC (T ; CC ) D sup TC (T ; CC ; x0 ; w0 ) j

n

(x0 ; w0 ) 2 X0n  W0

:

Some ideas on how to define meaningful notions of space and communication complexity are discussed in [59]. In the following discussion, we describe three coordination algorithms, which have been cast into this modeling framework and whose time complexity properties have been analyzed. Agree-and-Pursue Algorithm We begin our list of distributed algorithms with a simple law that is related to leader election algorithms, see Sect. “Leader Election”, and to cyclic pursuit algorithms as studied in the control literature. Despite the apparent simplicity, this example is remarkable in that it combines a leader election task (in the processor states) with a uniform robotic deployment task (in the physical state), arguably two of the most basic tasks in distributed algorithms and cooperative control, respectively. We consider n robots f [1] ; : : : ; [n] g in S1 , moving along on the unit circle with angular velocity equal to the control input. Each robot is described by the tuple (S1 ; [umax ; umax ]; S1 ; (0; e)), where e is the vector field on S1 describing unit-speed counterclockwise rotation. We assume that each robot can sense its own position and can communicate to any other robot within distance r along the circle. These data define the uniform robotic network Scircle . [Informal description] The processor state consists of dir (the robot’s direction of motion) taking values in fc; ccg (meaning clockwise and counterclockwise) and max-id (the largest UID received by the robot, initially set to the robot’s UID) taking values in I. At each communication round, each robot transmits its position and its processor state. Among the messages received from the robots moving towards its position, each robot picks the message with the largest value of max-id. If this value is larger than its own value, the agent resets its processor state with the selected message. Between communication rounds, each robot moves in the clockwise or counterclockwise direction depending on whether its processor state dir is c or cc. Each

robot moves kprop times the distance to the immediately next neighbor in the chosen direction, or, if no neighbors are detected, kprop times the communication range r. For this network and this law there are two tasks of interest. First, we define the direction agreement task Tdir : (S1 )n  W n ! ftrue; falseg by ( Tdir ( ; w) D

true;

if dir[1] D    D dir[n] ;

false;

otherwise;

where D ( [1] ; : : : ; [n] ), w D (w [1] ; : : : ; w [n] ), and w [i] D (dir[i] ; max-id[i] ), for i 2 I. Furthermore, for " > 0, we define the static equidistance task T"-eqdstnc : (S1 )n ! ftrue; falseg to be true if and only if ˇ    ˇ ˇ min distc [i] ; [ j]  min distcc [i] ; [ j] ˇ < "; j¤i

j¤i

for all i 2 I: In other words, T"-eqdstnc is true when, for every agent, the distance to the closest clockwise neighbor and to the closest counterclockwise neighbor are approximately equal. An implementation of this control and communication law is shown in Fig. 4. As parameters we select n D 45, r D 2/40, umax D 1/4 and kprop D 7/16. Along the evolution, all robots agree upon a common direction of motion and, after suitable time, they reach a uniform distribution. A careful analysis based on invariance properties and Lyapunov functions allows us to establish that, under appropriate conditions, indeed both tasks are achieved by the agree-and-pursue law [59]. Theorem 11 (Time complexity of agree-and-pursue law) For kprop 2 ]0; 12 [, in the limit as n ! C1 and " ! 0C , the network Scircle with umax (n)  kprop r(n), the law CC AGREE & PURSUE , and the tasks Tdir and T"-eqdstnc together satisfy: (i) TC(Tdir ; CC AGREE & PURSUE ) 2 (r(n)1 ); (ii) if ı(n) is lower bounded by a positive constant as n ! C1, then TC(T"-eqdstnc ; CC AGREE & PURSUE ) 2 ˝(n2 log(n")1 ) ; TC(T"-eqdstnc ; CC AGREE & PURSUE ) 2 O(n2 log(n"1 )) : If ı(n) is upper bounded by a negative constant, then CC AGREE & PURSUE does not achieve T"-eqdstnc in general.

Finally we compare these results with the complexity result known for the leader election problem.

Robotic Networks, Distributed Algorithms for

Robotic Networks, Distributed Algorithms for, Figure 4 The AGREE & PURSUE law. Disks and circles correspond to robots moving counterclockwise and clockwise, respectively. The initial positions and the initial directions of motion are randomly generated. The five pictures depict the network state at times 0, 9, 20, 100, and 800

Remark 12 (Comparison with leader election) Let us compare the agree-and-pursue control and communication law with the classical Le Lann–Chang–Roberts (LCR) algorithm for leader election discussed in Sect. “Leader Election”. The leader election task consists of electing a unique agent among all robots in the network; it is therefore different from, but closely related to, the coordination task Tdir . The LCR algorithm operates on a static network with the ring communication topology, and achieves leader election with time and total communication complexity, respectively, (n) and (n2 ). The agree-and-pursue law operates on a robotic network with the r(n)disk communication topology, and achieves Tdir with time and total communication complexity, respectively, (r(n)1 ) and O(n2 r(n)1 ). If wireless communication congestion is modeled by r(n) of order 1/n as in Remark 10, then the two algorithms have identical time complexity and the LCR algorithm has better communication complexity. Note that computations on a possibly disconnected, dynamic network are more complex than on a static ring topology. Aggregation Algorithms The rendezvous objective (also referred to as the gathering problem) is to achieve agreement over the location of the robots, that is, to steer each agent to a common location. An early reference on this problem is [2]; more recent references include [26,32,52,53]. We consider two scenarios which differ in the robots’ communication capabilities and the environment in which the robots move. First [26], we consider the problem of rendezvous for robots equipped with range-limited communication in obstacle-free environments. In this case, each robot is capable of sensing its position in the Euclidean space Rd and can communicate it to any other robot within a given distance r. This communication service is modeled by the r-disk graph, in which two robots are neighbors if and only if their Eu-

clidean distance is less than or equal to r. Second [36], we consider visually-guided robots. Here the robots are assumed to move in a nonconvex simple polygonal environment Q. Each robot can sense, within line of sight, any other robot as well as the distance to the boundary of the environment. The relationship between the robots can be characterized by the so-called visibility graph: two robots are neighbors if and only if they are mutually visible to each other. In both scenarios, the rendezvous problem cannot be solved with distributed information unless the robots’ initial positions form a connected communication graph. Arguably, a good property of any rendezvous algorithm is that of maintaining connectivity between robots. This connectivity-maintenance objective is interesting on its own. It turns out that this objective can be achieved through local constraints on the robots’ motion. Motion constraint sets that maintain connectivity are designed in [2,36] by exploiting the geometric properties of disk and visibility graphs. These discussions lead to the following algorithm that solves the rendezvous problems for both communication scenarios. The robots execute what is known as the Circumcenter Algorithm; here is an informal description. Each robot iteratively performs the following tasks: 1: acquire neighbors’ positions 2: compute connectivity constraint set 3: move toward the circumcenter of the point set comprised of its neighbors and of itself, while remaining inside the connectivity constraint set. One can prove that, under technical conditions, the algorithm does achieve the rendezvous task in both scenarios; see [26,36]. Additionally, when d D 1, it can be shown that the time complexity of this algorithm is (n); see [60].

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Robotic Networks, Distributed Algorithms for, Figure 5 Deployment algorithm for the area-coverage problem. Each of the 20 robots moves toward the centroid of its Voronoi cell. This strategy corresponds to the network following the gradient of Have . Areas of the polygon with greater importance are colored darker. Figures a and c show, respectively, the initial and final locations, with the corresponding Voronoi partitions. Figure b illustrates the gradient descent flow

Deployment Algorithms The problem of deploying a group of robots over a given region of interest can be tackled with the following simple heuristic. Each robot iteratively performs the following tasks: 1: acquire neighbors’ positions 2: compute own dominance region 3: move towards the center of own dominance region This short description can be made accurate by specifying what notions of dominance region and of center are to be adopted. In what follows we mention two examples and refer to [24,25,27,37] for more details. First, we consider the area-coverage deployment problem in a convex environment Q. The objective is to maximize the area within close range of the mobile nodes. This models a scenario in which the nodes are equipped with some sensors that take measurements of some physical quantity in the environment, e. g., temperature or concentration. Assume that certain regions in the environment are more important than others and describe this by a density function . This problems leads to the coverage performance metric Z Have (p1 ; : : : ; p n ) D min f (kq  p i k)(q)dq D

Q i2f1;:::;ng n XZ

f (kq  p i k)(q)dq :

iD1

this gradient scheme. Following the gradient of Have corresponds, in the algorithm described above, to defining (1) the dominance regions to be the Voronoi cells generated by the robots, and (2) the center of a region to be the centroid of the region (if f (x) D x 2 ). Because the closedloop system is a gradient flow for the cost function, performance is locally, continuously optimized. As a special case, when the environment is a segment and  D 1, the time complexity of the algorithm can be shown to be O(n3 log(n"1 )), where " is an accuracy threshold below which we consider the task accomplished. Second, we consider the problem of deploying to maximize the likelihood of detecting a source. For example, consider devices equipped with acoustic sensors attempting to detect a sound-source (or, similarly, antennas detecting RF signals, or chemical sensors localizing a pollutant source). For a variety of criteria, when the source emits a known signal and the noise is Gaussian, we know that the optimal detection algorithm involves a matched filter, that detection performance is a function of signal-to-noise-ratio, and, in turn, that signal-to-noise ratio is inversely proportional to the sensor-source distance. In this case, the appropriate cost function is 



Hworst p1 ; : : : ; p n D max

min

q2Q i2f1;:::;ng

f (kq  p i k)

D max f (kq  p i k) ; q2Vi

Vi

Here pi is the position of the ith node, f measures the performance of an individual sensor, and fV1 ; : : : ; Vn g is the Voronoi partition of the environment Q generated by the positions fp1 ; : : : ; p n g. If we assume that each node obeys a first-order dynamical behavior, then a simple gradient scheme can be easily implemented in a spatially-distributed manner. Figure 5 shows an implementation of

and a greedy motion coordination algorithm is for each node to move toward the circumcenter of its Voronoi cell. A detailed analysis [24] shows that the detection likelihood is inversely proportional to the circumradius of each node’s Voronoi cell, and that, if the nodes follow the algorithm described above, then the detection likelihood increases monotonically as a function of time.

Robotic Networks, Distributed Algorithms for

Bibliographical Notes In this section we present a necessarily incomplete discussion of some relevant literature that we have not yet mentioned in the previous sections. First, we review some literature on emergent and self-organized swarming behaviors in biological groups with distributed agent-to-agent interactions. Interesting dynamical systems arise in biological networks at multiple levels of resolution, all the way from interactions among molecules and cells [62] to the behavioral ecology of animal groups [66]. Flocks of birds and schools of fish can travel in formation and act as one unit (see [72]), allowing these animals to defend themselves against predators and protect their territories. Wildebeest and other animals exhibit complex collective behaviors when migrating, such as obstacle avoiding, leader election, and formation keeping (see [41,89]). Certain foraging behaviors include individual animals partitioning their environment into nonoverlapping zones (see [7]). Honey bees [85], gorillas [93], and whitefaced capuchins [13] exhibit synchronized group activities such as initiation of motion and change of travel direction. These remarkable dynamic capabilities are achieved apparently without following a group leader; see [7,13,41,66,72,85,93] for specific examples of animal species and [21,28] for general studies. With regards to distributed motion coordination algorithms, much progress has been made on pattern formation [9,46,86,94], flocking [67,96], self-assembly [49], swarm aggregation [38], gradient climbing [75], cyclic pursuit [14,58,90], vehicle routing [56,87], and connectivity maintenance problems [83,103]. Much research has been devoted to distributed task allocation problems. The work in [39] proposes a taxonomy of task allocation problems. In papers such as [1,40,63,84], advanced heuristic methods are developed, and their effectiveness is demonstrated through simulation or real world implementation. Distributed auction algorithms are discussed in [18,63] building on the classic works in [10,11]. A distributed MILP solver is proposed in [1]. A spatially distributed receding-horizon scheme is proposed in [33,73]. There has also been prior work on target assignment problems [6,91,102]. Target allocation for vehicles with nonholonomic constraints is studied in [76,81,82]. References with a focus on robotic networks include the survey in [99], the text [5] on behavior-based robotics, and the recent special issue [4] of the IEEE Transaction on Robotics and Automation. An important contribution towards a network model of mobile interacting robots is introduced in [94]. This model consists of a group of identical “distributed anonymous mobile robots” characterized

as follows: no explicit communication takes place between them, and at each time instant of an “activation schedule,” each robot senses the relative position of all other robots and moves according to a pre-specified algorithm. Communication complexity for control and communication algorithms A related model is presented in [32], where as few capabilities as possible are assumed on the agents, with the objective of understanding the limitations of multiagent networks. A brief survey of models, algorithms, and the need for appropriate complexity notions is presented in [79]. Recently, a notion of communication complexity for control and communication algorithms in multi-robot systems is analyzed in [48], see also [50]. Finally, with regards to linear distributed algorithms we mention the following references, on top of the ones discussed in Sect. “Averaging Algorithms”. Various results are available on continuous-time consensus algorithms [34,54,55,64,70,77], consensus over random networks [43,95,100], consensus algorithms for general functions [8,23], connections with the heat equation and partial difference equation [31], convergence in time-delayed and asynchronous settings [3,12], quantized consensus problems [47,80], applications to distributed signal processing [69,92,101], characterization of the convergence rates and time complexity [16,17,51,71]. Finally, two recent surveys are [68,78]. Future Directions Robotic networks incorporate numerous subsystems. Their design is challenging because they integrate heterogeneous hardware and software components. Additionally, the operation of robotic networks is subject to computing, energy, cost, and safety constraints. The interaction with the physical world and the uncertain response from other members of the network are also integral parts to consider in the management of robotic networks. Traditional centralized approaches are not valid to satisfy the scalability requirements of these systems. Thus, in order to successfully deploy robotic and communication networks, it is necessary to expand our present knowledge about how to integrate and efficiently control them. The present chapter has offered a glimpse into these problems. We have presented verification and complexity tools to evaluate the cost of cooperative strategies that achieve a variety of tasks. Networks of robotic agents are an example of the class of complex, networked systems which pervades our world. Understanding how to design robotic swarms requires the development of new fundamental theories that can explain the behavior of general networks evolving with time.

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Some of the desired properties of these systems are robustness, ease of control, predictability with time, guaranteed performance, and quality of service. Self-organization and distributed management would also allow for the minimal supervision necessary for scalability. However, devising systems that meet all these criteria is not an easy task. A small deviation by an agent or a change in certain parameters may produce a dramatic change in the overall network behavior. Relevant questions that pertain these aspects are currently being approached from a range of disciplines such as systems and control, operations research, random graph theory, statistical physics, and game theory over networks. Future work will undoubtedly lead to a cross-fertilization of these and other areas that will help design efficient robotic sensor networks. Acknowledgments This material is based upon work supported in part by NSF CAREER Award CMS-0643679, NSF CAREER Award ECS-0546871, and AFOSR MURI Award FA9550-07-10528. Bibliography 1. Alighanbari M, How JP (2006) Robust decentralized task assignment for cooperative UAVs. In: AIAA conference on guidance, navigation and control, Keystone, CO, August 2006 2. Ando H, Oasa Y, Suzuki I, Yamashita M (1999) Distributed memoryless point convergence algorithm for mobile robots with limited visibility. Trans IEEE Robotics Autom 15(5):818– 828 3. Angeli D, Bliman PA (2006) Stability of leaderless discrete-time multi-agent systems. Math Control Signal Syst 18(4):293–322 4. Arai T, Pagello E, Parker LE (2002) Guest editorial: Advances in multirobot systems. Trans IEEE Robotics Autom 18(5):655– 661 5. Arkin RC (1998) Behavior-Based Robotics. MIT Press, Cambridge, MA 6. Arslan G, Marden JR, Shamma JS (2007) Autonomous vehicle-target assignment: A game theoretic formulation. J ASME Dyn Syst Meas Control 129(5):584–596 7. Barlow GW (1974) Hexagonal territories. Anim Behav 22:876– 878 8. Bauso D, Giarré L, Pesenti R (2006) Nonlinear protocols for optimal distributed consensus in networks of dynamic agents. Syst Control Lett 55(11):918–928 9. Belta C, Kumar V (2004) Abstraction and control for groups of robots. Trans IEEE Robotics 20(5):865–875 10. Bertsekas DP, Castañón DA (1991) Parallel synchronous and asynchronous implementations of the auction algorithm. Parallel Comput 17:707–732 11. Bertsekas DP, Castañón DA (1993) Parallel primal-dual methods for the minimum cost flow problem. Comput Optim Appl 2(4):317–336

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Scaling Limits of Large Systems of Non-linear Partial Differential Equations

Scaling Limits of Large Systems of Non-linear Partial Differential Equations D. BENEDETTO, M. PULVIRENTI Dipartimento di Matematica, Università di Roma ‘La Sapienza’, Roma, Italy Article Outline Glossary Definition of the Subject Introduction Weak-Coupling Limit for Classical Systems Weak-Coupling Limit for Quantum Systems Weak-Coupling Limit in the Bose–Einstein and Fermi–Dirac Statistics Weak-Coupling Limit for a Single Particle: The Linear Theory Future Directions Bibliography Glossary Scaling limits A scaling limit denotes a procedure to reduce the degree of complexity of a large particle system. It consists in scaling the space-time variables and, possibly, other quantities (like the interaction potential or the density), in order to obtain a more handable description of the same system. The initial space-time coordinates are called microscopic, while the new ones, those well suited for the description of kinetic or hydrodynamical systems, are called macroscopic. Boltzmann equation It is an integro-differential kinetic equation for the one particle distribution function in the classical phase space (see Eq. (64) below). It arises in some physical regimes, namely for a rarefied gas and for a weakly interacting quantum dense gas. Uehling–Uhlenbeck equation It is a Boltzmann type equation taking into account corrections due to the Bose–Einstein or the Fermi–Dirac statistics (see Eq. (69) below). It holds in the weak-coupling limit. Fokker–Planck–Landau equation It is a kinetic equation diffusive in velocity (see Eq. (28) below). It arises in the context of a weakly interacting classical dense gas. Hydrodynamical equations They are evolution equations for macroscopic quantities like density, mean velocity, temperature, and so on. Low density limit Sometimes called Boltzmann–Grad limit, it is a scaling limit in which the density is van-

ishing. Applied to classical particle systems it gives the Boltzmann equation. Weak coupling limit It is a scaling limit in which the density is constant but the interaction vanishes suitably. Applied to a quantum particle systems it gives a Boltzmann equation. Applied to a classical particle systems it gives the Fokker–Planck–Landau equation. Hydrodynamic limit In this scaling we simply pass from micro to macro variables. We look at the behavior of suitable mean values, which are functions of the space and the time. We expect them to behave in a hydrodynamical way, namely to satisfy a set of hydrodynamical equations. Wigner transform It is the description of a quantum state as a function in the classical phase space. Definition of the Subject Many interesting systems in Physics and Applied Sciences are constituted by a large number of identical components so that they are difficult to analyze from a mathematical point of view. On the other hand, quite often, we are not interested in a detailed description of the system but rather to his global behavior. Therefore it is necessary to look for all procedures leading to simplified models, retaining all the interesting features of the original system, cutting away unnecessary informations. This is exactly the methodology of the Statistical Mechanics and the Kinetic Theory when dealing with large particle systems. In this contribution we want to approach a problem of this type, namely we try to outline the difficulties in making rigorous the limiting procedure leading the microscopic description of a large particle system (based on the fundamental laws like Newton or Schrödinger equations) to a more handable kinetic or hydrodynamic picture. Such a transition depends on the space-time scale we choose to describe the phenomenon. To fix the ideas we start by considering a system constituted by N identical classical point particles of unitary mass. The Newton equations are: X d2 qi D F(q i  q j ) ; 2 d

(1)

jD1:::N: j¤i

where fq1 : : : q N g, q i 2 R3 are the position of the particles and  is the time. Here F D r denotes the interparticle (conservative) force and  the two-body interaction potential. We now introduce a small parameter " > 0 expressing the ratio between the macroscopic and the microscopic scales. For instance " could denote the inverse ratio of a typical distance between two molecules of a gas, mea-

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sured in microscopic unities, and the same distance measured in meters. It can be as well the ratio between typical macroscopic and microscopic times. We now introduce the new variables x D "q

t D " :

Since we are interested in the macroscopic properties of the system, namely in the evolution of macroscopic quantities that are those varying on the (x; t) scale and hence almost constant in the (q; ) scale, it is natural to rescale also Eq. (1). We write  xi  x j  d2 1 X xi D F : 2 dt " "

(2)

jD1:::N: j¤i

Up to now we did nothing else than an innocent change of variables. However to obtain a non trivial description, N has to diverge when " ! 0, and we have to specify how. Also additional hypotheses on the strength of the interaction, according to the physical situation at hand, may be necessary. There are various possible scaling limits we shall discuss below.  The hydrodynamic limit Here the system is dense, namely N D O("3 ). For all x 2 ˝ R3 , we consider a small box  around x. In  there are a very large number of particles. We denote the initial density, mean velocity and temperature by ( 0 (x); u0 (x); T0 (x)). At a positive time t the particles starts to interact. For a microscopic time  the particles in  interact practically among themselves only (supposing a short range interaction) and, if an ergodic property would hold, we expect that they reach a thermal equilibrium state. We remind that t D " is essentially vanishing, therefore we expect that at macroscopic time t  0 the particle are distributed according to the Gibbs measure characterized by the parameters ( 0 (x); u0 (x); T0 (x)), being mass momentum and energy locally conserved. On a larger time scale (t > 0) mass momentum and energy are not locally conserved anymore. If a local equilibrium structure is preserved, the parameters of the equilibria ( (x; t); u(x; t); T(x; t)) will evolve according to five equations which are the exactly the well known Euler equations for compressible gas. This complex mechanism is very far to be proven rigorously, even for short times and initial regular data. A heuristic derivation of the Euler equations in terms of particle dynamics was first given by Morrey [35]. We address the reader to [39] and [19] for further references and comments.

 The low-density (Boltzmann–Grad) limit Now we are dealing with a rarefied gas assuming N D O("2 ). This is the reason why this scaling is called low-density. We assume also, just for simplicity, that the interaction range of  is one. Consider now a given test particle. If the gas is more or less homogeneous, the number of particles interacting with this particle in a given (macroscopic) time interval is O(N"2 ) D O(1). Therefore we expect a finite number of collisions per unit time. Moreover each collision is almost instantaneous so that each particle undergoes a jump process in velocity. We also assume that initially the particles are independently distributed according to the one-particle distribution on the phase space f0 (x; v) (v is the velocity). Of course the dynamics creates correlations so that, strictly speaking, we expect that the particles, for every positive time t, are not independently distributed anymore. However the probability that a given pair of particles interact is vanishing in the limit " ! 0. Therefore the statistical independence (called propagation of chaos) is recovered in the same limit. In other words we expect, for the time evolved distribution f (x; v; t), a nonlinear evolution equation, which is the celebrated Boltzmann equation discovered in 1872. This scaling limit, often called the Boltzmann–Grad limit [25], has been proved rigorously by Lanford [31] for a short time interval (see also [28] for a special case in which the limit holds globally in time). We address the reader to refs. [14,39] and [19] for additional comments and references.  The weak-coupling limit Now the gas is dense because we assume N D O("3 ), however the particles are weakly interacting. This condition is expressed by rescaling the interaction potential by setting !

p " :

After that the equation of motion (in macroscopic variables) become: x  x  d2 1 X i j xi D  p r : 2 dt " "

(3)

jD1:::N: j¤i

To have a vague idea of the behavior of a test particle in this limit, we observe first that the velocity change due to a single collision is p p v  Ft  O("/ ") D O( ") :

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

Here t D O(") is the time in which the collision takes place,  is assumed short-range in microscopic varip ables and then 1 / " is the order of magnitude of the force F. The number of collisions per unit time is "2 "3 D "1 , so that X jvj2 D O(1) : Therefore, provided that the propagation of chaos is ensured, v D v(t) is expected to be not absolutely continuous in the limit and the one-particle distribution function to satisfy a diffusion equation in velocity, the Landau–Fokker–Planck equation. The statistical independence at a positive time t is expected because the effect of the interaction between two given particle is going to vanish in the limit " ! 0. Note that the physical meaning of the propagation of chaos here is quite different from that arising in the contest of the Boltzmann equation. Here two particles can interact but the effect of the collision is small, while in a low-density regime the effect of a collision between two given particles is large but quite unlikely. The above scalings make sense as well for quantum systems. What we expect in this case? Notice that the transition from micro to macro variables increase the frequency of the quantum oscillations so that we are always in presence of a semiclassical limit as regards the kinetic energy part of the Hamiltonian. However the potential varies on the same scale of the oscillations. As a consequence we face the following scenario. The hydrodynamic limit of quantum systems yields the Euler equations for the equilibria parameters ; u; T. The only difference with the classical case is that the relationship between the pressure and the hydrodynamical fields, is that dictated by the Quantum Statistical Mechanics. Indeed the local equilibrium is achieved at a microscopic level and this is the only quantum effect we expect to survive in the limit. The situation is conceptually similar for the low density limit: here we have the classical Boltzmann equation for the one-particle distribution function. The only quantum macroscopic effect is that the cross-section must be computed in terms of the quantum scattering process (see [6]). Introduction The hydrodynamical and the low density limits are relatively more popular then the weak-coupling limit, which will be, for that reason, the object of the present contribu-

tion. More precisely we analyze large classical and quantum particle systems in the weak-coupling regime. We show how we expect such systems to be well described by the Landau–Fokker–Planck and the Boltzmann equations, for the classical and the quantum case respectively. We also describe the effects of the Fermi–Dirac and Bose– Einstein statistics. Unfortunately our arguments we present here are largely formal because the rigorous results we know up to now, are few. A rigorous proof of the validity of the weakcoupling limit, or a rigorous derivation of the corresponding kinetic equations, is a challenging and still open problem. In Sect. “Weak-Coupling Limit for Classical Systems” we introduce and discuss the problem for classical systems, presenting a formal proof of the validity of the Landau equation. In Sect. “Weak-Coupling Limit for Quantum Systems” we pass to analyze the case of a quantum system, neglecting the correlations due to the statistics. The case of Bosons and Fermions is discussed in Sect. “Weak-Coupling Limit in the Bose–Einstein and Fermi–Dirac Statistics”. Finally, in Sect. “Weak-Coupling Limit for a Single Particle: the Linear Theory” we briefly introduce and discuss the corresponding linear problems, namely the behavior of a single (classical or quantum) particle in a random distribution of obstacles. Section “Future Directions” is devoted to concluding remarks. Weak-Coupling Limit for Classical Systems We consider a classical system of N identical particles of unitary mass. Rescaled positions and velocities are denoted by x1 : : : x N and v1 : : : v N . After having also rescaled the potential in the following way p  ! " ; the Newton equations reads as d xi D vi dt

 xi  x j  d 1 X r vi D  p : (4) dt " " jD1:::N: j¤i

We also assume that N D O("3 ), namely the density is O(1). We are interested in the statistical behavior of the system so that we introduce a probability distribution on the phase space of the system, W N D W N (X N ; VN ), that is the state at time zero. Here (X N ; VN ) denote the set of positions and velocities: X N D (x1 ; : : : ; x N )

VN D (v1 ; : : : ; v N ) :

1507

1508

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

Then from Eq. (4) we obtain the following Liouville equation  1  (@ t CVN r N )W N (X N ; VN ) D p TN" W N (X N ; VN ) (5) " PN where VN  r N D iD1 v i  rx i and (@ t C VN  r N ) is the usual free stream operator. Also, we have introduced the operator X   " (Tk;` W N (X N ; VN ) ; (6) (TN" W N (X N ; VN ) D 0
˝j

f j0 D f0

with x  x  k `  (rv k  rv ` )W N : "

" W N D r Tk;`

(7)

To investigate the limit " ! 0 ; N D "3 , it is convenient to introduce the BBKGY hierarchy (from Bogoliubov, Born, Green, Kirkwood and Yvon, see e. g. [2] and [19]), for the j-particle distributions defined as Z Z Z Z f jN (X j ; Vj ) D dx jC1 : : : dx N dv jC1 : : : dv N (8) N W (X j ; x jC1 : : : x N ; Vj ; v jC1 : : : v N ) for j D 1; : : : ; N  1. Obviously, we set f NN D W N . From now on we shall suppose that, due to the fact that the particles are identical, the function W N and f jN which we have introduced are all symmetric in the exchange of particles. A partial integration of the Liouville equation (5) and standard manipulations give us the following hierarchy of equations: (for 1  j  N): @t C

j X

! v k r k

f jN

kD1

The operator

C "jC1

1 Nj N D p T j" f jN C p C "jC1 f jC1 : (9) " "

is defined as:

X

C "k; jC1 ;

(10)

kD1

and C "k; jC1 f jC1 (x1 : : : x j ; v1 : : : v j ) Z Z x  x  k ` D  dx jC1 dv jC1 F " rv k f jC1 (x1 ; x2 ; : : : ; x jC1 ; v1 ; : : : ; v jC1 ) :

;

(12)

where f 0 is a given one-particle distribution function. This means that any pair of particles are statistically uncorrelated at time zero. Of course such a statistical independence is destroyed at time t > 0 and Eq. (9) shows that the time evolution of f1N is determined by the knowledge of f2N which turns out to be dependent on f3N and so on. However, since the interaction between two given particle is going to vanish in the limit " ! 0, we can hope that such statistical independence, namely the factorization property (12), is recovered in the same limit. Therefore we expect that in the limit " ! 0 the oneparticle distribution function f1N converges to the solution of a suitable nonlinear kinetic equation f which we are going to investigate. If we expand f jN (t) as a perturbation of the free flow S(t) defined as (S(t) f j )(X j ; Vj ) D f j (X j  Vj t; Vj ) ;

(13)

we find Nj f jN (t) D S(t) f j0 C p "

Z

t

0

1 Cp "

N S(t  t1 )C "jC1 f jC1 (t1 )dt1

Z

t 0

S(t  t1 )T j" f jN (t1 )dt1 : (14)

j

C "jC1 D

numbering uses the fact that all particles are identical). The total operator C "jC1 takes into account all such collisions. The dynamics of the j-particle subsystem is governed by three effects: the free-stream operator, the “recollisions”, i. e. the collisions “inside” the subsystem, given by the T term, and the “creations”, i. e. the collisions with particles “outside” the subsystem, given by the C term. We finally fix the initial value f f j0 g NjD1 of the solution N f f j (t)g NjD1 assuming that f f j0 g NjD1 is factorized, that is, for all j D 1; : : : N

(11)

C "k; jC1 describes the “collision” of particle k, belonging to the j-particle subsystem, with a particle outside the subsystem, conventionally denoted by the number j C 1 (this

We now try to keep informations on the limit behavior of f jN (t). Assuming for the moment that the time evolved j-particle distributions f jN (t) are smooth (in the sense that the derivatives are uniformly bounded in "), then Z j Z X N C "jC1 f jC1 dr dv jC1 F(r) (X j ; Vj ; t1 ) D "3 (15) kD1 rv k f jC1 (X j ; x k  "r; Vj ; v jC1 ; t1 ) : Assuming now, quite reasonably, that Z drF(r) D 0 ;

(16)

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

we find that N C "jC1 f jC1 (X j ; Vj ; t1 ) D O("4 ) N provided that Dv2 f jC1 is uniformly bounded. Since 7 Nj p D O(" 2 ) "

we see that the second term in the right hand side of (14) does not give any contribution in the limit. Moreover Z

t

0

S(t  t1 )T j" f jN (t1 )dt1 D

XZ i¤k

t

dt1

0

! (x i  x k )  (v i  v k )(t  t1 ) F g(X j ; Vj ; t1 ) ; " (17) where g is a smooth function. Obviously the above time integral is O(") so that also the last term in the right hand side of (14) does not give any contribution in the limit. Then we are facing the alternative: either the limit is trivial or the time evolved distributions are not smooth. However we believe that the limit is not trivial (actually we expect to get a diffusion equation, according to the previous discussion) and a rigorous proof of this fact seems problematic. The difficulty in obtaining a-priori estimates induce us to exploit the full series expansion of the solution, namely f1N (t) D

XX n0 G n

Z

Z

t

K(G n )

t n1

dt2 : : :

dt1 0

Z

t1

0

0

dt n (18)

  S(t  t1 )O1 S(t1  t2 ) : : : O n S(t n ) f m0 : Here Oj is either an operator C or T expressing a creation of a new particle or a recollision between two particles respectively. Gn is a graph namely a sequence of indices (r1 ; l1 ); (r2 ; l2 ); : : : (r n ; l n ) where (r j ; l j ); r j < l j are the pair of indices of the particles involved in the interaction at time tj . The number of particles created in the process is m  1. It is convenient to represent the generic graph in the following way, as in Fig. 1. The legs of the graph denotes the particles and the nodes the creation of new particles (operators C). Recollisions (operators T) are represented by horizontal link. For instance the graph in figure is (1; 2); (1; 3); (1; 3); (2; 3); m D 3 and the integrand in Eq. (18) is in this case  S(t  t1 )C1;2 S(t1  t2 )C1;3 S(t2  t3 )T1;3 (19) S(t3  t4 )T2;3 S(t4 ) f30 :

Scaling Limits of Large Systems of Non-linear Partial Differential Equations, Figure 1 The collision sequence (1; 2); (1; 3); (1; 3); (2; 3)

Note that the knowledge of the graph determines completely the sequence of operators in the right hand side of (20). Finally the factor K(G n ) takes into account the divergences:   n K(G n ) D O " 2 "3(m1) :

(20)

We are not able to analyze the asymptotic behavior of each term of the expansion (18) however we can compute the limit for " ! 0 of the few terms up to the second order (in time). We have: Z N 1 t g N (x1 ; v1 ; t) D f 0 (x1  v1 t; v1 ) C p S(t  t1 ) " 0 Z (N  1) (N  2) X t1 " S(t1 ) f20 dt1 C dt2 C1;2 " " jD1;2 0 Z N 1 t " S(t  t1 )C1;2 S(t1  t2 )C "j;3 S(t2 ) f30 C dt1 " 0 Z t1 " "  dt2 S(t  t1 )C1;2 S(t1  t2 )T1;2 S(t2 ) f20 : 0

(21) Here the right hand side of (21) defines g N . The second and third term in (21) corresponding to the graphs shown in Fig. 2 are indeed vanishing as follows by the use of the previous arguments. The most interesting term is the last one, shown in Fig. 3.

1509

1510

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

The formal limit is of (21) is Z t Z dt1 g(x1 ; v1 ; t) D dv2 S(t  t1 )rv 1 a(v1  v2 ) 0

Scaling Limits of Large Systems of Non-linear Partial Differential Equations, Figure 2 The vanishing terms in the expansion of gN

 (rv 1  rv 2 ) S(t1 ) f20 ;

where (using F(r) D F(r)) the matrix a is given by: Z a(w) D Z

Z

C1

ds F(r) ˝ F(r  ws) D

dr 0

C1

1 2

Z dr

Z C1 1 1 ds 2 (2)3 1 Z 1 ˆ 2 D dk k ˝ k (k) (8)2

ds F(r) ˝ F(r  ws)



Z1 ˆ 2 e i(wk)s  dk k ˝ k (k) Scaling Limits of Large Systems of Non-linear Partial Differential Equations, Figure 3 The first non vanishing terms in the expansion of gN

(25)

 ı(w  k) D

A (jwj2 Id  w ˝ w); jwj3 (26)

To handle this term we denote by w D v1  v2 the relative velocity and note that, for a given function u: " u(x1 ; x2 ; v1 ; v2 ) S(t1  t2 )T1;2

(x1  x2 )  w(t1  t2 ) D F "

!

R

ˆ where (k) D ei l x (x)dx. Here the interaction potential  has been assumed spherically symmetric and: Z C1 1 ˆ 2: AD dr r3 (r) (27) 8 0 Looking at Eq. (25) we are led to introduce the following nonlinear equation:

 [(rv 1  rv 2 )u]

 (x1  v1 (t1  t2 ); x2  v2 (t1  t2 ); v1 ; v2 ) ! (x1  x2 )  w(t  t1 )  (rv 1  rv 2 C (t1  t2 ) D F "  (rx 1  rx 2 ))S(t1  t2 )u(x1 ; x2 ; v1 ; v2 ) :

(@ t C v  rx ) f D Q L ( f ; f )

with the collision operator QL given by: Z h i Q L ( f ; f )(v) D dv1 rv a(vv1 ) (rv rv 1 ) f (v) f (v1 ) :

(22) Therefore the last term in the r.h.s. of (22) is Z t1 Z Z Z N 1 t dt1 S(t  t1 ) dt2 dx2 dv2 " 0 0   x  x  (x1  x2 )  w(t1  t2 ) 1 2  rv 1 F F " "

(23)

 (rv 1  rv 2 C (t1  t2 )(rx 1  rx 2 ))  S(t1 ) f20 (x1 ; x2 ; v1 ; v2 ) : Setting now r D N

x 1 x 2 "

g (x1 ; v1 ; t) D (N  1)" Z  dv2 F(r)  rv 1

3

(29) Here x plays the role of a parameter and hence his dependence is omitted. Equation (28) is called the Fokker–Planck–Landau equation (Landau equation in the sequel) and has been introduced by Landau in the study of a dense, weakly interacting gas (see [32]). From (28) we obtain the following (infinite) hierarchy of equations (@ t C Vj  r X j ) f j D C jC1 f jC1

and s D Z

t 1 t 2 "

Z

t

dt1 0

f j (t) D f (t)˝ j

Z ds

dr

where f (t) solves Eq. (28). Accordingly C jC1 P k C k; jC1 where

0

F(r  ws)  (rv 1  rv 2 C "s(rx 1  rx 2 )) p  S(t1  "s) f20(x1 ; x2 ; v1 ; v2 ) C O( ") :

(30)

for the quantities:

then

t1 "

(28)

(24)

C k; jC1 f jC1 (x1 : : : x j ; v1 : : : v j ) Y D f (x r ; vr )Q L ( f ; f )(x k ; v k ) : r¤k

(31) D

(32)

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

limit from a particle system. In doing this we essentially followed the monograph [2]. This is not however the usual way in which the Landau equation is introduced in the literature. Indeed it is usually recovered from the Boltzmann equation. when the density increases and the grazing collisions become dominant. In particular, the case of the homogeneous Boltzmann equation has been investigated in [1,24,44] (see also [15] for the Coulomb potential case, and see the general survey [43]). In [1] the authors show that, under suitable assumptions on the cross-section, the diffusion Fokker–Planck–Landau equation (28) can indeed be derived. The diffusion operator is the form (29) but with a matrix a given by Scaling Limits of Large Systems of Non-linear Partial Differential Equations, Figure 4 The collision-recollision sequence (1; 2); (1; 2); (1; 3); (1; 3); (2; 4); (2; 4)

Therefore f has the following series expansion representation Z t1 Z t n1 XZ t dt1 dt2 : : : dt n f (t) D 0 0 (33) n0 0  0  S(t  t1 )C2 S(t1  t2 )C3 : : : C n1 S(t n ) f nC1 : The previous calculation shows the formal convergence of g N to the term with n D 1 of the expansion (33), namely we have an agreement between the particle system (18) and the solution to the Landau equation (33) at least up to the first order in time (second order in the potential). Although the above arguments can be made rigorous, under suitable assumption on the initial condition f 0 and the potential , it seems difficult to show the convergence of the whole series. On the other hand it is clear that the graphs which should contribute in the limit are those formed by a collision-recollision sequence, like that shown in Fig. 4. For those terms it is probably possible to show the convergence. For instance the case in figure has the asymptotics Z

Z

t

dt1 0

Z

t1

with ˛ smooth function. Next in [24] and [44] steps forward were performed to arrive to cover the case ˛(jwj)  1/jwj for small jwj, with  < 1. We remark that the diffusion coefficient found in this way is different from that derived here in (26). We conclude by observing that the Landau equation describes a genuine kinetic evolution. Mass, momentum and energy are conserved, while the kinetic entropy is decreasing. The equilibrium is Maxwellian. The difficulties in the validation problem are certainly related to the transition from a reversible (hamiltonian) dynamics to an irreversible one, as for the Boltzmann equation. Here we tried to compare the two series expansions as in the Lanford’s validation proof for the Boltzmann equation. This is a sort of Cauchy–Kowalevski brute force argument which works, as well, for negative times. It is clear that, in this way, we cannot go beyond a short time result and even this seems not trivial at all. Other approaches making use of the diffusive nature of the motion of a test particle are, at moment, absent. However the weak-coupling limit of a single particle in a random distribution of scatterers is well understood as we shall discuss later in Sect. “The Weak Coupling Limit for a Single Particle: the Linear Theory”.

t2

dt2 0

a(w) D ˛(jwj)(jwj2 Id  w ˝ w) ;

0

 dt3 S(t  t1 )C1;2 S(t1  t2 ) (34) 0 C1;3 S(t2  t3 )C2;4 S(t3 ) f4 :

However the proof that all other graphs are vanishing in the limit is not easy. Even more difficult is a uniform control of the series expansion (18), even for short times. As we shall see in the next section, something more can be obtained for quantum systems under the same scaling limit. Some comments are in order. In the present section we showed how the Landau equation is expected to be derived in the weak-coupling

Weak-Coupling Limit for Quantum Systems We consider the quantum analog of the system considered in Sect. “Introduction”, namely N identical quantum particles with unitary mass in R3 . In the present section the statistical nature of the particles will be ignored. The interaction between particles is still a two-body potential  so that the total potential energy is taken as U(x1 : : : x N ) D

X i< j

(x i  x j ) :

(35)

1511

1512

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

The associated Schrödinger equation reads i@ t  (X N ; t) D  12  N  (X N ; t)CU(X N ) (X N ; t); (36) P where  N D N iD1  i , i is the Laplacian with respect to the xi variables, X N D (x1 ; : : : ; x N ) and ¯ is normalized to unity. As for the classical system considered in Sect.“ Introduction” we rescale the equation and the potential by p (37) x ! "x ; t ! "t ;  ! " :

"2  N  " (X N ; t)CU" (X N )  " (X N ; t); 2 (38)

X Z  " (Tk;` W N (X N ; VN ) D i  D˙1

e

where: U" (x1 : : : x N ) D

 Z  1 1 " dYN dVN0 (Tk;` W N (X N ; VN ) D i (2)3N  h x  x y k  y`  k ` i YN (VN VN0 )  e  " 2  i x  x y k  y` k ` C W N (X N ; VN0 ) :  " 2

p X  xi  x j  "  : "

(43)

Equivalently, we may write

The resulting equation is, i"@ t  " (X N ; t) D 

" describes the interaction of particle k with where each Tk;` particle `

i "h (x k x ` )

W (x1 ; : : : ; x N ; v1 ; : : : ; v k  

i< j

WN

satisfies a transport-like equaThe Wigner transform tion, completely equivalent to the Schrödinger equation:

h ; 2

h : : : ; v` C  ; : : : ; v N ) : (44) 2

(39)

We want to analyze the limit " ! 0 in the above equations, when N D "3 . Note that this limit looks, at a first sight, similar to a semiclassical (or high frequency) limit. It is not so: indeed the potential varies on the same scale of the typical oscillations of the wave functions so that the scattering process is a genuine quantum process. Obviously, due to the oscillations, we do not expect that the wave function does converge to something in the limit. The right quantity to look at was introduced by Wigner in 1922 [45] to deal with kinetic problems. It is called the Wigner transform (of  " ) and is defined as  3N Z 1 " dYN e i YN VN  W N (X N ; VN ) D 2 " " (X N C YN )  " (X N  YN ) : (40) 2 2

N

dh ˆ (h) (2)3

" is a pseudodifferential operator which forNote that Tk;` mally converge, at fixed ", for „ ! 0 (here „ D 1) to its classical analog given by Eq. (7). Note also that in (44), “collisions” may take place between distant particles (x k ¤ x` ). However, such distant collisions are penalized by the highly oscillatory factor exp(ih(x k  x` ) / "). These oscillations turn out to play a crucial role throughout the analysis, and they explain why collisions tend to happen when x k D x` in the limit " ! 0. The formalism we have introduced is formally similar to the classical case so that we proceed as before by transforming Eq. (38) into a hierarchy of equations. We introduce the partial traces of the Wigner transform W N , denoted by f jN . They are defined through the following formula, valid for j D 1; : : : ; N  1:

f jN (X j ; Vj ) D

Z

Z dx jC1 : : :

Z

Z dx N

dv jC1 : : :

dv N

W N (X j ; x jC1 : : : x N ; Vj ; v jC1 : : : v N ) : (45)

 1  (@ t C VN  r N )W (X N ; VN ) D p TN" W N (X N ; VN ): " (41) N

The operator TN" on the right-hand-side of (38) plays the same role of the classical operator denoted with the same symbol in Sect. “Introduction”. It is X   " (TN" W N (X N ; VN ) D (Tk;` W N (X N ; VN ); (42) 0
Obviously, we set f NN D W N . The function f jN is the kinetic object that describes the state of the j particles subsystem at time t. As before, the wave function  , as well W N and f jN , are assumed to be symmetric in the exchange of particle, a property that is preserved in time. Proceeding then as in the derivation of the BBKGY hierarchy for classical systems, we readily transform Eq. (38)

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

is weakly vanishing, by a stationary phase argument (see [4]). Therefore, as for the classical case, we analyze the asymptotics of the collision-recollision term (shown in Fig. 1):

into the following hierarchy: j X

(@ t C

v k  r k ) f jN (X j ; Vj )

kD1

1 Nj N D p T j" f jN C p C "jC1 f jC1 ; " "

(46)



N 1 "

Z

Z

t

dt1 0

0

t1

d1 S(tt1 )C1;2 S(t1 1 )T1;2 S(1 ) f20 : (52)

where j

X

C "jC1 D

C "k; jC1 ;

(47)

kD1

X

Z 

D˙1

dh (2)3

Z dx jC1

 h N ˆ e i " (x k x jC1 ) f jC1 dv jC1 (h) x1 ; x2 ; : : : ; x jC1 ;  h h v1 ; : : : ; v k   ; : : : ; v jC1 C  : (48) 2 2

Z 

As before the initial value f f j0 g NjD1 is assumed completely factorized: for all j D 1; : : : ; N, we suppose f j0

˝j f0

D

;

(49) f0

where f 0 is a one-particle Wigner function, and is assumed to be a probability distribution. In the limit " ! 0, we expect that the j-particle distribution function f jN (t), that solves the hierarchy (46) with initial data (49), tends to be factorized for all times: f jN (t) f (t)˝ j (propagation of chaos). As for the classical case, if f jC1 is smooth: X Z dh N ˆ (h) (X j ; Vj ) D i"3  C "k; jC1 f jC1 (2)3 D˙1  Z Z N  dr dv jC1 e i hr f jC1 X j ; x k  "r; v1 ; : : : ;  h h D O("4 ) : (50) v k   ; : : : ; v jC1 C  2 2 Indeed, setting " D 0 in the integrand, the integration over r produces ı(h). As a consequence the integrand is independent on  and the sum vanishes. Therefore the integral is O("). Also 1 p " i

Z 0

t

Z t Z t1 Z Z N 1 X dx2 dv2 0 dt1 d1 " 0 0 ; 0 D˙1 Z Z   dk ˆ dh i "h  x 1 x 2 v 1 (tt 1 ) ˆ  (h) (k) e (2)3 (2)3    k e i "  x 1 x 2 v 1 (tt1 )(v 1 v 2  h)(t1 1 ) f20 x1  v1 t



and C "k; jC1 is defined by N C "k; jC1 f jC1 (X j ; Vj ) D i

Explicitly it looks as:

h k k h C t1 C  0 1 ; x2  v2 t1   t1   0 1 ; 2 2 2 2 k k h h : (53) v1     0 ; v2 C  C  0 2 2 2 2 By the change of variables: t1  1 D "s1 ; (i. e. 1 D t1  "s1 ) ;  D (h C k)/" ; (54) we have (54) D (N  1) "3

; 0 D˙1

Z

X D˙1

Z

t



dt1 0

dh i "h (x r x k )(v r v k )(tt 1 ) ˆ  (h)e 3 (2)  f jN (X j  Vj (t  t1 ); Vj ; t1 )

(51)

0

Z

Z

t

dt1 0

t 1 /"

ds1 0

d dk ˆ ˆ (k C "1 ) (k) dx2 dv2 3 (2) (2)3   0 (: : : ) ;  e i x 1 x 2 v 1 (tt1 ) ei s 1 k(v 1 v 2 (kC")) f jC1



(55) In the limit " ! 0, the above formula gives the asymptotics Z t Z X dk 2 ˆ dv2 (52)  0 dt1 j(k)j 3 "!0 (2) 0 ; 0 D˙1 ! Z C1

 0

 dt1 S(t  t1 )Tr;k f jN (t1 ) D

X

ei s 1 k(v 1 v 2 C k) ds1  f20 x1 v1 t (  0 )

k k t1 ; x1  v1 (t  t1 )  v2 t1 C (   0 ) t1 ; 2 ! 2 k k : (56) v1 C (   0 ) ; v2  (   0 ) 2 2

In [4], we completely justify formula (56) and its forthcoming consequences.

1513

1514

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

Now, we turn to identifying the limiting value obtained in (56). To do so, we observe that symmetry arguments allow us to replace the integral in s by its real part: Z 1 Re ei s 1 k(v 1 v 2 C k) ds1 D ı(k(v1 v2 C k)): (57) 0

Using this we realize that the contribution  D  0 in (56) gives rise to the gain term: Z

Z

t

dt1 0

Z

   d! B !; v1  v2 f20 x1  v1 (t  t1 )   v10 t1 ; x2  v2 (t  t1 )  v20 t1 ; v10 ; v20 ; (58) dv2

where the integral in ! is on the surface of the unitary sphere in R3 , and B(!; v) D

1 ˆ (!  v))j2 ; j!  vj j(! 8 2

(59)

and the velocities v10 , v20 are v10 D v1  !(!  (v2  v1 ) ;

v20 D v2 C !(!  (v2  v1 ) :

Similarly, the term  D  0 in (5) yields the loss term: Z t Z Z dt1 dv2 d! B(!; v1  v2 ) 0  f20 (x1

X (60)

 v1 t; x2  v2 (t  t); v1 ; v2 ) :

By the same arguments used in the previous section we can conclude that the full series expansion (20) (of course for the present quantum case) agrees, up to the second order in the potential, with Z t S(t) f0 C dt1 S(t  t1 )Q(S(t1 ) f0 ; S(t1 ) f0 ) (61) 0

where 1 4 2

Z

Z

2 ˆ dh j(h)j ı((h  (v  v1 C h)) Z Z  [ f (v C h) f (v1  h)  f (v) f (v1 )] D dv1 d!

Q( f ; f ) D

dv1

 B(!; v  v1 )[ f 0 f10  f f1 ] ;

(62)

with f1 D f (v1 ) ; f 0 D f (v 0 ) ; f10 D f (v10 ) ; v 0 D v C h D v  !(!  (v  v1 )) ; v10

(63)

D v1  h D v1 C !(!  (v  v1 )) :

In other words the kinetic equation which comes out is the Boltzmann equation with cross-section B: (@ t C v  rx ) f D Q( f ; f ) ;

where Q is given by Eq. (62). We note once more that the ı function in Eq. (62) expresses the energy conservation, while the momentum conservation is automatically satisfied. Note also that the cross-section B is the only quantum factor in the purely classical expression (62). It retains the quantum features of the elementary “collisions”. An important comment is in order. Why the kinetic equation for quantum systems is of Boltzmann type in contrast with the classical case where we got a diffusion? The answer is related to the asymptotics of a single scattering (see [36,37] and [8]). For quantum systems we have a finite probability of having any angle scattering, while for a classical particle, we surely have a small deviation from the free motion. Therefore a quantum particle, in this asymptotic regime, is going to perform a jump process (in velocity) rather than a diffusion. From a mathematical view point we observe that [4] proves more than agreement up to second order. We indeed consider the subseries (of the full series expansion expressing f jN (t)) formed by all the collision–recollision terms (as that shown in Fig. 3). In other words, we consider the subseries of f jN (t) given by

(64)

X

"4n

Z

t

Z

0

Z

n1

tn

dt n 0

t1

dt1 0

n1 r 1 ;:::;r n ;l 1 ;:::;l n

 S(t1  1 )Tr"1 ;l 1

Z

0

d1 S(t  t1 )Cr"1 ;l 1

dn S(n1  t n )Cr"n ;l n

0  S(t n  n )Tr"n ;l n S(n ) f jCnC1 :

(65)

Here the sum runs over all possible choices of the particles number r’s and l’s, namely we sum over the subset of graph of the form in Fig. 4. We establish in [4] that the subseries (65) is indeed absolutely convergent, for short times, uniformly in ". Moreover, we prove that it approaches the corresponding complete series expansion obtained by solving iteratively the Boltzmann equation with collision operator given by Eq. (62) extending and making rigorous the above argument. Under reasonable smoothness hypotheses on the potential and on the initial distribution, assuming in addition ˆ that (0) D 0, we also proved in [7], that all other terms than those considered in the subseries (65) are indeed vanˆ ishing in the limit. The condition (0) D 0 is probably only technical: it takes care automatically of some divergences which are difficult to deal with differently. Unfortunately this result, even under these severe assumptions on the potential, is not yet conclusive because we are not able to show a uniform bound on the full series. Thus a mathematical justification of the quantum Boltzmann equation is a still an open and difficult problem.

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

It is not surprising that we know more for the quantum case in comparison with the classical problem introduced in Sect. “Introduction”. Now the operators involved are differences while, for the classical case, we have to deal with derivatives. Introducing explicitly the ¯ dependence, we easily realize that the results discussed in the present section are not uniform in ¯. Weak-Coupling Limit in the Bose–Einstein and Fermi–Dirac Statistics In this section we approach the same problem as in Sect. “Weak-Coupling Limit for Classical Systems”, for Bosons or Fermions, namely for particles obeying the Fermi–Dirac or Bose–Einstein statistics. As we shall see, the kinetic equation we expect to hold in the limit, is the so called Uehling and Uhlembeck equation, which is a Boltzmann type equation, with cubic collision operator. We note that the effects of the quantum correction here, enter also in the structure of the operator. In this case, the starting point is still the rescaled Schrödinger equation (38), or the equivalent hierarchy (46). The only new point is that we cannot take a totally decorrelated initial datum as in (49). Indeed, the Fermi–Dirac or Bose–Einstein statistics yield correlations even at time zero. In this perspective, the most uncorrelated states one can introduce, and that do not violate the Fermi–Dirac or Bose–Einstein statistics, are the so called quasi-free states. They have, in terms of the Wigner formalism, the following form f j (x1 ; v1 ; : : : ; x j ; v j ) D

X

s() f j (x1 ; v1 ; : : : ; x j ; v j ) ;

2P j

(66) where each f j has the value f j (x1 ; v1 ; : : : ; x j ; v j ) D e

i(y 1 v 1 CCy j v j )

j Y kD1

 2i w k (y k Cy (k) )

e



f

Z

Z dy1 : : : dy j

dw1 : : : dw j

i

e " w k (x k x (k) )  x k C x(k) y k  y(k) C" ; wk ; 2 4 (67)

and f is a given one-particle Wigner function. Here P j denotes the group of all the permutations of j objects and  its generic element. Note that the uncorrelated case treated in Sect. “WeakCoupling Limit for Classical Systems”, is recovered by the contribution due the permutation  D identity.

To verify that (66) and (67) really describe admissible states for the statistics, consider the inverse Wigner transform of (66) and (67). The density matrix obtained in this way does verify the permutation invariance required by the statistics. Moreover the quasi-free states converge weakly to the completely factorized states as " ! 0. This is physically obvious because the quantum statistics become irrelevant in the semiclassical limit. However the dynamics take place on the scale " so that the effects of the statistics are present in the limit. Indeed it is expected that the one-particle distribution function f1N (t) converges to the solution of the following cubic Boltzmann equation: (@ t C v  rx ) f (x; v; t) D Q ( f )(x; v; t) ;

(68)

Z Q ( f )(x; v; t) D dv1 d! B (!; v  v1 )   f (x; v 0 ) f (x; v10 )(1 C 8 3 f (x; v))(1 C 8 3 f (x; v1 ))  f (x; v) f (x; v1 )(1 C 8 3 f (x; v 0 )(1 C 8 3 f (x; v10 )) ; (69) (for the notations see Eq. (63)). Here D C1 or D 1, for the Bose–Einstein or the Fermi–Dirac statistics respectively. Finally, B is the symmetrized or antisymmetrized cross-section derived from B (see (59)) in a natural way: B (!; v) D

1 j!  vj 16 2  ˆ (!  v)) C (v ˆ  ! (!  v)) 2 :  (!

As we see, the modification of the statistics transforms the quadratic Boltzmann equation of the Maxwell– Boltzmann case, into a cubic one (fourth order terms cancel). Also, the statistics affects the form of the cross-section and B has to be (anti)symmetrized into B . The collision operator (69) has been introduced by Nordheim in 1928 [38] and by Uehling–Uhlenbeck in 1933 on the basis of purely phenomenological considerations [42]. Plugging in the hierarchy (46) an initial datum satisfying (66), we can follow the same procedure as for the Maxwell–Boltzmann statistics, namely we write the full perturbative series expansion expressing f jN (t) in terms of the initial datum and try to analyze its asymptotic behavior. As we did before, we first restrict our attention to those terms of degree less than two in the potential. The analysis up to second order is performed in [5]. We actually recover here Eq. (68), (69) with the suitable B . Now the number of terms to control is much larger due to the sum over all permutations that enters the

1515

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Scaling Limits of Large Systems of Non-linear Partial Differential Equations

Scaling Limits of Large Systems of Non-linear Partial Differential Equations, Figure 5 The three graphs of second order with the statistical permutations

definition (66) of the initial state. Also, the asymptotics is much more delicate. In particular, we stress the fact that the initial datum brings its own highly oscillatory factors in the process, contrary to the Maxwell–Boltzmann case where the initial datum is uniformly smooth, and where the oscillatory factors simply come from the collision operators T and C. ˆ In [5] we consider the graphs of second order in , show in Fig. 5, which, because of the permutation of initial state, yields various terms: two of them are bilinear in the initial condition f 0 (C12 T12 ), and give in the limit the part of Q quadratic in f , twelve are cubic in f0 (C12 C13 ; C12 C23 ). Due to a non-stationary phase argument the two terms with permutation  D id D (123) vanish, as that the term with  D (321) for C12 C13 , and the term with  D (132) for C12 C23 . The two terms with  D (321) give rise to truly diverging contributions (negative powers of "), however their sum is seen to cancel. Last, permutations without fixed point ( D (312); (231)) and  D (132) for C12 C13 , and  D (321) for C12 C23 , give the part of Q cubic in f . This ends up the analysis of terms up to second order in the potential. The computation is heavy and hence we address the reader to [5] for the details. We mention that a similar second order analysis, using commutator expansions in the framework of the second quantization formalism, has been performed in [27] (following [26]) in the case of the van Hove limit for lattice systems (that is the same as the weak-coupling limit, yet without rescaling the distances). For more recent formal results in this direction, but in the context of the weakcoupling limit, we also quote [21].

We finally observe that the initial value problem for Eq. (68) is somehow trivial for Fermions. Indeed we have the a priori bounds f  1 / (8)3 making everything easy. For Bosons the situation is much more involved even for the spatially homogeneous case. The statistics favor large value of f and it is not clear whether the equation can explain dynamical condensation. See, for the mathematical side, [33,34]. Under the weak coupling limit we derived two very different equations, the Fokker–Planck–Landau equation in the classical case, and a Boltzmann equation with a quantum cross section in the quantum case. For the original large particle system a classical or a quantum description depends on the value of the physical scales of the macroscopic variables with respect to ¯. The same happen for the two asymptotic equations. The dependence on ¯ of the collision term (69) is given by

 Z Z  1 ˆ ! (!w) QU U ( f )(v) D dv d!  1 „ 16 2 „4   2 w! (!w) ˆ f(1 ˙ (2„)3 f )(1 ˙ (2„)3 f1 ) C  „  f 0 f10  (1 ˙ (2„)3 f 0 )(1 ˙ (2„)3 f10 ) f f1 g : (70) ˆ (!  (v  v1 )) / „) The quantum-cross section term (! make !  (v  v1 ) D O(„), so that the collision operator concentrates on grazing collisions. In this sense, the classical limit „ ! 0 for (70) is a natural grazing collision limit. More formally, the operator (70) tends to the operator QL given in Eq. (29) (see [9]); nevertheless no results are available for the limit of the solutions of the equations.

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

We conclude this section with some consideration on the low-density limit. In the classical case the low-density limit (or the Boltzmann–Grad limit) yields the usual Boltzmann equation for classical systems and this result has been proved for short times [31]. It is natural to investigate what happens, in the same scaling limit, to a quantum system. Here, due to the fact that the density is vanishing, the particles are too rare to make the statistical correlations effective. As a consequence, we expect that the Bose–Einstein, and Fermi– Dirac statistics, as well fully uncorrelated states, all give rise to the same Boltzmann equation along the low-density limit with the true quantum cross-section, given by the sum of the Born series, under smallness assumption for the potential. The analysis of the partial series of the dominant terms (uniform bounds and convergence as for the weakcoupling limit) has been performed in [6]. Weak-Coupling Limit for a Single Particle: The Linear Theory Consider the time evolution of a single particle, in the lowdensity or in the weak-coupling regime, under the action of a random configuration of obstacles c D fc1 : : : c N g

R3N . More precisely, after the rescaling, the basic equations are: X x˙ (t) D v(t) ; v˙(t) D  r" (x(t)  c j ) (71) j

for a classical particle and, for a quantum particle, i"@ t

D

X "2 " (x(t)  c j )  C 2

;

(72)

j

p where " D ( x" ) and " D "( x" ) for the low-density and weak-coupling limits respectively. Here  denotes a given smooth potential. We are interested in the behavior of f" (x; v; t) D E[ fc (x; v; t)]

(73)

where fc (t) is the time evolved classical distribution function or the Wigner transform of according to Eq. (71) or (72) respectively. Finally E denotes the expectation with respect to the obstacle distribution, for which, a natural choice could be the Poisson distribution of density " which is also scaled according to " D "2 ;

" D "3

for the low-density and weak-coupling respectively.

For the low-density scaling (this is the so called Lorentz model), we obtain, for classical systems, a linear Boltzmann equation (see [3,13,17,23,40]). It is also known that the system does not homogenize to a jump process given by a linear Boltzmann equation in case of a periodic distribution of obstacles [10]. For recent results in this direction see [11]. For the weak-coupling limit of a classical particle we obtain a linear Landau equation as it is shown in [29] and [16]. Let us spend a few words on these results. Each solution (x(t); y(t)) D (xc (t); vc (t)) of Eq. (71) is a sample of a stochastic process. Looking at the behavior of vc (t) we suddenly realize that it does not enjoy the Markov property because, once an obstacle produces a change in velocity, we know its existence and this creates time correlations with the future behavior of the particle. However if a recollision with the same obstacle is an unlikely event, we could consider the velocity change due to the collisions, as independent and Poisson distributed (as the obstacles) random variables. In this case it is not difficult to prove the convergence to a diffusion process, solution to the following stochastic differential equation dx D vdt ; dv D 2A jvjv 3 dt C

q

2A jvj



I

v˝v jvj2



dw ;

(74)

where dw is a brownian motion, A qis the 2A  constant given by Eq. (27), and jvj p   2 I  v ˝ v/jvj is the matrix 2a, with a given by Eq. (26). Notice that each collision preserves the energy so that the above process lives on the surface of the sphere jvj D const. Therefore the main point in the proof of the weakcoupling limit is to show that, in the same limit, the process under consideration is stochastically equivalent to a Markov process with the right properties. Moreover, as a consequence, we have also that f" (t) ! f (t) being f solution to the following linear Landau equation (@ t C v  rx ) f D divv [a(v)rv f ] ;

(75)

Alternatively we could proceed differently, in the same (Cauchy–Kovalevskii) spirit of our previous analysis for the nonlinear case. Namely we first represent the time evolved distribution fc (t) in terms of a series expansion. Then, taking the expectation of each term of the series, we exploit its limit by using the same arguments as in Sect. “Introduction”. Finally the convergence of the series (uniformly in ") should also be proven. This is certainly a possible program which is, however, conceptually

1517

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Scaling Limits of Large Systems of Non-linear Partial Differential Equations

and technically weaker than that based on the control of the particle trajectory illustrated above. Indeed the absolute control of the series expansion yielding the solution to Eq. (75) does not use the positivity of the diffusion coefficient so that, at best, we could recover the result for a short time only and in spaces of analytic functions, not at all natural in our context. This remark applies as well to the nonlinear case for which, however, we do not have any result, even for short times. As regards the corresponding weak-coupling quantum problem, the easiest case is when  is a Gaussian process. The kinetic equation is still a linear Boltzmann equation. The first result, holding for short times, has been obtained in [41] (see also [30]). More recently this result has been extended to arbitrary times [22]. The technique of [22] can be also applied to deal with a Poisson distribution of obstacles [12] Obviously the cross section appearing in the Boltzmann equation is the one computed in the Born approximation. Finally in [20] the low-density case has been successfully approached. The result is a linear Boltzmann equation with the full cross-section. Future Directions The program of obtaining macroscopic equations from the microscopic dynamics is an ambitious and difficult problem. It arised in 1900 with the Hilbert’s speech at the congress of mathematicians in Paris, as the sixth problem. After more than hundred years, such a program is far to be achieved. It is probably true that new ideas and techniques are needed. As we said before in discussing the linear problems, it may be quite possible that classical ideas in the field of partial differential equations are not enough to approach this kind of problems successfully. On the other hand a deeper and more detailed analysis of the particle dynamics is technically difficult and philosophically paradoxical. Indeed from the practical side we introduce a partial differential equation to reduce the difficulties of the study of particle evolution, while its justification may require a deeper control of the underlying microscopic dynamics. Bibliography Primary Literature 1. Arseniev AA, Buryak OE (1990) On a connection between the solution of the Boltzmann equation and the solution of the Landau–Fokker–Planck equation (Russian). Mat Sb 181(4):435– 446 (translation in Math USSR-Sb 69(2):465–478 1991)

2. Balescu R (1975) Equilibrium and Nonequilibrium Statistical Mechanics. Wiley, New York 3. Boldrighini C, Bunimovich LA, Ya Sinai G (1983) On the Boltzmann equation for nthe Lorentz gas. J Stat Phys 32:477–501 4. Benedetto D, Castella F, Esposito R, Pulvirenti M (2004) Some Considerations on the derivation of the nonlinear Quantum Boltzmann Equation. J Stat Phys 116(114):381–410 5. Benedetto D, Castella F, Esposito R, Pulvirenti M (2005) On The Weak–Coupling Limit for Bosons and Fermions. Math Mod Meth Appl Sci 15(12):1–33 6. Benedetto D, Castella F, Esposito R, Pulvirenti M (2006) Some Considerations on the derivation of the nonlinear Quantum Boltzmann Equation II: the low-density regime. J Stat Phys 124(2–4):951–996 7. Benedetto D, Castella F, Esposito R, Pulvirenti M (2008) From the N-body Schrödinger equation to the quantum Boltzmann equation: a term-by-term convergence result in the weak coupling regime. Commun Math Phys 277(1):1–44 8. Benedetto D, Esposito R, Pulvirenti M (2004) Asymptotic analysis of quantum scattering under mesoscopic scaling. Asymptot Anal 40(2):163–187 9. Benedetto D, Pulvirenti M (2007) The classical limit for the Uehling–Uhlenbeck operator. Bull Inst Math Acad Sinica 2(4):907–920 10. Burgain J, Golse F, Wennberg B (1998) On the distribution of free path lenght for the periodic Lorentz gas. Comm Math Phys 190:491–508 11. Caglioti E, Golse F (2003) On the distribution of free path lengths for the periodic Lorentz gas. III Comm Math Phys 236(2):199–221 12. Chen T (2005) Localization lengths and Boltzmann limit for the Anderson model at small disorders in dimension 3. J Stat Phys 120(1–2):279–337 13. Caglioti E, Pulvirenti M, Ricci V (2000) Derivation of a linear Boltzmann equation for a periodic Lorentz gas. Mark Proc Rel Fields 3:265–285 14. Cercignani C, Illner R, Pulvirenti M (1994) The mathematical theory of dilute gases, Applied Mathematical Sciences, vol 106. Springer, New York 15. Degond P, Lucquin–Desreux B (1992) The Fokker–Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math Models Methods Appl Sci 2(2):167–182 16. Dürr D, Goldstain S, Lebowitz JL (1987) Asymptotic motion of a classical particle in a random potential in two dimension: Landau model. Comm Math Phys 113:209–230 17. Desvillettes L, Pulvirenti M (1999) The linear Boltzmann eqaution for long-range forces: a derivation for nparticle systems. Math Moduls Methods Appl Sci 9:1123–1145 18. Di Perna RJ, Lions PL (1989) On the Cauchy problem for the Boltzmann equatioin. Ann Math 130:321–366 19. Esposito R, Pulvirenti M (2004) From particles to fluids. In: Friedlander S, Serre D (eds) Handbook of Mathematical Fluid Dynamics, vol 3. Elsevier, North Holland, pp 1–83 20. Eng D, Erdös L (2005) The linear Boltzmann equation as the low-density limit of a random Schrödinger equation. Rev Math Phys 17(6):669–743 21. Erdös L, Salmhofer M, Yau HT (2004) On the quantum Boltzmann equation. J Stat Phys 116:367–380 22. Erdös L, Yau HT (2000) Linear Boltzmann equation as a weakcoupling limit of a random Schrödinger equation. Comm Pure Appl Math 53:667–735

Scaling Limits of Large Systems of Non-linear Partial Differential Equations

23. Gallavotti G (1972) Rigorous theory of the Boltzmann equation in the Lorentz gas in Meccanica Statistica. reprint Quaderni CNR 50:191–204 24. Goudon T (1997) On Boltzmann equations and Fokker–Planck asymptotics: influence of grazing collisions. J Statist Phys 89(3– 4):751–776 25. Grad H (1949) On the kinetic Theory of rarefied gases. Comm Pure Appl Math 2:331–407 26. Hugenholtz MN (1983) Derivation of the Boltzmann equation for a Fermi gas. J Stat Phys 32:231–254 27. Ho NT, Landau LJ (1997) Fermi gas in a lattice in the van Hove limit. J Stat Phys 87:821–845 28. Illner R, Pulvirenti M (1986) Global Validity of the Boltzmann equation for a two-dimensional rare gas in the vacuum. Comm Math Phys 105:189–203 (Erratum and improved result, Comm Math Phys 121:143–146) 29. Kesten H, Papanicolaou G (1981) A limit theorem for stochastic acceleration. Comm Math Phys 78:19–31 30. Landau LJ (1994) Observation of quantum particles on a large space-time scale. J Stat Phys 77:259–309 31. Lanford III O (1975) Time evolution of large classical systems. In: Moser EJ (ed) Lecture Notes in Physics 38. Springer, pp 1– 111 32. Lifshitz EM, Pitaevskii LP (1981) Course of theoretical physics “Landau–Lifshit”, vol 10. Pergamon Press, Oxford-Elmsford 33. Xuguang LU (2005) The Boltzmann equation for Bose–Einstein particles: velocity concentration and convergence to equilibrium. J Stat Phys 119(5–6):1027–1067 34. Xuguang LU (2004) On isotropic distributional solutions to the Boltzmann equation for Bose–Einstein particles. J Stat Phys 116(5–6):1597–1649 35. CB Jr Morrey (1955) On the derivation of the equations of hydrodynamics from statistical mechanics. Comm Pure Appl Math 8:279–326 36. Nier F (1996) A semi-classical picture of quantum scattering. Ann Sci Ec Norm Sup 29(4):149–183 37. Nier F (1995) Asymptotic analysis of a scaled Wigner equation and quantum scattering. Transp Theory Statist Phys 24(4– 5):591–628 38. Nordheim LW (1928) On the Kinetic Method in the New Statis-

39. 40. 41.

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tics and Its Application in the Electron Theory of Conductivity. Proc Royal Soc Lond Ser A 119(783):689–698 Spohn H (1991) Large scale dynamics of interacting particles, Texts and monographs in physics. Springer Spohn H (1978) The Lorentz flight process converges to a random flight process. Comm Math Phys 60:277–290 Spohn H (1977) Derivation of the transport equation for electrons moving through random impurities. J Stat Phys 17:385– 412 Uehling EA, Uhlembeck GE (1933) Transport Phenomena in Einstein–Bose and Fermi–Dirac Gases. Phys Rev 43:552– 561 Villani C (2002) A review of mathematical topics in collisional kinetic theory. In: Friedlander S, Serre D (eds) Handbook of Mathematical Fluid Dynamics, vol 1. Elsevier, North Holland, pp 71–307 Villani C (1998) On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch Rational Mech Anal 143(3):273–307 Wigner E (1932) On the quantum correction for the thermodynamical equilibrium. Phys Rev 40:742–759

Books and Reviews Balescu R (1975) Equilibrium and Nonequilibrium Statistical Mechanics. Wiley, New York Cercignani C, Boltzmann L (1998) The man who trusted atoms. Oxford University Press, Oxford Cercignani C, Illner R, Pulvirenti M (1994) The mathematical theory of dilute gases. Applied Mathematical Sciences, vol 106. Springer, New York Esposito R, Pulvirenti M (2004) From particles to fluids. In: Friedlander S, Serre D (eds) Handbook of Mathematical Fluid Dynamics, vol 3. Elsevier, North Holland, pp 1–83 Spohn H (1991) Large scale dynamics of interacting particles, Texts and monographs in physics. Springer, Heidelberg Villani C (2002) A review of mathematical topics in collisional kinetic theory. In: Friedlander S, Serre D (eds) Handbook of Mathematical Fluid Dynamics, vol 1. Elsevier, North Holland, pp 71– 307

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Shallow Water Waves and Solitary Waves

Shallow Water Waves and Solitary Waves W ILLY HEREMAN Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, USA Article Outline Glossary Definition of the Subject Introduction Completely Integrable Shallow Water Wave Equations Shallow Water Wave Equations of Geophysical Fluid Dynamics Computation of Solitary Wave Solutions Water Wave Experiments and Observations Future Directions Acknowledgments Bibliography Glossary Deep water A surface wave is said to be in deep water if its wavelength is much shorter than the local water depth. Internal wave A internal wave travels within the interior of a fluid. The maximum velocity and maximum amplitude occur within the fluid or at an internal boundary (interface). Internal waves depend on the densitystratification of the fluid. Shallow water A surface wave is said to be in shallow water if its wavelength is much larger than the local water depth. Shallow water waves Shallow water waves correspond to the flow at the free surface of a body of shallow water under the force of gravity, or to the flow below a horizontal pressure surface in a fluid. Shallow water wave equations Shallow water wave equations are a set of partial differential equations that describe shallow water waves. Solitary wave A solitary wave is a localized gravity wave that maintains its coherence and, hence, its visibility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and propagate with constant speed and constant shape. Soliton Solitons are solitary waves that have an elastic scattering property: they retain their shape and speed after colliding with each other. Surface wave A surface wave travels at the free surface of a fluid. The maximum velocity of the wave and the maximum displacement of fluid particles occur at the free surface of the fluid.

Tsunami A tsunami is a very long ocean wave caused by an underwater earthquake, a submarine volcanic eruption, or by a landslide. Wave dispersion Wave dispersion in water waves refers to the property that longer waves have lower frequencies and travel faster.

Definition of the Subject The most familiar water waves are waves at the beach caused by wind or tides, waves created by throwing a stone in a pond, by the wake of a ship, or by raindrops in a river (see Fig. 1). Despite their familiarity, these are all different types of water waves. This article only addresses shallow water waves, where the depth of the water is much smaller than the wavelength of the disturbance of the free surface. Furthermore, the discussion is focused on gravity waves in which buoyancy acts as the restoring force. Little attention will we paid to capillary effects, and capillary waves for which the primary restoring force is surface tension are not covered. Although the history of shallow water waves [15,20,22] goes back to French and British mathematicians of the eighteenth and early nineteenth century, Stokes [71] is considered one of the pioneers of hydrodynamics (see [21]). He carefully derived the equations for the motion of incompressible, inviscid fluid, subject to a constant vertical gravitational force, where the fluid is bounded below by an impermeable bottom and above by a free surface. Starting from these fundamental equations and by making further simplifying assumptions, various shallow water

Shallow Water Waves and Solitary Waves, Figure 1 Capillary surface waves from raindrops. Photograph courtesy of E. Scheller and K. Socha

Shallow Water Waves and Solitary Waves

wave models can be derived. These shallow water models are widely used in oceanography and atmospheric science. This article discusses shallow water wave equations commonly used in oceanography and atmospheric science. They fall into two major categories: Shallow water wave models with wave dispersion are discussed in Sect. “Completely Integrable Shallow Water Wave Equations” Most of these are completely integrable equations that admit smooth solitary and cnoidal wave solutions for which computational procedures are outlined in Sect. “Computation of Solitary Wave Solutions”. Sect. “Shallow Water Wave Equations of Geophysical Fluid Dynamics” covers classical shallow water wave models without dispersion. Such hyperbolic systems can admit shocks. Sect. “Water Wave Experiments and Observations” addresses a few experiments and observations. The article concludes with future directions in Sect. “Future Directions”. Introduction The initial observation of a solitary wave in shallow water was made by John Scott Russell, shown in Fig. 2. Russell was a Scottish engineer and naval architect who was conducting experiments for the Union Canal Company to design a more efficient canal boat. In Russell’s [63] own words: “I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation.” Russell built a water tank to replicate the phenomenon and research the properties of the solitary wave he had observed. Details can be found in a biography of John Scott Russell (1808–1882) by Emmerson [27], and in review articles by Bullough [15], Craik [20], and Darrigol [22], who pay tribute to Russell’s research of water waves.

Shallow Water Waves and Solitary Waves, Figure 2 John Scott Russell. Source: [27]. Courtesy of John Murray Publishers

In 1895, the Dutch professor Diederik Korteweg and his doctoral student Gustav de Vries [47] derived a partial differential equation (PDE) which models the solitary wave that Russell had observed. Parenthetically, the equation which now bears their name had already appeared in seminal work on water waves published by Boussinesq [13,14] and Rayleigh [59]. The solitary wave was considered a relatively unimportant curiosity in the field of nonlinear waves. That all changed in 1965, when Zabusky and Kruskal realized that the KdV equation arises as the continuum limit of a one dimensional anharmonic lattice used by Fermi, Pasta, and Ulam [29] to investigate “thermalization” – or how energy is distributed among the many possible oscillations in the lattice. Zabusky and Kruskal [78] simulated the collision of solitary waves in a nonlinear crystal lattice and observed that they retain their shapes and speed after collision. Interacting solitary waves merely experience a phase shift, advancing the faster and retarding the slower. In analogy with colliding particles, they coined the word “solitons” to describe these elastically colliding waves. A narrative of the discovery of solitons can be found in [77]. Since the 1970s, the KdV equation and other equations that admit solitary wave and soliton solutions have

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Shallow Water Waves and Solitary Waves, Figure 3 Recreation of a solitary wave on the Scott Russell Aqueduct on the Union Canal. Photograph courtesy of Heriot–Watt University

been the subject of intense study (see, e. g., [23,30,60]). Indeed, scientists remain intrigued by the physical properties and elegant mathematical theory of the shallow water wave models. In particular, the so-called completely integrable models which can be solved with the Inverse Inverse scattering transform (IST). For details about the IST method the reader is referred to Ablowitz et al. [3], Ablowitz and Segur [2], and Ablowitz and Clarkson [1]. The completely integrable models discussed in the next section are infinite-dimensional Hamiltonian systems, with infinitely many conservation laws and higher-order symmetries, and admit soliton solutions of any order. As an aside, in 1995, scientists gathered at Heriot– Watt University for a conference and successfully recreated a solitary wave but of smaller dimensions than the one observed by Russell 161 years earlier (see Fig. 3). Completely Integrable Shallow Water Wave Equations Starting from Stokes’ [71] governing equations for water waves, completely integrable PDEs arise at various levels of approximation in shallow water wave theory. Four length scales play a crucial role in their derivation. As shown in Fig. 4, the wavelength  of the wave measures the distance between two successive peaks. The amplitude a measures the height of the wave, which is the varying distance between the undisturbed water to the peak of the wave. The depth of the water h is measured from the (flat) bottom of the water up to the quiescent free surface. The fourth length scale is along the Y-axis which is along the crest of the wave and perpendicular to the (X, Z)-plane. Assuming wave propagation in water of uniform (shallow) depth, i. e. h is constant, and ignoring dissipation,

Shallow Water Waves and Solitary Waves, Figure 4 Coordinate frame and periodic wave on the surface of water

the model equations discussed in this section have a set of common features and limitations which make them mathematically tractable [68]. They describe (i) long waves (or shallow water), i. e. h  , (ii) with relatively small amplitude, i. e. a  h, (iii) traveling in one direction (along the X-axis) or weakly two-dimensional (with a small component in the Y-direction). Furthermore, the small effects must be comparable in size. For example, in the derivation of the KdV and Boussinesq equations one assumes that " D a/h D O(h2 /2 ), where " is a small parameter ("  1), and O indicates the order of magnitude. The Korteweg–de Vries Equation The KdV equation was originally derived to describe shallow water waves of long wavelength and small amplitude. In the derivation, Korteweg and de Vries assumed that all motion is uniform in the Y-direction, along the crest of the wave. In that case, the surface elevation (above the equilibrium level h) of the wave, propagating in the X-direction, is a function only of the horizontal position X (along the canal) and of time T, i. e. Z D (X; T): In terms of the physical parameters, the KdV equation reads p @ p @ 3 gh @ C gh C  @T @X 2 h @X   3 1 @  T 1 2p D0;  C h gh 2 3 gh2 @X 3

(1)

where h is the uniform water depth, g is the gravitational acceleration (about 9:81m/sec2 at sea level), is the density, and T stands for the surface tension. The dimensionless parameter T / gh2 is called the Bond number which

Shallow Water Waves and Solitary Waves

measures the relative strength of surface tension and the gravitational force. Keeping only the first two terms inp(1), the speed of the associated linear (long) wave is c D gh. This is indeed the maximum attainable speed of propagation of gravityinduced water waves of infinitesimal amplitude. The speed of propagation of the small-amplitude solitary waves described by (1) is slightly higher. According to Russell’s emp pirical formula the speed equals g(h C k); where k is the height of the peak of the solitary wave above the surface of undisturbed water. As Bullough [15] has shown, Russell’s approximate speed and the true speed of solitary waves only differ by a term of O(k 2 /h2 ): The KdV equation can be recast in dimensionless variables as u t C ˛uu x C u x x x D 0 ;

(2)

where subscripts denote partial derivatives. The parameter ˛ can be scaled to any real number. Commonly used values are ˛ D ˙1 or ˛ D ˙6. The term ut describes the time evolution of the wave propagating in one direction. Therefore, (2) is called an evolution equation. The nonlinear term ˛uu x accounts for steepening of the wave, and the linear dispersive term uxxx describes spreading of the wave. The linear first-order p @ term gh @X in (1) can be removed by an elementary transformation. Conversely, a linear term in ux can be added to (2). The nonlinear steepening of the water wave can be balanced by dispersion. If so, the result of these counteracting effects is a stable solitary wave with particle-like properties. A solitary wave has a finite amplitude and propagates at constant speed and without change in shape over a fairly long distance. This is in contrast to the concentric group of small-amplitude capillary waves, shown in Fig. 1, which disperse as they propagate. The closed-form expression of a solitary wave solution is given by 12k 2 !  4k 3 (3) C sech2 (kx  ! t C ı) ˛k ˛ ! C 8k 3 12k 2 D  tanh2 (kx  ! t C ı) ; (4) ˛k ˛ where the wave number k, the angular frequency !, and ı are arbitrary constants. Requiring that limx!˙1 u(x; t) D 0 for all time leads to ! D 4k 3 . Then (3) and (4) reduce to u(x; t) D

12k 2 sech2 (kx  4k 3 t C ı) ˛ 12k 2 D [1  tanh2 (kx  4k 3 t C ı)] : ˛

u(x; t) D

Shallow Water Waves and Solitary Waves, Figure 5 Solitary wave (red) and periodic cnoidal (blue) wave profiles

The position of the hump-type wave at t D 0 is depicted in Fig. 5 for ˛ D 6; k D 2, and ı D 0. As time changes, the solitary wave with amplitude 2k 2 D 8 travels to the right at speed v D !/k D 4k 2 D 16: The speed is exactly twice the peak amplitude. So, the higher the wave the faster it travels, but it does so without change in shape. The reciprocal of the wavenumber k is a measure of the width of the sechsquare pulse. As shown by Korteweg and de Vries [47], Eq. (2) also has a simple periodic solution, u(x; t) D !  4k 3 (2m  1) 12k 2 m 2 C cn (kx  ! t C ı; m) ; ˛k ˛

which they called the cnoidal wave solution since they involve the Jacobi elliptic cosine function, cn, with modulus m (0 < m < 1). The wavenumber k gives the characteristic width of each oscillation in the “cnoid.” Three cycles of the cnoidal wave are depicted in Fig. 5 at t D 0. The graph corresponds to ˛ D 6; k D 2; m D 3/4; ! D 16; and ı D 0. Using the property lim m!1 cn(; m) D sech(); one readily verifies that (6) reduces to (3) as m tends to 1. Pictorially, the individual oscillations then stretch infinitely far apart leaving a single-pulse solitary wave. The celebrated KdV equation appears in all books and reviews about soliton theory. In addition, the equation has been featured in, e. g., Miura [52] and Miles [51]. Regularized Long-Wave Equations A couple of alternatives to the KdV equation have been proposed. A first alternative, u t C u x C ˛uu x  u x x t D 0 ;

(5)

(6)

(7)

was proposed by Benjamin, Bona, and Mahony [7]. Hence,

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(7) is referred to as the BBM or regularized long-wave (RLW) equation. Equation (7), which has a solitary wave solution, u(x; t) D

!  k  4k 2 ! 12k! C sech2 (kx ! tCı); (8) ˛k ˛

was also derived by Peregrine [57] to describe the behavior of an undular bore (in water), which comprises a smooth wavefront followed by a train of solitary waves. An undular bore can be interpreted as the dispersive analog of a shock wave in classical non-dispersive, dissipative hydrodynamics [26]. The linear dispersion relation for the KdV equation, ! D k(1  k 2 ), can be obtained by substituting u(x; t) D exp[i(kx  ! t C ı)] into u t C u x C u x x x D 0. The linear phase velocity, v p D !/k D 1  k 2 , becomes negative for jkj > 1, thereby contradicting the assumption of unidirectional propagation. Furthermore, the group velocity v g D d!/dk D 1  3k 2 has no lower bound which implies that there is no limit to the rate at which shorter ripples propagate in the negative x-direction. The BBM equation, where ! D k/(1 C k 2 ); v p D 1/(1 C k 2 ), and v g D (1  k 2 )/(1 C k 2 )2 ; was proposed to get around these objections and to address issues related to proving the existence of solutions of the KdV equation. The dispersion relation of (7) has more desirable properties for high wave numbers, but the group velocity becomes negative for jkj > 1: In addition, the KdV and BBM equations are first order in time making it impossible to specify both u and ut as initial data. To circumvent these limitations, a second alternative, u t C u x C ˛uu x C u x t t D 0 ;

over a long time scale, provided the initial data is properly imposed (see also [9]). Fine-tuning the dispersion relation of the KdV equation comes at a cost. In contrast to (2), the RLW and TRLW equations are no longer completely integrable. Perhaps that is why these equations never became as popular as the KdV equation. The Boussinesq Equation The classical Boussinesq equation, T T c 2  X X 

was derived by Boussinesq [12] to describe gravity-induced surface p waves as they propagate at constant (linear) speed c D gh in a canal of uniform depth h. In contrast to the KdV equation, (11) has a secondorder time-derivative term. Ignoring all but the first two terms in (11), one obtains the linear wave equation, T T  c 2  X X D 0; which describes both left-running and rightrunning waves. However, (11) is not bi-directional because in the derivation Boussinesq used the constraint T D c X , which limits (11) to waves traveling to the right. In dimensionless form, the Boussinesq equation reads u t t  c 2 u x x  ˛u2x  ˛uu x x  ˇu x x x x D 0 :

u(x; t) D

u(x; t) D

!  k  4k! 2 12! 2 C sech2 (kx ! tCı): (10) ˛k ˛

The TRLW equation shares many of the properties of both the KdV and BBM equations, at the costpof a more complicated dispersion relation, ! D (1 ˙ 1 C 4k 2 )/2k with two branches. Only one of these branches is valid because the derivation of the TRLW equation shows that (9) is unidirectional, despite the two time derivatives in u x t t . Bona and Chen [8] have shown that the initial value problem for the TRLW equation is well-posed, and that for small-amplitude, long waves, solutions of (9) agree with solutions of either (2) or (7). As a matter of fact, all three equations agree to the neglected order of approximation

(12)

The values of the parameters c; ˛ > 0; and ˇ do not matter, but the sign of ˇ matters. Typically, one sets c D 1; ˛ D 3, and ˇ D ˙1. A simple solitary wave solution of (12) is given by

(9)

was proposed by Joseph and Egri [45] and Jeffrey [43]. It is called the time regularized long-wave (TRLW) equation and its solitary wave solution is given by

 c 2 h2 3c 2  2  X C  X X   X X X X D 0; (11) h 3

! 2  c 2 k 2  4ˇk 4 ˛k 2 12ˇk 2 C sech2 (kx  ! t C ı) : ˛

(13)

The equation with ˇ D 1 is a scaled version of (11) and thus most relevant to shallow water wave theory. Mathematically, (12) with ˇ D 1 is ill-posed, (even without the nonlinear terms), which means that the initial value problem cannot be solved for arbitrary data. This shortcoming does not happen for (12) with ˇ D 1, which is therefore nicknamed the “good” Boussinesq equation [50]. Nonetheless, the classical and good Boussinesq equations are completely integrable. The “improved” or “regularized” Boussinesq equation (see, e. g., [10]) has ˇ D 1 but uxxtt instead of u x x x x , which improves the properties of the dispersion relation. Like (12), the regularized version describes uni-directional waves. The regularized Boussinesq equation and other al-

Shallow Water Waves and Solitary Waves

ternative equations listed in the literature (see, e. g., Madsen and Schäffer [49]) are not completely integrable. Bona et al. [10,11] analyzed a family of Boussinesq systems of the form w t C u x C (uw)x C ˛u x x x  ˇw x x t D 0 ; u t C w x C uu x C w x x x  ıu x x t D 0 ;

1D Shallow Water Wave Equation The so-called one-dimensional (1D) shallow water wave equation, Z x v t (y; t)d y D 0 ; (15) v x x t C ˛vv t  v t  v x C ˇv x 1

can be derived from the classical shallow water wave theory (see Sect. “Shallow Water Wave Equations of Geophysical Fluid Dynamics”) in the Boussinesq approximation. In that approximation one assumes that vertical variations of the static density, 0 , are neglected, except the buoyancy term proportional to d 0 /dz; which is, in fact, responsible for the existence of solitary waves. The integral term in (15) can be removed by introducing the potential u. Indeed, setting v D u x , Eq. (15) can be written as

u t C 2u x C 3uu x  ˛ 2 u x x t C u x x x  2˛ 2 u x u x x  ˛ 2 uu x x x D 0 ;

(17)

Closed-form solutions of (15), and in particular of (17), have been computed by Clarkson and Mansfield [19].

(18)

also models waves in shallow water. In (18), u is the fluid velocity in the x-direction or, equivalently, the height of the water’s free surface above a flat bottom, and ; and ˛ are constants. Retaining only the first four terms in (18) gives the BBM Eq. (7). Setting ˛ D 0 reduces (18) to the KdV equation. The CH equation admits solitary wave solutions, but in contrast to the hump-type solutions of the KdV and Boussinesq equations, they are implicit in nature (see, e. g., [44]). In the limit  ! 0, Eq. (18) with D 0; ˛ D 1 has a cusp-type solution of the form u(x; t) D c exp(jx  ct  x0 j). The solution is called a peakon since it has a peak (or corner) where the first derivatives are discontinuous. The solution travels at speed c > 0 which equals the height of the peakon. The Kadomtsev–Petviashvili Equation In their 1970 study [46] of the stability of line solitons, Kadomtsev and Petviashvili (KP) derived a 2D-generalization of the KdV equation which now bears their name. In dimensionless variables, the KP equation is (u t C ˛uu x C u x x x )x C  2 u y y D 0 ;

(19)

where y is the transverse direction. In the derivation of the KP equation, one assumes that the scale of variation in the y-direction (along the crest of the wave as shown in Fig. 4) is much longer than the wavelength along the x-direction. The solitary wave and periodic (cnoidal) solutions of (19) are, respectively, given by u(x; t) D

k!  4k 4 C  2 l 2 12k 2 C sech2 (kxCl y! tCı); ˛k 2 ˛ (20)

and

(16)

The equation is completely integrable and can be solved with the IST if and only if either ˛ D ˇ [41] or ˛ D 2ˇ [3]. When ˛ D ˇ; Eq. (16) can be integrated with respect to x and thus replaced by u x x t C ˛u x u t  u t  u x D 0 :

The CH equation, named after Camassa and Holm [16, 17],

(14)

which follow from the Euler equations as first-order approximations in the parameters "1 D a/h  1; "2 D h2 /2  1, where the Stokes number, S D "1 /"2 D a2 /h3  1. In (14) w(x, t) is the non-dimensional deviation of the water surface from its undisturbed position; u(x, t) is the non-dimensional horizontal velocity field at a height h (with 0   1) above the flat bottom of the water. The constant parameters ˛ through ı in (14) satisfy the following consistency conditions: ˛ C ˇ D 12 ( 2  13 ) and

C ı D 12 (1  2 )  0. Solitary wave solutions of various special cases of (14) have been computed by Chen [18]. Boussinesq systems arise when modeling the propagation of long-crested waves on large bodies of water (such as large lakes or the ocean). The Boussinesq family (14) encompasses many systems that appeared in the literature. Special cases and properties of well-posedness of (14) are addressed by Bona et al. [10,11].

u x x x t C ˛u x u x t  u x t  u x x C ˇu x x u t D 0 :

The Camassa–Holm Equation

u(x; t) D

k!  4k 4 (2m  1)   2 l 2 ˛k 2 12k 2 m 2 C cn (kx C l y  ! t C ı; m) : ˛

(21)

As shown in Fig. 6, near a flat beach the periodic waves appear as long, quasilinear stripes with a cn-squared cross section. Such waves are typically generated by wind and tides.

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Shallow Water Waves and Solitary Waves, Figure 6 Periodic plane waves in shallow water, off the coast of Lima, Peru. Photograph courtesy of A. Segur Shallow Water Waves and Solitary Waves, Figure 7 Setup for the geophysical shallow water wave model

The equation with  2 D 1 is referred to as KP1, whereas (19) with  2 D 1 is called KP2, which describes shallow water waves [67]. Both KP1 and KP2 are completely integrable equations but their solution structures are fundamentally different (see, e. g., [65], pp. 489–490). Shallow Water Wave Equations of Geophysical Fluid Dynamics The shallow water equations used in geophysical fluid dynamics are based on the assumption D/L  1, where D and L are characteristic values for the vertical and horizontal length scales of motion. For example, D could be the average depth of a layer of fluid (or the entire fluid) and L could be the wavelength of the wave. The geophysical fluid dynamics community (see, e. g., [55,72,73]) uses the following 2D shallow water equations,

ics and that the number of independent variables has been reduced by one. Indeed, z no longer explicitly appears in (22)–(24), where u, v, and h only depend on x, y, and t. A shortcoming of the model is that it does not take into account the density stratification which is present in the atmosphere (as well as in the ocean). Nonetheless, (22)–(24) are commonly used by atmospheric scientists to model flow of air at low speed. More sophisticated models treat the ocean or atmosphere as a stack of layers with variable thickness. Within each layer, the density is either assumed to be uniform or may vary horizontally due to temperature gradients. For example, Lavoie’s rotating shallow water wave equations (see [24]), u t C (u  r )u C 2˝  u D r (h ) C 12 hr ;

(25)

u t C uu x C vu y C gh x  2˝v D gb x ;

(22)

h t C r  (hu) D 0 ;

(26)

v t C uv x C vv y C gh y C 2˝u D gb y ;

(23)

t C u  (r ) D 0 ;

(27)

h t C (hu)x C (hv) y D 0;

(24)

to describe water flows with a free surface under the influence of gravity (with gravitational acceleration g) and the Coriolis force due to the earth’s rotation (with angular velocity ˝.) As usual, u D (u; v) denotes the horizontal velocity of the fluid and h(x; y; t) is its depth. As shown in Fig. 7, h(x; y; t) is the distance between the free surface z D s(x; y; t) and the variable bottom b(x; y). Hence, s(x; y; t) D b(x; y) C h(x; y; t). Eqs. (22) and (23) express the horizontal momentum-balance; (24) expresses conservation of mass. Note that the vertical component of the fluid velocity has been eliminated from the dynam-

consider only one active layer with layer depth h(x; y; t); but take into account the forcing due to a horizontally varying potential temperature field (x; y; t). Vector u D u(x; y; t)i C v(x; y; t) j denotes the fluid velocity and ˝ D ˝ k is the angular velocity vector of the Earth’s rotation. @ @ r D @x i C @y j is the gradient operator, and i, j, and k are unit vectors along the x, y, and z-axes. Lavoie’s equations are part of a family of multi-layer models proposed by Ripa [61] to study, for example, the effects of solar heating, fresh water fluxes, and wind stresses on the upper equatorial ocean. A study of the validity of various layered models has been presented by Ripa [62]. The more sophisticated the models become the harder

Shallow Water Waves and Solitary Waves

they become to treat with analytic methods so one has to apply numerical methods. Numerical aspects of various shallow water models in atmospheric research and beyond are discussed in e. g. [48,75,76].

As shown in Sect. “Completely Integrable Shallow Water Wave Equations”, solitary wave solutions of the KdV and Boussinesq equations (and like PDEs), can be expressed as polynomials of the hyperbolic secant (sech) or tangent (tanh) functions, whereas their simplest period solutions involve the Jacobi elliptic cosine (cn) function. There are several methods to compute exact, analytic expressions for solitary and periodic wave solutions of nonlinear PDEs. Two straightforward methods, namely the direct integration method and the tanh-method, will be discussed. Both methods seek traveling wave solutions. By working in a traveling frame of reference the PDE is replaced by an ordinary differential equation (ODE) for which one seeks closed-form solutions in terms of special functions. In the terminology of dynamical systems, the solitary wave solutions correspond to heteroclinic or homoclinic trajectories in the phase plane of a first-order dynamical system corresponding to the underlying ODE (see, e. g., [6]). The periodic solutions are bounded in the phase plane by these special trajectories, which correspond to the limit of infinite period and modulus one. Other more powerful methods, such as the aforementioned IST and Hirota’s method (see, e. g., [40]) deal with the PDE directly. These methods allow one to compute closed-form expressions of soliton solutions (in particular, solitary wave solutions) addressed elsewhere in the encyclopedia.

Z

 0

p

Exact expressions for solitary wave solutions can be obtained by direct integration. The steps are illustrated for the KdV equation given in (2). Assuming that the wave travels to the right at speed v D !/k, Eq. (2) can be put into a traveling frame of reference with independent variable  D k(x  vt  x0 ). This reduces (2) to an ODE, v 0 C ˛ 0 C k 2  000 D 0; for () D u(x; t): A first integration with respect to  yields

˜ C B˜ a 2  b 3 C A

(29)

where A is a constant of integration. Multiplication of (28) by  0 , followed by a second integration with respect to ;

D˙

 0

d ;

(30)

where a D v/k 2 ; b D ˛/3k 2 ; A˜ D 2A/k 2 ; and B˜ D 2B/k 2 : The evaluation of the elliptic integral in (30) depends on the relationship between the roots of the function ˜ C B: ˜ In turn, the nature of the f () D a 2  b 3 C A ˜ Two cases lead to roots depends on the choice of A˜ and B. physically relevant solutions. Case 1: If the three roots are real and distinct, then the integral can be expressed in terms of the inverse of the cn function (see, e. g., [25] for details). This leads to the cnoidal wave solution given in (6). Case 2: If the three roots are real and (only) two of them coincide, then the tanh-squared solution follows. This happens when A˜ D B˜ D 0. Integrating both sides of (30) then gives Z

 0

d

2 p D  p Arctanh a  a  b

"p

a  b p a

# CC

D ˙(  0 ) : (31) where, without loss of generality, C and 0 can be set to zero. Solving (31) for  yields   p  p  a a a a 1  tanh2  D sech2  : b 2 b 2 (32)

Returning to the original variables, one gets 3v () D sech2 ˛

p  v  ; 2k

(33)

or u(x; t) D

(28)

Z

d

() D

Direct Integration Method

˛ 2  C k 2  00 D A ; 2

˛ k2 v   2 C  3 C  0 2 D A C B ; 2 6 2

where B is an integration constant. Separation of variables and integration then leads to

Computation of Solitary Wave Solutions

 v C

yields

3v sech2 ˛

 p v (x  vt  x0 ) ; 2

(34)

where v is arbitrary. Setting v D !/k D 4k 2 , where k is arbitrary, and ı D kx0 , one can verify that (34) matches (5).

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The Tanh Method If one is only interested in tanh- or sech-type solutions, one can circumvent explicit integration (often involving elliptic integrals) and apply the so-called tanh-method. A detailed description of the method has been given by Baldwin et al. [5]. The method has been fully implemented in Mathematica, a popular symbolic manipulation program, and successfully applied to many nonlinear differential equations from soliton theory and beyond. The tanh-method is based on the following observation: all derivatives of the tanh function can be expressed as polynomials in tanh. Indeed, using the identity cosh2   sinh2  D 1 one computes tanh0  D sech2  D 1  tanh2 ; tanh00  D 2 tanh  C 2 tanh3 , etc. Therefore, all derivatives of T() D tanh  are polynomials in T. For example, T 0 D 1  T 2 ; T 00 D 2T C 2T 3 ; and T 000 D 2T C 8T 2  6T 4 . By applying the chain rule, the PDE in u(x, t) is then transformed into an ODE for U(T) where T D tanh  D tanh(kx! tCı) is the new independent variable. Since all derivatives of T are polynomials of T, the resulting ODE has polynomial coefficients in T. It is therefore natural to seek a polynomial solution of the ODE. The problem thus becomes algebraic. Indeed, after computing the degree of the polynomial solution, one finds its unknown coefficients by solving a nonlinear algebraic system. The method is illustrated using (2). Applying the chain rule (repeatedly), the terms of (2) become u t D !(1  T 2 )U 0 ; u x D k(1  T 2 )U 0 , and  u x x x D k 3 (1  T 2 ) 2(1  3T 2 )U 0

 6T(1  T 2 )U 00 C (1  T 2 )2 U 000 ;

(35)

where U(T) D U(tanh(kx  ! t C ı)) D u(x; t); U 0 D dU/dT; etc. Substitution into (2) and cancellation of a common 1 T 2 factor yields  !U 0 C ˛kUU 0  2k 3 (1  3T 2 )U 0  6k 3 T(1  T 2 )U 00 C k 3 (1  T 2 )2 U 000 D 0 :

(36)

This ODE for U(T) has polynomial coefficients in T. One therefore seeks a polynomial solution

U(T) D

N X

an T n ;

(37)

nD0

where the integer exponent N and the coefficients an must be computed.

Substituting T N into (36) and balancing the highest powers in T gives N D 2. Then, substituting U(T) D a0 C a1 T C a2 T 2

(38)

into (36), and equating to zero the coefficients of the various power terms in T, yields a1 (˛a2 C 2k 2 ) D 0 ; ˛a2 C 12k 2 D 0 ; a1 (˛ka0  2k 3  !) D 0 ;

(39)

˛ka12 C 2˛ka0 a2  16k 3 a2  2! a2 D 0 : The unique solution of this nonlinear system for the unknowns a0 ; a1 and a2 is a0 D

8k 3 C ! 12k 2 ; a1 D 0; a2 D  : ˛k ˛

(40)

Finally, substituting (40) into (38) and using T D tanh(kx  ! t C ı) yields (4). The solitary wave solutions and cnoidal wave solutions presented in Sect. “Completely Integrable Shallow Water Wave Equations” have been automatically computed with a Mathematica package [5] that implements the tanhmethod and variants. A review of numerical methods to compute solitary waves of arbitrary amplitude can be found in VandenBroeck [74]. Water Wave Experiments and Observations Through a series of experiments in a hydrodynamic tank, Hammack investigated the validity of the BBM equation [35] and KdV equation as models for long waves in shallow water [36,37,38] and long internal waves [69]. Their research addressed the question: Would an initial displacement of water, as it propagates forward, eventually evolve in a train of localized solitary waves (solitons) and an oscillatory tail as predicted by the KdV equation? Based on the experimental data, they concluded that (i) the KdV dynamics only occurs if the waves travel over a long distance, (ii) a substantial amount of water must be initially displaced (by a piston) to produce a soliton train, (iii) the water volume of the initial wave determines the shape of the leading wave in the wave train, and (iv) the initial direction of displacement (upward or downward piston motion) determines what happens later. Quickly raising the piston causes a train of solitons to emerge; quickly lowering it causes all wave energy to distribute into the oscillatory tail, as predicted by the theory. Several other researchers have tested the validity of the KdV equation and variants in laboratory experiments (see,

Shallow Water Waves and Solitary Waves

e. g., [39,60]). Bona et al. [9] give an in-depth evaluation of the BBM Eq. (7) with and without dissipative term u x x . Their study includes (i) a numerical scheme with error estimates, (ii) a convergence test of the computer code, (iii) a comparison between the predictions of the theoretical model and the results of laboratory experiments. The authors note that it is important to include dissipative effects to obtain reasonable agreement between the forecast of the model and the empirical results. Water tank experiments in conjunction with the analysis of actual data, allows researchers to judge whether or not the KdV equation can be used to model the dynamics of tsunamis (see [67]). Tsunami research intensified after the December 2004 tsunami devastated large coastal regions of India, Indonesia, Sri Lanka, and Thailand, and killed nearly 300,000 people. Apart from shallow water waves near beaches, the KdV equation and its solitary wave solution also apply to internal waves in the ocean. Internal solitary waves in the open ocean are slow waves of large amplitude that travel at the interface of stratified layers of different density. Stratification based on density differences is primarily due to variations in temperature or concentration (e. g. due to salinity gradients). For example, absorption of solar radiation creates a near surface thin layer of warmer water (of lower density) above a thicker layer or colder, denser water. The smaller the density change, the lower the wave frequency, and the slower the propagation speed. If the upper layer is thinner than the lower one, then the internal wave is a wave of depression causing a downward displacement of the fluid interface. Internal solitary waves are ubiquitous in stratified waters, in particular, whenever strong tidal currents occur near irregular topography. Such waves have been studied since the 1960s. An early, well-documented case deals with internal waves in the Andaman Sea, where Perry and Schimke [58] found groups of internal waves up to 80 m high and 2000 m long on the main thermocline at 500 m in 1500 m deep water. Their measurements were confirmed by Osborne and Burch [53] who showed that internal waves in the Andaman Sea are generated by tidal flows and can travel over hundreds of kilometers. Strong internal waves can affect biological life and interfere with underwater navigation. Understanding the behavior of internal waves can aid in the design of offshore production facilities for oil and natural gas. The near-surface current associated with the internal wave locally modulates the height of the water surface. Hence, the internal wave leaves a “signature” or “footprint” at the sea surface in the form of a packet of solitary waves (sometimes called current rips or tide rips). These

Shallow Water Waves and Solitary Waves, Figure 8 Three solitary wave packets generated by internal waves from sills in the Strait of Gibraltar. Original image STS41G-34-81 courtesy of the Earth Sciences and Image Analysis Laboratory, NASA Johnson Space Center (http://eol.jsc.nasa.gov). Ortho-rectified, color adjusted photograph courtesy of Global Ocean Associates

visual manifestations appear as long, quasilinear stripes in satellite imagery or photographs taken during space flights. Over 50 case studies and hundreds of images of oceanic internal waves can be found in “An Atlas of Internal Solitary-like Waves and Their Properties” [42]. Figure 8 shows a photograph of three solitary waves packets which are the surface signature of internal waves in the Strait of Gibraltar. The photograph was taken from the Space Shuttle on October 11, 1984. Spain is to the North, Morocco to the South. Alternate solitary wave packets move toward the northeast or the southeast. The amplitude of these waves is of the order of 50 m; their wavelength is in the range of 500–2000 m. The separation between the packets is approximately 30 km. Waves of longer wavelengths and higher amplitudes have traveled the furthest. The number of oscillations within each packet increases as time goes on. Solitary wave packets can reach 200 km into the Western Mediterranean sea and live for more than two days before dissipating. A in-depth study of solitary waves in the Strait of Gibraltar can be found in Farmer and Armi [28]. The KdV model is applicable to stratified fluids with two layers and internal solitary waves if (i) the ratio of the amplitude a to the upper layer depth h is small, and (ii) the wavelength  is long compared with the upper layer depth. More precisely, a/h D O(h2 /2 )  1. A detailed discussion of internal solitary waves and additional references can be found in Garrett and Munk [31],

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Grimshaw [32,33,34], Helfrich and Melville [39], Apel et al. [4], and Pelinovsky et al. [56]. The last three papers discuss a variety of other theoretical models including the extended KdV equation (also known as Gardner’s equation or combined KdV-modified KdV equation) which contains both quadratic and cubic nonlinearities. Solitary wave solutions of the extended KdV equation can be found in Scott ([65], p. 856), Helfrich and Melville [39], and Apel et al. [4]. A review of laboratory experiments with internal solitary waves was published by Ostrovsky and Stepanyants [54]. As discussed in the review paper by Staquet and Sommeria [70], internal gravity waves also occur in the atmosphere, where they are often caused by wind blowing over topography and cumulus convective clouds. Internal gravity waves reveal themselves as unusual cloud patterns, which are the counterpart of the solitary wave packets on the ocean’s surface. Future Directions For many shallow water wave applications, the full Euler equations are too complex to work with. Instead, various approximate models have been proposed. Arguably, the most famous shallow water wave equations are the KdV and Boussinesq equations. The KdV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magnetohydrodynamic waves in plasmas, waves in elastic rods, mid-latitude and equatorial planetary waves, acoustic waves on a crystal lattice, and more (see, e. g., [64,65,66]). The KdV equation has played a pivotal role in the development of the Inverse Scattering Transform and soliton theory, both of which had a lasting impact on twentiethcentury mathematical physics. Historically, the classical Boussinesq equation was derived to describe the propagation of shallow water waves in a canal. Boussinesq systems arise when modeling the propagation of long-crested waves on large bodies of water (e. g. large lakes or the ocean). As Bona et al. [10] point out, a plethora of formally-equivalent Boussinesq systems can be derived. Yet, such systems may have vastly different mathematical properties. The study of the well-posedness of the nonlinear models is of paramount importance and is the subject of ongoing research. Shallow water wave theory allows one to adequately model waves in canals, surface waves near beaches, and internal waves in the ocean (see [4]). Due to their widespread occurrence in the ocean (see [42]), solitary waves and “solitary wave packets” (solitons) are of inter-

est to oceanographers and geophysicists. The (periodic) cnoidal wave solutions are used by coastal engineers in studies of sediment movement, erosion of sandy beaches, interaction of waves with piers and other coastal structures. Apart from their physical relevance, the knowledge of solitary and cnoidal wave solutions of nonlinear PDEs facilitates the testing of numerical solvers and aids in stability analysis. Shallow water wave models are widely used in atmospheric science as a paradigm for geophysical fluid motions. They model, for example, inertia-gravity waves with fast time scale dynamics, and wave-vortex interactions and Rossby waves associated with slow advective-timescale dynamics. This article has reviewed commonly used shallow water wave models, with the hope of bridging two research communities: one that focuses on nonlinear equations with dispersive effects; the other on nonlinear hyperbolic equations without dispersive terms. Of common concern are the testing of the theoretical models on measured data and further validation of the equations with numerical simulations and laboratory experiments. A fusion of the expertise of both communities might advance research on water waves and help to answer open questions about wave breaking, instability, vorticity, and turbulence. Of paramount importance is the prevention of natural disasters, ecological ravage, and damage to man-made structures due to a better understanding of the dynamics of tsunamis, steep waves, strong internal waves, rips, tidal currents, and storm surges. Acknowledgments This material is based upon work supported by the National Science Foundation (NSF) of the USA under Award No. CCF-0830783. The author is grateful to Harvey Segur, Jerry Bona, Michel Grundland, and Douglas Poole for advise and comments. William Navidi, Benoit Huard, Anna Segur, Katherine Socha, Christopher Jackson, Michel Peyrard, and Christopher Eilbeck are thanked for help with figures and photographs. Bibliography Primary Literature 1. Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, Cambridge 2. Ablowitz MJ, Segur H (1981) Solitons and the inverse scattering transform. SIAM, Philadelphia, Pennsylvania

Shallow Water Waves and Solitary Waves

3. Ablowitz MJ et al (1974) The inverse scattering transform: Fourier analysis for nonlinear problems. Stud Appl Math 53:249–315 4. Apel JR et al (2007) Internal solitons in the ocean and their effect on underwater sound. J Acoust Soc Am 121:695–722 5. Baldwin D et al (2004) Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs. J Symb Comp 37:669–705 6. Balmforth NJ (1995) Solitary waves and homoclinic orbits. Annu Rev Fluid Mech 27:335–373 7. Benjamin TB et al (1972) Model equations for long waves in nonlinear dispersive systems. Phil Trans Roy Soc London Ser A 272:47–78 8. Bona JL, Chen H (1999) Comparison of model equations for small-amplitude long waves. Nonl Anal 38:625–647 9. Bona JL et al (1981) An evaluation of a model equation for water waves. Phil Trans Roy Soc London Ser A 302:457–510 10. Bona JL et al (2002) Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: Derivation and linear theory. J Nonl Sci 12:283–318 11. Bona JL et al (2004) Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II: The nonlinear theory. Nonlinearity 17:925–952 12. Boussinesq J (1871) Théorie de l’intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. C R Acad Sci Paris 72:755–759 13. Boussinesq J (1872) Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J Math Pures Appl 17:55–108 14. Boussinesq J (1877) Essai sur la théorie des eaux courantes. Académie des Sciences de l’Institut de France, Mémoires présentés par divers savants (ser 2) 23:1–680 15. Bullough RK (1988) “The wave” “par excellence”, the solitary, progressive great wave of equilibrium of the fluid–an early history of the solitary wave. In: Lakshmanan M (ed) Solitons: introduction and applications. Springer, Berlin, pp 7–42 16. Camassa R, Holm D (1993) An integrable shallow water equation with peakon solitons. Phys Rev Lett 71:1661–1664 17. Camassa R, Holm D (1994) An new integrable shallow water equation. Adv Appl Mech 31:1–33 18. Chen M (1998) Exact solutions of various Boussinesq systems. Appl Math Lett 11:45–49 19. Clarkson PA, Mansfield EL (1994) On a shallow water wave equation. Nonlinearity 7:975–1000 20. Craik ADD (2004) The origins of water wave theory. Annu Rev Fluid Mech 36:1–28 21. Craik ADD (2005) George Gabriel Stokes and water wave theory. Annu Rev Fluid Mech 37:23–42 22. Darrigol O (2003) The spirited horse, the engineer, and the mathematician: Water waves in nineteenth-century hydrodynamics. Arch Hist Exact Sci 58:21–95 23. Dauxois T, Peyrard M (2004) Physics of solitons. Cambridge University Press, Cambridge, UK. Transl of Peyrard M, Dauxois T (2004) Physique des solitons. Savoirs Actuels, EDP Sciences, CNRS Editions 24. Dellar P (2003) Common Hamiltonian structure of the shallow water equations with horizontal temperature gradients and magnetic fields. Phys Fluids 15:292–297

25. Drazin PG, Johnson RS (1989) Solitons: an introduction. Cambridge University Press, Cambridge, UK 26. El GA (2007) Korteweg–de Vries equation: solitons and undular bores. In: Grimshaw RHJ (ed) Solitary waves in fluids. WIT Press, Boston, pp 19–53 27. Emmerson GS (1977) John Scott Russell: A great Victorian engineer and naval architect. John Murray Publishers, London 28. Farmer DM, Armi L (1988) The flow of Mediterranean water through the Strait of Gibraltar. Prog Oceanogr 21:1–105 29. Fermi F et al (1955) Studies of nonlinear problems I. Los Alamos Sci Lab Rep LA-1940. Reproduced in: AC Newell (ed) (1974) Nonlinear wave motion. AMS, Providence, RI 30. Filippov AT (2000) The versatile soliton. Birkhäuser-Verlag, Basel 31. Garrett C, Munk W (1979) Internal waves in the ocean. Annu Rev Fluid Mech 11:339–369 32. Grimshaw R (1997) Internal solitary waves. In: Liu PLF (ed) Advances in coastal and ocean engineering, vol III. World Scientific, Singapore, pp 1–30 33. Grimshaw R (2001) Internal solitary waves. In: Grimshaw R (ed) Environmental stratified flows. Kluwer, Boston, pp 1–28 34. Grimshaw R (2005) Korteweg–de Vries equation. In: Grimshaw R (ed) Nonlinear waves in fluids: Recent advances and modern applications. Springer, Berlin, pp 1–28 35. Hammack JL (1973) A note on tsunamis: their generation and propagation in an ocean of uniform depth. J Fluid Mech 60:769–800 36. Hammack JL, Segur H (1974) The Korteweg–de Vries equation and water waves, part 2. Comparison with experiments. J Fluid Mech 65:289–314 37. Hammack JL, Segur H (1978a) The Korteweg–de Vries equation and water waves, part 3. Oscillatory waves. J Fluid Mech 84:337–358 38. Hammack JL, Segur H (1978b) Modelling criteria for long water waves. J Fluid Mech 84:359–373 39. Helfrich KR, Melville WK (2006) Long nonlinear internal waves. Annu Rev Fluid Mech 38:395–425 40. Hirota R (2004) The direct method in soliton theory. Cambridge University Press, Cambridge, UK 41. Hirota R, Satsuma J (1976) N-soliton solutions of model equations for shallow water waves. J Phys Soc Jpn 40:611–612 42. Jackson CR (2004) An atlas of internal solitary-like waves and their properties, 2nd edn. Global Ocean Associates, Alexandria, VA 43. Jeffrey A (1978) Nonlinear wave propagation. Z Ang Math Mech (ZAMM) 58:T38–T56 44. Johnson RS (2003) On solutions of the Camassa–Holm equation. Proc Roy Soc London Ser A 459:1687–1708 45. Joseph RI, Egri R (1977) Another possible model equation for long waves in nonlinear dispersive systems. Phys Lett A 61:429–432 46. Kadomtsev BB, Petviashvili VI (1970) On the stability of solitary waves in weakly dispersive media. Sov Phys Doklady 15:539– 541 47. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos Mag (Ser 5) 39:422–443 48. LeVeque RJ (2002) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge, UK 49. Madsen PA, Schäffer HA (1999) A review of Boussinesq-type equations for surface gravity waves. In: Liu PLF (ed) Advances

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in coastal and ocean engineering, vol V. World Scientific, Singapore, pp 1–95 McKean HP (1981) Boussinesq’s equation on the circle. Comm Pure Appl Math 34:599–691 Miles JW (1981) The Korteweg–de Vries equation: a historical essay. J Fluid Mech 106:131–147 Miura RM (1976) The Korteweg–de Vries equation: a survey of results. SIAM Rev 18:412–559 Osborne AR, Burch TL (1980) Internal solitons in the Andaman sea. Science 208:451–460 Ostrovsky LA, Stepanyants YA (2005) Internal solitons in laboratory experiments: Comparison with theoretical models. Chaos 15:037111 Pedlosky J (1987) Geophysical fluid dynamics, 2nd edn. Springer, Berlin Pelinovsky E et al (2007) In: Grimshaw RHJ (ed) Solitary waves in fluids. WIT Press, Boston, pp 85–110 Peregrine DH (1966) Calculations of the development of an undular bore. J Fluid Mech 25:321–330 Perry RB, Schimke GR (1965) Large amplitude internal waves observed off the north-west coast of Sumatra. J Geophys Res 70:2319–2324 Rayleigh L (1876) On waves. Philos Mag 1:257–279 Remoissenet M (1999) Waves called solitons: Concepts and experiments, 3rd edn. Springer, Berlin Ripa P (1993) Conservation laws for primitive equations models with inhomogeneous layers. Geophys Astrophys Fluid Dyn 70:85–111 Ripa P (1999) On the validity of layered models of ocean dynamics and thermodynamics with reduced vertical resolution. Dyn Atmos Oceans Fluid Dyn 29:1–40 Russell JS (1844) Report on waves, 14th meeting of the British Association for the Advancement of Science. John Murray, London, pp 311–390 Scott AC (2003) Nonlinear science: Emergence and dynamics of coherent structures. Oxford University Press, Oxford, UK Scott AC (ed) (2005) Encyclopedia of Nonlinear science. Routledge, New York Scott AC et al (1973) The soliton: a new concept in applied science. Proc IEEE 61:1443–1483 Segur H (2007a) Waves in shallow water, with emphasis on the tsunami of 2004. In: Kundu A (ed) Tsunami and nonlinear waves. Springer, Berlin Segur H (2007b) Integrable models of waves in shallow water. In: Pinski M, Birnir B (eds) Probability, geometry and integrable systems. Cambridge University Press, Cambridge, UK Segur H, Hammack JL (1982) Soliton models of long internal waves. J Fluid Mech 118:285–304 Staquet C, Sommeria J (2002) Internal gravity waves: From instabilities to turbulence. Annu Rev Fluid Mech 34:559–593 Stokes GG (1847) On the theory of oscillatory waves. Trans Camb Phil Soc 8:441–455 Toro EF (2001) Shock-capturing methods for free-surface shallow flows. Wiley, New York Vallis GK (2006) Atmospheric and oceanic fluid dynamics: fundamentals and large scale circulation. Cambridge University Press, Cambridge, UK Vanden-Broeck J-M (2007) Solitary waves in water: numerical

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Books and Reviews Boyd JP (1998) Weakly nonlinear solitary waves and beyond-allorder asymtotics. Kluwer, Dordrecht Calogero F, Degasperis A (1982) Spectral transform and solitons I. North Holland, Amsterdam Dickey LA (2003) Soliton equations and Hamiltonian systems, 2nd edn. World Scientific, Singapore Dodd RK et al (1982) Solitons and nonlinear wave equations. Academic Press, London Eilenberger G (1981) Solitons: Mathematical methods for physicists. Springer, Berlin Faddeev LD, Takhtajan LA (1987) Hamiltonian methods in the theory of solitons. Springer, Berlin Fordy A (ed) (1990) Soliton theory: A survey of results. Manchester University Press, Manchester Grimshaw RHJ (ed) (2007) Solitary waves in fluids. WIT Press, Boston Infeld E, Rowlands G (2000) Nonlinear waves, solitons, and chaos. Cambridge University Press, New York Johnson RS (1977) A modern introduction to the mathematical theory of water waves. Cambridge University Press, Cambridge, UK Lamb GL (1980) Elements of soliton theory. Wiley Interscience, New York Miles JW (1980) Solitary waves. Annu Rev Fluid Mech 12:11–43 Mei CC et al (2005) Theory and applications of ocean surface waves. World Scientific, Singapore Mitropol’sky YZ (2001) Dynamics of internal gravity waves in the ocean. Kluwer, Dordrecht Newell AC (1983) The history of the soliton. J Appl Mech 50:1127– 1137 Newell AC (1985) Solitons in mathematics and physics. SIAM, Philadelphia, PA Makhankov VG (1990) Soliton phenomenology. Kluwer, Dordrecht, The Netherlands Novikov SP et al (1984) Theory of solitons: The inverse scattering method. Consultants Bureau(Plenum Press), New York Pedlosky J (2003) Waves in the ocean and atmosphere: introduction to wave dynamics. Springer, Berlin Russell JS (1885) The wave of translation in the oceans of water, air and ether. Trübner, London Stoker JJ (1957) Water waves. Wiley Interscience, New York Whitham GB (1974) Linear and nonlinear waves. Wiley Interscience, New York

Smooth Ergodic Theory

Smooth Ergodic Theory AMIE W ILKINSON Northwestern University, Evanston, USA Article Outline Glossary Definition of the Subject Introduction The Volume Class The Fundamental Questions Lebesgue Measure and Local Properties of Volume Ergodicity of the Basic Examples Hyperbolic Systems Beyond Uniform Hyperbolicity The Presence of Critical Points and Other Singularities Future Directions Bibliography Glossary Conservative, dissipative Conservative dynamical systems (on a compact phase space) are those that preserve a finite measure equivalent to volume. Hamiltonian dynamical systems are important examples of conservative systems. Systems that are not conservative are called dissipative. Finding physically meaningful invariant measures for dissipative maps is a central object of study in smooth ergodic theory. Distortion estimate A key technique in smooth ergodic theory, a distortion estimate for a smooth map f gives a bound on the variation of the jacobian of f n in a given region, for n arbitrarily large. The jacobian of a smooth map at a point x is the absolute value of the determinant of derivative at x, measured in a fixed Riemannian metric. The jacobian measures the distortion of volume under f in that metric. Hopf argument A technique developed by Eberhard Hopf for proving that a conservative diffeomorphism or flow is ergodic. The argument relies on the Ergodic Theorem for invertible transformations, the density of continuous functions among integrable functions, and the existence of stable and unstable foliations for the system. The argument has been used, with various modifications, to establish ergodicity for hyperbolic, partially hyperbolic and nonuniformly hyperbolic systems. Hyperbolic A compact invariant set  M for a diffeomorphism f : M ! M is hyperbolic if, at every point in , the tangent space splits into two subspaces,

one that is uniformly contracted by the derivative of f , and another that is uniformly expanded. Expanding maps and Anosov diffeomorphisms are examples of globally hyperbolic maps. Hyperbolic diffeomorphisms and flows are the archetypical smooth systems displaying chaotic behavior, and their dynamical properties are well-understood. Nonuniform hyperbolicity and partial hyperbolicity are two generalizations of hyperbolicity that encompass a broader class of systems and display many of the chaotic features of hyperbolic systems. Sinai–Ruelle–Bowen (SRB) measure The concept of SRB measure is a rigorous formulation of what it means for an invariant measure to be “physically meaningful”. An SRB measure attracts a large set of orbits into its support, and its statistical features are reflected in the behavior of these attracted orbits. Definition of the Subject Smooth ergodic theory is the study of the statistical and geometric properties of measures invariant under a smooth transformation or flow. The study of smooth ergodic theory is as old as the study of abstract ergodic theory, having its origins in Bolzmann’s Ergodic Hypothesis in the late 19th Century. As a response to Boltzmann’s hypothesis, which was formulated in the context of Hamiltonian Mechanics, Birkhoff and von Neumann defined ergodicity in the 1930s and proved their foundational ergodic theorems. The study of ergodic properties of smooth systems saw an advance in the work of Hadamard and E. Hopf in the 1930s their study of geodesic flows for negatively curved surfaces. Beginning in the 1950s, Kolmogorov, Arnold and Moser developed a perturbative theory producing obstructions to ergodicity in Hamiltonian systems, known as Kolmogorov–Arnold–Moser (KAM) Theory. Beginning in the 1960s with the work of Anosov and Sinai on hyperbolic systems, the study of smooth ergodic theory has seen intense activity. This activity continues today, as the ergodic properties of systems displaying weak forms of hyperbolicity are further understood, and KAM theory is applied in increasingly broader contexts. Introduction This entry focuses on the basic arguments and principles in smooth ergodic theory, illustrating with simple and straightforward examples. The classic texts [1,2] are a good supplement. The discussion here sidesteps the topic of  Kolmogorov–Arnold–Moser (KAM) Theory, which has played an important role in the development of smooth

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ergodic theory. For reasons of space, detailed discussion of several active areas in smooth ergodic theory is omitted, including: higher mixing properties (Kolmogorov, Bernoulli, etc.), finer statistical properties (fast decay of correlations, Central Limit Theorem, large deviations), smooth thermodynamic formalism (transfer operators, pressure, dynamical zeta functions, etc.), the smooth ergodic theory of random dynamical systems, as well as any mention of infinite invariant measures. The text [3] covers many of these topics, and the texts [4,5,6] treat random smooth ergodic theory in depth. An excellent discussion of many of the recent developments in the field of smooth ergodic theory is [7]. This entry assumes knowledge of the basic concepts in ergodic theory and of basic differential topology. The texts [8] and [9] contain the necessary background. The Volume Class For simplicity, assume that M is a compact, boundaryless C 1 Riemannian manifold, and that f : M ! M is an orientation-preserving, C1 map satisfying m(D x f ) > 0, for all x 2 M, where m(D x f ) D

inf

v2Tx M;kvkD1

kD x f (v)k :

If f is a diffeomorphism, then this assumption is automatically satisfied, since in that case m(D x f ) D kD f (x) f 1 k1 > 0. For non-invertible maps, this assumption is essential in much of the following discussion. The Inverse Function Theorem implies that any map f satisfying these hypotheses is a covering map of positive degree d  1. These assumptions will avoid the issues of infinite measures and the behavior of f near critical points and singularities of the derivative. For most results discussed in this entry, this assumption is not too restrictive. The existence of critical points and other singularities is, however, a complication that cannot be avoided in many important applications. The ergodic-theoretic analysis of such examples can be considerably more involved, but contains many of the elements discussed in this entry. The discussion in Sect. “Beyond Uniform Hyperbolicity” indicates how some of these additional technicalities arise and can be overcome. For simplicity, the discussion here is confined almost exclusively to discrete time evolution. Many, though not all, of the the results mentioned here carry over to flows and semiflows using, for example, a cross-section construction (see Chap. 1 in [2]). Every smooth map f : M ! M satisfying these hypotheses preserves a natural measure class, the measure

class of a finite, smooth Riemannian volume on M. Fix such a volume  on M. Then there exists a continuous, positive jacobian function x 7! jacx f on M, with the property that for every sufficiently small ball B M, and every measurable set A B one has: Z jacx f d(x) : ( f (A)) D B

The jacobian of f at x is none other than the absolute value of the determinant of the derivative D x f (measured in the given Riemannian metric). To see that the measure class of  is preserved by f , observe that the Radon–Nikodym P df  derivative d (x) at x is equal to y2 f 1 (x) (jac y f )1 > 0. Hence f  is equivalent to , and f preserves the measure class of . In many contexts, the map f has a natural invariant measure in the measure class of volume. In this case, f is said to be conservative. One setting in which a natural invariant smooth measure appears is Hamiltonian dynamics. Any solution to Hamilton’s equations preserves a smooth volume called the Liouville measure. Furthermore, along the invariant, constant energy hypermanifolds of a Hamiltonian flow, the Liouville measure decomposes smoothly into invariant measures, each of which is equivalent to the induced Riemannian volume. In this way, many systems of physical or geometric origin, such as billiards, geodesic flows, hard sphere gases, and evolution of the n-body problem give rise to smooth conservative dynamical systems. Note that even though f preserves a smooth measure class, it might not preserve any measure in that measure class. Consider, for example, a diffeomorphism f : S 1 ! S 1 of the circle with exactly two fixed points, p and q, f 0 (p) > 1 > f 0 (q) > 0. Let  be an f -invariant probability measure. Let I be a neighborhood of p. Then T1 n (I) D fpg, but on the other hand, ( f n (I)) D nD1 f (I) > 0, for all n. This implies that (fpg) > 0, and so  does not lie in the measure class of volume. This is an example of a dissipative map. A map f is called dissipative if every f -invariant measure with full support has a singular part with respect to volume. As was just seen, if a diffeomorphism f has a periodic sink, then f is dissipative; more generally, if a diffeomorphism f has a periodic point p of period k such that jac p f k ¤ 1, then f is dissipative. The Fundamental Questions For a given smooth map f : M ! M, there are the following fundamental questions. 1. Is f conservative? That is, does there exist an invariant measure in the class of volume? If so, is it unique?

Smooth Ergodic Theory

2. When f is conservative, what are its statistical properties? Is it ergodic, mixing, a K-system, Bernoulli, etc.? Does it obey a Central Limit Theorem, fast decay of correlations, large deviations estimates, etc.? 3. If f is dissipative, does there exist an invariant measure, not in the class of volume, but (in some sense) natural with respect to volume? What are the statistical properties of such a measure, if it exists? There are several plausible ways to “answer” these questions. One might fix a given map f of interest and ask these questions for that specific f . What tends to happen in the analysis of a single map f is that either:  the question can be answered using “soft” methods, and so the answer applies not only to f but to perturbations of f , or even to generic or typical f inside a class of maps; or  the proof requires “hard” analysis or precise asymptotic information and cannot possibly be answered for a specific f , but can be answered for a large set of f t in a typical (or given) parametrized family f f t g t2(1;1) of smooth maps containing f D f0 . Both types of results appear in the discussion that follows. Lebesgue Measure and Local Properties of Volume Locally, any measure in the measure class of volume is, after a smooth change of coordinates, equivalent to Lebesgue measure in Rn . In fact, more is true: Moser’s Theorem implies that locally any Riemannian volume is, after a smooth change of coordinates, equal to Lebesgue measure in Rn . Hence to study many of the local properties of volume, it suffices to study the same properties for Lebesgue measure. One of the basic properties of Lebesgue measure is that every set of positive Lebesgue measure can be approximated arbitrarily well in measure from the outside by an open set, and from the inside by a compact set. A consequence of this property, of fundamental importance in smooth ergodic theory, is the following statement. Fundamental Principle #1: Two disjoint, positive Lebesgue measure sets cannot mix together uniformly at all scales. As an illustration of this principle, consider the following elementary exercise in measure theory. First, some notation. If  is a measure and A and B are -measurable sets with (B) > 0, the density of A in B is defined by: (A: B) D

(A \ B) : (B)

Proposition 1 Let P1 ; P2 ; : : : be sequence of (mod 0) finite partitions of the circle S1 into open intervals, with the properties: a) any element of Pn is a (mod 0) union of elements of PnC1 , and b) the maximum diameter of elements of Pn tends to 0 as n ! 1. Let A be any set of positive Lebesgue measure in S1 . Then there exists a sequence of intervals I1 ; I2 ; : : : , with I n 2 Pn such that: lim (A: I n ) D 1 :

n!1

Proof Assume that Lebesgue measure has been normalized so that (S 1 ) D 1. Fix a (mod 0) cover of S 1 n A by S pairwise disjoint elements {J i } of the union 1 nD1 P n with the properties: (J1 )  (J2 )     ; and  X [ 1 1 Ji D (J i ) < 1 :  iD1

iD1

For n 2 N, let U n be the union of all the intervals J i Sn that are contained in Pn , and let Vn D iD1 U n . This defines an increasing sequence of natural numbers i1 D S i nC1 1 J n and 1 < i2 < i3 <    such that U n D iDi n S i nC1 1 Vn D iD1 J n . For each n, the interval I n will be chosen to be an element Pn , disjoint from V n , in which the density of S1 0 as n ! 1). Since iDnC1 U i is very small (approaching S U (S 1 n A) \ I n is contained in 1 iDnC1 i , this choice of I n will ensure that the density of A in I n is large (approaching 1 as n ! 1). To make this choice of I n , note first that the density of S1 1 iDnC1 U i inside of S n Vn is: P1 S ( 1 iDi nC1 (J i ) iDnC1 U i ) D an : D P 1 i nC1 1 (S n Vn ) 1  iD1 (J i ) P Note that, since 1 (J i )1; one has a n ! 0 as n ! 1. iD1 S 1 Since the density of 1 iDnC1 U i inside of S n Vn is at most an , there is an interval I n in Pn , disjoint from V n , such that S the density of 1 iDnC1 U i inside of I n is at most an . Then lim (A: I n )  lim 1  a n D 1 :

n!1

n!1



In smooth ergodic theory, it is often useful to use a variation on Proposition 1 (generally, in higher dimensions) in which the partitions Pn are nested, dynamically-defined partitions. A simple application of this method can be used to prove that the doubling map on the circle is ergodic with respect to Lebesgue measure, which is done in Sect. “Lebesgue Measure and Local Properties of Volume”.

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Notice that this proposition does not claim that the intervals I n are nested. If one imposes stronger conditions on the partitions Pn , then one can draw stronger conclusions. A very useful theorem in this respect is the Lebesgue Density Theorem. A point x 2 M is a Lebesgue density point of a measurable set X  M if lim m(X : Br (x)) D 1 ;

r!0

where Br (x) is the Riemannian ball of radius r centered at x. Notice that the notion of Lebesgue density point depends only on the smooth structure of M, because any two Riemannian metrics have the same Lebesgue density points. The Lebesgue Density Theorem states that if A is a measurable set and b A is the set of Lebesgue density points of A, then m(A  b A) D 0. Ergodicity of the Basic Examples This section contains proofs of the ergodicity of two basic examples of conservative smooth maps: irrational rotations on the circle and the doubling map on the circle. See  Ergodic Theory: Basic Examples and Constructions for a more detailed description of these maps. These proofs serve as an elementary illustration of some of the fundamental techniques and principles in smooth ergodic theory. Rotations on the circle. Denote by S1 the circle R/Z, which is an additive group, and by  normalized LebesgueHaar measure on S1 . Fix a real number ˛ 2 R. The rotation R˛ : S 1 ! S 1 is the translation defined by R˛ (x) D x C˛. Since translations preserve Lebesgue-Haar measure, the map R˛ is conservative. Note that R˛ is a diffeomorphism and an isometry with respect to the canonical flat metric (length) on S1 . Proposition 2 If ˛ … Q, then the rotation R˛ : S 1 ! S 1 is ergodic with respect to Lebesgue measure. Proof Let A be an R˛ -invariant set in S1 , and suppose that 0 < (A) < 1. Denote by Ac the complement of A in S1 . Fix " > 0. Proposition 1 implies that there exists an interval I S 1 such that the density of A in I is large: (A: I) > 1  ". Similarly, one may choose an interval J such that (Ac : J) > 1  ". Without loss of generality, one may choose I and J to have the same length. Since ˛ is irrational, R˛ has a dense orbit, which meets the interval I. Since R˛ is an isometry, this implies that there is an integer n such that (R˛n (I)  J) < "(I). Since (I) D (J), this readily implies that j(A: R˛n (I))  (A: J)j < ". Also, since A is invariant, and R˛ is invertible and preserves measure, one has: (A: R˛n (I)) D (R˛n (A) : R˛n (I)) D (A: I) > 1  " :

But for " sufficiently small, this contradicts the facts that (A: J) D 1(Ac : J) < " and j(A: R˛n (I))(A: J)j < ".  Note that this is not a proof of the strongest possible statement about R˛ (namely, minimality and unique ergodicity). The point here is to show how “soft” arguments are often sufficient to establish ergodicity; this proof uses no more about R˛ than the fact that it is a transitive isometry. Hence the same argument shows: Theorem 1 Let f : M ! M be a transitive isometry of a Riemannian manifold M. Then f is ergodic with respect to Riemannian volume. One can isolate from this proof a useful principle: Fundamental Principle #2: Isometries preserve Lebesgue density at all scales, for arbitrarily many iterates. This principle implies, for example, that a smooth action by a compact Lie group on M is ergodic along typical (nonsingular) orbits. This principle is also useful in studying area-preserving flows on surfaces and, in a refined form, unipotent flows on homogeneous spaces. In the case of surface flows, ergodicity questions can be reduced to a study of interval exchange transformations. See the entry  Ergodic Theory: Basic Examples and Constructions for a detailed discussion of interval exchange transformations and flows on surfaces.  Ergodic Theory, Introduction to contains detailed information on unipotent flows. Doubling map on the circle. Let T2 : S 1 ! S 1 be the doubling map defined by T2 (x) D 2x. Then T 2 is a degree2 covering map and endomorphism of S1 with constant 2 )  D 12 C 12 D 1, T 2 prejacobian jac x T2  2. Since d(Td serves Lebesgue-Haar measure. The doubling map is the simplest example of a hyperbolic dynamical system, a topic treated in depth in the next section. As with the previous example, the focus here is on the property of ergodicity. It is again possible to prove much stronger results about T 2 , such as Bernoullicity, by other methods. Instead, here is a soft proof of ergodicity that will generalize readily to other contexts. Proposition 3 The doubling map T2 : S 1 ! S 1 is ergodic with respect to Lebesgue measure. Proof Let A be a T 2 -invariant set in S1 with (A) > 0. Let p 2 S 1 be the fixed point of T 2 , so that T2 (p) D p. For each n 2 N, the preimages of p under T2n define a (mod 0) partition Pn into 2n open intervals of length 2n ; the elements of Pn are the connected components of S 1 n T2n (fpg). Note that the sequence of partitions P1 ; P2 ; : : : is nested, in the sense of Proposition 1. Restricted to any

Smooth Ergodic Theory

interval J 2 Pn , the map T 2 n is a diffeomorphism onto S 1 nfpg with constant jacobian jacx (T2n ) D (T2n )0 (x) D 2n . Since A is invariant, it follows that T2n (A) D A. Fix " > 0. Proposition 1 implies that there exists an n 2 N and an interval J 2 Pn such that (A: J) > 1  ". Note that T2n (A \ J) A. But then (A)  (T2n (A \ J)) Z D jacx (T2n ) d(x) A\J

v 2 E u (x) H) kD x f n (v)k  C 1 n kvk ;

D 2n (A \ J) D 2n (A: J)(J)

and v 2 E s (x) H) kD x f n (v)k  Cn kvk :

> 2n (1  ")(J) D 1  " : Since " was arbitrary, one obtains that (A) D 1.

This section defines hyperbolic maps and attractors, provides examples, and investigates their ergodic properties. See [10,11] and  Hyperbolic Dynamical Systems for a thorough discussion of the topological and smooth properties of hyperbolic systems. A hyperbolic structure on a compact f -invariant set  M is given by a Df -invariant splitting T M D E u ˚ E s of the tangent bundle over  and constants C;  > 1 such that, for every x 2  and n 2 N :



In this proof, the facts that the intervals in Pn have constant length 2n and that the jacobian of T 2 n restricted to such an interval is constant and equal to 2n are not essential. The key fact really used in this proof is the assertion that the ratio: (T2n (A \ J) : T2n (J)) (A: J) is bounded, independently of n. In this case, the ratio is 1 for all n because T 2 has constant jacobian. It is tempting to try to extend this proof to other expanding maps on the circle, for example, a C1 , preserving map f : S 1 ! S 1 with dC 1 ( f ; T2 ) small. Many of the aspects of this proof carry through mutatis mutandis for such an f , save for one. A C1 -small perturbation of T 2 will in general no longer have constant jacobian, and the variation of the jacobian of f n on a small interval can be (and often is) unbounded. The reason for this unboundedness is a lack of control of the modulus of continuity of f 0 . Hence this argument can fail for C1 perturbations of T 2 . On the other hand, the argument still works for C2 perturbations of T 2 , even when the jacobian is not constant. The principle behind this fact can be loosely summarized: Fundamental Principle #3: On controlled scales, iterates of C2 expanding maps distort Lebesgue density in a controlled way. This principle requires further explanation and justification, which will come in the following section. The C2 hypothesis in this principle accounts for the fact that almost all results in smooth ergodic theory assume a C2 hypothesis (or something slightly weaker). Hyperbolic Systems One of the most developed areas of smooth ergodic theory is in the study of hyperbolic maps and attractors.

A hyperbolic attractor for a map f : M ! M is given by an open set U M such that: f (U) U, and such that the T set  D n0 f n (U) carries a hyperbolic structure. The set  is called the attractor, and U is an attracting region. A map f : M ! M is hyperbolic if M decomposes (mod 0) into a finite union of attracting regions for hyperbolic attractors. Typically one assumes as well that the restriction of f to each attractor i is topologically transitive. Every point p in a hyperbolic set  has smooth stable manifold W s (p) and unstable manifold W u (p), tangent, respectively, to the subspaces E s (p) and E u (p). The set W s (p) is precisely the set of q 2 M such that d( f n (p); f n (q)) tends to 0 as n ! 1, and it follows that f (W s(p)) D W s ( f (p)). When f is a diffeomorphism, the unstable manifold W u (p) is uniquely defined and is the stable manifold of f 1 . When f is not invertible, local unstable manifolds exist, but generally are not unique. If  is a transitive hyperbolic attractor, then every unstable manifold of every point p 2  is dense in . Examples of Hyperbolic Maps and Attractors Expanding Maps The previous section mentioned briefly the Cr perturbations of the doubling map T 2 . Such perturbations (as well as T 2 itself) are examples of expanding maps. A map f : M ! M is expanding if there exist constants  > 1 and C > 0 such that, for every x 2 M, and every nonzero vector v 2 Tx M : kD x f n (v)k  Cn kvk ; with respect to some (any) Riemannian metric on M. An expanding map is clearly hyperbolic, with U D M, Es the trivial bundle, and E u D TM. Any disk in M is a local unstable manifold for f . Anosov Diffeomorphisms A diffeomorphism f : M ! M is called Anosov if the tangent bundle splits as a direct sum TM D E u ˚ E s of two Df -invariant subbundles, such that Eu is uniformly expanded and Es is uni-

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formly contracted by Df . Similarly, a flow ' t : M ! M is called Anosov if the tangent bundle splits as a direct sum TM D E u ˚ E 0 ˚ E s of three D' t -invariant sub˙ Eu is uniformly bundles, such that E0 is generated by ', expanded and Es is uniformly contracted by D' t . Like expanding maps, an Anosov diffeomorphism is an Anosov attractor with  D U D M. A simple example of a conservative Anosov diffeomorphism is a hyperbolic linear automorphism of the torus. Any matrix A 2 SL(n; Z) induces an automorphism of Rn preserving the integer lattice Zn , and so descends to an automorphism f A : T n ! T n of the n-torus T n D Rn /Z n . Since the determinant of A is 1, the diffeomorphism f A preserves Lebesgue-Haar measure on T n . In the case where none of the eigenvalues of A have modulus 1, the resulting diffeomorphism f A is Anosov. The stablebundle Es at x 2 T n is the parallel translate to x of the sum of the contracted generalized eigenspaces of A, and the unstable bundle Eu at x is the translated sum of expanded eigenspaces. In general, the invariant subbundles Eu and Es of an Anosov diffeomorphism are integrable and tangent to a transverse pair of foliations W u and W s , respectively (see, e. g [12]. for a proof of this). The leaves of W s are uniformly contracted by f , and the leaves of W u are uniformly contracted by f 1 . The leaves of these foliations are as smooth as f , but the tangent bundles to the leaves do not vary smoothly in the manifold. The regularity properties of these foliations play an important role in the ergodic properties of Anosov diffeomorphisms. The first Anosov flows to be studied extensively were the geodesic flows for manifolds of negative sectional curvatures. As these flows are Hamiltonian, they are conservative. Eberhard Hopf showed in the 1930s that such geodesic flows for surfaces are ergodic with respect to Liouville measure [14]; it was not until the 1960s that ergodicity of all such flows was proved by Anosov [15]. The next section describes, in the context of Anosov diffeomorphisms, Hopf’s method and important refinements due to Anosov and Sinai. DA Attractors A simple way to produce a non-Anosov hyperbolic attractor on the torus is to start with an Anosov diffeomorphism, such as a linear hyperbolic automorphism, and deform it in a neighborhood of a fixed point, turning a saddle fixed point into a source, while preserving the stable foliation. If this procedure is carried out carefully enough, the resulting diffeomorphism is a dissipative hyperbolic diffeomorphism, called a derived from Anosov (DA) attractor. Other examples of hyperbolic attractors are the Plykin attractor and the solenoid. See [10].

Distortion Estimates Before describing the ergodic properties of hyperbolic systems, it is useful to pause for a brief discussion of distortion estimates. Distortion estimates are behind almost every result in smooth ergodic theory. In the hyperbolic setting, distortion estimates are applied to the action of f on unstable manifolds to show that the volume distortion of f along unstable manifolds can be controlled for arbitrarily many iterates. The example mentioned at the end of the previous section illustrates the ideas in a distortion estimate. Suppose that f : S 1 ! S 1 is a C2 expanding map, such as a C2 small perturbation of T 2 . Then there exist constants  > 1 and C > 0 such that ( f n )0 (x) > Cn for all x and n. Let d be the degree of f . If I is a sufficiently small open interval in S1 , then for each n, f n (I) is a union of d disjoint intervals. Furthermore, each of these intervals has diameter at most C 1 n times the diameter of I. It is now possible to justify the assertion in Fundamental Principle #3 in this context. Lemma There exists a constant K  1 such that, for all n 2 N, and for all x; y 2 f n (I), one has: K 1 

( f n )0 (x) K: ( f n )0 (y)

Proof Since f is C2 and f 0 is bounded away from 0, the function ˛(x) D log( f 0 (x)) is C1 . In particular, ˛ is Lipschitz continuous: there exists a constant L > 0 such that, for all x; y 2 S 1 , j˛(x)  ˛(y)j < Ld(x; y). For n  0, let ˛n (x) D log(( f n )0 (x)). The Chain Rule implies that P i ˛n (x) D n1 iD0 ˛( f (x)). The expanding hypothesis on f implies that for all x; y 2 f n (I) and for i D 0; : : : ; n, one has d( f i (x); f i (y))  C 1  in d( f n (x); f n (y))  C 1  in . Hence j˛n (x)  ˛n (y)j 

n1 X

j˛( f i (x))  ˛( f i (y))j

iD0

L

n1 X

d( f i (x); f i (y))

iD0

L

n1 X

C 1  in

iD0

< LC 1 1 (1  1 )1 : Setting K D exp(LC 1 1 (11 )1 ), one now sees that ( f n )0 (x)/( f n )0 (y) lies in the interval [K 1 :K], proving the claim. 

Smooth Ergodic Theory

In this distortion estimate, the function ˛ : M ! R is called a cocycle. The same argument applies to any Lipschitz continuous (or even Hölder continuous) cocycle.

As a passing comment, the ergodicity of  and positivity of imply that  is the unique f -invariant measure absolutely continuous with respect to . With more work, one can show that  is exact. See [2] for details.

Ergodicity of Expanding Maps The ergodic properties of C2 expanding maps are completely understood. In particular, every conservative expanding map is ergodic, and every expanding map is conservative. The proofs of these facts use Fundamental Principles #1 and 3 in a fairly direct way. Every C2 conservative expanding map is ergodic with respect to volume. The proof is a straightforward adaptation of the proof of Proposition 3 (see, e. g [2].). Here is a description of the proof for M D S 1 . As remarked earlier, the proof of Proposition 3 adapts easily to a general expanding map f : S 1 ! S 1 once one shows that for every f -invariant set A, and every connected component J of f n (S 1 n fpg), the quantity ( f n (A \ J) : f n (J)) (A: J) is bounded independently of n. This is a fairly direct consequence of the distortion estimate in Lemma 6.1 and is left as an exercise. The same distortion estimates show that every C2 expanding map is conservative, preserving a probability measure  in the measure class of volume. Here is a sketch of the proof for the case M D S 1 . To prove this, consider the push-forward n D fn . Then n is equivalent to Lebesgue, and its Radon–Nikodym derivative dn n d is the density function

n (x) D

X y2 f n (x)

1 : jac y f n

n RSince f  is a probability measure, it follows that S 1 n d D 1. A simple argument using the distortion estimate above (and summing up over all dn branches of f n at x) shows that there exists a constant c  1 such that for all x; y 2 S 1 ,

c 1 

n (x) c:

n (y)

Since the integral of n is 1, the functions n are uniformly bounded away from 0 and 1. It is easy to see that the meaP P sure n D n1 niD1 fi  has density n1 niD1 i . Let  be any subsequential weak* limit of n ; then  is absolutely continuous, with density bounded away from 0 and 1. With a little more care, one can show that is actually Lipschitz continuous.

Ergodicity of Conservative Anosov Diffeomorphisms Like conservative C2 expanding maps, conservative C2 Anosov diffeomorphisms are ergodic. This subsection outlines a proof of this fact. Unlike expanding maps, however, Anosov diffeomorphisms need not be conservative. The subsection following this one describe a type of invariant measure that is “natural” with respect to volume, called a Sinai-Ruelle-Bowen (or SRB) measure. The central result for hyperbolic systems states that every hyperbolic attractor carries an SRB measure. The Hopf Argument In the 1930s Hopf [14] proved that the geodesic flow for a compact, negatively-curved surface is ergodic. His method was to study the Birkhoff averages of continuous functions along leaves of the stable and unstable foliations of the flow. This type of argument has been used since then in increasingly general contexts, and has come to be known as the Hopf Argument. The core of the Hopf Argument is very simple. To any f : M ! M one can associate the stable equivalence relation s , where x s y iff lim n!1 d( f n (x); f n (y)) D 0. Denote by W s (x) the stable equivalence class containing x. When f is invertible, one defines the unstable equivalence relation to be the stable equivalence relation for f 1 , and one denotes by W u (x) the unstable equivalence class containing x. The first step in the Hopf Argument is to show that Birkhoff averages for continuous functions are constant along stable and unstable equivalence classes. Let  : M ! R be an integrable function, and let  D lim sup n!1

n 1X  ı fi : n

(1)

iD1

Observe that if  is continuous, then for every x 2 M and x 0 2 W s (x), lim n!1 j( f n (x))  ( f i (x 0 ))j D 0. It follows immediately that  f (x) D  f (x 0 ). In particular, if the limit in (1) exists at x, then it exists and is constant on W s (x). Fundamental Principle #4: Birkhoff averages of continuous functions are constant along stable equivalence classes. The next step of Hopf’s argument confines itself to the situation where f is conservative and Anosov. In this case, f

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is invertible, the stable equivalence classes are precisely the leaves of the stable foliation W s , and the unstable equivalence classes are the leaves of the unstable foliation W u . Since f is conservative, the Ergodic Theorem implies that for every L2 function , the function  f is equal (mod 0) to the projection of  onto the f -invariant functions in L2 . Since this projection is continuous, and the continuous functions are dense in L2 , to prove that f is ergodic, it suffices to show that the projection of any continuous function is trivial. That is, it suffices to show that for every continuous , the function  f is constant (a.e.). To this end, let  : M ! R be continuous. Since the f -invariant functions coincide with the f 1 -invariant functions, one obtains that  f D  f 1 a.e. The previous argument shows  f is constant along W s -leaves and  f 1 is constant along W u -leaves. The desired conclusion is that  f is a.e. constant. It suffices to show this in a local chart, since the manifold M is connected. In a local chart, after a smooth change of coordinates, one obtains a pair of transverse foliations F1 , F2 of the cube [1; 1]n by disks, and a measurable function : [1; 1]n ! R that is constant along the leaves of F1 and constant along the leaves of F2 . When the foliations F1 and F2 are smooth (at least C1 ), one can perform a further smooth change of coordinates so that F1 and F2 are transverse coordinate subspace foliations. In this case, Fubini’s theorem implies that any measurable function that is constant along two transverse coordinate foliations is a.e. constant. This completes the proof in the case that the foliations W s and W u are smooth. In Hopf’s original argument, the stable and unstable foliations were assumed to be C1 foliations (a hypotheses satisfied in the examples he considered, due to low-dimensionality. See also [16], where a pinching condition on the curvature, rather than low dimensionality, implies this C1 condition on the foliations.) Absolute Continuity For a general Anosov diffeomorphism or flow, the stable and unstable foliations are not C1 , and so the final step in Hopf’s orginal argument does not apply. The fundamental advance of Anosov and AnosovSinai was to prove that the stable and unstable foliations of an Anosov diffeomorphism (conservative or not) satisfy a weaker condition than smoothness, called absolute continuity. For conservative systems, absolute continuity is enough to finish Hopf’s argument, proving that every C2 conservative Anosov diffeomorphism is ergodic [15,17]. For a definition and careful discussion of absolute continuity of a foliation F , see [13]. Two consequences of the absolute continuity of F are:

1. (AC1) If A M is any measurable set, then (A) D 0 () F (x) (A) D 0 ; for   a.e. x 2 M ; where F (x) denotes the induced Riemannian volume on the leaf of F through x. 2. (AC2) If  is any small, smooth disk transverse to a local leaf of F , and T  is a 0-set in  (with respect to the induced Riemannian volume on ), then the union of the F leaves through points in T has Lebesgue measure 0 in M. The proof that W s and W u are absolutely continuous has a similar flavor to the proof that an expanding map has a unique absolutely continuous invariant measure (although the cocycles involved are Hölder continuous, rather than Lipschitz), and the facts are intimately related. With absolute continuity of the stable and unstable foliations in hand, one can now prove: Theorem 2 (Anosov) Let f be a C2 , conservative Anosov diffeomorphism. Then f is ergodic. Proof By the Hopf Argument, it suffices to show that if s and u are L2 functions with the following properties: 1. 2. 3.

s u s

is constant along leaves of W s , is constant along leaves of W u , and D u a.e.,

then s (and so u as well) is constant a.e. This is proved using the absolute continuity of W u and W s . Since M is connected, one may argue this locally. Let G be the full measure set of p 2 M such that s D u . Absolute continuity of W s (more precisely, consequence (AC1) of absolute continuity described above) implies that for almost every p 2 M, G has full measure in W s (p). Pick such a p. Then for almost every q 2 W s (p), s (q) D u (p); defining G0 to be the union over all q 2 W s (p) \ G of W u (q), one obtains that s is constant on G \ G 0 . But now, since W s (p)\G has full measure in W s (p), the absolute continuity of W u (consequence (AC2) above) implies that G0 has full measure in a neighborhood of p. Hence s is a.e. constant in a neighborhood of p, completing the proof.  SRB Measures In the absence of a smooth invariant measure, it is still possible for a map to have an invariant measure that behaves naturally with respect to volume. In computer simulations one observes such measures when one picks a point x at random and plots many iterates of x; in many systems, the resulting picture is surprisingly insensitive to the ini-

Smooth Ergodic Theory

tial choice of x. What appears to be happening in these systems is that the trajectory of almost every x in an open set U is converging to the support of a singular invariant probability measure . Furthermore, for any open set V, the proportion of forward iterates of x spent in V appears to converge to (V ) as the number of iterates tends to 1. In the 1960s and 70s, Sinai, Ruelle and Bowen rigorously established the existence of these physically observable measures for hyperbolic attractors [18,19,20]. Such measures are now known as Sinai-Ruelle-Bowen (SRB) measures, and have been shown to exist for non-hyperbolic maps with some hyperbolic features. This subsection describes the construction of SRB measures for hyperbolic attractors. An f -invariant probability measure  is called an SRB (or physical) measure if there exists an open set U M containing the support of  such that, for every continuous function  : M ! R and -a.e. x 2 U, Z n 1X i lim ( f (x)) D  d : n!1 n M iD1

The maximal open set U with this property is called the basin of f . To exclude the possibility that the SRB measure is supported on a periodic sink, one often adds the condition that at least one of the Lyapunov exponents of f with respect  is positive. Other definitions of SRB measure have been proposed (see [21]). Note that every ergodic absolutely continuous invariant measure with positive density in an open set is an SRB measure. Note also that an SRB measure for f is not in general an SRB measure for f 1 , unless f preserves an ergodic absolutely continuous invariant measure. Every transitive Anosov diffeomorphism carries a unique SRB measure. To prove this, one defines a sequence of probability measures n on M as follows. Fix a point p 2 M, and define 0 to be the normalized restriction of Riemannian volume to a ball Bu in W u (p). Set P n D n1 niD1 fi 0 . Distortion estimates show that the density of n on its support inside W u is bounded, above and below, independently of n. Passing to a subsequential weak* limit, one obtains a probability measure  on M with bounded densities on W u -leaves. To show that  is an SRB measure, choose a point q 2 M in the support of . Since  has positive density on unstable manifolds, almost every point in a neighborhood of q in W u (q) is a regular point for f (that is, a point where the forward Birkhoff averages of every continuous function exist). A variation on the Hopf Argument, using the absolute continuity of W s , shows that  is an ergodic SRB measure.

A similar argument shows that every transitive hyperbolic attractor admits an ergodic SRB measure. In fact this SRB measure has much stronger mixing properties, namely, it is Bernoulli. To prove this, one first constructs a Markov partition [22] conjugating the action of f to a Bernoulli shift. This map sends the SRB measure to a Gibbs state for a mixing Markov shift (see  Pressure and Equilibrium States in Ergodic Theory). A result that subsumes all of the results in this section is: Theorem 3 (Sinai, Ruelle, Bowen) Let  M be a transitive hyperbolic attractor for a C2 map f : M ! M. Then f has an ergodic SRB measure  supported on . Moreover: the disintegration of  along unstable manifolds of  is equivalent to the induced Riemannian volume, the Lyapunov exponents of  are all positive, and  is Bernoulli. Beyond Uniform Hyperbolicity The methods developed in the smooth ergodic theory of hyperbolic maps have been extended beyond the hyperbolic context. Two natural generalizations of hyperbolicity are:  partial hyperbolicity, which requires uniform expansion of Eu and uniform contraction of Es , but allows central directions at each point, in which the expansion and contraction is dominated by the behavior in the hyperbolic directions; and  nonuniform hyperbolicity, which requires hyperbolicity along almost every orbit, but allows the expansion of Eu and the contraction of Es to weaken near the exceptional set where there is no hyperbolicity. This section discusses both generalizations. Partial Hyperbolicity Brin and Pesin [23] and independently Pugh and Shub [24] first examined the ergodic properties of partially hyperbolic systems soon after the work of Anosov and Sinai on hyperbolic systems. One says that a diffeomorphism f : M ! M of a compact manifold M is partially hyperbolic if there is a nontrivial, continuous splitting of the tangent bundle, TM D E s ˚ E c ˚ E u , invariant under Df , such that Es is uniformly contracted, Eu is uniformly expanded, and Ec is dominated, meaning that for some n  1 and for all x 2 M : kD x f n j E s k < m(D x f n j E c )  kD x f n j E c k < m(D x f n j E u ): Partial hyperbolicity is a C1 -open condition: any diffeomorphism sufficiently C1 -close to a partially hyperbolic diffeomorphism is itself partially hyperbolic. For an exten-

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sive discussion of examples of partially hyperbolic dynamical systems, see the survey article [25] and the book [26]. Among these examples are: the time-1 map of an Anosov flow, the frame flow for a compact manifold of negative sectional curvature, and many affine transformations of compact homogeneous spaces. All of these examples preserve the volume induced by a Riemannian metric on M. As in the Anosov case, the stable and unstable bundles Es and Eu of a partially hyperbolic diffeomorphism are tangent to foliations, again denoted by W s and W u respectively [23]. Brin-Pesin and Pugh-Shub proved that these foliations are absolutely continuous. A partially hyperbolic diffeomorphism f : M ! M is accessible if any point in M can be reached from any other along an su-path, which is a concatenation of finitely many subpaths, each of which lies entirely in a single leaf of W s or a single leaf of W u . Accessibility is a global, topological property of the foliations W u and W s that is the analogue of transversality of W u and W s for Anosov diffeomorphisms. In fact, the transversality of these foliations in the Anosov case immediately implies that every Anosov diffeomorphism is accessible. Fundamental Principle #4 suggests that accessibility might be related to ergodicity for conservative systems. Conservative Partially Hyperbolic Diffeomorphisms Motivated by a breakthrough result with Grayson [27], Pugh and Shub conjectured that accessibility implies ergodicity, for a C2 , partially hyperbolic conservative diffeomorphism [28]. This conjecture has been proved under the hypothesis of center bunching [29], which is a mild spectral condition on the restriction of Df to the center bundle Ec . Center bunching is satisfied by most examples of interest, including all partially hyperbolic diffeomorphisms with dim(E c ) D 1. The proof in [29] is a modification of the Hopf Argument using Lebesgue density points and a delicate analysis of the geometric and measure-theoretic properties of the stable and unstable foliations. In the same article, Pugh and Shub also conjectured that accessibility is a widespread phenomenon, holding for an open and dense set (in the Cr topology) of partially hyperbolic diffeomorphisms. This conjecture has been proved completely for r D 1 [30], and for all r, with the additional assumption that the central bundle Ec is one dimensional [31]. Together, these two conjectures imply the third, central conjecture: in [28]: Conjecture 1 (Pugh–Shub) For any r  2, the Cr , conservative partially hyperbolic diffeomorphisms contain a Cr open and dense set of ergodic diffeomorphisms.

The validity of this conjecture in the absence of center bunching is currently an open question. Dissipative Partially Hyperbolic Diffeomorphisms There has been some progress in constructing SRB-type measures for dissipative partially hyperbolic diffeomorphisms, but the theory is less developed than in the conservative case. Using the same construction as for Anosov diffeomorphisms, one can construct invariant probability measures that are smooth along the W u foliation [32]. Such measures are referred to as u-Gibbs measures. Since the stable bundle Es is not transverse to the unstable bundle Eu , the Anosov argument cannot be carried through to show that u-Gibbs measures are SRB measures. Nonetheless, there are conditions that imply that a u-Gibbs measure is an SRB measure: for example, a u-Gibbs measure is SRB if it is the unique u-Gibbs measure [33], if the bundle E s ˚ E c is nonuniformly contracted [34], or if the bundle E u ˚ E c is nonuniformly expanded [35]. The proofs of the latter two results use Pesin Theory, which is explained in the next subsection. SRB measures have also been constructed in systems where Ec is nonuniformly hyperbolic [36], and in (noninvertible) partially hyperbolic covering maps where Ec is 1-dimensional [37]. It is not known whether accessibility plays a role in the existence of SRB measures for dissipative, non-Anosov partially hyperbolic diffeomorphisms. Nonuniform Hyperbolicity The concept of Lyapunov exponents gives a natural way to extend the notion of hyperbolicity to systems that behave hyperbolically, but in a nonuniform manner. The fundamental principles described above, suitably modified, apply to these nonuniformly hyperbolic systems and allow for the development of a smooth ergodic theory for these systems. This program was initially proposed and carried out by Yakov Pesin in the 1970s [38] and has come to be known as Pesin theory. Oseledec’s Theorem implies that if a smooth map f satisfying the condition m(D x f ) > 0 preserves a probability measure , then for -a.e. x 2 M and every nonzero vector v 2 Tx M, the limit (x; v) D lim

n!1

n 1X log kD x f i (v)k n iD1

exists. The number (x; v) is called the Lyapunov exponent at x in the direction of v. For each such x, there are finitely many possible values for the exponent (x; v), and the function x 7! (x; ) is measurable. See the discussion of Oseledec’s Theorem in  Ergodic Theorems.

Smooth Ergodic Theory

Let f be a smooth map. An f -invariant probability measure  is hyperbolic if the Lyapunov exponents of a.e. point are all nonzero. Observe that any invariant measure of a hyperbolic map is a hyperbolic measure. A conservative diffeomorphism f : M ! M is nonuniformly hyperbolic if the invariant measure equivalent to volume is hyperbolic. The term “nonuniform” is a bit misleading, as uniformly hyperbolic conservative systems are also nonuniformly hyperbolic. Unlike uniform hyperbolicity, however, nonuniform hyperbolicity allows for the possibility of different strengths of hyperbolicity along different orbits. Nonuniformly hyperbolic diffeomorphisms exist on all manifolds [39,40], and there are C1 -open sets of nonuniformly hyperbolic diffeomorphisms that are not Anosov diffeomorphisms [41]. In general, it is a very difficult problem to establish whether a given map carries a hyperbolic measure that is nonsingular with respect to volume. Hyperbolic Blocks As mentioned above, the derivative of f along almost every orbit of a nonuniformly hyperbolic system looks like the derivative down the orbit of a uniformly hyperbolic system; the nonuniformity can be detected only by examining a positive measure set of orbits. Recall that Lusin’s Theorem in measure theory states that every Borel measurable function is continuous on the complement of an arbitrarily small measure set. A sort of analogue of Lusin’s theorem holds for nonuniformly hyperbolic maps: every C2 , nonuniformly hyperbolic diffeomorphism is uniformly hyperbolic on a (noninvariant) compact set whose complement has arbitrarily small measure. The precise formulation of this statement is omitted, but here are some of its salient features. If  is a hyperbolic measure for a C2 diffeomorphism, then attached to -a.e. point x 2 M are transverse, smooth stable and unstable manifolds for f . The collection of all stable manifolds is called the stable lamination for f , and the collection of all unstable manifolds is called the unstable lamination for f . The stable lamination is invariant under f , meaning that f sends the stable manifold at x into the stable manifold for f (x). The stable manifold through x is contracted uniformly by all positive iterates of f in a neighborhood of x. Analogous statements hold for the unstable manifold of x, with f replaced by f 1 . The following quantites vary measurably in x 2 M:  the (inner) radii of the stable and unstable manifolds through x,  the angle between stable and unstable manifolds at x, and  the rates of contraction in these manifolds.

The stable and unstable laminations of a nonuniformly hyperbolic system are absolutely continuous. The precise definition of absolute continuity here is slightly different than in the uniformly and partially hyperbolic setting, but the consequences (AC1) and (AC2) of absolute continuity continue to hold. Ergodic Properties of Nonuniformly Hyperbolic Diffeomorphisms Since the stable and unstable laminations are absolutely continuous, the Hopf Argument can be applied in this setting to show: Theorem 4 (Pesin) Let f be C2 , conservative and nonuniformly hyperbolic. Then there exists a (mod 0) partition P of M into countably many f -invariant sets of positive volume such that the restriction of f to each P 2 P is ergodic. The proof of this theorem is also exposited in [42]. The countable partition can in examples be countably infinite; nonuniform hyperbolicity alone does not imply ergodicity. The Dissipative Case As mentioned above, establishing the existence of a nonsingular hyperbolic measure is a difficult problem in general. In systems with some global form of hyperbolicity, such as partial hyperbolicity, it is sometimes possible to “borrow” the expansion from the unstable direction and lend it to the central direction, via a small perturbation. Nonuniformly hyperbolic attractors have been constructed in this way [43]. This method is also behind the construction of a C1 open set of nonuniformly hyperbolic diffeomorphisms in [41]. For a given system of interest, it is sometimes possible to prove that a given invariant measure is hyperbolic by establishing an approximate form of hyperbolicity. The idea, due to Wojtkowski and called the cone method, is to isolate a measurable bundle of cones in TM defined over the support of the measure, such that the cone at a point x is mapped by D x f into the cone at f (x). Intersecting the images of these cones under all iterates of Df , one obtains an invariant subbundle of TM over the support of f that is nonuniformly expanded. Lai-Sang Young has developed a very general method [44] for proving the existence of SRB measures with strong mixing properties in systems that display “some hyperbolicity”. The idea is to isolate a region X in the manifold where the first return map is hyperbolic and distortion estimates hold. If this can be done, then the map carries a mixing, hyperbolic SRB measure. The precise rate of mixing is determined by the properties of the return-time function to X; the longer the return times, the slower the rate of mixing.

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More results on the existence of hyperbolic measures are discussed in the next section. An important subject in smooth ergodic theory is the relationship between entropy, Lyapunov exponents, and dimension of invariant measures of a smooth map. Significant results in this area include the Pesin entropy formula [45], the Ruelle entropy inequality [46], the entropy-exponents-dimension formula of Ledrappier– Young [47,48], and the proof by Barreira-Pesin-Schmeling that hyperbolic measures have a well-defined dimension [49].  Hyperbolic Dynamical Systems contains a discussion of these results; see this entry there for further information. The Presence of Critical Points and Other Singularities Now for a discussion of the aforementioned technical difficulties that arise in the presence of singularities and critical points for the derivative. Singularities, that is, points where Df (or even f ) fails to be defined, arise naturally in the study of billiards and hard sphere gases. The first subsection discusses some progress made in smooth ergodic theory in the presence of singularities. Critical points, that is, points where Df fails to be invertible, appear inescapably in the study of noninvertible maps. This type of complication already shows up for noninvertible maps in dimension 1, in the study of unimodal maps of the interval. The second subsection discusses the technique of parameter exclusion, developed by Jakobson, which allows for an ergodic analysis of a parametrized family of maps with criticalities. The technical advances used to overcome these issues in the interval have turned out to have applications to dissipative, nonhyperbolic, diffeomorphisms in higher dimension, where the derivative is “nearly critical” in places. The last subsection describes extensions of the parameter exclusion technique to these near-critical maps. Hyperbolic Billiards and Hard Sphere Gases In the 1870s the physicist Ludwig Boltzmann hypothesized that in a mechanical system with many interacting particles, physical measurements (observables), averaged over time, will converge to their expected value as time approaches infinity. The underlying dynamical system in this statement is a Hamiltonian system with many degrees of freedom, and the “expected value” is with respect to Liouville measure. Loosely phrased in modern terms, Boltzmann’s hypothesis states that a generic Hamiltonian system of this form will be ergodic on constant energy

submanifolds. Reasoning that the time scales involved in measurement of an observable in such a system are much larger than the rate of evolution of the system, Boltzmann’s hypothesis allowed him to assume that physical quantities associated to such a system behave like constants. In 1963, Sinai revived and formalized this ergodic hypothesis, stating it in a concrete formulation known as the Boltzmann-Sinai Ergodic Hypothesis. In Sinai’s formulation, the particles were replaced by N hard, elastic spheres, and to compactify the problem, he situated the spheres on a k-torus, k D 2; 3. The Boltzmann-Sinai Ergodic Hypothesis is the conjecture that the induced Hamiltonian system on the 2kN-dimensional configuration space is ergodic on constant energy manifolds, for any N  2. Sinai verified this conjecture for N D 2 by reducing the problem to a billiard map in the plane. As background for Sinai’s result, a brief discussion of planar billiard maps follows. Let D Rs be a connected region whose boundary @D is a collection of closed, piecewise smooth simple curves the plane. The billiard map is a map defined (almost everywhere) on @D [; ]. To define this map, one identifies each point (x; ) 2 @D [; ] with an inward-pointing tangent vector at x in the plane, so that the normal vector to @D at x corresponds to the pair (x; /2). This can be done in a unique way on the smooth components of @D. Then f (x; ) is obtained by following the ray originating at (x; ) until it strikes the boundary @D for the first time at (x 0 ; 0 ). Reflecting this vector about the normal at x0 , define f (x; ) D (x 0 ;   0 ). It is not hard to see that the billiard map is conservative. The billiard map is piecewise smooth, but not in general smooth: the degree of smoothness of f is one less than the degree of smoothness of @D. In addition to singularities arising from the corners of the table, there are singularities arising in the second derivative of f at the tangent vectors to the boundary. In the billiards studied studied by Sinai, the boundary @D consists of a union of concave circular arcs and straight line segments. Similar billiards, but with convex circular arcs, were first studied by Bunimovich [51]. Sinai and Bunimovich proved that these billiards are ergodic and nonuniformly hyperbolic. For the Boltzmann-Sinai problem with N  3, the relevant associated dynamical system is a higher dimensional billiard table in euclidean space, with circular arcs replaced by cylindrical boundary components. In a planar billiard table with circular/flat boundary, the behavior of vectors encountering a flat segment of boundary is easily understood, as is the behavior of vectors meeting a circular segment in a neighborhood of the nor-

Smooth Ergodic Theory

mal vector. If the billiard map is ergodic, however, every open set of vectors will meet the singularities in the table infinitely many times. To establish the nonuniform hyperbolicity of such billiard tables via conefieds, it is therefore necessary to understand precisely the fraction of time orbits spend near these singularities. Furthermore, to use the Hopf argument to establish ergodicity, one must avoid the singularities in the second derivative, where distortion estimates break down. The techniques for overcoming these obstacles involve imposing restrictions on the geometry of the table (even more so for higher dimensional tables), and are well beyond the scope of this paper. The study of hyperbolic billiards and hard sphere gases has a long and involved history. See the articles [50] and [52] for a survey of some of the results and techniques in the area. A discussion of methods in singular smooth ergodic theory, with particular applications to the Lorentz attractor, can be found in [53]. Another, more classical, reference is [54], which contains a formulation of properties on a critical set, due to Katok–Strelcyn, that are useful in establishing ergodicity of systems with singularities.

method has come to be known as parameter exclusion and has seen application far beyond the logistic family. As with billiards, it is possible to formulate geometric conditions on the map f t that control both expansion (hyperbolicity) and distortion on a positive measure set. As these conditions involve understanding infinitely many iterates of f t , they are impossible to verify for a given parameter value t. Using an inductive formulation of this condition, Jakobson showed that the set of parameters t near 4 that fail to satisfy the condition at iterate n have exponentially small measure (in n). He thereby showed that for a positive Lebesgue measure set of parameter values t, the map f t has an absolutely continuous invariant measure [55]. This measure is ergodic (mixing) and has a positive Lyapunov exponent. The delicacy of Jakobson’s approach is confirmed by the fact that for an open and dense set of parameter values, almost every orbit is attracted to a periodic sink, and so f t has no absolutely continuous invariant measure [56,57]. Jakobson’s method applies not only to the logistic family but to a very general class of C3 oneparameter families of maps on the interval.

Interval Maps and Parameter Exclusion

Near-Critical Diffeomorphisms

The logistic family of maps f t : x 7! tx(1  x) defined on the interval [0; 1] is very simple to define but exhibits an astonishing variety of dynamical features as the parameter t varies. For small positive values of t, almost every point in I is attracted under the map f t to the sink at 1. For values of t > 4, the map has a repelling hyperbolic Cantor set. As the value of t increases between 0 and 4, the map f t undergoes a cascade of period-doubling bifurcations, in which a periodic sink of period 2n becomes repelling and a new sink of period 2nC1 is born. At the accumulation of period doubling at t  3:57, a periodic point of period 3 appears, forcing the existence of periodic points of all periods. The dynamics of f t for t close to 4 has been the subject of intense inquiry in the last 20 years. The map f t , for t close to 4, shares some of the features of the doubling map T 2 ; it is 2-to-1, except at the critical point 12 , and it is uniformly expanding in the complement of a neighborhood of this critical point. Because this neighborhood of the critical point is not invariant, however, the only invariant sets on which f t is uniformly hyperbolic have measure zero. Furthermore, the second derivative of f t vanishes at the critical point, making it impossible to control distortion for orbits that spend too much time near the critical point. Despite these serious obstacles, Michael Jakobson [55] found a method for constructing absolutely continuous invariant measures for maps in the logistic family. The

Jakobson’s method in one dimension proved to extend to certain highly dissipative diffeomorphisms. The seminal paper in this extension is due to Benedicks and Carleson; the method has since been extended in a series of papers [59,60,61] and has been formulated in an abstract setting [62]. This extension turns out to be highly nontrivial, but it is possible to describe informally the similarities between the logistic family and higher-dimensional “near critical” diffeomorphisms. The diffeomorphisms to which this method applies are crudely hyperbolic with a one dimensional unstable direction. Roughly this means that in some invariant region of the manifold, the image of a small ball under f will be stretched significantly in one direction and shrunk in all other directions. The directions of stretching and contraction are transverse in a large proportion of the invariant region, but there are isolated “near critical” subregions where expanding and contracting directions are nearly tangent. The dynamics of such a diffeomorphism are very close to 1-dimensional if the contraction is strong enough, and the diffeomorphism resembles an interval map with isolated critical points, the critical points corresponding to the critical regions where stable and unstable directions are tangent. An illustration of this type of dynamics is the Hénon family of maps f a;b : (x; y) 7! (1  ax 2 C by; x), the orig-

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inal object of study in Benedicks-Carlesson’s work. When the parameter b is set to 0, the map f a;b is no longer a diffeomorphism, and indeed is precisely a projection composed with the logistic map. For small values of b and appropriate values of a, the Hénon map is strongly dissipative and displays the near critical behavior described in the previous paragraph. In analogy to Jakobson’s result, there is a positive measure set of parameters near b D 0 where f a;b has a mixing, hyperbolic SRB measure. See [63] for a detailed exposition of the parameter exclusion method for Hénon-like maps. Future Directions In addition to the open problems discussed in the previous sections, there are several general questions and problems worth mentioning:  What can be said about systems with everywhere vanishing Lyapunov exponents? Open sets of such systems exist in arbitrary dimension. Pesin theory carries into the nonuniformly hyperbolic setting the basic principles from uniformly hyperbolic theory (in particular, Fundamental Principles #3 and 4 above). To what extent do properties of isometric and unipotent systems (for example, Fundamental Principle #2) extend to conservative systems all of whose Lyaponov exponents vanish?  Can one establish the existence of and analyze in general the conservative systems on surfaces that have two positive measure regimes: one where Lyapunov exponents vanish, and the other where they are nonzero? Such systems are conjectured exist in the presence of KAM phenomena surrounding elliptic periodic points.  On a related note, how common are conservative systems whose Lyapunov exponents are nonvanishing on a positive measure set? See [64] for a discussion.  Find a broad description of those dissipative systems that admit finitely (or countably) many physical measures. Are such systems dense among all dissipative systems, or possibly generic among a restricted class of systems? See [41,65] for several questions and conjectures related to this problem.  Extend the methods in the study of systems with singularities to other specific systems of interest, including the infinite dimensional systems that arise in the study of partial differential equations.  Carry the methods of smooth ergodic theory further into the study of smooth actions of discrete groups (other than the integers) on manifolds. When do such actions admit (possibly non-invariant) “physical” measures?

There are other interesting open areas of future inquiry, but this gives a good sample of the range of possibilities. Bibliography 1. Arnol’d VI, Avez A (1986) Ergodic problems of classical mechanics. Translated from the French by Avez, A. Benjamin WA, Inc., New York Amsterdam 2. Mañé R (1987) Ergodic theory and differentiable dynamics. Translated from the Portuguese by Silvio L. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol 8. Springer, Berlin 3. Baladi V (2000) Positive transfer operators and decay of correlations. Advanced Series in Nonlinear Dynamics, 16. World Scientific Publishing Co., Inc., River Edge 4. Kifer Y (1986) Ergodic theory of random transformations. Progress in Probability and Statistics, vol 10. Birkhäuser, Boston 5. Kifer Y (1988) Random perturbations of dynamical systems. Progress in Probability and Statistics, 16. Birkhäuser, Boston 6. Liu PD, Qian M (1995) Smooth ergodic theory of random dynamical systems. Lecture Notes in Mathematics, vol 1606. Springer, Berlin 7. Bonatti C, Dìaz LJ, Viana M (2005) Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective. Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, III. Springer, Berlin 8. Cornfeld IP, Fomin SV, Sina˘ı YG (1982) Ergodic theory. Translated from the Russian by A. B. Sosinski˘ı. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 245. Springer, New York 9. Hirsch MW (1979) Differential topology. Graduate Texts in Mathematics, No. 33. Springer, New York Heidelberg 10. Robinson C (1995) Dynamical systems. Stability, symbolic dynamics, and chaos. Studies in Advanced Mathematics. CRC Press, Boca Raton 11. Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and Leonardo Mendoza. Encyclopedia of Mathematics and its Applications, vol 54. Cambridge University Press, Cambridge 12. Hirsch MW, Pugh CC, Shub M (1977) Invariant manifolds. Lecture Notes in Mathematics, vol 583. Springer, Berlin New York 13. Brin M, Stuck G (2002) Introduction to dynamical systems. Cambridge University Press, Cambridge 14. Hopf E (1939) Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung. Ber Verh Sachs Akad Wiss Leipzig 91:261–304 (German) 15. Anosov DV (1967) Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute of Mathematics, No. 90 1967. Translated from the Russian by S. Feder. American Mathematical Society, Providence 16. Sina˘ı JG (1961) Geodesic flows on compact surfaces of negative curvature. Dokl Akad Nauk SSSR 136:549–552 (Russian); translated as Soviet Math Dokl 2:106–109 17. Anosov DV, Sina˘ı JG (1967) Certain smooth ergodic systems. (Russian) Uspehi Mat Nauk 22 no. 5(137):107–172 18. Sina˘ı JG (1972) Gibbs measures in ergodic theory. (Russian) Uspehi Mat Nauk 27 no. 4(166):21–64 19. Ruelle D (1976) A measure associated with axiom-A attractors. Amer J Math 98(3):619–654

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20. Bowen R (1975) Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, vol 470. Springer, Berlin New York 21. Young LS (2002) What are SRB measures, and which dynamical systems have them? Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays. J Statist Phys 108(5– 6):733–754 22. Bowen R (1970) Markov partitions for Axiom A diffeomorphisms. Amer J Math 92:725–747 23. Brin MI, Pesin JB (1974) Partially hyperbolic dynamical systems. Izv Akad Nauk SSSR Ser Mat 38:170–212 (Russian) 24. Pugh C, Shub M (1972) Ergodicity of Anosov actions. Invent Math 15:1–23 25. Burns K, Pugh C, Shub M, Wilkinson A (2001) Recent results about stable ergodicity. Smooth ergodic theory and its applications. ( Seattle, 1999 ), 327–366. Proc Sympos Pure Math, 69, Amer Math Soc, Providence 26. Pesin YB (2004) Lectures on partial hyperbolicity and stable ergodicity. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich 27. Grayson M, Pugh C, Shub M (1994) Stably ergodic diffeomorphisms. Ann of Math 140(2):295–329 28. Pugh C, Shub M (1996) Stable ergodicity and partial hyperbolicity. International Conference on Dynamical Systems (Montevideo, 1995), 182–187, Pitman Res Notes Math Ser, 362, Longman, Harlow 29. Burns K, Wilkinson A, On the ergodicity of partially hyperbolic systems. Ann of Math. To appear 30. Dolgopyat D, Wilkinson A (2003) Stable accessibility is C 1 dense. Geometric methods in dynamics. II. Astérisque No. 287 31. Rodríguez HA, Rodríguez HF, Ures R (2008) Partially hyperbolic systems with 1D-center bundle. Invent Math 172(2) 32. Pesin YB, Sina˘ı YG (1983) Gibbs measures for partially hyperbolic attractors. Ergod Theor Dynam Syst 2(3–4):417–438 33. Dolgopyat D (2004) On differentiability of SRB states for partially hyperbolic systems. Invent Math 155(2):389–449 34. Bonatti C, Viana M (2000) SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J Math 115:157–193 35. Alves JF, Bonatti C, Viana M (2000) SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent Math 140(2):351–398 36. Burns K, Dolgopyat D, Pesin Y, Pollicott Mark Stable ergodicity for partially hyperbolic attractors with negative central exponents. Preprint 37. Tsujii M (2005) Physical measures for partially hyperbolic surface endomorphisms. Acta Math 194(1):37–132 38. Pesin JB (1977) Characteristic Ljapunov exponents, and smooth ergodic theory. Uspehi Mat Nauk 32(196):55–112, 287 (Russian) 39. Katok A (1979) Bernoulli diffeomorphisms on surfaces. Ann Math 110(2):529–547 40. Dolgopyat D, Pesin Y (2002) Every compact manifold carries a completely hyperbolic diffeomorphism. Ergod Theor Dynam Syst 22(2):409–435 41. Shub M, Wilkinson A (2000) Pathological foliations and removable zero exponents. Invent Math 139(3):495–508 42. Pugh C, Shub M (1989) Ergodic attractors. Trans Amer Math Soc 312(1):1–54 43. Viana M (1997) Multidimensional nonhyperbolic attractors. Inst Hautes Études Sci Publ Math No 85:63–96

44. Young LS (1998) Statistical properties of dynamical systems with some hyperbolicity. Ann Math 147(2):585–650 45. Pesin JB (1976) Characteristic Ljapunov exponents, and ergodic properties of smooth dynamical systems with invariant measure. Dokl Akad Nauk SSSR 226 (Russian) 46. Ruelle D (1978) An inequality for the entropy of differentiable maps. Bol Soc Brasil Mat 9(1):83–87 47. Ledrappier F, Young LS (1985) The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula. Ann Math 122(2):509–539 48. Ledrappier F, Young LS (1985) The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension. Ann Math 122(2):540–574 49. Barreira L, Pesin Y, Schmeling J (1999) Dimension and product structure of hyperbolic measures. Ann Math 149(2):755–783 50. Szász D (2000) Boltzmann’s ergodic hypothesis, a conjecture for centuries? Hard ball systems and the Lorentz gas, Encycl Math Sci, vol 101. Springer, Berlin, pp 421–448 51. Bunimoviˇc LA (1974) The ergodic properties of certain billiards. (Russian) Funkcional. Anal i Priložen 8(3):73–74 52. Chernov N, Markarian R (2003) Introduction to the ergodic theory of chaotic billiards. Second edition. Publicações Matemáticas do IMPA. IMPA Mathematical Publications 24o Col´quio Brasileiro de Matemática. 26th Brazilian Mathematics Colloquium Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro 53. Araújo V, Pacifico MJ (2007) Three Dimensional Flows. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications] 26o Col´quioquio Brasileiro de Matemática. [26th Brazilian Mathematics Colloquium] Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro 54. Katok A, Strelcyn JM, Ledrappier F, Przytycki F (1986) Invariant manifolds, entropy and billiards; smooth maps with singularities. Lecture Notes in Mathematics, 1222. Springer, Berlin 55. Jakobson MV (1981) Absolutely continuous invariant measures for one-parameter families of one-dimensional maps. Comm Math Phys 81(1):39–88 56. Graczyk J, S´ wiatek G (1997) Generic hyperbolicity in the logistic family. Ann Math 146(2):1–52 57. Lyubich M (1999) Feigenbaum–Coullet–Tresser universality and Milnor’s hairiness conjecture. Ann Math 149(2):319–420 58. Benedicks M, Carleson L (1991) The dynamics of the Hénon map. Ann Math 133(2):73–169 59. Mora L. Viana M (1993) Abundance of strange attractors. Acta Math 171(1):1–71 60. Benedicks M, Young LS (1993) Sina˘ı–Bowen–Ruelle measures for certain Hénon maps. Invent Math 112(3):541–576 61. Benedicks M, Viana M (2001) Solution of the basin problem for Hénon-like attractors. Invent Math 143(2):375–434 62. Wang QD, Young LS (2008) Toward a theory of rank one attractors. Ann Math 167(2) 63. Viana M, Lutstsatto S (2003) Exclusions of parameter values in Hénon-type systems. (Russian) Uspekhi Mat Nauk 58 (2003), no. 6(354), 3–44; translation in Russian Math. Surveys 58(6):1053–1092 64. Bochi J, Viana M (2004) Lyapunov exponents: how frequently are dynamical systems hyperbolic? Modern dynamical systems and applications. Cambridge Univ Press, Cambridge, pp 271– 297 65. Palis J (2005) A global perspective for non-conservative dynamics. Ann Inst H Poincaré Anal Non Linéaire 22(4):485–507

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Soliton Perturbation JI -HUAN HE Modern Textile Institute, Donghua University, Shanghai, China Article Outline Glossary Definition of the Subject Introduction Methods for Soliton Solutions Future Directions Bibliography Glossary Soliton A soliton is a nonlinear pulse-like wave that can exist in some nonlinear systems. The isolated wave can propagate without dispersing its energy over a large region of space; collision of two solitons leads to unchanged forms, solitons also exhibit particlelike properties. Soliton perturbation theory The soliton perturbation theory is used to study the solitons that are governed by the various nonlinear equations in presence of the perturbation terms. Homotopy perturbation method The homotopy perturbation method is a useful tool to the search for solitons without the requirement of presence of small perturbations. In this method, a homotopy is constructed with a homotopy parameter, p. When p D 0, it becomes a nonlinear wave equation such as a KdV equation with a known soliton solution; when p D 1, it turns out to be the original nonlinear equation. To change p from zero to unity, one must only change from a trial soliton to the solved soliton. Variational iteration method The variational iteration method is a new method for obtaining soliton-type solutions of various nonlinear wave equations. The method begins with a soliton-type solution with some unknown parameters which can be determined after few iterations. The iteration formulation is constructed by a general Lagrange multiplier which can be identified optimally via variational theory. Exp-function method The exp-function method is a new method for searching for both soliton-type solutions and periodic solutions of nonlinear systems. The method assumes that the solutions can be expressed in arbitrary forms of the exp-function.

Definition of the Subject The soliton is a kind of nonlinear wave. There are many equations of mathematical physics which have solutions of the soliton type. The first observation of this kind of wave was made in 1834 by John Scott Russell [1]. In 1895, the famous KdV equation, which possesses soliton solutions, was obtained by D. J. Korteweg and H. de Vries [2], who established a mathematical basis for the study of various solitary phenomena. From a modern perspective, the soliton is used as a constructive element to formulate the complex dynamical behavior of wave systems throughout science: from hydrodynamics to nonlinear optics, from plasmas to shock waves, from tornados to the Great Red Spot of Jupiter, from traffic flow to the Internet, from Tsunamis to turbulence [3]. More recently, solitary waves are of key importance in the quantum fields: on extremely small scales and at very high observational resolution equivalent to a very high energy, space–time resembles a stormy ocean and particles and their interactions have soliton-type solutions [4]. Introduction The soliton was first discovered in 1834 by John Scott Russell, who observed that a canal boat stopping suddenly gave rise to a solitary wave which traveled down the canal for several miles, without breaking up or losing strength. Russell named this phenomenon the ‘soliton’. In a highly informative as well as entertaining article [1] J.S. Russell gave an engaging historical account of the important scientific observation: I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview

Soliton Perturbation

with that singular and beautiful phenomenon which I have called the Wave of Translation. His ideas did not earn attention until 1965 when N.J. Zabusky and M.D. Kruskal began to use a finite difference approach to the study of KdV equation [5], and various analytical methods also led to a complete understanding of Solitons, especially the inverse scattering transform proposed by Gardner, Greene, Kruskal, and Miura [6] in 1967. The significance of Russell’s discovery was then fully appreciated. It was discovered that many phenomena in physics, electronics and biology can be described by the mathematical and physical theory of the ‘Soliton’. The particle-like properties of solitons [7] also caught much attention, and were proposed as models for elementary particles [8]. More recently it has been realized that some of the quantum fields which are used to describe particles and their interactions also have solutions of the soliton type [9].

In most cases the nonlinear term R(u) in Eq. (1) plays an import role in understanding various solitary phenomena, and the coefficient " is not limited to a “small parameter”.

Variational Approach Recently, variational theory and homotopy technology have been successfully applied to the search for soliton solutions [17,18] without requiring the small parameter assumption. Both variational and homotopy technologies can lead to an extremely simple and elementary, but rigorous, derivation of soliton solutions. Considering the KdV equation @u @u @3 u  6u C 3 D0; @t @x @x

(4)

we seek its traveling wave solutions in the following frame

Methods for Soliton Solutions The investigation of soliton solutions of nonlinear evolution equations plays an important role in the study of nonlinear physical phenomena. There are many analytical approaches to the search for soliton solutions, such as soliton perturbation, tanh-function method, projective approach, F-expansion method, and others [10,11,12,13,14].

u(x; t) D U()

v(x; t) D V();

 D x  ct ;

(5)

where c is angular frequency. Substituting Eq. (5) into Eq. (4) yields  cu0  6uu0 C u000 D 0 ;

(6)

Soliton Perturbation We consider the following perturbed nonlinear evolution equation [15,16] u T C N(u) D "R(u);

0 < "  1:

(1)

When " D 0, we have the un-perturbed equation u T C N(u) D 0 ;

(2)

which is assumed to have a soliton solution. When " ¤ 0, but 0 < "  1, we can use perturbation theory [15,16], and look for approximate solutions of Eq. (1), which are close to the soliton solutions of Eq. (2). Using multiple time scales (a slow time  and a fast time t, such that @T D @ t C "@ ), we assume that the soliton solution can be expressed in the form u(x; T) D u0 (; ) C "u1 (; ; t) C "2 u2 (; ; t) C    (3) where  D x  ct, and  is a slow time and t is a fast time. Substituting Eq. (3) into Eq. (1) and then equating likepowers of ", we can obtain a series of linear equations for u i (i D 0; 1; 2; 3; : : :).

where a prime denotes the differential with respect to . Integrating Eq. (6) yields the result  cu  3u2 C u00 D 0 :

(7)

By the semi-inverse method [19], the following variational formulation is established Z JD 0

1

1 1 2 cu C u3 C 2 2



du d

2 ! d :

(8)

The semi-inverse method is a powerful mathematical tool to the search for variational formulae for real-life physical problems. By the Ritz method, we search for a solitary wave solution in the form u D p sech2 (q) ; where p and q are constants to be further determined.

(9)

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Soliton Perturbation

According to the variational iteration method, its iteration formulation can be constructed as follows

Substituting Eq. (9) into Eq. (8) results in Z 1

1 2 cp sech4 (q) C p3 sech6 (q) JD 2 0  1 C (4p2 q2 sech4 (q) tanh2 (q) d 2 Z Z p3 1 cp2 1 sech4 (z)dz C sech6 (z)dz D 2q 0 q 0 Z 1 ˚ sech4 (z) tanh4 (z) dz C 2p2 q

u nC1 (x; t)

To search for its compacton-like solution, we assume the solution has the form (10)

2

 5c  8p C 4q D 0 :

u0 (x; t) D

a sin2 (kx C w t) ; b C c sin2 (kx C w t)

(19)

where a, b, k, and w are unknown constants further to be determined after few iterations [17]. (11) (12)

or simplifying 5c C 12p C 4q2 D 0 ;

˚ (u n ) t C u2n (u n )x C (u n )x x x dt : (18)

Making J stationary with respect to p and q results in @J 2cp 24p2 8pq D C C D 0; @p 3q 15q 15 @J 8p3 4p2 cp2 C D 2  D 0; @q 3q 15q2 15

t 0

0

8p3 4p2 q cp2 C C : D 3q 15q 15

Z

D u n (x; t) 

(13) (14)

From Eqs. (13) and (14), we can easily obtain the following relations: r 1 c p D  c; q D : (15) 2 4

Homotopy Perturbation Method The homotopy perturbation method [25] provides a simple mathematical tool for searching for soliton solutions without any small perturbation [18,26]. Considering the following nonlinear equation @u @u @3 u C au C b 3 C N(u) D 0; @t @x @x

a > 0; b > 0 ; (20)

we can construct a homotopy in the form

 @u @u @3 u (1  p) C 6u C 3 @t @x @x

 @u @u @3 u Cp C au C b 3 C N(u) D 0 : @t @x @x

(21)

So the solitary wave solution can be approximated as r c c 2 u D  sech (16) (x  ct  0 ) ; 2 4

When p D 0, we have

which is the exact solitary wave solution of KdV equation (4). The preceding analysis has the virtue of utter simplicity. The suggested variational approach can be readily applied to the search for solitary wave solutions of other nonlinear problems, and the present example can be used as paradigms for many other applications in searching for solitary wave solutions of real-life physics problems.

a well-known KdV equation whose soliton solution is known. When p D 1, Eq. (21) turns out to be the original equation. According to the homotopy perturbation method, we assume

Variational Iteration Method The variational iteration method [20] is an alternative approach to soliton solutions without the requirement of establishing a variational formulation for the discussed problems [17,21,22,23,24]. As an illustrating example, we consider the K(3,1) equation in the form [17]: u t C u2 u x C u x x x D 0 :

(17)

@u @u @3 u C 6u C 3 D0; @t @x @x

u D u0 C pu1 C p2 u2 C   

(22)

(23)

Substituting Eq. (23) into Eq. (21), and proceeding with the same process as the traditional perturbation method does, we can easily solve u0 ; u1 and other components. The solution can be expressed finally in the form u D u0 C u1 C u2 C   

(24)

The homotopy perturbation method always stops before the second iteration, so the solution can be expressed as u D u0 C u1 for most cases.

(25)

Soliton Perturbation

Parameter-Expansion Method

Future Directions

The parameter-expansion method [27,28,29,30] does not require one to construct a homotopy. To illustrate its solution procedure, we re-write Eq. (20) in the form

It is interesting to point out the connection of catastrophe theory to loop soliton chaos, and finally to chaotic Cantorian spacetime [34,35]. El Naschie [34,35] studied the Eguchi–Hanson gravitational instanton solution and its interpretation by ‘t Hooft in the context of a quantum gravitational Hilbert space, as an event and a possible solitonic “extended” particle. Transferring a certain solitonic solution of Einstein’s field equations in Euclidean “real” space–time to the mathematical infinitely-dimensional Hilbert space, it is possible to observe a new non-standard process by which a definite mass can be assigned to massless particles. Thus by invoking Einstein’s gravity in “solitonic” gauge theory and vice versa, an alternative explanation for how massless particles acquire mass is found, which is also in harmony with the basic structure of our standard model as it stands at present [34,35].

@u @3 u @u C au C b 3 C 1  N(u) D 0 : @t @x @x

(26)

Supposing that the parameters a, b, and 1 can be expressed in the forms a D a0 C pa1 C p2 a2 C   

(27)

b D b0 C pb1 C p2 b2 C   

(28)

1 D pc1 C p2 c2 C   

(29)

where p is a bookkeeping parameter, p D 1. Substituting Eqs. (23), (27), (28) and (29) into Eq. (26) and proceeding the same way as the perturbation method, we can easily obtain the needed solution.

Bibliography Exp-function Method The exp-function method [31,32,33] provides us with a straightforward and concise approach to obtaining generalized solitonary solutions and periodic solutions and the solution procedure, with the help of Matlab or Mathematica, is utterly simple. Consider a general nonlinear partial differential equation of the form F(u; u x ; u y ; uz ; u t ; u x x ; u y y ; uzz ; u t t ; u x y ; u x t ; u y t ; : : :) D 0:

(30)

Using a transformation  D ax C by C cz C dt ;

(31)

we can re-write Eq. (30) in the form of the following nonlinear ordinary differential equation: G(u; u0 ; u00 ; u000 ; : : :) D 0 ;

(32)

where a prime denotes a derivation with respect to . According to the exp-function method, the traveling wave solutions can be expressed in the form Pl

u() D P jnDk

a n exp(n)

mDi b m exp(m)

;

(33)

where i, j, k, and l are positive integer which could be freely chosen, an and bm are unknown constants to be determined. The solution procedure is illustrated in [32].

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solitons coexisting with bright soliton in nonlinear fiber arrays. Int J Nonlinear Sci Num Simul 6(4):371–385; Abdusalam HA (2005) On an improved complex tanh-function method. Int J Nonlinear Sci Num Simul 6(2):99–106 El-Sabbagh MF, Ali AT (2005) New exact solutions for (3+1)-dimensional Kadomtsev–Petviashvili equation and generalized (2+1)-dimensional Boussinesq equation. Int J Nonlinear Sci Num Simul 6(2):151–162 Shen JW, Xu W (2004) Bifurcations of smooth and non-smooth travelling wave solutions of the Degasperis-Procesi equation. Int J Nonlinear Sci Num Simul 5(4):397–402 Sheng Z (2007) Further improved F-expansion method and new exact solutions of Kadomstev–Petviashvili equation. Chaos Solit. Fract 32(4):1375–1383 Yu H, Yan J (2006) Direct approach of perturbation theory for kink solitons. Phys Lett A 351(1–2):97–100 Herman RL (2005) Exploring the connection between quasistationary and squared eigenfunction expansion techniques in soliton perturbation theory. Nonlinear Anal 63(5–7):e2473– e2482 He JH, Wu XH (2006) Construction of solitary solution and compacton-like solution by variational iteration method. Chaos Solit Fract 29(1):108–113 He JH (2005) Application of homotopy perturbation method to nonlinear wave equations. Chaos Solit Fract 26(3):695–700 He JH (2004) Variational principles for some nonlinear partial differential equations with variable coefficients. Chaos Solit Fract 19(4):847–851 He JH (1999) Variational iteration method – a kind of non-linear analytical technique: Some examples. Int J Non-Linear Mech 34(4):699–708 Abulwafa EM, Abdou MA, Mahmoud AA (2007) Nonlinear fluid flows in pipe-like domain problem using variational-iteration method. Chaos Solit Fract 32(4):1384–1397 Inc M (2007) Exact and numerical solitons with compact support for nonlinear dispersive K(m,p) equations by the variational iteration method. Phys A 375(2):447–456 Soliman AA (2006) A numerical simulation and explicit solutions of KdV-Burgers’ and Lax’s seventh-order KdV equations. Chaos Solit Fract 29(2):294–302 Abdou MA, Soliman AA (2005) Variational iteration method

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for solving Burger’s and coupled Burger’s equations. J Comput Appl Math 181(2):245–251 He JH (2000) A coupling method of a homotopy technique and a perturbation technique for non-linear problems. Int J NonLinear Mech 35(1):37–43 Ganji DD, Rafei M (2006) Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method. Phys Lett A 356(2):131–137 Shou DH, He JH (2007) Application of Parameter-expanding Method to Strongly Nonlinear Oscillators. Int J Nonlinear Sci Numer Simul 8:113–116 He JH (2001) Bookkeeping parameter in perturbation methods. Int J Nonlinear Sci Numer Simul 2:257–264 He JH (2002) Modified Lindstedt-Poincare methods for some strongly non-linear oscillations. Part I: expansion of a constant. Int J Non-Linear Mech 37:309–314 Xu L (2007) He’s parameter-expanding methods for strongly nonlinear oscillators. J Comput Appl Math 207(1):148–157 He J-H, Wu X-H (2006) Exp-function method for nonlinear wave equations. Chaos Solit Fract 30(3):700–708 He J-H, Abdou MA (2007) New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos Solit Fract 34(5):1421–1429 Wu X-H, He J-H (2008) EXP-function method and its application to nonlinear equations. Chaos Solit Fract 38(3):903–910 El Naschie MS (2004) Gravitational instanton in Hilbert space and the mass of high energy elementary particles. Chaos Solit Fract 20(5):917–923 El Naschie MS (2004) How gravitational instanton could solve the mass problem of the standard model of high energy particle physics. Chaos Solit Fract 21(1):249–260

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Solitons and Compactons

Solitons and Compactons JI -HUAN HE1 , SHUN-DONG Z HU2 1 Modern Textile Institute, Donghua University, Shanghai, China 2 Department of Science, Zhejiang Lishui University, Lishui, China Article Outline Glossary Definition of the Subject Introduction Solitons Compactons Generalized Solitons and Compacton-like Solutions Future Directions Cross References Bibliography Glossary Soliton A soliton is a stable pulse-like wave that can exist in some nonlinear systems. The soliton, after a collision with another soliton, eventually emerges unscathed. Compacton A compacton is a special solitary traveling wave that, unlike a soliton, does not have exponential tails. Generalized soliton A generalized soliton is a soliton with some free parameters. Generally a generalized soliton can be expressed by exponential functions. Compacton-like solution A compcton-like solution is a special wave solution which can be expressed by the squares of sinusoidal or cosinoidal functions.

Introduction Solitary waves were first observed by John Scott Russell in 1895, and were studied by D. J. Korteweg and H. de Vries in 1895 . Compactons are special solitons with finite wavelength. It was Philip Rosenau and his colleagues who first found compactons in 1993. Please refer to “ Soliton Perturbation” for detailed information. Solitons A soliton is a special solitary traveling wave that after a collision with another soliton eventually emerges unscathed. Solitons are solutions of partial differential equations that model phenomena like water waves or waves along a weakly anharmonic mass-spring chain. The Korteweg–de Vries (KdV) equation is the generic model for the study of nonlinear waves in fluid dynamics, plasma and elastic media. KdV equation is one of the most fundamental equations in nature and plays a pivotal role in nonlinear phenomena. We consider the KdV equation in the form @u @3 u @u C 6u C 3 D0: @t @x @x Its solitary traveling wave solution can be solved as ˚ u(x; t) D 12 c sech2 12 c 1/2 (x  ct) :

(2)

The bell-like solution as illustrated in Fig. 1 is called a soliton. We re-write Eq. (2) in an equivalently form: u() D p sec h2 (q) D

e2q

4p : C e2q C 2

(3)

where u(x; t) D u();  D x  ct, c is the wave velocity.

Definition of the Subject Soliton and compacton are two kinds of nonlinear waves. They play an indispensable and vital role in all ramifications of science and technology, and are used as constructive elements to formulate the complex dynamical behavior of wave systems throughout science: from hydrodynamics to nonlinear optics, from plasmas to shock waves, from tornados to the Great Red Spot of Jupiter, from traffic flow to Internet, from Tsunamis to turbulence. More recently, soliton and compacton are of key importance in the quantum fields and nanotechnology especially in nanohydrodynamics.

(1)

Solitons and Compactons, Figure 1 Bell-like solitary wave

1553

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The second Eq. (9) has a positive coefficient of u, and in this case the general solution reads

It is obvious that lim u() D 0

and

!1

lim u() D 0 :

!1

(4)

The soliton has exponential tails, which are the basic character of solitary waves. The soliton obeys a superposition-like principle: solitons passing through one another emerge unmodified, see Fig. 2.

Now consider a modified version of KdV equation in the form 2

u t C (u )x C (u )x x x D 0 :

This solution describes an oscillation at the angular velocity !. Equation (7) behaves sometimes like an oscillator when 1  cv 1/2 > 0, i. e., u D v 1/2 has a periodic solution, we assume v can be expressed in the form

 cu C u2 C (u2 )  D D ;

(6)

(7)

where u2 D v. In case c D 0, we have periodic solution: v() D A cos  C B sin . Periodic solution of nonlinear oscillators can be approximated by sinusoidal function. It helps understanding if an equation can be classified as oscillatory by direct inspection of its terms. We consider two common order differential equations whose exact solutions are important for physical understanding: 2

u k u D0;

(8)

(9)

Both equations have linear terms with constant coefficients. The crucial difference between these two very simple equations is the sign of the coefficient of u in the second term. This determines whether the solutions are exponential or oscillatory. The general solution of Eq. (8) is u D Ae k t C Bek t :

C A2 cos4 !  cA cos2 ! D 0 :

(13)

We, therefore, have 12A2 ! 2  cA D 0

(14)

16A2 ! 2 C A2 D 0 : Solving the above system, Eq. (14), yields !D

1 ; 4

AD

4 c: 3

(15)

We obtain the solution in the form 

1 4c u D v 1/2 D cos2 (x  ct) : 3 4

(16)

By a careful inspection, v can tend to a very small value or even zero, as a result, 1  cv 1/2 tends to negative infinite, and Eq. (7) behaves like Eq. (8) with k ! 1, the exponential tails vanish completely at the edge of the bell-shape (see Fig. 3): (  4c cos2 14 (x  ct) ; jx  ctj  2 3 uD (17) 0; otherwise. This is a compact wave. Unlike solitons (Fig. 4), compacton does not have exponential tails (Fig. 3). Generalized Solitons and Compacton-like Solutions

and u00 C ! 2 u D 0 :

Substituting Eq. (12) into Eq. (7) results in

(5)

where D is an integral constant. To avoid singular solutions, we set D D 0. We re-write Eq. (6) in the form v  C v  cv 1/2 D 0 ;

(12)

12A2 ! 2 cos2 !  16A2 ! 2 cos4 !

Introducing a complex variable  defined as  D x  ct, where c is the velocity of traveling wave, integrating once, we have

00

(11)

v D u2 D A2 cos4 ! :

Compactons

2

u D A cos ! t C B sin ! t :

(10)

Solitary solutions have tails, which can be best expressed by exponential functions. We can assume that a solitary solution can be expressed in the following general form d P

u() D

nDc q P mDp

a n exp(n) ;

(18)

b m exp(m)

where c; d; p; and q are positive integers which are unknown to be further determined, a n and b m are unknown

Solitons and Compactons

Solitons and Compactons, Figure 2 Collision of two solitary waves

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Solitons and Compactons

We suppose that the solution of Eq. (20) can be expressed as u() D

a c exp(c) C    C ad exp(d) : b p exp(p) C    C bq exp(q)

(21)

To determine values of c; d; p and q, we balance the linear term of highest order in Eq. (20) with the highest order nonlinear term. According to the homogeneous balance principle, we obtain the result c D p and d D q. For simplicity, we set c D p D 1 and d D q D 1, so Eq. (21) reduces to u() D

a1 exp() C a0 C a1 exp() : exp(p) C b0 C b1 exp()

(22)

Substituting Eq. (22) into Eq. (20), and by the help of Matlab, clearing the denominator and setting the coefficients of power terms like exp( j); j D 1; 2;    to zero yield a system of algebraic equations, solving the obtained system, we obtain the following exact solutions:   8 b02 3k 2 C 2a12 3k 2 b0 ˆ ˆ ˆ a 0 D a 1 b0 C ; a1 D ; < a1 8a1   ˆ b02 3k 2 C 2a12 ˆ ˆ ; ! D ka12  k 3 ; : b1 D 8a12 (23)

Solitons and Compactons, Figure 3 Compaton wave without tails

where a1 and b0 are free parameters, which depends upon the initial conditions and/or boundary conditions. The property that stability may depend on initial/boundary conditions is characteristic only for nonlinear systems. The relationship between wave speed and frequency is ! D ka12  k 3 :

Note that the value of a1 is determined from the initial/boundary conditions, so frequency or wave speed may not independent of initial/boundary conditions. Then, the closed form solution of Eq. (19) reads

Solitons and Compactons, Figure 4 Solitary wave with two tails

constants. The unknown constants can be easily determined using Matlab, the method is called the Exp-function method. We consider the modified KdV equation in the form: u t C u2 u x C u x x x D 0 :

(19)

Using a transformation: u(x; t) D u();  D kx C ! t, we have !u0 C ku2 u0 C k 3 u000 D 0 ;

(24)

(20)

where prime denotes the differential with respect to .

u(x; t) 

a1 exp[kx  (ka12 C k 3 )t] C a1 b0

 2 b 2 (3k 2 C2a 2 ) C 3ka 1b 0 C 0 8a 1 1 exp[kx C (ka12 C k 3 )t] D  exp[kx  (ka12 C k 3 )t] C b0  b 2 (3k 2 C2a 2 ) C 0 8a 2 1 exp[kx C (ka12 C k 3 )t] 1

Da1 C 

3k 2 b 0 8

exp[kx  (ka12 C k 3 )t] C b0 C

b 02 (3k 2 C2a 12 ) 8a 12

:

 exp[kx C (ka12 C k 3 )t] (25)

Solitons and Compactons

Solitons and Compactons, Figure 5 Propagation of a solution with respect to time

Solitons and Compactons, Figure 6 Periodic solution

Generally a1 , b0 and k are real numbers, and the obtained solution, Eq. (25), is a generalized soliton solution. p If we choose k D 1; a1 D 1; b0 D 8/5, Eq. (25) becomes p 3 1/40 u(x; t) D 1C p : (26) exp[x  2t] C 8/5 C exp[x C 2t]

Substituting Eq. (29) into Eq. (27) results in a periodic solution, which reads ˙3K 2 u(x; t) D a1 C

q

2 3K 2 C2a 12

cos[Kx  (Ka12  K 3 )t] ˙

q

2 3K 2 C2a 12

(30) The bell-like solution is illustrated in Fig. 5. In case k is an imaginary number, the obtained solitary solution can be converted into periodic solution or compact-like solution. We write k D iK, Eq. (25) becomes

or a generalized compact-like solution:

u(x; t) D

u(x; t) D a1 2  3K8 b 0

C 

ˆ ˆ ˆ :

˙3K 2

r

2 3K 2 C2a 21

cos[ K x( K a 12 K 3 ) t ]˙

r

!

;

2 3K 2 C2a 21

a1 C 3K 2 ; otherwise ˇ  ˇ  ˇKx  Ka 2  K 3 t ˇ   1 2

;

(1 C p) exp[Kx  (Ka12  K 3 )t] C b0

8 ˆ ˆ ˆ < a1 C

 b 2 (3k 2 C2a 2 ) Ci(1  p) 0 8a 2 1 exp[Kx C (Ka12  K 3 )t]

(31)

1

(27) where p D

b 02 (3K 2 C2a 12 ) 8a 12

.

If we search for a periodic solution or compact-like solution, the imaginary part in the denominator of Eq. (27) must be zero, that requires that 1 p D1

b02 (3K 2 C 2a12 ) D0: 8a12

Solving b0 from Eq. (28), we obtain s 8 b0 D ˙ : 3K 2 C 2a12

(28)

(29)

where a1 and K are free parameters, and it requires pthat 2a12 > 3K 2 . If we choose k D 1; a1 D 1; b0 D 8/5, Eq. (30) becomes p 3 2 u(x; t) D 1 C p : (32) cos[x  3t] C 2 The periodic solution is illustrated in Fig. 6. Now we give an heuristical explanation of why Eq. (19) behaves sometimes periodically and sometimes compaton-like. We re-written Eq. (20) in form u00 C

! 1 u C 2 u3 D 0: k3 3k

(33)

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Solitons and Compactons

It is a well-known Duffing equation with a periodic solution for all ! > 0 and k > 0. Actually in our study ! can be negative, we re-write Eq. (33) in the form u00 

! 1 u C 2 u3 D 0 ; k3 3k

! > 0:

(34)

This equation, however, has not always a periodic solution. We use the parameter-expansion method to find its period and the condition to be an oscillator. In order to carry out a straightforward expansion like that in the classical perturbation method, we need to introduce a parameter, , because none appear explicitly in this equation. To this end, we seek an expansion in the form 2

3

u D u0 C u1 C  u2 C  u3 C    :

! D ˝ 2 C m1  C m2 2 C : : : k3

1 D n1  C n2 2 C : : : ; 3k 2

(37)

 00 u0 C u1 C 2 u2 C : : :    C ˝ 2 C m1  C m2 2 C : : : u0 C u1 C 2 u2 C : : :   3  C n1  C n2 2 C : : :  u0 C u1 C 2 u2 C : : : D 0 (38) and equating coefficients of like powers of , we obtain Coefficient of 0 (39)

Coefficient of 1 u100 C ˝ 2 u1 C m1 u0 C n0 u03 D 0 :

(40)

The solution of Eq. (39) is u0 D A cos ˝ t :

(41)

Substituting u0 into (40) gives u100 C ˝ 2 u1 C A(m1 C 34 n0 A2 ) cos ˝ t C 14 n0 A3 cos 3˝ t D 0 :

or

AD0:

(43)

If the first-order approximate solution is searched for, then we have 

! D ˝ 2 C m1 k3

1 D n1 : 3k 2

(44) (45)

We finally obtain the following relationship ˝2 D 

! 1 C 2 A2 : 3 k 4k

(46)

To behave like an oscillator requires that 

! 1 C 2 A2 > 0 3 k 4k

(47)

or (36)

where mi and ni are unknown constants to be further determined. Interpretation of why such expansions work well is given by [1]. Substituting Eqs.(35)–(37) to (34), we have

u000 C ˝ 2 u0 D 0 :

3 m1 C n0 A2 D 0 4

(35)

The parameter  is used as a bookkeeping device and is set equal to unity. The coefficients of the linear term and nonlinear term can be, respectively, expanded in a similar way: 

No secular term in u1 requires that

(42)

! 1 < A2 : k 4

(48)

The amplitude A may strongly depend upon initial/boundary conditions which may determine the wave type of a nonlinear equation. Now we approximate Eq. (34) in the form 1 u C 2 k 00

  A2 cos2 ˝ t ! u D0:  C k 3

(49)

In case j˝ tj ! /2, the above equation behaves exponentially, resulting in a compact-like wave as discussed above. Future Directions It is interesting to identify the conditions for a nonlinear equation to have solitary, or periodic, or compacton-like solutions. In most open literature, many papers on soliton and compacton are focused themselves on a special solution with either a soliton or a compacton without considering the initial/boundary conditions, which might be vital important for its actual wave type. Solitons and compactons for difference-differential equations (e. g. Lotka–Volterra-like problems) have been caught much attention due to the fact that discrete spacetime may be the most radical and logical viewpoint of reality (refer to E-infinity theory detailed concept). For small scales, e. g., nano scales, the continuum assumption becomes invalid, and difference equations have to be used for space variables.

Solitons and Compactons

Fractional differential model is another compromise between the discrete and the continuum, and can best describe solitons and compactons. Many interesting phenomena arise in nanohydrodynamics recently, such as remarkably excellent thermal and electric conductivity, and extremely extraordinary fast flow in nanotubes. Consider a single compacton wave along a nanotube, and its wavelength is as same as the diameter of the nanotubes, under such a case, almost no energy is lost during the transportation, resulting in extremely extraordinary fast flow in the nanotubes. The physical understanding of the transformation k D iK is also worth further studying. Cross References  Soliton Perturbation Bibliography Primary Literature 1. He JH (2006) New interpretation of homotopy perturbation method. Int J Mod Phys 20(18):2561–2568

Some Famous Papers on Solitons and Compactons 2. Burger S, Bongs K, Dettmer S et al (1999) Dark solitons in Bose– Einstein condensates. Phys Rev Lett 83(25):5198–5201 3. Denschlag J, Simsarian JE, Feder DL et al (2000) Generating solitons by phase engineering of a Bose–Einstein condensate. Science 287(5450):97–101 4. Diakonov D, Petrov V, Polyakov M (1997) Exotic anti-decuplet of baryons: prediction from chiral solitons. Z Phys A 359(3):305–314 5. Duff MJ, Khuri RR, Lu JX (1995) Sting Solitons. Phys Rep 259(4– 5):213–326 6. Eisenberg HS, Silberberg Y, Morandotti R et al (1998) Discrete spatial optical solitons in waveguide arrays. Phys Rev Lett 81(16):3383–3386 7. Fleischer JW, Segev M, Efremidis NK, et al (2003) Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices. Nature 422(6928):147–150 8. He JH (2005) Application of homotopy perturbation method to nonlinear wave equations. Chaos Soliton Fract 26(3):695–700 9. He JH, Wu XH (2006) Construction of solitary solution and compacton-like solution by variational iteration method. Chaos Soliton Fract 29(1):108–113 10. Khaykovich L, Schreck F, Ferrari G et al (2002) Formation of a matter-wave bright soliton. Science 296(5571):1290–1293 11. Rosenau P (1997) On nonanalytic solitary waves formed by a nonlinear dispersion. Phys Lett A 230(5–6):305–318 12. Rosenau P (2000) Compact and noncompact dispersive patterns. Phys Lett A 275(3):193–203 13. Strecker KE, Partridge GB, Truscott AG et al (2002) Formation and propagation of matter-wave soliton trains. Nature 417(6885):150–153

14. Torruellas WE, Wang Z, Hagan DJ et al (1995) Observation of 2-dimensional spatial solitary waves in a quadratic medium. Phys Rev Lett 74(25):5036–5039

Review Article 15. He JH (2006) Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 20(10):1141–1199 and 20(18):2561–2568

Exp-function Method 16. He JH, Wu XH (2006) Construction of solitary solution and compacton-like solution by variational iteration method. Chaos Soliton Fract 29(1):108–113 17. He JH, Wu XH (2006) Exp-function method for nonlinear wave equations. Chaos Soliton Fract 30(3):700–708 18. Zhu SD (2007) Exp-function method for the Hybrid–Lattice system. Int J Nonlinear Sci 8(3):461–464 19. Zhu SD (2007) Exp-function method for the discrete mKdV lattice. Int J Nonlinear Sci 8(3):465–468

Parameter-Expansion Method 20. He JH (2001) Bookkeeping parameter in perturbation methods. Int J Nonlinear Sci Numer Simul 2(3):257–264 21. He JH (2002) Modified Lindstedt–Poincare methods for some strongly non-linear oscillations Part I: Expansion of a constant. Int J Nonlinear Mech 37(2):309–314 22. He JH (2006) Non-perturbative methods for strongly nonlinear problems. dissertation.de-Verlag im Internet GmbH, Berlin 23. Shou DH, He JH (2007) Application of parameter-expanding method to strongly nonlinear oscillators. Int J Nonlinear Sci Numer Simul 8(1):121–124 24. Xu L (2007) Application of He’s parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire. Phys Lett A 368(3–4):259–262 25. Xu L (2007) Determination of limit cycle by He’s parameter-expanding method for strongly nonlinear oscillators. J Sound Vib 302(1–2):178–184

Nanohydrodynamics and Nano-effect 26. He JH, Wan Y-Q, Xu L (2007) Nano-effects, quantum-like properties in electrospun nanofibers. Chaos Solitons Fract 33(1), 26–37 27. Majumder M, Chopra N, Andrews R et al (2005) Nanoscale hydrodynamics – Enhanced flow in carbon nanotubes. Nature 438(7064):44–44

E-Infinity Theory 28. El Naschie MS (2007) A review of applications and results of E-infinity theory. Int J Nonlinear Sci Numer Simul 8(1):11–20 29. El Naschie MS (2007) Deterministic quantum mechanics versus classical mechanical indeterminism. Int J Nonlinear Sci Numer Simul 8(1):5–10

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Fractional-Order Differential Equations 30. Draganescu GE (2006) Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives. J Math Phys 47(8):082902 31. He JH (1998) Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput Methods Appl Mech Eng 167(1–2):57–68

32. Momani S, Odibat Z (2007) Homotopy perturbation method for nonlinear partial differential equations of fractional order. Phys Lett A 365(5–6):345–350 33. Odibat ZM, Momani S (2006) Application of variational iteration method to Nonlinear differential equations of fractional order. Int J Nonlinear Sci Numer Simul 7(1):27–34 34. Wang Q (2007) Homotopy perturbation method for fractional KdV equation. Appl Math Comput 190(2):1795–1802

Solitons: Historical and Physical Introduction

Solitons: Historical and Physical Introduction FRANÇOIS MARIN Laboratoire Ondes et Milieux Complexes, Fre CNRS 3102, Le Havre Cedex, France Article Outline Glossary Definition of the Subject Introduction Historical Discovery of Solitons Physical Properties of Solitons and Associated Applications Mathematical Methods Suitable for the Study of Solitons Future Directions Bibliography Glossary Breaking waves As waves increase in height through the shoaling process, the crest of the wave tends to speed up relative to the rest of the wave. Waves break when the speed of the crest exceeds the speed of the advance of the wave as a whole. Crystal lattice A geometric arrangement of the points in space at which the atoms, molecules, or ions of a crystal occur. Deep water Water sufficiently deep that surface waves are little affected by the ocean bottom. Water deeper than one-half the surface wave length is considered deepwater. Fluxon Quantum of magnetic flux. Freak waves Single waves which result from a local focusing of wave energy. They are of considerable danger to mariners because of their unexpected nature. Geostrophic adjustment The process by which an unbalanced atmospheric flow field is modified to geostrophic equilibrium, generally by a mutual adjustment of the atmospheric wind and pressure fields depending on the initial horizontal scale of the disturbance. Geostrophic equilibrium A state of motion of an inviscid fluid in which the horizontal Coriolis force exactly balances the horizontal pressure force at all points of the field. Hydraulic jump A sudden turbulent rise in water level, such as often occurs at the foot of a spillway when the velocity of rapidly flowing water is instantaneously slowed.

Katabatic wind Most widely used in mountain meteorology to denote a downslope flow driven by cooling at the slope surface during periods of light larger-scale winds. Lightning Lightning is a transient, high-current electric discharge. Plasma Hot, ionized gas. Shallow water Water depths less than or equal to one half of the wavelength of a wave. Solitary wave Localized wave that propagates along one space direction only, with undeformed shape. Soliton Large-amplitude pulse of permanent form whose shape and speed are not altered by collision with other solitary waves, the exact solution of a nonlinear equation. Spillway A feature in a dam allowing excess water to pass without overtopping the dam. Thermocline A layer in which the temperature decreases significantly (relative to the layers above and below) with depth. Synoptic scale Used with respect to weather systems ranging in size from several hundred kilometers to several thousand kilometers. Thunder The sound emitted by rapidly expanding gases along the channel of a lightning discharge. Thunderstorm In general, a local storm, invariably produced by a cumulonimbus cloud and always accompanied by lightning and thunder, usually with strong gusts of wind, heavy rain, and sometimes with hail. Tidal bore Tidal wave that propagates up a relatively shallow and sloping estuary or river, in a solitary wave form. The leading edge presents an abrupt rise in level, sometimes with continuous breaking and often immediately followed by several large undulations. The tidal bore is usually associated with high tidal range and a sharp narrowing and shoaling at the entrance. Also called pororoca (Brazilian) and mascaret (French). Troposphere The portion of the atmosphere from the earth’s surface to the tropopause, that is the lowest 10– 20 km of the atmosphere. Tsunami Long period ocean wave generated by an earthquake or a volcanic explosion. Definition of the Subject The interest in nonlinear physics has grown significantly over the last fifty years. Although numerous nonlinear processes had been previously identified the mathematic tools of nonlinear physics had not yet been developed. The available tools were linear, and nonlinearities were avoided or treated as perturbations of linear theories. The solitary

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water wave, experimentally discovered in 1834 by John Scott Russell, led to numerous discussions. This humpshape localized wave that propagates along one space-direction with undeformed shape has spectacular stability properties. John Scott Russell carried out many experiments to obtain the properties of this wave. The theories which were based on linear approaches concluded that this kind of wave could not exist. The controversy was resolved by J. Boussinesq [5] and by Lord Rayleigh [64] who showed that if dissipation is neglected, the increase in local wave velocity associated with finite amplitude is balanced by the decrease associated with dispersion, leading to a wave of permanent form. A model equation describing the unidirectional propagation of long waves in water of relatively shallow depth with a localized solution representing a single hump as discovered by Russell was obtained by Korteweg and de Vries [42]. This equation has become very famous and is now known as the Korteweg– de Vries equation or KdV equation. These results were not considered very important when they were obtained. A remarkable numerical discovery was made by E. Fermi, J. Pasta and S. Ulam [18] as they studied the flow of incoherent energy in a solid modeled by a one-dimensional lattice of equal masses connected by nonlinear springs. The initial injected energy was not shared among all the degrees of freedom of the lattice, but returned almost entirely to the original excited mode. The explanation was given only ten years later by Zabusky and Kruskal [82] from numerical solutions of the KdV equation used to model in the continuum approximation a nonlinear atomic lattice with periodic boundary conditions. This led to the concept of solitons and ultimately to the development of integrable systems. The term soliton was chosen as it is a localized wave which propagates preserving its shape and velocity as a particle would propagate. A soliton is then a large-amplitude pulse of permanent form whose shape and speed are not altered by collision with other solitary waves, and it is the exact solution of a nonlinear equation. The mathematical features of solitons are often well developed in soliton literature since they are at the origin of very nice theoretical developments such as the powerful inverse scattering transform which solves a complex nonlinear equation through a series of linear steps. Nevertheless, the physics of solitons is very rich and it is a very actual research topic in numerous fields, for example in hydrodynamics, in optics, in electricity, in solid physics, in chemistry and in biology. Nonlinearity and dispersion are very common in macroscopic physics as well as in microscopic physics. Nonlinearity tends to localize the signals while dispersion spreads them. These two opposite effects sometimes compensate and the regime of solitons takes place. Real physical sys-

tems are only approximately described by the equations of the theory of solitons, but a remarkable feature of solitons is their very high stability relative to perturbations. The soliton concept is now firmly established. They are widely accepted as a structural basis for viewing and understanding the dynamic behavior of complex nonlinear systems. Introduction The first experimental observation of a solitary wave was made in August 1834 by a Scottish engineer named John Scott Russell (1808–1882). He was mounted on a horse along the Union Canal linking Edinburgh with Glasgow when he saw a rounded smooth well-defined heap detach itself from the prow of a boat brought to rest and continue its course without change of shape or diminution of speed. Scott–Russell reported his observation in the British Association Report [71] as follows: “I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation”. Russell wrote that this August day in 1834 was the happiest in his life. In July 1995, an international gathering of scientists witnessed a re-creation of the solitary wave observed by Russell on an aqueduct of the Union Canal. This aqueduct was named after this occasion Scott Russell aqueduct; the scientists were attending a conference on nonlinear waves in physics and biology at Heriot-Watt University (Edinburgh), near the canal. Throughout his life Russell remained convinced that “his” wave that he initially called the “Wave of Translation” and later the “Great Solitary Wave” was of fundamental importance. He therefore carried out extensive experiments in wave-tanks and established the velocity formula for solitary waves: p VS D g(d C AS ) (1)

Solitons: Historical and Physical Introduction

where g is the acceleration due to gravity, d the undisturbed water depth and AS the amplitude of the solitary wave. The “quasi-cycloidal” form of the wave was found to be independent of the way it was generated. Russell [69] described the induced movement of fluid particles: “By the transit of the wave the particles of the fluid are raised from their places, transferred forwards in the direction of the motion of the wave, and permanently deposited at rest in a new place at a considerable distance from their original position”. He also deduced from his numerous experiments that two solitary waves can cross each other “without change of any kind”. Figure 1 shows a schematic view of his experiments in tanks, adapted from Remoissenet [65]. An initial elevation of water may induce one or two solitary waves depending on the relation between its height and length (Fig. 1a); the case where an

initial depression does not evolve into solitary waves but leads to an oscillatory wave train of gradually increasing length and decreasing amplitude is shown in Fig. 1b. In Sect. “Historical Discovery of Solitons”, the historical discovery of solitons is presented. The physical concept of solitons and the associated applications are described in Sect. “Physical Properties of Solitons and Associated Applications”. In Sect. “Mathematical Methods Suitable for the Study of Solitons”, the mathematical methods suitable for the study of solitons are considered. Finally, Sect. “Future Directions” is devoted to future directions.

Solitons: Historical and Physical Introduction, Figure 1 Schematic view of Scott Russell’s experiments

Solitons: Historical and Physical Introduction, Figure 2 Resistance as a function of towing velocity according to Russell

Historical Discovery of Solitons After Russell had carried out his observation of the solitary wave in August 1834, he was put in charge by the Union Canal Company of determining the efficiency of canals for the transport of steam-driven barges. He worked on the resistance of boats towed through water of finite depth. A resistance proportional to the square of the velocity Vb of the boat was predicted by the theory at that time. Convinced of the imperfection of this theory, Russell performed numerous experiments in the summers of 1834 and 1835 [12]. Several boats were towed in canals of various depths at velocities ranging from 3 to 15 miles per hour. The resistance was measured by a dynamometer. Resistance curves such as the one depicted in Fig. 2 were obtained, with a resistance local maximum Rm for a boat velocity Vcr and a growing deviation from the theoretical curve for increasing values of boat velocity. The critical velocity Vcr was

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Solitons: Historical and Physical Introduction

found to depend on the water depth. Russell noticed that this critical velocity was the same as the velocity of the solitary wave for the given water depth d, and gave a qualitative explanation for the diminution of flow resistance for increasing values of the boat velocity just above the critical velocity Vcr . For this description, he considered the variation of the shape of the water surface around the moving boat with the velocity. For velocities smaller than the critical value, the water level is raised around the prow where a wave of displacement is generated, leading to an inclination of the boat. The resulting effect is an augmentation of the section of immersion and of the corresponding flow resistance. When the velocity of the boat reaches the critical value, the wave of displacement has the velocity of a solitary wave and the vessel travels on this solitary wave. For velocities greater than the critical value, the boat stays on the wave summit leading to a smaller flow resistance. Russell [70] tried a quantitative explanation of this process, but this paper contained several errors. However, Russell was a very fine observer; he was far ahead of his time in considering the fundamental importance of solitary waves. He mentioned [70] that “a large or high wave had a greater velocity than a small one. When a small wave preceded a large one, the latter invariably overtook the other, and when the large wave was before the less, their mutual distance invariably became greater”. He showed that the shape of solitary waves depends on their height, the width of the wave decreasing for increasing values of height. It is also fascinating that Russell had observed the collision properties of solitary waves [8]. Nevertheless, the observations of Russell induced numerous critical comments. Airy [2] wrote in his treatise “Tides and Waves”: “We are not disposed to recognize this wave (discovered by Scott Russell) as deserving the epithets “great” or “primary”, and we conceive that ever since it was known that the theory of shallow waves of great length was contained in the equation @2 X/@t 2 D gd@2 X/@x 2 the theory of the solitary wave has been perfectly well known”. In the equation quoted by Airy, t is the time and x is the horizontal coordinate of a fluid particle. The wave velocity in shallow water at the first order of approximation, when the ratio H/d between the wave height p H and the water depth may be neglected is V0 D gd, as found by Lagrange in 1786; this velocity differs from the formulation proposed by Russell for solitary waves (Eq. (1)). Airy mentioned further: “Some experiments were made by Mr. Russell on what he calls a negative wave. But (we know not why) he appears not to have been satisfied with these experiments and had omitted them in his abstract. All of the theorems of our IVth section, without exception, apply to these as well as to positive waves, the sign of the co-

efficient only being changed”. Also, according to Airy [2] and Lamb [46], long waves could not propagate in a canal of rectangular section without changing their shape. The controversy arose because the dispersion is neglected by the nonlinear shallow water theory. Stokes [73] considered Russell results and tried to obtain them analytically; however, his linear or weakly nonlinear approach did not permit him to retrieve the results of Russell, which made him doubtful about them. In order to distinguish his “great wave” from other waves, Russell [71] introduced four orders of waves: 1. Waves of translation. These waves involve mass transfer. The solitary waves observed by Russell are included in this order. 2. Oscillatory waves. These are the waves which can be the most often observed; they do not involve mass transfer. 3. Capillary waves. The surface tension effects are important for those waves. 4. Corpuscular waves. Those waves are rapid successions of solitary waves. Russell was most concerned with the first order, but he also carried out experiments with waves of the second and third order. The fourth order (corpuscular waves) suggested by Russell had been seriously criticized, and the manuscripts he submitted on this order were never published due to ignorance of mechanics principles. The great difficulties encountered by Russell to prove the importance of his findings strongly disappointed him. Tired of discussions, he stopped his work on solitary waves and started to build large steam ships with great success. A detailed biography of Russell may be found in [16]. A French mathematician, Joseph Boussinesq, knew of the existence of Russell’s observations, and also of detailed experiments performed by Bazin [3] in the long branch of the canal de Bourgogne close to Dijon (France), which confirmed Russell’s observations. Boussinesq tried to obtain a solution of Euler’s equations compatible with Russell’s and Bazin’s results. He developed the horizontal and vertical velocity components of fluid velocity in a rectangular channel in power of the distance from the bed. Including nonlinear terms which had been neglected by Lagrange, Boussinesq [5] obtained a solution with the properties of the solitary wave observed by Russell. In particular, the shape of the wave was found to be given by: # "r 3AS 2  D AS sech (x  VS t) (2) 4d 3 where  is the free surface displacement and sech x D 1/ cosh x; the velocity of the wave VS is given by Eq. (1).

Solitons: Historical and Physical Introduction

The article of Boussinesq [5] had been almost ignored in England, and five years after its publication Lord Rayleigh [64] independently obtained the solitary wave profile. Following Rayleigh’s work and including the effects of surface tension, the Amsterdam professor of mathematics Diederik Johannes Korteweg and his doctoral student Gustav de Vries derived a model equation [42] which describes the unidirectional propagation of long waves in water of relatively shallow depth. This famous equation is now known as the Korteweg–de Vries equation or KdV equation, and it has the following form: @ @3  3 V0 @ 1 @ C V0 C  C V0 d 2 3 D 0 : @t @x 2 d @x 6 @x

(3)

Korteweg and de Vries showed that this nonlinear wave equation has a localized solution which represents a single hump of positive elevation corresponding to the solitary wave observed by Scott Russell. The KdV equation had been in fact formulated in 1877 by Boussinesq in an impressive paper [6], but this author did not use directly this formulation. Russell’s experiments had found their theory and one could think that many scientists would rapidly extend the results of this theory. However, the KdV equation had a quiet life for many decades. In the early 1950s, the physicists Enrico Fermi, John Pasta and Stan Ulam could use one of the first computers, the “Maniac” to work on systems without closed analytic solutions. In particular, they studied the way a crystal evolves towards the thermal equilibrium through the numerical simulation of the dynamics of a one-dimensional lattice consisting of N equal masses (atoms) connected by nonlinear springs. The problem is described by the Hamiltonian HD

N1 X iD0

N1

1 2 X1 p C K(x iC1  x i )2 2 i 2 iD0

C

N1 K˛ X (x iC1  x i )3 3

(4)

iD0

where x i is the atom i displacement, p i the corresponding momentum, K the constant of the quadratic potential, and ˛( 1) a coefficient of nonlinear interaction. The boundary condition x0 D x N D 0 is assumed. Considering a normal mode decomposition r Ak D

N1 2 X x i sin(ik/N) ; N

(5)

iD0

the Hamiltonian (Eq. (4)) may be written in the following

way HD

 X 1 X  ˙2 A k C ! k2 A2k C ˛ c k l m A k A l A m (6) 2 k

k;l ;m

where the pulsations ! k are given by ! k2 D 4K sin2 (k/ 2N), and c k l m are constants. When the masses have very small displacements from the equilibrium positions, the dynamics of the lattice may be described by a superposition of normal modes. It was supposed that if only one mode (the fundamental mode k D 1 in the circumstances) is excited in a crystal lattice, nonlinear interactions would lead to energy equipartition or thermal equilibrium. This was suggested by the results at the beginning of the calculations since the modes 2; 3; : : : were successively excited. However, after 157 fundamental periods, almost all the energy returned to the fundamental mode. Furthermore, the energy returned almost periodically to the original excited mode leading to a near recurrence process, now known as the FPU paradox. Fermi et al. [18] did not get the expected result and the modeled crystal lattice did not approach energy equipartition. This remarkable discovery is at the origin of the comprehension of the concept of the soliton and of the developments of “numerical experiments”. Fermi died just after this work; this had a rather inhibitory effect on the significance of this result and no paper was published after the preprint “Studies of nonlinear problems” [18]. The results were communicated to the scientific community by Ulam through several conferences. Complementary studies [20,39] confirmed that the introduction of nonlinearity in a system does not guarantee the equipartition of energy. In 1960, the KdV equation was rederived by Gardner and Morikawa who worked on collisionless hydromagnetic waves. These authors attracted the attention of Zabrusky and Kruskal on the KdV equation, and the FPU paradox was solved a few years later in terms of solitons [82]. Zabrusky and Kruskal considered the following motion equations corresponding to the Hamiltonian (Eq. (4)) x¨ i D K(x iC1 C x i1  2x i )  C K˛ (x iC1  x i )2  (x i  x i1 )2 :

(7)

Taking into account that the recurrence occurs before the excitation of high-order modes, these authors carried out an asymptotic analysis in the continuum approximation and obtained an equation including dispersive and nonlinear terms which correspond to the KdV equation. This equation, obtained in a totally different physical context as the one to explain the observations of Scott Russell, is at the origin of the explanation of the FPU paradox. As depicted in Fig. 3, the numerical experiments of Zabrusky

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dure may be considered as the nonlinear analogue of the Fourier transform method suitable to linear dispersive systems. Physical Properties of Solitons and Associated Applications Properties of Solitons Let us consider the KdV equation (Eq. (3)). Introducing the variable X D x  V0 t, this equation may be written in the following way

Solitons: Historical and Physical Introduction, Figure 3 Evolution of an initially periodic profile from Zabusky and Kruskal numerical analysis of the KdV equation. The breaking time for the wave profile is tb

and Kruskal showed that the initial sinusoidal condition evolves into steep fronts and then into a finite number of short pulses which are solitons. These solitons travel around the lattice with velocities depending on their amplitude; they collide preserving their individual shapes and velocities with only a small change in their phases. As time increases, there is an instant at which the solitons collide at the same point, and the initial state comes close to recurrence. The recurrence period calculated by Zabrusky and Kruskal closely approximates the actual FPU recurrence period, showing that the KdV equation is a suitable approximation of the FPU system. The system had to be considered as totally nonlinear to be solved, and the linear normal modes of the system could not be considered. The pulse-like waves were called solitons by Zabrusky and Kruskal to indicate the remarkable quasi-particle properties of these very stable solitary waves. They were at first called solitrons to introduce them into the family of particle names such as electron and neutron, but the name solitron was a trade mark, and was therefore not suitable. The concept of solitons has spread over numerous branches of physics, such as optical physics, biological physics, astronomy, particle physics, condensed matter, fluid dynamics, ferromagnetism. The numerical experiments of Zabrusky and Kruskal led to the development of integrable systems. Taniuti and Wei [76] and Su and Gardner [74] have shown that a large class of nonlinear evolution equations may be reduced to consideration of the KdV equation which may be regarded as a very important canonical form. A famous method, the so-called Inverse Scattering Transform (IST) was proposed by Gardner et al. [24] to integrate the KdV equation. This proce-

1 @ 3 @ d 2 @3  D0: C  C V0 @t 2d @X 6 @X 3

(8)

Using the dimensionless variables  D /d,  D X/X0 and  D t/t0 where X 0 and t0 are respectively a typical length and time, the KdV equation may be formulated in its standard form: @ @ @3  C 6 C 3 D0: @ @ @

(9)

This equation has many solutions, as all nonlinear equations. Among these solutions, there are the spatially localized solutions: "r # A 2  D A sech (  2A) ; (10) 2 where A is a positive constant. These solutions quantitatively correspond to the observations of Scott Russell. In particular, the width of the soliton decreases for increasing values of its amplitude, as depicted in Fig. 4. In dimensional variables and in a fixed referential, the corresponding solution is given by Eq. (2). The KdV equation appears widely in physics, each time that waves propagate in a weakly nonlinear and weakly dispersive medium. The nonlinear term (@/@) in this equation causes the steepness of the wave form. In a dispersive medium, the Fourier components of a pulse propagate at different velocities, inducing a spreading of the pulse. The dispersive term @3 /@ 3 in the KdV equation makes the wave form spread. The soliton solution of the KdV equation results from the balance of nonlinearity and dispersion. This balance is generally very stable, which explains the numerous applications of the theory of solitons, even if the real physical situations (where friction and defects take place) are only approximately described by the soliton equations, and in particular by the KdV equation. It would be strictly speaking more appropriate

Solitons: Historical and Physical Introduction

Solitons: Historical and Physical Introduction, Figure 4 Comparison between two solutions of the KdV equation with different amplitude A

to use the terminology quasi-soliton in physics; however, bearing in mind that the word soliton has a quite ideal connotation in the context of systems with exact solutions, we will use the word soliton. The conservation of the shape and of the velocity of the soliton after a collision is a manifestation of stability. The high stability of solitons relative to perturbations, combined with the fact that they can spontaneously emerge in a physical system in which energy is supplied, lead to numerous applications of solitons in macroscopic and microscopic physics. Nevertheless, a soliton may sometimes vanish. For example, this is the case when such a wave propagates in a media where the water depth decreases during the wave propagation. The dispersive effect progressively decreases when the nonlinear effect increases, leading to the wave breaking as waves on a beach. The vanishing of a soliton may also result from energy dissipation. Marin et al. [52] have shown that sand ripples may form when solitons propagate in shallow water over a sandy bed, and that a strong interaction between the free surface and the bed occurs. This interaction leads to a decrease of the soliton amplitude, which may result under certain circumstances in the disappearance of the soliton. Let us now consider some of the most typical soliton applications. Solitons in Fluid Mechanics In the hydrodynamic field, a very nice case where solitons take place is the tidal bore, also called hydraulic jump in translation, and known as the mascaret in France. A tidal bore is a positive wave of translation which can occur in a river with a gently slopping bottom and a broad funnelshaped estuary, where a difference of more than six meters between high and low water takes place [50]. The tidal bore appears as a single wave or a series of waves which

propagate upstream. At a given instant, it is localized at the upstream extremity of the flood tide propagation in the inter-tidal zone. The flow direction is towards the sea before the tidal bore arrival, and may be in the opposite direction after its occurrence. Such bores are distributed widely throughout the world. The most powerful can be seen in the Amazon river in South America, where it is called pororoca, which means gigantic noise. An approximately 5 m-high wave propagates with a velocity of about 30 m/s, and surfers take advantage of this exceptional wave to organize competitions. In England, bores may be contemplated on several rivers, the most famous occurring in the Severn river, near Bristol. In France, bores take place mainly in the Mont Saint-Michel bay, in the Gironde, Dordogne and Garonne rivers. The mascaret of the Seine river almost disappeared between 1960 and 1970 because of the construction of dikes and the dredging works which had been carried out in order for large boats to reach the city of Rouen. Increasing the water depth induces a decrease of the nonlinear effects which become insufficient to balance the dispersion. In Canada, several bores are observed in Fundy bay. In Asia, an important bore occurs in the Qiantang river, while the one in the river Pungue in Africa can propagate upstream over more than 80 km. Two types of bores may be observed: the breaking bore and the undulated bore, depending on the value p of the Froude number F r defined by Fr D (U1 C U)/ gd1 , where U 1 is the flow velocity at the depth d1 before the passage of the bore and U the bore velocity (Fig. 5). In Fig. 5, d2 depicts the water depth after the passage of the bore and U2 the flow velocity at the depth d2 . For a value of F r lower than about 1.5, the bore is undulated, and for a value of F r greater than 1.5, the breaking bore occurs. Most of the bores are undulated bores. The tidal bore is a very important turbulent phenomenon for the inter-tidal zone of a river, inducing a significant mixing of waters and complex motions of sediment on the river bed [14]. River bores have been known for many centuries but only qualitative observations were carried out. Quantitative measurements came much later, mainly because solitons are inherently nonlinear when only linear processes were considered. Long waves such as tsunamis often behave like solitary waves [48]. In particular, the run-up and shoreward inundation are often simulated using solitary waves [49]. The tsunamis may be described by the KdV equation. The theory of very long shallow water waves is valid since the tsunamis have a wavelength which is generally greater than 100 km and they propagate in oceans whose mean depth is about p4000 m. Their propagation velocity is greater than V0 D gd, that is greater than approximately 700 km/h. When they approach a coast, their velocity de-

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Solitons: Historical and Physical Introduction, Figure 5 Sketch of the propagation of a positive wave of translation

creases and their height increases to satisfy the equation of energy conservation, and they finally break inducing often very significant damages on the shore. Most tsunamis have a seismic origin, such as the dramatic one which occurred in the Indian Ocean on 26 December 2004 and which led to the death of more than 220 000 people. These solitons may also result from a quick arrival in the sea of volcanic products, through a similar process to that in Scott Russell’s experiments depicted in Fig. 1. This was the case for the tsunami generated in 1883 by the eruption of Krakatau in Indonesia, when approximately 36 000 people died. The inundation phase for a tsunami highly depends on the bottom bathymetry. The breaking can be progressive and structural damages are mainly caused by inundation [13]. The breaking can also be explosive and induce the formation of a plunging jet. When the breaking takes place in very shallow water, the tsunami amplitude becomes so high that an undulation appears on the long wave, which develops into a progressive bore [10]. This turbulent front is of the same type as the wave which occurs when a dam breaks, and the water can rise very rapidly, typically from 0 to 3 meters in 1.5 minutes. The event of December 2004 has shown that the available models are not able to predict accurately the wave run-up heights under severe conditions. Helal and El-Eissa [37] presented the connection between shallow water waves and the KdV equation. More recently, Mei and Li [55] considered the propagation of long waves over a randomly rough seabed. These authors have shown analytically and numerically that, assuming a randomness height to mean depth ratio comparable to the one of mean depth to the characteristic wavelength, disorder causes diffusion, leading to spatial attenuation of the wave amplitude. Mei and Li [55] proposed a modified KdV equation which includes terms representing the effects of disorder on amplitude attenuation. After a propagation over a region of finite length, the transmitted wave is a pulse which disintegrates into several small solitons of vanishing energy.

Hydrodynamic solitons may also be observed in deep water. Stokes showed in 1847 through the use of a small amplitude expansion of a sinusoidal wave, that periodic waves of finite amplitude are possible in deep water. The Stokes waves are unstable to infinitesimal modulation perturbations [4]. Zakharov and Shabat [83] were the first to show that an initial wave packet may evolve into a number of envelope solitons and a dispersive tail, the envelope solitons consisting of a carrier wave modulated by an envelope signal. This was experimentally verified by Yuen and Lake [81], by carrying out detailed experiments in deep water in a wave tank. The stability properties of envelope solitons were also considered. The freak waves, also called rogue waves, extreme or giant waves, may also have soliton-like shape. These waves appear surprisingly in the sea (“wave from nowhere”) in deep or shallow water, and can cause severe damages to ships and fixed ocean structures. They are characterized by their brief occurrence in space and time, resulting from a local focusing of wave energy [19]. The main features of their physical processes may be obtained performing numerical simulations in the framework of classical nonlinear evolution equations, such as the KdV equation [41]. Oceanographers have also observed internal solitary waves in many regions around the world’s oceans [61]. Most of the observed solitons propagate at the interface between the thermocline and the deep sea [56]. They are mainly excited by tidal flows over bottom topography [25], and they contribute to significant vertical mixing. Brandt et al. [7] developed a rotationless Boussinesq-like model for the generation and propagation of internal waves; the results of the simulations are in good agreement with large solitary waves recorded with airborne Radar images. Michallet and Barthélemy [56] carried out experiments in a 3-m-long flume to study interfacial long-waves in a twoimmiscible-fluid system involving water and petrol. The comparison between the experiments and nonlinear theories shows that the KdV solitary waves match the experiments for small amplitude waves for all layer thickness ratios. When the crest is close to the critical level, that is approximately located at mid-depth, the large amplitude waves are well predicted by a modified KdV equation including both quadratic and cubic nonlinear terms (KdVmKdV equation). However, only very few laboratory or field measurements of internal solitary waves are available, due to the prohibitive cost of obtaining precise measurements in the field and to the difficulty of generating and measuring these waves in the laboratory. Blood pressure pulses may be considered as solitons [63]. There is a balance between the nonlinearity due to the blood flow and the dispersion connected to the

Solitons: Historical and Physical Introduction

elasticity of the artery. Previous studies [32,33,80] have shown that the dynamics of weakly nonlinear pressure waves in a thin nonlinear elastic tube filled with an incompressible fluid may be governed by the KdV equation. A Boussinesq-like equation was obtained by Paquerot and Remoissenet [62] to describe the blood pressure propagation in large arteries, in the limit of an ideal fluid and for slowly varying arterial parameters. Several methods have been tested to generate hydrodynamic solitons in a flume; the method used by Scott Russell has been described in Sect. “Introduction” (Fig. 1). Hammack and Segur [29] showed experimentally and theoretically that from any net positive volume of water above the still water level, at least one solitary wave will emerge followed by a train of dispersive waves. Maxworthy [54] and Weidman and Maxworthy [79] pushed the liquid in the horizontal direction by a single displacement of a piston. The generation of solitary waves using a piston-type wave maker has been studied in detail by Goring [26]. Guizien and Barthélemy [28] have proposed an experimental procedure to generate solitary waves in a flume using a piston type wave maker from Rayleigh’s (1876) solitary wave solution. The advantage of this method is that solitary waves are rapidly established with very little loss of amplitude in the initial stage of the propagation of the solitary waves. The generation of solitons using a wave flume in resonant mode, without an absorbing beach, has been considered by Marin et al. [52]. Surface waves are produced in shallow water by an oscillating paddle at one end of the flume; a near-perfect reflection takes place at the other end. The frequency of the oscillating paddle was chosen close to the resonant frequency of the mode whose wavelength is equal to the flume length. For small values of the amplitude of displacement of the oscillating paddle, only standing harmonic waves are generated in the flume. For values of this amplitude greater than a critical value, pulses propagating from one end of the channel to the other end are excited on the background of the standing harmonic wave. From one to four pulses propagating in each direction of the flume could be generated over the time period of the flow, depending on the frequency and the amplitude of horizontal displacement of the oscillating paddle. These pulses were identified as solitons (one pulse) and bound states of solitons (several pulses). The generation of solitons in the flume results from the excitation of high harmonics. The spatiotemporal properties of solitons generated in this way were studied in detail by Ezersky et al. [17]. Space-time diagrams have been constructed to highlight the spatiotemporal dynamics of nonlinear fields for two solitons colliding in the resonator. Period doublings and the multistability of the nonlinear waves, i. e. the occurrence of different

regimes for the same values of the control parameters but under different initial conditions, have been shown. It is important to keep in mind that the solitary waves correspond to the limit of cnoidal waves of period tending to infinity. The velocity of cnoidal waves Vcn is known to depend on the so-called elliptic parameter m and is given by the formula [11]

  p AS AS 3E(m) gd (11) C 1 Vcn D 1  2d md 2K(m) where K(m) and E(m) are complete elliptic functions of the first and second kind. The parameter m, which depends on the wave amplitude to width ratio, is responsible for the shape of the wave with m D 0 corresponding to the harmonic wave, and m D 1 to the soliton. Strictly speaking, the waves generated in a flume used in resonant mode are not exactly solitons, but cnoidal waves with an elliptic parameter very close to 1; the value of this parameter was found to be 0.9996 for the tests carried out by Ezersky et al. [17]. The shapes of pulses are almost indistinguishable for m D 1 and m D 0:9996, but the pulse repetition rates differ appreciably, the period tending to infinity as m ! 1. Atmospheric solitons also exist, such as the Morning Glory Cloud of the Gulf of Carpentaria in northern Australia, which can be seen as a spectacular propagating roll cloud. Atmospheric solitons are horizontally propagating nonlinear internal gravity waves that can travel over large distances with minimal change in form [68]. The solitary waves that have been observed in the atmosphere can be divided into two classes: those that propagate in a fairly shallow stratified layer near the ground and those that occupy the entire troposphere. The generation mechanisms appear to be quite different for these two classes of waves. The waves that occupy the lower part of the troposphere mainly involve a gravity current such as a thunderstorm outflow, a katabatic wind, a sea breeze front, or a downslope windstorm, interacting with a low-level stable layer. The motion of the gravity current produces perturbations that are trapped in the low-level stable layer and eventually evolve over time into a series of solitary waves. For the tropospheric scale waves, they are very probably generated by synoptic scale features such as large-scale convective systems and geostrophic adjustments. For the two types of atmospheric solitons, there must exist some feature in the atmosphere that serves to prevent the wave energy from propagating away in the vertical direction. These trapping mechanisms are either deep layers of the atmosphere with very low values of the buoyancy frequency, or critical layers. The KdV equation, eventually enhanced with higherorder nonlinearity, gives a reasonable agreement with the observations [68].

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The system (Eq. 16) of nonlinear equations cannot be solved analytically. The continuum limit is used to get approximate solutions; setting the position x D nı where ı is a hypothetical cellule length, we get @2 V ı 2 @2 V ı 4 @V @2 V 2  D C a : @t 2 LC0 @x 2 12LC0 @x 4 @t 2 Solitons: Historical and Physical Introduction, Figure 6 Elementary circuit of an electrical network with linear inductance’s L and nonlinear capacitance’s C

Solitons in Nonlinear Transmission Lines In another area of physics, transmission lines with nonlinear elements are found to propagate in a soliton mode. Nonlinear electrical transmission lines are simple experimental devices to observe and consider quantitatively the propagation and properties of solitons. The propagation of these waves is used for picosecond impulse generation and broadband millimeter-wave frequency multiplication [66]. Let us consider the elementary LC network depicted in Fig. 6, with linear inductors L and nonlinear capacitors C. The differential capacitance C(V n ) is supposed to depend nonlinearly on the voltage across the nth capacitor V n : C(Vn ) D

dQ n (Vn ) dVn

(12)

where Q n (Vn ) is the charge stored in the nth capacitor. Following Kirchhoff’s law, we have: d n ; Vn1  Vn D dt

I n  I nC1

dQ n D dt

where the magnetic flux  n is related to the current I n by the relation:  n D LI n . Using Eqs. (12) and (13), it is easy to obtain: d2 Q n 1 D (VnC1 C Vn1  2Vn ) ; dt 2 L

n D 1; 2; : : : (14)

Assuming the following capacitance-voltage relation, C(Vn ) D C0 (1  2aVn )

(15)

where a is a small nonlinear coefficient, we find: LC0

d2 Vn d2 Vn2  LC0 a 2 dt dt 2 D (VnC1 C Vn1  2Vn ) ;

n D 1; 2; : : :

This weakly dispersive and nonlinear wave equation describes waves that can travel both to the left and to the right; the dispersive nature is due to the discreteness of the electrical network. A localized wave solution of the Eq. (17) that does not change its shape as it propagates with constant velocity v may be found [65]: 0q 1 2 2   2 2 3(v  v0 ) v C B 3(v  v0 ) VD n  t A (18) sech2 @ 2 2av v0 ı p where v0 D ı/ LC. This solitary wave solution represents a pulse with amplitude Vm D

3 v 2  v02 2a v 2

(16)

(19)

which depends on the velocity v. The width Lw at half height of this pulse is given by Lw D 1:76 q

v0

:

(20)

3(v 2  v02 )

The waveforms of this solitary wave and of the KdV soliton are similar, and the width Lw depends on the amplitude V m : Vm L2w D C t

(13)

(17)

(21)

where Ct is a constant. Solitons in Plasmas Solitons may propagate in plasmas; this has received much attention because of a possible relevance to final state configurations in fusion devices. The combination of dispersion and nonlinearity in plasmas may be described by the KdV equation. A direct analogy between the equations for shallow water and those for plasmas has been mentioned by Gardner and Morikawa [23]. However, the KdV equation is not always adapted to the excitations which can be generated in plasmas by electromagnetic waves, since wave packets are often produced instead of pulses characterizing the solutions of the KdV equation. Another type of soliton equation is then more suitable, the nonlinear Schrödinger equation (NLS); this second class of soliton equation will be considered in Subsect. “Solitons in Optical Fibers”.

Solitons: Historical and Physical Introduction

Solitons in a Chain of Pendulums Hydrodynamic solitons, solitons in nonlinear transmission lines are nontopological solitons, because the system returns to its initial state after the passage of the wave. There is also another type of soliton, the kink solitons which are topological solitons, since the structure of the system is sometimes modified after the passage of the wave. A mechanical system consisting of a chain of pendulums, each pendulum being elastically connected to its neighbors by springs, is a typical example for which topological solitons can merge. It is easy to show that this mechanical system may be described by the following equation: 2 @2 2@  c C !02 sin D 0 0 @t 2 @x 2

(22)

where is the angle of rotation of the pendulums, x is the axis along which the pendulums are distributed, c02 D l12 ˇ/I i ; l1 being the distance between two pendulums, ˇ the torque constant of a section of spring between two pendulums, I i the moment of inertia of a single pendulum of mass m and length l2 , and !02 D mg l2 /I i . The first term in Eq. (22) represents the inertial effects of the pendulums, the second term corresponds to the restoring torque due to the coupling between pendulums, and the third term represents the gravitational torque. The Eq. (22) is called the Sine–Gordon (SG) equation [47]; it can be totally integrated and admits exact solitons solutions. It has become a very famous equation containing dispersion and nonlinearity which is used to model various phenomena in physics. It constitutes the third class of nonlinear equation leading to solitons. Fluxons in a Josephson Tunnel Junction Solitons arise also in a Josephson tunnel junction, which is a junction between two superconductors. These junctions are very attractive for information processing and storage [78]. The physical quantity of interest in the long Josephson junction is a quantum of magnetic flux, or a fluxon, which has a soliton behavior. The relevant nonlinear equation which describes the physical processes is the Sine–Gordon equation. This equation is particularly suitable in solid physics. Solitons in Optical Fibers One of the fundamental applications of solitons concerns optical fibers. In optical fiber communication using linear pulses, dispersion and losses in the fiber limit the information capacity which can be transported and the distance of

transmission. Hasegawa and Tappert [31] first proposed to balance the dispersive effect by nonlinearity using solitonbased communications. The Kerr effect, in other words the nonlinear change of the refractive index of the fiber material, is used and an initial optical pulse may form a nonlinear stable pulse, an optical soliton, which is in fact an envelope soliton. Hasegawa and Tappert proposed the previously mentioned nonlinear Schrödinger equation for the description of solitons propagating in optical fibers. The first observations of these solitons were made in 1980 by Mollenauer, Stolen, and Gordon [58]. In real media, the light intensity of the soliton decreases due to losses in the fiber, and suitable reshaping of the pulse is required. Numerous theoretical and experimental studies on nonlinear guided waves have been carried out, because of their wide range of applications [30,43]. Inserting an optical fiber in a laser cavity, a “soliton laser” may be obtained, a femtosecond laser which is now in use. A real revolution in international communications is taking place at present, based on optical soliton technology. It is anecdotal that a fiber-optic cable linking Edinburgh and Glasgow now runs beneath the path from which John Scott Russell made his initial observations, and along the aqueduct which now bears his name. The NLS equation may be obtained from the Sine– Gordon equation. Previously considered systems as water waves on the free surface or chains of pendulums have also weak amplitude solutions, plane waves which have the form: D

ei(qx!t) C c:c:

(23)

where is the wave amplitude, q is the wave number, ! is the pulsation and c.c. denotes the complex conjugate of a complex number. When the wave amplitude is sufficiently increased for the nonlinearity to take place, modulation can spontaneously arise because harmonics are generated by the nonlinearity. The initial wave may split in waves packets whose properties correspond to the ones of solitons. These solitons consisting of a carrier wave which is modulated by an envelope signal are called envelope solitons. It can be shown [63] that the evolution of the envelope is described by the NLS equation: i

@2 @ C P 2 C Q j j2 @t @x

D0

(24)

where P and Q are coefficients which depend on the physical problem, as the significance of the variables t and x. The NLS equation is formally similar to the Schrödinger equation of quantum mechanics: i„

„2 @2 @ U C @t 2m @x 2

D0:

(25)

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The potential U in Eq. (25) is proportional to the absolute square of the wave envelope , m is the mass of the quasiparticle and „ D h/2 where h is the Planck constant. The potential represents the self-trapping of the wave energy.

tons solution of the NLS equation [63]. Intense researches are presently being carried out on this subject. The great size of biological molecules allows collective behaviors of atom groups, which are with nonlinearity important components for the existence of solitons.

Solitons in Solid Physics Solitons are involved in the atomic structure of matter. At the beginning of the twentieth century, a central question in solid state physics was how a plastic deformation of a metal takes place in the corresponding crystal lattice. This problem induced numerous discussions, and it was suggested that if one atom in the lattice is pulled away, a neighbor atom follows the one which had been moved, jumping into the new hole, that is the new free lattice space. Then, a dislocation runs through the crystal. This proposition was improved in 1939 by Frenkel and Kontorova [22] who developed a model where not only each atom moves alone over one lattice distance, but also many atoms form a dislocation line generating a wave of translation. The process of traveling of the dislocation line through the crystal was assumed to occur without losses of energy. This is equivalent to Russell’s solitary wave of translation which travels over long distance without change of form and velocity. This led Frenkel and Kontorova to make an analogy between the dislocation line and the soliton, and they obtained the one-soliton solution of the Sine–Gordon equation. The solitons permit us to interpret properties of dielectric materials. In other respects, magnetic materials are interesting examples to experimentally verify in a very accurate way the theory of solitons at the atomic scale [57]. The concept of soliton in polymer physics constitutes a very nice case of an interdisciplinary approach. Their occurrence was suggested in 1988 by theoretician physicists [34]. Many studies have since then been carried out, in chemistry and experimental physics. Alan J. Heeger, Alan G. MacDiarmid and Hideki Shirakawa received the Nobel prize in chemistry in October 2000 for their works on the electric conduction of conductive plastics which is performed by solitons. Solitons in Biology Interest in the physical modeling of biological processes has grown significantly since the beginning of the 1990s. Nonlinear localization phenomena have been recently shown in systems very close to biological systems [15]. The notion of the soliton is now used to explain the dynamics of biological macromolecules such as proteins and the molecule ADN; an approach to the dynamics of the molecule ADN can be obtained using the envelope soli-

Mathematical Methods Suitable for the Study of Solitons Solitons may appear in many fields, such as fluid mechanics, solid state physics, plasma physics, optical fibers and biology, as shown in Sect. “Physical Properties of Solitons and Associated Applications”. The description of these physical systems often leads to dispersive and nonlinear equations which are not presently solvable. It is then important to refer to the great classes of soliton equations, which are idealized models for numerous systems. These great classes of soliton equations correspond to the Korteweg–de Vries equation, known as the KdV equation (Eq. (9)), the Sine–Gordon equation (SG; Eq. (22)), and the nonlinear Schrödinger equation (NLS, Eq. (24)). These three famous equations can be totally integrated. They are not completely dissociated. For instance, it is possible to obtain the NLS equation from the SG equation and from the KdV equation. The soliton solutions represent only a first approximation of the physical properties of real systems. Dissipation or a weak spatial or temporal variation of the physical parameters in real physical systems are examples of phenomena which are very common and which are not taken into account in the soliton equations. The modeling has to be adapted to the physical system and to the excitation conditions of this system. In the case of plasmas, the KdV equation is more suitable for a description of the system when intense pulses excite it, and the NLS equation is more adapted for sinusoidal perturbations. There are two broad theoretical approaches available to obtain the solutions of the great classes of soliton equations, the analytic and the algebraic methods. Analytic approaches include the powerful inverse scattering transform (IST) [24] and the remarkable Hirota method [38]. One-soliton solutions or multi-soliton solutions can be obtained. Moreover, the IST method can predict the number of solitons that can emerge from an initial disturbance applied to a physical system [59]. This method shows how the solitons have a role in nonlinear normal modes as the Fourier modes for a linear equation [44]. There are also the Bäcklund transformation method [67], the direct linearizing transform method [1], the Painlevé analysis [45], and the method of position-like solutions [40]. The algebraic methods involve the Lie group theoretic method [60], the

Solitons: Historical and Physical Introduction

direct algebraic method [53], and the tangent hyperbolic method [51]. The theoretical methods cannot always be used since they need some conditions to make them applicable. Numerical methods can be applied [36]. Zabusky and Kruskal [82] were the pioneers in studying in 1965 the KdV equation numerically, by using the leapfrog method as an explicit finite difference scheme. Many methods have since then been used: a split-step Fourier method [77], the hopscotch method [27], a pseudo-spectral method [21], spectral methods (the Galerkin and Chebyshev methods) [35,72], the finite element method [75], and Fourier collocation calculations [9].

ological molecules. Other important application fields for solitons concern magneto-electronics and secure communications. The formation of spatio-temporal patterns on perturbations of soliton systems are important problems to be studied in the years to come. The existence of instabilities in the solitary wave solutions of the great classes of soliton equations have to be tackled. Nonlinear excitations in systems whose spatial dimension is greater than one raise many questions. Among these questions, there are the problems of the possible stable structures, their collision properties and the effect of external forces. New applications will certainly emerge from these studies. Bibliography

Future Directions

Primary Literature

Solitons may occur in the macroscopic and microscopic scales of physics. This results from their very general existence conditions, the coexistence of dispersion and nonlinearity. The concept of solitons is not restricted to a specific field of physics, and the investigation of solitons remains a very active research area. We have previously mentioned (Sect. “Physical Properties of Solitons and Associated Applications”) that one important application of solitons concerns optic fibers. The propagation of optical solitons through nonlinear fibers has been extensively studied; however, in real media, the dynamics of these solitons and the conditions for their generation are significantly affected by various inhomogeneities in the media. The problem of nonlinear wave propagation in the form of solitons in inhomogeneous optical fiber media is actually not well understood, despite the wide range of applications [30]. The investigation of nonlinear processes in coupled optical waveguides is another research direction for the future design of optical computers and sensor elements. Recent advances have shown that solitons have a great potential for the improvement of optical systems which demand fast and reliable data transfer. In the hydrodynamic field, the tsunamis often behave like solitary waves. The dramatic tsunami which occurred in the Indian Ocean on 26 December 2004 has clearly shown that the actual numerical models do not accurately predict the water propagation on the coastal plains. An important direction for future research is the (3D) three-dimensional simulation of tsunami breaking along a coast. The 3D simulation of the interaction of a tsunami with different beach profiles, with and without obstacles has also to be considered. This will permit the design of protecting devices and the setting up of security zones. In biology, nonlinear localization phenomena have been proved in model systems close to biological systems, but the challenge is open for bi-

1. Ablowitz MJ, Clarkson PA (1991) Solitons, nonlinear evolutions equations and inverse scattering. Cambridge University Press, Cambridge 2. Airy GB (1845) Tides and waves. Encycl Metropolitana 5:291– 396 3. Bazin H (1865) Recherches expérimentales relatives aux remous et à la propagation des ondes. In: Darcy H and Bazin H (eds) Recherches hydrauliques, imprimerie impériale, Paris 4. Benjamin TB, Feir JE (1967) The disintegration of wavetrains on deep water. Part 1 Theory. J Fluid Mech 27:417–430 5. Boussinesq J (1871) Théorie de l’intumescence liquide appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire. CR Acad Sci 72:755–759 6. Boussinesq J (1877) Essai sur la théorie des eaux courantes. MSE 23:1–680 7. Brandt P, Rubino A, Alpers W, Backhaus JO (1997) Internal waves in the Strait of Messina studied by a numerical model and synthetic aperture radar images from the ERS 1/2 satellites. J Phys Oceanogr 27:648–663 8. Bullough RK (1988) The Wave Par Excellence, the solitary progressive great wave of equilibrium of the fluid: an early history of the solitary wave. In: Lakshmanan M (ed) Solitons: Introduction and applications. Springer Ser Nonlinear Dyn. Springer, New York, pp 150–281 9. Canuto C, Hussaini MY, Quarteroni A, Zang TA (1988) Spectral methods in fluid dynamics. Springer, Berlin 10. Chanson H (2005) Le tsunami du 26 décembre 2004: un phénomène hydraulique d’ampleur internationale. Premiers constats. Houille Blanche 2:25–32 11. Cooker MJ, Weidman PD, Bale DS (1997) Reflection of high amplitude wave at a vertical wall. J Fluid Mech 342:141–158 12. Darrigol O (2003) The spirited horse, the engineer, and the mathematician: water waves in nineteenth-century hydrodynamics. Arch Hist Exact Sci 58:21–95 13. Dias F, Dutykh D (2007) Dynamics of tsunami waves. Extreme Man-Made and Natural Hazards in Dynamics of Structures. In: Ibrahimbegovic A, Kozar I (eds) Proc NATO Adv Res Workshop on Extreme Man-Made and Natural Hazards in Dynamics of Structures. Springer, Opatija, Croatia 14. Donnelly C, Chanson H (2002) Environmental impact of a tidal bore on tropical rivers. In: Proc 5th Int River Management Symp. Brisbane, Australia

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15. Edler J, Hamm P (2002) Self-trapping of the amide I band in a peptide model crystal. J Chem Phys 117:2415–2424 16. Emerson GS (1977) John Scott Russell: a great Victorian engineer and naval architect. John Murray, London 17. Ezersky AB, Polukhina OE, Brossard J, Marin F, Mutabazi I (2006) Spatiotemporal properties of solitons excited on the surface of shallow water in a hydrodynamic resonator. Phys Fluids 18:067104 18. Fermi E, Pasta J, Ulam S (1955) Studies of nonlinear problems. Los Alamos report, LA-1940. published later In: Segré E (ed)(1965) Collected Papers of Enrico Fermi. University of Chicago Press 19. Fochesato C, Grilli S, Dias F (2007) Numerical modelling of extreme rogue waves generated by directional energy focusing. Wave Motion 44:395–416 20. Ford JJ (1961) Equipartition of energy for nonlinear systems. J Math Phys 2:387–393 21. Fornberg B, Whitham GB (1978) A numerical and theoretical study of certain nonlinear wave phenomena. Philos Trans R Soc London 289:373–404 22. Frenkel J, Kontorova T (1939) On the theory of plastic deformation and twinning. J Phys 1:137–149 23. Gardner CS, Morikawa GK (1960) Similarity in the asymptotic behaviour of collision-free hydromagnetic waves and water waves. Technical Report NYO-9082, Courant Institute of Mathematical Sciences. New York University, New York 24. Gardner CS, Green JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg-de Vries equation. Phys Rev Lett 19:1095–1097 25. Gerkema T, Zimmerman JTF (1994) Generation of nonlinear internal tides and solitary waves. J Phys Oceanogr 25:1081– 1094 26. Goring DG (1978) Tsunamis – The propagation of long waves onto a shelf. Ph D thesis. California Inst Techn, Pasadena, California 27. Greig IS, Morris JL (1976) A hopscotch method for the KdV equation. J Comput Phys 20:64–80 28. Guizien K, Barthélemy E (2002) Accuracy of solitary wave generation by a piston wave maker. J Hydraulic Res 40(3):321–331 29. Hammack JL, Segur H (1974) The Korteweg-de Vries equation and water waves. Part 2. Comparisons with experiments. J Fluid Mech 65:289–314 30. Hao R, Li L, Li ZH, Xue W, Zhou GS (2004) A new approach to exact soliton solutions and soliton interaction for the nonlinear Schrödinger equation with variable coefficients. Opt Commun 236:79–86 31. Hasegawa A, Tappert F (1973) Transmission of stationary nonlinear optical pulses in dispersive dielectric fiber: II. Normal dispersion. Appl Phys Lett 23:171–172 32. Hashizume Y (1985) Nonlinear pressure waves in a fluid-filled elastic tube. J Phys Soc Japan 54:3305–3312 33. Hashizume Y (1988) Nonlinear pressure wave propagation in arteries. J Phys Soc Japan 57:4160–4168 34. Heeger AJ, Kivelson S, Schrieffer JR, Su WP (1988) Solitons in conducting polymers. Rev Modern Phys 60:781–850 35. Helal MA (2001) Chebyshev spectral method for solving KdV equation with hydrodynamical application. Chaos Solit Fractals 12:943–950 36. Helal MA (2002) Review: Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics. Chaos, Solitons and Fractals 13:1917–1929

37. Helal MA, El-Eissa HN (1996) Shallow water waves and KdV equation (oceanographic application). PUMA 7(3–4):263–282 38. Hirota R (1971) Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys Rev Lett 27:1192– 1194 39. Jackson EA (1963) Nonlinear coupled oscillators. I. Perturbation theory: ergodic problems. J Math Phys 4:551–558 40. Jaworski M, Zagrodzinski J (1995) Position and position-like solution of KdV and Sine–Gordon equations. Chaos Solit Fractals 5(12):2229–2233 41. Kharif C, Pelinovsky E (2003) Physical mechanisms of the rogue wave phenomenon. Eur J Mech B/Fluids 22:603–634 42. Korteweg DJ, De Vries G (1895) On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil Mag 39(5):442–443 43. Kruglov VI, Peacok AC, Harvey JD (2003) Exact Self–Similar Solutions of the Generalized Nonlinear Schrödinger Equation with Distributed Coefficients. Phys Rev Lett 90(11):113902 http://prola.aps.org/abstract/PRL/v90/i11/e113902 44. Lakshmanan M (1997) Nonlinear physics: integrability, chaos and beyond. J Franklin Inst 334B(5/6):909–969 45. Lakshmanan M, Sahadevan R (1993) Painlevé analysis, Lie symmetries and integrability of coupled nonlinear oscillators of polynomial type. Phys Rep 224:1–93 46. Lamb H (1879) Treatise on the motion of fluids. Hydrodynamics, 6th edn 1952. Cambridge University Press, Cambridge 47. Lamb H (1971) Analytical descriptions of ultrashort optical pulse propagation in a resonant medium. Rev Mod Phys 43:99– 124 48. Liu PL-F, Synolakis CE, Yeh HH (1991) Report on the International Workshop on long-wave run-up. J Fluid Mech 229:675– 688 49. Lo EYM, Shao S (2002) Simulation of near-shore solitary wave mechanisms by an incompressible SPH method. Appl Ocean Res 24:275–286 50. Lynch DK (1982) Tidal bores. Sci Am 247:134–143 51. Malfliet W (1992) Solitary wave solutions of nonlinear wave equations. Am J Phys 60(7):650–654 52. Marin F, Abcha N, Brossard J, Ezersky AB (2005) Laboratory study of sand bedforms induced by solitary waves in shallow water. J Geophys Res 110(F4):F04S17 53. Martnez Alonso L, Olmedilla Morino E (1995) Algebraic geometry and soliton dynamics. Chaos Solit Fractals 5(12):2213–2227 54. Maxworthy T (1976) Experiments on collision between solitary waves. J Fluid Mech 76:177–185 55. Mei CC, Li Y (2004) Evolution of solitons over a randomly rough seabed. Phys Rev E 70:016302 56. Michallet H, Barthélemy E (1998) Experimental study of interfacial solitary waves. J Fluid Mech 366:159–177 57. Mireska HJ, Steiner M (1991) Solitary excitations in one-dimensional magnets. Adv Phys 40:196–356 58. Mollenauer LF, Stolen RH, Gordon JP (1980) Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys Rev Lett 45:1095–1098 59. Newell AC (1985) Solitons in mathematics and physics. SIAM, Philadelphia 60. Olver PJ (1986) Applications of Lie groups to differential equations. Graduate Texts in Mathematics, vol 107. Springer, Berlin 61. Ostrovsky LA, Stepanyants YA (1989) Do internal solitons exist in the ocean? Rev Geophys 27:293–310

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62. Paquerot JF, Remoissenet M (1994) Dynamics of nonlinear blood pressure waves in large arteries. Phys Lett A 194: 77–82 63. Peyrard M, Dauxois T (2004) Physique des solitons. EDP Sciences. CNRS Editions, Paris 64. Rayleigh L (1876) On waves. Phil Mag 5(1):257–279 65. Remoissenet M (1999) Waves called solitons – Concepts and experiments. Springer, Berlin 66. Rodwell MJ, Allen ST, Yu RY, Case MG, Bhattacharya U, Reddy M, Carman E, Kamegawa M, Konishi Y, Pusl J, Pullela R (1994) Active and nonlinear wave propagation devices in ultrafast electronics and optoelectronics. Proc IEEE 82:1037–1058 67. Rogers C, Shadwick WF (1982) Bäcklund transformations and applications. Academic, New York 68. Rottman JW, Grimshaw R (2001) Atmospheric internal solitary waves. In: Environmental stratified flows. Kluwer, Boston, pp 61–88 69. Russell JS (1837) Report on the committee on waves. In: Murray J (ed) Bristol, Brit Ass Rep, London, pp 417–496 70. Russell JS (1839) Experimental researches into the laws of certain hydrodynamical phenomena that accompany the motion of floating bodies, and have not previously been reduced into conformity with the known laws of the resistance of fluids. Trans Royal Soc Edinb 14:47–109 71. Russell JS (1844) Report on waves. In: Murray J (ed) Brit Ass Rep Adv Sci 14, London, pp 311–390 72. Sanz–Serna JM, Christie I (1981) Petrov-Galerkin method for nonlinear dispersive waves. J Comput Phys 39:94–102 73. Stokes GS (1847) On the theory of oscillatory waves. Trans Cambridge Phil Soc 8:441–473 74. Su CH, Gardner CS (1969) Korteweg-de Vries equation and generalizations. III. Derivation of the Korteweg-de Vries equation and Burgers equation. J Math Phys 10:536–539 75. Taha TR, Ablowitz MJ (1984) Analytical and numerical aspects of certain nonlinear evolution equations, (III) numerical, KdV equation. J Comput Phys 55:231–253 76. Taniuti T, Wei CC (1968) J Phys Soc Jpn 24:941–946 77. Tappert F (1974) Numerical solution of the KdV equation and

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its generalisation by split-step Fourier method. Lec Appl Math Am Math Soc 15:215–216 Ustinov AV (1998) Solitons in Josephson junctions. Phys D 123:315–329 Weidman PD, Maxworthy T (1978) Experiments on strong interaction between solitary waves. J Fluid Mech 85:417–431 Yomosa S (1987) Solitary waves in large blood vessels. J Phys Soc Japan 56:506–520 Yuen HC, Lake BM (1975) Nonlinear deep water waves: theory and experiments. Phys Fluids 18:956–960 Zabusky NJ, Kruskal MD (1965) Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243 Zakharov VE, Shabat AB (1972) Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media. Sov Phys JETP 34:62–69

Books and Reviews Agrawal GP (2001) Nonlinear Fiber Optics. Academic Press, Elsevier Akmediev NN, Ankiewicz A (1997) Solitons, Nonlinear Pulses and Beams. Chapman and Hall, London Braun OM, Kivshar YS (2004) The Frenkel–Kontorova Model. Concepts, Methods and Applications. Springer, Berlin Bullough RK, Caudrey P (1980) Solitons. Springer, Heidelberg Davydov AS (1985) Solitons in Molecular Systems. Reidel, Dordrecht Dodd RK, Eilbeck JC, Gibbon JD, Morris HC (1982) Solitons and Nonlinear Wave Equations. Academic Press, London Drazin PG, Johnson RS (1993) Solitons: an Introduction. Cambridge University Press, Cambridge Eilenberger G (1981) Solitons: Mathematical Methods for Physicists. Springer, Berlin Hasegawa A (1989) Optical Solitons in Fibers. Springer, Heidelberg Infeld E, Rowlands G (2000) Nonlinear waves, Solitons and Chaos. Cambridge University Press, Cambridge Lamb GL (1980) Elements of Soliton Theory. Wiley, New York Toda M (1978) Theory of Nonlinear Lattices. Springer, Berlin

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Solitons Interactions TARMO SOOMERE Center for Nonlinear Studies, Institute of Cybernetics, Tallinn University of Technology, Tallinn, Estonia Article Outline Glossary Definition of the Subject Introduction: Key Equations, Milestones, and Methods Extended Definitions Elastic Interactions of One-Dimensional and Line Solitons Geometry of Oblique Interactions of KP Line Solitons Soliton Interactions in Laboratory and Nature Effects in Higher Dimensions Applications of Line Soliton Interactions Future Directions Bibliography Glossary Solitary wave A localized structure which travels with constant speed and shape in otherwise homogeneous medium. Soliton In the classical nomenclature, a soliton is a nonlinear spatially or temporally localized entity (solitary wave, impulse, wave packet, kink, etc.) of permanent form that retains its identity and shape in interactions with other similar entities. Also frequently used term for solutions of the relevant nonlinear partial differential equations approximately describing such physical entities. Soliton (colloquially as well as in certain domains) Used for denoting any long-lived localized structures (e. g. solitary waves, self-trapped optical beams, localized vortices) which exhibit low energy radiation and approximately keep their shape. Soliton solution A solution, usually in closed form, of the nonlinear partial differential equations that exhibit properties of a single soliton. A multi-soliton solution exhibits properties and collective behavior of groups of solitons. Elastic interaction A generalization of the concept of elastic collisions of ideal mechanical bodies: interaction of localized entities in which the dynamically significant properties of the counterparts such as the total energy, linear and angular momentum, and topological charge or spin are conserved. For low-dimensional and scalar solitons denotes the case in which all

the properties of interacting solitons are completely restored after interaction. Inelastic interaction Interaction of soliton-like structures that modifies one or more dynamically significant properties of the counterparts. Phase shift A change of the position of a soliton owing to interactions with other solitons compared to the location in which it would be in the absence of other solitons. Resonant interaction A specific form of soliton interactions leading to the fusion of two or more solitons into a new soliton. Generally associated with an infinite phase shift of the counterparts. Definition of the Subject The concept of solitons reflects one of the most important developments in science of the 20th century: the nonlinear description of the world. It is customary to start an introduction to solitons by recalling that it is not easy to give a comprehensive and precise definition of a soliton. Frequently, a soliton is explained as a spatially localized (solitary) wave with spectacular stability properties. Although the combination of simultaneously being a localized structure propagating while mostly keeping its shape as waves do, and surviving for a long time in realistic (that is, nonlinear) conditions, already is a fascinating property in itself, yet another key quality defines whether a particular entity is a soliton. The distinction is made based on the way in which two or more of these objects interact with each other. The classical nomenclature associates the term soliton with (i) nonlinear and (ii) localized entities, which (iii) have a permanent form and (iv) retain their identity in interactions with other entities from the same class (e. g. Drazin and Johnson [37]; also the entry  Solitons and Compactons). The fundamental property of a soliton is thus to retain its identity in nonlinear interactions. In low-dimensional systems (understood here as environments admitting solitons in one spatial dimension and line solitons in two dimensions), a soliton’s amplitude, shape, and velocity are restored after each interaction, whereas in more complex systems this request embraces all dynamically significant qualities. In other words, a structure is a soliton if and only if its interactions are fully elastic. Thus, the interaction of solitons is always an intrinsic topic whenever solitons are considered. In the nonlinear world, linear superposition does not hold and nonlinear interaction takes place. Mathematically, classes of entities are called linear, whenever any linear combination of entities belongs to the same class.

Solitons Interactions

Among structures corresponding to the solutions of certain equations, exclusively those that satisfy linear equations are denoted as linear ones. Any linear combination of solutions of a linear equation also solves this equation. The principle of linear superposition is that the resultant structure is simply the sum of the parties and it implies that the individual linear wave shapes are perfectly restored after meeting with each other. The property of elasticity of soliton interactions has a fundamentally different nature, because, as a rule, a linear superposition of soliton solutions is not a solution of the governing nonlinear equation. This property distinguishes nonlinear solitary traveling waves (that is, spatially or temporally localized pulses, whereas the relevant single-wave solution of the underlying equation may be completely stable) from solitons, which may be observed and analyzed as separate entities but which always express property (iv) should similar entities appear in the system. The propagation and interaction of both solitons and solitary waves in realistic conditions (that is, in the presence of dissipation, external fields, and inhomogeneities of the medium) frequently results in a certain loss of energy (usually in the form of radiation of relatively short waves); in certain cases collapse or fission of the structure, or a net exchange of energy between the waves occurs. Even if a solitary wave travels without radiation losses in an ideal environment, it may only be called a soliton if neither radiation loss nor net energy exchange occurs when it meets a similar structure; otherwise a perfect re-emerging of each soliton after the interaction would be impossible. More generally, no net changes of any dynamically or topologically significant quantities are permitted in elastic collisions. Only changes of certain dynamically less significant properties such as the exact location (phase) of the solitons are common in elastic interactions. Solitary waves or structures possessing a selection of properties of solitons are sometimes called quasi-solitons. Although the first evidence of solitons was extracted from observations in nature, the theory of solitons and their interactions has been developed very much in terms of the mathematical theory of the relevant nonlinear partial differential equations (PDEs) of the motion. The definitions of solitons and their interactions are commonly formulated in terms of solutions to such equations. It is even customary to speak of solitary waves and solitons, and of their interactions, having in mind the relevant exact solutions to these equations. Therefore, in mathematical terms, a soliton – either a solitary wave or a more complex entity – is a solution of a nonlinear PDE that elastically interacts with other similar solutions. In the contemporary nomenclature it is also customary to associate solitons and

their interactions with numerical (multi-soliton) solutions to these equations. Interactions between solitons are the most fascinating features of soliton phenomena. The most instructive quality of soliton interactions is the universality of many of their features that exists in spite of the extremely diverse physical origins of the counterparts. The presentation of the basic conceptual ideas involving soliton interactions, a discussion of several particular cases of such interactions and an overview of observations of the relevant phenomena in natural conditions are mostly limited to effects occurring in the elastic interactions of one-dimensional and line solitons, in which the solitons behave very much like ideal mechanical bodies and which serve as the kernel of the classical concept of solitons and their interactions. Some spectacular features of nearly elastic interactions in higher dimensions are interpreted in terms of generalizations of the classical elastic interactions. Finally, a selection of practical applications of specific features of soliton interactions is again discussed from the viewpoint of line solitons. Introduction: Key Equations, Milestones, and Methods Integrable Equations The quality of some nonlinear PDEs to admit soliton solutions is associated with the property of integrability. Integrable equations usually admit an infinite number of integrals (constants) of motion. Many nonintegrable equations possess localized shape-preserving traveling wave solutions that resemble solitons. However, only integrable equations have the universal property of possessing exact multi-soliton solutions that reflect perfectly elastic interactions between individual solitons. Thus, the integrability of the underlying equation is a primary constituent of the classical soliton interactions. Arnold [10] defines, for example, a soliton as a solitary wave (solution) of an integrable PDE having additional stability and robustness features which are inherited directly from the integrability of this PDE. Nonlinear interactions even between the greatly different (but physically relevant) solutions of integrable equations are also elastic. Among the variety of integrable PDEs that admit multi-soliton solutions, examples related to the Korteweg–de Vries (KdV) equation, the (focusing) nonlinear Schrödinger (NLS) equation, the integrable Boussinesq equation, the sine-Gordon (SG), and the Kadomtsev– Petviashvili (KP) equation will be used below. Below we shall refer to the listed equations as the classical soliton equations. Since the KdV and the KP equations represent

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a large number of different physical systems and their solutions can be easily visualized in terms of the easily accessible environment of shallow-water waves, solutions to and results from studies of these equations will be largely used for illustrations of the explanations below. The rapid development of the theory of solitons has led to the discovery of many integrable equations which show multi-soliton solutions and describe soliton interactions (see, e. g. Arnold [10] and the entry  Partial Differential Equations that Lead to Solitons). A number of analytical methods have been developed to obtain the (multi-) soliton solutions of integrable equations, starting from the late 1960s. A major tool is the Inverse Scattering Transform (IST), first developed by Gardner, Green, Kruskal and Miura [47,50] for the KdV equation. The method consists of associating to the evolution equation a Sturm– Liouville problem. In the case of the KdV equation the Sturm–Liouville equation is just the time independent Schrödinger equation of quantum mechanics, where the potential is the wave function of the KdV equation. The single- and multi-soliton solutions are associated with the discrete spectrum of eigenvalues. A generalization of the Inverse Scattering Transform, known as the AKNS method (named after Ablowitz, Kaup, Newell and Segur [2] who developed the method), applies to a large class of integral equations. See the entry  Inverse Scattering Transform and the Theory of Solitons for detailed information. At the same time, Hirota [62] found that soliton solutions can be obtained by reducing, through a suitable transformation, the evolution equation to a bilinear form. Using some properties of the bilinear operator, it is possible to find the multi-soliton solution of an integrable equation. The proof of integrability of a particular PDE is usually nontrivial. The experience with the Hirota method, applied to a variety of nonlinear wave equations, is that exact multi-soliton solutions are found in many cases when it is not known if the equation is integrable. It is natural to expect that the existence of such solutions reflects the integrability in some sense; however, no proof of this conjecture is known at this time. Mathematically speaking, it is always possible to build an integrable equation representing certain features of an observed phenomenon. Nevertheless, only a few of these equations can be properly derived, normally using asymptotic expansions, from the corresponding primitive equations describing physical phenomena. Even the classical soliton equations only approximately describe the natural phenomena. A more exact description of the natural processes requires the introduction of additional contributions to these equations. Such contributions usually destroy the property of integrability and lead to nonelasticity

of interactions of soliton-like entities. Consequently, most of the times solitons are just a good approximation of solitary waves in nature. This limitation is to some extent balanced by the fact that in many cases integrable soliton-admitting equations adequately represent the basic features of physical phenomena even far beyond their formal scope of validity. In spite of such intrinsic limitations, the tools developed in soliton theory have allowed researchers to reach a very deep understanding of some physical phenomena which would have hardly been explained by other means. Milestones Several milestones of the solitons’ history ( Solitons: Historical and Physical Introduction) are particularly significant to soliton interactions. Already the famous first observations in 1834 of shallow-water solitons [132] implicitly involved certain effects created through soliton interactions. The observed wave ahead of a ship probably had a straight crest, as the one reproduced in 1995 in the Union Canal near Edinburgh (see, e. g. p. 11 in [34]). It is a remarkable feature of ship-induced solitons in relatively narrow and shallow channels that a straight-crested upstream soliton exists ahead of a two-dimensional (2D) disturbance, whereas the solitons generated ahead of a ship sailing in wider shallow water areas have curved crests. The straight crest of the Russell’s soliton and of her sister structures are thus not intrinsically one-dimensional (1D) structured solitons. They are formed owing to a specific mechanism of soliton interactions as described below. The KP equation, allowing a simple but adequate description of the crest-straightening phenomenon, was derived even later than the word soliton was coined. A substantial step towards the concept of solitons was made in the mid-1950s by Fermi, Pasta and Ulam [40] who numerically analyzed the evolution of phonons in an anharmonic lattice. From the mathematical point of view this problem can be described by a discretization of the KdV equation. Surprisingly at the time, the process did not lead to an energy equipartition among the modes, although nonlinearity generally tends to create such an equipartition. Instead, a sort of recurrence, an early evidence of the elastic soliton interactions, was observed. The basic features of soliton interactions were first demonstrated for media described by the KdV equation [64,174]. The evolution of a sinusoidal initial wave with periodic boundary conditions gives rise to an ensemble of solitary waves that move with different speeds and gradually overtake each other. Originally eight of them were mentioned; later revisitations increased the count to

Solitons Interactions

nine [106]. The difference in the number of solitons is not principal since small-amplitude solitons only become evident through additional phase shifts in extremely longterm simulatons [39,133]. One surprise of the system was that after a very long time the whole profile reappears. The other, conceptually much more striking effect, was the persistence of the waves: after a certain phase of interaction with each other, they continued thereafter as if there had been no interaction at all. This persistence, which reflects the essence of the soliton interactions, led Zabusky and Kruskal to invent the name “soliton”. Solitons and new features of their interactions were later discovered in a large number of different physical systems. The universal principle generalizing the physical forces affecting the birth of all solitons is that the propagating disturbance (a pulse in a fiber, a wave-packet on a water surface, a self-focusing optical beam, a localized vortex, or a vortex dipole, etc.) is captured in a potential well jointly induced by its motion and by virtue of the nonlinearity of the underlying system. The solitons can thus be interpreted as the bound states of the relevant potential wells, and soliton interactions – as interactions between the bound states of a jointly induced potential well, or between bound states of different wells located at close proximity. This concept of potential well becomes vividly evident in the inverse scattering theory due to the conservation of the eigenvalues of the underlying spectral problem. This universality explains why, in spite of the immense diversity of the solitons and the underlying physical systems, the interactions (collisions) between solitons in all of these systems follow the same principles. Classical Soliton-Admitting Equations and Appearance of Solitons The particular appearance of soliton interactions may vary a lot, depending on the physical system and the nature of the solitons. The Russell soliton is an example of nontopological solitons that often become evident in the form of traveling waves and that cannot exist in rest. On the contrary, single topological solitons (that interpolate between two different states of the system which have the same energy) may exist in rest. A localized entity consists of an infinite number of Fourier harmonics and generally experiences (linear) spreading or wave radiation. A localized entity of permanent form therefore may only appear if the spreading effects are balanced by a certain restoring force that evidently can only be of a nonlinear nature. In other words, all solitons correspond to a certain balance between spreading and nonlinear effects, the latter becoming evident, for

example, as nonlinear steepening of the wave shape in the KdV system, or focusing of diffractive structure in nonlinear optics. The physical meaning of spreading depends on the particular system. Dispersion is responsible for the spreading of pulses in all media in which the group velocity depends on the wave properties. The classical examples of dispersive solitons are surface [132], internal [33] and Rossby solitons [128] in geophysical systems. Similar structures occur in a large variety of environments such as drift solitons or ion-acoustic solitons in plasma [169], or solitons in optical fibers [60]. Many examples of dispersive solitons and their interactions occur in the framework of the KdV equation and its generalizations. It is a characteristic equation governing weakly nonlinear, spatially one-dimensional (1D) waves whose phase speed attains a simple maximum for infinitely long waves 3c0 c0 h 2 ˜˜ x C ˜ x x x D 0 ; 2h 6 in nondimensional variables  t C6x Cx x x D 0; (1) ˜ t C

where  is the relative water surface elevation in the shallow water environment (Fig. 1), h is the still water depth, and c0 is the maximum phase speed (celerity) of linear waves at this depth. Its generalization to the case of solitons propagating in slightly different directions ( t C 6x C x x x ) x C 3 y y D 0 ;

(2)

was derived by Kadomtsev and Petviashvili [66]. Named the KP equation after them, it describes features of a socalled weakly 2D environment (also called 1.5-dimensional systems). The nondimensional (x, y, t, ) and phys˜ in Eq. (2) are ical variables (x˜ ,py˜, ˜t , ı ) as follows: ı related p p "3 gh ˜t /h, x D ı "(x˜  ˜t gh) h, y D " y˜ h, t ıD  D 3˜ (2"h) C O(") whereas " D j˜max j h  1. Since the KdV equation only admits nontopological elevation solitons, the KdV/KP framework misses several basic features of soliton interactions, such as collisions of solitons with different topological charge and interactions of solitons of different type. These classes of dispersive solitons are represented by the sine-Gordon (SG) equation t t  c02 x x C !02 sin D 0 ;

(3)

where in many applications has the meaning of a certain 2-periodic angle and !0 is the minimum angular frequency of linear waves. This equation, with its name stemming from a play of words regarding its form which

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Solitons Interactions

Solitons Interactions, Figure 1 A selection of solitons. From left to right: top – KdV traveling wave soliton, KP plane (line) soliton, SG breather; middle – kink, antikink, and a kink-antikink pair; bottom – bright and gray envelope soliton, and a top view photograph of a 10-µm-wide optical spatial soliton propagating in a strontium barium niobate photorefractive crystal and the same beam diffracting naturally. The bottom right image is from [152]. Optical spatial solitons and their interactions: Universality and diversity. Reprinted with permission from AAAS

is similar to the Klein–Gordon equation, arose already in the 19th century in studies of certain surfaces. It grew greatly in importance when it was realized that it led to kink and antikink solutions with the collisional properties of solitons [116]. Its major field of application is solid state physics, yet it also appears in a number of other physical environments such as the propagation of fluxons in Josephson junctions (a junction between two superconductors), the motion of rigid pendulums attached to a stretched wire, and dislocations in crystals. Its kink solutions (Fig. 1) 0

1

B !0 x  vt  x0 C S D 4 arctan exp @˙ q A; c0 1  v 2 /c 2 0

(4)

where x0 is the initial position of the single soliton at the x-axis and v < c0 is the speed of translation of the soliton, represent so-called topological solitons that interpolate between two different states of the system, which have the same energy and may exist in rest (e. g. a frozen (anti)kink, v D 0). The SG equation is a simple model admitting breather solitons and soliton pairs with different

topological charges. The bulk topological charge is an invariant of the system. In a more complex manner the topological charge becomes evident in the theory of optical spatial solitons, where it can be interpreted in terms of spin of elementary particles. The kink and antikink solutions also represent the possibility of the existence of solitons and antisolitons. Both attractive and repulsive interactions can be vividly demonstrated in this framework and, after all, interactions of physically meaningful solitons of different types are possible. Another key equation for dispersive waves is the nonlinear Schrödinger (NLS) equation that has a nondimensional form i

t

Cp

xx

C qj j

2

D0

(5)

and only has soliton solutions in the focusing case pq > 0. This is a universal equation describing the evolution of complex wave envelopes in a dispersive weakly nonlinear medium. It applies, among other phenomena, to deep water waves. The studies of its soliton solutions and their interactions also date back to the 1960s [175,176]. Its major application in the context of soliton interactions is nonlinear optics, where its different versions describe the

Solitons Interactions

propagation and interactions of both dispersive solitons (e. g. envelope solitons in optical fibers) and diffractive optical spatial solitons. The basic reason for the existence of optical solitons is that the optical properties (refractive index or absorption) of some materials are modified by the presence of light. This feature introduces nonlinearity into the system and alters the propagation of optical pulses either in space or in time. A dispersive optical soliton is formed when the linear dispersion effects are balanced by a sort of lensing (of short light pulses of permanent form that are called temporal solitons in the optical nomenclature) in an appropriate material (for example, an optical fiber). They were predicted by Hasegawa and Tappert [60] and first observed by Mollenauer et al. [97]. The basic properties of their interactions [59] are analogous to those occurring for KdV or SG solitons. There is continuous interest for their studies because of their applications, e. g. in long-distance optical communication systems. The generic example of diffractive solitons, evolution and interactions of a part of which are described by the NLS equation in two space dimensions, are optical spatial solitons. The natural tendency to broaden in space owing to the diffraction of optical beams (Fig. 1) can be balanced, for example, due to the optical Kerr effect that consists of a local refractive index change induced by light. As a first approximation, this change is assumed to take place instantaneously and to be proportional to the local intensity of light. In the focusing case (when an increase of the intensity of light causes an increase in the refractive index), the light-induced lensing can make an optical beam stable with respect to perturbations in both its width and intensity. The resulting beam in a 2D or 3D medium is an example of a diffractive soliton. Interaction of such beams occurs owing to overlapping of the modified regions of the medium. Usually such an interaction is local and has a long range only under specific conditions [129]. The first studies suggesting the existence of such solitons in nonlinear optical Kerr media date back to the 1960s [11,27]. In some cases (such as the 1D Kerr solitons in a planar medium) they are classical solitons and described by a fully integrable equation (the NLS equation in two space dimensions [176]). Self-trapped light beams and more complex structures in a 2D and 3D medium attracted the attention of researchers starting from the 1980s, when the appropriate materials were found [13]. In multi-dimensional media, both diffraction and dispersion affect the propagation of solitary waves. A classical example of a diffractive-dispersive system is the motion of short-crested waves on the water surface. Most of the analysis of soliton interactions in this system has been made

for effectively 1D solitons with infinitely long crests (socalled plane or line solitons); the effects of diffraction being implicitly neglected in their propagation and interactions. A practically realized example of diffractive-dispersive solitons are short pulses of incoherent (or white) light that is self-trapped both in the direction of its propagation and in the transversal direction(s), and that propagate changing neither their shape nor their length [94]. Extended Definitions The classical nomenclature of low-dimensional solitons and their interactions was formulated several decades ago, and it is not surprising that many later discoveries have led to attempts to extend its content. A part of the extensions add several consistent features (such as the conservation of linear and angular momentum, and topological charge or spin during the soliton interactions); however, in several schools substantial generalizations of the term soliton and of the interpretation of the soliton interactions have been introduced. The only component of the classical nomenclature kept in all the interpretations is the essential role of nonlinearity. Since it is virtually impossible to reflect all the extensions, mostly the material and results following the classical definition are presented below. The solitons are commonly interpreted as a subset of traveling solitary waves. The classical nomenclature favors structures that are stationary in a proper coordinate system; yet it does not exclude the resting topological solitons (e. g. single-kink solutions to the sine-Gordon equation). A principal extension of the classical definition of solitons that caused quite a serious discussion in the 1970s [101] consists of accounting for the resonant interactions. They may result in the fusion of several solitons into a new soliton or, equivalently, in the decay of solitons into counterparts [177]. Although the number of solitons is not conserved, all the parties of the resonant system are solitons at the limit of exact resonance and no energy radiation occurs. It is commonly accepted now as a variation (or a limiting case) of elastic interactions. The classical soliton interactions preserve the shape of the solitons. The shape is understood in a wide sense; e. g. the shape of the envelope of the NLS solitons is preserved as well as the time-dependent shape of breathers that have an internal oscillation. This quality, which is universal for all scalar solitons, has a more general interpretation in the case of vector (composite) solitons. For example, the otherwise elastic interactions of Manakov vector solitons exhibit a shape-changing nature [125] in an integrable system consisting of two coupled NLS equations [82]. This property has a paramount practical importance and bright per-

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spectives e. g. in optical soliton-based computations [155], being one of the key options of practical applications of specific features of soliton interactions. An obvious reason for a wider view on soliton interactions is that the relatively simple integrable equations admitting soliton solutions stem from certain asymptotic expansions and only approximately describe the processes in nature. A more exact representation of the physical processes through inclusion of higher order terms of those expansions generally destroys the integrability of these equations. In many cases, though, the perturbing terms are small. The influence of a small dissipative perturbation is usually obvious: it brakes the moving solitons and/or damps the oscillating solitons. Such structures are sometimes called dissipative solitons although such a name is somewhat self-contradicting. Effects due to conservative (Hamiltonian) perturbations are much richer in content. Usually they do not destroy or brake solitons, but they may, e. g., render otherwise elastic soliton interactions to inelastic ones owing to extra wave radiation. A wellknown example of the effect of perturbations is dissipation-induced annihilation of a kink-antikink pair in a perturbed sine-Gordon equation that otherwise would lead to a breather [114]. A monumental survey of the relevant problems is presented in [77]; see also entry  Soliton Perturbation. Since none of the physical and numerical environments perfectly matches the governing equations, solitary waves both in realistic conditions and in numerical simulations always are approximations to solitons. In numerical experiments practically ideal solitons can be represented and errors basically occur due to purely computational inaccuracies. Laboratory experiments with solitons encounter problems with perfect excitation of even single solitons and moreover with controlling their interaction properties; however the relevant results are of fundamental importance in establishing the adequacy, the limits and the practical applicability of the properties of solitons and their interactions. In the theory of elementary particles frequently a soliton is defined merely as a localized solution (resp. entity) of permanent form and the constraint of its survival in collisions is released (e. g. Rebbi [127]). A popular and deep example is the quantum field theory, where e. g. the Yang–Mills field equations admit solutions localized in space (which represent very heavy elementary particles), and also solutions, localized in time as well as in space (instantons). This position has certain support by the conjecture that the classical soliton equations can be interpreted as reductions of self-dual Yang–Mills equations and thus the classical solitons form a subset of the above solutions.

A similar position is also widely adopted in studies of optical beams and vortices. A large part of such beams are described by nonintegrable equations [135] and thus lie beyond the classical soliton nomenclature. Many features of their behavior and interactions, however, demonstrate a striking similarity with that of classical solitons, and exhibit universal properties [152], resembling the similar universality of the classical solitons interactions. Soliton-like beams organized by different nonlinear effects were mostly called solitary waves until the mid-1990s. The modern nonlinear optics nomenclature treats all self-trapped optical beams as solitons [136] although their interactions are not always elastic. A selection of results of studies of their interactions that give a flavor of the richness of soliton interaction phenomena in 3D media is presented below. The constraint of energy conservation both in the motion and the interactions of the solitons is frequently disregarded in studies of long-lived structures which exhibit reasonable radiation [42] but that still survive in certain collisions. On the other hand, other highly interesting solitary structures such as (Rossby) dipole modons [83,84] do not radiate energy but only survive under certain conditions. A certain amount of results and concepts in this domain is presented to highlight the overlapping of the properties of soliton interactions and interactions of other long-living structures. Elastic Interactions of One-Dimensional and Line Solitons The term soliton was originally introduced for nonlinear disturbances, the interaction of which resembles the collision of particles and is fully elastic (Table 1), so that the number of solitons is always conserved and their amplitudes are fully restored afterwards. In low-dimensional cases such as those described by the KdV equation, the amplitudes, directions and propagation velocities of each soliton always recover their initial values (Fig. 2). The shape of each soliton is prescribed by the structure of (single-) soliton solutions to this equation and, as the simplest evidence of the recurrence of the system, it is not surprising that the shape is perfectly restored as well. Three phases of the interactions of solitons can be distinguished either in time or in space (Fig. 2). First, welldefined, clearly separated solitons approach each other. In the second (interaction) phase, they usually lose their identity and merge into a composite structure. After a while, the solitons emerge again. The appearance of the composite structure depends on the particular physical system; as a rule it is neither a linear superposition of (properly

Solitons Interactions

Solitons Interactions, Table 1 Properties of solitons in elastic interactions of 1D and line solitons. Analogous rules hold for angular momentum and spin in elastic collisions in higher dimensions Property Number of solitons Energy Amplitude Shape (steepness) Phase or location

1D interactions and nonresonant interactions of line solitons Conserved except in a short phase of interaction Restored after interactions for each soliton

Exceptions in resonant interactions Changed Merging possible; total energy conserved. Durable changes may occur

Substantial local changes may occur in the interaction region; restored after interactions Finite phase shifts commonly occur; yet no phase shifts in certain envi- Infinite phase shifts ronments. The total phase shift is exactly equal to the sum of partial shifts that would result from separate collisions with each of the solitons Geometry Substantial changes in the interaction region, restored after interaction Durable changes may occur Propagation direction, Conserved for each soliton Conservation of bulk linear linear momentum momentum Topological charge Conserved for each soliton

Solitons Interactions, Figure 2 Temporal evolution of KdV solitons described by the two-soliton solution u(x; t) D 12  (3 C 4 cosh(2x  8t) C cosh(4x  64t))/ ([3 cosh(x  28t) C cosh(3x  36t)]2 ) [37]. The amplitudes of the counterparts are u1 max D 8 and u2 max D 2. Notice the gradual decrease of the taller soliton when it catches the shorter, and its gradual increase when it moves further ahead of the shorter one, and the relatively small amplitude u˜ D 6 of the composite structure

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shifted) counterparts nor a new soliton. Therefore solitons survive the collision event, even though they completely merge for a while. The variety of the composites range from the complete vanishing of the counterparts (e. g. kink-antikink collision in the SG equation) over certain interactions of KdV solitons, in which the individual identities are almost conserved and two humps are always visible, to a fourfold elevation of the water surface in a resonant interaction of the KP solitons. In some environments, one cannot say whether the solitons propagate through each other as waves do or collide as particles do. Both interpretations have a solid ground. The interaction of unidirectional KdV solitons resembles the collision of two particles whereas, e. g. in headon interactions of Rossby dipole solitons[84] the identity of both counterparts can be continuously tracked and the interaction process resembles a sort of complex overtaking. Attraction and Repulsion Another feature of soliton interactions resembling collisions of real massive and/or electrically charged particles that exert certain forces on each other is that soliton interactions may show an attractive or repulsive nature. This feature can be demonstrated in the framework of the kinkantikink solution of the sine-Gordon equation S A D 4 arctan

c0 sinh vt ; v cosh x where  D

1 !0 q c0 1  v 2 /c 2 0

(6)

and that only exists if v > 0. The motion of the counterparts speeds up when they move towards each other, and slows down when they move apart after passing each other. If one examines the positions of the centers of the counterparts of this solution, the sequences of their positions in the vicinity of their meeting point are not the same as they would be if they had been alone in the system, or far from the other counterpart. If the solitons were massive or charged particles, one would say that an attractive force exists between them which decreases as their separation increases. On the contrary, the speed of two kinks as well as two antikinks approaching each other slows down as their separation decreases, which is characteristic for repulsive interaction. Another classical reflection of “forces” between solitons is the effect of attraction or repulsion of (envelope) solitons in optical fibers [53]. In two dimensions, the attractive interactions of two obliquely interacting line solitons become evident as an

attraction of the relevant wave crests. In the case of optical spatial solitons the attraction affects their centroids. For repulsive interactions the opposite holds. The relevant effects are vividly demonstrated in the framework of the KP equation for the description of drift vortex solitons in environment, otherwise described by the Hasegawa– Mima equation [158]. This environment is equivalent to the one described by the Charney–Obukhov equation for large-scale motions in geophysical hydrodynamics admitting Rossby waves and solitons. Transient Amplitude Changes, Durable Phase Shifts and Recurrence Patterns Already the first study of solitons [174] revealed some other universal aspects of soliton interactions. First, certain durable phase shifts may occur, the magnitude of which depends on the interaction details. For example, during the collision presented in Fig. 2, the taller soliton is shifted forward by an amount x1 D 12 log 3 and the shorter one backwards by x2 D log 3 compared to the case when the solitons are alone in the system. The abovediscussed “forces”, if present in the system, may be interpreted as the cause of the phase shifts. The total phase shift of a soliton induced by elastic collisions with any number of solitons is exactly equal to the sum of (partial) shifts that would result from separate collisions with each of the solitons involved. This property is commonly referred to as the absence of many-particle effects. In the case of line solitons the phase shifts become evident in the form of durable shifts of the counterparts during the interaction (Fig. 3). In the negative phase shift case the solitons are delayed compared to their position in the absence of interactions, whereas in the positive phase shift case they seem to be accelerated. Second, the amplitudes of the interacting solitons may gradually change as they approach to each other (Fig. 2). The amplitude of the composite structure is generally not equal to the sum of amplitudes of the interacting solitons. In interactions of a small number of KdV solitons running along an infinite medium, the behavior of each counterpart and its influence on the partners can always be identified from the shape of the elevation since after some time the solitons become separated enough to detect all of them and their perfectly restored properties in the limit of infinite separation. More insight into the properties of the soliton interaction is provided by long-term simulations of ensembles of solitons excited from a periodical signal (Fig. 4). The total number of interacting parties is infinite then, the soli-

Solitons Interactions

Solitons Interactions, Figure 3 Idealized patterns of crests of interacting KP solitons (bold lines), their position in the absence of interaction (dashed lines) and the crest of the residue s12 (bold dashed line, see Eq. (7)) corresponding to the negative phase shift case (adapted from [120])

Solitons Interactions, Figure 4 Time-slice plot over two 2 periods of the evolution of the ensemble of the KdV solitons generated from a sinusoidal initial condition in space for 0  t  97 at log d D 2:3. Reprinted from [134]

tons are tightly packed and such a separation not necessarily occurs. In some cases the presence of several shorter solitons can only be identified through their contributions to the phase shifts of the partners since they are seldom visually identifiable; such solitons are called hidden solitons [39]. The trajectories of the (crests of the) counterparts may show quite a complex pattern and may exhibit both simple recurrence (at which the initial system is approximately restored) as well as super-recurrence (understood as a nearly perfect restoration of the initial system over longer times). Practically periodical in time rhombus-like patterns (suggesting that interactions of quite a small number of

solitons govern the properties of the system) optionally occur in cases when a super-recurrence is possible (Fig. 5), whereas in other cases these patterns show gradual variation also over extremely long times. There seems to exist a critical value d in the range 1:8 < log d < 1:9 of the dispersion parameter d in the KdV equation (presented in the form  t C x C dx x x D 0), which distinguishes if a super-recurrence exist. If d < d , no superrecurrence can be detected even within extremely long calculation times [39,134]. Durable Local Amplitude Changes in Oblique Interactions of Line Solitons The interaction of unidirectional KdV solitons does not create any large changes in the wave amplitudes ([37], see also  Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the). However, amplitude amplification may occur under certain conditions when line solitons propagating in slightly different directions meet each other. Extensive evidence comes from coastal engineering, where it was known already a long time ago that ocean waves approaching breakwaters and seawalls under a certain angle caused unexpectedly large overtopping. This phenomenon was systematically studied already in the 1950s [117], well before the word soliton was coined. When a shallow-water wave is obliquely launched along an ideally reflecting wall, the reflection does not necessarily follow the rules of geometrical optics. For a certain range of incidence angles, the crests of the approaching and the reflected wave fuse together near the wall and the process resembles the Mach reflection. The common crest, an analogue of the Mach stem, is generally unsteady and gradually lengthens for sine waves [15] and Stokes waves [173]. This type of reflection also occurs during perfect reflection of random [89] and solitary waves of different nature [45,92,93,115,159]. For a certain set of parameters of the approaching KdV soliton, the resulting structure is equivalent to half of the pattern created by two interacting KP solitons of equal amplitude (Fig. 3). The prominent feature of both the (Mach) reflection and oblique interactions of shallow-water solitons is that the height of the common crest may considerably exceed the sum of the heights of the interacting solitons. For interacting waves with equal amplitudes the hump can be up to four times as high as the incoming waves. Originally established in the context of the reflection of Boussinesq solitons ([92,93]; named Mach reflection by Melville 1980 [91]), the amplitude amplification in a certain region occurs in all oblique attractive interactions

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Solitons Interactions, Figure 5 Patterns of locations of crests of the KdV solitons generated from sinusoidal initial conditions at log d D 2:3209. Reused with permission from [39]

of line KP solitons (that is, in interactions with a negative phase shift). Since the interaction pattern is stationary in an appropriately moving coordinate system for a range of the parameters of the interacting solitons, it may lead to durable local changes of the amplitude of the resulting pattern of elevation. Resonance While the theory of integrable systems requires that the 1D solitons retain their identity in interactions, collisions of solitons in multiple dimensions may lead to the formation of new solitons. The phenomenon called resonant interaction leads to the emergence of new structures that combine the energy of the counterparts and to modifications of the number of solitons and of some of their geometrical features. This possibility was first recognized by Newell and Redekopp [101] in the context of the KP and NLS equations. Miles [93] gives an early answer to their question about the practical consequences of this phenomenon in terms of an essential increase of the wave amplitude occurring during the resonant Mach reflection. Originally named “breakdown of the Zakharov–Shabat theory of integrable systems with more than one spatial dimension”, it highlights the more complex nature of soliton interactions in more than one dimension, where large phase shifts are possible.

Remarkably, the resonance conditions 1 D 1 C 2 , !1 D !1 C !2 , where  i D (k i ; l i ), i D 1; 2; 1 are the wave vectors and ! i are the frequencies of the solutions of the corresponding linearized KP equation, are precisely the same as those for the resonant interaction of three weakly nonlinear waves. Conceptually, however, such interactions more resemble so-called double resonance, in which not only the above resonance conditions are met but also the group velocities of two or more counterparts are equal and otherwise sporadic energy exchange within the resonant triplets is replaced by much more intense interactions (e. g. [145]). The patterns of line soliton crests (Fig. 3) suggest that only attractive interactions may result in resonance. The resonant interactions between line solitons correspond to infinitely large phase shifts and infinitely strong attraction, and lead to durable changes of the geometry of the system. In the situation depicted in Fig. 3, the resonance would lead to an infinitely long common crest and thus render the “four-arm” pattern to a “three-arm” one. In the KP framework, the resonance phenomenon is connected with the fact that its multi-soliton solutions representing the interaction of line solutions only exist for a subset of the space of parameters of the interacting solitons (Fig. 6). Resonance occurs when the parameters of the interacting solitons lie at a certain part of the border of this subset.

Solitons Interactions

Solitons Interactions, Figure 6 Negative and positive phase shift areas for the two-soliton solution of the KP equation for fixed l1;2 . The phase shift 12 D  ln A12 has a discontinuity along the line k1 C k2 D  at which jA12 j D 1 and the resonant interactions occur. No two-soliton solution exists in the area where A12 < 0. An analogue of linear superposition occurs when A12 D 0. Adapted from [120]

A new soliton born at the exact resonance may resonantly interact with yet another soliton. This mechanism has been employed in [17] to construct a family of exact solutions to the KP equation corresponding to resonance of several solitons. Such solutions consist of unequal numbers of incoming and outgoing line solitons, the parameters of which can be obtained from asymptotic analysis [16]. This class contains a variety of multisoliton solutions, many of which exhibit nontrivial spatial interaction patterns that are generally characteristic to multisoliton solutions to the KP equation [118]. Geometry of Oblique Interactions of KP Line Solitons Patterns A relatively simple but instructive, easily visualizable and rich-in-content model demonstrating the discussed features of interactions of line solitons is the KP equation. It admits simple explicit formulae for multi-soliton solutions and extensive analytical treatment of their geometrical properties. The two-soliton solution to the KP equation can be written as a sum  D s1 C s2 C s12 of terms, reflecting to some extent the two interacting solitons s1 , s2 and

Solitons Interactions, Figure 7 Patterns of idealized (solid/dashed straight lines) and factual (dashed curves) crests of the incoming solitons s1 , s2 , the crest of the residue s12 (green line), the visible wave crests and troughs (bold and bold dashed curves, respectively), and isolines of the two-soliton solution in the case l D l1 D l2 D 0:3, k1 D 0:6507, k2 D 0:4599 in normalized coordinates (x; y). The interaction center is the crossing point of all factual crests. Contour interval is 14 max(a1 ; a2 ) D 0:0689. Circles show the common points of the troughs and crests of the two-soliton solution. Reprinted from [146]

residue s12  p p  2 s1;2 D A12 k1;2  2 cosh '2;1 x C ln A12 ;  s12 D 2 2 (k1  k2 )2 C A12 (k1 C k2 )2 ;

(7)

'1 C '2 C ln A12 '1  '2 C cosh :  D cosh 2 2 Here ' i D k i x C l i y C ! i t, a i D 12 k 2i , i D 1; 2, are the phases and amplitudes of the interacting solitons, the ‘frequencies’ ! i satisfy the dispersion relation k i ! i C k 4i C 3l i2 D 0 ıof the linearized KP equation, A12 D [2  (k1  k2 )2 ] [2  (k1 C k2 )2 ] is the phase shift parameter and  D l1 k11  l2 k21 . The two-soliton solution only exists in a part of the parameter space R(4) (k1 ; k2 ; l1 ; l2 ). An interaction may result in eitherıthe positive or the negative phase shifts ı1;2 D  ln A12 j1;2 j of the counterparts (Fig. 6). The negative phase shift (cf. Fig. 3) can be attributed to an attractive interaction and the positive phase shift to a repulsive interaction. The residue s12 (with an amplitude a12 ) is at times considered as a virtual 2D “interaction soliton” [118], which grows into the resonant soliton at the limit of exact resonance A12 ! 1 (cf. Fig. 7). This virtual structure can be heuristically interpreted as supported by both dispersive and diffractive effects, the latter being balanced by the oblique motion of the interacting solitons.

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The patterns of both idealized (Fig. 3) and factual (Fig. 7) wave crests are symmetric with respect to a particular point called interaction center, and are stationary in a properly moving coordinate frame. If the amplitudes of the solitons are equal, the pattern is equivalent to the reflection of the incoming soliton from the wall following the motion of the interaction center. For unequal amplitude solitons the equivalence is not complete because mass and energy flux occur through such a wall. Amplitudes The phase shifts ı1;2 of the counterparts only depend on the amplitudes of the incoming solitons and the angle ˛12 D 2 arctan( 12 ) between their crests. For the negative phase shift case A12 > 1 (that is typical in interactions of solitons with comparable amplitudes, Fig. 5), an interaction pattern emerges, the height of which exceeds the sum of the heights of two incoming solitons. The maximum surface ı elevation for equal amplitude solitons is amax D 4a1;2 (1 C A1/2 12 ); thus, the nonlinear superposition of solitons may lead to a fourfold amplification of the surface elevation. In a highly idealized case of interactions of five solitons the maximum surface elevation may exceed the amplitude of the incoming solitons by more than an order. The largest elevation occurs if the interacting solitons are in exact resonance A12 D 1, when psolitons inter˜ For unsect under a physical angle ˛˜ 12 D 2 arctan 3/h. equal amplitude solitons the maximum elevation amax for finite A12 and the amplitude of the resonant soliton a1 at A12 D 1 are [46,146] amax D a12 C 2A1/2 12  a1

(k1 C k2 )2 D : 2

a1 C a2 2 ; A1/2 12 C 1

(8)

The near-resonant high hump is particularly narrow and its front is very steep. The maximum slope of the front of the two-soliton solution may be eight times as large as the slope of the incoming solitons [150]. For unequal amplitude solitons, the amplification of slope of the front of the interaction pattern is roughly proportional to the amplitude amplification. The extraordinary steepness of the front of the near-resonant hump, although intriguing, is not unexpected, because the resonant soliton is higher and therefore narrower than the incoming solitons. The length L12 of the idealized common crest (Fig. 3) is L12 ln A12 and therefore is modest unless the interacting solitons are near-resonant. The length of the area, where the total elevation exceeds the sum of the amplitudes of the counterparts, may be considerably longer; however it

Solitons Interactions, Figure 8 Surface elevation in the vicinity of the interaction area, corresponding to the incoming solitons with equal amplitudes intersecting under different angles [120]

is also roughly proportional to ln A12 [150]. For largely different amplitudes of the interacting solitons, the amplitude amplification remains modest but the spatial extent of substantial influence of nonlinear interaction is roughly as large as if the amplitudes were equal [151]. Their nearresonant interaction becomes evident as a wave with its crest oriented and propagating differently compared with the counterparts. The described features are universal for that part of the wave vector space, in which the two-soliton solution with a negative phase shift exists. In the case of a positive phase shift (that is typical for interactions of solitons with largely different amplitudes) the highest elevation does not exceed the amplitude of the larger soliton. Numerical simulations of the process of formation of the composite structure show that a high wave hump is formed quite fast in the interaction region and its height soon considerably exceeds the sum of the heights of the counterparts [58,124]. A high and gradually lengthening wave hump is also formed in cases when no exact twosoliton solution of the KP equations exists. The interaction in such cases is inelastic: neither the orientation of the crests nor the height of the solitons before and after interaction match each other, and a certain amount of wave radiation takes place. The described features become evident in a number of simulations of soliton interactions in different environments [105,168]. Soliton Interactions in Laboratory and Nature Ion-Acoustic Solitons The most productive studies of soliton interactions in the 1970s and the 1980s were performed in the framework of

Solitons Interactions

ion-acoustic waves. Their behavior is approximately described by the KP equation. An analytical solution describing interactions and resonance of two plane ion-acoustic solitons of small amplitude in the 3D collisionless plasma was given in [171]. This solution is wider than the twosoliton solution of the KP equation in the sense that the nearly unidirectional approximation is not used. The first successful experiment proving that the 2D interaction of solitons has a wider spectrum of features compared with the 1D collisions is reported in [178]. A collision of two equal amplitude spherical solitons led to significant changes and to the emergence of a new 2D object. This interaction was not perfectly elastic: the solitons were not only shifted in phase but also reduced in amplitude. Soon afterwards, a near-resonant interaction of two KP solitons, with a large-amplitude wave hump of the counterparts oriented across the principal propagation direction, was observed near the interaction center, whereas the counterparts experienced clear phase shifts [43]. The amplitude amplification was found to be close to twofold the sum of the amplitudes of the incoming solitons [103]. The first adequate evidence of the nonlinear increase in amplitude during collisions of unequal amplitude KP solitons was also obtained in this framework [46]. For largely different amplitudes the measured maximum amplitudes were less than the predicted ones. A probable reason for the discrepancy is that the near-resonant wave hump occurs and covers a substantial area (equivalently, can be reliably estimated with the use of relatively large probes) only if the incoming solitons are very close to resonance. The common part of the crests of interacting solitons in the reported experiments was pretty short suggesting that the solitons were not exactly in resonance. Moreover, the complete identification of all the crests in the interactions of solitons with considerably different amplitudes is not possible because the crests are partially invisible (Fig. 7, [146]). The generic reason for experimental difficulties in this and similar experiments is the abovediscussed feature that large-amplitude structures tend to mask the presence of shorter ones. Shallow-Water and Internal Solitons Phase shifts, formation of a common crest of the interacting soliton-like waves, and accompanying amplitude and orientation changes are commonly observable in very shallow areas (Fig. 9). A famous photo by Terry Toedtemeier (1978), in which two solitonic shallow-water waves have significant phase shifts and quite a long common crest on a beach in the US state of Oregon, has been reproduced in a number of sources (e. g. p. 24 in [34]).

Solitons Interactions, Figure 9 Interaction patterns of solitonic surface waves in very shallow water near Kauksi resort on Lake Peipsi, Estonia. Photo and copyright by Lauri Ilison, July 2003. First published in [151]

Although [93] predicted up to fourfold amplification of the surface elevation, much weaker amplitude amplification was found in early experiments [91]. An amplification close to the theoretical one was detected as a byproduct of studies of shallow-water channel superconductivity [25]. Russell’s “great wave of translation” [132] probably had a practically straight crest, reflecting a well-known feature of the generation of solitons in relatively narrow channels: a nearly perfectly 1D upstream soliton is created also by a 2D disturbance. The KdV model, which is frequently used to describe this phenomenon, is intrinsically 1D and certainly results in straight wave crests. An analysis of the simplified problem of ship motion in the KP equation and the Boussinesq equation shows, in accordance with common experience, that the wave crests are curved [71]. In fact, the essentially 2D waves with paraboloidal crests generally emerge in front of the disturbance. Their curvature monotonously decreases as the waves propagate [85,87]. Although the sidewall of the channel is not essential for the radiation of the upstream solitons, still it plays a crucial role in the transformation of already radiated waves. The straight-crested solitons are formed in the process of the (Mach) reflection of solitons with curved crests from the sidewalls [113]. This feature can be distinguished in many numerical studies, where an initially curved soliton starts to straighten when it comes in touch with the sidewall [149]. Since the reflection of solitons, which in this case is equivalent to soliton interactions, plays the key important role in creating straight-crested waves in channels, with a certain exaggeration it can be said that Russell’s soliton exists due to soliton interactions.

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Probably the most common solitonic phenomenon in nature next to the shallow water waves is represented by internal solitary waves in the oceans and in the atmosphere ([56,107,111,130], see also the entry  Non-linear Internal Waves). In many cases, the KdV equation is an appropriate model for internal waves and the effects occurring during their interactions are equivalent to those described above. Large-scale internal solitary waves and their groups exhibit many features unique to soliton interactions and are often called simply internal solitons. Intersecting solitary wave packets on Synthetic Aperture Radar (SAR) images frequently exhibit phase shifts while crossing each other [51], similarly to phase shifts occurring in the KP model, where the agreement between theory and experiments is quite good. Apel et al. [9] summarize advances in the descriptions and observations of internal solitons and their interactions. The other environment for internal solitons, which provides some instructive aspects and is eligible, e. g., for the description of small-scale internal solitons in a twolayer medium, is the Benjamin–Ono (BO) equation [1] 2 1   3 Z  x0; t ˇ @2 t Ccx C˛x C P:V : 4 dx 0 5 D 0:  @x 2 x  x0 1

(9) First, the BO solitons do not acquire a phase shift after a collision. Second, this environment demonstrates that generalization of integrable 1D soliton-admitting equations to even a weakly 2D case may lead to the loss of integrability, thus the preservation of this feature in the KP equation is not universal. The weakly 2D analogues of the BO equation [1] apparently are not integrable [9] and do not admit simple analytic multi-soliton solutions. Numerically simulated oblique interactions of weakly 2D BO solitons show that a phenomenon resembling the resonant interaction of the KP solitons and an accompanied amplitude amplification do occur but the newly generated wave in the zone of nonlinear interaction is far from the BO soliton [105,167]. Solitons at Planetary Scale Large-scale internal waves and solitons in nature are affected by the joint influence of the Earth’s rotation and sphericity, and perfect internal line solitons generally cannot exist in the ocean. The physical reason of this “antisoliton theorem” is that due to rotational dispersion, there is always a phase synchronism between a source moving at an arbitrary speed and linear perturbations in such envi-

ronments. This leads to unavoidable wave radiation from a large-scale solitary internal wave [49]; however this effect is negligible for relatively small-scale solitons, the typical time scales of which in the oceans are a few tens of minutes. The existence of nonstationary solitary waves is still possible [52]. They exhibit a phenomenon similar to the recurrence of solitons – the repeating decay and reemergence process, and formation of a nearly localized wave packet consisting of a long-wave envelope and shorter, faster solitary-like waves that propagate through the envelope [61]. The robust, long-lived structure may contain as much as 50% of the energy in the initial solitary wave. Interacting packets may either pass through one another, or merge to form a longer packet. Their further studies may reveal interesting features concerning the functioning of oceans and other large stratified water bodies. Yet in the majority of practically interesting cases such long-term behavior is masked by topographic effects that modify the internal waves much more dramatically. Another medium of particular practical importance as well as an instructive one from the viewpoint of soliton interactions is the large-scale geophysical flow in the oceans and the atmosphere. It is frequently treated as anisotropic quasi-2D turbulence, which has the property of energy concentration in large-scale structures. Eddies in such flows are stabilized by the background rotation and also affected by the joint effect of the Earth’s rotation and sphericity. The latter becomes evident through the North-South variation of the Coriolis force, the socalled (planetary) beta-effect. In this environment, largescale vortices often maintain their coherence for surprisingly long times and show an amazing variety of interactions; perhaps the most well-known example being the Jovian Great Red Spot. Form-preserving, uniformly translating, horizontally localized solutions (of the equations of planetary dynamics) are called modons. In nondissipative quasigeostrophic dynamics they exhibit a variety of properties of solitons. They carry a certain amount of fluid with them, which distinguishes them from (solitary) waves that cause, if at all, a finite displacement of the medium. The word modon was coined a decade after soliton was first used [157]. The standard quasi-geostrophic models predict that the accompanying Rossby-wave radiation should cause a decay of isolated eddies. Several models explain the persistence of monopolar vortices using external features of the flow, or interactions with a background shear that suppresses the Rossby-wave radiation [19,126,144]. A number of generalizations of the quasi-geostrophic model pre-

Solitons Interactions

dict that, within a certain range of parameters, a nonlinear anticyclonic vortex of a special form can exist for a long time [121,153,170] due to the mutual compensation of weak (scalar) nonlinearity and weak dispersion. Some authors call such structures (Petviashvili-type) Rossby solitons, although their collisions are not elastic. Long-living coherent vortices and their interactions have been studied in many experiments with rapidly rotating parabolic vessels [102]. The increase of the effective gravity due to the centrifugal acceleration towards the side walls modifies the governing equations, so that the (Petviashvili-type) Rossby soliton does not exist in the paraboloidal geometry [104]; yet anticyclonic vortices have a very long lifetime [154]. Another option to study the evolution of such vortices is a tank with a sloping bottom to imitate the planetary beta-effect (e. g. [41,90], among others; for an overview of earlier studies see [63]). The majority of interactions of solitary vortices in such an environment are inelastic. The 2D localized vortices of comparable size and intensity, and like sign attract each other and generally coalesce soon. Only practically equivalent vortices may form a quasi-stable pair rotating around each other. The vortices of different signs either repulse from each other or form a relatively stable dipole [54,55]. The simultaneous evolution of a large number of solitonlike vortices has mostly been studied theoretically or numerically (e. g. in terms of a cluster of point vortices or hetons, [57]). The simplest stable barotropic modon in flat-bottom beta-plane dynamics is the Larichev–Reznik dipole [83]. It is a traveling dipolar vortex with a characteristic north/ south antisymmetry. It propagates eastward at any speed, or westward at speeds greater than the longest-wave speed. The speed of translation is therefore out of the range of the phase speed of Rossby waves, which is the reason why this modon does not radiate waves (unlike large-scale internal solitons). The significance of such structures in everyday life and climate dynamics (where they may be responsible, e. g., for certain unusual weather events since the vortex pair may behave completely differently compared with a single cyclone or anticyclone) is far from being well understood. A part of the numerically simulated direct head-on and overtaking collisions of Larichev–Reznik modons are elastic [84]. An instructive feature of the head-on collisions is that the identity of all four vortices can be sometimes tracked during all the interaction phase (Fig. 10). The process resembles a collision of soft particle pairs connected by an elastic chain. Their oblique collisions generally are not elastic and may lead to the formation of new dipoles, tripoles, or to the destruction of the structures.

Effects in Higher Dimensions Optical Spatial Solitons Starting from the mid-1980s, various optical solitary entities have become a key arena for studies of the properties and interactions of solitary structures in multiple dimensions. The interest in these studies is continuously supported by a variety of existing and emerging applications of such solitons and their interactions in communication and computation technology. Another reason for the rapid progress in these studies is the relative ease of their generation in suitable materials, combined with the precision of contemporary optical experiments and the possibilities of precise control over almost every parameter [136,152]. They offered a lot of the further conceptual progress of soliton interactions through making possible 3D propagation of solitary entities in laboratory conditions, albeit in many cases these entities did not match the classical nomenclature of solitons. They were called solitary waves in the field for several decades, because the governing equations are not necessarily integrable and the interactions not necessarily elastic. Still their interaction at times shows amazingly elastic nature and they are commonly called optical spatial solitons (OSS) in the modern nonlinear optics nomenclature [136]. A large number of various kinds of OSSs have been identified [76,135]. Observations of spatial Kerr solitons became possible from the mid-1980s, when so-called slab waveguides were built [3,13]. Their behavior substantially depends on the dimensions of the soliton and the waveguide. For example, the bright Kerr solitons are stable only in planar systems whereas the 2D solitons were found to collapse. Quasi-stable 2D solitons in Kerr media described by the cubic NLS in two spatial dimensions exist in the form of so-called necklace-ring beams [143]. Optical spatial solitons occurring due to the saturation of the nonlinear change in the refractive index were first demonstrated in the 1970s [18] and realized in a solid medium in the 1990s [74]. The net effect of the multiple physical effects involved in photorefractive materials is that the underlying nonlinearities are also saturable. Such solitons were also discovered in the early 1990s [137] and were found to be stable in both slab waveguides [75] and in bulk media [38]. So-called quadratic solitons (that consist of multifrequency waves coupled by means of a second-order nonlinearity) were predicted in the mid-1970s [70] and realized experimentally in the 1990s [164]. They can be thought of as vector solitons because they involve mutual self-trapping of two or more components [152]. Their interactions

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Solitons Interactions, Figure 10 Head-on collision of Larichev–Reznik dipoles [67]

are similar to those occurring in other saturable nonlinear media. Coherent and Incoherent Collisions Coherent interactions of OSSs occur when the nonlinear medium responds quickly to the (interference) effects in the overlapping area of the beams. The increase of the intensity of light in the overlapping region of two parallel launched equivalent in-phase solitons in the focusing medium leads to an increase in the refractive index in that region. This in turn forces the centroid of each soliton toward it. Hence the solitons attract each other. The spatial distribution of the intensity of light resembles the temporal distribution of the amplitude of attracting interact-

ing line solitons. In the case of antiphase beams, an opposite process occurs and such solitons exert repulsion. Coherent collisions of Kerr solitons have been demonstrated and the attraction for in-phase and the repulsion for antiphase solitons were clearly observed at the beginning of the 1990s [4,5,139]. The collision of nonequivalent coherent OSSs results in an energy exchange between the beams and in a repulsive force that makes the beams diverge. Some reactions (e. g. photorefractive and thermal) of the optical medium are relatively slow. Solitons within which the phase varies randomly across the beam were first demonstrated in [95] and later extended to the beams of incoherent white light [94]. So-called incoherent soliton interactions occur when the response of the medium only follows the average intensity of light. Interactions of inco-

Solitons Interactions

herent bright solitons are always attractive (Andersen and Lisak 1995). The long-distance behavior of the counterparts in attractive interactions depends on the properties of the medium and the orientation of the beams. Pure Kerr solitons launched along nearly parallel directions result in periodic paths of the solitons’ centroids. As in the case of the recurrence of the KdV solitons, Kerr solitons exactly return to their initial conditions after each cycle. Solitons launched under large enough converging angles exhibit a slight lateral deflection (which is equivalent to the phase shift of the KdV solitons). The solitons launched under large enough diverging angles never collide. If the energy transfer were neglected, the force between the equivalent 1D Kerr solitons would vary from the maximum attractive between in-phase solitons to the maximum repulsive for antiphase solitons. This process leads to the (generally nonperiodic) energy transfer, where one soliton may gather net energy from the other. The details of the trajectories and other properties of the interacting solitons can be quite complex [152]. Interactions of 2D OSSs (e. g. in media with saturating nonlinearities) have a much larger variety of scenarios [152]; for example, interactions of incoherent photorefractive solitons may be both attractive and repulsive [156]. A number of resonance-like inelastic phenomena have been observed in which the number of solitons is not conserved. The fusion of two or more solitons into one structure resembles the resonance phenomenon. Here it happens owing to the gradual change of the orientation of the waveguides of attracting solitons launched under small relative angles into a new waveguide [48]. Differently from the resonance of line solitons, this happens for a certain range of initial parameters. The threshold is the maximum total internal reflection angle in the induced waveguide [140,142]. For larger collision angles, the solitons interact as described above. The solitons can fuse together already on the first merging or after a certain number of oscillations of their trajectories with decreasing amplitudes and periods [12,78,160,161]. A sort of inverse of this process is the breakup (fission, [142]) of optical solitons into new solitons upon their interaction [163]. Annihilation of solitons upon collision also may occur, when three solitons collided and only two emerged from the collision process [79]. Three-Dimensional Effects In 3D space the trajectories of interacting beam solitons do not necessarily lie on a single plane and the system possesses nonzero initial angular momentum. The trajecto-

ries of solitons that are launched along skew lines passing close to each other may bend in three dimensions. The solitons spiraling around each other form a double helix orbit [123], much like two celestial objects or two moving charged particles moving along nearly parallel trajectories do. The interaction generally leads to spiraling-fusion or spiraling-repulsion of the trajectories [14,160,161]. Since the mutual rotation encounters a centrifugal force (which is always repulsive), a perfect double-helix DNAlike system, an interesting class of elastic interactions, requires soliton attraction. It may be formed from identical in-phase coherent solitons; yet such a double helix is unstable with respect to perturbations of both the relative phase and amplitude of one of the solitons. This effect resembles processes occurring during the interactions of vortices of different and like sign in quasi-2D rotating fluids. Attractive interactions of incoherent solitons have made possible the observation of spiraling solitons in saturable media [141]. The two solitons of equal power orbit periodically about each other. Such a pair conserves angular momentum and is more or less stable for a certain range of the initial orientation and distance between the beams. Although the underlying saturable nonlinearities are described by nonintegrable equations, the spiraling process does not lead to measurable energy radiation [20]. When the initial distance between the solitons is increased, they do not form a long-living interacting system. The double spiral for beams located initially very close to each other converges fast and the counterparts eventually fuse. The pair apparently has a limited lifetime. A likely reason for its fusion or off-spiraling is the potential coherence of the light in the beams. Interactions of Vector Solitons Vector (composite) solitons consist of two or more components (also called modes) that mutually self-trap in a nonlinear medium. The simplest vector solitons, first suggested by Manakov [88], consist of two orthogonally polarized components in a nonlinear Kerr medium in which self-phase modulation is identical to cross-phase modulation. They are described by an integrable system and interact elastically. An appropriate nonlinear material and experimental conditions for their demonstration were developed a long time after their discovery [68]. Similar solitons may exist if each field component is at a different frequency, and the frequency difference between components is much larger than the inverse of the nonlinearity relaxation time [138,166], or if the field components are incoherent with one another [22,30]. The energy exchange

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Solitons Interactions, Figure 11 a The experimentally observed soliton spiraling process. The arrows indicate the initial trajectories. b, e, and g show different input conditions and c, f, and h are the outputs after 6.3 mm and d and i after 13 mm. The triangles indicate the centers of the corresponding diffracting beams. From [152]

between the components of the interacting vector solitons takes place without radiative losses [7]. Vector solitons can also consist of different modes of their jointly induced waveguide [29,165]. These multimode solitons may possess multiple humps, may contain both bright and dark components [28,166], may have a certain internal dynamics, and were found to be stable (or weakly unstable) in large regions of their parameter space [96,109,110]. Interactions between vector solitons have, additionally to the above-discussed generic properties of solitons interactions (e. g. [86]), also some unique features. The shape transformations of colliding multimode solitons can lead to two different multimode solitons emerging from the collision process [6,69,80]. This “polarization switching” was predicted initially for Manakov solitons [125]. Composite multimode multihump solitons may carry topological charge in one of the vector components [21, 98]. This charge carried by a soliton of finite energy can be interpreted as spin of real particles. Elastic collisions of such 3D solitons should, ideally, conserve not only energy, and linear and angular momentum [122], but also the equivalent of spin. Two colliding composite solitons carrying opposite spins may form a metastable state that later decays into two or three new solitons. If the solitons interact under a certain critical angle, angular momentum is transferred from spin to orbital angular momentum. Finally, the shape transformation of the vortex component occurs at large collision angles, for which scalar solitons of

all types simply go through one another unaffected or only have phase shifts [99,100]. The recent progress in optical vortices, vortex solitons, and their interactions is reviewed in [35]. Applications of Line Soliton Interactions A generic use of soliton interactions in every case where solitary waves or topological solitons may exist, follows from the definition of a soliton. It consists of establishing if the structure is a solitonic one or not from the properties of their interactions. The practical use of specific features of soliton interactions was started largely in parallel with the first technologies based on the use of solitons (e. g. optical solitonbased communications) more than a decade ago. The most promising is the technology of Optical Computing, which may be partially based on the interactions of vector solitons [155] and which may open completely new horizons in the architecture of computers. Soliton Interactions on a Water Surface Several applications based on properties of elastic soliton interactions (or changes of certain fields during such interactions) have been proposed in the framework of water waves. From the purely geometrical properties of oblique interactions of plane solitons one may extract, e. g., the water wave height. The relations for the phase shifts and for

Solitons Interactions

the intersection angle can be reduced to the following transcendental equation: q   ı 2 2  2 (ı2  ı1 )2 a12 : (10) ı1 2a1 1 C 2 /4 D ˙ ln 22 ı2 2  2 (ı2 C ı1 )2 a12 If the intersection angle ˛12 , the phase shifts ı1;2 and their signs (or the wave propagation directions) are known, equation (10) uniquely defines the heights of the interacting solitons [118,119]. The phenomenon of drastic increase of surface elevation and slope of wave fronts described in Sect. “Geometry of Oblique Interactions of KP Line Solitons” can be attributed to formation of “freak” or “giant” waves in the ocean, the height and steepness of which are considerably larger than expected based on the classical wave statistics [73]. Their sudden appearance is a feature of paramount importance to and a generic source of danger for navigation and in coastal and offshore engineering. Interactions of envelope (Schrödinger) solitons and breather solutions of the NLS [108] in deep water are their potential source, although the model based on envelope soliton interactions underestimates the probability of large wave formation compared with the fully nonlinear model [31]. Interacting solitary wave groups that emerge from a long wave packet can produce freak wave events and may lead to a threefold increase of the wave slope [32]. The problem of extremely large-amplitude internal waves is by no means less important, as they can pose acute (currently neither well understood nor accounted for) danger for submarines, oil and gas platforms, oil risers and pipelines, and to other engineering constructions in relatively deep water. The problem of “freak” internal waves and their generation due to internal soliton interactions has only recently attracted the attention of researchers [81]. There is a clear potential in the use of the advances concerning the geometry of the line soliton interactions in studies of both internal and shallow water solitons. While phase shifts usually have no dynamical significance, unexpectedly large elevations, extreme slopes, or changes in orientation of wave crests owing to the (Mach) reflection or oblique interactions of solitonic waves frequently cause an acute danger. The particularly high (stem) waves may lead to hits by high waves arriving from an unexpected direction. They may break during propagation into shallow water and attack armor blocks from an unprotected side [89] or overtop the breakwaters in unexpected locations [172]. These effects may be particularly pronounced in the case of ship wakes that often approach seawalls or breakwaters from other directions than wind waves

do [148]. The possibility of drastic steepening of the front of the near-resonant structure is immaterial in many physical systems but is a crucial component of danger from freak waves [73], since specifically steep and high waves present an acute danger (e. g. [162]). Even if the steepness of the wave front is not large in absolute terms, relatively steep and long, tsunami-like waves tend to become asymmetric and exhibit unexpectedly large runup heights [36]. The amplitude amplification, however, evidently becomes effective relatively seldom, because two or more systems of long-crested solitonic waves must approach a certain area from different directions, and extreme elevations and slopes only occur if the amplitudes of the interacting solitons, the angle between their crests and the water depth are specifically balanced. The long life-time of the resulting wave hump in favorable conditions [73] may drastically increase the probability of the occurrence of abnormally high waves. Ship-Induced Solitons and Wave Resistance An increasing source of solitonic waves is the fast ship traffic in relatively shallow areas. The generation of shallowwater solitons is most effective for large ships sailing p at speeds roughly equal to the maximum phase speed gh of surface gravity waves [87]. The low decay rates and exceptional compactness of the solitonic ship wakes has led to a significant impact on the safety of people, property and craft, unusually high hydrodynamic loads in a part of the nearshore, and to a considerable remote impact of the ship traffic in shallow areas [112,147]. Such a wake has probably caused a fatal accident as far as about 10 km from the sailing line already in 1912 [149]. The interactions of shipinduced solitons may lead to dangerous waves in the vicinity of coastal fairways and harbor entrances in otherwise sheltered areas [120]. A intriguing use of soliton interactions, combining effects occurring during an elastic oblique interaction of shallow-water solitons, followed by an annihilation of solitons and forced antisolitons, has been made for reducing the wave resistance in channels [25] and for catamaran design [26]. In a channel, the bow wave reflected from a side wall is used to cancel the stern wave, whereas the bow wave of one hull of the catamaran is used to cancel the stern wave of the other hull. Its practical use is possible for ships sailing at near-critical speeds, the wake of which may consist of a single soliton. The adequate calculation of the phase shift occurring during its Mach reflection (equivalently, during the interaction of two bow solitons) is a key component in achieving a perfect annihilation of properly timed bow solitons and the sterns’ waves of depression.

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The use of this effect by adjusting the channel width, the water depth and the ship’s speed probably is the first intentional use of the features occurring during soliton interactions in the design of a certain technology [23,24]. At the exact design conditions the reflected bow wave completely cancels the stern wave and leads to a sort of channel superconductivity [25]. The same effect may occur between two ships moving in parallel [65]. Finally, it is interesting to note that already J.S. Russell may have been aware of this possibility. He describes efforts of a spirited horse which, pulling a boat in a canal, had drawn the boat up into its own wave leading to a significant reduction in resistance. The boat owner had noted this, and the event led to ‘high-speed’ service on some canals in the 1820s and 1830s [131]. We shall probably never know if the decrease of wave resistance occurred due to simple crossing of the critical speed or owing to the channel superconductivity; however, even the remote possibility that the practical use of soliton interactions may have been reported even before the first report about solitons, is remarkable. Future Directions Although the theory of soliton interactions has been extensively developed during four decades and its basic features for low-dimensional environments have been well understood, it is still in the stage of rapid development. Its first practical applications appeared only a decade ago and there is evidently room for relevant developments in (geo)physical applications. Analysis of long-term evolution of ensembles of solitons is still a challenge and reveals ever new interesting features even in the simplest solitonadmitting systems such as the KdV equation. Although several formal procedures of building explicit multi-soliton solutions to many equations have been known since the 1970s, studies of properties of (optionally resonant) interactions of several line solitons only started at the turn of the millennium. Many fascinating features of elastic interactions of solitons of different kinds in more complex systems will obviously become evident in the nearest future. The results in this direction may largely widen understanding of the integrability of the underlying equations. Extremely rapid progress may be expected in studies into properties of long-lived soliton-like structures, in particular, of optical spatial solitons. The classification of their nomenclature and interactions is far from being completed. This environment is the key framework for identifying the new features of soliton interactions in higher dimensions, even though interactions of such structures are not necessarily elastic.

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Solitons, Introduction to

Solitons, Introduction to MOHAMED A. HELAL Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt The concept of solitons reflects one of the most important developments in science at the second half of the 20th century: the nonlinear description of the world. It is not easy to give a comprehensive and precise definition of a soliton. Frequently, a soliton is explained as a spatially localized wave in a medium that can interact strongly with other solitons but will afterwards regain its original form. It is a nonlinear pulse-like wave that can exist in some nonlinear systems. The soliton wave can propagate without dispersing its energy over a large region of space; collision of two solitons leads to unchanged forms, solitons also exhibit particle-like properties. The most remarkable property of solitons is that they do not disperse and thus conserve their form during propagation and collision.  Solitons Interactions. The nonlinear science has been growing for approximately fifty years. However, numerous nonlinear processes had been previously identified, but the nonlinear mathematical tools were not developed. The available tools were linear, and nonlinearities were avoided or treated as perturbations of linear theories. The first experimental observation of a solitary wave was made in August 1834 by a Scottish engineer named John Scott Russell (1808–1882). Scott Russell reported his observation; to the Fourteenth Meeting of the British Association for Advancement of Science, held in 1844; in a long report entitled: “Report on waves” as follows: “I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I

lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation”. This hump-shape localized wave that propagates along one space-direction with undeformed shape has spectacular stability properties. John Scott Russell carried out many experiments to get the properties of this wave.  Solitons: Historical and Physical Introduction. A model equation describing the unidirectional propagation of long waves in water of relatively shallow depth with a localized solution (representing a single hump as discovered by Russell) was obtained by Korteweg and de Vries. This equation has become very famous and is now known as the Korteweg–de Vries equation or KdV equation.  Water Waves and the Korteweg–de Vries Equation. There exists a certain class of nonlinear partial differential equations that leads to solitons. The Korteweg– de Vries equation, Kadomtsev–Petviashvili (KP) equation, Klein–Gordon (KG) equation, Sine–Gordon (SG) equation, nonlinear Schrodinger (NLS) equation, Korteweg–de Vries Burger’s (KdVB) equation, etc. . . are some known equations that lie in this specific class of this NLPDE. This class of nonlinear equations has a special type of the traveling wave solutions which are either solitary waves or solitons.  Partial Differential Equations that Lead to Solitons. The Korteweg–de Vries equation is a canonical equation in nonlinear wave physics demonstrating the existence of solitons and their elastic interactions. This KdV equation attracted many authors to publish an enormous number of publications. Different analytical methods for solving this well-known equation are presented in  Korteweg–de Vries Equation (KdV), Different Analytical Methods for Solving the. One of the most famous analytical methods for solving the KdV equation is the Inverse Scattering Transformation. This method was first introduced by the well-known scientists Gardener, Green, Kruskal, and Miura. In 1967, Gardner et al. published a magnificent and original paper on the analytical solution to the initial value problem due to a disturbance of finite amplitude in an infinite domain. This systematic method is not easy, and needs several long steps to get the complete solution.  Inverse Scattering Transform and the Theory of Solitons. There are various algebraic and geometric characteristics that the KdV equation possesses. More significantly, a lot of physically important solutions to the KdV equation can be presented explicitly through a simple, specific form, called the Hirota bilinear form. More compli-

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cated types of solutions leading to N-solitons are presented in  Korteweg–de Vries Equation (KdV), History, Exact N-Soliton Solutions and Further Properties of the. It was suitable to search for an easier method to solve this famous and well-known nonlinear partial differential equation (KdV). Many semi-analytical methods have been developed and used to solve the KdV equation. The most popular and famous semi-analytical method that was introduced to solve this nonlinear partial differential equation (KdV), was the Adomian Decomposition Method.  Adomian Decomposition Method Applied to Non-linear Evolution Equations in Soliton Theory. Other semi-analytical methods like the variational iteration method (VIM), homotopy analysis method (HAM), and homotopy perturbation method (HPM) are usable to solve these types of NLPDE such as the Korteweg–de Vries equation and even the modified Korteweg–de Vries equation (mKdV).  Korteweg–de Vries Equation (KdV) and Modified Korteweg–de Vries Equations (mKdV), Semi– analytical Methods for Solving the. It is indispensable to introduce other methods to illustrate the solution of this NLPDE. The numerical technique is a powerful tool which is always needed to present and compare the results in the easiest manner. There exist different numerical tools for solving the KdV and mKdV equations, and even for any evolution equations  Korteweg–de Vries Equation (KdV), Some Numerical Methods for Solving the. In hydrodynamical problems, the motion of incompressible inviscid fluid subject to a constant vertical gravitational force (where the fluid is bounded below by an impermeable bottom and above by a free surface) with some physical assumptions lead to the shallow water model. The shallow water wave equations are usually used in oceanography and atmospheric science. The shallow water approximation theory leads to the KdV equation. This means that the soliton solution as well as the solitary waves are the straight forward solutions for many physical problems.  Shallow Water Waves and Solitary Waves. The study of the water waves in the Pacific Ocean leads directly to a special type of long gravity waves named tsunamis. This word is a Japanese one, which means harbor’s wave. A tsunami is essentially a long wavelength water wave train or a series of waves generated in a body of water

(mostly in oceans) that vertically displaces the water column. Earthquakes, landslides, volcanic eruptions, nuclear explosions and the impact of cosmic bodies can generate tsunamis. Tsunamis, as they approach coastlines, can rise enormously and savagely attack and inundate to cause huge damage to properties and cost thousands of lives. The theoretical study of waves in oceanography is very complicated. The simplest mathematical model includes the KdV equation. This KdV represents the governing equation that lead to tsunamis.  Solitons, Tsunamis and Oceanographical Applications of. The rapid change in water density with depth is called Pycnocline. This happens in open or closed seas, as well as in oceans. In the frame of shallow water approximation theory that governs the oceans, and due to the pycnocline the nonlinear internal gravity waves are presented. They arise from perturbations to hydrostatic equilibrium, where balance is maintained between the force of gravity and the buoyant restoring force. An illustrative study of nonlinear waves (that was generated inside a stratified fluid occupying a semi infinite channel of finite and constant depth by a wave maker situated in motion at the finite extremity of the channel) is presented in this section.  Non-linear Internal Waves. The soliton perturbation theory is used to study solitons that are governed by the various nonlinear equations in the presence of the perturbation terms.  Soliton Perturbation. A compacton is a special solitary traveling wave that, unlike a soliton, does not have exponential tails. Another expression and terminology called compacton-like soliton is a special wave solution which can be expressed by the squares of sinusoidal or cosinusoidal functions. Soliton and compacton are two kinds of nonlinear waves. They play an indispensable and vital role in all branches of science and technology, and are used as constructive elements to formulate the complex dynamical behavior of wave systems throughout science: from hydrodynamics to nonlinear optics, from plasmas to shock waves, from tornados to the Great Red Spot of Jupiter, and from tsunamis to turbulence. More recently, soliton and compacton have been of key importance in the quantum fields and nanotechnology especially in nano-hydrodynamics.  Solitons and Compactons.

Solitons, Tsunamis and Oceanographical Applications of

Solitons, Tsunamis and Oceanographical Applications of M. LAKSHMANAN Center for Nonlinear Dynamics, Bharathidasan University, Tiruchirapalli, India Article Outline Glossary Definition of the Subject Introduction Shallow Water Waves and KdV Type Equations Deep Water Waves and NLS Type Equations Tsunamis as Solitons Internal Solitons Rossby Solitons Bore Solitons Future Directions Bibliography Glossary Soliton A class of nonlinear dispersive wave equations in (1+1) dimensions having a delicate balance between dispersion and nonlinearity admit localized solitary waves which under interaction retain their shapes and speeds asymptotically. Such waves are called solitons because of their particle like elastic collision property. The systems include Korteweg–de Vries, nonlinear Schrödinger, sine-Gordon and other nonlinear evolution equations. Certain (2+1) dimensional generalizations of these systems also admit soliton solutions of different types (plane solitons, algebraically decaying lump solitons and exponentially decaying dromions). Shallow and deep water waves Considering surface gravity waves in an ocean of depth h, they are called shallow-water waves if h  , where  is the wavelength (or from a practical point of view if h < 0:07). In the linearizedpcase, for shallow water waves the phase gh, where g is the acceleration due to speed c D gravity. Water waves are classified as deep (practically) if h > 0:28pand the corresponding wave speed is given by c D g/k, k D 2 .  Tsunami Tsunami is essentially a long wavelength water wave train, or a series of waves, generated in a body of water (mostly in oceans) that vertically displaces the water column. Earthquakes, landslides, volcanic eruptions, nuclear explosions and impact of cosmic bodies can generate tsunamis. Propagation of tsunamis is in many cases in the form of shallow wa-

ter waves and sometimes can be of the form of solitary waves/solitons. Tsunamis as they approach coastlines can rise enormously and savagely attack and inundate to cause devastating damage to life and property. Internal solitons Gravity waves can exist not only as surface waves but also as waves at the interface between two fluids of different density. While solitons were first recognized on the surface of water, the commonest ones in oceans actually happen underneath, as internal oceanic waves propagating on the pycnocline (the interface between density layers). Such waves occur in many seas around the globe, prominent among them being the Andaman and Sulu seas. Rossby solitons Rossby waves are typical examples of quasigeostrophic dynamical response of rotating fluid systems, where long waves between layers of the atmosphere as in the case of the Great Red Spot of Jupiter or in the barotropic atmosphere are formed and may be associated with solitonic structures. Bore solitons The classic bore (also called mascaret, poroca and aeger) arises generally in funnel shaped estuaries that amplify incoming tides, tsunamis or storm surges, the rapid rise propagating upstream against the flow of the river feeding the estuary. The profile depends on the Froude number, a dimensionless ratio of intertial and gravitational effects. Slower bores can take on oscillatory profile with a leading dispersive shockwave followed by a train of solitons. Definition of the Subject Surface and internal gravity waves arising in various oceanographic conditions are natural sources where one can identify/observe the generation, formation and propagation of solitary waves and solitons. Unlike the standard progressive waves of linear dispersive type, solitary waves are localized structures with long wavelengths and finite energies and propagate without change of speed or form and are patently nonlinear entities. The earliest scientifically recorded observation of a solitary wave was made by John Scott Russel in August 1834 in the Union Canal connecting the Scottish cities of Glasgow and Edinburgh. The theoretical formulation of the underlying phenomenon was provided by Korteweg and de Vries in 1895 who deduced the now famous Korteweg–de Vries (KdV) equation admitting solitary wave solutions. With the insightful numerical and analytical investigations of Martin Kruskal and coworkers in the 1960s the KdV solitary waves have been shown to possess the remarkable property that under collision they pass through each other without change of shape or speed except for a phase shift and so they are

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Solitons, Tsunamis and Oceanographical Applications of, Figure 1 KdV equation: a One soliton solution (solitary wave) at a fixed time, say t D 0, b Two soliton solution (depicting elastic collision)

solitons. Since then a large class of soliton possessing nonlinear dispersive wave equations such as the sine-Gordon, modified KdV (MKdV) and nonlinear Schrödinger (NLS) equations occurring in a wide range of physical phenomena have been identified. Several important oceanographic phenomena which correspond to nonlinear shallow water wave or deep water wave propagation have been identified/interpreted in terms of soliton propagation. These include tsunamis, especially earthquake induced ones like the 1960 Chilean or 2004 Indian Ocean earthquakes, internal solitary waves arising in stratified stable fluids such as the ones observed in Andaman or Sulu seas, Rossby waves including the Giant Red Spot of Jupiter and tidal bores occurring in estuaries of rivers. Detailed observations/laboratory experiments and theoretical formulations based on water wave equations resulting in the nonlinear evolution equations including KdV, Benjamin–Ono, Intermediate Long Wave (ILW), Kadomtsev–Petviashivili (KP), NLS, Davey– Stewartson (DS) and other equations clearly establish the relevance of soliton description in such oceanographic events. Introduction Historically, the remarkable observation of John Scott Russel [1,2] of the solitary wave in the Union Canal connecting the cities of Edinburgh and Glasgow in the month of August 1834 may be considered as the precursor to the realization of solitons in many oceanographic phenomena. While riding on a horse back and observing the motion of boat drawn by a pair of horses which suddenly stopped but not so the mass of water it had set in motion, the wave (which he called the ‘Great Wave of Translation’) in the form of large solitary heap of water surged forward and travelled a long distance without change of form or diminution of speed. The wave observed by Scott Russel is nothing but a solitary wave having a remarkable staying power and a patently nonlinear entity. Korteweg and de Vries in 1895, starting from the basic equations of hydrodynamics and

considering unidirectional shallow water wave propagation in rectangular channels, deduced [3] the now ubiquitous KdV equation as the underlying nonlinear evolution equation. It is a third order nonlinear partial differential equation in (1+1) dimensions with a delicate balance between dispersion and nonlinearity. It admits elliptic function cnoidal wave solutions and in a limiting form exact solitary wave solution of the type observed by John Scott Russel thereby vindicating his observations and putting to rest all the controversies surrounding them. It was the many faceted numerical and analytical study of Martin Kruskal and coworkers [4,5] which firmly established by 1967 that the KdV solitary waves have the further remarkable feature that they are solitons having elastic collision property (Fig. 1). It was proved decisively that the KdV solitary waves on collision pass through each other except for a finite phase shift, thereby retaining their forms and speeds asymptotically as in the case of particle like elastic collisions. The inverse scattering transform (IST) formalism developed for this purpose clearly shows that the KdV equation is a completely integrable infinite dimensional nonlinear Hamiltonian system and that it admits multisoliton solutions [6,7,8,9]. Since then a large class of nonlinear dispersive wave equations such as the sine-Gordon (s-G), modified Korteweg–de Vries (MKdV), NLS, etc. equations in (1+1) dimensions modeling varied physical phenomena have also been shown to be completely integrable soliton systems [6,7,8,9]. Interesting (2+1) dimensional versions of these systems such as Kadomtsev–Petviashvile (KP), Davey–Stewartson (DS) and Nizhnik–Novikov–Veselov (NNV) equations have also been shown to be integrable systems admitting basic nonlinear excitations such as line (plane) solitons, algebraically decaying lump solitons and exponentially localized dromion solutions [7,8]. It should be noted that not all solitary waves are solitons while the converse is always true. An example of a solitary wave which is not a soliton is the one which occurs in double-well  4 wave equation which radiates energy on collision with another such wave. However even

Solitons, Tsunamis and Oceanographical Applications of

such solitary waves having finite energies are sometimes referred to as solitons in the condensed matter, particle physics and fluid dynamics literature because of their localized structure. As noted above solitary waves and solitons are abundant in oceanographic phenomena, especially involving shallow water wave and deep water wave propagations. However, these events are large scale phenomena very often difficult to measure experimentally, rely mostly on satellite and other indirect observations and so controversies and differing interpretations do exist in ascribing exact solitonic or solitary wave properties to these phenomena. Yet it is generally realized that many of these events are closely identifiable with solitonic structures. Some of these observations deserve special attention. Tsunamis When large scale earthquakes, especially of magnitude 8.0 in Richter scale and above, occur in seabeds at appropriate geological faults tsunamis are generated and can propagate over large distances as small amplitude and long wavelength structures when shallowness condition is satisfied. Though the tsunamis are hardly felt in the midsea, they take monstrous structures when they approach land masses due to conservation of energy. The powerful Chilean earthquake of 1960 [10] led to tsunamis which propagated for almost fifteen hours before striking Hawaii islands and a further seven hours later they struck the Japanese islands of Honshu and Hokkaido. More recently the devastating Sumatra–Andaman earthquake of 2004 in the Indian Ocean generated tsunamis which not only struck the coastlines of Asian countries including Indonesia, Thailand, India and Srilanka but propagated as far as Somalia and Kenya in Africa, killing more than a quarter million people. There is very good sense in ascribing soliton description to such tsunamis. Internal Solitons Peculiar striations, visible on satellite photographs of the surface of the Andaman and Sulu seas in the far east (and in many other oceans around the globe), have been interpreted as secondary phenomena accompanying the passage of “internal solitons”. These are solitary wavelike distortions of the boundary layer between the warm upper layer of sea water and cold lower depths. These internal solitons are travelling ridges of warm water, extending hundreds of meters down below the thermal boundary, and carry enormous energy. Osborne and Burch [11] investigated the underwater currents which were experienced by an oil rig in the Andaman sea, which was drilling

at a depth of 3600 ft. One drilling rig was apparently spun through ninety degrees and moved one hundred feet by the passage of a soliton below. Rossby Waves and Solitons Rossby waves [12] are long waves between layers of the atmosphere, created by the rotation of the planet. In particular, in the atmosphere of a rotating planet, a fluid particle is endowed with a certain rotation rate, determined by its latitude. Consequently its motion in the north-south direction is constrained by the conservation of angular momentum as in the case of internal waves where gravity inhibits the vertical motion of a density stratified fluid. There is an analogy between internal waves and Rossby waves under suitable conditions. The KdV equation has been proposed as a model for the evolution of Rossby solitons [13] and NLS equation for the evolution of Rossby wave packets. The Great Red Spot of the planet of Jupiter is often associated with a Rossby soliton. Bore Solitons Rivers which flow to the open oceans are very often affected by tidal effects, tsunamis or storm surges. Tidal motions generate intensive water flows which can propagate upstream on tens of kilometers in the form of step-wise perturbation (hydraulic jumps) analogous to shock waves in acoustics. This phenomenon is known as a bore or mascaret (in French). Examples of such bores include the tidal bore of Seine river in France, Hooghly bore of the Ganges in India, the Amazon river bore in Brazil and Hangzhou bore in China. Bore disintegration into solitons is a possible phenomenon in such bores [14]. Besides these there are other possible oceanographic phenomena such as capillary wave solitons [15], resonant three-wave or four wave interaction solitons [16], etc., where also soliton picture is useful. In this article we will first point out how in shallow channels the long wavelength wave propagation is described by KdV equation and its generalizations (Sect. “Shallow Water Waves and KdV Type Equations”). Then we will briefly point out how NLS family of equations arise naturally in the description of deep water waves (Sect. “Deep Water Waves and NLS Type Equations”). Based on these details, we will point how the soliton picture plays a very important role in the understanding of tsunami propagation (Sect. “Tsunamis as Solitons”), generation of internal solitons (Sect. “Internal Solitons”), formation of Rossby solitons (Sect. “Rossby Solitons”) and disintegration of bores into solitons (Sect. “Bore Solitons”).

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Euler’s Equation As the density of the fluid D 0 D constant, using Newton’s law for the rate of change of momentum, we can write dV/dt D @V /@t C (V  r ) V D  10 r p  g j, where p D p(x; y; t) is the pressure at the point (x; y) and g is the acceleration due to gravity, which is acting vertically downwards (here j is the unit vector along the vertical direction). Since V D r  we obtain (after one integration)  t C 12 (r )2 C Solitons, Tsunamis and Oceanographical Applications of, Figure 2 One-dimensional wave motion in a shallow channel

Shallow Water Waves and KdV Type Equations Kortweg and de Vries [3] considered the wave phenomenon underlying the observations of Scott Russel from first principles of fluid dynamics and deduced the KdV equation to describe the unidirectional shallow water wave propagation in one dimension. Consider the one-dimensional (x-direction) wave motion of an incompressible and inviscid fluid (water) in a shallow channel of height h, and of sufficient width with uniform cross-section leading to the formation of a solitary wave propagating under gravity. The effect of surface tension is assumed to be negligible. Let the length of the wave be l and the maximum value of its amplitude, (x; t), above the horizontal surface be a (see Fig. 2). Then assuming a  h (shallow water) and h  l (long waves), one can introduce two natural small parameters into the problem D a/h and ı D h/l. Then the analysis proceeds as follows [3,6]. Equation of Motion: KdV Equation The fluid motion can be described by the velocity vector V (x; y; t) D u(x; y; t)i C v(x; y; t) j, where i and j are the unit vectors along the horizontal and vertical directions, respectively. As the motion is irrotational, we have r  V D 0. Consequently, we can introduce the velocity potential (x; y; t) by the relation V D r . Conservation of Density The system obviously admits the conservation law for the mass density (x, y, t) of the fluid, d /dt D t C r  ( V ) D 0. As is a constant, we have r  V D 0. Consequently  obeys the Laplace equation r 2 (x; y; t) D 0 :

(1)

p C gy D 0 :

0

(2)

Boundary Conditions The above two Eqs. (1) and (2) for the velocity potential (x; y; t) of the fluid have to be supplemented by appropriate boundary conditions, by taking into account the fact (see Fig. 2) that (a) the horizontal bed at y D 0 is hard and (b) the upper boundary y D y(x; t) is a free surface. As a result (a) the vertical velocity at y D 0 vanishes, v(x; 0; t) D 0, which implies  y (x; 0; t) D 0 :

(3)

(b) As the upper boundary is free, let us specify it by y D hC(x; t) (see Fig. 2). Then at the point x D x1 , y D y1  dy y(x; t), we can write dt1 D @ C @  dx 1 D  t Cx u1 D v1 . @t @x dt Since v1 D 1y , u1 D 1x , we obtain 1y D  t C x 1x :

(4)

(c) Similarly at y D y1 , the pressure p1 D 0. Then from (2), it follows that u1t C u1 u1x C v1 v1x C gx D 0 :

(5)

Thus the motion of the surface of water wave is essentially specified by the Laplace Eq. (1) and Euler Eq. (2) along with one fixed boundary condition (3) and two variable nonlinear boundary conditions (4) and (5). One has to then solve the Laplace equation subject to these boundary conditions. Taylor Expansion of (x; y; t) in y Making use of the fact ı D h/l  1, h  l, we assume y(D h C (x; t)) to be small to introduce the Taylor expansion (x; y; t) D

1 X

y n  n (x; t) :

(6)

nD0

Substituting the above series for  into the Laplace Eq. (1), solving recursively for  n (x; t)’s and making use of the boundary condition (4),  y (x; 0; t) D 0, one can

Solitons, Tsunamis and Oceanographical Applications of

show that u1 D 1x D f  12 y12 f x x C higher order in y1 ; v1 D 1y D

y1 f x C 16 y13 f x x x C

(7)

higher order in y1 ; (8)

where f D @0 /@x. We can then substitute these expressions into the nonlinear boundary conditions (4) and (5) to obtain equations for f and . Introduction of Small Parameters  and ı So far the analysis has not taken into account fully the shallow nature of the channel (a/h D  1) and the solitary nature of the wave (a/l D a/hh/l D ı  1,  1, ı  1), which are essential to realize the Scott Russel phenomenon. For this purpose one can stretch the independent and dependent variables in the above equations through appropriate scale changes, but retaining the overall form of the equations. To realize this one can introduce the natural scale changes x D l x 0 ;  D a0 ; t D

l 0 t ; c0

(9)

where c0 is a parameter to be determined. Then in order to retain the form of (7) and (8) we require u1 D c0 u10 ; v1 D ıc0 v10 ; f D c0 f 0 ;    y1 D h C (x; t) D h 1 C 0 x 0 ; t 0 :

(10)

 2 u10 D f 0  12 ı 2 1 C 0 f x0 0 x 0 D f 0  12 ı 2 f x0 0 x 0 ; (11) where we have omitted terms proportional to ı 2 as small compared to terms of the order ı 2 . Similarly from (8), we obtain   v10 D  1 C 0 f x0 0 C 16 ı 2 f x0 0 x 0 x 0 : (12) Now considering the nonlinear boundary condition (4) in the form v1 D  t C x u1 , it can be rewritten as (13)

Similarly considering the other boundary condition (5) and making use of the above transformations, it can be rewritten, after neglecting terms of the order 2 ı 2 , as f t00 C f 0 f x0 0 C

ga 0  0  12 ı 2 f x0 0 x 0 t 0 D 0 : c02 x

(14)

Now choosing the arbitrary parameter c0 as c02 D gh so that 0x 0 term is of order unity, (14) becomes f t00 C 0x 0 C f 0 f x0 0  12 ı 2 f x0 0 x 0 t 0 D 0 :

 t C f x C  f x C f x  16 ı 2 f x x x D 0 ;

(16)

f t C x C f f x  12 ı 2 f x x t D 0 :

(17)

Note that the small parameters and ı 2 have occurred in a natural way in (16), (17). Perturbation Analysis Since the parameters and ı 2 are small in (16), (17), we can make a perturbation expansion of f in these parameters: f D f (0) C f (1) C ı 2 f (2) C higher order terms ; (18) where f (i) , i D 0; 1; 2; : : : are functions of  and its spatial derivatives. Note that the above perturbation expansion is an asymptotic expansion. Substituting this into Eqs. (16) and (17) and regrouping and comparing different powers proportional to ( , ı 2 ) and solving them successively one can obtain (see for example [6] for further details) in a self consistent way, f (0) D ; f (1) D  14 2 ;

Then

0t 0 C f x0 0 C 0 f x0 0 C f 0 0x 0  16 ı 2 f x0 0 x 0 x 0 D 0 :

p (Note that c0 D gh is nothing but the speed of the water wave in the linearized limit). Omitting the primes for convenience, the evolution equation for the amplitude of the wave and the function related to the velocity potential reads

(15)

f (2) D 13 x x :

(19)

Using these expressions into (18) and substituting it in (16) and (17), we ultimately obtain the KdV equation in the form  t C x C 32 x C

ı2 x x x D 0 ; 6

(20)

describing the unidirectional propagation of long wavelength shallow water waves. The Standard (Contemporary) Form of KdV Equation Finally, changing to a moving frame of reference,  D x  t;  D t, and introducing the new variables u D (3 /2ı 2 );  0 D (6/ı 2 ) and redefining the variables  0 as t and  as x, we finally arrive at the ubiquitous form of the KdV equation as u t C 6uu x C u x x x D 0 :

(21)

The Korteweg–de Vries Eq. (21) admits cnoidal wave solution and in the limiting case solitary wave solution as well. This form can be easily obtained [6,17] by looking for a wave solution of the form u D 2 f (),  D (x  ct), and reducing the KdV equation into a third order nonlinear ordinary differential equation of the form c

@f @3 f @f C 12 f C 3 D0: @ @ @

(22)

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Integrating Eq. (22) twice and rearranging, we can obtain  2 @f D 4 f 3 C c f 2  2d f  2b  P( f ) ; (23) @ where b and d are integration constants. Calling the three real roots of the cubic equation P( f ) D 0 as ˛1 ; ˛2 and ˛3 such that  2 @f D 4( f  ˛1 )( f  ˛2 )( f  ˛3 ); (24) @ the solution can be expressed in terms of the Jacobian elliptic function as 2

f () D f (x  ct) D ˛3  (˛3  ˛2 )sn p  ˛3  ˛1 (x  ct) ; m ;

(25a)

where (˛1 C ˛2 C ˛3 ) D

c ; 4

m2 D

˛3  ˛2 ; ˛3  ˛1 ı D constant :

(25b)

Here m is the modulus parameter of the Jacobian elliptic function. Eq. (25a) represents in fact the so called cnoidal wave (because of its elliptic function form). In the limiting case m D 1, the form (25a) reduces to p ˛3  ˛1 (x  ct) : (26) f D ˛2 C (˛3  ˛2 ) sech2 Choosing now ˛1 D 0, ˛2 D 0, and using (25b) we have 

p c c 2 f D sech (x  ct) : (27) 4 2 Then the solitary wave solution to the KdV Eq. (21) can be written in the form p c c (x  ct C ı) ; u(x; t) D sech2 2 2 ı D constant : (28) Note that the velocity of the solitary wave is directly proportional to the amplitude: larger the wave the higher is the speed. More importantly, the KdV solitary wave is a soliton: it retains its shape and speed upon collision with another solitary wave of different amplitude, except for a phase shift, see Fig. 1 [6,7,8]. In fact for an arbitrary initial condition, the solution of the Cauchy initial value problem consists of N-number of solitons of different amplitudes in the background of small amplitude dispersive waves. All these results ultimately lead to the result that the KdV equation is a completely integrable, infinite dimensional, nonlinear Hamiltonian system. It possesses [6,7,8] (i)

a Lax pair of linear differential operators and is solvable through the so called inverse scattering transform (IST) method,

(ii) infinite number of conservation laws and associated infinite number of involutive integrals of motion, (iii) N-soliton solution, (iv) Hirota bilinear form, (v) Hamiltonian structure and a host of other interesting properties (see for example [6,7,8,9]). KdV Related Integrable and Nonintegrable NLEEs Depending on the actual physical situation, the derivation of the shallow water wave equation can be suitably modified to obtain other forms of nonlinear dispersive wave equations in (1+1) dimensions as well as in (2+1) dimensions relevant for the present context. Without going into the actual derivations, some of the important equations possessing solitary waves are listed below [6,7,8,9]. 1. Boussinesq equation [7] u t C uu x C gx  13 h2 u tx x D 0;

(29a)

 t C [u(h C )]x D 0

(29b)

2. Benjamin–Bona–Mahoney (BBM) equation [18] u t C u x C uu x  u x x t D 0

(30)

3. Camassa–Holm equation [19] u t C 2u x C 3uu x  u x x t D 3u x u x x C uu x x x (31) 4. Kadomtsev–Petviashville (KP) equation [7] (u t C 6uu x C u x x x )x C 3 2 u y y D 0 ( 2 D –1 : KP-I,  2 D C1 : KP-II).

(32)

In the derivation of the above equations, generally the bottom of water column or fluid bed is assumed to be flat. However in realistic situations the water depth varies as a function of the horizontal coordinates. In this situation, one often encounters inhomogeneous forms of the above wave equations. Typical example is the variable coefficient KdV equation [14]: u t C f (x; t)uu x C g(x; t)u x x x D 0;

(33)

where f and g are functions of x, t. More general forms can also be deduced depending upon the actual situations, see for example [14].

Solitons, Tsunamis and Oceanographical Applications of

Deep Water Waves and NLS Type Equations Deep water waves are strongly dispersive in contrast to the weakly dispersive nature of the shallow water waves (in the linear limit). Various authors (see for details [8]) have shown that nonlinear Schrödinger family of equations models the evolution of a packet of surface waves in this case. There are several oceanographic situations where such waves can arise [8]: (i) A localized storm at sea can generate a wide spectrum of waves, which then propagates away from the source region in all horizontal directions. If the propagating waves have small amplitudes and encounter no wind away from the source region, these waves can eventually sort themselves into nearly one-dimensional packets of nearly monochromatic waves. For appropriately chosen scales, the underlying evolution of each of these packets can be shown to satisfy the nonlinear Schrödinger equation and its generalizations in (2+1) dimensions. (ii) Nearly monochromatic, nearly one-dimensional waves can cover a broad range of surface waves in the sea that results due to a steady wind of long duration and fetch. Then the generalization of the NLS equation in (2+1) dimensions can describe the waves that result in when the wind stops. In all the above situations one looks for the solution of the equations of motion (1)-(5) but generalized in three dimensions which consists mainly in the form of a small amplitude, nearly monochromatic, nearly one-dimensional wave train. Assuming that this wave train travels in the x-direction with a mean wave number  D (k; l) with a characteristic amplitude ‘a’ of the disturbance and a characteristic variation ık in k, one can deduce the NLS equation in (2+1) dimensions under the following conditions: (i) (ii) (iii) (iv) (v)

small amplitudes such that ˆ D ı   a  1 slowly varying modulations, ık  1 nearly one dimensional waves, jlkj  1 balance of all three effects, ık D jlkj  O(ˆ ) for finite and deep water waves, (kh)2  ˆ

In the lowest order approximation (linear approximation) of water waves, the prediction is that a localized initial state will generally evolve into wave packets with a dominant wave number  and corresponding frequence !, given by the p dispersion relation ! D (g C  3 )tanh h;  D jj D k 2 C l 2 within which each wave propagates with the phase speed c D !/k, while the envelope propagates with the group velocity c g D d!/d. After a sufficiently long time the wave packet tends to disperse around the dominant wave number.

This tendency for dispersion can be offset by cumulative nonlinear effects. In the absence of surface tension, the outcome for unidirectional waves can be shown to be describable by the NLS equation. If the surface wave in the lowest order is   Aexpi(kx ! t)+c.c, where  is the velocity potential, then to leading order the wave amplitude evolves as i(A t C c g A x ) C 12 A x x C jAj2 A D 0:

(34)

The coefficients here are given by @2 ! ; @k 2 ! k2 (8C 2 S 2 C g  2T 2 ) D 16S 4 ! (2!C 2 C kc g )2 ; C 8C 2 S 2 (gh  c 2g ) D

(35)

where C D cosh(kh), S D sinh(kh), T D S/C. Equation (34) has been obtained originally by Zakharov in 1968 for deep water waves [20] and by Hasimoto and Ono for waves of finite depth in 1972 [21]. The NLS equation is also a soliton possessing integrable system and is solvable by the IST method [6,7,8,9]. For  > 0 (focusing case), the envelope solitary (soliton) wave solution (also called bright solitons in the optical physics context) is given by A(x; t) D a sech (x  c g t)exp(i˝ t);

(36)

C 21 a2 .

where a 2 D  2 , ˝ D When the effects of modulation in the transverse y-direction are taken into account, so that the wave amplitude is now given by A(x; y; t), the NLS equation is replaced by the Benney–Roskes system [22] also popularly known as the Davey–Stewartson equations [23], i(A t C c g A x ) C 12 A x x C 12 ıA y y C jAj2 A C UA D 0 ;

(37a)

˛U x x C U y y C ˇ(jAj2 ) y y D 0 ; (37b)  c2  c where ı D kg , ˛ D 1 g hg , ghˇ D 8C!2 S 2 (2!C 2 C kc g )2 . Here U(x; y; t) is the wave induced mean flow. In the deep water wave limit, kh ! 1, and U ! 0, ˇ ! 0 and one has the nonintegrable (2+1) dimensional NLS equation. On the other hand in the shallow water limit, one has the integrable Davey–Stewartson (DS) equations. For details see [7,8]. The DS-I equation admits algebraically decaying lump solitons and exponentially decaying dromions [24] besides the standard line solitons for appropriate choices of parameters. A typical dromion solution of DS-I equation is shown in Fig. 3.

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Solitons, Tsunamis and Oceanographical Applications of, Figure 3 Exponentially localized dromion solution of the Davey– Stewartson equation at a fixed time (t D 0) for suitable choice of parameters

Solitons, Tsunamis and Oceanographical Applications of, Figure 4 26 December 2004 Indian Ocean tsunami (adapted from the website www.blogaid.org.uk with the courtesy of Andy Budd)

Tsunamis as Solitons The term ‘tsunami’ (tsu:harbour, nami:wave in Japanese) which was perhaps an unknown word even for scientists in countries such as India, Srilanka, Thailand, etc. till recently has become a house-hold word since that fateful morning of December 26, 2004. When a powerful earthquake of magnitude 9.1–9.3 on the Richter scale, epicentered off the coast of Sumatra, Indonesia, struck at 07:58:53, local time, described as the 2004 Indian Ocean earthquake or Sumatra–Andaman earthquake (Fig. 4), it triggered a series of devastating tsunamis as high as 30 meters that spread throughout the Indian Ocean killing about 275,000 people and inundating coastal communities across South and Southeast Asia, including parts of Indonesia, Srilanka, India and Thailand and even reaching as far as the east coast of Africa [25]. The catastrophe is considered to be one of the deadliest disasters in modern history.

Since this earthquake and consequent tsunamis, several other earthquakes of smaller and larger magnitudes keep occurring off the coast of Indonesia. Even as late as July 17, 2006 an earthquake of magnitude 7.7 on the Richter scale struck off the town of Pandering at 15.19 local time and set off a tsunami of 2m high which had killed more than 300 people. These tsunamis, which can become monstrous tidal waves when they approach coastline, are essentially triggered due to the sudden vertical rise of the seabed by several meters (when earthquake occurs) which displaces massive volume of water. The tsunamis behave very differently in deep water than in shallow water as pointed out below. By no means the tsunami of 2004 and later ones are exceptional; More than two hundred tsunamis have been recorded in scientific literature since ancient times. The most notable earlier one is the tsunami triggered by the powerful earthquake (9.6 magnitude) off southern Chile on May 22, 1960 [10] which traveled almost 22 hours before striking Japanese islands. It is clear from the above events that the tsunami waves are fairly permanent and powerful ones, having the capacity to travel extraordinary distances without practically diminishing in size or speed. In this sense they seem to have considerable resemblance to shallow water nonlinear dispersive waves of KdV type, particularly solitary waves and solitons. It is then conceivable that tsunami dynamics has close connection with soliton dynamics. Basics of Tsunami Waves As noted above tsunami waves of the type described above are essentially triggered by massive earthquakes which lead to vertical displacement of a large volume of water. Other possible reasons also exist for the formation and propagation of tsunami waves: underwater nuclear explosion, larger meteorites falling into the sea, volcano explosions, rockslides, etc. But the most predominant cause of tsunamis appear to be large earthquakes as in the case of the Sumatra–Andaman earthquake of 2004. Then there are three major aspects associated with the tsunami dynamics [26]: 1. Generation of tsunamis 2. Propagation of tsunamis 3. Tsunami run up and inundation There exist rather successful models to approach the generation aspects of tsunamis when they occur due to the earthquakes [27]. Using the available seismic data it is possible to reconstruct the permanent deformation of the sea bottom due to earthquakes and simple models have been

Solitons, Tsunamis and Oceanographical Applications of

developed (see for example, [28]). Similarly the tsunami run up and inundation problems [29] are extremely complex and they require detailed critical study from a practical point of view in order to save structures and lives when a tsunami strikes. However, here we will be more concerned with the propagation of tsunami waves and their possible relation to wave propagation associated with nonlinear dispersive waves in shallow waters. In order to appreciate such a possible connection, we first look at the typical characteristic properties of tsunami waves as in the case of 2004 Indian Ocean tsunami waves or 1960 Chilean tsunamis. The Indian Ocean Tsunami of 2004 Considering the Indian Ocean 2004 tsunami, satellite observations after a couple of hours after the earthquake establish an amplitude of approximately 60 cms in the open ocean for the waves. The estimated typical wavelength is about 200 kms [30]. The maximum water depth h is between 1 and 4 kms. Consequently, one can identify in an average sense the following small parameters ( and ı 2 ) of roughly equal magnitude: D

h2 a  104  1; ı 2 D 2  104  1 h l

(38)

As a consequence, it is possible that a nonlinear shallow water wave theory where dispersion (KdV equation) also plays an important role (as discussed in Sect. “Shallow Water Waves and KdV Type Equations”) has considerable relevance [26]. However, we also wish to point out here that there are other points of view: Constantin and Johnson [31] estimate  0:002 and ı  0:04 and conclude that for both nonlinearity and dispersion to become significant the quantity ı 3/2 wavelength estimated as 90,000 kms is too large and shallow water equations with variable depth (without dispersion) should be used. However, it appears that these estimates can vary over a rather wide range and with suitable estimates it is possible that the range of 10,000–20,000 kms could be also possible ranges and hence taking into account the fact that both the Indian Ocean 2004 and Chilean 1960 tsunamis have traveled over 10 hours or more (in certain directions) before encountering land mass appears to allow for the possibility of nonlinear dispersive waves as relevant features for the phenomena. Segur [32] has argued that in the 2004 tsunamis, the propagation distances from the epicenter of the earthquake to India, Srilanka, or Thailand were too short for KdV dynamics to develop. In the same way one can argue that the waves that hit Somalia and Kenya in

the east coast of Africa (or as in the case of Chilean earthquake see also [32]) have traveled sufficiently long distance for KdV dynamics to become important [33]. In any case one can conclude that at least for long distance tsunami propagation solitary wave and soliton picture of KdV like equations become very relevant. Internal Solitons For a long time seafarers passing through the Strait of Malacca on their journeys between India and the Far East have noticed that in the Andaman sea, between Nicobar islands and the north east coast of Sumatra, often bands of strongly increased surface roughness (ripplings or bands of choppy water) occur [11,34]. Similar observations have been reported in other seas around the globe from time to time. In recent times there has been considerable progress in understanding these kind of observations in terms of internal solitons in the oceans [7]. These studies have been greatly facilitated by photographs taken from satellites and space-crafts orbiting the earth, for example by synthetic aperture radar (SAR) images of ERS-1/2 satellites [34,35]. Peculiar striations of 100 km long, separated by 6 to 15 km and grouped in packets of 4 to 8, visible on satellite photographs (see Fig. 5) of the surface of the Andaman and Sulu seas in the Far East, have been interpreted as secondary phenomena accompanying the passage of ‘internal solitons’, which are solitary wavelike distortions of the boundary layer between warm upper layer of sea water and cold lower depths. These internal solitons are traveling edges of warm water, extending hundreds of meters down below the thermal boundary [7]. They carry enormous energy with them which is perhaps the reason for unusually strong underwater currents experienced by deep-sea drilling rigs. Thus these internal solitons are potentially hazardous to sub-sea oil and gas explorations. The ability to predict them can improve substantially the cost effectiveness and safety of offshore drilling. A systematic study of the underwater currents experienced by an oil rig in the Andaman sea which was drilling at a depth of 3600 ft was carried out by Osborne and Burch in 1980 [11]. They spent four days measuring underwater currents and temperatures. The striations seen on satellite photographs turned out to be kilometer-wide bands of extremely choppy water, stretching from horizon to horizon, followed by about two kilometers of water “as smooth as a millpond”. These bands of agitated water are called “tide rips”, they arose in packets of 4 to 8, spaced about 5 to 10 km apart (they reached the research vessel at approximately hourly intervals) and this pattern was repeated with the regularity of tidal phenomenon.

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Solitons, Tsunamis and Oceanographical Applications of, Figure 5 SAR image of a 200 km × 200 km large section of the Andaman Sea acquired by the ERS-2 satellite on April 15, 1996 showing sea surface manifestations of two internal solitary wave packets, see [34]. Figure reproduced from ESA website www.earth.esa. int/workshops/ers97/papers/alpers3 with the courtesy of European Space Agency and W. Alpers

As described in [7], Osborne and Burch found that the amplitude of each succeeding soliton was less than the previous one, is precisely what is expected for solitons (note that the velocity of a solitary wave solution of KdV equation increases with amplitude, vide Eq. (22)). Thus if a number of solitons are generated together, then we expect them eventually to be arranged in an ordered sequence of decreasing amplitude. From the spacing between successive waves in a packet and the rate of separation calculated from the KdV equation, Osborne and Burch were able to estimate the distance the packet had traveled from its source and thus identify possible source regions [7]. They concluded that the solitons are generated by tidal currents off northern Sumatra or between the islands of the Nicobar chain that extends beyond it and that their observations have good general agreement with the predictions for internal solitons as given by the KdV equation. Numerous recent observations and predictions

of solitons in the Andaman sea have clearly established that it is a site where extraordinarily large internal solitons are encountered [34,35]. Further, Apel and Holbrook [36] undertook a detailed study of internal waves in the Sulu sea. Satellite photographs had suggested that the source of these waves was near the southern end of the Sulu sea and their research ship followed one wave packet for more than 250 miles over a period of two days – an extraordinary coherent phenomenon [7]. These internal solitons travel at speeds of about 8 kilometers per hour (5 miles per hour), with amplitude of about 100 meters and wavelength of about 1700 meters. Similar observations elsewhere have confirmed the presence of internal solitons in oceans including the strait of Messina, the strait of Gibraltar, off the western side of Baja California, the Gulf of California, the Archipelago of La Maddalena and the Georgia strait [7]. There has also been a number of experimental studies of internal solitons in laboratory tanks in the last few decades [37]. These experiments provide detailed quantitative information usually unavailable in the field conditions, and are also an efficient tool for verifying various theoretical models. As a theoretical formulation of internal solitons [7], consider two incompressible, immiscible fluids, with densities 1 and 2 and depths h1 and h2 respectively such that the total depth h D h1 C h2 . Let the lighter fluid of height h1 be lying over a heavier fluid of height h2 , in a constant gravitational field (Fig. 6). The lower fluid is assumed to rest on a horizontal impermeable bed, and the upper fluid is bounded by a free surface. Then as in Sect. “Shallow Water Waves and KdV Type Equations”, we denote the characteristic amplitude of wave by ‘a’ and the characteristic wavelength l D k 1 . Then the various nonlinear wave equations to describe the formation of internal solitons follow by suitable modification of the formulation in Sect. “Shallow Water Waves and KdV Type Equations”, assuming viscous effects to be negligible. Each of these equations is completely integrable and admits soliton solutions [7]. (a) KdV equation (Eq. (21)) follows when (i)

the waves are of long wavelength ı D

h l

 1,

(ii) the amplitude of the waves are small, D and

a h

 1,

(iii) the two effects are comparable ı 2 D O( ) (b) Intermediate-Long-Wave (ILW) equation [38] u t C u x C 2uu x C Tu x x D 0;

(39)

Solitons, Tsunamis and Oceanographical Applications of

Solitons, Tsunamis and Oceanographical Applications of, Figure 6 Formation of internal soliton (note that under suitable conditions small amplitude surface soliton can also be formed)

where Tu is the singular integral operator 1

 Z  1 (T f )(x) D (y  x) f (y)dy  coth 2l 2l

(40)

1

R1 with 1 the Cauchy principal value integral is obtained under the assumption that there is a thin (upper) layer, D hh12  1, the amplitude of the waves is small, a  h1 , the above two effects balance, ha1 D O( ), the characteristic wavelength is comparable to the total depth of the fluid, l D kh D O(1) and (e) the waves are long waves in comparison with the thin layer, kh1  1. (a) (b) (c) (d)

(c) Benjamin–Ono equation [39] u t C 2uu x C Hu x x D 0;

(41)

where Hu is the Hilbert transform (H f )(x) D

1 Z f (y) 1  dy;  yx

(42)

1

is obtained under the assumption (a) there is a thin (upper) layer h1  h2 , (b) the waves are long waves in comparison with the thin layer, kh1  1, (c) the waves are short in comparison with the total depth of the fluid, kh  1 and (d) the amplitude of the waves is small, a  h1 .

It may be noted that in the shallow water limit, as ı ! 0, the ILW equation reduces to the KdV equation, while the Benjamin–Ono equation reduces to it in the deep water wave limit as ı ! 1. Each of these equations have their own ranges of validity and admit solitary wave and soliton solutions to represent internal solitons of the oceans. Rossby Solitons The atmospheric dynamics is an important subject of human concern as we live within the atmosphere and are continuously affected by the weather and its rather complex behavior. The motion of the atmosphere is intimately connected with that of the ocean with which it exchanges fluxes of momentum, heat and moisture [40]. Then the atmospheric dynamics is dominated by the rotation of the earth and the vertical density stratification of the surrounding medium, leading to newer effects. Similar effects are also present in other planetary dynamics as well. In the atmosphere of a rotating planet, a fluid particle is endowed with a certain rotation rate (Coriolis frequency), determined by its latitude. Consequently its motion in the north-south direction is inhibited by conservation of angular momentum. The large scale atmospheric waves caused by the variation of the Coriolis frequency with latitude are called Rossby waves. In Sect. “Internal Solitons” we saw that KdV equation and its modifications model internal waves. Since there is a resemblance between internal waves and Rossby waves, it is expected that KdV like equations can model Rossby waves as well [8]. Under the simplest assumptions like long waves, incompressible fluid, ˇ-plane approximations, etc. Benney [41]

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Solitons, Tsunamis and Oceanographical Applications of, Figure 7 A coupled high/low pressure systems in the form of a Rossby soliton formed off the coast of California/Washington on Valentine’s Day 2005. The low pressure front hovers over Los Angeles dumping 30 inches of rain on the city in two weeks. The high pressure system lingers off the coast of Washington state providing unseasonally warm and sunny weather. This particular Rossby soliton proved exceptionally stable because of its remarkable symmetry. In an accompanying animated gif in this website one can watch the very interesting phenomenon as the jet stream splits around the soliton suctioning warm wet air from the off the coast of Mexico to Arizona, leaving behind a welcome drenching of rain. The figure and caption have been adapted from the website http://mathpost. la.asu.edu/~rubio/rossby_soliton/rs.html with the courtesy of the National Oceanic and Atmospheric Administration (NOAA) and Antonio Rubio

had derived the KdV equation as a model for Rossby waves in the presence of east-west zonal flow. Boyd [42] had shown that long, weakly nonlinear, equatorial Rossby waves are governed either by KdV or MKdV equation. Recently it has been shown by using the eight years of Topex/Poseidon altimeter observations [43] that a detailed characterization of major components of Pacific dynamics confirms the presence of equatorial Rossby solitons. Also observational and numerical studies of propagation of nonlinear Rossby wave packets in the barotropic atmosphere by Lee and Held [44,45] have established their presence, notably in the Northern Hemisphere as storm tracks, but more clearly in the Southern Hemisphere (see Fig. 7). They also found that the wavepackets both in the real atmosphere and in the numerical models behave like the envelope solitons of the nonlinear Schrödinger equation (Fig. 7). An interesting application of KdV equation to describe Rossby waves is the conjecture of Maxworthy and Redekopp [46] that the planet Jupiter’s Great Red Spot

might be a solitary Rossby wave. Photographs taken by the spacecraft Voyager of the cloud pattern show that the atmospheric motion on Jupiter is dominated by a number of east-west zonal currents, corresponding to the jet streams of the earth’s atmosphere [8]. Several oval-shaped spots are also seen, including the prominent Great Red Spot in the southern hemisphere. The latter one has been seen approximately this latitude for hundreds of years and maintains its form despite interactions with other atmospheric objects. One possible explanation is that the Red Spot is a solitary wave/soliton of the KdV equation, deduced from the quasigeostrophic form of potential vorticity equation for an incompressible fluid. A second test of the model is the combined effect of interaction of the Red Spot and Hollow of the Jupiter atmosphere on the South Tropical Disturbance which occurred early in the 20th century and lasted several decades. Maxworthy and Redekopp [46] have interpreted this interaction as that of two soliton collision of KdV equation, with the required phase shift [6,7,8].

Solitons, Tsunamis and Oceanographical Applications of

The above connection between the KdV solitary wave and the rotating planet may be seen more concretely by the following argument as detailed in [8]. One can start with the quasigeostrophic form of the potential vorticity equation for an incompressible fluid in the form   @ @ @ @  @y @x @x @y  

2 2 @ @ @ @ @ K2 C (ˇ  U 00 ) D 0; 2 2 C 2 C @x @y @z @z @x (43)



@ @ Cv @t @x





C

where x, y and z represent the east, north and vertical directions, while the function U is related toR the horizony tal stream function ˚ as ˚(x; y; z; t) D y 0 U()d C (x; y; z; t) in terms of a zonal shear flow and a perturbation. In (43), in the ˇ-plane approximation, the Coriolis parameter is given as f D 2˝ sin 0 C ˇ y and the function K(z) is given by K(z) D 2˝ sin 0 l2 /N(z)d, where N(z) is the Brunt–Väisälä frequency and l2 is the characteristic length scales in the north-south direction while d is the length scale in the vertical direction. Note that K compares the effects of rotation and density of variation. Then  D l2 /l1 represents the ratio of the length scales in the north-south and east-west directions. In the linear ( ! 0), long wave ( ! 0) limit, P the potential function has the form D n A n (x  c n t) n (y)p n (z), where cn is deduced by solving two related eigenvalue problems [8], (K 2 p0n )0 C k 2n p n D 0; p n (0) D p n (1) D 0 and  n00  k 2n  n C [(ˇ  U 00 )/(U  c n )] n D 0,  n (y s ) D  n (y N ) D 0. If the various modes are separable on a short time scale, one can then deduce an evolution equation for the individual modes, by eliminating secular terms at higher order in the expansion. Depending on the nature and existence of a stable density of stratification, which is characterized by the function N(z) mentioned above, either KdV or mKdV equation can be deduced for a given mode and thereby establishing the soliton connection in the present problem. In spite of the complexity of the phenomenon underlying Rossby solitons there is clear evidence of the significance of solitonic picture.

in the water height and speed which will propagate upstream [47]. Far less dangerous but very similar is the bore (mascaret in French), a tidal wave which can propagate in a river for considerable distances. Typical examples of bores occur in Seine river in France and the Hooghli river in West Bengal in India. A bore existed in the Seine river upto 1960 and disappeared when the river was dredged. The tide amplitude here is one of the largest in the world. The Hughli river is a branch of the Ganges that flows through Kolkata where a bore of 1m is present, essentially due to the shallowness of the river. Another interesting example is that at the time of the 1983 Japan sea tsunami, waves in the form of a bore ascended many rivers [48]. In some cases, bores had the form of one initial wave with a train of smaller waves and in other cases only a step with flat water surface behind was observed (Fig. 8). The Hangzhou bore in China is a tourist attraction. Other well known bores occur in the Amazon in Brazil and in Australia. Another interesting situation where bore solitons were observed was in the International H2 O Project (IHOP), as a density current (such as cold air from thunderstorm) intrudes into a fluid of lesser density that occurs beneath a low level inversion in the atmosphere. A spectacular bore and its evolution into a beautiful amplitude-ordered train of solitary waves were observed and sampled during the early morning of 20 June 2002 by the Leandre-II abroad the P-3 aircraft. The origin of this bore was traceable to a propagating cold outflow boundary from a mesoscale convective system in extreme western Kansas [49]. In the process of propagation the bore undergoes dissipation, dispersive disintegration, enhancement due to decrease of the river width and depth, influence of nonlinear

Bore Solitons Rivers which flow into the open oceans are usually affected by tidal flows, tsunami or storm surge. For a typical estuary as one moves towards the mouth of the river, the depth increases and width decreases. When a tidal wave, tsunami or storm surge hits such an estuary, it can be seen as a hydraulic jump (step-wise perturbations like a shock-wave)

Solitons, Tsunamis and Oceanographical Applications of, Figure 8 Tidal bore at the mouth of the Araguari River in Brazil. The bore is undular with over 20 waves visible. (Adapted from the book “Gravity Currents in the Environment and the Laboratory” by John E. Simpson, Cambridge University Press with the courtesy of D.K. Lynch, John E. Simpson and Cambridge University Press)

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effects and so on. The profile depends on the Froude number which is a dimensionless ratio of inertial and gravitational effects. Theoretical models have been developed to study these effects based on KdV and its generalizations [14]. For example, bore disintegration into solitons in channel of constant parameters can be studied in signal coordinates in terms of the KdV like equation for the perturbation of water surface, 1 x C (1  ˛) t  ˇ t t t D 0 ; (44) c0 p where c0 D gh, ˛ D 3/2h, ˇ D h2 /6c03 , h being the depth of the river, with the bore represented by a Heaviside step function as the boundary condition at x D 0. Other effects then can be incorporated into a variable KdV equation of the form (26). Future Directions We have indicated a few of the most important oceanographical applications of solitons including tsunamis, internal solitons, Rossby solitons and bore solitons. There are other interesting phenomena like capillary wave solitons [15], resonant three and four wave interaction solitons [16], etc. which are also of considerable interest depending on whether the wave propagation corresponds to shallow, intermediate or deep waters. Whatever be the situation, it is clear that experimental observations as well as their theoretical formulation and understandings are highly challenging complex nonlinear evolutionary problems. The main reason is that the phenomena are essentially large scale events and detailed experimental observations require considerable planning, funding, technology and manpower. Often satellite remote sensing measurements need to be carried out as in the case of internal solitons and Rossby solitons. Events like tsunami propagation are rare and time available for making careful measurements are limited and heavily dependent on satellite imaging, warning systems and after event measurements. Consequently developing and testing theoretical models are extremely hazardous and difficult. Yet the basic modeling in terms of solitary wave/soliton possessing nonlinear dispersive wave equations such as the KdV and NLS family of equations and their generalizations present fascinating possibilities to understand these large scale complex phenomena and opens up possibilities of prediction. Further understanding of such nonlinear evolution equations, both integrable and nonintegrable systems particularly in higher dimensions, can help to understand the various phenomena clearly and provide means of predicting events like tsunamis and damages which occur due to

internal solitons and bores. Detailed experimental observations can also help in this regard. It is clear that what has been understood so far is only the qualitative aspects of these phenomena and much more intensive work is needed to understand the quantitative aspects to predict them.

Bibliography Primary Literature 1. Russel JS (1844) Reports on Waves. 14th meeting of the British Association for Advancement of Science. John Murray, London, pp 311–390 2. Bullough RK (1988) The Wave Par Excellence. The solitary progressive great wave of equilibrium of fluid: An early history of the solitary wave. In: Lakshmanan M (ed) Solitons: Introduction and Applications. Springer, Berlin 3. Korteweg DJ, de Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos Mag 39:422–443 4. Zabusky NJ, Kruskal MD (1965) Interactions of solitons in a collisionless plasma and recurrence of initial states. Phys Rev Lett 15:240–243 5. Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the Korteweg–de Vries equation. Phys Rev Lett 19:1095–97 6. Lakshmanan M, Rajasekar S (2003) Nonlinear Dynamics: Integrability and Chaos. Springer, Berlin 7. Ablowitz MJ, Clarkson PA (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge 8. Ablowitz MJ, Segur H (1981) Solitons and Inverse Scattering Transform. Society for Industrial and Applied Mathematics, Philadelphia 9. Scott AC (1999) Nonlinear Science: Emergence and Dynamics of Coherent Structures. Oxford University Press, New York 10. Dudley WC, Miu L (1988) Tsunami! University of Hawaii Press, Honololu 11. Osborne AR, Burch TL (1980) Internal solitons in the Andaman Sea. Science 258:451–460 12. Rossby GG (1939) Relation between variations in the intensity of the zonal circulation of the atmosphere. J Mar Res 2:38–55 13. Redekopp L (1977) On the theory of solitary Rossby waves. J Fluid Mech 82:725–745 14. Caputo JG, Stepanyants YA (2003) Bore formation and disintegration into solitons in shallow inhomogeneous channels. Nonlinear Process Geophys 10:407–424 15. Longuet–Higgins MS (1993) Capillary gravity waves of solitary type and envelope solitons in deep water. J Fluid Mech 252:703–711 16. Philips OM (1974) Nonlinear dispersive waves. Ann Rev Fluid Mech 6:93–110 17. Helal MA, Molines JM (1981) Nonlinear internal waves in shallow water. A theoretical and experimental study. Tellus 33:488–504 18. Benjamin TB, Bona JL, Mahoney JJ (1972) Model equations for long waves in nonlinear dispersive systems. Philos Trans A Royal Soc 272:47–78

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19. Camassa R, Holm D (1992) An integrable shallow water equation with peaked solitons. Phys Rev Lett 71:1661–64 20. Zakharov VE (1968) Stability of periodic waves of finite amplitude on the surface of a deep fluid. J Appl Mech Tech Phys 2:190–194 21. Hasimoto H, Ono H (1972) Nonlinear modulation of gravity waves. J Phys Soc Jpn 33:805–811 22. Benney DJ, Roskes G (1969) Wave instabilities. Stud Appl Math 48:377–385 23. Davey A, Stewartson K (1974) On three dimensional packets of surface waves, Proc Royal Soc Lond A 338:101–110 24. Fokas AS, Santini PM (1990) Dromions and a boundary value problem for the Davey–Stewartson I equation. Physica D 44:99–130 25. Kundu A (ed) (2007) Tsunami and Nonlinear Waves. Springer, Berlin 26. Dias F, Dutykh D (2007) Dynamics of tsunami waves. In: Ibrahimbegovic A, Kozar I (eds) Extreme man-made and natural hazards in dynamics of structures. NATO security through Science Series. Springer, Berlin, pp 201–224 27. Dutykh D, Dias F (2007) Water waves generated by a moving bottom. In: Kundu A (ed) Tsunami and Nonlinear Waves. Springer, Berlin, pp 65–94 28. Okada Y (1992) Internal deformation due to shear and tensile faults in a half space. Bull Seism Soc Am 82:1018–1040 29. Carrier GF, Wu TT, Yeh H (2003) Tsunami runup and drawdown on a plane beach. J Fluid Mech 475:79–99 30. Banerjee P, Politz FF, Burgman R (2005) The size and duration of the Sumatra–Andaman earthquake from far-field static offsets. Science 308:1769–1772 31. Constantin A, Johnson RS (2006) Modelling tsunamis. J Phys A39:L215-L217 32. Segur H (2007) Waves in shallow water, with emphasis on the tsunami of (2004). In: Kundu A (ed) Tsunami and Nonlinear Waves. Springer, Berlin, pp 3–29 33. Lakshmanan M (2007) Integrable nonlinear wave equations and possible connections to tsunami dynamics. In: Kundu A (ed) Tsunami and Nonlinear Waves. Springer, Berlin, pp 31–49 34. Alpers W, Wang-Chen H, Cook L (1997) Observation of internal waves in the Andaman Sea by ERS SAR. IEEE Int 4:1518–1520 35. Hyder P, Jeans DRG, Cauqull E, Nerzic R (2005) Observations and predictability of internal solitons in the northern Andaman Sea. Appl Ocean Res 27:1–11 36. Apel JR, Holbroook JR (1980) The Sulu sea internal soliton experiment, 1. Background and overview. EOS Trans AGU 61:1009 37. Ostrovsky LA, Stepanyants YA (2005) Internal solitons in laboratory experiments: Comparison with theoretical models. Chaos 15(1–28):037111 38. Joseph RI (1977) Solitary waves in a finite depth fluid. J Phys A10:L225-L227 39. Benjamin TB (1967) Internal waves of permanent form in fluids of great depth. J Fluid Mech 29:559–592

40. Kundu PK, Cohen IM (2002) Fluid Mechanics, Second Edition. Academic Press, San Diego 41. Benney DJ (1966) Long nonlinear waves in fluid flows. Stud Appl Math 45:52–63 42. Boyd JP (1980) Equatorial solitary waves. Part I: Rossby solitons. J Phys Oceanogr 10:1699–1717 43. Susanto RD, Zheng Q, Xiao-Hai Y (1998) Complex singular value decomposition analysis of equatorial waves in the Pacific observed by TOPEX/Poseidon altimeter. J Atmospheric Ocean Technol 15:764–774 44. Lee S, Held I (1993) Baroclinic wave packets in models and observations. J Atmospheric Sci 50:1413–1428 45. Tan B (1996) Collision interactions of envelope Rossby solitons in a baratropic atmosphere. J Atmospheric Sci 53:1604–1616 46. Maxworthy T, Redekopp LG (1976) Theory of the Great Red Spot and other observed features of the Jovian atmosphere. Icarus 29:261–271 47. Caputo JG, Stepanyants YA (2007) Tsunami surge in a river: a hydraulic jump in an inhomogeneous channel. In: Kundu A (ed) Tsunami and Nonlinear Waves. Springer, Berlin, pp 97–112 48. Tsuji T, Yanuma T, Murata I, Fujiwara C (1991) Tsunami ascending in rivers as an undular bore. Nat Hazard 4:257–266 49. Koch SE, Pagowski M, Wilson JW, Fabry F, Flamant C, Feltz W, Schwemmer G, Geerts B (2005) The structure and dynamics of atmospheric bores and solitons as determined from remote sensing and modelling experiments during IHOP, AMS 32nd Conference on Radar Meteorology, Report JP6J.4

Books and Reviews Lamb H (1932) Hydrodynamics. Dover, New York Miles JW (1980) Solitary waves. Ann Rev Fluid Mech 12:11–43 Stoker JJ (1957) Water Waves. Interscience, New York Dauxois T, Peyrard M (2006) Physics of Solitons. Cambridge University Press, Cambridge Johnson RS (1997) An Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge Hammack JL (1973) A note on tsunamis: their generation and propagation in an ocean of uniform depth. J Fluid Mech 60:769–800 Drazin PG, Johnson RS (1989) Solitons: An Introduction. Cambridge University Press, Cambridge Mei CC (1983) The Applied Dynamics of Ocean Surface Waves. Wiley, New York Scott AC et al (1973) The soliton: a new concept in applied science. Proc IEEE 61:1443–83 Scott AC (ed) (2005) Encyclopedia of Nonlinear Science. Routledge, New York Sulem C, Sulem P (1999) The Nonlinear Schrödinger Equation. Springer, Berlin Helfrich KR, Melville WK (2006) Long nonlinear internal waves. Ann Rev Fluid Mech 38:395–425 Bourgault D, Richards C (2007) A laboratory experiment on internal solitary waves. Am J Phys 75:666–670

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trum if the maximal spectral type of U is singular with respect to (equivalent to, absolutely continuous with b respect to) a Haar measure of A. Koopman representation of a dynamical system T Let ´ MARIUSZ LEMA NCZYK A be a l.c.s.c. (and not compact) Abelian group and Faculty of Mathematics and Computer Science, T : a ! 7 T a representation of A in the group a Nicolaus Copernicus University, Toru´n, Poland Aut (X; B; ) of (measure-preserving) automorphisms of a standard probability Borel space (X; B; ). Article Outline The Koopman representation U D UT of T in L2 (X; B; ) is defined as the unitary representation Glossary a 7! U Ta 2 U(L2 (X; B; )), where U Ta ( f ) D f ı Ta . Definition of the Subject Ergodicity, weak mixing, mild mixing, mixing and Introduction rigidity of T A measure-preserving A-action T D Maximal Spectral Type of a Koopman Representation, ) is called ergodic if 0  1 2 b A is a sim(T a a2A Alexeyev’s Theorem ple eigenvalue of U . It is weakly mixing if UT has T Spectral Theory of Weighted Operators 2 (X; B; ) a continuous spectrum on the subspace L The Multiplicity Function 0 of zero mean functions. T is said to be rigid if there Rokhlin Cocycles is a sequence (a n ) going to infinity in A such that the Rank-1 and Related Systems sequence (U Ta n ) goes to the identity in the strong (or Spectral Theory of Dynamical Systems weak) operator topology; T is said to be mildly mixing of Probabilistic Origin if it has no non-trivial rigid factors. We say that T is Inducing and Spectral Theory mixing if the operator equal to zero is the only limit Special Flows and Flows on Surfaces, point of fU Ta j L 2 (X;B;) : a 2 Ag in the weak operator Interval Exchange Transformations 0 topology. Future Directions Spectral disjointness Two A-actions S and T are called Acknowledgments spectrally disjoint if the maximal spectral types of their Bibliography Koopman representations UT and US on the corresponding L20 -spaces are mutually singular. Glossary b satSCS property We say that a Borel measure  on A isfies the strong convolution singularity property (SCS Spectral decomposition of a unitary representation If property) if, for each n  1, in the disintegraU D (U a ) a2A is a continuous unitary representation (given map (1 ; : : : ; n ) 7! 1  : : :  n ) tion of a locally compact second countable (l.c.s.c.) R by the (n) () the conditional measures   d  ˝n D b Abelian group A in a separable Hilbert space H then  A  L1 a decomposition H D are atomic with exactly n! atoms ( (n) stands for the iD1 A(x i ) is called spectral if x 1  x 2  : : : (such a sequence of measures is nth convolution  : : :  ). An A-action T satisfies also called spectral); here A(x) :D spanfU a x : a 2 Ag the SCS property if the maximal spectral type of UT on L20 is a type of an SCS measure. is called the cyclic space generated by x 2 H and x stands for the spectral measure of x. Kolmogorov group property An A-action T satisfies the Maximal spectral type and the multiplicity function of U Kolmogorov group property if UT UT  UT . The maximal spectral type U of U is the type of x 1 Weighted operator Let T be an ergodic automorphism in any spectral decomposition of H; the multiplicity of (X; B; ) and  : X ! T be a measurable funcfunction MU : b tion. The (unitary) operator V D V;T acting on A ! f1; 2; : : :g [ fC1g is defined U P b a.e. and MU () D 1 L2 (X; B; ) by the formula V;T ( f )(x) D (x) f (Tx) iD1 1Yi (), where Y1 D A and Yi D supp dx i /dx 1 for i  2. is called a weighted operator. A representation U is said to have simple spectrum if H Induced automorphism Assume that T is an automorphism of a standard probability Borel space (X; B; ). is reduced to a single cyclic space. The multiplicity is Let A 2 B, (A) > 0. The induced automorphism TA uniform if there is only one essential value of MU . The is defined on the conditional space (A; B A ; A ), where essential supremum of MU is called the maximal specB A is the trace of B on A, A (B) D (B)/(A) for tral multiplicity. U is said to have discrete spectrum if H has an orthonormal base consisting of eigenvectors of B 2 B A and TA (x) D T k A (x) x, where k A (x) is the U; U has singular (Haar, absolutely continuous) specsmallest k  1 for which T k x 2 A.

Spectral Theory of Dynamical Systems

Spectral Theory of Dynamical Systems

AT property of an automorphism An automorphism T of a standard probability Borel space (X; B; ) is called approximatively transitive (AT for short) if for every " > 0 and every finite set f1 ; : : : ; f n of non-negative L1 -functions on (X; B; ) we can find f 2 L1 (X; B; ) P also non-negative such that k f i  j ˛ i j f ıT n j kL 1 < " for all i D 1; : : : ; n (for some ˛ i j  0, n j 2 N). Cocycles and group extensions If T is an ergodic automorphism, G is a l.c.s.c. Abelian group and ' : X ! G is measurable then the pair (T; ') generates a cocycle ' () () : Z  X ! G, where 8 for n > 0 ; < '(x) C : : : C '(T n1 x) (n) 0 for n D 0 ; ' (x) D : ('(T n x) C : : : C '(T 1 x)) for n < 0 : (That is (' (n) ) is a standard 1-cocycle in the algebraic sense for the Z-action n( f ) D f ı T n on the group of measurable functions on X with values in G; hence the function ' : X ! G itself is often called a cocycle.) Assume additionally that G is compact. Using the cocycle ' we define a group extension T' on (X  G; B ˝ B(G);  ˝ G ) (G stands for Haar measure of G), where T' (x; g) D (Tx; '(x) C g). Special flow Given an ergodic automorphism T on a standard probability Borel space (X; B; ) and a positive integrable function f : X ! RC we put X f D f(x; t) 2 X  R : 0  t < f (x)g ; B f D B ˝ B(R)j X f ;

and define  f as normalized  ˝ R j X f . By a spef cial flow we mean the R-action T f D (Tt ) t2R under which a point (x; s) 2 X f moves vertically with the unit speed, and once it reaches the graph of f , it is identified with (Tx; 0). Markov operator A linear operator J : L2 (X; B; ) ! L2 (Y; C ; ) is called Markov if it sends non-negative functions to non-negative functions and J1 D J  1 D 1. Unitary actions on Fock spaces If H is a separable Hilbert space then by H ˇn we denote the subspace of n-tensors of H ˝n symmetric under all permutations of coordinates, n  1; then the Hilbert space L ˇn is called a symmetric Fock space. F(H) :D 1 nD0 H L1 ˇn 2 U(F(H)) If V 2 U(H) then F(V) :D nD0 V where V ˇn D V ˝n jH ˇn . Definition of the Subject Spectral theory of dynamical systems is a study of special unitary representations, called Koopman representations

(see the glossary). Invariants of such representations are called spectral invariants of measure-preserving systems. Together with the entropy, they constitute the most important invariants used in the study of measure-theoretic intrinsic properties and classification problems of dynamical systems as well as in applications. Spectral theory was originated by von Neumann, Halmos and Koopman in the 1930s. In this article we will focus on recent progresses in the spectral theory of finite measure-preserving dynamical systems. Introduction Throughout A denotes a non-compact l.c.s.c. Abelian group (A will be most often Z or R). The assumption of second countability implies that A is metrizable,  compact and the space C0 (A) is separable. Moreover the dual group b A is also l.c.s.c. Abelian. General Unitary Representations We are interested in unitary, that is with values in the unitary group U(H) of a Hilbert space H, (weakly) continuous representations V : A 3 a 7! Va 2 U(H) of such groups (the scalar valued maps a 7! hVa x; yi are continuous for each x; y 2 H). b B(A); b ), where B(A) b stands for the Let H D L2 (A; b (whenever b  -algebra of Borel sets of A and  2 M C (A) X is a l.c.s.c. space, by M(X) we denote the set of complex Borel measures on X, while M C (X) stands for the subset of positive (finite) measures). Given a 2 A, for b B(A); b ) put f 2 L2 (A; b ; Va ( f )() D i(a)()  f () D (a)  f () ( 2 A) b where i : A ! b A is the canonical Pontriagin isomor phism of A with its second dual. Then V  D (Va ) a2A b is dense in is a unitary representation of A. Since C0 (A) b ), the latter space is separable. Therefore also diL2 (A; L  i of such type representations will be rect sums 1 iD1 V unitary representations of A in separable Hilbert spaces. b ) is a closed Lemma 1 (Wiener Lemma) If F L2 (A;  b Va -invariant subspace for all a 2 A then F D 1Y L2 (A; b ) for some Borel subset Y A. b B(A); Notice however that since A is not compact (equivalently, b is not discrete), we can find  continuous and thereA fore V  has no irreducible (non-zero) subrepresentation. From now on only unitary representations of A in separable Hilbert spaces will be considered and we will show how to classify them.

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A function r : A ! C is called positive definite if N X

r(a n  a m )z n z m  0

Koopman Representations (1)

n;mD0

for each N > 0, (a n ) A and (z n ) C. The central result about positive definite functions is the following theorem (see e. g. [173]). Theorem 1 (Bochner–Herglotz) Let r : A ! C be continuous. Then r is positive definite if and only if there exists b such that (a unique)  2 M C (A) Z r(a) D (a) d() for each a 2 A : b A If now U D (U a ) a2A is a representation of A in H then for each x 2 H the function r(a) :D hU a x; xi is continuous and satisfies (1), so it is positive definite. By the Bochner– Herglotz Theorem there exists a unique measure U;x D b (called the Spectral measure of x) such that x 2 M C (A) Z b  x (a) :D i(a)() dx () D hU a x; xi b A for each a 2 A. Since the partial map U a x 7! b x ) is isometric and equivariant, there exists i(a) 2 L2 (A; a unique extension of it to a unitary operator W : A(x) ! b x ) giving rise to an isomorphism of UjA(x) and L2 (A;  x V . Then the existence of a spectral decomposition is proved by making use of separability and a choice of maximal cyclic spaces at every step of an induction procedure. Moreover, a spectral decomposition is unique in the following sense. L Theorem 2 (Spectral Theorem) If H D 1 iD1 A(x i ) D L1 0 ) are two spectral decompositions of H then A(x iD1 i x i  x 0i for each i  1. It follows that the representation U is entirely determined by types (the sets of equivalent measures to a given one) of a decreasing sequence of measures or, equivalently, U is determined by its maximal spectral type U and its multiplicity function MU . Notice that eigenvalues of U correspond to Dirac meab is an eigenvalue (i. e. for some kxk D 1, sures:  2 A U a (x) D (a)x for each a 2 A) if and only if U;x D ı . Therefore U has a discrete spectrum if and only if the maximal spectral type of U is a discrete measure. We refer the reader to [64,105,127,147,156] for presentations of spectral theory needed in the theory of dynamical systems – such presentations are usually given for A D Z but once we have the Bochner–Herglotz Theorem and the Wiener Lemma, their extensions to the general case are straightforward.

We will consider measure-preserving representations of A. It means that we fix a probability standard Borel space (X; B; ) and by Aut (X; B; ) we denote the group of automorphisms of this space, that is T 2 Aut (X; B; ) if T: X ! X is a bimeasurable (a.e.) bijection satisfying (A) D (TA) D (T 1 A) for each A 2 B. Consider then a representation of A in Aut (X; B; ) that is a group homomorphism a 7! Ta 2 Aut (X; B; ); we write T D (Ta ) a2A . Moreover, we require that the associated Koopman representation UT is continuous. Unless explicitly stated, A-actions are assumed to be free, that is we assume that for -a.e. x 2 X the map a 7! Ta x is 1 1. In fact, since constant functions are obviously invariant for U Ta , that is the trivial character 1 is always an eigenvalue of UT , the Koopman representation is considered only on the subspace L20 (X; B; ) of zero mean functions. We will restrict our attention only to ergodic dynamical systems (see the glossary). It is easy to see that T is ergodic if and only if whenever A 2 B and A D Ta (A) (-a.e.) for all a 2 A then (A) equals 0 or 1. In case of ergodic Koopman representations, all eigenvalues are simple. In particular, (ergodic) Koopman representations with discrete spectra have simple spectra. The reader is referred to monographs mentioned above as well as to [26,158,177,196,204] for basic facts on the theory of dynamical systems. The passage T 7! UT can be seen as functorial (contravariant). In particular a measure-theoretic isomorphism of A-systems T and T 0 implies spectral isomorphism of the corresponding Koopman representations; hence spectral properties are measure-theoretic invariants. Since unitary representations are completely classified, one of the main questions in the spectral theory of dynamical systems is to decide which pairs ([ ]; M) can be realized by Koopman representations. The most spectacular, still unsolved, is the Banach problem concerning ([T ]; M  1). Another important problem is to give complete spectral classification in some classes of dynamical systems (classically, it was done in the theory of Kolmogorov and Gaussian dynamical systems). We will also see how spectral properties of dynamical systems can determine their statistical (ergodic) properties; a culmination given by the fact that a spectral isomorphism may imply measure-theoretic similitude (discrete spectrum case, Gaussian–Kronecker case). We conjecture that a dynamical system T either is spectrally determined or there are uncountably many pairwise non-isomorphic systems spectrally isomorphic to T . We could also consider Koopman representations in L p for 1  p ¤ 2. However whenever W : L p (X; B; ) !

Spectral Theory of Dynamical Systems

L p (Y; C ; ) is a surjective isometry and W ı U Ta D U S a ı W for each a 2 A then by the Lamperti Theorem (e. g. [172]) the isometry W has to come from a nonsingular pointwise map R : Y ! X and, by ergodicity, R “preserves” the measure  and hence establishes a measure-theoretic isomorphism [94] (see also [127]). Thus spectral classification of such Koopman representations goes back to the measure-theoretic classification of dynamical systems, so it looks hardly interesting. This does not mean that there are no interesting dynamical questions for p ¤ 2. Let us mention still open Thouvenot’s question (formulated in the 1980s) for Z-actions: For every ergodic T acting on (X; B; ), does there exist f 2 L1 (X; B; ) such that the closed linear span of f ı T n , n 2 Z equals L1 (X; B; )? Iwanik [79,80] proved that if T is a system with positive entropy then its L p -multiplicity is 1 for all p > 1. Moreover, Iwanik and de Sam Lazaro [85] proved that for Gaussian systems (they will be considered in Sect. “Spectral Theory of Dynamical Systems of Probabilistic Origin”) the L p -multiplicities are the same for all p > 1 (see also [137]). Markov Operators, Joinings and Koopman Representations, Disjointness and Spectral Disjointness, Entropy We would like to emphasize that spectral theory is closely related to the theory of joinings (see  Joinings in Ergodic Theory for needed definitions). The elements of the set J(S; T ) of joinings of two A-actions S and T are in a 1-1 correspondence with Markov operators J D J between the L2 -spaces equivariant with the corresponding Koopman representations (see the glossary and  Joinings in Ergodic Theory). The set of ergodic self-joinings of an ergodic A-action T is denoted by J2e (T ). Each Koopman representation UT consists of Markov operators (indeed, U Ta is clearly a Markov operator). In fact, if U 2 U(L2 (X; B; )) is Markov then it is of the form U R , where R 2 Aut (X; B; ) [133]. This allows us to see Koopman representations as unitary Markov representations, but since a spectral isomorphism does not “preserve” the set of Markov operators, spectrally isomorphic systems can have drastically different sets of self-joinings. We will touch here only some aspects of interactions (clearly, far from completeness) between the spectral theory and the theory of joinings. In order to see however an example of such interactions let us recall that the simplicity of eigenvalues for ergodic systems yields a short “joining” proof of the classical isomorphism theorem of Halmos-von Neumann in

the discrete spectrum case: Assume that S D (S )2A and T D (Ta ) a2A are ergodic A-actions on (X; B; ) and (Y; C ; ) respectively. Assume that both Koopman representations have purely discrete spectrum and that their group of eigenvalues are the same. Then S and T are measure-theoretically isomorphic. Indeed, each ergodic joining of T and S is the graph of an isomorphism of these two systems (see [127]; see also Goodson’s proof in [66]). Another example of such interactions appear in the study Rokhlin’s multiple mixing problem and its relation with the pairwise independence property (PID) for joinings of higher order. We will not deal with this subject here, referring the reader to  Joinings in Ergodic Theory (see however Sect. “Lifting Mixing Properties”). Following [60], two A-actions S and T are called disjoint provided the product measure is the only element in J(S; T ). It was already noticed in [72] that spectrally disjoint systems are disjoint in the Furstenberg sense; indeed, Im(J j L 2 ) D f0g since T ;J f  S; f . 0 Notice that whenever we take 2 J2e (T ) we obtain a new ergodic A-action (Ta  Ta ) a2A defined on the probability space (X  X; ). One can now ask how close spectrally to T is this new action? It turns out that except of the obvious fact that the marginal -algebras are factors, (T  T ; ) can have other factors spectrally disjoint with T : the most striking phenomenon here is a result of Smorodinsky and Thouvenot [198] (see also [29]) saying that each zero entropy system is a factor of an ergodic self-joining system of a fixed Bernoulli system (Bernoulli systems themselves have countable Haar spectrum). The situation changes if

D  ˝ . In this case for f ; g 2 L2 (X; B; ) the spectral measure of f ˝ g is equal to T ; f T ;g . A consequence of this observation is that an ergodic action T D (Ta ) a2A is weakly mixing (see the glossary) if and only if the product measure  ˝  is an ergodic self-joining of T . The entropy which is a basic measure-theoretic invariant does not appear when we deal with spectral properties. We will not give here any formal definition of entropy for amenable group actions referring the reader to [153]. Assume that A is countable and discrete. We always assume that A is Abelian, hence it is amenable. For each dynamical system T D (Ta ) a2A acting on (X; B; ), we can find a largest invariant sub- field P B, called the Pinsker  -algebra, such that the entropy of the corresponding quotient system is zero. Generalizing the classical Rokhlin-Sinai Theorem (see also [97] for Zd -actions), Thouvenot (unpublished) and independently Dooley and Golodets [31] proved this theorem for groups even more general than those considered here: The spectrum of UT on L2 (X; B; )  L2 (P ) is Haar with uniform infinite multiplicity. This general result is quite intricate and based on

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methods introduced to entropy theory by Rudolph and Weiss [179] with a very surprising use of Dye’s Theorem on orbital equivalence of all ergodic systems. For A which is not countable the same result was recently proved in [17] in case of unimodular amenable groups which are not increasing union of compact subgroups. It follows that spectral theory of dynamical systems essentially reduces to the zero entropy case. Maximal Spectral Type of a Koopman Representation, Alexeyev’s Theorem Only few general properties of maximal spectral types of Koopman representations are known. The fact that a Koopman representation preserves the space of real functions implies that its maximal spectral type is the type of a symmetric (invariant under the map  7! ) measure. Recall that the Gelfand spectrum (U) of a representation U D (U a ) a2A is defined as the of approximab if for tive eigenvalues of U, that is  (U) 3  2 A a sequence (x n ) bounded and bounded away from zero, kU a x n  (a)x n k ! 0 for each a 2 A. The spectrum is a closed subset in the topology of pointwise converb In case gence, hence in the compact-open topology of A. of A D Z, the above set (U) is equal to fz 2 C : U  z  Id is not invertibleg. Assume now that A is countable and discrete (and Abelian). Then there exists a Fölner sequence (B n )n1 whose elements tile A [153]. Take a free and ergodic action T D (Ta ) a2A on (X; B; ). By [153] for each " > 0 we can find a set Yn 2 B such that the sets Tb Yn are pairS wise disjoint for b 2 B n and ( b2B n Tb Yn ) > 1  ". b by considering functions of the form For each  2 A, P fn D (b)1 Tb Yn we obtain that  2 (UT ). It b2B n follows that the topological support of the maximal spectral type of the Koopman representation of a free and ergodic action is full [105,127,147]. The theory of Gaussian systems shows in particular that there are symmetric measures on the circle whose topological support is the whole circle but which cannot be maximal spectral types of Koopman representations. An open well-known question remains whether an absolutely continuous measure is the maximal spectral type of a Koopman representation if and only if is equivalent b (this is unknown for A D Z). to a Haar measure of A Another interesting question was recently raised by A. Katok (private communication): Is it true that the topological supports of all measures in a spectral sequence of a Koopman representation are full? If the answer to this question is positive then for example the essential supreb mum of MUT is the same on all balls of A.

Notice that the fact that all spectral measures in a spectral sequence are symmetric means that UT is isomorphic to UT 1 . A. del Junco [89] showed that generically for Z-actions, T and its inverse are not measure-theoretically isomorphic (in fact he proved disjointness). Let T be an A-action on (X; B; ). One can ask wether a “good” function can realize the maximal spectral type of UT . In particular can we find a function f 2 L1 (X; B; ) that realizes the maximal spectral type? The answer is given in the following general theorem (see [139]). Theorem 3 (Alexeyev’s Theorem) Assume that U D (U a ) a2A is a unitary representation of A in a separable Hilbert space H. Assume that F H is a dense linear subspace. Assume moreover that with some F-norm  – stronger than the norm k  k given by the scalar product – F becomes a Fréchet space. Then, for each spectral measure  ( U ) there exists y 2 F such that  y   . In particular, there exists y 2 F that realizes the maximal spectral type. By taking H D L2 (X; B; ) and F D L1 (X; B; ) we obtain the positive answer to the original question. Alexeyev [14] proved the above theorem for A D Z using analytic functions. Refining Alexeyev’s original proof, Fr¸aczek [52] showed the existence of a sufficiently regular function realizing the maximal spectral type depending only on the “regularity” of the underlying probability space, e. g. when X is a compact metric space (compact manifold) then one can find a continuous (smooth) function realizing the maximal spectral type. By the theory of systems of probabilistic origin (see Sect. “Spectral Theory of Dynamical Systems of Probabilistic Origin”), in case of simplicity of the spectrum, one can easily point out spectral measures whose types are not realized by (essentially) bounded functions. However, it is still an open question whether for each Koopman representation UT there exists a sequence ( f i ) i1

L1 (X; B; ) such that the sequence ( f i ) i1 is a spectral sequence for UT . For A D Z it is unknown whether the maximal spectral type of a Koopman representation can be realized by a characteristic function. Spectral Theory of Weighted Operators We now pass to the problem of possible essential values for the multiplicity function of a Koopman representation. However, one of known techniques is a use of cocycles, so before we tackle the multiplicity problem, we will go through recent results concerning spectral theory of compact group extensions automorphisms which in turn entail a study of weighted operators (see the glossary). Assume that T is an ergodic automorphism of a standard Borel probability space (X; B; ). Let  : X ! T

Spectral Theory of Dynamical Systems

be a measurable function and let V D V;T be the corresponding weighted operator. To see a connection of weighted operators with Koopman representations of compact group extensions consider a compact (metric) Abelian group G and a cocycle ' : X ! G. Then U T' (see the glossary) acts on L2 (X  G;  ˝ G ). But M L ; where L D L2 (X; )˝; L2 (X G; ˝G ) D 2b G where L is a U T' -invariant (clearly, closed) subspace. Moreover, the map f ˝  7! f settles a unitary isomorphism of U T' j L  with the operator Vı';T . Therefore, spectral analysis of such Koopman representations reduces to the spectral analysis of weighted operators Vı';T for all  2 b G. Maximal Spectral Type of Weighted Operators over Rotations The spectral analysis of weighted operators V;T is especially well developed in case of rotations, i. e. when we additionally assume that T is an ergodic rotation on a compact monothetic group X: Tx D x C x0 , where x0 is a topologically cyclic element of X (and  will stand for Haar measure x of X). In this case Helson’s analysis [74] applies (see also [68,82,127,160]) leading to the following conclusions:  The maximal spectral type V ;T is either discrete or continuous.  When V ;T is continuous it is either singular or Lebesgue.  The spectral multiplicity of V;T is uniform. We now pass to a description of some results in case when Tx D x C ˛ is an irrational rotation on the additive circle X D [0; 1). It was already noticed in the original paper by Anzai [16] that when  : X ! T is an affine cocycle ((x) D exp(nxCc), 0 ¤ n 2 Z) then V;T has a Lebesgue spectrum. It was then considered by several authors (originated by [123], see also [24,26]) to which extent this property is stable when we perturb our cocycle. Since the topological degree of affine cocyles is different from zero, when perturbing them we consider smooth perturbations by cocycles of degree zero. Theorem 4 ([82]) Assume that Tx D x C ˛ is an irrational rotation. If  : [0; 1) ! T is of non-zero degree, absolutely continuous, with the derivative of bounded variation then V;T has a Lebesgue spectrum. In the same paper, it is noticed that if we drop the assumption on the derivative then the maximal spectral type

of V;T is a Rajchman measure (i. e. its Fourier transform vanishes at infinity). It is still an open question, whether one can find  absolutely continuous with non-zero degree and such that V;T has singular spectrum. “Below” absolute continuity, topological properties of the cocycle seem to stop playing any role – in [82] a continuous, degree 1 cocycle  of bounded variation is constructed such that (x) D (x)/(Tx) for a measurable  : [0; 1) ! T (that is  is a coboundary) and therefore V;T has purely discrete spectrum (it is isomorphic to U T ). For other results about Lebesgue spectrum for Anzai skew products see also [24,53,81] (in [53] Zd -actions of rotations and so called winding numbers instead of topological degree are considered). When the cocycle is still smooth but its degree is zero the situation drastically changes. Given an absolutely continuous function f : [0; 1) ! R M. Herman [76], using the Denjoy–Koksma inequality (see e. g. [122]), showed R1 (q ) that f0 n ! 0 uniformly (here f0 D f  0 f d[0;1) and (q n ) stands for the sequence of denominators of ˛). It follows that Te2 i f is rigid and hence has a singular spectrum. B. Fayad [37] shows that this result is no longer true if one dimensional rotation is replaced by a multi-dimensional rotation (his counterexample is in the analytic class). See also [130] for the singularity of spectrum for functions f whose Fourier transform satisfies o(1/jnj) condition or to [84], where it is shown that sufficiently small variation implies singularity of the spectrum. A natural class of weighted operators arises when we consider Koopman operators of rotations on nil-manifolds. We only look at the particular example of such a rotation on a quotient of the Heisenberg group (R3 ; ) (a general spectral theory of nil-actions was mainly developed by W. Parry [157]) – these actions have countable Lebesgue spectrum in the orthocomplement of the subspace of eigenfunctions) that is take the nil-manifold R3 / Z3 on which we define the nil-rotation S((x; y; z) Z3 ) D (˛; ˇ; 0) (x; y; z) Z3 D x C ˛; y C ˇ; z C ˛ y Z3 , where ˛; ˇ and 1 are rationally independent. It can be shown that S is isomorphic to the skew product defined on [0; 1)2  T by   T' : (x; y; z) 7! x C ˛; y C ˇ; z  e2 i'(x;y) ; where '(x; y) D ˛ y  (x C ˛; y C ˇ) C (x; y) with (x; y) D x[y]. Since nil-cocycles can be considered as a certain analog of affine cocycles for one-dimensional rotations, it would be nice to explain to what kind of perturbations the Lebesgue spectrum property is stable. Yet another interesting problem which is related to the spectral theory of extensions given by so called Rokhlin cocycles (see Sect. “Rokhlin Cocycles”) arises, when given

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f : [0; 1) ! R, we want to describe spectrally the one-parameter set of weighted operators Wc :D Ve2 ic f ;T ; here T is a fixed irrational rotation by ˛. As proved by quite sophisticated arguments in [84], if we take f (x) D x then for all non-integer c 2 R the spectrum of Wc is continuous and singular (see also [68] and [145] where some special ˛’s are considered). It has been open for some time if at all one can find f : [0; 1) ! R such that for each c ¤ 0, the operator Wc has a Lebesgue spectrum. The positive answer is given in [205]: for example if f (x) D x (2C") (" > 0) and ˛ has bounded partial quotients then Wc has a Lebesgue spectrum for all c ¤ 0. All functions with such a property considered in [205] are non-integrable. It would be interesting to find an integrable f with the above property. We refer to [66] and the references therein for further results especially for transformations of the form (x; y) 7! (x C˛; 1[0;ˇ ) (x)C y) on [0; 1)Z/2Z. Recall however that earlier Katok and Stepin [104] used this kind of transformations to give a first counterexample to the Kolmogorov group property (see the glossary) for the spectrum. The Multiplicity Problem for Weighted Operators over Rotations In case of perturbations of affine cocycles, this problem was already raised by Kushnirenko [123]. Some significant results were obtained by M. Guenais. Before we state her results let us recall a useful criterion to find an upper bound for the multiplicity: If there exist c > 0 and a sequence (Fn )n1 of cyclic subspaces of H such that for each y 2 H, kyk D 1 we have lim inf n!1 kpro j F n yk2  c, then esssup(M U )  1/c which follows from a wellknown lemma of Chacon [23,26,111,127]. Using this and a technique which is close to the idea of local rank one (see [44,111]) M. Guenais [69] proved a series of results on multiplicity which we now list. Theorem 5 Assume that Tx D x C ˛ and let  : [0; 1) ! T be a cocycle. (i) If (x) D e2 i cx then M V ;T is bounded by jcj C 1. (ii) If  is absolutely continuous and  is of topological degree zero, then V;T has a simple spectrum. (iii) if  is of bounded variation, then M V ;T  max(2; 2 Var()/3). Remarks on the Banach Problem We already mentioned in Introduction the Banach problem in ergodic theory, which is simply the question whether there exists a Koopman representation for A D Z with simple Lebesgue spectrum. In fact no example of a Koopman representation with Lebesgue spectrum of fi-

nite multiplicity is known. Helson and Parry [75] asked for the validity of a still weaker version: Can one construct T such that U T has a Lebesgue component in its spectrum whose multiplicity is finite? Quite surprisingly in [75] they give a general construction yielding for each ergodic T a cocycle ' : X ! Z/2Z such that the unitary operator U T' has a Lebesgue spectrum in the orthocomplement of functions depending only on the X-coordinate. Then Mathew and Nadkarni [144] gave examples of cocycles over so called dyadic adding machine for which the multiplicity of the Lebesgue component was equal to 2. In [126] this was generalized to so called Toeplitz Z/2Z-extensions of adding machines: for each even number k we can find a two-point extension of an adding machine so that the multiplicity of the Lebesgue component is k. Moreover, it was shown that Mathew and Nadkarni’s constructions from [144] in fact are close to systems arising from number theory (like the famous Rudin– Shapiro sequence, e. g. [160]), relating the result about multiplicity of the Lebesgue component to results by Kamae [96] and Queffelec [160]. Independently of [126], Ageev [8] showed that one can construct 2-point extensions of the Chacon transformation realizing (by taking powers of the extension) each even number as the multiplicity of the Lebesgue component. Contrary to all previous examples, Ageev’s constructions are weakly mixing. Still an open question remains whether for A D Z one can find a Koopman representation with the Lebesgue component of multiplicity 1 (or even odd). In [70], M. Guenais studies the problem of Lebesgue spectrum in the classical case of Morse sequences (see [107] as well as [124], where the problem of spectral classification in this class is studied). All dynamical sytems arising from Morse sequences have simple spectra [124]. It follows that if a Lebesgue component appears in a Morse dynamical system, it has multiplicity one. Guenais [70] using a Riesz product technique relates the Lebesgue spectrum problem with the still open problem of whether a construction of “flat” trigonometric polynomials with coefficients ˙1 is possible. However, it is proved in [70] that such a construction can be carried out on some compact Abelian groups and it leads, for an Abelian countable torsion group A, to a construction of an ergodic action of A with simple spectrum and a Haar component in its spectrum. Lifting Mixing Properties We now give one more example of interactions between spectral theory and joinings (see Introduction) that gives rise to a quick proof of the fact that r-fold mixing prop-

Spectral Theory of Dynamical Systems

erty of T (r  2) lifts to a weakly mixing compact group extension T' (the original proof of this fact is due to D. Rudolph [175]). Indeed, to prove r-fold mixing for a mixing( = 2-mixing) transformation S (acting on (Y; C ; )) one has to prove that each weak limit of off-diagonal selfjoinings (given by powers of S, see  Joinings in Ergodic Theory) of order r is simply the product measure  ˝r . We need also a Furstenberg’s lemma [62] about relative unique ergodicity (RUE) of compact group extensions T' : If  ˝ G is an ergodic measure for T' then it is the only (ergodic) invariant measure for T' whose projection on the first coordinate is . Now the result about lifting r-fold mixing to compact group extensions follows directly from the fact that whenever T' is weakly mixing, ( ˝ G )˝r is an ergodic measure (this approach was shown to me by A. del Junco). In particular if T is mixing and T' is weakly mixing then for each  2 b G n f1g, the maximal spectral type of Vı';T is Rajchman. See Sect. “Rokhlin Cocycles” for a generalization of the lifting result to Rokhlin cocycle extensions. The Multiplicity Function In this chapter only A D Z is considered (for other groups, even for R, much less is known; see however the case of so called product Zd -actions [50]). Contrary to the case of maximal spectral type, it is rather commonly believed that there are no restrictions for the set of essential values of Koopman representations.

(H G is a closed subgroup and v is a continuous group automorphism of G). Then an algebraic lemma has been proved in [125] saying that each set M containing 1 is of the form M(G; v; H) and the techniques to construct “good” cocycles and a passage to “natural factors” yielded the following: For each M f1; 2; : : :g f1g containing 1 there exists an ergodic automorphim such that the set of essential values for its Koopman representation equals M. See also [166] for the case of non-Abelian group extensions. A similar in spirit approach (that means, a passage to a family of factors) is present in a recent paper of Ageev [13] in which he first applies Katok’s analysis (see [98,102]) for spectral multiplicities of the Koopman representation associated with Cartesian products T k for a generic transformation T. In a natural way this approach leads to study multiplicities of tensor products of unitary operators. Roughly, the multiplicity is computed as the number of atoms (counted modulo obvious symmetries) for conditional measures (see [98]) of a product measure over its convolution. Ageev [13] proved that for a typical automorphism T the set of the values of the multiplicity function for U T k equals fk; k(k  1); : : : ; k!g and then he just passes to “natural” factors for the Cartesian products by taking sets invariant under a fixed subgroup of permutations of coordinates. In particular, he obtains all sets of the form f2; 3; : : : ; ng on L20 . He also shows that such sets of multiplicities are realizable in the category of mixing transformations.

Cocycle Approach

Rokhlin’s Uniform Multiplicity Problem

We will only concentrate on some results of the last twenty years. In 1983, E.A. Robinson [164] proved that for each n  1 there exists an ergodic transformation whose maximal spectral multiplicity is n. Another important result was proved in [165] (see also [98]), where it is shown that given a finite set M N containing 1 and closed under the least common multiple one can find (even a weakly mixing) T so that the set of essential values of the multiplicity function equals M. This result was then extended in [67] to infinite sets and finally in [125] (see also [11]) to all subsets M N containing 1. In fact, as we have already noticed in the previous section the spectral theory for compact Abelian group extensions is reduced to a study of weighted operators and then to comparing maximal spectral types for such operators. This leads to sets of the form n M(G; v; H) D ](f ı v i : i 2 Zg \ anih(H)) : o  2 anih(H)

The Rokhlin multiplicity problem (see the recent book by Anosov [15]) was, given n  2, to construct an ergodic transformation with uniform multiplicity n on L20 . A solution for n D 2 was independently given by Ageev [9] and Ryzhikov [188] (see also [15] and [66]) and in fact it consists in showing that for some T (actually, any T with simple spectrum for which 1/2(Id CU T ) is in the weak operator closure of the powers of U T will do) the multiplicity of T  T is uniformly equal to 2 (see also Sect. “Future Directions”). In [12], Ageev proposed a new approach which consists in considering actions of “slightly non-Abelian” groups; and showing that for a “typical” action of such a group a fixed “direction” automorphism has a uniform multiplicity. Shortly after publication of [12], Danilenko [27], following Ageev’s approach, considerably simplified the original proof. We will present Danilenko’s arguments. Fix n  1. Denote e i D (0; : : : ; 1; : : : ; 0) 2 Z n ; i D 1; : : : ; n. We define an automorphism L of Zn setting

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L(e i ) D e iC1 , i D 1; : : : ; n  1 and L(e n ) D e 1 . Using L we define a semi-direct product G : D Zn Ì Z defining multiplication as (u; k)  (w; l) D (u C L k w; k C l). Put e0 D (0; 1), e i D (e i ; 0), i D 1; : : : ; n (and Le i D (Le i ; 0)). Moreover, denote e nC1 D e0n D (0; n). Notice that e0  e i  e01 D Le i for i D 1; : : : ; n (L(e nC1 ) D e nC1 ). Theorem 6 (Ageev, Danilenko) For every unitary representation U of G in a separable Hilbert space H, for which U e 1 L r e 1 has no non-trivial fixed points for 1  r < n, the essential values of the multiplicity function for U e nC1 are contained in the set of multiples of n. If, in addition, U e 0 has a simple spectrum, then U e nC1 has uniform multiplicity n. It is then a certain work to show that the assumption of the second part of the theorem is satisfied for a typical action of the group G. Using a special (C; F)-construction with all the cut-and-stack parameters explicit Danilenko [27] was also able to show that each set of the form k  M, where k  1 and M is an arbitrary subset of natural numbers containing 1, is realizable as the set of essential values of a Koopman representation. Some other constructions based on the solution of the Rokhlin problem for n D 2 and the method of [125] are presented in [103] leading to sets different than those pointed above; these sets contain 2 as their minimum. Rokhlin Cocycles We consider now a certain class of extensions which should be viewed as a generalization of the concept of compact group extensions. We will focus on Z-actions only. Assume that T is an ergodic automorphism of (X, B; ). Let G be a l.c.s.c. Abelian group. Assume that this group acts on (Y; C ; ), that is we have a G-action S D (S g ) g2G on (Y; C ; ). Let ' : X ! G be a cocycle. We then define an automorphism T';S of the space (X  Y; B ˝ C ;  ˝ ) by T';S (x; y) D (Tx; S'(x) (y)): Such an extension is called a Rokhlin cocycle extension (the map x 7! S'(x) is called a Rokhlin cocycle). Such an operation generalizes the case of compact group extensions; indeed, when G is compact the action of G on itself by rotations preserves Haar measure. (It is quite surprising, that when only we admit G nonAbelian, then, as shown in [28], each ergodic extension of T has a form of a Rokhlin cocycle extension.) Ergodic and spectral properties of such extensions are examined in several papers: [63,65,129,131,132,133,167,176]. Since

in these papers rather joining aspects are studied (among other things in [129] Furstenberg’s RUE lemma is generalized to this new context), we will mention here only few results, mainly spectral, following [129] and [133]. We will constantly assume that G is non-compact. As ' : X ! G is then a cocycle with values in a non-compact group, the theory of such cocycles is much more complicated (see e. g. [193]), and in fact the theory of Rokhlin cocycle extensions leads to interesting interactions between classical ergodic theory, the theory of cocycles and the theory of non-singular actions arising from cocycles taking values in non-compact groups – especially, the Mackey action associated to ' plays a crucial role here (see the problem of invariant measures for T';S in [132] and [28]); see also monographs [1,98,101,193]. Especially, two Borel subgroups of b G are important here: G :  ı ' D c  / ı T for a measurable ˙' D f 2 b  : X ! T and c 2 T g : and its subgroup ' given by c D 1. ' turns out to be the group of L1 -eigenvalues of the Mackey action (of G) associated to the cocycle '. This action is the quotient action of the natural action of G (by translations on the second coordinate) on the space of ergodic components of the skew product T' – the Mackey action is (in general) not measure-preserving, it is however non-singular. We refer the reader to [2,78,147] for other properties of those subgroups. Theorem 7 ([132,133]) R (i) T';S j L 2 (XY;˝)L 2 (X;) D Gˆ Vı';T dS . (ii) T';S is ergodic if and only if T is ergodic and S (' ) D 0. (iii) T';S is weakly mixing if and only if T is weakly mixing and S has no eigenvalues in ˙' . (iv) if T is mixing, S is mildly mixing, ' is recurrent and not cohomologous to a cocycle with values in a compact subgroup of G then T';S remains mixing. (v) If T is r-fold mixing, ' is recurrent and T';S is mildly mixing then T';S is also r-fold mixing. (vi) If T and R are disjoint, the cocycle ' is ergodic and S is mildly mixing then T';S remains disjoint with R. Let us recall [61,195] that an A-action S D (S a ) a2A is mildly mixing (see the glossary) if and only if the A-action (S a   a ) a2A remains ergodic for every properly ergodic non-singular A-action  D ( a ) a2A . Coming back to Smorodinsky–Thouvenot’s result about factors of ergodic self-joinings of a Bernoulli automorphism we would like to emphasize here that the disjointness result (vi) above was used in [132] to give

Spectral Theory of Dynamical Systems

an example of an automorphism which is disjoint from all weakly mixing transformations but which has an ergodic self-joining whose associated automorphism has a non-trivial weakly mixing factor. In a sense this is opposed to Smorodinsky–Thouvenot’s result as here from self-joinings we produced a “more complicated” system (namely the weakly mixing factor) than the original system. It would be interesting to develop the theory of spectral multiplicity for Rokhlin cocycle extensions as it was done in the case of compact group extensions. However a difficulty is that in the compact group extension case we deal with a countable direct sum of representations of the form Vı';T while in the non-compact case we have to consider an integral of such representations. Rank-1 and Related Systems Although properties like mixing, weak (and mild) mixing as well as ergodicity, are clearly spectral properties, they have “good” measure-theoretic formulations (expressed by a certain behavior on sets). Simple spectrum property is another example of a spectral property, and it was a popular question in the 1980s whether simple spectrum property of a Koopman representation can be expressed in a more “measure-theoretic” way. We now recall rank1 concept which can be seen as a notion close to Katok’s and Stepin’s theory of cyclic approximation [104] (see also [26]). Assume that T is an automorphism of a standard probability Borel space (X; B; ). T is said to have rank one property if there exists an increasing sequence of Rokhlin towers tending to the partition into points (a Rokhlin tower is a family fF; TF; : : : ; T n1 Fg of pairwise disjoint sets, while “tending to the partition into points” means that we can approximate every set in B by unions of levels of towers in the sequence). Baxter [20] showed that the maximal spectral type of such a T is realized by a characteristic function. Since the cyclic space generated by the characteristic function of the base contains characteristic functions of all levels of the tower, by the definition of rank one, the increasing sequence of cyclic spaces tends to the whole L2 -space, therefore rank one property implies simplicity of the spectrum for the Koopman representation. It was a question for some time whether rank-1 is just a characterization of simplicity of the spectrum, disproved by del Junco [88]. We refer the reader to [46] as a good source for basic properties of rank-1 transformations. Similarly to the rank one property, one can define finite rank automorphisms (simply by requiring that an approximation is given by a sequence of a fixed number of

towers) – see e. g. [152], or even, a more general property, namely the local rank one property can be defined, just by requiring that the approximating sequence of single towers fills up a fixed fraction of the space (see [44,111]). Local rank one (so the more finite rank) property implies finite multiplicity [111]. Mentzen [146] showed that for each n  1 one can construct an automorphism with simple spectrum and having rank n; in [138] there is an example of a simple spectrum automorphism which is not of local rank one. Ferenczi [45] introduced the notion of funny rank one (approximating towers are Rokhlin towers with “holes”). Funny rank one also implies simplicity of the spectrum. An example is given in [45] which is even not loosely Bernoulli (see Sect. “Inducing and Spectral Theory”, we recall that local rank one property implies loose Bernoullicity [44]). The notion of AT (see the glossary) has been introduced by Connes and Woods [25]. They proved that AT property implies zero entropy. They also proved that funny rank one automorphisms are AT. In [32] it is proved that the system induced by the classical Morse-Thue system is AT (it is an open question by S. Ferenczi whether this system has funny rank one property). A question by Dooley and Quas is whether AT implies funny rank one property. AT property implies “simplicity of the spectrum in L1 ” which we already considered in Introduction (a “generic” proof of this fact is due to J.-P. Thouvenot). A persistent question was formulated in the 1980s whether rank one itself is a spectral property. In [49] the authors maintained that this is not the case, based on an unpublished preprint of the first named author of [49] in which there was a construction of a Gaussian–Kronecker automorphism (see Sect. “Spectral Theory of Dynamical Systems of Probabilistic Origin”) having rank-1 property. This latter construction turned out to be false. In fact de la Rue [181] proved that no Gaussian automorphism can be of local rank one. Therefore the question whether: Rank one is a spectral property remains one of the interesting open questions in that theory. Downarowicz and Kwiatkowski [33] proved that rank-1 is a spectral property in the class of systems generated by generalized Morse sequences. One of the most beautiful theorems about rank-1 automorphisms is the following result of J. King [110] (for a different proof see [186]). Theorem 8 (WCT) If T is of rank one then for each element S of the centralizer C(T) of T there exists a sequence n (n k ) such that U T k ! U S strongly. A conjecture of J. King is that in fact for rank-1 automorphisms each indecomposable Markov operator J D J

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( 2 J2e (T)) is a weak limit of powers of U T (see [112], also [186]). To which extent the WCT remains true for actions of other groups is not clear. In [214] the WCT is proved in case of rank one flows, however the main argument seems to be based on the fact that a rank one flow has a non-zero time automorphism Tt0 which is of rank one, which is not true. After the proof of the WCT by Ryzhikov in [186] there is a remark that the rank one flow version of the theorem can be proved by a word for word repetition of the arguments. He also proves that if the flow (Tt ) t2R is mixing, then T1 does not have finite rank. On the other hand, for A D Z2 , Downarowicz and Kwiatkowski [34] gave recently a counterexample to the WCT. Even though it looks as if rank one construction is not complicated, mixing in this class is possible; historically the first mixing constructions were given by D. Ornstein [151] in 1970, using probability type arguments for a choice of spacers. Once mixing was shown, the question arose whether absolutely continuous spectrum is also possible, as this would give automatically the positive answer to the Banach problem. However Bourgain [21], relating spectral measures of rank one automorphisms with some classical constructions of Riesz product measures, proved that a certain subclass of Ornstein’s class consists of automorphisms with singular spectrum (see also [5] and [6]). Since in Ornstein’s class spacers are chosen in a certain “non-constructive” way, quite a lot of attention was devoted to the rank one automorphism defined by cutting a tower at the nth step into r n D n subcolumns of equal “width” and placing i spacers over the ith subcolumn. The mixing property conjectured by M. Smorodinsky, was proved by Adams [7] (in fact Adams proved a general result on mixing of a class of staircase transformations). Spectral properties of rank-1 transformations are also studied in [114], where the authors proved that P 2 D C1 (r stands for the numwhenever 1 n nD1 r n ber of subcolumns at the nth step of the construction of a rank-1 automorphism) then the spectrum is automatically singular. H. Abdalaoui [5] gives a criterion for singularity of the spectrum of a rank one transformation; his proof uses a central limit theorem. It seems that still the question whether rank one implies singularity of the spectrum remains the most important question of this theory. We have already seen in Sect. “Spectral Theory of Weighted Operators” that for a special class of rank one systems, namely those with discrete spectra ([87]), we have a nice theory for weighted operators. It would be extremely interesting to find a rank one automorphism with continuous spectrum for which a substitute of Helson’s analysis exists.

B. Fayad [39] constructs a rank one differentiable flow, as a special flow over a two-dimensional rotation. In [40] he gives new constructions of smooth flows with singular spectra which are mixing (with a new criterion for a Rajchman measure to be singular). In [35] a certain smooth change of time for an irrational flows on the 3-torus is given, so that the corresponding flow is partially mixing and has the local rank one property. Spectral Theory of Dynamical Systems of Probabilistic Origin Let us just recall that when (Yn )1 nD1 is a stationary process then its distribution  on RZ is invariant under the shift S on RZ : S((x n )n2Z ) D (y n )n2Z , where y n D x nC1 ; n 2 Z. In this way we obtain an automorphism S defined on (RZ ; B(RZ ); ). For each automorphism T we can find f : X ! R measurable such that the smallest  -algebra making the stationary process ( f ı T n )n2Z measurable is equal to B, therefore, for the purpose of this article, by a system of probabilistic origin we will mean (S; ) obtained from a stationary infinitely divisible process (see e. g. [142,192]). In particular, the theory of Gaussian dynamical systems is indeed a classical part of ergodic theory (e. g. [149,150,211,212]). If (X n )n2Z is a stationary real centered Gaussian process and  denotes the spectral measure of the process, i. e. b  (n) D E(X n  X 0 ), n 2 Z, then by S D S we denote the corresponding Gaussian system on the shift space (recall also that for each symmetric measure  on T there is exactly one stationary real centered Gaussian process whose spectral measure is  ). Notice that if  has an atom, then in the cyclic space generated by X0 there exists an eigenfunction Y for S – if now S were ergodic, jYj would be a constant function which is not possible by the nature of elements in Z(X 0 ). In what follows we assume that  is continuous. It follows that U S  restricted to Z(X 0 ) is spectrally the same as V D V  acting on L2 (T ;  ), and we obtain that (U S  ; L2 (RZ ;  )) can be represented as the symmetric Fock space built over H D L2 (T ;  ) and U S  D F(V) – see the glossary (H ˇn is called the n-th chaos). In other words the spectral theory of Gaussian dynamical systems is reduced to the spectral theory of special tensor products unitary operators. Classical results (see [26]) which can be obtained from this point of view are for example the following: (A) ergodicity implies weak mixing, (B) the multiplicity function is either 1 or is unbounded, (C) the maximal spectral type of U S  is equal to exp( ), hence Gaussian systems enjoy the Kolmogorov group property.

Spectral Theory of Dynamical Systems

However we can also look at a Gaussian system in a different way, simply by noticing that the variables e2 i f (f is a real variable), where f 2 Z(X 0 ) generate L2 (RZ ;  ). Now calculating the spectral measure of e2 i f is not difficult and we obtain easily (C). Moreover, integrals of type R 2 i f 2 i f ıT n 2 i f ıT nCm 0e 1 2 e d can also be calculated, e whence in particular we easily obtain Leonov’s theorem on the multiple mixing property of Gaussian systems [141]. One of the most beautiful parts of the theory of Gaussian systems concerns ergodic properties of S when  is concentrated on a thin Borel set. Recall that a closed subset K T is said to be a Kronecker set if each f 2 C(K) is a uniform limit of characters (restricted to K). Each Kronecker set has no rational relations. Gaussian–Kronecker automorphisms are, by definition, those Gaussian systems for which the measure  (always assumed to be continuous) is concentrated on K [ K, K a Kronecker set. The following theorem has been proved in [51] (see also [26]).

if for each ergodic self-joining 2 J2e (S ) and arbitrary f ; g 2 H the variable

Theorem 9 (Foia¸s–Stratila Theorem) If T is an ergodic automorphism and f is a real-valued element of L20 such that the spectral measure  f is concentrated on K [ K, where K is a Kronecker set, then the process ( f ı T n )n2Z is Gaussian.

(: : : ; x1 ; x0 ; x1 ; : : :) 7! (: : : ; x1 ; x0 ; x1 ; : : :)

This theorem is indeed striking as it gives examples of weakly mixing automorphisms which are spectrally determined (like rotations). A relative version of the Foia¸s– Stratila Theorem has been proved in [129]. The Foia¸s–Stratila Theorem implies that whenever a spectral measure  is Kronecker, it has no realization of the form  f with f bounded. We will see however in Sect. “Future Directions” that for some automorphisms T (having the SCS property) the maximal spectral type T has the property that ST has a simple spectrum. Gaussian–Kronecker automorphisms are examples of automorphisms with simple spectra. In fact, whenever  is concentrated on a set without rational relations, then S has a simple spectrum (see [26]). Examples of mixing automorphisms with simple spectra are known [149], however it is still unknown (Thouvenot’s question) whether the Foia¸s–Stratila property may hold in the mixing class. F. Parreau [154] using independent Helson sets gave an example of mildly mixing Gaussian system with the Foia¸s– Stratila property. In [165] there is a remark that the set of finite essential values of the multiplicity function of U S  forms a (multiplicative) subsemigroup of N. However, it seems that there is no “written” proof of this fact. Joining theory of a class of Gaussian system, called GAG, is developed in [136]. A Gaussian automorphism S with the Gaussian space H L20 (RZ ;  ) is called a GAG

(RZ  RZ ; ) 3 (x; y) 7! f (x) C g(y) is Gaussian. For GAG systems one can describe the centralizer and factors, they turn out to be objects close to the probability structure of the system. One of the crucial observations in [136] was that all Gaussian systems with simple spectrum are GAG. It is conjectured (J.P. Thouvenot) that in the class of zero entropy Gaussian systems the PID property holds true. For the spectral theory of classical factors of a Gaussian system see [137]; also spectrally they share basic spectral properties of Gaussian systems. Recall that historically one of the classical factors namely the  -algebra of sets invariant for the map

was the first example with zero entropy and countable Lebesgue spectrum (indeed, we need a singular measure  such that   is equivalent to Lebesgue measure [150]). For factors obtained as functions of a stationary process see [83]. T. de la Rue [181] proved that Gaussian systems are never of local rank-1, however his argument does not apply to classical factors. We conjecture that Gaussian systems are disjoint from rank-1 automorphisms (or even from local rank-1 systems). We now turn the attention to Poissonian systems (see [26] for more details). Assume that (X; B; ) is a standard Borel space, where  is infinite,  -finite. The new configuration space e X is taken as the set of all countable subsets fx i : i  1g of X. Once a set A 2 B, of finite measure is given one can define a map N A : e X ! N([f1g) just counting the number of elements belongB; e ) is ing to A. The measure-theoretic structure (e X; e given so that the maps N A become random variables with Poisson distribution of parameter (A) and such that whenever A1 ; : : : ; A k X are of finite measure and are pairwise disjoint then the variables N A 1 ; : : : ; N A k are independent. Assume now that T is an automorphism of (X; B; ). B; e ) It induces a natural automorphism on the space (e X; e defined by e T(fx i : i  1g D fTx i : i  1g. The automorphism e T is called the Poisson suspension of T (see [26]). Such a suspension is ergodic if and only if no set of positive and finite -measure is T-invariant. Moreover ergodicity of e T implies weak mixing. In fact the spectral structure of Ue T is very similar to the Gaussian one: namely the first

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chaos equals L2 (X; B; ) (we emphasize that this is about the whole L2 and not only L20 ) on which Ue T acts as U T and X; e ) together with the action of Ue the L2 (e T has the structure of the symmetric Fock space F(L2 (X; B; )) (see the glossary). We refer to [22,86,168,169] for ergodic properties of systems given by symmetric ˛-stable stationary processes, or more generally infinitely divisible processes. Again, they share spectral properties similar to the Gaussian case: ergodicity implies weak mixing, while mixing implies mixing of all orders. In [171], E. Roy clarifies the dynamical “status” of such systems. He uses Poisson suspension automorphisms and the Maruyama representation of an infinitely divisible process mixed with basic properties of automorphisms preserving infinite measure (see [1]) to prove that as a dynamical system, a stationary infinitely divisible process (without the Gaussian part), is a factor of the Poisson suspension over the Lévy measure of this process. In [170] a theory of ID-joinings is developed (which should be viewed as an analog of the GAG theory in the Gaussian class). Parreau and Roy [155] give an example of a Poisson suspension with a minimal possible set of ergodic self-joinings. Many natural problems still remain open here, for example (assuming always zero entropy of the dynamical system under consideration): Are Poisson suspensions disjoint from Gaussian systems? What is the spectral structure for dynamical systems generated by symmetric ˛stable process? Are such systems disjoint whenever ˛1 ¤ ˛2 ? Are Poissonian systems disjoint from local rank one automorphisms (cf. [181])? In [91] it is proved that Gaussian systems are disjoint from so called simple systems (see  Joinings in Ergodic Theory and [93,208]); we will come back to an extension of this result in Sect. “Future Directions”. It seems that flows of probabilistic origin satisfy the Kolmogorov group property for the spectrum. One can therefore ask how different are systems satisfying the Kolmogorov group property from systems for which the convolutions of the maximal spectral type are pairwise disjoint (see also Sect. “Future Directions” and the SCS property). We also mention here another problem which will be taken up in Sect. “Special Flows and Flows on Surfaces, Interval Exchange Transformations” – Is it true that flows of probabilistic origin are disjoint from smooth flows on surfaces? Recently A. Katok and A. Windsor announced that it is possible to construct a Kronecker measure so that the corresponding Gaussian system (Z-action (!)) has a smooth representation on the torus. Yet one more (joining) property seems to be characteristic in the class of systems of probabilistic origin, namely they satisfy so called ELF property (see [30] and  Join-

ings in Ergodic Theory). Vershik asked whether the ELF property is spectral – however the example of a cocycle from [205] together with Theorem 7 (i) yields a certain Rokhlin extension of a rotation which is ELF and has countable Lebesgue spectrum in the orthocomplement of the eigenfunctions (see [206]); on the other hand any affine extension of that rotation is spectrally the same, while it cannot have the ELF property. Prikhodko and Thouvenot (private communication) have constructed weakly mixing and non-mixing rank one automorphisms which enjoy the ELF property. Inducing and Spectral Theory Assume that T is an ergodic automorphism of a standard probability Borel space (X; B; ). Can “all” dynamics be obtained by inducing (see the glossary) from one fixed automorphism was a natural question from the very beginning of ergodic theory. Because of Abramov’s formula for entropy h(TA ) D h(T)/(A) it is clear that positive entropy transformations cannot be obtained from inducing on a zero entropy automorphism. However here we are interested in spectral questions and thus we ask how many spectral types we obtain when we induce. It is proved in [59] that the family of A 2 B for which TA is mixing is dense for the (pseudo) metric d(A1 ; A2 ) D (A1 4A2 ). De la Rue [182] proves the following result: For each ergodic transformation T of a standard probability space (X; B; ) the set of A 2 B for which the maximal spectral type of U TA is Lebesgue is dense in B. The multiplicity function is not determined in that paper. Recall (without giving a formal definition, see [152]) that a zero entropy automorphism is loosely Bernoulli (LB for short) if and only if it can be induced from an irrational rotation (see also [43,99]). The LB theory shows that not all dynamical systems can be obtained by inducing from an ergodic rotation. However an open question remained whether LB systems exhaust spectrally all Koopman representations. In a deep paper [180], de la Rue studies LB property in the class of Gaussian– Kronecker automorphisms, in particular he constructs S which is not LB. Suppose now that T is LB and for some A 2 B, U TA is isomorphic to U S . Then by the Foia¸s– Stratila Theorem, TA is isomorphic to S, and hence TA is not LB. However an induced automorphism from an LB automorphism is LB, a contradiction. Special Flows and Flows on Surfaces, Interval Exchange Transformations We now turn our attention to flows. The cases of the geodesic flow, horocycle flows on homogenous spaces of SL(2; R) and nilflows are classical (we refer the reader

Spectral Theory of Dynamical Systems

to [105] with a nice description of the first two cases, while for nilflows we refer to [157]: these classes of flows on homogenous spaces have countable Lebesgue spectrum, in the third case – in the orthocomplement of the eigenspace). On the other hand the classical cyclic approximation theory of Katok and Stepin [104] (see [26]) leads to examples of smooth flows on the torus with simple continuous singular spectra. Given an ergodic automorphism T on (X; B; ) and a positive integrable function f : X ! RC consider the corresponding special flow T f (see the glossary). Obviously, such a flow is ergodic. Special flows were introduced to ergodic theory by von Neumann in his fundamental work [148] in 1932. Also in that work he explains how to compute eigenvalues for special flows, namely: r 2 R is an eigenvalue of T f if and only if the following functional equation e2 ir f (x) D

(x) (Tx)

has a measurable solution  : X ! T . We recall also that the classical Ambrose-Kakutani theorem asserts that practically each ergodic flow has a special representation ([26], see also Rudolph’s theorem on special representation therein). A classical situation when we obtain “natural” special representations is while considering smooth flows on surfaces (we refer the reader to Hasselblatt’s and Katok’s monograph [73]). They have transversals on which the Poincaré map is piecewise isometric, and this entails a study of interval exchange transformations (IET), see [26,108,163]. Formally, to define IET of m intervals we need a permutation  of f1; : : : ; mg and a probability vector  D (1 ; : : : ; m ) (with positive entries). Then we define T D T; of [0; 1) by putting T; (x) D x C ˇ  i  ˇ i for x 2 [ˇ i ; ˇ iC1 ) ; P P  where ˇ i D j
a better understanding of the dynamics of the Rauzy map Veech [209] introduced the space of zippered rectangles. A zippered rectangle associated to the Rauzy class R is m , h 2 Rm , a 2 Rm , a quadruple (; h; a; ), where  2 RC C C  2 R and the vectors h and a satisfy some equations and inequalities. Every zippered rectangle (; h; a; ) determines a Riemann structure on a compact connected surface. Denote by ˝(R) the space of all zippered rectangles, corresponding to a given Rauzy class R and satisfying the condition h; hi D 1. In [209], Veech defined a flow (P t ) t2R on the space ˝(R) putting P t (; h; a; ) D (e t ; et h; et a; ) and extended the Rauzy map. On so called Veech moduli space of zippered rectangles, the flow (P t ) becomes the Teichmüller flow and it preserves a natural Lebesgue measure class; by passing to a transversal and projecting the measure on the space of IETs R  m1 Veech has proved the following fundamental theorem ([209], see also [143]) which is a generalization of the fact that Gauss measure 1/(ln 2)1/(1 C x)dx is invariant for the Gauss map which sends t 2 (0; 1) into the fractional part of its inverse. Theorem 10 (Veech, Masur, 1982) Assume that R is a Rauzy class. There exists a  -finite measure R on R  m1 which is P -invariant, equivalent to “Lebesgue” measure, conservative and ergodic. In [209] it is proved that a.e. (in the above sense) IET is then of rank one (and hence is ergodic and has a simple spectrum). He also formulated as an open problem whether we can achieve the weak mixing property a.e. This has been recently answered in positive by A. Avila and G. Forni [19] (for  which is not a rotation). Katok [100] proved that IET have no mixing factors (in fact his proof shows more: the IET class is disjoint with the class of mixing transformations). By their nature, IET transformations are of finite rank (see [26]) so they are of finite multiplicity. They need not be of simple spectrum (see remarks in [105] pp. 88–90). It remains an open question whether an IET can have a non-singular spectrum. Joining properties in the class of exchange of 3 and more intervals are studied in [47,48]. An important question of Veech [208] whether a.e. IET is simple is still open. When we consider a smooth flow on a surface preserving a smooth measure, whose only singularity (we assume that we have only finitely many singularities) are simple (non-degenerated) saddles then such a flow has a special representation over an interval exchange automorphism under a smooth function which has finitely many logarithmic singularities (see [73]). In the article by

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Arnold [18] the quasi-periodic Hamiltonian case is considered: H : R2 ! R satisfies H(xCm; yCn) D H(x; y)C n˛1 Cm˛2 , ˛1 /˛2 … Q, and we then consider the following system of differential equations on T 2 dx @H dy @H D ; D : dt @y dt @x

(2)

As Arnold shows the dynamical system arising from the system (2) has one ergodic component which has a special representation over the irrational rotation by ˛ :D ˛1 /˛2 under a smooth function with finitely many logarithmic singularities (all other ergodic components are periodic orbits and separatrices). By changing a speed as it is done in [57] so that critical points of the vector field in (2) become singular points, Arnold’s special representation is transformed to a special flow over the same irrational rotation however under a piecewise smooth function. If the sum of jumps is not zero then in fact we come back to von Neumann’s class of special flows considered in [148]. Similar classes of special flows (when the roof function is of bounded variation) are obtained from ergodic components of flows associated to billiards in convex polygones with rational angles [106]. Kochergin [115] showed that special flows over irrational rotations and under bounded variation functions are never mixing. This has been recently strengthened in [54] to the following: If T is an irrational rotation and f is of bounded variation then the special flow T f is spectrally disjoint from all mixing flows. In particular all such flows have singular spectra. Moreover, in [54] it is proved that whenever the Fourier transform of the roof function f is of order O(1/n) then T f is disjoint from all mixing flows (see also [55]). In fact in the papers [54,55,56,57] the authors discuss the problem of disjointness of those special flows with all ELF-flows conjecturing that no flow of probabilistic origin has a smooth realization on a surface. In [140] the analytic case is considered leading to a “generic” result on disjointness with the ELF class generalizing the classical Shklover’s result on the weak mixing property [197]. Kochergin [117] proved the absence of mixing for flows where the roof function has finitely many singularities, whenever the sum of “left logarithmic speeds” and the sum of “right logarithmic speeds” are equal – this is called a symmetric logarithmic case, however some Diophantine restriction is put on ˛. In [128], where also the absence of mixing is considered for the symmetric logarithmic case, it was conjectured (and proved for arbitrary rotation) that a necessary condition for mixing of a special flow T f (with arbitrary T and f ) is the condition that the sequence of distributions (( f0(n) ) )n tends to ı1 in the space of prob-

ability measures on R. K. Schmidt [194] proved it using the theory of cocycles and extending a result from [3] on tightness of cocycles. A. Katok [100] proved the absence of mixing for special flows over IET when the roof function is of bounded variation (see also [187]). Katok’s theorem was strengthened in [56] to the disjointness theorem with the class of mixing flows. On the other hand there is a lot of (difficult) results pointing out classes of special flows over irrational rotations which are mixing, especially (but not only) in the class of non-symmetric logarithmic singularities: [36,38] (B. Fayad was able to give a speed of convergence to zero for Fourier coefficients), [109,119,120]. Recently mixing property has been proved in a non-symmetric case in [203] when the base transformation is a special class of IETs. The eigenvalue problem (mainly how many frequencies can have the group of eigenvalues) for special flows over irrational rotations is studied in [41,42,71]. A. Avila and G. Forni [19] proved that a.e. translation flow on a surface (of genus at least two) is weakly mixing (which is a drastic difference with the linear flow case of the torus, where the spectrum is always discrete). The problem of whether mixing flows indicated in this chapter are mixing of all orders is open (it is also unknown whether they have singular spectra). One of several possible approaches (proposed by B. Fayad and J.-P. Thouvenot) toward positive solution of this problem would be to show that such flows enjoy so called Ratner’s property (R-property). This property may be viewed as a particular way of divergence of orbits of close points; it was shown to hold for horocycle flows by M. Ratner [162]. We refer the reader to [162] and the survey article [201] for the formal definitions and basic consequences of R-property. In particular, R-property implies “rigidity” of joinings and it also implies the PID property; hence mixing and Rproperty imply mixing of all orders. In [57,58] a version of R-property is shown for the class of von Neumann special flows (however ˛ is assumed to have bounded partial quotients). This allowed one to prove there that such flows are even mildly mixing (mixing is excluded by a Kochergin’s result). We conjecture that an R-property may also hold for special flows over multidimensional rotations with roof functions given by nil-cocycles which we mentioned in Sect. “Spectral Theory of Weighted Operators”. If indeed the R-property is ubiquitous in the class of smooth flows on surfaces it may also be useful to show that smooth flows on surfaces are disjoint with flows of probabilistic origin – see [91,92,135,190,202]. B. Fayad [40] gives a criterion that implies singularity of the maximal spectral type for a dynamical system

Spectral Theory of Dynamical Systems

on a Riemannian manifold. As an application he gives a class of smooth mixing flows (with singular spectra) on T 3 obtained from linear flows by a time change (again this is a drastic difference with dimension two, where a smooth time change of a linear flow leads to non-mixing flows [26]). The spectral multiplicity problem for special flows (with sufficiently regular roof functions) over irrational rotations seems to be completely untouched (except for the case of a sufficiently smooth f – the spectrum of T f is then simple [26]). It would be nice to have examples of such flows with finite bigger than one multiplicity. In particular, is it true that the von Neumann class of special flows have finite multiplicity? This was partially solved by A. Katok (private communication) on certain subspaces in L2 , but not on the whole L2 -space. Problem. Given Tx D x C ˛ (with ˛ irrational) can we find f : [0; 1) ! RC sufficiently regular (e. g. with finitely many “controllable” singularities) such that T f has a Lebesgue spectrum? Of course the above is related to the question whether at all one can find a smooth flow on a surface with a Lebesgue spectrum (for Z-actions we can even see positive entropy diffeomorphisms on the torus). We mention at the end that if we drop here (and in other problems) the assumption of regularity of f then the answers will be always positive because of the LB theory; in particular there is a section of any horocycle flow (it has the LB property [161]) such that in the corresponding special representation T f the map T is an irrational rotation. Using a Kochergin’s result [118] on cohomology (see also [98,176]) the L1 -function f is cohomologous to a positive function g which is even continuous, thus T f is isomorphic to T g . Future Directions We have already seen several cases where spectral properties interact with measure-theoretic properties of a system. Let us mention a few more cases which require further research and deeper understanding. We recall that the weak mixing property can be understood as a property complementary to discrete spectrum (more precisely to the distality [62]), or similarly mild mixing property is complementary to rigidity. This can be phrased quite precisely by saying that T is not weakly (mildly) mixing if and only if it has a non-trivial factor with discrete spectrum (it has a non-trivial rigid factor). It has been a question for quite a long time if in a sense mixing can be “built” on the same principle. In other words we

seek a certain “highly” non-mixing factor. It was quite surprising when in 2005 F. Parrreau (private communication) gave the positive answer to this problem. Theorem 11 (Parreau) Assume that T is an ergodic automorphism of a standard probability space (X; B; ). Assume moreover that T is not mixing. Then there exists a non-trivial factor (see below) of T which is disjoint with all mixing automorphisms. In fact, Parreau proved that each factor of T given by B1 ( ) (this  -algebra is described in [136]), where n U T k ! J , is disjoint from all mixing transformations. This proof leads to some other results of the same type, for example: Assume that T is an ergodic automorphism of a standard probability space. Assume that there exists a non-trivial automorphism S with a singular spectrum which is not disjoint with T. Then T has a non-trivial factor which is disjoint with any automorphism with a Lebesgue spectrum. The problem of spectral multiplicity of Cartesian products for “typical” transformation studied by Katok [98] and then its solution in [13] which we already considered in Sect. “The Multiplicity Function” lead to a study of those T for which (CS)

 (m) ?  (n) whenever m ¤ n ;

where  D T just stands for the reduced maximal spectral type of U T (which is constantly assumed to be a continuous measure), see also Stepin’s article [199]. The usefulness of the above property (CS) in ergodic theory was already shown in [90], where a spectral counterexample machinery was presented using the following observation: If A is a T 1 -invariant sub- -algebra such that the maximal spectral type on L2 (A) is absolutely continuous with respect to T then A is contained in one of the coordinate sub- -algebras B. Based on that in [90] it is shown how to construct two weakly isomorphic action which are not isomorphic or how to construct two nondisjoint automorphisms which have no common nontrivial factors (such constructions were previously known for so called minimal self-joining automorphisms [174]). See also [200] for extensions of those results to Zd -actions. Prikhodko and Ryzhikov [159] proved that the classical Chacon transformation enjoys the (CS) property. The SCS property defined in the glossary is stronger than the (CS) condition above; the SCS property implies that the corresponding Gaussian system ST has a simple spectrum. Ageev [10] shows that Chacon’s transformation satisfies the SCS property; moreover in [13] he shows that the SCS property is satisfied generically and he gives a construction of a rank one mixing SCS-system (see also [191]).

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In [134] it is proved that some special flows considered in Sect. “Special Flows and Flows on Surfaces, Interval Exchange Transformations” (including the von Neumann class, however with ˛ having unbounded partial quotients) have the SCS property. Since the corresponding Gaussian systems have simple spectra, it would be interesting to decide whether T (for an SCS-automorphism) can be concentrated on a set without rational relations. It is quite plausible that the SCS property is commonly seen for smooth flows on surfaces. Katok and Thouvenot (private communication) considered systems called infinitely divisible. These are systems T on (X; B; ) which have a family of factors B! inS n dexed by ! 2 1 nD0 f0; 1g (B" D B) such that B!0 ? B!1 ; B!0 _ B!1 D B! and for each  2 f0; 1gN , \n2N B[0;n] D f;; Xg. They showed (unpublished) that there are discrete spectrum transformations which are ID, and that there are rank one transformations with continuous spectra which are also ID (clearly Gaussian systems are ID). It was until recently that a relationship between ID automorphisms and systems coming from stationary ID processes was unclear. In [135] it is proved that dynamical systems coming from stationary ID processes are factors of ID automorphisms; moreover, ID automorphisms are disjoint with all systems having the SCS property. It would be nice to decide whether Koopman representations associated to ID automorphisms satisfy the Kolmogorov group property.

Acknowledgments Research supported by the EU Program Transfer of Knowledge “Operator Theory Methods for Differential Equations” TODEQ and the Polish Ministry of Science and High Education.

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Stability and Feedback Stabilization

Stability and Feedback Stabilization EDUARDO D. SONTAG Department of Mathematics, Rutgers University, New Brunswick, USA Article Outline Glossary Definition of the Subject Introduction Linear Systems Nonlinear Systems: Continuous Feedback Discontinuous Feedback Sensitivity to Small Measurement Errors Future Directions Bibliography Glossary Stability A globally asymptotically stable equilibrium is a state with the property that all solutions converge to this state, with no large excursions. Stabilization A system is stabilizable (with respect to a given state) if it is possible to find a feedback law that renders that state a globally asymptotically stable equilibrium. Lyapunov and control-Lyapunov functions A controlLyapunov functions is a scalar function which decreases along trajectories, if appropriate control actions are taken. For systems with no controls, one has a Lyapunov function. Definition of the Subject The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it represents a first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or the attainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring the stabilization of some quantity (typically, some sort of “error” signal). A very special case, when there are no inputs, is that of stability. Introduction This article discusses the problem of stabilization of an equilibrium, which we take without loss of generality to be the origin, for a finite-dimensional system x˙ D f (x; u).

The objective is to find a feedback law u D k(x) which renders the origin of the “closed-loop” system x˙ D f (x; k(x)) globally asymptotically stable. The problem of stabilization of equilibria is one of the central issues in control. In addition to its intrinsic interest, it represents a first step towards the solution of more complicated problems, such as the stabilization of periodic orbits or general invariant sets, or the attainment of other control objectives, such as tracking, disturbance rejection, or output feedback, all of which may be interpreted as requiring the stabilization of some quantity (typically, some sort of “error” signal). A very special case (when there are no inputs u) is that of stability. After setting up the basic definitions, we consider linear systems. Linear systems are widely used as models for physical processes, and they also play a major role in the general theory of local stabilization. We briefly review pole assignment and linear-quadratic optimization as approaches to obtaining feedback stabilizers. In general, there is a close connection between the existence of continuous stabilizing feedbacks and smooth control-Lyapunov functions, (cfl’s), which constitute an extension of the classical concept of Lyapunov functions from dynamical system theory. We discuss the role of clf’s in design methods and “universal” formulas for feedback controls. For nonlinear systems, it has been known since the late 1970s that, in general, there are topological obstructions to the existence of even continuous stabilizers. We review these obstructions, using tools from degree theory. Finally, we turn to discontinuous stabilization and the associated issue of defining precisely a “solution” for a differential equation with discontinuous right-hand side. We introduce techniques from nonsmooth analysis and differential games, in order to deal with discontinuous controllers. In particular, we discuss the effect of measurement errors on the performance of such controllers. Preliminaries In this article, we restrict attention to continuous-time deterministic systems whose states evolve in finite-dimensional Euclidean spaces Rn . (This excludes many other equally important objects of study in control theory: systems which evolve on infinite dimensional spaces and are described by PDE’s, systems evolving on manifolds which serve to model state constraints, discrete-time systems described by difference equations, and stochastic systems, among others.) In order to streamline the presentation, we suppose throughout that controls take values in U D Rm (constraints in controls would lead to proper

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subsets U ). A control (other names: input, forcing function) is any measurable locally essentially bounded map u() : [0; 1) ! U D Rm . In general, we use the notation jxj for Euclidean norms, and use kuk to indicate the essential supremum of a function u(). For basic terminology and facts about control systems, see [25]. Given a map f : Rn  Rm ! Rn which is locally Lipschitz and satisfies f (0; 0) D 0, we consider the associated forced system of ordinary differential equations x˙ (t) D f (x(t); u(t)) :

(1)

The maximal solution x() of (1) which corresponds to a given initial state x(0) D x 0 and to a given control u is defined on some maximal interval [0; tmax (x 0 ; u)), and is denoted by x(t; x 0 ; u). In the special case when f does not depend on u, we have an unforced system, or system with no inputs x˙ (t) D f (x(t)) :

(2)

Unforced systems are associated to a controlled system (1) in two different ways. The first is when one substitutes a feedback law u D k(x) in (1) to obtain a “closed-loop” system x˙ D f (x; k(x)). The second is when one considers instead the autonomous system x˙ D f (x; 0) which models the behavior of (1) in the absence of any controls. All definitions stated for unforced systems are implicitly applied also to systems with inputs (1) by setting u  0; for instance, we define the global asymptotic stability (GAS) property for (2), but we say that (1) is GAS if x˙ D f (x; 0) is. For systems with no inputs (2) we write just x(t; x 0 ) instead of x(t; x 0 ; u). Stability and Asymptotic Controllability Stability is one of the most important objectives in control theory, because a great variety of problems can be recast in stability terms. This includes questions of driving a system to a desired configuration (e. g., an inverted pendulum on a cart, to its upwards position), or the problem of tracking a reference signal (such as a pilot’s command to an aircraft). We focus in this talk on global asymptotic stabilization. Recall that the class of K1 functions consists of all ˛ : R0 ! R0 which are continuous, strictly increasing, unbounded, and satisfy ˛(0) D 0. The class of KL functions consists of those ˇ : R0  R0 ! R0 with the properties that (1) ˇ(; t) 2 K1 for all t, and (2) ˇ(r; t) decreases to zero as t ! 1. We will also use N to denote the set of all nondecreasing functions  : R0 ! R0 . Expressed in terms of such

comparison functions, the property of global asymptotic stability (GAS) of the origin for a system with no inputs (2) becomes:   (9ˇ 2 KL) jx(t; x 0 )j  ˇ jx 0 j; t 8x 0 ; 8t  0 : This definition is equivalent to a more classical “"-ı” definition usually provided in textbooks, which defines GAS as the combination of stability and global attractivity. For one implication, simply observe that   jx(t; x 0 )j  ˇ jx 0 j; 0 provides the stability (or “small overshoot”) property, while   jx(t; x 0 )j  ˇ jx 0 j; t ! 0 t!1

gives attractivity. The converse implication is an easy exercise. More generally, we define what it means for the system with inputs (1) to be (open loop, globally) asymptotically controllable (AC) (to the origin). The definition amounts to requiring that for each initial state x 0 there exists some control u D u x 0 () defined on [0; 1), such that the corresponding solution x(t; x 0 ; u) is defined for all t  0, and converges to zero as t ! 1, with “small” overshoot. Moreover, we wish to rule out the possibility that u(t) becomes unbounded for x near zero. The precise formulation is as follows. ˇ ˇ (9ˇ 2 KL)(9 2 N ) 8x 0 2 Rn 9u(); kuk   (ˇx 0 ˇ);   jx(t; x 0 ; u)j  ˇ jx 0 j; t

8t  0:

In particular, (global) asymptotic stability amounts to the existence of ˇ 2 KL such that jx(t; x 0 ; u)j  ˇ (jx 0 j; t) holds for all t  0. A very special case is that of exponential stability, in which an estimate of the type jx(t; x 0 ; u)j  Met jx 0 j holds. For linear systems (see below), asymptotic stability and exponential stability coincide. It is a puzzling fact that for general systems, one can find continuous coordinate changes that make asymptotically stable systems exponentially stable [8,13] (a fact of little practical utility, since finding such coordinate changes is as hard as establishing stability to being with). Feedback Stabilization A map k : Rn ! U is a feedback stabilizer for the system with inputs (1) if k is locally bounded (that is, k is bounded on each bounded subset of R), k(0) D 0, and the closed-loop system x˙ D f (x; k(x))

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Stability and Feedback Stabilization

is GAS, i. e. there is some ˇ 2 KL so that jx(t)j  ˇ (jx(0)j ; t) for all solutions and all t  0. (A technical difficulty with this definition lies the possible lack of solutions of (1) when k is not regular enough. We ignore this for now, but will most definitely return to this issue later.) For example, if (1) is a model of an undamped spring/mass system, where u represents the net effect of external forces, one obvious way to asymptotically stabilize the system is to introduce damping. In control-theoretic terms, this means that we choose u(t) D k(x(t)) to be a negative linear function of the velocity. Physically, one may implement a feedback controller by means of an analog device. In the example of the spring/mass system, one could achieve this by adding friction or connecting a dashpot. Alternatively, in modern control technology, one uses a digital computer to measure the state x and compute the appropriate control action to be applied. (There are many implementation issues which arise in digital control and are ignored in our theoretical formulation “u D k(x)”, among them the effect of delays in the actual computation of the control u(t) to be applied at time t, and the effect of quantization due to the finite precision of measuring devices and the digital nature of the computer. These issues are addressed in the literature, although a comprehensive theoretical framework is still lacking.) Observe that, obviously, if there exists a feedback stabilizer for (1), then (1) is also AC (we just use u(t) :D k(x(t; x 0 )) as u x 0 ). Thus, it is very natural to ask whether the converse holds: is every asymptotically controllable system also feedback stabilizable? Linear Systems A linear system is a system (1) for which the map f is linear. In other words, there are two linear transformations A: Rn ! Rn and B : Rm ! Rn so that the equations take the form x˙ D Ax C Bu :

(4)

Such a system is completely specified once that we are given A and B, which we identify by abuse of notation with their respective matrices A 2 Rnn and B 2 Rnm with respect to the canonical bases in Rn and Rm . We also say “the system (A, B)” when referring to (4). It is natural to look specifically for linear feedbacks k : Rn ! Rm which stabilize a linear system (just as a linear term, inversely proportional to velocity, stabilizes an undamped harmonic oscillator). (In fact, this is no loss of generality, since it can be easily proved for linear systems [25] that if a feedback stabilizer u D k(x) exists, then there also exists a linear feedback stabilizer.) We write

u D Fx, when expressing k(x) D Fx in matrix terms with respect to the canonical bases. Substituting this control law into (4) results in the equation x˙ D (A C BF)x. Thus, the mathematical problem reduces to: given A 2 Rnn and B 2 Rnm , find F 2 Rmn such that A C BF is Hurwitz. (Recall that a Hurwitz matrix is one all whose eigenvalues have negative real parts, and that the origin of the system x˙ D Hx is globally asymptotically stable if and only if H is a Hurwitz matrix.) The fundamental stabilization result for linear systems is as follows [25]: Theorem 1 A linear system is asymptotically controllable if and only if it admits a linear feedback stabilizer. A Remark on Linearization If the dynamics map f in (1) is continuously differentiable, we may expand to first order f (x; u) D Ax C Bu C o(x; u). Let us suppose that the linearized system (A, B) is AC, and pick a linear feedback stabilizer u D Fx, whose existence is guaranteed by Theorem 1. Then, the same feedback law k(x) D Fx, when fed back into the original system (1), results in x˙ D f (x; Fx) D (A C BF)x C o(x). Thus, k locally stabilizes the origin for the nonlinear system. Of course, the assumption that the linearization is AC is not always satisfied. Systems in which inputs enter multiplicatively, such as those controlling reaction rates in chemical problems, lead to degenerate linearizations. In addition, even if the linearized system (A, B) is AC, in general a linear stabilizer u D Fx will not work as a global stabilizer. For example, the system x˙ D x C x 2 C u can be locally stabilized with u :D 2x, but any linear feedback u D  f x ( f > 1) results in an additional equilibrium away from the origin (at x D f  1). Nonlinear feedback must be used (obviously, in this, example, u D 2x  x 2 works for global stabilization). Returning to linear systems, let us note that Theorem 1 is of great interest because (1) there is a simple algebraic test to check the AC property, and (2) there are several practically useful algorithms for obtaining a stabilizing F, including pole placement and optimization, which we sketch next. Pole Placement The first technique for stabilization is purely algebraic. In order to simplify this exposition, we will suppose that the system (4) is not just AC but is in fact controllable, meaning that every state can be steered, in finite time, to every other state. (Any AC system (4) can be decomposed

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into two components, of which one is already GAS and the other one is controllable, cf. [25], so this represents no loss of generality.) Controllability is characterized by the property – generically satisfied for pairs (A, B) – that ˇ ˇ h ˇ ˇ i ˇ ˇ ˇ ˇ rank BˇABˇA2 Bˇ : : : ˇAn1 B D n (note that the composite matrix shown has n rows and nm columns). Two pairs (A, B) and (e A; e B) are said to be feedback equivalent if there exist T 2 GL(n; R), F0 2 Rmn , and V 2 GL(m; R) so that (A C BF0 )T D T e A and e BV D T B :

(5)

Feedback equivalence corresponds to changes of basis in the state and control-value spaces (invertible matrices T and V, respectively) and feedback transformations u D F0 x C u0 , where u0 is a new control. (An equivalent way to describe feedback equivalence is by the requirement that two pairs should be in the same orbit under the action of a “feedback group” which is obtained as a suitable semidirect product of GL(n; R), (Rmn ; C), and GL(m; R).) Controllability is preserved under feedback equivalence. Moreover, if (5) holds and if one finds a matrix e F so that e ACe Be F is Hurwitz, then A C BF D T(e ACe Be F)T 1 is also Hurwitz, where F :D F0 CVe FT 1 . Thus, the task of finding a stabilizing feedback F can be reduced to the same problem for any pair (e A; e B) which is feedback equivalent to the given pair (A, B). One then proceeds to show that there always exists an equivalent pair (e A; e B) which has a form simple enough that the existence of e F is trivial to establish. In order to find such a pair, it is useful to study the classification of controllable pairs under feedback equivalence. This classification is closely related to Kronecker’s theory of “matrix pencils” applied to polynomial matrices [I  A; B] D [I; 0] C [A; B] modulo matrix equivalence, cf. [25]. The orbits under feedback equivalence are in one-to-one correspondence with the possible partitions of n D 1 C: : :Cr into the sum of r positive integers, r  m, and in each orbit one can find a pair (e A; e B) which is in “controller canonical form”, for which F can be trivially found. For simplicity, let is just discuss here the very special case of single-input systems (m D 1). For this case, the action of the feedback group is transitive, and each controllable system is feed-

back equivalent to the following special system: 0 B B B A :D B B @

0 0 :: : 0 0

1 0 :: : 0 0

0 1 :: : 0 0

::: ::: :: : ::: :::

0 0 :: : 1 0

1

0

C C C C C A

B B B B :D B B @

0 0 :: : 0 1

1 C C C C: C A

For this system, a stabilizing feedback is trivial to obtain. Indeed, take the polynomial p() D ( C 1)n D n  ˛n n1  : : :  ˛2   ˛1 . With F D (˛1 ; ˛2 ; ˛3 ; : : : ; ˛n ), 0 B B B A C BF :D B B @

0 0 : ‘:: 0 ˛1

1 0 :: : 0 ˛2

0 1 :: : 0 ˛3

::: ::: :: : ::: :::

0 0 :: : 1 ˛n

1 C C C C C A

has characteristic polynomial (C1)n , and hence is a Hurwitz matrix, as required for stabilization. Observe that, instead of the particular p() which we used, we could have picked any polynomial all whose roots have negative real parts, and the same argument applies. The conclusion is that, not only can we make AC BF Hurwitz, but we can assign to it any desired set of n eigenvalues (as long as they form a set closed under conjugation). This is the reason that the technique is called eigenvalue placement (or “pole placement” because the eigenvalues of A are the poles of the resolvent (I  A)1 ). See Chap. 5 in [25] for a detailed treatment of the pole placement problem. Variational Approach A second technique for stabilization is based on optimal control techniques. We first pick any two symmetric positive definite matrices R 2 Rmm and Q 2 Rnn (for instance the identity matrices of the respective sizes). Next, we consider the problem of minimizing, for each initial state x 0 at time t D 0, the infinite-horizon cost Z 1 ˚ Jx 0 (u) :D u(t)0 Ru(t) C x(t)0 Qx(t) dt 0

over all controls u : [0; 1) ! Rm which make Jx 0 (u) < 1, where x(t) D x(t; x 0 ; u) and prime indicates transpose. The main result from linear-quadratic optimal control (cf. Sect. 8.4 in [25]) implies that, for AC systems, there is a unique solution u to this problem, which is given in the following form: there is a matrix F 2 Rmn such that solving x˙ D (ACBF)x with x(0) D x 0 gives that u(t) :D Fx(t) minimizes Jx 0 (). Moreover, this F stabilizes the system (which is intuitively to be expected, since Jx 0 (u) < 1

Stability and Feedback Stabilization

implies that solutions x(t) must be in L2 ), and F can be computed by the formula F :D R1 B0 P ;

(6)

where P is a symmetric and positive definite solution of the Matrix Algebraic Riccati Equation PBR1 B0 P  PA  A0 P  Q D 0 :

(7)

A Sufficient Nonlinear Condition Although of limited applicability, it is worth remarking that there is a partial extension to nonlinear systems of the stabilization method which was just described. For simplicity, we specialize our discussion to control-affine systems, i. e., those for which the input appears only in an affine form. This class is sufficient for the study of most forced mechanical systems. The Eq. (1) becomes: x˙ D g0 (x) C

m X

u i g i (x) D g0 (x) C G(x)u :

iD1

(See Sect. 8.5 in [25] for general f , and for proofs). We now pick two continuous functions Q : Rn ! R0 and R : Rn ! Rnn , so that R(x) is a symmetric positive definite matrix for each x. In general, we say that a continuous function V : Rn ! R0 is positive definite if V(x) D 0 only if x D 0, and it is proper (or “weakly coercive”) if for each a  0 the set fxjV(x)  ag is compact, or, equivalently, V (x) ! 1 as jxj ! 1 (radial unboundedness). Given any such V which is also differentiable, we denote the vector function whose components are the directional derivatives of V in the directions of the various control vector fields g i , i  1 by:   LG V(x) :D rV(x)G(x) D L g 1 V(x); : : : ; L g m V(x) ; and also write L g 0 V (x) :D rV(x)g0 (x). We consider the following PDE on such functions V: 8x

1 Q(x)CL g 0 V(x) LG V(x)R(x)1 (LG V(x))0 D 0: 4 (8)

This reduces to the Algebraic Riccati Eq. (7) in the special case of linear systems, quadratic V(x) D x 0 Px and Q(x) D x 0 Qx, and constant matrices R(x)  R. We also take the following generalization of the feedback law (6): 1 k(x) :D  R(x)1 (LG V (x))0 : 2

(9)

Finally, we assume that Q is a positive definite function. One then has [25]: Theorem 2 Suppose that V is a twice continuously differentiable, positive definite, and proper solution of the PDE (8). Then, k defined by (9) stabilizes the system. This theorem arises from the following optimization problem: for each state x 0 2 Rn , minimize the cost Z 1 Jx 0 (u) :D u(t)0 R(x(t))u(t) C Q(x(t))dt ; 0

where x(t) D x(t; x 0 ; u), over all those controls u : [0; 1) ! U for which the solution x(t; x 0 ; u) of (1) is defined for all t  0 and satisfies lim t!1 x(t) D 0. Under the above assumptions, and as for linear systems, one also concludes that for each state x 0 the solution of x˙ D f (x; k(x)) with initial state x(0) D x 0 exists for all t  0, the control u(t) D k(x(t)) is optimal, and V (x 0 ) is the optimal cost from initial state x 0 . Moreover, the formula for k arises from the Hamilton–Jacobi–Bellman equation of optimal control theory, because ˚ k(x) D argmin rV(x)  f (x; u) C u0 R(x)u C Q(x) u

when f (x; u) D g0 (x) C G(x)u. There are applications where this method has proven useful. Unfortunately, however, and in contrast to the linear case, in general there exists no positive definite, proper, and C2 solution V of the above PDE. On the other hand, the formula (9) does appear, with variations, in other contexts, including generalizations of the idea of adding damping to systems, cf. Sect. 5.9 in [25], and, more generally, the use of auxiliary positive definite and proper functions V, in similar roles, will be central to the controlLyapunov ideas discussed later. Nonlinear Systems: Continuous Feedback One of the central topics which we will address here concerns possibly discontinuous feedback laws k. Before turning to that subject, however, we study continuous feedback. When dealing with linear systems, linear feedback is natural, and indeed sufficient from a theoretical standpoint, as shown by the results just reviewed. However, for our general study, major technical questions arise in even deciding on just what degree of regularity should be imposed on the feedback maps k. It turns out that the precise requirements away from 0, say asking whether k is merely continuous or smooth, are not very critical; it is often the case that one can “smooth out” a continuous feedback (or, even, make it real-analytic,

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via Grauert’s Theorem) away from the origin. So, in order to avoid unnecessary complications in exposition due to nonuniqueness, let us call a feedback k regular if it is locally Lipschitz on Rn n f0g. For such k, solutions of initial value problems x˙ D f (x; k(x)), x(0) D x 0 , are well defined (at least for small time intervals [0; ")) and, provided k is a stabilizing feedback, are unique (cf. Exercise 5.9.9 in [25]). On the other hand, behavior at the origin cannot be “smoothed out” and, at zero, the precise degree of smoothness plays a central role in the theory [12]. For instance, consider the system

and let  : (x; u) 7! x be the projection into the first coordinate in R. Then, (10) is equivalent to:  O D R n f0g : (One can easily include the requirement “kuk   (jx 0 j)” by asking that for each interval [K; K] R there must be some compact set C K R2 so that [K; K]  (C K ). For simplicity, we ignore this technicality.) In these terms, a stabilizing feedback is nothing else than a locally bounded map k : R ! R such that k(0) D 0 and so that k is a section of  on R n f0g: (x; k(x)) 2 O 8x 6D 0 :

x˙ D x C u3 : The continuous (and, in p fact, smooth away from zero) feedback u D k(x) :D  3 2x globally stabilizes the system (the closed-loop system becomes x˙ D x). However, there is no possible stabilizing feedback which is differentiable at the origin, since u D k(x) D O(x) implies that   x˙ D x C O x 3

For a regular feedback, we ask that k be locally Lipschitz on R n f0g. Clearly, there is no reason for Lipschitz, or for that matter, just continuous, sections of  to exist. As an illustration, take the system   x˙ D x (u  1)2  (x  1) (u C 1)2 C (x  2) :

about x D 0, which means that the solution starting at any positive and small point moves to the right, instead of towards the origin. (A general result, assuming that A has no purely imaginary eigenvalues, cf. [25], Section 5.8, is that if – and only if – x˙ D Ax C Bu C o(x; u) can be locally asymptotically stabilized using a feedback which is differentiable at the origin, the linearization x˙ D Ax C Bu must be AC itself. In the example that we gave, this linearization is just x˙ D x, which is not AC.) We now turn to the question of existence of regular feedback stabilizers. We first study a comparatively trivial case, namely systems with one state variable and one input. After that, we turn to multidimensional systems.

Let

The Special Case n D m D 1 There are algebraic obstructions to the stabilization of x˙ D f (x; u) if the input u appears nonlinearly in f . Ignoring the requirement that there be a  2 N so that controls can be picked with kuk  (jx 0 j), asymptotic controllability is, for n D m D 1, equivalent to: (8x 6D 0)(9u) x f (x; u) < 0

(10)

(this is proved in [28]; it is fairly obvious, but some care must be taken to deal with the fact that one is allowing arbitrary measurable controls; the argument proceeds by first approximating such controls by piecewise constant ones). Let us introduce the following set: O :D f(x; u)jx f (x; u) < 0g ;

˚



O1 D (u C 1)2 < (2  x) and O2 D f(u1)2 < (x 1)g

(these are the interiors of two disjoint parabolas). Here, O has three connected components, namely O2 , O1 intersected with x < 0, and O1 intersected with x < 0. It is clear that, even though  O D R, there is no continuous curve (graph of u D k(x)) which is always in O and projects onto R n f0g. On the other hand, there exist many possible feedback stabilizers provided that we allow one discontinuity. It is also possible to provide examples, even with f (x, u) smooth, for which an infinite number of switches are needed in any possible stabilizing feedback law. Finally, it may even be possible to stabilize semiglobally with a regular feedback, meaning that for each compact subset K of the state-space there is a regular, even smooth, feedback law u D k K (x) such that all states in K get driven asymptotically to the origin, but yet it may be impossible to find a single u D k(x) which works globally. See [26] for details. When feedback laws are required to be continuous at the origin, new obstructions arise. The case of systems with n D m D 1 is also a good way to introduce this subject. The first observation is that stabilization about the origin (even if just local) means that we must have, near zero: 8 < > 0 if x < 0 f (x; k(x)) < 0 if x > 0 : : D 0 if x D 0

Stability and Feedback Stabilization

In fact, all that we need is that f (x1 ; k(x1 )) < 0 for some x1 > 0 and f (x2 ; k(x2 )) > 0 for some x2 < 0. This guarantees, via the intermediate-value theorem that, if k is continuous, the projection ("; ") ! R ;

There is a non-slipping constraint on movement: the velocity (x˙1 ; x˙2 )0 must be parallel to the vector (cos ; sin )0 . This leads to the following equations: x˙1 D u1 cos

x 7! f (x; k(x))

is onto a neighborhood of zero, for each " > 0. It follows, in particular, that the image of ("; ")  ("; ") ! R ;

(x; u) 7! f (x; u)

also contains a neighborhood of zero, for any " > 0 (that is, the map (x; u) 7! f (x; u) is open at zero). This last property is intrinsic, being stated in terms of the original data f (x, u) and not depending upon the feedback k. Brockett’s condition, to be described next, is a far-reaching generalization of this argument; in its proof, degree theory replaces the use of the intermediate value theorem. Obstructions and Necessary Degree Conditions If there are “obstacles” in the state-space, or more precisely if the state-space is a proper subset of Rn , discontinuities in feedback laws cannot in general be avoided, since the domain of attraction of an asymptotically stable vector field must be diffeomorphic to Euclidean space. But even if states evolve in Euclidean spaces, similar obstructions may arise. These are due not to the topology of the state space, but to “virtual obstacles” implicit in the form of the system equations. These obstacles occur when it is impossible to move instantaneously in certain directions, even if it is possible to move eventually in every direction, the phenomenon of “nonholonomy”. As an illustration, let us consider a model for the “shopping cart” shown in Fig. 1 (“knife-edge” or “unicycle” are other names for this example). The state is given by the orientation , together with the coordinates x1 ; x2 of the midpoint between the back wheels. The front wheel is a castor, free to rotate.

x˙2 D u1 sin ˙ D u2 where we may view u1 as a “drive” command and u2 as a steering control. (In practice, one would implement these controls by means of differential forces on the two back corners of the cart.) The feedback transformation z1 :D , z2 :D x1 cos C x2 sin , z3 :D x1 sin x2 cos , v1 :D u2 , and v2 :D u1  u2 z3 brings the system into the system with equations z˙1 D v1 , z˙2 D v2 , z˙3 D z2 v1 known as “Brockett’s example” or “nonholonomic integrator” (yet another change can bring the third equation into the form z˙3 D z1 v2  z2 v1 ). We view the system as having state space R3 . Although a physically more accurate state space would be the manifold R2  S1 , the necessary condition to be given is of a local nature, so the global structure is unimportant. This system is (obviously) completely controllable (formally, controllability can be checked using the Lie algebra rank condition, as in e. g. [25], Exercise 4.3.16), and in particular is AC. However, we may expect that discontinuities are unavoidable due to the non-slip constraint, which does not allow moving from, for example the position x1 D 0, D 0, x2 D 1 in a straight line towards the origin. Indeed, one then has [3]: Theorem 3 If there is a stabilizing feedback which is regular and continuous at zero, then the map (x; u) 7! f (x; u) is open at zero. The test fails here, since no points of the form (0; "; ) belong to the image of the map R5 ! R3 : (x1 ; x2 ; ; u1 ; u2 )0 7! f (x; u) D (u1 cos ; u1 sin ; u2 )0 for 2 (/2; /2), unless " D 0. More generally, it is impossible to continuously stabilize any system without drift x˙ D u1 g1 (x) C : : : C u m g m (x) D G(x)u

Stability and Feedback Stabilization, Figure 1 Shopping cart

if m < n and rank[g1 (0); : : : ; g m (0)] D m (this includes all totally nonholonomic mechanical systems). Indeed, under these conditions, the map (x; u) 7! G(x)u cannot contain a neighborhood of zero in its image, when restricted

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Stability and Feedback Stabilization

to a small enough neighborhood of zero. Indeed, let us first rearrange the rows of G:   G1 (x) G(x) Ý G2 (x) so that G1 (x) is of size m  m and is nonsingular for all states x that belong to some neighborhood N of the origin. Then,    0 2 Im N  Rm ! Rn : (x; u) 7! G(x)u ) a D 0 a

Provided that a fixed-point theorem applies to continuous maps B ! B, this implies that F(x)  p must vanish somewhere in B, that is, the equation F(x) D p can be solved. (Because, for each small h > 0, the time-h flow  of F  p has a fixed point x h 2 B, i. e. (h; x h ) D x h , so picking a convergent subsequence x h ! x¯ gives that 0 D ((h; x h )  x h )/(h) ! F(x¯ )  p.) A fixed point theorem can indeed be applied, because B is a retract of Rn (use the flow itself); note that this argument gives a weaker conclusion than the degree condition. Control-Lyapunov Functions

(since G1 (x)u D 0 ) u D 0 ) G2 (x)u D 0 too). If the condition rank[g1 (0); : : : ; g m (0)] D m is violated, we cannot conclude a negative result. For instance, the system x˙1 D x1 u, x˙2 D x2 u has m D 1 < 2 D n but it can be stabilized by means of the feedback law u D (x12 C x22 ). Observe that for linear systems, Brockett’s condition says that rank[A; B] D n which is the Hautus controllability condition (see e. g. [25], Lemma 3.3.7) at the zero mode. Idea of the Proof One may prove Brockett’s condition in several ways. A proof based on degree theory is probably easiest, and proceeds as follows (for details see for instance [25], Sect 5.9). The basic fact, due to Krasnosel’ski, is that if the system x˙ D F(x) D f (x; k(x)) has the origin as an asymptotically stable point and F is regular (since k is), then the degree (index) of F with respect to zero is (1)n , where n is the system dimension. In particular, the degree is also nonzero with respect to points p in a neighborhood of 0, which means that the equation F(x) D p can be solved for small p, and hence f (x; u) D p can be solved as well. The proof that the degree is (1)n follows by exhibiting a homotopy, namely

   t 1 x ; x0  x0 ; F t (x 0 ) D t 1t between F0 D F and F1 (x) D x, and noting that the degree of the latter is obviously (1)n . An alternative proof uses Lyapunov functions. Asymptotic stability implies the existence of a smooth Lyapunov function V for x˙ D F(x) D f (x; k(x)), so, on the boundary @B of a sublevel set B D fxjV(x)  cg we have that F points towards the interior of B. Thus, for p small, F(x)  p still points to the interior, which means that B is invariant with respect to the perturbed vector field x˙ D F(x)  p.

The method of control-Lyapunov functions (“clf’s”) provides a powerful tool for studying stabilization problems, both as a basis of theoretical developments and as a method for actual feedback design. Before discussing clf’s, let us quickly review the classical concept of Lyapunov functions, through a simple example. Consider first a damped spring-mass system y¨ C y˙ C y D 0, or, in state-space form with x1 D y and x2 D y˙, x˙1 D x2 , x˙2 D x1  x2 . One way to verify global asymptotic stability of the equilibrium x D 0 is to pick the (Lyapunov) function V(x1 ; x2 ) :D 32 x12 C x1 x2 C x22 , and observe that rV(x): f (x) D  jxj2 < 0 if x 6D 0, which means that ˇ ˇ dV(x(t)) D  ˇx(t)2 ˇ < 0 dt along all nonzero solutions, and thus the energy-like function V decreases along all trajectories, which, since V is a nondegenerate quadratic form, implies that x(t) decreases, and in fact x(t) ! 0. Of course, in this case one could compute solutions explicitly, or simply note that the characteristic equation has all roots with negative real part, but Lyapunov functions are a general technique. (In fact, the classical converse theorems of Massera and Kurzweil [17,19] show that, whenever a system is GAS, there always exists a smooth Lyapunov function V.) Now let us modify this example to deal with a control system, and consider a forced (but undamped) harmonic oscillator x¨ C x D u, i. e. x˙1 D x2 , x˙2 D x1 C u. The damping feedback u D x2 stabilizes the system, but let us pretend that we do not know that. If we take the same V as before, now the derivatives along trajectories are, using “V˙ (x; u)” to denote rV(x): f (x; u) and omitting arguments t in x(t) and u(t): ˙ u) D x12 C x1 x2 C x22  (x1 C 2x2 )u : V(x; This expression is affine in u. Thus, if x is a state such that x1 C 2x2 6D 0, then we may pick a control value u (which

Stability and Feedback Stabilization

depends on this current state x) such that V˙ < 0. On the other hand, if x1 C 2x2 D 0, then the expression reduces to V˙ D 5x22 (for any u), which is negative unless x2 (and hence also x1 D 2x2 ) vanishes. In conclusion, for each x 6D 0 there is some u so that V˙ (x; u) < 0. This is, except for some technicalities to be discussed, the characterizing property of controlLyapunov functions. For any given compact subset B in Rn , we now pick some compact subset U 0 U so that 8x 2 B; x 6D 0 ; 9u 2 U 0 ˙ u) < 0 : such that V(x;

(11)

In principle, then, we could then stabilize the system, for states in B, by using the steepest descent feedback law: k(x) :D argmin rV(x)  f (x; u)

(12)

u2U0

(“argmin” means “pick any u at which the min is attained”; ˙ u) attains a minwe restricted U to be assured that V(x; imum). Note that the stabilization problem becomes, in these terms, a set of static nonlinear programming problems: minimize a function of u, for each x. Global stabilization is also possible, by appropriately picking U 0 as a function of the norm of x; later we discuss a precise formulation. Control–Lyapunov functions, if understood non-technically as the basic paradigm “look for a function V(x) with the properties that V(x)  0 if and only if x  0, and so that for each x 6D 0 it is possible to decrease V(x) by some control action,” constitute a very general approach to control (sometimes expressed in a dual fashion, as maximization of some measure of success). They appear in such disparate areas as A.I. game-playing programs (position evaluations), energy arguments for dissipative systems, program termination (Floyd/Dijkstra “variant”), and learning control (“critics” implemented by neuralnetworks). More relevantly to this paper, the idea underlies much of modern feedback control design, as illustrated for instance by the books [7,11,15,16,25]. Differentiable clf’s: Precise Definition A differentiable control-Lyapunov function (clf) is a differentiable function V : Rn ! R0 which is proper, positive definite, and infinitesimally decreasing, meaning that there exists a positive definite continuous function W : Rn ! R0 , and there is some  2 N , so that sup

min

x2R n juj(jxj)

rV(x)  f (x; u) C W(x)  0 :

(13)

This is basically the same as condition (11), with U 0 D the ball of radius (jxj) picked as a function of x. The main

difference is that, instead of saying “rV(x)  f (x; u) < 0 for x 6D 0” we write rV(x) f (x; u)  W(x), where W is negative when x 6D 0. The two definitions are equivalent, but the “Hamiltonian” version used here is the correct one for the generalizations to be given, to nonsmooth V. The basic result is due to Artstein [2]: P Theorem 4 A control-affine system x˙ D g0 (x)C u i g i (x) admits a differentiable clf if and only if it admits a regular stabilizing feedback. The proof of sufficiency is easy: if there is such a k, then the converse Lyapunov theorem, applied to the closed-loop system F(x) D f (x; k(x)), provides a smooth V such that L F V(x) D rV(x)F(x) < 0 8x 6D 0 : This gives that for all nonzero x there is some u (bounded on bounded sets, because k is locally bounded by definition ˙ of feedback) so that V(x; u) < 0; and one can put this in the form (13). The necessity is more interesting. The original proof in [2] proceeds by a nonconstructive argument involving partitions of unity, but it is also possible [24,25] to exhibit explicitly a feedback, written as a function:   k rV(x)  g0 (x); : : : ; rV (x)  g m (x) of the directional derivatives of V along the vector fields defining the system (universal formulas for stabilization). Taking for simplicity m D 1, one such formula is: p a(x) C a(x)2 C b(x)4 (0 if b D 0) k(x) :D  b(x) where a(x) :D rV(x)  g0 (x) and b(x) :D rV(x)  g1 (x). The expression for k is analytic in a,b when x 6D 0, because the clf property means that a(x) < 0 whenever b(x) D 0 [24,25]. Thus, the question of existence of regular feedback, for control-affine systems, reduces to the search for differentiable clf’s, and this gives rise to a vast literature dealing with the construction of such V’s, see [7,15,16,25] and references therein. Many other theoretical issues are also answered by Artstein’s theorem. For example, via Kurzweil’s converse theorem one has that the existence of k merely continuous on Rn nf0g suffices for the existence of smooth (infinitely differentiable) V, and from here one may in turn find a k which is smooth on Rn n f0g. In addition, one may easily characterize the existence of k continuous at zero as well as regular: this is equivalent to the small control property: for each " > 0 there is some ı > 0 so that 0 < jxj < ı implies that minjuj" rV(x)  f (x; u) < 0 (if this property holds, the universal formula automatically

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Stability and Feedback Stabilization

provides such a k). We should note that Artstein provided a result valid for general, not necessarily control-affine systems x˙ D f (x; u); however, the obtained “feedback” has values in sets of relaxed controls, and is not a feedback law in the classical sense. Later, we discuss a different generalization. Differentiable clf’s will in general not exist, because of obstructions to regular feedback stabilization. This leads us naturally into the twin subjects of discontinuous feedbacks and non-differentiable clf’s. Discontinuous Feedback The previous results and examples show that, in order to develop a satisfactory general theory of stabilization, one in which one proves the implication “asymptotic controllability implies feedback stabilizability,” we must allow discontinuous feedback laws u D k(x). But then, a major technical difficulty arises: solutions of the initialvalue problem x˙ D f (x; k(x)), x(0) D x 0 , interpreted in the classical sense of differentiable functions or even as (absolutely) continuous solutions of the integral equation Rt x(t) D x 0 C 0 f (x(s); k(x(s)))ds, do not exist in general. The only general theorems apply to systems x˙ D F(x) with continuous F. For example, there is no solution to x˙ D  sign x, x(0) D 0, where sign x D 1 for x < 0 and sign x D 1 for x  0. So one cannot even pose the stabilization problem in a mathematically consistent sense. There is, of course, an extensive literature addressing the question of discontinuous feedback laws for control systems and, more generally, differential equations with discontinuous right-hand sides. One of the best-known candidates for the concept of solution of (3) is that of a Filippov solution [6,9], which is defined as the solution of a certain differential inclusion with a multivalued righthand side which is built from f (x; k(x)). Unfortunately, there is no hope of obtaining the implication “asymptotic controllability implies feedback stabilizability” if one interprets solutions of (3) as Filippov solutions. This is a consequence of results in [5,22], which established that the existence of a discontinuous stabilizing feedback in the Filippov sense implies the Brockett necessary conditions, and, moreover, for systems affine in controls it also implies the existence of regular feedback (which we know is in general impossible). A different concept of solution originates with the theory of discontinuous positional control developed by Krasovskii and Subbotin in the context of differential games in [14], and it is the basis of the new approach to discontinuous stabilization proposed in [4], to which we now turn.

Limits of High-Frequency Sampling By a sampling schedule or partition  D ft i g i0 of 0; C1 we mean an infinite sequence 0 D t0 < t1 < t2 < : : : with lim i!1 t i D 1. We call d() :D sup(t iC1  t i ) i0

the diameter of . Suppose that k is a given feedback law for system (1). For each , the -trajectory starting from x 0 of system (3) is defined recursively on the intervals [t i ; t iC1 ), i D 0; 1; : : :, as follows. On each interval t i ; t iC1 ), the initial state is measured, the control value u i D k(x(t i )) is computed, and the constant control u  u i is applied until time t iC1 ; the process is then iterated. That is, we start with x(t0 ) D x 0 and solve recursively x˙ (t) D f (x(t); k(x(t i ))) ; t 2 t i ; t iC1 ) ;

i D 0; 1; 2; : : :

using as initial value x(ti ) the endpoint of the solution on the preceding interval. The ensuing -trajectory, which we denote as x (; x 0 ), is defined on some maximal nontrivial interval; it may fail to exist on the entire interval [0; C1) due to a blow-up on one of the subintervals t i ; t iC1 ). We say that it is well defined if x (t; x 0 ) is defined on all of [0; C1). Definition The feedback k : Rn ! U stabilizes the system (1) if there exists a function ˇ 2 KL so that the following property holds: For each 0<" 0 such that, for every sampling schedule  with d() < ı, and for each initial state x 0 with jx 0 j  K, the corresponding -trajectory of (3) is welldefined and satisfies ˇ ˇ ˇx (t; x 0 )ˇ  max fˇ (K; t) ; "g 8t  0 : (14) In particular, we have ˇ ˇ ˚ ˇ ˇ  ˇx (t; x 0 )ˇ  max ˇ ˇx 0 ˇ ; t ; "

8t  0

(15)

whenever 0 < " < jx 0 j and d() < ı("; jx 0 j) (just take K :D jx 0 j). Observe that the role of ı is to specify a lower bound on intersampling times. Roughly, one is requiring that t iC1  t i C (jx(t i )j) for each i, where is an appropriate positive function.

Stability and Feedback Stabilization

Our definition of stabilization is physically meaningful, and is very natural in the context of sampled-data (computer control) systems. It says in essence that a feedback k stabilizes the system if it drives all states asymptotically to the origin and with small overshoot when using any fast enough sampling schedule. A high enough sampling frequency is generally required when close to the origin, in order to guarantee small displacements, and also at infinity, so as to preclude large excursions or even blowups in finite time. This is the reason for making ı depend on " and K. This concept of stabilization can be reinterpreted in various ways. One is as follows. Pick any initial state x 0 , and consider any sequence of sampling schedules ` whose diameters d(` ) converge to zero as ` ! 1 (for instance, constant sampling rates with t i D i/`, i D 0; 1; 2; : : :). Note that the functions x` :D x` (; x 0 ) remain in a bounded set, namely the ball of radius ˇ(jx 0 j ; 0) (at least for ` large enough, for instance, any ` so that d(` )ı(jx 0 j /2; jx 0 j)). Because f (x; k(x)) is bounded on this ball, these functions are equicontinuous, and (Arzela– Ascoli’s Theorem) we may take a subsequence, which we denote again as fx` g, so that x` ! x as ` ! 1 (uniformly on compact time intervals) for some absolutely continuous (even Lipschitz) function x : [0; 1) ! Rn . We may think of any limit function x() that arises in this fashion as a generalized solution of the closed-loop Eq. (3). That is, generalized solutions are the limits of trajectories arising from arbitrarily high-frequency sampling when using the feedback law u D k(x). Generalized solutions, for a given initial state x 0 , may not be unique – just as may happen with continuous but non-Lipschitz feedback – but there is always existence, and, moreover, for any generalized solution, jx(t)j  ˇ(jx 0 j ; t) for all t  0. This is precisely the defining estimate for the GAS property. Moreover, if k happens to be regular, then the unique solution of x˙ D f (x; k(x)) in the classical sense is also the unique generalized solution, so we have a reasonable extension of the concept of solution. (This type of interpretation is somewhat analogous, at least in spirit, to the way in which “relaxed” controls are interpreted in optimal trajectory calculations, namely through high-frequency switching of approximating regular controls.) The definition of stabilization was given in [4] in a slightly different form; see [26] for a discussion of the equivalence. Stabilizing Feedbacks Exist The main result is [4]: Theorem 5 The system (1) admits a stabilizing feedback if and only if it is asymptotically controllable.

Necessity is clear. The sufficiency statement is proved by construction of k, and is based on the following ingredients:  Existence of a nonsmooth control-Lyapunov function V.  Regularization on shells of V.  Pointwise minimization of a Hamiltonian for the regularized V. In order to sketch this construction, we start by quickly reviewing a basic concept from nonsmooth analysis. Proximal Subgradients Let V be any continuous function Rn ! R (or even, just lower semicontinuous and with extended real values). A proximal subgradient of V at the point x 2 Rn is any vector  2 Rn such that, for some  > 0 and some neighborhood O of x, ˇ ˇ V(y)  V(x) C   (y  x)   2 ˇ y  x 2 ˇ 8y 2 O : In other words, proximal subgradients are the possible gradients of supporting quadratics at the point x. The set of all proximal subgradients at x is denoted @ p V(x). Nonsmooth Control–Lyapunov Functions A continuous (but not necessarily differentiable) V : Rn ! R0 is a control-Lyapunov function (clf) if it is proper, positive definite, and infinitesimally decreasing in the following generalized sense: there exist a positive definite continuous W : Rn ! R0 and a  2 N so that sup

max

min

x2R n 2@ p V (x) juj(jxj)

  f (x; u) C W(x)  0 :

(16)

This is the obvious generalization of the differentiable case in (13); we are still asking that one should be able to make rV(x)  f (x; u) < 0 by an appropriate choice of u D u x , for each x 6D 0, except that now we replace rV (x) by the proximal subgradient set @p V (x). An equivalent property is to ask that V be a viscosity supersolution of the corresponding Hamilton–Jacobi–Bellman equation. For nonsmooth clf’s, the main basic result is [23,26]: Theorem 6 The system (1) is asymptotically controllable if and only if it admits a continuous clf. The proof is based on first constructing an appropriate W, and then letting V be Rthe optimal cost (Bellman function) 1 for the problem min 0 W(x(s))ds. However, some care has to be taken to insure that V is continuous, and the cost has to be adjusted in order to deal with possibly unbounded minimizers. An important and very useful refinement of this result is the fact that a locally Lipschitz clf can also be shown to exist [21].

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Stability and Feedback Stabilization

Regularization Once V is known to exist, the next step in the construction of a stabilizing feedback is to obtain Lipschitz approximations of V. For this purpose, one considers the Iosida–Moreau inf-convolution of V with a quadratic function:

 1 V˛ (x) :D inf V(y) C 2 jy  xj2 2˛ y2R n where the number ˛ > 0 is picked constant on appropriate regions. One has that V˛ (x) % V(x), uniformly on compacts. Since V˛ is locally Lipschitz, Rademacher’s Theorem insures that V˛ is differentiable almost everywhere. The feedback k is then made equal to a pointwise minimizer k˛ of the Hamiltonian, at the points of differentiability (compare with (12) for the case of differentiable V): k˛ (x) :D argmin rV˛ (x)  f (x; u) ; u2U0

where ˛ and the compact U 0 D U 0 (˛) are chosen constant on certain compacts and this choice is made in between level curves. The critical fact is that V˛ is itself a clf for the original system, at least when restricted to the region where it is needed. More precisely, on each shell of the form C D fx 2 Rn jr  jxj  Rg ; there are positive numbers m and ˛0 and a compact subset U 0 such that, for each 0 < ˛  ˛0 , each x 2 C, and every  2 @p V˛ (x), min   f (x; u) C m  0 :

u2U0

See [26]. (Actually, this description is oversimplified, and the proof is a bit more delicate. One must define, on appropriate compact sets k(x) :D argmin ˛ (x)  f (x; u) ; u2U0

where ˛ (x) is carefully chosen. At points x of nondifferentiability, ˛ (x) is not a proximal subgradient of V˛ , since @p V˛ (x) may well be empty. One uses, instead, the fact that ˛ (x) happens to be in @p V (x 0 ) for some x 0  x. See [4] for details.) Sensitivity to Small Measurement Errors We have seen that every asymptotically controllable system admits a feedback stabilizer k, generally discontinuous, which renders the closed-loop system x˙ D f (x; k(x)) GAS. On the other hand, one of the main motivations for

the use of feedback is in order to deal with uncertainty, and one possible source of uncertainty are measurement errors in state estimation. The use of discontinuous feedback means that undesirable behavior – chattering – may arise. In fact, one of the main reasons for the focus on continuous feedback is precisely in order to avoid such behaviors. Thus, we turn now to an analysis of the effect of measurement errors. Suppose first that k is a continuous function of x. Then, if the error e is small, using the control u0 D k(x C e) instead of u D k(x) results in behavior which remains close to the intended one, since k(x C e)  k(x); moreover, if e  x then stability is preserved. This property of robustness to small errors when k is continuous can be rigorously established by means of a Lyapunov proof, based on the observation that, if V is a Lyapunov function for the closed-loop system, then continuity of f (x; k(x C e)) on e means that rV(x)  f (x; k(x C e))  rV(x)  f (x; k(x)) < 0 : Unfortunately, when k is not continuous, this argument breaks down. However, it can be modified so as to avoid invoking continuity of k. Assuming that V is continuously differentiable, one can argue that rV(x)  f (x; k(x C e))  rV(x C e)  f (x; k(x C e)) < 0 (using the Lyapunov property at the point x C e instead of at x). This observation leads to a theorem, formulated below, which says that a discontinuous feedback stabilizer, robust with respect to small observation errors, can be found provided that there is a C 1 clf. In general, as there are no C 1 , but only continuous, clf’s, one may not be able to find any feedback law that is robust in this sense. There are many well-known techniques for avoiding chattering, and a very common one is the introduction of deadzones where no action is taken. The feedback constructed in [4], with no modifications needed, can always be used in a manner robust with respect to small observation errors, using such an approach. Roughly speaking, the general idea is as follows. Suppose that the true current state, let us say at time t D t i , is x, but that the controller uses u D k(x˜ ), where x˜ D x C e, and e is small. Call x 0 the state that results at the next sampling time, t D t iC1 . By continuity of solutions on initial conditions, jx 0  x˜ 0 j is also small, where x˜ 0 is the state that would have resulted from applying the control u if the true state had been x˜ . By continuity, it follows that V˛ (x)  V˛ (x˜ ) and also V˛ (x 0 )  V˛ (x˜ 0 ). On the other hand, the construction in [4] provides that V˛ (x˜ 0 ) < V˛ (x˜ )  d(t iC1  t i ), where d is some positive constant (this is valid while we are far from the origin). Hence, if e is sufficiently small compared to the intersample time t iC1 t i , it will necessarily be

Stability and Feedback Stabilization

the case that V˛ (x 0 ) must also be smaller than V˛ (x). This discussion may be formalized in several ways; see [26] for a precise statement. If we insist upon fast sampling, a necessary condition arises, as was proved in the recent paper [18] (which, in turn, represented an extension of the work by Hermes [10] for classical solutions under observation error). We next discuss the main result from that paper. We consider systems x˙ (t) D f (x(t); k(x(t) C e(t)) C d(t))

(17)

in which there are observation errors as well as, now, possible actuator errors d(). Actuator errors d() : [0; 1) ! U are Lebesgue measurable and locally essentially bounded, and observation errors e(): [0; 1) ! Rn are locally bounded. We define solutions of (17), for each sampling schedule , in the usual manner, i. e., solving recursively on the intervals t i ; t iC1 ), i D 0; 1; : : :, the differential equation   x˙ (t) D f x(t); k(x(t i ) C e(t i )) C d(t) (18) with x(0) D x 0 . We write x(t) D x (t; x 0 ; d; e) for the solution, and say it is well-defined if it is defined for all t  0. Rn

! U stabilizes the sysDefinition The feedback k : tem (17) if there exists a function ˇ 2 KL so that the following property holds: For each

">0

(where B denotes the unit ball in Rn ) is strongly asymptotically stable. One may then apply converse Lyapunov theorems for upper semicontinuous compact convex differential inclusions to deduce the existence of V. We now summarize exactly which implications hold, writing “robust” to mean stabilization of the system subject to observation and actuator noise: C1V

()

+ C0V

()

9 robust k + 9k

()

AC :

Future Directions

0<" 0 and  D ("; K) such that, for every sampling schedule  with d() < ı, each initial state x 0 with jx 0 j  K, and each e; d such that j e(t)j   for all t  0 and j d(t)j   for almost all t  0, the corresponding -trajectory of (17) is well-defined and satisfies ˇ ˇ ˇx (t; x 0 ; d; e)ˇ  max fˇ (K; t) ; "g

g i (x) we may conclude that if there is a discontinuous feedback stabilizer that is robust with respect to small noise, then there is also a regular one, and even one that is smooth on Rn n f0g. For non control-affine systems, however, there may exist a discontinuous feedback stabilizer that is robust with respect to small noise, yet there is no regular feedback. Briefly, the sufficiency part of Theorem 7 proceeds by taking a pointwise minimization of the Hamiltonian, for a given C 1 clf, i. e. k(x) is defined as any u with juj   (jxj) which minimizes rV(x)  f (x; u). The necessity part is based on the following technical fact: if the perturbed system can be stabilized, then the differential inclusion \ co f (x; k(x C "B)) x˙ 2 F(x) :D

8t  0 :

(19)

In particular, taking K :D jx 0 j, one has that ˇ ˇ ˚ ˇ ˇ  ˇx (t; x 0 ; d; e)ˇ  max ˇ ˇx 0 ˇ ; t ; " 8t  0

There are several alternative approaches to feedback stabilization, notably the very appealing approach to discontinuous stabilization throgh patchy feedbacks [1], as well as other related “hybrid” approaches [20]. It is also extremely important to understand the effect of “large” disturbances on the behavior of feedback systems. This study leads one to the very active area of input to state stability (ISS) and related notions (output to state stability as a model of detectability, input to output stability for the study of regulation problems, and so forth), see [27]. Bibliography

whenever 0 < " < jx 0 j, d() < ı("; jx 0 j), and for all t, j e(t)j  ("; jx 0 j), and j d(t)j  ("; jx 0 j). The main result in [18] is as follows. Theorem 7 There is a feedback which stabilizes the system (17) if and only if there is a C 1 clf for the unperturbed system (1). It is interesting to note that, as a corollary of Artstein’s P ui Theorem, for control-affine systems x˙ D g0 (x) C

Primary Literature 1. Ancona F, Bressan A (2003) Flow stability of patchy vector fields and robust feedback stabilization. SIAM J Control Optim 41:1455–1476 2. Artstein Z (1983) Stabilization with relaxed controls. Nonlinear Anal Theory Methods Appl 7:1163–1173 3. Brockett R (1983) Asymptotic stability and feedback stabilization. In: Differential geometric control theory. Birkhauser, Boston, pp 181–191

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4. Clarke FH, Ledyaev Y, Sontag E, Subbotin A (1997) Asymptotic controllability implies feedback stabilization. IEEE Trans Autom Control 42(10):1394–1407 5. Coron J-M, Rosier L (1994) A relation between continuous time-varying and discontinuous feedback stabilization. J Math Syst Estim Control 4:67–84 6. Filippov A (1985) Differential equations with discontinuous right-hand side. Nauka, Moscow. English edition: Kluwer, Dordrecht (1988) 7. Freeman R, Kokotovi´c P (1996) Robust nonlinear control design. Birkhäuser, Boston 8. Grüne L, Sontag E, Wirth F (1999) Asymptotic stability equals exponential stability, and ISS equals finite energy gain—if you twist your eyes. Syst Control Lett 38(2):127–134 9. Hájek O (1979) Discontinuous differential equations, i & ii. J Diff Equ 32(2):149–170, 171–185 10. Hermes H (1967) Discontinuous vector fields and feedback control. In: Differential equations and dynamical systems. Proc Internat Sympos, Mayaguez 1965. Academic Press, New York, pp 155–165 11. Isidori A (1995) Nonlinear control systems: An introduction. Springer, Berlin 12. Kawski M (1989) Stabilization of nonlinear systems in the plane. Syst Control Lett 12:169–175 13. Kawski M (1995) Geometric homogeneity and applications to stabilization. In: Krener A, Mayne D (eds) Nonlinear control system design symposium (NOLCOS), Lake Tahoe, June 1995. Elsevier 14. Krasovskii N, Subbotin A (1988) Game-theoretical control problems. Springer, New York 15. Krsti´c M, Deng H (1998) Stabilization of uncertain nonlinear systems. Springer, London 16. Krsti´c M, Kanellakopoulos I, Kokotovi´c PV (1995) Nonlinear and adaptive control design. Wiley, New York 17. Kurzweil J (1956) On the inversion of Ljapunov’s second theorem on stability of motion. Am Math Soc Transl 2:19–77 18. Ledyaev Y, Sontag E (1999) A Lyapunov characterization of robust stabilization. Nonlinear Anal 37(7):813–840 19. Massera JL (1956) Contributions to stability theory. Ann Math 64:182–206 20. Prieur C (2006) Robust stabilization of nonlinear control sys-

21. 22.

23. 24.

25.

26.

27.

28.

tems by means of hybrid feedbacks. Rend Sem Mat Univ Pol Torino 64:25–38 Rifford L (2002) Semiconcave control-Lyapunov functions and stabilizing feedbacks. SIAM J Control Optim 41:659–681 Ryan E (1994) On Brockett’s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback. SIAM J Control Optim 32:1597–1604 Sontag E (1983) A Lyapunov-like characterization of asymptotic controllability. SIAM J Control Optim 21(3):462–471 Sontag E (1989) A “universal” construction of Artstein’s theorem on nonlinear stabilization. Syst Control Lett 13(2):117– 123 Sontag E (1998) Mathematical control theory. In: Deterministic finite-dimensional systems. Texts in Applied Mathematics, vol 6. Springer, New York Sontag E (1999) Stability and stabilization: Discontinuities and the effect of disturbances. In : Nonlinear analysis, differential equations and control (Montreal, 1998). NATO Sci Ser C Math Phys Sci, vol 528. Kluwer, Dordrecht, pp 551–598 Sontag E (2006) Input to state stability: Basic concepts and results. In: Nistri P, Stefani G (eds) Nonlinear and optimal control theory. Springer, Berlin, pp 163–220 Sontag E, Sussmann H (1980) Remarks on continuous feedback. In: Proc IEEE Conf Decision and Control, Albuquerque, Dec 1980. pp 916–921

Books and Reviews Clarke FH, Ledyaev YS, Stern RJ, Wolenski P (1998) Nonsmooth analysis and control theory. Springer, New York Freeman R, Kokotovi´c PV (1996) Robust nonlinear control design. Birkhäuser, Boston Isidori A (1995) Nonlinear control systems, 3rd edn. Springer, London Isidori A (1999) Nonlinear control systems II. Springer, London Khalil HK (1996) Nonlinear systems, 2nd edn. Prentice-Hall, Upper Saddle River Krsti´c M, Deng H (1998) Stabilization of uncertain nonlinear systems. Springer, London Sepulchre R, Jankovic M, Kokotovi´c PV (1997) Constructive nonlinear control. Springer, New York

Stability Theory of Ordinary Differential Equations

Stability Theory of Ordinary Differential Equations CARMEN CHICONE Department of Mathematics, University of Missouri-Columbia, Columbia, USA Article Outline Glossary Definition of the Subject Introduction Mathematical Formulation of the Stability Concept and Basic Results Stability in Conservative Systems and the KAM Theorem Averaging and the Stability of Perturbed Periodic Orbits Structural Stability Attractors Generalizations and Future Directions Bibliography Glossary Ordinary differential equation An equation for an unknown vector of functions of a single variable that involves derivatives of the unknown functions. The order of a differential equation is the highest order of the derivatives that appear. The most important class of differential equations are first-order systems of ordinary differential equations that can be written in the form u˙ D f (u; t), where f is a given smooth function f : U  J ! Rn , U is an open subset of Rn , and J is an open subset of R. The unknown functions are the components of u, and the vector of their first-order derivatives with respect to the independent variable t ˙ A solution of this differential equation is denoted by u. is a function u : K ! Rn , where K is an open subset of J such that du dt (t) D f (u(t); t) for all t 2 K. Dynamical system A set and a law of evolution for its elements. The first-order differential equation u˙ D f (u; t), where f : U  J ! Rn , is the law of evolution for the set U  J; it defines a continuous (time) dynamical system: Given (v; s) 2 U  J, the solution t 7! (t; s; v) such that (s; s; v) D v determines the evolution of the state v: the state v at time s evolves to the state (t; s; v) at time t. Similarly, a continuous function f : X ! X on a metric space X defines a discrete dynamical system. The state x 2 X evolves to f k (x) (which denotes the value of f composed with itself k times and evaluated at x) after k time-steps. The images of t 7! (t; s; v) and k 7! f k (x) are called the orbits of the corresponding states v and x.

Stability theory The mathematical analysis of the behavior of the distances between an orbit (or set of orbits) of a dynamical system and all other nearby orbits. Definition of the Subject An orbit or set of orbits of a dynamical system is stable if all solutions starting nearby remain nearby for all future times. This concept is of fundamental importance in applied mathematics: the stable solutions of mathematical models of physical processes correspond to motions that are observed in nature. Introduction Stability theory began with a basic question about the natural world: Is the solar system stable? Will the present configuration of the planets and the sun remain forever; or, might some planets collide, radically change their orbits, or escape from the solar system? With the advent of Isaac Newton’s second law of motion and the law of universal gravitation, the motions of the planets in the solar system were understood to correspond to the solutions of the Newtonian system of ordinary differential equations that modeled the positions and velocities of the planets and the sun according to their mutual gravitational attractions. Short-term approximate predictions (up to a few years in the future) verified this theory; but, due to the complexity of the differential equations of motion, the problem of long-term prediction was not solved; it still occupies a central place in mathematical research. Does the Newtonian model predict the stability of the solar system? During the 18th Century, Pierre Simon Laplace [52, 53] asserted a proof of the stability of the solar system. He considered the changes in the semi-major axes and eccentricities of the elliptical motions of the planets around the sun. Using (reasonable) approximations of the Newtonian equations of motion, he showed that for his approximate model these orbital elements do not change over long periods of time due to the disturbances caused by the gravitational attractions of the other bodies in the solar system. If true for the full Newtonian equations of motion, these assertions would imply the stability of the Newtonian solar system. In fact, no proof of the stability of the solar system is known (see [69]). Laplace’s results merely provide evidence in favor of stability of the solar system; on the other hand, this work was a primary stimulus for the later development of a general theory of stability. One of the first rigorous results in stability theory was stated by Joseph-Louis Lagrange [49] and proved by Leje-

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une Dirichlet [23,24]; it states that an isolated minimum of the potential energy of a conservative mechanical system is the position of a stable equilibrium point. Joseph Liouville [56,57,58,59] discussed the problem of the stability of rotating fluid bodies. The further development of this theory was suggested to Aleksandr Mikhailovich Lyapunov as a thesis topic by his advisor Pafnuty Chebyshev. This led to the fundamental and foundational work of Lyapunov on stability theory [63]. Henri Poincaré’s introduction of the qualitative theory of differential equations [71] influenced Lyapunov’s treatment of stability theory and laid much of the foundation for the modern theory of nonlinear dynamical systems. In addition, Poincaré’s work on celestial mechanics [72,73,74,75,76] discusses stability theory. Mathematical Formulation of the Stability Concept and Basic Results Consider the first-order ordinary differential equation u˙ D f (u; t) ;

(1)

Stability Theory of Ordinary Differential Equations, Figure 1 The figure depicts a stable rest point v. The orbit starting at w, an arbitrary point whose distance from v is less than ı, remains within distance  for all positive time

where the dependent variable u is an n-dimensional real vector, u˙ denotes the derivative of u with respect to the independent variable t, and f (a mapping, from the cross product of an open subset U of n-dimensional space Rn and an open subset J of the real line R, into Rn ) is at least class C1 . The existence and uniqueness theorem for differential equations (see, for example, [19]) states that if (v; s) 2 UJ, then there is a unique solution t 7! (t; s; v) of the differential equation such that (s; s; v) D v. Moreover, the function is as smooth as the function f . Definition 1 For v 2 Rn , the notation jvj denotes the length of v. A solution t 7! (t; s; v) of the differential Eq. (1) is called stable if it is defined for all time t  s and for every positive number there is a positive number ı such that j(t; s; w)  (t; s; v)j < whenever t  s and jw  vj < ı (see Fig. 1). A stable solution t 7! (t; s; v) is called orbitally asymptotically stable if there is a choice of ı such that the distance between the point (t; s; w) and the set f (t; s; v) : t  sg—called the forward orbit of the solution—converges to zero as t ! 1. A stable solution t 7! (t; s; v) is called asymptotically stable if there is a choice of ı such that the distance j(t; s; w)  (t; s; v)j converges to zero as t ! 1 whenever jw  vj < ı (see Fig. 2). A solution is called unstable if it is not stable.

Stability Theory of Ordinary Differential Equations, Figure 2 The figure depicts an orbitally asymptotically stable elliptical periodic orbit together with two additional orbits that approach the periodic orbit, one from the outside and one from the inside of the region bounded by the periodic orbit

While the definitions just given are widely accepted, the words “asymptotic stability” are often used to mean orbital asymptotic stability. For most situations orbital asymptotic stability is the desired concept. It is clear from the definitions that the concept of asymptotic stability is much stronger than the concept of orbital asymptotic stability for stable solutions whose orbits consist of more than one point. The position of the evolving state along an asymptotically stable orbit is approached by an open set of evolving states. On the other hand, for a solution to be orbitally asymptotically stable only the distances between the ele-

Stability Theory of Ordinary Differential Equations

ments of this open set of evolving states and the point set consisting of the forward orbit of the stable solution is required to approach zero. There is no requirement that the open set of evolving states all converge to the evolving state on the stable orbit in unison as time increases without bound. An important concept introduced by Poincaré, the return map, is useful in the study of orbital asymptotic stability; it will be discussed below. To illustrate the concept of stability, let us consider the (dimensionless) harmonic oscillator x¨ C x D 0 :

(2)

This second-order differential equation is recast as the first-order system x˙ D y ;

y˙ D x

(3)

by introducing the velocity y :D x˙ as a new variable. It is a classical mechanical system with kinetic energy 12 x˙ 2 and potential energy 12 x 2 . According to the principle of Lagrange, the equilibrium solution (x; y) D (0; 0) (which corresponds to an isolated minimum of the potential energy) is stable. This result is easily verified by inspection of the general solution of system (3) that is given by    cos t sin t  (t; ; ) D ; (4)  sin t cos t  where (; ) is an arbitrary point in R2 . All non-equilibrium solutions are periodic. In fact, the orbit p of the solution starting at (; ) is a circle with radius  2 C 2 . Given an open subset containing the origin, the disk bounded by one of these circles defines an open subset. Every orbit with initial value inside this disk remains inside the disk (and hence the given open set) for all t  0. On the other hand, the origin is not orbitally asymptotically stable; indeed, the periodic solutions do not approach the origin in positive time. The origin is also an equilibrium of the damped harmonic oscillator x¨ C x˙ C x D 0 ;

(5)

where > 0. Again, inspection of the general solution shows that the origin is asymptotically stable and therefore orbitally asymptotically stable. In effect, all terms in the formula for the general solution contain the factor e t/2 , which converges to zero as t ! 1. For a nonautonomous example (that is, a differential Eq. (1) such that the partial derivative of f with respect to t is not zero), consider the first-order scalar differential equation x˙ D x C sin t

(6)

Stability Theory of Ordinary Differential Equations, Figure 3 A graph of the asymptotically stable solution (7) of the scalar first-order differential Eq. (6) is depicted together with the graphs of three additional solutions given by t 7! (t; s; v) for (s; v) equal to (1:5; 0:75), (3:0; 0:75), and (2:0; 0:5)

whose solutions are given by   1 1 (t; s; v) D v  (sin s C cos s) est C (sin tCcos t) : 2 2 The solution starting at the state 1/2 at time zero,   1 1  t; 0; D (sin t C cos t) ; 2 2

(7)

is asymptotically stable. This result is true due to the presence of the exponential factor in the general solution; it decreases to zero for each fixed s as t ! 1 (cf. Fig. 3). The origin is an unstable equilibrium for the scalar first-order differential equation x˙ D x 2 , whose solutions are given by (t; s; v) D

v : 1  (t  s)v

Indeed, for every ı > 0 (no matter how small) there is a some v 2 R such that jvj < ı and v > 0. The corresponding solution converges to infinity as t ! (1 C sv)/v. Because explicit solutions of differential equations are rare, the main subject of stability theory is the determination of criteria for stability that do not require knowledge of the general solution of the differential equation. While the theory of stability is mature with many branches of development, the main results (originally obtained by Poincaré and Lyapunov) follow from two fundamental ideas: linearization and Lyapunov functions. The Principle of Linearized Stability A linear first-order differential equation is a differential equation of the form u˙ D A(t)u ;

(8)

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where A(t) denotes an n  n matrix-valued function of the independent variable t. Most of the analysis of autonomous linear differential equations can be reduced to linear algebra. Indeed, a function of the form et v, where  is a complex number and v ¤ 0 is a complex vector with n components, is a complex solution of the differential equation u˙ D Au ;

(9)

if and only if Av D v; that is,  is an eigenvalue of the matrix A and v is a corresponding eigenvector. The real and imaginary parts of a complex solution are automatically real solutions of the real differential Eq. (9). Also, linear combinations of solutions of linear equations are again solutions. Thus, in case there is a basis of eigenvectors, all solutions of the linear system can be expressed as linear combinations (superpositions) of the special exponential solutions. The general solution, for an arbitrary system matrix A, can always be obtained from linear combinations of solutions of the form p(t)et v, where p(t) is a polynomial of degree at most n  1 and Av D v. In fact, there is always a linear change of variables w D Bu, given by some invertible matrix B, such that the new system w˙ D Jw ;

w˙ D f u ( (t); t)w ;

(t; s; v) D e(ts)A v ;

(10)

where 1 X

(12)

where the subscript denotes the partial derivative with respect to u. The linearized equation may be viewed as an approximation of the differential equation for the deviation ı because the linearized equation is also obtained by Taylor expanding the function ı 7! f (ı C ; t)  f ( ; t) to first-order at ı D 0 and ignoring the remainder term. The principle of linearized stability states that the original solution is stable whenever the zero solution of the linearized equation is stable. The principle of linearized stability is not a theorem; rather, it serves as the underlying idea for the theory of stability by linearization. A basic result of this theory is the following theorem (see [63]). Theorem 4 Let u˙ D Au C B(t)u C g(u; t) ;

where the transformed system matrix J :D BAB1 is in Jordan canonical form (see, for example, [78]). This block diagonal system can be solved by linear combinations of solutions of the form p(t)et v, and the solution of the original system (9) is obtained by the inverse change of variables. Alternatively, it is not difficult to prove that the general solution of the system (9) is given by

e tA :D I C

Definition 3 The linearized equation along the solution of the differential Eq. (1) is

u(t0 ) D u0 ;

u 2 Rn

be a smooth initial value problem with solution t 7! If

(t).

(1) A is a constant matrix and all its eigenvalues have negative real parts, (2) B is an n  n matrix valued function, continuously dependent on t, such that the matrix norm kB(t)k converges to zero as t ! 1, (3) g is smooth and there are constants a > 0 and k > 0 such that jg(v; t)j  kjvj2 for all t  0 and jvj > a, then there are constants C > 1, ˇ > 0, and  > 0 such that

(tA) k

(11)

kD1

Rn .

and v is an arbitrary point in Solutions of the form t 7! p(t)et v converge to zero as t ! 1 whenever the real part of the eigenvalue  is less than zero. This observation leads to a basic result: Theorem 2 If all eigenvalues of the matrix A have negative real parts, then the zero solution of the autonomous system (9) is asymptotically stable. To examine the stability of a solution of the (perhaps nonlinear) differential Eq. (1), consider a second solution , define the deviation ı D  , and note that ı˙ D f ( ; t)  f ( ; t) D f (ı C ; t)  f ( ; t)

j (t)j  Cju0 je(tt0 ) ;

t  t0

whenever ju0 j  ˇ/C. In particular, the zero solution is asymptotically stable. Theorem 4 is used for the stability analysis of rest points of autonomous first-order differential equations. If f : Rn ! Rn and f () D 0, for some  2 Rn , then the constant function (t; ) D  is a solution of the autonomous differential equation u˙ D f (u) :

(13)

In this case,  is called a rest point for the differential equation. By Taylor expansion at the rest point—this time

Stability Theory of Ordinary Differential Equations

keeping the remainder term, the differential equation is recast in the form

˙ D D f () C R(; ) ; where Df denotes the derivative of f and :D    for an arbitrary solution . If f is class C2 , we have that jR(; v)j  kjvj2 : Thus, under the hypothesis that all eigenvalues of the constant system matrix D f () have negative real parts, Theorem 4 implies that the rest point  is asymptotically stable. The hypothesis that f is class C2 can be relaxed (see, for example, [19]): Theorem 5 If f : Rn ! Rn is class C1 ,  2 Rn , f () D 0, and all eigenvalues of the matrix D f () have negative real parts, then the rest point  is asymptotically stable. If the matrix D f () has an eigenvalue with positive real part, then the rest point  is unstable. The damped harmonic oscillator (5) is equivalent to the first-order system of (linear) differential equations u˙ D Au ;

(14)

where the system matrix is  A :D

0 1

1 

 :

(15)

If > 0, the system matrix has eigenvalues 12 ( ˙ p 2  4) whose real parts are always negative. Thus, the rest point at the origin is asymptotically stable in agreement with physical intuition. The power of the linearization method is demonstrated by its applications to systems whose general solutions are not known explicitly; for example, the nonlinear oscillator x¨ C x C x C g(x) D 0 ;

(16)

where g is an arbitrary (smooth) function such that g(0) D Dg(0) D 0. In this case, the origin is a rest point of the equivalent first-order system and the system matrix of the linearized system at the origin is again the matrix (15); therefore, by Theorem 5, the origin is asymptotically sta˙ D (0; 0) of ble. This result applies to the rest point ( ; ) the damped pendulum ¨ C ˙ C sin D 0 :

(17)

Stability Theory of Ordinary Differential Equations, Figure 4 The figure depicts portions of six orbits in the phase portrait of the damped pendulum (17) with  D 0:2: the rest points at ˙ equal to (; 0), (0; 0), and (; 0); and, the orbits starting (; ) at (; 0:001), ( 1:25; 1:3), and ( 1:243; 1:4). The last two orbits pass close to the (saddle type) unstable rest point at (; 0)

The differential Eq. (17) also has a rest point at ( ; ˙ ) D (; 0) where the system matrix of the linearized differential equation is 

0 1

1 

 :

(18)

If > 0, this matrix has an eigenvalue with positive real part; therefore, this rest point—which corresponds to the upward vertical equilibrium of the physical pendulum—is unstable (cf. Fig. 4). Definition 6 An n  n matrix is called infinitesimally hyperbolic if all of its eigenvalues have non-zero real parts. A rest point of an autonomous system is called hyperbolic if the system matrix of the linearized equation at this rest point is infinitesimally hyperbolic. The rest point is called nondegenerate if the system matrix is invertible. An important result on the principle of linearized stability for rest points is the Grobman–Hartman theorem (see [30,31,34,35] and [19,22]): Theorem 7 If v is a hyperbolic rest point for the autonomous differential equation u˙ D f (u), then there is an open set V containing v and a homeomorphism H with domain V such that the orbits of this differential equation in V are mapped by H to orbits of the linearized system w˙ D D f (v)w. In other words, the qualitative behavior of an autonomous differential equation in a sufficiently small neighborhood of a hyperbolic rest point is the same as the behavior of its linearization at this rest point.

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The related problem of the existence of a change of variables (a diffeomorphism) taking a given system to its linearization in a neighborhood of a rest point is more difficult to resolve. Fundamental work in this area is due to Poincaré [74], Carl Siegel [88], Henri Dulac [25], Shlomo Sternberg [90,91] and A.D. Brjuno [16] (see also [9,89]). The non-existence of certain relationships (resonances) among the eigenvalues of the system matrix of the linearization plays an important role in the theory. Definition 8 The vector of eigenvalues  D (1 ; 2 ; : : : ; n ) of the n  n-matrix A is called resonant if there is an n-vector of nonnegative integers M D (m1 ; m2 ; : : : ; P m n ) such that nkD1 jm k j  2 and at least one of the eigenvalues k is given by  k D hM; i ; where the angle brackets denote the usual inner product. The eigenvalues are nonresonant if no such relationship occurs. The first theorem of the subject was proved by Poincaré: Theorem 9 If the eigenvalues of the linearization of a vector field given by a formal power series are nonresonant, then there is a change of variables in the form of a formal power series that transforms the vector field to its linearization. In case the linearization is resonant, there is a formal change of variables that removes all of the terms in the formal series that defines the vector field except those that are resonant (that is, the corresponding monomials are given by products of variables whose powers form vectors with the properties of M in the definition of resonance). Definition 10 The vector of eigenvalues  D (1 ; 2 ; : : : ; n ) of the n  n-matrix A belongs to the Poincaré domain if the convex hull of its components in the complex plane does not contain the origin; otherwise, the vector belongs to the Siegel domain. The vector  is called type (C; ) for C > 0 and  > 0, if the inequality C j k  hM; ij  P ( k jm k j) is satisfied for each component k of  and all vectors M as in Definition 8. Theorem 11 If the eigenvalues of the linearization of an analytic vector field are nonresonant and in the Poincaré domain, then there is an analytic change of coordinates that transforms the vector field to its linearization. If the eigenvalues of the linearization of an analytic vector field are of

type (C; ), then there is an analytic change of coordinates that transforms the vector field to its linearization. For modern proofs of Poincaré’s and Siegel’s theorems see [71] or [87]. Sternberg proved Siegel’s result for vector fields in the Siegel domain for sufficiently smooth vector fields (see [90,91]). To successfully apply linearized stability for rest points of autonomous systems, the fundamental problem is to determine the real parts of the eigenvalues of a square matrix; especially, it is important to determine conditions that ensure all eigenvalues have negative real parts. The most important result in this direction is the Routh-Hurwitz criterion (see [40,84]): Theorem 12 Suppose that the characteristic polynomial of the real matrix A is written in the form n C a1 n1 C    C a n1  C a n ; let a m D 0 for m > n, and define the determinants k for k D 1; 2; : : : ; n by 1  0 B  0 C C B B  0 C  k :D det B C: B :: :: C @ : : A a2k1 a2k2 a2k3 a2k4 a2k5 a2k6    a k 0

a1 a3 a5 :: :

1 a2 a4 :: :

0 a1 a3 :: :

0 1 a2 :: :

0 0 a1 :: :

0 0 1 :: :

If  k > 0 for k D 1; 2; : : : n, then all roots of the characteristic polynomial have negative real parts. The principle of linearized stability is not valid in general. In particular, the stability of the linearized equation at a rest point of an autonomous differential equation does not always determine the stability of the rest point. For example, consider the scalar differential equation u˙ D u m , where m > 1 is a positive integer. The origin u D 0 is a rest point for this differential equation and the corresponding linearized system is w˙ D 0. The origin (and every other point on the real line) is a stable rest point of this linear differential equation. On the other hand, by solving the original equation, it is easy to see that the origin is asymptotically stable in case m is odd and unstable in case m is even. Thus, the stability of the linearization does not determine the stability of the rest point. There is no theorem known that can be used to determine the stability of a general nonautonomous first-order linear differential Eq. (8). In particular, there is no obvious relation between the (time-dependent) eigenvalues of a time-dependent matrix and the stability of the zero solution of the corresponding first-order linear differential

Stability Theory of Ordinary Differential Equations

equation. Consider, as an example, the -periodic system u˙ D A(t)u, where   1 C 32 cos2 t 1  32 sin t cos t A(t) D ; (19) 1  32 sin t cos t 1 C 32 sin2 t which was constructed by Lawrence Marcus and Hidehiko Yamabe [66].p The real parts of the eigenvalues of A(t) (given by 14 (1 ˙ 7 i)) are negative; but, u(t) D e t/2



 cos t sin t



is a solution that grows without bound. While a general theory of linearized stability for nonconstant solutions of differential equations remains an area of research, much is known in case the non-constant solution is periodic. Suppose that t 7! (t) is a periodic solution with period T of the differential Eq. (1) defined on Rn . Linearization leads to the differential equation

˙ D f u ( (t); t) ;

(20)

where the matrix (valued function) A(t) :D f u ( (t); t) is periodic with period T. Thus, the theory of linearized stability for periodic solutions starts with the stability theory for the periodic linear differential Eq. (8). The most basic results in this setting are due to Gaston Floquet [27]. To state them, let us first recall that a linear differential equation admits matrix solutions; that is, matrices whose columns are vector solutions. A fundamental matrix solution is an n  n-matrix solution with linearly independent columns; or, equivalently, a matrix solution whose values are all invertible as n  n-matrices. Floquet’s main results are stated in the following theorem (see [19] and recall the definition of the matrix exponential in display (11)). Theorem 13 If ˚(t) is a fundamental matrix solution of the T-periodic system (8), then ˚ (t C T) D ˚(t)˚

1

(0)˚ (T)

for all t 2 R. In addition, there is a matrix B (which may be complex) such that eT B D ˚ 1 (0)˚ (T) and a T-periodic matrix function t 7! P(t) (which may be complex valued) such that ˚ (t) D P(t)e tB for all t 2 R. Also, there is a real matrix R and a real 2T-periodic matrix function t ! Q(t) such that ˚(t) D Q(t)e tR for all t 2 R.

The stability of the zero solution of the periodic linear differential equation is intimately connected with the eigenvalues of the matrix R in Floquet’s theorem. Indeed, the time-dependent (real) change of variables u D Q(t)v transforms the T-periodic differential Eq. (8) to the autonomous linear system v˙ D Rv : Definition 14 The representation ˚ (t) D P(t)e tB of the fundamental matrix solution ˚ in Floquet’s theorem is called a Floquet normal form. The eigenvalues of the matrix e tB are called the characteristic (or Floquet) multipliers of the corresponding linear system. A complex number  is called a characteristic (or Floquet) exponent if there is a characteristic multiplier such that eT D . Theorem 15 (1) The characteristic multipliers and the characteristic exponents do not depend on the choice of the fundamental matrix solution of the T-periodic differential Eq. (8). (2) If the characteristic multipliers of the periodic system (8) all have modulus less than one; equivalently, if all characteristic exponents have negative real parts, then the zero solution is asymptotically stable. (3) If the characteristic multipliers of the periodic system (8) all have modulus less than or equal to one; equivalently, if all characteristic exponents have nonpositive real parts, and if the algebraic multiplicity equals the geometric multiplicity of each characteristic multiplier with modulus one; equivalently, if the algebraic multiplicity equals the geometric multiplicity of each characteristic exponent with real part zero, then the zero solution is stable. (4) If at least one characteristic multiplier of the periodic system (8) has modulus greater than one; equivalently, if a characteristic exponent has positive real part, then the zero solution is unstable. While Theorem 15 gives a complete description of the stability of the zero solution, no general method is known to determine the characteristic multipliers (cf. [98]). The power of the theorem is an immediate corollary: A finite set of numbers (namely, the characteristic multipliers) determine the stability of the zero solution, at least in the case where none of them have modulus one. The characteristic multipliers can be approximated by numerical integration. And, in special cases, their moduli can be determined by mathematical analysis. One of the most important applications for Floquet theory and the method of linearization, is the analysis of

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Hill’s equation x¨ C a(t)x D 0 ;

(21)

or, equivalently, the first-order linear system x˙ D y ;

y˙ D a(t)x ;

(22)

where x is a scalar and a is a T-periodic function (see [19, 64]). This equation arose from George Hill’s study of lunar motion. It is ubiquitous in stability theory. The stability of the zero solution of Hill’s system (22) can be reduced to the identification of a single number; namely, the magnitude of the trace of the principal fundamental matrix solution ˚(t) at t D T (that is, the fundamental matrix solution ˚ (t) such that ˚ (0) D I, where I is the n  n identity matrix). Theorem 16 Suppose that ˚ (t) is the principal fundamental matrix solution of system (22). If jtrace ˚ (T)j < 2, then the zero solution is stable. If jtrace ˚ (T)j > 2, then the zero solution is unstable. This result, of course, requires knowledge of the solutions of the system. On the other hand, a beautiful theorem of Lyapunov [55] gives a stability criterion using only properties of the function a: Theorem 17 If a : R ! R is a positive T-periodic function such that Z T a(t) dt  4 ; T 0

then all solutions of the Hill’s equation x¨ C a(t)x D 0 are bounded. In particular, the trivial solution is stable. The principle of linearized stability for periodic solutions of nonlinear systems motivates a theory similar to the stability theory for rest points. There is, however, one essential difference that will be explained for a T-periodic solution of the autonomous first-order differential equation u˙ D f (u):

f ( (0)); therefore, the number 1 is a characteristic multiplier. It corresponds to motion in the direction of the periodic solution and does not contribute to the stability or instability of the periodic solution. This characteristic multiplier must be treated separately in the stability theory. Theorem 18 If the number 1 occurs with algebraic multiplicity one in the set of characteristic multipliers of the linearized differential equation at a periodic solution and all other characteristic multipliers have modulus less than one, then the periodic solution is orbitally asymptotically stable. If at least one characteristic multiplier has modulus larger than one, then the periodic solution is unstable. Poincaré introduced a geometric approach to stability for periodic solutions. His idea is simple and useful: Consider a periodic solution and some point, say (0), on the orbit of this solution in Rn . Construct a hypersurface S in Rn that contains (0) and is transverse to the orbit of

at this point. By the implicit function theorem, there is an open neighborhood ˙ of (0) in S such that the solution of the differential equation starting at every point in ˙ returns to S. This defines a local diffeomorphism P : ˙ ! S, called the Poincaré (or return) map, which assigns p to its point of first return on S (cf. Fig. 5). The eigenvalues of the derivative of P at (0) are exactly the characteristic multipliers associated with . Thus, if all eigenvalues of this derivative have modulus less than one, then is orbitally asymptotically stable. The derivative of the Poincaré map can be computed using the linearized equations of the corresponding differential equation along its the periodic orbit. While explicit formulas for the desired derivative are only available for special cases, the Poincaré map provides an indispensable tool for

(23)

The linearized equation at is w˙ D D f ( (t))w. Let (t) D f ( (t)) and note that ˙ D D f ( (t)) f ( (t)) D D f ( (t))(t) ; (t) that is, t 7! f ( (t)) is a solution of the linearized equation. This solution is given by f ( (t)) D ˚(t) f ( (0)) ; where ˚(t) is the principal fundamental matrix solution of the linearized equation. If follows that ˚ (T) f ( (0)) D

Stability Theory of Ordinary Differential Equations, Figure 5 A schematic diagram of a Poincaré map P defined on a Poincaré section ˙ near a periodic orbit containing the point (0)

Stability Theory of Ordinary Differential Equations

organizing the study of stability for periodic orbits in general. It should be clear that the discrete dynamical system defined by P codes all the information about the qualitative behavior of the solutions of the differential equation that start near the orbit of . The existence of the Poincaré map is thus the main connection between discrete and continuous dynamical systems. Often, but not always, the dynamical properties of diffeomorphisms (or maps) are simpler to analyze than the corresponding properties of differential equations. Thus, theorems are often proved first for maps and later for differential equations. For instance, the Grobman–Hartman theorem was first proved for diffeomorphisms. This result can be applied to the Poincaré map of a periodic orbit to prove the existence of a homeomorphism that maps nearby orbits to the orbits of a corresponding linearization. For the special case of periodic solutions of autonomous first-order differential equations on the plane, there is exactly one important Floquet multiplier; it is identified in the following fundamental result. Theorem 19 If is a periodic solution with period T of the differential Eq. (23) defined on R2 , then the number Z D

T

 P(x) D

x 2 e4 1  x 2 C x 2 e4

 12 :

The periodic orbit  intersects the Poincaré section at x D 1 where we have P(1) D 1 and P 0 (1) D e4 < 1. A periodic solution of an autonomous system cannot be asymptotically stable. This fact depends on a special property of the solutions of autonomous systems: if the differential Eq. (1) is autonomous, then the function giving the solutions, t 7! (t; s; v), is a function of t  s. Hence, by replacing t  s by t, each solution can be expressed in the form t 7! (t; v) where (0; v) D v and (t; (s; v)) D (t C s; v) whenever both sides are defined. The family of functions v 7! (t; v), parametrized by t is called the flow of the differential Eq. (1). Suppose that t 7! (t; v) is an asymptotically stable periodic solution with period T. Let ı > 0 be as in Definition 1. If jw  vj < ı, then lim t!1 j(t; w)  (t; v)j D 0. For all sufficiently small s, we have that j(s; w)  vj < ı. Hence, lim t!1 j(t; (s; w))  (t; v)j D 0. By the triangle law, j(t C s; v)  (t; v)j  j(t C s; v)  (t C s; w)j C j(t C s; w)  (t; v)j

div f ( (t)) dt

 j(s; (t; v))  (s; (t; w)j C j(t C s; w)  (t; v)j:

0

(where div denotes the divergence) is (up to a positive scalar multiple) a characteristic exponent of . If  is negative, then is orbitally asymptotically stable. If  is positive, then is unstable.

x˙ D x  y  (x 2 C y 2 )x ;

y˙ D x C y  (x 2 C y 2 )y

has the periodic solution t ! (cos t; sin t) (depicted with a non-unit aspect ratio in Fig. 2). Also, the divergence of the vector field is 2  4(x 2 C y 2 ); it has value -2 along the corresponding periodic orbit  D f(x; y) : x 2 C y 2 D 1g. By Theorem 19,  is orbitally asymptotically stable. This is an example of a limit cycle; that is, an isolated periodic orbit. By changing to polar coordinates (r; ), the transformed system ˙ D 1

decouples, and its flow is given by 0 1   12 2 e2t r ; C tA :  t (r; ) D @ 1  r2 C r2 e2t

Since  7! (s; ) is smooth and a periodic orbit is compact, the law of the mean implies that there is a constant C > 0 such that j(t C s; v)  (t; v)j

The system of differential equations

r˙ D r(1  r2 ) ;

The positive x-axis is a Poincaré section with corresponding Poincaré map P given by

(24)

 Cj(t; v)  (t; w)j C j(t C s; w)  (t; v)j : Both summands on the right-hand side of the last inequality converge to zero as t ! 1. Hence, we have that lim j(kT C s; v)  (kT; v)j D 0 ;

k!1

where k denotes an integer variable. It follows that (s; v) D v for all s in some open set. The point v must be a rest point on a periodic orbit, in contradiction to the uniqueness of solutions. While a periodic solution of an autonomous system cannot be asymptotically stable, we can ask that an orbitally asymptotically stable periodic orbit satisfy a stronger type of stability. Definition 20 Let  denote the flow of the autonomous differential Eq. (23) and suppose that  is a periodic orbit corresponding to the solution t 7! (t; p). We say that

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q 2 Rn has asymptotic phase p with respect to  if the solution t 7! (t; q) is such that lim j(t; q)  (t; p)j D 0 :

t!1

We say that  is isochronous if there exists an open neighborhood of its orbit such that every point in this neighborhood has asymptotic phase with respect to the period solution. Theorem 21 If all eigenvalues of the derivative of a Poincaré map at the periodic solution t 7! (t; p) of (23) lie strictly inside the unit circle in the complex plane (equivalently, the corresponding periodic orbit is hyperbolic), then every point q in some open neighborhood of this periodic orbit has asymptotic phase. A theory of asymptotic phase that includes orbitally stable periodic orbits such that some eigenvalues of the derivative of an associated Poincaré map have modulus one was recently completed (see [21,26]). Lyapunov Functions In case the linearized differential equation along a solution is stable but not orbitally asymptotically stable, the principle of linearized stability usually cannot be justified. One of the great contributions of Lyapunov is the introduction of a method that can be used to determine the stability of such solutions. The fundamental ideas of Lyapunov are most clear for the stability analysis of rest points of autonomous systems (see [63]); but, the theory goes far beyond this case (see [51]). Definition 22 Let u0 be a rest point of the autonomous differential Eq. (23). A continuous function L : U ! R, where U  Rn is an open set with u0 2 U, is called a Lyapunov function at u0 if (1) L(u0 ) D 0, (2) L(x) > 0 for x 2 U n fu0 g, (3) the function L is continuously differentiable on the set ˙ U nfu0 g, and, on this set, L(x) :D grad L(x) f (x)  0. The function L is called a strict Lyapunov function if, in addition, ˙ (4) L(x) < 0 for x 2 U n fu0 g. Property (3) states that L does not increase along solutions. Theorem 23 If there is a Lyapunov function defined on an open neighborhood of a rest point of an autonomous firstorder differential equation, then the rest point is stable. If, in

addition, the Lyapunov function is a strict Lyapunov function, then the rest point is asymptotically stable. Lyapunov’s theorem was motivated by Lagrange’s principle: a rest point of a mechanical system corresponding to an isolated minimum of the potential energy is stable. To obtain this result, recall that the equation of motion of a (conservative) mechanical system for the motion of a particle is m x¨ D grad G(x) where m is the mass of the particle and x 2 Rn is its position. The kinetic energy of the particle is defined to be ˙ (where the angle brackets denote the usual K D m2 hx˙ ; xi inner product) and the potential energy is G, a quantity defined up to an additive constant. Consider the corresponding equivalent first-order system x˙ D y ;

y˙ D 

1 grad G(x) ; m

and suppose that x0 be an isolated minimum of G. The state (x; y) D (x0 ; 0) is a rest point of the mechanical system. By inspection, the total energy L(x; y) :D

m hy; yi C G(x)  G(x0 ) 2

(with an appropriate translation of the potential energy) is a Lyapunov function at the rest point. Thus, the rest point is stable. The proof of the first part of Lyapunov’s theorem is simple: Choose an open ball B with center at the rest point and radius sufficiently small so that its closure is in the domain of the Lyapunov function L. The continuous positive function L on the closed and bounded spherical boundary of B has a non-zero minimum value b. Because the continuous function L vanishes at the rest point, there is a second ball A that is concentric with B, has strictly smaller radius, and is such that the maximum value of L on all of A is smaller than b. A solution of the differential equation starting in A cannot reach the boundary of B because L does not increase along solutions. Thus, the rest point is stable. Lyapunov’s method extends to nonautonomous systems with an appropriate modification of the notion of a Lyapunov function. Suppose that the first-order differential Eq. (1) has a rest point in the sense that for some T 2 R, f (u0 ; t) D 0 for all t  T. A Lyapunov function is a class C1 function defined in a neighborhood of u0 for all t  T such that L(u0 ; t) D 0 for t  T, there is a continuous nonnegative function M defined on the same neighborhood of u0 such that L(u; t)  M(u) for all t  T

Stability Theory of Ordinary Differential Equations

and L˙ D L t C Lu  u˙  0 for all t  T. If such a Lyapunov function exists, then the rest point is stable. If, in addition, there is a positive function N with the same domain as M ˙ t)  N(u) for all t  T, then the rest such that L(u; point is asymptotically stable. Lyapunov’s direct method can be used to prove the principle of linearized stability for rest points of autonomous systems. To see how this might be done, consider the differential equation u˙ D Au C g(u) ;

u 2 Rn ;

where A is a real nn matrix and g : Rn ! Rn is a smooth function. Suppose that every eigenvalue of A has negative real part, and that for some a > 0, there is a constant k > 0 such that, using the usual norm on Rn , jg(x)j  kjxj2 whenever jxj < a. Let h; i denote the usual inner product on Rn , and let A denote the transpose of the real matrix A. Suppose that there is a real symmetric positive definite n  n matrix that also satisfies Lyapunov’s equation A B C BA D I and define L : Rn ! R by L(x) D hx; Bxi : Using Schwarz’s inequality, it can be proved that the restriction of L to a sufficiently small neighborhood of the origin is a strict Lyapunov function. The proof is completed by showing that there is a symmetric positive-definite solution of Lyapunov’s equation. In fact, Z 1  B :D e tA e tA dt 0

is a symmetric positive definite n  n matrix that satisfies Lyapunov’s equation. Alternatively, it is possible to prove the existence of a solution to Lyapunov’s equation using purely algebraic methods (see, for example, [97]). The instability of solutions can also be detected by Lyapunov’s methods. A simple result in this direction is the content of the following theorem. Theorem 24 Suppose that L is a smooth function defined on an open neighborhood U of the rest point u0 of the autonomous differential Eq. (23) such that L(u0 ) D 0 and ˙ V(u) > 0 on U n fu0 g. If L has a positive value somewhere in each open set containing u0 , then u0 is unstable. One indication of the subtlety of the determination of stability is the insolvability of the center-focus problem [41].

Consider a planar system of first-order differential equations in the form x˙ D y C P(x; y) ;

y˙ D x C Q(x; y) ;

where P and Q are analytic at the origin with leading-order terms at least quadratic in x and y. Is the origin a focus (that is, asymptotically stable or asymptotically unstable) or a center (that is, all orbits in some neighborhood of the origin are periodic)? There is no algorithm that can be used to solve this problem for all such systems in a finite number of steps. On the other hand, the center-focus problem is solved, by using Lyapunov’s stability theorem, if a certain sequence of numbers called Lyapunov quantities (which can be computed iteratively and algebraically) has a non-zero element. One problem is that the algorithm for computing the Lyapunov quantities may not terminate in a finite number of steps (see, for example, [19,89]). The center-focus problem is solved for some special cases. The most important theorem in this area of research is due to Nikolai Bautin (see [14] and [99,100] for a modern proof); he found (among other results) a complete solution of the center-focus problem for quadratic systems (that is, where P and Q are homogeneous quadratic polynomials). While the center-focus problem for quadratics had been solved by Dulac [25] and others [28,43,44,45] before his paper appeared; Bautin introduced a method, which involves the use of polynomial ideals, that has had far reaching consequences. Stability in Conservative Systems and the KAM Theorem Stability theory for conservative systems leads to many delicate problems, some of which—for example Lagrange’s problem—can be resolved by the method of Lyapunov functions. While the total energy, the total angular momentum, or other first integrals all have the property that their derivatives vanish along trajectories, these integrals do not always serve as Lyapunov functions because they often fail to be positive definite in punctured neighborhoods of rest points or periodic orbits. Thus, other methods are required. As in the quest to determine the stability of the solar system, a basic problem in mechanics is to determine the stability of periodic motions (or rest points) of conservative differential equations with respect to small perturbations. While no general solution is known, great progress has been made culminating in the Kolmogorov–Arnold– Moser (KAM) theorem (see [5,46,47,68,87,89,92]). A mechanical system with N degrees-of-freedom (that is, the positions are determined by N-coordinates) is called

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completely integrable if it has N independent first integrals whose Poisson brackets are in involution (see p. 271 in [8]). In this case, the system can be transformed to a first-order system of differential equations in the simple form I˙ D 0;

˙ D ˝(I) ;

(25)

where I is an N-dimensional variable (of actions), is an N-dimensional variable (of angles), and ˝ is a smooth function. The new coordinates are called action-angle variables (see, for generalizations, [39,42]). This system is in Hamiltonian form (I˙ D @H/@ , ˙ D @H/@I) with Hamiltonian H(I; ) :D !(I), where ! is an anti-derivative of ˝; and, of course, the Hamiltonian is constant along solutions of the system. The corresponding perturbed system is taken to have the form @F I˙ D (I; ); @

@F ˙ D ˝(I)  (I; ) @I

(26)

(or, more generally, a similar form with additional terms that are higher-order in ), where F is a smooth function that is 2-periodic in . This system is in Hamiltonian form with Hamiltonian H (I; ; ) :D !(I) C F(I; ) :

The unperturbed integrable system can be solved explicitly as I(t) D I0 ;

are called resonant; the others are called nonresonant. The KAM theorem states that most of the nonresonant tori in the unperturbed system (25) survive after small perturbations as invariant tori of the perturbed system (26). Theorem 25 If (in the Hamiltonian system (26)) H is sufficiently smooth, D˝(I) is invertible, and is sufficiently small, then almost all (in the sense of Lebesgue measure) nonresonant unperturbed invariant tori persist. There are similar theorems for time-dependent perturbations of integrable systems and for area-preserving maps (see [8]). For a mechanical system with N degrees-of-freedom, the perturbed invariant tori are N-dimensional manifolds embedded in the 2N-dimensional phase space. After restriction to a regular energy surface (that is, a level set of H corresponding to one of its regular values), these N-dimensional tori are embedded in a (2N  1)-dimensional manifold. For N  2, the invariant tori that exist by the KAM theorem separate the energy hypersurface into two components, one inside and one outside of each invariant torus. A motion starting in a bounded region whose boundary is one of these invariant tori cannot escape to the corresponding unbounded component. Since the invariant tori are dense, such motions are stable (cf. Fig. 6). More generally, each perturbed invariant torus is stable as an invariant set. This result can be used, for example, to prove the stability of certain periodic orbits in the restricted three-body problem (see [54]). On the other hand,

(t) D ˝(I0 )t C 0 ;

hence, by inspection, all orbits are periodic or quasi-periodic (when the angular variables are defined modulo 2) according to whether or not the vector of frequencies ˝(I0 ) satisfies a resonance relation, that is, hM; ˝(I0 )i D 0 for some N-vector M with integer components. Every orbit of the unperturbed system with one degree-of-freedom is periodic; but, for example for two degrees-of-freedom, orbits are periodic with period T only if there are integers m1 and m2 such that T˝1 (I0 ) D 2 m1 ;

T˝2 (I0 ) D 2 m2

or m1 ˝2 (I0 )  m2 ˝1 (I0 ) D 0. Geometrically, a choice of actions fixes an energy level and the angles correspond to an N-dimensional invariant torus. The flow on this torus is either periodic or quasi-periodic according to the existence of a corresponding resonance relation. For this reason, the tori with periodic flows

Stability Theory of Ordinary Differential Equations, Figure 6 The figure depicts portions of seven orbits of the (stroboscopic) Poincaré map  2 R2 7! (2; ) for the forced oscillator x˙ D y, y˙ D x  x 3 C 0:05 sin t: three fixed points at (x; y) equal to (1; 0), (0; 0), and (1; 0); an orbit starting at (x; y) D (1:0; 0:4) at time t D 0, which is close to a (3 : 1) resonance; an orbit starting at (1:0; 0:5), which is on an invariant torus that surrounds the resonant orbit; an orbit starting at (1:0; 0:6), which appears to be “chaotic”; and an orbit starting at (1; 1), which is on a large invariant torus. The (3 : 1) resonant orbit is stable

Stability Theory of Ordinary Differential Equations

for N > 2, the invariant tori no longer separate energy surfaces into bounded and unbounded components. Thus, it is possible for orbits to migrate outside nearby tori in a conjectured process called Arnold diffusion (see [6]). The validity of Arnold diffusion for the general Hamiltonian system (26) is an area of current research (see, for a review, [60]). Averaging and the Stability of Perturbed Periodic Orbits As we have seen, systems of the form I˙ D P(I; ) ;

˙ D ˝(I) C Q(I; ) ;

(27)

where P and Q are 2-periodic, often arise in mechanics. Here, the properties of system (27) are considered for general perturbations, which are not necessarily conservative. Note that, in case is small, the vector of actions (components of I) in the differential Eq. (27) evolves slowly relative to the evolution of the vector of angles . The averaging principle states that the evolution of the actions for such a system is well-approximated by the corresponding averaged equation given by ¯ I) ; I˙ D P(

(28)

where ¯ I ) :D P(

1 (2) N

Z TN

P(I ; ) d

and T N denotes the N-dimensional torus of angles. Exactly what is meant by “well-approximated” is clarified by the averaging theorem, which requires a severe restriction: the vector of angles is one-dimensional (cf. [61,86]). Theorem 26 If is a scalar variable and ˝ is bounded away from zero, then there is a near-identity change of variables of the form I D I C k(I; ) that is 2-periodic in which transforms system (27) into the form ¯ I ) C 2 P1 (I ; ; ) ; I˙ D P(

˙ D ˝(I ) C Q1 (I ; ; ) :

where P1 and P2 are 2-periodic in . Moreover, if the evolution of I starting at I 0 is given by I(t) and the evolution (according to the differential Eq. (28)) of the averaged action I starting at I 0 is given by I (t), then there are constants C > 0 and T(I0 ) > 0 such that jI (t)  I(t)j  C on the time interval 0  t < T(I0 ).

The existence and stability of perturbed period solutions is the content of the following theorem. Theorem 27 Consider the system I˙ D F(I; ) C 2 F2 (I; ; ) ;

˙ D ˝(I) C G(I; ; ) ; (29)

where I 2 R M , is a scalar, F, F 2 , and G are 2-periodic functions of , and there is some number c such that ˝(I)  c > 0. If the averaged system has a nondegenerate rest point (see Definition 6) and is sufficiently small, then system (29) has a periodic orbit. If in addition > 0 and the rest point is hyperbolic, then the periodic orbit has the same stability type as the hyperbolic rest point; that is, the dimensions of the corresponding stable and unstable manifolds are the same (see Definition 29). For a typical application of Theorem 27, consider the system of differential equations I˙ D (a C b sin k ) sin C 2 P(I; ; ); ˙ D cI C 2 Q(I; ; );

(30)

˙ D 1; where a, b, and c are non-zero constants, k is a positive integer, and both P and Q are 2-periodic in . The averaged system of differential equations I˙ D a sin

;

˙ D cI

has hyperbolic rest points at (I; ) D (0; 0) and (I; ) D (0; ). According to Theorem 27, if is sufficiently small, the system of differential Eq. (30) has corresponding hyperbolic periodic solutions. In case ac > 0, the solution of the perturbed system corresponding to D 0 is stable, and the solution corresponding to D  is unstable (in fact, it is a hyperbolic saddle). If ac < 0, the stability types are switched. Structural Stability An important aspect of the global theory of dynamical systems is the stability of the orbit structure as a whole. The motivation for the corresponding theory comes from applied mathematics. Mathematical models always contain simplifying assumptions. Dominant features are modeled; supposed small disturbing forces are ignored. Thus, it is natural to ask if the qualitative structure of the set of solutions—the phase portrait—of a model would remain the same if small perturbations were included in the model. The corresponding mathematical theory is called structural stability.

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Stability Theory of Ordinary Differential Equations

The correct mathematical formulation of the definition of structural stability requires the introduction of a topology on the space of dynamical systems under consideration. The natural setting for the theory is the space of class C1 vector fields on a smooth compact manifold. There is a one-to-one correspondence between vector fields and differential equations: the right-hand side of a first-order autonomous differential equation defines a vector field on Rn . The local representatives, with respect to coordinate charts, of a vector field on an n-dimensional manifold M are vector fields on Rn . The set of all smooth vector fields X(M) on M has the structure of a Banach space after the introduction of a Riemannian metric. Thus, two vector fields are close if the distance between them in this Banach space is small. Definition 28 A vector field X on a smooth manifold M is called structurally stable, if there is some neighborhood U of X in X(M) such that for each Y 2 U there is a homeomorphism of M that maps orbits of Y to orbits of X and preserves the orientation of orbits according to the direction of time. Such a homeomorphism is called a topological equivalence. In other words, all perturbations of a structurally stable vector field have the same qualitative orbit structures. At first impression, it might seem that the homeomorphism in the definition of structural stability should be a diffeomorphism. But, this requirement is too strong. To see why, consider a hyperbolic rest point u0 of the first-order autonomous differential Eq. (23) in Rn . Let u˙ D g(u) be a differential equation on Rn , and consider the perturbation of the differential Eq. (23) given by u˙ D f (u) C g(u). We have that f (u0 ) D 0. Thus, F(u; ) :D f (u) C g(u) is such that F(u0 ; 0) D 0 and DF(u0 ; 0) D D f (u0 ). Because of the hyperbolicity, the linear transformation D f (u0 ) is invertible. By an application of the implicit function theorem, there is a smooth function ˇ : J ! Rn , where J is an open interval in R containing the origin, such that F(ˇ( ); )) D 0 for all in J. In other words, every small perturbation of (the vector field) f has a rest point near u0 . Moreover, for sufficiently small (and in view of the continuity of eigenvalues with respect to the components of the corresponding matrix), the perturbed rest point ˇ( ) is hyperbolic with the same number of eigenvalues (counting multiplicities) with positive and negative real parts as the eigenvalues of D f (u0 ). But, of course, the perturbed eigenvalues are in general not the same as the eigenvalues of D f (u0 ). The hyperbolic rest point u0 is structurally stable in the sense that the local phase portraits at the corresponding rest points of sufficiently small perturbations of f are qual-

itatively the same as the phase portrait of the differential Eq. (23). This fact can be proved using the Grobman– Hartman theorem together with an argument to show that the phase portraits of hyperbolic linear first-order systems are topologically equivalent if their system matrices have the same numbers of eigenvalues (counting multiplicities) with positive and negative real parts. Suppose there were a diffeomorphism h taking orbits to orbits. It would correspond to a change of variables v D h(u) (a smooth conjugacy) taking the differential Eq. (23) to the differential equation v˙ D F(v), where the parameter is suppressed for simplicity. By differentiation of the equation v D h(u) with respect to the independent variable t, it follows that F(h(u)) D Dh(u) f (u). By linearization at the rest point u0 , we have DF(h(u0 ))Dh(u0 ) D Dh(u0 )D f (u0 ) : In particular the system matrix D f (h(u0 )) for the linearized system at the perturbed rest point is similar, via the invertible matrix Dh(u0 ), to the system matrix D f (u0 ) for the linearized system at the unperturbed rest point. Thus, the two rest points would have to have exactly the same eigenvalues, contrary to the general situation for which the eigenvalues of the perturbed linearization are different from the eigenvalues of the unperturbed linearization. The notion of hyperbolicity is an essential hypothesis in structural stability theory; indeed, hyperbolic rest points are structurally stable. Likewise hyperbolic periodic orbits, defined to be those periodic orbits such that there is an associated hyperbolic Poincaré map are structurally stable. This notion can be extended to general invariant sets. The essential requirement is that all of the solutions of the linearized differential equations along the solutions in the invariant set, which are not initially in the direction of the invariant set, either grow or decay exponentially with uniform exponential growth rates over the entire invariant set. The global theory of structural stability requires a global hypothesis to ensure that the rest points and period orbits are connected with orbits in general position. Definition 29 Let A be an invariant set (a union of orbits) of an autonomous first-order differential equation, vector field, or discrete dynamical system. The stable manifold of A, denoted W s (A) is the set of all solutions that converge to A as t ! 1. The unstable manifold W u (A) is the set of all solutions that converge to A as t ! 1. The stable manifold theorem asserts that the stable and unstable manifolds for rest points and periodic orbits of smooth vector fields are indeed (immersed) smooth manifolds (see, for example, [37,38]).

Stability Theory of Ordinary Differential Equations

Two immersed manifolds are called transversal if they do not intersect; or, if the sum of their tangent spaces at each point of their intersection is the tangent space of the ambient manifold. While periodicity is the most familiar form of recurrence for dynamical systems, a more subtle notion of recurrence is required for structural stability theorems. Definition 30 The point u is called a nonwandering point for an autonomous differential equation if, for each neighborhood U of u and each time t0 > 0, there is some time t > t0 such that at least one solution starting in U at time zero returns to U at time t. Likewise for a discrete dynamical system defined by a diffeomorphism h, the point u is nonwandering if for each neighborhood U of u there is some integer k > 0 such that h k (U) \ U is not empty. The set of all nonwandering points is called the nonwandering set. For vector fields on compact orientable two-dimensional manifolds, the classical structural stability theorem is due to M. Peixoto [70] and is a generalization of earlier work for vector fields in the plane by L. Pointryagin and A. Andronov [2,3]. Theorem 31 A class C1 vector field on a compact orientable two-dimensional manifold M is structurally stable if and only if (1) all rest points are hyperbolic, (2) all periodic solutions are hyperbolic, (3) stable and unstable manifolds of all pairs of saddle points (rest points whose linearizations have one positive and one negative eigenvalue) are disjoint, (4) the nonwandering set consists of only rest points and periodic solutions. Moreover, the structurally stable vector fields form an open and dense set in X(M). The notion of transversality in Peixoto’s theorem is embodied in the property (3). The generalization of this property is an essential ingredient of the theory. Definition 32 A dynamical system satisfies the strong transversality property if for every pair of points fu; vg in its nonwandering set, the corresponding invariant manifolds W s (u) and W u (y) intersect transversally (at all their points of intersection). Definition 33 A dynamical system satisfies Axiom A if its nonwandering set is hyperbolic and the periodic orbits are dense in this set. The most important result of the theory is the structural stability theorem:

Theorem 34 A class C1 diffeomorphism is structurally stable if and only if it satisfies Axiom A and the strong transversality property. Joel Robbin [77] proved that a C2 diffeomorphism is C1 structurally stable if it satisfies Axiom A and the strong transversality property. This result was improved by Clark Robinson [81] by relaxing the C2 requirement to C1 ; he also proved the analogous result for differential equations [79,80]. Ricardo Mañé [65] proved the converse. A simple example of a structurally stable diffeomorphism is the north-pole south-pole map of the Riemann sphere; it is given by z 7! 2z for the complex variable z. There are two hyperbolic rest points, an unstable fixed point at z D 0 and a stable fixed point at z D 1. All other points are attracted to 1 under forward iteration and to zero under backward (inverse) iteration. A more exotic example is provided by the hyperbolic toral automorphisms. To define these maps, note that a 2  2-integer matrix with determinant one determines a diffeomorphism of the plane that preserves the integer lattice. Thus, it projects to a map of the torus formed by the quotient space R2 /Z2 . In case the matrix is hyperbolic, the corresponding diffeomorphism on the torus has a dense set of periodic orbits and thus its nonwandering set is the entire torus. It can be proved that the linear hyperbolic structure of the transformation of the plane (that is, one stable and one unstable eigenspace) projects to a uniform hyperbolic structure on the entire torus; and, moreover, the strong transversality condition is satisfied. Thus, small perturbations of such a map have the same properties. This example was generalized by Dmitri Anosov and others to include a class of diffeomorphisms and vector fields in the continuous case now called Anosov dynamical systems. The most important example of a continuous flow with this property is the geodesic flow in the unit tangent bundle of a compact Riemannian manifold with negative sectional curvatures (see [4]). Attractors A natural outgrowth of the theory of stability for rest points and periodic solutions is its generalization to invariant sets. The fundamental objects of study are the attractors. Unfortunately, the definition of this concept is not universally accepted; different authors use different definitions. The following definition is perhaps the most popular: Definition 35 A closed invariant set A for a dynamical system is called an attracting set if it is contained in an open neighborhood U of the ambient space such that the

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intersection of the sequence of all forward motions of U under the action of the dynamical system is equal to A. An attracting set is called an attractor if, in addition, there is no closed invariant attracting subset of A.

Corollary 37 If the differential Eq. (23) on the plane is analytic and has a positively invariant annulus containing no rest points, then the annulus contains an orbitally asymptotically stable periodic solution (a stable limit cycle).

The existence of attractors is part of the stability theory of dynamical systems; the structure of attractors is the subject of another rich theory. For most of the history of differential equations the only known attractors (the classical attractors) were rest points, periodic orbits, and tori, which are all manifolds. The existence of attractors with fractal dimensions, called strange attractors, was noticed and proved only recently [62,82,85,95]. For differential equations in Rn , the existence of an attractor for autonomous systems is usually proved by demonstrating the existence of an (n  1)-dimensional sphere (or other separating hypersurface) such that the vector field corresponding to the differential equation points into the bounded region of space bounded by the sphere (more precisely, the inner product of the vector field and the inner normal is positive at all points on the sphere). As an example, consider the Lorenz equations on R3

As an example, consider the model of a damped pendulum with torque given by the first-order planar system

x˙ D (y  x); y˙ D rx  y  xz; z˙ D x y  bz; where , r, and b are positive constants. The origin is a rest point for the Lorenz system and, if r < 1, then L(x; y; z) :D

1 2 x C y 2 C z2 

is a Lyapunov function. In this case, the origin is globally asymptotically stable. More generally (that is, with no restriction on the size of r), there is a constant c > 0 such that the vector field points into the bounded region of space bounded by the ellipsoid rx 2 C y 2 C(z2r)2 D c. Thus, the Lorenz equations have an attractor. For some parameter values, for example  D 10, b D 8/3, and r D 28, the attractor is a fractal, called the Lorenz attractor (see [29,62,93,95]). For the special case of two-dimensional first-order autonomous systems, the existence and nature of attractors can be more precisely specified. The main result is the Poincaré-Bendixson theorem (see [15] and, for a proof, [1]): Theorem 36 An attractor of the class C1 differential Eq. (23) on the plane that contains no rest points is a periodic orbit.

˙ D v ;

v˙ D  sin C   v ;

(31)

where  and  are positive constants. By viewing the variable as an angular variable modulo 2, the phase space is the cylinder rather than the plane. Nonetheless, the Poincaré-Bendixson theory is valid on the cylinder because a bounded region of the cylinder can be flattened by a change of variables to a region of the plane. If jj > 1, then the system of differential Eq. (31) has a globally attracting periodic orbit. Three main observations can be used to construct a proof: there are no rest points, the quantity  sin C   v is negative for sufficiently large values of v > 0, and it is positive for negative values of v with sufficiently large absolute values. These facts imply the existence of an invariant annulus containing no rest points. The remainder of the proof uses Theorem 19 and the analyticity of solutions (see p. 98 in [19]). Generalizations and Future Directions The concepts of stability theory have been successfully generalized to infinite-dimensional dynamical systems (which arise from the analysis of partial differential equations) defined on Banach spaces and to abstract dynamical systems on metric spaces [13,32,33,36,83,94]. Research in this direction has produced some results on stability and the existence of attractors. No general theorem akin to Lyapunov’s Theorem 4 is known for infinite-dimensional dynamical systems. A main difficulty is that, for an unbounded operator A, the non-zero elements in the spectrum of eA may not all be given by the exponentials of elements in the spectrum of A (see, for a discussion, Chap. 2 in [20]). Thus, to determine the stability of a rest point by linearization, this spectral mapping property must be checked separately for most cases of interest. Part of the motivation for the determination of attractors in infinite-dimensional dynamical systems is the possibility of reducing the long-term dynamical properties to the analysis of a finite-dimensional dynamical system on the attractor. The fundamental organizing concept in this direction is the “inertial manifold”. Definition 38 An inertial manifold is a finite-dimensional Lipschitz manifold that is invariant under forward

Stability Theory of Ordinary Differential Equations

motions of the dynamical system and attracts all solutions of the dynamical system at an exponential rate. Inertial manifolds have been proved to exist for many dynamical systems defined by evolution-type partial differential equations, for example, reaction-diffusion equations with appropriate boundary conditions. Much current research is devoted to determining conditions that imply the existence of inertial manifolds (or their generalizations) for the equations of fluid dynamics. The conceptual framework and basic abstract theory of stability is mature and well-understood for finite-dimensional dynamical systems. This theory is currently being extended to an abstract theory for infinite-dimensional dynamical systems. The most important direction for further development is the application of this theory to specific differential equations (including partial differential equations) that arise as mathematical models in applied mathematics. Even the problem that originally motivated stability theory (that is, the stability of periodic and quasiperiodic motions of the N-body problem of classical mechanics) remains open to future research.

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Stochastic Noises, Observation, Identification and Realization with

Stochastic Noises, Observation, Identification and Realization with GIORGIO PICCI Department of Information Engineering, University of Padua, Padua, Italy Article Outline Glossary Introduction Stochastic Realization Wide-Sense Stochastic Realization Geometric Stochastic Realization Dynamical System Identification Future Directions Bibliography Glossary Wiener process A Wiener process is a continuous time stochastic process w D fw(t) ; t 2 Rg with continuous sample paths and stationary independent increments of zero mean and finite variance. The variance of the increments w(t)  w(s) must then have the form  2 (t  s) and w(t)  w(s) must have a Gaussian distribution with mean zero (Levy’s theorem [30]). The Wiener process is normalized (or standard) if  2 D 1. A p-dimensional Wiener process is a vector stochastic process having p components, w k ; k D 1; : : : ; p, which are independent Wiener processes. A wide-sense p-dimensional normalized Wiener process is a continuous time stochastic process w D fw(t) ; t 2 Rg which is continuous in mean square and has stationary orthogonal increments; i. e., Ef[w(t1 )  w(s1 )][w(t2 )  w(s2 )]0 g D I p j[s1 ; t1 ] \ [s2 ; t2 ] j where E denotes expectation, I p is the p  p identity matrix, the prime denotes transpose and j  j denotes length (Lebesgue measure). This is a weaker concept than that of a Wiener process. A Wiener process with w(0) D 0 is sometimes called a Brownian motion process. Usually a Brownian motion is only defined on the half line RC . Wiener processes and stochastic integrals with respect to a Wiener process, introduced by Wiener and Itô, are the basic building blocks of stochastic calculus [14,25]. Since stochastic integrals only depend on the increments of w, it is immaterial whether an arbitrary random vector is added to w. The equivalence class of

(wide sense) Wiener processes differing from each other by the addition of an arbitrary (constant) random vector of finite variance (e. g. w(0)), will still be called a Wiener process and denoted by the symbol dw. White noise Continuous time white noise is the formal time derivative of a Wiener process. This derivative must be understood in a suitable distributional sense since it fails to exist as a limit in any of the standard senses of probability theory. Much more standard is the notion of discrete-time (strict or wide-sense) normalized white noise process. It is a sequence of independent equally distributed (respectively orthogonal) random vectors w D fw(t) ; t 2 Zg with unit variance; i. e., ( I if s D t 0 Efw(t)w(s) g D Iı ts :D 0 if s ¤ t : This last condition alone characterizes wide-sense white noise. We shall always require that w should have zero mean. Stochastic differential and difference equations Stochastic differential equations are equations that define pathwise a continuous time stochastic process x by “local” evolution laws of the type dx(t) D f (x)dt C G(x)dw(t) ; where dw is a vector Wiener process. The equation should in reality be interpreted as an integral equation, the last term being a stochastic integral in the sense of Itô, see [14,25]. Under certain growth conditions on the coefficients the equation can be shown to have a unique solution which, in case f and G depend pointwise on x(t), is a continuous Markov process, in fact a diffusion process. In discrete time (t 2 Z) a stochastic difference equation is an object of the type x(t C 1) D f (x(t))dt C G(x(t))w(t) where now w is white noise. The solution of an equation of this type is a (discrete-time) Markov process. Stochastic dynamical systems In continuous time, a (finite-dimensional) stochastic system is a pair (x; y) of vector, say n and m dimensional, stochastic processes satisfying equations of the type dx(t) D f (x(t))dt C G(x(t))dw(t) dy(t) D h(x(t))dt C J(x(t))dw(t) : The process x, which is Markov, is called the state of the system while y is the output process. The differential notation in the second equation is merely to allow a possible additive white noise component in the

Stochastic Noises, Observation, Identification and Realization with

output process so y is actually the integral of the variable which would physically be called the output of the system. If there is no additive white noise component (J  0) this trick is unnecessary and the output variable is simply expressed as a memoryless function of the state, dy/dt D h(x(t)). A similar definition, with the obvious modifications, serves to introduce the concept of discrete-time stochastic systems. The probabilistic essence of the concept of stochastic system is more generally captured in terms of conditional independence of the sigma-algebras induced by the underlying processes. From this it can be clearly seen that the state process at time t plays the role of dynamic memory of the system: the future and the past of the output and of the state sigma-algebras are conditionally independent given the present state x(t). For a formal definition the reader is for example referred to p. 219 in [51]. We shall introduce these concepts in a wide-sense context later on. Introduction In this article we discuss some general ideas which motivate the use of stochastic dynamical models in applied sciences. We discuss the inherent mathematical problem of stochastic model building, called stochastic realization and the statistical problem of (dynamic) system identification, i. e. procedures for estimating dynamic stochastic models starting from observed data. We shall in particular discuss linear stochastic state-space models with random inputs, their construction and the relevant identification techniques. We shall assume that the reader has some general background on stationary stochastic processes, statistics [12] and linear system theory as for example exposed in the textbook [24]. There are no physical laws telling us how to obtain stochastic descriptions of nature. The laws of mechanics, electromagnetism, fluid dynamics etc. are, by nature, essentially deterministic. A fundamental empirical observation is that stochastic models generally come about as aggregate descriptions of complicated large deterministic systems. Think for example of describing mathematically the outcome of a dice-throwing experiment. Doing this by the laws of physics involves an enormous number of complicated factors such as the mechanics of a cubic-shaped rigid body flying in the air after an initial impulse applied on a specific region of one of its six faces. To describe the trajectory (i. e. the motion of the barycenter, the body orientation and the angular momenta) one should take into account, besides gravity, lift phenomena due to portance, drag, added-mass effects etc., describing the interaction

with the surrounding fluid. Then one should describe the discontinuous mechanics of the dice landing on an elastic surface etc. Modeling all of this by the known laws of physics is perhaps possible and we may in principle be able to set up a set of algebraic-differential equations allowing us to predict exactly (or “almost exactly”) which upper face of the dice will eventually show. This however, provided we knew an enormous number of geometrical and physical parameters of the system such as, for example, the dimensions of the dice, its initial position and orientation, its mass and moments of inertia, the location, direction and strength of the initial impulse, the density of the air surrounding the trajectory (depending on the temperature, humidity etc.), the possible air convection phenomena, the mechanical structure and geometry of the landing medium, etc. Indeed, this predictive model based on the (deterministic) laws of physics would require a very large number of equations and involve an unimaginable detailed prior knowledge of the parameters of the experiment. Trivially, the rudimentary stochastic model consisting of a probability distribution on the six possible outcomes f1; 2; : : : ; 6g, although not allowing an “exact”, i. e. “deterministic” prediction of the outcome, is incomparably simpler. One could venture to say that most stochastic models used in science and engineering come about, for the same reason, as simple aggregate descriptions of complicated deterministic systems. Thermodynamics, to name just one of the most conspicuous instances of this phenomenon, can be seen as stochastic aggregation of large ensembles of microscopic particles evolving in time according to the deterministic laws of mechanics [28,43]. Simplification is payed in terms of uncertainty and the predictions based on stochastic aggregate models are by nature uncertain. The modeling process hence requires us to quantify the uncertainty of aggregate modes. In a more precise sense, we should investigate their mathematical structure and describe how to construct them. This will be one of the main concerns of this article. A Brownian Particle in a Heat Bath The process of constructing aggregate models can be elucidated by a famous example, the Ford–Kac–Mazur model [20] which is probably the simplest explicitly solvable example of a problem of this nature. Consider the problem of describing the dynamics of a particle, called the Brownian particle, coupled to a “heat bath” consisting of a very large number of identical particles obeying the laws of classical Hamiltonian mechanics and moving under the influence of a quadratic potential. Mathematically, the ensemble can be described as a large

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number of coupled harmonic oscillators. The Brownian particle is the only particle of the ensemble which is assumed to be accessible to external observation. One assumes that the macroscopic observables of the system are the relative position q(t) and/or the momentum p(t) of the Brownian particle, at each instant of time. It is shown in [20] that in the limit when the number of particles in the heat bath tends to infinity, the motion of the Brownian particle is governed (exactly!) by the stochastic differential equations, ˙ D p(t) q(t) dp(t) D ap(t)dt C kdw(t) ;

(1) a>0

(2)

the second of which is the ubiquitous Langevin equation of statistical physics. This equation contains a dynamical friction term, (ap(t)), and a forcing function term w(t) which is described mathematically as a Wiener process. Physically, these terms can be interpreted as the influence of the surrounding medium on the observed particle, the friction accounting for the energy transferred from the particle to the heat bath and the forcing term w(t) representing the sum of infinitesimal “random shocks” inflicted to the particle by the surrounding medium. The stochastic process describing the temporal evolution of the observables q(t) and p(t) is a Markov diffusion process (in fact, both p(t) and the two-dimensional vector process [q(t) p(t)] are Markov). The evolution of the macroscopic observables is hence dissipative, because of the friction term, and irreversible, since diffusion processes evolve in time by the action of a semigroup which cannot be inverted to become a group. This is in accordance with the basic postulates of (macroscopic) thermodynamics. The Ford–Kac–Mazur example can be generalized [48,49,50,52]. Assume a microscopic system described by a Hamiltonian H on the phase space ˝, a smooth manifold of very large dimension 2N. The solution of the Hamilton equations determines the phase of the system, say configurations and momenta z(t) :D [q(t) p(t)]0 at each time t, uniquely in terms of the initial value z(0) 2 ˝. The correspondence defines a flow z(t) D ˚(t)z(0) on the phase space which leaves invariant the total energy, H(z(t)), of the system. The microscopic evolution is in this sense, reversible and conservative. However, microscopic phenomena are complex in the sense that they involve an enormous number of interacting components and it is impossible to keep track of the dynamics of such a complex system. One must then resort to a statistical description. A probability distribution on ˝ which is invariant for the Hamiltonian flow U(t) defines “thermal equilib-

rium”. It is known that in a finite-dimensional space any absolutely continuous U(t)-invariant probability measure admits a one parameter family of densities (z) of the Maxwell–Boltzmann type, equal to a normalization con1 stant times exp[ 2ˇ H(z)], ˇ > 0, being interpreted as the “absolute temperature”. For a quadratic Hamiltonian in a linear phase space these distributions are Gaussian. The choice of an initial phase z(0) of a system in thermal equilibrium can then be interpreted as a random choice of an elementary event in the elementary outcome space f˝; g. It follows that in thermal equilibrium any observable; i. e. any measurable function h : ˝ ! R on the phase space can be regarded as a random variable. In fact, once combined with the (measure-preserving) Hamiltonian flow z(t) D U(t)z(0), any measurable observable defines a stationary stochastic process. In general, the time evolution of any finite, say m-dimensional, family of observables fh1 ; : : : ; h m g can be identified with a vector-valued m-dimensional stationary process fy(t)g on ˝, with components y k (t; z(0)) :D h k (U(t)z(0))

(3)

where z(0) 2 ˝ is the elementary event and U(t) the measure preserving group of transformations on the underlying probability space. The main point of this story is that the “randomization” of the phase space may, under certain conditions, lead to a description of certain observable processes fy(t)g of (3) by a stochastic dynamical model of a much simpler structure than the microscopic deterministic Hamiltonian description. These could for example be representations of fy(t)g of the type y(t) D h(x(t)) where fx(t)g a finite dimensional, say Rn -valued Markov process and h is a suitable function h : Rn ! Rm . This description, in case the Markovian representation has a smaller complexity (dimension) than that of the microscopic description, would precisely meet both the general physical plausibility for a thermodynamic description and the mathematical requirement of reduction of complexity. Remark 1 This view generalizes the way of conceiving stochastic aggregation in the physical literature, e.g [31,32], where it is generally required that the observables themselves should be Markov and satisfy a Langevin equation. In general the requirement that the observables should themselves be Markov processes is seldom a reasonable one. For example, in the Brownian particle example the macroscopic observable q(t) by itself need not satisfy

Stochastic Noises, Observation, Identification and Realization with

any stochastic differential equation. Formally, q(t) is just a memoryless function of a two-dimensional Markov process x(t) :D [q(t) p(t)]0 and it is instead the “thermodynamic state” x(t) which satisfies a stochastic differential equation. Stochastic Realization Stochastic realization is the abstract version of the aggregation problem mentioned above. One studies the problem of representing an m-dimensional stationary process fy(t)g as a memoryless function of a Markov process. In this setting the physical phase space and the Hamiltonian group fU t g are generalized to an abstract probability space and to a measure preserving one-parameter group of transformations mapping ˝ onto itself (the shift group of the process). In this article we shall assume that fy(t)g is smooth with continuous values but the problem area also concerns stationary finite state-processes, see [5,19, 46,66], which have important applications in communications and biology. In the present context, the question is when does a given stationary, m-dimensional, stochastic process fy(t)g admit representations of the type y(t) D h(x(t)) where fx(t)g is a Markov diffusion process taking values on a finite-dimensional state space X and h is say a continuous function from X to Rm . Since a smooth Markov diffusion process fx(t)g is a solution of a stochastic differential equation (Chap. VI, Par. 3 in [14]), any such process must admit representations as the output of a stochastic dynamical system of the type dx(t) D f (x(t))dt C G(x(t))dw(t)

(4)

y(t) D h(x(t))

(5)

where fw(t)g is a vector m Wiener process. This stochastic dynamical system generalizes the Langevin equation representation. For a Gaussian mean square continuous stationary process, the representation simplifies to a linear model of the form dx(t) D Ax(t)dt C Bdw(t)

(6)

y(t) D Cx(t)

(7)

where A; B; C are constant matrices of appropriate dimensions. There may be a need for introducing an additive noise term also in the second (output) equation when the process y has stationary increments, a situation more general than stationarity, see [37]. Representations as the output of a stochastic dynamical system are called stochastic realizations or state-space

realizations of the process y and the Markov process x(t) is accordingly called the state process of y, or of the realization. Signal descriptions by a finite dimensional stochastic realization can be transformed into other equivalent forms such as ARMA models etc., which are also widely used in the engineering and econometric literature. It should be stressed that, irrespective of which particular equivalent form of the stochastic model, virtually all sequential signal processing, prediction and control algorithms (Kalman filtering to name just the most popular) require availability of a finite-dimensional realization of one form or another. Hence signal representation by a stochastic system is indeed a crucial prerequisite for system analysis and design in engineering and applied sciences. A substantial amount of literature on the stochastic realization problem has appeared in the last four decades. The stationary (and stationary increments) Gaussian case is now covered by a rather complete linear theory, and is surveyed e. g. in [35,37]. For a more up to date account, see the forthcoming book [41]. The nonlinear realization problem is still underdeveloped, see [45,51,62]. Wide-Sense Stochastic Realization We shall discuss only the wide-sense version of the realization problem, where random variables and processes are described in terms of first- and second-order moments. Since first- and second-order moments individuate a Gaussian distribution uniquely, the theory will in particular cover the Gaussian case. Hereafter, widesense stationarity will be simply referred to as stationarity. Also, a “wide-sense Markov process” will simply be called a “Markov process”. The wide-sense realization problem may look like a very particular modeling problem to deal with, but it is completely solvable and is general enough to reveal quite explicitly some of the important features of the solution set of stochastic realizations, some of which are quite unexpected. It is also motivated by its wide applicability since most times in practice the only reasonable way to describe a priori random phenomena is by second-order moments, say by covariances or spectra. We advise the reader that in many applied fields one may want to construct dynamical system models involving also exogenous input (or decision) variables. The stochastic realization problem of constructing linear state-space representations of a stationary process with inputs has been studied in [11,53,54,55]. In the last two references one may find also material related to the germane problem of identification with inputs.

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For reasons of space limitations we shall not touch upon this subject. A pointer to the relevant identification literature will be given in the final section of the article. We shall discuss in some detail only stochastic realization of discrete-time random processes. An analogous (and in fact somewhat simpler) treatment can be given for continuous time processes [35] but we choose discrete-time since this theory makes contact with system identification which we shall discuss later in the section entitled “Dynamical System Identification”. Given a vector (say m-dimensional) wide-sense stationary purely nondeterministic (purely nondeterministic will be abbreviated to p.n.d. in the following; this property is called linear regularity in the Russian literature, see [57]) process y D fy(t)g with t 2 Z (the integers), one wants to find representations of y in terms of a finite dimensional Markov process. In other words one wants to find linear representations of the stationary process y, of the form x(t C 1) D Ax(t) C Bw(t)

(8a)

y(t) D Cx(t) C Dw(t)

(8b)

and procedures for constructing them. Here fw(t)g is a vector, p-dimensional normalized white noise process, i. e. Efw(t)w(s)0 g D Iı(t  s) ; Efw(t)g D 0, ı being the Kronecker delta function. Note that a white noise w could be seen as a (degenerate) kind of Markov process. The reason for having the term Dw(t) in (8b) is to avoid the presence of such degenerate components in the dynamic equation for the state process x which would artificially increase its dimension. With this convention, in the extreme circumstance where y D w, the state dimension (of a minimal realization) is zero. The linear representation (8a) involves auxiliary variables, i. e. random quantities which are not given as a part of the original data, especially the n-dimensional state process x (a stationary Markov process) and the generating white noise w. The peculiar properties of these processes actually embody the desired system structure (8a). Constructing these auxiliary processes is an essential part of the realization problem. We shall restrict ourselves to representations (8a) for which  (A,B,C) is a minimal triplet; i. e. (A; B) is a reachable pair and (A; C) is an observable pair, see e. g. [24] for these notions;

 B  has independent columns. D These are classical conditions of nonredundancy of the model [24] and entail no loss of generality. From standard

spectral representation theory, see e. g., Chap. 1 in [57], it follows that Eq. (8a) admits stationary solutions if and only if the rows of the n  p matrix (e i I  A)1 B are square integrable functions of 2 [; ]. This is equivalent to the absence of poles of the function z ! (zI  A)1 B on the unit circle. Equivalently, the eigenvalues of A of modulus one, if any, must be “unreachable” for the pair (A; B). When eigenvalues on the unit circle are present, to guarantee stationarity they must be simple roots of the minimal polynomial of A. These eigenvalues, necessarily in even number say 2k, give then rise to a sum of k uncorrelated sinusoidal oscillations with random amplitude, the so-called purely deterministic component of the process. This purely deterministic component of the stationary Markov process x obeys a fixed undriven (i. e. deterministic) linear difference equation of the type x(t C 1) D Ao x(t), which is the restriction of (8a) to the modulus one eigenspace of A. The initial conditions are random variables independent of the driving noise w. Since there is no stochastic forcing term in this restricted state equation, it follows that the presence of a purely deterministic component in the Markov process necessarily implies that the pair (A; B) must be nonreachable. The first of the two conditions of nonredundancy listed above hence excludes the presence of a purely deterministic component in the state, and therefore also in the output process y. From this discussion it is immediately seen that there are stationary solutions x of the difference Eq. (8a) which are p.n.d. Markov processes if and only if A does not have eigenvalues of modulus one; i. e. j(A)j ¤ 1 :

(9)

When j(A)j < 1, the representation (8a) is called forward or causal. When j(A)j > 1 the representation is called backward or anticausal. Traditionally the causality of a representation, i. e. the condition j(A)j < 1 has been regarded as being equivalent to stationarity. In fact, stationarity permits a whole family of representations (8a) of the (same) process x, where the matrices A can have very different spectra. Formulas for computing a backward representation starting from a forward one or vice versa are given in [34,36]. The two representations (in particular the relative A matrices) are related by a particular family of linear state feedback transformations [47]. The covariance function of a zero-mean wide sense stationary process y is the m  m matrix function k 7!  k , defined by  k :D Efy(t C k)y(t)0 g D Efy(k)y(0)0 g

k 2Z:

This matrix function will be the initial data of our problem. Since y is p.n.d. the function  admits a Fourier transform

Stochastic Noises, Observation, Identification and Realization with

Stochastic Noises, Observation, Identification and Realization with, Figure 1 Shaping filter representation of a stochastic process

causality (j(A)j < 1) x(t) is a function of the infinite past fw(s); s < tg and hence is uncorrelated with w(t) so that the two terms in the right hand side of (8a) are also uncorrelated and their variances add. In fact, by reachability of the model, P is actually (symmetric and) positive definite. Now, introduce the decomposition ˚(z) D ˚C (z) C ˚C

called the spectral density matrix of y, given by ˚(z) D

1 X

 t zt

tD1

where z D e i ; 2 [; ]. It is easy to see that the spectral density matrix of a process admitting a state-space realization (8a) must be a rational function of z. This fact follows easily by eliminating the variable x in (8a) and thereby expressing y as the output of a linear filter as in Fig. 1 below whose transfer function W(z) D C(zI  A)1 B C D is a rational function of z, and then using the classical Kintchine and Wiener formula for the spectrum of a filtered stationary process [28,69] so that,  0 ˚ (z) D W(z)W 1z : (10) Hence we have Proposition 1 The transfer function W(z) D C(zI  A)1 B C D of any state space representation (8a) of the stationary process y, is a spectral factor of ˚ , in the sense that it satisfies the spectral factorization equation (10). It is easy and instructive to check this directly. Supposing that one has a causal realization (8a), one can compute for k  0, k1 h X CAk1s Bw(t C s)  k D E CAk x(t) C sD0

i 0 C Dw(t C k) Cx(t) C Dw(t) D CAk1 C¯ 0 C 12 0 ı(t) ;

(11) (12)

where C¯ 0 D Ex(t C1)y(t)0 and 0 D Ey(t)y(t)0 . By identifying terms, these matrices can be expressed directly in terms of the realization parameters A; B; C; D, by P D APA0 C BB0 ¯0

0

C D APC C BD 0

(13a) 0

0 D CPC C DD

(13b) 0

(13c)

the matrix P D Ex(t)x(t)0 being the covariance matrix of the state process. Note that P obeys Eq. (13a) since by

 1 0 z

(14)

where ˚C (z) D 1/20 C 1 z1 C 2 z2 C    is analytic outside of the unit circle. Clearly ˚C (z) must be the Fourier transform of the “causal” tract of the covariance function , computed in (12). Hence ˚C (z) D C(zI  A)1 C¯ 0 C 12 0 :

(15)

This shows that the spectrum of a process y described by a state-space model (8a), besides being a rational function of z, is directly expressible in parametric form in terms of the parameters of a causal realization. This explicit computation of the spectrum seems to be due to R.E. Kalman and B.D.O. Anderson [4,21]. “Classical” stochastic realization theory, developed in the late 1960s [3,4,17], deals with the inverse problem of computing the parameters A; B; C; D of a state space realization starting from the spectrum (or, equivalently, of the covariance function) of a stationary process y. Of course one assumes here that y is p.n.d. and has a spectral density ˚ which is a rational function of z D e j . This inverse problem is just a parametric version of the spectral factorization problem: Given a rational spectrum ˚ (z) one looks for rational spectral factors parametrized as W(z) D C(zI  A)1 B C D, solutions of (10). Since rational factors of a given spectral density matrix are in general infinitely many, one must preliminarily single out the interesting solutions by imposing certain restrictions. One very reasonable restriction which is imposed on the solutions of (10) is that the spectral factors should be of minimal degree. The degree of a proper rational matrix function is by definition the dimension n of the state of a minimal realization. Minimal degree (for short, minimal) spectral factors must have a degree which is exactly one half of the degree of the spectral density matrix ˚(z). The solution of the inverse problem is described in the following proposition. Proposition 2 Assume the spectrum is given in the parametric form (15) where j(A)j < 1 and (A; C; C¯0 ) is a minimal triplet of dimension n. Then the minimal degree spectral factors of ˚(z), analytic infjzj > 1g, are in one-to-one correspondence with the symmetric n  n matrices P solving

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the Linear Matrix Inequality (LMI) 

P  APA0 C¯ 0  APC 0 0 M(P) :D C¯  CPA0 0  CPC 0

(16)

where the symbol  means positive semidefinite. The correspondence is to be understood in the following sense: Corresponding to each solution P D P0 of (16), consider B of M(P), the full column rank matrix factor D

M(P) D

B D



[B0 D0 ]

(17)

(this factor is unique modulo right multiplication by orthogonal matrices) and form the rational matrix W(z) :D C(zI  A)1 B C D :

(18)

Then (18) is a minimal realization of a minimal analytic spectral factor of ˚(z) and all minimal analytic factors can be obtained in this way. Note that any matrix P solving the LMI satisfies the system (12) and hence can be interpreted as the state covariance of a causal realization of y. It follows that any P solving the LMI is necessarily positive definite. In fact it has been shown [17] that the set of symmetric solutions to the LMI (16) P :D fP D P 0 ; M(P)  0g

is a closed, bounded and convex set with maximal and minimal elements P ; PC 2 P which satisfy P  P  PC

for all P 2 P

where P1  P2 means that P2  P1  0 (is positive semidefinite). The existence of solutions to the LMI is equivalent to the property of ˚C (z) defined in (14) of being a positivereal matrix function, namely that ˚C should be analytic in fjzj > 1g and that <e ˚C (e j )  0. The second condition is automatically satisfied when ˚ admits spectral factors since 2<e ˚C (e j ) D W(e j )W(e j )  0 (the star meaning complex conjugate transpose). This property can be characterized in general, for not necessarily stable representations, by the so-called positive-real lemma discovered in the 1960s by Kalman, Yakubovich, and Popov [22,56,71] in the context of dissipative systems and stability theory. The first application of the Kalman–

Yakubovich–Popov theory to factorization of spectral density functions and to stochastic realization (called stationary covariance generation) is due to B.D.O. Anderson [3,4]. Far reaching generalizations to not necessarily stable systems have been pursued by J.C. Willems in his widely quoted paper [70]. The terminology “positive real” derives from network synthesis. Under certain conditions on the zeros of the spectrum, discussed e. g. in [18], the matrix (P) :D 0  CPC 0 is nonsingular (and hence positive definite) for all P 2 P . In this case (16) is equivalent to the nonnegativity of the Schur complement of (P) in the matrix M(P), which can be written P  APA0  (C¯ 0  APC 0 )(P)1 (C¯ 0  APC 0 )0  0 : (19) This is the Algebraic Riccati Inequality (ARI) of spectral factorization, first analyzed in the work of [3,17]. The solutions of the ARI with equality sign; i. e. the solutions of the Algebraic Riccati Equation (ARE) P  APA0  (C¯ 0  APC 0 )(P)1 (C¯ 0  APC 0 )0 D 0 (20) make M(P) of minimal rank m D dim y and hence correspond to square spectral factors (18). These spectral factors all have the same denominator matrix A and hence only differ by the location of their zeros. In particular one may show that P ; PC solve the ARE and the two spectral factors W and WC corresponding to P ; PC (are square and) have all of their zeros inside and respectively outside of the unit circle. In other words, W is the minimum phase and WC the maximum phase spectral factor. These spectral factors are essentially unique. All other square spectral factors are obtained (in rough terms) by “flipping” some of the zeros of W to their reciprocals outside of the unit circle. The maximum phase factor, WC , is obtained by flipping all of the zeros of W to their reciprocals. More information on the zero structure of the minimal spectral factors can be found in [39]. In the next section we shall analyze in more detail the correspondence between spectral factors and stochastic realizations. As we shall see, for square spectral factors this correspondence is one-to-one and hence this particular class of stochastic realizations, for which dim w D dim y D m can be classified according to their zero location. The LMI plays an important role in many areas of system theory such as stability theory, dissipative systems and is central in H 1 control and estimation theory. It seems to be much less appreciated in the scientific community that it plays a very basic role in modeling of stationary random signals as well. As we shall see in the next section certain

Stochastic Noises, Observation, Identification and Realization with

solutions of the LMI (or of the ARI) have special probabilistic properties and are related to Kalman-filter or “innovations-type” realizations. Geometric Stochastic Realization The classical stationary covariance realization theory of the previous section is purely distributional as it says nothing about representation of random quantities in a truly probabilistic sense (i. e. how to generate the random variables or the sample paths of a given process, not just its covariance function). This was implicitly pointed out by Kalman already in [23]. In the last decades a geometric coordinate-free approach to stochastic modeling has been put forward in a series of papers by Lindquist, Picci, Ruckebusch et al. [33,34,35,58,59,60] which aims at the representation of random processes in this more specific sense. This motivation is also present in the early papers by Akaike [1,2]. A main point of the geometric approach is to provide procedures for the construction of the random quantities defining a stochastic state-space realization. In particular the state space of a realization is defined in terms of the conditional independence relation between past and future of the signals involved. This relation is intrinsically coordinate-free and in the present setting involves only linear subspaces of a given ambient Hilbert space of random variables, typically made of linear functionals of the variables of the process y to be modeled (but in some situations other random data may be used to construct the model). Constructing the State Space of a Stationary Process The theme of this section will be a review of geometric realization theory for the stochastic process y. The geometric theory centers on the idea of Markovian Splitting Subspace for the process y. This concept is the probabilistic analog of the deterministic notion of state space of a dynamical system and captures at an abstract level the property of “dynamic memory” that the state variables have in deterministic dynamical system theory. Once a stochastic state space is given the construction of the auxiliary random quantities which enter in a state space model and in particular the state process is fairly obvious. The state vector x(t) of a particular realization can be regarded just as a particular basis for the state space, hence once a statespace is constructed, finding state equations is just a matter of choosing a basis and computing coordinates. Notation 1 In what follows the symbols _, C and ˚ will denote vector sum, direct vector sum and orthogonal vector sum of subspaces, the symbol x? will denote the orthogo-

nal complement of a (closed) subspace X of a Hilbert space with respect to some predefined ambient space. Given a collection fX˛ j ˛ 2 Ag of subsets of a Hilbert space H we shall denote by spanfX ˛ j ˛ 2 Ag the closure in H of the linear (real) vector space generated by the collection. The orthogonal projection onto the subspace X will be denoted by the symbol E( j X) or by the shorthand EX . The notation E(z j X) will be used also when z is vector-valued. The symbol will then denote the vector with components E(z k j X); k D 1; : : :. For vector quantities, jvj will denote Euclidean length (or absolute value in the scalar case). Let y be a zero mean stationary vector process with finite second-order moments defined on some underlying probability space f˝; A; g and let L2 f˝; A; g denote the Hilbert space of second-order random variables defined on ˝, endowed with the inner product < ;  >D E fg. Let H(y) be the (closed) linear subspace of L2 f˝; A; g, linearly generated by the variables of the process; i. e. H(y) :D spanfy k (t) ; k D 1; 2; : : : ; m; t 2 Zg : If y is generated by a linear model of the type (8a) this Hilbert space will in general be a proper subspace of the space H(w), generated by the input white noise of the model (8a). All random variables of the stochastic system (8a) belong to H(w) and for this reason H(w) is called the ambient space of the model. By stationarity of w, H(w) comes naturally equipped with a unitary operator U, called the shift of w, such that Uw i (t) D w i (t C 1) for i D 1; ˚ 2;t : : : ; p and all t 2 Z (stationarity). Note that the family U ; t 2 Z forms a one parameter group of unitary operators on the ambient space; in particular, U t D (U  ) t . The pair (H(w); U) is also called a stationary Hilbert space. Clearly the processes x and y will also be stationary with respect to U. The past and future subspaces (at time zero) of a stationary process are H (y) :D spanfy k (t) ; k D 1; 2; : : : ; m; t < 0g HC (y) :D spanfy k (t) ; k D 1; 2; : : : ; m; t  0g : Because of stationarity everything propagates in time by the action of the unitary group, e. g. the past and fut  ture at time t are obtained as H t (y) D U H (y) and C t C H t (y) D U H (y). For this reason the geometric definitions and constructions to follow, although valid for an arbitrary time instant will usually be referred to the time instant t D 0. Let X be a subspace of some large stationary Hilbert space H of wide-sense random variables containing H(y). Define X t :D U t X;

X t :D _st Xs ;

XC t :D _st Xs :

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Stochastic Noises, Observation, Identification and Realization with

Definition 1 A Markovian Splitting Subspace X for the process y is a subspace of H making the vector sums H(y) _ X and H(y)C _ XC conditionally orthogonal (i. e. conditionally uncorrelated) given X, denoted, H(y) _ X ? H(y)C _ XC j X:

(21)

The conditional orthogonality condition (21) can be equivalently written as E[H(y)C _ XC j H(y) _ X ] D

The following representation Theorem provides the link between Markovian splitting subspaces and scattering pairs. Theorem 1 ([35]) The intersection

E[H(y)C _ XC j X]

(22)

E[H(y) _ X j H(y)C _ XC ] D 



E[H(y) _ X j X]

(23)

which formalize the intuitive meaning of the splitting subspace X as a dynamic memory of the past (future) for the purpose of predicting the joint future (past). It follows in particular that X is Markovian (i. e. X ? XC j X) and any basis vector x :D [x1 ; x2 ; : : : ; x n ]0 in a (finite-dimensional) Markovian splitting subspace X generates a stationary Markov process x(t) :D U t x; t 2 Z which, as we shall see in a moment, serves as a state for the process y. The subspace X is called proper, or p.n.d. if  C C \ t Y t _ X t D f0g; and \ t H(y) t _ X t D f0g :

¯ : H(y)C _ XC D HC (w)

If X is proper, every Markov process x(t) :D U t x is purely nondeterministic. The subspaces S :D H(y) _ X and S :D YC _ XC

X DS\S

(25)

of any scattering pair of subspaces of H is a Markovian splitting subspace. Conversely every Markovian splitting subspace can be represented as the intersection of a scattering pair. The correspondence X $ (S; S) is one-to-one, the scattering pair corresponding to X being given by S D H(y) _ X

S D H(y)C _ XC :

(24)

associated to a Markovian Splitting subspace X, play an important role in the geometric theory of stochastic systems. They are called the scattering pair of X as they can be seen to form an incoming–outgoing pair in the sense of Lax–Phillips Scattering Theory [29]. Definition 2 Given a stationary Hilbert space (H; U) containing H(y), a scattering pair for the process y is a pair of subspaces (S; S) satisfying the following conditions for a stationary process,

(26)

The process of forming scattering pairs associated to X should be thought of as an “extension” of the past and future spaces of y. The rationale for this extension is that scattering pairs have an extremely simple splitting geometry due to the fact that S?Sj S\S

Obviously for the existence of proper splitting subspaces y must also be purely nondeterministic [57]. Properness is, by the Wold decomposition theorem, equivalent to the existence of two vector white noise processes w and w¯ such that, H(y) _ X D H (w) ;

1. U  S S and US S, i. e. S and S are invariant for the left and right shift semigroups (this means that S t is increasing and S t is decreasing with time). 2. S _ S D H 3. S  H(y) and S  H(y)C ? 4. S? S or, equivalently, S S

(27)

equivalent to S_SDS

?

˚ (S \ S) ˚ S?

(28)

which is called perpendicular intersection. It is easy to show that Property (4) in the definition of a scattering pair is actually equivalent to perpendicular intersection. This property of conditional orthogonality given the intersection can also be seen as a natural generalization of the Markov property. Indeed for a Markovian family of subspaces X D X \ XC and S D X ; S D XC intersect perpendicularly with intersection X. Note that A ? B j X ) A \ B X but the inclusion of the intersection in the splitting subspace X is only proper in general. For perpendicularly intersecting subspaces, the intersection is actually the unique minimal subspace making them conditionally orthogonal. Theorem 1 is the fundamental device for the construction and classification of Markovian splitting subspaces. ¯ t the (wandering) subspaces spanned Denote by W t ; W by the components, at time t, of the generating noises w(t) ¯ and w(t), of the scattering pair of X. Since S tC1 D S t ˚ W t ;

(29)

Stochastic Noises, Observation, Identification and Realization with

we can write, X tC1 D S tC1 \ S tC1 (S t \ S tC1 ) ˚ (W t \ S tC1 ) (30) Since S t is decreasing in time, we have S t \ S tC1 X t and by projecting the shifted basis Ux(t) :D x(t C 1), onto the last orthogonal direct sum above, the time evolution of any basis vector x(t) :D [x1 (t); x2 (t); : : : ; x n (t)]0 in X t , is described by a linear equation of the type x(t C 1) D Ax(t) C Bw(t) : It is also easy to see that, by the p.n.d. property, A must have all its eigenvalues strictly inside of the unit circle. Naturally, by decomposing instead S t1 D St ˚ Wt one could have obtained a backward difference equation model for the Markov process x, driven by the backward generating ¯ noise w. To complete the representation, note that by definition of the past space, y(t) 2 (S tC1 \ S t ). Inserting the decomposition (29) and projecting y(t), leads to a state-output equation of the form y(t) D Cx(t) C Dw(t) : Here one could also obtain a state-output equation driven ¯ the same noise driving the backby the backward noise w, ward state model obtained before. As we have just seen, any basis in a Markovian splitting subspaces produces a stochastic realization of y. It is easy to reverse the implication. In fact the following fundamental characterization holds. Theorem 2 ([33,37]) The state space X D spanfx1 (0); x2 (0); : : : ; x n (0)g of any stochastic realization (8a) is a Markovian splitting subspace for the process y. Conversely, given a finite-dimensional Markovian splitting subspace X, to any choice of basis x(0) D [ x1 (0); x2 (0); : : : ; x n (0) ]0 in X there corresponds a stochastic realization of y of the type (8a). Once a basis in X is available, there are obvious formulas ¯ of the corresponding to compute the parameters (A; C; C) stochastic realization, namely A D Ex(t C 1)x(t)0 P1

(31)

C D Ey(t)x(t)0 P1

(32)

C¯ D Ey(t  1)x(t)0

(33)

where P D Ex(t)x(t)0 is the state covariance matrix (i. e. the Gramian matrix of the basis). The matrices B and D, however, are related to the (unobservable) generating white noise w and require the solution of the LMI. This

abstract procedure which permits us to compute the parameters of a stochastic realization once the state has been constructed, can be rendered quite concrete and forms the conceptual basis of subspace identification. Stochastic realizations are called internal when H D H(y), i. e. the state space is built from the Hilbert space made just of the linear statistics of the process y. For identification the only realizations of interest are the internal ones. A central problem of geometric realization theory is to construct and to classify all minimal state spaces, i. e. the minimal Markovian splitting subspaces for the process y. The obvious ordering of subspaces of H by inclusion, induces an ordering on the family of Markovian splitting subspaces. The notion of minimality is most naturally defined with respect to this ordering. Note that this definition is independent of assumptions of finite-dimensionality and applies also to infinite-dimensional Markovian splitting subspaces, i. e. to situations where comparing dimension would not make much sense. Definition 3 A Markovian splitting subspace is minimal if it does not contain (properly) other Markovian splitting subspaces. The study of minimality forms an elegant chapter of stochastic system theory. There are several known geometric and algebraic characterizations of minimality of splitting subspaces and of the corresponding stochastic statespace realizations. Since, however, the discussion of this topic would take us too far from the main theme of the paper we shall refer the reader to the literature [35,37]. Contrary to the deterministic situation minimal Markovian splitting subspaces are nonunique. Two very important examples are the forward and backward predictor spaces (at time zero): 

X :D EH HC

C

XC :D EH H

(34)

for which we have the following characterization [35]. Proposition 3 The subspaces X and XC are minimal Markovian splitting subspaces contained in the past H , and, respectively, in the future HC , of the process y. A basis in the forward predictor space X originates a stationary state-space model in which the state variables are linear functionals of the past history of the process y, i. e. x(t) 2 H t (y). In other words the state coincides with its best estimate (the orthogonal projection), E[x(t) j H t (y)] given the past of y. It follows that the dynamical equations (8a) describe in this case a steady-state Kalman predictor and the input white noise w D w is the steady state innovation process of y.

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Stochastic Noises, Observation, Identification and Realization with

Before closing this section we should remark that a scattering picture emerges also in some of the early papers on the “unitary dilation” approach to statistical mechanics, e. g. [31,32]. It is remarkable that the abstract scattering picture described in this section makes contact with this literature. The dilation approach, however, deals (in our language) with the reverse problem of describing the stationary shift group starting from (Markov) processes which are described by a Langevin-type equation. This is a curious and (in our view) somewhat unnatural point of view which apparently does not lead to capture the phenomenon of nonuniqueness of the macroscopic description of the observables. Whether this has any physical relevance is however not clear to us. Dynamical System Identification System identification is the problem of describing an observed time series (e. g. a sequence of real numbers representing stock prices, temperatures in a room, sampled audio or video signals, etc.) by a linear dynamic model, in particular by a state-space model of the type (8a). It is a widely accepted viewpoint that system identification should be regarded as a statistical problem [40,42]. This is so because the experimental data which one wants to model very often come from complicated, often unknown, nonlinear physical phenomena subject to unknown interactions with the environment, and it would in general be impossible to attempt a “physical” modelization from first principles. Consequently the models describing the data should be stochastic. The general goal is to get accurate models which are as simple as possible. The problem can then be approached from (at least) two conceptually different viewpoints. Identification by Parametric Optimization This is the mainstream “optimization” approach, based on the principle of minimizing a suitable measure of discrepancy between the observed data and the data predicted by a probability law underlying a certain chosen model class. Well-known examples of distance functions are the likelihood function, or the average squared prediction-error of the observed data corresponding to a particular model. Except for rather trivial model classes, these distance functions depend nonlinearly on the model parameters and the minimization can only be done numerically. Hence the optimization approach leads to iterative algorithms in the parameter space, say in the space of minimal (A; B; C; D) matrix quadruples which parametrize a chosen model class. In spite of the fact that this has been almost the only accepted paradigm in system identification

for many decades, [42,61], this approach has several drawbacks, including the need of unique parametrization of multivariable models, the fact that the cost function generally has complicated local minima which, for moderate or large dimension of the model are very difficult to detect, and the inherent insensitivity of the cost to variations of some parameters and the corresponding ill-posedness of the estimation problem. It seems that these limitations are a consequence of the intrinsically “blind” philosophy which underlies setting the problem as a parameter optimization problem. Almost all problems of controller and/or estimator design could in principle be formulated just as parametric optimization after choosing a certain family of parametric structures for the controller or the estimator. Pushing this philosophy to the extreme, in principle one would not need the maximum principle, Kalman filtering, H 1 theory, etc. (in fact one would not need system and control theory altogether), since everything could be reduced to a nonlinear programming problem in a suitable space of controller or estimator parameters. It is dubious however, whether any real progress in the field of control and estimation could have occurred by following this type of paradigm. Identification by Stochastic Realization This is commonly referred to as “subspace methods” identification or “subspace identification” tout-court. Subspace identification is essentially a sample version of stochastic realization. Under a reasonable assumption of second-order ergodicity of the process y which has produced the observed data, one can set up a “sample” counterpart of the abstract Hilbert space operations of stochastic realization theory. In this setting the geometric operations of stochastic realization can be translated into statistical operations on the observed data. One obtains a statistical theory of model building which is in a sense perfectly isomorphic to the abstract probabilistic realization theory. The main point is the idea of constructing first a (sample) state space for the process y, starting from certain vector spaces, called the future and past spaces associated with the observations. According to the recipe of (34) the state space is constructed by orthogonal projection of the future onto the past, as seen in the previous section. Successively, a well conditioned basis is chosen in the state space e. g. by principal components (canonical correlation) analysis. Once a “robust” basis; i. e. a state vector, is chosen, the parameters A; C; C¯ of the model are computed by formulas analogous to (31),(32),(33). The final step of finding the B and D matrices requires solving a Riccati equation.

Stochastic Noises, Observation, Identification and Realization with

By similarity with realization theory, one recognizes that the inherent nonlinearity of model identification has to do with the quadratic nature of the spectral factorization problem. As we have seen, spectral factorization for statespace models involves the solution of a Riccati equation (or more generally of a linear matrix inequality), a problem which has been the object of intensive theoretical and numerical studies in the past three decades. The nonlinearity of the stochastic system identification problem is hence of a well-known and well-understood kind and is much better dealt with by the explicit methods of Riccati solution developed in system theory rather than by nonspecific optimization algorithms. This has allowed a systematic introduction of very reliable and efficient numerical linear algebra tools to solve the problem. The “subspace” approach has been suggested more or less implicitly by several authors in the past [6,13,15,44] but is first clearly presented in [63]. Various extensions to cover identification of systems with inputs have appeared since this paper and there is now a large literature on the subject. For references we refer the reader to the recently published book [27]. Since the optimization approach is well covered in the literature, we shall just analyze subspace identification below. The Hilbert Space of a Time Series The crucial conceptual step in subspace identification is a proper identification of the Hilbert space of random data in which modeling takes place. We shall briefly illustrate this point, following [38]. We shall initially consider an idealized situation in which the observed data (time series) fy0 ; y1 ; : : : ; y t ; : : :g

y t 2 Rm

(35)

is infinitely long. We shall assume that the limit for N ! 1 and for any   0, of the time averages

NCt X0 1 y tC y 0t !  N C 1 tDt

for arbitrary t0 , which can be read as a kind of “statistical regularity” of the (future) data. The ergodicity (36) is of course unverifiable in practice as it says something about data which have not been observed yet. Some assumption of this sort about the mechanism generating future data seems however to be necessary to even formulate the identification problem. We shall call  the true covariance of the time series fy t g. Now, for each t 2 ZC define the m  1 matrices y(t) :D [y t ; y tC1 ; y tC2 ; : : :]

(38)

and consider the sequence y :D fy(t) j t 2 ZC g. We shall make this sequence of semi-infinite matrices into an object isomorphic to the stationary processes y. Define the vector space Y of scalar semi-infinite real sequences obtained as finite linear combinations of the rows of y, Y :D

nX

a0k y(t k );

a k 2 R m ; t k 2 ZC

o

(39)

This space can be naturally made into an inner product space in the following way. First, define the bilinear form h; i on the generators by letting N

N!1

(36)

exists. It can be shown that this limit is a function  !  of positive type. In the continuous-time setting, functions admitting an “ergodic” limit of the sample correlation function (36), have been studied in depth by Wiener in his famous work on Generalized Harmonic Analysis. Although a systematic translation of the continuous-time results of Wiener into discrete-time seems not to be available in the literature, it is quite evident that a totally analogous set of results holds also for discrete-time signals. In

(37)

0

ha 0 y(k); b0 y( j)i :D lim

N

1 X y tC y 0t N C 1 tD0

particular it is rather easy to show, by adapting Wiener’s proof for continuous time, that the limits of the time averages (36) form a matrix function  of positive type, in other words a bona-fide stationary covariance matrix, see [67,68] which can then be identified with the “true” covariance function of an underlying stochastic process. The assumption actually implies that

1 X 0 a y tCk y 0tC j b N C 1 tD0 D a 0  k j b;

(40)

for a; b 2 Rm , and then extend it by linearity to all elements of Y . Let a :D fa k ; k 2 ZC g be a sequence of vectors a k 2 Rm , with compact support in ZC , and let a0 :D fa0k g. A generic element  of the vector space Y can be represented as X D a0k y(k) D a0 y k

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Stochastic Noises, Observation, Identification and Realization with

Let us assume that the infinite block-symmetric Toeplitz matrix 3 2 0 1 : : :  k : : : 6 01 0 1 : : : : : :7 7 6 6 : :: 7 :: T D 6 :: (41) 7 : : 7 6 0 5 4 0 k ::: constructed from the “true” covariance sequence f0 ; 1 ; : : : ;  k ; : : :g of the data, is positive definite. Since the bilinear form (40) on Y can be represented by the quadratic form X a0k  k j b j D a0 T b h; i D ha0 y; b0 yi D kj

it can be seen that the bilinear form is nondegenerate (unless  D 0 identically) and defines a bona-fide inner product. As usual one identifies elements whose difference has norm zero (this means h; i D 0 ,  D 0). By closing the vector space Y with respect to the norm induced by the inner product (40), one obtains a real Hilbert space of semi-infinite sequences which hereafter we shall still denote Y . This is the basic data space on which our models will be defined. The shift operator U operates on the family of semiinfinite matrices (38), by the rule Ua 0 y(t) D a 0 y(t C 1)

t 2Z;

a 2 Rm

It is easy to see that U is a linear map which is isometric with respect to the inner product (40) and can be extended by linearity to all of Y . This Hilbert space framework for time series was introduced in [38]. It is shown in this reference that the “stationary Hilbert space” (Y ; U) is isomorphic to the standard stochastic Hilbert space setup in the L2 theory of second-order stationary random processes. By virtue of this isomorphism one can formally think of the observed time series (35) as an ergodic sample path of some Gaussian stationary stochastic process y defined on a true probability space, having covariance matrices equal to the limit  of the sum (36) as N ! 1. Linear functions and operators on the “tail sequences” y(t) correspond to the same linear functions and operators on the random variables of the process y. In particular the second-order moments of y can be computed in terms of the tail sequences y, by substituting expectations with ergodic limits of the type (40). Since second-order properties are all what matters in our setting, one may even regard the tail sequence y of (38) as being the same object as the underlying stochastic process y. The usual probabilistic language can be adopted in the present setting provided one

identifies real random variables as semi-infinite strings of numbers having the “ergodic property” described at the beginning of this section. The inner product of two semiinfinite strings  and  in Y corresponds in particular to the expectation Efg; i. e. h; i D Efg:

(42)

This unification of language permits us to carry over in its entirety the geometric theory of stochastic realization derived in the abstract L2 setting to the time series framework. In particular, the past and future subspaces of the “processes” y at time t are defined as the closure of the linear vector spaces spanned by the relative past or future “random variables” y(t), in the metric of the Hilbert space Y . The only difference to keep in mind here is the different interpretation that representation formulas like (8a) have in this context. The equalities involved in the representation ( x(t C 1) D Ax(t) C Bw(t) (43) y(t) D Cx(t) C Dw(t) are now to be understood in the sense of equalities of elements of Y , i. e. as asymptotic equality of sequences in the sense of Cesàro limits. In particular the equality signs in the model (43) do not necessarily imply that the same relations hold for the sample values y t ; x t ; w t at a particular instant of time t. This is in a certain sense similar to the “with probability one” interpretation of the equality sign to be given to the model (43) in case the variables are bona-fide random variables in a probability space. Consider the orthogonal projection E[  j X] of a (row) random variable  onto a subspace X of the space Y . In the probabilistic L2 setting this has the wellknown interpretation of wide-sense conditional expectation given the random variables in X (a true conditional expectation, in the case of Gaussian distributions). In this setting the projection operator has an immediate and useful statistical meaning. Assume for simplicity that X is given as the rowspace of some matrix of generators X, then the projection E[  j X] has exactly the familiar aspect of the least squares formula expressing the best approximation of the vector  as a linear combination of the rows of X. For, writing E[  j X ] to denote the projection expressed (perhaps nonuniquely) in terms of the rows of X, the classical linear “conditional expectation” formula leads to E[  j X ] D  X 0 [X X 0 ]] X ;

(44)

which is the universally known “least squares” formula of statistics. The pseudoinverse ] can be substituted by a true inverse in case the rows of X are linearly independent.

Stochastic Noises, Observation, Identification and Realization with

The Question of Statistical Efficiency There are questions about the statistical significance (what are the uncertainty bounds on the parameters and on the estimated transfer functions etc.) which are easily and naturally addressed in the optimization framework but harder to be addressed in the subspace/realization approach to identification. The main difference with the mainstream ¯ is statistical approach is that the estimation of (A; C; C) not done directly by optimizing a likelihood or other distance functions but by just matching second-order moments, i. e. by solving the Eqs. (46). This way of proceeding can be seen as an instance of estimation by the method of moments described in the statistical textbooks e. g. p. 497 in [12], a very old idea used extensively by K. Pearson in the beginning of the last century. The underlying estimation principle is that the parameter estimates should match exactly the sample second-order moments and is close in spirit to the wide-sense setting that we are working in. It does not involve optimality or minimal distance criteria between the “true” and the model distributions. For this reason it is generally claimed in the literature that one should expect better results (in the sense of smaller asymptotic variance of the estimates) by optimization methods. However, a proof that subspace methods can be efficient (under certain conditions which will be too long to report here) has appeared recently [8]. Usable expressions for the asymptotic covariance of the parameter estimates are given in [9]. The arguments in the derivations are, however, far from trivial. Subspace Algorithms Most standard algorithms for subspace identification can be shown to be essentially equivalent to the following three-step procedure (see e. g. [6,38]), 1. The first step is the estimation of a finite sequence of sample covariance matrices fˆ 0 ; ˆ 1 ; : : : ; ˆ g

(45)

from the observed data. Since the data are necessarily finite,  must be also finite (and in general small compared to the data length). 2. The second step is identification of a rational model for the covariance sequence. This is a minimal partial realization (also called “rational extension”) problem. Given a finite set of experimental covariance data one is asked to find a minimal value of n and a minimal triplet ¯ of dimensions n  n, m  n and of matrices (A; C; C), m  n respectively, such that ˆ k D CAk1 C¯ 0

k D 1; : : : ; :

(46)

Recall that (A; C; C¯ 0 ) is a minimal triplet, in the sense of deterministic linear system theory, if the pairs of matrices (A; C) and (A; C¯ 0 ) are an observable and, respectively, a reachable pair, see e. g. [24]. There are wellknown algorithms for computing minimal partial realizations (see e. g. [72] for an efficient algorithm) thereby ¯ of producing “estimates” of the parameters (A; C; C) a minimal realization of a rational spectral density matrix of the process. 3. The third step is to compute, starting from the realization of the rational spectrum estimated in step two, a stationary state-space model. Typically one is interested in the innovation model. This is accomplished by computing the minimal solution P of the Linear Matrix Inequality (16), or, equivalently, the minimal solution of the associated algebraic Riccati equation. A warning is in order concerning the practical use of these algorithms in that some nontrivial mathematical questions related to positivity of the estimated spectrum are often overlooked in the implementation. This issue is thoroughly discussed in [38] and here we shall just give a short summary. ¯ interpolatIn determining a minimal triplet (A; C; C) ing the partial sequence (45) so that CAk1 C¯ 0 D ˆ k k D 1; 2; : : : ; , we also completely determine the infinite sequence fˆ 0 ; ˆ1 ; ˆ 2 ; ˆ 3 ; : : :g

(47)

by setting ˆ k D CAk1 C¯ 0 for k D  C 1;  C 2; : : : . This sequence is called a minimal rational extension of the finite sequence (45). The attribute “rational” is due to the fact that 1 ˚ˆ C (z) :D ˆ 0 C ˆ 1 z1 C ˆ 2 z2 C : : : 2 1 D ˆ 0 C C(zI  A)1 C¯ 0 2

(48)

is a rational function. In order for (47) to be a bona fide covariance sequence, however, it is necessary that the infinite block-Toeplitz matrix obtained by extending the finite matrix 2 ˆ 0 6 ˆ 01 6 T D 6 : 4 :: ˆ 0



ˆ 1 ˆ 0 :: : ˆ 0

1

ˆ 2 ˆ 1 :: : ˆ 2

  :: : :::

ˆ  ˆ 1 :: : ˆ 0

3 7 7 7 5

(49)

using the rational extension sequence (47), should be nonnegative definite. Equivalently, the spectral density func-

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tion corresponding to (47) ˆ ˚(z) D ˆ 0 C

1 X

  ˆ k z k C zk

kD1

 0 D ˚ˆ C (z) C ˚ˆ C z1

(50)

should be nonnegative definite on the unit circle. This is equivalent to the function ˚ˆ C (z) being positive real. Consequently, the partial realization needs to be done subject to the extra constraint of positivity. The constraint of positivity is a rather tricky one and in some of the methods described in the literature it is disregarded. For this reason some subspace algorithms may fail to provide a positive extension and hence may lead to ¯ for which there are no solutions of the LMI data (A; C; C) and hence to totally inconsistent results. It is important to appreciate the fact that the problem of positivity of the extension has little to do with the noise or sample variability of the covariance data and is present equally well for any finite covariance sequence extracted from a true (infinitely long) rational covariance sequence. For there is no guarantee that, even in this idealized situation, the order of a minimal rational extension (47) of the finite covariance subsequence would be sufficiently large to equal the order of the infinite sequence and hence to generate a positive extension. A minimal partial realization may well fail to be positive because its order is too low to guarantee positivity. It is shown in [38] that there are relatively “large” sets of data (45) for which this happens and the matrix A may even fail to be stable [7]. Future Directions Because of the pervasive presence of fast digital computers in modern society, we are witnessing a sharp change of paradigms in the analysis, design, prediction and control of both man made and natural (say biological, economical etc) systems. Because of the tremendous computing capabilities at our disposal nowadays, stochastic modeling, realization and identification have become an essential ingredient of modern applied sciences. In particular modeling and simulation of complex technological, biological, economic, and environmental systems is becoming more and more essential for understanding the behavior and for prediction and control of these systems. New problems continually arise and much work remains to be done. We just mention here problems where there is a need of modeling the effect of input or decision variables, possibly in the presence of feedback. Such Input–Output models have important applications in many applied areas such

as control of industrial processes, econometrics, etc. There is a large body of results concerning stochastic realization and subspace identification of systems with inputs which we could not discuss in this article, see [10,11,26,27,53, 54,55,64,65]. The area is important for applications and still needs research. Modeling and realization of finite state processes and of certain classes of highly non-Gaussian signals is needed in diverse applications such as source coding, image processing, and computer vision. Bibliography Primary Literature 1. Akaike H (1975) Markovian representation of stochastic processes by canonical variables. SIAM J Control 13:162–173 2. Akaike H (1974) Stochastic Theory of Minimal Realization. IEEE Trans Autom Control AC-19(6):667–674 3. Anderson BDO (1969) The inverse problem of Stationary Covariance Generation. J Stat Phys 1(1):133–147 4. Anderson BDO (1967) An algebraic solution to the spectral factorization problem. IEEE Trans Autom Control AC-12:410–414 5. Anderson BDO (1999) The realization problem for Hidden Markov Models. Math Control Signals Syst 12:80–120 6. Aoki M (1990) State Space Modeling of Time Series, 2nd edn. Springer, New York 7. Byrnes CI, Lindquist A (1982) The stability and instability of partial realizations. Syst Control Lett 2:2301–2312 8. Bauer D (2005) Comparing the CCA Subspace Method to Pseudo Maximum Likelihood Methods in the case of No Exogenous Inputs. J Time Ser Anal 26:631–668 9. Chiuso A, Picci G (2004) The Asymptotic Variance of Subspace Estimates. J Econom 118(1–2):257–291 10. Chiuso A, Picci G (2004) On the Ill-conditioning of subspace identification with inputs. Automatica 40(4):575–589 11. Chiuso A, Picci G (2005) Consistency Analysis of some Closedloop Subspace Identification Methods. Automatica 41:377– 391, special issue on System Identification 12. Cramèr H (1949) Mathematical Methods of Statistics. Princeton University Press, Princeton 13. Desai UB, Pal D, Kirkpatrick RD (1985) A Realization Approach to Stochastic Model Reduction. Int J Control 42:821–838 14. Doob JL (1953) Stochastic Processes. Wiley, New York 15. Faurre P (1969) Identification par minimisation d’une representation Markovienne de processus aleatoires. Symposium on Optimization, Nice 1969. Lecture Notes in Mathematics, vol 132. Springer, New York 16. Faurre P, Chataigner P (1971) Identification en temp reel et en temp differee par factorisation de matrices de Hankel. French– Swedish colloquium on process control, IRIA Roquencourt 17. Faurre P (1973) Realisations Markovienne de processus stationnaires. Report de Recherche no 13, INRIA, Roquencourt 18. Ferrante A, Picci G, Pinzoni S (2002) Silverman algorithm and the structure of discrete-time stochastic systems. Linear Algebra Appl (special issue on systems and control) 351–352:219– 242 19. Finesso L, Spreij PJC (2002) Approximate realization of finite Hidden Markov Chains. Proceedings of the 2002 IEEE Information Theory Workshop, Bangalore, pp 90–93

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20. Ford GW, Kac M, Mazur P (1965) Statistical mechanics of assemblies of coupled oscillators. J Math Phys 6:504–515 21. Kalman RE (1963) On a new Criterion of Linear Passive Systems. In: Proc. of the First Allerton Conference. University of Illinois, Urbana Ill, Nov 1963, pp 456–470 22. Kalman RE (1963) Lyapunov Functions for the problem of Lur’e in Automatic Control. Proc Natl Acad Sci 49:201–205 23. Kalman RE (1965) Linear stochastic filtering: reappraisal and outlook. In: Proc. Symposium on Syatem Theory, Politechnic Institute of Brooklin, pp 197–205 24. Kalman RE, Falb PL, Arbib MA (1969) Topics in Mathematical Systems Theory. McGraw–Hill, New York 25. Karatzas I, Shreve S (1987) Brownian motion and stochastic calculus. Springer, New York 26. Katayama T, Picci G (1999) Realization of Stochastic Systems with Exogenous Inputs and Subspace Identification Methods. Automatica 35(10):1635–1652 27. Katayama T (2005) Subspace methods for System Identification. Springer, New York 28. Kintchine A (1934) Korrelationstheorie Stationäre Stochastischen Prozessen. Math Annalen 109:604–615 29. Lax PD, Phillips RS (1967) Scattering Theory. Academic Press, New York 30. Levy P (1948) Processus stochastiques et Mouvement Brownien. Gauthier–Villars, Paris 31. Lewis JT, Thomas LC (1975) How to make a heat bath. In: Arthurs AM (ed) Functional Integration and its Applications. Clarendon, Oxford 32. Lewis JT, Maassen H (1984) Hamiltonian models of classical and quantum stochastic processes. In: Accardi L, Frigerio A, Gorini V (eds) Quantum Probability and Applications to the Quantum Theory of Irreversible processes. Springer Lecture Notes in Mathematics, vol 1055. Springer, New York, pp 245– 276 33. Lindquist A, Picci G, Ruckebusch G (1979) On minimal splitting subspaces and Markovian representation. Math Syst Theory 12:271–279 34. Lindquist A, Picci G (1979) On the stochastic realization problem. SIAM J Control Optim 17:365–389 35. Lindquist A, Picci G (1985) Realization theory for multivariate stationary Gaussian processes. SIAM J Control Optim 23:809– 857 36. Lindquist A,Pavon M (1984) On the structure of state space models of discrete-time vector processes. IEEE Trans Autom Control AC-29:418–432 37. Lindquist A, Picci G (1991) A geometric approach to modelling and estimation of linear stochastic systems. J Math Syst Estim Control 1:241–333 38. Lindquist A, Picci G (1996) Canonical correlation analysis approximate covariance extension and identification of stationary time series. Automatica 32:709–733 39. Lindquist A, Michaletzky G, Picci G (1995) Zeros of Spectral Factors, the Geometry of Splitting Subspaces, and the Algebraic Riccati Inequality. SIAM J Control Optim 33:365–401 40. Lindquist A, Picci G (1996) Geometric Methods for State Space Identification. In: Identification, Adaptation, Learning. (Lectures given at the NATO-ASI School, From Identification to Learning, Como, Italy, Aug 1994. Springer 41. Lindquist A, Picci G, Linear Stochastic Systems: A Geometric Approach. (in preparation)

42. Ljung L (1987) System Identification – Theory for the User. Prentice–Hall, New York 43. Martin-Löf A (1979) Statistical Mechanics and the foundations of Thermodynamics. Lecture Notes in Physics, vol 101. Springer, New York 44. Moonen M, De Moor B, Vanderberghe L, Vandewalle J (1989) On- and Off-Line Identification of Linear State–Space Models. Int J Control 49:219–232 45. Taylor TJ, Pavon M (1988) A solution of the nonlinear stochastic realization problem. Syst Control Lett 11:117–121 46. Picci G (1978) On the internal structure of finite-state stochastic processes. In: Mohler R, Ruberti A (eds) Recent developments in Variable Structure Systems. Springer Lecture Notes in Economics and Mathematical Systems. vol 162. Springer, New York, pp 288–304 47. Picci G, Pinzoni S (1994) Acausal Models and Balanced realizations of stationary processes. Linear Algebra Appl (special issue on Systems Theory) 205–206:957–1003 48. Picci G (1986) Application of stochastic realization theory to a fundamental problem of statistical physics. In: Byrnes CI, Lindquist A (eds) Modeling, Identification and Robust Control. North Holland, Amsterdam 49. Picci G, Taylor TJ (1990) Stochastic aggregation of linear Hamiltonian systems with microcanonical distribution. In: Kaashoek MA, van Schuppen JH, Ran ACM (eds) Realization and modeling in System Theory. Birkhäuser, Boston, pp 513–520 50. Picci G (1992) Markovian representation of linear Hamiltonian systems. In: Guerra F, Loffredo MI, Marchioro C (eds) Probabilistic Methods in Mathematical Physics. World Sciedntific, Singapore, pp 358–373 51. Picci G (1991) Stochastic realization theory. In: Antoulas A (ed) Mathematical System Theory. Springer, New York 52. Picci G, Taylor TSJ (1992) Generation of Gaussian Processes and Linear Chaos. In: Proc 31st IEEE Conf on Decision and Control. Tucson, pp 2125–2131 53. Picci G, Katayama T (1996) Stochastic realization with exogenous inputs and “Subspace Methods” Identification. Signal Process 52(2):145–160 54. Picci G (1996) Geometric methods in Stochastic Realization and System Identification. CWI Q (invited paper) 9:205–240 55. Picci G (1997) Oblique Splitting Susbspaces and Stochastic Realization with Inputs. In: Helmke U, Prätzel–Wolters D, Zerz E (eds) Operators, Systems and Linear Algebra. Teubner, Stuttgart, pp 157–174 56. Popov VM (1964) Hyperstability and Optimality of Automatic Systems with several Control functions. Rev Rumaine Sci Tech Ser Electrotech Energ 9:629–690 57. Rozanov NI (1963) Stationary Random Processes. Holden-Day, San Francisco 58. Ruckebusch G (1976) Répresentations markoviennes de processus gaussiens stationnaires. Comptes Rendues Acad Sci Paris Series A 282:649–651 59. Ruckebusch G (1978) A state space approach to the stochastic realization problem. Proc 1978 IEEE Intern Symp Circuits and Systems, pp 972–977 60. Ruckebusch G (1978) Factorisations minimales de densités spectrales et répresentations markoviennes. In: Proc 1re Colloque AFCET–SMF, Palaiseau 61. Söderström T, Stoica P (1989) System Identification. Prentice– Hall, New York

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62. van Schuppen JH (1989) Stochastic realization problems. In: Nijmeijer H, Schumacher JM (eds) Three decades of mathematical system theory. Lecture Notes in Control and Information Sciences, vol 135. Springer, New York, pp 480– 532 63. Van Overschee P, De Moor B (1993) Subspace algorithms for the stochastic identification problem. Automatica 29:649– 660 64. Van Overschee P, De Moor B (1994) N4SID: Subspace algorithms for the identification of combined deterministicstochastic systems. Automatica 30:75–93 65. Verhaegen M (1994) Identification of the deterministic part of MIMO State Space Models given in Innovations form from Input–Output data. Automatica 30:61–74 66. Vidyasagar M, (Hidden) Markov Models: Theory and Applications to Biology. Princeton University Press, Princeton 67. Wiener N (1930) Generalized Harmonic Analysis. Acta Math 55:117–258 68. Wiener N (1933) Generalized Harmonic Analysis. In: The Fourier Integral and Certain of its Applications. Cambridge University Press, Cambridge 69. Wiener N (1949) Extrapolation, Interpolation and Smoothing of Stationary Time Series. The M.I.T. Press, Cambridge 70. Willems JC (1971) Least Squares Stationary Optimal Control

and the Algebraic Riccati Equation. IEEE Trans Autom Control AC-16:621–634 71. Yakubovich VA (1963) Absolute stability of Nonlinear Control Systems in critical case: part I and II. Avtom Telemeh 24:293– 303, 717–731 72. Zeiger HP, McEwen AJ (1974) Approximate linear realization of given dimension via Ho’s algorithm. IEEE Trans Autom Control AC-19:153

Books and Reviews Doob JL (1942) The Brownian movement and stochastic equations. Ann Math 43:351–369 Faurre P, Clerget M, Germain F (1979) Opérateurs Rationnels Positifs. Dunod, Paris Kalman RE (1981) Realization of covariance sequences. Proc Toeplitz Memorial Conference, Tel Aviv, Birkhäuser Nelson E (1967) Dynamical theories of Brownian motion. Princeton University press, Princeton Picci G (1976) Stochastic realization of Gaussian processes. Proc IEEE 64:112–122 Picci G (1992) Stochastic model reduction by aggregation. In: Isidori A, Tarn TJ (eds) Systems Models and Feedback: Theory and Applicatons. Birkhäuser Verlag, Basel, pp 169–177

Symbolic Dynamics

Symbolic Dynamics BRIAN MARCUS Department of Mathematics, University of British Columbia, Vancouver, Canada Article Outline Glossary Definition of the Subject Introduction Origins of Symbolic Dynamics: Modeling of Dynamical Systems Shift Spaces and Sliding Block Codes Shifts of Finite Type and Sofic Shifts Entropy and Periodic Points The Conjugacy Problem Other Coding Problems Coding for Data Recording Channels Connections with Information Theory and Ergodic Theory Higher Dimensional Shift Spaces Future Directions Bibliography Glossary In this glossary, we give only brief descriptions of key terms. We refer to specific sections in the text for more precise definitions. Almost conjugacy (Sect. “Other Coding Problems”) A common extension of two shift spaces given by factor codes that are one-to-one almost everywhere. Automorphism (Sect. “The Conjugacy Problem”) An invertible sliding block code from a shift space to itself; equivalently, a shift-commuting homeomorphism from a shift space to itself; equivalently, a topological conjugacy from a shift space to itself. Dimension group (Sect. “The Conjugacy Problem”) A particular group associated to a shift of finite type. This group, together with a distinguished sub-semigroup and an automorphism, captures many invariants of topological conjugacy for shifts of finite type. Embedding (Sect. “Shift Spaces and Sliding Block Codes”) A one-to-one sliding block code from one shift space to another; equivalently, a one-to-one continuous shift-commuting mapping from one shift space to another. Factor map (Sect. “Shift Spaces and Sliding Block Codes”) An onto sliding block code from one shift space to another; equivalently, an onto continuous shift-commuting mapping from one shift space to another. Sometimes called Factor Code.

Finite equivalence (Sect. “Other Coding Problems”) A common extension of two shift spaces given by finite-to-one factor codes. Full shift (Sect. “Shift Spaces and Sliding Block Codes”) The set of all bi-infinite sequences over an alphabet (together with the shift mapping). Typically, the alphabet is finite. Higher dimensional shift space (Sect. “Higher Dimensional Shift Spaces”) A set of bi-infinite arrays of a given dimension, determined by a collection of finite forbidden arrays. Typically, the alphabet is finite. Markov partition (Sect. “Origins of Symbolic Dynamics: Modeling of Dynamical Systems”) A finite cover of the underlying phase space of a dynamical system, which allows the system to be modeled by a shift of finite type. The elements of the cover are closed sets, which are allowed to intersect only on their boundaries. Measure of maximal entropy (Sect. “Connections with Information Theory and Ergodic Theory”) A shiftinvariant measure of maximal measure-theoretic entropy on a shift space. Its measure-theoretic entropy coincides with the topological entropy of the shift space. Road problem (Sect. “Other Coding Problems”) A recently-solved classical problem in symbolic dynamics, graph theory and automata theory. Run-length limited shift (Sect. “Coding for Data Recording Channels”) The set of all bi-infinite binary sequences whose runs of zeros, between two successive ones, are bounded below and above by specific numbers. Shift equivalence (Sect. “The Conjugacy Problem”) An equivalence relation on defining matrices for shifts of finite type. This relation characterizes the corresponding shifts of finite type, up to an eventual notion of topological conjugacy. Shift space (Sect. “Shift Spaces and Sliding Block Codes”) A set of bi-infinite sequences determined by a collection of finite forbidden words; equivalently, a closed shift-invariant subset of a full shift. Shift of finite type (Sect. “Shifts of Finite Type and Sofic Shifts”) A set of bi-infinite sequences determined by a finite collection of finite forbidden words. Sliding block code (Sect. “Shift Spaces and Sliding Block Codes”) A mapping from one shift space to another determined by a finite sliding block window; equivalently, a continuous shift-commuting mapping from one shift space to another. Sofic shift (Sect. “Shifts of Finite Type and Sofic Shifts”) A shift space which is a factor of a shift of finite type;

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equivalently, a set of bi-infinite sequences determined by a finite directed labeled graph. State splitting (Sect. “The Conjugacy Problem”) A splitting of states in a finite directed graph that creates a new graph, whose vertices are the split states. The operation that creates the new graph from the original graph is a basic building block for all topological conjugacies between shifts of finite type. Strong shift equivalence (Sect. “The Conjugacy Problem”) An equivalence relation on defining matrices for shifts of finite type. In principle, this relation characterizes the corresponding shifts of finite type, up to topological conjugacy. Topological conjugacy (Sect. “Shift Spaces and Sliding Block Codes”) A bijective sliding block code from one shift space to another; equivalently, a shift-commuting homeomorphism from one shift space to another. Sometimes called conjugacy. Topological entropy (Sect. “Entropy and Periodic Points”) The asymptotic growth rate of the number of finite sequences of given length in a shift space (as the length goes to infinity). Zeta function (Sect. “Entropy and Periodic Points”) An expression for the number of periodic points of each given period in a shift space. Definition of the Subject Symbolic dynamics is the study of shift spaces, which consist of infinite or bi-infinite sequences defined by a shiftinvariant constraint on the finite-length sub-words. Mappings between two such spaces can be regarded as codes or encodings. Shift spaces are classified, up to various kinds of invertible encodings, by combinatorial, algebraic, topological and measure-theoretic invariants. The subject is intimately related to many other areas of research, including dynamical systems, ergodic theory, automata theory and information theory. Shift spaces and their associated shift mappings are used to model a rich and important class of smooth dynamical systems and ergodic measure-preserving transformations. These models have provided a valuable tool for classifying and understanding fundamental properties of dynamical systems. In addition, techniques from symbolic dynamics have had profound applications for data recording applications, such as algorithms and analysis of invertible encodings, and problems in matrix theory, such as characterization of the set of eigenvalues of a nonnegative matrix.

Dynamics: Modeling of Dynamical Systems” reviews the roots of symbolic dynamics in modeling of dynamical systems. Section “Shift Spaces and Sliding Block Codes” lays the foundation by defining the kinds of spaces and mappings considered in the subject. Section “Shifts of Finite Type and Sofic Shifts” focuses on distinguished special classes of spaces, known as shifts of finite type and sofic shifts. Section “Entropy and Periodic Points” introduces the most fundamental invariants, periodic points and topological entropy. Sections “The Conjugacy Problem” and “Other Coding Problems” survey progress on the conjugacy problem and other classification/coding problems for shifts of finite type and sofic shifts. In Sect. “Coding for Data Recording Channels”, we present applications to coding for data recording. Section “Connections with Information Theory and Ergodic Theory” provides a link with information theory and ergodic theory. Finally, Sect. “Higher Dimensional Shift Spaces” treats higher dimensional symbolic dynamics. While this article covers many of the most important topics in the subject, others have been omitted or treated lightly, due to space limitations. These include one-sided shift spaces, countable state symbolic systems, orbit equivalence, flow equivalence, the automorphism group, cellular automata, and substitution systems. References to work in these sub-areas can be found in the sources mentioned below. For introductory reading on symbolic dynamics and its applications, beyond this article, one can consult the textbooks Kitchens [65] and Lind and Marcus [78]. There are also excellent introductory survey articles, such as Boyle [23], Lind and Schmidt [79], and S. Williams [132]. In addition there are very good expositions which focus on other aspects of the subject. These include Beal [11], which focuses on connections between symbolic dynamics and automata theory, the lecture notes Marcus–Roth– Siegel [83], which focuses on constrained coding applications, and Immink [54], which focuses on applications to data storage. There are also several excellent collections of articles on special areas of the subject, such as [17,133], and [128]. The book [15] by Berstel and Perrin treats the subject of variable length codes and contains many ideas related to symbolic dynamics. Finally, there is an excellent recent survey by Boyle on Open Problems in Symbolic Dynamics [24].

Introduction

Origins of Symbolic Dynamics: Modeling of Dynamical Systems

This article is intended to give a picture of major topics in symbolic dynamics. Section “Origins of Symbolic

Symbolic dynamics began as an effort to model dynamical systems using sequences of symbols. A dynamical system

Symbolic Dynamics

for instance s0 D 0 because x belongs to the left half of the square, and s2 D 1 because T 2 (x) belongs to the right half of the square. Now, if x is represented by the sequence s(x), then T(x) is represented by the shift of s(x): s(T(x)) D : : : 110:01 : : : :

Symbolic Dynamics, Figure 1 Representing points symbolically

is a pair (X; T) where X is a set and T is a transformation from X to itself. For definiteness, we assume that T is invertible, although this is not a necessary restriction. Since T maps X to itself, we can iterate T : T 2 D T ı T; T 3 D T ı T ı T, etc. The orbit of a point x 2 X is the sequence of points: : : : ; T 2 (x); T 1 (x); x; T(x); T 2 (x); : : : In the theory of dynamical systems, one asks questions about orbits such as the following: Are there periodic orbits (i. e., x such that T n (x) D x for some n > 0)? Are there dense orbits (the orbit of x is dense if for any point y in X, T n (x) is “close” to y for some n)? How does the behavior of an orbit vary with x? How can we describe the collection of all orbits of the dynamical system? When is the dynamical system “chaotic”? For more information on dynamical systems, we refer the reader to [16,38,47]. The subject of dynamical systems has its roots in Classical Mechanics; in that setting, X is the set of all possible states of a system (e. g., the positions, momenta of all particles in a physical system), and the transformation T is the time evolution map, which maps the state of the system at one time to the state of the system at one time unit later. Symbolic dynamics provides a model for the orbits of a dynamical system (X; T) via a space of sequences. This is done by “quantizing” X into cells, associating symbols to the cells and representing points as bi-infinite sequences of symbols. For instance in Fig. 1, X is a square, and T is some transformation of the square. We have drawn a portion of the orbit of a point x 2 X for the dynamical system (X; T). We have also quantized X into two cells: the left half, called ‘0’, and the right half, called ‘1’. Then the point x is represented by the bi-infinite sequence s(x) D : : : s2 s1 :s0 s1 s2 : : : where sn is the label of the cell to which T n (x) belongs (here, we use the decimal point to separate coordinates s i ; i < 0 from s i ; i  0). So, for x as given in Fig. 1, we see that s(x) D : : : 11:001 : : :

So, T is represented ‘symbolically’ as the shift transformation. By representing all points of X as bi-infinite sequences, we obtain a symbolic dynamical system (Y;  ) where Y is a set of sequences (representing X) and  is the shift transformation (representing T). The “symbolic” refers to the symbols, and the “dynamical” refers to the action of the shift transformation. For this representation to be faithful, distinct points should be represented by distinct sequences, and this imposes extra conditions on how X is quantized into cells. Also, Y is typically a set of sequences constrained by certain rules, such as a certain symbol may only be followed by certain other symbols. In this way, one can use symbolic dynamics to study dynamical systems. Properties of orbits of the original dynamical system are reflected in properties of the resulting sequences. For instance, a point whose orbit is periodic becomes a periodic sequence, and the distribution of the orbit of a point x in X is reflected in the distribution of finite strings within s(x). Beginning with Hadamard [46] in 1898 and followed by Hedlund, Morse and others in the 1920s, 1930s and 1940s [48,49,92,93], this method was used to prove the existence of periodic, almost periodic and other interesting motions in classical dynamical systems, such as geodesic flows on surfaces of negative curvature; this was done by finding interesting sequences satisfying the constraints defined by the corresponding symbolic dynamical system. Later on, this was extended to general hyperbolic systems, where the symbolic dynamics is constructed using a Markov Partition, which is a disjoint collection of open sets whose closures cover X, each of which looks like a “rectangle”, with vertical (resp., horizontal) fibers contracted (resp., expanded) by T. Markov Partitions were developed by Adler and Weiss [5], Sinai [120] and Bowen [18,19]; see also [13] and Sect. 6.5 in [78]. In more recent years, symbolic dynamics has been used as a tool in classification problems for dynamical systems. Here, the problem of determining when one dynamical system is ‘equivalent’ to another becomes, via symbolic dynamics, a coding problem. Roughly speaking, two dynamical systems, (X1 ; T1 ) and (X2 ; T2 ), are equivalent if there is an invertible mapping from X 1 to X 2 which makes T 1 “look like” T 2 . If the corresponding symbolic dynamical systems are denoted (Y1 ;  ) and (Y2 ;  ), then an equiv-

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alence between (X1 ; T1 ) and (X2 ; T2 ) becomes a time-invariant, invertible encoding from Y 1 to Y 2 (time-invariant because the shift transformation represents the dynamics). Thus, the classification problem in dynamical systems leads to a coding problem between constrained sets of sequences. We have described dynamical systems as the discretetime iteration of a single mapping. However, continuoustime iterations have been studied since the inception of the subject. These are known as continuous-time flows, with the main example being the set of solutions to a system of ordinary differential equations. Indeed, the work of Hedlund and Morse mentioned above was done in this context. Shift Spaces and Sliding Block Codes Let A be an alphabet of symbols, which we assume to be finite. The principal objects of study in symbolic dynamics are certain kinds of collections of sequences of symbols from A. Typically, these sequences are infinite x D x0 x1 x2 : : :, but it is often more convenient to deal with bi-infinite sequences x D : : : x2 x1 x0 x1 x2 : : : For some problems, the results are similar in the infinite and bi-infinite categories, while for other problems, they are quite different. In this article, we focus on the bi-infinite setting. The symbol xi is the ith coordinate of x. When writing a specific sequence, we need to specify which is the 0th coordinate. As suggested in Sect. “Origins of Symbolic Dynamics: Modeling of Dynamical Systems”, this is done with a decimal point to separate the xi with i  0 from those with i < 0: x D : : : x2 x1 :x0 x1 x2 : : :. A block or word over A is a finite sequence of symbols from A. A block of length N is called an N-block. For blocks u; v, the block uv is the concatenation of u and v, and for a block w, the concatenation of N copies of w is denoted wN . The full A-shift AZ is the set of all bi-infinite sequences of symbols from A. The full r-shift is the full shift over the alphabet f0; 1; : : : ; r1g. The shift map  on a full shift maps a point x to the point y D (x) whose ith coordinate is y i D x iC1. The orbit of a point in a full shift is its orbit under the shift map. The full shift contains many different types of orbits. For instance, it contains a dense orbit (namely, any sequence which contains every block in the alphabet) and periodic orbits (namely, any sequence which is periodic). We are interested in sets that can be specified by a list (finite or infinite) of forbidden blocks. Namely, given a collection F of “forbidden blocks” over A, the subset X consisting of all sequences in AZ , none of whose subwords

belong to F , is called a shift space (or simply shift), and we write X D X F . When a shift space X is contained in a shift space Y, we say that X is a subshift of Y. Example 1 X is the set of all binary sequences with no two 1’s next to each other. Here X D X F , where F D f11g. This shift is called the golden mean shift, for reasons that will become apparent later. Example 2 X is the set of all binary sequences so that between any two 1’s there are an even number of 0’s. We can take for F the collection f102nC1 1 : n  0g : This example is naturally called the even shift. Example 3 X is the set of all binary sequences such that between any two successive 1’s number of 0’s is prime. We can take for F the collection f10n 1 : n is compositeg : This example is naturally called the prime shift. Alternatively (and equivalently), shift spaces can be defined as closed, shift-invariant subsets of full shifts. Here, “closed” means with respect to a metric, for which two points are close if they agree in a large “central block”; one such metric is (x; y) D 2k if x ¤ y, with k maximal such that x[k;k] D y[k;k] (with the conventions that

(x; y) D 0 if x D y and (x; y) D 2 if x0 ¤ y0 ). Let X be a subset of a full shift, and let B N (X) denote the set of all N-blocks that occur in elements of X. The language of X is B(X) D [ N B N (X). It can be shown that the language of a shift space determines the shift space uniquely, and so we can equally well describe a shift space by specifying the “occurring” or “allowed” blocks, rather than the forbidden blocks. For example, the golden mean shift is specified by the language of blocks in which 1’s are isolated. This establishes a connection with automata theory [6,11], which studies collections of blocks, rather than infinite or bi-infinite sequences. The languages that occur in symbolic dynamics, i. e. as B(X) for some shift space X, are simply those sets L of blocks that satisfy two simple properties: every sub-block of an element of L belongs to L, and every element w 2 L is extendable to a larger block awb 2 L, with a; b 2 A. Some examples are best described by allowed blocks. One example is the famous Morse shift. Example 4 Let A0 D 0 and inductively define blocks A nC1 D A n A n , where A n denotes bitwise complement.

Symbolic Dynamics

The shift space whose allowed blocks are the sublocks of the An is called the Morse shift. This shift space has an alternative description as follows. Since each An is a prefix of A nC1 , the sequence of blocks An determines a unique right-infinite sequence x C (with each An as a a prefix). If we denote x  as the left-infinite sequence obtained by writing x C backwards, then the Morse shift is the closure of the orbit of the point x  :x C . The sequence x C is known as the Prouhet–Thue–Morse (PTM) sequence, since Prouhet and Thue introduced it earlier, but for different purposes. While it is not immediately obvious, it can be shown that the PTM sequence is not periodic; moreover, the Morse shift is a minimal shift, which means that all its orbits are dense [91]. In contrast, while the full, golden mean, even and prime shifts all have dense orbits, each also has a dense set of periodic orbits. There are two simple, but very important, constructions in symbolic dynamics that construct from a given shift space a new version which in some sense looks deeper into the space at the cost of a larger alphabet and more complex description. Let X be a shift space over the alphabet A, and A(N) D B N (X). We can consider A(N) as an alphabet in its own right, and form the full shift (A(N) )Z . Define the Nth higher block code by (ˇ N (x)) i D x[i;iCN1] : Then the Nth higher block shift or higher block presentation of a shift space X is the image X [N] D ˇ N (X) in the full shift over A(N) . Similarly, define the Nth higher power code N : X ! (A(N) )Z by ( N (x)) i D x[i N;i NCN1] : The Nth higher power shift X N of X is the image X N D

N (X) of X. The difference between X [N] and X N is that the former is constructed by considering overlapping blocks and the latter by non-overlapping blocks. Next, we turn to mappings between shift spaces. Suppose that x D : : : x1 :x0 x1 : : : is a sequence in a shift space X over A. We can transform x into a new sequence y D : : : y1 :y0 y1 : : : over another alphabet C as follows. Fix integers m and n with m  n. To compute the ith coordinate yi of the transformed sequence, we use a function ˚ that depends on the “window” of coordinates of x from position i  m to position i C n. Here ˚ : B mCnC1 (X) ! C is a fixed block map, called an (m C n C 1)-block map from allowed (m C n C 1)-blocks in X to symbols in C , and so y i D ˚(x im x imC1 : : : x iCn ) D ˚(x[im;iCn] ) :

Symbolic Dynamics, Figure 2 Sliding block code

This is illustrated in Fig. 2. Let X be a shift space over A, and ˚ : B mCnC1 (X) ! C be a block map. Then the map  : X ! C Z defined by y D (x), with yi given by ˚ above, is called the sliding block code with memory m and anticipation n induced by ˚. We will denote the formation of  from ˚ by [m;n]  D ˚1 , or more simply by  D ˚1 if the memory and anticipation of  are understood. If not specified, the memory is taken to be 0. If Y is a shift space contained in C Z and (X)  Y, we write  : X ! Y. In analogy with the characterization of shift spaces as closed shift-invariant sets, sliding block codes can be characterized in a topological manner: namely, as the maps between shift spaces that are continuous and commute with the shift. This result is known as the Curtis–Hedlund– Lyndon theorem [50]. Example 5 Let A D f0; 1g D C , X D AZ , m D 0, n D 1, and ˚ (a0 a1 ) D a0 C a1 (mod 2). Let  D ˚1 : X ! X. Example 6 The sliding block code, generated by ˚(00) D 1; ˚ (01) D 0 D ˚(10), maps the golden mean shift onto the even shift. Example 7 There is a trivial sliding block code from the full 2-shift into the full 3-shift, generated by ˚(0) D 0; ˚(1) D 1. If a sliding block code  : X ! Y is onto, then  is called a factor code or factor map, and Y is a factor of X. If  : X ! Y is one-to-one, then  is called an embedding of X into Y. The sliding block code in Example 7 is an embedding but not a factor code, while the codes in Examples 5 and 6 are factor maps, but not embeddings. A major (and unrealistic) goal of symbolic dynamics is to classify in an explicit way shift spaces up to the following natural notion of equivalence. A sliding block code  : X ! Y is a conjugacy (or topological conjugacy) if it is invertible with sliding block inverse. Equivalently, a conjugacy is a bijective sliding block code and therefore simulta-

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neously a factor code and an embedding. If there is a conjugacy from one shift space X to another Y, we say that X and Y are conjugate, denoted X Š Y. As an example, the higher block map ˇ N is a conjugacy between a shift space X and its higher block shift X [N] . Via this code, we can “re-code” any sliding block code as a 1block code (though typically a conjugacy and its inverse cannot, by this artifice, be simultaneously re-coded to 1block codes). In this section, we have given examples of relatively simple sliding block codes. But the typical conjugacy, as well as factor code and embedding, can be much more complicated. Shifts of Finite Type and Sofic Shifts A shift of finite type (SFT) is a shift space that can be described by a finite set of forbidden blocks, i. e., a shift space X having the form X F for some finite set F of blocks. The terminology shift of finite type (or subshift of finite type) comes from dynamical systems (Smale [121]). An SFT is M-step (or has memory M) if it can be described by a collection of forbidden blocks all of which have length M C 1. It is easy to see that any SFT is M-step for some M. Since any shift space can be defined by many different collections of forbidden blocks, it is useful to have the following equivalent condition expressed in terms of allowed blocks: an SFT is M-step if and only if whenever u is an allowed block of length at least M, u0 is the suffix of u with length M and a is a symbol, then ua is allowed if and only if u0 a is allowed. In other words, in order to tell whether a symbol can be allowably concatenated to the end of an allowed word u, one need only look at the last M symbols of u. This is analogous to the “finite memory” property of M-step Markov chains. The golden mean shift X is is a 1-step SFT, since it was defined by a forbidden list consisting of exactly one block: F D f11g. Equivalently, it is only the last symbol of an allowed block that determines whether a given symbol can be concatenated at the end. In contrast, the even shift is not an SFT: for any M, the symbol 1 can be concatenated to the end of exactly one of the (allowed) words 10 M and 10 MC1 . Recall that the higher block code ˇ M is a conjugacy from X to X [M] . Via this code, any SFT can be recoded to a 1-step SFT. And so any sliding block code on a shift space can be recoded to a 1-block code on a 1-step SFT. It is useful to have a concrete description of 1-step SFT’s. In fact, these are precisely the shift spaces consisting of all bi-infinite sequences of vertices along paths on a finite di-

rected graph. These are called vertex shifts. We find it more convenient to work with sequences of edges instead. To be precise: Let G be a finite directed graph (or simply graph) with vertices (or states) V D V (G) and edges E D E (G). For an edge e, i(e) denotes the initial state and t(e) the terminal state. A path in G is a finite sequence of edges in G such that the terminal state of an edge coincides with the initial state of the following edge; a cycle in G is path that begins and ends at the same state. We will assume that G is essential, i. e., that every state has at least one outgoing edge and one incoming edge. The adjacency matrix A D A(G) is the matrix indexed by V with AIJ equal to the the number of edges in G with initial state I and terminal state J. Since a graph and its adjacency matrix essentially determine the same information, we will frequently associate a graph G with its adjacency matrix A and a nonnegative integer matrix A with a graph G. The edge shift X G or X A is the shift space over the alphabet A D E defined by XG D XA ˚ D  D ( i ) i2Z 2 E Z : each  iC1 follows  i : It can be readily verified that edge shifts are 1-step SFT’s. While edge shifts do not include all 1-step SFT’s, any 1step SFT can be recoded to an edge shift, and, compared with vertex shifts, edge shifts offer the advantage of a more compact description. For many purposes, one can study a general shift space X by breaking it into smaller, more well-behaved pieces. A shift space is irreducible if whenever u and w are allowed blocks, there is a “connecting” block v such that uvw is allowed. While shift spaces do not always decompose into disjoint unions of irreducible shifts, every SFT can be written as a finite disjoint union of irreducible SFT’s X i together with “transient” one-way connections from one X i to another. And irreducible edge shifts can be characterized in a particularly concrete form: namely, X G is irreducible if and only if G is irreducible, i. e., for every ordered pair of vertices I and J there is a path in G starting at I and ending at J. There is a stronger notion which is defined by a uniformity condition on the length of the connecting block. A shift space is mixing if whenever u and w are allowed blocks, there is an N, possibly depending on u and w, such that for all n  N, there is block v of length n such that uvw is allowed. And an edge shift X G is mixing if and only if G is primitive, i. e., there is an integer N such that for any n  N and any ordered pair of vertices I and J there is

Symbolic Dynamics

Symbolic Dynamics, Figure 3 Presentation of golden mean shift

Symbolic Dynamics, Figure 4 Presentation of even shift

a path in G of length n starting at I and terminating at J. It follows that for SFT’s in the definition of mixing, the uniform connecting length N can be chosen independent of the allowed blocks u and w. It can be shown that, in some sense, any irreducible SFT X can be broken down into a union of disjoint maximal mixing shifts; namely, X can be written as the disjoint union of finitely many sets X i ; i D 0; : : : ; p  1 such that (X i ) D X iC1 mod p and for each i,  p restricted to X i can be regarded as a mixing SFT. This is a consequence of Perron–Frobenius theory, upon which symbolic dynamics relies heavily; see Seneta [118] for an introduction to this theory. A sofic shift is the set of bi–infinite sequences obtained from a finite labeled directed graph G D (G; L); here, G is a finite directed graph and L is a labeling of the edges of G. The labeled graph is often called a presentation of the sofic shift. The golden mean shift and even shift are sofic, with presentations given in Figs. 3 and 4. SFT’s are sofic because any M-step SFT can be presented by a graph whose states are allowed M-blocks. Note also that any sofic shift is a factor of an SFT, namely via a (1-block) factor code L1 on the edge shift X G based on a presentation (G; L). In fact, the converse is true, and so the sofic shifts are precisely the shift spaces that are factors of SFT’s. This was the original definition of sofic shifts given by Weiss [130]. Typically, the labeling is right resolving, which means that at any given state, all outgoing edges have distinct labels (as in Figs. 3 and 4). Theorem 1 (a) Any sofic shift has a right resolving presentation. (b) Any irreducible sofic shift has a unique minimal right resolving presentation. Part (a) is a direct consequence of the subset construction in automata theory [6,11] which constructs a right resolv-

ing presentation from an arbitrary presentation; see also Coven and Paul [33,34]. Part (b) makes use of the the stateminimization algorithm from automata theory [6,11], but requires an idea beyond that found in automata theory (Fischer [39,40]). The unique presentation in part (b) may be regarded as a canonical presentation. We remark that for an irreducible (resp. mixing) sofic shift, the underlying graph of the unique minimal right resolving presentation is irreducible (resp., primitive). Sometimes it is useful to weaken the concept of right resolving to right closing, which means “right resolving with delay”; more precisely a labeling is right closing, with delay D if all paths of length D C 1 with the same initial state and the same label have the same initial edge. Also, we sometimes consider left resolving and left closing labellings (replace “outgoing” with “incoming” in the definition of right resolving, and replace “initial” with “terminal” in the definition of right closing). While the class of SFT’s is defined by a “finite-memory” property, the more general class of sofic shifts is defined by a “finite-state” property, in that the possible symbols that can occur at time 0 are determined by the past via one of finitely many states. In a presentation, the vertices can be viewed as state information which connects sequences in the past with sequences in the future. It is not difficult to show that neither the prime shift (Example 3) nor the Morse shift (Example 4) are sofic and therefore also not SFT. For the prime shift, this can be done by an application of the pumping lemma from automata theory [6] (or p. 68 of [78]). There are uncountably many shift spaces, but only countably many sofic shifts. So, it is not surprising that the behavior of sofic shifts is very special. However, they are very useful in modeling smooth dynamical systems (Sect. “Origins of Symbolic Dynamics: Modeling of Dynamical Systems”) and in information theory and applications to data recording (Sect. “Coding for Data Recording Channels”), where they arise as constrained systems, although this term is usually reserved for the set of (finite) blocks obtained from a finite directed labeled graph [83]. Entropy and Periodic Points An invariant of conjugacy is an object associated to a shift space that is preserved under conjugacy. It can be shown that many of the concepts that we have already introduced are invariants: irreducibility, mixing, as well as the properties of being a shift of finite type or sofic shift. Beyond these qualitative invariants, there are many quantitative invariants that can, in many cases, be computed explicitly. Foremost among these is topological entropy.

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The (topological) entropy (or simply entropy) of X is: log jB N (X)j h(X) D lim ; N!1 N here, j  j denotes cardinality, and for definiteness the log is taken to mean log2 but any base will do. A subadditivity argument shows that the limit does indeed exist [127]. It should be evident that h(X) is a measure of the “size” or “complexity” of X, as it is simply the asymptotic growth rate of the number of blocks that occur in X. Topological entropy for continuous dynamical systems was defined by Adler, Konheim and McAndrew [3], in analogy with measure-theoretic entropy. Since a k-block sliding block code from a shift space X to a shift space Y maps B NCk1 (X) into B N (Y), it is not hard to see that entropy is an invariant of conjugacy and that it cannot increase under factors and cannot decrease under embeddings. In many cases, one can explicitly compute entropy. For example, for the full r-shift X, jB N (X)j D r N , and so h(X) D log r. And from the defining sequence of the Morse shift, we see that the number of distinct 2 N -blocks is at most 4(2 N ), it follows that the growth rate cannot be exponential and so the entropy of the Morse shift is zero. For SFT’s and more generally for sofic shifts, entropy can be computed explicitly. The key to this computation is the following result, which is based on the Perron– Frobenius theorem. Theorem 2 [99,119] For any graph G, h(X G ) D log A(G) , where A(G) is the largest eigenvalue of A(G). The rough idea is that the number of N-blocks in X G is the number of paths of length N and thus also the sum of the entries of A(G) N , which is controlled by the largest eigenvalue. This is proven first for primitive graphs, whose adjacency matrices have a unique eigenvalue of maximum modulus and this eigenvalue is positive; in fact, for a primitive graph and a pair of states I; J, the number of paths N of length N from I to J grows like A(G) D 2 N h(X G ) . For general graphs, one uses the decomposition into primitive, and then irreducible, graphs. To extend this to a sofic shift Y, one uses a right resolving presentation (G; L) of Y. Since every block of Y is the label of at most most V (G) paths in G, it follows that: Theorem 3 Let G D (G; L) be a right-resolving labeled graph presenting a sofic shift Y. Then h(Y) D h(X G ). From Fig. 3, we see that the golden mean shift is obtained as a right resolving presentation of the graph with adja-

cency matrix:

AD

1 1

1 0

 :

A computation shows that A is the golden mean, and so the entropy of the golden mean shift is the log of the golden mean; this is one explanation of the meaning of the term golden mean shift. From Fig. 4, we see that the even shift has the same entropy. One cannot overstate the importance of entropy as an invariant. Yet, it is somewhat crude; it is perhaps not surprising that a single numerical invariant would not be sufficient to completely capture the many intricacies of shift spaces, even of sofic shifts or SFT’s. An invariant finer than entropy is the zeta function, which combines information regarding the numbers of periodic sequences of all periods, described as follows. For a shift space X, let p n (X) denote the number of points in X of period n (i. e., the number of x 2 X such that  n (x) D x). It is straightforward to show that each p n (X) is an invariant. Since distinct periodic sequences define distinct blocks, it follows that lim sup n!1

1 log p n (X)  h(X) : n

The inequality can be strict; for example, there are shift spaces (such as the direct product of the Morse shift with the full 2-shift) with positive entropy but no periodic points at all. However, for irreducible SFT’s and sofic shifts, the entropy h(X) can be recovered from the sequence p n (X). The key to understanding this is the fact that for an edge shift X D X A , p n (X) D tr(An ) and thus is the sum of the nth powers of the (non-zero) eigenvalues of A; in the case that A is primitive, the largest eigenvalue A strictly dominates the other eigenvalues, and thus for large n, log p n (X) D log tr(An ) n log A nh(X) : This shows that for a mixing SFT, the entropy equals the growth rate of numbers of periodic points. In fact, this result applies to all SFT’s and sofic shifts. Theorem 4 For a sofic shift X, lim sup n!1

1 log p n (X) D h(X) : n

The lim sup turns out be a limit in the case that X is a mixing sofic shift.

Symbolic Dynamics

Most of what we have stated for p n (X) here applies equally well to q n (X), the number of points of least period n in X. This follows from the fact that “most” periodic points of period n have least period n. The periodic point information can be conveniently combined into a single invariant, known as the zeta function. For a shift space X, ! 1 X p n (X) n  X (t) D exp : t n nD1 For an edge shift X A , one computes the zeta function to be the reciprocal of a polynomial: 1 1 D ;  X A (t) D r 1 t A (t ) det(I  tA) which is completely determined by the non-zero eigenvalues (with multiplicity) of A (Bowen and Lanford [21]). As an example, the zeta function of the golden mean shift is 1/(1  t  t 2 ). The discussion above applies equally well to SFT’s since they are conjugate to edge shifts. For a sofic shift, the zeta function turns out to be a rational function, i. e., quotient of two polynomials. This can be shown by analyzing properties of a right resolving presentation of the sofic shift. From this it turns out that the zeta function of the even shift is (1  t)/(1  t  t 2 ). The technique for computing zeta functions of sofic shifts was developed by Manning [80] (actually, Manning developed the technique to compute zeta functions of hyperbolic dynamical systems). So, for sofic shifts, all of the periodic point information is determined by a finite collection of complex numbers, namely the zeros and poles of the zeta function. Finally, we mention another simple invariant obtained from the periodic points. The period, per(X), of a shift space X is the gcd of lengths of periodic points in X, i. e., the gcd of the set of n such that p n (X) ¤ 0. If X D X G is an edge shift and G is irreducible, then per(X G ) D per(G), which is defined to be the gcd of cycle lengths in G and coincides with the gcd of the lengths of all cycles in G based at any given state. For an irreducible graph with period p and any state I, the number of cycles of length N, a multiple N of p, at I grows like A(G) D 2 N h(X G ) . Also, an irreducible graph G is primitive if per(G) D 1. The Conjugacy Problem The conjugacy problem for SFT’s and sofic shifts is a major open problem. After much effort, it remains unsolved today. Much of what we know goes back to R. Williams [131] in the 1970s. One of Williams’ main results was that any

conjugacy can be decomposed into simple building blocks, as follows. Let A and B be nonnegative integral matrices, with associated graphs G and H. An elementary equivalence from A to B is a pair (R; S) of rectangular nonnegative integral matrices satisfying A D RS ;

B D SR :

(1)

In this case we write (R; S) : A  B. A strong shift equivalence of lag ` from A to B is a sequence of ` elementary equivalences (R1 ; S1 ) : A D A0  A1 ; (R2 ; S2 ) : A1  A2 ; ::: ; (R` ; S` ) : A`1  A` D B : In this case we write A  B(lag `). We say that A is strong shift equivalent to B (and write A  B) if there is a strong shift equivalence of some lag from A to B. Via the matrix Equation (1), one creates a graph K whose state set is the disjoint union of V (G) and V (H); for each I 2 VG and J 2 VH , the graph has RIJ edges from I to J and SJI edges from J to I. The equation A D RS allows one to associate each edge e of G with a unique path r(e)s(e) of length two running from V (G) to V (H) to V (G). Similarly, the equation B D SR allows one to associate each edge e of H with a path s(e)r(e) of length two running from V (H) to V (G) to V (H). One can show that the 2-block sliding block code defined by ˚(e f ) D s(e) r( f ) defines a conjugacy from X A to X B , and so whenever A  B, X A Š X B . Williams proved this and its converse: Theorem 5 (R. Williams [131]) The edge shifts X A and X B are conjugate if and only if A and B are strong shift equivalent. In fact, Williams showed that any conjugacy can be decomposed into a composition of conjugacies defined by elementary equivalences. This can be interpreted in a way that shows that X A and X B are conjugate if and only if we can pass from G to H by a sequence of state splittings and amalgamations, defined as follows. Let G be a graph with states V and edges E . For each I 2 V , partition the outgoing edges from I into disjoint nonempty sets EI1 ; EI2 ; : : : ; EIm(I) . Let P denote the resulting partition of E , and let PI denote the partition P restricted to EI . The out-split graph H formed from G using P has states I 1 ; I 2 ; : : : ; I m(I) , where I ranges over the states in V , and edges ej , where e is any edge in E and

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Symbolic Dynamics, Figure 5 A state splitting

1  j  m(t(e)). If e 2 E goes from I to J, then e 2 EIi for some i, and we define the e j to have initial state I i and terminal state J j , The 2-block code generated by ˚(e f ) D e j , j where f 2 E t(e) , defines a conjugacy, called an out-splitting code, from X G to X H . Similarly, one defines an insplitting code. Inverses of these conjugacies are called outamalgamation and in-amalgamation codes. Figure 5 depicts an out-splitting. The graph (a) on the left has three states I; J; K and the partition elements that define the splitting are EI1 D fag; EI2 D fb; cg; E J1 D fdg; EK1 D feg; EK2 D f f g; the graph (b) is the resulting split graph. By interpreting state splitting and amalgamation in terms of adjacency matrices one can show that such operations generate elementary equivalences. And one can decompose elementary equivalences into splittings and amalgamations. It follows that: Theorem 6 (R. Williams [131]) Every conjugacy from one edge shift to another is the composition of splitting codes and amalgamation codes. This classification for edge shifts naturally extends to SFT’s since every SFT is conjugate to an edge shift. It also extends to sofic shifts, and we describe this in the context of irreducible sofic shifts. Recall that an irreducible sofic shift has a unique minimal right resolving presentation. Any labeled graph can be completely described by a symbolic adjacency matrix, which records the transitions (edges) in the underlying physical graph, as well as the labels of the edges. Namely, the symbolic adjacency matrix is indexed by the states of the underlying graph and the (I; J)-entry is the formal sum of the labels of edges from I to J. It turns out that the notions of elementary equivalence, and hence strong shift equivalence, can be extended to more general categories, in particular to symbolic adjacency matrices.

Theorem 7 (Krieger [72], Nasu [97]) Let X and Y be irreducible sofic shifts. Let A and B be the symbolic adjacency matrices of the minimal right-resolving presentations of X and Y, respectively. Then X and Y are conjugate if and only if A and B are strong shift equivalent. The classification, provided by these results, would be of limited use if the story ended here. Fortunately, Williams showed that strong shift equivalence yields a strong, delicate and somewhat computable necessary condition for conjugacy. Let A and B be nonnegative integral matrices and `  1. A shift equivalence of lag ` is a pair (R; S) of rectangular nonnegative integral matrices satisfying the shift equivalence equations AR D RB ;

SA D BS ;

A` D RS ;

B` D SR :

We denote this situation by (R; S) : A B(lag l). We say that A is shift equivalent to B, written A B, if there is a shift equivalence from A to B of some lag. It is not hard to see that an elementary equivalence is a shift equivalence of lag 1 and that shift equivalence is an equivalence relation. It follows that: Theorem 8 (Williams [131]) Strong shift equivalence implies shift equivalence. More precisely, if A  B(lag `), then A B(lag `). Recall from Sect. “Entropy and Periodic Points” that for an edge shift X A , the set of nonzero eigenvalues, with multiplicity, of A determines the zeta function and hence this set is an invariant of conjugacy. Using the shift equivalence equations, one can show more: the entire Jordan form corresponding to the nonzero eigenvalues is an invariant. This information depends only on properties of the adjacency matrix considered as a linear transformation (over R or Q). However, A is a nonnegative, integral matrix and both nonnegativity and integrality provide sub-

Symbolic Dynamics

stantially more information. One such invariant, that follows from shift equivalence and makes use of integrality is the Bowen–Franks group [20], BF(A) D Z r /Z r (I  A). Until recently all of the information contained in known conjugacy invariants, such as those above, was subsumed in shift equivalence. And Kim and Roush showed that shift equivalence is decidable [60,61], meaning that there is a finite decision procedure via a Turing machine that decides whether two given edge shifts, and therefore two given SFT’s, are conjugate (they also showed that a notion of shift equivalence for sofic shifts, formulated by Boyle and Krieger [26], is decidable [62]). So, a central focus of the subject was the question: is shift equivalence a complete invariant of conjugacy? The answer turns out to be No, as proven by Kim and Roush [63]; see also the survey article [125]. However, it is a complete invariant of a weaker form of conjugacy; we say that X A and X B are eventually conjugate if all sufficiently large powers are conjugate. It is not hard to show that if A B, then X A and X B are eventually conjugate, and the converse is true as well: Theorem 9 (Kim and Roush [60], Williams [131]) Edge shifts X A and X B are eventually conjugate if and only if A and B are shift equivalent. Also the Kim–Roush counterexamples and subsequent work do not bear on a special case of the conjugacy problem: if A is shift equivalent to the 1  1 matrix [n], is X A conjugate to the full n-shift? This question, known as the little shift equivalence problem, is particularly intriguing because in this case the condition A [n] is simply the statement that A has exactly one non-zero eigenvalue, namely n. Shift equivalence can be characterized in another way that has turned out to be very useful. Let A be an r  r integral matrix. Let RA denote the real eventual range of A, i.e, R A D R r Ar . The dimension group of A is defined: A D fv 2 R A : vAk 2 Zr for some k  0g : The dimension group automorphism ı A of A is the restriction of A to A , so that ıA (v) D vA for v 2 A . The dimension pair of A is (A ; ıA ). If A is also nonnegative, then we define the dimension semigroup of A to be C A

k

C r

D fv 2 R A : vA 2 (Z ) for some k  0g :

The dimension triple of A is (A ; C A ; ı A ). It can be shown that the dimension triple completely characterizes shift equivalence, i. e., two nonnegative integral matrices are shift equivalent if and only if their dimension groups are isomorphic by an isomorphism that preserves the dimension semigroup and intertwines the

dimension group automorphisms. Also, by associating equivalence classes of certain subsets of the shift space X A to elements of C A , one can interpret the dimension triple in terms of the action of the shift map on X A [27,68]. And the dimension triple arises prominently in the study of the automorphism group of an SFT, which we now briefly describe. The dimension group for SFT’s was developed by Krieger [68,69]. In many areas of mathematics, objects are studied by means of their symmetries. This holds true in symbolic dynamics, where symmetries are expressed by automorphisms. An automorphism of a shift space X is a conjugacy from X to itself. The set of all automorphisms of a shift space X is a group under composition, and is naturally called the automorphism group, denoted aut(X). The goals are to understand aut(X) as a group (What kinds of subgroups does it contain? How “big” is it?) and how it acts on X, e. g., given shift-invariant subsets U; V , such as finite sets of periodic points, when is there an automorphism of X that maps U to V?. One might hope that the automorphism group would shed new light on the conjugacy problem for SFT’s. Indeed, tools developed to study the automorphism group eventually paved the way for Kim and Roush to find examples of shift equivalent matrices that are not strong shift equivalent. On the other hand, the automorphism group cannot tell the entire story. For instance, aut(X A ) and aut(X A> ) are isomorphic, since any automorphism read backwards can be viewed as an automorphism of the transposed shift, yet X A and X A> may fail to be conjugate (for an example due to Kollmer, see p. 81 of [105]). It is not even known if the automorphism groups of the full 2-shift and the full 3-shift are isomorphic. A good deal of our understanding of the action of the automorphism group on an SFT comes from understanding its induced representation as an action of the dimension group; this action is known as the dimension representation. For a much more thorough exposition on aut(X), we refer the reader to Wagoner [125,126]. Other Coding Problems The difficulties encountered in attempts to solve the conjugacy problem motivated the formulation and study of weaker, but meaningful, notions of equivalence. For instance, we might say that two shift spaces are equivalent if one can be invertibly encoded to the other by some kind of “finite-state machine”. A precise version of this is as follows. Shift spaces X and Y are finitely equivalent if there is an SFT W together with finite-to-one factor codes  X :

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Symbolic Dynamics, Figure 6 Some road-colorings

W ! X and Y : W ! Y. We call W a common extension, and  X ,  Y the legs. Here, by “finite-to-one” we mean merely that each point has a finite number of inverse images. It can be shown that any finite-to-one factor code from one shift space to another must preserve entropy, and so entropy is an invariant of finite equivalence. For irreducible sofic shifts, entropy is a complete invariant: Theorem 10 (Parry [100]) Two irreducible sofic shifts are finitely equivalent if and only if they have the same entropy. Note that from this result and the fact that finite-to-one codes between general shift spaces preserve entropy, for irreducible sofic shifts, we could have just as well defined finite equivalence with the common extension W merely being a shift space. However, if W is an SFT, we get a more concrete coding interpretation as follows. First, we recode W to an edge shift X G and recode the legs,  X D (˚ X )1 and Y D (˚Y )1 , to one-block codes, and (with a bit more argument) we can assume that G is irreducible. In this set-up, the finite-to-one condition translates to the so-called “no-diamond” condition, which means that for any given pair of states I; J and finite sequence w, there is at most one path from I to J with label w [33,34]. Since, for any fixed state I, the number of cycles of length n, a multiple of p D per(G), at I grows like 2n h(W) , we have, in this set-up a means to invertibly encode a “large” set of allowed blocks in X to allowed blocks in Y: namely, fix state I, and a large n, which is a multiple of p; for any cycle of length n at state I encode the ˚ X -label of to the ˚ Y -label of . For encoding and decoding, one can dispense with state information if the legs are “almost invertible”. A factor code  is almost invertible if it is one-to-one on sequences that are typical in the following sense: x is typical if every allowed block appears infinitely often in x both to the left and the right. We then say that shift spaces X and Y are almost conjugate if there is an SFT W and almost invertible factor codes  X : W ! X, Y : W ! Y. We call (W;  X ; Y ) an almost conjugacy between X and Y.

For irreducible sofic shifts, it can be shown that any almost invertible factor code is finite-to-one, and so almost conjugacy implies finite equivalence. Thus, entropy is again an invariant, and together with a second very mild invariant, it is complete: Theorem 11 (Adler–Marcus [4]) Let X and Y be irreducible sofic shifts with minimal right resolving presentations (G; L) and (H; M). Then X and Y are almost conjugate if and only if h(X) D h(Y) and per(G) D per(H). In particular, if X and Y are mixing, then per(G) D 1 D per(H) and entropy itself is a complete invariant. Thus, with an an almost conjugacy, one can invertibly encode most sequences in X to those of Y without the need for auxiliary state information. Moreover, if X and Y are almost conjugate, then there is an almost conjugacy of X and Y in which one leg is right-resolving and the other leg is left-resolving (and the common extension is irreducible [4]). This gives an even more concrete interpretation to the encoding. The proofs of Theorems 10 and 11 are actually quite constructive. For illustration we consider a very special, but historically important, case. Let G be a graph with constant out-degree n. A road coloring ˚ is a labeling of G such that at each state of G, each symbol 0; : : : ; n  1 appears exactly once as the label of an outgoing edge. An n-ary word w is synchronizing if all paths that are labeled w end at the same state. Figure 6 gives examples of road-colorings, with n D 2. For a road-coloring, a binary word may be viewed as a sequence of instructions given to drivers starting at each of the states. A synchronizing word is a word that drives everybody to the same state. For instance, in Fig. 6a, the word 11 drives everybody to the state in the lower-left corner. But Fig. 6b does not have a synchronizing word because whenever a driver takes a ‘0’ road he stays where he is and whenever a driver takes a ‘1’ road he rotates by 120 degrees. The road-coloring in Fig. 6c is essentially the only road-coloring of its underlying graph, and there is no syn-

Symbolic Dynamics

chronizing word because drivers must always oscillate between the two states. Now, let X be the full n-shift and Y be an irreducible SFT with entropy log n. Suppose that we could find a presentation (G; L) of Y with G having constant out-degree n and L1 finite-to-one. Then, define ˚ to be any road coloring of G. Then ˚1 would be a finite-to-one (in fact, right resolving!) factor code from X G to X. And we would obtain a finite equivalence with X D X G , Y D L1 and  X D ˚1 . If, moreover, we could choose L and ˚ such that  Y and  X are almost invertible, then we would have an almost conjugacy. It turned that this could be arranged for  Y via a construction related to state splitting [2]. And if Y were mixing, then G could be chosen to be primitive and  Y almost invertible. In this setting,  X D ˚1 would be almost invertible iff ˚ has a synchronizing word; the sufficiency follows from the fact that every bi-infinite binary sequence which contains w infinitely often to the left would be the label of exactly one bi-infinite sequence of edges (to see this, use the synchronizing and road-coloring properties). The construction of such a labeling ˚ became known as the Road Problem, which remained open for thirty years. In the meantime, a weaker version of the the road problem was solved and applied to yield this special case of Theorem 11 (see [2]). Nevertheless, the problem remained an important problem in graph/automata theory and was only recently solved: Theorem 12 (Road theorem (Trachtman [122])) If G is a finite directed primitive graph with constant out-degree n, there a road-coloring of G which has a synchronizing word. Trachtman’s approach relies heavily on earlier work of Friedman [44] and Kari [57]. The primitivity assumption above is close to necessary. Clearly some kind of connectivity is required and in the presence of irreducibility, primitivity would be necessary since otherwise there would be a “phase” introduced in the graph that would never allow a word to synchronize, as in Fig. 6c. So far, we have focused on equivalences between symbolic systems. There has also been considerable attention paid to problems of embedding one system into another and factoring one onto another. One of the most striking results of this type is the Krieger embedding theorem. It is not hard to show that any proper subshift of an irreducible SFT must have strictly smaller entropy. Thus, a necessary condition for a proper embedding of a shift space into an irreducible SFT is that it have strictly smaller entropy. This condition, together with a trivially necessary condition on periodic

points, turns out to be sufficient. Recall that q n (X) denotes the number of points of least period n in X. Theorem 13 (Embedding Theorem (Krieger [70])) Let X and Y be irreducible shifts of finite type. Then there is a proper embedding of X into Y if and only if h(X) < h(Y) and for each n  1, q n (X)  q n (Y). In fact, Krieger’s theorem shows that these conditions are necessary and sufficient for a proper embedding of any shift space into a mixing shift space. The analogous problems for embedding into irreducible or mixing sofic shifts are still open, though there are partial results [22]. Using the embedding theorem and other tools from symbolic dynamics, Boyle and Handelman [25] obtained a stunning application to linear algebra: namely, a complete characterization of the non-zero spectra of primitive matrices over R. In fact, they obtained characterizations of non-zero spectra for primitive matrices over many other subrings of R. While they did not obtain a complete characterization over Z, they formulated a conjecture for Z and obtained many partial results towards that conjecture, which was later proven using other tools. The result, stated below, shows that three simple necessary conditions on a set of nonzero complex numbers are actually sufficient. In order to state these conditions, we need the following notation: Let  D f1 ; : : : ;  k g be a list of nonzero complex Q numbers (with multiplicity). Let f (t) D kiD1 (t   i ), P Pk k and trn () D iD1  i , with  being the d/n (n/d) Mobius Inversion function.  Integrality Condition: f (t) is a monic polynomial (with integer coefficients).  Perron Condition: There is a positive entry in , occurring just once, that strictly dominates in absolute value all other entries. We denote this entry by  .  Net Trace Condition: tr n ()  0 for all n  1. Theorem 14 (Kim–Ormes–Roush [64]) Let  be a list of nonzero complex numbers satisfying the Integrality, Perron, and Net Trace Conditions. Then there is a primitive integral matrix A for which  is the non-zero spectrum of A. These conditions are all indeed necessary for  to be the non-zero spectrum of a primitive integral matrix. The integrality condition is that  forms a complete set of algebraic conjugates; the Perron condition states that  must satisfy the conditions of the Perron–Frobenius theorem for primitive matrices; and the Net Trace condition assures that the number of periodic points of least period n would be nonnegative.

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We now turn from embeddings to factors. One special case, which is somewhat related to the Road Problem above and also important for data recording applications (Sect. “Coding for Data Recording Channels”) is: Theorem 15 [1,81] An SFT X factors onto the full n-shift iff h(X)  log(n). While this special case treats both the equal entropy case (h(X) D log(n)) and unequal entropy case (h(X) > log(n)), in general, the factor problem naturally divides into two cases: lower entropy factors and equal entropy factors. In either case, a trivial necessary condition for Y to be a factor of X is that whenever q n (X) ¤ 0, there exists a d/n such that qd (Y) ¤ 0. This condition is denoted P(X) & P(Y). Building on ideas from Krieger’s embedding theorem, this necessary condition was shown to be sufficient. Theorem 16 (Lower entropy factor theorem (Boyle [22])) Let X and Y be irreducible SFT’s with h(X) > h(Y). Then there is a factor code from X to Y if and only if P(X) & P(Y). As with the embedding theorem, the lower entropy factors problem for irreducible sofic shifts is still open. The equal entropy factors problem for SFT’s is quite different. Clearly, P(X) & P(Y) is a necessary condition. A second necessary condition involves the dimension group and is simplest to state in the case of mixing edge shifts X A and X B . We say that a subgroup  of the dimension group A is pure if whenever an integer multiple of an element v 2 A is in , then so is v; intuitively  does not have any “rational holes” in A . The condition is that there is a pure ı A -invariant subgroup  of A such that (B ; ıB ) is a quotient of (; ıA j ). In the equal entropy case, this condition and the trivial periodic point condition, P(X) & P(Y), subsume all known necessary conditions for the existence of a factor code from one mixing edge shift to another. It is not known if these two conditions are sufficient. For references on this problem, see [9,27,66].

of the peak and trough are degraded, and the positions at which they occur are distorted. In order to control intersymbol interference, it is desirable that 1’s not be too close together, or equivalently that runs of 0’s not be too short. On the other hand, for timing control, it is desirable that runs of 0’s not be too long; this is a consequence of the fact that only 1’s are observed: the length of a run of 0’s is inferred by connection to a clock via a feedback loop, and a long run of 0’s could cause the clock to drift more than one clock cell. This gives rise to run-length-limited shift spaces, X(d; k), where runs of 0’s are constrained to be bounded below by some positive integer d and bounded above by some positive k  d. More precisely, X(d; k) is defined by the constraints that 1’s occur infinitely often in each direction, and there are at least d 0’s, but no more than k 0’s, between successive 1’s. Note that X(d; k) is an SFT with forbidden list F D f0 kC1 ; 10 i 1; 0  i < dg. Figure 7 depicts a labeled graph presentation of X(1; 3). The SFT’s X(1; 3), X(2; 7), and X(2; 10) have been used in floppy disks, hard disks, and the compact audio disk, respectively. Now in order to record completely arbitrary information, we need to build an encoder which encodes arbitrary binary sequences into sequences that satisfy a given constraint (such as X(d; k)). The encoder is a finite-state machine, as depicted in Fig. 8. It maps arbitrary binary data sequences, grouped into blocks of length p (called p-blocks), into constrained sequences, grouped into blocks of length q (called q-blocks). The encoded q-block is a function of the p-block as well as an internal state. When concatenated together, the sequence of encoded q-blocks must satisfy the given constraint. Also, the encoded sequences should be decodable, meaning that given the initial encoder state, a string of p-blocks can be

Symbolic Dynamics, Figure 7 X(1, 3)

Coding for Data Recording Channels In magnetic recording, within any given clock cell, a ‘1’ is represented as a change in magnetic polarity, while a ‘0’ is represented as an absence of such a change. Two successive 1’s (separated by some number, m  0, of 0’s) are read as a voltage peak followed by a voltage trough (or vice versa). If the peak and trough occur too close together, intersymbol interference can occur: the amplitudes

Symbolic Dynamics, Figure 8 Encoder

Symbolic Dynamics

sliding-block decodable finite-state code into X if and only if p/q  h(X).

Symbolic Dynamics, Figure 9 Sliding block decoder

recovered from its encoded string of q-blocks, possibly allowing a fixed delay in time. If the constrained sequences satisfy the constraints of a sofic shift X, we say that such a code is a rate p:q finite-state code into X. In terms of symbolic dynamics, such a code consists of an edge shift X G , a right resolving factor code  1 from X G onto the full 2 p -shift and a right closing sliding block code  2 into X q (the right closing condition expresses the decodability condition). In most applications, it is important that a stronger decoding condition be imposed. Namely, a sliding block decodable rate p:q finite-state code into X consists of a finitestate code given by (X G ; 1 ; 2 ) and a sliding block code : X q ! X2 p such that ı 2 D 1 . This means that the decoded p-block depends only upon the local context of the received q-block – that is, decoding is accomplished by applying a time-invariant function to a window consisting of a bounded amount of memory and/or anticipation, but otherwise is state-independent (see Fig. 9, which depicts the situation where the memory is 1 and the anticipation is 2). The point is that whenever the window of the decoder passes beyond a raw channel error, that error cannot possibly affect future decoding; thus, sliding block decoders control error propagation. Symbolic dynamics has played an important role in providing a framework for constructing such codes as well as for establishing bounds on various figures of merit for such codes. Specifically, by modifying the constructions used in the proofs of Theorems 10 and 11, in the early 1980’s, Adler, Coppersmith and Hassner (ACH) established the following results: Theorem 17 (Finite-state coding theorem [1]) Let X be a sofic shift and p; q be positive integers. Then there is a rate p:q finite-state code into X if and only if p/q  h(X). Theorem 18 (Sliding block decoding theorem [1]) Let X be an SFT and p; q be positive integers. Then there is a rate p:q

We have described all of this in the context of the binary alphabet for data sequences. In fact, it works just as well for any finite alphabet, and the method used to prove Theorem 18 solved, at the same time, a special case of the factor problem for SFT’s: namely, Theorem 15 above. Sometimes, Theorems 17 and 18 are stated only for rate 1 : 1 codes but in the context of arbitrary finite alphabets (e. g. see Chap. 5 in [78]), this easily extends to the general p : q case by passing to powers. The ACH paper was the beginning of a rigorous theory of constrained coding. Theorem 18 has been extended to a large class of sofic shifts, and bounds have been established on such figures of merit as number of encoder states as well as the size of the decoding window of the sliding block decoder. While the state-splitting algorithm does construct codes with “relatively small” decoding windows, it is not yet understood how to construct codes with the smallest such windows. This is of substantial engineering interest since the smaller the decoding window, the smaller the error propagation. There is now a substantial literature on the construction of these types of codes, including the state-splitting algorithm as well as many other methods of encoder/decoder design and a wealth of examples that go well beyond the run length limited constraints. See for example the expositions [11,54], and [83] as well as the papers [7,8,10,12,31,41,42,43,53,56]. Connections with Information Theory and Ergodic Theory The concept of entropy was developed by Shannon [119] in information theory in the 1940’s and was adapted to ergodic theory in the 1950’s and to dynamical systems, and in particular symbolic dynamics, in the 1960’s. Shannon focused on entropy for random variables and finite sequences of random variables, but the concept naturally extends to stationary stochastic processes, in particular stationary Markov chains. Roughly speaking, for a stationary process , the entropy h() is the asymptotic growth rate of the number of allowed sequences, weighted by the joint stationary probabilities of  (for background on entropy and information theory see Cover and Thomas [35]). A Markov chain on a graph G is defined by assigning transition probabilities to the edges. Let P denote the stochastic matrix indexed by states of G, with PIJ equal to the sum of the transition probabilities of all edges from I to J. This, together with a stationary vector for P, completely defines the (joint distributions of the) Markov chain.

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So, a graph itself can be viewed as specifying only which transitions are possible. For that reason, one could view G as an “intrinsic Markov chain”, and Parry [99] used this terminology when he introduced SFT’s based on graphs and matrices. More generally, a stationary process on a shift space X is any stationary process which assigns positive probability only to allowed blocks in X. For an irreducible graph G with strictly positive transition probabilities on all edges, by Perron–Frobenius theory, P will always have a unique stationary vector. And there is a particular such Markov chain G on G that in some sense distributes probabilities on paths as uniformly as possible. This Markov chain is the most “random” possible stationary process (not just among Markov chains) on X G in the sense that it has maximal entropy; moreover, its entropy coincides with the topological entropy of X G . The Markov chain G is defined as follows: let  denote the largest eigenvalue of A(G) and w; v denote corresponding left, right eigenvectors, normalized such that w  v D 1; for any path of length n from state I to state J, we define the stationary probability of : G ( ) D

wI v J : n

It is clear from the formula that this distribution is fairly uniform, since all paths with the same initial state, terminal state and length have the same stationary probability. This defines a joint stationary distribution on allowed blocks of X G , and it is not hard to show that it is consistent and Markov, given by assigning transition probability v J /(v I ) to each edge from I to J. To summarize: Theorem 19 Let G be an irreducible graph.  Any stationary process  on G satisfies h()  log , and  The Markov chain G is the unique stationary process on G such that h(G ) D log . The construction of G is effectively due to Shannon [119] and uniqueness is due to Parry [99] The unique entropymaximizing stationary process on irreducible edge shifts naturally extends to irreducible SFT’s and irreducible sofic shifts (this process will be M-step Markov for an M-step SFT). Many results in symbolic dynamics were originally proved using G . One example is the fact that any factor code from one irreducible SFT to another of the same entropy must be finite-to-one [32]. This result, and many others, were later proven using methods that rely only on

the basic combinatorial structure of G and X G . Nevertheless, G provides much motivation and insight into symbolic dynamics problems, and it illustrates a connection with ergodic theory, which we now discuss. For background on ergodic theory (such as the concepts of measure-preserving transformations, homomorphisms, isomorphisms, ergodicity, mixing, and measuretheoretic entropy), we refer the reader to [107,113,127]. Observe that a stationary process on a shift space X can be viewed as a measure-preserving transformation: the transformation is the shift mapping and the measure on “cylinder sets”, consisting of x 2 X with prescribed coordinate values x i D a i ; : : : ; x j D a j , is defined as the probability of the word a i : : : a j ; the stationarity of the process translates directly into preservation of the measure. The symbolic dynamical concepts of irreducibility and mixing correspond naturally to the concepts of ergodicity and (measure-theoretic) mixing in ergodic theory. It is well-known [107,127] that the measure-preserving transformation (MPT) defined by a stationary Markov chain  on an irreducible (resp., primitive) graph G is ergodic (resp., mixing) if  assigns strictly positive conditional probabilities to all edges of G. Now, suppose that G and H are irreducible graphs and  : X G ! X H is a factor code. For a stationary measure  on G, we define a stationary measure  D () on X H by transporting  to X H : for a measurable set A in X H , define   (A) D   1 (A) : Then  defines a measure-preserving homomorphism from the MPT defined by  to the MPT defined by . Since measure-preserving homomorphisms between MPT’s cannot reduce measure-theoretic entropy, we have h()  h(). Suppose that now  is actually a conjugacy. Then it defines a measure-preserving isomorphism, and so h() D h(). If  D G , then by uniqueness, we have  D H . Thus  defines a measure-theoretic isomorphism between the MPT defined by G and the MPT defined by H . In fact, this holds whenever  is merely an almost invertible factor code. This establishes the following result: Theorem 20 Let G; H be irreducible graphs, and let G ; H be the stationary Markov chains of maximal entropy on G; H. If X G ; X H are almost conjugate (in particular, if they are conjugate), then the measure-preserving transformations defined by G and H are isomorphic. Hence conjugacies and almost conjugacies yield isomorphisms between measure-preserving transformations defined by stationary Markov chains of maximal entropy.

Symbolic Dynamics

In fact, the isomorphisms obtained in this way have some very desirable properties compared to the run-of-the-mill isomorphism. For instance, an isomorphism  between stationary processes typically has an infinite window; i. e., to know (x)0 , you typically need to know all of x, not just a central block x[n;n] (these are the kinds of isomorphisms that appear in general ergodic theory and in particular in Ornstein’s celebrated isomorphism theory [98]). In contrast, by definition, a conjugacy always has a finite window of uniform size. It turns out that an isomorphism obtained from an almost conjugacy, as well as its inverse, has finite expected coding length in the sense that to know (x)0 , you need to know only a central block x[n(x);n(x)] , where the function n(x) has finite expectation [4]. In particular, by Theorem 11, whenever X G and X H are mixing edge shifts with the same entropy, the measure-preserving transformations defined by G and H are isomorphic via an isomorphism with finite expected coding length. The notions of conjugacy, finite equivalence, almost conjugacy, embedding, factor code, and so on can all be generalized to the context of stationary measures, in particular to stationary Markov chains. For instance, a conjugacy between two stationary measures is a map that is simultaneously a conjugacy of the underlying shift spaces and an isomorphism of the associated measure-preserving transformations. Many results in symbolic dynamics have been generalized to the context of stationary Markov chains. There is a substantial literature on this, in particular on finitary isomorphisms with finite expected coding time, e. g., [71,84,101,114], and [90]. The expositions [102,105] give a nice introduction to the subject of strong finitary codings between stationary Markov chains. See also the research papers [45,85,86,103,104,123,124]. Higher Dimensional Shift Spaces In this section we introduce higher dimensional shift spaces. For a more thorough introduction, we refer the reader to Lind [76]. For the related subject of tiling systems, see Robinson [112], Radin [110], and Mozes [94,95]. d The d-dimensional full A-shift is defined to be AZ . Ordinarily, A is a finite alphabet, and here we restrict ourselves to this case. An element x of the full shift may be regarded as a function x : Zd ! A, or, more informally, as a “configuration” of alphabet choices at the sites of the integer lattice Zd . d For x 2 AZ and F  Zd , let xF denote the restriction of x to F. The usual metric on the one-dimensional d full shift naturally generalizes to a metric on AZ given by (x; y) D 2k , where k is the largest integer such that x[k;k]d D y[k;k]d (with the usual conventions when

x D y and x0 ¤ y0 ). In analogy with one dimension, according to this definition, two points are “close” if they agree on a large cube [k; k]d . We define higher dimensional shift spaces, with the following terminology. A shape is a finite subset F of Zd , and a pattern f on a shape F is a function f : F ! A. We say that X is a d-dimensional shift space (or d-dimensional shift) if it can be represented by a list F (finite or infinite) of “forbidden” patterns ˚ d X D X F D x 2 AZ :  n (x)F 62 F

for all n 2 Zd and all shapes F :

Just as in one dimension, we can equivalently define a shift space to be a closed (with respect to the metric ) d translation-invariant subset of AZ . Here “translation-inn variance” means that  (X) D X for all n 2 Zd , where  n is the translation in direction n defined by ( n (x))m D x mCn . We say that a pattern f on a shape F occurs in a shift space X if there is an x 2 X such that x F D f . Hence the analogue of the language of a shift space is the set of all occurring patterns. d A d-dimensional shift of finite type X is a subset of AZ defined by a finite list F of forbidden patterns. Just as in one dimension, a d-dimensional shift of finite type X can also be defined by specifying allowed patterns instead of forbidden patterns, and there is no loss of generality in requiring the shapes of the patterns to be the same. Thus we can specify a finite list L of patterns on a fixed shape F, and set ˚ d X D XLc D x 2 AZ : for all n 2 Zd ;  n (x)F 2 L : In fact, there is no loss in generality in assuming that F is a d-dimensional cube F D [0; k]d . Given a finite list L of patterns on a shape F, we say that a pattern f 0 on a shape F 0 is L-admissible (in the shift of finite type X D XLc ) if each of its sub-patterns, whose shape is a translate of F, belongs to L. Of course, any pattern which occurs in X is L-admissible. But an L-admissible pattern need not occur in X. The analogue of vertex shift (or 1-step shift of finite type) in higher dimensions is defined by a collection of d transition matrices A1 ; : : : ; A d all indexed by the same set of symbols A D f1; : : : ; mg. We set ˝(A1 ; : : : ; A d ) D ˚ d x 2 f1; : : : ; mgZ : A i (x n ; x nCe i ) D 1 for all n; i ; where ei is as usual the ith standard basis vector and A i (a; b) denotes the (a; b)-entry of Ai . Such a shift space

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is called a matrix subshift. When d D 2, this amounts to a pair of transition matrices A1 and A2 with identical vertex sets. The matrix A1 controls transitions in the horizontal direction and the matrix A2 controls transitions in the vertical direction. Note that any matrix subshift is a shift of finite type and, in particular, can be specified by a list L of patterns on the unit cube F D f(a1 ; : : : ; a n ) : a i 2 f0; 1gg; specifically, ˝(A1 ; : : : ; A n ) D X Lc where L is the set of all patterns f : F ! f1; : : : ; mg such that if n; n C e i 2 F, then A i ( f (n); f (n C e i )) D 1. When we speak of admissible patterns for a matrix subshift, we mean L-admissible patterns with this particular L. Just as in one dimension, we can recode any shift of finite type to a matrix subshift. Higher dimensional SFT’s can behave very differently from one dimension. For example, there is a simple method to determine if a one-dimensional edge shift, and therefore a one-dimensional shift of finite type, is nonempty, and there is an algorithm to tell, for a given finite list L, whether a given block occurs in X D XLc . The corresponding problems in higher dimensions, called the nonemptiness problem and the extension problem, turn out to be undecidable [14,111]; see also [67]. Even for twodimensional matrix subshifts X, these decision problems are undecidable. On the other hand, there are some special classes where these problems are decidable. This class includes any two-dimensional matrix subshift such that A1 commutes with A2 and A> 2 . For this class, any admissible pattern on a cube must occur, and so the nonemptiness and extension problems are decidable; see [87,88]. A point x in a d-dimensional shift X is periodic if its orbit f n (x) : n 2 Zd g is finite. Observe that this reduces to the usual notion of periodic point in one dimension. Now an ordinary (one-dimensional) nonempty shift of finite type is conjugate to an edge shift X G , where G has at least one cycle. Hence a one-dimensional shift of finite type is nonempty if and only if it has a periodic point. This turns out to be false in higher dimensions [14,111], and this fact is intimately related to the undecidability results mentioned above. While one can formulate a notion of zeta function for keeping track of numbers of periodic points, the zeta function is hard to compute, even for very special and explicit matrix subshifts, and, even in this setting, it is not a rational function[77]. In higher dimensions, the entropy of a shift is defined as the asymptotic growth rate of the number of occurring patterns in arbitrarily large cubes. In particular, for twodimensional shifts it is defined by h(X) D lim

n!1

1 log jX [0;n1][0;n1] j ; n2

where X[0;n1][0;n1] denotes the set of occurring pat-

terns on the square [0; n  1]  [0; n  1] that occur in X. Recall from Sect. “Entropy and Periodic Points” that it is easy to compute the entropy of a (onedimensional) shift of finite type using linear algebra. But in higher dimensions, there is no analogous formula and, in fact, other than the group shifts mentioned below, the entropies of only a very few higher dimensional shifts of finite type have been computed explicitly. Even for the twodimensional “golden mean” matrix subshift defined by the horizontal and vertical transition matrices

 1 1 A1 D A2 D 1 0 an explicit formula for the entropy is not known. However, there are good numerical approximations to the entropy of some matrix subshifts (e. g., [30,96]). And recently the set of numbers that can occur as entropies of SFT’s in higher dimensions has been characterized [51,52]; this characterization turns out to be remarkably different from the analogous characterization in one dimension [74]. One of the few 2-dimensional SFT’s for which entropy has been computed is the domino tiling system (see [58], Chap. 5 in [115]), which consists of all possible tilings of the plane using the 1  2 and 2  1 dominoes. This can be translated into an SFT with four symbols L; R; T; B, subject to the constraints:    

x i; j x i; j x i; j x i; j

D D D D

L ) x iC1; j D R , R ) x i1; j D L , T ) x i; j1 D B , B ) x i; jC1 D T .

The entropy of this SFT is given by a remarkable integral formula: Z Z 1 1 1 (4  2 cos(2 s)  2 cos(2 t)) ds dt 4 0 0 Further work along these lines can be found in [59,117]. One can formulate notions of irreducibility and mixing for higher dimensional shift spaces. It turns out that for SFT’s there are several notions of mixing that all coincide in one dimension, but are vastly different in higher dimensions. For instance, a higher dimensional shift space is strongly irreducible if there is an integer R > 0 such that for any two shapes F; F 0 of distance at least R, any occuring configurations on F and F 0 can be combined to form an occuring configuration on F [ F 0 [29,129]. For one-dimensional SFT’s, this is equivalent to mixing, but much stronger than mixing for two-dimensional SFT’s.

Symbolic Dynamics

Just as in one dimension, we have the notion of sliding block code for higher dimensional shifts. For finite alphabets A; B, a cube F Zd and a function ˚ : AF ! B, d d the mapping  D ˚1 : A Z ! B Z defined by ˚1 (x)n D ˚(x nCF ) is called a sliding block code. By restriction we have the notion of sliding block code from one d-dimensional shift space to another. As expected, for d-dimensional shifts X and Y, the sliding block codes  : X ! Y coincide exactly with the continuous translation-commuting maps from X to Y, i. e., the maps which are continuous with respect to the metric , defined above, and which satisfy  ı  n D  n ı  for all n 2 Zd . Thus it makes sense to consider the various coding problems, in particular the conjugacy, factor and embedding problems, in the higher dimensional setting, but these are very difficult. Even the question of determining when a higher dimensional SFT of entropy at least log n factors onto the full n-shift seems very difficult (in contrast to Theorem 15). However, there are some positive results for strongly irreducible SFT’s. For instance, it is known that any strongly irreducible SFT of entropy strictly larger than log n factors onto the full n-shift [37,55]. In fact, that result requires only a weaker assumption than strong irreducibility, but still much stronger than mixing; recent examples [28] show, among other things, that one cannot weaken that assumption to mere mixing. There are results on other coding problems as well. For instance, a version of the Embedding theorem in one dimension (Theorem 13) holds for SFT’s in two dimensions with a strong mixing property [73]; however, it is required that the shift to be embedded contains no points that are periodic in any single direction. And some other results on entropy of proper subshifts of strongly irreducible SFT’s carry over from one dimension to higher dimensions [106,109]. But in many cases where versions of the result carry over, the proofs are much different from those in one dimension. Measures of maximal entropy for two-dimensional SFT’s behave very differently from the one dimensional case. For instance, even with very strong mixing properties, such as strong irreducibility, there can be more than one measure of maximal entropy [29], and the relationships among entropy-preserving, finite-to-one, and almost invertibility for factor codes discussed in Sect. “Other Coding Problems” can be very different in higher dimensions [89]. Other differences with respect to entropy can be found in [108]. There is also the natural notion of higher dimensional sofic shifts, which can be defined as those shift spaces that

are factors of SFT’s. Recall that every one-dimensional sofic shift is a right-resolving, and hence entropy-preserving, factor of an SFT. It is not known if there is an analogue to this fact in higher dimensions, although recently there has been some progress: every sofic shift Y is a factor of an SFT with entropy arbitrarily close to h(Y) [36]. Finally, there is a subclass of d-dimensional SFT’s which is somewhat tractable, namely d-dimensional shifts with group structure in the following sense. Let A be a (finite) group. Then the full d-dimensional shift over A is also a group with respect to the coordinate-wise group structure. A (higher-dimensional) group shift is a subshift d of AZ which is also a subgroup. For a survey on results for this class, we refer the reader to [79].

Future Directions The future directions of the subject will likely be determined by progress on solutions to open problems. In the course of describing topics in this article, we have mentioned many open problems along the way. For a much more complete list on a wealth of sub-areas of symbolic dynamics, we refer the reader to the article [24]. While it is difficult to single out the most important challenges, certainly the problem of understanding multi-dimensional shift spaces, especially of finite type, is one of the most important.

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System Regulation and Design, Geometric and Algebraic Methods in

System Regulation and Design, Geometric and Algebraic Methods in ALBERTO ISIDORI 1 , LORENZO MARCONI 2 Department of Informatica e Sistemistica, University of Rome, La Sapienza, Italy 2 Center for Research on Complex Automated Systems (CASY), University of Bologna, Bologna, Italy

1

design problems, exogenous inputs are not available for measurement, nor are known ahead of time, but rather can only be seen as unspecified members of a given family of functions. Embedding a suitable “internal model” of such a family in the controller is a design strategy that has proven to be quite successful in handling uncertainties in the controlled plant as well as in the exogenous inputs. Introduction

Article Outline Glossary Definition Introduction The Generalized Tracking Problem The Steady-State Behavior of a System Necessary Conditions for Output Regulation Sufficient Conditions for Output Regulation Future Directions Bibliography Glossary Exosystem A dynamical system modeling the set of all exogenous inputs (command/disturbances) affecting a controlled plant. Internal model A model of the exogenous inputs (command/disturbances) affecting a controlled plant, embedded in the interior of the controller. Generalized tracking problem The problem of designing a controller able to asymptotically track/reject any exogenous command/disturbance in a fixed set of functions. Observer A device designed to asymptotically track the state of a dynamical system on the basis of measured observations. Steady state A family of behaviors, in a dynamical system, that are asymptotically approached, as actual time tends to infinity or as initial time tends to minus infinity. Definition A central problem in control theory is the design of feedback controllers so as to have certain outputs of a given plant to track prescribed reference trajectories. In any realistic scenario, this control goal has to be achieved in spite of a good number of phenomena which would cause the system to behave differently than expected. These phenomena could be endogenous, for instance parameter variations, or exogenous, such as additional undesired inputs affecting the behavior of the plant. In numerous

The problem of controlling the output of a system so as to achieve asymptotic tracking of prescribed trajectories and/or asymptotic rejection of disturbances is a central problem in control theory. There are essentially three different possibilities to approach the problem: tracking by dynamic inversion, adaptive tracking, tracking via internal models. Tracking by dynamic inversion consists in computing a precise initial state and a precise control input (or equivalently a reference trajectory of the state), such that, if the system is accordingly initialized and driven, its output exactly reproduces the reference signal. The computation of such control input, however, requires “perfect knowledge” of the entire trajectory which is to be tracked as well as “perfect knowledge” of the model of the plant to be controlled. Thus, this type of approach is not suitable in the presence of large uncertainties on plant parameters as well as on the reference signal. Adaptive tracking consists in tuning the parameters of a control input computed via dynamic inversion in such a way as to guarantee asymptotic convergence to zero of a tracking error. This method can successfully handle parameter uncertainties, but still presupposes the knowledge of the entire trajectory which is to be tracked (to be used in the design of the adaptation algorithm) and therefore an approach of this kind is not suited to the problem of tracking unknown trajectories. Of course, one might consider the problem of tracking a slowly varying reference trajectory as a stabilization problem in the presence of a slowly varying unknown parameter, but this would, in most cases, yield a very conservative solution. In most cases of practical interest, the trajectory to be tracked (or the disturbance to be rejected) is not available for measurement. Rather, it is only known that this trajectory is simply an (undefined) member in a set of functions, for instance the set of all possible solutions of an ordinary differential equation. These cases include the classical problem of the set point control, the problem of active suppression of harmonic disturbances of unknown amplitude, phase and even frequency, the synchronization of nonlinear oscillations, and similar others. It is in these cases that tracking via internal models proves particularly

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efficient, in its ability to handle simultaneously uncertainties in plant parameters as well as in the trajectory which is to be tracked. For linear multivariable systems, tracking problems of this kind (those in which the exogenous commands and/or disturbances are only known to be members in the set of solutions of a given ordinary differential equations) have been addressed in very elegant geometric terms by Davison, Francis, Wonham [8,10,11] and others. In particular, one of the most relevant contributions of [11] was a clear delineation of what is known as internal model principle, i. e. the fact that the property of perfect tracking is insensitive to plant parameter variations “only if the controller utilizes feedback of the regulated variable, and incorporates in the feedback path a suitably reduplicated model of the dynamic structure of the exogenous signals which the regulator is required to process”. Conversely, in a stableclosed loop system, if the controller utilizes feedback of the regulated variable and incorporates an internal model of the exogenous signals, the output regulation property is insensitive to plant parameter variations. A nonlinear enhancement of this theory, which uses a combination of geometry and nonlinear dynamical systems theory, was initiated in [15,16,19] in the context of solving the problem near an equilibrium, in the presence of exogenous signals which were produced by a Poisson stable system. In particular, in [19] it was shown how the use center manifold theory near an equilibrium determines the necessity of the existence of an internal model whenever one can solve an output regulation problem in spite of (small) parameter uncertainties (see also [3]). Since these pioneering contributions, the theory has experienced a tremendous growth, culminating in the development of design methods able to handle the case of parametric uncertainties affecting the autonomous (linear) system which generates the exogenous signals (such as in [9,23]), the case of nonlinear exogenous systems (such as in [5]), or a combination thereof (as in [22]). The purpose of these notes is to present a self-contained exposition of the fundamentals of these design methods. The Generalized Tracking Problem In this article, we address tracking problems that can be cast in the following terms. Consider a finite-dimensional, time-invariant, nonlinear system described by equations of the form x˙ D f (w; x; u) e D h(w; x) y D k(w; x) ;

(1)

in which x 2 Rn is a vector of state variables, u 2 Rm is a vector of inputs used for control purposes, w 2 Rs is a vector of inputs which cannot be controlled and include exogenous commands, exogenous disturbances and model uncertainties, e 2 R p is a vector of regulated outputs which include tracking errors and any other variable that needs to be steered to 0, y 2 Rq is a vector of outputs that are available for measurement and hence used to feed the device that supplies the control action. The problem is to design a controller, which receives y(t) as input and produces u(t) as output, able to guarantee that, in the resulting closed-loop system, x(t) remains bounded and lim e(t) D 0 ;

t!1

(2)

regardless of what the exogenous input w(t) actually is. The ability to successfully address this problem very much depends on how much the controller is allowed to know about the exogenous disturbance w(t). In the ideal situation in which w(t) is available to the controller in realtime, the design problem indeed looks much simpler. This is, though, only an extremely optimistic situation which does not represent, in any circumstance, a realistic scenario. The other extreme situation is the one in which nothing is known about w(t). In this, pessimistic, scenario the best result one could hope for is the fulfillment of some prescribed ultimate bound for je(t)j, but certainly not a sharp goal such as (2). A more comfortable, intermediate, situation is the one in which w(t) is only known to belong to a fixed family of functions of time, for instance the family of all solutions obtained from a fixed ordinary differential equation of the form w˙ D s(w)

(3)

as the corresponding initial condition w(0) is allowed to vary on a prescribed set. This situation is in fact sufficiently distant from the ideal but unrealistic case of perfect knowledge of w(t) and from the realistic but conservative case of totally unknown w(t). But, above all, this way of thinking at the exogenous inputs covers a number of cases of major practical relevance. There is, in fact, abundance of design problems in which parameter uncertainties, reference commands and/or exogenous disturbances can be modeled as functions of time that satisfy an ordinary differential equation. The control law is to be provided by a system modeled by equations of the form ˙ D '(; y) u D (; y)

(4)

System Regulation and Design, Geometric and Algebraic Methods in

with state  2 R . The initial conditions x(0) of the plant (1), w(0) of the exosystem (3) and (0) of the controller (4) are allowed to range over fixed compact sets X Rn , W

Rs and, respectively, ) R . All maps characterizing the model of the controlled plant, of the exosystem and of the controller are assumed to be sufficiently differentiable. The problem which will be studied, known as the generalized tracking problem (or problem of output regulation or also generalized servomechanism problem) is to design a feedback controller of the form (4) so as to obtain a closed loop system in which all trajectories are bounded and the regulated output e(t) asymptotically decays to 0 as t ! 1. More precisely, it is required that the composition of (1), (3) and (4), that is the autonomous system w˙ D s(w) x˙ D f (w; x; (; k(w; x))) ˙ D '(; k(w; x))

(5)

with output e D h(w; x) ; be such that:  The positive orbit of W  X  ) is bounded, i. e. there exists a bounded subset S of Rs  Rn  R such that, for any (w0 ; x0 ; 0 ) 2 W  X  ), the integral curve (w(t); x(t); (t)) of (5) passing through (w0 ; x0 ; 0 ) at time t D 0 remains in S for all t  0.  lim t!1 e(t) D 0, uniformly in the initial condition, i. e., for every " > 0 there exists a time ¯t , depending only on " and not on (w0 ; x0 ; 0 ), such that the integral curve (w(t); x(t); (t)) of (5) passing through (w0 ; x0 ; 0 ) at time t D 0 satisfies ke(t)k  " for all t  ¯t.

The Steady-State Behavior of a System Limit Sets The generalized tracking problem can be seen as the problem of forcing in the plant, by means of an appropriate control input u(t), a response x(t) that asymptotically compensates the effect, on the regulated variable e(t), of the exogenous input w(t). The classical way in which the problem is addressed for linear, time-invariant systems, when the exosystem is a neutrally stable linear system, is to seek a controller forcing in the associated closed-loop system (5) a (stable) “steady state” behavior entirely contained in the kernel of the map defining the tracking error e. Thus, it is natural to expect that a similar tool should also

be effective in the more general setting considered here. It appears, though, that a rigorous investigation of the concept of “steady state”, beyond the classical domain of linear system theory, had never been fully pursued. Motivated by the current practice in linear system theory, the “steady state” behavior of a dynamical system can be viewed as a kind limit behavior, approached either as the actual time t tends to C1 or, alternatively, as the initial time t0 tends to 1. Relevant, in this regard, are certain concepts introduced by G.D. Birkhoff in his classical 1927 essay, where he asserts that “with an arbitrary dynamical system : : : there is associated always a closed set of ‘central motions’ which do possess this property of regional recurrence, towards which all other motions of the system in general tend asymptotically” (see p. 190 in [2]). In particular, a fundamental role is played by the concept of !-limit set of a given point, which is defined as follows. Consider an autonomous ordinary differential equation x˙ D f (x)

(6)

with x 2 Rn , t 2 R. It is well known that, if f : Rn ! Rn is locally Lipschitz, for any x0 2 Rn , the solution of (6) with initial condition x(0) D x0 , denoted by x(t; x0 ), exists on some open interval of the point t D 0 and is unique. Assume, in particular, that x(t; x0 ) is defined for all t  0. A point x is said to be an !-limit point of the motion x(t; x0 ) if there exists a sequence of times ft k g, with lim k!1 t k D 1, such that lim x(t k ; x0 ) D x :

k!1

The !-limit set of a point x0 , denoted !(x0 ), is the union of all !-limit points of the motion x(t; x0 ). It is obvious from this definition that an !-limit point is not necessarily a limit of x(t; x0 ) as t ! 1, as the solution in question may not admit any limit as t ! 1. It happens though, that if the motion x(t; x0 ) is bounded, then x(t; x0 ) asymptotically approaches the set !(x0 ). This property is precisely described in what follows [2]. Lemma 1 Suppose there is a number M such that kx(t; x0 )k  M for all t  0. Then, !(x0 ) is a nonempty compact connected set, invariant under (6). Moreover, the distance of x(t; x0 ) from !(x0 ) tends to 0 as t ! 1. One of the remarkable features of !(x0 ), as indicated in this Lemma, is the fact that this set is invariant for (6). Invariance means that for any initial condition x¯0 2 !(x0 ) the solution x(t; x¯0 ) of (6) exists for all t 2 (1; C1) and that x(t; x¯0 ) 2 !(x0 ) for all such t. Put in different terms, the set !(x0 ) is filled by motions of (6) which are bounded

1713

1714

System Regulation and Design, Geometric and Algebraic Methods in

backward and forward in time. The other remarkable feature is that x(t; x0 ) approaches !(x0 ) as t ! 1, in the sense that the distance of the point x(t; x0 ) (the value at time t of the solution of (6) starting in x0 at time t D 0) from the set !(x0 ) tends to 0 as t ! 1. Since any motion x(t; x0 ) which is bounded in positive time asymptotically approaches the !-limit set !(x0 ) as t ! 1, one may be tempted to look, for a system (6) in which all motions are bounded in positive time, at the union of the limit sets of all points x0 , i. e. at the set ˝D

[

!(x0 )

x 0 2R n

and to say that the system is in steady state if its state x(t) evolves in the (invariant) set ˝. There is a major drawback, though, in taking this as definition of “steady state” behavior of a nonlinear system: the convergence of x(t; x0 ) to ˝ is not guaranteed to be uniform in x0 , even if the latter ranges over a compact set (see, e. g [7]). One of the main motivations for looking into the concept of steady state is the aim to shape the steady state response of a system to a given (or to a given family of) forcing inputs. But this motivation looses much of its meaning if the time needed to get within an "-distance from the steady state may grow unbounded as the initial state changes (even when the latter is picked within a fixed bounded set). In other words, uniform convergence to the steady state (which is automatically guaranteed in the case of linear systems) is an indispensable feature to be required in a nonlinear version of this notion. The set ˝, the union of all !-limit points of all points in the state space does not have this property of uniform convergence, but there is a larger set which does have this property. This larger set, known as the ! limit set of a set, is precisely defined as follows. Consider again system (6), let B be a subset of Rn and suppose x(t; x0 ) is defined for all t  0 and all x0 2 B. The !-limit set of B, denoted !(B), is the set of all points x for which there exists a sequence of pairs fx k ; t k g, with x k 2 B and lim k!1 t k D 1 such that lim x(t k ; x k ) D x :

k!1

It is clear from the definition that if B consists of only one single point x0 , all xk ’s in the definition above are necessarily equal to x0 and the definition in question reduces to the definition of !-limit set of a point, given earlier. It is also clear form this definition that, if for some x0 2 B the set !(x0 ) is nonempty, all points of !(x0 ) are points of !(B). In fact, all such points have the property indicated in the

definition, if all the xk ’s are taken equal to x0 . Thus, in particular, if all motions with x0 2 B are bounded in positive time, [ !(x0 ) !(B) : x 0 2B

However, the converse inclusion is not true in general. The relevant properties of the !-limit set of a set, which extend those presented earlier in Lemma 1, can be summarized as follows [14]. Lemma 2 Let B be a nonempty bounded subset of Rn and suppose there is a number M such that kx(t; x0 )k  M for all t  0 and all x0 2 B. Then !(B) is a nonempty compact set, invariant under (6). Moreover, the distance of x(t; x0 ) from !(B) tends to 0 as t ! 1, uniformly in x0 2 B. If B is connected, so is !(B). Thus, as it is the case for the !-limit set of a point, we see that the !-limit set of a bounded set, being compact and invariant, is filled with motions which exist for all t 2 (1; C1) and are bounded backward and forward in time. But, above all, we see that the set in question is uniformly approached by motions with initial state x0 2 B, a property that the set ˝ does not have. Note also that the set of all such trajectories is a “behavior”, in the sense of J.C. Willems [25]. The set !(B), as shown in the previous Lemma, asymptotically attracts, as t ! 1, all motions that start in B. Since the convergence to !(B) is uniform in x0 , it is also true that, whenever !(B) is contained in the interior of B, the set !(B) is asymptotically stable, in the sense of Lyapunov (see [14]). The Steady State Behavior of a Nonlinear System Consider now again system (6), with initial conditions in a closed subset X Rn . Suppose the set X is positively invariant, which means that for any initial condition x0 2 X, the solution x(t; x0 ) exists for all t  0 and x(t; x0 ) 2 X for all t  0. The motions of this system are said to be ultimately bounded if there is a bounded subset B with the property that, for every compact subset X 0 of X, there is a time T > 0 such that kx(t; x0 )k 2 B for all t  T and all x0 2 X0 . In other words, if the motions of the system are ultimately bounded, every motion eventually enters and remains in the bounded set B. Suppose the motions of (6) are ultimately bounded and let B0 ¤ B be any other bounded subset with the property that, for every compact subset X 0 of X, there is a time T > 0 such that kx(t; x0 )k 2 B0 for all t  T and all x0 2 X0 . Then, it is easy to check that !(B0 ) D !(B). Thus,

System Regulation and Design, Geometric and Algebraic Methods in

in view of the properties described in Lemma 2 above, the following definition can be adopted (see [7]). Definition Suppose the motions of system (6), with initial conditions in a closed and positively invariant set X, are ultimately bounded. A steady state motion is any motion with initial condition x(0) 2 !(B). The set !(B) is the steady state locus of (6) and the restriction of (6) to !(B) is the steady state behavior of (6). The notion introduced in this way recaptures the classical notion of steady state for linear systems and provides a new powerful tool to deal with similar issues in the case of nonlinear systems. Example In order to see how this notion includes the classical viewpoint, consider an n-dimensional, single-input, asymptotically stable linear system z˙ D Fz C Gu

(7)

forced by the harmonic input u(t) D u0 sin(! t C 0 ). A simple method to determine the periodic motion of (7) consists in viewing the forcing input u(t) as provided by an autonomous “signal generator” of the form w˙ D Sw in which  0 SD !

u D Qw

! 0

 Q D (1

0)

and in analyzing the state state behavior of the associated “augmented” system w˙ D Sw z˙ D Fz C GQw :

(8)

As a matter of fact, let ˘ be the unique solution of the Sylvester equation ˘ S D F˘ C GQ and observe that the graph of the linear map  : R2 ! Rn w 7! ˘ w is an invariant subspace for the system (8). Since all trajectories of (8) approach this subspace as t ! 1, the limit behavior of (8) is determined by the restriction of its motion to this invariant subspace. Revisiting this analysis from the viewpoint of the more general notion of steady state introduce above, let W R2 be a set of the form W D fw 2 R : kwk  cg 2

(9)

in which c is a fixed number, and suppose the set of initial conditions for (8) is W  Rn . This is in fact the case when the problem of evaluating the periodic response of (7) to harmonic inputs whose amplitude does not exceed a fixed number c is addressed. The set W is compact and invariant for the upper subsystem of (8) and, as it is easy to check, the !-limit set of W under the motion of the upper subsystem of (8) is the subset W itself. The set W  Rn is closed and positively invariant for the full system (8) and, moreover, since the lower subsystem of (8) is a linear asymptotically stable system driven by a bounded input, it is immediate to check that the motions of system (8), with initial conditions taken in W  Rn , are ultimately bounded. As a matter of fact, any bounded set B of the form B D f(w; z) 2 R2  Rn : w 2 W; kz  ˘ wk  dg in which d is any positive number, has the property indicated in the definition of ultimate boundedness. It is easy to check that !(B) D f(w; z) 2 R2  Rn : w 2 W; z D ˘ wg ; i. e. !(B) is the graph of the restriction of the map  to the set W. The restriction of (8) to the invariant set !(B) characterizes the steady state behavior of (7) under the family of all harmonic inputs of fixed angular frequency !, and amplitude not exceeding c. Example A similar result, namely the fact that the steady state locus is the graph of a map, can be reached if the “signal generator” is any nonlinear system, with initial conditions chosen in a compact invariant set W. More precisely, consider an augmented system of the form w˙ D s(w) z˙ D Fz C Gq(w) ;

(10)

in which w 2 W Rr , x 2 Rn , and assume that: (i) all eigenvalues of F have negative real part, (ii) the set W is a compact set, invariant for the the upper subsystem of (10). As in the previous example, the !-limit set of W under the motion of the upper subsystem of (10) is the subset W itself. Moreover, since the lower subsystem of (10) is a linear asymptotically stable system driven by the bounded input u(t) D q(w(t; w0 )), the motions of system (10), with initial conditions taken in WRn , are ultimately bounded. It is easy to check that the steady state locus of (10) is the graph of the map  : W ! Rn w 7! (w) ;

1715

1716

System Regulation and Design, Geometric and Algebraic Methods in

defined by Z

0

(w) D lim

T!1 T

eF Gq(w(; w)) d :

(11)

To see why this is the case, pick any initial condition (w0 ; z0 ) for (10) on the graph of  and compute the solution z(t) of the lower equation of (10) by means of the classical variation of constants formula, to obtain Z t eF(t) Gq(w(; w0 )) d z(t) D e F t z0 C 0

Since by hypothesis z0 D (w0 ), using (10) one obtains z(t) De

Z

Ft

Z C Z t

0

e

1 Z 0

D 1 Z 0

D 1

e F(t) Gq(w(; w0 )) d eF Gq(w( C t; w0 )) d eF Gq(w( ; w(t; w0 ))) d

D(w(t; w0 )) D (w(t)) which proves the invariance of the graph of  for (10). It is deduced from this that that any point of the graph of  is necessarily a point of the steady state locus of (10). To complete the proof of the claim it remains to show that no other point of W  Rn can be a point of the steady state locus. But this is a straightforward consequence of the fact that F has eigenvalues with negative real part. There are various ways in which the result discussed in the previous example can be generalized. For instance, it can be extended to describe the steady state response of a nonlinear system z˙ D f (z; u)

u D q(w)

(13)

with initial conditions chosen in a compact invariant set W and, moreover, suppose that, kq(w)k   for all w 2 W. If this is the case, the set W  Z is positively invariant for w˙ D s(w)

(14)

and the motions of the latter are ultimately bounded, with B D W  Z. The set !(B) may have a complicated structure but it is possible to show, by means arguments similar to those which are used in the proof of the Center Manifold theorem, that if Z and B are small enough the set in question can still be expressed as the graph of a map z D (w). In particular, the graph in question is precisely the center manifold of (14) at (0; 0) if s(0) D 0 and the matrix " # @s (0) SD @w

Gq(w(; w0 )) d

0

D

w˙ D s(w)

z˙ D f (z; q(w)) ;

eF Gq(w(; w0 )) d

1 t F(t)

its interior and a number  > 0 such that, if kz0 k 2 Z and ku(t)k   for all t  0, the solution of (12) with initial condition z(0) D z0 satisfies kz(t)k 2 Z for all t  0. Suppose that the input u to (12) is produced, as before, by a signal generator of the form

(12)

in the neighborhood of a locally exponentially stable equilibrium point. To this end, suppose that f (0; 0) D 0 and that the matrix " # @f (0; 0) FD @z has all eigenvalues with negative real part. Then, it is well known (see e. g. p. 275 [13]) that it is always possible to find a compact subset Z Rn , which contains z D 0 in

has all eigenvalues on the imaginary axis. A common feature of the examples discussed above is the fact that the steady state locus of a system of the form (14) can be expressed as the graph of a map z D (w). This means that, so long as this is the case, a system of this form has a unique well defined steady state response to the input u(t) D q(w(t)). As a matter of fact, the response in question is precisely z(t) D (w(t)). Of course, this may not always be the case and multiple steady state responses to a given input may occur. In general, the following property holds. Lemma 3 Let W be a compact set, invariant under the flow of (13). Let Z be a closed set and suppose that the motions of (14) with initial conditions in W  Z are ultimately bounded. Then, the steady state locus of (14) is the graph of a set-valued map defined on the whole of W. Necessary Conditions for Output Regulation Taking advantage of the notions introduced in the previous section, we are now in a position to highlight some general properties that any controller that solves a problem of output regulation must necessarily have. Recall

System Regulation and Design, Geometric and Algebraic Methods in

that, as defined earlier, the problem of output regulation is solved if, in the composite system (5):  the positive orbit of W  X  ) is bounded,  lim t!1 e(t) D 0, uniformly in the initial condition. The notions introduced in the previous section are instrumental to prove the following, elementary – but fundamental – result, which is a nonlinear enhancement of a Lemma of [10] on which all the theory of output regulation for linear systems is based. Lemma 4 Suppose the positive orbit of W  X  ) is bounded. Then lim e(t) D 0

since e D e1 , we deduce that the steady state locus of the closed loop system (5) is necessarily a subset of the set of all states in which e1 D 0. This being the case, it is seen from the form of the equations (16) that, when the closed loop system (5) is in steady state, necessarily also e2 D e3 D    D e r D 0 : As a consequence, the following conclusions hold:  The steady state locus !(W  Z  E  )) of the closedloop system is a subset of the set Rs  Rnr  f0g  R .  The restriction of the closed-loop system to its steady state locus !(W  Z  E  )) reduces to w˙ D s(w)

t!1

z˙ D f0 (w; z) ˙ D '(; 0) :

if and only if !(W  X  )) f(w; x; ) : h(w; x) D 0g:

(17)

(15)  For each (w; z; 0; : : : ; 0; ) 2 !(W  Z  E  ))

It is seen from this simple result that the problem of output regulation can be simply cast as the problem of shaping the steady state locus of the closed loop system, in such a way that property (15) holds. To proceed with the analysis in a more concrete fashion, we consider from now on the special case in which the controlled plant (4) is modeled by equations in normal form z˙ D f0 (w; z) C f1 (w; z; e1 )e1 e˙1 D e2 :: : (16)

e˙r1 D e r e˙r D q(w; z; e1 ; : : : ; e r ) C b(w; z; e1 ; : : : ; e r )u e D e1 y D col(e1 ; : : : ; e r ) ;

with state (z; e1 ; : : : ; e r ) 2 Rnr  Rr , control input u 2 R, regulated output e 2 R, measured output y 2 Rr . The functions f0 (); f1 (); q(); b(); s() in (16) and (3) are assumed to be at least continuously differentiable. It is also assumed that

0 D q(w; z; 0; : : : ; 0) C b(w; z; 0; : : : ; 0) (; 0) : (18) The prior analysis implicitly assumes that the positive orbit of W under the flow of exosystem is bounded, i. e. that the motions of the exosystem asymptotically approach the its own steady state locus !(W). In principle, !(W) may differ from W but there is no loss of generality in assuming from the very beginning that the two sets coincide. After all, the problem in question is a problem concerning how the closed-loop system behaves in steady state and there is no special interest in considering exosystems that are not “in steady state”. We make this assumption precise as follows. Assumption (i) The compact set W is invariant for (3). With this in mind we observe that, by Lemma 3, if the positive orbit of W  Z  E  ) under the flow of (5) is bounded, then !(W  Z  E  )) is the graph of a (possibly set-valued) map defined on the whole of W. Consider now the set Ass D f(w; z) : (w; z; 0; : : : ; 0; ) 2 !(W  Z  E  ));

for some  2 Rg and define the map

b(w; z; e1 ; : : : ; e r ) ¤ 0

8(w; z; e1 ; : : : ; e r ) :

The initial conditions of (16) range on a set Z  E, in which Z is a fixed compact subset of Rnr and E D f(e1 ; : : : ; e r ) 2 Rr : je i j  cg, with c a fixed number. Suppose that a controller of the form (4) solves the problem of output regulation. Then Lemma 4 applies and,

uss : Ass ! R (w; z) 7! 

q(w; z; 0; : : : ; 0) : b(w; z; 0; : : : ; 0)

By construction, the set Ass is the graph of a (possibly set-valued) map defined on the whole of W, which is

1717

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System Regulation and Design, Geometric and Algebraic Methods in

invariant for the dynamics of w˙ D s(w)

(19)

z˙ D f0 (w; z) ;

A :D !(W  Z)

that are precisely the zero dynamics of the “augmented system” (3)–(16), while the map uss () is the control that forces the motion of (3) –(16) to evolve on Ass . With this in mind, the conclusions reached above can be rephrased in the following terms. Suppose that a controller of the form (4) solves the problem of output regulation for (16) with exosystem (3). Then, there exists a (possibly set-valued) map defined on the whole of W whose graph Ass is invariant for the autonomous system (19). Moreover, for each (w0 ; z0 ) 2 Ass there is a point 0 2 R such that the integral curve of (19) issued from (w0 ; z0 ) and the integral curve of ˙ D '(; 0) issued from 0 satisfy uss (w(t); z(t)) D ((t)) ;

Assumption (ii) There exists a bounded subset B of C which contains the positive orbit of the set W  Z under the flow of (19) and the resulting omega-limit set

8t 2 R :

This is a nonlinear version of the celebrated internal model principle of [11].

satisfies (w; z) 2 C ;

j(w; z)jA  d0

The Control Structure On the basis of the ideas presented in the previous section we proceed now with the construction of a controller that solves the problem of output regulation. The “steady state” features of this controller are those identified at the end of the section, namely this controller has to be able to “generate” all controls of the form uss (w(t); z(t)) for any “steady state” trajectory w(t); z(t) of (19). The controller should incorporate a device that generates all such trajectories (the internal model), thus making sure that the “appropriate” state-state behavior takes place, and a device guaranteeing that convergence to this specific steady state behavior occurs. It is here that additional assumptions are needed. Note that, since W is invariant for w˙ D s(w), the closed cylinder C :D W  Rnr

is locally invariant for (19). Hence, it is natural regard (19) as a system defined on C and endow the latter with the subset topology.

z2Z

(20)

where d0 is a positive number. While in the analysis of the necessity we have only identified the existence of a compact set (actually, the graph of a map defined on W) which is invariant for (19), the new assumption (ii) implies, in its first part, the existence of a compact set A (still the graph of a map defined on W) which is not only invariant but also uniformly attractive of all trajectories of (19) issued from points of W  Z. The second part of the assumption, in turn, guarantees that this set is also stable in the sense of Lyapunov. In the next assumption we strengthen this property by also requiring the set A to be locally exponentially stable (this assumption is useful to straighten the subsequent analysis, but is not essential). Assumption (iii) There exist M  1,  > 0 such that (w0 ; z0 ) 2 C ;

Sufficient Conditions for Output Regulation

)

j(w0 ; z0 )jA  d0 t

j(w(t); z(t))jA  Me

j(w0 ; z0 )jA ;

) 8t  0

in which (w(t); z(t)) denotes the solution of (19) passing through (w0 ; z0 ) at time t D 0. To simplify the exposition, we address the special case in which the controlled system (16) has relative degree 1, and in which the coefficient b(w; z; e1 ) is identically equal to 1. In other words, we consider a system modeled by equations of the form z˙ D f0 (w; z) C f1 (w; z; e)e e˙ D q(w; z; e) C u

(21)

yDe: There is no loss of generality in considering a system having this simple form (21) because, as shown for instance in [9,22], the case of a more general system of the form (16) can easily be reduced, by appropriate manipulations, to this one. For convenience, rewrite the augmented system (3)–(21) as z˙ D f0 (z) C f1 (z; e)e e˙ D q0 (z) C q1 (z; e)e C u

(22)

System Regulation and Design, Geometric and Algebraic Methods in

having set z D (w; z): Consistently let Z :D W  Z denote the compact set where the initial condition z(0) is supposed to range. In these notations, assumptions (i) – (ii) (iii) express the property that, in the autonomous system z˙ D f0 (z) ;

(23)

the set A is asymptotically and locally exponentially stable, with a domain of attraction that contains the set Z. Suppose now that the control u is chosen as u D ke. The closed-loop system thus obtained can be regarded as a feedback interconnection of z˙ D f0 (z) C f1 (z; e)e

(24)

viewed as a system with input e and state z, and e˙ D q0 (z) C q1 (z; e)e  ke

(25)

viewed as a system with input z and state e. By assumption, system (24) possesses, when e D 0, an invariant set A which is asymptotically and locally exponentially stable, with a domain of attraction that contains the set Z of all admissible initial conditions. Thus, standard arguments (see, e. g [6].) can be invoked to claim that, if k is large enough, all trajectories of the interconnection (24)–(25) with initial conditions in Z  E remain bounded and the state (z; e) can be steered to an arbitrary small neighborhood of the set A  f0g. This does not solve the problem at issue, though, because the variable e(t) is not guaranteed to converge to zero (but only to converge to a neighborhood of zero, whose size can be made arbitrarily small by increasing the gain coefficient k). The condition for having e(t) ! 0 as t ! 1 is simply that the “coupling” term q0 (z) vanishes on the set A, but there is no reason for this to occur (see again, e. g.[6]). This is why a more elaborate, internal-model-based, controller is needed. System (16) being affine in the control input u, it seems natural to look for a controller having a similar structure, namely a controller of the form ˙ D '() C Gv u D () C v

(26)

with state  2 R , in which v is a residual control input, to be eventually chosen as a function of the measured output y. Here '(), G and () are functions to be determined. We will show in what follows that, if the triplet f'(); G; ()g possesses what we will define as asymptotic internal model property, the choice of the residual control v in (26) as v D ke solves the problem of output regulation, provided that the gain coefficient k is sufficiently high.

The Internal Model Controlling this system by means of (26) yields a closedloop system z˙ D f0 (z) C f1 (z; e)e e˙ D q0 (z) C q1 (z; e)e C () C v ˙ D '() C Gv

(27)

which, regarded as a system with input v and output e, has relative degree 1 and zero dynamics given by z˙ D f0 (z) ˙ D '()  G[ () C q0 (z)] :

(28)

System (27) can be put in normal form by means of the change of variables  D   Ge which yields z˙ D f0 (z) C f1 (z; e)e ˙ D '( C Ge)  G ( C Ge)  Gq0 (z)  Gq1 (z; e)e e˙ D q0 (z) C q1 (z; e)e C ( C Ge) C v : (29) Setting x D (z; ), this system can be further rewritten in the form x˙ D f (x) C `(x; e)e

(30)

e˙ D q(x) C r(x; e)e C v in which f (x) D

f0 (z)

!

'(x)  G[ (x) C q0 (z)]

q(x) D q0 (z) C (x) and `(x; e), r(x; e) are suitable continuous functions. Suppose now that the residual control v is chosen as v D ke. This yields a closed-loop system having a structure similar to the one considered in the previous subsection, namely a feedback interconnection of x˙ D f (x) C `(x; e)e

(31)

viewed as a system with input e and state x, and e˙ D q(x) C r(x; e)e  ke

(32)

viewed as a system with input x and state e. As claimed earlier, a high-gain control on e, namely a control v D ke

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with large k, would succeed in steering e(t) to zero if two conditions are fulfilled: (P1) the dynamics (28) possesses a compact invariant set which is asymptotically (and locally exponentially) stable, with a domain of attraction that contains the set Z ) of all admissible initial conditions, and (P2) the function q0 (z) C () vanishes on this invariant set. These are the properties that will be sought in what follows. Note that the fulfillment of these is determined only by properties of the autonomous system (23) and of the function

D q0 (z)

(33)

which, in the composite system (28), can be viewed as the output of (23) driving a system of the form ˙ D '()  G[ () C ] :

its steady state behavior. Property (P2), on the other hand, expresses the property that at each point of (z; ) 2 gr() q0 (z) D  () :

Thus, looking again at system (28), it is realized that gr() is in fact invariant for z˙ D f0 (z) ˙ D '() :

@(z) f0 (z) D '((z)) @z

The notion of steady state provides a useful interpretation of the properties in question. In fact, recall that, by assumption, all trajectories of system (23) with initial conditions in Z asymptotically converge to the compact invariant set A, and the latter is also locally exponentially stable. If property (P1) holds, all trajectories of the composite system (28) with initial conditions in Z  ) asymptotically converge to the limit set !(Z  )). Since (28) is a triangular system, it is readily seen that the set !(Z  )) is the graph of a set-valued map defined on A, i. e. that there exists a map  : z 2 A 7! (z) R ; such that !(Z  )) D f(z; ) : z 2 A;  2 (z)g : D gr() : The set gr() is the steady state locus of (28) and the restriction of the latter to this invariant set characterizes

8z 2 A;

(37)

while the fact that (35) holds at each point of (z; ) 2 gr() is expressed by the property that q0 (z) D  ((z))

Proposition 1 Pick compact sets Z, E and ) for the initial conditions of the closed-loop system (3), (21), (26). Assume that (i)-(ii)-(iii) hold and that the triplet f'(); G; ()g is an asymptotic internal model of (23) – (33). Then there exists k ? > 0 such that for all k  k ? the controller (26) with v D ke solves the generalized tracking problem.

(36)

Note that, if the map (z) is single-valued and C1 , its invariance for (36) is expressed by the property that

(34)

For convenience, we will say that triplet f'(); G; ()g is an asymptotic internal model of the pair (23) – (33) if properties (P1) and (P2) are satisfied. In this terminology, we can summarize as follows the conclusion obtained so far.

(35)

8z 2 A:

(38)

The Design of an Internal Model As we have seen in the earlier sections, the proposed controller, if the asymptotic internal model property holds, is able to force – in the closed loop system – convergence to a steady state in which the regulated variable is identically zero. As a consequence, the controller solves the generalized tracking problem. It remains to be shown, therefore, how the asymptotic internal model property can be obtained. To this end, it is convenient to observe that the properties required in (P1) and (P2) are quite similar to properties that are usually sought in the design of state observers. As a matter of fact it is seen from (37) and (38) that, for each z0 2 A, the function of time ˆ D (z(t; z0 )) (t) which is defined (and bounded) for all t 2 R satisfies ˆ d(t) ˆ D '((t)) dt

(39)

and, moreover ˆ

((t)) D q0 (z(t; z0 )) : In view of the latter, system (34) can be rewritten in the form ˆ  ()] ˙ D '() C G[ ()

(40)

and interpreted as a copy of the dynamics (39) of ˆ corˆ  ()] weighted by rected by an “innovation term” [ ()

System Regulation and Design, Geometric and Algebraic Methods in

an “output injection gain” G. This is the classical structure on an observer and the requirement in (P1) expresses the ˆ (the “observation property that the difference (t)  (t) error”, in our interpretation) should asymptotically decay to zero (with ultimate exponential decay). This interpretation is at the basis of a number of major recent advances in the design of regulators. In fact, in a number of recent papers, this interpretation has been pursued and, taking into consideration various approaches to the design of nonlinear observers, has lead to effective design methods (see [5,9,22]). Two of such design methods are highlighted in the remaining part of this section. The High-Gain Observer as an Internal Model (see [5]) The construction summarized in this section relies upon the following additional hypothesis. Assumption (iv) Suppose there exist an integer d > 0 and a locally Lipschitz function f : Rd ! R such that, for any z0 2 A, the solution z(t) of passing through z0 at time t D 0 is such that the function (t) :D q0 (z(t)) satisfies

(d) (t) D f ( (t); (1) (t); : : : ; (d1) (t)) for all t 2 R. Let  : W  Rn1 ! Rd be the map defined as (z) :D col(q0 (z); L f0 q0 (z); : : : ; Ld1 f 0 q0 (z))

(41)

and let f c : Rd ! R be a C1 function with compact support which agrees with f () on (A). Then, it easy to check that the properties indicated in (37) and (38) are fulfilled by choosing 1 0 2 C B :: C B C B : '() D B

() D 1 : (42) C; C B d A @ f c (1 ; 2 ; : : : ; d ) Comparing this construction with the earlier remarks we observe, in particular, that system z˙ D f0 (z)

D q0 (z)

As a matter of fact, the property in question is achieved by choosing 0

1 c0 B C G D D g @ ::: A cd1 where D g D diag(g; g 2 ;    ; g d ), g is a design parameter, and the ci ’s are such that the polynomial d C c0 d1 C    C cd1 D 0 is Hurwitz, as formally proved in Lemmas 1 and 2 of [5] to which the interested reader is referred for details. It is worth noting that the assumption in question clearly covers the interesting (and widely addressed in the recent past literature, see [15]) case in which the function f () is linear, namely the case in which (43) is immersed into a linear observable system. In this case, although the choice indicated above is clearly still valid, a more direct way of designing the regulator is to use f () instead of f c () in the definition of '(), and simply choose G in such a way that ˙ D '()  G () is a stable linear system. The Andrieu-Praly’s Observer as an Internal Model (see [22]) In this subsection we exploit certain results of the theory presented in [1] to weaken (and, to some extent, suppress) the Assumption (iv) presented at the beginning of the earlier subsection. Let (F; G) 2 Rdd  Rd1 be a controllable pair and set '() D F C G ();

with : Rd ! R a continuous function to be determined later. If this is the case, the composite system (28) becomes z˙ D f0 (z) ˙ D F  Gq0 (z) ;

is immersed into a system which is uniformly observable, in the sense of [12] (even though system (43) might not have had such a property). It is precisely this that makes it possible to choose G in such a way that the property indicated in (P1) can be achieved.

(45)

which is precisely a system of the form (9), considered in the Example. If the matrix F is Hurwitz, and we restrict z to belong to the set A, this system has a well defined steady state behavior, which is the graph of the map Z

0

(z) D (43)

(44)

1

eFs Gq0 (z(s; z))ds :

(46)

As shown in Example, the graph in question is invariant for system (45), is asymptotically (and locally exponentially) stable and with a domain of attraction that coincides with W Rd . Moreover, it can also be shown that there exists a number ` > 0 such that, if the eigenvalues of F have real part which is less `, the map (46) is C1 (see, e. g. [22]).

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If this is the case, to say that the graph of (46) is invariant for (45) is equivalent to say that @ f0 (z) D F(z)  Gq0 (z) @z

8z 2 A:

(47)

This being the case, it is immediate to check that properties (P1) and (P2) will be fulfilled if a function () can be found that renders (38) satisfied. As a matter of fact, bearing in mind (44), condition (37) becomes @ f0 (z) D F(z)  G ((z)) @z

8z 2 A

which, if condition (38) holds, reduces to (47). This, shows that a triplet having the asymptotic internal model property can be found if a function () exists which satisfies (38). It is here that the dimension d of the pair (F, G) plays a role, as formalized in the next proposition whose proof can be found in [22]. Proposition 2 Suppose d  2(s C n  r) C 2 : Then for almost all choices (see [22] for details) of a controllable pair (F, G), with F a Hurwitz matrix whose eigenvalues have real part which is less than `, the map (46) satisfies (z1 ) D (z2 )

)

q0 (z1 ) D q0 (z2 ) :

As a consequence there exist a continuous map : (A) ! R such that q0 (z) D  ((z))

8z 2 A:

(48)

The map () in (46) is defined only on A, but is not difficult to extend it to a C1 map defined on the whole set W  Rnr , as shown in [22]. Also the map () that makes (48) true can be extended to the whole Rd , but this extension is only known to be continuous. We have shown in this way that the existence of triplet (); G; () which has the internal model property can always be achieved, so long as the integer d is large enough. This result shows that the general design procedure outlined earlier in the article is always applicable (so long as the standing hypotheses (i)–(ii)–(iii) are applicable). From the constructive viewpoint, though, it must be observed that the result indicated in Proposition PR1 is only an existence result and that the function (), whose existence is guaranteed, is only known to be continuous. Obtaining continuous differentiability of such () and a constructive procedure are likely to require further hypotheses, which should be in any case weaker than Assumption (iv) considered earlier, whose study is subject of current investigation.

Future Directions One of the basic hypotheses of the theory described in the previous sections is that the zero dynamics of the augmented system consisting of the controlled plant and of the exosystem possess a compact attractor which is also locally asymptotically stable. This assumption, with an acceptable abuse of terminology, is usually referred to as the “minimum-phase” property. Another standing hypothesis is that the regulated variable coincides with the measured variable. The future directions of the research in this area are aimed at the removal of these assumptions. There are several directions in which the problem can be tackled. One way is the use of control structures by means of which the controlled system (plus a part of the controller) can be interpreted with systems having a different zero dynamics (in particular a zero dynamics possessing a compact attractor). This is the nonlinear equivalent of certain design procedures for linear systems based on the assignment of zeros. This technique has proven to be powerful in the stabilization of certain classes of nonlinear systems (see [18]) and is expected to be successful, with appropriate enhancements, in the design of regulators. The analysis of the necessary conditions for output regulation has also shown that, if the generalized tracking problem is solvable, the inverse dynamics of the augmented system consisting of the controlled plant and of the exosystem possesses a compact invariant set to which all initial conditions are asymptotically controllable (by means of the regulated variable viewed as a control). This set is not necessarily asymptotically stable (as it would be under the “minimum-phase” hypothesis) but could be asymptotically stabilizable (by either full-state or measurementbased feedback). Thus, another direction in which the research will evolve, in the development of control schemes for possibly non “minimum-phase” nonlinear systems, is based on the exploitation of the (weaker) assumption that in the zero dynamics of the augmented system there is a compact set that can be made invariant and asymptotically stable by means of feedback. This will yield a design procedure in which a virtual control (either full-state or measurements-based) is designed to stabilize that compact invariant set. The (possibly dynamic) controller obtained in this way will then be embedded into a regulator designed according to the principles developed in this article.

Bibliography 1. Andrieu V, Praly L (2006) On the existence of a KazantisKravaris/Luenberger observer. SIAM J Contr Optim 45:432–456 2. Birkhoff GD (1927) Dynamical systems. American Mathematical Society, Providence

System Regulation and Design, Geometric and Algebraic Methods in

3. Byrnes CI, Delli Priscoli F, Isidori A, Kang W (1997) Structurally stable output regulation of nonlinear systems. Automatica 33:369–385 4. Byrnes CI, Isidori A (2003) Limit Sets, Zero dynamics and internal models in the problem of nonlinear output regulation. IEEE Trans Autom Contr 48:1712–1723 5. Byrnes CI, Isidori A (2004) Nonlinear internal models for output regulation. IEEE Trans Autom Contr 49:2244–2247 6. Byrnes CI, Isidori A, Praly L (2003) On the asymptotic properties of a system Arising in non-equilibrium theory of output regulation, preprint of the Mittag-Leffler Institute. 18, Stockholm 7. Isidori A, Byrnes CI (2007) The steady-state response of a nonlinear system: ideas, tools and applications, preprint 8. Davison EJ (1976) The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans Autom Contr AC-21:25–34 9. Delli Priscoli F, Marconi L, Isidori A (2006) A new approach to adaptive nonlinear regulation. SIAM J Contr Optim 45:829–855 10. Francis BA (1977) The linear multivariable regulator problem. SIAM J Contr Optim 14:486–505 11. Francis BA, Wonham WM (1976) The internal model principle of control theory. Automatica 12:457–465 12. Gauthier JP, Kupka I (2001) Deterministic observation theory and applications. Cambridge University Press, Cambridge 13. Hahn W (1967) Stability of motions. Springer, New York 14. Hale JK, Magalhães LT, Oliva WM (2002) Dynamics in infinite dimensions. Springer, New York

15. Huang J, Lin CF (1994) On a robust nonlinear multivariable servomechanism problem. IEEE Trans Autom Contr 39:1510– 1513 16. Huang J, Rugh WJ (1990) On a nonlinear multivariable servomechanism problem. Automatica 26:963–972 17. Isidori A (1995) Nonlinear Control Systems. Springer, London 18. Isidori A (2000), A tool for semiglobal stabilization of uncertain non-minimum-phase nonlinear systems via output feedback. IEEE Trans Autom Contr AC-45:1817–1827 19. Isidori A, Byrnes CI (1990) Output regulation of nonlinear systems. IEEE Trans Autom Contr 25:131–140 20. Isidori A, Marconi L, Serrani A (2003) Robust autonomous guidance: An internal model-based approach. Springer, London 21. Khalil H (1994) Robust servomechanism output feedback controllers for feedback linearizable systems. Automatica 30:587– 1599 22. Marconi L, Praly L, Isidori A (2006) Output stabilization via nonlinear luenberger observers. SIAM J Contr Optim 45:2277–2298 23. Serrani A, Isidori A, Marconi L (2001) Semiglobal nonlinear output regulation with adaptive internal model. IEEE Trans Autom Contr 46:1178–1194 24. Teel AR, Praly L (1995) Tools for semiglobal stabilization by partial state and output feedback. SIAM J Control Optim 33:1443– 1485 25. Willems JC (1991) Paradigms and puzzles in the theory of dynamical systems, IEEE Transaction on Automatic Control 36:259–294

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Systems and Control, Introduction to MATTHIAS KAWSKI Department of Mathematics and Statistics, Arizona State University, Tempe, USA Control of dynamical systems has a long history: Watt’s automatic centrifugal governor designed to regulate steam engines in the 1780s is considered a precursor of modern feedback control. Similarly, Bernoulli’s work in the 1690s on the brachystochrone problem is a progenitor of optimal control. A distinguishing external feature of controlled dynamical systems is the presence of inputs that interact with the system in the form of deliberate controls, or unavoidable perturbations. Commonly control systems also have outputs (observations) that typically provide only incomplete information about the state of the system. Systems and control theory utilizes a broad array of mathematical disciplines – but a uniting, characteristic feature is the kind of questions being asked. In the 1950s the field became more formalized, it began to develop its own distinctive identity, and it has rapidly evolved ever since. Naturally, the linear theory developed first and quickly became a mature subject, with controllers now pervading everyday life, from thermostats, dozens of controllers in every automobile, to cell phones, attitude control of satellites, electric power networks, highway traffic control, to name just a few. The articles in this section focus on the more recent developments, articulating how the fundamental problems and questions of systems and control theory are addressed in ever new settings, on new tools, and new applications. Whereas in classical dynamical systems one tries to predict the uniquely determined future, a basic distinguishing question in control theory asks whether it is possible to steer the system to any desired target. This question about controllability has a simple and elegant answer in the case of linear systems. While much also is known for nonlinear systems, many challenges remain in both the finite dimensional setting [see  Finite Dimensional Controllability] and in the setting of distributed systems modeled by partial differential equations [see  Control of Non-linear Partial Differential Equations]. Assuming controllability using controls that are functions of time, the next question asks whether one can automate the control using feedback, that is, by making it a function of the state or of a measured output. Whereas for linear systems the theory is straightforward, unavoidable topological obstacles make nonlinear feedback stabilization problems much more challenging [see  Stability and Feedback Stabilization]. A notion dual to control-

lability is that of observability: Rather than asking about the map from inputs to the state, a system is observable if the history of the outputs (measurements) determines the state of the system [see  Observability (Deterministic Systems) and Realization Theory]. The typical engineering problem involves much more complex multi-tasking, dealing with multiple objectives in the presence of diverse constraints. Much of this well developed theory of output regulation, disturbance rejection, while attending to performance criteria, has recently been generalized to nonlinear systems [see  System Regulation and Design, Geometric and Algebraic Methods in]. Generally, if there is one, then there are many control strategies that achieve a certain task – naturally one likes to single out a strategy that is the best. This leads to optimal control theory, which supersedes much of the classical calculus of variations, and which has rich interfaces with Lagrangian and Hamiltonian mechanics [see  Maximum Principle in Optimal Control]. In many applications all one has to work with are sampled data for inputs and measured outputs. Moreover, commonly these are also corrupted by measurement errors and noises. A fundamental problem is to devise algorithms to identify the best model system that could generate these input-output pairs. Characteristically such algorithm should work in real time and improve the model upon newly available measurements. This well developed broad field includes geometric approaches to stochastic systems [see  Stochastic Noises, Observation, Identification and Realization with] as well as more formal learning theories [see  Learning, System Identification, and Complexity]. Systems and control theory is unified by a common core of questions, but employs a diverse array of models. Consequently it interfaces a wide range of mathematical disciplines, utilizes many different mathematical tools, but also fosters new development in diverse subdisciplines. An area that has evolved closely together with control is nonsmooth analysis with its rich theory of tangent objects to nonsmooth sets. Not only are controlled dynamical systems naturally modeled by differential inclusions, but control intrinsically is full of objects that are not differentiable in a classical sense. Most important is the value function in optimal control which encodes the minimal cost (time) to the target. Closely related to classical dynamic programming, this function ought to satisfy the Hamilton–Jacobi–Bellmann equation, a first order partial differential equation, but the value function generally is not differentiable in a traditional sense [see  Nonsmooth Analysis in Systems and Control Theory]. Complementary to the nonsmooth approach are differential ge-

Systems and Control, Introduction to

ometric approaches which include Lie theory and symplectic geometry. Closely related is a functional analytic operator calculus introduced into control by Agrachëv and Gamkrelidze which linearizes problems by imbedding them into infinite dimensional settings. This dramatically facilitates formal manipulations of nonlinear objects, and also gives new ways of looking at the underlying geometry [see  Chronological Calculus in Systems and Control Theory]. Many of the early successful engineering implementations of systems and control theory used models with both continuous time and states (differential equations). But with the advent of ubiquitous digital controllers as well as ever broadening fields of applications the theories for increasingly diverse models are becoming well developed, too. These include, systems with or mix of continuous and discrete states (suitable, for example, to model systems with hysteresis) [see  Hybrid Control Systems] and discrete time Hamiltonian and Lagrangian systems [see

 Discrete Control Systems] There are many new applications with quickly developing theories such as control of biological and biomedical applications, social dynamics, manufacturing systems, financial markets, and many more. Some of the most intense research in recent years has focused on systems with distributed intelligent agents (controllers) each of which has access to only local information [see  Robotic Networks, Distributed Algorithms for]. Compared to these emerging areas the geometric theory of mechanical systems is one of the most mature and thriving research areas with scores of exciting new problems. It serves as a role model for the depth of geometric insight and for its well-understood close interconnections with many other disciplines [see  Mechanical Systems: Symmetries and Reduction]. Acknowledgment This work was partially supported by the National Science Foundation through the grant DMS 05-09030.

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Topological Dynamics

Topological Dynamics ETHAN AKIN Mathematics Department, The City College, New York City, USA Article Outline Glossary Definition of the Subject Introduction and History Dynamic Relations, Invariant Sets and Lyapunov Functions Attractors and Chain Recurrence Chaos and Equicontinuity Minimality and Multiple Recurrence Future Directions Cross References Bibliography

Glossary Attractor An attractor is an invariant subset for a dynamical system such that points sufficiently close to the set remain close and approach the set in the limit as time tends to infinity. Equilibrium An equilibrium, or a fixed point, is a point which remains at rest for all time. Invariant set A subset is invariant if the orbit of each point of the set remains in the set at all times, both positive and negative. The set is + invariant, or forward invariant, if the forward orbit of each such point remains in the set. Lyapunov function A continuous, real-valued function on the state space is a Lyapunov function when it is non-decreasing on each orbit as time moves forward. Orbit The orbit of an initial position is the set of points through which the system moves as time varies positively and negatively through all values. The forward orbit is the subset associated with positive times. Recurrence A point is recurrent if it is in its own future. Different concepts of recurrence are obtained from different interpretations of this vague description. Repellor A repellor is an attractor for the reverse system obtained by changing the sign of the time variable. Transitivity A system is transitive if every point is in the future of every other point. Periodicity, minimality, topological transitivity and chain transitivity are different concepts of transitivity obtained from different interpretations of this vague description.

Definition of the Subject A dynamical system is model for the motion of a system through time. The time variable is either discrete, varying over the integers, or continuous, taking real values. The systems considered in topological dynamics are primarily deterministic, rather than stochastic, so the the future states of the system are functions of the past. As the name suggests, topological dynamics concentrates on those aspects of dynamical systems theory which can be grasped by using topological methods. Introduction and History The many branches of dynamical systems theory are outgrowths of the study of differential equations and their applications to physics, especially celestial mechanics. The classical subject of ordinary differential equations remains active as a topic in analysis, see, e. g. the older texts Coddington and Levinson [28] and Hartman [39] as well as Murdock [50]. One can observe the distinct fields of differentiable dynamics, measurable dynamics (that is, ergodic theory) and topological dynamics all emerging in the work of Poincaré on the Three Body Problem (for a history, see Barrow-Green [23]). The transition from the differential equations to the dynamical systems viewpoint can be seen in the two parts of the great book Nemitskii and Stepanov [52]. We can illustrate the difference by considering the initial value problem in ordinary differential equations: dx D (x) ; x(0) D p : (1) dt Here x is a vector variable in a Euclidean space X D Rn or in a manifold X, and the initial point p lies in X. The infinitesimal change (x) is thought of as a vector attached to the point x so that  is a vector field on X. The associated solution path is the function  such that as time t varies, x D (t; p) moves in X according to the above equation and with p D (0; p) so that p is associated with the initial time t D 0. The solution is a curve in the space X along which x moves beginning at the point p. A theorem of differential equations asserts that the function  exists and is unique, given mild smoothness conditions, e. g. Lipschitz conditions, on the function . Because the equation is autonomous, i. e.  may vary with x, but is assumed independent of t, the solutions satisfy the following semigroup identities, sometimes also called the Kolmogorov equations: (t; (s; p)) D (t C s; p) :

(2)

Suppose we solve Eq. (1), beginning at p, and after s units of time, we arrive at q D (s; p). If we again solve the

Topological Dynamics

equation, beginning now at q, then the identity (2) says that we continue to move along the old curve at the same speed. Thus, after t units of time we are where we would have been on the old solution at the same time, t C s units after time 0. The initial point p is a parameter here. For each solution path it remains constant, the fixed base point of the path. The solution path based at p is also called the orbit of p when we want to emphasize the role of the initial point. It follows from the semigroup identities that distinct solution curves, regarded as subsets of X, do not intersect and so X is subdivided, foliated, by these curves. Changing p may shift us from one curve to another, but the motion given by the original differential equation is always on one of these curves. The gestalt switch to the dynamical systems viewpoint occurs when we reverse the emphasis between t and p. Above we thought of p as a fixed parameter and t as the time variable along the solution path. Instead, we now think of the initial point, relabeled x, as our variable and the time value as the parameter. For each fixed t value we define the time-t map  t : X ! X by  t (x) D (t; x). For each point x 2 X we ask whither it has moved in t units of time. The function  : T  X ! X is called the flow of the system and the semigroup identities can be rewritten:  t ı  s D  tCs for all t; s 2 T :

(3) t

These simply say that the association t 7! is a group homomorphism from the additive group T of real numbers to the automorphism group of X. In particular, observe that the time-0 map  0 is the identity map 1X . We originally obtained the flow  by solving a differential equation. This requires differentiable structure on the underlying space which is why we specified that X be a Euclidean space or a manifold. The automorphism group is then the group of diffeomorphisms on X. For the subject of topological dynamics we begin with a flow, a continuous map  subject to the condition (3). In modern parlance a flow is just a continuous group action of the group T of additive reals on the topological space X. The automorphism group is the group of homeomorphisms on X. If we replace the group of reals by letting T be the group of integers then the action is entirely determined by the generator f Ddef  1 , the time 1 homeomorphism with  n obtained by iterating f n times if n is positive and iterating the inverse f 1 jnj times if n is negative. Above I mentioned the requirement that the vector field  be smooth in order that the flow function  be defined. However, I neglected to point out that in the ab-

sence of some sort of boundedness condition the solution path might go to infinity in a finite time. In such a case the function  would not be defined on the entire domain T  X but only on some open subset containing f0g  X. The problems related to this issue are handled in the general theory by assuming that the space X is compact. Of course, Euclidean space is only locally compact. In applying the general theory to systems on a noncompact space one usually restricts to some compact invariant subset or else compactifies, i. e. embeds the system in one on a larger, compact space. Already in Nemytskii and Stepanov [52] much attention is devoted to conditions so that a solution path has a compact closure, see also Bahtia and Szego [27]. As topological dynamics has matured the theory has been extended to cover the action of more general topological groups T, usually countable and discrete, or locally compact, or Polish (admits a complete, separable metric). This was already emphasized in the first treatise which explicitly concerned topological dynamics, Gottschalk and Hedlund [38]. In differentiable dynamics we return to the case where the space X is a manifold and the flow is smooth. The breadth and depth of the results then obtained make it much more than a subfield of topological dynamics, see, for example, Katok and Hasselblatt [45]. For measurable dynamics we weaken the assumption of continuity to mere measurability but assume that the space carries a measure invariant with respect to the flow, see, for example, Peterson [54] and Rudolph [56]. The measurable and topological theories are especially closely linked with a number of parallel results, see Glasner [36] as well as Alpern and Prasad [12]. The reader should also take note of Oxtoby [53], a beautiful little book which explicitly describes this parallelism using a great variety of applications. The current relationship between topological dynamics and dynamical systems theory in general is best understood by analogy with that between point-set, or general, topology and analysis. General topology proper, even excluding algebraic topology and homotopy theory, is a large specialty with a rich history and considerable current research (for some representative surveys see the Russian Encyclopedia volumes [13,14,15]). But much of this work is little known to nonspecialists. On the other hand, the fundamentals of point-set topology are part of the foundation upon which modern analysis is built. Compactness was a rather new idea when it was used by Jesse Douglas in his solution of the Plateau Problem, Douglas [32] (see Almgren [11]). Nowadays continuity, compactness and connectedness are in the vocabulary of every analyst. In addition, there recur unexpected applications of hitherto specialized topics.

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Topological Dynamics

For example, indecomposable continua, examined by Bing and his students, see Bing [25], are now widely recognized and used in the study of strange attractors, see Brown [26]. Similarly, the area of topological dynamics proper is large and some of the more technical results have found application. We will touch on some of these in the end, but for the most part this article will concentrate on those basic aspects which provide a foundation for dynamical systems theory in general. We will focus on chain recurrence and the theory of attractors, following the exposition of Akin [1] and Akin, Hurley and Kennedy [8]). To describe what we want to look for, let us begin with the simplest qualitative situation: the differential equation model (1) where the vector field  is the gradient of some smooth real-valued potential function U on X. Think of X as the Euclidean plane and the graph of U as a surface in space over X. The motion in X can be visualized on the surface above. On the surface it is always upward, perpendicular to the contour curves of constant height. For simplicity we will assume that U has isolated critical points. These critical points: local maxima, minima and saddles, are equilibria for the system, points at which the the gradient field  vanishes. We observe two kinds of behavior. The orbit of a critical point is constant, resting at equilibrium. The other kind exhibits what engineers call transient behavior. A non-equilibrium solution path moves asymptotically toward a critical point (or towards infinity). As it approaches its limit, the motion slows, becoming imperceptible, indistinguishable from rest at the limit point. The set of points whose orbits tend to a particular critical point e is called the stable set for e. Each local maximum e is an attractor or sink. The stable set for e is an open set containing e which is called the domain of attraction for e. Such a state is called asymptotically stable illustrated by the rest state of a cone on its base. The local minima are repellors or sources which are attractors for the system with time reversed. Solution paths near a repellor move away from it. The stable set for a repellor e consists of e alone. Consider a cone balanced on its point. A saddle point between two local maxima is like the highest point of a pass between two mountains. Separating the domains of attraction for the two peaks are solution paths which have limit the saddle point equilibrium. There is a kind of knife, used for cutting bread dough, which has a semi-circular blade. The saddle point equilibrium is a state like this knife balanced on the midpoint of its blade. A slight perturbation will cause it to fall down on one side or the other. But if you start the knife balanced elsewhere along its blade there remains the possibility – not achievable in practice – of its rolling back along

the blade toward balance at the midpoint equilibrium. Notice that these are first-order systems with no momentum. Imagine everything going on in thick, clear molasses. As this model illustrates, it is usually true that the stable set of a saddle point is a lower dimensional set in X and the union of the domains of attraction of the local maxima is a dense open subset of X. There do exist examples where the stable set of a saddle has nonempty interior. This is a pathology which we will hope to exclude by imposing various conditions. For example, the potential function U is called a Morse function when all of its critical points are nondegenerate. That is, the Hessian matrix of second partials is nonsingular at each critical point. For the gradient system of a Morse function each equilibrium is of a type called hyperbolic. For the saddle points of such a system the stable sets are manifolds of lower dimension. From the cone example, we omitted what physicists call neutral stability, the cone resting on its side. From our point of view this is another sort of pathology: an infinite, connected set of equilibria. Each of these equilibria is stable but not asymptotically stable. If we perturb the cone by lifting its point and turning it a bit, then it drops back toward an equilibrium near to but not necessarily identical with the original state (Remember, no momentum). We obtain a similar classification into sinks, saddles, etc. and complementary transient behavior when we remove the assumption that the system comes from the gradient of U and retain only the condition that U increases along nonequilibrium solution paths. The function U is then called a strict Lyapunov function for the system. Instead of the steepest path ascent of a mountain goat, we may observe the spiraling upward of a car on a mountain road. However, these gradient-like systems are too simple to represent a typical dynamical system. Lacking is the general behavior complementary to transience, namely recurrence. A point is recurrent – in some sense – if it is “in its own future” – in the appropriate sense. Beyond equilibrium the simplest kind of recurrence occurs on a periodic orbit. A periodic orbit returns infinitely often to each point on the orbit. As we will see, there are increasingly broad concepts of recurrence obtained by extending the notion of the “future” of a point. Clearly, a real-valued function cannot be strictly increasing along a periodic orbit. When we consider Lyapunov functions in general we will see that they remain constant along the orbit of each recurrent point. Nonetheless, the picture we are looking for can be related to the gradient landscape by replacing the critical points by blobs of various sizes. Each blob is a closed, in-

Topological Dynamics

variant set of a special type. A subset A is an invariant set for the system when A contains the entire orbit of each of its points. The special condition on each blob A which replaces an equilibrium is a kind of transitivity. This means that if p and q are points of A then each point is in the “future” of the other and so, in particular, each point is recurrent in the appropriate sense. Here again it remains to provide a meaning – or actually several different meanings – for this vague notion of “future”. In this more general situation the transient orbits need not converge to a point. Instead, each accumulates on a closed subset of one of these blobs. If there are only finitely many of the blobs then there is a classification of them as attractor, repellor, or saddle analogous to the description for equilibria in the gradient system. Within each blob the motion may be quite complicated. It is in attempting to describe such motions that the concept of chaos arises. For some applied fields this sort of thinking is relatively new. When I learned population genetics – admittedly that was over thirty years ago – most of the analysis consisted of identifying and classifying the equilibria, tricky enough in several variables. This is perfectly appropriate for gradient-like systems and is a good first step in any case. However, it has become apparent that more complicated recurrence may occur and so requires attention. While most applications use differential equations and the associated flows, it is more convenient to develop the discrete time theory and then to derive from it the results for flows. In what follows we will describe the results for a cascade, a homeomorphism f on a compact metric space X with the dynamics introduced by iteration. Focusing on this case, we will not discuss further real flows or noninvertible functions, and we will omit as well the extensions to noncompact state spaces and to compact, nonmetrizable spaces. Dynamic Relations, Invariant Sets and Lyapunov Functions It is convenient to assume that our state spaces are nonempty and metrizable. It is essential to assume that they are compact. Recall that if A is a subset of a compact metric space X then A is closed if and only if it is compact. Also, the continuous image of a compact set is compact and so for continuous maps between compact metric spaces the image of a closed set is closed. Perhaps less familiar are the following important results: Proposition 1 Let X be a compact metric space and fA n g be a decreasing sequence of closed subsets of X with intersection A.

(a) If U is an open subset of X with A  U then for sufficiently large n, A n  U. (b) If A n is nonempty for every n, then the intersection A is nonempty. (c) If h : X ! Y is a continuous map with Y a metric space then \ h(A n ) D h(A) : (4) n

Proof (a): We are assuming A nC1  A n for all n and T A D n A n . The complementary open sets Vn D X n A n are increasing and, together with U, they cover X. By compactness, fU; V1 ; : : : ; VN g covers X for some N and so fU; VN g suffice to cover X. Hence, A N is a subset of U as is An for any n  N. (b): If A is empty then U D ; is an open set containing A and so by (a), An is empty for sufficiently large n. (c): Since A  A n for all n, it is clear that h(A) is conT tained in n h(A n ). On the other hand, if y is a point of the latter intersection then fh1 (y) \ A n g is a decreasing sequence of nonempty compact sets. By (b) the intersection h1 (y) \ A is nonempty.  If fB n g is any sequence of closed subsets of X then we define the lim sup : lim sup B n Ddef n

\ [ n

Bk

(5)

kn

where Q denotes the closure of Q. It follows from (b) above that the lim sup of a sequence of nonempty sets is nonempty. It is easy to check that !   [ [ Bn D B n [ lim sup B n : (6) n

n

n

We want to study a homeomorphism on a space. By compactness this is just a bijective (= one-to-one and onto) continuous function from the space to itself. It will be convenient to use the more general language of relations. A function f : X ! Y is usually described as a rule associating to every point x in X a unique point y D f (x) in Y. In set theory the function f is defined to be the set of ordered pairs f(x; f (x)) : x 2 Xg. Thus, the function f is a subset of the product X  Y. It is what other people call the graph of the function. We will use this language so that, for example, the identity map 1X on X is the diagonal subset f(x; x) : x 2 Xg. The notation is extended by defining a relation from X to Y, written F : X ! Y, to be an arbitrary subset of X  Y. Then F(x) D fy : (x; y) 2 Fg: Thus, a relation is a function exactly when the set F(x) contains

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Topological Dynamics

a single point for every x 2 X. In the function case, we will use the symbol F(x) for both the set and the single point it contains, the latter being the usual meaning of F(x). As they are arbitrary subsets of X  Y we can perform set operations like union, intersection, closure and interior on relations. In addition, for F : X ! Y we define the inverse F 1 : Y ! X by F 1 Ddef f(y; x) : (x; y) 2 Fg :

(7)

If A  X then its image is F(A) Ddef fy : (x; y) 2 F for some x 2 Ag [ D F(x) D 2 ((A  Y) \ F) ;

(8)

x2A

where 2 : X  Y ! Y is the projection to the second coordinate. If G : Y ! Z is another relation then the composition G ı F : X ! Z is the relation given by G ı F Ddef f(x; z) : there exists y 2 Y such that (x; y) 2 F and (y; z) 2 Gg D 13 ((X  G) \ (F  Z)) ;

(9)

where 13 : X Y  Z ! X  Z is the projection map. This generalizes composition of functions and, as with functions, composition is associative. Clearly, (G ı F)1 D F 1 ı G 1 : We call F a closed relation when it is a closed subset of X  Y. Clearly, the inverse of a closed relation is closed and by compactness, the composition of closed relations is closed. If A is a closed subset of X and F is a closed relation then the image F(A) is a closed subset of Y. For relations being closed is analogous to being continuous for functions. In fact, a function is continuous if and only if, regarded as a relation, it is closed. This is another application of compactness. If Y D X, so that F : X ! X, then we call F a relation on X. For a positive integer n we define F n to be the n-fold composition of F with F 0 Ddef 1 X and F n Ddef (F 1 )n D (F n )1 . This is well-defined because composition is associative. Clearly, F m ıF n D F mCn when m and n have the same sign, i. e. when mn  0. On the other hand, the equations F ı F 1 D F 1 ı F D 1 X D F 0 all hold if and only if the relation F is a bijective function. The utility of this relation-speak, once one gets used to it, is that it allows us to extend to this more general situation our intuitions about a function as a way of moving from input here to output there. For example, if  0 then

we can use the metric d on X to define the relations on X V Ddef f(x; y) : d(x; y) < g ; V¯ Ddef f(x; y) : d(x; y)  g :

(10)

Thus, V (x) and V¯ (x) are the open ball and the closed ball centered at x with radius . We can think of these relations as ways of moving from a point x to a nearby point. Each V¯ is a closed, symmetric and reflexive relation. The triangle inequality is equivalent to the inclusion V¯ ı V¯ı  V¯ Cı . In general, a relation F on X is reflexive if 1 X  F, symmetric if F 1 D F and transitive if F ı F  F. Now we apply this relation notation to a homeomorphism f on X. For x 2 X the orbit sequence of x is the bi-infinite sequence f: : : ; f 2 (x); f 1 (x); x; f (x); f 2 (x); : : : g. This is just the discrete time analogue of the solution path discussed in the Introduction. We are thinking of it as a sequence and so as a function from the set Z of discrete times to the state space X with parameter the initial point x. Now we define the orbit relation O f Ddef

1 [

fn :

(11)

nD1

Thus, O f (x) D f f (x); f 2 (x); : : : g is a set, not a sequence, consisting of the states which follow the initial point x in time. Notice that for reasons which will be clear when we consider the cyclic sets below, we begin the union with n D 1 rather than n D 0 and so the initial point x itself need not be included in O f (x). If with think of the point f (x) as the immediate temporal successor of x then O f (x) is the set of points which occur on the orbit of x at some positive time. This is the first – and simplest – interpretation of the “future” of x with respect to the dynamical system obtained by iterating f . It is convenient to extend this notion by including the limit points of the positive orbit sequence. For x 2 X define ! f (x) D lim sup f f n (x)g ; n

(12)

R f (x) Ddef O f (x) D O f (x) [ ! f (x) :

Thus, from f we have defined the orbit relation O f and the orbit closure relation R f with R f D O f [ ! f . While R f (x) is closed for each x, the relation R f itself is usually not closed. As was mentioned above, among relations the closed relations are the analogues of continuous

Topological Dynamics

functions. We obtain closed relations by defining n

˝ f D lim sup f ; n

(13)

N f (x) Ddef O f D O f [ ˝ f :

Here we are taking the closure in X  X and so we obtain closed relations. The relation N f , defined by Auslander et al. [18,19,20] (see also Ura [63,64]), is called the prolongation of f : N f (x) is our next, broader, notion of the “future” of x. To compare all these suppose that x; y 2 X. Then y 2 O f (x) when there exists a positive integer n such that y D f n (x) while y 2 ! f (x) if there is a sequence of positive integers n i ! 1 such that f n i (x) ! y. On the other hand, y 2 ˝ f (x) iff there are sequences x i ! x and n i ! 1 such that f n i (x i ) ! y. Thus, y 2 R f (x) if for every > 0 we can run along the orbit of x and at some positive time make a smaller than jump to y. Similarly, y 2 N f (x) if for every > 0 we can make an initial small jump to a point x1 , run along the orbit of x1 and at some positive time make an small jump to y. The relations O f and R f are transitive but usually not closed. In general, when we pass to the closure, obtaining N f , we lose transitivity. When we consider Lyapunov functions we will see why it is natural to want both of these properties, closure and transitivity. The intersection of any collection of closed, transitive relations is a closed, transitive relation. Notice that X  X is such a relation. Thus, we obtain G f , the smallest closed, transitive relation which contains f by intersecting: G f Ddef

\˚

Q  X  X: Q D Q

and f ; Q ı Q  Q :

(14)

There is an alternative procedure, due to Conley, see [29], which constructs a closed, transitive relation, generally larger than G f , in a simple and direct fashion. A chain or 0-chain is a finite or infinite sequence fx n g such that x nC1 D f (x n ) along the way, i. e. a piece of the orbit sequence. Given  0 an -chain is a finite or infinite sequence fx n g such that each x nC1 at most distance away from the point f (xn ), i. e. x nC1 2 V¯ ( f (x n )). If the chain has at least two terms, but only finitely many, then the first and last terms are called the beginning and the end of the chain. The number of terms minus 1 is then called the length of the chain. We say that x chains to y, written y 2 C f (x) if for every > 0 there is an chain which begins at x and ends at y. That is, \ C f Ddef O(V¯ ı f ) : (15) >0

Compare this with (12) and (13). If y 2 R f (x) then we can get to y by moving along the orbit of x and then taking an arbitrarily small jump at the end. If y 2 N f (x) then we are allowed a small jump at the beginning as well as the end. Finally, if y 2 C f (x) then we are allowed a small jump at each iterative step. The chain relation is of great importance for applications. Suppose we are computing the orbit of a point on a computer. At each step there is usually some round-off error. Thus, what we take to be an orbit sequence is in reality an chain for some positive . It follows that, in general, what we can expect to compute directly about f is only that level of information which is contained in C f . As the intersection of transitive relations, C f is transitive. It is not hard to show directly that the chain relation C f is also closed. Since x1 D x; x2 D f (x) is a 0-chain beginning at x and ending at f (x), we have f  C f . It follows that G f  C f . This inclusion may be strict. The identity map f D 1 X is already a closed equivalence relation. Hence, G 1 X D 1 X . On the other hand, for any > 0 the relation OV is an open equivalence relation, and so each equivalence class is clopen (= closed and open). If X is connected then the entire space is a single equivalence class. Since this is true for every positive we have X connected H) C 1 X D X  X :

(16)

Since C f is transitive, the composites f(C f )n g form a decreasing sequence of closed transitive relations. We denote by ˝ C f the intersection of this sequence. That is, ˝ C f Ddef

1 \

(C f )n :

(17)

nD0

One can show that y 2 ˝ C f (x) if and only if for every > 0 and positive integer N there is an chain of length greater than N which begins at x and ends at y. In addition, the following identity holds (compare (12) and (13)): C f D O f [ ˝C f :

(18)

Thus, built upon f we have a tower of relations: f  Of  Rf  N f  Gf  C f :

(19)

These are the relations which capture the successively broader notions of the “future” of an input x. A useful identity which holds for A D O; R; N ; G and C is A f D f [ (A f ) ı f D f [ f ı (A f ) :

(20)

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Topological Dynamics

These are easy to check directly for all but G . For that one, observe that f [(G f )ı f and the other composite are closed and transitive and so each contains G f . It is also easy to show that for A D O; N ; G and C : A( f 1 ) D (A f )1 and so we can omit the parentheses in these cases. The analogue for R is usually false and we define ˛ f Ddef !( f

1

):

(21)

Thus, ˛ f (x) is the set of limit points of the negative time orbit sequence fx; f 1 (x); f 2 (x); : : : g and it is usually not true that ˛ f equals (! f )1 . For  D ˛; !; ˝ and ˝ C it is true that  f D f ı  f D f 1 ı  f D  f ı f D  f ı f 1 : (22) Now we are ready to consider the variety of recurrence concepts. Recall that a point x is recurrent – in some sense – if it lies in its own “future”. Thus, for any relation F on X we define the cyclic set jFj Ddef fx : (x; x) 2 Fg :

(23)

Clearly, if F is a closed relation then jFj is a closed subset of X. A point x lies in j f j when x D f (x) and so j f j is the set of fixed points for f , while x 2 jO f j when x D f n (x) for some positive integer n and so jO f j is the set of periodic points. It is easy to check that every periodic point is contained in j! f j and so j! f j D jR f j. These are called recurrent points or sometimes the positive recurrent points to distinguish them from j˛ f j, the set of negative recurrent points. Similarly, we have j˝ f j D jN f j, called the set of nonwandering points. The points of jG f j are called generalized nonwandering and those of jC f j D j˝ C f j are called chain recurrent. The set of periodic points and the sets of recurrent points need not be closed. The rest, associated with closed relations, are closed subsets. For an illustration of these ideas, observe that for x; y; z 2 X y; z 2 ! f (x) H) z 2 ˝ f (y) :

(24)

Just hop from y to a nearby point on the orbit of x, moving arbitrarily far along the orbit, you repeatedly arrive nearby z and then can hop to it. In particular, with y D z we see that every point y of ! f (x) is non-wandering. However, the points of ! f (x) need not be recurrent. That is, while y 2 ˝ f (y) for all y 2 ! f (x) it need not be true that y 2 ! f (y). In particular, R f (y) can be a proper subset of N f (y).

On the other hand, it is true that for most points x in X the orbit closure R f (x) is equal to the prolongation set N f (x). Recall that a subset of a complete metric space is called residual when it is the countable intersection of dense, open subsets. By the Baire Category Theorem a residual subset is dense. Theorem 1 If f is a homeomorphism on X then fx 2 X : ! f (x) D ˝ f (x)g D fx 2 X : R f (x) D N f (x)g is a residual subset of X. In particular, if every point is nonwandering, i. e. x 2 ˝ f (x) for all x, then the set of recurrent points is residual in X. However, if the set of nonwandering points is a proper subset of X, it need not be true that most of these points are recurrent. The closure of j! f j can be a proper subset of the closed set j˝ f j. From recurrence we turn to the notion of invariance. Let F be a relation on X and A be a closed subset of X. We call A + invariant for F if F(A)  A and invariant for F if F(A) D A. For a homeomorphism f on X, the set A is invariant for f if and only if it is + invariant for f and for f 1 . For example, (20) implies that for A D O; R; N ; G and C each of the sets A f (x) is + invariant for f and (22) implies that for  D ˛; !; ˝ and ˝ C each of the sets  f (x) is invariant for f . Finally, for A D O; R; N ; G and C each of the cyclic sets jA f j is invariant for f . If A is a nonempty, closed invariant subset for a homeomorphism f then the restriction f jA is a homeomorphism on A and we call this dynamical system the subsystem determined by A. In general, if F is a relation on X and A is any subset of X then we call the relation F\(AA) on A the restriction of F to A. For a homeomorphism f the families of + invariant subsets and of invariant subsets are each closed under the operations of closure and interior and under arbitrary unions and intersections. If A is + invariant then the sequence f f n (A)g is decreasing and the intersection is f invariant. Furthermore, this intersection contains every other f invariant subset of A and so is the maximum invariant subset of A. If A is + invariant for f then it is + invariant for O f . If, in addition, A is closed then it is + invariant for R f . However, + invariance with respect to the later relations in the tower (19) are successively stronger conditions, and the relations of (19) provide convenient tools for studying these conditions. We call a closed + invariant subset A a stable subset, or a Lyapunov stable subset, if it has a neighborhood basis of + invariant neighborhoods. That is, if G is open and A  G then there exists a + invariant open set U such that A  U  G.

Topological Dynamics

Theorem 2 A closed subset A is + invariant for N f if and only if it is a stable + invariant set for f . Proof If G is an open set which contains A and N f (A)  A then U D fx : N f (x)  Gg is an open set which contains A and which is + invariant by (20). The reverse implication is easy to check directly.  Invariance with respect to G f is characterized by using Lyapunov functions, which generalize the strict Lyapunov functions described in the introduction. For a closed relation F on X, a Lyapunov function L for F is a continuous, real-valued function on X such that (x; y) 2 F H) L(x)  L(y) ;

(25)

(some authors, e. g. Lyapunov, use the reverse inequality). For any continuous, real-valued function L on X the set f(x; y) : L(x)  L(y)g is a closed, transitive relation on X. To say that L is a Lyapunov function for F is exactly to say that this relation contains F. It follows that if L is a Lyapunov function for a homeomorphism f then it is automatically a Lyapunov function for the closed, transitive relation G f . That L be a Lyapunov function for C f is usually a stronger condition. For example, any continuous, real-valued function is a Lyapunov function for 1X , but by (16) if X is connected then constant functions are the only Lyapunov functions for C 1 X . The following result is a dynamic analogue of Urysohn’s Lemma in general topology and it has a similar proof. Theorem 3 A closed subset A is + invariant for G f if and only if there exists a Lyapunov function L : X ! [0; 1] for f such that A D L1 (1). A Lyapunov function for a homeomorphism f is non-decreasing on each orbit sequence x; f (x); f 2 (x); : : : . Suppose that x is a periodic point, i. e. x 2 jO f j. Then x D f n (x) for some positive integer n, and it follows that L must be constant on the orbit of x. Now suppose, more generally, that x is a generalized recurrent point, i. e. x 2 jG f j. By (20) ( f (x); x) 2 G f and so L( f (x)) D L(x) whenever L is a Lyapunov function for f (and hence for G f ). Recall that the set jG f j of generalized recurrent points is f invariant. It follows that f n (x) 2 jG f j for every integer n and so L( f nC1 (x)) D L( f n (x)) for all n. Thus, a Lyapunov function for a homeomorphism f is constant on the orbit of each generalized recurrent point x. Similarly, if x is a chain recurrent point, i. e. x 2 jC f j, then L( f (x)) D L(x) whenever L is a Lyapunov function for C f and so a Lyapunov function for C f is constant on the orbit of each chain recurrent point.

Of fundamental importance is the observation that one can construct Lyapunov functions for f (and for C f ) which are increasing on all orbit sequences which are not generalized recurrent (respectively, chain recurrent). Theorem 4 For a homeomorphism f on a compact metric space X there exist continuous functions L1 ; L2 : X ! [0; 1] such that L1 is a Lyapunov function for f , and hence for G f , and L2 is a Lyapunov function for C f and, in addition, x 2 jG f j () L1 (x) D L1 ( f (x)) ; x 2 jC f j () L2 (x) D L2 ( f (x)) :

(26)

Lyapunov functions which satisfy the conditions of (26) are called complete Lyapunov functions for f and for C f , respectively. If L is a Lyapunov function for f then we define the set of critical points for L ˚ (27) jLj Ddef x 2 X : L(x) D L( f (x)) : This language is a bit abusive because here criticality describes a relationship between L and f . It does not depend only upon L. However, we adopt this language to compare the general situation with the simpler strict Lyapunov function case in the introduction. Similarly, we call L(jLj)  R the set of critical values for L. The complementary points of X and R respectively are called regular points and regular values for L. Thus, the generalized recurrent points are always critical points for a Lyapunov function L and for a complete Lyapunov function these are the only critical points. For invariance with respect to C f we turn to the study of attractors. Attractors and Chain Recurrence For a homeomorphism f on X we say that a closed set U is inward if f (U) is contained in U ı , the interior of U. By compactness this implies that V¯ ı f (U)  U for some > 0. That is, U is + invariant for the relation V¯ ı f . Hence, any chain for f which begins in U remains in U. It follows that an inward set for f is C f + invariant. For example, assume that L is a Lyapunov function for f . Then for any s 2 R the closed set Us D fx : L(x)  sg is + invariant for f . Suppose now that s is a regular value for L. This means that for all x such that L(x) D s we have L( f (x)) > L(x) D s. On the other hand, for the remaining points x of U s we have L( f (x))  L(x) > s. Thus, f (U s ) is contained in the open set fx : L(x) > sg  Us and so U s is inward. It easily follows that L is a Lyapunov function for C f if the set of critical values is nowhere dense.

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Topological Dynamics

T n If U is inward for f then we define A D 1 nD0 f f (U)g to be the attractor associated with U. Since an inward set U is + invariant for f , the sequence f f n (U)g is decreasing. In fact, for U inward and n; m 2 Z we have n > m H) f n (U)  f m (U)ı :

(28)

The associated attractor is the maximum f invariant subset of U and f f n (U) : n 2 Zg is a sequence of inward neighborhoods of A, forming a neighborhood basis for the set A. That is, if G is any open which contains A then by Proposition 1(a), f n (U)  G for sufficiently large n. For example, the entire space X is an inward set and is its own associated attractor. In general, a set A is inward and equal to its own attractor if and only if A is a clopen, f invariant set. The power of the attractor idea comes from the equivalence of a number of descriptions of different apparent strength. We use the “weak” ones to test for an attractor and then apply the “strong” conditions. The following collects these alternative descriptions. Theorem 5 Let f be a homeomorphism on X and A be a closed f invariant subset of X. The following conditions are equivalent. (i) (ii) (iii) (iv) (v) (vi) (vii)

A is an attractor. That is, there exists an inward set U T n such that 1 nD0 f (U) D A. There exists a neighborhood G of A such that T1 n nD0 f (G)  A. A is N f + invariant and the set fx : ! f (x)  Ag is a neighborhood of A. The set fx : ˝ f (x)  Ag is a neighborhood of A. The set fx : ˝ C f (x)  Ag is a neighborhood of A. A is G f + invariant and the set A \ jG f j is clopen in the relative topology of the closed set jG f j. A is C f + invariant and the set A \ jC f j is clopen in the relative topology of the closed set jC f j.

Applying Theorem 2 to condition (iii) of Theorem 5 we see that if A is a stable set for f and, in addition, the orbit of every point in some neighborhood of x tends asymptotically toward A then A is an attractor. The latter condition alone does not suffice, although the strengthening in (iv) is sufficient. For example, the homeomorphism of [0; 1] defined by t 7! t 2 has f0g as an attractor. If we identify the two fixed points 0; 1 2 [0; 1] by mapping t to z D e 2 i t then we obtain a homeomorphism f on the unit circle X. For every z 2 X, we have ! f (z) D f1g, the unique fixed point. However, f1g is not an attractor for f . In fact, ˝ f (1) D X. The class of attractors is closed under finite union and finite intersection. Using infinite intersections we can characterize C f invariance.

Theorem 6 Let f be a homeomorphism on X and A be a closed f invariant subset of X. The following conditions are equivalent. When they hold we call A a quasi-attractor for f . (i) A is C f + invariant. (ii) A is the intersection of a (possibly infinite) set of attractors. (iii) The set of inward neighborhoods of A form a basis for the neighborhood system of A. From Theorem 2 again it follows that a quasi-attractor is stable. If A is a closed invariant set for a homeomorphism f then A is called an isolated invariant set if it is the maximum invariant subset of some neighborhood U of A. That is, C1 \ f n (U) : (29) AD nD1

In that case, U is called an isolating neighborhood for A. Notice that if U is a closed isolating neighborhood for A and the positive orbit of x remains in U, i. e. f n (x) 2 U for n D 0; 1; : : : then ! f (x)  A because ! f (x) is an invariant subset of U. Since an attractor is the maximum invariant subset of some inward set, it follows that an attractor is isolated. Conversely, by condition (iii) of Theorem 5 an invariant set is an attractor precisely when it is isolated and stable. In particular, a quasi-attractor is an attractor exactly when it is an isolated invariant set. An attractor for f 1 is called a repellor for f . If U is an inward set for f then X n U D X n (U ı ) is an inward T n (X n U) is the associated set for f 1 and B D 1 nD0 f repellor for f . We call B the repellor dual to the attractor T n 1 A D 1 nD0 f (U). Recall that an f invariant set is f invariant. In particular, attractors and repellors are both S n (U) D X n B f and f 1 invariant. The open set 1 nD0 f is called the domain of attraction for A. The name comes from the implication: x 2 X n B H) !(x)  ˝ C f (x)  A ; x 2 X n A H) ˛(x)  ˝ C f 1 (x)  B :

(30)

For example, the entire space X is both an attractor and a repellor with dual ;. If A is an attractor then we call the set of chain recurrent points in A, i. e. A \ jC f j; the trace of the attractor A. An attractor is determined by its trace via the equation C f (A \ jC f j) D A :

(31)

The trace of an attractor is a clopen subset of jC f j by part (vii) of Theorem 5 Conversely, suppose A0 is a subset of jC f j which is + invariant for the restriction of C f to jC f j.

Topological Dynamics

That is, x 2 A0 and y 2 C f (x) \ jC f j implies y 2 A0 . If A0 is clopen in jC f j, then C f (A0 ) is the attractor with trace A0 and C f 1 (jC f j n A0 ) is the dual repellor. By Theorem 6 if A0 is merely closed then C f (A0 ) is a quasi-attractor. With the relative topology the set jC f j D jC f 1 j of chain recurrent points is a compact metric space and so has only countably many clopen subsets (Every clopen subset is a finite union of members of a countable basis for the topology). It follows that, while there are often uncountably many inward sets, there are only countably many attractors. When restricted to jC f j D jC f 1 j the closed relation C f \ C f 1 is reflexive as well as symmetric and transitive. The individual equivalence classes are closed f invariant subsets of X called the chain components of f , or the basic sets of f . These chain components are the analogues of the individual critical points in gradient case described in the introduction. Any two points of a chain component are related by C f . This is a type of transitivity condition. As with recurrence, there are several – increasingly broad – notions of dynamic transitivity and these can be associated with the relations of (19). First, we consider when the entire system f on X is transitive in the some way. Then we say that a closed f invariant subset A is a transitive subset in this way, when the subsystem f jA on A is transitive in the appropriate way. A group action on a set is called transitive when one can move from any element of the set to any other by some element of the group. That is, the entire set is a single orbit of the group action. Notice that this use of the word is unrelated to transitivity of a relation. Recall that a relation F on a set X is transitive when F ı F  F. We are now considering when it happens that any two points of X are related by F. This just says that F D X  X which we will call the total relation on X. First, what does it mean to say that O f is total, that is to say all of X lies in a single orbit of f ? First, compactness implies that X is then finite and so the homeomorphism f is a permutation of the finite set X. Such permutation is a product of disjoint cycles and O f D X  X exactly when all of X consists of a single cycle. The associated invariant subsets are the periodic orbits of f , including the fixed points. Next, we consider when the orbit closure relation R f is total, that is, when every point is in the orbit closure of every other. Theorem 7 Let f be a homeomorphism on X. The following conditions are equivalent and when they hold we call f minimal.

For all x 2 X; O f (x) is dense in X. For all x 2 X; R f (x) D X, i. e. R f D X  X. For all x 2 X; ! f (x) D X, i. e. ! f D X  X. X is the only nonempty, closed f + invariant subset of X. (v) X is the only nonempty, closed f invariant subset of X.

(i) (ii) (iii) (iv)

Recall our convention that the state space of a dynamical system is nonempty, although we do allow the empty set as an invariant subset. For example, the empty set is the repellor/attractor dual to the attractor/repellor which is the entire space. Thus, a closed f invariant subset A of X is minimal when it is nonempty but contains no nonempty, proper f invariant subset. From compactness it follows via the usual Zorn’s Lemma argument that every nonempty, closed f + invariant subset of X contains a minimal, nonempty, closed f invariant subset. Theorem 8 Let f be a homeomorphism on X. The following conditions are equivalent and when they hold we call f topologically transitive. (i) For some x 2 X; O f (x) is dense in X, i. e. R f (x) D X. (ii) For all x 2 X; N f (x) D X, i. e. N f D X  X. (iii) For all x 2 X; ˝ f (x) D X, i. e. ˝ f D X  X. (iv) X is the only closed f + invariant subset with a nonempty interior. If f is topologically transitive then the set fx : ! f (x) D Xg is residual, i. e. it is the countable intersection of dense, open subsets of X. Every point in a minimal system has a dense orbit. If f is topologically transitive then the set of transitive points for f , Trans f Ddef fx : ! f (x) D Xg D fx : R f (x) D Xg; (32) is dense by the Theorem 8. However, its complement is either empty, which is the minimal case, or else is dense as well. Also, there is a usually rich variety of invariant subsets. It may happen, for instance, that the set of periodic points is dense. Most well-studied examples of chaotic dynamical systems are non-minimal, topologically transitive systems. In fact, Devaney used the conjunction of topological transitivity and density of periodic points in an infinite system as a definition of chaos. The broadest notion of transitivity is associated with the chain relation C f . Theorem 9 Let f be a homeomorphism on X. The following conditions are equivalent and when they hold we call f chain transitive. (i) For all x 2 X; C f (x) D X, i. e. C f D X  X.

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(ii) For all x 2 X; ˝ C f (x) D X, i. e. ˝ C f D X  X. (iii) X is the only nonempty inward set. (iv) X is the only nonempty attractor. Before proceeding further, we pause to observe that each of these three concepts is the same for f and for its inverse. Because the f invariant subsets are the same as the f 1 invariant subsets, we see that f 1 is minimal when f is. Since N ( f 1 ) D (N f )1 and C ( f 1 ) D (C f )1 , it follows as well that f 1 is topologically transitive or chain transitive when f satisfies the corresponding property. There are many naturally occurring chain transitive subsets. For every x 2 X the limit sets ˛ f (x) and ! f (x) are chain transitive subsets as are each of the chain components. Notice that if x and y are points of some chain component B then by definition of the equivalence relation C f \ C f 1 ; x and y can be connected by chains in X for any positive . To say that B is a chain transitive subset is to make the stronger statement that they can be connected via chains which remain in B. Contrast this positive result with the trap set by implication (24) which suggests that B D ! f (x) is a topologically transitive subset. However, (24), which says B  B  N f , does not imply that the restriction f jB is topologically transitive, i. e. B  B D N ( f jB). It may not be possible to get from near y to near z without a hop which takes you outside B. In fact, if f is any chain transitive homeomorphism on a space X then it is possible to embed X as a closed subset of a space Y and extend f to a homeomorphism g on Y in such a way that X D ! g(y) for some point y 2 Y. Any chain transitive subset for f is contained in a unique chain component of jC f j. In fact the chain components are precisely the maximal chain transitive subsets. In particular, each subset ˛ f (x) and ! f (x) is contained in some chain component of f . By using (16) one can show that each connected component of the closed subset jC f j is contained in some chain component as well. It follows that the space of chain components, that is, the space of C f \ C f 1 equivalence classes with the quotient topology from jC f j, is zero-dimensional. So this space is either countable or is the union of a Cantor set with a countable set. An individual chain component B is called isolated when it is a clopen subset of the chain recurrent set jC f j or, equivalently, if it is an isolated point in the space of chain components. Thus, B is isolated when it has a neighborhood U in X such that U \ jC f j D B. It can be proved that B is an isolated chain component precisely when it is an isolated invariant set, i. e. it admits a neighborhood U satisfying (29).

The individual chain components generalize the role played by the critical points in the gradient case. They are the blobs we described at the end of the introduction. We complete the analogy by identifying which chain components are like the relative maxima and minima among the critical points. Theorem 10 Let B be a nonempty, closed, f invariant set for a homeomorphism f on X. The following conditions are equivalent and when they hold we call B a terminal chain component. (i)

B is a chain transitive quasi-attractor.

(ii) B is a C f + invariant, chain transitive subset of X. (iii) B is a C f + invariant chain component. (iv) In the set of nonempty, closed C f + invariant subsets of X the set B is an element which is minimal with respect to inclusion. In particular, B is a chain transitive attractor if and only if it is an isolated, terminal chain component. From condition (iv) and Zorn’s Lemma it follows that every nonempty, closed C f + invariant subset of X contains a terminal chain component. In particular, ˝ C f (x) contains a terminal chain component for every x 2 X. Clearly, if there are only finitely many chain components then each terminal chain component is isolated and so is an attractor. Corollary 1 Let f be a homeomorphism on X and let B be a nonempty, closed, C f + invariant subset of X. If for some x 2 X; B  ! f (x) then B is a terminal chain component and B D ! f (x). Proof B contains some terminal chain component B1 and ! f (x) is contained in some chain component B2 . Thus, B1  B  ! f (x)  B2 . Since distinct chain components  are disjoint, B1 D B2 . Because C ( f 1 ) D (C f )1 , the chain components for f and f 1 are the same. A chain component is called an initial chain component for f when it is a terminal chain component for f 1 . Let L be a Lyapunov function for C f . L is constant on each chain component and Theorem 4 says that L can be constructed to be strictly increasing on the orbit of every point of X n jC f j. That is, for such a complete Lyapunov function the set jLj of critical points equals jC f j. In that case, the local maxima occur at terminal chain components and, as a partial converse, an isolated terminal chain component is a local maximum for any complete C f Lyapunov function.

Topological Dynamics

Theorem 11 For a homeomorphism f on X let L be a C f Lyapunov function and let B be a chain component with r the constant value of L on B. (a) If there exists a neighborhood U of B such that L(x) < r for all x 2 U n B, then B is a terminal chain component. (b) Assume L is a complete Lyapunov function. If B is an attractor then it is a terminal chain component and the domain of attraction U is an open set containing B such that L(x) < r for all x 2 U n B. In particular, if there are only finitely many chain components for f then by using L, a complete C f Lyapunov function which distinguishes the chain components, we obtain the picture promised at the end of the introduction. The local maxima of L occur at the terminal chain components which are attractors. The local minima are at the initial chain components which are repellors. The remaining chain components play the role of saddles. As in the gradient case, it can happen that there is an open set of points x such that ! f (x) is contained in one of these saddle chain components. There is a natural topological condition which, when it holds, excludes this pathology. Theorem 12 For a homeomorphism f on X, assume that N f D C f or, equivalently, that ˝ f D ˝ C f . If A is a stable closed f invariant subset of X then A is a quasi-attractor. For a residual set of points x in X, ! f (x) is a terminal chain component and ˛ f (x) is an initial chain component. Proof The result for a stable set A follows from the characterizations in Theorems 2 and 6 By Theorem 1 the set of x such that ! f (x) D ˝ f (x) is always residual. By assumption this agrees with the set of x such that ! f (x) D ˝ C f (x). Corollary 1 implies that for such x the set ! f (x) is a terminal chain component. For ˛ f (x) apply this result to f 1 .  If X is a compact manifold of dimension at least two, then the condition N f D C f holds for a residual set in the Polish group of homeomorphisms on X with the uniform topology. If, in addition, f has only finitely many chain components (and this is not a residual condition on f ) then it follows that the points which are in the domain of attraction of some terminal chain component and in the domain of repulsion of some initial chain component form a dense, open subset of X. Chaos and Equicontinuity Any chain transitive system can occur as a chain component, but in the most interesting cases the chain compo-

nents are topologically transitive subsets. In this section we consider the antithetical phenomena of equicontinuity and chaos in topologically transitive systems. If f is a homeomorphism on X and x 2 X then x is called a transitive point when its orbit O f (x) is dense in X. As in (32) we denote by Transf the set of transitive points for f . The system is minimal exactly when every point is transitive, i. e. when Trans f D X. To study equicontinuity we introduce a new metric df defined using the original metric d on X. ˚ d f (x; y) Ddef sup d( f n (x); f n (y)) :

n D 0; 1; 2; : : : :

(33)

It is easy to check that df is a metric, i. e. it satisfies the conditions of positivity and symmetry as well as the triangle inequality. However, it is usually not topologically equivalent to the original metric d, generating a topology which is usually strictly finer ( = more open sets). We call a point x 2 X an equicontinuity point for f when for every > 0 there exists a ı > 0 so that for all y 2 X d(x; y) < ı H) d f (x; y)  ;

(34)

or, equivalently, if for every > 0 there exists a neighborhood U of x with df diameter at most (the terms “neighborhood”, “open set”, etc. will always refer to the original topology given by d). Here the df diameter of a subset A  X is diamf (A) Ddef sup fd f (x; y) : x; y 2 Ag :

(35)

We denote the, possibly empty, set of equicontinuity points for f by Eqf . When every point is an equicontinuity point, the system associated with f is called equicontinuous, or we just say that f is equicontinuous. This coincides with the concept of equicontinuity of the set of functions f f n : n D 1; 2; : : : g. Recall that any finite set of continuous functions is equicontinuous, but equicontinuity for an infinite set is often, as in this case, a strong condition. Equicontinuity of f says exactly that the metrics d and df are topologically equivalent. For compact spaces topologically equivalent metrics are uniformly equivalent. Hence, if f is equicontinuous, then for every > 0 there exists a ı > 0 such that for all x; y 2 X implication (34) holds. When f is equicontinuous then we can replace d by df since in the equicontinuous case the latter is a metric giving the correct topology. The homeomorphism f is then an isometry of the metric, i. e. d f (x; y) D d f ( f (x); f (y)) for all x; y 2 X. This equality uses a famous little problem which keeps being rediscovered: If f is a surjective

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map on a compact metric space X with metric d such that d( f (x); f (y))  d(x; y) for every x; y 2 X then f is an isometry on X, see e. g. Alexopoulos [9] or Akin [3], Proposition 2.4(c). Conversely, if f is an isometry of a metric d on X (with the correct topology) then f is clearly equicontinuous. When X has a dense set of equicontinuity points we call the system almost equicontinuous. If f is almost equicontinuous but not equicontinuous then the ı in (34) will depend upon x 2 Eq f as well as upon . On the other hand, we say that the system has sensitive dependence upon initial conditions, or simply that f is sensitive when there exists > 0 such that diam f (U) > for every nonempty open subset U of X, or, equivalently, if there exists > 0 such that for any x 2 X and ı > 0, there exists y 2 X such that d(x; y) < ı but for some n > 0d( f n (x); f n (y)) > . Here the important issue is that is independent of the choice of x and ı. Sensitivity is a popular candidate for a definition of chaos in the topological context. Suppose for a sensitive homeomorphism f you are attempting to estimate analyze the orbit of x, but may make an – arbitrarily small – positive error for initial point input. Even if you are able to compute the iterates exactly, you still cannot be certain that you always remain close to the orbit you want. If you have chosen a bad point y as input then at some time n you will be at f n (y) more than distance away from the point f n (x) that you want. For transitive systems, the Auslander–Yorke Dichotomy Theorem holds (see [21]): Theorem 13 Let f be a topologically transitive homeomorphism on a compact metric space X. Exactly one of the following alternatives is true.  (Sensitivity) The homeomorphism f is sensitive and there are no equicontinuity points, i. e. Eq f D ;.  (Almost Equicontinuity) The set of equicontinuity points coincides with the set of transitive points, i. e. Eq f D Trans f , and so the set of equicontinuity points is residual in X. Proof For > 0 let Eq f ; be the union of all open sets with df diameter at most . So x 2 Eq f ; if and only if it has a neighborhood with df diameter at most . It is then easy to check that x 2 Eq f ; if and only if f (x) 2 Eq f ; . Thus, Eq f ; is an f invariant open set. If Eq f ; is nonempty and x is a transitive point then the orbit eventually enters Eq f ; . Since f n (x) 2 Eq f ; for some positive n, it follows that x 2 Eq f ; because the set is invariant. This shows that Eq f ; ¤ ; H) Trans f  Eq f ; :

(36)

If for every > 0 we have Eq f ; ¤ ; then Trans f  D Eq f . We omit the proof that only the transitive points can be equicontinuous in a topologically transitive system. If, instead, for some > 0 the set Eq f ; is empty then by definition f is sensitive. 

T

>0 Eq f ;

The system is minimal if and only if Trans f D X and so a topologically transitive system is equicontinuous if and only if it is both almost equicontinuous and minimal. In particular, we have: Corollary 2 Let f be a minimal homeomorphism on X. Either f is sensitive or f is equicontinuous. Banks et al. [22] showed that a topologically transitive homeomorphism on an infinite space with dense periodic points is always sensitive. On the other hand, there do exist topologically transitive homeomorphisms which are almost equicontinuous but not minimal and hence not equicontinuous. If f is an almost equicontinuous, topologically transitive homeomorphism then there is a sequence of positive integers n k ! 1 such that the sequence f f n k g converges uniformly to the identity 1X , see Glasner and Weiss [37] and Akin, Auslander and Berg [6]. In general, when such a convergent sequence of iterates exists the homeomorphism f is called uniformly rigid. Other notions of chaos are defined using ideas related to proximality. A pair (x; y) 2 X  X is called proximal for f if lim inf n!1 d( f n (x); f n (y)) D 0 ;

(37)

or, equivalently, if 1 X \ !( f  f )(x; y) ¤ ; where f  f is the homeomorphism on X  X defined by ( f  f )(x; y) D ( f (x); f (y)). If a pair is not proximal then it is called distal. The pair is called asymptotic if lim d( f n (x); f n (y)) D 0 :

n!1

(38)

We will call (x, y) a Li–Yorke pair if it is proximal but not asymptotic. That is, (37) holds but lim sup d( f n (x); f n (y)) > 0 :

(39)

n!1

Li and Yorke [47] called a subset A of X a scrambled set if every non-diagonal pair in A  A is a Li–Yorke pair. Following their definition f is called Li–Yorke chaotic when there is an uncountable scrambled subset. The homeomorphism f is called distal when every nondiagonal pair in X  X is a distal pair. If f is an isometry then clearly f is distal and so every equicontinuous homeomorphism is distal. There exist homeomorphisms f

Topological Dynamics

which are minimal and distal but not equicontinuous, as described in e. g. Auslander [17]. Since such an f is minimal but not equicontinuous, it is sensitive. On the other hand, a distal homeomorphism has no proximal pairs and so is certainly not Li–Yorke chaotic. There exists an almost equicontinuous, topologically transitive, non-minimal homeomorphism f such that a fixed point e 2 X is the unique minimal subset of X and so the pair (e, e) is the unique subset in X  X which is minimal for f  f . It follows that (e; e) 2 !( f  f )(x; y) for every pair (x; y) 2 X  X and so every pair is proximal. On the other hand, because f is uniformly rigid, every pair (x, y) is recurrent for f  f and it follows that no non-diagonal pair is asymptotic. Thus, the entire space X is scrambled and f is Li–Yorke chaotic but not sensitive. There is a sharpening of topological transitivity which always implies sensitivity as well as most other topological conditions associated with chaos. A homeomorphism f on X is called weak mixing if the homeomorphism f  f on X  X is topologically transitive. If (x, y) is a transitive point for f  f then since !( f  f )(x; y) D X  X it follows that d f (x; y) D M, where M is the diameter of the entire space X. Suppose f is weak mixing and U is any nonempty open subset of X. Since Trans f  f is dense in X  X, there is a transitive pair (x, y) in U  U. Hence, diam f (U) D M. When f is weak mixing, there exists an uncountable A  X such that every nondiagonal pair in A A is a transitive point for f  f , Iwanik [44] and Huang and Ye [42], see also Akin [5]. Since such a set A is clearly scrambled, it follows that f is Li–Yorke chaotic. For any two subsets U; V  X we define the hitting time set to be the set integers given by: N(U; V) Ddef fn  0 : f n (U) \ V ¤ ;g :

(40)

The topological transitivity condition ˝ f D X  X says that whenever U and V are nonempty open subsets the hitting time set N(U, V) is infinite. We call f mixing if for every pair of nonempty open subsets U, V there exists k such that n 2 N(U; V ) for all n  k. As the names suggest, mixing implies weak mixing. Example The most important example of a chaotic homeomorphism is the ubiquitous shift homeomorphism. We think a of fixed finite set A as an alphabet and for any positive integer k the sequences in A of length k, i. e. the elements of Ak , are called words of length k. When A is equipped with the discrete topology it is compact and so by the Tychonoff Product Theorem any product of infinitely many copies of A is compact when given the product topology. Using the group of integers as our index set, we let X D AZ . There is a metric compatible with this

topology. For x; y 2 X let d(x; y) Ddef inff2n : x i D y i for all i with jij < ng :

(41)

The metric d is an ultrametric. That is, it satisfies a strengthening of the triangle inequality. For all x; y; z 2 X: d(x; z)  max(d(x; y); d(y; z)) :

(42)

The ultrametric inequality is equivalent to the condition that for every positive the open relation V is an equivalence relation. For any word a 2 Ak and any integer j, the set fx 2 X : x iC j D a i : i D 1; : : : ; kg is a clopen subset of X called a cylinder set, which we will denote U a; j . For example, with k D 2n  1 and j D n the cylinder sets are precisely the open balls of radius when 2n <  2nC1 . From this we see that X is a Cantor set with cylinder sets as a countable basis of clopen sets. On X the shift homeomorphism s is defined by the equation s(x) i D x iC1 for all i 2 Z :

(43)

The fixed points jsj are exactly the constant sequences in X and the periodic points jOsj are the periodic sequences in X. That is, s t (x) D x for some positive integer t exactly when x iCt D x i for all i 2 Z. It is easy to see that s is mixing. For example, if a; b 2 A2n1 and j D n, then the hitting time set N(U a; j ; Ub; j ) contains every integer t larger than 2n. For if x in D a i and x inCt D b i for i D 1; : : : ; 2n  1 then x 2 U a; j and s t (x) 2 Ub; j . This illustrates why s is chaotic in the sense of unpredictable. Given x 2 X, if y 2 X satisfies d(x; y) D 2n , then moving right from position 0 the first n entries of y are known exactly. They agree with the entries of x. But after position n, x provides no information about y. The remaining entries on the right can be chosen arbitrarily. Since s is mixing it is certainly topologically transitive, but it is useful to characterize the transitive points of s. For any point x 2 X and positive integer n we can scan from the central position x0 , left and right n  1 steps and observe a word of length 2n  1. As we apply s the central position shifts right. When we have applied st it has shifted t steps. Scanning left and right we observe a new word. A point is a transitive point when for every n we can observe every word in A2n1 in this way by varying t. To construct a transitive point x we need only list the – countably many – finite words of every length and lay them out end to end to get the right side of x. The left side of x can be arbitrary.

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The shift homeomorphism is also expansive. In general, a homeomorphism f on a compact metric space X is called expansive when the diagonal 1X is an isolated invariant set for the homeomorphism f  f on X  X and so there exists > 0 such that V¯ is an isolating neighborhood in the sense of (29). That is, if x; y 2 X and d( f n (x); f n (y))  for all n 2 Z then x D y. Such a number > 0 is called an expansivity constant. With respect to the metric given by (41) it is easy to check that 12 is an expansivity constant for the shift homeomorphism s. The shift is interesting in its own right. In addition, the chaotic behavior of other important examples, especially expansive homeomorphisms, are studied by comparing them with the shift. For a compact metric space X let H(X) denote the automorphism group of X, i. e. the group of homeomorphisms on X, with the topology of uniform convergence. The metric on H(X) is defined by the equation, for f ; g 2 H(X) : d( f ; g) Ddef supfd( f (x); g(x)) : x 2 Xg :

(44)

We obtain a topologically equivalent, but complete, metric by using max(d( f ; g); d( f 1 :g 1 )). The automorphism group is a Polish (= admits a complete, separable metric) topological group. For f 2 H(X) we define the translation homeomorphisms ` f and rf on H(X) by ` f (g) Ddef f ı g

and

r f (g) Ddef g ı f :

(45)

For any x 2 X, the evaluation map ev x : H(X) ! X is the continuous map defined by ev x ( f ) D f (x). For any f 2 H(X) let Gf denote the closure in H(X) of the cyclic subgroup generated by f . That is, G f D f f n : n 2 Zg  H(X) :

(46)

Thus, Gf is a closed, abelian subgroup of H(X). Let Iso(X) denote the closed subgroup of isometries in H(X) (In contrast with H(X) this varies with the choice of metric). It follows from the Arzela–Ascoli Theorem that Iso(X) is compact in H(X). If f is an isometry on X then Gf is a compact, abelian subgroup of Iso(X). Furthermore,the translation homeomorphisms ` f and rf restrict to isometries on the compact space Iso(X) and Gf is a closed invariant subset. In fact, under either ` f or r f ; G f is just the orbit closure of f regarded as a point of Iso(X). Recall that if f is an equicontinuous homeomorphism, then we can replace the original metric by a topologically equivalent one, e. g. replace d by df , to get one for which f is an isometry. If f is a homeomorphism on X and g is a homeomorphism on Y then we say that a continuous function

 : Y ! X maps g to f when  ı g D f ı  : Y ! X. If, in addition,  is a homeomorphism then we call  an isomorphism from g to f . Theorem 14 Let f be a homeomorphism on X. Fix x 2 X. The evaluation map ev x : H(X) ! X maps ` f on H(X) to f on X. If f is an almost equicontinuous, topologically transitive homeomorphism then evx restricts to a homeomorphism from the closed subgroup Gf of H(X) onto Trans f the residual subset of X consisting of the transitive points for f . If f is a minimal isometry then evx restricts to a homeomorphism from the compact subgroup Gf of H(X) onto X and so is an isomorphism from ` f on Gf to f on X. The isometry result is classical, see e. g. Gottschalk and Hedlund [38]. It shows that minimal, equicontinuous homeomorphisms are just translations on compact, monothetic groups, where a topological group is monothetic when it has a dense cyclic subgroup. Similarly a topologically transitive, almost equicontinuous homeomorphism which is not minimal is obtained by “compactifying” a translation on a noncompact monothetic group, see Akin and Glasner [7]. In fact, the group cannot even be locally compact and so, for example, is not the discrete group of integers. While examples of such topologically transitive, almost equicontinuous but not equicontinuous systems are known, it is not known whether there are any finite dimensional examples. And it is known that they cannot occur on a zero-dimensional space, i. e. the Cantor set. If finite dimensional examples do not exist then every topologically transitive homeomorphism on a compact manifold is sensitive except for the equicontinuous ones which we now describe. Example We can identify the circle S with the quotient topological group R/Z. For a 2 R the translation L a D R a on R induces the rotation  a on the circle R/Z. If a 2 Z then this is the identity map. If a is rational then  a is periodic. But if a is irrational then  a is a minimal isometry on S. More generally, if f1; a1 ; : : : ; a n g is linearly independent over the field of rationals Q then on the torus X D S n the product homeomorphism  a 1       a n is a minimal isometry. Such systems are sometimes called quasiperiodic. Of course, the translation by 1 on the finite cyclic group Z/kZ of integers modulo k is just a version of a single periodic orbit, the unique minimal map on a finite space of cardinality k. Recall that if fX 1 ; X2 ; : : : g is a sequence of topological spaces and fp n : X nC1 ! X n g is a sequence of continuous maps, then the inverse limit is the closed subset LIM of the

Topological Dynamics

1 X : product space ˘nD1 n

LIM D lim fX n ; p n g Ddef fx : p n (x nC1 ) D x n for n D 1; 2; : : : g :

(47)

If the spaces are compact and the maps are surjective then the nth coordinate projection n maps the compact space LIM onto X n for every n (Hint: use Proposition 1). Also, if the spaces are topological groups and the maps are homomorphisms, then LIM is a closed subgroup of the product topological group. Finally, if f f n : X n ! X n g is a sequence of homeomorphisms such that pn maps f nC1 to f n for ev1 f restricts ery n then the product homeomorphism ˘nD1 n to a homeomorphism f on LIM and n maps f to f n for every n. Now let fk n g be an increasing sequence of positive integers such that kn divides knC1 for every n. Let X n denote the finite cyclic group Z/k n Z and let p n : X nC1 ! X n be the quotient homomorphism induced by the inclusion map k nC1 Z ! k n Z. Let f n denote the translation by 1 on X n . Define X to be the inverse limit of this system with f the restriction to X of the product homeomorphism. X is a topological group whose underlying space is zerodimensional and perfect, i. e. the Cantor set. The product homeomorphism is an isometry when we use the metric analogous to the one defined by (41). Furthermore, the restriction f to X is a minimal isometry. These systems are usually called adding machines or odometers. It can be proved that every equicontinuous minimal homeomorphism on a Cantor space is isomorphic to one of these. In general, every equicontinuous minimal homeomorphism is isomorphic to (1) a periodic orbit, (2) an adding machine, (3) a quasi-periodic motion on a torus, or (4) a product with each factor either an irrational rotation on a circle, an adding machine or a periodic orbit. The informal expression strange attractor is perhaps best defined as a topologically transitive attractor which is not equicontinuous, i. e. which is not one of these. The word “chaos” suggests instability and unpredictability. However, for many examples what is most apparent is stability. For the Henon attractor or the Lorenz attractor the word “the” is used because in simulations one begins with virtually any initial point, performs the iterations and, after discarding an initial segment, one observes a particular, repeatable picture. The set as a whole is a predictable feature, stable under perturbation of initial point, an varying continuously with the defining parameters. However, once the orbit is close enough to the attractor, essentially moving within the attractor itself, the motion is unpredictable, sensitive to arbitrarily small perturbations. What remains is statistical prediction. We cannot

exactly predict when the orbit will enter some small subset of the attractor, but we can describe approximately the amount of time it spends in the subset. Such analysis requires an invariant measure and this takes us to the boundary between topological and measurable dynamics. By a measure  on a compact metric space X we will mean a Borel probability measure on the space. Such a measure acts, via integration, on the Banach algebra C (X) of continuous real-valued functions on X. The set P (X) of such measures can thus be regarded as a convex subset of the dual space of C (X). Inheriting the weak topology on the dual space, P (X) becomes a compact, metrizable space. There is a natural inclusion map ı : X ! P (X) which associates to x 2 X the point mass at x, denoted ıx . The support of a measure  on X is the smallest closed set with measure 1. Denoted jj its complement can be obtained by taking the union out of a countable base for the topology of those members with measure 0. The measure has full support (or simply  is full) when jj D X. A measure is full when every nonempty open set has positive measure. A continuous map h : X ! Y induces a map h : P (X) ! P (Y) which associates to  the measure h  defined by h (A) D (h1 (A)) for every Borel subset A of Y. The continuous linear operator h is the dual of h : C (Y) ! C (X) given by u 7! u ı h. Furthermore, h is an extension of h. That is, h (ıx ) D ı h(x) . The supports are related by h(jj) D jh j :

(48)

In particular, if f is a homeomorphism on X then f  is a linear homeomorphism on P (X). extending f on X. A measure  such that f  D  is called an invariant measure for f . Thus, j f j, the set of fixed points for f  , is the set of invariant measures for f . Clearly, j f j is compact, convex subset of P (X). The classical theorem of Krylov and Bogolubov says that this set is nonempty. To prove it, one begins with an arbitrary point x 2 X and considers the sequence of Cesaro averages n

n ( f ; x) Ddef

1 X ı f n (x) : nC1

(49)

iD0

It can be shown that every measure in the nonempty set of limit points of this sequence lies in j f j. If the set of limit points consists of a single measure , i. e. the sequence converges to , then x is called a convergence point for the invariant measure . For  2 j f j we denote by Con() the – possibly empty – set of convergence points for .

1741

1742

Topological Dynamics

The extreme points of the compact convex set j f j are called the ergodic measures for f . That is,  is ergodic if it is not in the interior of some line segment connecting a pair of distinct invariant measures. Equivalently,  is ergodic if a Borel subset A of X is invariant, i. e. f 1 (A) D A, only when the measure (A) is either 0 or 1. It follows that if u : X ! R is a Borel measurable, invariant function, then for any ergodic measure  there is a set of measure 1 on which u is constant. Or more simply, u ı f D u implies u is constant almost everywhere with respect to . The central result in measurable dynamics is the Birkhoff Pointwise Ergodic Theorem which says, in this context: Theorem 15 Given a homeomorphism f on a compact metric space X, let u : X ! R be a bounded, Borel measurable function. There exists a bounded, invariant, Borel measurable function. uˆ : X ! R such that for every f invariant measure  Z

Z u d D

uˆ d

and lim

n!1

1 nC1

2j f  j n X

ˆ u( f n (x)) D u(x)

(50)

iD0

almost everywhere with respect to . In particular, if  is ergodic then Z ˆ u(x) D u d

(51)

almost everywhere with respect to . It is a consequence of the ergodic theorem that  ergodic H) (Con()) D 1 :

(52)

Thus, with respect to an ergodic measure  for almost every point x, n

lim

n!1

positive measure and so they cannot all be pairwise disjoint. Applying this to the subsystem obtained by restricting to jj it follows from Theorem 1 that the set of recurrent points in jj form a dense Gı subset of jj. If x 2 Con() then the orbit of x is dense in jj and so the subsystem on jj is topologically transitive whenever Con() is nonempty. By the Birkhoff Ergodic Theorem this applies whenever  is ergodic. Since the support is a nonempty, closed, invariant subspace it follows that if f is minimal then every invariant measure has full support. The homeomorphism f is called strictly ergodic when it is minimal and has a unique invariant measure , which is necessarily ergodic. In that case, it can be shown that every point is a convergence point for , that is, Con() D X. We note that in the dynamical systems context the topological notion of residual (that is, a dense Gı subset) is quite different from the measure theoretic idea (a set of full measure). For example, [ Con Ddef Con() (54)

1 X u( f n (x)) D nC1

Z u d ;

(53)

iD0

for every u 2 C (X). The left hand side is the time-average of the function u along the orbit with initial point x and it equals the space-average on the right. For any f invariant measure  it follows from (48) that the support, jj, is an f invariant subset of X. The Poincaré Recurrence Theorem says that if U is any open set with U \ jj ¤ ; then U is non-wandering. That is, for some positive integer n, U \ f n (U) ¤ ;. This is clear from the observation that the open sets f n (U) all have the same,

is the set of points whose associated Cesaro average sequence is a Cauchy sequence. It follows that Con is a Borel set. By (52) (Con) D 1 for every ergodic measure . Since every invariant measure is a limit of convex combinations of ergodic measures it follows that Con has measure 1 for every invariant measure . On the other hand, it can be shown that if f is the shift homeomorphism on X D AZ then for x in the dense Gı subset of X the set of limit points of the sequence fn ( f ; x)g is all of j f j, Denker et al. [30] or Akin (Chap. 9 in [1]). Since the shift has many different invariant measures it follows that Con is disjoint from this residual subset. In general, if a homeomorphism f has more than one full, ergodic measure then Con is of first category, Akin [1, Theorem 8.11]. In particular, if f is minimal but not strictly ergodic then Con is of first category. The plethora of invariant measures undercuts somewhat their utility for statistical analysis. Suppose that there are two different ergodic measures  and  with common support, some invariant subset A of X. By restricting to A, we can reduce to the case when A D X and so  and  are distinct full, ergodic measures. The sets Con() and Con() are disjoint and of measure 1 with respect to  and , respectively. For an open set U the average frequency with which an orbit of x 2 Con() lies is U is given by (U) and similarly for . These mutually singular measures lead to different statistics. One solution to this difficulty is to select what seems to be the best invariant measure in some sense, e. g. the

Topological Dynamics

measure of maximum entropy (see the article by King) if it should happen to be unique. However, as our introductory discussion illustrates, this somewhat misses the point. Return to the case of a chaotic attractor A, a closed invariant subset of X. It often happens that the state space X comes equipped with a natural measure  or at least a Radon–Nikodym equivalence class of measures, all with the same sets of measure 0. For example, if X is a manifold then  is locally Lebesgue measure. The measure  is usually not f invariant and it often happens that the set A of interest has  measure 0. What we want, an appropriate measure  for this situation, would be an invariant measure with support A, i. e.  2 j f j and jj D A ;

(55)

and an open set U containing A such that with respect to  almost every point of U is a convergence point for . That is, (U n Con()) D 0 :

(56)

Notice that for x 2 Con(), ! f (x) D jj D A and so by Corollary 1 A is a terminal chain component. That is, for such a measure to exist, the attractor A must be at least chain transitive. If, in addition, Con() \ A ¤ ; then A is topologically transitive. At least when strong hyperbolicity conditions hold this program can be carried out with  the Bowen measure for the invariant set, see Katok–Hasselblatt (Chapter 20 in [45]). Minimality and Multiple Recurrence In this section we provide a sketch of some important topics which were neglected in the above exposition. We first consider the study of minimal systems. In Theorem 7 we defined a homeomorphism f on a compact space X to be minimal when every point has a dense orbit. A subset A of X is a minimal subset when it is a nonempty, closed, invariant subset such that the restriction f jA defines a minimal homeomorphism on A. The term “minimal” is used because f jA is minimal precisely when A is minimal, with respect to inclusion, in the family of nonempty, closed, invariant subsets. By Zorn’s lemma every such subset contains a minimal subset. Since every system contains minimal systems, the classification of minimal systems provides a foundation upon which to build an understanding of dynamical systems in general. On the other hand, if you start with the space, homeomorphisms which are minimal – on the whole space – are rather hard to construct. Some spaces have the fixed-point

property, i. e. every homeomorphism on the space has a fixed point, and so admit no minimal homeomorphisms. The tori, which are monothetic groups, admit the equicontinuous minimal homeomorphisms described in the previous section. Fathi and Herman [34] constructed a minimal homeomorphism on the 3-sphere. For most other connected, compact manifolds it is not known whether they admit minimal homeomorphisms or not. It is even difficult to construct topologically transitive homeomorphisms, but for these a beautiful – but nonconstructive – argument due to Oxtoby shows that every such manifold admits topologically transitive homeomorphisms. He uses that Baire Category Theorem to show that if the dimension is at least two then the topologically transitive homeomorphisms are residual in the class of volume preserving homeomorphisms, see Chap. 18 in [53]. Since we understand the equicontinuous minimal systems, we begin by building upon them. This requires a change in out point of view. Up to now we have mostly considered the behavior of a single dynamical system (X, f ) consisting of a homeomorphism f on a compact metric space X. Regarding these as our objects of study we turn now to the maps between such systems. A homomorphism of dynamical systems, also called an action map,  : (X; f ) ! (Y; g) is a continuous map  : X ! Y such that gı D  ı f and so, inductively, g n ı D  ı f n for all n 2 Z. Thus,  maps the orbit of a point x 2 X to the orbit of (x) 2 Y. In general, for the map   : XX ! Y Y we have that   (A f )  A g

for A D O; R; N ; G ; C

  ( f )  g

for  D !; ˛; ˝; ˝ C :

(57)

That is,  maps the various dynamic relations associated with f to the corresponding relations for g. If  is bijective then it is called an isomorphism between the two systems and the inverse map  1 is an action map, continuous by compactness. If X is a closed invariant subset of Y with f D gjX then the inclusion map  is an action map and then (X, f ) is called the subsystem of (Y, g) determined by the invariant set X. On the other hand, if  is surjective then (Y, g) is called a factor, or quotient system, of (X, f ) and (X, f ) is called a lift of (Y, g). A surjective action map is called a factor map. For a factor map  : (X; f ) ! (Y; g) we define an important subset R() of X  X: R() Ddef f(x1 ; x2 ) 2 X  X : (x1 ) D (x2 )g D (  )1 (1Y ) :

(58)

1743

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Topological Dynamics

The subset R() is an ICER on X. That is, it is an invariant, closed equivalence relation. In general, if R is any ICER on X then on the space of equivalence classes the homeomorphism f induces a homeomorphism and the natural quotient map is a factor map of dynamical systems. The original factor system (Y, g) is isomorphic to the quotient system obtained from the ICER R(). Notice that if (Y, g) is the trivial system, meaning that Y consisting of a single point, then R() is the total relation X  X on X. We use the ICER R() to extend various definitions from dynamical systems to action maps between dynamical systems. For example, a factor map  : (X; f ) ! (Y; g) is called equicontinuous if for every > 0 there exists ı > 0 such that (x1 ; x2 ) 2 R() and d(x1 ; x2 ) < ı H) d f (x1 ; x2 ) < :

(59)

Comparing this with (34) we see that (X, f ) is equicontinuous if and only if the factor map to the trivial system is equicontinuous. Similarly, recall that (x1 ; x2 ) 2 X is a distal pair for (X, f ) if !( f  f )(x1 ; x2 ) is disjoint from the diagonal 1X , and the system (X, f ) is distal when every nondiagonal pair is distal. A factor map  : (X; f ) ! (Y; g) is distal when every nondiagonal pair in R() is distal. Again (X, f ) is a distal system if and only if the factor map to the trivial system is distal. It is easy to check that a distal lift of a distal system is distal. Since an equicontinuous factor map is distal, it follows that an equicontinuous lift of an equicontinuous system is distal. However, it need not be equicontinuous. Example If a is irrational then the rotation  a on the circle Y D R/Z, defined by x 7! a C x, is an equicontinuous minimal map. On the torus X D R/Z  R/Z we define f by f (x; y) Ddef (a C x; x C y) :

(60)

It can be shown that (X, f ) is minimal. The projection  to the first coordinate defines an equicontinuous factor map to Y;  a ) and so (X, f ) is distal. It is not, however, equicontinuous. The above projection map is an example of a group extension. Let G be a compact topological group, like R/Z. Given any dynamical system (Y, g) and any continuous map q : Y ! G we let X D Y  G and define f on X by: f (x; y) Ddef (g(x); L q(x) (y)) :

(61)

The homeomorphism commutes with 1Y  Rz for any group element z and from this it easily follows that the projection  to the first coordinate is an equicontinuous factor map. If (Y, g) is minimal then the restriction of  to any minimal subset of X defines an equicontinuous factor map from the associated minimal subsystem. It can be shown that any equicontinuous factor map between minimal systems can be obtained via a factor from such a group extension. The Furstenberg Structure Theorem says that any distal minimal system can be obtained by a – possibly transfinite – inverse limit construction, beginning with an equicontinuous system and such that each lift is an equicontinuous factor map. For the details of this and for the structure theorem due to Veech for more general minimal systems, we refer the reader to Auslander (Chaps. 7, 14 in [17]) respectively. The factor maps described above are projections from products. There exist examples which are not product projections even locally. For example, in Auslander (Chap. 1 in [17]) the author uses a construction due to Floyd to build an action map  between minimal systems which is not an isomorphism but which is almost one-to-one. That is, for a residual set of points x in the domain the set of preimages  1 ((x)) is a singleton. Such a factor map is the opposite of distal. It is a proximal map, meaning that every pair (x1 ; x2 ) 2 R() is a proximal pair. Also, one cannot base all one’s constructions upon equicontinuous minimal systems. A dynamical system (X, f ) is called weak mixing when the product (XX; f  f ) is topologically transitive. Inclusion (57) implies that if a dynamical system is minimal, topologically transitive or chain transitive then any factor satisfies the corresponding property. It follows that any factor of a weak mixing system is weak mixing. It is clear that only the trivial system is both weak mixing and equicontinuous. Hence for a weak mixing system the trivial system is the only equicontinuous factor. For a minimal system the converse is true: if the trivial system is the only equicontinuous factor then the system is weak mixing. Furthermore, nontrivial weak mixing, minimal systems do exist. Gottschalk and Hedlund in [38] introduced the idea using various special families of subsets of N, the set of nonnegative integers, in order to distinguish different sorts of recurrence. A family F is a collection of subsets of N which is hereditary upwards. That is, if A  B and A 2 F then B 2 F . The family is called proper when it is a proper subset of the entire power set of N, i. e. it is neither empty nor the entire power set. The heredity property implies that a family F is proper iff N 2 F and ; 62 F . For a family F the dual family, denoted F  (or sometimes k F ) is

Topological Dynamics

defined by: F  Ddef fB  N : B \ A ¤ ; for all A 2 F g

D fB  N : N n B 62 F g :

(62)

It is easy to check that F  D F and that F  is proper if and only if F is. A filter is a proper family which is closed under pairwise intersection. A family F is the dual of a filter when it satisfies what Furstenberg calls the Ramsey Property: A [ B 2 F H) A 2 F or B 2 F :

(63)

For example, a set is in the dual of the family of infinite sets if and only if it is cofinite, i. e. its complement is finite. The family of cofinite sets is a filter. A subset A  N is called thick when it contains arbitrarily long runs. That is, for every positive integer L there exists i such that i; i C 1; : : : :; i C L 2 A. Dual to the thick sets are the syndetic sets. A subset B is called syndetic or relatively dense if there exists a positive integer L such that every run of length L meets B. In (40) we defined the hitting time set N(U, V) for (X, f ) when U; V  X. When U is a singleton fxg we omit the braces and so have N(x; V) D fn  0 : f n (x) 2 Vg :

(64)

It is clear that (X, f ) is topologically transitive when N(U, V) is nonempty for every pair of nonempty open sets U, V. Furstenberg showed that the stronger property that (X, f ) be weak mixing is characterized by the condition that each such N(U, V) is thick. Recall that a point x 2 X is recurrent when x 2 ! f (x). The point is called a minimal point when it is an element of a minimal subset of X in which case this minimal subset is ! f (x). Clearly, point x 2 X is recurrent when N(x, U) is nonempty for every neighborhood U of x. Gottschalk and Hedlund proved that x is a minimal point if and only if every such N(x, U) is relatively dense. For this reason, minimal points are also called almost periodic points. Furstenberg reversed this procedure by using dynamical systems arguments to derive properties about families of sets and more generally to derive results in combinatorial number theory. If f1 ; : : : ; f k are homeomorphisms on a space X then x 2 X is a multiple recurrent point for f1 ; : : : :; f k if there exists a sequence of positive integers n i ! 1 such that the k sequences f f1n i (x)g; : : : ; f f kn i (x)g all have limit x. That is, the point (x; : : : ; x) is a recurrent point for the homeomorphism f1    f k on X k . The Furstenberg Multiple Recurrence Theorem says:

Theorem 16 If f1 ; : : : ; f k are commuting homeomorphisms on a compact metric space X, i. e. f i ı f j D f j ı f i for i; j D 1; : : : ; k, then there exists a multiple recurrent point for f1 ; : : : ; f k . Corollary 3 If f is a homeomorphism on a compact metric space X and x 2 X then for every positive integer k and every > 0 there exist positive integers m; n such that with y D f m (x) the distance between any two of the points y; f n (y); f 2n (y); : : : ; f kn (y) is less than . Proof This follows easily by applying the Multiple Recurrence Theorem to the restrictions of the homeomorphisms f ; f 2 ; : : : ; f k to the closed invariant subset ! f (x). Obtain a multiple recurrent point y 0 2 ! f (x) and then given > 0 choose y D f m (x) to approximate y 0 sufficiently closely.  These results are of great interest in themselves and they have been extended in various directions, see Bergelson– Leibman [24] for example. In addition, Furstenberg used the corollary to obtain a proof of Van der Waerden’s Theorem: Theorem 17 If B1 ; : : : ; B p is a partition of N then at least one of these sets contains arithmetic progressions of arbitrary length. Proof It suffices to show that for each k D 1; 2; : : : and some a D 1; : : : ; p the subset Ba contains an arithmetic progression of length kC1. For this will then apply to some fixed Ba for infinitely many k. Let A D f1; : : : ; pg and on X D AZ define the shift homeomorphism defined by (43). Using the metric given by (41) and  12 we observe that if x; y 2 X with d(x; y) < then x0 D y0 . Choose x 2 X such that x i D a if and only if i 2 B a for i 2 N. Apply Corollary 3 to x and . Choosing positive integers m; n such that the points f m (x); f mCn (x); f mC2n (x); : : : ; f mCkn (x) all lie within of each other we have that m; m C n; : : : ; m C kn all lie in Ba where a is the common value of the 0 coordinate of these points.  One of the great triumphs of modern dynamical systems theory is Furstenberg’s use of the ergodic theory version of these arguments to prove Szemerédi’s Theorem [35, Theorem 3.21]. Theorem 18 Let B be a subset of N with positive upper Banach density, that is lim supjIj!1 jB \ Ij/jIj > 0 ;

(65)

where I varies over bounded subintervals of N and jAj denotes the cardinality of A N. The set B contains arithmetic progressions of arbitrary length.

1745

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Topological Dynamics

For details and further results, we refer to Furstenberg’s beautiful book [35]. Future Directions As far as applied work is concerned, the resolution of the statistical problems, described at the end in the Sect. “Chaos and Equicontinuity”, concerning chaotic attractors would be most useful. In addition, it would be nice to know which compact manifolds admit minimal homeomorphisms. Cross References  Entropy in Ergodic Theory Bibliography The approach to topological dynamics which was described in the initial sections is presented in detail in Akin [1] and [2]. For the deeper, more specialized work in topological dynamics see Auslander [17], Ellis [33] and Akin [4]. A general survey of the field is given in de Vries [65]. Furstenberg [35] is a classic illustration of the applicability of topological dynamics to other fields. Clark Robinson’s text [55] is an excellent introduction to dynamical systems in general. More elementary introductions are Devaney [31] and Alligood et al. [10]. Hirsch et al. [40] provides a nice transition from differential equations to the general theory. Much of modern differentiable dynamics grows out of the work of Smale and his students, see especially the classic Smale [60], included in the collection [61]. The seminal paper Shub and Smale [58] provides a bridge between this work and the purely topological aspects of attractor theory, see also Shub [57]. The fashionable topic of chaos has generated a large sample of writing whose quality exhibits extremely high variance. For expository surveys I recommend Lorenz [48] and Stewart [62]. An excellent collection of relatively readable, classic papers is Hunt et al. [43]. For applications in biology see Hofbauer and Sigmund [41] and May [49]. Also, don’t miss Sigmund’s delightful book [59]. 1. Akin E (1993) The General Topology of Dynamical Systems. Am Math Soc 2. Akin E (1996) Dynamical systems: the topological foundations. In: Aulbach B, Colonius F (eds) Six Lectures on Dynamical Systems. World Scientific, Singapore, pp 1–43 3. Akin E (1996) On chain continuity. Discret Contin Dyn Syst 2:111–120 4. Akin E (1997) Recurrence in Topological Dynamics: Furstenberg Families and Ellis Actions. Plenum Press, New York

5. Akin E (2004) Lectures on Cantor and Mycielski sets for dynamical systems. In: Assani I (ed) Chapel Hill Ergodic Theory Workshops. Am Math Soc, Providence, pp 21–80 6. Akin E, Auslander J, Berg K (1996) When is a transitive map chaotic? In: Bergelson V, March K, Rosenblatt J (eds) Conference in Ergodic Theory and Probability. Walter de Gruyter, Berlin, pp 25–40 7. Akin E, Glasner E (2001) Residual properties and almost equicontinuity. J d’Analyse Math 84:243–286 8. Akin E, Hurley M, Kennedy JA (2003) Dynamics of Topologically Generic Homeomorphisms. Memoir, vol 783. Am Math Soc, Providence 9. Alexopoulos J (1991) Contraction mappings in a compact metric space. Solution No.6611. Math Assoc Am Mon 98:450 10. Alligood KT, Sauer TD, Yorke JA (1996) Chaos: An Introduction to Dynamical Systems. Spring Science Business Media, New York 11. Almgren FJ (1966) Plateau’s Problem. Benjamin WA, Inc., New York 12. Alpern SR, Prasad VS (2001) Typical dynamics of volume preserving homeomorphisms. Cambridge University Press, Cambridge 13. Arhangel’skii AV, Pontryagin LS (eds) (1990) General topology I. Springer, Berlin 14. Arhangel’skii AV (ed) (1995) General Topology III. Springer, Berlin 15. Arhangel’skii AV (ed) (1996) General Topology II. Springer, Berlin 16. Auslander J (1964) Generalized recurrence in dynamical systems. In: Contributions to Differential Equations, vol 3. John Wiley, New York, pp 55–74 17. Auslander J (1988) Minimal Flows and Their Extensions. NorthHolland, Amsterdam 18. Auslander J, Bhatia N, Siebert P (1964) Attractors in dynamical systems. Bol Soc Mat Mex 9:55–66 19. Auslander J, Siebert P (1963) Prolongations and generalized Liapunov functions. In: International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics. Academic Press, New York, pp 454–462 20. Auslander J, Siebert P (1964) Prolongations and stability in dynamical systems. Ann Inst Fourier 14(2):237–268 21. Auslander J, Yorke J (1980) Interval maps, factors of maps and chaos. Tohoku Math J 32:177–188 22. Banks J, Brooks J, Cairns G, Davis G, Stacey P (1992) On Devaney’s Definition of Chaos. Am Math Mon 99:332–333 23. Barrow-Green J (1997) Poincaré and the Three Body Problem. American Mathematical Society, Providence 24. Bergelson V, Leibman A (1996) Polynomial extensions of Van der Waerden’s and Szemeredi’s theorems. J Am Math Soc 9:725–753 25. Bing RH (1988) The Collected Papers of R. H. Bing. American Mathematical Society, Providence 26. Brown M (ed) (1991) Continuum Theory and Dynamical Systems. American Mathematical Society, Providence 27. Bhatia N, Szego G (1970) Stability Theory of Dynamical Systems. Springer, Berlin 28. Coddington E, Levinson N (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York 29. Conley C (1978) Isolated Invariant Sets and the Morse Index. American Mathematical Society, Providence

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30. Denker M, Grillenberger C, Sigmund K (1978) Ergodic Theory on Compact Spaces. Lect. Notes in Math, vol 527. Springer, Berlin 31. Devaney RL (1989) An Introduction to Chaotic Dynamical Systems, 2nd edn. Addison-Wesley Publishing Company, Redwood City 32. Douglas J (1939) Minimal surfaces of higher topological structure. Ann Math 40:205–298 33. Ellis R (1969) Lectures on Topological Dynamics. WA Benjamin Inc, New York 34. Fathi A, Herman M (1977) Existence de difféomorphismes minimaux. Soc Math de France, Astérique 49:37–59 35. Furstenberg H (1981) Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton Univ. Press, Princeton 36. Glasner E (2003) Ergodic Theory Via Joinings. American Mathematical Society, Providence 37. Glasner E, Weiss B (1993) Sensitive dependence on initial conditions. Nonlinearity 6:1067–1075 38. Gottschalk W, Hedlund G (1955) Topological Dynamics. American Mathematical Society, Providence 39. Hartman P (1964) Ordinary Differential Equations. John Wiley, New York 40. Hirsch MW, Smale S, Devaney RL (2004) Differential Equations, Dynamical Systems and an Introduction to Chaos, 2nd edn. Elsevier Academic Press, Amsterdam 41. Hofbauer J, Sigmund K (1988) The Theory of Evolution and Dynamical Systems. Cambridge Univ. Press, Cambridge 42. Huang W, Ye X (2002) Devaney’s chaos or 2-scattering implies Li–Yorke’s chaos. Topol Appl 117:259–272 43. Hunt B, Kennedy JA, Li T-Y, Nusse H (eds) (2004) The Theory of Chaotic Attractors. Springer, Berlin 44. Iwanik A (1989) Independent sets of transitive points. In: Dynamical Systems and Ergodic Theory. Banach Cent Publ 23:277–282 45. Katok A, Hasselblatt B (1995) Introduction to the Modern Theory of Dynamical Systems. Cambridge Univ. Press, Cambridge 46. Kuratowski K (1973) Applications of the Baire-category method to the problem of independent sets. Fundam Math 81:65–72

47. Li TY, Yorke J (1975) Period three implies chaos. Am Math Mon 82:985–992 48. Lorenz E (1993) The Essence of Chaos. University of Washington Press, Seattle 49. May RM (1973) Stability and Complexity in Model Ecosystems. Princeton Univ. Press, Princeton 50. Murdock J (1991) Perturbations. John Wiley and Sons, New York 51. Mycielski J (1964) Independent sets in topological algebras. Fundam Math 55:139–147 52. Nemytskii V, Stepanov V (1960) Qualitative Theory of Differential Equations. Princeton U Press, Princeton 53. Oxtoby J (1980) Measure and Category, 2nd edn. Springer, Berlin 54. Peterson K (1983) Ergodic Theory. Cambridge Univ. Press, Cambridge 55. Robinson C (1995) Dynamical Systems: Stability, Symbolic Dynamics and Chaos. CRC Press, Boca Raton 56. Rudolph D (1990) Fundamentals of Measurable Dynamics: Ergodic Theory on Lebesgue Spaces. Oxford Univ. Press, Oxford 57. Shub M (1987) Global Stability of Dynamical Systems. Springer, New York 58. Shub M, Smale S (1972) Beyond hyperbolicity. Ann Math 96:587–591 59. Sigmund K (1993) Games of Life: Explorations in Ecology, Evolution and Behavior. Oxford Univ. Press, Oxford 60. Smale S (1967) Differentiable dynamical systems. Bull Am Math Soc 73:747–817 61. Smale S (1980) The Mathematics of Time. Springer, Berlin 62. Stewart I (1989) Does God Play Dice? Basil Blackwell, Oxford 63. Ura T (1953) Sur les courbes défines par les équations différentielles dans l’espace à m dimensions. Ann Sci Ecole Norm Sup 70(3):287–360 64. Ura T (1959) Sur le courant extérieur à une région invariante. Funk Ehv 2:143–200 65. de Vries J (1993) Elements of Topological Dynamics. Kluwer, Dordecht

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Vehicular Traffic: A Review of Continuum Mathematical Models BENEDETTO PICCOLI , ANDREA TOSIN Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche, Rome, Italy Article Outline Glossary Definition of the Subject Introduction Macroscopic Modeling Kinetic Modeling Road Networks Future Directions Bibliography Glossary Mathematical model Simplified description of a real world system in mathematical terms, e. g., by means of differential equations or other suitable mathematical structures. Scale of observation/representation Level of detail at which a system is observed and mathematically modeled. Macroscopic scale Scale of observation focusing on the evolution in time and space of some macroscopic quantities of the system at hand; that is, quantities that can be experimentally observed and measured. Kinetic scale Scale of representation focusing on a statistical description of the evolution of the microscopic states of the system at hand. Vehicular traffic The overall dynamics produced by cars or other transports in motion along a road. Road network A set of roads connected to one another by junctions. Definition of the Subject Vehicular traffic is attracting a growing scientific interest because of its connections with other important problems, like, e. g., environmental pollution and congestion of cities. Rational planning and management of vehicle fluxes are key topics in modern societies under both economical and social points of view, as the increasing number of projects aimed at monitoring the quality of the road traffic demonstrates. In spite of their importance, however, these issues cannot be effectively handled by simple experimental approaches. On the one hand, observation and data recording may provide useful information

on the physics of traffic, highlighting some typical features like, e. g., clustering of the vehicles, the appearance of stop-and-go waves, the phase transition between the regimes of free and congested flow, the trend of the traffic in uniform flow conditions. The books by Kerner [48] and Leutzbach [54] extensively report about traffic phenomena, real traffic data, and their phenomenological interpretation. On the other hand, processing and organization of (usually huge amounts of) experimental measurements hardly allow one to catch the real unsteady traffic dynamics, which definitely makes this approach scarcely predictive. Therefore vehicular traffic is not only an engineering matter but also a challenging mathematical problem. This paper reviews some of the major macroscopic and kinetic mathematical models of vehicular traffic available in the specialized literature. Application to road networks is also discussed, and an extensive list of references is provided to both the original works presented here and other sources for further details. Microscopic models are only briefly recalled in the introduction for the sake of thoroughness, as they are actually beyond the scope of the review. However, suitable references are included for the interested readers. Introduction The mathematical modeling of vehicular traffic requires, first of all, the choice of the scale of representation. The relevant literature offers many examples of models at any scale, from the microscopic to the macroscopic through the kinetic one. Each of them implies some technical approximations, and suffers therefore from related drawbacks, either analytical or computational. The microscopic scale is like a magnifying glass focused on each single vehicle, whose dynamics is described by a system of ordinary second-order differential equations of the form  ˙k gN x¨ i D a i t; fx k g N kD1 ; f x kD1 ;

i D 1; : : : ; N ;

(1)

where t is time, x i D x i (t) the scalar position of the ith vehicle along the road, x˙ i D x˙ i (t) its velocity, and N 2 N the total number of vehicles. The function ai describes the acceleration of the ith vehicle, which in principle might be influenced by the positions and velocities of all other vehicles simultaneously present on the road. As a matter of fact, each vehicle is commonly assumed to be influenced by its heading vehicle only (follow-the-leader models), so that the acceleration ai depends at most on xi , x iC1 and on x˙ i , x˙ iC1 . It is immediately seen that the size of system (1) rapidly increases with the number N of vehicles

Vehicular Traffic: A Review of Continuum Mathematical Models

considered, which frequently makes the microscopic approach not competitive for computational purposes. Furthermore, from the analytical point of view it is often difficult to investigate the relevant global features of the system, also in connection with control and optimization problems. In this paper we will not be concerned specifically with microscopic modeling of vehicular traffic. The interested reader can find further details on the subject in Aw et al. [4], Chakroborty et al. [20], Gazis et al. [36], Helbing [41,42], Kerner and Klenov [49], Nagel et al. [56] Treiber et al. [66,67,68] and in the main references listed therein, as well as in the review by Hoogendoorn and Bovy [46]. Macroscopic and kinetic models aim at describing the big picture without looking specifically at each single subject of the system, hence they are computationally more efficient: Few partial differential equations, that can be solved numerically in a feasible time, are normally involved, and the global characteristics of the system are readily accessible. Nevertheless, now the modeling is, in a sense, less accurate than in the microscopic case, due to the fact that both approaches rely on the continuum hypothesis, clearly not physically satisfied by cars along a road: The number of vehicles should be large enough so that it makes sense to introduce concepts like macroscopic density, average speed, or kinetic distribution function as continuous functions of space and, in the latter case, also velocity. However, the continuum hypothesis can be profitably accepted as a technical approximation of the physical reality, regarding macroscopic quantities globally as measures of traffic features and as tools to depict the spatial and temporal evolution of traffic waves. Macroscopic modeling is based on the idea, originally due in the fifties to Lighthill and Whitham, and independently, to Richards, that the classical Euler and NavierStokes equations of fluid dynamics describing the flow of fluids could also describe the motion of cars along a road, provided a large-scale point of view is adopted so as to consider cars as small particles and their density as the main quantity at which one looks. This analogy remains nowadays in all macroscopic models of vehicular traffic, as terms like traffic pressure, traffic flow, traffic waves demonstrate. Kinetic modeling, first used by Prigogine in the sixties, relies instead on the principles of statistical mechanics introduced by Boltzmann to describe the unsteady evolution of a gas. The kinetic approach as well requires in principle a large number of particles in the system but, unlike the macroscopic approach, it is based on a microscopic modeling of their mutual interactions. The evolution of the system is described by means of a distribution function over

the microscopic state of the particles, whence the interesting macroscopic quantities are obtained via suitable microscopic average procedures. In this review we present and discuss some of the main continuum mathematical models of vehicular traffic available in the literature (see also Bellomo et al. [10]). Specifically, the paper is organized in four more sections that follow this Introduction. Section “Macroscopic Modeling” is devoted to macroscopic models of both first and second order, which describe the gross evolution of traffic via either the sole mass conservation equation supplemented by suitable closure relations or a coupled system of mass conservation and momentum balance equations. Section “Kinetic Modeling” reports about traffic modeling by methods of the kinetic theory, from the very first attempts recalled above to the more recent contributions in the field based on the discrete kinetic theory. These two sections account mainly for traffic models on single oneway roads, whereas Sect. “Road Networks” finally deals with traffic problems on road networks. Such topic provides, in the present context, a strong link between mathematical modeling and real world applications, as most prediction, control, and optimization problems concern precisely the management of car fluxes over complex systems of interconnected roads, e. g., urban networks or some specific sub-networks. In addition, it poses nontrivial theoretical questions, which motivate the development of innovative mathematical methods and tools susceptible of application also to different fields. These sections of the paper are conceived so as to give first the general mathematical framework, which is then detailed to obtain particular models. Section “Future Directions” finally sketches some research perspectives in the field. Macroscopic Modeling In the macroscopic approach to the modeling of vehicular traffic the flow of cars along a road is assimilated to the flow of fluid particles, for which suitable balance or conservation laws can be written. For this reason, macroscopic models are often called in the present context hydrodynamic models. In the macroscopic approach, cars are not followed individually. The point of view is rather that of the classical continuum mechanics, i. e., one looks at the evolution in time and space of some average quantities of interest such as the mass density or the mean velocity, referred to an infinitesimal reference volume individuated by a point in the geometrical space. In more detail, since the motion of cars along a road is usually modeled as one-dimensional, the spatial coordinate x is a scalar independent variable of

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the problem which ranges in an appropriate subset of R, possibly in the whole real axis. The main dependent variables introduced to describe the problem mathematically are the density of cars D

(t; x) and their average velocity u D u(t; x) at time t in the point x. From these quantities another important variable is derived, namely the flux q D q(t; x) given by q D u, which is of great interest for both theoretical and experimental purposes. Indeed, many real data, used to construct and to validate macroscopic models, refer precisely to flux measurements, because these can be performed with higher precision than measurements of other variables. Since, as said before, macroscopic models of vehicular traffic are inspired to the continuum equations of fluid dynamics, the Eulerian point of view is normally adopted, meaning that the above dependent variables evaluated at (t; x) yield the evolution at time t of the vehicles flowing through the fixed in space position x. In other words, the spatial coordinate is not linked to any reference configuration but rather to the actual geometrical space, hence at different times the values of the state variables computed in the point x refer in general to different continuum particles flowing through x. Of course, all models can in principle be converted in Lagrangian form, up to a proper change of variable. The basic evolution equation translates the principle of conservation of the vehicles: @ @ C ( u) D 0 : (2) @t @x According to this equation, the time variation in the amount of cars within any stretch of road comprised between two locations x1 < x2 is only due to the difference between the incoming flux at x1 and the outgoing flux at x2 . Integration of Eq. (2) over the interval [x1 ; x2 ] produces indeed Zx 2 d

(t; x)dx D q(t; x1 )  q(t; x2 ) ; dt x1

where the integral at the left-hand side defines precisely the mass of cars contained in [x1 ; x2 ] at time t. It can be questioned that Eq. (2) does not give rise by itself to a self-consistent mathematical model, as it involves simultaneously two variables, and u. To overcome this difficulty, one possibility is to devise suitable closures which allow one to express the velocity u, or equivalently the flux q, as a function of the density . This way Eq. (2) becomes an evolution equation for the sole density : @ @ C f ( ) D 0 ; @t @x

(3)

where q D f ( ) is the closure relation. Macroscopic models based on Eq. (3) are usually called first order models. We briefly recall here that a diffusion term is sometimes added to the right-hand side of Eq. (3) in order to smooth the resulting density field , which, from the theory of nonlinear conservation laws (see e. g., Bressan [18], Dafermos [27], Serre [63,64]), is known to possibly develop discontinuities in finite time even for smooth initial data. The resulting equation reads   @ @ @ @ C f ( ) D ( ) ; (4) @t @x @x @x where > 0 is a parameter accounting for the strength of the diffusion with respect to the other terms of the equation, and ( ) is the (possibly nonlinear) diffusion coefficient. Notice that if  is a positive constant then Eq. (4) reduces formally to the linear heat equation with nonlinear advection. However, unlike Eq. (3), models based on Eq. (4) do not preserve in general the total amount of vehicles. A second possibility consists instead in joining Eq. (2) to an evolution equation for the flux q inspired by the momentum balance of a continuum: @q @ C (qu) D A[ ; u; D ; Du] ; (5) @t @x where A is some material model for the generalized forces acting on the system and responsible for momentum variations. These forces express the macroscopic outcome of the microscopic interactions among the vehicles; as we will see later, in most models they are assumed to be determined by either the local density or the local velocity u of the cars, as well as by their respective variations in time and/or space which in the above equation are denoted by the operator D. Using q D u and taking Eq. (2) into account, Eq. (5) can be further manipulated to obtain @u @u Cu D a[ ; u; D ; Du] ; (6) @t @x where a :D A/ is a material model for the acceleration of the vehicles to be specified. Finally, the set of equations resulting from this second approach is 8 @ @ ˆ ˆ C ( u) D 0 ˆ < @t @x (7) ˆ ˆ @u @u ˆ : Cu D a[ ; u; D ; Du] @t @x which is self-consistent in the unknowns , u once the acceleration a has been conveniently designed. Mathematical models based on the system (7) are commonly termed in the literature second order models or, less frequently, two-equations models.

Vehicular Traffic: A Review of Continuum Mathematical Models

Lighthill–Whitham–Richards and Other First Order Models First order models are obtained from the conservation law (3) by resorting to a suitable closure relation which expresses the flow of cars q as a function of the density . A relation of the form q D f ( ) is named in this context fundamental diagram. If max > 0 denotes the maximum vehicle density allowed along the road according, for instance, to the capacity, i. e., the maximum sustainable occupancy of the latter, the function f is often required 1. to be monotonically increasing from D 0 up to a certain density value  2 (0; max ); 2. to be decreasing for    max ; 3. to have D  as unique maximum point in [0; max ]; 4. to be concave in the interval [0; max ]. These assumptions aim mainly at reproducing the typical behavior of experimentally computed fundamental diagrams (see Fig. 1 and, e. g., Bonzani and Mussone [16,17], Kerner [47,48,49]). In addition, some of them, for instance assumptions (3) and (4), are stated for theoretical purposes linked to the qualitative properties of the conservation law (3). We observe that sometimes assumption (4) is relaxed, allowing the function f to possibly change concavity in the decreasing branch. By prescribing the flux f one implicitly assigns also the relation between the average velocity u and the density (velocity diagram), indeed using q D u one has u( ) D

f ( ) :

Vehicular Traffic: A Review of Continuum Mathematical Models, Figure 1 Fundamental diagram based on experimental data referring to one-week traffic flow in viale del Muro Torto, Rome, Italy. Measures kindly provided by ATAC S.p.A.

Since the function f often vanishes for D 0, this formula commonly determines a finite value for u in the limit

! 0C . The prototype of all fluxes complying with the above assumptions is the parabolic profile firstly proposed by Lighthill and Whitham [55,71] and then, independently, by Richards [62], which gives rise to the so-called LWR model:  

; (8) f ( ) D umax 1 

max from which the following velocity diagram is obtained:  

u( ) D umax 1  (9)

max (see Fig. 2a). The parameter umax > 0 identifies the maximum velocity of the cars in a situation of completely free road ( D 0). Notice that Eq. (8) assumes zero flux of vehicles when both D 0 and D max ; correspondingly, the average velocity given by Eq. (9) is maximum in the first case and vanishes in the second case. A generalization of Eqs. (8), (9) is provided by the Greenshield model (Fig. 2b):  



n f ( ) D umax 1  ;

max

 n 

u( ) D umax 1  (n 2 N) ;

max while an example of fundamental diagram satisfying all the previous assumptions but which generates an unbounded velocity diagram for ! 0C is due to Greenberg (Fig. 2c):  

max f ( ) D u0 log ;

 

max ; u( ) D u0 log

where u0 > 0 is a tuning parameter. These formulas were specifically conceived to fit some experimental data on the traffic flow in the Lincoln tunnel in New York. Additional fundamental and velocity diagrams, for which some of the above discussed assumptions fail, are reported in the book by Garavello and Piccoli [34]. Moreover, other closure relations of the mass conservation equation for first order models are discussed in Bellomo and Coscia [8]. Here we simply mention a further diagram due to Bonzani and Mussone [17] (Fig. 2d): 

f ( ) D umax e˛ max  ; 

u( ) D umax e˛ max 

(10)

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Vehicular Traffic: A Review of Continuum Mathematical Models, Figure 2 Fundamental diagrams (solid lines) and velocity diagrams (dashed lines) for a LWR model, b Greenshield’s model, c Greenberg’s model, and d Bonzani–Mussone’s model

which depends on a phenomenological parameter ˛ > 0 linked to road and environmental conditions. In particular, the lower the ˛, the more the flow of vehicles is favored; according to the authors, comparisons with suitable sets of experimental data suggest the interval [1; 2:5] as a standard range for ˛. Notice that in this case the function f may have an inflection point in the interval (; max ), hence the hypothesis (4) is potentially violated. Fictitious Density Model In view of the analogy with the flow of fluid particles, the macroscopic modeling framework accounts for a fully mechanical evolution of the traffic. On the other hand, it can be argued that cars are actually not completely mechanical subjects, for the presence of the drivers is likely to affect their behavior in an essentially unconventional ‘humanlike’ fashion.

Still in the context of first order hydrodynamic models, De Angelis [32] tries to explicitly account for the action of the drivers on the evolution of the traffic by introducing the concept of fictitious density. The fictitious density  represents the density of cars along the road as perceived by the drivers, hence it expresses the personal feelings of the latter about the local conditions of traffic. This new variable, which is in general different from the actual density , is used to model the instantaneous reactions of the drivers. Specifically, the fictitious density is linked to the actual density and to its spatial gradient in such a way that if is increasing (respectively, decreasing) then  increases (respectively, decreases) in turn but more quickly than , while for approaching the limit threshold max also  tends to saturate to max . In other words, the sensitivity of the drivers should consist in feeling a fictitious congestion state of the road higher or lower than the real one,

Vehicular Traffic: A Review of Continuum Mathematical Models

and by consequence in anticipating certain behaviors that a purely mechanical agent would exhibit only in presence of the appropriate external conditions. The dependence of  on and @x proposed by De Angelis is   @

; (11)

 [ ; @x ] D C  1 

max @x where  > 0 is a parameter related to the sensitivity of the drivers, which weights the effect of the spatial variations of the density . It can be observed that for increasing actual density, that is @x > 0, the fictitious density is larger than , thus suggesting a more careful attitude of the system car-driver with respect to the purely mechanical case. Conversely, for decreasing actual density, namely @x < 0, the fictitious density is lower than , which implies a more aggressive behavior induced by the presence of the driver. Specific models are obtained by plugging Eq. (11) into the expression of the velocity diagram. This amounts essentially to devising a closure relation of the mass conservation equation (2) of the form u D u(  ), so that the evolution equation finally reads @ @ C ( u(  )) D 0 : @t @x

The former, ar , is called relaxation and models the tendency of each driver to travel at a certain desired speed depending on the level of congestion of the road: a r [ ; u] D

Ve ( )  u ; 

where u is the actual speed and Ve ( ) the desired one, while  > 0 is a parameter representing the relaxation time or, in other words, the reaction time of the drivers in adjusting their actual speed to Ve ( ). The desired speed in Eq. (13) may be thought of as the speed possibly observed in stationary homogeneous traffic conditions for a specific value of the density . As a matter of fact, the function Ve ( ) can be one of the velocity diagrams used in first order models (see Subsect. “Lighthill–Whitham– Richards and Other First Order Models”). The second term forming the acceleration, aa , is named anticipation and accounts for the reaction of the drivers to the variations in the traffic conditions ahead of them. In more detail, it models the tendency to accelerate or to decelerate in response to a decrease or an increase, respectively, of the density of cars, therefore it is linked to the density and to its spatial gradient @x as a a [ ; @x ] D 



on @x , this class of In view of the dependence of models often specializes in nonlinear, possibly degenerate, parabolic equations in the unknown . For instance, technical calculations with the Lighthill–Whitham–Richards diagram (9) give rise to "  2 # @ @ @  @2 C h( ) D f ( ) 2 C h( ) @t @x

max @x @x

while the flux f is given by Eq. (8). Different models of fictitious density, as well as their correlations with diffusion models based on Eq. (4), are considered by Bonzani [15]. The Payne–Whitham Model Payne [58] and Whitham’s [71] model is a second order model in which the acceleration a (cf. Eq. (6)) is constructed as the sum of two terms: a[ ; u; D ; Du] D ar [ ; u] C a a [ ; @x ] :

(12)

1 @ p( ) ;

@x

(14)

where the function p D p( ) is the analogous to the pressure in the fluid dynamics equations. A common hypothesis on this traffic pressure is that it is a nondecreasing function of the density; assuming moreover differentiability of p with respect to , the anticipation term can be formally rewritten in the equivalent form a a [ ; @x ] D 

where the function h is defined by   2 h( ) D umax 1 

max

(13)

p0 ( ) @ ;

@x

(15)

whence one sees that the ratio p0 ( )/ is a kind of weight on the gradient @x , which amplifies or reduces the effect of the latter on the basis of the local density of cars along the road. In view of Eqs. (13)–(15), the final form of Payne– Whitham’s model is as follows: 8 @ @ ˆ ˆ C ( u) D 0 ˆ ˆ < @t @x (16) ˆ ˆ @u Ve ( )  u p0 ( ) @ ˆ @u ˆ : Cu D  : @t @x 

@x According to the derivation of these equations proposed by Whitham himself in the book [71] as continuous

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approximation of a microscopic car-following model, one has the following representation of the traffic pressure: p0 ( ) D

1 ˇˇ 0 ˇˇ V ( ) : 2 e

More in general, the relation between p and can be expressed in the form (see e. g., Günther et al. [39], Klar and Wegener [50]) p( ) D  e ( ) ; where  e denotes the equilibrium value in homogeneous traffic conditions of the variance of the microscopic velocity of cars (cf. Eq. (25)). For the sake of thoroughness, we recall that sometimes a diffusive acceleration term of the form   @2 u a d @2x u D

@x 2 is introduced in the decomposition (12), the parameter  > 0 representing a sort of viscosity by analogy with the theory of Newtonian fluids. The Aw–Rascle Model The hydrodynamic model by Payne and Whitham suffers from some severe drawbacks which led Daganzo [28] to write a celebrated ‘requiem’ for this kind of second order approximation of traffic flow. Flaws of the model include the possible onset of negative speeds of the vehicles as well as the violation of the so-called anisotropy principle, i. e., the fact that a car should be influenced only by the traffic dynamics ahead of it, being practically insensitive to what happens behind. By computing the characteristic speeds 1 ; 2 of the hyperbolic system p (16) in the variables , u one finds indeed 1;2 D u ˙ p0 ( ), meaning that some information generated by the motion of the vehicles travels faster than the vehicles themselves, hence affects the forward traffic dynamics. Aw and Rascle [3] suggest that all of these inconsistencies are caused by a too direct application to cars of the analogy with fluid particles, to which, on the other hand, the principles of nonnegativity of the velocity and of anisotropy do not apply. In particular, they observe that the anticipation term aa should not be regarded as an Eulerian quantity, as it is specifically linked to the perception of the traffic conditions by the drivers while in motion within the flow of vehicles. An external (Eulerian) observer might see an increasing vehicle density in a certain fixed spatial position x while, at the same time, a driver, i. e., an internal (Lagragian) observer, flowing through x sees a decreasing density in front of her/him if, for example, cars ahead

move faster. In such a situation, the behavior predicted by the Payne–Whitham’s anticipation term, that is a deceleration, is obviously inconsistent with the presumable reaction of the driver, who is instead expected to maintain the current speed if not to accelerate. The problem is solved by Aw and Rascle by making the pressure p a material (i. e., Lagrangian) variable of the model, which technically amounts to expressing the anticipation term via the material derivative D t :D @ t C u@x of p: a a [ ; u; @ t ; @x ] D D t p( ) D @ t p( )  u@x p( ) :

(17)

The original Aw–Rascle’s (AR) model assumes in addition no relaxation term in the acceleration, thus it writes 8 @ @ ˆ ˆ ( u) D 0 < C @t @x (18) ˆ ˆ : @ (u C p( )) C u @ (u C p( )) D 0 @t @x for a smooth and increasing function p of the density , e. g. p( ) D  ;

> 0:

Notice that, owing to the conservation of the vehicle mass, the second equation in (18) can be regarded as the nonconservative form of a conservation equation for the generalized flux w, with w :D u C p( ). For this reason, this equation is frequently encountered in the literature as @ @ ( w) C ( wu) D 0 : @t @x

(19)

Computing the characteristic speeds of the hyperbolic system (18) in the variables ; u gives 1 D u, 2 D u 

p0 ( ), whence, in view of the assumption of nondecreasing pressure, one sees that no information can now be advected faster than cars. Further extensions of the AR model involve the introduction of a relaxation term (Greenberg [38], Rascle [61]) to let drivers travel at some maximum desired speed when the traffic conditions allow. In particular, Greenberg’s relaxation term takes the form a r [ ; u] D

Vm  (u C p( )) ; 

where Vm > 0 is a constant value representing the maximum possible speed along the road and  > 0 is the relaxation time, while Rascle’s relaxation term is more

Vehicular Traffic: A Review of Continuum Mathematical Models

similar to the corresponding Payne–Whitham’s term (cf. Eq. (13)): a r [ ; u] D

V( )  u ; 

though here the function V( ) need not represent an equilibrium velocity. The Phase Transition Model Numerous experimental observations (see e. g., Kerner [47,48,49]) report about strongly different qualitative and quantitative features of the flow of vehicles in free and congested regimes, corresponding respectively to low and high density of cars, far from or close to the road capacity. In particular, from experimentally computed fundamental diagrams it can be inferred that the assumption that the average speed be a one-to-one function of the density applies well to free traffic, while being completely inadequate to describe congested flows. Indeed, in the latter case the density no longer determines univocally the velocity u, or equivalently the flux q, of the vehicles, for relevant fluctuations are recorded so that the admissible pairs ( ; q) are scattered over a two-dimensional region of the state space. In the specialized literature, this phenomenon is called a phase transition. Starting from these experimental facts, Colombo [23, 24] proposed a mathematical model in which the existence of the phase transition is postulated and accounted for by splitting the state space ( ; q) in two regions, ˝ f and ˝ c , corresponding to the regimes of free and congested flow, respectively. In ˝ f a classical relation of the form u D u( ) is imposed, and the model simply consists in the conservation law (3) with f ( ) D u( ). In particular, the LWR velocity diagram (9) is chosen. Conversely, in ˝ c the density and the flux q are regarded as two independent state variables, and the model is similar to a second order model in which the mass conservation equation (2) is joined to an evolution equation for the flux of the form @q @ C ((q  q )u) D 0 : @t @x

(20)

Furthermore, the average velocity is specified as a function of both and q which, in a way, generalizes the LWR velocity diagram:  

q : (21) 1 u( ; q) D

max Notice that Eq. (20) closely recalls the momentum balance (5) of second order models with zero external force A. The parameter q > 0 is introduced to obtain that at high

car density and high car speed breaking produces shocks and accelerating produces rarefaction waves, while at low car density and speed the opposite occurs (see [23] for further details). Finally, Colombo’s phase transition model writes: Free flow: ( ; q) 2 ˝ f 8 @ @ ˆ < C ( u) D 0 @t @x ˆ : u D u( ) ; Congested flow: ( ; q) 2 ˝c 8 @ @ ˆ ˆ C ( u) D 0 ˆ ˆ @t @x ˆ < @ @q C ((q  q )u) D 0 ˆ ˆ ˆ @t @x ˆ ˆ : u D u( ; q) : The regions ˝ f , ˝ c are defined so as to be positively invariant with respect to their corresponding systems of equations: If the initial datum is entirely contained in one of the two regions, then the solution will remain in that region at all successive times of existence. In other words, if the traffic is initially free (respectively, congested), then it will remain free (respectively, congested) during the whole subsequent evolution. Technically, letting ˝ D [0; max ] [0; C1) 3 ( ; q) denote the state space: ˝ f D f( ; q) 2 ˝ : u( )  Vˆ f ; q D umax g;

Q 1  q q  q ˝c D ( ; q) 2 ˝ : u( ; q)  Vˆc ; 

max

 Q 2  q  ;

max where Vˆ f , Vˆc are two fixed speed thresholds such that above the former the flow is free while below the latter the flow is congested, with furthermore Vˆc < Vˆ f  umax in order for the two phases not to overlap, and Q1 2 (0; q ), Q2 2 (q ; C1) are tuning parameters related to various environmental conditions. Kinetic Modeling The kinetic approach to the modeling of vehicular traffic is based on the choice of an intermediate representation scale between the macroscopic and the microscopic ones, technically called mesoscopic. Rather than looking at each single car of the system, like in the microscopic approach, a distribution function, usually denoted by f , is introduced over the microscopic state (x; v) of the vehicles, x standing

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for position and v for velocity assumed one-dimensional with moreover v  0. The distribution function is such that the quantity f (t; x; v)dxdv represents a measure of the number of vehicles whose position at time t is comprised between x and x C dx, with a velocity comprised between v and v C dv. Consequently, under suitable integrability assumptions on f , if D x  R, Dv  RC denote the spatial domain and the velocity domain, respectively, the quantity Z Z N(t) D f (t; x; v)dxdv (22) Dv Dx

gives the total amount of vehicles present in the system at time t. It is worth pointing out that f is in general not a probability distribution function over (x; v), as the integral in Eq. (22) may not equal 1 at all times. However, if the total mass of cars is preserved then N(t) is constant in t, hence it is possible to rescale the distribution function so that it have a unit integral on D x  Dv for all times t of existence. The knowledge of the distribution function enables one to recover the usual macroscopic variables of interest as local averages over the microscopic states of the vehicles. For instance, the density of cars at time t in the point x is given by Z f (t; x; v)dv ; (23)

(t; x) D Dv

while the flux q and the average velocity u are obtained from the first momentum of f with respect to v as Z q(t; x) q(t; x) D v f (t; x; v)dv; u(t; x) D : (24)

(t; x) Dv

Higher order momenta are related to other macroscopic variables, such as the average kinetic energy and the variance of the velocity. In particular, the latter is given by Z 1 (t; x) D (v  u(t; x))2 f (t; x; v)dv (25)

(t; x) Dv

and is often linked, at a macroscopic level, to the traffic pressure responsible for the anticipation terms in the hydrodynamic equations of traffic (see e. g., the Payne–Whitham’s model, Subsect. “The Payne–Whitham Model”, and Günther et al. [39] or Klar and Wegener [50] for further details).

Modeling of the system is obtained by stating an evolution equation in time and space for the distribution function f . Unlike the macroscopic approach, such an equation is written considering the influence of microscopic interactions among the vehicles on the microscopic states. Concerning this, it is useful to introduce a specific terminology to distinguish in an interaction the vehicle which is likely to change its state from the one which potentially causes such a change. The former is technically called the candidate vehicle, while the latter is called the field vehicle. Practically all kinetic models of vehicular traffic present in the literature describe the interactions by appealing to the following general guidelines: 1. Cars are regarded as points, their dimensions being negligible. 2. Interactions are binary, in the sense that those involving simultaneously more than two vehicles are disregarded. 3. Interactions modify by themselves only the velocity of the vehicles, not their positions. More specifically, in an interaction the sole velocity of the candidate vehicle may vary, but that of the field vehicle remains instead unchanged. 4. Vehicles are anisotropic particles which react mainly to frontal than to rear stimuli (see Daganzo [28]), therefore candidate vehicles only interact with field vehicles located ahead of them. 5. There exists a probability of passing P such that when a candidate vehicle encounters a slower field vehicle it may overtake it instantaneously with probability P without modifying its own velocity. 6. Interactions are conservative, in the sense that they preserve the total number of vehicles of the system. The evolution equation for the distribution function is obtained from a balance principle which states that the time variation in the number of vehicles belonging to any subset of the state space D x  Dv is determined by the state transitions due to the interactions of the vehicles with each other. Therefore, in the absence of external actions on the system it writes @f @f Cv D J[ f ] ; @t @x

(26)

where J[ f ] D J[ f ](t; x; v) is an operator acting on the distribution function f , charged to describe the interactions and their effects on the states of the vehicles. Following the classical terminology coming from the collisional kinetic theory of the gas dynamics (see e. g., Villani [69]), J is frequently termed the collisional operator, though in the present context this is a slight abuse of speech because

Vehicular Traffic: A Review of Continuum Mathematical Models

interactions among vehicles are not like collisions among mechanical particles. In view of the assumption (6) above, the operator J is required to satisfy Z J[ f ](t; x; v)dv D 0; 8x 2 D x ; 8t  0 Dv

so that integrating Eq. (26) with respect to v and recalling Eqs. (23), (24) yields the macroscopic mass conservation equation (2). Specific models are obtained from Eq. (26) by detailing the form of the collisional operator. The Prigogine Model In his pioneering work on the mathematical modeling of vehicular traffic by methods of the kinetic theory, Prigogine [59,60] constructs the collisional operator through the contribution of two terms: J[ f ] D J r [ f ] C J i [ f2 ] :

(27)

and field vehicles. As usual in the kinetic framework, it is further split into two more operators: J i [ f2 ] D G[ f2 ]  L[ f2 ] : The gain operator G[ f2 ] accounts for the interactions of candidate vehicles having velocity v > v with field vehicles with velocity v, which force the former to slow down to v if they cannot overtake the latter, causing this way a gain of cars with velocity state v: C1 Z

G[ f2 ] D (1  P)

(v  v) f2 (t; x; v ; x; v)dv : (30) v

The loss operator L[ f2 ] accounts instead for the interactions of candidate vehicles having velocity v with field vehicles with velocity v < v, which force the former to slow down to v if they cannot overtake the latter, thus giving rise to a loss of cars with velocity state v: Zv L[ f2 ] D (1  P)

(v  v ) f2 (t; x; v; x; v )dv :

(31)

0

The first one, J r [ f ], is called relaxation term and models the tendency of each driver to adapt the state of her/his vehicle to a desired standard state. The latter is described by the desired distribution function f0 D f0 (t; x; v), which in essence corresponds to the driving program each driver aims at. In Prigogine’s model, the function f 0 is expressed as f0 (t; x; v) D (t; x) f˜0 (v) ;

(28)

where f˜0 is a prescribed probability distribution over the variable v, independent of x and t. The factorization (28) translates the hypothesis that the desired distribution of the velocity is, for each driver, unaffected by the local vehicle density. The relaxation term is then taken to be Jr [ f ] D 

1P P

for

f2 (t; x; v; x ; v ) D f (t; x; v) f (t; x ; v ) ; so that J i takes the form of a bilinear operator acting on f : C1 Z

f  f0 ; T

J i [ f ; f ] D (1  P) f (t; x; v)

where T is the relaxation time which depends on the probability of passing P as TD

In Eq. (30) it is assumed Dv D RC , that is no upper limitation is imposed on the velocity of the cars. The function f 2 appearing in Eqs. (30), (31) is the joint distribution function of candidate and field vehicles, such that f2 (t; x; v; x ; v ) gives a measure of the joint probability to find a candidate vehicle in the state (x; v) and simultaneously a field vehicle in the state (x ; v ). Introducing the hypothesis of vehicular chaos (see e. g., Hoogendoorn and Bovy [46]), which states that vehicles are actually uncorrelated due to the mixing caused by overtaking, one can express the function f 2 in terms of f as

P D1

:

max

(29)

In these formulas,  is a positive parameter of the model and max denotes the maximum vehicle density locally allowed along the road according to the road capacity. The second term appearing in the decomposition (27), J i [ f2 ], is called interaction term and is represented by an operator which models the interactions among candidate

(v  v) f (t; x; v )dv 0

and Prigogine’s model finally reads @f f (t; x; v)  f0 (t; x; v) @f Cv D @t @x T C1 Z (v  v) f (t; x; v )dv : C (1  P) f (t; x; v)

(32)

0

Notice that the forms of G and L in Eqs. (30), (31) implicitly assume localized interactions (x D x ) like in the classical Boltzmann collisional kinetic theory.

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The Paveri Fontana Model

with

One of the main criticisms to Prigogine’s model is that the desired speed distribution function f˜0 in Eq. (28) is prescribed a priori, and is thus independent of the evolution of the system. In order to correct this drawback, Paveri Fontana [57] conceived a model in which the desired speed is taken into account as a further state variable w ranging in the same domain Dv of the true speed v. At the same time, a generalized distribution function g D g(t; x; v; w) is introduced such that

G[g; g] D Z (1  P)

g(t; x; v; w)dxdvdw gives a measure of the number of vehicles that at time t have a position comprised between x and x C dx, a true speed comprised between v and vCdv, and a desired speed comprised between w and w C dw. Notice that the distribution function f and the desired distribution function f 0 are readily recovered by integrating out w and v, respectively, from g: Z f (t; x; v) D g(t; x; v; w)dw; Dv

Z

f0 (t; x; w) D

(33) g(t; x; v; w)dv:

Dv

Using these relations, an evolution equation for g is straightforwardly derived from the corresponding equation (26) for f : @g @g Cv D I[g] ; @t @x

where the first term is again a relaxation toward the desired speed having now the form (35)

while the second one describes the interactions among the vehicles. Specifically, in the same spirit as Prigogine’s model (cf. Eqs. (30), (31)), gain and loss terms can be identified such that I i [g; g] D G[g; g]  L[g; g]

(36)

L[g; g] D

Z

(v  v ) f (t; x; v )g(t; x; v; w)dv ;

(37)

v

(1  P) 0

the probability of passing P being defined as    

P D 1 H 1

max

c where H() is the Heaviside function and c 2 (0; max ) a critical density threshold above which overtaking is inhibited. Notice in particular that candidate vehicles are described via the generalized distribution function g in order to take their desired speed into account, while field vehicles are described by means of the distribution function f , in which the dependence on w has been integrated out, to focus rather on their actual speed. Moreover, Eqs. (36), (37) still assume localized interactions as confirmed by the fact that both f and g are evaluated at the same spatial position x. Putting Eqs. (34)–(37) together we deduce the final form of Paveri Fontana’s model: @g @g @ w  v  Cv D g @t @x @v T " C1 Z (v  v)g(t; x; v ; w)dv C (1  P) f (t; x; v) v

#

Zv

I[g] D I r [g] C I i [g; g] ;

@ w  v  g ; @v T

(v  v) f (t; x; v)g(t; x; v ; w)dv ; v

(34)

where I is a new collisional operator acting on the generalized distribution function. Analogously to Eq. (27), Paveri Fontana’s choice consists in splitting I in a twofold contribution:

I r [g] D 

C1

 g(t; x; v; w)

(v  v ) f (t; x; v )dv :

(38)

0

Integrating Eq. (38) over w and using the first of Eqs. (33) gives an evolution equation for the distribution function f : @f @f Cv @t @x 2 C1 3 Z v f (t; x; v) 5 @ 41 w g(t; x; v; w)dw  D @v T T 0 C1 Z

C (1  P) f (t; x; v)

(v  v) f (t; x; v )dv : 0

This equation differs from the corresponding Eq. (32) of Prigogine’s model only for the relaxation term at the right-

Vehicular Traffic: A Review of Continuum Mathematical Models

hand side, which here maintains the explicit dependence on the generalized distribution function g. Similarly, integration of Eq. (38) with respect to v, along with the second of Eqs. (33), yields the following evolution equation for the desired distribution function f 0 : 3 2 C1 Z @ f0 @ 4 vg(t; x; v; w)dv 5 D 0 C @t @x 0

which definitely confirms that f 0 depends now on the overall evolution of the system. Enskog-Like Models As noticed by some authors (see e. g., Klar and Wegener [50]), the localized interactions framework used in classical kinetic models of traffic (like Prigogine’s and Paveri Fontana’s ones) prevents backward propagation of the perturbations in the negative x direction. This is essentially due to the fact that the flow of vehicles is unidirectional, given the positivity constraint on the velocity v. On the other hand, the localized interactions assumption is a heritage of the Boltzmann collisional kinetic theory, where however the aforementioned drawback is not present because the flow of gas particles need not be unidirectional. To obviate this difficulty of the theory, Klar and coworkers suggest, in a series of papers [39,40,50,51,70] focusing among other things on this topic, see also the review by Klar and Wegener [52], to describe the collisional operator J of the kinetic equation (26) as: J[ f ] D G[ f ; f ]  f L[ f ] where the gain operator G[ f ; f ] is given by

G[ f ; f ] D

M “ X jD1

jv1  v2 j j (v; v1 ; v2 ; ) (39)

and the loss operator by L[ f ] D

M Z X

!1 D fv2 2 Dv : v2 > vg ; while the acceleration domains are ˚ ˝2 D (v1 ; v2 ) 2 Dv2 : v1 < v2 ; !2 D fv2 2 Dv : v2 < vg : Specifically, in the case of a deceleration it is assumed that the post-interaction velocity v of the candidate vehicle either remains unchanged, thus equal to the initial velocity v1 , if an overtaking of the field vehicle is possible, or stochastically reduces to a certain fraction of the velocity v2 of the field vehicle, if the candidate is forced to queue. This is expressed by the following form of the transition probability  1 : 1 (v; v1 ; v2 ; ) D

˝j

 f (t; x; v1 ) f (t; x C h j ; v2 )k(h j ; )dv1 dv2

j D 1; : : : ; M, are introduced to delocalize the interactions, similar to an Enskog-like kinetic setting, and to trigger different behaviors of the candidate vehicle (acceleration, deceleration) on the basis of the distance separating it from its leading field cars. Integration is performed over subsets of the velocity domain, ˝ j  Dv2 , ! j  Dv , associated with each of the thresholds hj . Finally, the function k weights the interactions according to the distance of the interacting pairs and the local congestion of the road. Its presence in the collisional operator can be formally justified by referring to the Enskog kinetic equations of a dense gas. The interested reader might want to consult the book by Bellomo et al. [9] and the paper by Cercignani and Lampis [19] for further details on this part of the theory. In the simplest case, two constant thresholds h1 , h2 are considered to model, respectively, a deceleration of the candidate vehicle when the distance from the field vehicle falls below h1 , and an acceleration when it grows instead above h2 . Correspondingly, the deceleration domains in G and L are given by ˚ ˝1 D (v1 ; v2 ) 2 Dv2 : v1 > v2 ;

jv  v2 j f (t; x C h j ; v2 )k(h j ; )dv2 : (40)

jD1 ! j

Notice that the interacting pairs are not supposed to occupy the same spatial position; in particular, the field vehicle is assumed to be located at x C h j , x being the position of the candidate vehicle. The thresholds h j > 0,

Pı(v  v1 ) C (1  P)

1 [ˇ v 2 ;v 2 ] (v) ; (1  ˇ)v2

where P is the probability of passing defined like in Eq. (29), which carries the dependence of  1 on , ˇ 2 [0; 1] is a parameter, and I is the indicator function of the set I. Conversely, in the case of an acceleration the post-interaction velocity v of the candidate vehicle is assumed to be uniformly distributed between the values v1 and v1 C ˛(vmax  v1 ), where ˛ depends on through P as ˛ D ˛0 P for a suitable parameter ˛0 > 0, and vmax is the maximum allowed velocity on the road. Therefore, the

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transition probability associated to this second threshold takes the form 2 (v; v1 ; v2 ; ) D

1 [v ;v C˛(v max v)] (v) : ˛(vmax  v) 1 1

The final form of Klar and coworkers’ model with two equal interaction thresholds (h1 D h2  h) writes: 2

X @f @f Cv D @t @x jD1

jv1  v2 j j (v; v1 ; v2 ; )

q(t; x) D

˝j

jD1 ! j

We observe that the spirit in which the collisional operator is constructed in model (42) somehow differs from Prigogine’s and Paveri Fontana’s models in that interactions are now defined by detailing the short-range reactions of each driver to the dynamics of the neighboring vehicles rather than by interpreting her/his overall behavior. Discrete Velocity Models Recently new mathematical models of vehicular traffic, based on the discrete kinetic theory, have been proposed in the literature, with the aim of taking into account, still at a mesoscopic level of description, some aspects of the strong granular nature of the flow of vehicles. Indeed, as reported by several authors (see e. g., Ben-Naim and Krapivsky [11,12], Berthelin et al. [13]), cars along a road tend to cluster, which gives granularity in space, with a nearly constant speed within each cluster, which makes also the velocity a discretely distributed variable. Referring to Delitala and Tosin [33], the main idea is to make the velocity variable v discrete by introducing in the domain Dv a grid Iv D fv i gniD1 of the form 0 D v1 < v2 <    < v n D 1

iD1

f i (t; x)ı(v  v i ) ;

u(t; x) D

n X

f i (t; x);

iD1 n X

v i f i (t; x); iD1 Pn v i f i (t; x) PiD1 n iD1 f i (t; x)

:

Moreover, plugging Eq. (41) into Eq. (26) yields the following system of evolution equations for the distribution functions f i : @ fi @ fi C vi D J i [f]; @t @x

i D 1; : : : ; n ;

(42)

where f D ( f1 ; : : : ; f n ) and J i [f] is the ith collisional operator. Modeling of J i [f] is performed by appealing to a discrete stochastic game theory (see Bertotti and Delitala [14]), after identifying gain and loss contributions: J i [f] D G i [f; f]  f i L i [f] with xC Z n X G i [f; f](t; x) D [ ](t; x )Aihk [ ](t; x ) h;kD1 x

 f h (t; x) f k (t; x )w(x  x)dx

(43)

and

L i [f](t; x) D

Z n xC X

[ ](t; x ) f k (t; x )w(x  x)dx :

kD1 x

(44)

and letting then v 2 Iv , while time and space are left continuous. Each vi is interpreted as a velocity class, encompassing a certain range of velocities v which are not individually distinguished. Correspondingly, the distribution function f is expressed as a linear combination of Dirac functions in the variable v, with coefficients depending on time and space: n X

(t; x) D

“

 f (t; x; v1 ) f (t; x C h; v2 )k(h; )dv1 dv2 2 Z X  f (t; x; v) jvv2 j f (t; x Ch; v2 )k(h; )dv2 :

f (t; x; v) D

where f i (t; x) gives the distribution of cars in the point x having at time t a velocity comprised in the ith velocity class. Using this representation, the following expressions for the macroscopic variables of interest are easily derived from Eqs. (23)–(24):

(41)

One of the main features of Delitala–Tosin’s model is that interactions are not local, but are distributed over a characteristic length  > 0, which can be interpreted as the visibility length of the drivers: A candidate vehicle located in x is supposed to be affected, in average, by all field vehicles comprised in the visibility zone [x; x C ] ahead of it. One can therefore speak of averaged binary interactions. Averaging is carried out by the weight function w, which is nonnegative and supported in the above visibility zone, with unit integral there.

Vehicular Traffic: A Review of Continuum Mathematical Models

Acceleration and deceleration of the vehicles are described, like in Klar and coworkers’ model, as short-range reactions of the drivers to the traffic dynamics in terms of velocity class transitions. Specifically, a table of games Aihk is introduced, which gives the probability that a candidate vehicle in the hth velocity class adjusts its speed to the ith class after an interaction with a field vehicle in the kth class. These probabilities are parameterized by the density of cars , so as to account for a dependence of the interactions on the local traffic congestion, and are subject to the additional requirements Aihk [ ]  0;

n X

Aihk [ ] D 1;

8i; h; k 2 f1; : : : ; ng;

iD1

8 2 [0; max ] ; where max is the road capacity. Finally, interactions are assumed to happen with more or less frequency according to the local degree of occupancy of the road, which is expressed in Eqs. (43), (44) by the interaction rate [ ] > 0. In particular, the latter can be thought of as a measure of the reactiveness of the drivers to the evolution of the traffic in their visibility zone. The framework depicted by Eqs. (42)–(44) can be formally derived from very general kinetic structures (see Arlotti et al. [1], Bellomo [7]) as shown by Tosin [65]. Specific models are then obtained by detailing the table of games and the interaction rate as functions of the macroscopic density of cars . In [33] the following form of [ ] is proposed: [ ] D

1 1  / max

which models an increasing reactiveness of the drivers as the road becomes more and more congested, while the table of games is constructed by distinguishing three main cases: (i)

If the candidate vehicle interacts with a faster field vehicle then it either maintains its current speed or possibly accelerates of one velocity class (followthe-leader strategy), depending on the available surrounding free space. (ii) If the candidate vehicle interacts with a slower field vehicle then it either maintains its current speed, provided it can overtake the leader, or slows down to the velocity class of the latter and queues. (iii) If the candidate vehicle interacts with an equally fast field vehicle then a mix of the two previous situations arises, according to the current evolution of the system. Technically, the expressions of the coefficients Aihk , where h, k denote the pre-interaction velocity classes of the

candidate and the field vehicle, respectively, are as follows: (i)

for h < k 8 ˆ <1  ˛(1  / max ) i A hk [ ] D ˛(1  / max ) ˆ : 0

if i D h if i D h C 1 otherwise;

(ii) for h > k 8 ˆ <1  ˛(1  / max ) i A hk [ ] D ˛(1  / max ) ˆ : 0

if i D k if i D h otherwise;

(iii) for h D k 8 ˆ ˛ / max ˆ ˆ ˆ <1  ˛ Aihk [ ] D ˆ ˆ˛(1  / max ) ˆ ˆ :0

if i D h  1 if i D h if i D h C 1 otherwise,

where ˛ 2 [0; 1] is a parameter related to the quality of the road (˛ D 0 stands for bad road, ˛ D 1 for good road). Notice that in case (ii) the coefficient Ahhk [ ], linked to the possibility for the candidate vehicle to maintain its speed because it can overtake the heading field vehicle, defines in practice a probability of passing  

P D˛ 1

max which closely recalls that of Prigogine’s model (cf. Eq. (29)). Finally, in case (iii) a modification of the table of games is needed for h D 1 or h D n, for in these cases the candidate vehicle cannot brake or accelerate, respectively, due to the lack of further lower or higher velocity classes. The deceleration or the acceleration are then simply merged into the tendency of the drivers to preserve the current speed. Figure 3 shows the dimensionless equilibrium diagrams for the flux q (fundamental diagram) and the average speed u vs. the vehicle density , obtained as stationary solutions to Delitala–Tosin’s discrete velocity model for two different values of the parameter ˛ appearing in the table of games. Notice the almost linear growth of q, with a corresponding nearly constant u close to the maximum dimensionless allowed value umax D 1, for low density, and their subsequent strongly nonlinear decrease to zero for high density. This behavior, in good agreement with the experimental results, suggests that the model is able to capture in average the phase transition between free and congested flow occurring in traffic (see Subsect. “The

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Vehicular Traffic: A Review of Continuum Mathematical Models

Vehicular Traffic: A Review of Continuum Mathematical Models, Figure 3 Fundamental diagram (a) and velocity diagram (b) computed as stationary dimensionless solutions to Delitala–Tosin’s model Eqs. (42)–(44) for two different values of the parameter ˛ of the table of games, corresponding to mediocre (˛ D 0:6) and good (˛ D 1) road conditions

Phase Transition Model” and also Kerner [47,48,49]). Further, we observe that when the quality of the road decreases the phase transition is correspondingly anticipated at lower values of . In the discrete velocity model by Coscia et al. [26] the velocity grid I v is conceived so as to have a variable step, which tends to zero for high vehicle concentrations. Specifically, the discretization consists of 2n  1 points given by i1 Ve ( ); i D 1; : : : ; 2n  1 ; n1 where Ve ( ) is an average equilibrium velocity, so that v1 D 0, v2n1 D vmax D 2Ve ( ), and the grid is symmetric with respect to its central point v n D Ve ( ). Since Ve ( ) ! 0 for ! max , we note that v i ( ) ! 0 each i D 1; : : : ; 2n1 as the density locally approaches its maximum admissible value. The authors recommend for Ve ( ) the use of the Bonzani–Mussone’s velocity diagram (10). Alternatively, one can use the classical diagram (9) of the LWR model. The mathematical structure of the discrete kinetic equations to generate particular models relies on the localized Boltzmann-like collisional framework: v i ( ) D

@ fi @ C (v i ( ) f i ) D G i [f; f]  f i L i [f] ; @t @x i D 1; : : : ; 2n  1 for gain and loss operators given respectively by G i [f; f] D

2n1 X h;kD1

jv h  v k jAihk f h f k ;

L i [f] D

2n1 X

jv i  v k j f k ;

kD1

where > 0 is a constant. The table of games Aihk is assumed to be constant in time and space, and is constructed so as to allow interactions only among vehicles belonging to sufficiently close velocity classes (jh  kj  1 in [26]). After an interaction, the candidate vehicle can only fall in a velocity class adjacent to its current one, thus the sole potentially non-zero coefficients Aihk are those for which i D h  1, i D h, and i D h C 1. In addition, slow and fast cars, which are distinguished by comparing their velocity class to the central class vn , are supposed to interact only with faster and slower cars, respectively, with a possible tendency to accelerate in the former case and to decelerate in the latter case. If a ; d 2 [0; 1] denote constant acceleration and deceleration probabilities, the table of games proposed by Coscia and coworkers reads as follows: (i) for h < n AhC1 h;hC1 D a ;

Aihk D 0 otherwise;

Ahh;h1 D 1 d ;

Aihk D 0 otherwise:

Ahh;hC1 D 1 a ; (ii) for h > n Ah1 h;h1 D d ;

A further particularization of these expressions is obtained by setting a D , d D  for 0    1, in order to identify a basic reaction probability 2 [0; 1] and a measure  of the relative strength of a and d .

Vehicular Traffic: A Review of Continuum Mathematical Models

Road Networks A road network is a finite set of N 2 N interconnected roads, which communicate with each other through exchanges of vehicle fluxes at junctions. Therefore, a network can be modeled as an oriented graph in which roads are assimilated to the arcs and junctions to the vertices. Each arcs is represented by an interval I i D [a i ; b i ] R, i D 1; : : : ; N, a i < b i with possibly either a i D 1 or b i D C1, whereas each vertex J is univocally identified by the sets of its incoming and outgoing arcs as J D ((i1 ; : : : ; i n ); ( j1 ; : : : ; j m )), where the first n-uple and the second m-uple contain the indexes of the incoming and the outgoing arcs, respectively (m; n  N). Hence, the complete model is given by a couple (I ; J ), where I D fI i g N iD1 is the collection of arcs and J the collection of vertices. The evolution of traffic on road networks is usually described at a macroscopic level via car densities and other conserved quantities, see Chitour and Piccoli [21], Coclite et al. [22], Garavello and Piccoli [34,35], Herty et al. [43,44], Holden and Risebro [45], Lebacque and Khoshyaran [53]. Similar ideas were also used for telecommunication networks, see D’Apice et al. [31], gas pipelines networks, see Banda et al. [5], Colombo and Garavello [25], and supply chains, see Armbruster et al. [2], D’Apice and Manzo [29], Göttlich et al. [37]. In all these cases one is faced with a system of conservation laws: @u i @ C f (u i ) D 0; @t @x

i D 1; : : : ; N ;

(45)

Rd

is the vector of where u i D u i (t; x) : RC  I i ! the conserved quantities on the ith arc of the graph and f : Rd ! Rd is the flux function. In the case of vehicular traffic, Eq. (45) may represent one of the first or second order models (if d D 1; 2, respectively) reviewed in Sect. “Macroscopic Modeling”. For instance, choosing the Lighthill–Whitham–Richards model one can identify u i with the density of cars i D i (t; x) : RC  I i ! R along the ith road, so that on each arc of the graph the following scalar conservation law is set: @ @ i C f ( i ) D 0; @t @x

i D 1; : : : ; N ;

(46)

where the flux f : R ! R is given by Eq. (8). Conversely, the Aw–Rascle model yields the 2  2 system 0

@ @t

1

  @ B 2

i C C A D 0; @ qi qi @x  q i p( i )

i q i  i p( i )

where now u i D ( i ; q i ) with q i D i w i the ith generalized flux (cf. Eq. (19)). To complete the model, system (45) has to be supplemented by the dynamics at the junctions of the network. A natural assumption is the conservation of the quantities u i , i D 1; : : : ; N at vertices, which is for instance obtained by requiring that u i be a solution at each given vertex: This means that it satisfies Eq. (45), possibly in weak form, for test functions defined on the whole graph and smooth through that vertex. However, in some cases the conservation is not strictly necessary from the modeling point of view. This may either be a modeling choice, like in Garavello and Piccoli [35], or due to some absorption at vertices, e. g. for the presence of queues (Göttlich et al. [37]). The conservation of the quantities u i at vertices is in general not sufficient to describe univocally the dynamics on the network. This was first shown by Holden and Risebro [45], who introduced an additional maximization of some functional at vertices to get uniqueness of the solution to the problem. Then various other choices were proposed, along with techniques to solve Cauchy problems on complete networks. Usually one assigns some traffic distribution rules from incoming to outgoing arcs, joined to traffic flux (or possibly other functionals) maximization. In some cases, extra rules must be used: For instance, for a junction of a road network with two incoming and one outgoing roads, one needs to describe the right of way of the two incoming roads. Definition of Solution and Dynamics at Vertices. Riemann Solvers In this subsection we briefly report about mathematical details needed to state the concept of solution for a system of conservation laws on a network, including the proper formal definition of the dynamics at the junctions. We will assume in the sequel that the network load be described on each arc I i by a vector-valued variable u i D u i (t; x) : RC  I i ! Rd satisfying the conservation law (45) for a given flux f : Rd ! Rd . The function u i is required to be a weak entropic solution to Eq. (45) on I i D [a i ; b i ], thus for every smooth positive test function ' D '(t; x) : RC  I i ! RC compactly supported in (0; C1)  (a i ; b i ) the following relation must hold: C1 Z Zb i 

ui 0

i D 1; : : : ; N ; (47)

@' @' C f (u i ) @t @x

 dxdt D 0

ai

with, in addition, entropy conditions fulfilled, see Bressan [18], Dafermos [27], Serre [63,64]. It is well known that for every initial datum with bounded total variation

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(possibly sufficiently small, especially in the case of systems of conservation laws) Eq. (45) on R admits a unique entropic weak solution depending continuously on the initial datum in L1loc . Moreover, if the initial datum is taken in L1 \ L1 then Lipschitz continuous dependence of the mapping t 7! u i (t; ) in L1 is achieved. To define solutions at vertices, fix a vertex J with n incoming arcs, say I1 ; : : : ; I n , and m outgoing arcs, say I nC1 ; : : : ; I nCm . A weak solution at J is a collection of functions u l : RC  I l ! Rd , l D 1; : : : ; n C m, such that 3 2 C1 b l  nCm X 6 Z Z  @' l @' l 7 C f (u l ) dxdt 5 D 0 (48) ul 4 @t @x l D1

0

' i (; b i ) D ' j (; a j );

@' j @' i (; b i ) D (; a j ) @x @x

for all i D 1; : : : ; n and all j D n C 1; : : : ; n C m. As a consequence, the solution satisfies the Rankine– Hugoniot condition at the vertex J, namely

iD1

Definition 1 For an initial datum u 0 D (u01 ; : : : ; u0nCm ) 2 Rd(nCm) prescribed at a vertex J, a Riemann solver at J is a mapping RS : Rd(nCm) ! Rd(nCm) that associates to u 0 a vector uˆ D (ˆu1 ; : : : ; uˆ nCm ) 2 Rd(nCm) such that: (i) On every incoming arc I i , i D 1; : : : ; n, the solution is given by the waves produced by the Riemann problem (u0i ; uˆ i ); (ii) On every outgoing arc I j , j D nC1; : : : ; nCm, the solution is given by the waves produced by the Riemann problem (ˆu j ; u0j ).

al

for every smooth test function ' l : RC  I j ! R compactly supported either in (0; C1)  (a l ; b l ] if l D 1; : : : ; n (incoming arcs) or in (0; C1)  [a l ; b l ) if l D n C 1; : : : ; n C m (outgoing arcs). Test functions are also required to stick smoothly across the vertex, i. e.,

n X

from that vertex. Correspondingly, the following definition of Riemann solver at a vertex is given:

f (u i (t; b i )) D

nCm X

f (u j (t; a C j ))

Any Riemann solver at J must fulfil the following consistency condition: RS(RS(u 0 )) D RS(u 0 ) : Additional hypotheses are usually stated for a Riemann solver at a vertex, namely: (H1) The waves generated from the vertex must have negative speeds on the incoming arcs and positive speeds on the outgoing ones. (H2) The solution to a Riemann problem at a vertex must satisfy Eq. (48). (H3) The mapping u0l 7! f (ˆu l ) has to be continuous each l D 1; : : : ; n C m.

jDnC1

for almost every t > 0. This Kirchhoff-like condition also guarantees the conservation of the solution at vertices. For a system of conservation laws on the real line, a Riemann problem consists in a Cauchy problem for an initial datum of Heaviside type, that is a piecewise constant function with only one discontinuity. Then one looks for self-similar solutions constituted by simple waves, which are the building blocks to deal with more general Cauchy problems via wave-front tracking algorithms (see Bressan [18], Garavello and Piccoli [34]). These solutions are formed by (a combination of) continuous waves, called rarefactions, and traveling discontinuities, called shocks, whose speeds are related to the eigenvalues of the Jacobian matrix of the flux f (namely, to the derivative of f in the scalar case, see Bressan [18]). A mapping yielding the solution to a Riemann problem as a function of the initial datum is said to be a Riemann solver. Analogously, a Riemann problem for a vertex in a network is the Cauchy problem corresponding to an initial datum which is constant on each arc entering or issuing

Hypothesis (H1) is a consistency condition on the description of the dynamics at the vertex. In fact, if (H1) does not hold then some waves generated by the Riemann solver may disappear through the vertex. Condition (H2) is necessary to have a weak solution at the vertex. However, in some cases (H2) may be violated if only some components of the solution are conserved across the junctions, see for instance Garavello and Piccoli [35]. Finally, (H3) is a regularity condition needed to have a well-posed theory. We simply remark here that continuity of the mapping u0 7! uˆ might not hold in case (H1) holds true. An important consequence of hypothesis (H1) is that some restrictions on the admissible values of the solutions of the problem, and of the corresponding fluxes, arise. For instance, consider a single conservation law for a bounded quantity u, say u 2 [0; umax ] for a certain fixed threshold umax > 0, and assume the following: (F) The flux f : [0; umax ] ! R is strictly concave, with in addition f (0) D f (umax ) D 0. Thus f has a unique maximum point  2 (0; umax ).

Vehicular Traffic: A Review of Continuum Mathematical Models

If  : [0; umax ] ! [0; umax ] is defined to be the mapping such that f ((%)) D f (%) for all % 2 [0; umax ], with however (%) ¤ % for all % 2 [0; umax ]nfg, then the following results are obtained:

on the unknown u i is prescribed. In particular, the latter is required to satisfy

Proposition 2 Consider a single conservation law for a bounded variable u 2 [0; umax ] between an incoming arc I i and an outgoing arc I j at a vertex J, and assume that the flux function satisfies assumption (F) above. Let RS : R2 ! R2 be a Riemann solver for the vertex J, where the initial datum u0 D (u0i ; u0j ) is prescribed, and set uˆ :D RS(u 0 ) D (uˆi ; uˆj ). Then i (˚    if 0  u0i   u0i [  u0i ; umax uˆi 2 if   u0i  umax ; [; umax ] 8 <[0;  ] if 0  u0j    i uˆj 2 n 0 o  : u j [ 0;  u0j if   u0j  umax :

in the sense specified by Bardos et al. [6].

Proposition 3 Consider a single conservation law like in the previous Proposition 2 and assume again the hypothesis (F) on the flux f . For an initial datum u0 D (u0i ; u0j ), a Riemann solver at a vertex J complying with the hypothesis (H1) above is univocally defined by prescribing the values f (ˆu) of the flux. Moreover, there exist maximal possible fluxes in the incoming and outgoing arcs I i , I j given respectively by (   if 0  u0i   f u0i max 0 f i (u ) :D f () if   u0i  umax ; ( f () if 0  u0j    0 f jmax (u0 ) :D f uj if   u0j  umax : After assigning a Riemann solver RS at a vertex J, the admissible solutions u l across J are defined as those functions of bounded total variation: TV(u l (t; ); I l ) < C1; l D 1; : : : ; n C m for almost every time t of existence, with the further property: RS(u J (t)) D u J (t) C where u J (t) :D (u1 (t; b1 ); : : : ; un (t; b n ); u nC1 (t; a nC1 ); C d(nCm) : : : ; unCm (t; a nCm )) 2 R . Finally, for an arc I i D [a i ; b i ] such that either a i > 1 and I i is not an outgoing arc of any vertex of the graph or b i < C1 and I i is not an incoming arc of any vertex of the graph, a boundary condition i D i (t) : RC ! Rd

u i (t; a i ) D

i (t)

or

u i (t; b i ) D

i (t); 8 t

>0

Constructing Solutions on a Network To solve the Cauchy problem on a network for a prescribed initial datum u0 D (u01 ; : : : ; u0N ), where each u0i D u0i (x) : I i ! Rd is a measurable function, and possibly a set of suitable boundary conditions amounts to finding N vector-valued functions u i D u i (t; x) : RC  I i ! Rd such that: The mapping t 7! ku i (t; )kL 1 is continuous each loc i D 1; : : : ; N; (ii) Each u i is a weak entropic solution to Eq. (45) on the corresponding arc I i ; (iii) At each vertex of the graph the proper collection of incoming and outgoing u i defines an admissible solution in the sense discussed in the previous section; (iv) u i (0; x) D u0i (x) for almost every x 2 I i . (i)

There is a general strategy to prove existence of admissible solutions on a network, which basically relies on the following steps: 1. Construct approximate solutions via wave-front tracking algorithms, using suitable Riemann solvers at vertices to deal with the interactions among different arcs; 2. Estimate the total variation of the flux at vertices, then on the whole network; 3. Use appropriate compactness properties to pass to the limit and recover the solution to the problem. The wave-front tracking algorithm invoked by step 1 can be roughly described as follows. First one approximates the initial datum u0i by a piecewise constant function, then uses classical self-similar solutions to Riemann problems on arcs combined with solutions generated by Riemann solvers at vertices. Rarefaction waves are split in rarefaction shocks fans, i. e., collections of small non-entropic shocks. When waves interact, either along arcs or at junctions, new Riemann problems are generated to be sequentially solved. In order to successfully perform this construction, one needs to estimate the number of waves and of interactions, which is achieved via suitable functionals (see Garavello and Piccoli [34,35] for details). Estimation of the total variation of the flux on the network (step 2) can be reduced to the case of a single junction, thanks to the following result:

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Vehicular Traffic: A Review of Continuum Mathematical Models

Theorem 4 Let K RdN be a fixed compact set and assume u D (u1 ; : : : ; u N ) 2 K. For each vertex J of the network, consider the new network which has J as unique junction, whose incoming and outgoing arcs are prolonged to infinity. If for this network there exists a constant C > 0 such that for every initial datum u 0 D (u01 ; : : : ; u0N ) with bounded total variation it results TV[ f (u(t))]  C  TV [ f (u0 )] ; then the same estimate holds for the flux on the entire network with a possibly time-dependent constant C t > 0. Finally, passing to the limit to obtain the true entropic solution to the original problem as required by step 3 is usually the most delicate task. Indeed, even if some compactness of the sequence f f (u )g2N holds (where fu g2N is the sequence generated by the wave-front tracking algorithm), still u may fail to converge. This issue was solved for the scalar case u i 2 R with concave flux f by proving that the number of waves crossing the maximum of the flux can be controlled. To be precise, assume that condition (F) holds true on the function f : [0; umax ] ! R, then define C Definition 5 A wave (u i ; u i ) in the ith arc I i is said to C  be a big shock if u i < u i and C sgn(u i  )  sgn(u i  ) < 0:

Definition 6 Fix a vertex J. An incoming arc I i is said to have a good datum at J at time t > 0 if u i (t; b i ) 2 [; umax ] and a bad datum otherwise. Conversely, an outgoing arc I j is said to have a good datum at J at time t > 0 if u j (t; a C j ) 2 [0;  ] and a bad datum otherwise. Then one can prove (see Garavello and Piccoli [34]): Lemma 7 If an arc I i of a vertex J has a good datum, then the datum remains good after all interactions with waves coming in J from other arcs. Furthermore, no big shocks are produced. Conversely, if I i has a bad datum then after any interaction with waves coming in J from other arcs either the datum of I i remains unchanged or a big shock is produced with a resulting good new datum. Once the number of big shocks is bounded via Lemma 7, one can invert the flux u 7! f (u), which allows one to pass to the limit also in the sequence fu g2N . The interested reader is referred to the book by Garavello and Piccoli [34] for further details also on this part of the theory.

Riemann Solvers at Vertexes In this section we focus specifically on vehicular traffic networks, and describe several Riemann solvers at vertices recently proposed in the literature for the Lighthill– Whitham–Richards model (46). In particular, we consider on each arc I i the density of cars as a bounded variable i ], where i

i 2 [0; max max stands for the capacity of the ith road, and we look for Riemann solvers at vertices complying with the hypothesis (H1). Notice that if the flux f is given on each arc by the parabolic profile (8) (with max i ) then condition (F) is straightforwardly replaced by max i /2. satisfied with  D max According to Proposition 3, a Riemann solver at a vertex J is defined by simply determining the proper values of the flux f . In the papers by Coclite et al. [22] and by D’Apice et al. [31] two different Riemann solvers at vertices, hereafter denoted by R1 and R2, are considered, based on the following algorithms:  Solver R1: (a) The traffic from incoming roads is distributed on outgoing roads according to fixed rates; (b) Under rule (a), the flow of cars through the junction is maximized.  Solver R2: The flow of cars through the junction is maximized over both incoming and outgoing roads. Concerning the solver R1, for a certain junction J let I1 ; : : : ; I n be the n incoming roads and, analogously, I nC1 ; : : : ; I nCm the m outgoing ones. Rule (a) corresponds to defining a stochastic matrix A D (˛ ji )

iD1;:::;n jDnC1;:::;nCm

(49)

whose entries ˛ ji represent the rate of traffic coming from I i and directed to I j . For this, the coefficients ˛ ji are required to satisfy 0 < ˛ ji < 1;

nCm X

˛ ji D 1 :

jDnC1

Notice that the second condition entails in particular that no cars are lost at the junction, indeed if

j D

n X

˛ ji i

iD1

is the total density of cars flowing through the jth outgoing road after passing the junction then it results nCm X jDnC1

j D

n X iD1

i :

Vehicular Traffic: A Review of Continuum Mathematical Models

Furthermore, it can be shown that rule (a) implies the fulfillment of hypothesis (H2) by the Riemann solver R1. In view of Proposition 3, the incoming and outgoing fluxes at J must take values in the sets ˝in D

n O [0; f imax ] Rn ; iD1

˝out D

nCm O

(50) [0;

f jmax ]

R : m

jDnC1

Moreover, owing to rule (a) the incoming fluxes must specifically belong to the region Vehicular Traffic: A Review of Continuum Mathematical Models, Figure 4 The two cases P 2 ˝in and P … ˝in

˝˜ in D f 2 ˝in : A 2 ˝out g ; A being the matrix defined in Eq. (49), which is a convex subset of ˝in determined by linear constraints. By consequence, rule (b) is equivalent to perform a maximization over the incoming fluxes only, the outgoing ones being univocally recoverable using rule (a). Finally, rules (a) and (b) give rise to a linear programming problem at vertex J, whose solution always exists and is in addition unique provided the gradient of the cost function (here, the vector with all unit components) is not orthogonal to the linear constraints describing the set ˝˜ in . The orthogonality condition cannot hold if n > m. To be specific, we consider in the sequel the simple case study (n; m) D (2; 1) corresponding to a junction with two incoming roads I1 ; I2 and one outgoing road I 3 . If not all traffic coming from I 1 and I 2 can flow to I 3 , then one should assign an additional yielding or priority rule: Rn

(c) There exists a priority vector p 2 such that the vector of incoming fluxes must be parallel to p. In the case under consideration, the matrix A reduces to the vector (1; 1), which gives only the trivial information that all vehicles flow toward the road I 3 . Rule (b) amounts instead to determining the through flux as  D minf f1max C f2max; f3max g : If  D f1max C f2max then one simply takes the maximal flux over both incoming arcs. Conversely, if the opposite happens, consider the set of all possible incoming fluxes ( 1 ; 2 ), which must belong to the region ˝in D [0; f1max ]  [0; f2max ], and define the following lines: r p D ftp : t 2 Rg;

r D f( 1 ; 2 ) : 1 C 2 D  g ;

where p is the vector invoked by rule (c). These lines intersect at a point P. As shown by Fig. 4, two situations may

arise, namely either P belongs to ˝in or P lies outside ˝in . In the first case the incoming fluxes are individuated by P, whereas in the second case they are determined by the projection Q of P onto the convex set ˝in \ r . The reasoning can be repeated also in the case of n incoming arcs. In Rn , the general definition of the set ˝in given by Eq. (50) applies; furthermore, the line rp is defined as above (for p 2 Rn ), while r is replaced by the hyperplane ( ) n X

i D  : H D ( 1 ; : : : ; n ) : iD1

The point P D r p \ H exists and is unique. If P 2 ˝in then it is used again to determine the incoming fluxes, otherwise one chooses its projection Q over ˝in \ H . Notice that the point Q is unique since ˝in \H is a closed convex subset of H  . It is to check that the hypothesis (H3) is verified for this Riemann solver. Concerning the solver (R2), one defines the maximal incoming and outgoing fluxes using again Proposition 3: in D

n X

f imax ;

out D

iD1

nCm X

f jmax ;

jDnC1

so that the through flux is simply determined by  D minfin ; out g : Then one uses rule (c) over both incoming and outgoing arcs to find the values of the flux over the arcs. Once again hypothesis (H1) is assumed, thus densities are

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Vehicular Traffic: A Review of Continuum Mathematical Models

uniquely determined. Property (H2) holds by definition, and (H3) is easily proved. Different Riemann solvers at vertices, along with related qualitative analyzes, are studied by D’Apice and Piccoli [30]. Future Directions Various research perspectives may be envisaged in the field of vehicular traffic, aimed on the one hand at an improvement of the mathematical framework, in order to include in the theory further specific features of this nonstandard system, and on the other hand at a coupling with other topics intimately correlated to traffic, so as to enlarge the scenario of applications to real world problems. Without claiming to be thorough, we simply outline here some possible basic ideas that require to be appropriately developed. In both macroscopic and kinetic models presented in this review vehicles are described at an essentially mechanical level. Indeed, the macroscopic approach relies on the analogy with fluid particles, hence on principles proper of continuum mechanics, whereas the kinetic approach accounts for a microscopic state of the cars fully characterized by mechanical variables only, such as position and velocity. However, it can be argued that cars are actually not completely mechanical subjects, for the presence of the drivers is likely to affect their behavior in a strongly unconventional way. The concept of fictitious density proposed by De Angelis (see Subsect. “Fictitious Density Model” and [32]) is an attempt to introduce this aspect of the problem in the macroscopic equations of traffic, but in a sense it is too deterministic to catch the real essence of the human behavior, which is instead quite unpredictable. On the other hand, at the kinetic level of representation no model has been so far introduced in the literature taking the action of the driver explicitly into account. Promising ideas toward this issue are expressed in the recent book by Bellomo [7] dealing with generalized kinetic methods for systems of active particles, namely particles whose microscopic state includes, besides the standard mechanical variables, also an additional internal variable named activity. In the case of vehicular traffic, the latter may represent the driving ability, or alternatively the personality, of the drivers, and can be conceived so as to affect the velocity transition probabilities in an essentially stochastic way. Further research perspectives concern, primarily at a macroscopic level of description, the development of new hydrodynamic models and algorithms for real-time reconstruction of traffic flows in a road network by few experimental information. Using few data, as well as few model parameters, is indeed necessary for compatibility

reasons with actual traffic measurements, which are usually pointwise scattered on the network. In addition, the development of fluid dynamic and gas dynamic models to simulate the emission and propagation of pollutants produced by cars in an urban context might provide powerful technological tools to rationalize the design and use of public and private transportation resources, and to reduce unpleasant effects of urban traffic on the environment. Bibliography 1. Arlotti L, Bellomo N, De Angelis E (2002) Generalized kinetic (Boltzmann) models: mathematical structures and applications. Math Models Methods Appl Sci 12(4):567–591 2. Armbruster D, Degond P, Ringhofer C (2006) A model for the dynamics of large queuing networks and supply chains. SIAM J Appl Math (electronic) 66(3):896–920 3. Aw A, Rascle M (2000) Resurrection of “second order” models of traffic flow. SIAM J Appl Math 60(3):916–938 4. Aw A, Klar A, Materne T, Rascle M (2002) Derivation of continuum traffic flow models from microscopic follow-the-leader models. SIAM J Appl Math (electronic) 63(1):259–278 5. Banda MK, Herty M, Klar A (2006) Gas flow in pipeline networks. Netw Heterog Media (electronic) 1(1):41–56 6. Bardos C, le Roux AY, Nédélec JC (1979) First order quasilinear equations with boundary conditions. Comm Partial Differential Equations 4(9):1017–1034 7. Bellomo N (2007) Modelling complex living systems. A kinetic theory and stochastic game approach. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston 8. Bellomo N, Coscia V (2005) First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow. C R Mec 333:843–851 9. Bellomo N, Lachowicz M, Polewczak J, Toscani G (1991) Mathematical topics in nonlinear kinetic theory II. The Enskog equation. Series on Advances in Mathematics for Applied Sciences, vol 1. World Scientific Publishing, Teaneck 10. Bellomo N, Delitala M, Coscia V (2002) On the mathematical theory of vehicular traffic flow I. Fluid dynamic and kinetic modelling. Math Models Methods Appl Sci 12(12):1801–1843 11. Ben-Naim E, Krapivsky PL (1998) Steady-state properties of traffic flows. J Phys A 31(40):8073–8080 12. Ben-Naim E, Krapivsky PL (2003) Kinetic theory of traffic flows. Traffic Granul Flow 1:155 13. Berthelin F, Degond P, Delitala M, Rascle M (2008) A model for the formation and evolution of traffic jams. Arch Ration Mech Anal 187(2):185–220 14. Bertotti ML, Delitala M (2004) From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences. Math Models Methods Appl Sci 14(7):1061–1084 15. Bonzani I (2000) Hydrodynamic models of traffic flow: drivers’ behaviour and nonlinear diffusion. Math Comput Modelling 31(6–7):1–8 16. Bonzani I, Mussone L (2002) Stochastic modelling of traffic flow. Math Comput Model 36(1–2):109–119 17. Bonzani I, Mussone L (2003) From experiments to hydrodynamic traffic flow models I. Modelling and parameter identification. Math Comput Model 37(12–13):1435–1442

Vehicular Traffic: A Review of Continuum Mathematical Models

18. Bressan A (2000) Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem. In: Oxford Lecture Series in Mathematics and its Applications, vol 20. Oxford University Press, Oxford 19. Cercignani C, Lampis M (1988) On the kinetic theory of a dense gas of rough spheres. J Stat Phys 53(3–4):655–672 20. Chakroborty P, Agrawal S, Vasishtha K (2004) Microscopic modeling of driver behavior in uninterrupted traffic flow. J Transp Eng 130(4):438–451 21. Chitour Y, Piccoli B (2005) Traffic circles and timing of traffic lights for cars flow. Discret Contin Dyn Syst Ser B 5(3):599–630 22. Coclite GM, Garavello M, Piccoli B (2005) Traffic flow on a road network. SIAM J Math Anal (electronic) 36(6):1862–1886 23. Colombo RM (2002) A 22 hyperbolic traffic flow model, traffic flow – modelling and simulation. Math Comput Model 35(5– 6):683–688 24. Colombo RM (2002) Hyperbolic phase transitions in traffic flow. SIAM J Appl Math 63(2):708–721 25. Colombo RM, Garavello M (2006) A well posed Riemann problem for the p-system at a junction. Netw Heterog Media (electronic) 1(3):495–511 26. Coscia V, Delitala M, Frasca P (2007) On the mathematical theory of vehicular traffic flow, II. Discrete velocity kinetic models. Int J Non-Linear Mech 42(3):411–421 27. Dafermos CM (2005) Hyperbolic conservation laws in continuum physics. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol 325, 2nd edn. Springer, Berlin 28. Daganzo CF (1995) Requiem for second-order fluid approximation of traffic flow. Transp Res 29B(4):277–286 29. D’Apice C, Manzo R (2006) A fluid dynamic model for supply chains. Netw Heterog Media (electronic) 1(3):379–398 30. D’Apice C, Piccoli B (2008) Vertex flow models for network traffic. Math Models Methods Appl Sci (submitted) 31. D’Apice C, Manzo R, Piccoli B (2006) Packet flow on telecommunication networks. SIAM J Math Anal (electronic) 38(3): 717–740 32. De Angelis E (1999) Nonlinear hydrodynamic models of traffic flow modelling and mathematical problems. Math Comput Model 29(7):83–95 33. Delitala M, Tosin A (2007) Mathematical modeling of vehicular traffic: a discrete kinetic theory approach. Math Models Methods Appl Sci 17(6):901–932 34. Garavello M, Piccoli B (2006) Traffic flow on networks. In: AIMS Series on Applied Mathematics, vol 1. American Institute of Mathematical Sciences (AIMS), Springfield 35. Garavello M, Piccoli B (2006) Traffic flow on a road network using the Aw–Rascle model. Comm Partial Differ Equ 31(1– 3):243–275 36. Gazis DC, Herman R, Rothery RW (1961) Nonlinear follow-theleader models of traffic flow. Oper Res 9:545–567 37. Göttlich S, Herty M, Klar A (2006) Modelling and optimization of supply chains on complex networks. Commun Math Sci 4(2):315–330 38. Greenberg JM (2001/02) Extensions and amplifications of a traffic model of Aw and Rascle. SIAM J Appl Math (electronic) 62(3):729–745 39. Günther M, Klar A, Materne T, Wegener R (2002) An explicitly solvable kinetic model for vehicular traffic and associated macroscopic equations. Math Comput Model 35(5–6):591–606

40. Günther M, Klar A, Materne T, Wegener R (2003) Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations. SIAM J Appl Math 64(2):468–483 41. Helbing D (1998) From microscopic to macroscopic traffic models. In: A perspective look at nonlinear media. Lecture Notes in Phys, vol 503. Springer, Berlin, pp 122–139 42. Helbing D (2001) Traffic and related self-driven manyparticle systems. Rev Mod Phys 73(4):1067–1141, doi:10.1103/ RevModPhys.73.1067 43. Herty M, Kirchner C, Moutari S (2006) Multi-class traffic models on road networks. Commun Math Sci 4(3):591–608 44. Herty M, Moutari S, Rascle M (2006) Optimization criteria for modelling intersections of vehicular traffic flow. Netw Heterog Media (electronic) 1(2):275–294 45. Holden H, Risebro NH (1995) A mathematical model of traffic flow on a network of unidirectional roads. SIAM J Math Anal 26(4):999–1017 46. Hoogendoorn SP, Bovy PHL (2001) State-of-the-art of vehicular traffic flow modelling. J Syst Cont Eng 215(4):283–303 47. Kerner BS (2000) Phase transitions in traffic flow. In: Helbing D, Hermann H, Schreckenberg M, Wolf DE (eds) Traffic and Granular Flow ’99. Springer, New York, pp 253–283 48. Kerner BS (2004) The physics of traffic. Springer, Berlin 49. Kerner BS, Klenov SL (2002) A microscopic model for phase transitions in traffic flow. J Phys A 35(3):L31–L43 50. Klar A, Wegener R (1997) Enskog-like kinetic models for vehicular traffic. J Stat Phys 87(1–2):91–114 51. Klar A, Wegener R (2000) Kinetic derivation of macroscopic anticipation models for vehicular traffic. SIAM J Appl Math 60(5):1749–1766 52. Klar A, Wegener R (2004) Traffic flow: models and numerics. In: Modeling and computational methods for kinetic equations. Model Simul Sci Eng Technol. Birkhäuser, Boston, pp 219–258 53. Lebacque JP, Khoshyaran MM (1999) Modelling vehicular traffic flow on networks using macroscopic models. In: Finite volumes for complex applications II. Hermes Sci Publ, Paris, pp 551–558 54. Leutzbach W (1988) Introduction to the Theory of Traffic Flow. Springer, New York 55. Lighthill MJ, Whitham GB (1955) On kinematic waves, II. A theory of traffic flow on long crowded roads. Proc Roy Soc Lond Ser A 229:317–345 56. Nagel K, Wagner P, Woesler R (2003) Still flowing: approaches to traffic flow and traffic jam modeling. Oper Res 51(5): 681–710 57. Paveri Fontana SL (1975) On Boltzmann-like treatments for traffic flow. Transp Res 9:225–235 58. Payne HJ (1971) Models of freeway traffic and control. Math Models Publ Syst Simul Council Proc 28:51–61 59. Prigogine I (1961) A Boltzmann-like approach to the statistical theory of traffic flow. In: Theory of traffic flow. Elsevier, Amsterdam, pp 158–164 60. Prigogine I, Herman R (1971) Kinetic theory of vehicular traffic. American Elsevier Publishing, New York 61. Rascle M (2002) An improved macroscopic model of traffic flow: Derivation and links with the Lighthill–Whitham model. Math Comput Model, Traffic Flow Model Simul 35(5–6): 581–590 62. Richards PI (1956) Shock waves on the highway. Oper Res 4: 42–51

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63. Serre D (1996) Hyperbolicité, entropies, ondes de choc. [Hyperbolicity, entropies, shock waves.] In: Systèmes de lois de conservation, I. Fondations. [Foundations.] Diderot Editeur, Paris 64. Serre D (1996) Structures géométriques, oscillation et problémes mixtes. [Geometric structures, oscillation and mixed problems.] In: Systèmes de lois de conservation, II. Fondations. [Foundations.] Diderot Editeur, Paris 65. Tosin A (2008) Discrete kinetic and stochastic game theory for vehicular traffic: Modeling and mathematical problems. Ph D thesis, Department of Mathematics, Politecnico di Torino 66. Treiber M, Helbing D (2003) Memory effects in microscopic traffic models and wide scattering in flow-density data. Phys Rev E 68(4):046–119, doi:10.1103/PhysRevE.68.046119

67. Treiber M, Hennecke A, Helbing D (2000) Congested traffic states in empirical observations and microscopic simulations. Phys Rev E 62(2):1805–1824, doi:10.1103/PhysRevE.62.1805 68. Treiber M, Kesting A, Helbing D (2006) Delays, inaccuracies and anticipation in microscopic traffic models. Physica A 360(1): 71–88 69. Villani C (2002) A review of mathematical topics in collisional kinetic theory. In: Handbook of mathematical fluid dynamics, vol I. North-Holland, Amsterdam, pp 71–305 70. Wegener R, Klar A (1996) A kinetic model for vehicular traffic derived from a stochastic microscopic model. Transp Theory Stat Phys 25(7):785–798 71. Whitham GB (1974) Linear and nonlinear waves. Wiley-Interscience, New York

Water Waves and the Korteweg–de Vries Equation

Water Waves and the Korteweg–de Vries Equation LOKENATH DEBNATH Department of Mathematics, University of Texas – Pan American, Edinburg, USA Article Outline Glossary Definition of the Subject Introduction The Euler Equation of Motion in Rectangular Cartesian and Cylindrical Polar Coordinates Basic Equations of Water Waves with Effects of Surface Tension The Stokes Waves and Nonlinear Dispersion Relation Surface Gravity Waves on a Running Stream in Water History of Russell’s Solitary Waves and Their Interactions The Korteweg–de Vries and Boussinesq Equations Solutions of the KdV Equation: Solitons and Cnoidal Waves Derivation of the KdV Equation from the Euler Equations Two-Dimensional and Axisymmetric KdV Equations The Nonlinear Schrödinger Equation and Solitary Waves Whitham’s Equations of Nonlinear Dispersive Waves Whitham’s Instability Analysis of Water Waves The Benjamin–Feir Instability of the Stokes Water Waves Future Directions Bibliography Glossary Axisymmetric (concentric) KdV equation This is a partial differential equation for the free surface (R; t) in the form (2R C R1  C 3 ) C 13    D 0. Benjamin–Feir instability of water waves This describes the instability of nonlinear water waves. Bernoulli’s equation This is a partial differential equation which determines the pressure in terms of the velocity potential in the form  t C 21 (r)2 C P Cgz D 0, where  is the velocity potential, P is the pressure, is the density and g is the acceleration due to gravity. Boussinesq equation This is a nonlinear partial differential equation in shallow water of depth h given by u t t  c2 ux x C

1  2 1 u x x D h2 u x x t t ; 2 3 p where c 2 D gh :

Cnoidal waves Waves are represented by the Jacobian elliptic function cn(z; m).

Continuity equation This is an equation describing the conservation of mass of a fluid. More precisely, this equation for an incompressible fluid is div u D @u @x @v C @y C @w D 0, where u D (u; v; w) is the velocity @z field, and x D (x; y; z). Continuum hypothesis It requires that the velocity u D (u; v; w), pressure p and density are continuous functions of position x D (x; y; z) and time t. Crapper’s nonlinear capillary waves Pure progressive capillary waves of arbitrary amplitude. Dispersion relation A mathematical relation between the wavenumber, frequency and/or the amplitude of a wave. Euler equations This is a nonlinear partial differential equation for an inviscid incompressible fluid flow governed by the velocity field u D (u; v; w) and pressure P(x; t) under the external force F. More precisely, @u C (u  r)u D  1 rP C F, where is the constant @t density of the fluid. Group velocity The velocity defined by the derivative of the frequency with respect to the wavenumber (c g D d!/dk). Johnson’s equation This is a nearly concentric KdV equation for (R; ; ) in cylindrical polar coordinates in the form   1 1 1 2R C  C 3 C    C 2  D 0 : R 3 R Kadomtsev–Petviashvili (KP) equation This is a two-dimensional KdV equation in the form   1 2 t C 3 C    C  y y D 0 : 3  KdV–Burgers equation This is a nonlinear partial differential equation in the form  t C c0  x C d   x C   x x x    x x D 0 ; where  D

2 1 6 c0 h0

:

Korteweg–de Vries (KdV) equation This is a nonlinear partial differential equation for a solitary wave (or soliton). This equation in a shallow water of depth h is governed by the free surface elevation (x; t) in the 

3 C c(1 C 2h )x C ch6 x x x D 0, where form @ p @t cD gh is the shallow water speed. Laplace equation This is partial differential equation of the form r 2  D  x x C  y y C zz where the  D (x; y; z) is the potential. Linear dispersion relation A mathematical relation between the wavenumber k and the frequency ! of waves (! D !(k)). 2

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Linear dispersive waves Waves with the given dispersion relation between the wavenumber and the frequency. Linear Schrödinger equation This can be written in the form i a t C 1/2 ! 00 (k)(@2 a)/(@x 2 ) D 0, where a D a(x; t) is the amplitude and ! D !(k). Linear wave equation in Cartesian coordinates In three dimensions this can be written in the form u t t D c 2 r 2 u, where c is a constant and r 2 D @2 / (@x 2 ) C @2 /(@y 2 ) C @/(@z2 ) is the three-dimensional Laplacian. Linear wave equation in cylindrical polar coordinates This can be written in the form u t t D urr C 1/r ur C1/r2 u . Navier–Stokes equations This is a nonlinear partial differential equation for an incompressible and viscous fluid flow governed by the velocity field u D (u; v; w) and pressure P(x; t) under the action of external force F. More precisely, (@u)/(@t) C (u  r)u D 1/ rP C  2 u C F, where is the density and  D (/ ) is the kinematic viscosity. Nonlinear dispersion relation A mathematical relation between the wavenumber k, frequency ! and the amplitude a that is D(!; k; a) D 0. Nonlinear dispersive waves Waves with the given dispersion relation between the wavenumber, frequency and the amplitude. Non-linear Schrödinger (NLS) equation This is a nonlinear partial differential equation for the nonlinear modulation of a monochromatic wave. The amplitude, a(x; t) of the modulation satisfies the equation i[(@a)/ (@t) C !00 (@a)/(@x)] C 1/2 !000 (@2 a)/(@x 2 ) C jaj2 a D 0, where !0 D !0 (k), and is a constant. Ocean waves Waves observed on the surface or inside the ocean. Phase velocity The velocity defined by the ratio of the frequency ! and the wavenumber k, (c p D !/k). Resonant or critical phenomenon Waves with unbounded amplitude. Sinusoidal (or exponential) wave A wave is of the form u(x; t) D a Re exp[i(kx  ! t)] D a cos(kx  ! t), where a is amplitude, k is the wavenumber (k D 2/ ),  is the wavelength and ! is the frequency. Solitary waves (or soliton) Waves describing a single hump of given height travel in a medium without change of shape. Stokes expansion This is an expansion of the frequency in terms of the wavenumber k and the amplitude a, that is, !(k) D !0 (k) C !2 (k)a 2 C : : : . Stokes wave Water waves with dispersion relation involving the wavenumber, frequency and amplitude.

Surface-capillary gravity waves Waves under the joint action of the gravitational field and surface tension. Surface gravity waves Water waves under the action of the gravitational field. Variational principle For three-dimensional water ’ waves, it is of the form ıI D ı D L dx dt D 0, where L is called the Lagrangian. Velocity potential A single valued function  D (x; t) defined by u D r. Water waves Waves observed on the surface or inside of a body of water. Waves on a running stream Waves observed on the surface or inside of a body of fluid which is moving with a given velocity. Whitham averaged variational’principle This can be formulated in the form ı L dx dt D 0, where L is called the Whitham average Lagrangian over the phase of the integral of the R 2 Lagrangian L defined by L(!; k; a; x; t) D 1/(2) 0 L d , where L is the Lagrangian. Whitham’s conservation equations These are first order nonlinear partial differential equations in the form (@k)/(@t) C (@!)/(@x) D 0, @/(@t)f f (k)A2 g C @/(@x)f f (k)C(k)A2 g D 0, where k D k(x; t) is the density of waves, ! D !(x; t) is the flux of waves, A D A(x; t) is the amplitude and f (k) is an arbitrary function. Whitham’s equation This first order nonlinear partial differential equation represents the conservation of waves. Mathematically, (@k/@t) C (@!/@x) D 0 where k D k(x; t) is the density of waves and ! D !(x; t) is the flux of waves. Whitham’s equation for slowly varying wavetrain This is written in the form @/(@t)L!  @/(@x i )L k i D 0, where L is the Whitham averaged Lagrangian. Whitham’s nonlinear Rnonlocal equations It is in the 1 form u t C duu x C 1 K(x s) us (s; t) ds D 0, where K(x) D F 1 fc(k) D !/kg and F 1 is the inverse Fourier transformation. Definition of the Subject A wave is usually defined as the propagation of a disturbance in a medium. The simplest example is the exponential or sinusoidal wave which has the form  u(x; t) D a Re exp i(kx  ! t) D a cos(kx  ! t) ;

(1)

where a is called the amplitude, Re stands for the real part, k (D 2/) is called the wavenumber and  is called the

Water Waves and the Korteweg–de Vries Equation

wavelength of the wave, and ! is called the frequency and it is a definite function of the wavenumber k and hence, ! D !(k) is determined by the particular equation of the problem. The quantity D kx  ! t is called the phase of the wave so that a wave of a constant phase propagate with kx  ! t D constant. The mathematical relation ! D !(k) or D(!; k) D 0 ;

(2)

is called the dispersion relation. The phase (or wave) velocity is defined by c(k) D

!(k) : k

(3)

! D !(k) (4)

where  (k) < 0,  (k) D 0, or  (k) > 0. In this case, the sinusoidal wave takes the form   u(x; t) D a exp (k)t exp i(kx   t) :

(5)

If (k) < 0, the amplitude of the wave decays to zero as t ! 1 and the waves are called dissipative. When (k) D 0, waves are dispersive. On the other hand, if (k) > 0 for some or all k, the solution grows exponentially as t ! 1. This case corresponds to instability. The group velocity of the wave (1) is defined by C(k) D

d! : dk

(6)

So, in general, c(k) ¤ C(k). It is convenient to modify the definition slightly. Waves are called dispersive if ! 0 (k) is not a constant, that is, ! 00 (k) ¤ 0. In higher dimensions, all the above ideas can be generalized without any difficulty. The sinusoidal waves in three space dimensions are defined by  u(x; t) D a Re exp i(  x  ! t) ;

! D !() or

(7)

D(!; ) D 0 :

(8)

This function D is also determined by the particular equation of the problem. In this case, D x! t is the phase function. Similarly, the phase velocity of the waves are defined as follows: c(k) D

This shows that the phase velocity, in general, depends on the wavenumber k (or wavelength ) so that waves with different wavelength propagate with different phase velocity. The waves are called dispersive if the phase velocity is not a constant, but depends on the wavenumber k. On the other hand, waves are called nondispersive if c(k) is constant, that is, independent of k. In general, the dispersion relation (2) can be written in the complex form

D (k) C i (k) ;

where a is the amplitude, x D (x; y; z) is the displacement vector,  D (k; l; m) is the wavenumber vector and ! is the frequency which is related to  by the dispersion relation

!() b ; 

(9)

b ) is the unit vector in the  direction where b  D (b k ;b l;m and the group velocity is defined by k C() D r !() D b

@! b @! @! Cl Cm b : @k @l @m

(10)

If the dispersion relation (2) depends on the amplitude a so that the dispersion relation becomes ! D !(k; a) or

D(!; k; a) D 0 :

(11)

In general, the linear plane waves are recognized by the existence of periodic wavetrains in the form u(x; t) D f ( ) D f (kx  ! t) ;

(12)

where f is a periodic function of the phase . For linear problems, solutions more general than (1) can be obtained by the principle of superposition to form Fourier integrals as Z 1  u(x; t) D U(k) exp i fkx  !(k)tg dk ; (13) 1

where the arbitrary function U(k) may be chosen to fit arbitrary initial or boundary conditions, provided the data are reasonable enough to admit Fourier transform U(k) D F fu(x; 0)g, and ! D !(k) is the dispersion relation (2) appropriate to the problem. On the other hand, the nonlinear dispersive waves are also recognized by the existence of periodic wavetrains (12) and the solution must include the amplitude parameter a and it also requires a nonlinear dispersion relation of the form (11). In shallow water of constant depth h, the free surface elevation (x; t) satisfies the Korteweg and de Vries (KdV) equation   2  3 ch t C c 1 C (14)  x C x x x D 0 ; 2h 6

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Water Waves and the Korteweg–de Vries Equation

p where c D gh is the shallow water velocity, g is the acceleration due to gravity, and the total depth H D h C(x; t). The first two terms ( t C c x ) describe the wave evolution at speed c, the third term with the coefficient (3c/2h) represents the nonlinear wave steepening and the last term with the coefficient (ch2 /6) describes linear dispersion. The KdV equation admits an exact solution in the form " 1 # 3a 2 2 X ; (15) (x; t) D a sech 4h3 where X D x  Ut, and U is the wave velocity given by  a U D C 1C : (16) 2h The solution (15) is called the solitary wave (or soliton) describing a single hump of height a above the undisturbed depth h and tending rapidly to zero away from X D 0. The solitary wave propagates to the right with velocity U(> c) which is directly proportional to the amplitude a and has 1 width b1 D (3a/4h3 ) 2 , that is, b1 is inversely proportional to the square root of the amplitude a. Another significant feature of the solitary wave is that it travels in the medium without change of shape. In general, the solution for (x; t) can be expressed in terms of the Jacobian elliptic function cn(X; m) # " 1 3b 2 2 (X) D a cn X; m ; (17) 4h3 1

where m D (a/b) 2 is the modulus of the cn function. Consequently, the solution (17) is called the cnoidal wave. The nonlinear Schrödinger (NLS) equation can be written in the standard form i

t

C

xx

C j j2

D0; 1 < x < 1;

t >0;

(18)

and is a constant. With X D x  Ut, we seek the solution in the form  D exp i(mX  nt) f (X) ; (19) where f (X) can be expressed in terms of the Jacobian elliptic function in the form 1

f (X) D (˛1 /˛2 ) 2 sn( X; ) ;

(20)

where ˛1 , ˛2 ,  D (˛2 ˇ2 /ˇ1 ˛2 ) and  D (˛1 ˇ2 )/(ˇ1 ˛2 ) are constants.

The limiting case of the solitary wave is possible and has the form  f (X) D

2˛

 12 sech

p ˛(x  Ut) ;

(21)

where ˛ and are positive constants. This solution represents a solitary wave solution which propagates without change of shape with constant velocity. However, unlike the KdV solitary waves, the amplitude and the velocity are independent parameters. It is important to note that the solution (21) is possible only in the unstable case > 0. This suggests that the end result of an unstable wavetrain subject to small modulation is a series of solitary waves. Introduction Water waves are the most common observable phenomena in Nature. The subject of water waves is most fascinating and highly mathematical, and varied of all areas in the study of wave motions in the physical world. The mathematical as well as physical problems deal with water waves and their breaking on beaches, with flood waves in rivers, with ocean waves from storms, with ship waves on water, with free oscillations of enclosed waters such as lakes and harbors. The study of water waves and their various ramifications remain central to fluid dynamics in general, and to the dynamics of oceans in particular. The mathematical theory of water waves is quite interesting in its own merit and intrinsically beautiful. It has provided the solid background and impetus for the development of the theory of nonlinear dispersive waves. Indeed, most of the fundamental ideas and results for nonlinear dispersive waves and solitons originated in the investigation of water waves. Historically, the problems of water waves in oceans originated from the classic work of Leonhard Euler (1707– 1783), A.G. Cauchy (1789–1857), S.D. Poisson (1781– 1840), Joseph Boussinesq (1842–1929), George Airy (1801–1892), Lord Kelvin (1824–1907), Lord Rayleigh (1842–1919), George Stokes (1819–1903), Scott Russell (1808–1882) and many others. Indeed, Euler formulated the boundary value problem to understand water wave phenomena and provided successful mathematical and physical description of inviscid and incompressible fluid motion in general. Based on Newton’s second law of motion, Euler first formulated his celebration equation of motion of an inviscid fluid about 250 years ago. Euler’s equation is still considered the basis of all inviscid fluid flows. In 1821, Claude Navier (1785–1836) included the effect of viscosity to the Euler equation, and first developed the equations of motion of viscous fluid. In 1845, Sir George Stokes provided a sound mathematical foundation of vis-

Water Waves and the Korteweg–de Vries Equation

cous fluid flows and rederived the equations of motion for an incompressible viscous fluid that is universally known as the Navier–Stokes equations in the form Du @u 1 D C (u  r)u D  rP C F C r 2 u ; Dt @t

(23)

(24)

Both the Euler Eq. (24) and the Navier–Stokes Eq. (22) form a closed set when the equation of mass conservation (or the continuity equation) div u D r  u D 0

Dv 1 @P @P ; D ; @x Dt

@y @P g; @z

(26)

where the total derivative is given by D @ @ @ @ D Cu Cv Cw ; Dt @t @x @y @z

(27)

and

u(x; t) D (u; v; w) is the fluid velocity, P(x; t) is the pressure field at a point x D (x; y; z), t is time, is a constant density, F(x; t) is a general body force per unit mass,  D (/ ) is known as kinematic viscosity,  is called the 2 dynamic  @ viscosity,  r D r  r is the Laplace operator and @ @ r D @x ; @y ; @z is the familiar gradient operator. In particular, when  D 0 ( D 0), the Navier–Stokes Eq. (22) reduces to the celebrated Euler equation of motion for an inviscid fluid Du @u 1 D C (u  r)u D  rP C F : Dt @t

Du 1 D Dt

1 Dw D Dt

(22)

where (D / D t) is the total (or convective) derivative given by D @ D C (u  r) ; Dt @t

and the continuity equation are

(25)

is added to (22) or (24) so that there are four equations for four unknown quantities u, v, w, and P. The study of these equations is based on the continuum hypothesis which requires that u, p and are continuous function of x D (x; y; z) and t. Remarkably, this 150-year old system of the Navier– Stokes Eqs. (22) and (25) provided the fundamental basis of modern fluid mechanics. However, there are certain major difficulties associated with the Navier–Stokes equations. First, there are no general results for the Navier– Stokes equations on existence of solutions, uniqueness, regularity, and continuous dependence on the initial conditions. Second, another difficulty arises from the strong nonlinear convective term, (u  r)u in Eq. (22). The Euler Equation of Motion in Rectangular Cartesian and Cylindrical Polar Coordinates With F D (0; 0; g), where g is the acceleration due to gravity and constant density , the three components of the Euler’s equation of motion in Cartesian coordinates

@u @v @w C C D0: @x @y @z

(28)

The basic assumption in continuum mechanics is that the motion of the fluid can be described mathematically as a topological deformation that depends continuously on the time t. Consequently, we assume the fluid under consideration to have a boundary surface S, fixed or moving, which separated it from other media. We consider the case of a body of water with air above it so that S is the interface between them. We represent that surface S by an equation S(x; y; z; t) D 0. The kinematic condition is derived from the fact that the normal fluid velocity of the surface (S t /jrSj) is equal to the normal velocity (u  n D u  rS/jrSj), that is, DS D S t C (u  r)S D 0 : Dt

(29)

This means that any fluid particle originally on the boundary surface, S will remain on it. It is often convenient to represent the free surface by the equation z D h(x; y; t) so that the equation for the boundary surface S is S D z  h(x; y; t) D 0 ;

(30)

where z is independent of other variables. Thus, the kinematic free surface condition follows from (29) in the form w  (h t C u h x C v h y ) D 0 on z D h(x; y; t) ;

t >0:

(31)

Since the gravitational force is only body force which acts in the negative z-direction, the continuity Eq. (28) and the Euler equation in the form @u 1 C (u  r)u D  rP  g m b; @t

(32)

1775

1776

Water Waves and the Korteweg–de Vries Equation

where m b is the unit vector in the positive z-direction, represent the fundamental equations for water wave motion. In general, the water wave motion is unsteady, and irrotational which physically means that the individual fluid particle do not rotate. Mathematically, this implies that vorticity ! D curl u D 0. So, there exists a single-valued velocity potential  so that u D r, where  D (x; t). Consequently, the continuity equation reduces to the Laplace equation r 2 D

1 2 ru  u  ! 2

(34)

combined with ! D curl u D 0 and u D r, the Euler Eq. (32) may be written as 

1 P 2 (35) r  t C (r) C C gz D 0 : 2

This can be integrated with respect to the space variables to give the equation for pressure P at every point of the fluid t 0;

(36)

where C(t) is an arbitrary function of time only (rC D 0). Since the pressure gradient affects the flow, a function of t alone added to the pressure field P has no effect on the motion. So, without loss of generality, we can set C(t) D 0 in Eq. (36). Thus, the pressure Eq. (36) becomes 1 P  t C (r)2 C C gz D 0 : 2

t  0:

(38)

(33)

So,  is a harmonic function. This is, indeed, a great advantage because the velocity field u can be obtained from a single potential function  which satisfies a linear partial differential equation. This equation with prescribed boundary conditions can readily be solved in many simple cases without difficulty. In view of a vector identity

1 P  t C (r)2 C C gz D C(t) ; 2

1 1  t C (r)2 C Pa C gz D C(t) 2

on S ;

@2  @2  @2  C C D0: @x 2 @y 2 @z2

(u  r)u D

is nondimensional the free surface elevation that tends to zero as t ! 0. We suppose that the bottom boundary z D b(x; y) is a rigid solid surface. Since the upper boundary is the surface exposed to a constant atmospheric pressure Pa , we have P D Pa on this surface, S. Thus, Eq. (35) assumes the form

(37)

This is the so-called the Bernoulli’s equation which determines the pressure in terms of the velocity potential . Thus, Eqs. (33) and (37) are used to determine the potential  (hence, three velocity components u, v, and w) and the pressure field P. We consider a body of inviscid, incompressible fluid occupying the region b(x; y)  z  h(x; y; t) D h0 Ca(x; y; t), where z D b(x; y) is the bottom boundary surface and z D h0 is the undisturbed (initial) typical constant depth, a is the typical amplitude and (x; y; t)

Absorbing 1 Pa and C(t) into  t , this equation may be rewritten as 1  t C (r)2 C gz D 0 ; 2

on S ;

t  0:

(39)

Since S is a upper boundary surface of the fluid, it contains the same fluid particles for all time t, that is, S is a material surface. Hence, it follows from (39) that 

D 1  t C (r)2 C gz D 0 ; Dt 2 on S ;

t  0:

(40)

Or, equivalently,

 

@ @ 1 C (r  r) C (r)2 C gz @t @t 2 1 D  t t C 2r  r( t ) C r  r(r)2 C gz 2 D0; on S ; t  0 : (41)

We next include the effects of surface tension force per unit length which does support a pressure difference across a curved surface so that P  Pa D (T/R), where R1 D (R11 C R21 ) is called the Gaussian curvature expressed as the sum of two principal radii of curvatures R11 and R21 given by @ @x @ D @y

R11 D R21

h h

1

h x (1 C h2x C h2y ) 2

i

i 1

h y (1 C h2x C h2y ) 2

; (42ab) :

Explicitly, R1 can be written in terms of h(x; y; t) and its partial derivatives as R1 D (R11 C R21 ) D

h x x (1 C h2y )  2 h x h y h x y C h y y (1 C h2x ) (1 C h2x C h2y )3/2

:

(43)

Water Waves and the Korteweg–de Vries Equation

For small deviation of the free surface z D h(x; y; t), its first partial derivatives hx and hy are small so that (43) takes the linearized form R1  (h x x C h y y ) :

(44)

Consequently, the linearized dynamic condition at the free surface z D h becomes  t C gz 

T (h x x C h y y ) D 0 :

(45)

For an inviscid fluid, like the free surface condition (29), there is a bottom boundary condition which follows from the fact that D  z  b(x; y; t) D 0 ; Dt

(46)

where z D b(x; y; t) is the equation of the fixed solid bottom boundary surface. Thus, Eq. (46) assumes the form w D b t C (u  r)b D b t C u b x C v b y on z D b(x; y; t) ;

(47)

where z D b(x; y; t) is given. However, there is a class of problems including sediment movement, where b is not known. For stationary bottom, b is independent of time so that (47) becomes w D u bx C v b y

on z D b(x; y) :

(48)

For one-dimensional problem, h D b(x) with u D (u; 0) so that (48) reduces to the simple bottom condition 0

w D u b (x) on z D b(x) :

(50)

where h0 is the undisturbed depth of water. The pressure field can be rewritten as P D Pa C g (h0  z) C (g h0 )p ;

(51)

where Pa is the constant atmospheric pressure, p is the pressure variable which measures the deviation from the hydrostatic pressure, g (h0  z), and g h0 is the typical pressure scale based on the pressure at depth h D h0 . We next introduce two fundamental parameters " and ı as "D

a h0

and ı D

h02 ; 2

1 (x; y) ;  c  0 t D t;  1 (u ; v  ) D (u; v) ; c0 p  p D : g h0

(x  ; y  ) D

(z ; b ) D

1 (z; b) ; h0 (53)

w D



c0 h0

 w

and (54)

In terms of these nondimensional flow variables and parameters, the Euler Eqs. (26) and the continuity Eq. (28) can be rewritten, dropping the asterisks, in the form   @ @ D (u; v) D  ; p; Dt @x @y

ı

Dw @p D ; Dt @z

(55)

where D @ @ @ @ D Cu Cv Cw ; Dt @t @x @y @z and @u @v @w C C D0: @x @y @z

(56)

(49)

It is convenient to introduce a typical amplitude parameter a by writing the free surface z D h(x; y; t) in the form h D h0 C a(x; y; t) ;

where  is the typical wavelength of the surface gravity wave, " and ı are called the amplitude and long wavelength parameters respectively. In terms of ptypical depth scale h0 ,  is the typical wavelength, c0 D g h0 is the typical horizontal velocity scale, (/c0 ) is the typical time scale, it is also convenient to introduce non-dimensional flow variables denoted by asterisks

(52)

The most general dynamic condition is P D Pa  TR on the free surface z D h D h0 C a. Without surface tension (T D 0), it becomes P D Pa on z D h0 C a. Using the pressure field (51), this free surface dynamic condition reduces to the nondimensional form p D " on z D 1 C ". Thus, the nondimensional kinematic condition (31) and the dynamic condition at the free surface are given by w  "( t C u x C v  y ) D 0 ; p D " on z D 1 C " :

(57ab)

The nondimensional form of the bottom boundary condition (48) remains the unchanged form. Consistent with the governing equations and free surface boundary conditions, we introduce a set of scaled flow variables (u; v; w; p) ! "(u; v; w; p)

(58)

1777

1778

Water Waves and the Korteweg–de Vries Equation

so that the Euler Eqs. (55) and the continuity Eq. (56) reduce to the form   @ D @ @p Dw (u; v) D  ; p; ı D ; (59) Dt @x @y Dt @z @u @v @w C C D0; (60) @x @y @z where   D @ @ @ @ D C" u Cv Cw : Dt @t @x @y @y

(61)

The free surface boundary conditions (57ab) remain the same. The horizontal bottom (b D 0) boundary condition is w D0;

on z D 0 ;

(62)

In cylindrical polar coordinates (r; ; z), the Euler equations and the continuity equation are given by 1 D u v2  D  Pr ; Dt r

1 Dw D  Pz  g ; Dt

Dv uv 1 1 C D P ; Dt r

r

where (64)

and 1 @v @w 1 @ (ru) C C D0: r @r r @ @z

(65)

In terms of the above nondimensional flow variables except for x and y that are replaced by r D 1 r, the above Eqs. (63)–(65) assume the following nondimensional form, dropping the asterisks, D u v2 @p  D : Dt r @r Dw @p ı D ; Dt @z

Dv uv 1 @p C D ; Dt r r @ (66)

(67)

and 1 @v @w 1 @ (ru) C C D0: r @r r @ @z

r 2  D  x x C  y y C zz D 0 ; b(x; y)  z  h0 C a  ;

(68)

t >0;

(69)

t >0;

(70)

 t C ( x x C  y  y )  z D 0 ; on z D h0 C a  ;

1 T  t C (r)2 C g   (R11 C R21 ) D 0 ; 2

z  ( x b x C  y b y ) D 0 ;

t >0;

on z D b(x; y) ;

(71) (72)

where R11 and R21 are given by (42ab), h0 is the typical depth of water and a is the typical amplitude of the surface gravity wave. In water of infinite depth with the origin at the free surface, the bottom boundary condition (72) is replaced by (r) ! 0 as z ! 1. Because of the presence of nonlinear terms in the free surface boundary conditions (70)–(71), the determination of  and  in the general case is a difficult task. Similarly, we can write the basic equations for the water wave problems in cylindrical polar coordinates. In terms of the nondimensional flow variables and the parameters " and ı stated above, the basic water wave Eqs. (69)–(72) without surface tension can be written in the nondimensional form ı( x x C  y y ) C zz D 0 ; on b  z  1 C " ;

where @ @ v @ @ D D Cu C Cw ; Dt @t @r r @ @z

We consider an irrotational unsteady motion of an inviscid and incompressible water occupying the region b(x; y)  z  h0 C a (x; y; t) in a constant gravitational field g which is in the negative z-direction. The equation of uneven bottom boundary is z D b(x; y). Including the effects of surface tension T, the basic equations, two free surface boundary conditions and the bottom boundary condition for the velocity potential  D (x; y; z; t) and the free surface elevation  D (x; y; t) are

on z D h0 C a  ; (63)

@ @ v @ @ D D Cu C Cw ; Dt @t @r r @ @z

Basic Equations of Water Waves with Effects of Surface Tension

t >0;

(73)

on z D 1 C " ;

(74)

" " 2  D0;  t C  C ( x2 C  2y ) C 2 2ı z on z D 1 C " ;

(75)

 ı  t C "( x x C  y  y )  z D 0 ;

Water Waves and the Korteweg–de Vries Equation

z  ı( x b x C  y b y ) D 0 ;

on z D b(x; y) ;

(76)

where ("/ı) D (a2 /h03 ) is another fundamental parameter in water-wave theory. Luke [47] first explicitly formulated a variational principle for two-dimensional water waves and proved that the basic Laplace equation, free surface and bottom boundary conditions can be derived from the Hamilton principle. We formulate the variational principle for three-dimensional water waves in the form “ L dx dt D 0 ; (77) ıI D ı D

where the Lagrangian L is assumed to be equal to the pressure so that Z

(x;t)

L D  h(x;y)

 1 2  t C (r) C gz dz ; 2



(78)

where D is an arbitrary region in the (x; t) space, and (x; z; t) is the velocity potential of an unbounded fluid lying between the rigid bottom z D h(x; y) and the free boundary surface z D (x; y; t). Using the standard procedure in the calculus of variations (see Debnath [19]), the following nonlinear system of equations for the classical water waves can be derived: r 2 D 0 ;

h < z <  ;

1 < (x; y) < 1 ; (79)

 t C ( x x C  y  y )  z D 0 ;

on z D  ;

(80)

 t C 12 (r)2 C gz D 0 ;

on z D  ;

(81)

z C  x h x C  y h y D 0 ;

on z D h :

(82)

These results are in perfect agreement with those of Luke for the two-dimensional waves on water of arbitrary but uniform depth h. In his pioneering work, Whitham [58,59] first developed a general approach to linear and nonlinear dispersive waves using a Lagrangian. It is now well known that most of the general ideas about dispersive waves have originated from the classical problems of water waves. Making reference to Debnath [19], we first state the solution of the linearized two-dimensional problem of the classical water waves on water of uniform depth h governed by the following equation, free surface and boundary conditions  x x C zz D 0 ;

h  z  0 ;

 t D z ;

 t C g D 0 ;

z D 0 ;

on z D h ;

t > 0;

(x; z; t) D    ig cosh k(z C h) exp i(! t  kx) ; Re a ! cosh kh  (x; t) D Re a exp i(! t  kx) ;

(86) (87)

where a D (C!/ig) cosh kh D max jj is the amplitude and C is an arbitrary constant. Using (84ab), we obtain the celebrated dispersion relation between the frequency ! and the wavenumber k in the form: ! 2 D gk tanh kh :

(88)

Physically, this relation describes the interaction between inertial and gravitational forces. This can also be rewritten in terms of the wave (or phase) velocity, c(k) as 1 ! D (gk 1 tanh kh) 2 k

    1 g 2 h 2 D : tanh 2 

c(k) D

(89)

This formula shows that the phase velocity c(k) depends on the gravity g and depth h as well as the wavenumber k or wavelength  D (2/k). Thus, water waves of different wavelengths travel with different wave (or phase) velocities. Such waves are called dispersive, as time passes, these waves disperse (or spread out) into various group of waves such that each group of waves such that each group would consist of waves having approximately the same wavelength. Figure 1 showing a plot of the phase velocity c given by (89) against the wavelength  reveals 1 a transition between the deep water limit c (g/2) 2 (parabolicpform) when  < (3:5)h and the shallow water limit c gh for  > (14)h.

(83)

on z D 0 ; t > 0 ; (84ab) t 0:

The solutions for (x; z; t) and (x; t) representing a sinusoidal wave propagating in the x-direction are given by

(85)

Water Waves and theKorteweg–de Vries Equation, Figure 1  The phase velocity c  against the wavelength  (from [46], Figure 52, page 217)

1779

1780

Water Waves and the Korteweg–de Vries Equation

The quantity ( d!/dk) represents the velocity of such a group in the direction of propagation and is called the group velocity, denoted by C(k) so that  g  d! D (tanh kh C kh sech2 kh) ; (90) C(k) D dk 2! which is, using (89)  D 12 c(k) 1 C 2kh cosech(2kh) :

(91)

Evidently, the group velocity is, in general, different from the phase velocity. Two limiting cases are of special interest: (i) Shallow water waves and (ii) Deep water waves. For shallow water, the wavelength;  D (2/k) is large compared with the depth h so that kh  1, and hence, tanh kh  kh and sin 2kh  2kh. In such a case, (88)– (91) reduce to p (92) ! 2 D gk h2 ; c(k) D gh D C(k) : Both phase and group velocities are independent of the wavenumber k. Thus, shallow water waves are nondispersive, and their phase velocity is equal to group velocity. Both vary as the square root of the depth h. In the other limiting case dealing with deep water waves, the wavelength is very small compared with the depth  so that kh  1. In the limit as kh ! 1, tanh kh ! 1, cosh k(z C h)/ cosh kh ! exp(kz), and the corresponding solutions for  and  become    iag kz  D Re e exp i(! t  kx) !  ag  D (93) e kz sin(kx  ! t) ; !   D Re a exp i(! t  kx) D a cos(kx  ! t) : (94) Evidently, for deep water waves, the dispersion relation (88), the phase velocity (89) and the group velocity (90) become ! 2 D gk D



2 g 

 ;

1 2

c(k) D (g/k) D C(k) D

1 c(k) : 2



g 2

 12 ;

(95abc)

Thus, deep water waves are dispersive and their phase velocity is proportional to the square root of their wavelengths. Also, the group velocity is equal to one-half of the phase velocity. Another equivalent form of the phase velocity c(k) follows from the dispersion relation (88) in the form  2   ! g !h ! D D tanh : (96) c(k) D k !k ! c

Water Waves and the Korteweg–de Vries Equation, Figure 2 The phase velocity c for waves of frequency ! on water of depth h (from [46], Figure 53, page 218)

The phase velocity c(k) for water waves of frequency ! against depth h is shown in Fig. 2. This figure shows a transition between the deep water limit c (g/!) when 2 (! h/c) water limit p  2 or h  2g/! and the shallow c gh when (! h/c)  1 or h  g/! 2 . Similarly, an alternative form of the group velocity follows from the dispersion relation (88) as C(k) D

d! d dc dc D (ck) D c C k D c : (97) dk dk dk d

Finally, we close the section by adding surface capillarygravity waves on water of constant depth h with the free surface at z D 0. For such waves, the linearized free surface conditions (70)–(71) with the effect of surface tension are modified as follows:  t D z ;

 t C g 

T x x D 0

on z D 0 :

(98)

These conditions can be combined to obtain ( t t C gz ) 

T  x xz D 0

on z D 0 :

(99)

In this case, the solutions for the velocity potential (x; z; t) and the free surface elevation (x; t) are exactly the same as (86) and (87) on water of constant depth h or as (93)–(94) for deep water. Evidently, the dispersion relation for capillary-gravity waves on water of constant depth h is given by   T k2 tanh kh : ! 2 D gk 1 C

g

(100)

The phase velocity is   12

 g T k2 : 1C tanh kh c(k) D k

g

(101)

Water Waves and the Korteweg–de Vries Equation

The corresponding dispersion relation for deep water (kh  1) is ! 2 D gk(1 C T  ) ;

(102)

where T  D (T k 2 /g ) D (4 2 T/g 2 ) is a parameter giving the relative importance of surface tension and gravity. The phase velocity is

1 i1 hg 2 g  2  : (1 C T ) D ( )(1 C T ) c(k) D k 2

(103)

The phase p velocity c(k) has a minimum value at k D k m D g /T (or T  D 1). The corresponding minimum p value for c D c m attained at k D k m D g /T (or T  D 1) is  c D cm D

4gT

 14 (104) 1

at wavelength  D m D 2(T/g ) 2 . The inequality k  k m holds for waves to be effectively pure gravity waves, since the surface tension is negligible by comparison. This inequality is equivalent to the wavelength  to be large compared with 1 m D (2/k m ) D 2(T/g ) 2 . The phase velocity c() for capillary-gravity waves against  in deep water is shown in Fig. 3. This figure shows the transition between the 1 pure capillary-wave value (2 T/ ) 2 and the pure grav1 ity-wave value (g/2) 2 . Also shown in this figure are (i) the minimum phase velocity cm attained at  D m , (ii) the gravity-wave curve (T  D 0) for  > m , and (iii) the capillary-wave curve with no gravitational effect g ! 0 or T  ! 1. Figure 3 also shows that for  > 4m the capillary-gravity wave curve is rapidly running closer with the gravity-wave curve. On the other hand, when

 < m , the rapid tendency is for wave speeds to increase again so that the capillary-gravity wave curve approaches the capillary wave curve as T  ! 1, which corresponds to very short waves called capillary waves (or ripples). In fact, when  < 14 m , results (102) and (103) for T  ! 1 give ! 2 D 1 T k 3

and

1

c(k) D ( 1 T k) 2 :

(105ab)

For such capillary waves, surface tension is the only significant restoring force. The Stokes Waves and Nonlinear Dispersion Relation In his 1847 classic paper, Stokes [54] first established the nonlinear solutions for periodic plane waves on deep water. We consider here some of the nonlinear effects neglected in the linearized theory. A more simple approach is to recall the exact free surface dynamic condition (41) with constant atmospheric pressure and the negligible surface tension so that (41) becomes 1 r  r(r)2 D 0 ; 2 on z D  ; for t  0 : (106)

( t t C gz ) C 2r  r t C

This condition is applied on the unknown free surface z D  given by (39), that is,

 1 1  D   t C (r)2 : (107) g 2 In the absence of nonlinear terms, results (106) and (107) reduce to (84ab) and (84b), respectively. We use Taylor series expansions of  and its derivatives from z D  to z D 0 in the form (x; y; ; t) D (x; y; 0; t)     @ 1 2 @2  C C : : : : (108)  C @z zD0 2 @z2 zD0 Using this expansion procedure for each of the derivatives in (106) and (107), we can generate a series of boundary conditions on the surface plane z D 0. Thus, we obtain first three conditions: L D  t t C gz D 0 C O( 2 ) ;

Water Waves and the Korteweg–de Vries Equation, Figure 3  The phase velocity c  given by (103) against  (from [46], Figure 56, page 224)

L C 2r  r t 

(109)

@ 1  t (L) g @z D 0 C O( 3 ) ;

(110)

1781

1782

Water Waves and the Korteweg–de Vries Equation

flatter. This feature becomes accentuated as the wave amplitude is increased. For the third-order free-surface condition (111), direct substitution of the plane wave potential (112) in the nonlinear terms in (111) eliminates all but one term. Therefore, the free-surface boundary condition for the thirdorder plane wave solution is

L C 2r  r t 1 C r  r(r)2 2 @ 1   t (L C 2r  r t ) g @z

 @ 1 1 1    t z t C (r)2 (L) g g 2 @z C

1 2 @2  (L) 2g 2 t @z2

D 0 C O( 4 ) ;

L C (111)

where the symbol O( ) is used to indicate the magnitude of the neglected terms. If we substitute the first-order velocity potential (93) for deep water in the second-order boundary condition (110), the second-order terms in (110) vanish. Thus, the first-order potential is a solution of the second-order boundary value problem, and we can write  ga  D (112) e kz sin(kx  ! t) C O(a3 ) : ! Similarly, using (108), we can incorporate the second-order effects in (x; z; t) so that (x; z; t)

 1 1  t C (r)2 g 2 zD

 1 1 D  t C (r)2 g 2 zD0

 @ 1 1 C C :::   t C (r)2 @z g 2

 zD0 1 1 1 D C : : : : (113)  t C (r)2   t z t g 2 g zD0

D

Direct substitution of (112) in (113) gives the following second-order result:  D a cos(kx  ! t) ˚ 1 C ka 2 2 cos2 (kx  ! t)  1 C : : : 2 1 D a cos C ka 2 cos 2 C : : : ; 2

(114)

where the phase D kx  ! t. Clearly, the second term in (114) represents the nonlinear effects on the free-surface profile, and it is positive both at the crests D 0; 2; 4; : : : and at the troughs D 0; 3; 5; : : :. The notable feature of this solution (114) is that the wave profile is no longer sinusoidal as shown in Fig. 2.7 of Debnath [19]. The actual shape of this wave profile is a curve known as a trochoid: The crests are steeper and the troughs are

1 r  r(r)2 D 0 C O( 4 ) : 2

(115)

The first-order solution (112) satisfies this third-order boundary condition so that the dispersion relation (95a) includes a second-order effect of the form ! 2 D gk(1 C a 2 k 2 ) C O(a3 k 3 ) :

(116)

This remarkable dependence of frequency on wave amplitude is usually known as amplitude (or nonlinear) dispersion. The modified phase velocity expression is  g  12 1 ! (1 C k 2 a2 ) 2 D k k  g  12 1  (1 C a2 k 2 ) : k 2

c(k) D

(117)

The significant change from the linearized theory is (116) or (117), which confirms that the phase velocity now depends on amplitude a as well as on wavelength; steep waves travel faster than less steep waves of the same wavelength. Surface gravity waves thus acquire amplitude dispersion as a second-order correction. The dependence of c on amplitude is known as the amplitude dispersion in contrast to the frequency dispersion as given by (95a). It may be noted that Stokes’ results (114), (116), and (117) can easily be approximated further to obtain solutions for long waves (or shallow water) and for short waves (or deep water). We next discuss the phenomenon of breaking of water waves which is one of the most common observable phenomena on an ocean beach. A wave coming from deep ocean changes shape as it moves across a shallow beach. Its amplitude and wavelength also are modified. The wavetrain is very smooth some distance offshore, but as it moves inshore, the front of the wave steepens noticeably until, finally, it breaks. After breaking, waves continue to move inshore as a series of bores or hydraulic jumps, whose energy is gradually dissipated by means of the water turbulence. Of the phenomena common to waves on beaches, breaking is the physically most significant and mathematically least known. In fact, it is one of the most intriguing longstanding problems of water waves. For waves of small amplitude in deep water, the maximum particle velocity is v D a! D ack. But the basic assumption of small amplitude theory implies that vc

Water Waves and the Korteweg–de Vries Equation

Water Waves and the Korteweg–de Vries Equation, Figure 4 The steepest wave profile

D ak  1. Therefore, wave breaking can never be predicted by the small amplitude wave theory. That possibility arises only in the theory of finite amplitude waves. It is to be noted that the Stokes expansions are limited to relatively small amplitude and cannot predict the wavetrain of maximum height at which the crests are found to be very sharp. For a wave profile of constant shape moving at a uniform velocity, it can be shown that the maximum total crest angle as the wave begins to break is 120ı as shown in Fig. 4. The upshot of the Stokes analysis reveals that the inclusion of higher-order terms in the representation of the surface wave profile distorts its shape away from the linear sinusoidal curve. The effects of nonlinearity are likely to make crests narrower (sharper) and the troughs flatter as depicted in Figure 2.7 of Debnath [19]. The resulting wave profile more accurately portrays the water waves that are observed in nature. Finally, the sharp crest angle of 120ı was first found by Stokes. On the other hand, in 1865, Rankine conjectured that there exists a wave of extreme height. In a moving reference frame, the Euler equations are Galilean invariant, and the Bernoulli Eq. (36) on the free surface of water with

D 1 becomes 2 1 2 jrj

C gz D E :

(118)

Thus, this equation represents the conservation of local energy, where the first term is the kinetic energy of the fluid and the second term is the potential energy due to gravity. For the wave of maximum height, E D gzmax , where zmax is the maximum height of the fluid. Thus, the velocity is zero at the maximum height so that there will be a stagnation point in the fluid flow. Rankine conjectured that a cusp is developed at the peak of the free surface with a vertical slope so that the angle subtended at the peak is 120ı as also conjectured by Stokes [54]. Toland [56] and Amick et al. [4] have proved rigorously the existence of a wave of greatest height and the Stokes conjecture for the wave of extreme form. However, Toland [56] also proved that if the singularity at the peak is not a cusp, that is, if there is no vertical slope at the peak of the free surface,

then the Stokes remarkable conjecture of the crest angle of 120ı is true. Subsequently, Amick et al. [4] confirmed that the singularity at the peak is not a cusp. Therefore, the full Euler equations exhibit singularities, and there is a limiting amplitude to the periodic waves. The nonlinear solutions for plane waves based on systematic power series in the wave amplitude are known as Stokes expansions. Stokes [54] showed that the free surface elevation  of a plane wavetrain on deep water can be expanded in powers of the amplitude a as  D a cos C

1 2 3 ka cos 2 C k 2 a3 cos 3 2 8 C ::: ;

(119)

where D kx  ! t and the square of the frequency is   5 ! 2 D gk 1 C a2 k 2 C a 4 k 4 C : : : : (120) 4 A question was raised about the convergence of the Stokes expansion in order to prove the existence of solution representing periodic water waves of permanent form. Considerable attention had been given to this problem by several authors. The problem was eventually resolved by Levi–Civita [43], who proved formally that the Stokes expansion for deep water converges provided the wave steepness (ak) is very small. Struik [55] extended the proof of Levi–Civita to small-amplitude waves on water of arbitrary, but constant depth. Subsequently, Kraskovskii [40,41] finally established the existence of permanent periodic waves for all amplitudes less than the extreme value at which the waves assume a sharp-crested form. Although the success of the perturbation expansion as a means of representing waves of finite amplitude depends on the convergence of Stokes’ expansion, convergence does not at all imply stability. In spite of the preceding attempts to establish the possibility of finite-amplitude water waves of permanent form, the question of their stability was altogether ignored until the 1960s. The independent work of Lighthill [44,45], Benjamin [5] and Whitham [60,61] finally established that Stokes waves on deep water are definitely unstable! This is one of the most remarkable discoveries in the history of the subject of the theory of water waves. As a complement to the Stokes theory of pure gravity waves, Crapper [16] first discovered the exact nonlinear solution for pure progressive capillary waves of arbitrary amplitude by using a complex variable method. He obtained a remarkably new result for the phase velocity  c(k) D

kT

1  1/4 2  2 H2 1C ; 4

(121)

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Water Waves and the Korteweg–de Vries Equation

depth increases, waves are gradually deformed and seem to behave more and more like a series of solitary waves or a train of cnoidal waves. As revealed by reliable observations, soon they become unstable and then break, forming a whitecap. However, little is known about the detailed nature of wave breaking except for the fact that it is a typical complicated nonlinear phenomenon. It has also been predicted that swell in shallow water tends to break when the crest-to-trough height is about 0.88 times the local water depth. Surface Gravity Waves on a Running Stream in Water

Water Waves and the Korteweg–de Vries Equation, Figure 5 Surface profile for nonlinear capillary waves up to maximum  possible wave, H/ D 0:73 (from [16])

where H D 2a is the wave height representing the vertical distance between crest and trough. This result clearly shows that the phase velocity decreases as the wave amplitude increases for a fixed wavelength. Result (121) is now known as Crapper’s nonlinear capillary wave solution. It has also been shown by Crapper that the capillary wave of greatest height occurs when H D 0:73, which has a striking contrast with the corresponding result due to Michell [48] for pure gravity waves, H D 0:142 D 17 . Figure 5 exhibits Crapper’s nonlinear capillary wave profiles. One of the important features of Crapper’s solution is that there is a maximum possible steepness for pure capillary waves, H/ D 0:73, at which the waves hit each other. In the case of pure gravity waves, Stokes suggested that the wave steepness would possibly be greatest when the crest actually becomes a point with the maximum included crest angle of 120ı . Miche [49] obtained the maximum wave height in water of finite depth h as   H D (0:14) tanh kh ; (122)  max which, in very shallow water (kh  1), gives   H D 0:88 : h max

(123)

This result is in excellent agreement with experimental observations. As the ratio of wave amplitude to local water

Debnath and Rosenblat [18] solved the initial value problem for the generation and propagation of two-dimensional surface waves at the free surface of a running stream on water of finite depth. With the aid of generalized functions, it is proved that the asymptotic solution of the initial value problem leads to the ultimate steady-state solution and the transient solution without the need to resort to the use of a radiation condition at infinity or equivalent device. The fundamental two-dimensional water wave equations on a running stream with velocity U in the x-direction in water of depth h (see Debnath [19], page 115) are given by  x x C zz D 0 ; h  z  0 ; 1 < x < 1;  t C Ux  z D 0 ;  t C U x C g  D 

on z D 0 ;

t > 0;

(124) (125)

P p(x)e i!t ;

z D 0;

z D 0 ;

t >0;

t >0;

on z D h ;

(126) (127)

where the term on the right hand side of (126) represents the arbitrary applied pressure with frequency !, P is a constant and p(x) is arbitrary function of x. Making reference to Debnath [19] for a detailed method of solution and asymptotic analysis of the solution of the initial value problem, we discuss only the inherent resonant (critical) phenomenon involved in this problem. For certain limiting values of the three quantities U, !, and h, there exists a critical speed U c such that 1g 4 ! for h ! 1 ;

(a) U D U c D

! and U are finite :

(128)

Water Waves and the Korteweg–de Vries Equation

and (b) U D U c D

p

gh

for ! D 0 ;

h and U are finite:

(129)

A special property of (a) and (b) is that in both situations there exists a critical value of the speed U, above and below which the steady-state wave system has respectively quite a different character; and that when U assumes its critical value, U c the solution is singular due presumably to a breakdown of the linearized theory. More precisely, when U > U c , these are two surface waves downstream of the origin traveling with speeds (!/s1 ) and (!/s2 ), respectively, in the position x-direction; there are none upstream, where s1 and s2 are two roots (see Debnath and Rosenblat [18]) of the equations 1

(! C kU) C (gk tanh kh) 2 D 0 :

(130ab)

On the other hand, when U < U c , these are two more surface waves which move with speeds (!/1 ) and (!/2 ), both in the negative x-direction, where 1 and 2 are arising from (130a) at positive values of k D 1 and k D 2 . The former of these exists only on the upstream side of the origin, and the latter only on the downstream side. This latter wave thus appears to originate at infinity, but this is only when it is viewed from the moving coordinate system. It is easily verified from equation (130a) that the speed (!/1 ) is always less than U, the speed of the frame, so that relative to axes at rest this wave travels in a positive direction. Depending on U, ! and h, there is a delimiting case when roots 1 and 2 coalesce into a double root. This is the critical case which demand a certain relationship between physical quantities involved, which can be p combined into velocities U, gh and (g/!). Indeed, when h ! 1, we have the case (a), and when ! D 0, the case (b) occurs. When ! D 0, the wave system is degenerate. In this case, s1 D s2 D 1 D 0 and the only wave that occurs is the one associated with the root 2 . As before, it exists on the downstream side, when U > U c , only, but is now a steady motion relative to the moving frame. This is the case discussed by Stoker ( [53], p. 214) who found a similar behavior. Combining the results (128) and (129) together, we see that there are three quantities involved phaving the dimeng sions of velocity, namely U, ! and gh . In general, all these three quantities are finite, and hence the critical situation can be expressed as a functional relationship between them of the form  p  F U; g/!; gh D 0 : (131)

This function F is found and isp seen to yield U as a singlevalued function of (g/!) and gh , with formulas (128) and (129) emerging as limiting cases. Debnath and Rosenblat [18] also showed that the free surface elevation (x; t) becomes singular when roots 1 and 2 coalesce into a double root,  . Their asymptotic evaluation of (x; t) for large jxj and t reveals that it represents two waves where the amplitude of one wave in1 creases like x, while the amplitude of other wave is of t 2 as t ! 1. This singular behavior is not unexpected and is in accord with the findings of Stoker [53] and Kaplan [37]. Mathematically, this critical situation corresponds to the coalescence of two roots (1 D 2 D  ), which is in turn is equivalent to the reinforcement by superposition of two like waves leading to a resonance-type effect. Physically, this situation would reveal itself through wave motions of large amplitude, which cannot come within the scope of linearized theory. Consequently, it would be necessary to include nonlinear terms in the original formulation of the problem in order to achieve a mathematically valid and physically reasonable solution. Akylas [2] developed a nonlinear theory near the critip cal speed U c D gh under the action of surface pressure p(x) traveling at a constant speed U. With slow time scale 1 T D "t and the slow space variable X D " 3 x, where " D ah , his asymptotic analysis reveals that the nonlinear response is of bounded amplitude A(X; t) and is governed by a forced Korteweg and de Vries (KdV) equation in the form A T C A X  2AA X  16 A X X X D  p(0)ı 0 (X) ; (132) p 2 where (U/ gh ) D 1 C " 3 , D O(1), p(k) is the Fourier transform of p(X) and ı(X) is the Dirac delta function. The main conclusion of this asymptotic analysis is that the far field disturbance is of relatively large amplitude, but it remains bounded. The main question whether the nonlinear response evolves to solitons, or disperses out or even gives a cnoidal wave, still remains open. However, it was shown by numerical study that Eq. (132) admits a series of solitons that are generated in front of the pressure field. For < 0, the linearized solution consists of a periodic sinusoidal wave in X < 0. The nonlinear evolution of the wave disturbance at the resonance condition ( D 0) remains bounded. The major conclusion of this study is that a series of solitary waves (or solitons) is successively generated when X < 0 and propagates in front of the pressure field. This prediction is in theoretical agreement with the nonlinear study of Wu and Wu [64], and also in good agreement with experimental findings of Huang et al. [32] who observed solitons in their experiments involving a ship moving in shallow water.

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Water Waves and the Korteweg–de Vries Equation

Akylas [3] also developed a nonlinear theory to extend Debnath and Rosenblat’s linearized study for the evolution of surface waves generated by a moving oscillatory pressure near the critical (or resonant) speed. Based on the slow time scale T D "t and the slow spatial variable p X D " x, his nonlinear asymptotic analysis reveals that the evolution for the wave envelope A(X; T) is governed by the forced nonlinear Schrödinger equation. For details, the reader is referred to Akylas [3] or Debnath [19]. Water Waves and the Korteweg–de Vries Equation, Figure 6 A solitary wave

History of Russell’s Solitary Waves and Their Interactions Historically, John Scott Russell first experimentally observed the solitary wave, a long water wave without change in shape, on the Edinburgh–Glasgow Canal in 1834. He called it the “great wave of translation” and then reported his observations at the British Association in his 1844 paper “Report on Waves.” Thus, the solitary wave represents, not a periodic wave, but the propagation of a single isolated symmetrical hump of unchanged form. His discovery of this remarkable phenomenon inspired him further to conduct a series of extensive laboratory experiments on the generation and propagation of such waves. Based on his experimental findings, Russell discovered, empirically, one of the most important relations between the speed U of a solitary wave and its maximum amplitude a above the free surface of liquid of finite depth h in the form U 2 D g(h C a) ;

(133)

where g is the acceleration due to gravity. His experiments stimulated great interest in the subject of water waves and his findings received a strong criticism from G.B. Airy [1] and G.G. Stokes [54]. In spite of his remarkable work on the existence of periodic wavetrains representing a typical feature of nonlinear dispersive wave systems, Stokes’ conclusion on the existence of the solitary wave was erroneous. However, Stokes [54] proposed that the free surface elevation of the plane wavetrains on deep water can be expanded in powers of the wave amplitude. His original result for the dispersion relation in deep water is given by (120). Despite these serious attempts to prove the existence of finite-amplitude water waves of permanent form, the independent question of their stability remained unanswered until the 1960s except for an isolated study by Korteweg and de Vries in 1895 on long surface waves in water of finite depth. But one of the most remarkable discoveries made in the 1960s was that the periodic Stokes waves on sufficiently deep water are definitely unstable.

This result seems revolutionary in view of the sustained attempts to prove the existence of Stokes waves of finite amplitude and permanent form. Scott Russell’s discovery of solitary waves contradicted the theories of water waves due to Airy and Stokes; they raised questions on the existence of Russell’s solitary waves and conjectured that such waves cannot propagate in a liquid medium without change of form. It was not until the 1870s that Russell’s prediction was finally and independently confirmed by both J. Boussinesq [9,10,11,12] and Lord Rayleigh [51]. From the equations of motion for an inviscid incompressible liquid, they derived formula (133). In fact, they also showed that the Russell’s solitary wave profile (see Fig. 6) z D (x; t) is given by  (x; t) D a sech2 ˇ(x  Ut) ;

(134)

˚ where ˇ 2 D 3a  4h2 (h C a) for any a > 0. Although these authors found the sech2 solution, which is valid only if a  h, they did not write any equation for  that admits (134) as a solution. However, Boussinesq made much more progress and discovered several new ideas, including a nonlinear evolution equation for such long water waves in the form  

 3 2 1 C h2 x x x ; (135)  t t D c 2 x x C 2 h xx 3 p where c D gh is the speed of the shallow water waves. This is known as the Boussinesq (bidirectional) equation, which admits the solution  (136) (x; t) D a sech2 (3a/h3 )1/2 (x C Ut) : This represents solitary waves traveling in both, the positive and negative x-directions. More than 60 years later, in 1895, two Dutchmen, D.J. Korteweg (1848–1941) and G. de Vries [39], formulated

Water Waves and the Korteweg–de Vries Equation

a mathematical model equation to provide an explanation of the phenomenon observed by Scott Russell. They derived the now-famous equation for the propagation of waves in one direction on the surface water of density in the form 

  c 1 3 (137) t D " C  X C  X X X ; h 2 2 where X is a coordinate chosen to be moving (almost) with p the wave, c D gh, " is a small parameter, and  Dh

h2 T  3 g



1 3 h ; 3

(138)

  when the surface tension T  13 g h2 is negligible. Equation (137) is known as the Korteweg–de Vries (KdV) equation. This is one of the simplest and most useful nonlinear model equations for solitary waves, and it represents the longtime evolution of wave phenomena in which the steepening effect of the nonlinear term is counterbalanced by the smoothening effect of the linear dispersion. It is convenient to introduce the change of variables  D (X  ; t) and X  D X C ("/h)ct which, dropping the asterisks, allows us to rewrite Eq. (137) in the form t D

c h



1 3  X C  X X X 2 2

 :

(139)

Modern developments in the theory and applications of the KdV solitary waves began with the seminal work published as a Los Alamos Scientific Laboratory Report in 1955 by Fermi, Pasta, and Ulam [23] on a numerical model of a discrete nonlinear mass-spring system. In 1914, Debye suggested that the finite thermal conductivity of an anharmonic lattice is due to the nonlinear forces in the springs. This suggestion led Fermi, Pasta, and Ulam to believe that a smooth initial state would eventually relax to an equipartition of energy among all modes because of nonlinearity. But their study led to the striking conclusion that there is no equipartition of energy among the modes. Although all the energy was initially in the lowest modes, after flowing back and forth among various low-order modes, it eventually returns to the lowest mode, and the end state is a series of recurring states. This remarkable fact has become known as the Fermi–Pasta–Ulam (FPU) recurrence phenomenon. Cercignani [13] and later on Palais [50] described the FPU experiment and its relationship to the KdV equation in some detail. This curious result of the FPU experiment inspired Martin Kruskal and Norman Zabusky [65] to formulate a continuum model for the nonlinear mass-spring system

Water Waves and the Korteweg–de Vries Equation, Figure 7 Development of solitary waves: a initial profile at t D 0, b profile at t D  1 , and c wave profile at t D (3:6) 1 (from [65])

to understand why recurrence occurred. In fact, they considered the initial-value problem for the KdV equation, u t C uu x C ı u x x x D 0 ; (140)   2 where ı D h` , ` is a typical horizontal length scale, with the initial condition u(x; 0) D cos  x ;

0 x 2;

(141)

and the periodic boundary conditions with period 2, so that u(x; p t) D u(x C 2; t) for all t. Their numerical study with ı D 0:022 produced remarkably new interesting results, which are shown in Fig. 7. They observed that, initially, the wave steepened in regions where it had a negative slope, a consequence of the dominant effects of nonlinearity over the dispersive term, ıu x x x . As the wave steepens, the dispersive effect then becomes significant and balances the nonlinearity. At later times, the solution develops a series of eight well-defined waves, each like sech2 functions with the taller (faster) waves ever catching up and overtaking the shorter (slower) waves. These waves undergo nonlinear interaction according to the KdV equation and then emerge from the interaction without change of form and amplitude, but with only a small change in their phases. So, the most remarkable feature is that these waves retain their identities after the nonlinear interaction. Another surprising fact is that the initial profile reappears, very similarly to the FPU recurrence phenomenon. In view of their preservation of shape and the resemblance to the particle-like character of these waves, Kruskal and Zabusky called these solitary waves, solitons, like photon, proton, electron, and other terms for elementary particles in physics. Historically, the famous 1965 paper of Zabusky and Kruskal [65] marked the birth of the new concept of the

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Water Waves and the Korteweg–de Vries Equation

soliton, a name intended to signify particle-like quantities. Subsequently, Zabusky [66] confirmed, numerically, the actual physical interaction of two solitons, and Lax [42] gave a rigorous analytical proof that the identities of two distinct solitons are preserved through the nonlinear interaction governed by the KdV equation. Physically, when two solitons of different amplitudes (and hence, of different speeds) are placed far apart on the real line, the taller (faster) wave to the left of the shorter (slower) wave, the taller one eventually catches up to the shorter one and then overtakes it. When this happens, they undergo a nonlinear interaction according to the KdV equation and emerge from the interaction completely preserved in form and speed with only a phase shift. Thus, these two remarkable features, (i) steady progressive pulse-like solutions and (ii) the preservation of their shapes and speeds, confirmed the particle-like property of the waves and, hence, the definition of the soliton. Subsequently, Gardner et al. [24,25] and Hirota [29,30,31] constructed analytical solutions of the KdV equation that provide the description of the interaction among N solitons for any positive integral N. After the discovery of the complete integrability of the KdV equation in 1967, the theory of the KdV equation and its relationship to the Euler equations of motion as an approximate model derived from the theory of asymptotic expansions became of major interest. From a physical point of view, the KdV equation is not only a standard nonlinear model for long water waves in a dispersive medium, it also arises as an approximate model in numerous other fields, including ion-acoustic plasma waves, magnetohydrodynamic waves, and anharmonic lattice vibrations. Experimental confirmation of solitons and their interactions has been provided successfully by Zabusky and Galvin [67], Hammack and Segur [26], and Weidman and Maxworthy [57]. Thus, these discoveries have led, in turn, to extensive theoretical, experimental, and computational studies over the last 40 years. Many nonlinear model equations have now been found that possess similar properties, and diverse branches of pure and applied mathematics have been required to explain many of the novel features that have appeared. The Korteweg–de Vries and Boussinesq Equations We consider an inviscid liquid of constant mean depth h and constant density without surface tension. We assume that the (x; y)-plane is the undisturbed free surface with the z-axis positive upward. The free surface elevation above the undisturbed mean depth h is given by z D (x; y; t), so that the free surface is at z D H D h C  and z D 0 is the horizontal rigid bottom (see Fig. 8).

Water Waves and the Korteweg–de Vries Equation, Figure 8 A shallow water wave model

It has already been recognized that the parameters " D (a/h) and  D ak, where a is the surface wave amplitude and k is the wavenumber, must both be small for the linearized theory of surface waves to be valid. To develop the nonlinear shallow water theory, it is convenient to introduce the following nondimensional flow variables based on a different length scale h (which could be the fluid depth): (x  ; y  ) D

1 (x; y) ; l

z D

 D

 ; a

  ct t D ; l   h  D  ; (142) al c

z ; h

p where c D gh is the typical horizontal velocity (or shallow water wave speed). We next introduce two fundamental parameters to characterize the nonlinear shallow water waves: "D

a h

and

ıD

h2 ; l2

(143)

p where " is called the amplitude parameter and ı is called the long wavelength or shallowness parameter. In terms of the preceding nondimensional flow variables and the parameters, the basic equations for water waves (73)–(76) can be written in the nondimensional form, dropping the asterisks: ı( x x C  y y ) C zz D 0 ;

0  z  1 C " ;

@ " 2 " C ( x2 C  2y ) C  CD0 @t 2 2ı z on z D 1 C " ;

(144)

(145)

 ı  t C "( x x C  y  y )  z D 0 on z D 1 C " ; z D 0

on z D 0 :

(146) (147)

Water Waves and the Korteweg–de Vries Equation

It is noted that the parameter  D ak does not enter explicitly in Eqs. (144)–(147), but an equivalent parameter D (a/l) is p associated with " and ı through

D (a/h)  (h/l) D " ı. If " is small, the terms involving " in (145) and (146) can be neglected to recover the linearized free surface conditions. However, the assumption that ı is small might be interpreted as the characteristic feature of the shallow water theory. So, we expand  in terms of ı without any assumption about ", and write 2

 D 0 C ı1 C ı 2 C : : : ;

(148)

and then substitute in (144)–(146). The lowest-order term in (144) is 0zz D 0 ;

(149)

which, with (147), yields 0z D 0, for all z, or 0 D 0 (x; y; t), which indicates that the horizontal velocity components are independent of the vertical coordinate z in lowest order. Consequently, we use the notation 0x D u(x; y; t)

and 0y D v(x; y; t) :

We next consider the free surface boundary conditions retaining all terms up to order ı, " in (145), and ı 2 , "2 , and ı" in (146). It turns out that conditions (145) and (146) become ı 0t  (u tx C v t y ) C  C 12 "(u2 C v 2 ) D 0 ; 2 ˚ ı  t C "(ux C v y ) C (1 C ")(u x C v y ) D

ı2  2 (r u)x C (r 2 v) y : 6

1 ı(u tx x C v tx y ) D 0 ; (158) 2 1 v t C "(uu y C vv y ) C  y  ı(u tx y C v t y y ) D 0 : (159) 2

u t C "(uu x C vv x ) C x 

Simplifying (157) yields    t C u(1 C ") x C v(1 C ") y D

The first- and second-order terms in (144) are given by (151)

1x x C 1y y C 2zz D 0 :

(152)

Integrating (151) with respect to z and using (150ab) gives 1z D z(u x C v y ) C C(x; y; t) ;

z2 (u x C v y ) ; 2

(154)

so that 1 D 0 at z D 0 and u and v are then the horizontal velocity components at the bottom boundary. We next substitute (154) in (151) and (152), and then integrate with condition (147) to determine the arbitrary function. Consequently, 1 3 2 z (r u)x C (r 2 v) y ; 6 1 4 2 z (r u)x C (r 2 v) y ; 2 D 24

(160)

Evidently, Eqs. (158)–(160) represent the nondimensional shallow water equations. Using the fact that 0 is irrotational, that is, u y D v x and neglecting terms O(ı) in (158)–(160), we obtain the fundamental shallow water equations u t C "(uu x C vu x ) C x D 0 ;

(161)

v t C "(uv x C vv y ) C  y D 0 ;

(162)

   t C u(1 C ") x C v(1 C ") y D 0 :

(163)

This system of three, coupled, nonlinear equations is closed and admits some interesting and useful solutions for u, v, and . It is equivalent to the boundary-layer equations in fluid mechanics. Finally, it can be linearized when " D (a/h)  1 to obtain the following dimensional equations: u t C g x D 0 ;

vt C g  y D 0 ;

 t C h(u x C v y ) D 0 :

(164abc)

Eliminating u and v from these equations gives

2z D

where r 2 is the two-dimensional Laplacian.

ı 2 (r u)x C (r 2 v) y : 6

(153)

where the arbitrary function C(x; y; t) becomes zero because of the bottom boundary condition (147). Integrating the resulting Eq. (153), again with respect to z and omitting the arbitrary constant, we obtain 1 D 

(157)

Differentiating (156) first with respect to x and then with respect to y gives two equations:

(150ab)

0x x C 0y y C 1zz D 0 ;

(156)

(155ab)

 t t D c 2 (x x C  y y ) :

(165)

This is a well-known two-dimensional wave equation. It corresponds to the nondispersive shallow water waves that

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Water Waves and the Korteweg–de Vries Equation

p propagate with constant velocity c p D gh. This velocity is simply the linearized version of g(h C ). The wave equation has the simple d’Alembert solution representing plane progressive waves (x; y; t) D f (k1 x C l1 y  1 ct) C g(k2 x C l2 y  2 ct) ; r2

(166)

(kr2

D C l r2 ), where f and g are arbitrary functions and r D 1; 2. We consider the one-dimensional case retaining both " and ı order terms in (158)–(160) so that these equations reduce to the Boussinesq [9,10,11,12] equations u t C "uu x C x  12 ı u tx x D 0 ;

(167)

  t C u(1 C ") x 

(168)

1 6

ı ux x x D 0 :

On the other hand, Eqs. (161)–(163), expressed in dimensional form, read u t C uu x C vu y C gH x D 0 ;

(169)

v t C uv x C vv y C gH y D 0 ;

(170)

H t C (uH)x C (vH) y D 0 ;

(171)

where H D (h C ) is the total depth and H x D x , since the depth h is constant. In particular, the one-dimensional version of the shallow water equations follows from (169)–(171) and is given by u t C uu x C gH x D 0 ;

(172)

H t C (uH)x D 0 :

(173)

This system of approximate shallow water equations is analogous to the exact governing equations of gas dynamics for the case of a compressible flow involving only one space variable (see Riabouchinsky [52]). It is convenient toprewrite these equations of  in terms  the wave speed c D gH by using dc D g/2c dH, so that they become u t C uu x C 2cc x D 0 ;

(174)

2c t C cu x C 2uc x D 0 :

(175)

The standard method of characteristics can easily be used to solve (174) and (175). Adding and subtracting these equations allows us to rewrite them in the characteristic form

 @ @ C (c C u) (u C 2c) D 0 ; (176) @t @x



 @ @ C (c  u) (u  2c) D 0 : @t @x

(177)

Equations (176) and (177) show that u C 2c propagates in the positive x-direction with velocity cCu, and u2c travels in the negative x-direction with velocity c  u, that is, both u C 2c and u  2c propagate in their respective directions with velocity c relative to the water. In other words, dx DuCc dt dx Duc u  2c D constant on curves C on which dt (178ab)

u C 2c D constant on curves CC on which

where CC and C are characteristic curves of the system of partial differential Eqs. (172) and (173). A disturbance propagates along these characteristic curves at speed c relative to the flow speed. The quantities (uC2c) are called the Riemann invariants of the system, and a simple wave is propagating to p the right into water of depth h, that is, u  2c D c0 D gh. Then, the solution is given by   3 (179) u D f () ; x D  C c0 C u t ; 2 where u(x; t) D f (x) at t D 0. However, we note that   3 u x D 1  u x t f 0 () ; 2 giving ux D

2 f 0 () : 2 C 3t f 0 ()

(180)

Thus, if f 0 () is anywhere less than zero, ux tends to infinity as t ! 2/(3 f 0 ). In terms of the free surface elevation, solution (179) implies that the wave profile progressively distorts itself, and, in fact, any forward-facing portion of such a wave continually steepens, or the higher parts of the wave tend to catch up with lower parts in front of them. Thus, all of these waves, carrying an increase of elevation, invariably break. The breaking of water waves on beaches is perhaps the most common and the most striking phenomenon in nature. An alternative system equivalent to the nonlinear evolution Eqs. (167) and (168) can be derived from the nonlinear shallow water theory, retaining both " and ı order terms with ı < 1. This system is also known as the Boussinesq equations, which, in dimensional variables, are given by   t C (h C )u x D 0 ; (181)

Water Waves and the Korteweg–de Vries Equation

u t C uu x C gx D

1 2 h ux x t : 3

(182)

They describe the evolution of long water waves that move in both positive and negative x-directions. Eliminating  and neglecting terms smaller than O("; ı) gives a single Boussinesq equation for u(x; t) in the form 1 1 u t t  c 2 u x x C (u2 )x t D h2 u x x t t : 2 3

(183)

The linearized Boussinesq equation for u and  follows from (181) and (182) as

2   2 @ 1 2 @4 u 2 @  c  h D0: (184)  @t 2 @x 2 3 @x 2 @t 2 This is in perfect agreement with the infinitesimal wave theory result expanded for small kh. Thus, the third derivative term in (182) may be identified with the frequency dispersion. Another equivalent version of the Boussinesq equation is given by   3 2 1 2  t t  c x x D C h2 x x x x : (185) 2 h xx 3 There are several features of this equation. It is a nonlinear partial differential equation that incorporates the basic idea of nonlinearity and dispersion. Boussinesq obtained three invariant physical quantities, Q, E, and M, defined by Z 1 Z 1  dx ; E D 2 dx ; QD 1 1 Z 1

  3  MD 2x  3 dx ; (186) h 1 provided that  ! 0 as jxj ! 1. Evidently, Q and E represent the volume and the energy of the solitary wave. The third quantity M is called the moment of instability, and the variational problem, ıM D 0 with E fixed, leads to the unique solitary-wave solution. Boussinesq also derived the results for the amplitude and volume of a solitary wave of given energy in the form   3 E 3/2 aD (187ab) ; Q D 2 h E 1/3 : 4 h The former result shows that the amplitude of a solitary wave in a channel varies inversely as the channel depth h. The Boussinesq equation can then be written in the normalized form u t t  u x x  32 (u2 )x x  u x x x x D 0 :

(188)

This particular form is of special interest because it admits inverse scattering formalism. Equation (188) has steady progressive wave solutions in the form u(x; t) D 4k 2 f (X) ;

X D kx  ! t ;

(189)

where the equation for f (X) can be integrated to obtain f X X D 6A C (4  6B) f  6 f 2 ;

(190)

where A is a constant of integration and ! 2 D k 2 C k 4 (4  6B) :

(191)

For the special case A D B D 0, a single solitary-wave solution is given by f (X) D sech2 (X  X0 ) ;

(192)

where X 0 is a constant of integration. This result can be used to construct a solution for a series of solitary waves, spaced 2 apart, in the form 1 X

f (X) D

sech2 (X  2n ) :

(193)

nD1

This is a 2 periodic function that satisfies (190) for certain values of A and B. We next assume that " and ı are comparable, so that all terms O("; ı) in (158)–(160) can be retained. For the case of the two-dimensional wave motion (v D 0 and @/@y D 0), these equations become u t C x C " u u x  12 ı u tx x D 0 ;

(194)

  t C u(1 C ") x  16 ı u x x x D 0 :

(195)

We now seek steady progressive wave solutions traveling to the positive x-direction only, so that u D u(x  Ut) and  D (x Ut). With the terms of zero order in " and ı and U D 1, we assume a solution of the form u D  C "P C ı Q ;

(196)

where P and Q are unknown functions to be determined. Consequently, Eqs. (194) and (195) become ( C "P C ıQ) t C x C "x  12 ı  tx x D 0 ;

(197)

  t C (1 C ")( C "P C ıQ) x  16 ı x x x D 0 : (198) These equations must be consistent so that we stipulate for the zero order  t D x ;

P D  14 2 ;

Q D 13 x x D  13 x t : (199)

1791

1792

Water Waves and the Korteweg–de Vries Equation

We use these results to rewrite both (197) and (198) with the assumption that " and ı are of equal order and small enough for their products and squares to be ignored, so that the ratio ("/ı) D (al 2 /h3 ) is of the order one. Consequently, we obtain a single equation for (x; t) in the form    t C 1 C 32 "  x C 16 ı x x x D 0 :

(200)

This is now universally known as the Korteweg and de Vries equation as they discovered it in their 1895 seminal work. We point out that ("/ı) D al 2/h3 is one of the fundamental parameters in the theory of nonlinear shallow water waves. Recently, Infeld [33] considered threedimensional generalizations of the Boussinesq and Korteweg–de Vries equations. Solutions of the KdV Equation: Solitons and Cnoidal Waves To find solutions of the KdV equation, it is convenient to rewrite it in terms of dimensional variables as   3 ch2 (201)  x C x x x D 0 ; t C c 1 C 2h 6 p where c D gh, and the total depth H D h C . The first two terms ( t C c x ) describe wave evolution at the shallow water speed c, the third term with coefficient (3c/2h) represents a nonlinear wave steepening, and the last term with coefficient (ch2 /6) describes linear dispersion. Thus, the KdV equation is a balance between time evolution, nonlinearity, and linear dispersion. The dimensional velocity u is obtained from (196) with (199) in the form   1 2 h c   C x x : uD h 4h 3

(202)

We seek a traveling wave solution of (201) in the frame X so that  D (X) and X D x  Ut with  ! 0, as jxj ! 1, where U is a constant speed. Substituting this solution in (201) gives (c  U)0 C

3c 0 ch2 000  C  D0; 2h 6

(203)

where 0 D d/dX. Integrating this equation with respect to X yields (c  U) C

3c 2 ch2 00  C  D A; 4h 6

where A is an integrating constant.

(204)

We multiply this equation by 20 and integrate again to obtain  2   c  ch d 2 (c  U)2 C 3 C 2h 6 dX D 2A C B ;

(205)

where B is also a constant of integration. We now consider a special case when  and its derivatives tend to zero at infinity and A D B D 0, so that (205) gives   d 2 3 D 3 2 (a  ) ; (206) dX h where



U a D 2h 1 c

 :

(207)

The right-hand side of (206) vanishes at  D 0 and  D a, and the exact solution of (206) represents Russull’s solitary wave in the form   3a 1/2 : (208ab) (X) D a sech2 (bX) ; b D 4h3 Thus, the explicit form of the solution is # "  3a 1/2 2 (x; t) D a sech (x  Ut) ; 4h3

(209)

where the velocity of the wave is  a U D c 1C : (210) 2h This is an exact solution of the KdV equation for all (a/h); however, the equation is derived with the approximation (a/h)  1. The solution (209) is called a soliton (or solitary wave) describing a single hump of height a above the undisturbed depth h and tending rapidly to zero away from X D 0. The solitary wave propagates to the right with velocity U(> c), which is directly proportional to the amplitude a and has width b1 D (3a/4h3 )1/2 , that is, b1 is inversely proportional to the square root of the amplitude a. Another significant feature of the soliton solution is that it travels in the medium without change of shape, which is hardly possible without retaining ı-order terms in the governing equation. A solitary wave profile has already been shown in Fig. 6. In the general case, when both A and B are nonzero, (205) can be rewritten as     U h3 d 2 D 3 C 2h  1 2 3 dX c 2h C (2A C B) D F() ; (211) c where F() is a cubic with simple zeros.

Water Waves and the Korteweg–de Vries Equation

We seek a real bounded solution for (X), which has a minimum value zero and a maximum value a and oscillates between the two values. For bounded solutions, all three zeros 1 , 2 , 3 must be real. Without loss of generality, we set 1 D 0 and 2 D a. Hence, the third zero must be negative so that 3 D (b  a) with b > a > 0. With these choices, F() D (a  )(  a C b) and Eq. (211) assumes the form   h3 d 2 D (a  )(  a C b) ; (212) 3 dX

(213)

which is obtained by comparing the coefficients of 2 in (211) and (212). Writing a   D p2 , it follows from Eq. (212) that   3 1/2 dp dX D  (214) 1/2 : 2 4h3 (a  p )(b  p2 ) p Substituting p D a q in (214) gives the standard elliptic integral of the first kind (see Dutta and Debnath [22] and Helal and Molines [28]) 

3b 4h3

1/2

Z XD 0

q

dq  1/2 ; 2 (1  q )(1  m2 q2 )  a 1/2 mD ; b

(215)

and then, function q can be expressed in terms of the Jacobian elliptic function, sn(z; m) # "  3b 1/2 X; m ; (216) q(X; m) D sn 4h3 where m is the modulus of sn(z; m). Finally, " )# (  3b 1/2 2 (X) D a 1  sn X 4h3 # "  3b 1/2 2 D a cn X ; 4h3

(217)

where cn(z; m) is also the Jacobian elliptic function with a period 2K(m), where K(m) is the complete elliptic integral of the first kind defined by Z /2 K(m) D (1  m2 sin2 )1/2 d ; (218) 0

and

cn2 (z)

C sn2 (z) D 1.

It is important to note that cn z is periodic, and hence, (X) represents a train of periodic waves in shallow water. Thus, these waves are called cnoidal waves with wavelength 

where

  2a  b ; U D c 1C 2h

Water Waves and the Korteweg–de Vries Equation, Figure 9 A cnoidal wave

4h3 D2 3b

1/2 K(m) :

(219)

The upshot of this analysis is that solution (217) represents a nonlinear wave whose shape and wavelength (or period) all depend on the amplitude of the wave. A typical cnoidal wave is shown in Fig. 9. Sometimes, the cnoidal waves with slowly varying amplitude are observed in rivers. More often, wavetrains behind a weak bore (called an undular bore) can be regarded as cnoidal waves. Two limiting cases are of special physical interest: (i) m ! 0 and (ii) m ! 1. In the first case, sn z ! sin z, cn z ! cos z as m ! 0 (a ! 0). This corresponds to small-amplitude waves where the linearized KdV equation is appropriate. So, in this limiting case, the solution (217) becomes (x; t) D

1  a 1 C cos(kx  ! t) ; 2  1/2 3b kD ; h3

where the corresponding dispersion relation is   1 ! D U k D ck 1  k 2 h2 : 6

(220)

(221)

This corresponds to the first two terms of the series expansion of (gk tanh kh)1/2 . Thus, these results are in perfect agreement with the linearized theory. In the second limiting case, m ! 1(a ! b), cn z ! sech z. Thus, the cnoidal wave solution tends to the classical KdV solitary-wave solution where the wavelength , given by (219), tends to infinity because K(a) D 1 and K(0) D /2. The solution identically reduces to (209) with (210). We next report the numerical computation of the KdV Eq. (201) due to Berezin and Karpman [8]. In terms of new variables defined by   3c ; (222) x  D x  ct ; t  D t ;  D 2h

1793

1794

Water Waves and the Korteweg–de Vries Equation

omitting the asterisks, Eq. (201) becomes  t C x C ˇx x x D 0 ; (223) 1 2 where ˇ D 6 ch . We examine the numerical solution of (223) with the initial condition x (x; 0) D 0 f ; (224) ` where 0 is constant and f () is a nondimensional function characterizing the initial wave profile. It is convenient to introduce the dimensionless variables D

x ; `

D

0 t ; `

uD

 0

(225)

a nonlinear interaction and, then, emerge from the interaction without any change in shape and amplitude. The end result is that the taller soliton reappears in front and the shorter one behind. This is essentially strong computational evidence of the stability of solitons. Using the transformation x  D "ˇ(x  ct) ;

t  D "3 t ;

1   D ˛ "2 

with ˛ˇ(D 3c/2h) D 6 and ˇ 3 (D ch2 /6) D 1, we write the KdV Eq. (201) in the normalized form, dropping the asterisks,  t C 6x C x x x D 0 :

(228)

We next seek a steady progressive wave solution of (228) in the form

so that Eqs. (223) and (224) reduce to u C uu C  2 u   D 0 ;

(226)

u(; 0) D f () ;

(227)

where the dimensionless parameter  is defined by  D 1 `(0 /ˇ) 2 . Berezin and Karpman [8] obtained the numerical solution of (226) with the Gaussian initial pulse of the form u(; 0) D f () D exp( 2 ) and values of the parameter  D 1:9 and  D 16:5. Their numerical solutions are shown in Fig. 10. As shown in Fig. 10 for case (a), the perturbation splits into a soliton and a wavepacket. In case (b), there are six solitons. It is readily seen that the peaks of the solitons lie nearly on a straight line. This is due to the fact that the velocity of the soliton is proportional to its amplitude, so that the distances traversed by the solitons would also be proportional to their amplitudes. Zabusky’s [66] numerical investigation of the interaction of two solitons reveals that the taller soliton, initially behind, catches up to the shorter one, they undergo

 D 2k 2 f (X) ;

X D kx  ! t :

(229)

Then, the equation for f (X) can be integrated once to obtain f 00 (X) D 6A C (4  6B) f  6 f 2 ;

(230)

where A is a constant of integration and the frequency ! is given by ! D k 3 (4  6B) :

(231)

A single-soliton solution corresponds to the special case A D B D 0 and is given by f (X) D sech2 (X  X 0 ) ;

(232)

where X 0 is a constant. Thus, for a series of solitons spaced 2 apart, we write f (X) D

1 X

sech2 (X  2n ) :

(233)

nD1

This is a 2 periodic function that satisfies (230) for some A and B. The general elliptic function solution of (228) can be obtained from the integral of (230), which can be written as f X2 D 4C C 12Af C (4  6B) f 2  4 f 3 ;

Water Waves and the Korteweg–de Vries Equation, Figure 10 The solutions of the KdV equation u(x; t) for large values of t with the values of the similarity parameter : a D 1:9, b D 16:5 (from [8])

(234)

where C is a constant of integration. Various asymptotic and numerical results lead to the relations (see Whitham [63]) CD

1 d2 B( ) 1 dA( ) ; D 2 d 4 d 2

(235ab)

Water Waves and the Korteweg–de Vries Equation

and the continuity Eq. (60) in (1 C 1) dimensions are given by

and the cubic in (234) can be factorized as   3  C C 3Af C 1  B f 2  f 3 2 D ( f1  f )( f  f2 )( f  f3 ) ;

(236)

where f r ()(r D 1; 2; 3) are determined from A(), B( ), and C(). If we set f1 > f2 > f3 and, then, the periodic solution oscillates between f 1 at X D 0 and f 2 at X D  , one particular form of the solution is given by p (237) f (X) D f2 C ( f1  f2 )cn2 ( ( f1  f3 ) X) ; where the modulus m of cn(z; m) is given by   f1  f2 2 m D : f1  f3

(238)

p f2 C ( f1  f2 )cn2 ( ( f1  f3 ) X) 1 X

sech2 (X  2n) ;

(239)

nD1

which can be verified by comparing the periods and poles of the two sides. Finally, the higher-order modified KdV equation v t C (p C 1)v p v x C v x x x x D 0 ;

p > 2;

(240)

admits single-soliton solutions in the form v(x; t) D a sech2/p (kx  ! t) :

(242)

 ı w t C "(uw x C wwz ) D p z ;

(243)

u x C wz D 0 :

(244)

The free surface and bottom boundary conditions are obtained from (57ab) and (62) in the form w D  t C "ux ;

p D;

on z D 1 C " ;

(245)

w D 0 on z D 0 :

Thus, it follows from (233) and (237) that the following identity holds:

D

u t C "(uu x C wuz ) D p x ;

(241)

However, in view of the fractional powers of the sech function, it seems, perhaps, unlikely that there will be any simple superposition formula.

(246) p It can easily be shown that, for any ı as " ! 0, there exists a region in the (x; t)-space where there is a balance between nonlinearity and dispersion which leads to the KdV equation. The region of interest is defined by a scaling of the independent flow variables as r r ı ı x and t ! t; (247) x! " " p for any " and ı. In order to ensure consistency in the continuity equation, it is necessary to introduce a scaling of w by r " w: (248) w! ı Consequently, the net effect of the scalings is to replace ı by " in Eqs. (242)–(246) so that they become u t C "(uu x C wuz ) D p x ;

(249)

 " w t C "(uw x C wwz ) D p z ;

(250)

u x C wz D 0 ;

(251)

w D  t C "ux ; and p D  on z D 1 C " ; (252) Derivation of the KdV Equation from the Euler Equations This problem was discussed in Sect. “The Korteweg–de Vries and Boussinesq Equations” by using the Laplace equation for the velocity potential under the assumption that ı D O(") as " ! 0. Here we follow Johnson [35] to present another derivation of the KdV equation from the Euler equation in (1C1) dimensions. This approach can be generalized to derive higher dimensional KdV equations. We consider the problem of surface gravity waves which propagate in the positive x-direction over stationary water of constant depth. The associated Euler Eqs. (59)

w D 0 on z D 0 :

(253)

In the limit as " ! 0, the first-order approximation of Eqs. (249) and (252) gives D p;

0z 1;

and u t C x D 0 :

(254)

It then follows from (251) that w D z u x which satisfies (253). The boundary condition (252) leads to  t C u x D 0 on z D 1, which can be combined with (254) to obtain the linear wave equation  t t  x x D 0 :

(255)

1795

1796

Water Waves and the Korteweg–de Vries Equation

For waves propagating in the positive x-direction, we introduce the far-field variables  Dxt

and  D "t ;

(256)

so that  D O(1) and  D O(1) give the far-field region of the problem. This is the region where nonlinearity balances the dispersion to produce the KdV equation. With the choice of the transformations (256), Eqs. (249)–(253) can be rewritten in the form

w0 C " 0 w0z C " w1 D 0  " 1 C "(0 C u0 0 ) C O("2 ) on z D 1 :

These conditions are to be used together with (257), (258), and (261). The equations in the next order are given by  u1 C u0 C u0 u0 C w0 u0z D p1 ; p1z D w0 ;

 u C "(u C uu C wuz ) D p ;  " w C "(w C uw C wwz ) D p z ;

(257)

u1 C w1z D 0 ;

(258)

p1 C 0 p0 D 1 ;

u C w z D 0 ;

(259)

w1 C 0 C w0z D 1 C 0 C u0 0

pD; on z D 1 C " ;

w D 0 on z D 0 :

(260) (261)

We seek an asymptotic series expansion of the solutions of the system (257)–(261) in the form (; ; ") D

q(; ; z; ") D

1 X nD0 1 X

on z D 0 :

(262)

(271) (272)

Noting that u0z D 0 ;

p0z D 0 ;

and

w0z D 0 ;

(273)

we obtain p1 D 12 (1  z2 ) 0  C 1 ;

"n n (; ) ;

(274)

and therefore,

n

" q n (; ; z) ;

nD0

where q (and the corresponding qn ) denotes each of the variables u, w, and p. Consequently, the leading-order problem is given by u0 D p0 ;

w1 D 0

(269) (270)

on z D 1 ; w D  C "( C u ) ;

(268)

p0z D 0 ;

u0 C w0z D 0 ;

(263)

w1z D  u1 D p1  u0  u0 u0  D  1 C 12 (1  z2 )0   C 0 C 0 0 : (275) Finally, we find h i w1 D  1 C 0 C 0 C 0 C 12 0   z C 16 z3 0   ; (276)

p0 D 0 ;

w C 0 D 0 on z D 1 ;

w D 0 on z D 0 :

(264) (265)

  (w1 )zD1 D  1 C 0 C 0 0 C 12 0   C 16 0  

These leading-order equations give p0 D 0 ;

u0 D 0 ;

which satisfies the bottom boundary condition on z D 0. The free surface boundary condition on z D 1 gives

D 1 C 0 C 2 0 0 ;

w0 C z 0 D 0 ; 0 z 1;

with u0 D 0 whenever 0 D 0. The boundary condition on w0 at z D 1 is automatically satisfied. Using the Taylor series expansion of u, w, and p about z D 1, the two surface boundary conditions on z D 1C" are rewritten on z D 1 and, hence, take the form p0 C " 0 p0z C " p1 D 0 C "1 C O("2 ) on z D 1 :

(277)

(266)

(267)

which yields the equation for 0 (; ) in the form 0 C 32 0 0 C

1 6

0   D 0 :

(278)

This is the Korteweg–de Vries (KdV) equation, which describes nonlinear plane gravity waves propagating in the x-direction. The exact solution of the general initial-value problem for the KdV equation can be obtained provided the initial data decay sufficiently rapidly as jj ! 1. We

Water Waves and the Korteweg–de Vries Equation

may raise the question of whether the asymptotic expansion for  (and hence, for the other flow variables) is uniformly valid as jj ! 1 and as  ! 1. For the case of  ! 1, this question is difficult to answer because the equations for n (n  1) are not easy to solve. However, all the available numerical evidence suggests that the asymptotic expansion of  is indeed uniformly valid as  ! 1 (at least for solutions which satisfy  ! 0 as jj ! 1). From a physical point of view, if the waves are allowed to propagate indefinitely, then other physical effects cannot be neglected. In the case of real water waves, the most common physical effects include viscous damping. In practice, the viscous damping seems to be sufficiently weak to allow the dispersive and nonlinear effects to dominate before the waves eventually decay completely.

This requirement can be incorporated in the governing equations by a scaling of the flow variables as p p y ! "y and v ! "v ; (283) and using the same far-field transformations as (256). Consequently, Eqs. (59)–(62) and the free surface condition (57ab) with the parameters ı replaced by " reduce to the form  u C "(u C uu C "vu y C wuz ) D p ;

(284)

 v C "(v C uv C "vv y C wvz ) D p y ;   " w C "(w C uw C "vw y C wwz )

(285)

D p z ;  u C "v y C wz D 0 ;

Two-Dimensional and Axisymmetric KdV Equations It is well known that the KdV equation describes nonlinear plane waves which propagate only in the x-direction. However, there are many physical situations in which waves move on a two-dimensional surface. So it is natural to include both x- and y-directions with the appropriate balance of dispersion and nonlinearity. One of the simplest examples is the classical two-dimensional linear wave equation

and p D  on z D 1 C " ; w D 0 on z D 0 :

u(x; t) D a exp i(! t    x) ;

(280)

where a is the wave amplitude, ! is the frequency, x D (x; y), and the wavenumber vector is  D (k; `). The dispersion relation is given by ! 2 D c 2 (k 2 C `2 ) :

(281)

For waves that propagate predominantly in the x-direction, the wavenumber component ` becomes small so that the approximate phase velocity is given by   1 `2 cp D c 1 C 2 k2

as ` ! 0 :

(289)

u0 D 0 ; w0 D z 0 ; 0 z 1;

v0 D 0y :

which describes the propagation of long waves. This equation has a solution in the form

(282)

It follows from (282) that the phase velocity suffers from a small correction provided by the wavenumber component ` in the y-direction. In order to ensure that this small correction is the same order as the dispersion and nonlinp earity, it it necessary to require ` D O( ") or `2 D O(").

(288)

We seek the same asymptotic series solution (262) valid as " ! 0 without any change of the leading order problem except that the flow variables involve y so that

(279)



(287)

w D  C "( C uu C "v y )

p0 D 0 ; u t t D c 2 (u x x C u y y ) ;

(286)

(290) (291)

At the next order, the only difference from Sect. “Solutions of the KdV Equation: Solitons and Cnoidal Waves” is in the continuity equation which becomes w1z D u1  v0y :

(292)

This change leads to the following equation:   w1 D  1 C 0 C 0 0 C 12 0   C v0y z C

1 6

z3 0   :

(293)

Using Eq. (291), the final result is the equation for the leading-order representation of the surface wave in the form 2 0 C 3 0 0 C 13 0  C v0y D 0 :

(294)

Consequently, differentiating (294) with respect to  and replacing v0 by 0y give the evolution equation for 0 (; ; y) in the form   2 0 C 3 0 0 C 13 0    C 0y y D 0 : (295)

1797

1798

Water Waves and the Korteweg–de Vries Equation

This is the two-dimensional or, more precisely, the (1 C 2)-dimensional KdV equation. Obviously, when there is no y-dependence, (295) reduces to the KdV Eq. (278). The two-dimensional KdV equation is also known as the Kadomtsev–Petviashvili (KP) equation (Kadomtsev and Petviashvili, [36]). This is another very special completely integrable equation, and it has an exact analytical solution that represents obliquely crossing nonlinear waves. Physically, any number of waves cross obliquely and interact nonlinearly. In particular, the nonlinear interaction of three waves leads to a resonance phenomenon. Such an interaction becomes more pronounced, leading to a strongly nonlinear interaction as the wave configuration is more nearly that of parallel waves. This situation can be interpreted as one in which the waves interact nonlinearly over a large distance so that a strong distortion is produced among these waves. The reader is referred to Johnson’s [35] book for more detailed information on oblique interaction of waves. We now consider the Euler Eqs. (63) and the continuity Eq. (65) in cylindrical polar coordinates (r; ; z). It is convenient to use the nondimensional flow variables, parameters, and scaled variables similar to those defined by (53), (54), and (58) with r D `r where r is a nondimensional variable so that the polar form of Euler Eqs. (66), continuity Eq. (65), the free surface boundary conditions (57ab) in nondimensional cylindrical polar form become D u "v 2 @p Dv "uv 1 @p  D ; C D ; Dt r @r Dt r r @ Dw @p ı D ; (296) Dt @z 1 @ 1 @v @w (ru) C C D0; (297) r @r r @ @z  v  w D  t C " u r C  r and p D  on z D 1 C " ; (298) w D 0 on z D 0 ;

(299)

where   D @ @ v @ @ D C" u C Cw ; Dt @t @r r @ @z

(300)

and " and ı are defined by (52). @  For the case of axisymmetric wave motions @ D0 , the governing equations become u t C "(uur C wuz ) D p r ;  ı w t C "(uwr C wwz ) D p z ;

(301) (302)

ur C

u C wz D 0 ; r

w D  t C "ur

and p D  on z D 1 C " ;

w D 0 on z D 0 :

(303) (304) (305)

In the limit as " ! 0, the linearized version of Eqs. (296)–(300) become 1 v t D  p ; ıw t D p z ; r 1 @ 1 @v @w (ru) C C D0; r @r r @ @z

u t D p r ;

w D t

and p D  on z D 1 ;

w D 0 on z D 0 ;

(306) (307) (308) (309)

For long waves (ı ! 0), Eqs. (306)–(309) lead to the classical wave equation  t t D rr C

1 1 r C 2  : r r

(310)

For axisymmetric surface waves, the wave Eq. (310) becomes  t t D rr C

1 r : r

(311)

This can be solved by using the Hankel transform (see Debnath and Bhatta [21]). It is convenient to introduce the characteristic variable  D r  t for outgoing waves and R D ˛r so that ˛ ! 0 which corresponds to large radius r. In other words, R D O(1), ˛ ! 0 gives r ! 1. The Eq. (311) reduces to the form   1 1 2  R C  C ˛ RR C R D 0 : (312) R R In the limit as ˛ ! 0, it follows that p R  D g() ;

(313)

where g() is an arbitrary function of . For outgoing waves, the correct solution takes the form for ˛ ! 0(r ! 1), Z 1 1 D p (314) g() d D p f () ; R R where  D 0 when f D 0. This shows that the amplitude of axisymmetric waves decays as the radius r ! 1(R ! 1). This behavior is

Water Waves and the Korteweg–de Vries Equation

totally different from the derivation of the KdV equation where the amplitude is uniformly O("). In the present axisymmetric case, the amplitude decreases as the radius r increases so that there is no far-field region where the balance between nonlinearity and dispersion occurs. In other words, the amplitude is so small that nonlinear terms play no role at the leading order. However, there exists a scaling of the flow variables which leads to the axisymmetric (concentric) KdV equation as shown by Johnson [35]. We recall the axisymmetric Euler equations of motion and boundary conditions (301)–(305) and introduce scalings in terms of large radial variable R (see Johnson [35]), D

"2 "6 (r  t) and R D 2 r : ı ı

(315)

We next apply the transformations of the flow variables   "2 "3  ; p ; u ; w  ; (316) (; p; u; w) D ı ı where large distance/time is measured by the scale (ı 2 /"6 ) so that 

ı2 "6



 12 D

"3 ı

 ;

which represents the scale of the amplitude of the waves. It is important to point out that the original wave amplitude parameter " can now be interpreted based on the amplitude of the wave for r D O(1) and t D O(1). Consequently, the governing equations and the boundary conditions (301)–(305) become, dropping the asterisks in the variables,  u C ˛(uu C wuz C ˛ u u R ) D (p C ˛ p R ) ; (317)  ˛ w C ˛(uw C wwz C ˛ u w R ) D p z ; (318)  1  u C w z C ˛ u R C u D 0 ; (319) R w D  C ˛(u C ˛ u R ) ;

pD

on z D 1 C ˛ ; wD0

on z D 0 ;

(320) (321)

where ˛ D (ı 1 "4 ) is a new parameter. These equations are similar in structure to those discussed above with parameter ", which is now replaced by ˛ in (317)–(321) so that the limit as ˛ ! 0 is required. This requirement is satisfied (for example " ! 0 with ı fixed), and the scaling introduced by (315) describes the region where the appropriate balance occurs so that the wave amplitude in this region is O(˛).

We now seek an asymptotic series solution in the form q(; R; z) D

1 X

˛ n q n (; R; z) ;

˛ !0;

(322)

nD0

where q represents each of , u, w, and p. In the leading order, we obtain the familiar equations p0 D 0 ;

u0 D 0 ;

w0 D z 0 ; 0  z  1:

It follows from the continuity Eq. (319) that   1 w1z D u1z  u0R C u0 : R

(323)

(324)

Without any more algebraic calculation, it turns out that 0 (; R) satisfies the nonlinear evolution equation 2 0R C

1 1 0 C 3 0 0 C 0   D 0 : R 3

(325)

This is usually referred to as the axisymmetric (concentric) KdV equation which includes a new term R1 0 . We may use the large time variable  D (ı 2 "6 )t so that R D  C ˛   (see Johnson [35] or [34]). Johnson [34] derived a new concentric KdV equation which incorporates weak dependence on the angular cop ordinate . In the derivation of KP Eq. (295), " was chosen as the scaling parameter in the y-direction. Similarly, the appropriate scaling on the angular variable may be p chosen as ˛. In the derivation of the concentric KdV Eq. (325), the parameter ˛ plays the role of " which is used as the small parameter in the asymptotic solution of the KdV equation. Following the work of Johnson [34,35], we choose the variables  and R defined by (315) and the scaled variable as p (326) D ˛  D (ı 1 "2 )  ; which introduces a small angular deviation from purely concentric effects. We also use the scaled velocity component in the -direction as v D (ı 3/2 "5 )v  ;

(327) 1 r

so that the scalings on u D r and v D  are found to be consistent. Using (315)–(316), Eqs. (296)–(300), and following Johnson [35], the equation in cylindrical polar coordinates can be obtained in the form:   1 1 1 2 0R C 0 C 3 0 0 C 0   C 2 0 D 0 : R 3 R  (328)

1799

1800

Water Waves and the Korteweg–de Vries Equation

This is known as Johnson’s (or nearly concentric KdV) equation, as it was first derived by Johnson [34]. In the absence of the -dependence, Eq. (328) reduces to the concentric KdV Eq. (325). The Nonlinear Schrödinger Equation and Solitary Waves

i

We describe below that the nonlinear modulation of a quasi-monochromatic wave described by the nonlinear Schrödinger equation. To take into account the nonlinearity and the modulation in the far-field approximation, the wavenumber k and frequency ! in the linear dispersion @ @ relation are replaced by k  i @x and ! C i @t , respectively. It is convenient to use the nonlinear dispersion relation in the form   @ @ (329) D ki ; ! C i ; jAj2 A D 0 : @x @t We consider the case of a weak nonlinearity and a slow variation of the amplitude, and hence, the amplitude A is assumed to be a slowly varying function of space and time. @ @ We next expand (329) with respect to jAj2 , i @x , and i @t to obtain   @ @ D(k; !; 0)  i D k  D! A @x @t   @2 @2 @2 1 D k k 2  2D k! C D!! 2 A  2 @x @x@t @t @D jAj2 A D 0 ; (330) C @jAj2 where the first term D(k; !; 0) corresponds to the linear dispersion so that D(k; !; 0) D 0 represents the linear dispersion relation. Introducing the transformation x  D x  C t, t  D t, assuming that A D O("), @x@ D O("), and @t@ D O("2 ), retaining all terms up to O("3 ), and dropping the asterisks, we find that @A @2 A i C p 2 C qjAj2 A D 0 ; @t @x where 1 pD 2



dC dk

 ;

1 qD D!



@D @jAj2

(331)

;

(332)

D! ; Dk

dC D (D k k C 2 C D!k C C 2 D!! )/D! ; dk have been used with D given by (329).

@A @2 A 1 C ! 00 (k) 2 D 0 ; @t 2 @x

(333)

(334)

when p D 12 C 0 (k) D 12 ! 00 (k) is used. More explicitly, if the nonlinear dispersion relation is given by ! D !(k; a2 ) ;

(335)

and if we expand ! in a Taylor series about k D k0 and jaj2 D 0, we obtain !  !0 C (k  k0 )!00

  @! 1 C (k  k0 )2 !000 C jaj2 : (336) 2 @jaj2 jaj2 D0     @ @ Replacing (!  !0 ) by i @t , k  k0 by i @x , and assuming that the resulting operators act on the amplitude function a(x; t), it turns out that a(x; t) satisfies the equation i(a t C !00 a x ) C where



D

@! @jaj2

1 00 ! a x x C jaj2 a D 0 ; 2 0

(337)

D constant :

(338)

 jaj2 D0

Equation (337) is known as the cubic nonlinear Schrödinger equation. If we choose a frame of reference moving with the linear group velocity !00 , that is,  D x  !00 t and  D t, the term involving ax will drop out from (337), and therefore, the amplitude a(x; t) satisfies the normalized nonlinear Schrödinger equation i a C

1 2

!000 a  C jaj2 a D 0 :

(339)

The corresponding dispersion relation is given by !D



and the following results CD

Equation (331) is known as the cubic nonlinear Schrödinger (NLS) equation. When the last cubic nonlinear term is neglected, Eq. (331) reduces to the corresponding linear Schrödinger (NLS) equation for A(x; t) in the form

1 2

!000 k 2  a2 :

(340)

According to the stability criterion established by Whitham [62], the wave modulation is stable if !000 < 0 or unstable if !000 > 0. To study the solitary wave solution, it is convenient to use the NLS equation in the standard form i

t

C

xx

C j j2

D0; 1 < x < 1;

t > 0:

(341)

Water Waves and the Korteweg–de Vries Equation

We then seek waves of permanent form by assuming the solution D f (X)e i(m Xnt) ;

X D x  Ut ;

(342)

for some functions f and constant wave speed U to be determined, and where m, n are constants. Substitution of (342) in (341) gives f 00 C i(2m  U) f 0 C (n  m2 ) f C j f j2 f D 0 : (343) We eliminate f 0 by setting 2m  U D 0, and then write n D m2  ˛, so that f can be assumed real. Thus, Eq. (343) becomes f 00  ˛ f C f 3 D 0 :

(344)

Multiplying this equation by 2 f 0 and integrating, we find that f

02

D A C ˛ f  f 4 D F( f ) ; 2 2

(345)

where F( f ) D (˛1  ˛2 f 2 )(ˇ1  ˇ2 f 2 ), so that ˛ D (˛1 ˇ2 C ˛2 ˇ1 ), A D ˛1 ˇ1 , D 2(˛2 ˇ2 ), and the ˛’s and ˇ’s are assumed to be real and distinct. Evidently, it follows from (345) that Z

f

p

XD 0

df (˛1  ˛2 f 2 )(ˇ1  ˇ2 f 2 )

:

(346)

Setting (˛2 /˛1 )1/2 f D u in this integral (346), we deduce the following elliptic integral of the first kind (see Dutta and Debnath [22] and Helal and Molines [28]): Z

f

X D 0

df

p ; (1  u2 )(1   2 u2 )

(347)

where  D (˛2 ˇ1 )1/2 and  D (˛1 ˇ2 )/(ˇ1 ˛2 ). Thus, the final solution can be expressed in terms of the Jacobian elliptic function u D sn( X; ) :

(348a)

Thus, the solution for f (X) is given by  f (X) D

˛1 ˛2

1/2 sn( X; ) :

(348b)

In particular, when A D 0, ˛ > 0, and > 0, we obtain a solitary wave solution. In this case, (345) can be rewritten Z fn  p

2 o 12 f2 1  ˛X D df : (349) f 2˛ 0

Substitution of ( /2˛)1/2 f D sech in this integral gives the exact solution  1/2 p 2˛ f (X) D sech ˛(x  Ut) : (350)

This represents a solitary wave solution that propagates without change of shape with constant velocity U. Unlike the solitary wave solution of the KdV equation, the amplitude and the velocity of the wave are independent parameters. It is noted that the solitary wave exists only for the unstable case ( > 0). This means that small modulations of the unstable wavetrain lead to a series of solitary waves. The well-known nonlinear dispersion relation (116) for deep water waves is   p 1 ! D gk 1 C a2 k 2 : (351) 2 Therefore, !0 ; !00 D 2k0

!000 D

!0 ; 4k02

and D  12 !0 k02 ; (352)

and the NLS equation for deep water waves is obtained from (337) in the form   !0 !0 1 a x  2 a x x  !0 k02 jaj2 a D 0 : (353) i at C 2k0 2 8k0 The normalized form of this equation in a frame of reference moving with the linear group velocity !00 is   !0 1 (354) a x x  !0 k02 jaj2 a D 0 : i at  2 2 8k0 Since !000 D (!02 /8) > 0, this equation confirms the instability of deep water waves. This is one of the most remarkable recent results in the theory of water waves. We next discuss the uniform solution and the solitary wave solution of the NLS Eq. (354). We look for solutions in the form a(x; t) D A(X) exp(i 2 t) ;

X D x  !00 t ;

(355)

and substitute this in Eq. (354) to obtain the following equation:   8k 2 1 A X X D 0 2 A C !0 k02 A3 : (356) !0 2 We multiply this equation by 2AX and, then, integrate to find   8 2 2 2 2 4 02 4 4

k0 A C 2k0 A A X D  A0 m C !0 D (A20  A2 )(A2  m02 A30 ) ;

(357)

1801

1802

Water Waves and the Korteweg–de Vries Equation

where A40 m02 is an integrating constant, 2k04 D 1, m02 D 1  m2 , and A20 D 4 2 /!0 k02 (m2  2), which relates A0 , , and m. Finally, we rewrite Eq. (357) in the form A20 dX D h 1

A2 A20

dA  2 A A20

 m02

i1/2 ;

(358a)

or, equivalently,

(x; t) D a(x; t) exp fi (x; t)g C c:c: ;

t

ds  1/2 ; (1  s 2 )(s 2  m02 ) s D (A/A0 ) :

(358b)

This can readily be expressed in terms of the Jacobi dn function (see Dutta and Debnath [22] and Helal and Molines [28]):  A D A0 dn A0 (X  X0 ); m ;

(359)

where m is the modulus of the dn function. In the limit, m ! 0, dn z ! 1, and 2 !  12 !0 k02 A20 . Hence, the solution becomes 



a(x; t) D A(t) D A0 exp  12 i !0 k02 A20 t :

(360)

On the other hand, when m ! 1, dn z ! sech z, and

2 !  14 !0 k02 A20 . Therefore, the solitary wave solution is   i 2 2 a(x; t) D A0 exp  !0 k0 A0 t 4   sech A0 (x  !00 t  X0 ) :

(361)

An analysis of this section reveals several remarkable features of the nonlinear Schrödinger equation. Like the KdV equation, the nonlinear Schrödinger equation is the lowest-order approximation for weakly and strongly nonlinear dispersive waves system. This equation can also be used to investigate instability phenomena in many other physical systems. Like the various forms of the KdV equation, the NLS equation arises in many physical problems, including nonlinear water waves and ocean waves, waves in plasma, propagation of heat pulses in a solid, self-trapping phenomena in nonlinear optics, nonlinear waves in a fluid-filled viscoelastic tube, and various nonlinear instability phenomena in fluids and plasmas. Whitham’s Equations of Nonlinear Dispersive Waves To describe a slowly varying nonlinear and nonuniform oscillatory wavetrain in a dispersive medium, we assume

(362)

where c:c: stands for the complex conjugate and a(x; t) is the complex amplitude (see Debnath [20]). The phase function (x; t) is given by (x; t) D xk(x; t)  t!(x; t) ;

Z A0 (X  X 0 ) D

the existence of a one-dimensional solution in the form (see Whitham [62] or Debnath [20]), so that

(363)

and k, !, and a are slowly varying functions of space variable x and time t. Because of the slow variations of k and !, it is reasonable to assume that these quantities still satisfy the linear Dispersion relation of the form ! D W(k) :

(364)

Differentiating (363) with respect to x and t, respectively, we obtain ˚ x D k C x  t W 0 (k) k x ;

(365)

˚ t D W(k) C x  t W 0 (k) k t :

(366)

In the neighborhood of stationary points defined by W 0 (k) D (x/t) > 0, these equations become x D k(x; t) and t D !(x; t) :

(367ab)

These results can be used as a definition of local wavenumber and local frequency of a slowly varying nonlinear wavetrain. In view of (367), relation (364) gives a nonlinear partial differential equation for the phase (x; t) in the form   @ @ CW D0: (368) @t @x The solution of this equation determines the geometry of the wave pattern. We eliminate from (367ab) to obtain the equation @k @! C D0: @t @x

(369)

This is known as the Whitham equation for the conservation of waves, where k represents the density of waves and ! is the flux of waves. Using the dispersion relation (364), Eq. (369) gives @k @k C C(k) D0; @t @x

(370)

Water Waves and the Korteweg–de Vries Equation

where C(k) D W 0 (k) is the group velocity. This represents the simplest nonlinear wave (hyperbolic) equation for the propagation of wavenumber k with group velocity C(k). Equations (370) and (364) reveal that ! also satisfies the first-order, nonlinear wave (hyperbolic) equation @! @! C W 0 (k) D0: @t @x

(371)

It follows from Eqs. (370) and (371) that both k and ! remain constant on the characteristic curves defined by dx D W 0 (k) D C(k) dt

(372)

in the (x; t)-plane. Since k or ! is constant on each characteristic, the characteristics are straight lines with slope C(k). Finally, it follows from the preceding analysis that any constant value of the phase propagates according to (x; t) = constant, and hence,   dx (373) x D 0 ; t C dt which gives, by (367ab), dx ! t D D D c(k) : dt x k

@ @  0 jAj2 C W (k)jAj2 D 0 : @t @x

(379)

This represents the equation for the conservation of wave energy where jAj2 and jAj2 W 0 (k) are the energy density and energy flux, respectively. It also follows that the energy propagates with group velocity W 0 (k). It has been shown that the wavenumber k also propagates with the group velocity. Thus, the group velocity plays a double role. The preceding analysis reveals another important fact that (364), (369), and (379) constitute a closed set of equations for the three functions k, !, and A. Indeed, these are the fundamental equations for nonlinear dispersive waves and are known as Whitham’s equations. In his pioneering work on nonlinear dispersive waves, Whitham [62] formulated a more general energy equation based on the amount of energy Q(t) between two points x1 and x2 Z x2 Q(t) D g(k)A2 dx ; (380) x1

(374)

Thus, the phase of the waves propagates with the phase speed c(k). On the other hand, Eq. (370) ensures that the wavenumber k propagates with group velocity C (k) D ( d!/dk) D W 0 (k). We next follow Whitham [62] or Debnath [20] to investigate how wave energy propagates in a dispersive medium. We consider the following integral involving the square of the wave amplitude (energy) between any two points x D x1 and x D x2 (0 < x1 < x2 ): Z x2 Z x2 jAj2 dx D AA dx (375) Q(t) D x1 x1 Z x2 F(k)F  (k) D 2 dx ; (376) tjW 00 (k)j x1 which, due to a change of variable x D tW 0 (k), Z k2 D 2 F(k)F  (k)dk ;

In the limit, as x2  x1 ! 0, this result reduces to the first-order, partial differential equation

where g(k) is an arbitrary proportionality factor associated with the square of the amplitude and energy. In a new coordinate system moving with the group velocity, that is, along the rays x D C(k)t, Eq. (380) reduces to the form Z x2 g(k)F(k)F  (k)dk ; (381) Q(t) D 2 x1

where ! D ˝(k) and ˝ 00 (k) > 0 and k1 and k2 are defined by x1 D C(k1 )t and x2 D C(k2 )t, respectively. Using the principle of conservation of energy, that is, stating that the energy between the points x1 and x2 traveling with the group velocities C(k1 ) and C(k2 ) remains invariant, it turns out from (380) that dQ D dt

(377)

k1

where x r D t W 0 (kr ), r D 1; 2. When kr is kept fixed as t varies, Q(t) remains constant so that Z dQ d x2 0D jAj2 dx D dt dt x 1 Z x2 @ D jAj2 dx C jAj22 W 0 (k2 )  jAj21 W 0 (k1 ) : (378) x 1 @t

Z

x2 x1

@ ˚ g(k)A2 dx C g(k2 )C(k2 )A2 (x2 ; t) @t  g(k1 )C(k1 )A2 (x1 ; t) D 0 ;

(382)

which is, in the limit as x2  x1 ! 0, @ ˚ @ ˚ g(k)A2 C g(k)C(k)A2 D 0 : @t @x

(383)

This may be treated as the more general energy equation. Quantities g(k)A2 and g(k)C(k)A2 represent the energy density and the energy flux, so that they are proportional

1803

1804

Water Waves and the Korteweg–de Vries Equation

to jAj2 and C(k)jAj2 , respectively. The flux of energy propagates with the group velocity C(k). Hence, the group velocity has a double role: it is the propagation velocity for the wavenumber k and for the energy g(k)jAj2 . Thus, (369) and (383) are known as Whitham’s conservation equations for nonlinear dispersive waves. The former represents the conservation of wave-number k and the latter is the conservation of energy (or more generally, the conservation of wave action). Whitham also derived the conservation equations more rigorously from a general and extremely powerful approach that is now known as Whitham’s averaged variational principle. For slowly varying dispersive wavetrains, the solution maintains the elementary form u D ˚ ( ; a), but !, , and a are no longer constants, so that is not a linear function of xi and t. The local wavenumber and local frequency are defined by ki D

@ ; @x i

!D

@ : @t

(384ab)

The Whitham averaged Lagrangian over the phase of the integral of the Lagrangian L is defined by Z 2 1 L(!; ; a; x; t) D L d (385) 2 0 and is calculated by assuming the uniform periodic solution u D ˚( ; a) in L. Whitham first formulated the averaged variational principle in the form “ ı L dx dt D 0 ; (386) to derive the equations for !, , and a. It is noted that the dependence of L on x and t reflects possible nonuniformity of the medium supporting the wave motion. In a uniform medium, L is independent of x and t, so that the Whitham function L D L(!; ; a). However, in a uniform medium, some additional variables also appear only through their derivatives. They represent potentials whose derivatives are important physical quantities. The Euler equations resulting from the independent variations of ıa and ı in (386) with L D L(!; ; a) are ıa : L a (!; ; a) D 0 ; ı :

@ @ L!  Lk D 0 : @t @x i i

(387) (388)

The -eliminant of (384ab) gives the consistency equations @k i @! D0; C @t @x i

@k j @k i  D0: @x j @x i

(389ab)

Thus, (387)–(389) represent the Whitham equations for describing the slowly varying wavetrain in a nonuniform medium and constitute a closed set from which the triad !, , and a can be determined. In linear problems, the Lagrangian L, in general, is a quadratic in u and its derivatives. Hence, if ˚ ( ) D a cos is substituted in (385), L must always take the form L(!; ; a) D D(!; )a 2 ;

(390)

so that the dispersion relation (L a D 0) must take the form D(!; ) D 0 :

(391)

We note that the stationary value of L is, in fact, zero for linear problems. In the simple case, L equals the difference between kinetic and potential energy. This proves the well-known principle of equipartition of energy, stating that average potential and kinetic energies are equal. Whitham’s Instability Analysis of Water Waves Section “The Nonlinear Schrödinger Equation and Solitary Waves” deals with Whitham’s new remarkable variational approach to the theory of slowly varying, nonlinear, dispersive waves. Based upon Whitham’s ideas and, especially, Whitham’s fundamental dispersion Eq. (388), Lighthill [44,45] developed an elegant and remarkably simple result determining whether very gradual – not necessarily small – variations in the properties of a wavetrain are governed by hyperbolic or elliptic partial differential equations. A general account of Lighthill’s work with special reference to the instability of wavetrains on deep water was described by Debnath [19]. This section is devoted to the Whitham instability theory with applications to water waves. According to Whitham’s nonlinear dispersive wave theory L a D 0 gives a dispersion relation that depends on wave amplitude a has the form ! D !(k; a) ;

(392)

where equations for k and a are no longer uncoupled and constitute a system of partial differential equations. The first important question is whether these equations are hyperbolic or elliptic. This can be answered by a standard and simple method combined with Whitham’s conservation Eqs. (389a) and (383). For moderately small amplitudes, we use the Stokes expansion of ! in terms of k and a2 in the form !(k) D !0 (k) C !2 (k)a 2 C : : : :

(393)

Water Waves and the Korteweg–de Vries Equation

We substitute this result in (389a) and (383), replace ! 0 (k) by its linear value !00 (k), and retain the terms of order a2 to obtain the equations for k and a2 in the form @k @k  0 @a 2 C ! (k) C !20 (k)a 2 C !2 (k) D 0 ; (394) @t @x @x @a2 @k @a 2 C !00 (k) C !000 (k)a 2 D0: (395) @t @x @x Neglecting the term O(a 2 ), these equations can be rewritten as @k @a 2 @k C !00 C !2 D0; @t @x @x @a2 @k @a 2 C !00 (k) C !000 a2 D0: @t @x @x

(396) (397)

These describe the modulations of a linear dispersive wavetrain and represent a coupled system due to the nonlinear dispersion relation (393) exhibiting the dependence of ! on both k and a. In matrix form, these equations read !00

!2

!000 a2

!00

!

@k @x @a 2 @x

! C

1

0

0

1

!

@k @t @a 2 @t

! D 0: (398)

Hence, the eigenvalues  are the roots of the determinant equation ˇ ˇ 0 ˇ !  !2 ˇˇ 0 ˇ ja i j  b i j j D ˇ D0; (399) !000 a2 !00   ˇ where aij and bij are the coefficient matrices of (398). This determinant equation gives the characteristic velocities   dx D C(k) D dt (400ab)  1/2 00 2 D C0 (k)Ca !2 (k)!0 (k) C O(a ) ; where C0 (k) D !00 (k) is the linear group velocity, and, in general, !000 (k) ¤ 0 for dispersive waves. The equations are hyperbolic or elliptic depending on whether !2 (k) !000 (k) > 0 or < 0. In the hyperbolic case, the characteristics are real, and the double characteristic velocity splits into two separate velocities and provides a generalization of the group velocity of nonlinear dispersive waves. In fact, the characteristic velocities (400ab) are used to define the nonlinear group velocities. The splitting of the double characteristic velocity into two different velocities is one of the most significant results of the Whitham theory. This means that any initial disturbance of finite extent will eventually split into two

separate disturbances. This prediction is radically different from that of the linearized theory, where an initial disturbance may suffer from a distortion due to dependence of the linear group velocity C0 (k) D !00 (k) on the wavenumber k, but would never split into two. Another significant consequence of nonlinearity in the hyperbolic case is that compressive modulations will suffer from gradual distortion and steepening in the typical hyperbolic manner discussed earlier. This leads to the multiple-valued solutions and hence, eventually, breaking of waves. In the elliptic case (!2 ; !000 < 0), the characteristics are imaginary. This leads to ill-posed problems in the theory of nonlinear wave propagation. Any small sinusoidal disturbances ina and k may be represented by solutions of the form exp ia fx  C(k)tg , where C(k) is given by (400ab) for the unperturbed values of a and k. Thus, when C(k) is complex, these disturbances will grow exponentially in time, and hence, the periodic wavetrains become definitely unstable. An application of this analysis to the Stokes waves on deep water reveals that the associated dispersion equation is elliptic in this case. For waves on deep water, the dispersion relation is   p 1 (401) ! D gk 1 C a2 k 2 C O(a 4 ) : 2 This result is compared with the Stokespexpansion (393) to p give !0 (k) D gk and !2 (k) D 12 k 2 gk. Hence, !000 !2 < 0, the velocities (400ab) are complex, and the Stokes waves on deep water are definitely unstable. The instability of deep water waves came as a real surprise to researchers in the field in view of the very long and controversial history of the subject. The question of instability went unrecognized for a long period of time, even though special efforts have been made to prove the existence of a permanent shape for periodic water waves for all amplitudes less than the critical value at which the waves assume a sharpcrested form. However, Lighthill’s [45] theoretical investigation into the elliptic case and Benjamin and Feir’s [6] theoretical and experimental findings have provided conclusive evidence of the instability of Stokes waves on deep water. For more details of nonlinear dispersive wave phenomena, the reader is also referred to Debnath [19]. In his pioneering work on nonlinear water waves, Whitham [62] observed that the neglect of dispersion in the nonlinear shallow water equations leads to the development of multivalued solutions with a vertical slope, and hence, eventually breaking occurs. It seems clear that the third derivative dispersive term in the KdV equation produces the periodic and solitary waves which are not found

1805

1806

Water Waves and the Korteweg–de Vries Equation

in the shallow water theory. However, the KdV equation cannot describe the observed symmetrical peaking of the crests with a finite angle. On the other hand, the Stokes waves include full effects of dispersion, but they are limited to small amplitude, and describe neither the solitary waves nor the peaking phenomenon. Although both breaking and peaking are without doubt involved in the governing equations of the exact potential theory, Whitham [62] developed a mathematical equation that can include all these phenomena. It has been shown earlier that the breaking of shallow water waves is governed by the nonlinear equation  t C c0 x C d x D 0 ;

d D 3c0 (2h0 )1 :

(402)

On the other hand, the linear equation corresponding to a general linear dispersion relation ! D c(k) k

(403)

is given by the integrodifferential equation in the form Z 1 t C K(x  s)s (s; t)ds D 0 ; (404) 1

where the kernel K is given by the inverse Fourier transform of c(k): Z 1 1 e i kx c(k)dk : (405) K(x) D F 1 fc(k)g D 2 1 Whitham combined the above ideas to formulate a new class of nonlinear nonlocal equations Z 1  t C d x C K(x  s)s (s; t)ds D 0 : (406) 1

This is well known as the Whitham equation, which can, indeed, describe symmetric waves that propagate without change of shape and peak at a critical height, as well as asymmetric waves that invariably break. Once a wave breaks, it usually continues to travel in the form of a bore as observed in tidal waves. The weak bores have a smooth but oscillatory structure, whereas the strong bores have a structure similar to turbulence with no coherent oscillations. Since the region where waves break is a zone of high energy dissipation, it is natural to include a second derivative dissipative term in the KdV equation to obtain  t C c0 x C d x C  x x x   x x D 0 ; 1 2 6 c0 h0 .

(407)

This is known as the KdV–Burgers where  D equation, which also arises in nonlinear acoustics for fluids with gas bubbles (see [38]).

Whitham’s [62] equations for the slow modulation of the wave amplitude a and the wavenumber k in the case of two-dimensional deep water waves are given by @  a2  @ a2 C )D0; (408) (C ! 0 @t @x !0 @k @! C D0; (409) @t @x p where !0 D gk is the first-order approximation for the wave frequency !(k) and C D (g/2!0 ) is the group velocity. Chu and Mei [14,15] observed that certain terms of the dispersive type, neglected in Whitham’s equations to the same order of approximation, must be included to extend the validity of these equations. Whitham’s theory is based on the direct use of Stokes’ dispersion relation for a uniform wavetrain,   ! D !0 1 C 12 "2 a2 k 2 ;

(410)

whereas Chu and Mei added terms of higher derivatives and dispersive type to the expression for !, so that

! D !0 1 C "

2



1 2 2 a k C 2



a !0



   2!0 a

:

tt

(411) They used the expression (411) to transform (408) and (409) in a frame of reference moving with the group velocity C(k) and obtained the following nondimensional equations: @a2 @ 2 C (a  x ) D 0 ; @t @x

 a2 @ ax x @2  2 C  x C C D0; 2 @x@t @x 4 16a

(412) (413)

where we have used Chu and Mei’s result W D 2 x and  is a small phase variation. Integrating (413) with respect to x and setting the constant of integration to be zero gives 1 1 ax x  t C  x2  a2  D0: 2 8 32a

(414)

A transformation  D a exp(4i) is used to simplify (412) and (414), which reduces to the nonlinear Schrödinger equation i  t C 18 x x C 12  j j2 D 0 :

(415)

This equation has also been derived and exploited by several authors, including Benney and Roskes [7], Hasimoto

Water Waves and the Korteweg–de Vries Equation

and Ono [27], and Davey and Stewartson [17] to examine the nonlinear evolution of Stokes’ waves on water. An alternative way to study nonlinear evolution of two- and three-dimensional wavepackets is to use the method of multiple scales in which the small parameter " is explicitly built into the expansion procedure. The small parameter " characterizes the wave steepness. This method has been employed by several authors in various fields and has also been used by Hasimoto and Ono [27] and Davey and Stewartson [17]. The Benjamin–Feir Instability of the Stokes Water Waves One of the simplest solutions of the nonlinear Schrödinger Eq. (354) is given by (360), that is,   A(t) D A0 exp  12 i!0 k02 A20 t ; (416) where A0 is a constant. This essentially represents the fundamental component of the Stokes wave. We consider a perturbation of (416) and express it in the form  a(x; t) D A(t) 1 C B(x; t) ; (417) where B(x; t) is the perturbation function. Substituting this result in (354) gives  i(1 C B)A t C iAB t 

!0 8k02

 AB x x

 1 D !0 k02 A20 (1 C B) C BB (1 C B) 2 C(B C B )B C (B C B ) A ;



i˝  C

! 0 `2 8k02



1 B2  !0 k02 A20 (B1 C B2 ) D 0 : (422) 2

We take the complex conjugate of (422) to transform it into the form   1 ! 0 `2 i˝ C B2  !0 k02 A20 (B1 C B2 ) D 0 : (423) 2 8k02 The pair of linear homogeneous Eqs. (421) and (423) for B1 and B2 admits a nontrivial eigenvalue for ˝ provided ˇ ˇ i˝ C !0 `2  1 ! k 2 A2  12 !0 k02 A20 ˇ 2 0 0 0 8k 02 ˇ 2 ˇ  12 !0 k02 A20 i˝ C !8k0 `2  12 !0 k02 A20 ˇ 0

(424) which is equivalent to 1 ˝ D 2 2



!0 ` 2k0

2 

k02 A20

`2  2 8k0

 :

B (x; t)

(425)

The growth rate ˝ is purely imaginary or real (and positive) depending on whether `2 > 8k04 A20 or `2 < 8k04 A20 . The former case represents a wave solution for B, and the latter corresponds to the Benjamin–Feir (or modulational) instability with a criterion in terms of the nondimensional wavenumber `˜ D (`/k0 ) as `˜2 < 8k02 A20 :

(418)

ˇ ˇ ˇ ˇ D 0; ˇ ˇ

Thus, the range of instability is given by p 0 < `˜ < `˜c D 2 2 k0 A0 :

(426)

(427)

where is the complex conjugate of the perturbed function B(x; t). Neglecting squares of B, Eq. (418) reduces to   !0 1 iB t  (419) B x x D !0 k02 A20 (B C B ) : 2 2 8k0

˜ the maximum instability occurs Since ˝ is a function of `, ˜ ˜ at ` D `max D 2k0 A0 , with a maximum growth rate given by 1 (Re ˝)max D !0 k02 A20 : (428) 2

We look for a solution for perturbed quantity B(x; t) in the form

To establish the connection with the Benjamin–Feir instability [6], we have to find the velocity potential for the fundamental wave mode multiplied by exp(kz). It turns out that the term proportional to B1 is the upper sideband, whereas that proportional to B2 is the lower sideband. The main conclusion of this analysis is that Stokes water waves are definitely unstable.

B(x; t) D B1 exp(˝ t C i`x) C B2 exp(˝  t  i`x) ;

(420)

where B1 and B2 are complex constants, ` is a real wavenumber, and ˝ is a growth rate (possibly a complex quantity) to be determined. Substituting the solution for B in (419) yields a pair of coupled equations:   ! 0 `2 1 i˝ C B1  !0 k02 A20 (B1 C B2 ) D 0 ; (421) 2 2 8k0

Future Directions Although this chapter has been devoted to water waves and the Korteweg and de Vries equations, there are some challenging problems dealing with solitary waves

1807

1808

Water Waves and the Korteweg–de Vries Equation

envelopes in the wake of a ship in oceans, and the KdV equation with variable coefficients in an ocean of variable depth. Despite some recent progress, there is no complete theory for ships in waves that takes into account of the effects of the finite ship volume, and possible interactions between Kelvin wake and soliton envelopes. In order to respond to experimental results with a rough bottom, theories based on an empirical formula for the bottom stress have been developed, but they are not yet completely satisfactory when compared to experiments. Of the systems that conserve energy, some are completely integrable in the sense of solitary wave theory, but many including those most important for applications are not. In some cases, useful analytical techniques are currently available, and others are waiting to be discovered. So there is a need for research in the area of nonlinear lattices. When we deal with real world problems, the conservation of energy does not hold because of dissipative effects and forcing terms. Consequently, the KdV equation and other relevant evolution equations need modification by including terms that describe such effects. So, there are challenging problems dealing with these modified evolution equations, their methods of solutions and the qualitative and quantitative understanding of the solutions. On the other hand, prospects for research in the transverse coupling between solitary waves and parallel systems are indeed bright. Bibliography Primary Literature 1. Airy GB (1845) Tides and Waves. In: Encyclopedia Metropolitana, Article 192 2. Akylas TR (1984) On the excitation of long nonlinear water waves by a moving pressure distribution. J Fluid Mech 141:455–466 3. Akylas TR (1984) On the excitation of nonlinear water waves by a moving pressure distribution oscillatory at resonant frequency. Phys Fluids 27:2803–2807 4. Amick CJ, Fraenkel LE, Toland JF (1982) On the Stokes Conjecture for the wave of extreme form. Acta Math Stockh 148:193– 214 5. Benjamin TB (1967) Instability of periodic wave trains in nonlinear dispersive systems. Proc Roy Soc London A 299:59–75 6. Benjamin TB, Feir JE (1967) The disintegration of wavetrains on deep water. Part 1. Theory. J Fluid Mech 27:417–430 7. Benney DJ, Roskes G (1969) Wave instabilities. Stud Appl Math 48:377–385 8. Berezin YA, Karpman VI (1966) Nonlinear evolution of disturbances in plasmas and other dispersive media. Soviet Phys JETP 24:1049–1056 9. Boussinesq MJ (1871) Théorie de l’intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire. Comptes Rendus 72:755–759

10. Boussinesq MJ (1871) Théorie generale des mouvenments qui sont propagés dans un canal rectangulaire horizontal. Comptes Rendus 72:755–759 11. Boussinesq MJ (1872) Théorie des ondes et des rumous qui se propagent le long d’un canal rectagulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J Math Pures Appl ser 2/17:55–108 12. Boussinesq MJ (1877) Essai sur la théorie des eaux courants. Mémoires Acad Sci Inst Fr ser 2 23:1–630 13. Cercignani C (1977) Solitons. Theory and application. Rev Nuovo Cimento 7:429–469 14. Chu VH, Mei CC (1970) On slowly-varying Stokes waves. J Fluid Mech 41:873–887 15. Chu VH, Mei CC (1971) The nonlinear evolution of Stokes waves in deep water. J Fluid Mech 47:337–351 16. Crapper GD (1957) An exact solution for progressive capillary waves of arbitrary amplitude. J Fluid Mech 2:532–540 17. Davey A, Stewartson K (1974) On three-dimensional packets of surface waves. Proc Roy Soc London A 338:101–110 18. Debnath L, Rosenblat S (1969) The ultimate approach to the steady state in the generation of waves on a running stream. Quart J Mech Appl Math XXII:221–233 19. Debnath L (1994) Nonlinear Water Waves. Academic Press, Boston 20. Debnath L (2005) Nonlinear Partial Differential Equations for Scientists and Engineers (Second Edition). Birkhauser, Boston 21. Debnath L, Bhatta D (2007) Integral Transforms and Their Applications. Second Edition. Chapman Hall/CRC Press, Boca Raton 22. Dutta M, Debnath L (1965) Elements of the Theory of Elliptic and Associated Functions with Applications. World Press Pub. Ltd., Calcutta 23. Fermi E, Pasta J, Ulam S (1955) Studies of nonlinear problems I. Los Alamos Report LA 1940. In: Newell AC (ed) Lectures in Applied Mathematics. American Mathematical Society, Providence 15:143–156 24. Gardner CS, Greene JM, Kruskal MD, Miura RM (1967) Method for solving the KdV equation. Phys Rev Lett 19:1095–1097 25. Gardner CS, Greene JM, Kruskal MD, Miura RM (1974) Korteweg–de Vries equation and generalizations, VI, Methods for exact solution. Comm Pure Appl Math 27:97–133 26. Hammack JL, Segur H (1974) The Korteweg–de Vries equation and water waves. Part 2. Comparison with experiments. J Fluid Mech 65:289–314 27. Hasimoto H, Ono H (1972) Nonlinear modulation of gravity waves. J Phys Soc Jpn 35:805–811 28. Helal MA, Molines JM (1981) Nonlinear internal waves in shallow water. A Theoretical and Experimental study. Tellus 33:488–504 29. Hirota R (1971) Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys Rev Lett 27:1192– 1194 30. Hirota R (1973) Exact envelope-soliton solutions of a nonlinear wave equation. J Math Phys 14:805–809 31. Hirota R (1973) Exact N-Solutions of the wave equation of long waves in shallow water and in nonlinear lattices. J Math Phys 14:810–814 32. Huang DB, Sibul OJ, Webster WC, Wehausen JV, Wu DM, Wu

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TY (1982) Ship moving in a transcritical range. Proc. Conf. on Behavior of Ships in Restricted Waters (Varna, Bulgaria) 2:26-1– 26-10 Infeld E (1980) On three-dimensional generalizations of the Boussinesq and Korteweg–de Vries equations. Quart Appl Math XXXVIII:277–287 Johnson RS (1980) Water waves and Korteweg–de Vries equations. J Fluid Mech 97:701–719 Johnson RS (1997) A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge University Press, Cambridge Kadomtsev BB, Petviashvili VI (1970) On the stability of solitary waves in weakly dispersive media. Sov Phys Dokl 15:539–541 Kaplan P (1957) The waves generated by the forward motion of oscillatory pressure distribution. In: Proc. 5th Midwest Conf. Fluid Mech, pp 316–328 Karpman VI (1975) Nonlinear Waves in Dispersive Media. Pergamon Press, London Korteweg DJ and de Vries G (1895) On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil Mag (5) 39:422–443 Kraskovskii YP (1960) On the theory of steady waves of not all small amplitude (in Russian). Dokl Akad Nauk SSSR 130:1237– 1252 Kraskovskii YP (1961) On the theory of permanent waves of finite amplitude (in Russian). Zh Vychisl Mat Fiz 1:836–855 Lax PD (1968) Integrals of nonlinear equations of evolution and solitary waves. Comm. Pure Appl Math 21:467–490 Levi–Civita T (1925) Détermination rigoureuse des ondes permanentes d’ampleur finie. Math Ann 93:264–314 Lighthill MJ (1965) Group velocity. J Inst Math Appl 1:1–28 Lighthill MJ (1967) Some special cases treated by the Whitham theory. Proc Roy Soc London A 299:38–53 Lighthill MJ (1978) Waves in Fluids. Cambridge University Press, Cambridge Luke JC (1967) A variational principle for a fluid with a free surface. J Fluid Mech 27:395–397 Michell JH (1893) On the highest waves in water. Phil Mag 36:430–435 Miche R (1944) Movements ondulatories de la mer en profondeur constants ou décreoissante. Forme limite de la houle lors de son déferlement. Application aux dignes maritimes. Ann Ponts Chaussées 114:25–78, 131–164, 270–292, 369– 406 Palais RS (1997) The symmetries of solitons. Bull Amer Math Soc 34:339–403 Rayleigh L (1876) On waves. Phil Mag 1:257–279 Riabouchinsky D (1932) Sur l’analogie hydraulique des mouvements d’un fluide compressible, Institut de France, Académie des Sciences. Comptes Rendus 195:998 Stoker JJ (1957) Water Waves. Interscience, New York Stokes G (1847) On the theory of oscillatory waves. Trans Camb Phil Soc 8:197–229 Struik DJ (1926) Détermination rigoureuse des ondes irrotationnelles périodiques dans un canal à profondeur finie. Math Ann 95:595–634

56. Toland JF (1978) On the existence of a wave of greatest height and Stokes’ conjecture. Proc. Roy. Soc. Lond. A 363:469–485 57. Weidman PD, Maxworthy T (1978) Experiments on strong interactions between solitary waves. J Fluid Mech 85:417–431 58. Whitham GB (1965) A general approach to linear and nonlinear dispersive waves using a Lagrangian. J Fluid Mech 22:273–283 59. Whitham GB (1965) Nonlinear dispersive waves, Proc Roy Soc London A 283:238–261 60. Whitham GB (1967) Nonlinear dispersion of water waves. J Fluid. Mech 27:399–412 61. Whitham GB (1967) Variational methods and application to water waves. Proc Roy Soc London A 299:6–25 62. Whitham GB (1974) Linear and Nonlinear Waves. John Wiley, New York 63. Whitham GB (1984) Comments on periodic waves and solitons. IMA J Appl Math 32:353–366 64. Wu DM, Wu TY (1982) Three-dimensional nonlinear long waves due to a moving pressure. In: Proc 14th Symp. Naval Hydrodyn. National Academy of Sciences, Washington DC, pp 103–129 65. Zabusky NJ, Kruskal MD (1965) Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys Rev Lett 15:240–243 66. Zabusky NJ (1967) A synergetic approach to problems of nonlinear dispersive wave propagation and interaction. In: Ames WF (ed) Proc. Symp. on Nonlinear Partial Differential Equations. Academic Press, Boston. pp 223–258 67. Zabusky NJ, Galvin CJ (1971) Shallow water waves, the Korteweg–de Vries equation and solitons. J Fluid Mech 48:811– 824

Books and Reviews Ablowitz MJ, Clarkson PA (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge Bhatnagar PL (1979) Nonlinear Waves in One-Dimensional Dispersive Systems. Oxford University Press, Oxford Crapper GD (1984) Introduction to Water Waves. Ellis Horwood Limited, Chichester Debnath L (1983) Nonlinear Waves. Cambridge University Press, Cambridge Dodd RK, Eilbeck JC, Gibbons JD, Morris HC (1982) Solitons and Nonlinear Wave Equations. Academic Press, London Drazin PG, Johnson RS (1989) Solitons: An Introduction. Cambridge University Press, Cambridge Infeld E, Rowlands G (1990) Nonlinear Waves, Solitons and Chaos. Cambridge University Press, Cambridge Lamb H (1932) Hydrodynamics, 6th edition. Cambridge University Press, Cambridge Leibovich S, Seebass AR (1972) Nonlinear Waves. Cornell University Press, Ithaca Myint-U T, Debnath L (2007) Linear Partial Different Equations for Scientists and Engineers (fourth edition). Birkhauser, Boston Russell JS (1844) Report on waves. In: Murray J (ed) Report on the 14th Meeting of the British Association for the Advancement of Science. London 311–390

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List of Glossary Terms

A A Hamiltonian system with symmetry 1212 A periodic orbit 1212 Absolute error 908 Absolutely continuous curves 953 Accuracy 924 Action-angle variables 810 Actions 1337 Adiabatic decoupling 1317 Adiabatic perturbation theory 1317 Adomian decomposition method 908 Adomian decomposition method, Homotopy analysis method, Homotopy perturbation method and Variational perturbation method 890 Adomian method 1 Adomian polynomials 908 Agmon metric 1376 Agree-and-pursue algorithm 1489 AKNS method 771 Alexander–Orbach conjecture 13 Algorithmic complexity 924 Allometric laws 488 Almost conjugacy 1689 Almost every, essentially 357 Almost everywhere 783 Almost everywhere (abbreviated a. e.) 313 Almost everywhere (a.e.) 205 Analytical method for solving an equation 837 Anomalous diffusion 13, 591 Approximate controllability 755 ARMA 924 Asymptotic series 1376 AT property of an automorphism 1618 Attractor 538, 740, 1726 Aubry set 684 Automorphism 241, 1689 Autoparametric resonance 183

Averaging algorithms 1489 Axisymmetric (concentric) KdV equation 1771 B Banach fixed point theorem 1236 Baroclinic fluid 1102 Barotropic fluid 1102 Basic notation 1009 Basin of attraction 538 Benjamin–Feir instability of water waves 1771 Bernoulli shift 225, 783 Bernoulli’s equation 1771 Bifurcation 48, 183, 657, 937, 1172, 1325 Biological swimmers 26 Blow up 160 Boltzmann equation 1505 Bore solitons 1603 Boussinesq equation 1771 Box 637 Box diameter 637 Box dimension 537 Branch switching 1172 Breaking waves 1561 Brouwer degree 1236 Brouwer fixed point theorem 1236 Brownian motion 591, 1126 C Canard 1475 Cantor set, Cantor dust, Cantor family, Cantor stratification 657 Catastrophe 32 Cauchy problem 160 ˇ Cech–Stone compactification of N, ˇN 313 Central configurations 1043 Chaos 657 Choreographical solution 1043

1812

List of Glossary Terms

Chromatin 607 Chronological algebra 88 Chronological calculus 88 Ck Conjugacy 369 Ck topology 740 Classical limit 1438 Classically allowed and forbidden regions – turning points 1376 Cluster 1400 (cn )-Conservative system 357 Cnoidal waves 1771 Coadjoint orbit 981 Cocycles and group extensions 1619 Codimension 32, 1172 Coexistence 1251 Collision and singularities 1043 Comb model 13 Compact visitation 13 Compacton 1553 Compacton-like solution 1553 Complex system 463 Complexity measures 1489 Compositional strand asymmetry 606 Conditional measure 783 Conductance exponent (t) 446 Conductance (G) 446 Confidence 924 Conjugacy 369 Conservation law 729 Conservative, dissipative 1533 Conservative system 357 Conservativeness 329 Continuation 1172, 1212 Continuity equation 1771 Continuous and discrete dynamical systems 48 Continuous stochastic process 1126 Continuum hypothesis 1771 Control function 102 Control system 395 Control systems and reachable sets 953 Controllability 88, 395 Controllability to trajectories 755 Cooperative control 1489 Correlation dimension 538 Correlation function 488 Cotangent bundle 981 Coupling of two measure spaces 783 Covariance 571 Crapper’s nonlinear capillary waves 1771 Critical equations 683 Critical exponent 1400

Critical part of the spectrum 48 Critical point 488 Crossover 463 Crystal lattice 1561 Curvature 313 Cylindrical Wiener process 1126 D Deep water 1520, 1561 Defining system 1172 Degree of a node 637 Dielectric breakdown model 429 Diffeomorphism 32 Differentiable rigidity 369 Diffusion 1400 Diffusion-limited aggregation 429 Dimension 537 Dimension group 1689 Diophantine approximation 302, 313 Diophantine condition, Diophantine frequency vector 657 Direct scattering problem 771 Discrete variational mechanics 143 Disjoint measure-preserving systems 796 Dispersion relation 1771 Dispersive estimate 160 Distance 637 Distortion estimate 1533 Distributed algorithm 1489 Distributed parameter system 102 Doubling map 357 Dynamic programming 1137 Dynamic programming principle 1115 Dynamical system 126, 241, 288, 964, 1082, 1172, 1422, 1475, 1653 E Effective medium theory (EMT) 446 Eigenvalue and spectrum 48 Elastic interaction 1576 Electronic structure problem 1317 Ellipticity/parabolicity 1115 Embedding 1689 Energy-momentum map 1438 Entropy 729, 1422 Entropy, measure-theoretic (or: metric entropy) Entropy, topological 63 " 837 Equidistributed 313 Equilibrium 1172, 1726

63

List of Glossary Terms

Equilibrium state 1422 Equivalence of dynamical systems 48 Equivariance 981 Equivariant system 937 Ergodic 313, 964 Ergodic average 241 Ergodic decomposition 357 Ergodic measure preserving transformation 783 Ergodic system 357 Ergodic theorem 241, 357, 783 Ergodic theory 302, 313, 1422 Ergodicity 63, 329 Ergodicity, weak mixing, mild mixing, mixing and rigidity of T 1618 Eukaryote 606 Euler equations 1771 Evolution equations 160, 1126 Evolutionarily stable equilibria (ESS) 1265 Exact controllability 755 Exact solution 890 Exosystem 1711 Exp-function method 1548 Exponentially stable Hamiltonian 1070

Free action 981 Full shift 1689 Function spaces notation 1009 Functions: bounded, continuous, uniformly Lipschitz continuous 383 Further normalization 1082 G Generalized gradients and subgradients 1137 Generalized soliton 1553 Generalized tracking problem 1711 Genetic locus 1265 Geodesic 783 Geodesic (flow) 313 Geodesics 571 Geometric integrator 143 Geostrophic adjustment 1561 Geostrophic equilibrium 1561 Gevrey spaces 1276 Gibbs state 1422 Global asymptotic stability 704 Global rigidity 369 Global solution 1009 Group velocity 1771

F Factor map 205, 1689 Fast convergent (Newton) method 810 Finite difference method 908 Finite equivalence 1689 Fitness landscape 1265 Flatness 395 Floquet theory 183 Flow map 383 Flux function 729 Fluxon 1561 Fokker–Planck–Landau equation 1505 Foliation 740 Fonts 205 Foreign exchange market 512 Formal power series 1290 Fractal 175, 429, 446, 512, 532, 559, 606 Fractal dimension 175, 532 Fractal geometry 288 Fractal set 488 Fractal system 463 Fractality 571 Fractals 1400, 1457 Fractional diffusion equation 13 Fractons 591 Freak waves 1561

H Haar measure (on a compact group) 313 Hamiltonian 1212 Hamiltonian dynamics 810 Hamiltonian PDE 1337 Hamiltonian system 126, 810 Hamilton–Jacobi equations 683 Harmonic measure 429 Hausdorff a-measure 357 Hausdorff dimension 302, 537 Heterogeneously catalyzed reactions 1457 Higher dimensional shift space 1689 Hilbert space 1351 Hill’s equation 1251 Hirota bilinear method 884 Histones 607 Homeomorphism, diffeomorphism 740 Homogeneous spaces 302 Homotopy analysis method 908 Homotopy perturbation method 908, 1548 Hopf argument 1533 Horocycle 783 Hybrid closed-loop system 704 Hybrid controller 704 Hydraulic jump 1561

1813

1814

List of Glossary Terms

Hydrodynamic limit 1505 Hydrodynamical equations 1505 Hyperbolic 1533 Hyperbolicity 63

KKS (Kostant-Kirillov-Souriau) form 981 Kolmogorov group property 1618 Kolmogorov typicality 63 Koopman representation of a dynamical system T Korteweg–de Vries equation 884, 890, 908 Korteweg–de Vries (KdV) equation 1771

I Ince–Strutt diagram 183 Induced automorphism 1618 Inelastic interaction 1576 Infinite dimensional control system 755 Infinite measure preserving system 357 Initial conditions 908 Initial value problem 383 Input-output map 1195 Instability pockets 1251 Instability tongues 1251 Integrability 771 Integrable Hamiltonian systems 810 Integrable system 126, 657 Internal model 1711 Internal solitons 1603 Internal wave 1520 Internal waves 1102 Invariance principle 704 Invariant manifold 48 Invariant measure 740, 783, 1422 Invariant set 1726 Invariant tori 810 Invariant torus 126 Inverse scattering problem 771 Inverse scattering transform 771, 884 Invertible system 357 Ising model 1400 Iteration 241

L

J Johnson’s equation 1771 Joining 796 Joining of two measure preserving transformations Joint spectrum 1438 Julia set 538 K Kadomtsev–Petviashvili (KP) equation 1771 KAM 810 KAM theory 657, 1301 Katabatic wind 1561 KdV–Burgers equation 1771 Kinetic scale 1748

1618

783

Lagrange multiplier 908 Laplace equation 1771 Lattice 369 Lattice; unimodular lattice 302 Lax method 771 LCR algorithm 1489 Leader election 1489 Lebesgue space 964 Leray–Schauder degree 1236 Leray–Schauder–Schaefer fixed point theorem 1236 Liapunov–Schmidt reduction 1212 Lie algebras of vector fields 1389 Lie group 143 Lie group action 981 Lie groups 953 Lightning 1561 Limit cycle 1172 Limit sets 740 Lindstedt series 1290 Linear dispersion relation 1771 Linear dispersive waves 1771 Linear Schrödinger equation 1772 Linear wave equation in Cartesian coordinates 1772 Linear wave equation in cylindrical polar coordinates 1772 Ljusternik–Schnirelmann category 1236 Local properties 48 Local rigidity 369 Local solution 1009 Localization exponent 175 Long-range power law correlations 488 Long-term correlations 463 Low density limit 1505 Lower and upper solutions 1236 Lyapunov and control-Lyapunov functions 1639 Lyapunov exponents 63 Lyapunov function 1137, 1726 Lyapunov stability theory 704

List of Glossary Terms

M

N

Macroscopic scale 1748 Magnetic terms 981 Manifolds 953 Map germ 32 Marginal of a probability measure on a product space 796 Markov chain 225 Markov intertwining 796 Markov operator 1619 Markov partition 1689 Markov shift 783 Markov shift (topological, countable state) 63 Mathematical model 1407, 1748 Mathieu equation 1251 Maximal spectral type and the multiplicity function of U 1618 Maximal inequality 241 Maximum entropy measure 63 Mean ergodic theorem 241 Measure 964 Measure of maximal entropy 1689 Measure preserving system 357 Measure preserving transformation 783 Measure rigidity 369 Measure space 205 Measure theoretic entropy 783 Measure-preserving map 205 Measure-preserving transformation 225 Measure-theoretic entropy 225, 313 Mechanical connection 981 Metric approach 683 Metric number theory 302 Microscopic swimmers 26 Minimal self-joinings 796 Mixing 313 Modal analysis 183 Modified Korteweg–de Vries 890 Molecular dynamics 1317 Momentum cocycle 981 Momentum mapping 981 Monodromy 1438 Monodromy matrix 183 Morphogenesis 488 -Compatible metric 357 Multifractal 429, 446 Multifractal system 463 Multiscale problem 1290

Nash equilibrium 1265 Navier–Stokes equations 26 Navier–Stokes equations 1772 n-Body problem 1043 Nearly integrable system 657 Nearly-integrable Hamiltonian systems 810 Nekhoroshev theorem (1977) 1070 Nilpotent linear transformation 1276 Noise 1126 Nonlinear dispersion relation 1772 Nonlinear dispersive waves 1772 Non-linear Schrödinger (NLS) equation 1772 Nonlinear waves 1102 Nonsingular dynamical system 329 Non-smooth dynamical system 1325 Non-stationarities 463 Normal form 1082, 1172, 1337 Normal form on invariant manifolds 1389 Normal form procedure 1152 Normal form truncation 657 Normalizing transformations and convergence 1389 Notation 205, 357, 383 Notations 160 O Observability 395, 1195 Observer 1195, 1711 Ocean waves 1772 Off-diagonal self-joinings 796 Open/closed loop 395 Operator 241 Orbit 538, 1726 Orbit of x 241 Ordinary differential equation 383, 1653 Oscillator 1475 Output function 395 P PAC 924 Palais–Smale condition for a C 1 function ' : X ! R 1236 Parametric excitation 183 Parametric resonance 183, 1251 Partial differential equation 102 Partition 205 Percolation 446, 559, 1400 Percolation threshold pc 446 Period-doubling bifurcation 937

1815

1816

List of Glossary Terms

Persistence 463 Persistent property 658 Perturbation 1276 Perturbation problem 658 Perturbation theory 1082, 1152, 1301, 1337 Phase shift 1576 Phase velocity 1772 Philosophy of science 1407 Physical measure 63 Pitchfork bifurcation 937 Plasma 1561 Poincaré operator 1236 Poincaré section/map 1212 Poincaré–Dulac normal form 1389 Pointwise and uniform convergence 383 Pointwise ergodic theorem 241 Poisson reduction 981 Poisson systems 1212 Pontryagin Maximum Principle 1137 Pontryagin Maximum Principle of optimal control Positive contraction 241 Positive definite sequence 357 Potential function 32 Power law distribution 512 Preservation of structure 1152 Pressure 1422 Prevalence 63 Principal connection 981 Probability space 205, 783 Probability vector 205 Process in a measure-preserving dynamical systems 796 Product transformation 225 Proper action 981 Pycnocline 1102 Q Quantum bifurcation 1438 Quantum Mechanics 571 Quantum phase transition 1438 Quantum-classical correspondence 1438 (Quasi) Ergodic hypothesis 313 Quasi integrable Hamiltonian 1070 Quasi-integrable system 126 Quasi-periodic 1251 Quasi-periodic motion 126 Quasi-periodic motions 810 R Radial distribution function

488

88

Random resistor network 446 Rational function, rational map 783 Reaction kinetics 1457 Realization 1195 Recurrence 313, 1726 Reduction principle 48 Regular solution 1009 Relative periodic orbit 1212 Relativity 571 Relaxation oscillation 1475 Renormalization group 126, 1290 Repellor 1726 Replication 607 Representations 1351 R-Equivalence 32 Resonance 658 Resonance vs. Non-Resonance 1337 Resonant eigenvalues 1389 Resonant interaction 1576 Resonant or critical phenomenon 1772 Reversible system 937 Reynolds number 26 Riemannian manifolds 953 Road network 1748 Road problem 1689 Rossby solitons 1603 Rotations on T 357 Run-length limited shift 1689 S Saddle-node bifurcation 937 Sample complexity 924 Scale free network 1265 Scale of observation/representation 1748 Scale-free network 637 Scaling law 463 Scaling limits 1505 Scattering data 771 Schauder fixed point theorem 1236 Schrödinger equation 1351 Schrödinger representation 1351 SCS property 1618 Self-affine system 463 Self-joining 796 Self-organized criticality 488 Semiclassical limit 1376 Semiconcave and semiconvex functions 683 Sensitivity on initial conditions 63 Separatrices 658 Shallow and deep water waves 1603

List of Glossary Terms

Shallow water 1102, 1520, 1561 Shallow water wave equations 1520 Shallow water waves 1520 Shape space 981 Shift equivalence 1689 Shift of finite type 1689 Shift space 1689 Shock 729 Short-term correlations 463 Simplicity 796 Sinai–Ruelle–Bowen measure 1422 Sinai–Ruelle–Bowen measures 63 Sinai–Ruelle–Bowen (SRB) measure 1533 Singular perturbations 1475 Singular reduction 981 Singularity 32 Singularity theory 658 Sinusoidal (or exponential) wave 1772 Sliding block code 1689 Small divisors/denominators 810 Small-world network 637, 1265 Smooth flow 740 Sobolev inequality 1236 Sobolev space 1337 Sobolev spaces 160 Sofic shift 1689 Solitary wave 1520, 1561, 1576 Solitary waves 1102 Solitary waves (or soliton) 1772 Soliton 771, 884, 890, 1520, 1548, 1553, 1561, 1576, 1603 Soliton (colloquially as well as in certain domains) 1576 Soliton perturbation theory 1548 Soliton solution 1576 Solitons 1, 837, 908, 1102 Space domain 102 Spacetime 571 Special flow 1619 Spectral decomposition of a unitary representation 1618 Spectral density of states 591 Spectral dimension 13, 591 Spectral disjointness 1618 Spectral (or fracton) dimension 175 Spillway 1561 Spin-orbit problem 1301 Spontaneous symmetry breaking 1438 Stability 810, 1639 Stability theory 1653 Stabilizability 395 Stabilization 1639 State equation 102 State splitting 1690

State variables 102 States and observables 1351 Stationary process 241 Statistical stability 63 Steady state 1711 Steady Stokes equations 26 Steady-State flow 1009 Stochastic differential and difference equations Stochastic dynamical system 1126 Stochastic dynamical systems 1672 Stochastic process 783, 964 Stochastic stability 63 Stokes expansion 1772 Stokes lines 1376 Stokes wave 1772 Strong shift equivalence 1690 Structural stability 63 Structurally stable 658 Subshift of finite type 63 Subshift, shift space 964 Sum resonance 1251 S-Unit theorems 313 Supervisor of hybrid controllers 704 Surface gravity waves 1772 Surface wave 1520 Surface-capillary gravity waves 1772 Swimming 26 Swimming microrobots 26 Symmetry 1082 Symmetry breaking 1439 Symmetry reduction 1082, 1152 Symplectic 143 Symplectic reduction 981 Symplectic structure 810 Syndetic set 357 Synoptic scale 1561 T Tangent and cotangent spaces 953 Target system 1407 Temporal regularization 704 Test function 1172 Thermocline 1561 Three-body problem 1301 Three-dimensional (or 3D) flow 1009 Thunder 1561 Thunderstorm 1561 Tidal bore 1561 Time evolution of the scattering data 771 Time series 463

1672

1817

1818

List of Glossary Terms

Topological conjugacy 1690 Topological entropy 313, 1690 Topological genericity 64 Torus 126 Trajectory 102 Transcription 607 Transitivity 1726 Tree 126 Troposphere 1561 Tsunami 1520, 1561, 1603 Tunnel effect 1376 Two-dimensional (or planar, or 2D) flow Types II, II1 , II1 and III 329 Types of solitons 1 Typical singularity 1325

Vector field 383 Vehicular traffic 1748 Velocity potential 1772 Verification functions 1137 Viscosity solutions 683, 1115 W

1009

U Uehling–Uhlenbeck equation 1505 Unfolding 32 Unfolding morphism 32 Uniform distribution 241 Uniform ellipticity/uniform parabolicity Unitary actions on Fock spaces 1619 Universal inputs 1195 Upper density 357 V Vapnik–Chervonenkis dimension 924 Variational approach 1043 Variational iteration method 908, 1548 Variational principle 1772 Variational principles 1212

1115

Walk dimension 13 Water waves 1772 Wave dispersion 1520 Wavelet transform 606 Wavelet transform modulus maxima method 606 Waves on a running stream 1772 Weak coupling limit 1505 Weak solutions 1115 Weighted operator 1618 (Well-posed) Hybrid dynamical system 704 White noise 1672 Whitham averaged variational principle 1772 Whitham’s conservation equations 1772 Whitham’s equation 1772 Whitham’s equation for slowly varying wavetrain 1772 Whitham’s nonlinear nonlocal equations 1772 Wiener process 1672 Wigner transform 1505 Wirtinger inequality 1236 WKB 1377 Z Zeno (and discrete) solutions Zeta function 1690 Zinbiel algebra 88

704

Index

A A priori unstable systems 833 Abel’s identity 392 Absence of many-particle effects 1584 Absolute continuity 747 Absolutely continuous 245 Absolutely continuous curves 953 Absolutely continuous function 392 Absolutely continuous invariant measure with non-maximum entropy 76 Absorbing state 1466 Abstract input-output map 1199 Accessibility 402 Accuracy 924, 925, 929–931 Acoustic wave 1206 Action 246, 584, 1224, 1618 affine 984 disjoint 1621 ergodic 1618 indefinite 1228 Mackey 1626 mildly mixing 1618 mixing 1618 nil 1623 product 1625 rigid 1618 spectrally disjoint 1618 weakly isomorphic 1633 weakly mixing 1618 Action-angle variable 672, 810, 1071, 1303, 1442, 1664 Action function 1225 definition 1001 Action levels 2–1-choreography symmetry 1056 Lagrange solutions in the 3 body problem 1061 line symmetry 1056 mountain pass solution in the 3 body problem 1061 Action map 1743

Adding machine 266 Adiabatic decoupling 1317 Adiabatic invariant 1077 Adiabatic perturbation theory 1317 Adic transformations 282 Adjoint action 1156 operator 1154 Adjoint bundle 1003 ADM 890, 892, 902, 904 Adomian decomposition method 1, 3, 890, 910, 913, 1602 Adomian polynomials 2, 3 Adomian polynomials 911 Adsorption-limited 1468 Aerogel 599 Affine action, mechanical systems 984 Affine structure 669 Affine systems 402 Aggregation cluster-cluster 416 diffusion-limited 416 kinetic 416 with injection 523 Aggregation algorithms 1499 Agmon metric 1376, 1384 Agree-and-pursue algorithm 1489, 1498 AKNS method 771, 774, 1578 AKNS pair 774 Alexander–Orbach conjecture 22 Algebra 32, 967 of vector fields 1090  -Algebra 967 Algebraic action, invariant measure classifying 789 Algebraic branching equation 1186 Algebraic dynamical systems 322 Algebraic graph-theory 1496 Algebraic Riccati equation 1678

1820

Index

Algorithm wave-front tracking 1765 Algorithmic complexity 76 Allometric law, biological fractals 493 Allometry, biological fractals 489 Almost conjugacy 1700 Almost equicontinuous 1738 Almost global existence 1338 Almost-invariant subspace 1321 Almost-periodic 1252, 1257 Almost sure invariance principle 75 Alternative method 940 Ambrosetti–Prodi problem 1244 Amenable 252 Amenable group Følner sequence 793 Amplitude 1477 Amplitude dispersion 1782 Amplitude parameter 1788 Analyzing wavelets 610, 612 Anharmonic forces 180 Anharmonic oscillator 1367 generalized 1367 quartic 1367, 1368 Annihilation reaction 1464 Anomalous diffusion 600 Anosov action 370 Anosov diffeomorphism 371, 1433 SRB measure 1434 Anosov flow 319, 371 Ant in the labyrinth 566 Anti-Stokes line 1380 Anti-symmetry, icosahedral group 1053 Anticipation 1753 Antisoliton 1590 Approximate controllability 760 Approximation Diophantine 302, 304, 306 metric Diophantine 302 with dependent quantities 306 Approximatively controllable 399 Arbitrary exponential rate observer 1198 Aristotelian physics 1290 ARMA (auto-regressive moving average) 924, 926, 934 Arnold 33, 820 work in reduction theory 991 Arnold diffusion 661, 827, 1070, 1665 Arnold tongues 946 Arnold’s scheme 820 Arnold’s transformation 821 Arzela–Ascoli’s lemma 386

Associated flow ergodic theory 338 Astronomy 1290 Asymmetric tree, fractal dimension 492 Asymptotic 1738 Asymptotic analysis 1479 Asymptotic behavior 580 of the transition semigroup 1129 Asymptotic equipartition property 1427 Asymptotic freedom 1292 Asymptotic phase 1662 Asymptotic series 1367 strong 1367 Atmospheric science 1521 Attachment model, biological fractals 499 Attracting heteroclinic 1254 Attracting tori 1254 Attractor reconstruction 293 Attractors 538, 740, 1434, 1668, 1726, 1734 Lorenz attractor 547, 548 Aubry–Mather set 1114 Aubry set 684, 692, 693 critical equation 696 dynamical properties 697 metric characterization 696 Auslander–Yorke dichotomy theorem 1738 AUTO-07p 1191 AUTO, numerical studies 1221 Autocorrelation 514 Autocorrelation function 421, 424, 426 fractal time series 503 Automorphism 242, 1618 approximatively transitive 1619 Chacon 1624 GAG 1629 Gaussian 1628 Gaussian–Kronecker 1629 induced 1618 infinitely divisible 1634 interval exchange 1631 loosely Bernoulli 1630 Poissonian 1629 rank one 1627 Automorphism group 1699 Autonomous systems dynamical systems 1084 Autoparametric resonance 183, 191, 1261 Autoparametric systems 1261 Averaged guiding function 1243 Averaging 1252, 1665

Index

Averaging algorithm 1489, 1495 Averaging principle 1665 Averaging theorem 1168 Averaging theory 822 dynamical system 92 mechanical systems 89 Axiom A 1667 Axiom A flow 319 Axisymmetric KdV equation 1799 B Backbone 414, 417, 565 Bacterial colony growth 496 Bak–Sneppen model 504 Baker–Campbell–Haussdorf formula 1086 Balance laws, fluids in the Eulerian description 1012 Ballistic aggregation 430 Ballistic deposition (BD) 497 Banach principle 244 Bang-bang principle 956 Baroclinic fluid 1102 Barotropic fluid 1102 Basin of a measure 68 Basin of attraction 538 Basis 975 BBKGY hierarchy 1508, 1512 BD see Ballistic deposition Behavioral science biological fractals 489 Benford’s law 315 Benjamin–Bona–Mahoney (BBM) equation 1608 Benjamin–Feir instability 1807 Bernoulli equation 1776 Bernoulli flow 790 Bernoulli property 70, 235 isomorphism 787 Bernoulli shift 784 isomorphic transformation 786, 788 Bernoulli system 969 Bernstein–Moser lemma 825 Berry–Esseen inequality 75 Beta-effect, planetary 1590 Beta function 1292 ˇ-Mixing 932 Bi-fractal 484 Bianchi identity 1003 Bias 1402 Bifractality 616, 617

Bifurcation 183, 657, 1153, 1173, 1215 diagram 1173 equation 1233 equations 937 Hamiltonian Hopf 1447 Hopf 1153, 1166 Hopf–Ne˘ımark–Sacker 1153, 1167 imperfect 1445 liquids 1019, 1024 mapping 941 organization of 1446 period doubling 189 period multiplying 190 pitchfork 189 point 985, 1019 quantum 1438 quasi-periodic Hamiltonian Hopf 1167 quasi-periodic Hopf 1166 saddle-node 1445 Taylor–Couette flow 1020 theory 189 Bilinear form 1578 Bilinear realization 1200 Bioinformatics 507 biological fractals 489 Biological evolution fractals 504 Biological fractals 495 physical models 494 Biology fractals 488 Birkhoff ergodic theorem 785 Birkhoff normal form 1338 Birkhoff pointwise ergodic theorem 67, 1742 Birkhoff theorem ergodic probability measures 289 Bismut–Elworthy formula 1131 Bispinor 587 Blobs 414, 417 Bloch’s method 1360 2 1361 3 1361 Body force, fluids 1012 3-Body problem 1053 mountain pass solutions 1059 planar 1056 planar symmetry groups 1055 Bohr recurrence 366 Boltzmann 327 Boltzmann equation 1514 Boltzmann’s ergodic hypothesis 67

1821

1822

Index

Bolza problem 118 Borel property 1166 Borel summability 1297 Borel summation method 1367 Borel theorem 1166 Borel transform 1367 Borel, E. 1165 Born approximation 1370, 1371 Born–Oppenheimer approximation 1317 Born series 1370, 1371 Bose–Einstein and Fermi–Dirac statistics 1515, 1517 Boson peak 181 Bound state 776–778, 1579 Boundary conditions Dirichlet type 104 Neumann type 104 Boundary of the Hill’s region 1229, 1230 Boundary singularity 1061 Boundary value problem 1014 Bounded distortion, Markov map 1433 Bounded domains, flow 1014 Bounded function 383 Bourbaki school 550 Boussinesq 1521, 1524 Boussinesq equation 1103, 1207, 1524, 1530, 1608, 1786, 1788, 1790, 1791 Boussinesq systems 1525, 1530 Bowen ball 290 Bowen–Ruelle–Sinai theorem 69 Bowen’s formula 1434 Box counting 639 Box counting method 418, 594 biological fractals 497 Box covering algorithms 652 compact-box-burning 653 greedy coloring algorithm 652 maximum-excluded-mass-burning 654 random box burning 653 Box dimension dB 639 Brain size scaling, biological fractals 493 Branch switching codimension 2 equilibria 1187 flip points 1187 homoclinic branch switching 1190 Hopf points 1187 simple branch points 1185 Branching structures, self-similar 490 Break-down thresholds 827 Breaking, symmetry 576 Brockett condition 405 Brown noise, biological fractals 503

Brownian motion 596, 1126 biological fractals 494 Brownian trajectory, biological fractals Bundle picture, mechanics 993 Bundle reduction cotangent 996 tangent and cotangent 993 Bundles Lagrange–Poincaré 994 principal 998 Burgers equation 4, 6, 1127, 1207 controllability 765 Butterfly effect 540

494

C C1 subsolutions Hamilton–Jacobi equations 701 Cable-stayed bridges 193 Caccioppoli’s inequality 108 Cahn–Hilliard equation 1128, 1209 Calculus 573 of variations 1137, 1142 Camassa–Holm equation 7, 1525, 1608 Campbell–Baker–Hausdorff formula 94 Canards 1477, 1482 Cancellation 1293 Canonical anticommutation relations see CAR Canonical coordinates 959 Canonical symplectic form 958 Canonical transformation 1397 Cantor dust 657 Cantor family 657 Cantor set 413, 657, 1167 biological fractals 490 Cantor stratification 657 Capacity dimension 592 Capillary waves 1781 CAR, representations 352 Carathéodory condition 393 Cardiovascular system, biological fractals 493 Carleman estimates 107, 109 Cartan structure equations 1003 Cartan subalgebras 961 Cartesian product nonsingular maps 348 Catalyst 1461 Catalytic oxidation of carbon monoxide 1465, 1466 Catalyzed reactions 1467

Index

Category Lusternik–Schnirelman 1226 Cauchy estimate 817 Cauchy–Euler approximate solution 386 Cauchy–Peano theorem 385 Cauchy problem 384, 1214, 1286 Cauliflower anatomy, biological fractals 491 fractal dimension 490 Cayley tree 568 Cayley’s formula 1295 ˇ Cech–Stone compactification 313 Celestial mechanics 1301, 1309 Cell problem 1122 Center 665 Center-focus problem 1663 Center manifold 1090, 1177, 1179 construction of 54 for infinite dimensional flows 58 in discrete dynamical 55 Center-saddle bifurcation 663 Centered moving average (CMA) method 475 Central limit theorem 75 Centralizer 1283, 1394 Chacón maps 334 Chain 1731 Chain components 1735 Change of variables 1286 Channel superconductivity 1589 Chaos 328, 657, 1262 Chaos and fractals, applications of 539 Chaotic attractors 1254 Chaotic dynamics 191 statistical theory 1422 Chaotic hypothesis entropy production 1435 Chaotic in the sense of Devaney 64 Chaotic in the sense of Li and Yorke 64 Chaotic motions 1290, 1293 particles 1044 Characteristic curves 1790 Characteristic exponents 1255 Characteristic factor 256 Characteristic length scale 593 Characteristic multiplier 1659 Charges, non-abelian 589 Chemical dimension 411 Chemical distance 411, 564 Chen–Fliess series 95, 97 Chenciner–Montgomery symmetry group 1057

Chenciner–Venturelli Hip–Hop 1051 Choreography n-body problem 1043, 1045 non-planar 1052 Choreography symmetry 1057 2–1-Choreography symmetry 1055 Chromatin 607, 609, 630 Chronological algebra 88, 90, 93 free 93 Chronological calculus 88, 89, 1725 Chronological exponential 91 Chronological identity 93 Chronological logarithm 93 Chronological product 93 Chronology 577 Circle group 947 Circle method, biological fractals 497 Ck Conjugacy 369 Ck,l,p,r Locally rigid 375 Classical integrable systems 826 Classical solution 1115 Classical subsolution 1116 Classification theorem 42 Clean value, reduction theorem 987 Climate records 422 Climate modeling 422 Closed loop equation 120 Closing conditions 373 Clouds 409 Cluster 560, 1296, 1401 finite 417, 561, 562 infinite 417, 560, 562 spanning 562 Clustering 480 of extreme events 423 Cnoidal wave 1521, 1523, 1525, 1530, 1774, 1792, 1793 Coadjoint orbit bundles 999 Coadjoint orbits 960, 986 as symplectic reduced spaces 988 cotangent bundle reduction 999 Coarse-grained 642 Coastline 409 Coboundary cocycle 340 Cocycle identity 984 Cocycle superrigidity 377 Cocycles 373, 1619 affine 1623 dynamical systems 340 momentum 983

1823

1824

Index

nil 1623 orbit classification 341 Rokhlin 1626 weak equivalence 340 Codimension 32, 42 codim(f ) 42 Coding DNA, fractal features 506 Coercivity 1225 Coexistence 1251, 1256, 1259 Cohomological functional equation 972 Collisions asymptotic estimates 1062 between two or more particles 1044, 1060 Collisions instants, isolatedness 1062 Colloid aggregation 430 Comb model 13, 16 Combinatorics 328 Communication complexity 1492 Compact visitation 16 Compactification 581 Compacton 1, 4, 9, 10, 1553, 1554, 1602 Comparison principle 1118, 1119, 1121 Compatibility condition 774 Complete basis 975 Complete integrability 1395 Completely dissipative 972 Completely integrable 960, 1521, 1522, 1664 Completely integrable equations 1521, 1522 Complex dynamics 279, 293 Complex social behavior 1471 Complexity 571, 1724 algorithmic 924, 925, 931 ergodic theory 1427 measures 1489 sample 924, 925, 929–931, 935 Compositional strand asymmetry 606 Compton scattering 1291 Computer-assisted 1310 Computer-assisted proof 827, 834 Concentric KdV equation 1799 Condition for optimality 104, 119 necessary 119 Palais–Smale 1232 weighted Palais–Smale 1232 Condition A 1393 Condition S( ) 403 Condition ! 1393 Condition ! # 1395 Conditional expectation 976, 977

Conditional measures 976 Conditionally periodic 658 Conductance matrix 449 spanning tree polynomial 450 Conductivity exponent 601 Cone field 745 Confidence 924, 925, 929, 930, 934 Configuration space 1214, 1222, 1225, 1227–1229 Conformal field theory 440 Conformal maps Hastings–Levitov scheme 432 Laurent coefficients 438 Conformal repeller 1434 Conical intersection 1320 Conjecture Littlewood’s 303, 305 Oppenheim 305 Oppenheim conjecture 303 Conjugacy 369 Conjugacy class 1285 Conjugate Hamiltonian 687 Conjugation function 128 Connecting orbits 1189 Consensus 1501 Conservation laws 729 n-body problem 1063 Conservative 972 Conserved quantities 731, 1088 Consistent algorithm 930, 931 Constant cosmological 581 of motion 1088 Constitutive equation fluids 1012 liquids 1012 CONTENT 1191 Continuation 1174, 1215, 1233 Moore–Penrose 1176 of an orbit 1219 pseudo-arclength 1175 Continued fraction 315 expansion 974 Continuity equation 1775 Continuous feedback 1643 Continuous function 383 Continuous stochastic process 1126 Continuous time 395 Continuum hypothesis 1749, 1775 Contraction 243, 573 principle 938

Index

Control 102, 104, 704, 1640 boundary 121 deterministic 1116 hybrid 704, 707, 710 juggling 707 linear convex 121 linear quadratic 120 mobile robot 722 networked 712 optimal 104, 118 piecewise constant 1197 positional 104, 118 reset 712 sample-and-hold 711 stochastic 1116, 1117 supervisors 717 temperature 705 Control function 104 Control input 395 Control-Lyapunov function 1646 Control space 42, 104 Control system 395, 757 trajectories 953 Control systems optimal 954 reachable sets 953 Controllability 88, 395, 761, 1196, 1642, 1724 approximate 104, 760 Burgers equation 765 exact 762 heat equation 760 linear partial differential equations 763 linear systems 758 nonlinear systems 764 one-dimensional heat equation 763 regional null 122 return method 766 semilinear wave equation 764 stabilization and optimal control 1114 wave equation 758 zero 762 Controllability Gramian 398 Controllability to zero 760 Controllability/observability duality 761, 763 Controllable 397, 402 Controllable along a trajectory 400 Controlled Lagrangians 155 Convergence 1083, 1096, 1389 Cooperative control 1489 Coordinate fractal 573

Coordinate transformation 41 Coordination task 1497 Coproduct 96 Core symmetry groups 1053 Coriolis term, bundle reduction 998 Correction, log-periodic 577 Corrections to the energy 1355, 1358 effective order 1358 1 1356, 1359 2 1356, 1359 3 1356, 1359 n 1355 Corrections to the wave function 1356 first order 1356 through order 1356 [2] 1356 Correlation exponent 468 Correlation functions 466, 468, 595 Correlation length 417, 562, 601 biological fractals 501 Correlations between degrees 643 Correspondence 585 Cost final 118 running 118 Cost functional 118 Cotangent bundle 958 reduction 986, 989, 993, 996 Countable state Markov shifts 78 Countably additive 967 Countably subadditive 967 Coupling 1478 nearest neighbor 1486 Covariance scale 578, 581 strong 585 Covariance function 466, 468, 480 Coverage method 594 Covering lemma 247 Covering number 924, 931 Covering space 1164 Cp -Norm 830 Crack 409 Crapper’s nonlinear capillary wave solution Crashes market 577 Crisis economic 577 Criterion Mahler’s compactness criterion 305

1784

1825

1826

Index

Critical curve 697, 698, 702 Critical density, percolation 499 Critical dimension ergodic theory 345 Critical equation Aubry set 696 Hamilton–Jacobi equations 692 Critical exponents 562 Critical opalescence, biological fractals 501 Critical parameter 684 Critical part of the spectrum 48 Critical phenomena 559 Critical point 1401, 1733 biological fractals 500 Critical point universality 1290 Critical speed 1786 Critical subsolution, Hamilton–Jacobi equations 692, 693, 698, 699 Critical value 1733 Cross section 1369, 1370 Crossover 467, 472 Crystallization 435 Cubic 41 Cubic Boltzmann equation, Uehling–Uhlenbeck equation 1515 Cubic nonlinear Schrödinger equation 1800 Current distribution 562 Curvature 313 as a two-form on the base 1003 of a connection 1002 Curve fitting 925, 927, 928 Curves 43, 543 fractal 575 Menger sponge 546 shortening 1230 space filling curve 543 von Koch curve 543, 545 Cuspons 1 Cut-off function 49 Cutting and stacking 281 Cyclic actions, symmetry groups 1046 Cyclic group C6k 1052 Cyclic group of order 2n 1049 Cyclic set 1732 Cyclic subspaces 978 Cylinder set, ergodic theory 1426 Cylindrical Wiener process 1126 D d-Dimensional full A-shift

1705

d-Dimensional shift of finite type X 1705 d-Dimensional shift space 1705 d’Alembert solution 1790 Damping distributed 109 indirect 114 memory 116 Dangling ends 414, 417, 565 David-star 413 De Vries, Gustav 1, 5, 1521 Dealer model 527 Decay exponential 112 of correlation 75, 237 Decimal 964 Decomposition of the motion 1224 Defining systems 1173 branch point continuation 1186 codimension 1 bifurcations of limit cycles 1183 formulation as a BVP 1189 Deformation rigidity 375 Degeneracy 1359 accidental 1363 degree of 1363 normal 1233 residual 1359 Degenerate eigenvalue 1357 Degenerate perturbation theory 1307 Degenerated 32, 44 Degree correlations 648 Degree distribution 638 Degree of freedom 657, 811 Dendritic crystal growth 433 Density 255 probability 586 Density fluctuations 1461, 1464 Density of cars 1750 Dependency constant 776 Deployment algorithms 1500 Derivation, Navier–Stokes equations 1011 Derivatives covariant 589 Detection of bifurcations 1180 codimension 2 bifurcations 1184 homoclinic bifurcations 1190 test functions for codimension 1 bifurcations 1180 test functions for codimension 1 cycle bifurcations 1181 Determining equations 1084

Index

Deterministic fractal 592, 1403, 1459 Deterministic perturbations 79 Detrending 470, 471, 474 Development human 577 Deviations principle, ergodic theory 1423 Devil’s staircase 180 Diagram Bonzani and Mussone 1751 fundamental 1751 Greenberg 1751 Greenshield 1751 LWR 1751 velocity 1751 Diagrammatic rule 130 node factor 131 propagator 131 Dielectric breakdown model 429 Diffeomorphism 32, 37, 1285, 1433 construction of 1218 local 1218 Differentiability 573 Differentiable dynamics 327, 1432 Differentiable rigidity 369 Differential equation system 1043 Differential equations 541, 1661 existence and uniqueness theorem 1654 symmetry of 1083 Differential game 1149 Differential inclusions 1142, 1724 Diffraction 1581 Diffusion 6, 13, 649, 1402 anomalous 13 normal 13 paradoxical 18 Diffusion constant 1463 Diffusion equation 15, 596 normal 15 on fractal lattices 23 Diffusion exponent 1460 Diffusion-limited aggregation (DLA) 429, 494, 495, 596, 1471 advection-diffusion 440 branch competition 442 crossover 439 fixed scale transformations 442 noise reduction 441 numerical methods 434 theory of DLA 441 Turkevich–Sher relation 441 Diffusion-limited CCA (DLCA) 596

Diffusion-limited cluster-cluster aggregation (DLCA) 596 Diffusion-limited reaction 1461, 1471 Digraph 1490 Dihedral action, symmetry groups 1046 Dihedral group G=D2n 1049 4-Dihedral periodic minimizers 1054 6-Dihedral symmetric periodic minimizers 1054 Dilation symmetry 175 Dilations 571, 573 Dimension 18, 537, 543, 552, 1404 box 537, 552 chemical 410 correlation 538, 552 critical 582 fractal 17, 410, 537, 575, 577 Hausdorff 302, 309, 537, 546 information 552 Lyapunov 537, 552 metric 545 spectral 21 topological 537, 544, 545, 553, 575 variable 581 Walk’s 18 Dimension group 1699 Dimension-like characteristics, topological entropy Dimension spectrum 297 Dimension theory higher-dimensional dynamical systems 294 low-dimensional dynamical systems 294 Dimensionality 645 Diophantine approximation 328 Diophantine condition 133, 657, 1164, 1166 Bryuno condition 133 Melnikov condition 138 standard 133 Diophantine frequency vector 657 Diophantine property 1296, 1297 Diophantine vector 812, 1073, 1308 Dirac system 166 Direct integration 1527 Direct scattering problem 771, 773, 776 Directed networks 1405 Directed percolation 460, 1467 Dirichlet boundary conditions 756, 1340 Dirichlet condition 1120, 1121 Discontinous system escaping vector field 1327 regularization 1328 sliding vector field 1327 Discontinuous feedback 1648

290

1827

1828

Index

Discontinuous system 1326 Discrete Euler–Lagrange 145 Discrete Euler–Poincaré 145 Discrete Hamilton’s principle 145 Discrete Lagrange–d’Alembert principle 152 Discrete Lagrangian and Hamiltonian mechanics 144 Discrete Laplacian 448 Discrete mechanical systems 996 Discrete scale invariance (DSI) 1459 Discrete systems 943 Discrete time 395 Discrete variational principle 143 Discretization 1419 Diseased lungs, biological fractals 493 Disjointness 798 ergodic theory 349 Disorder strong 561 weak 561 Disordered system 181 Dispersion 4, 162, 1114, 1579 linear 5 management 200 nonlinear 5, 9 relation 1773, 1779, 1780, 1793, 1797, 1801, 1802 Dispersive equation 162 Dispersive estimate 162 Dissipation 110 McGehee coordinates 1063 relation 111, 112 Dissipative surface diffeomorphisms 291 Distal 1738 Distributed algorithm 1489, 1491, 1725 Distribution function 1755 Distribution of distances, biological fractals 490 Disturbance 40 Divergence 583 dk 643 DLA see Diffusion limited aggregation, 565 DNA and chromatin 636 fractal features 504 replication 621 sequence 504, 606–608 strand-asymmetry 616 walk 608, 616 Domain 1401 of attraction 1728 Dominated splitting 750 Doob’s theorem 1130 Double resonance 1586

Double series representation 1103 Double well model splitting 1382 symmetric double well 1385 Doubling map 226 Downarowicz–Serafin entropy realization theorem 76 Drainage networks 532 Drift 402, 405 Dual problem 106 Duality controllability/observability 761 Duffing’s equation 1245 Dynamic fractals 502 Dynamic programming 119, 1137, 1143, 1145, 1724 Dynamic programming principle 1116, 1120 Dynamical exponents 563 Dynamical models 1412, 1413, 1416 Dynamical systems 327, 540, 542, 548, 1082, 1653 cocycles 340 continuous 705, 707 discrete 707 fractal geometry 288 hybrid 704, 705, 707 linearizable dynamical system 1092 multifractal analysis 296 simple pendulum 547 Dynamical systems, higher-dimensional 294 Dynamics 1281 complex 293 fast 1484 reduction of 986 scale 579 slow 1484 Dynamorphisms 1416 Dyson series 1373 Dürer Pentagon 412 E Earthquake 533, 577 Eccentric anomaly 1304 Eckmann–Ruelle conjecture 295 Ecology biological fractals 489 spactial correlations 501 Effective Hamiltonian 1322 Effective medium theorie Curie–Weiss effective field 450 Effective medium theory 446 effective homogeneous medium 451 Egorov theorem 1386, 1387

Index

Eigenvalue 1218, 1221, 1278, 1389 Eight-shaped orbit, n-body problem 1050 Eight shaped three-body solution, n-body problem 1047 Eikonal equation 1117 Einstein formula, biological fractals 494 Einstein relation 649 Einstein, Albert 1291 Elastic backbone 565 Elastodynamics 1349 Electrical networks, complex 1096 Electrodynamics 1290 Electronic ecology 502 Eliasson 1294 Ellipsoidal geodesics 957 Elliptic 662 Ellipticity 1115, 1116 uniform 1115 Embedding 1693 Embedding space 592 Embedding theorem 1701 Empirical mode decomposition 475 Endogenous variable 42 Endomorphism ergodic 242 Endpoint 167 Energy 109, 116, 1221 total 115 Energy band 1319 Energy density 1803 Energy equality, Navier–Stokes equations 1032 Energy equipartition 1077, 1578 Energy flux 1803 Energy optimization biological fractals 493 Energy shell 673 Entrainment 1478, 1482, 1486 mutual 1482 Entropy 251, 313, 328, 359, 522, 729, 733 completely positive 235 ergodic theory 344, 1424, 1425 Kolmogorov–Sinai 786 measure-theoretic 63, 313 of a measure 290 topological 63, 313 Entropy formula 753 Entropy of symbolic extensions 77 Entropy production 1435 ergodic theory 1424 Entropy spectrum 296 Epicycles 1290

Epidemic simulations 1469 Epidemic spreading, percolation 499 Epidemiology biological fractals 489 "-Free Rokhlin lemma 331 "p (n,k) 32 Equation 586, 1221, 1522 Boussinesq 1577 continuity 586 Dirac 588 dispersive 162 dynamics 579, 584 Einstein 567 Euler 586 Euler–Lagrange 579, 1225 evolution 1578 geodesic 582 Hamiltonian 1213, 1228 integrable 1577 Kadomtsev–Petviashvili (KP) 1577 Kirchhoff 566 Klein–Gordon 587 Korteweg–de Vries (KdV) 1577, 1607 motion 582 nonlinear 1578 nonlinear Schrödinger (NLS) 1577 Pauli 588 Schrödinger 584 sine-Gordon (SG) 1577 Equation with infinitely many variables 1134 Equicontinuity point 1737 Equicontinuous 1737 Equicontinuous function 383 Equidistributed 313, 314, 316 Equidistribution of periodic points 68 Equilibrium 1726 Aubry set 693 ergodic theory 1427 Equilibrium distribution ergodic theory 1424 Equilibrium state 328 ergodic theory 1422, 1427 Equipartition property 1427 Equivariance 944 infinitesimal 983 Equivariant bifurcation theory 1083 Equivariant branching lemma 943 Equivariant momentum maps 983 Equivariant orbits 1046 Erd˝os and Rényi 568

1829

1830

Index

Ergodic 228, 313, 328, 785 Ramsey theory 316 theorem 314 unique 316 Ergodic decomposition 230 Ergodic decomposition theorem 328, 331 Ergodic flows 332 Ergodic hypothesis 225 Ergodic invariant measure 289 Ergodic measures 970 Ergodic probability 289 Ergodic Ramsey theory 364 Ergodic theorem 328 Ergodic theory 289, 327, 783 associated flow 338 critical dimension 345 disjointness 349 entropy 344 fractal geometry 288 full groups 338 Hausdorff dimension 289 infinitely divisible stationary processes 349 L1 -spectrum 342 maharam extension 338 mixing 335 mixing recurrence 335 multiple recurrence 335, 336 non-singular transformations 329 nonsingular joinings 346 normalizer of the full group 339 orbit 338 orbit classification of type III systems 338 outer conjugacy problem 339 poisson suspensions 350 polynomial recurrence 336 pressure and equilibrium states 1422 quasi-invariance 342 symmetric stable processes 349 topological group 337 topological pressure 291 Ergodicity 63, 316, 1130 unique 230 Euclid’s Elements 550 Euler equations 1775, 1795, 1804 Euler–Hill symmetry 1056 Euler–Lagrange equation in integrated form 956 Euler–Lagrange equations reduced 994 Euler–Poincaré equations 991 Euler–Poincaré reduction 990 Even shift 1692

Evolution 577, 1265 theory, biological fractals 489 Evolution equation 103 approximately controllable 105 exactly controllable 105 null controllable 105 Evolutionarily stable strategy 1269 Evolutionary game theory 1266 Exact 235 Exact controllability 762 Exact dimensional 70 Exact solution 890, 892, 893, 895, 902, 903 Exactly controllable 399 Example 1227 of a Hamiltonian 1221 Exchange of stability 943 Excluded volume effect 596 Exogenous variable 42 Expansions 964 Expansive 1740 Experiment two-slit 583 Exponent complex 577 Diophantine 304, 306 multiplicative 308 multiplicative Diophantine 308 Exponential stability 1074, 1640 Exponentially small 1166 Extended phase space 1084 Extension group 1619 Extension problem 1706 Extension theorem 388 Exterior covariant derivative 1002 External (or barotropic) modes 1106 External parameters 1160 External perimeter 417, 565 Extrema 43 Extreme event 480 Extreme value statistics 481 F Factor code 1693 Factor map 976, 1693 Factors 280 Family 40 Faraday waves 199 Fast system 1328 Feedback 110, 1138, 1144, 1147, 1724 discontinuous 1138, 1147

Index

distributed 110 globally distributed 110 locally distributed 110 memory type 112 nonlinear 112 optimal 1149 sample-and-hold 1147 stabilizing 1147 synthesis 1144 Feedback controllers 155 Feedback equivalent 1642 Feedback group 1642 Feedback law 395, 1639 Feedback stabilization 1724 Feedback stabilizer 1640 Feller 1129 Fermi 1291 Fermi–Pasta–Ulam recurrence phenomenon 1787 Fermi systems 1296 Feynman 572 Feynman graphs 1291, 1296 Feynman–Hellmann theorem 1357 Fiber bundle 1096 Fibonacci numbers, biological fractals 489 Field 589 gauge 588, 589 Filling scheme 244 Filtration 559 Financial markets seismic activity 425 Financial records 409 Fine structure 1361 Finitary isomorphism 790 Finite alphabet, Markov chain 1432 Finite-difference method 916 Finite-dimensional method Navier–Stokes equations 1015, 1022 Finite lattice, ergodic theory 1425 Finite portion transport 524 Finite rank, ergodic theory 334 Finite-size scaling 453 Finite state code 1703 Finite statistical description 68 Finite system, ergodic theory 1424 Finite time estimates 924, 932, 935 Finite type, subshift 1429 Finitely additive 967 Finitely equivalent 1699 Finitely fixed transformation 791 First-order nonlinear partial differential equations 1113

First passage 458 time 649 First-return map 973 Fisher equation 1207 Fisher–Tippet–Gumbel distribution 482 Fisher’s theorem 1267 Fitness landscape 1266 Five invariant manifolds 51 Fixed point 943 Flatness 395, 405 Fleming–Viot model 1128 Flip bifurcation 945 Floods 534 Floquet analysis 188 Floquet exponent 673 Floquet form 658 Floquet multiplier 186, 664, 1661 Floquet normal form 1659 Floquet theory 183, 1254, 1257 Flory’s mean field theory 1404 Flow 657, 1214, 1215, 1661 horocycle 1630 in bounded domains 1014 induced 1223 isomorphism theory 790 nil 1630 past a circular cylinder 1021 past an obstacle 1039 rank one 1628 smooth, with singular spectra 1628 special 1619 special von Neumann 1632 three-dimensional flow 1022 translation 1632 Flow map 390 Flow resistance 1564 Fluctuation analysis (FA) 469, 471, 474 detrended 471 Fourier-detrended 474 modified detrended 474 Fluctuation analysis (MFDFA), multifractal detrended 478 Fluctuation exponent 470 Fluctuations 1268 Fluid mechanics, Navier–Stokes equations 1011 Fluid motion 541 Fluid–structure interaction 199 Fluids balance laws 1011 body force 1012 constitutive equation 1012

1831

1832

Index

incompressible 1012 mechanics modeling 1011 Newtonian 1011 steady-state flow 1013 stretching tensor 1012 Flux 729, 731, 1750 Flux function 729 Fluxon 1561 Fokker–Planck–Landau equation 1510 Fold catastrophe 42 Fold point 1484 Folklore theorem 974 Force scale 579 Forced Korteweg and de Vries equation 1785 Forced nonlinear Schrödinger equation 1786 Ford–Kac–Mazur model 1673 Foreign exchange market 515 Forest fire biological fractals 498 Forest fire model 500 Formal power series 128, 1153 Formal series 1290, 1298 Formulation Lagrangian 1214 Poisson 1214 Fourier filtering 482 Fourier integral operator 1385 Fourier transform 448 FPU paradox 1565 Fractal 429, 532, 591, 593, 606, 635, 1457, 1458 biology 488 definition 489 deterministic 410, 411 distribution 532 geometry 429 random 410, 415 real biological 497 scaling 438 time series 502 Fractal dimension 175, 430, 465, 490, 532, 563, 592, 639, 1401, 1459, 1460 biology 489 measurements 497 Fractal dynamics 175 Fractal features 504 Fractal geometry 176, 409 dynamical systems 288 ergodic theory 288 Fractal growth processes 429 Fractal media 1471

Fractal metabolic rates 493 Fractal noise 503 Fractal reaction kinetics 1457 Fractal structure 175 Fractal tiling 524 Fractality 573, 642 Fractional Brownian motion 419, 611 Fractional Brownian signals 613 Fracton 601 Fracton dimension 601, 602 Fracton regime 177 Fracture 440 Fragmentation 533 Fredholm alternative 948 Fredholm operator 950 Free action, mechanical systems 984 Frequency dispersion 1782 Frequency domain 117 Frequency-halving bifurcation 674 Frequency spectrum 1295 Frequency vector 128, 812 Bryuno vector 133 Diophantine vector 133 Full A-shift 1692 Full group normalizer, ergodic theory 339 Full groups, ergodic theory 338 Full r-shift 1692 Fuller’s phenomenon 956 Function 383, 392, 1225 fractal 575 Hamiltonian 1213, 1221, 1222, 1228 hyperregular 1228 invariant 1222 Lagrange 579 Lagrangian 1214, 1228 Function-analytic method, Navier–Stokes equations 1016, 1022 Function germs 32 Functional multivalued 1228 penalized 1232 Tonelli 1229, 1230 Functional derivative 1339 Fundamental matrix of solutions 1660 Fungi, biological fractals 497 Furstenberg structure theorem 1744 Furstenberg’s class 349 Furstenberg’s conjecture 379 Furstenberg’s multiple recurrence theorem 965 Further normalization 1082

Index

Følner sequence 252 amenable group 793 G G-Equivariant minimizers, n-body problem 1066 G-Invariance 1222 G-Space, Hamiltonian 983 g-Twisted action 1226 Galactic dynamics 1096 Galerkin method 1015 Galois correspondence 1414, 1420 Gas dynamics 731 Gas like regions, critical point 501 Gauge theory of mechanics 993 Gauss map 974 Gaussian random variable 1130 Gaussian systems 270 Gelation 559 Gene detection 619 Gene organization 626, 628 Generalized central limit theorem 522 Generalized gradients 1137, 1139 Generalized hip–Hops, n-body problem 1051 Generalized notions of solutions 1113 Generalized orbits 1060 Generalized polynomial 361 Generalized regularized long wave equation 1206 Generalized solitons and compacton-like solutions 1554 Generalized solution, n-body 1060 Generalized Sundman–Sperling estimates, n-body problem 1063 Generalized Zakharov–Kuznetsov equation 1208 Generated 32 Generating function 812 Generators 589 Generic 229, 657 Generic chaos 64 Generic dynamics 80 Generic point see Regular point Genericity 1165 Geodesic flow 371, 1667 Geodesics 313, 318, 319 fractal 583 Geometric complexity 288 Geometric condition, multiplier 111 Geometric coordinate-free approach to stochastic modeling 1679 Geometric integration 143 Geometric mechanics 981

Geometric singular perturbation theory 1325 Geometric structures, reduction of 996 Geometrical condition piecewise multiplier 111 Geometry curved 579 dynamical systems 288 noncommutative 581 space-time 581 Geometry of group action 1094 Geophysical fluid dynamics 1526 Gevrey 1165–1167 Gevrey-2 1072 Gevrey index 1282 Gevrey spaces 1276 Ghettos 1400 Gibbs distribution discrete 1428 ergodic theory 1428 Gibbs–Markov 74 Gibbs measure 328 Gibbs property, ergodic theory 1424 Gibbs state ergodic theory 1422, 1430, 1431 property 1431 summable potential 1430 Gibbsian systems 360 Glimm–Effros theorem 332 Global asymptotic stability, controllable 1640 Global attractor, Navier–Stokes equations 1036 Global rigidity 369 Global simplicity 78 Goh condition 962 Golden mean shift 1692 Golden number 1257 Golubitsky 33 Gradient 1338 Graph self-joinings (GSJ) 347 Grashof number 1038 Gravity loop quantum 581 quantum 581 Great multiplier 1072 Green matrix 1238 Green’s function 1368 Gronwall’s lemma 391 Gross Pitaevskii 1349 Group 1363 action 1222 character 1222 Galilean 576

1833

1834

Index

Heisenberg 1623 homology 1227 homotopy 1226, 1231 invariance 1363 Lie 1222 Lorentz 580 representation of 1363 semisymplectic 1222 symplectic 1222 temporal character 1222 Group action 1412 Group shift 1707 Group velocity 1579, 1780 Gröbner basis 1168 GSJ see Graph selfjoining Guckenheimer 33 Guiding function 1242, 1243 Gyroscopic systems 1259, 1261 Gyroscopic term bundle reduction 998 H ¯-Pseudodifferential calculus 1385 elliptic symbol 1386 H 1 1225 Haar measure 313 Hadamard 1291 Hales–Jewett coloring theorem 365 Half relaxed limits 1121 Hall set 97 HAM 890, 892, 893, 895, 896, 900, 902 Hamachi’s example, nonsingular Bernoulli transformations 334 Hamilton equations 811, 966 Hamilton–Jacobi–Bellman equation 1116, 1117 Hamilton–Jacobi–Bellman–Isaacs equation 1118 Hamilton–Jacobi equations 120, 683, 684, 822, 1134, 1137, 1144 C1 subsolutions 701 critical equation 692 critical subsolution 692, 693 critical subsolutions 699 intrinsic metric 694 semiconcave functions 701 solutions 687 strongly semiconcave 689 strongly semiconvex 689 subsolution 691 subsolutions 685, 690 time-dependent 698

viscosity solution 700 viscosity solutions 688 viscosity test functions 693 Hamilton–Poincaré equations 994 Hamiltonian 119, 684, 810, 811, 958, 1144, 1161, 1165–1167, 1258, 1442 conjugate 687 dynamics 810 effective 1440 flow 811 left-invariant 959 lift 958 vector field 958 Hamiltonian chaos 1259 Hamiltonian dynamical systems 1083 Hamiltonian function 1257 Hamiltonian G-space 983 Hamiltonian Hopf bifurcation 665 Hamiltonian system 189, 271, 657, 810, 966, 1096, 1259, 1262 billiard systems 271 KAM-systems and stably non-ergodic behavior 272 singular 1044 Hamiltonian vector field 1083, 1341 Harmonic 446 Harmonic function 448, 458 Harmonic instability 187 Harmonic measure 429 growth probability 433 Harmonic oscillator 956, 1655 damped harmonic oscillator 1657 Hartman–Grobman theorem 742 Hausdorff dimension 592 conformal repeller 1434 ergodic theory 289 Hautus 398 Heat equation 103 boundary control 756 controllability 756, 760 one-dimensional 763 Hele–Shaw problem 604 Hepp 1294 Hepp’s theorem 1292 Hereman, W. 11 Hessian matrix 32 Heteroclinic orbit 663 Heterodimensional cycle 749 Heterogeneous chemical reactions 1462 Heterogeneously catalyzed reactions 1457, 1465 High density regions, crictical point 501 Higher rank Z Anosov actions 372

Index

Hilbert space 977 transform 247 uniqueness method 109 Hill equation 185, 1251, 1252, 1254, 1256, 1257, 1660 Hip–Hops, n-body problem 1050 Hirota method 1527, 1578 bilinear 2 Hitting time set 1739 Hofbauer’s example 1432 Holonomy invariance of measures of maximum entropy 68 Homoclinic bifurcations 291 Homoclinic orbit 664 Homoclinic tangency 749 Homogenization 13, 1122 Homological equation 1072, 1154, 1156, 1342, 1390 Homological operator 1085 Homology 1227 Homomorphism of measure algebras 974 Homotopy 1226 analysis method 890, 911, 916 perturbation method 912 Hopf algebra 98 Hopf-algebra 97 Hopf bifurcation 657, 949, 1178, 1477 Hopf decomposition 330 Hopf fibration 990 Hopping rate 180 Horizon free 118 infinite 118 Horizontal lift 1001 Horizontal projection 1001 Horizontal space, definition 1001 Horizontal vector fields definition 1002 Horocycle flow 372, 784 Horseshoe 539, 549 HPM 890, 892, 893, 897, 900, 902 Hurewicz pointwise ergodic theorem 331 Hurst exponent 419, 465, 469, 477, 613 generalized 477 Hybrid arc 708 Hybrid control 1149 Hybrid control systems 1725 Hybrid time domain 708 Hybrid, well-posed 707 Hydrodynamic limit 1506

Hydrodynamics 26 Hyman, J. 9 Hyperbolic 1657 infinitesimal 1657 Hyperbolic attractor 748 Hyperbolic equilibrium 658 Hyperbolic linear system 742 Hyperbolic measure 295 pointwise dimension 295 Hyperbolic set 744 Hyperbolic system 732 Hyperbolic toral automorphisms Hyperbolicity 63, 1666 Hyperfine structure 1361 Hypothesis on C1 -norm 58 Hysteresis 1261 Hénon-like maps 73 Hölder exponent 477, 610 Hölder norms 833

1667

I ICER 1744 60-Icosahedral periodic minimizers 1054 Ideal 32 maximal 90 Identification 1724 Ill-posed problems 1805 Impact fracture 513 Impermeable 1105 Implicit function theorem (IFT) 812, 829, 938, 1213, 1218, 1219 continuation 1219, 1220 Impulsive system 725 Ince–Strutt diagram 183, 186 Incompressible fluids 1012 Incompressible viscous fluids, Navier–Stokes equations 1010 Indian Ocean earthquake 1610 Indian rod trick 197 Indian rope trick 196 Indifferent fixed point 1433 Induced transformation 280, 785 Inertia tensor, locked 1001 Inertial manifold 1668 Infimum 976 Infinite-dimensional control system 756 Infinite-dimensional controllability 755 examples 756 Infinite-dimensional torus 1343

1835

1836

Index

Infinite ergodic theory 279 Infinite iterated function systems 292 Infinitely divisible stationary processes, ergodic theory 349 Infinitesimal decrease 1139, 1146 Infinitesimal equivariance 983 Infinitesimal generator 59 Infinitesimal nonequivariance two-cocycle 984 Infinitesimal rigidity 376 Infinitesimally free action 985 Infinitesimally hyperbolic 1657 Information loss 587 which-way 583 Initial-boundary value problem 1013, 1026 Initial submanifold 989 Initial value problem 384, 1726 Input 1640 Instability pockets 1251 Instability tongues 1251, 1255 Insulator–conductor 561 Integrability 1398, 1577 Integrable affine structure 669 Integrable system 657 integrable 1082 Integral inequality 111 nonlinear 113 Integral (of motion) 657, 960 Interaction attractive 1580 elastic 1576 inelastic 1576 nonlinear 1576 repulsive 1580 resonant 1576 Interaction center 1588 Intermediate normalization 1356, 1362 n 1356 Internal model 1712, 1718, 1719 asymptotic internal model property 1719, 1720 design of 1720 principle 1712, 1718 Internal (or baroclinic) modes 1106 Internal solitary wave 1520, 1529 Internal waves 1102 Interval exchange maps 266 Interval exchange transformation 226 Interval maps 72, 269, 1433 ˇ-shifts 270 Continued fraction map 269 f -expansions 270

Farey map 270 piecewise C2 expanding maps 269 Smooth expanding interval maps 269 Intrinsic metric Hamilton–Jacobi equations 694 Invariance ergodic theory 1426 scale 576 Invariance principle 709 Invariance relation 1094 elementary 1094 Invariant manifold 49, 1095, 1396 Invariant measure 740, 1129, 1426, 1741 entropy 1426 ergodic theory 289, 1425 Invariant set 1726 Invariants 1732 method of 996 Inverse function theorem 829 Inverse limit 282 Inverse problem 1109 Inverse scattering problem 771, 773 Inverse scattering transform (IST) 771, 1522, 1525, 1527, 1562, 1578, 1601 Involution 1253 Irreducible shift space 1694 Irreversibility 571 Irreversible phase transitions (IPTs) 1465 Ishii’s lemma 1118, 1119 Ising 1400 Ising chain 1432 Ising model 1461 Iso-energetic KAM tori 826 Iso-energetic Poincaré-mapping 664 Isobaric 1105 Isochors, biological fractals 504 Isochronous 1662 Isolated chain component 1736 Isolated invariant set 1734 Isolating neighborhood 1734 Isomorphic 974 Isomorphic transformation, Bernoulli shift 788 Isomorphism 328 Bernoulli property 787 Bernoulli shifts 786 finitary 790 Ornstein’s theorem 787 Isomorphism invariant 784, 785 Isomorphism theory 783 action of amenable groups 793 transformation factor 792 Isosceles symmetry 1056

Index

Isotropy subgroup 985 IST method 773 Italian symmetry 1053 Iterated function systems 291 Iterated integral functionals 95 Iteration of functions 543 ITPFI transformations 341 J Jackson–Moser–Zehnder, lemma 825 Jacobi elliptic 1523, 1527 Jacobi elliptic function 1523, 1527 Jacobi, C.G. 957 Jacobian ideal 32, 41 Jacobi’s elimination of the node 990 Jahn–Teller effect 1318 Jensen’s inequality 1428 John Scott Russell experiments 1562 Johnson’s equation 1800 Join 976 Joining 283, 328, 798 composition of joinings 802 ELF property 807 filtering problem 807 Markov intertwining 801 minimal self-joinings 802 pairwise independent joining 806 relatively independent joining 800 self-joinings 802 simple 804 Jordan canonical form 1277 Jost solution 776, 777 Julia set 414, 538, 551, 792 K K-Automorphism 336 K-Property 235 ergodic theory 335 Kadomtsev–Petviashvili equation 1208, 1525, 1608, 1798 Kakutani equivalence, isomorphism theory 791 Kalman 397 Kalman controllability decomposition 397, 404 Kalman’s observer, state-linear systems 1201 KAM 1308, 1311 KAM see also Kolmogorov–Arnold–Moser (KAM) KAM method 377 KAM stability 1076 KAM theorem 202, 1663 KAM theory 127, 657, 1302, 1338

Kaplan–Yorke conjecture 295 Kato bound 1366 Kato–Rellich theorem 1366 Kawahara equation 1208 KdV 884–888 KdV–Burgers equation 1806 KdV equation 891–902, 908, 1108, 1562, 1607 third-order 5 Kepler’s equation 129, 1299, 1305 Kerr effect 1571, 1581 Kinetic shaping 155 Kink 1, 4, 6 KKS see Kostant, Kirillov, Souriau Kleiber’s law, biological fractals 493 Klein–Gordon equation 1206 K(n,n) equation 5, 9 Kneading invariants 78 Koch curve 410, 411 Kolmogorov 33, 813, 814, 820, 1291 normal form 813 set 820 transformation 814 Kolmogorov equations 1726 Kolmogorov group, property 1618 Kolmogorov operator 1129 Kolmogorov–Sinai entropy 753, 786 Kolmogorov typicality 63 Kolmogorov’s method 1293 Koopman operator 228 Korteweg and de Vries 1601 Korteweg–de Vries equation 890, 901, 1206, 1337, 1340, 1522, 1523, 1530, 1773, 1787, 1788, 1792, 1796 Korteweg, D.J. 1, 5, 1521, 1522 Kostant, Kirillov, Souriau coadjoint form 986 Krengel’s entropy 344 Krieger generator theorem 786 Kronecker factor 234 Kruskal, M.D. 5 Krylov–Bogoliubov theorem 67, 1129 L L1 -spectrum, ergodic theory 342 Lacunary sequence 360 Lagrange multiplier 912 Lagrange–Poincaré bundles 994 Lagrange–Poincaré equations 994 Lagrange symmetry 1057 Lagrange’s principle 1655, 1662 Lagrangian 697 Lagrangian equilibrium 1077

1837

1838

Index

Lagrangian point 1309 Lagrangian reduction 993 Lagrangian torus 661 Lamb shift 1291 Landau 1292 Landesman–Lazer condition 1246 Landslide 534 Langevin equation 1674 dissipative 1674 irreversible 1674 Markov diffusion process 1674 Langmuir–Hinshelwood mechanism 1465 Laplace 1290 Laplace equation 1776 1-Laplace equation 1116, 1118, 1123 Laplace’s limit 1299 Laplacian growth 432 Large deviations problems 1130 Lattice 369, 1401 unimodular lattice 302, 305 Lattice dynamics 1565 Lattice-Gas reaction models 1467 Lavoie’s 1526 Lavoie’s equations 1526 Law composition 576 scaling 577 Lax method 771, 773 Lax pair 773 LCR algorithm 1489 Le Lann–Chang–Roberts (LCR) algorithm 1494 Leader election 1489, 1493 Learning 1724 Least-squares method 926 Lebesgue space 974 Ledrappier–Young theorem 295 Legendre polynomials 1304 Legendre transformation and reduction theory 994 Length Compton 586 fractal 574 Planck 581 Length-scale 581, 642 invariant 581 Level crossing 1354, 1365 Li–Yorke pair 1738 Libration 1312 Librational Tori 1310 Lie 1155 Lie algebra 1159, 1389 bundle 1003

free 96 Nilpotent Lie algebra 1091 semisimple Lie algebra 1091 subalgebra 1159 Lie bracket 401 Lie groups 143, 953, 1083 derivative 1155, 1159 group 1159 left-invariant 959 Lie algebra 959 right-invariant vector fields 959 subgroup 1159 Lie–Poisson bracket 990, 992 Lie–Poisson bundle 1003 Lie–Poisson reduction 990 Lie series, exponential 97 Lie transforms 1086, 1258 Light bullets 1206 Lim sup 1729 Limit point bifurcation 1178 Limit point of cycles 1178 Limit set 740, 1254, 1713 of a given point 1713 of a set 1714 Lindstedt 826 algorithm 1291 series 127, 1294, 1295, 1297 Line symmetry 1055 Linear damping 1260 Linear functional, multiplicative 90 Linear matrix inequality 1678 Linear plane waves 1773 linear Schrödinger equation 1800 Linear superposition 1576 Linear system 397, 1641 controllability 758 Linearization 1138, 1153, 1283 continuous 1155 formal 1153 holomorphic 1166 smooth 1155 Linearization principle 399 Linearized equation 1656 controllability 764 Linearized stability 1655 Link number 520 Lin’s method 940 Liouville equation 1508 Liouville theorem 966 Lippmann–Schwinger equation 1369 Lipschitz continuous function 383, 685–687, 694

Index

Liquid 1012 constitutive equation 1012 non-Newtonian 1014 Oseen fundamental tensor 1022 planar motions 1013 pure shear flow 1014 steady bifurcation 1018 three-dimensional flow 1022 two-dimensional flow 1013, 1024 Liquid like regions critical point 501 Littlewood’s conjecture 379 Liénard differential equation 1239 Local bifurcation 665 Local entropy 77 Local ergodicity 70 Local limit theorem 75 Local product structure 746 Local rigidity 369 Local uniqueness of invariant mainfolds 57 Local unstable manifold 745 Local well posedness 161 Localization length 178 Localized coefficients 1342, 1345 Locally compact groups, unitary representations 352 Locally free actions 985 Locally minimizing solution, n-body 1060 Locked inertia tensor 1001 Loewner evolution 437 Logarithmic type potentials, n-body problem 1064 Logistic map dynamic fractals 502 Long-range correlations 466 biological fractals 500 Long-term correlations 473 Long-term memory 420, 423, 424 Long-time behavior, Navier–Stokes 1036 Long wavelength parameter 1788 Loosely Bernoulli transformation 791 Lorenz attractor 1668 Lorenz equation 1668 Lorenz-like strange attractor 751 Louis Michel 1094 Low density limit 1506, 1507, 1517 Low density regions, critical point 501 Low-dimensional dynamical systems 294 Lower and upper solutions 1243–1245 Lower dimensional tori 813, 827 Luenberger observer 1198

Lung biological fractals 492 diseased 493 Lusternik–Schnirelmann category 1247 Lyapunov exponent 63, 295, 1481 Lyapunov function 1137, 1139, 1146, 1662, 1726, 1733 converse theorems 715 patchy control 718, 719 semiconcave 1147, 1148 strict 1728 Lyapunov–Schmidt method 937 Lyapunov spectrum 297 Lyapunov’s equation 1663 Lyapunov’s theorem 1662 Lévy flight foraging 501 Lévy flights 434 Lévy walk models 501 M Mach reflection 1585 Mach stem 1585 MACSYMA 1258 Magnetic field 535 Magnetic term 990 bundle reduction 998 Magnetization 562 Magnetohydrodynamics (MHD) 1349 Magnitude 474 series 474 Maharam extension, ergodic theory 338 Majorizing series 1392 Malthus, Robert 502 Mandelbrot–Given fractal 414, 592, 603 Mandelbrot set 414, 551 Manifold 953 differentiable 1224 Hilbert 1225 Poisson 1223 Riemannian 1225 Riemannian manifolds 953 symplectic 1212–1214, 1222, 1228 tangent and cotangent spaces 953 Map 306, 308 Gauss map 303, 304 momentum 982 nondegenerate 306, 308 Poinaré 1217 Poincaré 1217 Mapping 710 locally bounded 710

1839

1840

Index

outer semicontinuous 710 regularization 711 set-valued 705 Mapping rules, DNA 505 Marchal’s P12 –symmetry 1052 Marchenko integral equation 779 Marchenko kernel 779 Marchenko method 779 Market 1400 Markov chain finite alphabet 1432 Markov map, uniformly expanding 1433 Markov odometers 333 Markov partition 746, 1423, 1691 Markov process 248, 969 Markov shift 63, 784, 969 isomorphism theory 792 Markovian splitting subspace 1680 minimal 1681 Matching 1479 MATCONT 1191 Material points, fluid 1011 Material surfaces, fluid 1011 Material volumes, fluid 1011 Mathematical finance 1128 Mathematical model 1409 Mathematical physics, and mechanical systems 986 Mathematical structure 1414 Mather 33 Mathieu equation 185, 1251–1253, 1255, 1256, 1258 Matrix monodromy 1218 symplectic 1213, 1215 Matrix action 784 Matrix pencil 1642 Matrix subshift 1706 Maurer–Cartan equations 1004 Maximal entropy measure 1431 Maximal inequality 244 Maximal KAM tori 812 Maximum entropy measure 63 Maximum entropy measures, existence of 71 Maximum principle 88, 958, 1137, 1138, 1141, 1142, 1724 abnormal 958 extremals 958 normal 958 Pontryagin 1142 Maximum spectral type 978 Mañé 78 McGehee coordinates, dissipation 1063

Mean anomaly 1304 Mean curvature flow 1116, 1118 Mean ergodic theorems 331 Mean square displacement 601 Measurable 327 Measurable dynamical systems isomorphism 799 Measurable map 976 Measure 306, 307, 327, 591, 967 absolutely friendly 307 ergodic theory 1427 extremal 306, 307 Federer 306 friendly 307 Rajchman 1628 spectral 1620 spectral, of the process 1628 strongly extremal measure 308 Measure algebra 974 Measure algebra endomorphism 965 Measure-preserving dynamical system 797 commutant 802 empirical distribution 805 Gaussian dynamical systems 807 K-system 799 Krieger’s finite generator theorem 805 mixing 803 multifold mixing 806 Ornstein’s isomorphism theorem 805 pointwise convergence 805 Poisson suspension 807 rank-one transformations 802 spectral types 799 weak mixing 799 Measure-preserving map 327 Measure-preserving system 965 Measure-preserving transformation 784 Measure rigidity 369 Measure space 327 Measure-theoretic entropy 786 Measure-theoretically isomorphic 227 Measures of maximum entropy 68 Mechanical connection 991, 993, 1001 extremal characterization 1002 Mechanical systems 981 and mathematical physics 986 discrete 996 nonholonomic 996 Mechanics gauge theory 993

Index

geometric 981 stochastic 583 Melnikov conditions 827 Melnikov theory 1262 Metabolic rates fractal 493 Metric fractal 582 Jacobi 1229 Riemannian 1230 Metropolis 1401 Microlocal analysis 110 Mild solution 105, 1131 Minimal rational extension 1685 Minimal realizations 1199 Minimal self joining transformation 789 Minimal system 1735 observability 1196 Minimal time problem 1143, 1149 Minimizer n-body problem 1050 planar equivariant 1051 three-dimensional equivariant 1052 Minimum level, observability 1201 Minimum-phase 1722 Mixing 232, 322 higer order 236 mild-mixing 234 strong-mixing 232 weak-mixing 232 Mixing recurrence, ergodic theory 335 Modal analysis 183 Model Aw–Rascle 1754 Colombo 1755 De Angelis 1752 Delitala and Tosin 1760 discrete velocity 1760 Enskog-like 1759 fictitious density 1752 first order 1750 Klar and coworkers 1759 Lighthill–Whitham–Richards 1751 LWR 1751 mathematical 1748 Paveri Fontana 1758 Payne–Whitham 1753 Prigogine 1757 recursive scale-free trees 645 second order 1750

Song–Havlin–Makse 643 Swiss cheese 561 Model of replication 625 Modified decomposition method 4 Modified Korteweg-de Vries (mKdV) equation 772, 1795 Modons 1590 Modularity 646 Module 1084 Molecular dynamics (MD) 1317 Moment map 1094 Momentum cocycles 983 Momentum mapping 672 Momentum maps 982, 1222 in symplectic reduction 995 nonequivariant 983 Momentum one-cocycle 983 Monge–Ampère equation 1118 Monodromy 657, 1438 classical 1448 fractional 1448 Hamiltonian 1442 quantum 1448 Monodromy matrix 183, 186 Monofractal 426, 606, 612 Monte Carlo 1463, 1464, 1467, 1469 Montgomery example 962 Morphism 32, 41 Morphotype transition 497 Morrey 1291 Morse 33 Morse function 1728 Morse shift 1693 Morse theory 1247, 1248, 1441 Moser 33, 824 Motion coordination 1501 periodic 1212 Mountain pass solution, 3-body problem 1059, 1061 Mountains 409 Moyal product 1386 Multibody system 191 Multicellular organism DNA fractal features 506 Multifractal analysis 606, 630 dynamical systems 296 Multifractal decomposition 296 Multifractal formalism 607, 609 Multifractal records 425 Multifractal spectra 296, 614

1841

1842

Index

Multifractals 426, 429, 447, 467, 476, 483, 553, 606, 612 binomial 483 distribution 456 Dq 431 f(˛) 431 log binomial 456 multifractal measure 431 multifractal scaling 455 Multiple recurrence 317 ergodic theory 335, 336 polynomial 317 Multiple scale transformation of variables 1107 Multiplet 589, 1361 Multiplicative ergodic theorem 251, 752 Multiplicative noise 1127 Multiplicity algebraic 1278 Multiplicity function 1618 Multiplier 1218, 1221 Multiplier method 109, 110 Multiscale analysis 133, 1290, 1293, 1298 Multiscale problems 1294 Multisymplectic reduction 996 Music fractal noise 503 N n-Body problem 1043, 1213, 1308 absence of collision 1064 basic definitions 1045 choreographies 1043 conservation laws 1063 D4q × C2 group 1050 D6 × C3 group 1051 Dq × C2 group 1050 eight shaped three-body solution 1047 G-equivariant minimizers 1066 generalized solution 1060 generalized Sundman–Sperling estimates 1063 locally minimizing solution 1060 Neumann boundary conditions 1066 rotating circle property 1048 Standard variation 1066 N-Soliton solution 779, 780 Nash equilibrium (NE) 1269 Natural extension 283 Navier–Stokes derivation of some fundamental a priori estimates 1027

long-time behavior 1036 uniqueness and continuous dependence on the initial data 1029 Navier–Stokes equations 26, 109, 553, 1009, 1010, 1012, 1014, 1026, 1114, 1126, 1775 derivation 1011 global attractor 1036 less regular solutions 1031 partial regularity results 1031 regularity 1029 times of irregularity 1029 turbulence 1037 weak solutions 1014 weak solutions and related properties 1032 Nearest neighbor method 559 Nearly-integrable Hamiltonian 657, 813 Neimark–Sacker 1179 bifurcation 946, 1254 Nekhoroshev estimates 1168 stability 1076 theorem 1341 theory 661, 1309 Network 567, 637, 1270, 1763 arc 1763 fractal 637 scale-free 637 self-similar 637 small world 638 vertex 1763 Neumann boundary conditions 1340 n-body problem 1066 Neumann condition 1121 Neumann series 818, 831, 1371 Neuron biolobical fractals 498 fractals 503 Newhouse 79 Newhouse phenomenon 291 Newmann–Moser problem 957 Newton 1290 equations 1213 scheme 814, 816 tangent scheme 829 Newton–Fourier method 544 Newton method 1393 Newtonian fluids 1011 Nilpotent perturbation 1276 Nilsystem 364 NLS equation 772 Noble numbers 1310

Index

Noise fractal time series 503 1/f noise 514 Noise terms 4 Nonabelian routh reduction 994 Nonatomic 974 Noncanonical poisson brackets 992 Noncoding DNA, fractal features 506 Non-degeneracy condition 1308 Non-degenerate 819 Non-deterministic fractals 1459 Non-differentiable 571, 585 Non-emptiness problem 1706 Non-equilibrium steady state 1435 Non-equivariance one-cocycle 983 Non-equivariant momentum maps 983 Non-equivariant reduction 988 Non-holonomic integrator 1138, 1645 Non-holonomic mechanical systems 996 Non-homogeneous boundary conditions, Navier–Stokes equations 1017 Non-invertible transformation, isomorphism theory 791 Nonlinear correlations 426 Nonlinear dispersion 1800 Nonlinear dispersion relation 1781 Nonlinear dispersive waves 1773 Nonlinear group velocities 1805 Nonlinear partial differential equations (PDEs) 1113, 1577, 1601 Nonlinear Schrödinger equation 171, 1207, 1337, 1340, 1343, 1570, 1774, 1806 Nonlinear Schrödinger equations 1128 Nonlinear systems controllability 764 observability 1199 Nonlinear transmission lines 1570 Nonlinear wave equation 169 Nonlinear waves 1102 Nonlinearity 539, 553, 579 Non-Newtonian 1014 Non-planar choreography 1052 Non-planar minimizers 1058, 1060 Non-planar symmetric orbit 1053 Non-resonance 1237 Non-resonance condition 948 Non-resonant motion 1294 Nonsingular automorphism 245 Nonsingular Bernoulli transformations 334 Nonsingular Chacón map 334 Nonsingular coding, ergodic theory 348

Nonsingular dynamical systems 329 Nonsingular ergodic theory 329 Nonsingular joinings, ergodic theory 346 Nonsingular maps, Cartesian products 348 Nonsingular odometers 333 Nonsingular systems spectral theory 342 unitary operator 344 Nonsingular transformation 970 ergodic theory 329, 330 hopf decomposition 330 ITPFI 341 smooth 342 Nonsmooth analysis 1137–1139, 1724 Non-standard analysis 1477 Non-stationarity 465, 467, 473 Non-uniform hyperbolicity 750, 751 Non-uniformly expanding maps 73 Non-uniformly hyperbolic systems 276 Nonwandering 1667 set 1667 Normal 1137 Normal diffusion 600 Normal form 42, 98, 1082, 1089, 1152, 1178, 1253, 1254, 1257, 1258, 1277, 1389, 1442 Birkhoff 1162, 1163, 1166 near equilibrium points 1216 semi-local 1162 takens 1162 theory 1279 truncation 658 Normal frequency 671, 828 Normal modes 678 nonlinear 1216 Normal stress effect, liquids 1014 Normalization 1153, 1252–1254 Normalize 1158 Normalized 1157 Normally hyperbolic invariant manifolds 56 Norming constant 776–778 Nucleosome 607, 609 Nucleotide strand asymmetries 630 Null controllable 399 Number theory 328 Numerical approximation 121 Numerical integration 99 Numerical methods 890, 892, 910 Numerical studies 1227

1843

1844

Index

O Oblique condition 1121 Observability 761, 1195, 1724 complete systems 1196 deterministic systems 1195 estimation error 1197 geometric setting 1195 indistinguishability relation 1197 inequality 106 input-output map 1197 linear systems 1198 minimal systems 1196 nonlinear systems 1196 piecewise constant controls 1197 rank condition 1198 symmetric systems 1196 weak indistinguishability relations 1197 Observable 404 pressure of 1427 Observation space 1196 Observer synthesis problem 1196 Observers 1197 arbitrary exponential rate 1198 injectable systems 1202 linear systems 1202 persistent-excitation 1202 skew-adjoint state linear systems 1202 Oceanography 1521, 1529, 1530 24-Octahedral periodic minimizers 1054 Odometers 1741 Markov 333 nonsingular 333 Omega-limit 699 ˝-Limit sets 714 ˝-Stability 748 One-cocycles 983 One-dimensional heat equation, controllability One-forms, principle connections 1000 One-fracton relaxation 179 One-particle distribution 1506 Operator 573 closed linear 103 collisional 1756 covariant derivative 584 dilation 582 Floquet 1217, 1221 gain 1757 loss 1757

763

Markov 1619 weighted 1618 Operator norm 830 Oppenheim’s conjecture 378 Optimal control 144, 1138, 1724 Optimal controls, existence 118 Optimal exponent 1075 Optimality principle 120 Optimization 1137 Orbit 327, 1653, 1726 brake 1229, 1231 closure relation 1730 coadjoint 986 generalized 1060 in closed non-simply connectedHill regions with boundary 1231 in compact Hill regions with boundary 1231 in compact Hill regions without boundary 1230 internal periodic 1229 non-planar symmetric 1053 nondegenerate 1220 normal 1220 periodic 1219 relation 1730 relative periodic 1223 sequence 1730 Orbit classification, type III systems 338 Orbit reduced space 989 Orbit reduction theorem, symplectic 989 Orbit reduction, mechanical systems 985, 986, 988 Orbit space 944 Orbit theory, ergodic 338 Orbital stability 1478 Order fractional 575 Order parameter 562 Ordinary differential equation (ODE) 383, 1653 Organization center 32, 42 Organization principles DNA 504 Origin of replication 607, 621 Ornstein theory 328 Ornstein’s isomorphism theorem 787 Ornstein’s proof 784 Orthogonal collocation 1176 Orthogonality theorems 1363 Oscillations nonlinear 1485 relaxation 1475 self-sustained 1475

Index

Oscillator harmonic 1439, 1475 Oseen fundamental tensor, liquids 1022 Oseledets theorem 70 Out-split graph 1697 Outer measure 968 Output function 395 Output-injection, observers 1202 Outstanding problems 1299 P P-Dimension 934 PAC learning 928, 929 Packing number 924, 931 Padé approximants 1368, 1374 Padé-approximants 1259 Padé approximate 1103 Painlevé’s theorem n-body 1061 Pairwise independence, property 1621 Palais–Smale condition 1236, 1245, 1247 Paley–Littlewood partition of unity 165 Palis conjecture 69 Palis–Yoccoz theorem 292 Parabolic 663 Parabolic equations degenerate 122 nonlinear 109 Parabolic systems 292 Parabolicity, uniform 1115 Parametric excitation 183, 1252, 1254, 1258 Parametric resonance 183, 1251 combination 194 Parametrically excited pendulum 1259 Parry’s entropy 344 Partial differential equations 1282 Partial hyperbolicity 750 Partially hyperbolic difeomorphism 373 Partially hyperbolic system 74, 273 group extensions 276 time-One maps of Anosov Flows 276 Particle collisions in the n-body problem 1044 fluid 1011 quantum 575 Partition function 476 Potts model 449 Past a sphere at increasing Reynolds numbers, flow 1023 Path 572

Peakon 1, 4, 7 Pendulum damped pendulum 1657 pendulum with torque 1668 Pendulum equation 1245–1247 Pendulum model 1409 Percolating networks 176 Percolation 416, 429, 446, 1401, 1460 biological fractals 498 bond 561 conducting backbone 453 Coniglio’s theorem 456 continuum 561 exponent 564 hopping 561 invasion 559, 560 nodes, links, and blobs picture 452 quasi-one-dimensional “bubble” model 457 site 561 site-bond 561 threshold 559, 560 Percolation cluster (IPC) 22, 1460 finite 22 Percolation critical threshold 500 Percolation network 593 Perihelion 1305 Period 1477 minimal 1215 refractory 1478 Period-doubling 1179 Period-doubling bifurcation 664, 945 Period solutions, perturbed 1665 Periodic like trajectories 1338 Periodic orbit 1302, 1311 Periodic point 943 measure 1431 Periodic solution 939, 1163 subharmonic 1164 Perron’s method 1119, 1120 Persistence 465, 1153 Persistent 1153 Persistent property 660 Perturbation 1152, 1153, 1276 geometrical 1480 problem 1153 regular 1479 singular 1476, 1480 Perturbation problem 658 Perturbation theory 1082, 1301, 1352, 1358 free fall 1352

1845

1846

Index

Rayleigh 1353 vibrating string 1352 Pesin entropy formula 70 Pesin theory 70 Phase condition 1174 Phase locking 1261 Phase portrait 1173, 1439 Phase separation 1401 Phase shift 1482, 1576 Phase shift operator 947 Phase space 658, 1084 Phase speed (celerity) 1579 Phase transition 464, 559, 561, 1755 ergodic theory 1423 Phase velocity 1773, 1780 Philosophical structuralism 1414, 1415 Phonon regime 177 Physical measure 748 Physical models, biological fractals 494 Physical systems 546 Physically relevant measures and strange Attractors 276 Hénon Diffeomorphisms 279 Intermittent Maps 278 Unimodal Maps 277 Picard iteration 388 Picard–Lindelöf theorem 388 PIN see Protein interaction networks Pinched torus 672 Pinsker -algebra 235 Pitchfork 1259 Pitchfork bifurcation 942 Planar 3-body problem, symmetry groups 1056 Planar equivariant minimizer 1051 Planar flow of a viscous liquid past a cylinder 1024 Planar motions, liquids 1013 Planar symmetry groups 1055 3-body problem 1055 Planetary problem 1311 Playground swing 183, 202 Pochhammer–Chree equation 1207 Poincaré 33, 1291 Domain 1658 Poincaré domain 1280 Poincaré–Dulac normal form 1086 Poincaré–Dulac theorem 1347 Poincaré-Lindstedt 1252, 1254, 1255 Poincaré–Lindstedt 1252 Poincaré-Lyapounov-Nekhoroshev theorem 1092 Poincaré map 943, 1178, 1660, 1666 Poincaré mapping 660 Poincaré Operator 1238

Poincaré operator 1236, 1238 Poincaré recurrence principle 358 Poincaré Recurrence Theorem 328 Poincaré renormalized form 1093 Poincaré section 1661 Poincaré transformation 1085 Poincaré’s theorem 1291 Point critical 1225, 1234 minimum 1225 multiperiodic 1234 Point of view Eulerian 1750 Lagrangian 1750 Point reduction, mechanical systems 985, 988 Pointwise convergent sequence 383 Pointwise dimension ergodic theory 290 hyperbolic measure 295 Pointwise ergodic theorems 331 Poiseuille flow, biological fractals 492 Poisson brackets 960, 1212, 1214, 1341 noncanonical 992 Poisson–Lie structure 960 Poisson manifold 960, 992 Poisson suspensions, ergodic theory 350 Pole placement 1641 Pole shifting theorem 404 Polynomial 41 Polynomial recurrence ergodic theory 336 Pontryagin’s maximum principle 119, 152 Population dynamical fractals 502 Population dynamics biological fractals 489 Porous media equation 1128 Positive-real lemma 1678 Positively invariant set 1239, 1243 Postulate Born 586 von Neumann 586 Potential 773, 777 gauge 588 Potential shaping 155 Potential well 1579 Potts model 449 Power law 575 Power series 1389 expansions 819 Power spectrum 468, 482, 514 Precession 1305, 1307

Index

Pressure ergodic theory 1422, 1424 observable 1427 variational principle 1428 Prevalence 63 Primary resonance 1258 Principal bundle, connections of 998 Principal connections 991, 998 definition 1000 Principle anisotropy 1754 conservation of the vehicles 1750 covariance 588 Hamiltonian variational 1233 of the least action 1225 variational 1224 Prisoner’s dilemma 1271 Probability ergodic 289 of passing 1757 of velocity class transitions 1761 presence 586 Probability monads 1418 Problem Banach 1620 fixed-energy 1228 of output regulation 1713 Rokhlin’s multiple mixing 1621 Rokhlin’s uniform multiplicity 1625 Process, T, T 1 236 Products 280 Projection 1358 Propagation of chaos 1506, 1507 Proper action, mechanical systems 985 Property (T) 376 Protein interaction networks (PIN) 498 Proximal 1738 Proximal subgradient 1649 Pucci operator 1115, 1118 Pure shear flow, liquids 1014 Pycnocline 1102, 1602 PyDSTOOL 1191 Q Quadratic family 72 Qualitative theory 1337 Quantum 571 Quantum electrodynamics Quantum particle systems Quantum phase transition

1291 1114 1438

Quasi-attractor 1734 Quasi-degenerate case 1361 Quasi-finite type 78 Quasi-homogeneous potentials, n-body problem 1065 Quasi-invariance 971 ergodic theory 342 Quasi-linear system 1348 Quasi-periodic 657, 812, 1251, 1252, 1257, 1262, 1740 Quasi-periodic motion 128, 810, 1290, 1291 Quasi-periodic solution 812 Quasi-periodic torus 1164 Quasi-solitons 1577 Quenching 192, 1478 R R-Partition 975 Radar 542 Radiation condition 1105 Radiation losses 1577 Raghunathan conjecture 378 Random fractal 592, 593, 1464 Random graph 1404 Random midpoint displacement method 482 Random multiplicative process 522 Random W-cascades 611, 614 Random walk 14, 415, 458, 469, 514, 596, 1460, 1461 biological fractals 494 boundaries 350 escape probability 459 exit probability 458 on random walk 17 Pólya’s Theorem 459 recurrence 350, 460 transience and recurrence 459 with non-stationary increments 350 Rank 32 Rank-one attractors 73 Rank-one transformations 334 Rankine–Hugoniot equation 733 Rational map isomorphism theory 792 of the Riemann sphere 784 Ratner’s, property 1632 Rayleigh–Schrödinger 1353 RS perturbation theory 1353, 1354 RCP see Rotating circle property Reaction-diffusion 1471 Reaction-diffusion equation 59, 1127 Reaction kinetics 1457 in fractals 1458

1847

1848

Index

Reaction-limited CCA (RLCA) 596 Reaction order 1462 Real biological fractals 497 Realization input-output map 1199 paracompact manifold 1199 state space 1679 Realization theory 1195, 1199 Recurrence 314, 1584, 1726 Recurrence theorem 242 Recurrent 973 Red bonds 414, 417, 565 Redistribution of energy levels 1451 Reduced Euler–Lagrange equations 994 Reduced space 981 symplectic 984 Reduced system 1153 Reducibility 827 Reduction 1153, 1158, 1162, 1390, 1439, 1725 at zero 998 Lagrangian 993 Liapunov–Schmidt 1233 multisymplectic 996 nonequivariant 988 of dynamics 986 of geometric structures 996 semidirect product 994 singular 995 Reduction lemma 987 Reduction principle in stability theory 57 Reduction theorem 54 remarks on 987 symplectic 985 Reduction theory definition 981 historical overview 990 symplectic 982 Reference frame 573 Reflection coefficient 776, 777 Reflection symmetry 1259 Region closed 1231 compact 1230, 1231 Hill 1228–1231 nonsimply connected 1231 with boundary 1231 without boundary 1230 Regular 32 Regular lattice 559 Regular perturbations 1366

Regular point reduction theorem 988 symmetry algebra 985 Regular potential, Gibbs property 1430 Regularized long-wave (RLW) equation 1523 Regulation 1724 Relation 1729 Compton 586 transitive 1730 Relative entropy 1424 ergodic theory 1431 Relative equilibria 676 n-body problem 1045 Relatively finitely determined factor isomorphism theory 793 Relatively very weak Bernoulli factor isomorphism theory 793 Relativity 571 doubly-special 581 scales 575 special scale 580 Relaxation 1753 Relaxation methods 453 Relaxation time 1753 biological fractals 503 Relay systems 1326 Relication origins 621 Remainder 1286 Renormalization 559, 638, 1131, 1294 Renormalization group (RG) 126, 1290, 1296, 1299 Renormalization theory 1291, 1298 Repellor 1726, 1734 Replication 606, 607 Replication domains 626, 628, 629 Replication origin 617, 622, 625, 627, 628, 630 Replication terminations 625 Replication timing 626–628 Replicator equation 1269 Replicon 621, 622 Replicon model 627 Representation 1363 CAR 352 equivalent 1363 inequivalent 1363 irreducible 1363 Koopman 1618 Maruyama 1630 measure-preserving 1620 of the center manifold 51 reducible 1364 unitary 1619

Index

Repulsion between hubs 647 Rescaled-range analysis 469 Residual degeneracy 1363 Resilience 648 Resistance 566, 649 Resistor network 446 conductance exponent 453 Kirchhoff’s law 449 maximum voltage 456 optimal defects 456 random 448 Resolution 572 Resonance 658, 822, 1153, 1154, 1158, 1163, 1237, 1253, 1281, 1658 sporadic resonance 1094 Resonance condition 1216 Resonance phenomenon 1798 Resonance relation 1086, 1094, 1664 Resonance tongue interaction 198 Resonance tongues 187, 1255 Resonance-type effect 1785 Resonant (critical) phenomenon 1784 Resonant monomial 1389 Resonant motion 1297 Resonant normal form 1343 Resonant perturbation theory 1305 Resonant scalar monomial 1094 Resonant zone 1071 Rest point 1656, 1657 nondegenerate rest point 1657 Restricted orbit equivalence, isomorphism theory 791 Resummation 135, 1292, 1296, 1298 resummed series 136, 1298 Return interval 423–426, 480 Return map 1655, 1660 Return method 122, 401 controllability 766 Return time map, isomorphism theory 790 Return times ergodic probability measure 294 Reversible 1160, 1167, 1253, 1254 Reversible systems 189

-Hyperbolic 57 Riccati equation 121, 1241, 1643 Riemann invariants 1790 Riemann problem 733, 1764 Riemann solver 1764 Right closing factor code 1695 Right-equivalent 32, 37 Right invariant system, input-output mapping 1201 Right resolving factor code 1695

Ring 32 Risk estimation 426 Rivers 409 RLW equation 7 Road theorem 1701 Robotic networks 1496, 1501, 1725 Robustness 648, 709, 742 Rokhlin’s lemma 282, 331 Rokhlin’s theorem 785 Rosenau, P. 9 Rotating circle property (RCP) 1048 Rotation of a compact group 266 Rotation number 129, 293 best approximant 133 Rotor 1259, 1261 Rotor dynamics 1259 Roughness 551 Routh–Hurwitz criterion 1658 Routh reduction, nonabelian 994 Routh, work in reduction theory 990 Routhian 994 Rudolph structure theorem 788 Rudolph’s counterexample machine 789 Ruelle inequality 70 Ruelle’s Perron–Frobenius theorem 1430 Ruelle’s pressure formula 291 Rule Leibniz 584 Run-length-limited shift spaces 1702 Running couplings 1292, 1298 Russell, John Scott 1, 1521 Rüssmann 824 non-degeneracy condition 826 S S-Unit theorem 322 Saddle 664 Saddle-node 1259 Saddle-node or limit point bifurcation Saddle point 1247 Saddle point theorem 1247 Sandbox method 418 SAT-property, random walk 351 Scale 572, 1297 kinetic 1749 macroscopic 1749 microscopic 1748 minimal 581

677, 942

1849

1850

Index

of representation 1748 Planck 572, 581 Scale-free networks 568, 1404 Scale-free property 637 Scale index 1298 Scale invariance 409, 512, 591, 1459 Scale transformation 15 Scaling 23, 577, 1232 Scaling exponent 464, 468, 476 Rényi 476 Scaling factor 1297 Scaling laws 464, 1292 Scattering 773, 1368 amplitude 1369, 1370 coefficient 776 data 771, 773, 777, 779 potential 1369 solutions 1369 Schaeffer 33 Schmeling–Troubetzkoy theorem 296 Schmitz–Schreiber method 483 Schrödinger equation 162, 163, 772, 774, 1317 nonlinear 171 Schrödinger operators 827 Schur’s lemma 1364 Schwarzian derivative 549 Scrambled set 1738 Second law of thermodynamics 358 Second-order phase transition 563 Section Poincaré 1224 Secular term 1107 Segregation 1463 Self-affine fractal 418, 535, 594 Self-avoiding walk 415 biological fractals 495 Self-energy clusters 1297 Self-energy resummations 1298 Self-modulation model 526 Self-organization 571 Self-organized critical (SOC) system 499 Self-organized criticality (SOC) 504 Self-propulsion 26, 30 Self-similar 560, 573, 644 Self-similar approximants 1368, 1374 Self-similar branching structures 490 Self-similarity 410, 559, 606 biological fractals 497 Semi-analytical methods homotopy analysis method (HAM) 1602

homotopy perturbation method (HPM) 1602 variational iteration method (VIM) 1602 Semi-jets 1117 Semialgebra 967 Semiclassical limit 1376 Semiclassical methods 1377 Semiconcave functions, Hamilton–Jacobi equations 701 Semiconcavity 689 Semicontinuous envelope 1119 Semiconvexity 689 Semidirect product reduction 994 Semilinear wave equation, controllability 764 Sensitive dependence 1738 Sensitive initial conditions 540 Sensitivity of initial conditions 553 Sensitivity on initial conditions 63 Separation of time scales 1317 of variables 957 Separatrices 658 Sequence 383 Morse 1624 Rudin–Shapiro 1624 spectral 1618 Series expansion 91 Set critical 1225 Kronecker 1629 of analyticity 1396 of k-recurrence 363 of recurrence 360 Shadowing lemma 746 Shadowing property 745 Shallow water 1102, 1602 Shallow water theory 1105 Shallow water wave and solitary waves 1530 Shallow water wave equation 1525, 1526, 1789 Shannon–McMillan–Breiman theorem 785, 1426 Parry’s generalization 345 Sharma–Tasso–Olver equation 1209 Shattered 930 Shift equivalence 1698 Shift homeomorphism 1740 Shift map 784 Shift of finite type 1694 Shift operator 944 Shift space 1431, 1692 ergodic theory 1425 Shift transformation 968

Index

Shifts 267 Chacon system 268 coded systems 269 context-free systems 269 Prouhet–Thue–Morse 268 Sofic systems 268 Sturmian systems 268 Toeplitz systems 268 Ship dynamics 1096 Shock 729, 732 Shooting method 154 Shortest path dimensions 564 Shuffle product 96 Siegel 1291 Siegel–Bryuno lemma 135 Siegel domain 1658 Siegel’s method 1293 Sierpinski carpet 411, 412, 1459, 1467 Sierpinski gasket 411, 513, 563, 603, 1458, 1463, 1465, 1469 Sierpinski sponge 411, 412 Siersma’s trick 42  -Finite ergodic invariant measures, classifying 351 Signs, series 474 Silica aerogel 178 Šilnikov orbit 1262 Similarity dimension 592 Simple 1152, 1156 Simple choreographies, minimizing properties 1058 Simplified 1157 Simplify 1152, 1156 Sinai–Ruelle–Bowen (SRB) measure 63, 1433, 1434 existence of 70 Sine-Gordon equation 4, 772, 1571 Single well potential 1381 Singular hamiltonian systems 1044 Singular perturbation problem in U (SP-problem) 1332 Singular reduction 995 mechanical systems 986 Singular trajectories 962 Singular value decomposition 475 Singularity 32, 606, 610 and symmetry 985 focus-focus 1442 spectrum 606 theory 674 Sinusoidal forcing 1481 SIR see Susceptible-infective-removed (SIR) Size-consistency 1362 Skew-adjoint, state linear systems 1202

Skew DNA walks 617, 621 Skew product 280 action 1416 group extensions of dynamical systems 280 random Dynamical Systems 280 system 139 Skew profile 617, 618, 627 Sliding block code 969, 1693 Slow manifold 1484 Slow system 1328 Smale work in reduction theory 991 Smale horseshoe 743 Small denominator 1072, 1086, 1294, 1296, 1338, 1342, 1391 Small divisor 810, 816, 1164, 1302 Small divisor problem 133 Small noise 1130 Small parameter " 1106 Small time locally controllable 402 Small world 567 Small-world network 1404 Smallness assumption 819 Smooth dynamical system 965 Smooth flow 740 Smooth nonsingular transformations 342 Smoothness of the invariant manifold 53 Snowdrift game 1273 SOC see Self-organized critical system Sofic shift 1695 Solenoid 673 Solenoid attractor 744 Solitary wave solutions 1, 1102, 1205–1207, 1520, 1523–1525, 1530, 1572, 1576, 1601, 1774, 1786, 1792, 1801 Soliton 1, 5, 200, 771, 884–887, 889–892, 1102, 1205, 1206, 1520, 1561, 1576, 1601, 1608, 1774, 1787, 1792 annihilation 1593 antikink 1580 breather 1580 collision of two solitons 1601 diffractive 1579 dispersive 1579 envelope 1581 fission 1593 fusion 1593 hidden 1585 historical discovery 1563 interaction 1587 internal 1579

1851

1852

Index

ion-acoustic 1579 kink 1580 line 1581 mathematical methods 1563 nontopological 1579 optical spatial 1580 physical properties 1572 plane 1581 Rossby 1579 Russell 1579 shallow-water 1578 ship-induced 1595 spiraling 1593 surface 1579 topological 1579 vector (composite) 1581 Soliton pertubation 1548 exp-function method 1551 homotopy perturbation method 1550 parameter-expansion method 1551 variational approach 1549 variational iteration method 1550 Soliton pertubation theory 1602 Soliton solution 771, 772, 779, 1576 single- and multi-soliton solutions 1578 two-soliton solution 1587 Solomon 1405 Solution complete 709 discrete 709 Hamilton–Jacobi equations 687 long-time behavior 698 maximal 709 nontrivial 708 periodic 1213, 1228, 1234 Zeno 709 Solution path 1726 Solution space 1408, 1410, 1412 Sophus Lie 1083 Source localization 723 Space configuration 1226 cyclic 1618 homogeneous space 302 K(,1) 1227 L2 1225 of paths 1226 of relative loop 1227 of relative loops 1226 Sobolev 1225

symmetric fock 1619 Veech moduli 1631 Space complexity 1492 Space three-body problem 1057 Space-time fractality 583 Spatial correlation 501 Spatial discrete scale invariance 1471 Spatial DSI 1470 Special symplectic reduction method 995 Spectral analysis 468 Spectral dimension 175, 601, 602, 1461, 1462, 1464 Spectral factorization problem 1677 Spectral mapping problem 59 Spectral mapping property 1668 Spectral methods 117 Spectral property 228 Spectral theory, nonsingular systems 342 Spectrum 978, 1279 absolutely continuous 1618 countable Lebesgue spectrum 235 discrete 1618 Haar 1618 rational 234 simple 1618 singular 1618 Spin-orbit interaction 1309, 1310 Spontaneous symmetry breaking 1094 Spreading of an epidemic 559 SRB measure 966 Stability 714, 742 global (pre-)asymptotic 714 Liapunov 1215 linear 1215 of periodic orbits 1215 spectral 1215 stability of rotating fluid bodies 1654 stability of the solar system 1653 Stability criterion 1800 Stability theory 1653 Stabilizability 395 Stabilization 104, 109, 720, 1639 Stable 42, 1654 asymptotically stable 1654 orbitally asymptotically stable 1654 Stable distributions 521 Stable laws 75 Stable manifold 658, 743, 1666 Stable manifold theorem 743, 745 Stable set 743, 1728 Stable subset 1732

Index

Stag hunt 1271 Standard map 129 Standard symplectic form 811 Standard variation, n-body problem 1066 Stark effect 1364 State adjoint 119 of the system, controllability 755 State constraint 1121 State-linear realization 1200 State-linear skew-adjoint realization 1200 State-linear systems, observation space 1200 State observers 1720 high-gain observer 1721 State space 42 State-space realizations 1675 State variable 104 Stationary Markov process 242 Stationary process 797, 966 past and future subspaces 1679 scattering pair 1680 Stationary stochastic process 242, 1674 distribution 242 Statistical learning 924, 925, 928, 935 Statistical stability 63 Steady bifurcation liquids 1018 three-dimensional flow 1023 Steady flow, a liquid past a body 1021 Steady state 1713, 1714 behavior 1715 locus 1715 motion 1715 nonequilibrium 1435 Steady-state flow 1013 Steady-state solutions Navier–Stokes equations 1013 Steep function 1074 Stefan problem 432 Stochastic 1495 Stochastic convolution 1130 Stochastic differential and difference equations Stochastic differential games 1118 Stochastic dynamical system 1130, 1672 on Hilbert spaces 1114 Stochastic matrix 969 Stochastic partial differential equation 1126 Stochastic perturbations 80, 1130 Stochastic processes 1460 Stochastic quantization 1128

1672

Stochastic realization 1675 generating white noise w 1676 state process x 1676 stochastic realization of discrete-time random processes 1676 Stochastic stability 63, 748 Stokes approximation viscous liquids 1024 Stokes equations 26 Stokes expansion 1783, 1804 Stokes line 1376, 1380 Stokes paradox, viscous liquids 1025 Stokes rule 1381 Stokes water waves 1807 Stokes waves 1781 Strand asymmetries 626 Strange attractor 360, 539, 547, 548, 552, 1254 Hénon scheme 549 Stratification 657 Stratified fluid 1102 Stress vector, Navier–Stokes equations 1012 Stretched exponential 424, 480 Stretching tensor, fluids 1012 Strichartz estimates 166 Strictly hyperbolic system 732 Strong feller 1129 Strong interactions 1290 Strong law of large numbers 66 Strong Lp inequality 245 Strong mixing 253 Strong shift equivalence 1697 Strong transversality property 1667 Strongly irreducible 1706 Strongly mixing 1130 Strongly semiconcave, Hamilton–Jacobi equations 689 Strongly semiconvex, Hamilton–Jacobi equations 689 Structural stability 63, 747, 1326, 1665 Structural unstable 40 Structurally stable 371, 658, 1153, 1666 Structure 1152, 1159 multi-symplectic 1234 preservation 1159, 1163 symplectic 1213, 1234, 1235 Structure condition 1118 Structure factor biological fractals 501 Sub- and supersolutions 1243 Sub-Riemannian geometry 1114 Subadditive ergodic theory 1426 Subdiffusion 14

1853

1854

Index

Subgradient 1137 ergodic theory 1428 limiting 1139, 1140 proximal 1139, 1140 viscosity 1141 Subharmonic instabilities 187 Subharmonic solution 944 Submanifold, initial 989 Subsequence ergodic theorem 253 Subshift ergodic theory 1423 finite type 1429 Subshift of finite type 63 Subsolutions Hamilton–Jacobi equations 685, 690, 691 regularity results 690 Subspace algorithms 1685 Subspace identification 1682 Sum resonance 1251, 1254, 1257, 1260 Summable potential Gibbs state 1430 Summation at the smallest term 1073 Summation rule 1290 Sundman–Sperling estimates, generalized 1063 Super-localized wave functions 176 Super-recurrence 1585 Superconductor–normal conductor 561 Supercritical bifurcation, liquids 1019 Superdiffusion 14 Superexponential stability 1077 Superrigidity 377 Superstrings 581 Supersymmetry 581 Surface 43, 1529 Surface wave 1520 Susceptible-infective-removed (SIR) 500 Suspension flows 281 Swarming behaviors 1501 Swimming 26, 27 efficiency 29 optimal 27, 29 strategies 27 Symbolic dynamical system 1691 Symbolic dynamics 66, 746, 1423, 1425 Symmetric stable processes, ergodic theory 349 Symmetry 1082, 1084, 1152, 1251, 1363, 1364, 1390, 1725 and perturbation theory 1364 and singularities 985 approximate symmetry 1088 axial 1160

axially 1158 breaking 1439 generalized symmetry 1096 Hamiltonian systems with 1221 mechanical systems 981 rotational 1157, 1161, 1162 toroidal 1153, 1158, 1162, 1164 torus 1157 Symmetry algebra 985 Symmetry breaking spontaneous 1439 Symmetry groups 1046, 1258 core 1053 planar 3-body problem 1056 Symmetry reduction 1082, 1094, 1228 Symplectic 143, 812, 1071, 1159 matrix 812 Symplectic form 670, 830 Symplectic manifold 811, 958 Symplectic orbit reduction theorem 989 Symplectic orthogonal 987 Symplectic reduced space 984 as coadjoint orbits 988 Symplectic reduction 986 special method 995 theorem 984, 985 theory 982 without momentum maps 995 Synchronous network 1491 System identification 924, 926–928, 932, 933, 935, 1682 Systems adjoint 119 affine 94, 96 approximately controllable 105 approximating 97 discrete time 99 dynamical 94 exactly controllable 105 fully nonlinear 99 Hamiltonian 1213, 1219 Lagrangian with symmetries 1226 lifted 98 linear 98 natural mechanical 1225 nonlinear 90, 98 null controllable 105 of balance laws 730 of conservation laws 730 universal 96 well-posed 707 with discontinuities 71

Index

Systems affine 401 Szemerédi’s Theorem 1745 Szemerédi’s theorem 365 Szemerédi’s theorem 964 T T-Equivalent 971 T-Exponential 1373 T-Nonshrinkable 971 Table of games 1761 Tail entropy 77 Tangent bundle 958 Tangent bundle reduction 993 Tanh method 1528 Target 118 Task allocation 1501 Taylor–Couette flow bifurcation 1020 Taylor vortices, liquids 1020 Temporal physiological series 507 Temporal regularization 713 Terminal chain component 1736 Terminus of replication 607, 621 12-Tetrahedral periodic minimizers 1054 Theorem Alexeyev’s 1622 Ambrose–Kakutani 1631 averaging 1665 Bochner–Herglotz 1620 Dirichlet’s 303, 309 Floquet’s 1659 flowbox 1153 Foia¸s–Stratila 1629 Hartman–Grobman 1155, 1657 KAM 1663 Khintchine–Groshev 304, 305, 310 Krein 1218 Leonov’s 1629 Lusternik–Fet 1230 Lyapunov 1216, 1660 Marsden–Weinstein 1222 mountain 1228 Noether 584, 1153, 1222 Poincaré 1153 Poincaré-Bendixson 1668 Poincaré’s 1155 Poincaré’s theorem 1658 relative equilibria 1223 Siegel’s 1658 singular reduction 1223

Souriau 1222 spectral 1620 stable manifold 1666 Sternberg’s 1155 structural stability 1667 weak closure 1627 Weyl 813 Theory Cantorized singularity 1166 ergodic theory 302 gauge 588 KAM 1165, 1168, 1217 metric number theory 302 normal form 1168 parametrized KAM 1167 perturbation 1152 singularity 1153, 1164, 1165 Yang–Mills 589 Thermal conductivity 180 Thermal equilibrium 1674 Thermodynamic formalism 1422 ergodic theory 1424 Thermodynamic state 1675 Thom 33 Three-body problem 542, 1043, 1301 Three-body solution, eight shaped 1047 Three-dimensional equivariant minimizer 1052 Three-dimensional flow in a bounded domain 1038 liquids 1022 steady bifurcation 1023 Tidal bore 1561 Time re-scaling 1230 Time complexity 1492, 1497 Time-dependent Hamilton–Jacobi equations 698 Time-dependent vectorfields 1253 Time discrete scale invariance 1470, 1471 Time DSI 1464 Time invariant 397 Time ordered product 1373 Time regularized long-wave (TRLW) equation 1524 Time scales 1475 Time series biological fractals 502 Time-varying 398 TNS action 374 Topography 535 Topological conjugacy 1693 Topological dynamical system 965 Topological dynamics 327

1855

1856

Index

Topological entropy 753, 1696 dimension-like characteristics ergodic theory Topological genericity 64 Topological group, ergodic theory 337 Topological pressure, ergodic theory 291 Topological rigidity 378 Topological trapping 826 Topologically transitive 1735 Topology 541, 643 of uniform convergence 91 strong and weak 955 usual 91 weak 90 Torus 126, 830 invariant 129 lower-dimensional 137 maximal 137 resonant 137 Total relation 1735 Total stability 827 Totally ergodic 229 Tower transformations 334 Tower, Rokhlin 1627 Traces of the attractor 1734 Tracking 720, 1711 adaptive tracking 1711 dynamic inversion 1711 tracking via internal models 1711 Trajectory 390 Brownian 494 optimal 118 Transcription 606, 607 Transcritical bifurcation 943 Transfer map 373 Transfer operator Ruelle’s Perron–Frobenius theorem 1430 Transfer principle 247 Transfinite fractals 646 Transformation 1157, 1221, 1280 canonical 1161 equivariant 1160 finitely fixed 791 isomorphic to Bernoulli shifts 788 loosely Bernoulli 791 minimal self joining 789 non-invertible 791 normalizing 1152 not isomorphic to Bernoulli shifts 788 scale 574, 576, 589

290

 -preserving 1160 [T; T 1 ] 785 Transformation process, cutting and stacking Transfractals 646 transfractal dimension 646 Transition 575, 577 Transition semigroup 1128 Transitive points 1735 Transitivity 1726 Transmission coefficient 776, 777, 780 Transversal 1153, 1667 Transversality condition 943, 1141 Traveling wave solutions 890, 892 Tree 129, 1295 cluster 133 finitely determined transformation 792 fractal dimension 492 matrix 449 number 1295 renormalized 136 self-energy cluster 133 tree of life 577 value 1295 very week Bernoulli transformation 792 Trigger 1478 Trivial symmetry 1055 Triviality conjecture 1292 Trochoid 1782 True anomaly 1304 Truncation 1152, 1157, 1281 Tschirnhaus-transformation 41 Tsunami 1520, 1529, 1561, 1602 Tuned damper 192 Tunneling between wells 1384 Turbulence 541 Navier–Stokes equations 1037 Turning points 1376 Twist mappings 824, 832 Two-body problem 1043 Two-cocycle 984 Two-cocycle identity 984 Two-dimensional flow, liquids 1013, 1024 Two-dimensional KdV 1798 Two-dimensional wave equation 1789 Two-forms, curvature 1003 Two fracton relaxation 179 Two-sphere 149 Two-valuedness 583

789

Index

U Uehling–Uhlenbeck equation 1515 Ultimatum game 1274 Uncertainty 572 Undular bore 1524, 1793 Unfolding 40 Unfolding dimension 32 Unfolding morphism 32 Unfolding of homoclinic tangencies 74 Uniform continuous function 383 Uniform convergent sequence 383 Uniform hyperbolicity 741 Uniformly distributed 245 Uniformly expanding maps 69 Uniformly expanding Markov map 1433 Uniformly hyperbolic diffeomorphisms 69 Uniformly hyperbolic systems 276, 746 Anosov Systems 274 Axiom A Systems 275 Geodesic Flow on Manifold of Negative Curvature 273 Horocycle Flow 273 Markov Partitions and Coding 274 Uniformly rigid 1738 Unimodal maps 72 Unipotent matrix 378 Uniquely ergodic 245, 378, 970 Unitarily equivalent 977 Unitary operator 977 nonsingular system 344 Unitary representations, locally compact groups Universal 32, 41 Universal formula for stabilization 1647 Universality 452 Unperturbed system 1152 Unstability in the quadratic family 79 Unstable manifold 658, 743 Upper density 255 V Validation 1419 Value critical 1225 Value function 119, 1116, 1724 Van-der-Corput trick 362 Van der Pol 1475 Van-der-Pol equation 548 Van-der-Waals equation 1291 Van der Waerden’s theorem 1745 Vanishing viscosity method 1115, 1117, 1122

352

Variable stochastic 574 Variation of parameters 92, 98 Variational formulation, Navier–Stokes equations 1014 Variational iteration method 890, 912, 915 Variational principle 68, 76, 313 ergodic theory 1427, 1428, 1431 Variations formula 92 VC-dimension 930, 931, 934 Vector field 90, 384, 659 autonomous 1215 C0 equivalent 1325 equivariant 1222 germ-equivalent 1326 Vector space 32 Vehicle candidate 1756 field 1756 Venttsel condition 1123 Verhulst equation 1240, 1241 Verification functions 1137, 1143 Versal 42 Versal unfolding 1258 Vertical projection 1001 Vertical space, definition 1001 Vertical vectors, definition 1002 Vibrational density of states 177 VIM 890, 892, 899, 902, 904 Virial series 1291 Viscosity bacterial colonies 497 Viscosity solutions 121, 689, 1113, 1115, 1116, 1137, 1145 Hamilton–Jacobi equations 688, 700 semicontinuous 1122 Viscosity subsolution 1119 Lp 1123 Viscosity test functions Hamilton–Jacobi equations 693 Viscous fingering 434 Volatility clustering 527 Volcanic eruption 534 Voltage distribution 454 Volterra series 89, 97 Volume preserving 1159 von Neumann algebras, ergodic theory 351 von Neumann mean ergodic theorem 331 von Neumann theorem 1130 Voter model 1468

1857

1858

Index

W Walk 1401 biological fractals 495 random 572 Wandering 971 Wasserman, Stanley 33 Water tank experiments 1528, 1529 Water wave equations 1778, 1784 Water wave observations 1528, 1529 Water wave problem 1349 Watson’s theorem 1368 Wave 1523, 1525, 1529, 1772 “freak” or “giant” 1595 function 585 linear 1577 nonlinear 1577 traveling 1577 Wave dispersion 1520, 1523, 1524, 1530 Wave equation 103, 108, 109, 163, 164, 1128, 1337 boundary control 756 controllability 756, 758 nonlinear 169 strongly stable 110 (uniformly) exponentially stable 110 Wave function 588 biquaternionic 588 nondifferentiable 587 Wave maker 1103 Wave packet 163 Wavelet analysis 470 Wavelet transform 606, 607, 609 Wavelet transform modulus maxima (WTMM) method 477, 606, 612 Wavelet transform skeleton 606 Wavelets 606, 636 Weak-coupling limit 1506, 1515 for a single particle 1517 for classical systems 1507, 1515 for quantum systems 1511 Weak KAM theory 683, 684 Weak L1 inequality 244 Weak-mixing 66 Weak observability 1197 Weak solution 732, 1115 , viscosity 688 Weak subsolution, definition 686 Weak topology, symbolic dynamics 1426 Weakly-minimal realization 1199 Weakly wandering 971 Weierstrass 1291 Well-posedness 1113 , local 161 Weyl chamber flow 372

Weyl quantization 1386 Weyl’s criterion 361 White noise 514, 1130 biological fractals 503 Whitham averaged Lagrangian 1804 Whitham equation 1802, 1804 Whitham’s averaged variational principle 1804 Whitham’s conservation equations 1804 Whitham’s equation 1803, 1806 Whitney 33 Whitney smoothness 825, 1167 Whitney umbrella 1258 Wiener Lemma 1619 Wigner equation 1512 Wigner transform 1512 Wigner’s theorem 1356 example of 1355 Wilson 1292 WKB approximation 1377 connection formula 1379 semiclassical solutions in the classically allowed region 1378 semiclassical solutions in the classically forbidden region 1379 Words 96 Wronskian 392 WT skeleton 610–612, 622 WTMM method 617 Y Yang–Mills construction, mechanics Yang–Mills theories 1293 Yomdin’s theory 77 Young’s dimension formula 294

993

Z Zabusky, N.J. 2 Zakharov–Shabat system 772 Zeeman 33 Zeeman effect 1364 Zeeman’s tolerance stability conjecture Zeipel’s theorem n-body 1062 structure in the collision set 1062 Zero controllability 761, 762 Zero dynamics 1718 Zero-noise limit 73 Zeta function 319, 1697 Zinbiel algebra 88, 96

79